CINXE.COM

<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>The class of Aronszajn lines under epimorphisms</title>   <meta content="width=device-width, initial-scale=1, shrink-to-fit=no" name="viewport"/> <link href="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/css/bootstrap.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv-fonts.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/latexml_styles.css" rel="stylesheet" type="text/css"/> <script src="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/js/bootstrap.bundle.min.js"></script> <script src="https://cdnjs.cloudflare.com/ajax/libs/html2canvas/1.3.3/html2canvas.min.js"></script> <script src="/static/browse/0.3.4/js/addons_new.js"></script> <script src="/static/browse/0.3.4/js/feedbackOverlay.js"></script> <meta content="Aronszajn line, Aronszajn tree, Countryman line, forcing, linear order, basis, strongly surjective" lang="en" name="keywords"/> <base href="/html/2503.13728v1/"/></head> <body> <nav class="ltx_page_navbar"> <nav class="ltx_TOC"> <ol class="ltx_toclist"> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S1" title="In The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1 </span>Introduction</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S2" title="In The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2 </span>Aronszajn and Countryman lines</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S3" title="In The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3 </span>Strongly surjective Aronszajn lines</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S4" title="In The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4 </span>Aronszajn line decompositions</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S5" title="In The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5 </span>An infinite antichain</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6" title="In The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">6 </span>An infinite decreasing chain</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.SS1" title="In 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">6.1 </span>Moore’s forcing</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.SS2" title="In 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">6.2 </span>Forcing epimorphisms</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S7" title="In The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">7 </span>A two element basis for the Aronszajn lines</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S7.SS1" title="In 7. A two element basis for the Aronszajn lines ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">7.1 </span>On a basis for all uncountable linear orders</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S8" title="In The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">8 </span>On a question on Countryman lines</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S9" title="In The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">9 </span>Concluding remarks and questions</span></a></li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_1line ltx_leqno"> <h1 class="ltx_title ltx_title_document">The class of Aronszajn lines under epimorphisms</h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Lucas Polymeris </span></span> <span class="ltx_author_before"> and </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Carlos Martinez-Ranero </span></span> </div> <div class="ltx_dates">(Date: March 17, 2025)</div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract.</h6> <p class="ltx_p" id="id19.19">A linear order <math alttext="A" class="ltx_Math" display="inline" id="id1.1.m1.1"><semantics id="id1.1.m1.1a"><mi id="id1.1.m1.1.1" xref="id1.1.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="id1.1.m1.1b"><ci id="id1.1.m1.1.1.cmml" xref="id1.1.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="id1.1.m1.1c">A</annotation><annotation encoding="application/x-llamapun" id="id1.1.m1.1d">italic_A</annotation></semantics></math> is called <em class="ltx_emph ltx_font_italic" id="id19.19.1">strongly surjective</em> if for every non empty suborder <math alttext="B\preceq A" class="ltx_Math" display="inline" id="id2.2.m2.1"><semantics id="id2.2.m2.1a"><mrow id="id2.2.m2.1.1" xref="id2.2.m2.1.1.cmml"><mi id="id2.2.m2.1.1.2" xref="id2.2.m2.1.1.2.cmml">B</mi><mo id="id2.2.m2.1.1.1" xref="id2.2.m2.1.1.1.cmml">⪯</mo><mi id="id2.2.m2.1.1.3" xref="id2.2.m2.1.1.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="id2.2.m2.1b"><apply id="id2.2.m2.1.1.cmml" xref="id2.2.m2.1.1"><csymbol cd="latexml" id="id2.2.m2.1.1.1.cmml" xref="id2.2.m2.1.1.1">precedes-or-equals</csymbol><ci id="id2.2.m2.1.1.2.cmml" xref="id2.2.m2.1.1.2">𝐵</ci><ci id="id2.2.m2.1.1.3.cmml" xref="id2.2.m2.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="id2.2.m2.1c">B\preceq A</annotation><annotation encoding="application/x-llamapun" id="id2.2.m2.1d">italic_B ⪯ italic_A</annotation></semantics></math>, there is an epimorphism from <math alttext="A" class="ltx_Math" display="inline" id="id3.3.m3.1"><semantics id="id3.3.m3.1a"><mi id="id3.3.m3.1.1" xref="id3.3.m3.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="id3.3.m3.1b"><ci id="id3.3.m3.1.1.cmml" xref="id3.3.m3.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="id3.3.m3.1c">A</annotation><annotation encoding="application/x-llamapun" id="id3.3.m3.1d">italic_A</annotation></semantics></math> onto <math alttext="B" class="ltx_Math" display="inline" id="id4.4.m4.1"><semantics id="id4.4.m4.1a"><mi id="id4.4.m4.1.1" xref="id4.4.m4.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="id4.4.m4.1b"><ci id="id4.4.m4.1.1.cmml" xref="id4.4.m4.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="id4.4.m4.1c">B</annotation><annotation encoding="application/x-llamapun" id="id4.4.m4.1d">italic_B</annotation></semantics></math> (denoted by <math alttext="B\trianglelefteq A" class="ltx_Math" display="inline" id="id5.5.m5.1"><semantics id="id5.5.m5.1a"><mrow id="id5.5.m5.1.1" xref="id5.5.m5.1.1.cmml"><mi id="id5.5.m5.1.1.2" xref="id5.5.m5.1.1.2.cmml">B</mi><mo id="id5.5.m5.1.1.1" xref="id5.5.m5.1.1.1.cmml">⁢</mo><mi id="id5.5.m5.1.1.3" mathvariant="normal" xref="id5.5.m5.1.1.3.cmml">⊴</mi><mo id="id5.5.m5.1.1.1a" xref="id5.5.m5.1.1.1.cmml">⁢</mo><mi id="id5.5.m5.1.1.4" xref="id5.5.m5.1.1.4.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="id5.5.m5.1b"><apply id="id5.5.m5.1.1.cmml" xref="id5.5.m5.1.1"><times id="id5.5.m5.1.1.1.cmml" xref="id5.5.m5.1.1.1"></times><ci id="id5.5.m5.1.1.2.cmml" xref="id5.5.m5.1.1.2">𝐵</ci><ci id="id5.5.m5.1.1.3.cmml" xref="id5.5.m5.1.1.3">⊴</ci><ci id="id5.5.m5.1.1.4.cmml" xref="id5.5.m5.1.1.4">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="id5.5.m5.1c">B\trianglelefteq A</annotation><annotation encoding="application/x-llamapun" id="id5.5.m5.1d">italic_B ⊴ italic_A</annotation></semantics></math>). We show, answering some questions of Dániel T. Soukup, that under <math alttext="\mathsf{MA}_{\aleph_{1}}" class="ltx_Math" display="inline" id="id6.6.m6.1"><semantics id="id6.6.m6.1a"><msub id="id6.6.m6.1.1" xref="id6.6.m6.1.1.cmml"><mi id="id6.6.m6.1.1.2" xref="id6.6.m6.1.1.2.cmml">𝖬𝖠</mi><msub id="id6.6.m6.1.1.3" xref="id6.6.m6.1.1.3.cmml"><mi id="id6.6.m6.1.1.3.2" mathvariant="normal" xref="id6.6.m6.1.1.3.2.cmml">ℵ</mi><mn id="id6.6.m6.1.1.3.3" xref="id6.6.m6.1.1.3.3.cmml">1</mn></msub></msub><annotation-xml encoding="MathML-Content" id="id6.6.m6.1b"><apply id="id6.6.m6.1.1.cmml" xref="id6.6.m6.1.1"><csymbol cd="ambiguous" id="id6.6.m6.1.1.1.cmml" xref="id6.6.m6.1.1">subscript</csymbol><ci id="id6.6.m6.1.1.2.cmml" xref="id6.6.m6.1.1.2">𝖬𝖠</ci><apply id="id6.6.m6.1.1.3.cmml" xref="id6.6.m6.1.1.3"><csymbol cd="ambiguous" id="id6.6.m6.1.1.3.1.cmml" xref="id6.6.m6.1.1.3">subscript</csymbol><ci id="id6.6.m6.1.1.3.2.cmml" xref="id6.6.m6.1.1.3.2">ℵ</ci><cn id="id6.6.m6.1.1.3.3.cmml" type="integer" xref="id6.6.m6.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="id6.6.m6.1c">\mathsf{MA}_{\aleph_{1}}</annotation><annotation encoding="application/x-llamapun" id="id6.6.m6.1d">sansserif_MA start_POSTSUBSCRIPT roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> there is a strongly surjective Countryman line. We also study the general structure of the class of Aronszajn lines under <math alttext="\trianglelefteq" class="ltx_Math" display="inline" id="id7.7.m7.1"><semantics id="id7.7.m7.1a"><mi id="id7.7.m7.1.1" mathvariant="normal" xref="id7.7.m7.1.1.cmml">⊴</mi><annotation-xml encoding="MathML-Content" id="id7.7.m7.1b"><ci id="id7.7.m7.1.1.cmml" xref="id7.7.m7.1.1">⊴</ci></annotation-xml><annotation encoding="application/x-tex" id="id7.7.m7.1c">\trianglelefteq</annotation><annotation encoding="application/x-llamapun" id="id7.7.m7.1d">⊴</annotation></semantics></math>, and compare it with the well known embeddability relation <math alttext="\preceq" class="ltx_Math" display="inline" id="id8.8.m8.1"><semantics id="id8.8.m8.1a"><mo id="id8.8.m8.1.1" xref="id8.8.m8.1.1.cmml">⪯</mo><annotation-xml encoding="MathML-Content" id="id8.8.m8.1b"><csymbol cd="latexml" id="id8.8.m8.1.1.cmml" xref="id8.8.m8.1.1">precedes-or-equals</csymbol></annotation-xml><annotation encoding="application/x-tex" id="id8.8.m8.1c">\preceq</annotation><annotation encoding="application/x-llamapun" id="id8.8.m8.1d">⪯</annotation></semantics></math>. Under <math alttext="\mathsf{PFA}" class="ltx_Math" display="inline" id="id9.9.m9.1"><semantics id="id9.9.m9.1a"><mi id="id9.9.m9.1.1" xref="id9.9.m9.1.1.cmml">𝖯𝖥𝖠</mi><annotation-xml encoding="MathML-Content" id="id9.9.m9.1b"><ci id="id9.9.m9.1.1.cmml" xref="id9.9.m9.1.1">𝖯𝖥𝖠</ci></annotation-xml><annotation encoding="application/x-tex" id="id9.9.m9.1c">\mathsf{PFA}</annotation><annotation encoding="application/x-llamapun" id="id9.9.m9.1d">sansserif_PFA</annotation></semantics></math>, the class of Aronszajn lines and the class of countable linear orders enjoy similar nice properties when viewed under the embeddability relation; both are well-quasi-ordered and have a finite basis. We show that this analogy does not extend perfectly to the <math alttext="\trianglelefteq" class="ltx_Math" display="inline" id="id10.10.m10.1"><semantics id="id10.10.m10.1a"><mi id="id10.10.m10.1.1" mathvariant="normal" xref="id10.10.m10.1.1.cmml">⊴</mi><annotation-xml encoding="MathML-Content" id="id10.10.m10.1b"><ci id="id10.10.m10.1.1.cmml" xref="id10.10.m10.1.1">⊴</ci></annotation-xml><annotation encoding="application/x-tex" id="id10.10.m10.1c">\trianglelefteq</annotation><annotation encoding="application/x-llamapun" id="id10.10.m10.1d">⊴</annotation></semantics></math> relation; while it is known that the countable linear orders are still well-quasi-ordered under <math alttext="\trianglelefteq" class="ltx_Math" display="inline" id="id11.11.m11.1"><semantics id="id11.11.m11.1a"><mi id="id11.11.m11.1.1" mathvariant="normal" xref="id11.11.m11.1.1.cmml">⊴</mi><annotation-xml encoding="MathML-Content" id="id11.11.m11.1b"><ci id="id11.11.m11.1.1.cmml" xref="id11.11.m11.1.1">⊴</ci></annotation-xml><annotation encoding="application/x-tex" id="id11.11.m11.1c">\trianglelefteq</annotation><annotation encoding="application/x-llamapun" id="id11.11.m11.1d">⊴</annotation></semantics></math>, we show that already in <math alttext="\mathsf{ZFC}" class="ltx_Math" display="inline" id="id12.12.m12.1"><semantics id="id12.12.m12.1a"><mi id="id12.12.m12.1.1" xref="id12.12.m12.1.1.cmml">𝖹𝖥𝖢</mi><annotation-xml encoding="MathML-Content" id="id12.12.m12.1b"><ci id="id12.12.m12.1.1.cmml" xref="id12.12.m12.1.1">𝖹𝖥𝖢</ci></annotation-xml><annotation encoding="application/x-tex" id="id12.12.m12.1c">\mathsf{ZFC}</annotation><annotation encoding="application/x-llamapun" id="id12.12.m12.1d">sansserif_ZFC</annotation></semantics></math> the class of Aronszajn lines has an infinite antichain, and under <math alttext="\mathsf{MA}_{\aleph_{1}}" class="ltx_Math" display="inline" id="id13.13.m13.1"><semantics id="id13.13.m13.1a"><msub id="id13.13.m13.1.1" xref="id13.13.m13.1.1.cmml"><mi id="id13.13.m13.1.1.2" xref="id13.13.m13.1.1.2.cmml">𝖬𝖠</mi><msub id="id13.13.m13.1.1.3" xref="id13.13.m13.1.1.3.cmml"><mi id="id13.13.m13.1.1.3.2" mathvariant="normal" xref="id13.13.m13.1.1.3.2.cmml">ℵ</mi><mn id="id13.13.m13.1.1.3.3" xref="id13.13.m13.1.1.3.3.cmml">1</mn></msub></msub><annotation-xml encoding="MathML-Content" id="id13.13.m13.1b"><apply id="id13.13.m13.1.1.cmml" xref="id13.13.m13.1.1"><csymbol cd="ambiguous" id="id13.13.m13.1.1.1.cmml" xref="id13.13.m13.1.1">subscript</csymbol><ci id="id13.13.m13.1.1.2.cmml" xref="id13.13.m13.1.1.2">𝖬𝖠</ci><apply id="id13.13.m13.1.1.3.cmml" xref="id13.13.m13.1.1.3"><csymbol cd="ambiguous" id="id13.13.m13.1.1.3.1.cmml" xref="id13.13.m13.1.1.3">subscript</csymbol><ci id="id13.13.m13.1.1.3.2.cmml" xref="id13.13.m13.1.1.3.2">ℵ</ci><cn id="id13.13.m13.1.1.3.3.cmml" type="integer" xref="id13.13.m13.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="id13.13.m13.1c">\mathsf{MA}_{\aleph_{1}}</annotation><annotation encoding="application/x-llamapun" id="id13.13.m13.1d">sansserif_MA start_POSTSUBSCRIPT roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> an infinite decreasing chain as well. We show that some of the analogy survives by proving that under <math alttext="\mathsf{PFA}" class="ltx_Math" display="inline" id="id14.14.m14.1"><semantics id="id14.14.m14.1a"><mi id="id14.14.m14.1.1" xref="id14.14.m14.1.1.cmml">𝖯𝖥𝖠</mi><annotation-xml encoding="MathML-Content" id="id14.14.m14.1b"><ci id="id14.14.m14.1.1.cmml" xref="id14.14.m14.1.1">𝖯𝖥𝖠</ci></annotation-xml><annotation encoding="application/x-tex" id="id14.14.m14.1c">\mathsf{PFA}</annotation><annotation encoding="application/x-llamapun" id="id14.14.m14.1d">sansserif_PFA</annotation></semantics></math>, for some carefully constructed Countryman line <math alttext="C" class="ltx_Math" display="inline" id="id15.15.m15.1"><semantics id="id15.15.m15.1a"><mi id="id15.15.m15.1.1" xref="id15.15.m15.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="id15.15.m15.1b"><ci id="id15.15.m15.1.1.cmml" xref="id15.15.m15.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="id15.15.m15.1c">C</annotation><annotation encoding="application/x-llamapun" id="id15.15.m15.1d">italic_C</annotation></semantics></math>, <math alttext="C" class="ltx_Math" display="inline" id="id16.16.m16.1"><semantics id="id16.16.m16.1a"><mi id="id16.16.m16.1.1" xref="id16.16.m16.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="id16.16.m16.1b"><ci id="id16.16.m16.1.1.cmml" xref="id16.16.m16.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="id16.16.m16.1c">C</annotation><annotation encoding="application/x-llamapun" id="id16.16.m16.1d">italic_C</annotation></semantics></math> and <math alttext="C^{\star}" class="ltx_Math" display="inline" id="id17.17.m17.1"><semantics id="id17.17.m17.1a"><msup id="id17.17.m17.1.1" xref="id17.17.m17.1.1.cmml"><mi id="id17.17.m17.1.1.2" xref="id17.17.m17.1.1.2.cmml">C</mi><mo id="id17.17.m17.1.1.3" xref="id17.17.m17.1.1.3.cmml">⋆</mo></msup><annotation-xml encoding="MathML-Content" id="id17.17.m17.1b"><apply id="id17.17.m17.1.1.cmml" xref="id17.17.m17.1.1"><csymbol cd="ambiguous" id="id17.17.m17.1.1.1.cmml" xref="id17.17.m17.1.1">superscript</csymbol><ci id="id17.17.m17.1.1.2.cmml" xref="id17.17.m17.1.1.2">𝐶</ci><ci id="id17.17.m17.1.1.3.cmml" xref="id17.17.m17.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="id17.17.m17.1c">C^{\star}</annotation><annotation encoding="application/x-llamapun" id="id17.17.m17.1d">italic_C start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math> form a <math alttext="\trianglelefteq" class="ltx_Math" display="inline" id="id18.18.m18.1"><semantics id="id18.18.m18.1a"><mi id="id18.18.m18.1.1" mathvariant="normal" xref="id18.18.m18.1.1.cmml">⊴</mi><annotation-xml encoding="MathML-Content" id="id18.18.m18.1b"><ci id="id18.18.m18.1.1.cmml" xref="id18.18.m18.1.1">⊴</ci></annotation-xml><annotation encoding="application/x-tex" id="id18.18.m18.1c">\trianglelefteq</annotation><annotation encoding="application/x-llamapun" id="id18.18.m18.1d">⊴</annotation></semantics></math>-basis for the class of Aronszajn lines. Finally we show that this does not extend to all uncountable linear orders by proving that there is never a finite <math alttext="\trianglelefteq" class="ltx_Math" display="inline" id="id19.19.m19.1"><semantics id="id19.19.m19.1a"><mi id="id19.19.m19.1.1" mathvariant="normal" xref="id19.19.m19.1.1.cmml">⊴</mi><annotation-xml encoding="MathML-Content" id="id19.19.m19.1b"><ci id="id19.19.m19.1.1.cmml" xref="id19.19.m19.1.1">⊴</ci></annotation-xml><annotation encoding="application/x-tex" id="id19.19.m19.1c">\trianglelefteq</annotation><annotation encoding="application/x-llamapun" id="id19.19.m19.1d">⊴</annotation></semantics></math>-basis for the uncountable real orders.</p> </div> <div class="ltx_keywords"> <h6 class="ltx_title ltx_title_keywords">Key words and phrases: </h6>Aronszajn line, Aronszajn tree, Countryman line, forcing, linear order, basis, strongly surjective </div> <div class="ltx_classification"> <h6 class="ltx_title ltx_title_classification">2010 Mathematics Subject Classification: </h6>03E35, 03E04, 06A05 </div> <div class="ltx_acknowledgements">The authors would like to thank Justin Moore for reading an earlier version of this material and pointing out <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S8.Thmtheorem5" title="Remark 8.5. ‣ 8. On a question on Countryman lines ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Remark</span> <span class="ltx_text ltx_ref_tag">8.5</span></a> </div> <div class="ltx_acknowledgements">The first named author was supported by ANID-Subdirección de Capital Humano/Magíster Nacional/2024 - 22241722 </div> <div class="ltx_acknowledgements">The second named author was partially supported by Proyecto VRID-Investigación No. 220.015.024-INV </div> <section class="ltx_section" id="S1"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">1. </span>Introduction</h2> <div class="ltx_para" id="S1.p1"> <p class="ltx_p" id="S1.p1.19">Given linear orders <math alttext="A" class="ltx_Math" display="inline" id="S1.p1.1.m1.1"><semantics id="S1.p1.1.m1.1a"><mi id="S1.p1.1.m1.1.1" xref="S1.p1.1.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S1.p1.1.m1.1b"><ci id="S1.p1.1.m1.1.1.cmml" xref="S1.p1.1.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.1.m1.1c">A</annotation><annotation encoding="application/x-llamapun" id="S1.p1.1.m1.1d">italic_A</annotation></semantics></math> and <math alttext="B" class="ltx_Math" display="inline" id="S1.p1.2.m2.1"><semantics id="S1.p1.2.m2.1a"><mi id="S1.p1.2.m2.1.1" xref="S1.p1.2.m2.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S1.p1.2.m2.1b"><ci id="S1.p1.2.m2.1.1.cmml" xref="S1.p1.2.m2.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.2.m2.1c">B</annotation><annotation encoding="application/x-llamapun" id="S1.p1.2.m2.1d">italic_B</annotation></semantics></math>, an <em class="ltx_emph ltx_font_italic" id="S1.p1.19.1">epimorphism</em> from <math alttext="B" class="ltx_Math" display="inline" id="S1.p1.3.m3.1"><semantics id="S1.p1.3.m3.1a"><mi id="S1.p1.3.m3.1.1" xref="S1.p1.3.m3.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S1.p1.3.m3.1b"><ci id="S1.p1.3.m3.1.1.cmml" xref="S1.p1.3.m3.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.3.m3.1c">B</annotation><annotation encoding="application/x-llamapun" id="S1.p1.3.m3.1d">italic_B</annotation></semantics></math> onto <math alttext="A" class="ltx_Math" display="inline" id="S1.p1.4.m4.1"><semantics id="S1.p1.4.m4.1a"><mi id="S1.p1.4.m4.1.1" xref="S1.p1.4.m4.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S1.p1.4.m4.1b"><ci id="S1.p1.4.m4.1.1.cmml" xref="S1.p1.4.m4.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.4.m4.1c">A</annotation><annotation encoding="application/x-llamapun" id="S1.p1.4.m4.1d">italic_A</annotation></semantics></math> is any montone surjective function <math alttext="f:B\twoheadrightarrow A" class="ltx_Math" display="inline" id="S1.p1.5.m5.1"><semantics id="S1.p1.5.m5.1a"><mrow id="S1.p1.5.m5.1.1" xref="S1.p1.5.m5.1.1.cmml"><mi id="S1.p1.5.m5.1.1.2" xref="S1.p1.5.m5.1.1.2.cmml">f</mi><mo id="S1.p1.5.m5.1.1.1" lspace="0.278em" rspace="0.278em" xref="S1.p1.5.m5.1.1.1.cmml">:</mo><mrow id="S1.p1.5.m5.1.1.3" xref="S1.p1.5.m5.1.1.3.cmml"><mi id="S1.p1.5.m5.1.1.3.2" xref="S1.p1.5.m5.1.1.3.2.cmml">B</mi><mo id="S1.p1.5.m5.1.1.3.1" stretchy="false" xref="S1.p1.5.m5.1.1.3.1.cmml">↠</mo><mi id="S1.p1.5.m5.1.1.3.3" xref="S1.p1.5.m5.1.1.3.3.cmml">A</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p1.5.m5.1b"><apply id="S1.p1.5.m5.1.1.cmml" xref="S1.p1.5.m5.1.1"><ci id="S1.p1.5.m5.1.1.1.cmml" xref="S1.p1.5.m5.1.1.1">:</ci><ci id="S1.p1.5.m5.1.1.2.cmml" xref="S1.p1.5.m5.1.1.2">𝑓</ci><apply id="S1.p1.5.m5.1.1.3.cmml" xref="S1.p1.5.m5.1.1.3"><ci id="S1.p1.5.m5.1.1.3.1.cmml" xref="S1.p1.5.m5.1.1.3.1">↠</ci><ci id="S1.p1.5.m5.1.1.3.2.cmml" xref="S1.p1.5.m5.1.1.3.2">𝐵</ci><ci id="S1.p1.5.m5.1.1.3.3.cmml" xref="S1.p1.5.m5.1.1.3.3">𝐴</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.5.m5.1c">f:B\twoheadrightarrow A</annotation><annotation encoding="application/x-llamapun" id="S1.p1.5.m5.1d">italic_f : italic_B ↠ italic_A</annotation></semantics></math>. Let <math alttext="A\trianglelefteq B" class="ltx_Math" display="inline" id="S1.p1.6.m6.1"><semantics id="S1.p1.6.m6.1a"><mrow id="S1.p1.6.m6.1.1" xref="S1.p1.6.m6.1.1.cmml"><mi id="S1.p1.6.m6.1.1.2" xref="S1.p1.6.m6.1.1.2.cmml">A</mi><mo id="S1.p1.6.m6.1.1.1" xref="S1.p1.6.m6.1.1.1.cmml">⁢</mo><mi id="S1.p1.6.m6.1.1.3" mathvariant="normal" xref="S1.p1.6.m6.1.1.3.cmml">⊴</mi><mo id="S1.p1.6.m6.1.1.1a" xref="S1.p1.6.m6.1.1.1.cmml">⁢</mo><mi id="S1.p1.6.m6.1.1.4" xref="S1.p1.6.m6.1.1.4.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p1.6.m6.1b"><apply id="S1.p1.6.m6.1.1.cmml" xref="S1.p1.6.m6.1.1"><times id="S1.p1.6.m6.1.1.1.cmml" xref="S1.p1.6.m6.1.1.1"></times><ci id="S1.p1.6.m6.1.1.2.cmml" xref="S1.p1.6.m6.1.1.2">𝐴</ci><ci id="S1.p1.6.m6.1.1.3.cmml" xref="S1.p1.6.m6.1.1.3">⊴</ci><ci id="S1.p1.6.m6.1.1.4.cmml" xref="S1.p1.6.m6.1.1.4">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.6.m6.1c">A\trianglelefteq B</annotation><annotation encoding="application/x-llamapun" id="S1.p1.6.m6.1d">italic_A ⊴ italic_B</annotation></semantics></math> denote that either <math alttext="A=\varnothing" class="ltx_Math" display="inline" id="S1.p1.7.m7.1"><semantics id="S1.p1.7.m7.1a"><mrow id="S1.p1.7.m7.1.1" xref="S1.p1.7.m7.1.1.cmml"><mi id="S1.p1.7.m7.1.1.2" xref="S1.p1.7.m7.1.1.2.cmml">A</mi><mo id="S1.p1.7.m7.1.1.1" xref="S1.p1.7.m7.1.1.1.cmml">=</mo><mi id="S1.p1.7.m7.1.1.3" mathvariant="normal" xref="S1.p1.7.m7.1.1.3.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p1.7.m7.1b"><apply id="S1.p1.7.m7.1.1.cmml" xref="S1.p1.7.m7.1.1"><eq id="S1.p1.7.m7.1.1.1.cmml" xref="S1.p1.7.m7.1.1.1"></eq><ci id="S1.p1.7.m7.1.1.2.cmml" xref="S1.p1.7.m7.1.1.2">𝐴</ci><emptyset id="S1.p1.7.m7.1.1.3.cmml" xref="S1.p1.7.m7.1.1.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.7.m7.1c">A=\varnothing</annotation><annotation encoding="application/x-llamapun" id="S1.p1.7.m7.1d">italic_A = ∅</annotation></semantics></math> or there is an epimorphism from <math alttext="B" class="ltx_Math" display="inline" id="S1.p1.8.m8.1"><semantics id="S1.p1.8.m8.1a"><mi id="S1.p1.8.m8.1.1" xref="S1.p1.8.m8.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S1.p1.8.m8.1b"><ci id="S1.p1.8.m8.1.1.cmml" xref="S1.p1.8.m8.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.8.m8.1c">B</annotation><annotation encoding="application/x-llamapun" id="S1.p1.8.m8.1d">italic_B</annotation></semantics></math> onto <math alttext="A" class="ltx_Math" display="inline" id="S1.p1.9.m9.1"><semantics id="S1.p1.9.m9.1a"><mi id="S1.p1.9.m9.1.1" xref="S1.p1.9.m9.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S1.p1.9.m9.1b"><ci id="S1.p1.9.m9.1.1.cmml" xref="S1.p1.9.m9.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.9.m9.1c">A</annotation><annotation encoding="application/x-llamapun" id="S1.p1.9.m9.1d">italic_A</annotation></semantics></math>. Epimorphisms between linear orders where studied in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib12" title="">12</a>]</cite> and later revisited in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib4" title="">4</a>]</cite> and <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib5" title="">5</a>]</cite>. This relation is somewhat a dual of the <em class="ltx_emph ltx_font_italic" id="S1.p1.19.2">embeddability</em> relation: say that <math alttext="A" class="ltx_Math" display="inline" id="S1.p1.10.m10.1"><semantics id="S1.p1.10.m10.1a"><mi id="S1.p1.10.m10.1.1" xref="S1.p1.10.m10.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S1.p1.10.m10.1b"><ci id="S1.p1.10.m10.1.1.cmml" xref="S1.p1.10.m10.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.10.m10.1c">A</annotation><annotation encoding="application/x-llamapun" id="S1.p1.10.m10.1d">italic_A</annotation></semantics></math> <em class="ltx_emph ltx_font_italic" id="S1.p1.19.3">embeds</em> into <math alttext="B" class="ltx_Math" display="inline" id="S1.p1.11.m11.1"><semantics id="S1.p1.11.m11.1a"><mi id="S1.p1.11.m11.1.1" xref="S1.p1.11.m11.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S1.p1.11.m11.1b"><ci id="S1.p1.11.m11.1.1.cmml" xref="S1.p1.11.m11.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.11.m11.1c">B</annotation><annotation encoding="application/x-llamapun" id="S1.p1.11.m11.1d">italic_B</annotation></semantics></math> (denoted by <math alttext="A\preceq B" class="ltx_Math" display="inline" id="S1.p1.12.m12.1"><semantics id="S1.p1.12.m12.1a"><mrow id="S1.p1.12.m12.1.1" xref="S1.p1.12.m12.1.1.cmml"><mi id="S1.p1.12.m12.1.1.2" xref="S1.p1.12.m12.1.1.2.cmml">A</mi><mo id="S1.p1.12.m12.1.1.1" xref="S1.p1.12.m12.1.1.1.cmml">⪯</mo><mi id="S1.p1.12.m12.1.1.3" xref="S1.p1.12.m12.1.1.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p1.12.m12.1b"><apply id="S1.p1.12.m12.1.1.cmml" xref="S1.p1.12.m12.1.1"><csymbol cd="latexml" id="S1.p1.12.m12.1.1.1.cmml" xref="S1.p1.12.m12.1.1.1">precedes-or-equals</csymbol><ci id="S1.p1.12.m12.1.1.2.cmml" xref="S1.p1.12.m12.1.1.2">𝐴</ci><ci id="S1.p1.12.m12.1.1.3.cmml" xref="S1.p1.12.m12.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.12.m12.1c">A\preceq B</annotation><annotation encoding="application/x-llamapun" id="S1.p1.12.m12.1d">italic_A ⪯ italic_B</annotation></semantics></math>) if there is a monotone injective mapping <math alttext="f:A\to B" class="ltx_Math" display="inline" id="S1.p1.13.m13.1"><semantics id="S1.p1.13.m13.1a"><mrow id="S1.p1.13.m13.1.1" xref="S1.p1.13.m13.1.1.cmml"><mi id="S1.p1.13.m13.1.1.2" xref="S1.p1.13.m13.1.1.2.cmml">f</mi><mo id="S1.p1.13.m13.1.1.1" lspace="0.278em" rspace="0.278em" xref="S1.p1.13.m13.1.1.1.cmml">:</mo><mrow id="S1.p1.13.m13.1.1.3" xref="S1.p1.13.m13.1.1.3.cmml"><mi id="S1.p1.13.m13.1.1.3.2" xref="S1.p1.13.m13.1.1.3.2.cmml">A</mi><mo id="S1.p1.13.m13.1.1.3.1" stretchy="false" xref="S1.p1.13.m13.1.1.3.1.cmml">→</mo><mi id="S1.p1.13.m13.1.1.3.3" xref="S1.p1.13.m13.1.1.3.3.cmml">B</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p1.13.m13.1b"><apply id="S1.p1.13.m13.1.1.cmml" xref="S1.p1.13.m13.1.1"><ci id="S1.p1.13.m13.1.1.1.cmml" xref="S1.p1.13.m13.1.1.1">:</ci><ci id="S1.p1.13.m13.1.1.2.cmml" xref="S1.p1.13.m13.1.1.2">𝑓</ci><apply id="S1.p1.13.m13.1.1.3.cmml" xref="S1.p1.13.m13.1.1.3"><ci id="S1.p1.13.m13.1.1.3.1.cmml" xref="S1.p1.13.m13.1.1.3.1">→</ci><ci id="S1.p1.13.m13.1.1.3.2.cmml" xref="S1.p1.13.m13.1.1.3.2">𝐴</ci><ci id="S1.p1.13.m13.1.1.3.3.cmml" xref="S1.p1.13.m13.1.1.3.3">𝐵</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.13.m13.1c">f:A\to B</annotation><annotation encoding="application/x-llamapun" id="S1.p1.13.m13.1d">italic_f : italic_A → italic_B</annotation></semantics></math>. In this case we also say that <math alttext="A" class="ltx_Math" display="inline" id="S1.p1.14.m14.1"><semantics id="S1.p1.14.m14.1a"><mi id="S1.p1.14.m14.1.1" xref="S1.p1.14.m14.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S1.p1.14.m14.1b"><ci id="S1.p1.14.m14.1.1.cmml" xref="S1.p1.14.m14.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.14.m14.1c">A</annotation><annotation encoding="application/x-llamapun" id="S1.p1.14.m14.1d">italic_A</annotation></semantics></math> is a <em class="ltx_emph ltx_font_italic" id="S1.p1.19.4">suborder of</em> <math alttext="B" class="ltx_Math" display="inline" id="S1.p1.15.m15.1"><semantics id="S1.p1.15.m15.1a"><mi id="S1.p1.15.m15.1.1" xref="S1.p1.15.m15.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S1.p1.15.m15.1b"><ci id="S1.p1.15.m15.1.1.cmml" xref="S1.p1.15.m15.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.15.m15.1c">B</annotation><annotation encoding="application/x-llamapun" id="S1.p1.15.m15.1d">italic_B</annotation></semantics></math> or that <math alttext="B" class="ltx_Math" display="inline" id="S1.p1.16.m16.1"><semantics id="S1.p1.16.m16.1a"><mi id="S1.p1.16.m16.1.1" xref="S1.p1.16.m16.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S1.p1.16.m16.1b"><ci id="S1.p1.16.m16.1.1.cmml" xref="S1.p1.16.m16.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.16.m16.1c">B</annotation><annotation encoding="application/x-llamapun" id="S1.p1.16.m16.1d">italic_B</annotation></semantics></math> contains a copy of <math alttext="A" class="ltx_Math" display="inline" id="S1.p1.17.m17.1"><semantics id="S1.p1.17.m17.1a"><mi id="S1.p1.17.m17.1.1" xref="S1.p1.17.m17.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S1.p1.17.m17.1b"><ci id="S1.p1.17.m17.1.1.cmml" xref="S1.p1.17.m17.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.17.m17.1c">A</annotation><annotation encoding="application/x-llamapun" id="S1.p1.17.m17.1d">italic_A</annotation></semantics></math>. Note that <math alttext="A\trianglelefteq B" class="ltx_Math" display="inline" id="S1.p1.18.m18.1"><semantics id="S1.p1.18.m18.1a"><mrow id="S1.p1.18.m18.1.1" xref="S1.p1.18.m18.1.1.cmml"><mi id="S1.p1.18.m18.1.1.2" xref="S1.p1.18.m18.1.1.2.cmml">A</mi><mo id="S1.p1.18.m18.1.1.1" xref="S1.p1.18.m18.1.1.1.cmml">⁢</mo><mi id="S1.p1.18.m18.1.1.3" mathvariant="normal" xref="S1.p1.18.m18.1.1.3.cmml">⊴</mi><mo id="S1.p1.18.m18.1.1.1a" xref="S1.p1.18.m18.1.1.1.cmml">⁢</mo><mi id="S1.p1.18.m18.1.1.4" xref="S1.p1.18.m18.1.1.4.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p1.18.m18.1b"><apply id="S1.p1.18.m18.1.1.cmml" xref="S1.p1.18.m18.1.1"><times id="S1.p1.18.m18.1.1.1.cmml" xref="S1.p1.18.m18.1.1.1"></times><ci id="S1.p1.18.m18.1.1.2.cmml" xref="S1.p1.18.m18.1.1.2">𝐴</ci><ci id="S1.p1.18.m18.1.1.3.cmml" xref="S1.p1.18.m18.1.1.3">⊴</ci><ci id="S1.p1.18.m18.1.1.4.cmml" xref="S1.p1.18.m18.1.1.4">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.18.m18.1c">A\trianglelefteq B</annotation><annotation encoding="application/x-llamapun" id="S1.p1.18.m18.1d">italic_A ⊴ italic_B</annotation></semantics></math> implies <math alttext="A\preceq B" class="ltx_Math" display="inline" id="S1.p1.19.m19.1"><semantics id="S1.p1.19.m19.1a"><mrow id="S1.p1.19.m19.1.1" xref="S1.p1.19.m19.1.1.cmml"><mi id="S1.p1.19.m19.1.1.2" xref="S1.p1.19.m19.1.1.2.cmml">A</mi><mo id="S1.p1.19.m19.1.1.1" xref="S1.p1.19.m19.1.1.1.cmml">⪯</mo><mi id="S1.p1.19.m19.1.1.3" xref="S1.p1.19.m19.1.1.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p1.19.m19.1b"><apply id="S1.p1.19.m19.1.1.cmml" xref="S1.p1.19.m19.1.1"><csymbol cd="latexml" id="S1.p1.19.m19.1.1.1.cmml" xref="S1.p1.19.m19.1.1.1">precedes-or-equals</csymbol><ci id="S1.p1.19.m19.1.1.2.cmml" xref="S1.p1.19.m19.1.1.2">𝐴</ci><ci id="S1.p1.19.m19.1.1.3.cmml" xref="S1.p1.19.m19.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.19.m19.1c">A\preceq B</annotation><annotation encoding="application/x-llamapun" id="S1.p1.19.m19.1d">italic_A ⪯ italic_B</annotation></semantics></math>, while the converse is not true in general.</p> </div> <div class="ltx_para" id="S1.p2"> <p class="ltx_p" id="S1.p2.6">In this work we give an analysis of the structure of the class of Aronszajn lines under <math alttext="\trianglelefteq" class="ltx_Math" display="inline" id="S1.p2.1.m1.1"><semantics id="S1.p2.1.m1.1a"><mi id="S1.p2.1.m1.1.1" mathvariant="normal" xref="S1.p2.1.m1.1.1.cmml">⊴</mi><annotation-xml encoding="MathML-Content" id="S1.p2.1.m1.1b"><ci id="S1.p2.1.m1.1.1.cmml" xref="S1.p2.1.m1.1.1">⊴</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.1.m1.1c">\trianglelefteq</annotation><annotation encoding="application/x-llamapun" id="S1.p2.1.m1.1d">⊴</annotation></semantics></math>, and answer several questions that in our opinion present themselves naturally once understood the context and analogies at play. For this it will be particularly important to have an idea of the development of the <math alttext="\preceq" class="ltx_Math" display="inline" id="S1.p2.2.m2.1"><semantics id="S1.p2.2.m2.1a"><mo id="S1.p2.2.m2.1.1" xref="S1.p2.2.m2.1.1.cmml">⪯</mo><annotation-xml encoding="MathML-Content" id="S1.p2.2.m2.1b"><csymbol cd="latexml" id="S1.p2.2.m2.1.1.cmml" xref="S1.p2.2.m2.1.1">precedes-or-equals</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.2.m2.1c">\preceq</annotation><annotation encoding="application/x-llamapun" id="S1.p2.2.m2.1d">⪯</annotation></semantics></math> relation on linear orders, which has been widely studied, and has motivated many deep and important subjects in Set Theory. Examples of this are the formulation of Martin’s Axiom (<math alttext="\mathsf{MA}" class="ltx_Math" display="inline" id="S1.p2.3.m3.1"><semantics id="S1.p2.3.m3.1a"><mi id="S1.p2.3.m3.1.1" xref="S1.p2.3.m3.1.1.cmml">𝖬𝖠</mi><annotation-xml encoding="MathML-Content" id="S1.p2.3.m3.1b"><ci id="S1.p2.3.m3.1.1.cmml" xref="S1.p2.3.m3.1.1">𝖬𝖠</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.3.m3.1c">\mathsf{MA}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.3.m3.1d">sansserif_MA</annotation></semantics></math>), which was conceived by looking at the solution to Suslin’s Problem, and the Proper Forcing Axiom (<math alttext="\mathsf{PFA}" class="ltx_Math" display="inline" id="S1.p2.4.m4.1"><semantics id="S1.p2.4.m4.1a"><mi id="S1.p2.4.m4.1.1" xref="S1.p2.4.m4.1.1.cmml">𝖯𝖥𝖠</mi><annotation-xml encoding="MathML-Content" id="S1.p2.4.m4.1b"><ci id="S1.p2.4.m4.1.1.cmml" xref="S1.p2.4.m4.1.1">𝖯𝖥𝖠</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.4.m4.1c">\mathsf{PFA}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.4.m4.1d">sansserif_PFA</annotation></semantics></math>), whose formulation was influenced by Baumgartner’s theorem on <math alttext="\aleph_{1}" class="ltx_Math" display="inline" id="S1.p2.5.m5.1"><semantics id="S1.p2.5.m5.1a"><msub id="S1.p2.5.m5.1.1" xref="S1.p2.5.m5.1.1.cmml"><mi id="S1.p2.5.m5.1.1.2" mathvariant="normal" xref="S1.p2.5.m5.1.1.2.cmml">ℵ</mi><mn id="S1.p2.5.m5.1.1.3" xref="S1.p2.5.m5.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S1.p2.5.m5.1b"><apply id="S1.p2.5.m5.1.1.cmml" xref="S1.p2.5.m5.1.1"><csymbol cd="ambiguous" id="S1.p2.5.m5.1.1.1.cmml" xref="S1.p2.5.m5.1.1">subscript</csymbol><ci id="S1.p2.5.m5.1.1.2.cmml" xref="S1.p2.5.m5.1.1.2">ℵ</ci><cn id="S1.p2.5.m5.1.1.3.cmml" type="integer" xref="S1.p2.5.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.5.m5.1c">\aleph_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.5.m5.1d">roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-dense real orders, and later motivated modern methods such as the method of Minimal Walks and new set theoretic hypothesis of broad interest such as the Open Coloring Axiom. We proceed to give some historical and mathematical context on this development, and in such doing we shall see that an analogy between the class of Aronszajn lines and the countable linear orders is suggested under <math alttext="\mathsf{PFA}" class="ltx_Math" display="inline" id="S1.p2.6.m6.1"><semantics id="S1.p2.6.m6.1a"><mi id="S1.p2.6.m6.1.1" xref="S1.p2.6.m6.1.1.cmml">𝖯𝖥𝖠</mi><annotation-xml encoding="MathML-Content" id="S1.p2.6.m6.1b"><ci id="S1.p2.6.m6.1.1.cmml" xref="S1.p2.6.m6.1.1">𝖯𝖥𝖠</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.6.m6.1c">\mathsf{PFA}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.6.m6.1d">sansserif_PFA</annotation></semantics></math>.</p> </div> <section class="ltx_subsection" id="S1.SSx1"> <h3 class="ltx_title ltx_title_subsection">Historical and mathematical context</h3> <div class="ltx_para" id="S1.SSx1.p1"> <p class="ltx_p" id="S1.SSx1.p1.23">Let <math alttext="\mathcal{R}" class="ltx_Math" display="inline" id="S1.SSx1.p1.1.m1.1"><semantics id="S1.SSx1.p1.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S1.SSx1.p1.1.m1.1.1" xref="S1.SSx1.p1.1.m1.1.1.cmml">ℛ</mi><annotation-xml encoding="MathML-Content" id="S1.SSx1.p1.1.m1.1b"><ci id="S1.SSx1.p1.1.m1.1.1.cmml" xref="S1.SSx1.p1.1.m1.1.1">ℛ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p1.1.m1.1c">\mathcal{R}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p1.1.m1.1d">caligraphic_R</annotation></semantics></math> be either <math alttext="\trianglelefteq" class="ltx_Math" display="inline" id="S1.SSx1.p1.2.m2.1"><semantics id="S1.SSx1.p1.2.m2.1a"><mi id="S1.SSx1.p1.2.m2.1.1" mathvariant="normal" xref="S1.SSx1.p1.2.m2.1.1.cmml">⊴</mi><annotation-xml encoding="MathML-Content" id="S1.SSx1.p1.2.m2.1b"><ci id="S1.SSx1.p1.2.m2.1.1.cmml" xref="S1.SSx1.p1.2.m2.1.1">⊴</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p1.2.m2.1c">\trianglelefteq</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p1.2.m2.1d">⊴</annotation></semantics></math> or <math alttext="\preceq" class="ltx_Math" display="inline" id="S1.SSx1.p1.3.m3.1"><semantics id="S1.SSx1.p1.3.m3.1a"><mo id="S1.SSx1.p1.3.m3.1.1" xref="S1.SSx1.p1.3.m3.1.1.cmml">⪯</mo><annotation-xml encoding="MathML-Content" id="S1.SSx1.p1.3.m3.1b"><csymbol cd="latexml" id="S1.SSx1.p1.3.m3.1.1.cmml" xref="S1.SSx1.p1.3.m3.1.1">precedes-or-equals</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p1.3.m3.1c">\preceq</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p1.3.m3.1d">⪯</annotation></semantics></math>. Clearly <math alttext="\mathcal{R}" class="ltx_Math" display="inline" id="S1.SSx1.p1.4.m4.1"><semantics id="S1.SSx1.p1.4.m4.1a"><mi class="ltx_font_mathcaligraphic" id="S1.SSx1.p1.4.m4.1.1" xref="S1.SSx1.p1.4.m4.1.1.cmml">ℛ</mi><annotation-xml encoding="MathML-Content" id="S1.SSx1.p1.4.m4.1b"><ci id="S1.SSx1.p1.4.m4.1.1.cmml" xref="S1.SSx1.p1.4.m4.1.1">ℛ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p1.4.m4.1c">\mathcal{R}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p1.4.m4.1d">caligraphic_R</annotation></semantics></math> is a preorder in the class of linear orders. We say that <math alttext="A" class="ltx_Math" display="inline" id="S1.SSx1.p1.5.m5.1"><semantics id="S1.SSx1.p1.5.m5.1a"><mi id="S1.SSx1.p1.5.m5.1.1" xref="S1.SSx1.p1.5.m5.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S1.SSx1.p1.5.m5.1b"><ci id="S1.SSx1.p1.5.m5.1.1.cmml" xref="S1.SSx1.p1.5.m5.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p1.5.m5.1c">A</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p1.5.m5.1d">italic_A</annotation></semantics></math> and <math alttext="B" class="ltx_Math" display="inline" id="S1.SSx1.p1.6.m6.1"><semantics id="S1.SSx1.p1.6.m6.1a"><mi id="S1.SSx1.p1.6.m6.1.1" xref="S1.SSx1.p1.6.m6.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S1.SSx1.p1.6.m6.1b"><ci id="S1.SSx1.p1.6.m6.1.1.cmml" xref="S1.SSx1.p1.6.m6.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p1.6.m6.1c">B</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p1.6.m6.1d">italic_B</annotation></semantics></math> are <em class="ltx_emph ltx_font_italic" id="S1.SSx1.p1.7.1"><math alttext="\mathcal{R}" class="ltx_Math" display="inline" id="S1.SSx1.p1.7.1.m1.1"><semantics id="S1.SSx1.p1.7.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S1.SSx1.p1.7.1.m1.1.1" xref="S1.SSx1.p1.7.1.m1.1.1.cmml">ℛ</mi><annotation-xml encoding="MathML-Content" id="S1.SSx1.p1.7.1.m1.1b"><ci id="S1.SSx1.p1.7.1.m1.1.1.cmml" xref="S1.SSx1.p1.7.1.m1.1.1">ℛ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p1.7.1.m1.1c">\mathcal{R}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p1.7.1.m1.1d">caligraphic_R</annotation></semantics></math>-equivalent</em> if <math alttext="A\mathrel{\mathcal{R}}B" class="ltx_Math" display="inline" id="S1.SSx1.p1.8.m7.1"><semantics id="S1.SSx1.p1.8.m7.1a"><mrow id="S1.SSx1.p1.8.m7.1.1" xref="S1.SSx1.p1.8.m7.1.1.cmml"><mi id="S1.SSx1.p1.8.m7.1.1.2" xref="S1.SSx1.p1.8.m7.1.1.2.cmml">A</mi><mo class="ltx_font_mathcaligraphic" id="S1.SSx1.p1.8.m7.1.1.1" xref="S1.SSx1.p1.8.m7.1.1.1.cmml">ℛ</mo><mi id="S1.SSx1.p1.8.m7.1.1.3" xref="S1.SSx1.p1.8.m7.1.1.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.SSx1.p1.8.m7.1b"><apply id="S1.SSx1.p1.8.m7.1.1.cmml" xref="S1.SSx1.p1.8.m7.1.1"><ci id="S1.SSx1.p1.8.m7.1.1.1.cmml" xref="S1.SSx1.p1.8.m7.1.1.1">ℛ</ci><ci id="S1.SSx1.p1.8.m7.1.1.2.cmml" xref="S1.SSx1.p1.8.m7.1.1.2">𝐴</ci><ci id="S1.SSx1.p1.8.m7.1.1.3.cmml" xref="S1.SSx1.p1.8.m7.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p1.8.m7.1c">A\mathrel{\mathcal{R}}B</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p1.8.m7.1d">italic_A caligraphic_R italic_B</annotation></semantics></math> and <math alttext="B\mathrel{\mathcal{R}}A" class="ltx_Math" display="inline" id="S1.SSx1.p1.9.m8.1"><semantics id="S1.SSx1.p1.9.m8.1a"><mrow id="S1.SSx1.p1.9.m8.1.1" xref="S1.SSx1.p1.9.m8.1.1.cmml"><mi id="S1.SSx1.p1.9.m8.1.1.2" xref="S1.SSx1.p1.9.m8.1.1.2.cmml">B</mi><mo class="ltx_font_mathcaligraphic" id="S1.SSx1.p1.9.m8.1.1.1" xref="S1.SSx1.p1.9.m8.1.1.1.cmml">ℛ</mo><mi id="S1.SSx1.p1.9.m8.1.1.3" xref="S1.SSx1.p1.9.m8.1.1.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.SSx1.p1.9.m8.1b"><apply id="S1.SSx1.p1.9.m8.1.1.cmml" xref="S1.SSx1.p1.9.m8.1.1"><ci id="S1.SSx1.p1.9.m8.1.1.1.cmml" xref="S1.SSx1.p1.9.m8.1.1.1">ℛ</ci><ci id="S1.SSx1.p1.9.m8.1.1.2.cmml" xref="S1.SSx1.p1.9.m8.1.1.2">𝐵</ci><ci id="S1.SSx1.p1.9.m8.1.1.3.cmml" xref="S1.SSx1.p1.9.m8.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p1.9.m8.1c">B\mathrel{\mathcal{R}}A</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p1.9.m8.1d">italic_B caligraphic_R italic_A</annotation></semantics></math>. Let <math alttext="\mathfrak{C}" class="ltx_Math" display="inline" id="S1.SSx1.p1.10.m9.1"><semantics id="S1.SSx1.p1.10.m9.1a"><mi id="S1.SSx1.p1.10.m9.1.1" xref="S1.SSx1.p1.10.m9.1.1.cmml">ℭ</mi><annotation-xml encoding="MathML-Content" id="S1.SSx1.p1.10.m9.1b"><ci id="S1.SSx1.p1.10.m9.1.1.cmml" xref="S1.SSx1.p1.10.m9.1.1">ℭ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p1.10.m9.1c">\mathfrak{C}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p1.10.m9.1d">fraktur_C</annotation></semantics></math> be a subclass of the linear orders. We say that <math alttext="A" class="ltx_Math" display="inline" id="S1.SSx1.p1.11.m10.1"><semantics id="S1.SSx1.p1.11.m10.1a"><mi id="S1.SSx1.p1.11.m10.1.1" xref="S1.SSx1.p1.11.m10.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S1.SSx1.p1.11.m10.1b"><ci id="S1.SSx1.p1.11.m10.1.1.cmml" xref="S1.SSx1.p1.11.m10.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p1.11.m10.1c">A</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p1.11.m10.1d">italic_A</annotation></semantics></math> is <em class="ltx_emph ltx_font_italic" id="S1.SSx1.p1.12.2"><math alttext="\mathcal{R}" class="ltx_Math" display="inline" id="S1.SSx1.p1.12.2.m1.1"><semantics id="S1.SSx1.p1.12.2.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S1.SSx1.p1.12.2.m1.1.1" xref="S1.SSx1.p1.12.2.m1.1.1.cmml">ℛ</mi><annotation-xml encoding="MathML-Content" id="S1.SSx1.p1.12.2.m1.1b"><ci id="S1.SSx1.p1.12.2.m1.1.1.cmml" xref="S1.SSx1.p1.12.2.m1.1.1">ℛ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p1.12.2.m1.1c">\mathcal{R}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p1.12.2.m1.1d">caligraphic_R</annotation></semantics></math>-minimal</em> in <math alttext="\mathfrak{C}" class="ltx_Math" display="inline" id="S1.SSx1.p1.13.m11.1"><semantics id="S1.SSx1.p1.13.m11.1a"><mi id="S1.SSx1.p1.13.m11.1.1" xref="S1.SSx1.p1.13.m11.1.1.cmml">ℭ</mi><annotation-xml encoding="MathML-Content" id="S1.SSx1.p1.13.m11.1b"><ci id="S1.SSx1.p1.13.m11.1.1.cmml" xref="S1.SSx1.p1.13.m11.1.1">ℭ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p1.13.m11.1c">\mathfrak{C}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p1.13.m11.1d">fraktur_C</annotation></semantics></math> if <math alttext="B\mathrel{\mathcal{R}}A" class="ltx_Math" display="inline" id="S1.SSx1.p1.14.m12.1"><semantics id="S1.SSx1.p1.14.m12.1a"><mrow id="S1.SSx1.p1.14.m12.1.1" xref="S1.SSx1.p1.14.m12.1.1.cmml"><mi id="S1.SSx1.p1.14.m12.1.1.2" xref="S1.SSx1.p1.14.m12.1.1.2.cmml">B</mi><mo class="ltx_font_mathcaligraphic" id="S1.SSx1.p1.14.m12.1.1.1" xref="S1.SSx1.p1.14.m12.1.1.1.cmml">ℛ</mo><mi id="S1.SSx1.p1.14.m12.1.1.3" xref="S1.SSx1.p1.14.m12.1.1.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.SSx1.p1.14.m12.1b"><apply id="S1.SSx1.p1.14.m12.1.1.cmml" xref="S1.SSx1.p1.14.m12.1.1"><ci id="S1.SSx1.p1.14.m12.1.1.1.cmml" xref="S1.SSx1.p1.14.m12.1.1.1">ℛ</ci><ci id="S1.SSx1.p1.14.m12.1.1.2.cmml" xref="S1.SSx1.p1.14.m12.1.1.2">𝐵</ci><ci id="S1.SSx1.p1.14.m12.1.1.3.cmml" xref="S1.SSx1.p1.14.m12.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p1.14.m12.1c">B\mathrel{\mathcal{R}}A</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p1.14.m12.1d">italic_B caligraphic_R italic_A</annotation></semantics></math> implies <math alttext="A\mathrel{\mathcal{R}}B" class="ltx_Math" display="inline" id="S1.SSx1.p1.15.m13.1"><semantics id="S1.SSx1.p1.15.m13.1a"><mrow id="S1.SSx1.p1.15.m13.1.1" xref="S1.SSx1.p1.15.m13.1.1.cmml"><mi id="S1.SSx1.p1.15.m13.1.1.2" xref="S1.SSx1.p1.15.m13.1.1.2.cmml">A</mi><mo class="ltx_font_mathcaligraphic" id="S1.SSx1.p1.15.m13.1.1.1" xref="S1.SSx1.p1.15.m13.1.1.1.cmml">ℛ</mo><mi id="S1.SSx1.p1.15.m13.1.1.3" xref="S1.SSx1.p1.15.m13.1.1.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.SSx1.p1.15.m13.1b"><apply id="S1.SSx1.p1.15.m13.1.1.cmml" xref="S1.SSx1.p1.15.m13.1.1"><ci id="S1.SSx1.p1.15.m13.1.1.1.cmml" xref="S1.SSx1.p1.15.m13.1.1.1">ℛ</ci><ci id="S1.SSx1.p1.15.m13.1.1.2.cmml" xref="S1.SSx1.p1.15.m13.1.1.2">𝐴</ci><ci id="S1.SSx1.p1.15.m13.1.1.3.cmml" xref="S1.SSx1.p1.15.m13.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p1.15.m13.1c">A\mathrel{\mathcal{R}}B</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p1.15.m13.1d">italic_A caligraphic_R italic_B</annotation></semantics></math> for every <math alttext="B\in\mathfrak{C}" class="ltx_Math" display="inline" id="S1.SSx1.p1.16.m14.1"><semantics id="S1.SSx1.p1.16.m14.1a"><mrow id="S1.SSx1.p1.16.m14.1.1" xref="S1.SSx1.p1.16.m14.1.1.cmml"><mi id="S1.SSx1.p1.16.m14.1.1.2" xref="S1.SSx1.p1.16.m14.1.1.2.cmml">B</mi><mo id="S1.SSx1.p1.16.m14.1.1.1" xref="S1.SSx1.p1.16.m14.1.1.1.cmml">∈</mo><mi id="S1.SSx1.p1.16.m14.1.1.3" xref="S1.SSx1.p1.16.m14.1.1.3.cmml">ℭ</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.SSx1.p1.16.m14.1b"><apply id="S1.SSx1.p1.16.m14.1.1.cmml" xref="S1.SSx1.p1.16.m14.1.1"><in id="S1.SSx1.p1.16.m14.1.1.1.cmml" xref="S1.SSx1.p1.16.m14.1.1.1"></in><ci id="S1.SSx1.p1.16.m14.1.1.2.cmml" xref="S1.SSx1.p1.16.m14.1.1.2">𝐵</ci><ci id="S1.SSx1.p1.16.m14.1.1.3.cmml" xref="S1.SSx1.p1.16.m14.1.1.3">ℭ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p1.16.m14.1c">B\in\mathfrak{C}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p1.16.m14.1d">italic_B ∈ fraktur_C</annotation></semantics></math>. An <em class="ltx_emph ltx_font_italic" id="S1.SSx1.p1.17.3"><math alttext="\mathcal{R}" class="ltx_Math" display="inline" id="S1.SSx1.p1.17.3.m1.1"><semantics id="S1.SSx1.p1.17.3.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S1.SSx1.p1.17.3.m1.1.1" xref="S1.SSx1.p1.17.3.m1.1.1.cmml">ℛ</mi><annotation-xml encoding="MathML-Content" id="S1.SSx1.p1.17.3.m1.1b"><ci id="S1.SSx1.p1.17.3.m1.1.1.cmml" xref="S1.SSx1.p1.17.3.m1.1.1">ℛ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p1.17.3.m1.1c">\mathcal{R}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p1.17.3.m1.1d">caligraphic_R</annotation></semantics></math>-basis</em> for <math alttext="\mathfrak{C}" class="ltx_Math" display="inline" id="S1.SSx1.p1.18.m15.1"><semantics id="S1.SSx1.p1.18.m15.1a"><mi id="S1.SSx1.p1.18.m15.1.1" xref="S1.SSx1.p1.18.m15.1.1.cmml">ℭ</mi><annotation-xml encoding="MathML-Content" id="S1.SSx1.p1.18.m15.1b"><ci id="S1.SSx1.p1.18.m15.1.1.cmml" xref="S1.SSx1.p1.18.m15.1.1">ℭ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p1.18.m15.1c">\mathfrak{C}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p1.18.m15.1d">fraktur_C</annotation></semantics></math> is a subset <math alttext="F" class="ltx_Math" display="inline" id="S1.SSx1.p1.19.m16.1"><semantics id="S1.SSx1.p1.19.m16.1a"><mi id="S1.SSx1.p1.19.m16.1.1" xref="S1.SSx1.p1.19.m16.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="S1.SSx1.p1.19.m16.1b"><ci id="S1.SSx1.p1.19.m16.1.1.cmml" xref="S1.SSx1.p1.19.m16.1.1">𝐹</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p1.19.m16.1c">F</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p1.19.m16.1d">italic_F</annotation></semantics></math> of <math alttext="\mathfrak{C}" class="ltx_Math" display="inline" id="S1.SSx1.p1.20.m17.1"><semantics id="S1.SSx1.p1.20.m17.1a"><mi id="S1.SSx1.p1.20.m17.1.1" xref="S1.SSx1.p1.20.m17.1.1.cmml">ℭ</mi><annotation-xml encoding="MathML-Content" id="S1.SSx1.p1.20.m17.1b"><ci id="S1.SSx1.p1.20.m17.1.1.cmml" xref="S1.SSx1.p1.20.m17.1.1">ℭ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p1.20.m17.1c">\mathfrak{C}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p1.20.m17.1d">fraktur_C</annotation></semantics></math> such that for every <math alttext="A\in\mathfrak{C}" class="ltx_Math" display="inline" id="S1.SSx1.p1.21.m18.1"><semantics id="S1.SSx1.p1.21.m18.1a"><mrow id="S1.SSx1.p1.21.m18.1.1" xref="S1.SSx1.p1.21.m18.1.1.cmml"><mi id="S1.SSx1.p1.21.m18.1.1.2" xref="S1.SSx1.p1.21.m18.1.1.2.cmml">A</mi><mo id="S1.SSx1.p1.21.m18.1.1.1" xref="S1.SSx1.p1.21.m18.1.1.1.cmml">∈</mo><mi id="S1.SSx1.p1.21.m18.1.1.3" xref="S1.SSx1.p1.21.m18.1.1.3.cmml">ℭ</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.SSx1.p1.21.m18.1b"><apply id="S1.SSx1.p1.21.m18.1.1.cmml" xref="S1.SSx1.p1.21.m18.1.1"><in id="S1.SSx1.p1.21.m18.1.1.1.cmml" xref="S1.SSx1.p1.21.m18.1.1.1"></in><ci id="S1.SSx1.p1.21.m18.1.1.2.cmml" xref="S1.SSx1.p1.21.m18.1.1.2">𝐴</ci><ci id="S1.SSx1.p1.21.m18.1.1.3.cmml" xref="S1.SSx1.p1.21.m18.1.1.3">ℭ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p1.21.m18.1c">A\in\mathfrak{C}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p1.21.m18.1d">italic_A ∈ fraktur_C</annotation></semantics></math>, there is <math alttext="X\in F" class="ltx_Math" display="inline" id="S1.SSx1.p1.22.m19.1"><semantics id="S1.SSx1.p1.22.m19.1a"><mrow id="S1.SSx1.p1.22.m19.1.1" xref="S1.SSx1.p1.22.m19.1.1.cmml"><mi id="S1.SSx1.p1.22.m19.1.1.2" xref="S1.SSx1.p1.22.m19.1.1.2.cmml">X</mi><mo id="S1.SSx1.p1.22.m19.1.1.1" xref="S1.SSx1.p1.22.m19.1.1.1.cmml">∈</mo><mi id="S1.SSx1.p1.22.m19.1.1.3" xref="S1.SSx1.p1.22.m19.1.1.3.cmml">F</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.SSx1.p1.22.m19.1b"><apply id="S1.SSx1.p1.22.m19.1.1.cmml" xref="S1.SSx1.p1.22.m19.1.1"><in id="S1.SSx1.p1.22.m19.1.1.1.cmml" xref="S1.SSx1.p1.22.m19.1.1.1"></in><ci id="S1.SSx1.p1.22.m19.1.1.2.cmml" xref="S1.SSx1.p1.22.m19.1.1.2">𝑋</ci><ci id="S1.SSx1.p1.22.m19.1.1.3.cmml" xref="S1.SSx1.p1.22.m19.1.1.3">𝐹</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p1.22.m19.1c">X\in F</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p1.22.m19.1d">italic_X ∈ italic_F</annotation></semantics></math> such that <math alttext="X\mathrel{\mathcal{R}}A" class="ltx_Math" display="inline" id="S1.SSx1.p1.23.m20.1"><semantics id="S1.SSx1.p1.23.m20.1a"><mrow id="S1.SSx1.p1.23.m20.1.1" xref="S1.SSx1.p1.23.m20.1.1.cmml"><mi id="S1.SSx1.p1.23.m20.1.1.2" xref="S1.SSx1.p1.23.m20.1.1.2.cmml">X</mi><mo class="ltx_font_mathcaligraphic" id="S1.SSx1.p1.23.m20.1.1.1" xref="S1.SSx1.p1.23.m20.1.1.1.cmml">ℛ</mo><mi id="S1.SSx1.p1.23.m20.1.1.3" xref="S1.SSx1.p1.23.m20.1.1.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.SSx1.p1.23.m20.1b"><apply id="S1.SSx1.p1.23.m20.1.1.cmml" xref="S1.SSx1.p1.23.m20.1.1"><ci id="S1.SSx1.p1.23.m20.1.1.1.cmml" xref="S1.SSx1.p1.23.m20.1.1.1">ℛ</ci><ci id="S1.SSx1.p1.23.m20.1.1.2.cmml" xref="S1.SSx1.p1.23.m20.1.1.2">𝑋</ci><ci id="S1.SSx1.p1.23.m20.1.1.3.cmml" xref="S1.SSx1.p1.23.m20.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p1.23.m20.1c">X\mathrel{\mathcal{R}}A</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p1.23.m20.1d">italic_X caligraphic_R italic_A</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S1.SSx1.p2"> <p class="ltx_p" id="S1.SSx1.p2.12">We now review the properties of <math alttext="\preceq" class="ltx_Math" display="inline" id="S1.SSx1.p2.1.m1.1"><semantics id="S1.SSx1.p2.1.m1.1a"><mo id="S1.SSx1.p2.1.m1.1.1" xref="S1.SSx1.p2.1.m1.1.1.cmml">⪯</mo><annotation-xml encoding="MathML-Content" id="S1.SSx1.p2.1.m1.1b"><csymbol cd="latexml" id="S1.SSx1.p2.1.m1.1.1.cmml" xref="S1.SSx1.p2.1.m1.1.1">precedes-or-equals</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p2.1.m1.1c">\preceq</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p2.1.m1.1d">⪯</annotation></semantics></math> in some classes of linear orders. Consider first the class of countable (infinite) linear orders. One easily sees that <math alttext="\omega" class="ltx_Math" display="inline" id="S1.SSx1.p2.2.m2.1"><semantics id="S1.SSx1.p2.2.m2.1a"><mi id="S1.SSx1.p2.2.m2.1.1" xref="S1.SSx1.p2.2.m2.1.1.cmml">ω</mi><annotation-xml encoding="MathML-Content" id="S1.SSx1.p2.2.m2.1b"><ci id="S1.SSx1.p2.2.m2.1.1.cmml" xref="S1.SSx1.p2.2.m2.1.1">𝜔</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p2.2.m2.1c">\omega</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p2.2.m2.1d">italic_ω</annotation></semantics></math> and <math alttext="\omega^{\star}" class="ltx_Math" display="inline" id="S1.SSx1.p2.3.m3.1"><semantics id="S1.SSx1.p2.3.m3.1a"><msup id="S1.SSx1.p2.3.m3.1.1" xref="S1.SSx1.p2.3.m3.1.1.cmml"><mi id="S1.SSx1.p2.3.m3.1.1.2" xref="S1.SSx1.p2.3.m3.1.1.2.cmml">ω</mi><mo id="S1.SSx1.p2.3.m3.1.1.3" xref="S1.SSx1.p2.3.m3.1.1.3.cmml">⋆</mo></msup><annotation-xml encoding="MathML-Content" id="S1.SSx1.p2.3.m3.1b"><apply id="S1.SSx1.p2.3.m3.1.1.cmml" xref="S1.SSx1.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S1.SSx1.p2.3.m3.1.1.1.cmml" xref="S1.SSx1.p2.3.m3.1.1">superscript</csymbol><ci id="S1.SSx1.p2.3.m3.1.1.2.cmml" xref="S1.SSx1.p2.3.m3.1.1.2">𝜔</ci><ci id="S1.SSx1.p2.3.m3.1.1.3.cmml" xref="S1.SSx1.p2.3.m3.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p2.3.m3.1c">\omega^{\star}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p2.3.m3.1d">italic_ω start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_note ltx_role_footnote" id="footnote1"><sup class="ltx_note_mark">1</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">1</sup><span class="ltx_tag ltx_tag_note">1</span>For a linear order <math alttext="A" class="ltx_Math" display="inline" id="footnote1.m1.1"><semantics id="footnote1.m1.1b"><mi id="footnote1.m1.1.1" xref="footnote1.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="footnote1.m1.1c"><ci id="footnote1.m1.1.1.cmml" xref="footnote1.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="footnote1.m1.1d">A</annotation><annotation encoding="application/x-llamapun" id="footnote1.m1.1e">italic_A</annotation></semantics></math>, <math alttext="A^{\star}" class="ltx_Math" display="inline" id="footnote1.m2.1"><semantics id="footnote1.m2.1b"><msup id="footnote1.m2.1.1" xref="footnote1.m2.1.1.cmml"><mi id="footnote1.m2.1.1.2" xref="footnote1.m2.1.1.2.cmml">A</mi><mo id="footnote1.m2.1.1.3" xref="footnote1.m2.1.1.3.cmml">⋆</mo></msup><annotation-xml encoding="MathML-Content" id="footnote1.m2.1c"><apply id="footnote1.m2.1.1.cmml" xref="footnote1.m2.1.1"><csymbol cd="ambiguous" id="footnote1.m2.1.1.1.cmml" xref="footnote1.m2.1.1">superscript</csymbol><ci id="footnote1.m2.1.1.2.cmml" xref="footnote1.m2.1.1.2">𝐴</ci><ci id="footnote1.m2.1.1.3.cmml" xref="footnote1.m2.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote1.m2.1d">A^{\star}</annotation><annotation encoding="application/x-llamapun" id="footnote1.m2.1e">italic_A start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math> denotes its reverse, i.e., the linear order defined by <math alttext="a<_{A^{\star}}b" class="ltx_Math" display="inline" id="footnote1.m3.1"><semantics id="footnote1.m3.1b"><mrow id="footnote1.m3.1.1" xref="footnote1.m3.1.1.cmml"><mi id="footnote1.m3.1.1.2" xref="footnote1.m3.1.1.2.cmml">a</mi><msub id="footnote1.m3.1.1.1" xref="footnote1.m3.1.1.1.cmml"><mo id="footnote1.m3.1.1.1.2" xref="footnote1.m3.1.1.1.2.cmml"><</mo><msup id="footnote1.m3.1.1.1.3" xref="footnote1.m3.1.1.1.3.cmml"><mi id="footnote1.m3.1.1.1.3.2" xref="footnote1.m3.1.1.1.3.2.cmml">A</mi><mo id="footnote1.m3.1.1.1.3.3" xref="footnote1.m3.1.1.1.3.3.cmml">⋆</mo></msup></msub><mi id="footnote1.m3.1.1.3" xref="footnote1.m3.1.1.3.cmml">b</mi></mrow><annotation-xml encoding="MathML-Content" id="footnote1.m3.1c"><apply id="footnote1.m3.1.1.cmml" xref="footnote1.m3.1.1"><apply id="footnote1.m3.1.1.1.cmml" xref="footnote1.m3.1.1.1"><csymbol cd="ambiguous" id="footnote1.m3.1.1.1.1.cmml" xref="footnote1.m3.1.1.1">subscript</csymbol><lt id="footnote1.m3.1.1.1.2.cmml" xref="footnote1.m3.1.1.1.2"></lt><apply id="footnote1.m3.1.1.1.3.cmml" xref="footnote1.m3.1.1.1.3"><csymbol cd="ambiguous" id="footnote1.m3.1.1.1.3.1.cmml" xref="footnote1.m3.1.1.1.3">superscript</csymbol><ci id="footnote1.m3.1.1.1.3.2.cmml" xref="footnote1.m3.1.1.1.3.2">𝐴</ci><ci id="footnote1.m3.1.1.1.3.3.cmml" xref="footnote1.m3.1.1.1.3.3">⋆</ci></apply></apply><ci id="footnote1.m3.1.1.2.cmml" xref="footnote1.m3.1.1.2">𝑎</ci><ci id="footnote1.m3.1.1.3.cmml" xref="footnote1.m3.1.1.3">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote1.m3.1d">a<_{A^{\star}}b</annotation><annotation encoding="application/x-llamapun" id="footnote1.m3.1e">italic_a < start_POSTSUBSCRIPT italic_A start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_b</annotation></semantics></math> iff <math alttext="b<_{A}a" class="ltx_Math" display="inline" id="footnote1.m4.1"><semantics id="footnote1.m4.1b"><mrow id="footnote1.m4.1.1" xref="footnote1.m4.1.1.cmml"><mi id="footnote1.m4.1.1.2" xref="footnote1.m4.1.1.2.cmml">b</mi><msub id="footnote1.m4.1.1.1" xref="footnote1.m4.1.1.1.cmml"><mo id="footnote1.m4.1.1.1.2" xref="footnote1.m4.1.1.1.2.cmml"><</mo><mi id="footnote1.m4.1.1.1.3" xref="footnote1.m4.1.1.1.3.cmml">A</mi></msub><mi id="footnote1.m4.1.1.3" xref="footnote1.m4.1.1.3.cmml">a</mi></mrow><annotation-xml encoding="MathML-Content" id="footnote1.m4.1c"><apply id="footnote1.m4.1.1.cmml" xref="footnote1.m4.1.1"><apply id="footnote1.m4.1.1.1.cmml" xref="footnote1.m4.1.1.1"><csymbol cd="ambiguous" id="footnote1.m4.1.1.1.1.cmml" xref="footnote1.m4.1.1.1">subscript</csymbol><lt id="footnote1.m4.1.1.1.2.cmml" xref="footnote1.m4.1.1.1.2"></lt><ci id="footnote1.m4.1.1.1.3.cmml" xref="footnote1.m4.1.1.1.3">𝐴</ci></apply><ci id="footnote1.m4.1.1.2.cmml" xref="footnote1.m4.1.1.2">𝑏</ci><ci id="footnote1.m4.1.1.3.cmml" xref="footnote1.m4.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote1.m4.1d">b<_{A}a</annotation><annotation encoding="application/x-llamapun" id="footnote1.m4.1e">italic_b < start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_a</annotation></semantics></math>.</span></span></span> are the unique (modulo order isomorphism) <math alttext="\preceq" class="ltx_Math" display="inline" id="S1.SSx1.p2.4.m4.1"><semantics id="S1.SSx1.p2.4.m4.1a"><mo id="S1.SSx1.p2.4.m4.1.1" xref="S1.SSx1.p2.4.m4.1.1.cmml">⪯</mo><annotation-xml encoding="MathML-Content" id="S1.SSx1.p2.4.m4.1b"><csymbol cd="latexml" id="S1.SSx1.p2.4.m4.1.1.cmml" xref="S1.SSx1.p2.4.m4.1.1">precedes-or-equals</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p2.4.m4.1c">\preceq</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p2.4.m4.1d">⪯</annotation></semantics></math>-minimal elements in this class. Recall that <math alttext="\omega" class="ltx_Math" display="inline" id="S1.SSx1.p2.5.m5.1"><semantics id="S1.SSx1.p2.5.m5.1a"><mi id="S1.SSx1.p2.5.m5.1.1" xref="S1.SSx1.p2.5.m5.1.1.cmml">ω</mi><annotation-xml encoding="MathML-Content" id="S1.SSx1.p2.5.m5.1b"><ci id="S1.SSx1.p2.5.m5.1.1.cmml" xref="S1.SSx1.p2.5.m5.1.1">𝜔</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p2.5.m5.1c">\omega</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p2.5.m5.1d">italic_ω</annotation></semantics></math> is the first infinite ordinal, isomorphic to the natural numbers. It is also easily checked that every infinite linear order contains a copy of <math alttext="\omega" class="ltx_Math" display="inline" id="S1.SSx1.p2.6.m6.1"><semantics id="S1.SSx1.p2.6.m6.1a"><mi id="S1.SSx1.p2.6.m6.1.1" xref="S1.SSx1.p2.6.m6.1.1.cmml">ω</mi><annotation-xml encoding="MathML-Content" id="S1.SSx1.p2.6.m6.1b"><ci id="S1.SSx1.p2.6.m6.1.1.cmml" xref="S1.SSx1.p2.6.m6.1.1">𝜔</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p2.6.m6.1c">\omega</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p2.6.m6.1d">italic_ω</annotation></semantics></math> or <math alttext="\omega^{\star}" class="ltx_Math" display="inline" id="S1.SSx1.p2.7.m7.1"><semantics id="S1.SSx1.p2.7.m7.1a"><msup id="S1.SSx1.p2.7.m7.1.1" xref="S1.SSx1.p2.7.m7.1.1.cmml"><mi id="S1.SSx1.p2.7.m7.1.1.2" xref="S1.SSx1.p2.7.m7.1.1.2.cmml">ω</mi><mo id="S1.SSx1.p2.7.m7.1.1.3" xref="S1.SSx1.p2.7.m7.1.1.3.cmml">⋆</mo></msup><annotation-xml encoding="MathML-Content" id="S1.SSx1.p2.7.m7.1b"><apply id="S1.SSx1.p2.7.m7.1.1.cmml" xref="S1.SSx1.p2.7.m7.1.1"><csymbol cd="ambiguous" id="S1.SSx1.p2.7.m7.1.1.1.cmml" xref="S1.SSx1.p2.7.m7.1.1">superscript</csymbol><ci id="S1.SSx1.p2.7.m7.1.1.2.cmml" xref="S1.SSx1.p2.7.m7.1.1.2">𝜔</ci><ci id="S1.SSx1.p2.7.m7.1.1.3.cmml" xref="S1.SSx1.p2.7.m7.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p2.7.m7.1c">\omega^{\star}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p2.7.m7.1d">italic_ω start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math>, thus these form a <math alttext="\preceq" class="ltx_Math" display="inline" id="S1.SSx1.p2.8.m8.1"><semantics id="S1.SSx1.p2.8.m8.1a"><mo id="S1.SSx1.p2.8.m8.1.1" xref="S1.SSx1.p2.8.m8.1.1.cmml">⪯</mo><annotation-xml encoding="MathML-Content" id="S1.SSx1.p2.8.m8.1b"><csymbol cd="latexml" id="S1.SSx1.p2.8.m8.1.1.cmml" xref="S1.SSx1.p2.8.m8.1.1">precedes-or-equals</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p2.8.m8.1c">\preceq</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p2.8.m8.1d">⪯</annotation></semantics></math>-basis for the class of countable linear orders. What more can be said about the structure of the countable linear orders under <math alttext="\preceq" class="ltx_Math" display="inline" id="S1.SSx1.p2.9.m9.1"><semantics id="S1.SSx1.p2.9.m9.1a"><mo id="S1.SSx1.p2.9.m9.1.1" xref="S1.SSx1.p2.9.m9.1.1.cmml">⪯</mo><annotation-xml encoding="MathML-Content" id="S1.SSx1.p2.9.m9.1b"><csymbol cd="latexml" id="S1.SSx1.p2.9.m9.1.1.cmml" xref="S1.SSx1.p2.9.m9.1.1">precedes-or-equals</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p2.9.m9.1c">\preceq</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p2.9.m9.1d">⪯</annotation></semantics></math>? For starters the class has a top element <math alttext="\mathbb{Q}" class="ltx_Math" display="inline" id="S1.SSx1.p2.10.m10.1"><semantics id="S1.SSx1.p2.10.m10.1a"><mi id="S1.SSx1.p2.10.m10.1.1" xref="S1.SSx1.p2.10.m10.1.1.cmml">ℚ</mi><annotation-xml encoding="MathML-Content" id="S1.SSx1.p2.10.m10.1b"><ci id="S1.SSx1.p2.10.m10.1.1.cmml" xref="S1.SSx1.p2.10.m10.1.1">ℚ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p2.10.m10.1c">\mathbb{Q}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p2.10.m10.1d">blackboard_Q</annotation></semantics></math>: every countable linear order embeds into <math alttext="\mathbb{Q}" class="ltx_Math" display="inline" id="S1.SSx1.p2.11.m11.1"><semantics id="S1.SSx1.p2.11.m11.1a"><mi id="S1.SSx1.p2.11.m11.1.1" xref="S1.SSx1.p2.11.m11.1.1.cmml">ℚ</mi><annotation-xml encoding="MathML-Content" id="S1.SSx1.p2.11.m11.1b"><ci id="S1.SSx1.p2.11.m11.1.1.cmml" xref="S1.SSx1.p2.11.m11.1.1">ℚ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p2.11.m11.1c">\mathbb{Q}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p2.11.m11.1d">blackboard_Q</annotation></semantics></math>. This follows from a theorem of Cantor, and its proof is maybe the first example of what is now called the back-and-forth method in logic. Even more can be said, a celebrated theorem of Laver <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib13" title="">13</a>]</cite>, that answers a conjecture by Fraïssé, says that the class of countable linear order is well-quasi-ordered by <math alttext="\preceq" class="ltx_Math" display="inline" id="S1.SSx1.p2.12.m12.1"><semantics id="S1.SSx1.p2.12.m12.1a"><mo id="S1.SSx1.p2.12.m12.1.1" xref="S1.SSx1.p2.12.m12.1.1.cmml">⪯</mo><annotation-xml encoding="MathML-Content" id="S1.SSx1.p2.12.m12.1b"><csymbol cd="latexml" id="S1.SSx1.p2.12.m12.1.1.cmml" xref="S1.SSx1.p2.12.m12.1.1">precedes-or-equals</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p2.12.m12.1c">\preceq</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p2.12.m12.1d">⪯</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="S1.Thmtheorem1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem1.1.1.1">Definition 1.1</span></span><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem1.2.2">.</span> </h6> <div class="ltx_para" id="S1.Thmtheorem1.p1"> <p class="ltx_p" id="S1.Thmtheorem1.p1.7">A class <math alttext="\mathfrak{C}" class="ltx_Math" display="inline" id="S1.Thmtheorem1.p1.1.m1.1"><semantics id="S1.Thmtheorem1.p1.1.m1.1a"><mi id="S1.Thmtheorem1.p1.1.m1.1.1" xref="S1.Thmtheorem1.p1.1.m1.1.1.cmml">ℭ</mi><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem1.p1.1.m1.1b"><ci id="S1.Thmtheorem1.p1.1.m1.1.1.cmml" xref="S1.Thmtheorem1.p1.1.m1.1.1">ℭ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem1.p1.1.m1.1c">\mathfrak{C}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem1.p1.1.m1.1d">fraktur_C</annotation></semantics></math> is <em class="ltx_emph ltx_font_italic" id="S1.Thmtheorem1.p1.7.1">well-quasi-ordered</em> by a preorder relation <math alttext="\mathcal{R}" class="ltx_Math" display="inline" id="S1.Thmtheorem1.p1.2.m2.1"><semantics id="S1.Thmtheorem1.p1.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem1.p1.2.m2.1.1" xref="S1.Thmtheorem1.p1.2.m2.1.1.cmml">ℛ</mi><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem1.p1.2.m2.1b"><ci id="S1.Thmtheorem1.p1.2.m2.1.1.cmml" xref="S1.Thmtheorem1.p1.2.m2.1.1">ℛ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem1.p1.2.m2.1c">\mathcal{R}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem1.p1.2.m2.1d">caligraphic_R</annotation></semantics></math>, if it contains no uncountable antichain (an uncountable subset of <math alttext="\mathfrak{C}" class="ltx_Math" display="inline" id="S1.Thmtheorem1.p1.3.m3.1"><semantics id="S1.Thmtheorem1.p1.3.m3.1a"><mi id="S1.Thmtheorem1.p1.3.m3.1.1" xref="S1.Thmtheorem1.p1.3.m3.1.1.cmml">ℭ</mi><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem1.p1.3.m3.1b"><ci id="S1.Thmtheorem1.p1.3.m3.1.1.cmml" xref="S1.Thmtheorem1.p1.3.m3.1.1">ℭ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem1.p1.3.m3.1c">\mathfrak{C}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem1.p1.3.m3.1d">fraktur_C</annotation></semantics></math> of pairwise <math alttext="\mathcal{R}" class="ltx_Math" display="inline" id="S1.Thmtheorem1.p1.4.m4.1"><semantics id="S1.Thmtheorem1.p1.4.m4.1a"><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem1.p1.4.m4.1.1" xref="S1.Thmtheorem1.p1.4.m4.1.1.cmml">ℛ</mi><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem1.p1.4.m4.1b"><ci id="S1.Thmtheorem1.p1.4.m4.1.1.cmml" xref="S1.Thmtheorem1.p1.4.m4.1.1">ℛ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem1.p1.4.m4.1c">\mathcal{R}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem1.p1.4.m4.1d">caligraphic_R</annotation></semantics></math>-incomparable elements), and no infinite decreasing sequence <math alttext="\cdots A_{2}\mathrel{\mathcal{R}}A_{1}\mathrel{\mathcal{R}}A_{0}" class="ltx_Math" display="inline" id="S1.Thmtheorem1.p1.5.m5.1"><semantics id="S1.Thmtheorem1.p1.5.m5.1a"><mrow id="S1.Thmtheorem1.p1.5.m5.1.1" xref="S1.Thmtheorem1.p1.5.m5.1.1.cmml"><mrow id="S1.Thmtheorem1.p1.5.m5.1.1.2" xref="S1.Thmtheorem1.p1.5.m5.1.1.2.cmml"><mi id="S1.Thmtheorem1.p1.5.m5.1.1.2.2" mathvariant="normal" xref="S1.Thmtheorem1.p1.5.m5.1.1.2.2.cmml">⋯</mi><mo id="S1.Thmtheorem1.p1.5.m5.1.1.2.1" xref="S1.Thmtheorem1.p1.5.m5.1.1.2.1.cmml">⁢</mo><msub id="S1.Thmtheorem1.p1.5.m5.1.1.2.3" xref="S1.Thmtheorem1.p1.5.m5.1.1.2.3.cmml"><mi id="S1.Thmtheorem1.p1.5.m5.1.1.2.3.2" xref="S1.Thmtheorem1.p1.5.m5.1.1.2.3.2.cmml">A</mi><mn id="S1.Thmtheorem1.p1.5.m5.1.1.2.3.3" xref="S1.Thmtheorem1.p1.5.m5.1.1.2.3.3.cmml">2</mn></msub></mrow><mo class="ltx_font_mathcaligraphic" id="S1.Thmtheorem1.p1.5.m5.1.1.3" xref="S1.Thmtheorem1.p1.5.m5.1.1.3.cmml">ℛ</mo><msub id="S1.Thmtheorem1.p1.5.m5.1.1.4" xref="S1.Thmtheorem1.p1.5.m5.1.1.4.cmml"><mi id="S1.Thmtheorem1.p1.5.m5.1.1.4.2" xref="S1.Thmtheorem1.p1.5.m5.1.1.4.2.cmml">A</mi><mn id="S1.Thmtheorem1.p1.5.m5.1.1.4.3" xref="S1.Thmtheorem1.p1.5.m5.1.1.4.3.cmml">1</mn></msub><mo class="ltx_font_mathcaligraphic" id="S1.Thmtheorem1.p1.5.m5.1.1.5" xref="S1.Thmtheorem1.p1.5.m5.1.1.5.cmml">ℛ</mo><msub id="S1.Thmtheorem1.p1.5.m5.1.1.6" xref="S1.Thmtheorem1.p1.5.m5.1.1.6.cmml"><mi id="S1.Thmtheorem1.p1.5.m5.1.1.6.2" xref="S1.Thmtheorem1.p1.5.m5.1.1.6.2.cmml">A</mi><mn id="S1.Thmtheorem1.p1.5.m5.1.1.6.3" xref="S1.Thmtheorem1.p1.5.m5.1.1.6.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem1.p1.5.m5.1b"><apply id="S1.Thmtheorem1.p1.5.m5.1.1.cmml" xref="S1.Thmtheorem1.p1.5.m5.1.1"><and id="S1.Thmtheorem1.p1.5.m5.1.1a.cmml" xref="S1.Thmtheorem1.p1.5.m5.1.1"></and><apply id="S1.Thmtheorem1.p1.5.m5.1.1b.cmml" xref="S1.Thmtheorem1.p1.5.m5.1.1"><ci id="S1.Thmtheorem1.p1.5.m5.1.1.3.cmml" xref="S1.Thmtheorem1.p1.5.m5.1.1.3">ℛ</ci><apply id="S1.Thmtheorem1.p1.5.m5.1.1.2.cmml" xref="S1.Thmtheorem1.p1.5.m5.1.1.2"><times id="S1.Thmtheorem1.p1.5.m5.1.1.2.1.cmml" xref="S1.Thmtheorem1.p1.5.m5.1.1.2.1"></times><ci id="S1.Thmtheorem1.p1.5.m5.1.1.2.2.cmml" xref="S1.Thmtheorem1.p1.5.m5.1.1.2.2">⋯</ci><apply id="S1.Thmtheorem1.p1.5.m5.1.1.2.3.cmml" xref="S1.Thmtheorem1.p1.5.m5.1.1.2.3"><csymbol cd="ambiguous" id="S1.Thmtheorem1.p1.5.m5.1.1.2.3.1.cmml" xref="S1.Thmtheorem1.p1.5.m5.1.1.2.3">subscript</csymbol><ci id="S1.Thmtheorem1.p1.5.m5.1.1.2.3.2.cmml" xref="S1.Thmtheorem1.p1.5.m5.1.1.2.3.2">𝐴</ci><cn id="S1.Thmtheorem1.p1.5.m5.1.1.2.3.3.cmml" type="integer" xref="S1.Thmtheorem1.p1.5.m5.1.1.2.3.3">2</cn></apply></apply><apply id="S1.Thmtheorem1.p1.5.m5.1.1.4.cmml" xref="S1.Thmtheorem1.p1.5.m5.1.1.4"><csymbol cd="ambiguous" id="S1.Thmtheorem1.p1.5.m5.1.1.4.1.cmml" xref="S1.Thmtheorem1.p1.5.m5.1.1.4">subscript</csymbol><ci id="S1.Thmtheorem1.p1.5.m5.1.1.4.2.cmml" xref="S1.Thmtheorem1.p1.5.m5.1.1.4.2">𝐴</ci><cn id="S1.Thmtheorem1.p1.5.m5.1.1.4.3.cmml" type="integer" xref="S1.Thmtheorem1.p1.5.m5.1.1.4.3">1</cn></apply></apply><apply id="S1.Thmtheorem1.p1.5.m5.1.1c.cmml" xref="S1.Thmtheorem1.p1.5.m5.1.1"><ci id="S1.Thmtheorem1.p1.5.m5.1.1.5.cmml" xref="S1.Thmtheorem1.p1.5.m5.1.1.5">ℛ</ci><share href="https://arxiv.org/html/2503.13728v1#S1.Thmtheorem1.p1.5.m5.1.1.4.cmml" id="S1.Thmtheorem1.p1.5.m5.1.1d.cmml" xref="S1.Thmtheorem1.p1.5.m5.1.1"></share><apply id="S1.Thmtheorem1.p1.5.m5.1.1.6.cmml" xref="S1.Thmtheorem1.p1.5.m5.1.1.6"><csymbol cd="ambiguous" id="S1.Thmtheorem1.p1.5.m5.1.1.6.1.cmml" xref="S1.Thmtheorem1.p1.5.m5.1.1.6">subscript</csymbol><ci id="S1.Thmtheorem1.p1.5.m5.1.1.6.2.cmml" xref="S1.Thmtheorem1.p1.5.m5.1.1.6.2">𝐴</ci><cn id="S1.Thmtheorem1.p1.5.m5.1.1.6.3.cmml" type="integer" xref="S1.Thmtheorem1.p1.5.m5.1.1.6.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem1.p1.5.m5.1c">\cdots A_{2}\mathrel{\mathcal{R}}A_{1}\mathrel{\mathcal{R}}A_{0}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem1.p1.5.m5.1d">⋯ italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT caligraphic_R italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT caligraphic_R italic_A start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> of non <math alttext="\mathcal{R}" class="ltx_Math" display="inline" id="S1.Thmtheorem1.p1.6.m6.1"><semantics id="S1.Thmtheorem1.p1.6.m6.1a"><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem1.p1.6.m6.1.1" xref="S1.Thmtheorem1.p1.6.m6.1.1.cmml">ℛ</mi><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem1.p1.6.m6.1b"><ci id="S1.Thmtheorem1.p1.6.m6.1.1.cmml" xref="S1.Thmtheorem1.p1.6.m6.1.1">ℛ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem1.p1.6.m6.1c">\mathcal{R}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem1.p1.6.m6.1d">caligraphic_R</annotation></semantics></math>-equivalent members of <math alttext="\mathfrak{C}" class="ltx_Math" display="inline" id="S1.Thmtheorem1.p1.7.m7.1"><semantics id="S1.Thmtheorem1.p1.7.m7.1a"><mi id="S1.Thmtheorem1.p1.7.m7.1.1" xref="S1.Thmtheorem1.p1.7.m7.1.1.cmml">ℭ</mi><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem1.p1.7.m7.1b"><ci id="S1.Thmtheorem1.p1.7.m7.1.1.cmml" xref="S1.Thmtheorem1.p1.7.m7.1.1">ℭ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem1.p1.7.m7.1c">\mathfrak{C}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem1.p1.7.m7.1d">fraktur_C</annotation></semantics></math>.</p> </div> </div> <div class="ltx_para" id="S1.SSx1.p3"> <p class="ltx_p" id="S1.SSx1.p3.1">Being well-quasi-ordered can be interpreted as having a structure that is as similar as possible to the structure of the class of ordinals under <math alttext="\preceq" class="ltx_Math" display="inline" id="S1.SSx1.p3.1.m1.1"><semantics id="S1.SSx1.p3.1.m1.1a"><mo id="S1.SSx1.p3.1.m1.1.1" xref="S1.SSx1.p3.1.m1.1.1.cmml">⪯</mo><annotation-xml encoding="MathML-Content" id="S1.SSx1.p3.1.m1.1b"><csymbol cd="latexml" id="S1.SSx1.p3.1.m1.1.1.cmml" xref="S1.SSx1.p3.1.m1.1.1">precedes-or-equals</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p3.1.m1.1c">\preceq</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p3.1.m1.1d">⪯</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S1.SSx1.p4"> <p class="ltx_p" id="S1.SSx1.p4.7">What about the class of uncountable linear orders? Since being well-quasi-ordered implies the existence of a finite <math alttext="\preceq" class="ltx_Math" display="inline" id="S1.SSx1.p4.1.m1.1"><semantics id="S1.SSx1.p4.1.m1.1a"><mo id="S1.SSx1.p4.1.m1.1.1" xref="S1.SSx1.p4.1.m1.1.1.cmml">⪯</mo><annotation-xml encoding="MathML-Content" id="S1.SSx1.p4.1.m1.1b"><csymbol cd="latexml" id="S1.SSx1.p4.1.m1.1.1.cmml" xref="S1.SSx1.p4.1.m1.1.1">precedes-or-equals</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p4.1.m1.1c">\preceq</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p4.1.m1.1d">⪯</annotation></semantics></math>-basis, let us start from there. The first difference is that not every uncountable linear order contains a copy of <math alttext="\omega_{1}" class="ltx_Math" display="inline" id="S1.SSx1.p4.2.m2.1"><semantics id="S1.SSx1.p4.2.m2.1a"><msub id="S1.SSx1.p4.2.m2.1.1" xref="S1.SSx1.p4.2.m2.1.1.cmml"><mi id="S1.SSx1.p4.2.m2.1.1.2" xref="S1.SSx1.p4.2.m2.1.1.2.cmml">ω</mi><mn id="S1.SSx1.p4.2.m2.1.1.3" xref="S1.SSx1.p4.2.m2.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S1.SSx1.p4.2.m2.1b"><apply id="S1.SSx1.p4.2.m2.1.1.cmml" xref="S1.SSx1.p4.2.m2.1.1"><csymbol cd="ambiguous" id="S1.SSx1.p4.2.m2.1.1.1.cmml" xref="S1.SSx1.p4.2.m2.1.1">subscript</csymbol><ci id="S1.SSx1.p4.2.m2.1.1.2.cmml" xref="S1.SSx1.p4.2.m2.1.1.2">𝜔</ci><cn id="S1.SSx1.p4.2.m2.1.1.3.cmml" type="integer" xref="S1.SSx1.p4.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p4.2.m2.1c">\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p4.2.m2.1d">italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> or <math alttext="\omega_{1}^{\star}" class="ltx_Math" display="inline" id="S1.SSx1.p4.3.m3.1"><semantics id="S1.SSx1.p4.3.m3.1a"><msubsup id="S1.SSx1.p4.3.m3.1.1" xref="S1.SSx1.p4.3.m3.1.1.cmml"><mi id="S1.SSx1.p4.3.m3.1.1.2.2" xref="S1.SSx1.p4.3.m3.1.1.2.2.cmml">ω</mi><mn id="S1.SSx1.p4.3.m3.1.1.2.3" xref="S1.SSx1.p4.3.m3.1.1.2.3.cmml">1</mn><mo id="S1.SSx1.p4.3.m3.1.1.3" xref="S1.SSx1.p4.3.m3.1.1.3.cmml">⋆</mo></msubsup><annotation-xml encoding="MathML-Content" id="S1.SSx1.p4.3.m3.1b"><apply id="S1.SSx1.p4.3.m3.1.1.cmml" xref="S1.SSx1.p4.3.m3.1.1"><csymbol cd="ambiguous" id="S1.SSx1.p4.3.m3.1.1.1.cmml" xref="S1.SSx1.p4.3.m3.1.1">superscript</csymbol><apply id="S1.SSx1.p4.3.m3.1.1.2.cmml" xref="S1.SSx1.p4.3.m3.1.1"><csymbol cd="ambiguous" id="S1.SSx1.p4.3.m3.1.1.2.1.cmml" xref="S1.SSx1.p4.3.m3.1.1">subscript</csymbol><ci id="S1.SSx1.p4.3.m3.1.1.2.2.cmml" xref="S1.SSx1.p4.3.m3.1.1.2.2">𝜔</ci><cn id="S1.SSx1.p4.3.m3.1.1.2.3.cmml" type="integer" xref="S1.SSx1.p4.3.m3.1.1.2.3">1</cn></apply><ci id="S1.SSx1.p4.3.m3.1.1.3.cmml" xref="S1.SSx1.p4.3.m3.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p4.3.m3.1c">\omega_{1}^{\star}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p4.3.m3.1d">italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math>, for example <math alttext="\mathbb{R}" class="ltx_Math" display="inline" id="S1.SSx1.p4.4.m4.1"><semantics id="S1.SSx1.p4.4.m4.1a"><mi id="S1.SSx1.p4.4.m4.1.1" xref="S1.SSx1.p4.4.m4.1.1.cmml">ℝ</mi><annotation-xml encoding="MathML-Content" id="S1.SSx1.p4.4.m4.1b"><ci id="S1.SSx1.p4.4.m4.1.1.cmml" xref="S1.SSx1.p4.4.m4.1.1">ℝ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p4.4.m4.1c">\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p4.4.m4.1d">blackboard_R</annotation></semantics></math> does not. Is there a finite <math alttext="\preceq" class="ltx_Math" display="inline" id="S1.SSx1.p4.5.m5.1"><semantics id="S1.SSx1.p4.5.m5.1a"><mo id="S1.SSx1.p4.5.m5.1.1" xref="S1.SSx1.p4.5.m5.1.1.cmml">⪯</mo><annotation-xml encoding="MathML-Content" id="S1.SSx1.p4.5.m5.1b"><csymbol cd="latexml" id="S1.SSx1.p4.5.m5.1.1.cmml" xref="S1.SSx1.p4.5.m5.1.1">precedes-or-equals</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p4.5.m5.1c">\preceq</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p4.5.m5.1d">⪯</annotation></semantics></math>-basis for the uncountable real orders? Consistently no: if <math alttext="\mathsf{CH}" class="ltx_Math" display="inline" id="S1.SSx1.p4.6.m6.1"><semantics id="S1.SSx1.p4.6.m6.1a"><mi id="S1.SSx1.p4.6.m6.1.1" xref="S1.SSx1.p4.6.m6.1.1.cmml">𝖢𝖧</mi><annotation-xml encoding="MathML-Content" id="S1.SSx1.p4.6.m6.1b"><ci id="S1.SSx1.p4.6.m6.1.1.cmml" xref="S1.SSx1.p4.6.m6.1.1">𝖢𝖧</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p4.6.m6.1c">\mathsf{CH}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p4.6.m6.1d">sansserif_CH</annotation></semantics></math> holds then every <math alttext="\preceq" class="ltx_Math" display="inline" id="S1.SSx1.p4.7.m7.1"><semantics id="S1.SSx1.p4.7.m7.1a"><mo id="S1.SSx1.p4.7.m7.1.1" xref="S1.SSx1.p4.7.m7.1.1.cmml">⪯</mo><annotation-xml encoding="MathML-Content" id="S1.SSx1.p4.7.m7.1b"><csymbol cd="latexml" id="S1.SSx1.p4.7.m7.1.1.cmml" xref="S1.SSx1.p4.7.m7.1.1">precedes-or-equals</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p4.7.m7.1c">\preceq</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p4.7.m7.1d">⪯</annotation></semantics></math>-basis for the uncountable real orders is infinite (see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib3" title="">3</a>]</cite>).</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="S1.Thmtheorem2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem2.1.1.1">Definition 1.2</span></span><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem2.2.2">.</span> </h6> <div class="ltx_para" id="S1.Thmtheorem2.p1"> <p class="ltx_p" id="S1.Thmtheorem2.p1.7">For an infinite cardinal <math alttext="\kappa" class="ltx_Math" display="inline" id="S1.Thmtheorem2.p1.1.m1.1"><semantics id="S1.Thmtheorem2.p1.1.m1.1a"><mi id="S1.Thmtheorem2.p1.1.m1.1.1" xref="S1.Thmtheorem2.p1.1.m1.1.1.cmml">κ</mi><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem2.p1.1.m1.1b"><ci id="S1.Thmtheorem2.p1.1.m1.1.1.cmml" xref="S1.Thmtheorem2.p1.1.m1.1.1">𝜅</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem2.p1.1.m1.1c">\kappa</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem2.p1.1.m1.1d">italic_κ</annotation></semantics></math>, a linear order <math alttext="A" class="ltx_Math" display="inline" id="S1.Thmtheorem2.p1.2.m2.1"><semantics id="S1.Thmtheorem2.p1.2.m2.1a"><mi id="S1.Thmtheorem2.p1.2.m2.1.1" xref="S1.Thmtheorem2.p1.2.m2.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem2.p1.2.m2.1b"><ci id="S1.Thmtheorem2.p1.2.m2.1.1.cmml" xref="S1.Thmtheorem2.p1.2.m2.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem2.p1.2.m2.1c">A</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem2.p1.2.m2.1d">italic_A</annotation></semantics></math> is called <em class="ltx_emph ltx_font_italic" id="S1.Thmtheorem2.p1.3.1"><math alttext="\kappa" class="ltx_Math" display="inline" id="S1.Thmtheorem2.p1.3.1.m1.1"><semantics id="S1.Thmtheorem2.p1.3.1.m1.1a"><mi id="S1.Thmtheorem2.p1.3.1.m1.1.1" xref="S1.Thmtheorem2.p1.3.1.m1.1.1.cmml">κ</mi><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem2.p1.3.1.m1.1b"><ci id="S1.Thmtheorem2.p1.3.1.m1.1.1.cmml" xref="S1.Thmtheorem2.p1.3.1.m1.1.1">𝜅</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem2.p1.3.1.m1.1c">\kappa</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem2.p1.3.1.m1.1d">italic_κ</annotation></semantics></math>-dense</em> if <math alttext="A" class="ltx_Math" display="inline" id="S1.Thmtheorem2.p1.4.m3.1"><semantics id="S1.Thmtheorem2.p1.4.m3.1a"><mi id="S1.Thmtheorem2.p1.4.m3.1.1" xref="S1.Thmtheorem2.p1.4.m3.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem2.p1.4.m3.1b"><ci id="S1.Thmtheorem2.p1.4.m3.1.1.cmml" xref="S1.Thmtheorem2.p1.4.m3.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem2.p1.4.m3.1c">A</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem2.p1.4.m3.1d">italic_A</annotation></semantics></math> has no left or right endpoint and for each <math alttext="a<_{A}b" class="ltx_Math" display="inline" id="S1.Thmtheorem2.p1.5.m4.1"><semantics id="S1.Thmtheorem2.p1.5.m4.1a"><mrow id="S1.Thmtheorem2.p1.5.m4.1.1" xref="S1.Thmtheorem2.p1.5.m4.1.1.cmml"><mi id="S1.Thmtheorem2.p1.5.m4.1.1.2" xref="S1.Thmtheorem2.p1.5.m4.1.1.2.cmml">a</mi><msub id="S1.Thmtheorem2.p1.5.m4.1.1.1" xref="S1.Thmtheorem2.p1.5.m4.1.1.1.cmml"><mo id="S1.Thmtheorem2.p1.5.m4.1.1.1.2" xref="S1.Thmtheorem2.p1.5.m4.1.1.1.2.cmml"><</mo><mi id="S1.Thmtheorem2.p1.5.m4.1.1.1.3" xref="S1.Thmtheorem2.p1.5.m4.1.1.1.3.cmml">A</mi></msub><mi id="S1.Thmtheorem2.p1.5.m4.1.1.3" xref="S1.Thmtheorem2.p1.5.m4.1.1.3.cmml">b</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem2.p1.5.m4.1b"><apply id="S1.Thmtheorem2.p1.5.m4.1.1.cmml" xref="S1.Thmtheorem2.p1.5.m4.1.1"><apply id="S1.Thmtheorem2.p1.5.m4.1.1.1.cmml" xref="S1.Thmtheorem2.p1.5.m4.1.1.1"><csymbol cd="ambiguous" id="S1.Thmtheorem2.p1.5.m4.1.1.1.1.cmml" xref="S1.Thmtheorem2.p1.5.m4.1.1.1">subscript</csymbol><lt id="S1.Thmtheorem2.p1.5.m4.1.1.1.2.cmml" xref="S1.Thmtheorem2.p1.5.m4.1.1.1.2"></lt><ci id="S1.Thmtheorem2.p1.5.m4.1.1.1.3.cmml" xref="S1.Thmtheorem2.p1.5.m4.1.1.1.3">𝐴</ci></apply><ci id="S1.Thmtheorem2.p1.5.m4.1.1.2.cmml" xref="S1.Thmtheorem2.p1.5.m4.1.1.2">𝑎</ci><ci id="S1.Thmtheorem2.p1.5.m4.1.1.3.cmml" xref="S1.Thmtheorem2.p1.5.m4.1.1.3">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem2.p1.5.m4.1c">a<_{A}b</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem2.p1.5.m4.1d">italic_a < start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_b</annotation></semantics></math>, <math alttext="{[a,b]}_{A}" class="ltx_Math" display="inline" id="S1.Thmtheorem2.p1.6.m5.2"><semantics id="S1.Thmtheorem2.p1.6.m5.2a"><msub id="S1.Thmtheorem2.p1.6.m5.2.3" xref="S1.Thmtheorem2.p1.6.m5.2.3.cmml"><mrow id="S1.Thmtheorem2.p1.6.m5.2.3.2.2" xref="S1.Thmtheorem2.p1.6.m5.2.3.2.1.cmml"><mo id="S1.Thmtheorem2.p1.6.m5.2.3.2.2.1" stretchy="false" xref="S1.Thmtheorem2.p1.6.m5.2.3.2.1.cmml">[</mo><mi id="S1.Thmtheorem2.p1.6.m5.1.1" xref="S1.Thmtheorem2.p1.6.m5.1.1.cmml">a</mi><mo id="S1.Thmtheorem2.p1.6.m5.2.3.2.2.2" xref="S1.Thmtheorem2.p1.6.m5.2.3.2.1.cmml">,</mo><mi id="S1.Thmtheorem2.p1.6.m5.2.2" xref="S1.Thmtheorem2.p1.6.m5.2.2.cmml">b</mi><mo id="S1.Thmtheorem2.p1.6.m5.2.3.2.2.3" stretchy="false" xref="S1.Thmtheorem2.p1.6.m5.2.3.2.1.cmml">]</mo></mrow><mi id="S1.Thmtheorem2.p1.6.m5.2.3.3" xref="S1.Thmtheorem2.p1.6.m5.2.3.3.cmml">A</mi></msub><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem2.p1.6.m5.2b"><apply id="S1.Thmtheorem2.p1.6.m5.2.3.cmml" xref="S1.Thmtheorem2.p1.6.m5.2.3"><csymbol cd="ambiguous" id="S1.Thmtheorem2.p1.6.m5.2.3.1.cmml" xref="S1.Thmtheorem2.p1.6.m5.2.3">subscript</csymbol><interval closure="closed" id="S1.Thmtheorem2.p1.6.m5.2.3.2.1.cmml" xref="S1.Thmtheorem2.p1.6.m5.2.3.2.2"><ci id="S1.Thmtheorem2.p1.6.m5.1.1.cmml" xref="S1.Thmtheorem2.p1.6.m5.1.1">𝑎</ci><ci id="S1.Thmtheorem2.p1.6.m5.2.2.cmml" xref="S1.Thmtheorem2.p1.6.m5.2.2">𝑏</ci></interval><ci id="S1.Thmtheorem2.p1.6.m5.2.3.3.cmml" xref="S1.Thmtheorem2.p1.6.m5.2.3.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem2.p1.6.m5.2c">{[a,b]}_{A}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem2.p1.6.m5.2d">[ italic_a , italic_b ] start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT</annotation></semantics></math> has size <math alttext="\kappa" class="ltx_Math" display="inline" id="S1.Thmtheorem2.p1.7.m6.1"><semantics id="S1.Thmtheorem2.p1.7.m6.1a"><mi id="S1.Thmtheorem2.p1.7.m6.1.1" xref="S1.Thmtheorem2.p1.7.m6.1.1.cmml">κ</mi><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem2.p1.7.m6.1b"><ci id="S1.Thmtheorem2.p1.7.m6.1.1.cmml" xref="S1.Thmtheorem2.p1.7.m6.1.1">𝜅</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem2.p1.7.m6.1c">\kappa</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem2.p1.7.m6.1d">italic_κ</annotation></semantics></math>.</p> </div> </div> <div class="ltx_para" id="S1.SSx1.p5"> <p class="ltx_p" id="S1.SSx1.p5.11">An important theorem of Baumgartner <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib2" title="">2</a>]</cite> says that under <math alttext="\mathsf{PFA}" class="ltx_Math" display="inline" id="S1.SSx1.p5.1.m1.1"><semantics id="S1.SSx1.p5.1.m1.1a"><mi id="S1.SSx1.p5.1.m1.1.1" xref="S1.SSx1.p5.1.m1.1.1.cmml">𝖯𝖥𝖠</mi><annotation-xml encoding="MathML-Content" id="S1.SSx1.p5.1.m1.1b"><ci id="S1.SSx1.p5.1.m1.1.1.cmml" xref="S1.SSx1.p5.1.m1.1.1">𝖯𝖥𝖠</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p5.1.m1.1c">\mathsf{PFA}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p5.1.m1.1d">sansserif_PFA</annotation></semantics></math> all <math alttext="\aleph_{1}" class="ltx_Math" display="inline" id="S1.SSx1.p5.2.m2.1"><semantics id="S1.SSx1.p5.2.m2.1a"><msub id="S1.SSx1.p5.2.m2.1.1" xref="S1.SSx1.p5.2.m2.1.1.cmml"><mi id="S1.SSx1.p5.2.m2.1.1.2" mathvariant="normal" xref="S1.SSx1.p5.2.m2.1.1.2.cmml">ℵ</mi><mn id="S1.SSx1.p5.2.m2.1.1.3" xref="S1.SSx1.p5.2.m2.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S1.SSx1.p5.2.m2.1b"><apply id="S1.SSx1.p5.2.m2.1.1.cmml" xref="S1.SSx1.p5.2.m2.1.1"><csymbol cd="ambiguous" id="S1.SSx1.p5.2.m2.1.1.1.cmml" xref="S1.SSx1.p5.2.m2.1.1">subscript</csymbol><ci id="S1.SSx1.p5.2.m2.1.1.2.cmml" xref="S1.SSx1.p5.2.m2.1.1.2">ℵ</ci><cn id="S1.SSx1.p5.2.m2.1.1.3.cmml" type="integer" xref="S1.SSx1.p5.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p5.2.m2.1c">\aleph_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p5.2.m2.1d">roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-dense real orders are isomorphic. Noting that each uncountable real order contains an <math alttext="\aleph_{1}" class="ltx_Math" display="inline" id="S1.SSx1.p5.3.m3.1"><semantics id="S1.SSx1.p5.3.m3.1a"><msub id="S1.SSx1.p5.3.m3.1.1" xref="S1.SSx1.p5.3.m3.1.1.cmml"><mi id="S1.SSx1.p5.3.m3.1.1.2" mathvariant="normal" xref="S1.SSx1.p5.3.m3.1.1.2.cmml">ℵ</mi><mn id="S1.SSx1.p5.3.m3.1.1.3" xref="S1.SSx1.p5.3.m3.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S1.SSx1.p5.3.m3.1b"><apply id="S1.SSx1.p5.3.m3.1.1.cmml" xref="S1.SSx1.p5.3.m3.1.1"><csymbol cd="ambiguous" id="S1.SSx1.p5.3.m3.1.1.1.cmml" xref="S1.SSx1.p5.3.m3.1.1">subscript</csymbol><ci id="S1.SSx1.p5.3.m3.1.1.2.cmml" xref="S1.SSx1.p5.3.m3.1.1.2">ℵ</ci><cn id="S1.SSx1.p5.3.m3.1.1.3.cmml" type="integer" xref="S1.SSx1.p5.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p5.3.m3.1c">\aleph_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p5.3.m3.1d">roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-dense suborder, Baumgartner’s theorem implies that under <math alttext="\mathsf{PFA}" class="ltx_Math" display="inline" id="S1.SSx1.p5.4.m4.1"><semantics id="S1.SSx1.p5.4.m4.1a"><mi id="S1.SSx1.p5.4.m4.1.1" xref="S1.SSx1.p5.4.m4.1.1.cmml">𝖯𝖥𝖠</mi><annotation-xml encoding="MathML-Content" id="S1.SSx1.p5.4.m4.1b"><ci id="S1.SSx1.p5.4.m4.1.1.cmml" xref="S1.SSx1.p5.4.m4.1.1">𝖯𝖥𝖠</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p5.4.m4.1c">\mathsf{PFA}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p5.4.m4.1d">sansserif_PFA</annotation></semantics></math> the uncountable real orders have a single element basis. Fix such a set <math alttext="X" class="ltx_Math" display="inline" id="S1.SSx1.p5.5.m5.1"><semantics id="S1.SSx1.p5.5.m5.1a"><mi id="S1.SSx1.p5.5.m5.1.1" xref="S1.SSx1.p5.5.m5.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S1.SSx1.p5.5.m5.1b"><ci id="S1.SSx1.p5.5.m5.1.1.cmml" xref="S1.SSx1.p5.5.m5.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p5.5.m5.1c">X</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p5.5.m5.1d">italic_X</annotation></semantics></math>. Does this imply that under <math alttext="\mathsf{PFA}" class="ltx_Math" display="inline" id="S1.SSx1.p5.6.m6.1"><semantics id="S1.SSx1.p5.6.m6.1a"><mi id="S1.SSx1.p5.6.m6.1.1" xref="S1.SSx1.p5.6.m6.1.1.cmml">𝖯𝖥𝖠</mi><annotation-xml encoding="MathML-Content" id="S1.SSx1.p5.6.m6.1b"><ci id="S1.SSx1.p5.6.m6.1.1.cmml" xref="S1.SSx1.p5.6.m6.1.1">𝖯𝖥𝖠</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p5.6.m6.1c">\mathsf{PFA}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p5.6.m6.1d">sansserif_PFA</annotation></semantics></math>, <math alttext="\omega_{1}" class="ltx_Math" display="inline" id="S1.SSx1.p5.7.m7.1"><semantics id="S1.SSx1.p5.7.m7.1a"><msub id="S1.SSx1.p5.7.m7.1.1" xref="S1.SSx1.p5.7.m7.1.1.cmml"><mi id="S1.SSx1.p5.7.m7.1.1.2" xref="S1.SSx1.p5.7.m7.1.1.2.cmml">ω</mi><mn id="S1.SSx1.p5.7.m7.1.1.3" xref="S1.SSx1.p5.7.m7.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S1.SSx1.p5.7.m7.1b"><apply id="S1.SSx1.p5.7.m7.1.1.cmml" xref="S1.SSx1.p5.7.m7.1.1"><csymbol cd="ambiguous" id="S1.SSx1.p5.7.m7.1.1.1.cmml" xref="S1.SSx1.p5.7.m7.1.1">subscript</csymbol><ci id="S1.SSx1.p5.7.m7.1.1.2.cmml" xref="S1.SSx1.p5.7.m7.1.1.2">𝜔</ci><cn id="S1.SSx1.p5.7.m7.1.1.3.cmml" type="integer" xref="S1.SSx1.p5.7.m7.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p5.7.m7.1c">\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p5.7.m7.1d">italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\omega_{1}^{\star}" class="ltx_Math" display="inline" id="S1.SSx1.p5.8.m8.1"><semantics id="S1.SSx1.p5.8.m8.1a"><msubsup id="S1.SSx1.p5.8.m8.1.1" xref="S1.SSx1.p5.8.m8.1.1.cmml"><mi id="S1.SSx1.p5.8.m8.1.1.2.2" xref="S1.SSx1.p5.8.m8.1.1.2.2.cmml">ω</mi><mn id="S1.SSx1.p5.8.m8.1.1.2.3" xref="S1.SSx1.p5.8.m8.1.1.2.3.cmml">1</mn><mo id="S1.SSx1.p5.8.m8.1.1.3" xref="S1.SSx1.p5.8.m8.1.1.3.cmml">⋆</mo></msubsup><annotation-xml encoding="MathML-Content" id="S1.SSx1.p5.8.m8.1b"><apply id="S1.SSx1.p5.8.m8.1.1.cmml" xref="S1.SSx1.p5.8.m8.1.1"><csymbol cd="ambiguous" id="S1.SSx1.p5.8.m8.1.1.1.cmml" xref="S1.SSx1.p5.8.m8.1.1">superscript</csymbol><apply id="S1.SSx1.p5.8.m8.1.1.2.cmml" xref="S1.SSx1.p5.8.m8.1.1"><csymbol cd="ambiguous" id="S1.SSx1.p5.8.m8.1.1.2.1.cmml" xref="S1.SSx1.p5.8.m8.1.1">subscript</csymbol><ci id="S1.SSx1.p5.8.m8.1.1.2.2.cmml" xref="S1.SSx1.p5.8.m8.1.1.2.2">𝜔</ci><cn id="S1.SSx1.p5.8.m8.1.1.2.3.cmml" type="integer" xref="S1.SSx1.p5.8.m8.1.1.2.3">1</cn></apply><ci id="S1.SSx1.p5.8.m8.1.1.3.cmml" xref="S1.SSx1.p5.8.m8.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p5.8.m8.1c">\omega_{1}^{\star}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p5.8.m8.1d">italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="X" class="ltx_Math" display="inline" id="S1.SSx1.p5.9.m9.1"><semantics id="S1.SSx1.p5.9.m9.1a"><mi id="S1.SSx1.p5.9.m9.1.1" xref="S1.SSx1.p5.9.m9.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S1.SSx1.p5.9.m9.1b"><ci id="S1.SSx1.p5.9.m9.1.1.cmml" xref="S1.SSx1.p5.9.m9.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p5.9.m9.1c">X</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p5.9.m9.1d">italic_X</annotation></semantics></math> form a <math alttext="\preceq" class="ltx_Math" display="inline" id="S1.SSx1.p5.10.m10.1"><semantics id="S1.SSx1.p5.10.m10.1a"><mo id="S1.SSx1.p5.10.m10.1.1" xref="S1.SSx1.p5.10.m10.1.1.cmml">⪯</mo><annotation-xml encoding="MathML-Content" id="S1.SSx1.p5.10.m10.1b"><csymbol cd="latexml" id="S1.SSx1.p5.10.m10.1.1.cmml" xref="S1.SSx1.p5.10.m10.1.1">precedes-or-equals</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p5.10.m10.1c">\preceq</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p5.10.m10.1d">⪯</annotation></semantics></math>-basis for the uncountable linear orders? The answer is no, because the following object can be proved to exists in <math alttext="\mathsf{ZFC}" class="ltx_Math" display="inline" id="S1.SSx1.p5.11.m11.1"><semantics id="S1.SSx1.p5.11.m11.1a"><mi id="S1.SSx1.p5.11.m11.1.1" xref="S1.SSx1.p5.11.m11.1.1.cmml">𝖹𝖥𝖢</mi><annotation-xml encoding="MathML-Content" id="S1.SSx1.p5.11.m11.1b"><ci id="S1.SSx1.p5.11.m11.1.1.cmml" xref="S1.SSx1.p5.11.m11.1.1">𝖹𝖥𝖢</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p5.11.m11.1c">\mathsf{ZFC}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p5.11.m11.1d">sansserif_ZFC</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="S1.Thmtheorem3"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem3.1.1.1">Definition 1.3</span></span><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem3.2.2">.</span> </h6> <div class="ltx_para" id="S1.Thmtheorem3.p1"> <p class="ltx_p" id="S1.Thmtheorem3.p1.3">An <em class="ltx_emph ltx_font_italic" id="S1.Thmtheorem3.p1.3.1">Aronszajn line</em> is an uncountable linear order <math alttext="A" class="ltx_Math" display="inline" id="S1.Thmtheorem3.p1.1.m1.1"><semantics id="S1.Thmtheorem3.p1.1.m1.1a"><mi id="S1.Thmtheorem3.p1.1.m1.1.1" xref="S1.Thmtheorem3.p1.1.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem3.p1.1.m1.1b"><ci id="S1.Thmtheorem3.p1.1.m1.1.1.cmml" xref="S1.Thmtheorem3.p1.1.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem3.p1.1.m1.1c">A</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem3.p1.1.m1.1d">italic_A</annotation></semantics></math> that does not contain copies of <math alttext="\omega_{1}" class="ltx_Math" display="inline" id="S1.Thmtheorem3.p1.2.m2.1"><semantics id="S1.Thmtheorem3.p1.2.m2.1a"><msub id="S1.Thmtheorem3.p1.2.m2.1.1" xref="S1.Thmtheorem3.p1.2.m2.1.1.cmml"><mi id="S1.Thmtheorem3.p1.2.m2.1.1.2" xref="S1.Thmtheorem3.p1.2.m2.1.1.2.cmml">ω</mi><mn id="S1.Thmtheorem3.p1.2.m2.1.1.3" xref="S1.Thmtheorem3.p1.2.m2.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem3.p1.2.m2.1b"><apply id="S1.Thmtheorem3.p1.2.m2.1.1.cmml" xref="S1.Thmtheorem3.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S1.Thmtheorem3.p1.2.m2.1.1.1.cmml" xref="S1.Thmtheorem3.p1.2.m2.1.1">subscript</csymbol><ci id="S1.Thmtheorem3.p1.2.m2.1.1.2.cmml" xref="S1.Thmtheorem3.p1.2.m2.1.1.2">𝜔</ci><cn id="S1.Thmtheorem3.p1.2.m2.1.1.3.cmml" type="integer" xref="S1.Thmtheorem3.p1.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem3.p1.2.m2.1c">\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem3.p1.2.m2.1d">italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\omega_{1}^{\star}" class="ltx_Math" display="inline" id="S1.Thmtheorem3.p1.3.m3.1"><semantics id="S1.Thmtheorem3.p1.3.m3.1a"><msubsup id="S1.Thmtheorem3.p1.3.m3.1.1" xref="S1.Thmtheorem3.p1.3.m3.1.1.cmml"><mi id="S1.Thmtheorem3.p1.3.m3.1.1.2.2" xref="S1.Thmtheorem3.p1.3.m3.1.1.2.2.cmml">ω</mi><mn id="S1.Thmtheorem3.p1.3.m3.1.1.2.3" xref="S1.Thmtheorem3.p1.3.m3.1.1.2.3.cmml">1</mn><mo id="S1.Thmtheorem3.p1.3.m3.1.1.3" xref="S1.Thmtheorem3.p1.3.m3.1.1.3.cmml">⋆</mo></msubsup><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem3.p1.3.m3.1b"><apply id="S1.Thmtheorem3.p1.3.m3.1.1.cmml" xref="S1.Thmtheorem3.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S1.Thmtheorem3.p1.3.m3.1.1.1.cmml" xref="S1.Thmtheorem3.p1.3.m3.1.1">superscript</csymbol><apply id="S1.Thmtheorem3.p1.3.m3.1.1.2.cmml" xref="S1.Thmtheorem3.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S1.Thmtheorem3.p1.3.m3.1.1.2.1.cmml" xref="S1.Thmtheorem3.p1.3.m3.1.1">subscript</csymbol><ci id="S1.Thmtheorem3.p1.3.m3.1.1.2.2.cmml" xref="S1.Thmtheorem3.p1.3.m3.1.1.2.2">𝜔</ci><cn id="S1.Thmtheorem3.p1.3.m3.1.1.2.3.cmml" type="integer" xref="S1.Thmtheorem3.p1.3.m3.1.1.2.3">1</cn></apply><ci id="S1.Thmtheorem3.p1.3.m3.1.1.3.cmml" xref="S1.Thmtheorem3.p1.3.m3.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem3.p1.3.m3.1c">\omega_{1}^{\star}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem3.p1.3.m3.1d">italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math>, or any uncountable set of reals.</p> </div> </div> <div class="ltx_para" id="S1.SSx1.p6"> <p class="ltx_p" id="S1.SSx1.p6.4">Aronszajn lines can be constructed from Aronszajn trees, objects first proved to exists by Aronszajn (see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib11" title="">11</a>]</cite>). Specker <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib20" title="">20</a>]</cite> rediscovered the notion of Aronszajn line, which is why they are also called Specker orders or Specker types. Shelah <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib17" title="">17</a>]</cite> proved the existence of an Aronszajn line <math alttext="C" class="ltx_Math" display="inline" id="S1.SSx1.p6.1.m1.1"><semantics id="S1.SSx1.p6.1.m1.1a"><mi id="S1.SSx1.p6.1.m1.1.1" xref="S1.SSx1.p6.1.m1.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S1.SSx1.p6.1.m1.1b"><ci id="S1.SSx1.p6.1.m1.1.1.cmml" xref="S1.SSx1.p6.1.m1.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p6.1.m1.1c">C</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p6.1.m1.1d">italic_C</annotation></semantics></math>, such that <math alttext="C" class="ltx_Math" display="inline" id="S1.SSx1.p6.2.m2.1"><semantics id="S1.SSx1.p6.2.m2.1a"><mi id="S1.SSx1.p6.2.m2.1.1" xref="S1.SSx1.p6.2.m2.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S1.SSx1.p6.2.m2.1b"><ci id="S1.SSx1.p6.2.m2.1.1.cmml" xref="S1.SSx1.p6.2.m2.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p6.2.m2.1c">C</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p6.2.m2.1d">italic_C</annotation></semantics></math> and <math alttext="C^{\star}" class="ltx_Math" display="inline" id="S1.SSx1.p6.3.m3.1"><semantics id="S1.SSx1.p6.3.m3.1a"><msup id="S1.SSx1.p6.3.m3.1.1" xref="S1.SSx1.p6.3.m3.1.1.cmml"><mi id="S1.SSx1.p6.3.m3.1.1.2" xref="S1.SSx1.p6.3.m3.1.1.2.cmml">C</mi><mo id="S1.SSx1.p6.3.m3.1.1.3" xref="S1.SSx1.p6.3.m3.1.1.3.cmml">⋆</mo></msup><annotation-xml encoding="MathML-Content" id="S1.SSx1.p6.3.m3.1b"><apply id="S1.SSx1.p6.3.m3.1.1.cmml" xref="S1.SSx1.p6.3.m3.1.1"><csymbol cd="ambiguous" id="S1.SSx1.p6.3.m3.1.1.1.cmml" xref="S1.SSx1.p6.3.m3.1.1">superscript</csymbol><ci id="S1.SSx1.p6.3.m3.1.1.2.cmml" xref="S1.SSx1.p6.3.m3.1.1.2">𝐶</ci><ci id="S1.SSx1.p6.3.m3.1.1.3.cmml" xref="S1.SSx1.p6.3.m3.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p6.3.m3.1c">C^{\star}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p6.3.m3.1d">italic_C start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math> do not have uncountable suborders in common, thus, any <math alttext="\preceq" class="ltx_Math" display="inline" id="S1.SSx1.p6.4.m4.1"><semantics id="S1.SSx1.p6.4.m4.1a"><mo id="S1.SSx1.p6.4.m4.1.1" xref="S1.SSx1.p6.4.m4.1.1.cmml">⪯</mo><annotation-xml encoding="MathML-Content" id="S1.SSx1.p6.4.m4.1b"><csymbol cd="latexml" id="S1.SSx1.p6.4.m4.1.1.cmml" xref="S1.SSx1.p6.4.m4.1.1">precedes-or-equals</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p6.4.m4.1c">\preceq</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p6.4.m4.1d">⪯</annotation></semantics></math>-basis for the class of Aronszajn lines contains at least two elements. He proved this by constructing what is now called a Countryman line, an object conjectured (to not exist) by Countryman in an unpublished text.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="S1.Thmtheorem4"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem4.1.1.1">Definition 1.4</span></span><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem4.2.2">.</span> </h6> <div class="ltx_para" id="S1.Thmtheorem4.p1"> <p class="ltx_p" id="S1.Thmtheorem4.p1.4">A <em class="ltx_emph ltx_font_italic" id="S1.Thmtheorem4.p1.4.1">Countryman line</em> is an uncountable linear order such that under the product order (<math alttext="(x,y)\leq(x^{\prime},y^{\prime})" class="ltx_Math" display="inline" id="S1.Thmtheorem4.p1.1.m1.4"><semantics id="S1.Thmtheorem4.p1.1.m1.4a"><mrow id="S1.Thmtheorem4.p1.1.m1.4.4" xref="S1.Thmtheorem4.p1.1.m1.4.4.cmml"><mrow id="S1.Thmtheorem4.p1.1.m1.4.4.4.2" xref="S1.Thmtheorem4.p1.1.m1.4.4.4.1.cmml"><mo id="S1.Thmtheorem4.p1.1.m1.4.4.4.2.1" stretchy="false" xref="S1.Thmtheorem4.p1.1.m1.4.4.4.1.cmml">(</mo><mi id="S1.Thmtheorem4.p1.1.m1.1.1" xref="S1.Thmtheorem4.p1.1.m1.1.1.cmml">x</mi><mo id="S1.Thmtheorem4.p1.1.m1.4.4.4.2.2" xref="S1.Thmtheorem4.p1.1.m1.4.4.4.1.cmml">,</mo><mi id="S1.Thmtheorem4.p1.1.m1.2.2" xref="S1.Thmtheorem4.p1.1.m1.2.2.cmml">y</mi><mo id="S1.Thmtheorem4.p1.1.m1.4.4.4.2.3" stretchy="false" xref="S1.Thmtheorem4.p1.1.m1.4.4.4.1.cmml">)</mo></mrow><mo id="S1.Thmtheorem4.p1.1.m1.4.4.3" xref="S1.Thmtheorem4.p1.1.m1.4.4.3.cmml">≤</mo><mrow id="S1.Thmtheorem4.p1.1.m1.4.4.2.2" xref="S1.Thmtheorem4.p1.1.m1.4.4.2.3.cmml"><mo id="S1.Thmtheorem4.p1.1.m1.4.4.2.2.3" stretchy="false" xref="S1.Thmtheorem4.p1.1.m1.4.4.2.3.cmml">(</mo><msup id="S1.Thmtheorem4.p1.1.m1.3.3.1.1.1" xref="S1.Thmtheorem4.p1.1.m1.3.3.1.1.1.cmml"><mi id="S1.Thmtheorem4.p1.1.m1.3.3.1.1.1.2" xref="S1.Thmtheorem4.p1.1.m1.3.3.1.1.1.2.cmml">x</mi><mo id="S1.Thmtheorem4.p1.1.m1.3.3.1.1.1.3" xref="S1.Thmtheorem4.p1.1.m1.3.3.1.1.1.3.cmml">′</mo></msup><mo id="S1.Thmtheorem4.p1.1.m1.4.4.2.2.4" xref="S1.Thmtheorem4.p1.1.m1.4.4.2.3.cmml">,</mo><msup id="S1.Thmtheorem4.p1.1.m1.4.4.2.2.2" xref="S1.Thmtheorem4.p1.1.m1.4.4.2.2.2.cmml"><mi id="S1.Thmtheorem4.p1.1.m1.4.4.2.2.2.2" xref="S1.Thmtheorem4.p1.1.m1.4.4.2.2.2.2.cmml">y</mi><mo id="S1.Thmtheorem4.p1.1.m1.4.4.2.2.2.3" xref="S1.Thmtheorem4.p1.1.m1.4.4.2.2.2.3.cmml">′</mo></msup><mo id="S1.Thmtheorem4.p1.1.m1.4.4.2.2.5" stretchy="false" xref="S1.Thmtheorem4.p1.1.m1.4.4.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem4.p1.1.m1.4b"><apply id="S1.Thmtheorem4.p1.1.m1.4.4.cmml" xref="S1.Thmtheorem4.p1.1.m1.4.4"><leq id="S1.Thmtheorem4.p1.1.m1.4.4.3.cmml" xref="S1.Thmtheorem4.p1.1.m1.4.4.3"></leq><interval closure="open" id="S1.Thmtheorem4.p1.1.m1.4.4.4.1.cmml" xref="S1.Thmtheorem4.p1.1.m1.4.4.4.2"><ci id="S1.Thmtheorem4.p1.1.m1.1.1.cmml" xref="S1.Thmtheorem4.p1.1.m1.1.1">𝑥</ci><ci id="S1.Thmtheorem4.p1.1.m1.2.2.cmml" xref="S1.Thmtheorem4.p1.1.m1.2.2">𝑦</ci></interval><interval closure="open" id="S1.Thmtheorem4.p1.1.m1.4.4.2.3.cmml" xref="S1.Thmtheorem4.p1.1.m1.4.4.2.2"><apply id="S1.Thmtheorem4.p1.1.m1.3.3.1.1.1.cmml" xref="S1.Thmtheorem4.p1.1.m1.3.3.1.1.1"><csymbol cd="ambiguous" id="S1.Thmtheorem4.p1.1.m1.3.3.1.1.1.1.cmml" xref="S1.Thmtheorem4.p1.1.m1.3.3.1.1.1">superscript</csymbol><ci id="S1.Thmtheorem4.p1.1.m1.3.3.1.1.1.2.cmml" xref="S1.Thmtheorem4.p1.1.m1.3.3.1.1.1.2">𝑥</ci><ci id="S1.Thmtheorem4.p1.1.m1.3.3.1.1.1.3.cmml" xref="S1.Thmtheorem4.p1.1.m1.3.3.1.1.1.3">′</ci></apply><apply id="S1.Thmtheorem4.p1.1.m1.4.4.2.2.2.cmml" xref="S1.Thmtheorem4.p1.1.m1.4.4.2.2.2"><csymbol cd="ambiguous" id="S1.Thmtheorem4.p1.1.m1.4.4.2.2.2.1.cmml" xref="S1.Thmtheorem4.p1.1.m1.4.4.2.2.2">superscript</csymbol><ci id="S1.Thmtheorem4.p1.1.m1.4.4.2.2.2.2.cmml" xref="S1.Thmtheorem4.p1.1.m1.4.4.2.2.2.2">𝑦</ci><ci id="S1.Thmtheorem4.p1.1.m1.4.4.2.2.2.3.cmml" xref="S1.Thmtheorem4.p1.1.m1.4.4.2.2.2.3">′</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem4.p1.1.m1.4c">(x,y)\leq(x^{\prime},y^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem4.p1.1.m1.4d">( italic_x , italic_y ) ≤ ( italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> iff <math alttext="x\leq_{C}x^{\prime}" class="ltx_Math" display="inline" id="S1.Thmtheorem4.p1.2.m2.1"><semantics id="S1.Thmtheorem4.p1.2.m2.1a"><mrow id="S1.Thmtheorem4.p1.2.m2.1.1" xref="S1.Thmtheorem4.p1.2.m2.1.1.cmml"><mi id="S1.Thmtheorem4.p1.2.m2.1.1.2" xref="S1.Thmtheorem4.p1.2.m2.1.1.2.cmml">x</mi><msub id="S1.Thmtheorem4.p1.2.m2.1.1.1" xref="S1.Thmtheorem4.p1.2.m2.1.1.1.cmml"><mo id="S1.Thmtheorem4.p1.2.m2.1.1.1.2" xref="S1.Thmtheorem4.p1.2.m2.1.1.1.2.cmml">≤</mo><mi id="S1.Thmtheorem4.p1.2.m2.1.1.1.3" xref="S1.Thmtheorem4.p1.2.m2.1.1.1.3.cmml">C</mi></msub><msup id="S1.Thmtheorem4.p1.2.m2.1.1.3" xref="S1.Thmtheorem4.p1.2.m2.1.1.3.cmml"><mi id="S1.Thmtheorem4.p1.2.m2.1.1.3.2" xref="S1.Thmtheorem4.p1.2.m2.1.1.3.2.cmml">x</mi><mo id="S1.Thmtheorem4.p1.2.m2.1.1.3.3" xref="S1.Thmtheorem4.p1.2.m2.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem4.p1.2.m2.1b"><apply id="S1.Thmtheorem4.p1.2.m2.1.1.cmml" xref="S1.Thmtheorem4.p1.2.m2.1.1"><apply id="S1.Thmtheorem4.p1.2.m2.1.1.1.cmml" xref="S1.Thmtheorem4.p1.2.m2.1.1.1"><csymbol cd="ambiguous" id="S1.Thmtheorem4.p1.2.m2.1.1.1.1.cmml" xref="S1.Thmtheorem4.p1.2.m2.1.1.1">subscript</csymbol><leq id="S1.Thmtheorem4.p1.2.m2.1.1.1.2.cmml" xref="S1.Thmtheorem4.p1.2.m2.1.1.1.2"></leq><ci id="S1.Thmtheorem4.p1.2.m2.1.1.1.3.cmml" xref="S1.Thmtheorem4.p1.2.m2.1.1.1.3">𝐶</ci></apply><ci id="S1.Thmtheorem4.p1.2.m2.1.1.2.cmml" xref="S1.Thmtheorem4.p1.2.m2.1.1.2">𝑥</ci><apply id="S1.Thmtheorem4.p1.2.m2.1.1.3.cmml" xref="S1.Thmtheorem4.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S1.Thmtheorem4.p1.2.m2.1.1.3.1.cmml" xref="S1.Thmtheorem4.p1.2.m2.1.1.3">superscript</csymbol><ci id="S1.Thmtheorem4.p1.2.m2.1.1.3.2.cmml" xref="S1.Thmtheorem4.p1.2.m2.1.1.3.2">𝑥</ci><ci id="S1.Thmtheorem4.p1.2.m2.1.1.3.3.cmml" xref="S1.Thmtheorem4.p1.2.m2.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem4.p1.2.m2.1c">x\leq_{C}x^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem4.p1.2.m2.1d">italic_x ≤ start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="y\leq_{C}y^{\prime}" class="ltx_Math" display="inline" id="S1.Thmtheorem4.p1.3.m3.1"><semantics id="S1.Thmtheorem4.p1.3.m3.1a"><mrow id="S1.Thmtheorem4.p1.3.m3.1.1" xref="S1.Thmtheorem4.p1.3.m3.1.1.cmml"><mi id="S1.Thmtheorem4.p1.3.m3.1.1.2" xref="S1.Thmtheorem4.p1.3.m3.1.1.2.cmml">y</mi><msub id="S1.Thmtheorem4.p1.3.m3.1.1.1" xref="S1.Thmtheorem4.p1.3.m3.1.1.1.cmml"><mo id="S1.Thmtheorem4.p1.3.m3.1.1.1.2" xref="S1.Thmtheorem4.p1.3.m3.1.1.1.2.cmml">≤</mo><mi id="S1.Thmtheorem4.p1.3.m3.1.1.1.3" xref="S1.Thmtheorem4.p1.3.m3.1.1.1.3.cmml">C</mi></msub><msup id="S1.Thmtheorem4.p1.3.m3.1.1.3" xref="S1.Thmtheorem4.p1.3.m3.1.1.3.cmml"><mi id="S1.Thmtheorem4.p1.3.m3.1.1.3.2" xref="S1.Thmtheorem4.p1.3.m3.1.1.3.2.cmml">y</mi><mo id="S1.Thmtheorem4.p1.3.m3.1.1.3.3" xref="S1.Thmtheorem4.p1.3.m3.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem4.p1.3.m3.1b"><apply id="S1.Thmtheorem4.p1.3.m3.1.1.cmml" xref="S1.Thmtheorem4.p1.3.m3.1.1"><apply id="S1.Thmtheorem4.p1.3.m3.1.1.1.cmml" xref="S1.Thmtheorem4.p1.3.m3.1.1.1"><csymbol cd="ambiguous" id="S1.Thmtheorem4.p1.3.m3.1.1.1.1.cmml" xref="S1.Thmtheorem4.p1.3.m3.1.1.1">subscript</csymbol><leq id="S1.Thmtheorem4.p1.3.m3.1.1.1.2.cmml" xref="S1.Thmtheorem4.p1.3.m3.1.1.1.2"></leq><ci id="S1.Thmtheorem4.p1.3.m3.1.1.1.3.cmml" xref="S1.Thmtheorem4.p1.3.m3.1.1.1.3">𝐶</ci></apply><ci id="S1.Thmtheorem4.p1.3.m3.1.1.2.cmml" xref="S1.Thmtheorem4.p1.3.m3.1.1.2">𝑦</ci><apply id="S1.Thmtheorem4.p1.3.m3.1.1.3.cmml" xref="S1.Thmtheorem4.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S1.Thmtheorem4.p1.3.m3.1.1.3.1.cmml" xref="S1.Thmtheorem4.p1.3.m3.1.1.3">superscript</csymbol><ci id="S1.Thmtheorem4.p1.3.m3.1.1.3.2.cmml" xref="S1.Thmtheorem4.p1.3.m3.1.1.3.2">𝑦</ci><ci id="S1.Thmtheorem4.p1.3.m3.1.1.3.3.cmml" xref="S1.Thmtheorem4.p1.3.m3.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem4.p1.3.m3.1c">y\leq_{C}y^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem4.p1.3.m3.1d">italic_y ≤ start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT italic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>), <math alttext="C^{2}" class="ltx_Math" display="inline" id="S1.Thmtheorem4.p1.4.m4.1"><semantics id="S1.Thmtheorem4.p1.4.m4.1a"><msup id="S1.Thmtheorem4.p1.4.m4.1.1" xref="S1.Thmtheorem4.p1.4.m4.1.1.cmml"><mi id="S1.Thmtheorem4.p1.4.m4.1.1.2" xref="S1.Thmtheorem4.p1.4.m4.1.1.2.cmml">C</mi><mn id="S1.Thmtheorem4.p1.4.m4.1.1.3" xref="S1.Thmtheorem4.p1.4.m4.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem4.p1.4.m4.1b"><apply id="S1.Thmtheorem4.p1.4.m4.1.1.cmml" xref="S1.Thmtheorem4.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S1.Thmtheorem4.p1.4.m4.1.1.1.cmml" xref="S1.Thmtheorem4.p1.4.m4.1.1">superscript</csymbol><ci id="S1.Thmtheorem4.p1.4.m4.1.1.2.cmml" xref="S1.Thmtheorem4.p1.4.m4.1.1.2">𝐶</ci><cn id="S1.Thmtheorem4.p1.4.m4.1.1.3.cmml" type="integer" xref="S1.Thmtheorem4.p1.4.m4.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem4.p1.4.m4.1c">C^{2}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem4.p1.4.m4.1d">italic_C start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> is the union of countably many chains (sets of pairwise comparable elements).</p> </div> </div> <div class="ltx_para" id="S1.SSx1.p7"> <p class="ltx_p" id="S1.SSx1.p7.13">It can be proven that any Countryman line is necessarily Aronszajn, and Shelah conjectured that consistently <math alttext="C" class="ltx_Math" display="inline" id="S1.SSx1.p7.1.m1.1"><semantics id="S1.SSx1.p7.1.m1.1a"><mi id="S1.SSx1.p7.1.m1.1.1" xref="S1.SSx1.p7.1.m1.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S1.SSx1.p7.1.m1.1b"><ci id="S1.SSx1.p7.1.m1.1.1.cmml" xref="S1.SSx1.p7.1.m1.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p7.1.m1.1c">C</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p7.1.m1.1d">italic_C</annotation></semantics></math> and <math alttext="C^{\star}" class="ltx_Math" display="inline" id="S1.SSx1.p7.2.m2.1"><semantics id="S1.SSx1.p7.2.m2.1a"><msup id="S1.SSx1.p7.2.m2.1.1" xref="S1.SSx1.p7.2.m2.1.1.cmml"><mi id="S1.SSx1.p7.2.m2.1.1.2" xref="S1.SSx1.p7.2.m2.1.1.2.cmml">C</mi><mo id="S1.SSx1.p7.2.m2.1.1.3" xref="S1.SSx1.p7.2.m2.1.1.3.cmml">⋆</mo></msup><annotation-xml encoding="MathML-Content" id="S1.SSx1.p7.2.m2.1b"><apply id="S1.SSx1.p7.2.m2.1.1.cmml" xref="S1.SSx1.p7.2.m2.1.1"><csymbol cd="ambiguous" id="S1.SSx1.p7.2.m2.1.1.1.cmml" xref="S1.SSx1.p7.2.m2.1.1">superscript</csymbol><ci id="S1.SSx1.p7.2.m2.1.1.2.cmml" xref="S1.SSx1.p7.2.m2.1.1.2">𝐶</ci><ci id="S1.SSx1.p7.2.m2.1.1.3.cmml" xref="S1.SSx1.p7.2.m2.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p7.2.m2.1c">C^{\star}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p7.2.m2.1d">italic_C start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math> form a <math alttext="\preceq" class="ltx_Math" display="inline" id="S1.SSx1.p7.3.m3.1"><semantics id="S1.SSx1.p7.3.m3.1a"><mo id="S1.SSx1.p7.3.m3.1.1" xref="S1.SSx1.p7.3.m3.1.1.cmml">⪯</mo><annotation-xml encoding="MathML-Content" id="S1.SSx1.p7.3.m3.1b"><csymbol cd="latexml" id="S1.SSx1.p7.3.m3.1.1.cmml" xref="S1.SSx1.p7.3.m3.1.1">precedes-or-equals</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p7.3.m3.1c">\preceq</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p7.3.m3.1d">⪯</annotation></semantics></math>-basis for the Aronszajn lines. Later this became known as the Five Basis Conjecture: under <math alttext="\mathsf{PFA}" class="ltx_Math" display="inline" id="S1.SSx1.p7.4.m4.1"><semantics id="S1.SSx1.p7.4.m4.1a"><mi id="S1.SSx1.p7.4.m4.1.1" xref="S1.SSx1.p7.4.m4.1.1.cmml">𝖯𝖥𝖠</mi><annotation-xml encoding="MathML-Content" id="S1.SSx1.p7.4.m4.1b"><ci id="S1.SSx1.p7.4.m4.1.1.cmml" xref="S1.SSx1.p7.4.m4.1.1">𝖯𝖥𝖠</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p7.4.m4.1c">\mathsf{PFA}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p7.4.m4.1d">sansserif_PFA</annotation></semantics></math>, If <math alttext="C" class="ltx_Math" display="inline" id="S1.SSx1.p7.5.m5.1"><semantics id="S1.SSx1.p7.5.m5.1a"><mi id="S1.SSx1.p7.5.m5.1.1" xref="S1.SSx1.p7.5.m5.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S1.SSx1.p7.5.m5.1b"><ci id="S1.SSx1.p7.5.m5.1.1.cmml" xref="S1.SSx1.p7.5.m5.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p7.5.m5.1c">C</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p7.5.m5.1d">italic_C</annotation></semantics></math> is any Countryman line, and <math alttext="R" class="ltx_Math" display="inline" id="S1.SSx1.p7.6.m6.1"><semantics id="S1.SSx1.p7.6.m6.1a"><mi id="S1.SSx1.p7.6.m6.1.1" xref="S1.SSx1.p7.6.m6.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="S1.SSx1.p7.6.m6.1b"><ci id="S1.SSx1.p7.6.m6.1.1.cmml" xref="S1.SSx1.p7.6.m6.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p7.6.m6.1c">R</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p7.6.m6.1d">italic_R</annotation></semantics></math> any <math alttext="\aleph_{1}" class="ltx_Math" display="inline" id="S1.SSx1.p7.7.m7.1"><semantics id="S1.SSx1.p7.7.m7.1a"><msub id="S1.SSx1.p7.7.m7.1.1" xref="S1.SSx1.p7.7.m7.1.1.cmml"><mi id="S1.SSx1.p7.7.m7.1.1.2" mathvariant="normal" xref="S1.SSx1.p7.7.m7.1.1.2.cmml">ℵ</mi><mn id="S1.SSx1.p7.7.m7.1.1.3" xref="S1.SSx1.p7.7.m7.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S1.SSx1.p7.7.m7.1b"><apply id="S1.SSx1.p7.7.m7.1.1.cmml" xref="S1.SSx1.p7.7.m7.1.1"><csymbol cd="ambiguous" id="S1.SSx1.p7.7.m7.1.1.1.cmml" xref="S1.SSx1.p7.7.m7.1.1">subscript</csymbol><ci id="S1.SSx1.p7.7.m7.1.1.2.cmml" xref="S1.SSx1.p7.7.m7.1.1.2">ℵ</ci><cn id="S1.SSx1.p7.7.m7.1.1.3.cmml" type="integer" xref="S1.SSx1.p7.7.m7.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p7.7.m7.1c">\aleph_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p7.7.m7.1d">roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-dense set of reals, then <math alttext="\omega_{1}" class="ltx_Math" display="inline" id="S1.SSx1.p7.8.m8.1"><semantics id="S1.SSx1.p7.8.m8.1a"><msub id="S1.SSx1.p7.8.m8.1.1" xref="S1.SSx1.p7.8.m8.1.1.cmml"><mi id="S1.SSx1.p7.8.m8.1.1.2" xref="S1.SSx1.p7.8.m8.1.1.2.cmml">ω</mi><mn id="S1.SSx1.p7.8.m8.1.1.3" xref="S1.SSx1.p7.8.m8.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S1.SSx1.p7.8.m8.1b"><apply id="S1.SSx1.p7.8.m8.1.1.cmml" xref="S1.SSx1.p7.8.m8.1.1"><csymbol cd="ambiguous" id="S1.SSx1.p7.8.m8.1.1.1.cmml" xref="S1.SSx1.p7.8.m8.1.1">subscript</csymbol><ci id="S1.SSx1.p7.8.m8.1.1.2.cmml" xref="S1.SSx1.p7.8.m8.1.1.2">𝜔</ci><cn id="S1.SSx1.p7.8.m8.1.1.3.cmml" type="integer" xref="S1.SSx1.p7.8.m8.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p7.8.m8.1c">\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p7.8.m8.1d">italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\omega_{1}^{\star}" class="ltx_Math" display="inline" id="S1.SSx1.p7.9.m9.1"><semantics id="S1.SSx1.p7.9.m9.1a"><msubsup id="S1.SSx1.p7.9.m9.1.1" xref="S1.SSx1.p7.9.m9.1.1.cmml"><mi id="S1.SSx1.p7.9.m9.1.1.2.2" xref="S1.SSx1.p7.9.m9.1.1.2.2.cmml">ω</mi><mn id="S1.SSx1.p7.9.m9.1.1.2.3" xref="S1.SSx1.p7.9.m9.1.1.2.3.cmml">1</mn><mo id="S1.SSx1.p7.9.m9.1.1.3" xref="S1.SSx1.p7.9.m9.1.1.3.cmml">⋆</mo></msubsup><annotation-xml encoding="MathML-Content" id="S1.SSx1.p7.9.m9.1b"><apply id="S1.SSx1.p7.9.m9.1.1.cmml" xref="S1.SSx1.p7.9.m9.1.1"><csymbol cd="ambiguous" id="S1.SSx1.p7.9.m9.1.1.1.cmml" xref="S1.SSx1.p7.9.m9.1.1">superscript</csymbol><apply id="S1.SSx1.p7.9.m9.1.1.2.cmml" xref="S1.SSx1.p7.9.m9.1.1"><csymbol cd="ambiguous" id="S1.SSx1.p7.9.m9.1.1.2.1.cmml" xref="S1.SSx1.p7.9.m9.1.1">subscript</csymbol><ci id="S1.SSx1.p7.9.m9.1.1.2.2.cmml" xref="S1.SSx1.p7.9.m9.1.1.2.2">𝜔</ci><cn id="S1.SSx1.p7.9.m9.1.1.2.3.cmml" type="integer" xref="S1.SSx1.p7.9.m9.1.1.2.3">1</cn></apply><ci id="S1.SSx1.p7.9.m9.1.1.3.cmml" xref="S1.SSx1.p7.9.m9.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p7.9.m9.1c">\omega_{1}^{\star}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p7.9.m9.1d">italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="C" class="ltx_Math" display="inline" id="S1.SSx1.p7.10.m10.1"><semantics id="S1.SSx1.p7.10.m10.1a"><mi id="S1.SSx1.p7.10.m10.1.1" xref="S1.SSx1.p7.10.m10.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S1.SSx1.p7.10.m10.1b"><ci id="S1.SSx1.p7.10.m10.1.1.cmml" xref="S1.SSx1.p7.10.m10.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p7.10.m10.1c">C</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p7.10.m10.1d">italic_C</annotation></semantics></math>, <math alttext="C^{\star}" class="ltx_Math" display="inline" id="S1.SSx1.p7.11.m11.1"><semantics id="S1.SSx1.p7.11.m11.1a"><msup id="S1.SSx1.p7.11.m11.1.1" xref="S1.SSx1.p7.11.m11.1.1.cmml"><mi id="S1.SSx1.p7.11.m11.1.1.2" xref="S1.SSx1.p7.11.m11.1.1.2.cmml">C</mi><mo id="S1.SSx1.p7.11.m11.1.1.3" xref="S1.SSx1.p7.11.m11.1.1.3.cmml">⋆</mo></msup><annotation-xml encoding="MathML-Content" id="S1.SSx1.p7.11.m11.1b"><apply id="S1.SSx1.p7.11.m11.1.1.cmml" xref="S1.SSx1.p7.11.m11.1.1"><csymbol cd="ambiguous" id="S1.SSx1.p7.11.m11.1.1.1.cmml" xref="S1.SSx1.p7.11.m11.1.1">superscript</csymbol><ci id="S1.SSx1.p7.11.m11.1.1.2.cmml" xref="S1.SSx1.p7.11.m11.1.1.2">𝐶</ci><ci id="S1.SSx1.p7.11.m11.1.1.3.cmml" xref="S1.SSx1.p7.11.m11.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p7.11.m11.1c">C^{\star}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p7.11.m11.1d">italic_C start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="R" class="ltx_Math" display="inline" id="S1.SSx1.p7.12.m12.1"><semantics id="S1.SSx1.p7.12.m12.1a"><mi id="S1.SSx1.p7.12.m12.1.1" xref="S1.SSx1.p7.12.m12.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="S1.SSx1.p7.12.m12.1b"><ci id="S1.SSx1.p7.12.m12.1.1.cmml" xref="S1.SSx1.p7.12.m12.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p7.12.m12.1c">R</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p7.12.m12.1d">italic_R</annotation></semantics></math> form a <math alttext="\preceq" class="ltx_Math" display="inline" id="S1.SSx1.p7.13.m13.1"><semantics id="S1.SSx1.p7.13.m13.1a"><mo id="S1.SSx1.p7.13.m13.1.1" xref="S1.SSx1.p7.13.m13.1.1.cmml">⪯</mo><annotation-xml encoding="MathML-Content" id="S1.SSx1.p7.13.m13.1b"><csymbol cd="latexml" id="S1.SSx1.p7.13.m13.1.1.cmml" xref="S1.SSx1.p7.13.m13.1.1">precedes-or-equals</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p7.13.m13.1c">\preceq</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p7.13.m13.1d">⪯</annotation></semantics></math>-basis for the uncountable linear orders. After about thirty years, this was finally proved by Moore.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S1.Thmtheorem5"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem5.1.1.1">Theorem 1.5</span></span><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem5.2.2">.</span> </h6> <div class="ltx_para" id="S1.Thmtheorem5.p1"> <p class="ltx_p" id="S1.Thmtheorem5.p1.5"><span class="ltx_text ltx_font_italic" id="S1.Thmtheorem5.p1.5.5">(Moore <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib15" title="">15</a>]</cite>) Assume <math alttext="\mathsf{PFA}" class="ltx_Math" display="inline" id="S1.Thmtheorem5.p1.1.1.m1.1"><semantics id="S1.Thmtheorem5.p1.1.1.m1.1a"><mi id="S1.Thmtheorem5.p1.1.1.m1.1.1" xref="S1.Thmtheorem5.p1.1.1.m1.1.1.cmml">𝖯𝖥𝖠</mi><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem5.p1.1.1.m1.1b"><ci id="S1.Thmtheorem5.p1.1.1.m1.1.1.cmml" xref="S1.Thmtheorem5.p1.1.1.m1.1.1">𝖯𝖥𝖠</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem5.p1.1.1.m1.1c">\mathsf{PFA}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem5.p1.1.1.m1.1d">sansserif_PFA</annotation></semantics></math>. If <math alttext="C" class="ltx_Math" display="inline" id="S1.Thmtheorem5.p1.2.2.m2.1"><semantics id="S1.Thmtheorem5.p1.2.2.m2.1a"><mi id="S1.Thmtheorem5.p1.2.2.m2.1.1" xref="S1.Thmtheorem5.p1.2.2.m2.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem5.p1.2.2.m2.1b"><ci id="S1.Thmtheorem5.p1.2.2.m2.1.1.cmml" xref="S1.Thmtheorem5.p1.2.2.m2.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem5.p1.2.2.m2.1c">C</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem5.p1.2.2.m2.1d">italic_C</annotation></semantics></math> is any Countryman line, then <math alttext="C" class="ltx_Math" display="inline" id="S1.Thmtheorem5.p1.3.3.m3.1"><semantics id="S1.Thmtheorem5.p1.3.3.m3.1a"><mi id="S1.Thmtheorem5.p1.3.3.m3.1.1" xref="S1.Thmtheorem5.p1.3.3.m3.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem5.p1.3.3.m3.1b"><ci id="S1.Thmtheorem5.p1.3.3.m3.1.1.cmml" xref="S1.Thmtheorem5.p1.3.3.m3.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem5.p1.3.3.m3.1c">C</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem5.p1.3.3.m3.1d">italic_C</annotation></semantics></math> and <math alttext="C^{\star}" class="ltx_Math" display="inline" id="S1.Thmtheorem5.p1.4.4.m4.1"><semantics id="S1.Thmtheorem5.p1.4.4.m4.1a"><msup id="S1.Thmtheorem5.p1.4.4.m4.1.1" xref="S1.Thmtheorem5.p1.4.4.m4.1.1.cmml"><mi id="S1.Thmtheorem5.p1.4.4.m4.1.1.2" xref="S1.Thmtheorem5.p1.4.4.m4.1.1.2.cmml">C</mi><mo id="S1.Thmtheorem5.p1.4.4.m4.1.1.3" xref="S1.Thmtheorem5.p1.4.4.m4.1.1.3.cmml">⋆</mo></msup><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem5.p1.4.4.m4.1b"><apply id="S1.Thmtheorem5.p1.4.4.m4.1.1.cmml" xref="S1.Thmtheorem5.p1.4.4.m4.1.1"><csymbol cd="ambiguous" id="S1.Thmtheorem5.p1.4.4.m4.1.1.1.cmml" xref="S1.Thmtheorem5.p1.4.4.m4.1.1">superscript</csymbol><ci id="S1.Thmtheorem5.p1.4.4.m4.1.1.2.cmml" xref="S1.Thmtheorem5.p1.4.4.m4.1.1.2">𝐶</ci><ci id="S1.Thmtheorem5.p1.4.4.m4.1.1.3.cmml" xref="S1.Thmtheorem5.p1.4.4.m4.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem5.p1.4.4.m4.1c">C^{\star}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem5.p1.4.4.m4.1d">italic_C start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math> form a <math alttext="\preceq" class="ltx_Math" display="inline" id="S1.Thmtheorem5.p1.5.5.m5.1"><semantics id="S1.Thmtheorem5.p1.5.5.m5.1a"><mo id="S1.Thmtheorem5.p1.5.5.m5.1.1" xref="S1.Thmtheorem5.p1.5.5.m5.1.1.cmml">⪯</mo><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem5.p1.5.5.m5.1b"><csymbol cd="latexml" id="S1.Thmtheorem5.p1.5.5.m5.1.1.cmml" xref="S1.Thmtheorem5.p1.5.5.m5.1.1">precedes-or-equals</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem5.p1.5.5.m5.1c">\preceq</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem5.p1.5.5.m5.1d">⪯</annotation></semantics></math>-basis for the Aronszajn lines.</span></p> </div> </div> <div class="ltx_para" id="S1.SSx1.p8"> <p class="ltx_p" id="S1.SSx1.p8.2">Shortly after, Moore <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib16" title="">16</a>]</cite> also proved that under <math alttext="\mathsf{PFA}" class="ltx_Math" display="inline" id="S1.SSx1.p8.1.m1.1"><semantics id="S1.SSx1.p8.1.m1.1a"><mi id="S1.SSx1.p8.1.m1.1.1" xref="S1.SSx1.p8.1.m1.1.1.cmml">𝖯𝖥𝖠</mi><annotation-xml encoding="MathML-Content" id="S1.SSx1.p8.1.m1.1b"><ci id="S1.SSx1.p8.1.m1.1.1.cmml" xref="S1.SSx1.p8.1.m1.1.1">𝖯𝖥𝖠</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p8.1.m1.1c">\mathsf{PFA}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p8.1.m1.1d">sansserif_PFA</annotation></semantics></math> there is also a <math alttext="\preceq" class="ltx_Math" display="inline" id="S1.SSx1.p8.2.m2.1"><semantics id="S1.SSx1.p8.2.m2.1a"><mo id="S1.SSx1.p8.2.m2.1.1" xref="S1.SSx1.p8.2.m2.1.1.cmml">⪯</mo><annotation-xml encoding="MathML-Content" id="S1.SSx1.p8.2.m2.1b"><csymbol cd="latexml" id="S1.SSx1.p8.2.m2.1.1.cmml" xref="S1.SSx1.p8.2.m2.1.1">precedes-or-equals</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p8.2.m2.1c">\preceq</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p8.2.m2.1d">⪯</annotation></semantics></math>-top element in the class of Aronszajn lines.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S1.Thmtheorem6"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem6.1.1.1">Theorem 1.6</span></span><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem6.2.2">.</span> </h6> <div class="ltx_para" id="S1.Thmtheorem6.p1"> <p class="ltx_p" id="S1.Thmtheorem6.p1.2"><span class="ltx_text ltx_font_italic" id="S1.Thmtheorem6.p1.2.2">(Moore <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib16" title="">16</a>]</cite>) Assume <math alttext="\mathsf{PFA}" class="ltx_Math" display="inline" id="S1.Thmtheorem6.p1.1.1.m1.1"><semantics id="S1.Thmtheorem6.p1.1.1.m1.1a"><mi id="S1.Thmtheorem6.p1.1.1.m1.1.1" xref="S1.Thmtheorem6.p1.1.1.m1.1.1.cmml">𝖯𝖥𝖠</mi><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem6.p1.1.1.m1.1b"><ci id="S1.Thmtheorem6.p1.1.1.m1.1.1.cmml" xref="S1.Thmtheorem6.p1.1.1.m1.1.1">𝖯𝖥𝖠</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem6.p1.1.1.m1.1c">\mathsf{PFA}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem6.p1.1.1.m1.1d">sansserif_PFA</annotation></semantics></math>. There is a universal Aronszajn line <math alttext="\eta_{C}" class="ltx_Math" display="inline" id="S1.Thmtheorem6.p1.2.2.m2.1"><semantics id="S1.Thmtheorem6.p1.2.2.m2.1a"><msub id="S1.Thmtheorem6.p1.2.2.m2.1.1" xref="S1.Thmtheorem6.p1.2.2.m2.1.1.cmml"><mi id="S1.Thmtheorem6.p1.2.2.m2.1.1.2" xref="S1.Thmtheorem6.p1.2.2.m2.1.1.2.cmml">η</mi><mi id="S1.Thmtheorem6.p1.2.2.m2.1.1.3" xref="S1.Thmtheorem6.p1.2.2.m2.1.1.3.cmml">C</mi></msub><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem6.p1.2.2.m2.1b"><apply id="S1.Thmtheorem6.p1.2.2.m2.1.1.cmml" xref="S1.Thmtheorem6.p1.2.2.m2.1.1"><csymbol cd="ambiguous" id="S1.Thmtheorem6.p1.2.2.m2.1.1.1.cmml" xref="S1.Thmtheorem6.p1.2.2.m2.1.1">subscript</csymbol><ci id="S1.Thmtheorem6.p1.2.2.m2.1.1.2.cmml" xref="S1.Thmtheorem6.p1.2.2.m2.1.1.2">𝜂</ci><ci id="S1.Thmtheorem6.p1.2.2.m2.1.1.3.cmml" xref="S1.Thmtheorem6.p1.2.2.m2.1.1.3">𝐶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem6.p1.2.2.m2.1c">\eta_{C}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem6.p1.2.2.m2.1d">italic_η start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S1.SSx1.p9"> <p class="ltx_p" id="S1.SSx1.p9.9">The construction of <math alttext="\eta_{C}" class="ltx_Math" display="inline" id="S1.SSx1.p9.1.m1.1"><semantics id="S1.SSx1.p9.1.m1.1a"><msub id="S1.SSx1.p9.1.m1.1.1" xref="S1.SSx1.p9.1.m1.1.1.cmml"><mi id="S1.SSx1.p9.1.m1.1.1.2" xref="S1.SSx1.p9.1.m1.1.1.2.cmml">η</mi><mi id="S1.SSx1.p9.1.m1.1.1.3" xref="S1.SSx1.p9.1.m1.1.1.3.cmml">C</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SSx1.p9.1.m1.1b"><apply id="S1.SSx1.p9.1.m1.1.1.cmml" xref="S1.SSx1.p9.1.m1.1.1"><csymbol cd="ambiguous" id="S1.SSx1.p9.1.m1.1.1.1.cmml" xref="S1.SSx1.p9.1.m1.1.1">subscript</csymbol><ci id="S1.SSx1.p9.1.m1.1.1.2.cmml" xref="S1.SSx1.p9.1.m1.1.1.2">𝜂</ci><ci id="S1.SSx1.p9.1.m1.1.1.3.cmml" xref="S1.SSx1.p9.1.m1.1.1.3">𝐶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p9.1.m1.1c">\eta_{C}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p9.1.m1.1d">italic_η start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT</annotation></semantics></math> is as follows: fix a Countryman line <math alttext="C" class="ltx_Math" display="inline" id="S1.SSx1.p9.2.m2.1"><semantics id="S1.SSx1.p9.2.m2.1a"><mi id="S1.SSx1.p9.2.m2.1.1" xref="S1.SSx1.p9.2.m2.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S1.SSx1.p9.2.m2.1b"><ci id="S1.SSx1.p9.2.m2.1.1.cmml" xref="S1.SSx1.p9.2.m2.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p9.2.m2.1c">C</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p9.2.m2.1d">italic_C</annotation></semantics></math>, and take the lexicographic ordering of the <math alttext="\omega" class="ltx_Math" display="inline" id="S1.SSx1.p9.3.m3.1"><semantics id="S1.SSx1.p9.3.m3.1a"><mi id="S1.SSx1.p9.3.m3.1.1" xref="S1.SSx1.p9.3.m3.1.1.cmml">ω</mi><annotation-xml encoding="MathML-Content" id="S1.SSx1.p9.3.m3.1b"><ci id="S1.SSx1.p9.3.m3.1.1.cmml" xref="S1.SSx1.p9.3.m3.1.1">𝜔</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p9.3.m3.1c">\omega</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p9.3.m3.1d">italic_ω</annotation></semantics></math>-sequences on <math alttext="C^{\star}+\{0\}+C" class="ltx_Math" display="inline" id="S1.SSx1.p9.4.m4.1"><semantics id="S1.SSx1.p9.4.m4.1a"><mrow id="S1.SSx1.p9.4.m4.1.2" xref="S1.SSx1.p9.4.m4.1.2.cmml"><msup id="S1.SSx1.p9.4.m4.1.2.2" xref="S1.SSx1.p9.4.m4.1.2.2.cmml"><mi id="S1.SSx1.p9.4.m4.1.2.2.2" xref="S1.SSx1.p9.4.m4.1.2.2.2.cmml">C</mi><mo id="S1.SSx1.p9.4.m4.1.2.2.3" xref="S1.SSx1.p9.4.m4.1.2.2.3.cmml">⋆</mo></msup><mo id="S1.SSx1.p9.4.m4.1.2.1" xref="S1.SSx1.p9.4.m4.1.2.1.cmml">+</mo><mrow id="S1.SSx1.p9.4.m4.1.2.3.2" xref="S1.SSx1.p9.4.m4.1.2.3.1.cmml"><mo id="S1.SSx1.p9.4.m4.1.2.3.2.1" stretchy="false" xref="S1.SSx1.p9.4.m4.1.2.3.1.cmml">{</mo><mn id="S1.SSx1.p9.4.m4.1.1" xref="S1.SSx1.p9.4.m4.1.1.cmml">0</mn><mo id="S1.SSx1.p9.4.m4.1.2.3.2.2" stretchy="false" xref="S1.SSx1.p9.4.m4.1.2.3.1.cmml">}</mo></mrow><mo id="S1.SSx1.p9.4.m4.1.2.1a" xref="S1.SSx1.p9.4.m4.1.2.1.cmml">+</mo><mi id="S1.SSx1.p9.4.m4.1.2.4" xref="S1.SSx1.p9.4.m4.1.2.4.cmml">C</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.SSx1.p9.4.m4.1b"><apply id="S1.SSx1.p9.4.m4.1.2.cmml" xref="S1.SSx1.p9.4.m4.1.2"><plus id="S1.SSx1.p9.4.m4.1.2.1.cmml" xref="S1.SSx1.p9.4.m4.1.2.1"></plus><apply id="S1.SSx1.p9.4.m4.1.2.2.cmml" xref="S1.SSx1.p9.4.m4.1.2.2"><csymbol cd="ambiguous" id="S1.SSx1.p9.4.m4.1.2.2.1.cmml" xref="S1.SSx1.p9.4.m4.1.2.2">superscript</csymbol><ci id="S1.SSx1.p9.4.m4.1.2.2.2.cmml" xref="S1.SSx1.p9.4.m4.1.2.2.2">𝐶</ci><ci id="S1.SSx1.p9.4.m4.1.2.2.3.cmml" xref="S1.SSx1.p9.4.m4.1.2.2.3">⋆</ci></apply><set id="S1.SSx1.p9.4.m4.1.2.3.1.cmml" xref="S1.SSx1.p9.4.m4.1.2.3.2"><cn id="S1.SSx1.p9.4.m4.1.1.cmml" type="integer" xref="S1.SSx1.p9.4.m4.1.1">0</cn></set><ci id="S1.SSx1.p9.4.m4.1.2.4.cmml" xref="S1.SSx1.p9.4.m4.1.2.4">𝐶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p9.4.m4.1c">C^{\star}+\{0\}+C</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p9.4.m4.1d">italic_C start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT + { 0 } + italic_C</annotation></semantics></math> which are eventually <math alttext="0" class="ltx_Math" display="inline" id="S1.SSx1.p9.5.m5.1"><semantics id="S1.SSx1.p9.5.m5.1a"><mn id="S1.SSx1.p9.5.m5.1.1" xref="S1.SSx1.p9.5.m5.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S1.SSx1.p9.5.m5.1b"><cn id="S1.SSx1.p9.5.m5.1.1.cmml" type="integer" xref="S1.SSx1.p9.5.m5.1.1">0</cn></annotation-xml></semantics></math>. Note that replacing <math alttext="C" class="ltx_Math" display="inline" id="S1.SSx1.p9.6.m6.1"><semantics id="S1.SSx1.p9.6.m6.1a"><mi id="S1.SSx1.p9.6.m6.1.1" xref="S1.SSx1.p9.6.m6.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S1.SSx1.p9.6.m6.1b"><ci id="S1.SSx1.p9.6.m6.1.1.cmml" xref="S1.SSx1.p9.6.m6.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p9.6.m6.1c">C</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p9.6.m6.1d">italic_C</annotation></semantics></math> by <math alttext="\omega" class="ltx_Math" display="inline" id="S1.SSx1.p9.7.m7.1"><semantics id="S1.SSx1.p9.7.m7.1a"><mi id="S1.SSx1.p9.7.m7.1.1" xref="S1.SSx1.p9.7.m7.1.1.cmml">ω</mi><annotation-xml encoding="MathML-Content" id="S1.SSx1.p9.7.m7.1b"><ci id="S1.SSx1.p9.7.m7.1.1.cmml" xref="S1.SSx1.p9.7.m7.1.1">𝜔</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p9.7.m7.1c">\omega</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p9.7.m7.1d">italic_ω</annotation></semantics></math> in the previous construction one gets an isomorphic copy of <math alttext="\mathbb{Q}" class="ltx_Math" display="inline" id="S1.SSx1.p9.8.m8.1"><semantics id="S1.SSx1.p9.8.m8.1a"><mi id="S1.SSx1.p9.8.m8.1.1" xref="S1.SSx1.p9.8.m8.1.1.cmml">ℚ</mi><annotation-xml encoding="MathML-Content" id="S1.SSx1.p9.8.m8.1b"><ci id="S1.SSx1.p9.8.m8.1.1.cmml" xref="S1.SSx1.p9.8.m8.1.1">ℚ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p9.8.m8.1c">\mathbb{Q}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p9.8.m8.1d">blackboard_Q</annotation></semantics></math>. This suggests an analogy between the countable linear orders and the class of Aronszajn lines under <math alttext="\mathsf{PFA}" class="ltx_Math" display="inline" id="S1.SSx1.p9.9.m9.1"><semantics id="S1.SSx1.p9.9.m9.1a"><mi id="S1.SSx1.p9.9.m9.1.1" xref="S1.SSx1.p9.9.m9.1.1.cmml">𝖯𝖥𝖠</mi><annotation-xml encoding="MathML-Content" id="S1.SSx1.p9.9.m9.1b"><ci id="S1.SSx1.p9.9.m9.1.1.cmml" xref="S1.SSx1.p9.9.m9.1.1">𝖯𝖥𝖠</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p9.9.m9.1c">\mathsf{PFA}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p9.9.m9.1d">sansserif_PFA</annotation></semantics></math>. Martínez-Ranero extended this analogy further by proving that the analogous of Laver’s theorem holds in the class of Aronszajn lines.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S1.Thmtheorem7"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem7.1.1.1">Theorem 1.7</span></span><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem7.2.2">.</span> </h6> <div class="ltx_para" id="S1.Thmtheorem7.p1"> <p class="ltx_p" id="S1.Thmtheorem7.p1.2"><span class="ltx_text ltx_font_italic" id="S1.Thmtheorem7.p1.2.2">(Martínez-Ranero <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib14" title="">14</a>]</cite>) Assume <math alttext="\mathsf{PFA}" class="ltx_Math" display="inline" id="S1.Thmtheorem7.p1.1.1.m1.1"><semantics id="S1.Thmtheorem7.p1.1.1.m1.1a"><mi id="S1.Thmtheorem7.p1.1.1.m1.1.1" xref="S1.Thmtheorem7.p1.1.1.m1.1.1.cmml">𝖯𝖥𝖠</mi><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem7.p1.1.1.m1.1b"><ci id="S1.Thmtheorem7.p1.1.1.m1.1.1.cmml" xref="S1.Thmtheorem7.p1.1.1.m1.1.1">𝖯𝖥𝖠</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem7.p1.1.1.m1.1c">\mathsf{PFA}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem7.p1.1.1.m1.1d">sansserif_PFA</annotation></semantics></math>. The class of Aronszajn lines is well-quasi-ordered by <math alttext="\preceq" class="ltx_Math" display="inline" id="S1.Thmtheorem7.p1.2.2.m2.1"><semantics id="S1.Thmtheorem7.p1.2.2.m2.1a"><mo id="S1.Thmtheorem7.p1.2.2.m2.1.1" xref="S1.Thmtheorem7.p1.2.2.m2.1.1.cmml">⪯</mo><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem7.p1.2.2.m2.1b"><csymbol cd="latexml" id="S1.Thmtheorem7.p1.2.2.m2.1.1.cmml" xref="S1.Thmtheorem7.p1.2.2.m2.1.1">precedes-or-equals</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem7.p1.2.2.m2.1c">\preceq</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem7.p1.2.2.m2.1d">⪯</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S1.SSx1.p10"> <p class="ltx_p" id="S1.SSx1.p10.9">Now let us ask about <math alttext="\trianglelefteq" class="ltx_Math" display="inline" id="S1.SSx1.p10.1.m1.1"><semantics id="S1.SSx1.p10.1.m1.1a"><mi id="S1.SSx1.p10.1.m1.1.1" mathvariant="normal" xref="S1.SSx1.p10.1.m1.1.1.cmml">⊴</mi><annotation-xml encoding="MathML-Content" id="S1.SSx1.p10.1.m1.1b"><ci id="S1.SSx1.p10.1.m1.1.1.cmml" xref="S1.SSx1.p10.1.m1.1.1">⊴</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p10.1.m1.1c">\trianglelefteq</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p10.1.m1.1d">⊴</annotation></semantics></math>. Again it is not hard to see that <math alttext="\omega" class="ltx_Math" display="inline" id="S1.SSx1.p10.2.m2.1"><semantics id="S1.SSx1.p10.2.m2.1a"><mi id="S1.SSx1.p10.2.m2.1.1" xref="S1.SSx1.p10.2.m2.1.1.cmml">ω</mi><annotation-xml encoding="MathML-Content" id="S1.SSx1.p10.2.m2.1b"><ci id="S1.SSx1.p10.2.m2.1.1.cmml" xref="S1.SSx1.p10.2.m2.1.1">𝜔</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p10.2.m2.1c">\omega</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p10.2.m2.1d">italic_ω</annotation></semantics></math>, <math alttext="\omega^{\star}" class="ltx_Math" display="inline" id="S1.SSx1.p10.3.m3.1"><semantics id="S1.SSx1.p10.3.m3.1a"><msup id="S1.SSx1.p10.3.m3.1.1" xref="S1.SSx1.p10.3.m3.1.1.cmml"><mi id="S1.SSx1.p10.3.m3.1.1.2" xref="S1.SSx1.p10.3.m3.1.1.2.cmml">ω</mi><mo id="S1.SSx1.p10.3.m3.1.1.3" xref="S1.SSx1.p10.3.m3.1.1.3.cmml">⋆</mo></msup><annotation-xml encoding="MathML-Content" id="S1.SSx1.p10.3.m3.1b"><apply id="S1.SSx1.p10.3.m3.1.1.cmml" xref="S1.SSx1.p10.3.m3.1.1"><csymbol cd="ambiguous" id="S1.SSx1.p10.3.m3.1.1.1.cmml" xref="S1.SSx1.p10.3.m3.1.1">superscript</csymbol><ci id="S1.SSx1.p10.3.m3.1.1.2.cmml" xref="S1.SSx1.p10.3.m3.1.1.2">𝜔</ci><ci id="S1.SSx1.p10.3.m3.1.1.3.cmml" xref="S1.SSx1.p10.3.m3.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p10.3.m3.1c">\omega^{\star}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p10.3.m3.1d">italic_ω start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="\omega+1" class="ltx_Math" display="inline" id="S1.SSx1.p10.4.m4.1"><semantics id="S1.SSx1.p10.4.m4.1a"><mrow id="S1.SSx1.p10.4.m4.1.1" xref="S1.SSx1.p10.4.m4.1.1.cmml"><mi id="S1.SSx1.p10.4.m4.1.1.2" xref="S1.SSx1.p10.4.m4.1.1.2.cmml">ω</mi><mo id="S1.SSx1.p10.4.m4.1.1.1" xref="S1.SSx1.p10.4.m4.1.1.1.cmml">+</mo><mn id="S1.SSx1.p10.4.m4.1.1.3" xref="S1.SSx1.p10.4.m4.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.SSx1.p10.4.m4.1b"><apply id="S1.SSx1.p10.4.m4.1.1.cmml" xref="S1.SSx1.p10.4.m4.1.1"><plus id="S1.SSx1.p10.4.m4.1.1.1.cmml" xref="S1.SSx1.p10.4.m4.1.1.1"></plus><ci id="S1.SSx1.p10.4.m4.1.1.2.cmml" xref="S1.SSx1.p10.4.m4.1.1.2">𝜔</ci><cn id="S1.SSx1.p10.4.m4.1.1.3.cmml" type="integer" xref="S1.SSx1.p10.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p10.4.m4.1c">\omega+1</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p10.4.m4.1d">italic_ω + 1</annotation></semantics></math> and <math alttext="1+\omega^{\star}" class="ltx_Math" display="inline" id="S1.SSx1.p10.5.m5.1"><semantics id="S1.SSx1.p10.5.m5.1a"><mrow id="S1.SSx1.p10.5.m5.1.1" xref="S1.SSx1.p10.5.m5.1.1.cmml"><mn id="S1.SSx1.p10.5.m5.1.1.2" xref="S1.SSx1.p10.5.m5.1.1.2.cmml">1</mn><mo id="S1.SSx1.p10.5.m5.1.1.1" xref="S1.SSx1.p10.5.m5.1.1.1.cmml">+</mo><msup id="S1.SSx1.p10.5.m5.1.1.3" xref="S1.SSx1.p10.5.m5.1.1.3.cmml"><mi id="S1.SSx1.p10.5.m5.1.1.3.2" xref="S1.SSx1.p10.5.m5.1.1.3.2.cmml">ω</mi><mo id="S1.SSx1.p10.5.m5.1.1.3.3" xref="S1.SSx1.p10.5.m5.1.1.3.3.cmml">⋆</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S1.SSx1.p10.5.m5.1b"><apply id="S1.SSx1.p10.5.m5.1.1.cmml" xref="S1.SSx1.p10.5.m5.1.1"><plus id="S1.SSx1.p10.5.m5.1.1.1.cmml" xref="S1.SSx1.p10.5.m5.1.1.1"></plus><cn id="S1.SSx1.p10.5.m5.1.1.2.cmml" type="integer" xref="S1.SSx1.p10.5.m5.1.1.2">1</cn><apply id="S1.SSx1.p10.5.m5.1.1.3.cmml" xref="S1.SSx1.p10.5.m5.1.1.3"><csymbol cd="ambiguous" id="S1.SSx1.p10.5.m5.1.1.3.1.cmml" xref="S1.SSx1.p10.5.m5.1.1.3">superscript</csymbol><ci id="S1.SSx1.p10.5.m5.1.1.3.2.cmml" xref="S1.SSx1.p10.5.m5.1.1.3.2">𝜔</ci><ci id="S1.SSx1.p10.5.m5.1.1.3.3.cmml" xref="S1.SSx1.p10.5.m5.1.1.3.3">⋆</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p10.5.m5.1c">1+\omega^{\star}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p10.5.m5.1d">1 + italic_ω start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math> form a <math alttext="\trianglelefteq" class="ltx_Math" display="inline" id="S1.SSx1.p10.6.m6.1"><semantics id="S1.SSx1.p10.6.m6.1a"><mi id="S1.SSx1.p10.6.m6.1.1" mathvariant="normal" xref="S1.SSx1.p10.6.m6.1.1.cmml">⊴</mi><annotation-xml encoding="MathML-Content" id="S1.SSx1.p10.6.m6.1b"><ci id="S1.SSx1.p10.6.m6.1.1.cmml" xref="S1.SSx1.p10.6.m6.1.1">⊴</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p10.6.m6.1c">\trianglelefteq</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p10.6.m6.1d">⊴</annotation></semantics></math>-basis for the countable linear orders, and that <math alttext="\mathbb{Q}" class="ltx_Math" display="inline" id="S1.SSx1.p10.7.m7.1"><semantics id="S1.SSx1.p10.7.m7.1a"><mi id="S1.SSx1.p10.7.m7.1.1" xref="S1.SSx1.p10.7.m7.1.1.cmml">ℚ</mi><annotation-xml encoding="MathML-Content" id="S1.SSx1.p10.7.m7.1b"><ci id="S1.SSx1.p10.7.m7.1.1.cmml" xref="S1.SSx1.p10.7.m7.1.1">ℚ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p10.7.m7.1c">\mathbb{Q}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p10.7.m7.1d">blackboard_Q</annotation></semantics></math> is the <math alttext="\trianglelefteq" class="ltx_Math" display="inline" id="S1.SSx1.p10.8.m8.1"><semantics id="S1.SSx1.p10.8.m8.1a"><mi id="S1.SSx1.p10.8.m8.1.1" mathvariant="normal" xref="S1.SSx1.p10.8.m8.1.1.cmml">⊴</mi><annotation-xml encoding="MathML-Content" id="S1.SSx1.p10.8.m8.1b"><ci id="S1.SSx1.p10.8.m8.1.1.cmml" xref="S1.SSx1.p10.8.m8.1.1">⊴</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p10.8.m8.1c">\trianglelefteq</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p10.8.m8.1d">⊴</annotation></semantics></math>-top element again. More surprisingly Landraitis proved in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib12" title="">12</a>]</cite> (and later Camerlo, Carroy and Marcone in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib4" title="">4</a>]</cite>) that this class is also well-quasi-ordered by <math alttext="\trianglelefteq" class="ltx_Math" display="inline" id="S1.SSx1.p10.9.m9.1"><semantics id="S1.SSx1.p10.9.m9.1a"><mi id="S1.SSx1.p10.9.m9.1.1" mathvariant="normal" xref="S1.SSx1.p10.9.m9.1.1.cmml">⊴</mi><annotation-xml encoding="MathML-Content" id="S1.SSx1.p10.9.m9.1b"><ci id="S1.SSx1.p10.9.m9.1.1.cmml" xref="S1.SSx1.p10.9.m9.1.1">⊴</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p10.9.m9.1c">\trianglelefteq</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p10.9.m9.1d">⊴</annotation></semantics></math>. We believe that the following questions present themselves naturally.</p> </div> <div class="ltx_theorem ltx_theorem_question" id="Thmquestion1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmquestion1.1.1.1">Question 1</span></span><span class="ltx_text ltx_font_bold" id="Thmquestion1.2.2">.</span> </h6> <div class="ltx_para" id="Thmquestion1.p1"> <p class="ltx_p" id="Thmquestion1.p1.2">Assume <math alttext="\mathsf{PFA}" class="ltx_Math" display="inline" id="Thmquestion1.p1.1.m1.1"><semantics id="Thmquestion1.p1.1.m1.1a"><mi id="Thmquestion1.p1.1.m1.1.1" xref="Thmquestion1.p1.1.m1.1.1.cmml">𝖯𝖥𝖠</mi><annotation-xml encoding="MathML-Content" id="Thmquestion1.p1.1.m1.1b"><ci id="Thmquestion1.p1.1.m1.1.1.cmml" xref="Thmquestion1.p1.1.m1.1.1">𝖯𝖥𝖠</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmquestion1.p1.1.m1.1c">\mathsf{PFA}</annotation><annotation encoding="application/x-llamapun" id="Thmquestion1.p1.1.m1.1d">sansserif_PFA</annotation></semantics></math>. Is there a finite <math alttext="\trianglelefteq" class="ltx_Math" display="inline" id="Thmquestion1.p1.2.m2.1"><semantics id="Thmquestion1.p1.2.m2.1a"><mi id="Thmquestion1.p1.2.m2.1.1" mathvariant="normal" xref="Thmquestion1.p1.2.m2.1.1.cmml">⊴</mi><annotation-xml encoding="MathML-Content" id="Thmquestion1.p1.2.m2.1b"><ci id="Thmquestion1.p1.2.m2.1.1.cmml" xref="Thmquestion1.p1.2.m2.1.1">⊴</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmquestion1.p1.2.m2.1c">\trianglelefteq</annotation><annotation encoding="application/x-llamapun" id="Thmquestion1.p1.2.m2.1d">⊴</annotation></semantics></math>-basis for the class of Aronszajn lines?</p> </div> </div> <div class="ltx_theorem ltx_theorem_question" id="Thmquestion2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmquestion2.1.1.1">Question 2</span></span><span class="ltx_text ltx_font_bold" id="Thmquestion2.2.2">.</span> </h6> <div class="ltx_para" id="Thmquestion2.p1"> <p class="ltx_p" id="Thmquestion2.p1.2">Assume <math alttext="\mathsf{PFA}" class="ltx_Math" display="inline" id="Thmquestion2.p1.1.m1.1"><semantics id="Thmquestion2.p1.1.m1.1a"><mi id="Thmquestion2.p1.1.m1.1.1" xref="Thmquestion2.p1.1.m1.1.1.cmml">𝖯𝖥𝖠</mi><annotation-xml encoding="MathML-Content" id="Thmquestion2.p1.1.m1.1b"><ci id="Thmquestion2.p1.1.m1.1.1.cmml" xref="Thmquestion2.p1.1.m1.1.1">𝖯𝖥𝖠</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmquestion2.p1.1.m1.1c">\mathsf{PFA}</annotation><annotation encoding="application/x-llamapun" id="Thmquestion2.p1.1.m1.1d">sansserif_PFA</annotation></semantics></math>. Is the class of Aronszajn lines well-quasi-ordered by <math alttext="\trianglelefteq" class="ltx_Math" display="inline" id="Thmquestion2.p1.2.m2.1"><semantics id="Thmquestion2.p1.2.m2.1a"><mi id="Thmquestion2.p1.2.m2.1.1" mathvariant="normal" xref="Thmquestion2.p1.2.m2.1.1.cmml">⊴</mi><annotation-xml encoding="MathML-Content" id="Thmquestion2.p1.2.m2.1b"><ci id="Thmquestion2.p1.2.m2.1.1.cmml" xref="Thmquestion2.p1.2.m2.1.1">⊴</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmquestion2.p1.2.m2.1c">\trianglelefteq</annotation><annotation encoding="application/x-llamapun" id="Thmquestion2.p1.2.m2.1d">⊴</annotation></semantics></math>?</p> </div> </div> <div class="ltx_para" id="S1.SSx1.p11"> <p class="ltx_p" id="S1.SSx1.p11.1">In <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib5" title="">5</a>]</cite> the following notion is introduced to study the <math alttext="\trianglelefteq" class="ltx_Math" display="inline" id="S1.SSx1.p11.1.m1.1"><semantics id="S1.SSx1.p11.1.m1.1a"><mi id="S1.SSx1.p11.1.m1.1.1" mathvariant="normal" xref="S1.SSx1.p11.1.m1.1.1.cmml">⊴</mi><annotation-xml encoding="MathML-Content" id="S1.SSx1.p11.1.m1.1b"><ci id="S1.SSx1.p11.1.m1.1.1.cmml" xref="S1.SSx1.p11.1.m1.1.1">⊴</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p11.1.m1.1c">\trianglelefteq</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p11.1.m1.1d">⊴</annotation></semantics></math> relation.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="S1.Thmtheorem8"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem8.1.1.1">Definition 1.8</span></span><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem8.2.2">.</span> </h6> <div class="ltx_para" id="S1.Thmtheorem8.p1"> <p class="ltx_p" id="S1.Thmtheorem8.p1.4">A linear order <math alttext="A" class="ltx_Math" display="inline" id="S1.Thmtheorem8.p1.1.m1.1"><semantics id="S1.Thmtheorem8.p1.1.m1.1a"><mi id="S1.Thmtheorem8.p1.1.m1.1.1" xref="S1.Thmtheorem8.p1.1.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem8.p1.1.m1.1b"><ci id="S1.Thmtheorem8.p1.1.m1.1.1.cmml" xref="S1.Thmtheorem8.p1.1.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem8.p1.1.m1.1c">A</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem8.p1.1.m1.1d">italic_A</annotation></semantics></math> is called <em class="ltx_emph ltx_font_italic" id="S1.Thmtheorem8.p1.4.1">strongly surjective</em> if <math alttext="B\trianglelefteq A" class="ltx_Math" display="inline" id="S1.Thmtheorem8.p1.2.m2.1"><semantics id="S1.Thmtheorem8.p1.2.m2.1a"><mrow id="S1.Thmtheorem8.p1.2.m2.1.1" xref="S1.Thmtheorem8.p1.2.m2.1.1.cmml"><mi id="S1.Thmtheorem8.p1.2.m2.1.1.2" xref="S1.Thmtheorem8.p1.2.m2.1.1.2.cmml">B</mi><mo id="S1.Thmtheorem8.p1.2.m2.1.1.1" xref="S1.Thmtheorem8.p1.2.m2.1.1.1.cmml">⁢</mo><mi id="S1.Thmtheorem8.p1.2.m2.1.1.3" mathvariant="normal" xref="S1.Thmtheorem8.p1.2.m2.1.1.3.cmml">⊴</mi><mo id="S1.Thmtheorem8.p1.2.m2.1.1.1a" xref="S1.Thmtheorem8.p1.2.m2.1.1.1.cmml">⁢</mo><mi id="S1.Thmtheorem8.p1.2.m2.1.1.4" xref="S1.Thmtheorem8.p1.2.m2.1.1.4.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem8.p1.2.m2.1b"><apply id="S1.Thmtheorem8.p1.2.m2.1.1.cmml" xref="S1.Thmtheorem8.p1.2.m2.1.1"><times id="S1.Thmtheorem8.p1.2.m2.1.1.1.cmml" xref="S1.Thmtheorem8.p1.2.m2.1.1.1"></times><ci id="S1.Thmtheorem8.p1.2.m2.1.1.2.cmml" xref="S1.Thmtheorem8.p1.2.m2.1.1.2">𝐵</ci><ci id="S1.Thmtheorem8.p1.2.m2.1.1.3.cmml" xref="S1.Thmtheorem8.p1.2.m2.1.1.3">⊴</ci><ci id="S1.Thmtheorem8.p1.2.m2.1.1.4.cmml" xref="S1.Thmtheorem8.p1.2.m2.1.1.4">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem8.p1.2.m2.1c">B\trianglelefteq A</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem8.p1.2.m2.1d">italic_B ⊴ italic_A</annotation></semantics></math> whenever <math alttext="B" class="ltx_Math" display="inline" id="S1.Thmtheorem8.p1.3.m3.1"><semantics id="S1.Thmtheorem8.p1.3.m3.1a"><mi id="S1.Thmtheorem8.p1.3.m3.1.1" xref="S1.Thmtheorem8.p1.3.m3.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem8.p1.3.m3.1b"><ci id="S1.Thmtheorem8.p1.3.m3.1.1.cmml" xref="S1.Thmtheorem8.p1.3.m3.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem8.p1.3.m3.1c">B</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem8.p1.3.m3.1d">italic_B</annotation></semantics></math> is a suborder of <math alttext="A" class="ltx_Math" display="inline" id="S1.Thmtheorem8.p1.4.m4.1"><semantics id="S1.Thmtheorem8.p1.4.m4.1a"><mi id="S1.Thmtheorem8.p1.4.m4.1.1" xref="S1.Thmtheorem8.p1.4.m4.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem8.p1.4.m4.1b"><ci id="S1.Thmtheorem8.p1.4.m4.1.1.cmml" xref="S1.Thmtheorem8.p1.4.m4.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem8.p1.4.m4.1c">A</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem8.p1.4.m4.1d">italic_A</annotation></semantics></math>.</p> </div> </div> <div class="ltx_para" id="S1.SSx1.p12"> <p class="ltx_p" id="S1.SSx1.p12.6">Easy examples of strongly surjective orders are <math alttext="\omega" class="ltx_Math" display="inline" id="S1.SSx1.p12.1.m1.1"><semantics id="S1.SSx1.p12.1.m1.1a"><mi id="S1.SSx1.p12.1.m1.1.1" xref="S1.SSx1.p12.1.m1.1.1.cmml">ω</mi><annotation-xml encoding="MathML-Content" id="S1.SSx1.p12.1.m1.1b"><ci id="S1.SSx1.p12.1.m1.1.1.cmml" xref="S1.SSx1.p12.1.m1.1.1">𝜔</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p12.1.m1.1c">\omega</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p12.1.m1.1d">italic_ω</annotation></semantics></math>, <math alttext="\omega^{\star}" class="ltx_Math" display="inline" id="S1.SSx1.p12.2.m2.1"><semantics id="S1.SSx1.p12.2.m2.1a"><msup id="S1.SSx1.p12.2.m2.1.1" xref="S1.SSx1.p12.2.m2.1.1.cmml"><mi id="S1.SSx1.p12.2.m2.1.1.2" xref="S1.SSx1.p12.2.m2.1.1.2.cmml">ω</mi><mo id="S1.SSx1.p12.2.m2.1.1.3" xref="S1.SSx1.p12.2.m2.1.1.3.cmml">⋆</mo></msup><annotation-xml encoding="MathML-Content" id="S1.SSx1.p12.2.m2.1b"><apply id="S1.SSx1.p12.2.m2.1.1.cmml" xref="S1.SSx1.p12.2.m2.1.1"><csymbol cd="ambiguous" id="S1.SSx1.p12.2.m2.1.1.1.cmml" xref="S1.SSx1.p12.2.m2.1.1">superscript</csymbol><ci id="S1.SSx1.p12.2.m2.1.1.2.cmml" xref="S1.SSx1.p12.2.m2.1.1.2">𝜔</ci><ci id="S1.SSx1.p12.2.m2.1.1.3.cmml" xref="S1.SSx1.p12.2.m2.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p12.2.m2.1c">\omega^{\star}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p12.2.m2.1d">italic_ω start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\mathbb{Q}" class="ltx_Math" display="inline" id="S1.SSx1.p12.3.m3.1"><semantics id="S1.SSx1.p12.3.m3.1a"><mi id="S1.SSx1.p12.3.m3.1.1" xref="S1.SSx1.p12.3.m3.1.1.cmml">ℚ</mi><annotation-xml encoding="MathML-Content" id="S1.SSx1.p12.3.m3.1b"><ci id="S1.SSx1.p12.3.m3.1.1.cmml" xref="S1.SSx1.p12.3.m3.1.1">ℚ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p12.3.m3.1c">\mathbb{Q}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p12.3.m3.1d">blackboard_Q</annotation></semantics></math>. The authors of <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib5" title="">5</a>]</cite> ask if there are uncountable strongly surjective orders. They prove that consistently yes: if the conclusion of Baumgartner’s theorem holds, then any <math alttext="\aleph_{1}" class="ltx_Math" display="inline" id="S1.SSx1.p12.4.m4.1"><semantics id="S1.SSx1.p12.4.m4.1a"><msub id="S1.SSx1.p12.4.m4.1.1" xref="S1.SSx1.p12.4.m4.1.1.cmml"><mi id="S1.SSx1.p12.4.m4.1.1.2" mathvariant="normal" xref="S1.SSx1.p12.4.m4.1.1.2.cmml">ℵ</mi><mn id="S1.SSx1.p12.4.m4.1.1.3" xref="S1.SSx1.p12.4.m4.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S1.SSx1.p12.4.m4.1b"><apply id="S1.SSx1.p12.4.m4.1.1.cmml" xref="S1.SSx1.p12.4.m4.1.1"><csymbol cd="ambiguous" id="S1.SSx1.p12.4.m4.1.1.1.cmml" xref="S1.SSx1.p12.4.m4.1.1">subscript</csymbol><ci id="S1.SSx1.p12.4.m4.1.1.2.cmml" xref="S1.SSx1.p12.4.m4.1.1.2">ℵ</ci><cn id="S1.SSx1.p12.4.m4.1.1.3.cmml" type="integer" xref="S1.SSx1.p12.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p12.4.m4.1c">\aleph_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p12.4.m4.1d">roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-dense set of reals is strongly surjective. They also proved that any such order contains no copies of <math alttext="\omega_{1}" class="ltx_Math" display="inline" id="S1.SSx1.p12.5.m5.1"><semantics id="S1.SSx1.p12.5.m5.1a"><msub id="S1.SSx1.p12.5.m5.1.1" xref="S1.SSx1.p12.5.m5.1.1.cmml"><mi id="S1.SSx1.p12.5.m5.1.1.2" xref="S1.SSx1.p12.5.m5.1.1.2.cmml">ω</mi><mn id="S1.SSx1.p12.5.m5.1.1.3" xref="S1.SSx1.p12.5.m5.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S1.SSx1.p12.5.m5.1b"><apply id="S1.SSx1.p12.5.m5.1.1.cmml" xref="S1.SSx1.p12.5.m5.1.1"><csymbol cd="ambiguous" id="S1.SSx1.p12.5.m5.1.1.1.cmml" xref="S1.SSx1.p12.5.m5.1.1">subscript</csymbol><ci id="S1.SSx1.p12.5.m5.1.1.2.cmml" xref="S1.SSx1.p12.5.m5.1.1.2">𝜔</ci><cn id="S1.SSx1.p12.5.m5.1.1.3.cmml" type="integer" xref="S1.SSx1.p12.5.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p12.5.m5.1c">\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p12.5.m5.1d">italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> or <math alttext="\omega_{1}^{\star}" class="ltx_Math" display="inline" id="S1.SSx1.p12.6.m6.1"><semantics id="S1.SSx1.p12.6.m6.1a"><msubsup id="S1.SSx1.p12.6.m6.1.1" xref="S1.SSx1.p12.6.m6.1.1.cmml"><mi id="S1.SSx1.p12.6.m6.1.1.2.2" xref="S1.SSx1.p12.6.m6.1.1.2.2.cmml">ω</mi><mn id="S1.SSx1.p12.6.m6.1.1.2.3" xref="S1.SSx1.p12.6.m6.1.1.2.3.cmml">1</mn><mo id="S1.SSx1.p12.6.m6.1.1.3" xref="S1.SSx1.p12.6.m6.1.1.3.cmml">⋆</mo></msubsup><annotation-xml encoding="MathML-Content" id="S1.SSx1.p12.6.m6.1b"><apply id="S1.SSx1.p12.6.m6.1.1.cmml" xref="S1.SSx1.p12.6.m6.1.1"><csymbol cd="ambiguous" id="S1.SSx1.p12.6.m6.1.1.1.cmml" xref="S1.SSx1.p12.6.m6.1.1">superscript</csymbol><apply id="S1.SSx1.p12.6.m6.1.1.2.cmml" xref="S1.SSx1.p12.6.m6.1.1"><csymbol cd="ambiguous" id="S1.SSx1.p12.6.m6.1.1.2.1.cmml" xref="S1.SSx1.p12.6.m6.1.1">subscript</csymbol><ci id="S1.SSx1.p12.6.m6.1.1.2.2.cmml" xref="S1.SSx1.p12.6.m6.1.1.2.2">𝜔</ci><cn id="S1.SSx1.p12.6.m6.1.1.2.3.cmml" type="integer" xref="S1.SSx1.p12.6.m6.1.1.2.3">1</cn></apply><ci id="S1.SSx1.p12.6.m6.1.1.3.cmml" xref="S1.SSx1.p12.6.m6.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p12.6.m6.1c">\omega_{1}^{\star}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p12.6.m6.1d">italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math> (such an order is called <em class="ltx_emph ltx_font_italic" id="S1.SSx1.p12.6.1">short</em>), and thus one is naturally led to ask for an Aronszajn uncountable strongly surjective order.</p> </div> <div class="ltx_para" id="S1.SSx1.p13"> <p class="ltx_p" id="S1.SSx1.p13.2">Shortly after, Soukup <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib19" title="">19</a>]</cite> studied this question in depth. His results include that consistently with <math alttext="\mathsf{CH}" class="ltx_Math" display="inline" id="S1.SSx1.p13.1.m1.1"><semantics id="S1.SSx1.p13.1.m1.1a"><mi id="S1.SSx1.p13.1.m1.1.1" xref="S1.SSx1.p13.1.m1.1.1.cmml">𝖢𝖧</mi><annotation-xml encoding="MathML-Content" id="S1.SSx1.p13.1.m1.1b"><ci id="S1.SSx1.p13.1.m1.1.1.cmml" xref="S1.SSx1.p13.1.m1.1.1">𝖢𝖧</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p13.1.m1.1c">\mathsf{CH}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p13.1.m1.1d">sansserif_CH</annotation></semantics></math> there are no uncountable strongly surjective orders, and that consistently there are Aronszajn strongly surjective orders. In Particular that <math alttext="\diamondsuit^{+}" class="ltx_Math" display="inline" id="S1.SSx1.p13.2.m2.1"><semantics id="S1.SSx1.p13.2.m2.1a"><msup id="S1.SSx1.p13.2.m2.1.1" xref="S1.SSx1.p13.2.m2.1.1.cmml"><mi id="S1.SSx1.p13.2.m2.1.1.2" mathvariant="normal" xref="S1.SSx1.p13.2.m2.1.1.2.cmml">♢</mi><mo id="S1.SSx1.p13.2.m2.1.1.3" xref="S1.SSx1.p13.2.m2.1.1.3.cmml">+</mo></msup><annotation-xml encoding="MathML-Content" id="S1.SSx1.p13.2.m2.1b"><apply id="S1.SSx1.p13.2.m2.1.1.cmml" xref="S1.SSx1.p13.2.m2.1.1"><csymbol cd="ambiguous" id="S1.SSx1.p13.2.m2.1.1.1.cmml" xref="S1.SSx1.p13.2.m2.1.1">superscript</csymbol><ci id="S1.SSx1.p13.2.m2.1.1.2.cmml" xref="S1.SSx1.p13.2.m2.1.1.2">♢</ci><plus id="S1.SSx1.p13.2.m2.1.1.3.cmml" xref="S1.SSx1.p13.2.m2.1.1.3"></plus></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p13.2.m2.1c">\diamondsuit^{+}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p13.2.m2.1d">♢ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> implies the existence of a strongly surjective lexicographically ordered Suslin tree. He then asks the following natural questions (see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib19" title="">19</a>, Problems 40, 41 and 46]</cite>).</p> </div> <div class="ltx_theorem ltx_theorem_question" id="Thmquestion3"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmquestion3.1.1.1">Question 3</span></span><span class="ltx_text ltx_font_bold" id="Thmquestion3.2.2">.</span> </h6> <div class="ltx_para" id="Thmquestion3.p1"> <p class="ltx_p" id="Thmquestion3.p1.1">Does <math alttext="\mathsf{MA}_{\aleph_{1}}" class="ltx_Math" display="inline" id="Thmquestion3.p1.1.m1.1"><semantics id="Thmquestion3.p1.1.m1.1a"><msub id="Thmquestion3.p1.1.m1.1.1" xref="Thmquestion3.p1.1.m1.1.1.cmml"><mi id="Thmquestion3.p1.1.m1.1.1.2" xref="Thmquestion3.p1.1.m1.1.1.2.cmml">𝖬𝖠</mi><msub id="Thmquestion3.p1.1.m1.1.1.3" xref="Thmquestion3.p1.1.m1.1.1.3.cmml"><mi id="Thmquestion3.p1.1.m1.1.1.3.2" mathvariant="normal" xref="Thmquestion3.p1.1.m1.1.1.3.2.cmml">ℵ</mi><mn id="Thmquestion3.p1.1.m1.1.1.3.3" xref="Thmquestion3.p1.1.m1.1.1.3.3.cmml">1</mn></msub></msub><annotation-xml encoding="MathML-Content" id="Thmquestion3.p1.1.m1.1b"><apply id="Thmquestion3.p1.1.m1.1.1.cmml" xref="Thmquestion3.p1.1.m1.1.1"><csymbol cd="ambiguous" id="Thmquestion3.p1.1.m1.1.1.1.cmml" xref="Thmquestion3.p1.1.m1.1.1">subscript</csymbol><ci id="Thmquestion3.p1.1.m1.1.1.2.cmml" xref="Thmquestion3.p1.1.m1.1.1.2">𝖬𝖠</ci><apply id="Thmquestion3.p1.1.m1.1.1.3.cmml" xref="Thmquestion3.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="Thmquestion3.p1.1.m1.1.1.3.1.cmml" xref="Thmquestion3.p1.1.m1.1.1.3">subscript</csymbol><ci id="Thmquestion3.p1.1.m1.1.1.3.2.cmml" xref="Thmquestion3.p1.1.m1.1.1.3.2">ℵ</ci><cn id="Thmquestion3.p1.1.m1.1.1.3.3.cmml" type="integer" xref="Thmquestion3.p1.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmquestion3.p1.1.m1.1c">\mathsf{MA}_{\aleph_{1}}</annotation><annotation encoding="application/x-llamapun" id="Thmquestion3.p1.1.m1.1d">sansserif_MA start_POSTSUBSCRIPT roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> implies the existence of uncountable strongly surjective linear orders?</p> </div> </div> <div class="ltx_theorem ltx_theorem_question" id="Thmquestion4"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmquestion4.1.1.1">Question 4</span></span><span class="ltx_text ltx_font_bold" id="Thmquestion4.2.2">.</span> </h6> <div class="ltx_para" id="Thmquestion4.p1"> <p class="ltx_p" id="Thmquestion4.p1.1">Can a strongly surjective linear order be Countryman?</p> </div> </div> <div class="ltx_theorem ltx_theorem_question" id="Thmquestion5"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmquestion5.1.1.1">Question 5</span></span><span class="ltx_text ltx_font_bold" id="Thmquestion5.2.2">.</span> </h6> <div class="ltx_para" id="Thmquestion5.p1"> <p class="ltx_p" id="Thmquestion5.p1.1">Can there be real and Aronszajn uncountable strongly surjective orders simultaneously?</p> </div> </div> <div class="ltx_para" id="S1.SSx1.p14"> <p class="ltx_p" id="S1.SSx1.p14.1">The analogy of the countable linear orders with the class of Aronszajn lines under <math alttext="\mathsf{PFA}" class="ltx_Math" display="inline" id="S1.SSx1.p14.1.m1.1"><semantics id="S1.SSx1.p14.1.m1.1a"><mi id="S1.SSx1.p14.1.m1.1.1" xref="S1.SSx1.p14.1.m1.1.1.cmml">𝖯𝖥𝖠</mi><annotation-xml encoding="MathML-Content" id="S1.SSx1.p14.1.m1.1b"><ci id="S1.SSx1.p14.1.m1.1.1.cmml" xref="S1.SSx1.p14.1.m1.1.1">𝖯𝖥𝖠</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx1.p14.1.m1.1c">\mathsf{PFA}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx1.p14.1.m1.1d">sansserif_PFA</annotation></semantics></math> previously mentioned also suggests the following question.</p> </div> <div class="ltx_theorem ltx_theorem_question" id="Thmquestion6"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmquestion6.1.1.1">Question 6</span></span><span class="ltx_text ltx_font_bold" id="Thmquestion6.2.2">.</span> </h6> <div class="ltx_para" id="Thmquestion6.p1"> <p class="ltx_p" id="Thmquestion6.p1.2">Assume <math alttext="\mathsf{PFA}" class="ltx_Math" display="inline" id="Thmquestion6.p1.1.m1.1"><semantics id="Thmquestion6.p1.1.m1.1a"><mi id="Thmquestion6.p1.1.m1.1.1" xref="Thmquestion6.p1.1.m1.1.1.cmml">𝖯𝖥𝖠</mi><annotation-xml encoding="MathML-Content" id="Thmquestion6.p1.1.m1.1b"><ci id="Thmquestion6.p1.1.m1.1.1.cmml" xref="Thmquestion6.p1.1.m1.1.1">𝖯𝖥𝖠</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmquestion6.p1.1.m1.1c">\mathsf{PFA}</annotation><annotation encoding="application/x-llamapun" id="Thmquestion6.p1.1.m1.1d">sansserif_PFA</annotation></semantics></math>. Is <math alttext="\eta_{C}" class="ltx_Math" display="inline" id="Thmquestion6.p1.2.m2.1"><semantics id="Thmquestion6.p1.2.m2.1a"><msub id="Thmquestion6.p1.2.m2.1.1" xref="Thmquestion6.p1.2.m2.1.1.cmml"><mi id="Thmquestion6.p1.2.m2.1.1.2" xref="Thmquestion6.p1.2.m2.1.1.2.cmml">η</mi><mi id="Thmquestion6.p1.2.m2.1.1.3" xref="Thmquestion6.p1.2.m2.1.1.3.cmml">C</mi></msub><annotation-xml encoding="MathML-Content" id="Thmquestion6.p1.2.m2.1b"><apply id="Thmquestion6.p1.2.m2.1.1.cmml" xref="Thmquestion6.p1.2.m2.1.1"><csymbol cd="ambiguous" id="Thmquestion6.p1.2.m2.1.1.1.cmml" xref="Thmquestion6.p1.2.m2.1.1">subscript</csymbol><ci id="Thmquestion6.p1.2.m2.1.1.2.cmml" xref="Thmquestion6.p1.2.m2.1.1.2">𝜂</ci><ci id="Thmquestion6.p1.2.m2.1.1.3.cmml" xref="Thmquestion6.p1.2.m2.1.1.3">𝐶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmquestion6.p1.2.m2.1c">\eta_{C}</annotation><annotation encoding="application/x-llamapun" id="Thmquestion6.p1.2.m2.1d">italic_η start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT</annotation></semantics></math> (or any other universal Aronszajn line) strongly surjective?</p> </div> </div> </section> <section class="ltx_subsection" id="S1.SSx2"> <h3 class="ltx_title ltx_title_subsection">Main results</h3> <div class="ltx_para" id="S1.SSx2.p1"> <p class="ltx_p" id="S1.SSx2.p1.18">Let <math alttext="A" class="ltx_Math" display="inline" id="S1.SSx2.p1.1.m1.1"><semantics id="S1.SSx2.p1.1.m1.1a"><mi id="S1.SSx2.p1.1.m1.1.1" xref="S1.SSx2.p1.1.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S1.SSx2.p1.1.m1.1b"><ci id="S1.SSx2.p1.1.m1.1.1.cmml" xref="S1.SSx2.p1.1.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx2.p1.1.m1.1c">A</annotation><annotation encoding="application/x-llamapun" id="S1.SSx2.p1.1.m1.1d">italic_A</annotation></semantics></math> be an Aronszajn line, a <em class="ltx_emph ltx_font_italic" id="S1.SSx2.p1.2.1">decomposition for <math alttext="A" class="ltx_Math" display="inline" id="S1.SSx2.p1.2.1.m1.1"><semantics id="S1.SSx2.p1.2.1.m1.1a"><mi id="S1.SSx2.p1.2.1.m1.1.1" xref="S1.SSx2.p1.2.1.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S1.SSx2.p1.2.1.m1.1b"><ci id="S1.SSx2.p1.2.1.m1.1.1.cmml" xref="S1.SSx2.p1.2.1.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx2.p1.2.1.m1.1c">A</annotation><annotation encoding="application/x-llamapun" id="S1.SSx2.p1.2.1.m1.1d">italic_A</annotation></semantics></math></em> is a <math alttext="\subseteq" class="ltx_Math" display="inline" id="S1.SSx2.p1.3.m2.1"><semantics id="S1.SSx2.p1.3.m2.1a"><mo id="S1.SSx2.p1.3.m2.1.1" xref="S1.SSx2.p1.3.m2.1.1.cmml">⊆</mo><annotation-xml encoding="MathML-Content" id="S1.SSx2.p1.3.m2.1b"><subset id="S1.SSx2.p1.3.m2.1.1.cmml" xref="S1.SSx2.p1.3.m2.1.1"></subset></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx2.p1.3.m2.1c">\subseteq</annotation><annotation encoding="application/x-llamapun" id="S1.SSx2.p1.3.m2.1d">⊆</annotation></semantics></math>-increasing sequence <math alttext="D:=\langle D_{\xi}:\xi<\omega_{1}\rangle" class="ltx_math_unparsed" display="inline" id="S1.SSx2.p1.4.m3.1"><semantics id="S1.SSx2.p1.4.m3.1a"><mrow id="S1.SSx2.p1.4.m3.1b"><mi id="S1.SSx2.p1.4.m3.1.1">D</mi><mo id="S1.SSx2.p1.4.m3.1.2" lspace="0.278em" rspace="0.278em">:=</mo><mrow id="S1.SSx2.p1.4.m3.1.3"><mo id="S1.SSx2.p1.4.m3.1.3.1" stretchy="false">⟨</mo><msub id="S1.SSx2.p1.4.m3.1.3.2"><mi id="S1.SSx2.p1.4.m3.1.3.2.2">D</mi><mi id="S1.SSx2.p1.4.m3.1.3.2.3">ξ</mi></msub><mo id="S1.SSx2.p1.4.m3.1.3.3" lspace="0.278em" rspace="0.278em">:</mo><mi id="S1.SSx2.p1.4.m3.1.3.4">ξ</mi><mo id="S1.SSx2.p1.4.m3.1.3.5"><</mo><msub id="S1.SSx2.p1.4.m3.1.3.6"><mi id="S1.SSx2.p1.4.m3.1.3.6.2">ω</mi><mn id="S1.SSx2.p1.4.m3.1.3.6.3">1</mn></msub><mo id="S1.SSx2.p1.4.m3.1.3.7" stretchy="false">⟩</mo></mrow></mrow><annotation encoding="application/x-tex" id="S1.SSx2.p1.4.m3.1c">D:=\langle D_{\xi}:\xi<\omega_{1}\rangle</annotation><annotation encoding="application/x-llamapun" id="S1.SSx2.p1.4.m3.1d">italic_D := ⟨ italic_D start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT : italic_ξ < italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⟩</annotation></semantics></math> of countable subsets of <math alttext="A" class="ltx_Math" display="inline" id="S1.SSx2.p1.5.m4.1"><semantics id="S1.SSx2.p1.5.m4.1a"><mi id="S1.SSx2.p1.5.m4.1.1" xref="S1.SSx2.p1.5.m4.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S1.SSx2.p1.5.m4.1b"><ci id="S1.SSx2.p1.5.m4.1.1.cmml" xref="S1.SSx2.p1.5.m4.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx2.p1.5.m4.1c">A</annotation><annotation encoding="application/x-llamapun" id="S1.SSx2.p1.5.m4.1d">italic_A</annotation></semantics></math>, that covers <math alttext="A" class="ltx_Math" display="inline" id="S1.SSx2.p1.6.m5.1"><semantics id="S1.SSx2.p1.6.m5.1a"><mi id="S1.SSx2.p1.6.m5.1.1" xref="S1.SSx2.p1.6.m5.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S1.SSx2.p1.6.m5.1b"><ci id="S1.SSx2.p1.6.m5.1.1.cmml" xref="S1.SSx2.p1.6.m5.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx2.p1.6.m5.1c">A</annotation><annotation encoding="application/x-llamapun" id="S1.SSx2.p1.6.m5.1d">italic_A</annotation></semantics></math> and that is continuous, i.e., that at limits is the union. A well known fact is that any Aronszajn line has size <math alttext="\aleph_{1}" class="ltx_Math" display="inline" id="S1.SSx2.p1.7.m6.1"><semantics id="S1.SSx2.p1.7.m6.1a"><msub id="S1.SSx2.p1.7.m6.1.1" xref="S1.SSx2.p1.7.m6.1.1.cmml"><mi id="S1.SSx2.p1.7.m6.1.1.2" mathvariant="normal" xref="S1.SSx2.p1.7.m6.1.1.2.cmml">ℵ</mi><mn id="S1.SSx2.p1.7.m6.1.1.3" xref="S1.SSx2.p1.7.m6.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S1.SSx2.p1.7.m6.1b"><apply id="S1.SSx2.p1.7.m6.1.1.cmml" xref="S1.SSx2.p1.7.m6.1.1"><csymbol cd="ambiguous" id="S1.SSx2.p1.7.m6.1.1.1.cmml" xref="S1.SSx2.p1.7.m6.1.1">subscript</csymbol><ci id="S1.SSx2.p1.7.m6.1.1.2.cmml" xref="S1.SSx2.p1.7.m6.1.1.2">ℵ</ci><cn id="S1.SSx2.p1.7.m6.1.1.3.cmml" type="integer" xref="S1.SSx2.p1.7.m6.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx2.p1.7.m6.1c">\aleph_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx2.p1.7.m6.1d">roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, and thus that these decompositions always exist. The <em class="ltx_emph ltx_font_italic" id="S1.SSx2.p1.18.2">complementary intervals</em> of <math alttext="A\setminus D_{\xi}" class="ltx_Math" display="inline" id="S1.SSx2.p1.8.m7.1"><semantics id="S1.SSx2.p1.8.m7.1a"><mrow id="S1.SSx2.p1.8.m7.1.1" xref="S1.SSx2.p1.8.m7.1.1.cmml"><mi id="S1.SSx2.p1.8.m7.1.1.2" xref="S1.SSx2.p1.8.m7.1.1.2.cmml">A</mi><mo id="S1.SSx2.p1.8.m7.1.1.1" xref="S1.SSx2.p1.8.m7.1.1.1.cmml">∖</mo><msub id="S1.SSx2.p1.8.m7.1.1.3" xref="S1.SSx2.p1.8.m7.1.1.3.cmml"><mi id="S1.SSx2.p1.8.m7.1.1.3.2" xref="S1.SSx2.p1.8.m7.1.1.3.2.cmml">D</mi><mi id="S1.SSx2.p1.8.m7.1.1.3.3" xref="S1.SSx2.p1.8.m7.1.1.3.3.cmml">ξ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S1.SSx2.p1.8.m7.1b"><apply id="S1.SSx2.p1.8.m7.1.1.cmml" xref="S1.SSx2.p1.8.m7.1.1"><setdiff id="S1.SSx2.p1.8.m7.1.1.1.cmml" xref="S1.SSx2.p1.8.m7.1.1.1"></setdiff><ci id="S1.SSx2.p1.8.m7.1.1.2.cmml" xref="S1.SSx2.p1.8.m7.1.1.2">𝐴</ci><apply id="S1.SSx2.p1.8.m7.1.1.3.cmml" xref="S1.SSx2.p1.8.m7.1.1.3"><csymbol cd="ambiguous" id="S1.SSx2.p1.8.m7.1.1.3.1.cmml" xref="S1.SSx2.p1.8.m7.1.1.3">subscript</csymbol><ci id="S1.SSx2.p1.8.m7.1.1.3.2.cmml" xref="S1.SSx2.p1.8.m7.1.1.3.2">𝐷</ci><ci id="S1.SSx2.p1.8.m7.1.1.3.3.cmml" xref="S1.SSx2.p1.8.m7.1.1.3.3">𝜉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx2.p1.8.m7.1c">A\setminus D_{\xi}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx2.p1.8.m7.1d">italic_A ∖ italic_D start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT</annotation></semantics></math> are the intervals of <math alttext="A" class="ltx_Math" display="inline" id="S1.SSx2.p1.9.m8.1"><semantics id="S1.SSx2.p1.9.m8.1a"><mi id="S1.SSx2.p1.9.m8.1.1" xref="S1.SSx2.p1.9.m8.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S1.SSx2.p1.9.m8.1b"><ci id="S1.SSx2.p1.9.m8.1.1.cmml" xref="S1.SSx2.p1.9.m8.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx2.p1.9.m8.1c">A</annotation><annotation encoding="application/x-llamapun" id="S1.SSx2.p1.9.m8.1d">italic_A</annotation></semantics></math> that are maximal with respect to not having points in <math alttext="D_{\xi}" class="ltx_Math" display="inline" id="S1.SSx2.p1.10.m9.1"><semantics id="S1.SSx2.p1.10.m9.1a"><msub id="S1.SSx2.p1.10.m9.1.1" xref="S1.SSx2.p1.10.m9.1.1.cmml"><mi id="S1.SSx2.p1.10.m9.1.1.2" xref="S1.SSx2.p1.10.m9.1.1.2.cmml">D</mi><mi id="S1.SSx2.p1.10.m9.1.1.3" xref="S1.SSx2.p1.10.m9.1.1.3.cmml">ξ</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SSx2.p1.10.m9.1b"><apply id="S1.SSx2.p1.10.m9.1.1.cmml" xref="S1.SSx2.p1.10.m9.1.1"><csymbol cd="ambiguous" id="S1.SSx2.p1.10.m9.1.1.1.cmml" xref="S1.SSx2.p1.10.m9.1.1">subscript</csymbol><ci id="S1.SSx2.p1.10.m9.1.1.2.cmml" xref="S1.SSx2.p1.10.m9.1.1.2">𝐷</ci><ci id="S1.SSx2.p1.10.m9.1.1.3.cmml" xref="S1.SSx2.p1.10.m9.1.1.3">𝜉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx2.p1.10.m9.1c">D_{\xi}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx2.p1.10.m9.1d">italic_D start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT</annotation></semantics></math>. Since <math alttext="D_{\xi}" class="ltx_Math" display="inline" id="S1.SSx2.p1.11.m10.1"><semantics id="S1.SSx2.p1.11.m10.1a"><msub id="S1.SSx2.p1.11.m10.1.1" xref="S1.SSx2.p1.11.m10.1.1.cmml"><mi id="S1.SSx2.p1.11.m10.1.1.2" xref="S1.SSx2.p1.11.m10.1.1.2.cmml">D</mi><mi id="S1.SSx2.p1.11.m10.1.1.3" xref="S1.SSx2.p1.11.m10.1.1.3.cmml">ξ</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SSx2.p1.11.m10.1b"><apply id="S1.SSx2.p1.11.m10.1.1.cmml" xref="S1.SSx2.p1.11.m10.1.1"><csymbol cd="ambiguous" id="S1.SSx2.p1.11.m10.1.1.1.cmml" xref="S1.SSx2.p1.11.m10.1.1">subscript</csymbol><ci id="S1.SSx2.p1.11.m10.1.1.2.cmml" xref="S1.SSx2.p1.11.m10.1.1.2">𝐷</ci><ci id="S1.SSx2.p1.11.m10.1.1.3.cmml" xref="S1.SSx2.p1.11.m10.1.1.3">𝜉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx2.p1.11.m10.1c">D_{\xi}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx2.p1.11.m10.1d">italic_D start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT</annotation></semantics></math> is countable, and <math alttext="A" class="ltx_Math" display="inline" id="S1.SSx2.p1.12.m11.1"><semantics id="S1.SSx2.p1.12.m11.1a"><mi id="S1.SSx2.p1.12.m11.1.1" xref="S1.SSx2.p1.12.m11.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S1.SSx2.p1.12.m11.1b"><ci id="S1.SSx2.p1.12.m11.1.1.cmml" xref="S1.SSx2.p1.12.m11.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx2.p1.12.m11.1c">A</annotation><annotation encoding="application/x-llamapun" id="S1.SSx2.p1.12.m11.1d">italic_A</annotation></semantics></math> is Aronszajn, there are Countably many complementary intervals of <math alttext="A\setminus D_{\xi}" class="ltx_Math" display="inline" id="S1.SSx2.p1.13.m12.1"><semantics id="S1.SSx2.p1.13.m12.1a"><mrow id="S1.SSx2.p1.13.m12.1.1" xref="S1.SSx2.p1.13.m12.1.1.cmml"><mi id="S1.SSx2.p1.13.m12.1.1.2" xref="S1.SSx2.p1.13.m12.1.1.2.cmml">A</mi><mo id="S1.SSx2.p1.13.m12.1.1.1" xref="S1.SSx2.p1.13.m12.1.1.1.cmml">∖</mo><msub id="S1.SSx2.p1.13.m12.1.1.3" xref="S1.SSx2.p1.13.m12.1.1.3.cmml"><mi id="S1.SSx2.p1.13.m12.1.1.3.2" xref="S1.SSx2.p1.13.m12.1.1.3.2.cmml">D</mi><mi id="S1.SSx2.p1.13.m12.1.1.3.3" xref="S1.SSx2.p1.13.m12.1.1.3.3.cmml">ξ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S1.SSx2.p1.13.m12.1b"><apply id="S1.SSx2.p1.13.m12.1.1.cmml" xref="S1.SSx2.p1.13.m12.1.1"><setdiff id="S1.SSx2.p1.13.m12.1.1.1.cmml" xref="S1.SSx2.p1.13.m12.1.1.1"></setdiff><ci id="S1.SSx2.p1.13.m12.1.1.2.cmml" xref="S1.SSx2.p1.13.m12.1.1.2">𝐴</ci><apply id="S1.SSx2.p1.13.m12.1.1.3.cmml" xref="S1.SSx2.p1.13.m12.1.1.3"><csymbol cd="ambiguous" id="S1.SSx2.p1.13.m12.1.1.3.1.cmml" xref="S1.SSx2.p1.13.m12.1.1.3">subscript</csymbol><ci id="S1.SSx2.p1.13.m12.1.1.3.2.cmml" xref="S1.SSx2.p1.13.m12.1.1.3.2">𝐷</ci><ci id="S1.SSx2.p1.13.m12.1.1.3.3.cmml" xref="S1.SSx2.p1.13.m12.1.1.3.3">𝜉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx2.p1.13.m12.1c">A\setminus D_{\xi}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx2.p1.13.m12.1d">italic_A ∖ italic_D start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT</annotation></semantics></math>. Now define <math alttext="\mathscr{L}(A,D)" class="ltx_Math" display="inline" id="S1.SSx2.p1.14.m13.2"><semantics id="S1.SSx2.p1.14.m13.2a"><mrow id="S1.SSx2.p1.14.m13.2.3" xref="S1.SSx2.p1.14.m13.2.3.cmml"><mi class="ltx_font_mathscript" id="S1.SSx2.p1.14.m13.2.3.2" xref="S1.SSx2.p1.14.m13.2.3.2.cmml">ℒ</mi><mo id="S1.SSx2.p1.14.m13.2.3.1" xref="S1.SSx2.p1.14.m13.2.3.1.cmml">⁢</mo><mrow id="S1.SSx2.p1.14.m13.2.3.3.2" xref="S1.SSx2.p1.14.m13.2.3.3.1.cmml"><mo id="S1.SSx2.p1.14.m13.2.3.3.2.1" stretchy="false" xref="S1.SSx2.p1.14.m13.2.3.3.1.cmml">(</mo><mi id="S1.SSx2.p1.14.m13.1.1" xref="S1.SSx2.p1.14.m13.1.1.cmml">A</mi><mo id="S1.SSx2.p1.14.m13.2.3.3.2.2" xref="S1.SSx2.p1.14.m13.2.3.3.1.cmml">,</mo><mi id="S1.SSx2.p1.14.m13.2.2" xref="S1.SSx2.p1.14.m13.2.2.cmml">D</mi><mo id="S1.SSx2.p1.14.m13.2.3.3.2.3" stretchy="false" xref="S1.SSx2.p1.14.m13.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SSx2.p1.14.m13.2b"><apply id="S1.SSx2.p1.14.m13.2.3.cmml" xref="S1.SSx2.p1.14.m13.2.3"><times id="S1.SSx2.p1.14.m13.2.3.1.cmml" xref="S1.SSx2.p1.14.m13.2.3.1"></times><ci id="S1.SSx2.p1.14.m13.2.3.2.cmml" xref="S1.SSx2.p1.14.m13.2.3.2">ℒ</ci><interval closure="open" id="S1.SSx2.p1.14.m13.2.3.3.1.cmml" xref="S1.SSx2.p1.14.m13.2.3.3.2"><ci id="S1.SSx2.p1.14.m13.1.1.cmml" xref="S1.SSx2.p1.14.m13.1.1">𝐴</ci><ci id="S1.SSx2.p1.14.m13.2.2.cmml" xref="S1.SSx2.p1.14.m13.2.2">𝐷</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx2.p1.14.m13.2c">\mathscr{L}(A,D)</annotation><annotation encoding="application/x-llamapun" id="S1.SSx2.p1.14.m13.2d">script_L ( italic_A , italic_D )</annotation></semantics></math> (resp <math alttext="\hat{\mathscr{L}}(A,D)" class="ltx_Math" display="inline" id="S1.SSx2.p1.15.m14.2"><semantics id="S1.SSx2.p1.15.m14.2a"><mrow id="S1.SSx2.p1.15.m14.2.3" xref="S1.SSx2.p1.15.m14.2.3.cmml"><mover accent="true" id="S1.SSx2.p1.15.m14.2.3.2" xref="S1.SSx2.p1.15.m14.2.3.2.cmml"><mi class="ltx_font_mathscript" id="S1.SSx2.p1.15.m14.2.3.2.2" xref="S1.SSx2.p1.15.m14.2.3.2.2.cmml">ℒ</mi><mo id="S1.SSx2.p1.15.m14.2.3.2.1" xref="S1.SSx2.p1.15.m14.2.3.2.1.cmml">^</mo></mover><mo id="S1.SSx2.p1.15.m14.2.3.1" xref="S1.SSx2.p1.15.m14.2.3.1.cmml">⁢</mo><mrow id="S1.SSx2.p1.15.m14.2.3.3.2" xref="S1.SSx2.p1.15.m14.2.3.3.1.cmml"><mo id="S1.SSx2.p1.15.m14.2.3.3.2.1" stretchy="false" xref="S1.SSx2.p1.15.m14.2.3.3.1.cmml">(</mo><mi id="S1.SSx2.p1.15.m14.1.1" xref="S1.SSx2.p1.15.m14.1.1.cmml">A</mi><mo id="S1.SSx2.p1.15.m14.2.3.3.2.2" xref="S1.SSx2.p1.15.m14.2.3.3.1.cmml">,</mo><mi id="S1.SSx2.p1.15.m14.2.2" xref="S1.SSx2.p1.15.m14.2.2.cmml">D</mi><mo id="S1.SSx2.p1.15.m14.2.3.3.2.3" stretchy="false" xref="S1.SSx2.p1.15.m14.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SSx2.p1.15.m14.2b"><apply id="S1.SSx2.p1.15.m14.2.3.cmml" xref="S1.SSx2.p1.15.m14.2.3"><times id="S1.SSx2.p1.15.m14.2.3.1.cmml" xref="S1.SSx2.p1.15.m14.2.3.1"></times><apply id="S1.SSx2.p1.15.m14.2.3.2.cmml" xref="S1.SSx2.p1.15.m14.2.3.2"><ci id="S1.SSx2.p1.15.m14.2.3.2.1.cmml" xref="S1.SSx2.p1.15.m14.2.3.2.1">^</ci><ci id="S1.SSx2.p1.15.m14.2.3.2.2.cmml" xref="S1.SSx2.p1.15.m14.2.3.2.2">ℒ</ci></apply><interval closure="open" id="S1.SSx2.p1.15.m14.2.3.3.1.cmml" xref="S1.SSx2.p1.15.m14.2.3.3.2"><ci id="S1.SSx2.p1.15.m14.1.1.cmml" xref="S1.SSx2.p1.15.m14.1.1">𝐴</ci><ci id="S1.SSx2.p1.15.m14.2.2.cmml" xref="S1.SSx2.p1.15.m14.2.2">𝐷</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx2.p1.15.m14.2c">\hat{\mathscr{L}}(A,D)</annotation><annotation encoding="application/x-llamapun" id="S1.SSx2.p1.15.m14.2d">over^ start_ARG script_L end_ARG ( italic_A , italic_D )</annotation></semantics></math>) to be the set of <math alttext="\xi<\omega_{1}" class="ltx_Math" display="inline" id="S1.SSx2.p1.16.m15.1"><semantics id="S1.SSx2.p1.16.m15.1a"><mrow id="S1.SSx2.p1.16.m15.1.1" xref="S1.SSx2.p1.16.m15.1.1.cmml"><mi id="S1.SSx2.p1.16.m15.1.1.2" xref="S1.SSx2.p1.16.m15.1.1.2.cmml">ξ</mi><mo id="S1.SSx2.p1.16.m15.1.1.1" xref="S1.SSx2.p1.16.m15.1.1.1.cmml"><</mo><msub id="S1.SSx2.p1.16.m15.1.1.3" xref="S1.SSx2.p1.16.m15.1.1.3.cmml"><mi id="S1.SSx2.p1.16.m15.1.1.3.2" xref="S1.SSx2.p1.16.m15.1.1.3.2.cmml">ω</mi><mn id="S1.SSx2.p1.16.m15.1.1.3.3" xref="S1.SSx2.p1.16.m15.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S1.SSx2.p1.16.m15.1b"><apply id="S1.SSx2.p1.16.m15.1.1.cmml" xref="S1.SSx2.p1.16.m15.1.1"><lt id="S1.SSx2.p1.16.m15.1.1.1.cmml" xref="S1.SSx2.p1.16.m15.1.1.1"></lt><ci id="S1.SSx2.p1.16.m15.1.1.2.cmml" xref="S1.SSx2.p1.16.m15.1.1.2">𝜉</ci><apply id="S1.SSx2.p1.16.m15.1.1.3.cmml" xref="S1.SSx2.p1.16.m15.1.1.3"><csymbol cd="ambiguous" id="S1.SSx2.p1.16.m15.1.1.3.1.cmml" xref="S1.SSx2.p1.16.m15.1.1.3">subscript</csymbol><ci id="S1.SSx2.p1.16.m15.1.1.3.2.cmml" xref="S1.SSx2.p1.16.m15.1.1.3.2">𝜔</ci><cn id="S1.SSx2.p1.16.m15.1.1.3.3.cmml" type="integer" xref="S1.SSx2.p1.16.m15.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx2.p1.16.m15.1c">\xi<\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx2.p1.16.m15.1d">italic_ξ < italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> such that some (resp. every) complementary interval has a left endpoint. <math alttext="\mathscr{R}" class="ltx_Math" display="inline" id="S1.SSx2.p1.17.m16.1"><semantics id="S1.SSx2.p1.17.m16.1a"><mi class="ltx_font_mathscript" id="S1.SSx2.p1.17.m16.1.1" xref="S1.SSx2.p1.17.m16.1.1.cmml">ℛ</mi><annotation-xml encoding="MathML-Content" id="S1.SSx2.p1.17.m16.1b"><ci id="S1.SSx2.p1.17.m16.1.1.cmml" xref="S1.SSx2.p1.17.m16.1.1">ℛ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx2.p1.17.m16.1c">\mathscr{R}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx2.p1.17.m16.1d">script_R</annotation></semantics></math> and <math alttext="\hat{\mathscr{R}}" class="ltx_Math" display="inline" id="S1.SSx2.p1.18.m17.1"><semantics id="S1.SSx2.p1.18.m17.1a"><mover accent="true" id="S1.SSx2.p1.18.m17.1.1" xref="S1.SSx2.p1.18.m17.1.1.cmml"><mi class="ltx_font_mathscript" id="S1.SSx2.p1.18.m17.1.1.2" xref="S1.SSx2.p1.18.m17.1.1.2.cmml">ℛ</mi><mo id="S1.SSx2.p1.18.m17.1.1.1" xref="S1.SSx2.p1.18.m17.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="S1.SSx2.p1.18.m17.1b"><apply id="S1.SSx2.p1.18.m17.1.1.cmml" xref="S1.SSx2.p1.18.m17.1.1"><ci id="S1.SSx2.p1.18.m17.1.1.1.cmml" xref="S1.SSx2.p1.18.m17.1.1.1">^</ci><ci id="S1.SSx2.p1.18.m17.1.1.2.cmml" xref="S1.SSx2.p1.18.m17.1.1.2">ℛ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx2.p1.18.m17.1c">\hat{\mathscr{R}}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx2.p1.18.m17.1d">over^ start_ARG script_R end_ARG</annotation></semantics></math> are defined analogously using right endpoints.</p> </div> <div class="ltx_para" id="S1.SSx2.p2"> <p class="ltx_p" id="S1.SSx2.p2.13">With this notation, we say that <math alttext="A" class="ltx_Math" display="inline" id="S1.SSx2.p2.1.m1.1"><semantics id="S1.SSx2.p2.1.m1.1a"><mi id="S1.SSx2.p2.1.m1.1.1" xref="S1.SSx2.p2.1.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S1.SSx2.p2.1.m1.1b"><ci id="S1.SSx2.p2.1.m1.1.1.cmml" xref="S1.SSx2.p2.1.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx2.p2.1.m1.1c">A</annotation><annotation encoding="application/x-llamapun" id="S1.SSx2.p2.1.m1.1d">italic_A</annotation></semantics></math> is <em class="ltx_emph ltx_font_italic" id="S1.SSx2.p2.13.1">non stationary</em> if there is a decomposition <math alttext="D" class="ltx_Math" display="inline" id="S1.SSx2.p2.2.m2.1"><semantics id="S1.SSx2.p2.2.m2.1a"><mi id="S1.SSx2.p2.2.m2.1.1" xref="S1.SSx2.p2.2.m2.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S1.SSx2.p2.2.m2.1b"><ci id="S1.SSx2.p2.2.m2.1.1.cmml" xref="S1.SSx2.p2.2.m2.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx2.p2.2.m2.1c">D</annotation><annotation encoding="application/x-llamapun" id="S1.SSx2.p2.2.m2.1d">italic_D</annotation></semantics></math> for <math alttext="A" class="ltx_Math" display="inline" id="S1.SSx2.p2.3.m3.1"><semantics id="S1.SSx2.p2.3.m3.1a"><mi id="S1.SSx2.p2.3.m3.1.1" xref="S1.SSx2.p2.3.m3.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S1.SSx2.p2.3.m3.1b"><ci id="S1.SSx2.p2.3.m3.1.1.cmml" xref="S1.SSx2.p2.3.m3.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx2.p2.3.m3.1c">A</annotation><annotation encoding="application/x-llamapun" id="S1.SSx2.p2.3.m3.1d">italic_A</annotation></semantics></math> such that <math alttext="\mathscr{L}(A)" class="ltx_Math" display="inline" id="S1.SSx2.p2.4.m4.1"><semantics id="S1.SSx2.p2.4.m4.1a"><mrow id="S1.SSx2.p2.4.m4.1.2" xref="S1.SSx2.p2.4.m4.1.2.cmml"><mi class="ltx_font_mathscript" id="S1.SSx2.p2.4.m4.1.2.2" xref="S1.SSx2.p2.4.m4.1.2.2.cmml">ℒ</mi><mo id="S1.SSx2.p2.4.m4.1.2.1" xref="S1.SSx2.p2.4.m4.1.2.1.cmml">⁢</mo><mrow id="S1.SSx2.p2.4.m4.1.2.3.2" xref="S1.SSx2.p2.4.m4.1.2.cmml"><mo id="S1.SSx2.p2.4.m4.1.2.3.2.1" stretchy="false" xref="S1.SSx2.p2.4.m4.1.2.cmml">(</mo><mi id="S1.SSx2.p2.4.m4.1.1" xref="S1.SSx2.p2.4.m4.1.1.cmml">A</mi><mo id="S1.SSx2.p2.4.m4.1.2.3.2.2" stretchy="false" xref="S1.SSx2.p2.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SSx2.p2.4.m4.1b"><apply id="S1.SSx2.p2.4.m4.1.2.cmml" xref="S1.SSx2.p2.4.m4.1.2"><times id="S1.SSx2.p2.4.m4.1.2.1.cmml" xref="S1.SSx2.p2.4.m4.1.2.1"></times><ci id="S1.SSx2.p2.4.m4.1.2.2.cmml" xref="S1.SSx2.p2.4.m4.1.2.2">ℒ</ci><ci id="S1.SSx2.p2.4.m4.1.1.cmml" xref="S1.SSx2.p2.4.m4.1.1">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx2.p2.4.m4.1c">\mathscr{L}(A)</annotation><annotation encoding="application/x-llamapun" id="S1.SSx2.p2.4.m4.1d">script_L ( italic_A )</annotation></semantics></math> and <math alttext="\mathscr{R}(A)" class="ltx_Math" display="inline" id="S1.SSx2.p2.5.m5.1"><semantics id="S1.SSx2.p2.5.m5.1a"><mrow id="S1.SSx2.p2.5.m5.1.2" xref="S1.SSx2.p2.5.m5.1.2.cmml"><mi class="ltx_font_mathscript" id="S1.SSx2.p2.5.m5.1.2.2" xref="S1.SSx2.p2.5.m5.1.2.2.cmml">ℛ</mi><mo id="S1.SSx2.p2.5.m5.1.2.1" xref="S1.SSx2.p2.5.m5.1.2.1.cmml">⁢</mo><mrow id="S1.SSx2.p2.5.m5.1.2.3.2" xref="S1.SSx2.p2.5.m5.1.2.cmml"><mo id="S1.SSx2.p2.5.m5.1.2.3.2.1" stretchy="false" xref="S1.SSx2.p2.5.m5.1.2.cmml">(</mo><mi id="S1.SSx2.p2.5.m5.1.1" xref="S1.SSx2.p2.5.m5.1.1.cmml">A</mi><mo id="S1.SSx2.p2.5.m5.1.2.3.2.2" stretchy="false" xref="S1.SSx2.p2.5.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SSx2.p2.5.m5.1b"><apply id="S1.SSx2.p2.5.m5.1.2.cmml" xref="S1.SSx2.p2.5.m5.1.2"><times id="S1.SSx2.p2.5.m5.1.2.1.cmml" xref="S1.SSx2.p2.5.m5.1.2.1"></times><ci id="S1.SSx2.p2.5.m5.1.2.2.cmml" xref="S1.SSx2.p2.5.m5.1.2.2">ℛ</ci><ci id="S1.SSx2.p2.5.m5.1.1.cmml" xref="S1.SSx2.p2.5.m5.1.1">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx2.p2.5.m5.1c">\mathscr{R}(A)</annotation><annotation encoding="application/x-llamapun" id="S1.SSx2.p2.5.m5.1d">script_R ( italic_A )</annotation></semantics></math> are non stationary subsets of <math alttext="\omega_{1}" class="ltx_Math" display="inline" id="S1.SSx2.p2.6.m6.1"><semantics id="S1.SSx2.p2.6.m6.1a"><msub id="S1.SSx2.p2.6.m6.1.1" xref="S1.SSx2.p2.6.m6.1.1.cmml"><mi id="S1.SSx2.p2.6.m6.1.1.2" xref="S1.SSx2.p2.6.m6.1.1.2.cmml">ω</mi><mn id="S1.SSx2.p2.6.m6.1.1.3" xref="S1.SSx2.p2.6.m6.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S1.SSx2.p2.6.m6.1b"><apply id="S1.SSx2.p2.6.m6.1.1.cmml" xref="S1.SSx2.p2.6.m6.1.1"><csymbol cd="ambiguous" id="S1.SSx2.p2.6.m6.1.1.1.cmml" xref="S1.SSx2.p2.6.m6.1.1">subscript</csymbol><ci id="S1.SSx2.p2.6.m6.1.1.2.cmml" xref="S1.SSx2.p2.6.m6.1.1.2">𝜔</ci><cn id="S1.SSx2.p2.6.m6.1.1.3.cmml" type="integer" xref="S1.SSx2.p2.6.m6.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx2.p2.6.m6.1c">\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx2.p2.6.m6.1d">italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>. In proving <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S1.Thmtheorem6" title="Theorem 1.6. ‣ Historical and mathematical context ‣ 1. Introduction ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">1.6</span></a>, Moore <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib16" title="">16</a>]</cite> also proved that under <math alttext="\mathsf{MA}_{\aleph_{1}}" class="ltx_Math" display="inline" id="S1.SSx2.p2.7.m7.1"><semantics id="S1.SSx2.p2.7.m7.1a"><msub id="S1.SSx2.p2.7.m7.1.1" xref="S1.SSx2.p2.7.m7.1.1.cmml"><mi id="S1.SSx2.p2.7.m7.1.1.2" xref="S1.SSx2.p2.7.m7.1.1.2.cmml">𝖬𝖠</mi><msub id="S1.SSx2.p2.7.m7.1.1.3" xref="S1.SSx2.p2.7.m7.1.1.3.cmml"><mi id="S1.SSx2.p2.7.m7.1.1.3.2" mathvariant="normal" xref="S1.SSx2.p2.7.m7.1.1.3.2.cmml">ℵ</mi><mn id="S1.SSx2.p2.7.m7.1.1.3.3" xref="S1.SSx2.p2.7.m7.1.1.3.3.cmml">1</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S1.SSx2.p2.7.m7.1b"><apply id="S1.SSx2.p2.7.m7.1.1.cmml" xref="S1.SSx2.p2.7.m7.1.1"><csymbol cd="ambiguous" id="S1.SSx2.p2.7.m7.1.1.1.cmml" xref="S1.SSx2.p2.7.m7.1.1">subscript</csymbol><ci id="S1.SSx2.p2.7.m7.1.1.2.cmml" xref="S1.SSx2.p2.7.m7.1.1.2">𝖬𝖠</ci><apply id="S1.SSx2.p2.7.m7.1.1.3.cmml" xref="S1.SSx2.p2.7.m7.1.1.3"><csymbol cd="ambiguous" id="S1.SSx2.p2.7.m7.1.1.3.1.cmml" xref="S1.SSx2.p2.7.m7.1.1.3">subscript</csymbol><ci id="S1.SSx2.p2.7.m7.1.1.3.2.cmml" xref="S1.SSx2.p2.7.m7.1.1.3.2">ℵ</ci><cn id="S1.SSx2.p2.7.m7.1.1.3.3.cmml" type="integer" xref="S1.SSx2.p2.7.m7.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx2.p2.7.m7.1c">\mathsf{MA}_{\aleph_{1}}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx2.p2.7.m7.1d">sansserif_MA start_POSTSUBSCRIPT roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> any two <math alttext="\aleph_{1}" class="ltx_Math" display="inline" id="S1.SSx2.p2.8.m8.1"><semantics id="S1.SSx2.p2.8.m8.1a"><msub id="S1.SSx2.p2.8.m8.1.1" xref="S1.SSx2.p2.8.m8.1.1.cmml"><mi id="S1.SSx2.p2.8.m8.1.1.2" mathvariant="normal" xref="S1.SSx2.p2.8.m8.1.1.2.cmml">ℵ</mi><mn id="S1.SSx2.p2.8.m8.1.1.3" xref="S1.SSx2.p2.8.m8.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S1.SSx2.p2.8.m8.1b"><apply id="S1.SSx2.p2.8.m8.1.1.cmml" xref="S1.SSx2.p2.8.m8.1.1"><csymbol cd="ambiguous" id="S1.SSx2.p2.8.m8.1.1.1.cmml" xref="S1.SSx2.p2.8.m8.1.1">subscript</csymbol><ci id="S1.SSx2.p2.8.m8.1.1.2.cmml" xref="S1.SSx2.p2.8.m8.1.1.2">ℵ</ci><cn id="S1.SSx2.p2.8.m8.1.1.3.cmml" type="integer" xref="S1.SSx2.p2.8.m8.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx2.p2.8.m8.1c">\aleph_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx2.p2.8.m8.1d">roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-dense and non stationary Countryman lines are isomorphic or reverse isomorphic. As our first result, we apply this theorem to prove that under <math alttext="\mathsf{MA}_{\aleph_{1}}" class="ltx_Math" display="inline" id="S1.SSx2.p2.9.m9.1"><semantics id="S1.SSx2.p2.9.m9.1a"><msub id="S1.SSx2.p2.9.m9.1.1" xref="S1.SSx2.p2.9.m9.1.1.cmml"><mi id="S1.SSx2.p2.9.m9.1.1.2" xref="S1.SSx2.p2.9.m9.1.1.2.cmml">𝖬𝖠</mi><msub id="S1.SSx2.p2.9.m9.1.1.3" xref="S1.SSx2.p2.9.m9.1.1.3.cmml"><mi id="S1.SSx2.p2.9.m9.1.1.3.2" mathvariant="normal" xref="S1.SSx2.p2.9.m9.1.1.3.2.cmml">ℵ</mi><mn id="S1.SSx2.p2.9.m9.1.1.3.3" xref="S1.SSx2.p2.9.m9.1.1.3.3.cmml">1</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S1.SSx2.p2.9.m9.1b"><apply id="S1.SSx2.p2.9.m9.1.1.cmml" xref="S1.SSx2.p2.9.m9.1.1"><csymbol cd="ambiguous" id="S1.SSx2.p2.9.m9.1.1.1.cmml" xref="S1.SSx2.p2.9.m9.1.1">subscript</csymbol><ci id="S1.SSx2.p2.9.m9.1.1.2.cmml" xref="S1.SSx2.p2.9.m9.1.1.2">𝖬𝖠</ci><apply id="S1.SSx2.p2.9.m9.1.1.3.cmml" xref="S1.SSx2.p2.9.m9.1.1.3"><csymbol cd="ambiguous" id="S1.SSx2.p2.9.m9.1.1.3.1.cmml" xref="S1.SSx2.p2.9.m9.1.1.3">subscript</csymbol><ci id="S1.SSx2.p2.9.m9.1.1.3.2.cmml" xref="S1.SSx2.p2.9.m9.1.1.3.2">ℵ</ci><cn id="S1.SSx2.p2.9.m9.1.1.3.3.cmml" type="integer" xref="S1.SSx2.p2.9.m9.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx2.p2.9.m9.1c">\mathsf{MA}_{\aleph_{1}}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx2.p2.9.m9.1d">sansserif_MA start_POSTSUBSCRIPT roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>, any <math alttext="\aleph_{1}" class="ltx_Math" display="inline" id="S1.SSx2.p2.10.m10.1"><semantics id="S1.SSx2.p2.10.m10.1a"><msub id="S1.SSx2.p2.10.m10.1.1" xref="S1.SSx2.p2.10.m10.1.1.cmml"><mi id="S1.SSx2.p2.10.m10.1.1.2" mathvariant="normal" xref="S1.SSx2.p2.10.m10.1.1.2.cmml">ℵ</mi><mn id="S1.SSx2.p2.10.m10.1.1.3" xref="S1.SSx2.p2.10.m10.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S1.SSx2.p2.10.m10.1b"><apply id="S1.SSx2.p2.10.m10.1.1.cmml" xref="S1.SSx2.p2.10.m10.1.1"><csymbol cd="ambiguous" id="S1.SSx2.p2.10.m10.1.1.1.cmml" xref="S1.SSx2.p2.10.m10.1.1">subscript</csymbol><ci id="S1.SSx2.p2.10.m10.1.1.2.cmml" xref="S1.SSx2.p2.10.m10.1.1.2">ℵ</ci><cn id="S1.SSx2.p2.10.m10.1.1.3.cmml" type="integer" xref="S1.SSx2.p2.10.m10.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx2.p2.10.m10.1c">\aleph_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx2.p2.10.m10.1d">roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-dense and non stationary Countryman line is strongly surjective. This answers Soukup questions <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#Thmquestion3" title="Question 3. ‣ Historical and mathematical context ‣ 1. Introduction ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">3</span></a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#Thmquestion4" title="Question 4. ‣ Historical and mathematical context ‣ 1. Introduction ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">4</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#Thmquestion5" title="Question 5. ‣ Historical and mathematical context ‣ 1. Introduction ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">5</span></a>. For the last one note that one can force <math alttext="\mathsf{MA}_{\aleph_{1}}" class="ltx_Math" display="inline" id="S1.SSx2.p2.11.m11.1"><semantics id="S1.SSx2.p2.11.m11.1a"><msub id="S1.SSx2.p2.11.m11.1.1" xref="S1.SSx2.p2.11.m11.1.1.cmml"><mi id="S1.SSx2.p2.11.m11.1.1.2" xref="S1.SSx2.p2.11.m11.1.1.2.cmml">𝖬𝖠</mi><msub id="S1.SSx2.p2.11.m11.1.1.3" xref="S1.SSx2.p2.11.m11.1.1.3.cmml"><mi id="S1.SSx2.p2.11.m11.1.1.3.2" mathvariant="normal" xref="S1.SSx2.p2.11.m11.1.1.3.2.cmml">ℵ</mi><mn id="S1.SSx2.p2.11.m11.1.1.3.3" xref="S1.SSx2.p2.11.m11.1.1.3.3.cmml">1</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S1.SSx2.p2.11.m11.1b"><apply id="S1.SSx2.p2.11.m11.1.1.cmml" xref="S1.SSx2.p2.11.m11.1.1"><csymbol cd="ambiguous" id="S1.SSx2.p2.11.m11.1.1.1.cmml" xref="S1.SSx2.p2.11.m11.1.1">subscript</csymbol><ci id="S1.SSx2.p2.11.m11.1.1.2.cmml" xref="S1.SSx2.p2.11.m11.1.1.2">𝖬𝖠</ci><apply id="S1.SSx2.p2.11.m11.1.1.3.cmml" xref="S1.SSx2.p2.11.m11.1.1.3"><csymbol cd="ambiguous" id="S1.SSx2.p2.11.m11.1.1.3.1.cmml" xref="S1.SSx2.p2.11.m11.1.1.3">subscript</csymbol><ci id="S1.SSx2.p2.11.m11.1.1.3.2.cmml" xref="S1.SSx2.p2.11.m11.1.1.3.2">ℵ</ci><cn id="S1.SSx2.p2.11.m11.1.1.3.3.cmml" type="integer" xref="S1.SSx2.p2.11.m11.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx2.p2.11.m11.1c">\mathsf{MA}_{\aleph_{1}}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx2.p2.11.m11.1d">sansserif_MA start_POSTSUBSCRIPT roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> and the conclusions of Baumgartner’s theorem on <math alttext="\aleph_{1}" class="ltx_Math" display="inline" id="S1.SSx2.p2.12.m12.1"><semantics id="S1.SSx2.p2.12.m12.1a"><msub id="S1.SSx2.p2.12.m12.1.1" xref="S1.SSx2.p2.12.m12.1.1.cmml"><mi id="S1.SSx2.p2.12.m12.1.1.2" mathvariant="normal" xref="S1.SSx2.p2.12.m12.1.1.2.cmml">ℵ</mi><mn id="S1.SSx2.p2.12.m12.1.1.3" xref="S1.SSx2.p2.12.m12.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S1.SSx2.p2.12.m12.1b"><apply id="S1.SSx2.p2.12.m12.1.1.cmml" xref="S1.SSx2.p2.12.m12.1.1"><csymbol cd="ambiguous" id="S1.SSx2.p2.12.m12.1.1.1.cmml" xref="S1.SSx2.p2.12.m12.1.1">subscript</csymbol><ci id="S1.SSx2.p2.12.m12.1.1.2.cmml" xref="S1.SSx2.p2.12.m12.1.1.2">ℵ</ci><cn id="S1.SSx2.p2.12.m12.1.1.3.cmml" type="integer" xref="S1.SSx2.p2.12.m12.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx2.p2.12.m12.1c">\aleph_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx2.p2.12.m12.1d">roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-dense set of reals simultaneously. Also both follow from <math alttext="\mathsf{PFA}" class="ltx_Math" display="inline" id="S1.SSx2.p2.13.m13.1"><semantics id="S1.SSx2.p2.13.m13.1a"><mi id="S1.SSx2.p2.13.m13.1.1" xref="S1.SSx2.p2.13.m13.1.1.cmml">𝖯𝖥𝖠</mi><annotation-xml encoding="MathML-Content" id="S1.SSx2.p2.13.m13.1b"><ci id="S1.SSx2.p2.13.m13.1.1.cmml" xref="S1.SSx2.p2.13.m13.1.1">𝖯𝖥𝖠</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx2.p2.13.m13.1c">\mathsf{PFA}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx2.p2.13.m13.1d">sansserif_PFA</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S1.SSx2.p3"> <p class="ltx_p" id="S1.SSx2.p3.3">Similar to Baumgartner’s argument to show that there are many non isomorphic Aronszajn line, we show that <math alttext="\mathscr{L}" class="ltx_Math" display="inline" id="S1.SSx2.p3.1.m1.1"><semantics id="S1.SSx2.p3.1.m1.1a"><mi class="ltx_font_mathscript" id="S1.SSx2.p3.1.m1.1.1" xref="S1.SSx2.p3.1.m1.1.1.cmml">ℒ</mi><annotation-xml encoding="MathML-Content" id="S1.SSx2.p3.1.m1.1b"><ci id="S1.SSx2.p3.1.m1.1.1.cmml" xref="S1.SSx2.p3.1.m1.1.1">ℒ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx2.p3.1.m1.1c">\mathscr{L}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx2.p3.1.m1.1d">script_L</annotation></semantics></math> and <math alttext="\mathscr{R}" class="ltx_Math" display="inline" id="S1.SSx2.p3.2.m2.1"><semantics id="S1.SSx2.p3.2.m2.1a"><mi class="ltx_font_mathscript" id="S1.SSx2.p3.2.m2.1.1" xref="S1.SSx2.p3.2.m2.1.1.cmml">ℛ</mi><annotation-xml encoding="MathML-Content" id="S1.SSx2.p3.2.m2.1b"><ci id="S1.SSx2.p3.2.m2.1.1.cmml" xref="S1.SSx2.p3.2.m2.1.1">ℛ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx2.p3.2.m2.1c">\mathscr{R}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx2.p3.2.m2.1d">script_R</annotation></semantics></math> can be used to construct <math alttext="\trianglelefteq" class="ltx_Math" display="inline" id="S1.SSx2.p3.3.m3.1"><semantics id="S1.SSx2.p3.3.m3.1a"><mi id="S1.SSx2.p3.3.m3.1.1" mathvariant="normal" xref="S1.SSx2.p3.3.m3.1.1.cmml">⊴</mi><annotation-xml encoding="MathML-Content" id="S1.SSx2.p3.3.m3.1b"><ci id="S1.SSx2.p3.3.m3.1.1.cmml" xref="S1.SSx2.p3.3.m3.1.1">⊴</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx2.p3.3.m3.1c">\trianglelefteq</annotation><annotation encoding="application/x-llamapun" id="S1.SSx2.p3.3.m3.1d">⊴</annotation></semantics></math>-incomparable Aronszajn lines.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S1.Thmtheorem9"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem9.1.1.1">Theorem 1.9</span></span><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem9.2.2">.</span> </h6> <div class="ltx_para" id="S1.Thmtheorem9.p1"> <p class="ltx_p" id="S1.Thmtheorem9.p1.8"><span class="ltx_text ltx_font_italic" id="S1.Thmtheorem9.p1.8.8">Suppose that <math alttext="A" class="ltx_Math" display="inline" id="S1.Thmtheorem9.p1.1.1.m1.1"><semantics id="S1.Thmtheorem9.p1.1.1.m1.1a"><mi id="S1.Thmtheorem9.p1.1.1.m1.1.1" xref="S1.Thmtheorem9.p1.1.1.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem9.p1.1.1.m1.1b"><ci id="S1.Thmtheorem9.p1.1.1.m1.1.1.cmml" xref="S1.Thmtheorem9.p1.1.1.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem9.p1.1.1.m1.1c">A</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem9.p1.1.1.m1.1d">italic_A</annotation></semantics></math> and <math alttext="B" class="ltx_Math" display="inline" id="S1.Thmtheorem9.p1.2.2.m2.1"><semantics id="S1.Thmtheorem9.p1.2.2.m2.1a"><mi id="S1.Thmtheorem9.p1.2.2.m2.1.1" xref="S1.Thmtheorem9.p1.2.2.m2.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem9.p1.2.2.m2.1b"><ci id="S1.Thmtheorem9.p1.2.2.m2.1.1.cmml" xref="S1.Thmtheorem9.p1.2.2.m2.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem9.p1.2.2.m2.1c">B</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem9.p1.2.2.m2.1d">italic_B</annotation></semantics></math> are Aronszajn lines with respective decompositions <math alttext="D" class="ltx_Math" display="inline" id="S1.Thmtheorem9.p1.3.3.m3.1"><semantics id="S1.Thmtheorem9.p1.3.3.m3.1a"><mi id="S1.Thmtheorem9.p1.3.3.m3.1.1" xref="S1.Thmtheorem9.p1.3.3.m3.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem9.p1.3.3.m3.1b"><ci id="S1.Thmtheorem9.p1.3.3.m3.1.1.cmml" xref="S1.Thmtheorem9.p1.3.3.m3.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem9.p1.3.3.m3.1c">D</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem9.p1.3.3.m3.1d">italic_D</annotation></semantics></math> and <math alttext="E" class="ltx_Math" display="inline" id="S1.Thmtheorem9.p1.4.4.m4.1"><semantics id="S1.Thmtheorem9.p1.4.4.m4.1a"><mi id="S1.Thmtheorem9.p1.4.4.m4.1.1" xref="S1.Thmtheorem9.p1.4.4.m4.1.1.cmml">E</mi><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem9.p1.4.4.m4.1b"><ci id="S1.Thmtheorem9.p1.4.4.m4.1.1.cmml" xref="S1.Thmtheorem9.p1.4.4.m4.1.1">𝐸</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem9.p1.4.4.m4.1c">E</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem9.p1.4.4.m4.1d">italic_E</annotation></semantics></math>. If <math alttext="A\trianglerighteq B" class="ltx_Math" display="inline" id="S1.Thmtheorem9.p1.5.5.m5.1"><semantics id="S1.Thmtheorem9.p1.5.5.m5.1a"><mrow id="S1.Thmtheorem9.p1.5.5.m5.1.1" xref="S1.Thmtheorem9.p1.5.5.m5.1.1.cmml"><mi id="S1.Thmtheorem9.p1.5.5.m5.1.1.2" xref="S1.Thmtheorem9.p1.5.5.m5.1.1.2.cmml">A</mi><mo id="S1.Thmtheorem9.p1.5.5.m5.1.1.1" xref="S1.Thmtheorem9.p1.5.5.m5.1.1.1.cmml">⁢</mo><mi id="S1.Thmtheorem9.p1.5.5.m5.1.1.3" mathvariant="normal" xref="S1.Thmtheorem9.p1.5.5.m5.1.1.3.cmml">⊵</mi><mo id="S1.Thmtheorem9.p1.5.5.m5.1.1.1a" xref="S1.Thmtheorem9.p1.5.5.m5.1.1.1.cmml">⁢</mo><mi id="S1.Thmtheorem9.p1.5.5.m5.1.1.4" xref="S1.Thmtheorem9.p1.5.5.m5.1.1.4.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem9.p1.5.5.m5.1b"><apply id="S1.Thmtheorem9.p1.5.5.m5.1.1.cmml" xref="S1.Thmtheorem9.p1.5.5.m5.1.1"><times id="S1.Thmtheorem9.p1.5.5.m5.1.1.1.cmml" xref="S1.Thmtheorem9.p1.5.5.m5.1.1.1"></times><ci id="S1.Thmtheorem9.p1.5.5.m5.1.1.2.cmml" xref="S1.Thmtheorem9.p1.5.5.m5.1.1.2">𝐴</ci><ci id="S1.Thmtheorem9.p1.5.5.m5.1.1.3.cmml" xref="S1.Thmtheorem9.p1.5.5.m5.1.1.3">⊵</ci><ci id="S1.Thmtheorem9.p1.5.5.m5.1.1.4.cmml" xref="S1.Thmtheorem9.p1.5.5.m5.1.1.4">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem9.p1.5.5.m5.1c">A\trianglerighteq B</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem9.p1.5.5.m5.1d">italic_A ⊵ italic_B</annotation></semantics></math> then <math alttext="\hat{\mathscr{L}}(A,D)\setminus\hat{\mathscr{L}}(B,E)" class="ltx_Math" display="inline" id="S1.Thmtheorem9.p1.6.6.m6.4"><semantics id="S1.Thmtheorem9.p1.6.6.m6.4a"><mrow id="S1.Thmtheorem9.p1.6.6.m6.4.5" xref="S1.Thmtheorem9.p1.6.6.m6.4.5.cmml"><mrow id="S1.Thmtheorem9.p1.6.6.m6.4.5.2" xref="S1.Thmtheorem9.p1.6.6.m6.4.5.2.cmml"><mover accent="true" id="S1.Thmtheorem9.p1.6.6.m6.4.5.2.2" xref="S1.Thmtheorem9.p1.6.6.m6.4.5.2.2.cmml"><mi class="ltx_font_mathscript" id="S1.Thmtheorem9.p1.6.6.m6.4.5.2.2.2" xref="S1.Thmtheorem9.p1.6.6.m6.4.5.2.2.2.cmml">ℒ</mi><mo id="S1.Thmtheorem9.p1.6.6.m6.4.5.2.2.1" xref="S1.Thmtheorem9.p1.6.6.m6.4.5.2.2.1.cmml">^</mo></mover><mo id="S1.Thmtheorem9.p1.6.6.m6.4.5.2.1" xref="S1.Thmtheorem9.p1.6.6.m6.4.5.2.1.cmml">⁢</mo><mrow id="S1.Thmtheorem9.p1.6.6.m6.4.5.2.3.2" xref="S1.Thmtheorem9.p1.6.6.m6.4.5.2.3.1.cmml"><mo id="S1.Thmtheorem9.p1.6.6.m6.4.5.2.3.2.1" stretchy="false" xref="S1.Thmtheorem9.p1.6.6.m6.4.5.2.3.1.cmml">(</mo><mi id="S1.Thmtheorem9.p1.6.6.m6.1.1" xref="S1.Thmtheorem9.p1.6.6.m6.1.1.cmml">A</mi><mo id="S1.Thmtheorem9.p1.6.6.m6.4.5.2.3.2.2" xref="S1.Thmtheorem9.p1.6.6.m6.4.5.2.3.1.cmml">,</mo><mi id="S1.Thmtheorem9.p1.6.6.m6.2.2" xref="S1.Thmtheorem9.p1.6.6.m6.2.2.cmml">D</mi><mo id="S1.Thmtheorem9.p1.6.6.m6.4.5.2.3.2.3" stretchy="false" xref="S1.Thmtheorem9.p1.6.6.m6.4.5.2.3.1.cmml">)</mo></mrow></mrow><mo id="S1.Thmtheorem9.p1.6.6.m6.4.5.1" xref="S1.Thmtheorem9.p1.6.6.m6.4.5.1.cmml">∖</mo><mrow id="S1.Thmtheorem9.p1.6.6.m6.4.5.3" xref="S1.Thmtheorem9.p1.6.6.m6.4.5.3.cmml"><mover accent="true" id="S1.Thmtheorem9.p1.6.6.m6.4.5.3.2" xref="S1.Thmtheorem9.p1.6.6.m6.4.5.3.2.cmml"><mi class="ltx_font_mathscript" id="S1.Thmtheorem9.p1.6.6.m6.4.5.3.2.2" xref="S1.Thmtheorem9.p1.6.6.m6.4.5.3.2.2.cmml">ℒ</mi><mo id="S1.Thmtheorem9.p1.6.6.m6.4.5.3.2.1" xref="S1.Thmtheorem9.p1.6.6.m6.4.5.3.2.1.cmml">^</mo></mover><mo id="S1.Thmtheorem9.p1.6.6.m6.4.5.3.1" xref="S1.Thmtheorem9.p1.6.6.m6.4.5.3.1.cmml">⁢</mo><mrow id="S1.Thmtheorem9.p1.6.6.m6.4.5.3.3.2" xref="S1.Thmtheorem9.p1.6.6.m6.4.5.3.3.1.cmml"><mo id="S1.Thmtheorem9.p1.6.6.m6.4.5.3.3.2.1" stretchy="false" xref="S1.Thmtheorem9.p1.6.6.m6.4.5.3.3.1.cmml">(</mo><mi id="S1.Thmtheorem9.p1.6.6.m6.3.3" xref="S1.Thmtheorem9.p1.6.6.m6.3.3.cmml">B</mi><mo id="S1.Thmtheorem9.p1.6.6.m6.4.5.3.3.2.2" xref="S1.Thmtheorem9.p1.6.6.m6.4.5.3.3.1.cmml">,</mo><mi id="S1.Thmtheorem9.p1.6.6.m6.4.4" xref="S1.Thmtheorem9.p1.6.6.m6.4.4.cmml">E</mi><mo id="S1.Thmtheorem9.p1.6.6.m6.4.5.3.3.2.3" stretchy="false" xref="S1.Thmtheorem9.p1.6.6.m6.4.5.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem9.p1.6.6.m6.4b"><apply id="S1.Thmtheorem9.p1.6.6.m6.4.5.cmml" xref="S1.Thmtheorem9.p1.6.6.m6.4.5"><setdiff id="S1.Thmtheorem9.p1.6.6.m6.4.5.1.cmml" xref="S1.Thmtheorem9.p1.6.6.m6.4.5.1"></setdiff><apply id="S1.Thmtheorem9.p1.6.6.m6.4.5.2.cmml" xref="S1.Thmtheorem9.p1.6.6.m6.4.5.2"><times id="S1.Thmtheorem9.p1.6.6.m6.4.5.2.1.cmml" xref="S1.Thmtheorem9.p1.6.6.m6.4.5.2.1"></times><apply id="S1.Thmtheorem9.p1.6.6.m6.4.5.2.2.cmml" xref="S1.Thmtheorem9.p1.6.6.m6.4.5.2.2"><ci id="S1.Thmtheorem9.p1.6.6.m6.4.5.2.2.1.cmml" xref="S1.Thmtheorem9.p1.6.6.m6.4.5.2.2.1">^</ci><ci id="S1.Thmtheorem9.p1.6.6.m6.4.5.2.2.2.cmml" xref="S1.Thmtheorem9.p1.6.6.m6.4.5.2.2.2">ℒ</ci></apply><interval closure="open" id="S1.Thmtheorem9.p1.6.6.m6.4.5.2.3.1.cmml" xref="S1.Thmtheorem9.p1.6.6.m6.4.5.2.3.2"><ci id="S1.Thmtheorem9.p1.6.6.m6.1.1.cmml" xref="S1.Thmtheorem9.p1.6.6.m6.1.1">𝐴</ci><ci id="S1.Thmtheorem9.p1.6.6.m6.2.2.cmml" xref="S1.Thmtheorem9.p1.6.6.m6.2.2">𝐷</ci></interval></apply><apply id="S1.Thmtheorem9.p1.6.6.m6.4.5.3.cmml" xref="S1.Thmtheorem9.p1.6.6.m6.4.5.3"><times id="S1.Thmtheorem9.p1.6.6.m6.4.5.3.1.cmml" xref="S1.Thmtheorem9.p1.6.6.m6.4.5.3.1"></times><apply id="S1.Thmtheorem9.p1.6.6.m6.4.5.3.2.cmml" xref="S1.Thmtheorem9.p1.6.6.m6.4.5.3.2"><ci id="S1.Thmtheorem9.p1.6.6.m6.4.5.3.2.1.cmml" xref="S1.Thmtheorem9.p1.6.6.m6.4.5.3.2.1">^</ci><ci id="S1.Thmtheorem9.p1.6.6.m6.4.5.3.2.2.cmml" xref="S1.Thmtheorem9.p1.6.6.m6.4.5.3.2.2">ℒ</ci></apply><interval closure="open" id="S1.Thmtheorem9.p1.6.6.m6.4.5.3.3.1.cmml" xref="S1.Thmtheorem9.p1.6.6.m6.4.5.3.3.2"><ci id="S1.Thmtheorem9.p1.6.6.m6.3.3.cmml" xref="S1.Thmtheorem9.p1.6.6.m6.3.3">𝐵</ci><ci id="S1.Thmtheorem9.p1.6.6.m6.4.4.cmml" xref="S1.Thmtheorem9.p1.6.6.m6.4.4">𝐸</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem9.p1.6.6.m6.4c">\hat{\mathscr{L}}(A,D)\setminus\hat{\mathscr{L}}(B,E)</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem9.p1.6.6.m6.4d">over^ start_ARG script_L end_ARG ( italic_A , italic_D ) ∖ over^ start_ARG script_L end_ARG ( italic_B , italic_E )</annotation></semantics></math> and <math alttext="\hat{\mathscr{R}}(A,D)\setminus\hat{\mathscr{R}}(B,E)" class="ltx_Math" display="inline" id="S1.Thmtheorem9.p1.7.7.m7.4"><semantics id="S1.Thmtheorem9.p1.7.7.m7.4a"><mrow id="S1.Thmtheorem9.p1.7.7.m7.4.5" xref="S1.Thmtheorem9.p1.7.7.m7.4.5.cmml"><mrow id="S1.Thmtheorem9.p1.7.7.m7.4.5.2" xref="S1.Thmtheorem9.p1.7.7.m7.4.5.2.cmml"><mover accent="true" id="S1.Thmtheorem9.p1.7.7.m7.4.5.2.2" xref="S1.Thmtheorem9.p1.7.7.m7.4.5.2.2.cmml"><mi class="ltx_font_mathscript" id="S1.Thmtheorem9.p1.7.7.m7.4.5.2.2.2" xref="S1.Thmtheorem9.p1.7.7.m7.4.5.2.2.2.cmml">ℛ</mi><mo id="S1.Thmtheorem9.p1.7.7.m7.4.5.2.2.1" xref="S1.Thmtheorem9.p1.7.7.m7.4.5.2.2.1.cmml">^</mo></mover><mo id="S1.Thmtheorem9.p1.7.7.m7.4.5.2.1" xref="S1.Thmtheorem9.p1.7.7.m7.4.5.2.1.cmml">⁢</mo><mrow id="S1.Thmtheorem9.p1.7.7.m7.4.5.2.3.2" xref="S1.Thmtheorem9.p1.7.7.m7.4.5.2.3.1.cmml"><mo id="S1.Thmtheorem9.p1.7.7.m7.4.5.2.3.2.1" stretchy="false" xref="S1.Thmtheorem9.p1.7.7.m7.4.5.2.3.1.cmml">(</mo><mi id="S1.Thmtheorem9.p1.7.7.m7.1.1" xref="S1.Thmtheorem9.p1.7.7.m7.1.1.cmml">A</mi><mo id="S1.Thmtheorem9.p1.7.7.m7.4.5.2.3.2.2" xref="S1.Thmtheorem9.p1.7.7.m7.4.5.2.3.1.cmml">,</mo><mi id="S1.Thmtheorem9.p1.7.7.m7.2.2" xref="S1.Thmtheorem9.p1.7.7.m7.2.2.cmml">D</mi><mo id="S1.Thmtheorem9.p1.7.7.m7.4.5.2.3.2.3" stretchy="false" xref="S1.Thmtheorem9.p1.7.7.m7.4.5.2.3.1.cmml">)</mo></mrow></mrow><mo id="S1.Thmtheorem9.p1.7.7.m7.4.5.1" xref="S1.Thmtheorem9.p1.7.7.m7.4.5.1.cmml">∖</mo><mrow id="S1.Thmtheorem9.p1.7.7.m7.4.5.3" xref="S1.Thmtheorem9.p1.7.7.m7.4.5.3.cmml"><mover accent="true" id="S1.Thmtheorem9.p1.7.7.m7.4.5.3.2" xref="S1.Thmtheorem9.p1.7.7.m7.4.5.3.2.cmml"><mi class="ltx_font_mathscript" id="S1.Thmtheorem9.p1.7.7.m7.4.5.3.2.2" xref="S1.Thmtheorem9.p1.7.7.m7.4.5.3.2.2.cmml">ℛ</mi><mo id="S1.Thmtheorem9.p1.7.7.m7.4.5.3.2.1" xref="S1.Thmtheorem9.p1.7.7.m7.4.5.3.2.1.cmml">^</mo></mover><mo id="S1.Thmtheorem9.p1.7.7.m7.4.5.3.1" xref="S1.Thmtheorem9.p1.7.7.m7.4.5.3.1.cmml">⁢</mo><mrow id="S1.Thmtheorem9.p1.7.7.m7.4.5.3.3.2" xref="S1.Thmtheorem9.p1.7.7.m7.4.5.3.3.1.cmml"><mo id="S1.Thmtheorem9.p1.7.7.m7.4.5.3.3.2.1" stretchy="false" xref="S1.Thmtheorem9.p1.7.7.m7.4.5.3.3.1.cmml">(</mo><mi id="S1.Thmtheorem9.p1.7.7.m7.3.3" xref="S1.Thmtheorem9.p1.7.7.m7.3.3.cmml">B</mi><mo id="S1.Thmtheorem9.p1.7.7.m7.4.5.3.3.2.2" xref="S1.Thmtheorem9.p1.7.7.m7.4.5.3.3.1.cmml">,</mo><mi id="S1.Thmtheorem9.p1.7.7.m7.4.4" xref="S1.Thmtheorem9.p1.7.7.m7.4.4.cmml">E</mi><mo id="S1.Thmtheorem9.p1.7.7.m7.4.5.3.3.2.3" stretchy="false" xref="S1.Thmtheorem9.p1.7.7.m7.4.5.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem9.p1.7.7.m7.4b"><apply id="S1.Thmtheorem9.p1.7.7.m7.4.5.cmml" xref="S1.Thmtheorem9.p1.7.7.m7.4.5"><setdiff id="S1.Thmtheorem9.p1.7.7.m7.4.5.1.cmml" xref="S1.Thmtheorem9.p1.7.7.m7.4.5.1"></setdiff><apply id="S1.Thmtheorem9.p1.7.7.m7.4.5.2.cmml" xref="S1.Thmtheorem9.p1.7.7.m7.4.5.2"><times id="S1.Thmtheorem9.p1.7.7.m7.4.5.2.1.cmml" xref="S1.Thmtheorem9.p1.7.7.m7.4.5.2.1"></times><apply id="S1.Thmtheorem9.p1.7.7.m7.4.5.2.2.cmml" xref="S1.Thmtheorem9.p1.7.7.m7.4.5.2.2"><ci id="S1.Thmtheorem9.p1.7.7.m7.4.5.2.2.1.cmml" xref="S1.Thmtheorem9.p1.7.7.m7.4.5.2.2.1">^</ci><ci id="S1.Thmtheorem9.p1.7.7.m7.4.5.2.2.2.cmml" xref="S1.Thmtheorem9.p1.7.7.m7.4.5.2.2.2">ℛ</ci></apply><interval closure="open" id="S1.Thmtheorem9.p1.7.7.m7.4.5.2.3.1.cmml" xref="S1.Thmtheorem9.p1.7.7.m7.4.5.2.3.2"><ci id="S1.Thmtheorem9.p1.7.7.m7.1.1.cmml" xref="S1.Thmtheorem9.p1.7.7.m7.1.1">𝐴</ci><ci id="S1.Thmtheorem9.p1.7.7.m7.2.2.cmml" xref="S1.Thmtheorem9.p1.7.7.m7.2.2">𝐷</ci></interval></apply><apply id="S1.Thmtheorem9.p1.7.7.m7.4.5.3.cmml" xref="S1.Thmtheorem9.p1.7.7.m7.4.5.3"><times id="S1.Thmtheorem9.p1.7.7.m7.4.5.3.1.cmml" xref="S1.Thmtheorem9.p1.7.7.m7.4.5.3.1"></times><apply id="S1.Thmtheorem9.p1.7.7.m7.4.5.3.2.cmml" xref="S1.Thmtheorem9.p1.7.7.m7.4.5.3.2"><ci id="S1.Thmtheorem9.p1.7.7.m7.4.5.3.2.1.cmml" xref="S1.Thmtheorem9.p1.7.7.m7.4.5.3.2.1">^</ci><ci id="S1.Thmtheorem9.p1.7.7.m7.4.5.3.2.2.cmml" xref="S1.Thmtheorem9.p1.7.7.m7.4.5.3.2.2">ℛ</ci></apply><interval closure="open" id="S1.Thmtheorem9.p1.7.7.m7.4.5.3.3.1.cmml" xref="S1.Thmtheorem9.p1.7.7.m7.4.5.3.3.2"><ci id="S1.Thmtheorem9.p1.7.7.m7.3.3.cmml" xref="S1.Thmtheorem9.p1.7.7.m7.3.3">𝐵</ci><ci id="S1.Thmtheorem9.p1.7.7.m7.4.4.cmml" xref="S1.Thmtheorem9.p1.7.7.m7.4.4">𝐸</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem9.p1.7.7.m7.4c">\hat{\mathscr{R}}(A,D)\setminus\hat{\mathscr{R}}(B,E)</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem9.p1.7.7.m7.4d">over^ start_ARG script_R end_ARG ( italic_A , italic_D ) ∖ over^ start_ARG script_R end_ARG ( italic_B , italic_E )</annotation></semantics></math> are non stationary subsets of <math alttext="\omega_{1}" class="ltx_Math" display="inline" id="S1.Thmtheorem9.p1.8.8.m8.1"><semantics id="S1.Thmtheorem9.p1.8.8.m8.1a"><msub id="S1.Thmtheorem9.p1.8.8.m8.1.1" xref="S1.Thmtheorem9.p1.8.8.m8.1.1.cmml"><mi id="S1.Thmtheorem9.p1.8.8.m8.1.1.2" xref="S1.Thmtheorem9.p1.8.8.m8.1.1.2.cmml">ω</mi><mn id="S1.Thmtheorem9.p1.8.8.m8.1.1.3" xref="S1.Thmtheorem9.p1.8.8.m8.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem9.p1.8.8.m8.1b"><apply id="S1.Thmtheorem9.p1.8.8.m8.1.1.cmml" xref="S1.Thmtheorem9.p1.8.8.m8.1.1"><csymbol cd="ambiguous" id="S1.Thmtheorem9.p1.8.8.m8.1.1.1.cmml" xref="S1.Thmtheorem9.p1.8.8.m8.1.1">subscript</csymbol><ci id="S1.Thmtheorem9.p1.8.8.m8.1.1.2.cmml" xref="S1.Thmtheorem9.p1.8.8.m8.1.1.2">𝜔</ci><cn id="S1.Thmtheorem9.p1.8.8.m8.1.1.3.cmml" type="integer" xref="S1.Thmtheorem9.p1.8.8.m8.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem9.p1.8.8.m8.1c">\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem9.p1.8.8.m8.1d">italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S1.SSx2.p4"> <p class="ltx_p" id="S1.SSx2.p4.5">The idea is that if <math alttext="f:A\twoheadrightarrow B" class="ltx_Math" display="inline" id="S1.SSx2.p4.1.m1.1"><semantics id="S1.SSx2.p4.1.m1.1a"><mrow id="S1.SSx2.p4.1.m1.1.1" xref="S1.SSx2.p4.1.m1.1.1.cmml"><mi id="S1.SSx2.p4.1.m1.1.1.2" xref="S1.SSx2.p4.1.m1.1.1.2.cmml">f</mi><mo id="S1.SSx2.p4.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S1.SSx2.p4.1.m1.1.1.1.cmml">:</mo><mrow id="S1.SSx2.p4.1.m1.1.1.3" xref="S1.SSx2.p4.1.m1.1.1.3.cmml"><mi id="S1.SSx2.p4.1.m1.1.1.3.2" xref="S1.SSx2.p4.1.m1.1.1.3.2.cmml">A</mi><mo id="S1.SSx2.p4.1.m1.1.1.3.1" stretchy="false" xref="S1.SSx2.p4.1.m1.1.1.3.1.cmml">↠</mo><mi id="S1.SSx2.p4.1.m1.1.1.3.3" xref="S1.SSx2.p4.1.m1.1.1.3.3.cmml">B</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SSx2.p4.1.m1.1b"><apply id="S1.SSx2.p4.1.m1.1.1.cmml" xref="S1.SSx2.p4.1.m1.1.1"><ci id="S1.SSx2.p4.1.m1.1.1.1.cmml" xref="S1.SSx2.p4.1.m1.1.1.1">:</ci><ci id="S1.SSx2.p4.1.m1.1.1.2.cmml" xref="S1.SSx2.p4.1.m1.1.1.2">𝑓</ci><apply id="S1.SSx2.p4.1.m1.1.1.3.cmml" xref="S1.SSx2.p4.1.m1.1.1.3"><ci id="S1.SSx2.p4.1.m1.1.1.3.1.cmml" xref="S1.SSx2.p4.1.m1.1.1.3.1">↠</ci><ci id="S1.SSx2.p4.1.m1.1.1.3.2.cmml" xref="S1.SSx2.p4.1.m1.1.1.3.2">𝐴</ci><ci id="S1.SSx2.p4.1.m1.1.1.3.3.cmml" xref="S1.SSx2.p4.1.m1.1.1.3.3">𝐵</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx2.p4.1.m1.1c">f:A\twoheadrightarrow B</annotation><annotation encoding="application/x-llamapun" id="S1.SSx2.p4.1.m1.1d">italic_f : italic_A ↠ italic_B</annotation></semantics></math> is en epimorphism, then modulo a club subset of levels, <math alttext="f" class="ltx_Math" display="inline" id="S1.SSx2.p4.2.m2.1"><semantics id="S1.SSx2.p4.2.m2.1a"><mi id="S1.SSx2.p4.2.m2.1.1" xref="S1.SSx2.p4.2.m2.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S1.SSx2.p4.2.m2.1b"><ci id="S1.SSx2.p4.2.m2.1.1.cmml" xref="S1.SSx2.p4.2.m2.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx2.p4.2.m2.1c">f</annotation><annotation encoding="application/x-llamapun" id="S1.SSx2.p4.2.m2.1d">italic_f</annotation></semantics></math> must map left (resp. right) endpoints of the complementary intervals of <math alttext="A\setminus D_{\xi}" class="ltx_Math" display="inline" id="S1.SSx2.p4.3.m3.1"><semantics id="S1.SSx2.p4.3.m3.1a"><mrow id="S1.SSx2.p4.3.m3.1.1" xref="S1.SSx2.p4.3.m3.1.1.cmml"><mi id="S1.SSx2.p4.3.m3.1.1.2" xref="S1.SSx2.p4.3.m3.1.1.2.cmml">A</mi><mo id="S1.SSx2.p4.3.m3.1.1.1" xref="S1.SSx2.p4.3.m3.1.1.1.cmml">∖</mo><msub id="S1.SSx2.p4.3.m3.1.1.3" xref="S1.SSx2.p4.3.m3.1.1.3.cmml"><mi id="S1.SSx2.p4.3.m3.1.1.3.2" xref="S1.SSx2.p4.3.m3.1.1.3.2.cmml">D</mi><mi id="S1.SSx2.p4.3.m3.1.1.3.3" xref="S1.SSx2.p4.3.m3.1.1.3.3.cmml">ξ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S1.SSx2.p4.3.m3.1b"><apply id="S1.SSx2.p4.3.m3.1.1.cmml" xref="S1.SSx2.p4.3.m3.1.1"><setdiff id="S1.SSx2.p4.3.m3.1.1.1.cmml" xref="S1.SSx2.p4.3.m3.1.1.1"></setdiff><ci id="S1.SSx2.p4.3.m3.1.1.2.cmml" xref="S1.SSx2.p4.3.m3.1.1.2">𝐴</ci><apply id="S1.SSx2.p4.3.m3.1.1.3.cmml" xref="S1.SSx2.p4.3.m3.1.1.3"><csymbol cd="ambiguous" id="S1.SSx2.p4.3.m3.1.1.3.1.cmml" xref="S1.SSx2.p4.3.m3.1.1.3">subscript</csymbol><ci id="S1.SSx2.p4.3.m3.1.1.3.2.cmml" xref="S1.SSx2.p4.3.m3.1.1.3.2">𝐷</ci><ci id="S1.SSx2.p4.3.m3.1.1.3.3.cmml" xref="S1.SSx2.p4.3.m3.1.1.3.3">𝜉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx2.p4.3.m3.1c">A\setminus D_{\xi}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx2.p4.3.m3.1d">italic_A ∖ italic_D start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT</annotation></semantics></math> to left endpoints of the complementary intervals of <math alttext="B\setminus E_{\xi}" class="ltx_Math" display="inline" id="S1.SSx2.p4.4.m4.1"><semantics id="S1.SSx2.p4.4.m4.1a"><mrow id="S1.SSx2.p4.4.m4.1.1" xref="S1.SSx2.p4.4.m4.1.1.cmml"><mi id="S1.SSx2.p4.4.m4.1.1.2" xref="S1.SSx2.p4.4.m4.1.1.2.cmml">B</mi><mo id="S1.SSx2.p4.4.m4.1.1.1" xref="S1.SSx2.p4.4.m4.1.1.1.cmml">∖</mo><msub id="S1.SSx2.p4.4.m4.1.1.3" xref="S1.SSx2.p4.4.m4.1.1.3.cmml"><mi id="S1.SSx2.p4.4.m4.1.1.3.2" xref="S1.SSx2.p4.4.m4.1.1.3.2.cmml">E</mi><mi id="S1.SSx2.p4.4.m4.1.1.3.3" xref="S1.SSx2.p4.4.m4.1.1.3.3.cmml">ξ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S1.SSx2.p4.4.m4.1b"><apply id="S1.SSx2.p4.4.m4.1.1.cmml" xref="S1.SSx2.p4.4.m4.1.1"><setdiff id="S1.SSx2.p4.4.m4.1.1.1.cmml" xref="S1.SSx2.p4.4.m4.1.1.1"></setdiff><ci id="S1.SSx2.p4.4.m4.1.1.2.cmml" xref="S1.SSx2.p4.4.m4.1.1.2">𝐵</ci><apply id="S1.SSx2.p4.4.m4.1.1.3.cmml" xref="S1.SSx2.p4.4.m4.1.1.3"><csymbol cd="ambiguous" id="S1.SSx2.p4.4.m4.1.1.3.1.cmml" xref="S1.SSx2.p4.4.m4.1.1.3">subscript</csymbol><ci id="S1.SSx2.p4.4.m4.1.1.3.2.cmml" xref="S1.SSx2.p4.4.m4.1.1.3.2">𝐸</ci><ci id="S1.SSx2.p4.4.m4.1.1.3.3.cmml" xref="S1.SSx2.p4.4.m4.1.1.3.3">𝜉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx2.p4.4.m4.1c">B\setminus E_{\xi}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx2.p4.4.m4.1d">italic_B ∖ italic_E start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT</annotation></semantics></math>. We then apply this theorem to construct an infinite <math alttext="\trianglelefteq" class="ltx_Math" display="inline" id="S1.SSx2.p4.5.m5.1"><semantics id="S1.SSx2.p4.5.m5.1a"><mi id="S1.SSx2.p4.5.m5.1.1" mathvariant="normal" xref="S1.SSx2.p4.5.m5.1.1.cmml">⊴</mi><annotation-xml encoding="MathML-Content" id="S1.SSx2.p4.5.m5.1b"><ci id="S1.SSx2.p4.5.m5.1.1.cmml" xref="S1.SSx2.p4.5.m5.1.1">⊴</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx2.p4.5.m5.1c">\trianglelefteq</annotation><annotation encoding="application/x-llamapun" id="S1.SSx2.p4.5.m5.1d">⊴</annotation></semantics></math>-antichain of Countryman lines, giving a negative answer to <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#Thmquestion2" title="Question 2. ‣ Historical and mathematical context ‣ 1. Introduction ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">2</span></a>.</p> </div> <div class="ltx_para" id="S1.SSx2.p5"> <p class="ltx_p" id="S1.SSx2.p5.10">Fix <math alttext="C" class="ltx_Math" display="inline" id="S1.SSx2.p5.1.m1.1"><semantics id="S1.SSx2.p5.1.m1.1a"><mi id="S1.SSx2.p5.1.m1.1.1" xref="S1.SSx2.p5.1.m1.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S1.SSx2.p5.1.m1.1b"><ci id="S1.SSx2.p5.1.m1.1.1.cmml" xref="S1.SSx2.p5.1.m1.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx2.p5.1.m1.1c">C</annotation><annotation encoding="application/x-llamapun" id="S1.SSx2.p5.1.m1.1d">italic_C</annotation></semantics></math> and <math alttext="C^{\prime}" class="ltx_Math" display="inline" id="S1.SSx2.p5.2.m2.1"><semantics id="S1.SSx2.p5.2.m2.1a"><msup id="S1.SSx2.p5.2.m2.1.1" xref="S1.SSx2.p5.2.m2.1.1.cmml"><mi id="S1.SSx2.p5.2.m2.1.1.2" xref="S1.SSx2.p5.2.m2.1.1.2.cmml">C</mi><mo id="S1.SSx2.p5.2.m2.1.1.3" xref="S1.SSx2.p5.2.m2.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S1.SSx2.p5.2.m2.1b"><apply id="S1.SSx2.p5.2.m2.1.1.cmml" xref="S1.SSx2.p5.2.m2.1.1"><csymbol cd="ambiguous" id="S1.SSx2.p5.2.m2.1.1.1.cmml" xref="S1.SSx2.p5.2.m2.1.1">superscript</csymbol><ci id="S1.SSx2.p5.2.m2.1.1.2.cmml" xref="S1.SSx2.p5.2.m2.1.1.2">𝐶</ci><ci id="S1.SSx2.p5.2.m2.1.1.3.cmml" xref="S1.SSx2.p5.2.m2.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx2.p5.2.m2.1c">C^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx2.p5.2.m2.1d">italic_C start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> be <math alttext="\aleph_{1}" class="ltx_Math" display="inline" id="S1.SSx2.p5.3.m3.1"><semantics id="S1.SSx2.p5.3.m3.1a"><msub id="S1.SSx2.p5.3.m3.1.1" xref="S1.SSx2.p5.3.m3.1.1.cmml"><mi id="S1.SSx2.p5.3.m3.1.1.2" mathvariant="normal" xref="S1.SSx2.p5.3.m3.1.1.2.cmml">ℵ</mi><mn id="S1.SSx2.p5.3.m3.1.1.3" xref="S1.SSx2.p5.3.m3.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S1.SSx2.p5.3.m3.1b"><apply id="S1.SSx2.p5.3.m3.1.1.cmml" xref="S1.SSx2.p5.3.m3.1.1"><csymbol cd="ambiguous" id="S1.SSx2.p5.3.m3.1.1.1.cmml" xref="S1.SSx2.p5.3.m3.1.1">subscript</csymbol><ci id="S1.SSx2.p5.3.m3.1.1.2.cmml" xref="S1.SSx2.p5.3.m3.1.1.2">ℵ</ci><cn id="S1.SSx2.p5.3.m3.1.1.3.cmml" type="integer" xref="S1.SSx2.p5.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx2.p5.3.m3.1c">\aleph_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx2.p5.3.m3.1d">roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-dense Countryman lines. For <math alttext="C\trianglerighteq C^{\prime}" class="ltx_Math" display="inline" id="S1.SSx2.p5.4.m4.1"><semantics id="S1.SSx2.p5.4.m4.1a"><mrow id="S1.SSx2.p5.4.m4.1.1" xref="S1.SSx2.p5.4.m4.1.1.cmml"><mi id="S1.SSx2.p5.4.m4.1.1.2" xref="S1.SSx2.p5.4.m4.1.1.2.cmml">C</mi><mo id="S1.SSx2.p5.4.m4.1.1.1" xref="S1.SSx2.p5.4.m4.1.1.1.cmml">⁢</mo><mi id="S1.SSx2.p5.4.m4.1.1.3" mathvariant="normal" xref="S1.SSx2.p5.4.m4.1.1.3.cmml">⊵</mi><mo id="S1.SSx2.p5.4.m4.1.1.1a" xref="S1.SSx2.p5.4.m4.1.1.1.cmml">⁢</mo><msup id="S1.SSx2.p5.4.m4.1.1.4" xref="S1.SSx2.p5.4.m4.1.1.4.cmml"><mi id="S1.SSx2.p5.4.m4.1.1.4.2" xref="S1.SSx2.p5.4.m4.1.1.4.2.cmml">C</mi><mo id="S1.SSx2.p5.4.m4.1.1.4.3" xref="S1.SSx2.p5.4.m4.1.1.4.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S1.SSx2.p5.4.m4.1b"><apply id="S1.SSx2.p5.4.m4.1.1.cmml" xref="S1.SSx2.p5.4.m4.1.1"><times id="S1.SSx2.p5.4.m4.1.1.1.cmml" xref="S1.SSx2.p5.4.m4.1.1.1"></times><ci id="S1.SSx2.p5.4.m4.1.1.2.cmml" xref="S1.SSx2.p5.4.m4.1.1.2">𝐶</ci><ci id="S1.SSx2.p5.4.m4.1.1.3.cmml" xref="S1.SSx2.p5.4.m4.1.1.3">⊵</ci><apply id="S1.SSx2.p5.4.m4.1.1.4.cmml" xref="S1.SSx2.p5.4.m4.1.1.4"><csymbol cd="ambiguous" id="S1.SSx2.p5.4.m4.1.1.4.1.cmml" xref="S1.SSx2.p5.4.m4.1.1.4">superscript</csymbol><ci id="S1.SSx2.p5.4.m4.1.1.4.2.cmml" xref="S1.SSx2.p5.4.m4.1.1.4.2">𝐶</ci><ci id="S1.SSx2.p5.4.m4.1.1.4.3.cmml" xref="S1.SSx2.p5.4.m4.1.1.4.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx2.p5.4.m4.1c">C\trianglerighteq C^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx2.p5.4.m4.1d">italic_C ⊵ italic_C start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> to hold, an obvious necessary condition is that <math alttext="C^{\prime}\preceq C" class="ltx_Math" display="inline" id="S1.SSx2.p5.5.m5.1"><semantics id="S1.SSx2.p5.5.m5.1a"><mrow id="S1.SSx2.p5.5.m5.1.1" xref="S1.SSx2.p5.5.m5.1.1.cmml"><msup id="S1.SSx2.p5.5.m5.1.1.2" xref="S1.SSx2.p5.5.m5.1.1.2.cmml"><mi id="S1.SSx2.p5.5.m5.1.1.2.2" xref="S1.SSx2.p5.5.m5.1.1.2.2.cmml">C</mi><mo id="S1.SSx2.p5.5.m5.1.1.2.3" xref="S1.SSx2.p5.5.m5.1.1.2.3.cmml">′</mo></msup><mo id="S1.SSx2.p5.5.m5.1.1.1" xref="S1.SSx2.p5.5.m5.1.1.1.cmml">⪯</mo><mi id="S1.SSx2.p5.5.m5.1.1.3" xref="S1.SSx2.p5.5.m5.1.1.3.cmml">C</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.SSx2.p5.5.m5.1b"><apply id="S1.SSx2.p5.5.m5.1.1.cmml" xref="S1.SSx2.p5.5.m5.1.1"><csymbol cd="latexml" id="S1.SSx2.p5.5.m5.1.1.1.cmml" xref="S1.SSx2.p5.5.m5.1.1.1">precedes-or-equals</csymbol><apply id="S1.SSx2.p5.5.m5.1.1.2.cmml" xref="S1.SSx2.p5.5.m5.1.1.2"><csymbol cd="ambiguous" id="S1.SSx2.p5.5.m5.1.1.2.1.cmml" xref="S1.SSx2.p5.5.m5.1.1.2">superscript</csymbol><ci id="S1.SSx2.p5.5.m5.1.1.2.2.cmml" xref="S1.SSx2.p5.5.m5.1.1.2.2">𝐶</ci><ci id="S1.SSx2.p5.5.m5.1.1.2.3.cmml" xref="S1.SSx2.p5.5.m5.1.1.2.3">′</ci></apply><ci id="S1.SSx2.p5.5.m5.1.1.3.cmml" xref="S1.SSx2.p5.5.m5.1.1.3">𝐶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx2.p5.5.m5.1c">C^{\prime}\preceq C</annotation><annotation encoding="application/x-llamapun" id="S1.SSx2.p5.5.m5.1d">italic_C start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⪯ italic_C</annotation></semantics></math>. And we know that <math alttext="\mathsf{MA}_{\aleph_{1}}" class="ltx_Math" display="inline" id="S1.SSx2.p5.6.m6.1"><semantics id="S1.SSx2.p5.6.m6.1a"><msub id="S1.SSx2.p5.6.m6.1.1" xref="S1.SSx2.p5.6.m6.1.1.cmml"><mi id="S1.SSx2.p5.6.m6.1.1.2" xref="S1.SSx2.p5.6.m6.1.1.2.cmml">𝖬𝖠</mi><msub id="S1.SSx2.p5.6.m6.1.1.3" xref="S1.SSx2.p5.6.m6.1.1.3.cmml"><mi id="S1.SSx2.p5.6.m6.1.1.3.2" mathvariant="normal" xref="S1.SSx2.p5.6.m6.1.1.3.2.cmml">ℵ</mi><mn id="S1.SSx2.p5.6.m6.1.1.3.3" xref="S1.SSx2.p5.6.m6.1.1.3.3.cmml">1</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S1.SSx2.p5.6.m6.1b"><apply id="S1.SSx2.p5.6.m6.1.1.cmml" xref="S1.SSx2.p5.6.m6.1.1"><csymbol cd="ambiguous" id="S1.SSx2.p5.6.m6.1.1.1.cmml" xref="S1.SSx2.p5.6.m6.1.1">subscript</csymbol><ci id="S1.SSx2.p5.6.m6.1.1.2.cmml" xref="S1.SSx2.p5.6.m6.1.1.2">𝖬𝖠</ci><apply id="S1.SSx2.p5.6.m6.1.1.3.cmml" xref="S1.SSx2.p5.6.m6.1.1.3"><csymbol cd="ambiguous" id="S1.SSx2.p5.6.m6.1.1.3.1.cmml" xref="S1.SSx2.p5.6.m6.1.1.3">subscript</csymbol><ci id="S1.SSx2.p5.6.m6.1.1.3.2.cmml" xref="S1.SSx2.p5.6.m6.1.1.3.2">ℵ</ci><cn id="S1.SSx2.p5.6.m6.1.1.3.3.cmml" type="integer" xref="S1.SSx2.p5.6.m6.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx2.p5.6.m6.1c">\mathsf{MA}_{\aleph_{1}}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx2.p5.6.m6.1d">sansserif_MA start_POSTSUBSCRIPT roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>, either <math alttext="C^{\prime}\preceq C" class="ltx_Math" display="inline" id="S1.SSx2.p5.7.m7.1"><semantics id="S1.SSx2.p5.7.m7.1a"><mrow id="S1.SSx2.p5.7.m7.1.1" xref="S1.SSx2.p5.7.m7.1.1.cmml"><msup id="S1.SSx2.p5.7.m7.1.1.2" xref="S1.SSx2.p5.7.m7.1.1.2.cmml"><mi id="S1.SSx2.p5.7.m7.1.1.2.2" xref="S1.SSx2.p5.7.m7.1.1.2.2.cmml">C</mi><mo id="S1.SSx2.p5.7.m7.1.1.2.3" xref="S1.SSx2.p5.7.m7.1.1.2.3.cmml">′</mo></msup><mo id="S1.SSx2.p5.7.m7.1.1.1" xref="S1.SSx2.p5.7.m7.1.1.1.cmml">⪯</mo><mi id="S1.SSx2.p5.7.m7.1.1.3" xref="S1.SSx2.p5.7.m7.1.1.3.cmml">C</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.SSx2.p5.7.m7.1b"><apply id="S1.SSx2.p5.7.m7.1.1.cmml" xref="S1.SSx2.p5.7.m7.1.1"><csymbol cd="latexml" id="S1.SSx2.p5.7.m7.1.1.1.cmml" xref="S1.SSx2.p5.7.m7.1.1.1">precedes-or-equals</csymbol><apply id="S1.SSx2.p5.7.m7.1.1.2.cmml" xref="S1.SSx2.p5.7.m7.1.1.2"><csymbol cd="ambiguous" id="S1.SSx2.p5.7.m7.1.1.2.1.cmml" xref="S1.SSx2.p5.7.m7.1.1.2">superscript</csymbol><ci id="S1.SSx2.p5.7.m7.1.1.2.2.cmml" xref="S1.SSx2.p5.7.m7.1.1.2.2">𝐶</ci><ci id="S1.SSx2.p5.7.m7.1.1.2.3.cmml" xref="S1.SSx2.p5.7.m7.1.1.2.3">′</ci></apply><ci id="S1.SSx2.p5.7.m7.1.1.3.cmml" xref="S1.SSx2.p5.7.m7.1.1.3">𝐶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx2.p5.7.m7.1c">C^{\prime}\preceq C</annotation><annotation encoding="application/x-llamapun" id="S1.SSx2.p5.7.m7.1d">italic_C start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⪯ italic_C</annotation></semantics></math>, or <math alttext="C^{\prime}\preceq C^{\star}" class="ltx_Math" display="inline" id="S1.SSx2.p5.8.m8.1"><semantics id="S1.SSx2.p5.8.m8.1a"><mrow id="S1.SSx2.p5.8.m8.1.1" xref="S1.SSx2.p5.8.m8.1.1.cmml"><msup id="S1.SSx2.p5.8.m8.1.1.2" xref="S1.SSx2.p5.8.m8.1.1.2.cmml"><mi id="S1.SSx2.p5.8.m8.1.1.2.2" xref="S1.SSx2.p5.8.m8.1.1.2.2.cmml">C</mi><mo id="S1.SSx2.p5.8.m8.1.1.2.3" xref="S1.SSx2.p5.8.m8.1.1.2.3.cmml">′</mo></msup><mo id="S1.SSx2.p5.8.m8.1.1.1" xref="S1.SSx2.p5.8.m8.1.1.1.cmml">⪯</mo><msup id="S1.SSx2.p5.8.m8.1.1.3" xref="S1.SSx2.p5.8.m8.1.1.3.cmml"><mi id="S1.SSx2.p5.8.m8.1.1.3.2" xref="S1.SSx2.p5.8.m8.1.1.3.2.cmml">C</mi><mo id="S1.SSx2.p5.8.m8.1.1.3.3" xref="S1.SSx2.p5.8.m8.1.1.3.3.cmml">⋆</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S1.SSx2.p5.8.m8.1b"><apply id="S1.SSx2.p5.8.m8.1.1.cmml" xref="S1.SSx2.p5.8.m8.1.1"><csymbol cd="latexml" id="S1.SSx2.p5.8.m8.1.1.1.cmml" xref="S1.SSx2.p5.8.m8.1.1.1">precedes-or-equals</csymbol><apply id="S1.SSx2.p5.8.m8.1.1.2.cmml" xref="S1.SSx2.p5.8.m8.1.1.2"><csymbol cd="ambiguous" id="S1.SSx2.p5.8.m8.1.1.2.1.cmml" xref="S1.SSx2.p5.8.m8.1.1.2">superscript</csymbol><ci id="S1.SSx2.p5.8.m8.1.1.2.2.cmml" xref="S1.SSx2.p5.8.m8.1.1.2.2">𝐶</ci><ci id="S1.SSx2.p5.8.m8.1.1.2.3.cmml" xref="S1.SSx2.p5.8.m8.1.1.2.3">′</ci></apply><apply id="S1.SSx2.p5.8.m8.1.1.3.cmml" xref="S1.SSx2.p5.8.m8.1.1.3"><csymbol cd="ambiguous" id="S1.SSx2.p5.8.m8.1.1.3.1.cmml" xref="S1.SSx2.p5.8.m8.1.1.3">superscript</csymbol><ci id="S1.SSx2.p5.8.m8.1.1.3.2.cmml" xref="S1.SSx2.p5.8.m8.1.1.3.2">𝐶</ci><ci id="S1.SSx2.p5.8.m8.1.1.3.3.cmml" xref="S1.SSx2.p5.8.m8.1.1.3.3">⋆</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx2.p5.8.m8.1c">C^{\prime}\preceq C^{\star}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx2.p5.8.m8.1d">italic_C start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⪯ italic_C start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math>. The next theorem shows that under <math alttext="\mathsf{MA}_{\aleph_{1}}" class="ltx_Math" display="inline" id="S1.SSx2.p5.9.m9.1"><semantics id="S1.SSx2.p5.9.m9.1a"><msub id="S1.SSx2.p5.9.m9.1.1" xref="S1.SSx2.p5.9.m9.1.1.cmml"><mi id="S1.SSx2.p5.9.m9.1.1.2" xref="S1.SSx2.p5.9.m9.1.1.2.cmml">𝖬𝖠</mi><msub id="S1.SSx2.p5.9.m9.1.1.3" xref="S1.SSx2.p5.9.m9.1.1.3.cmml"><mi id="S1.SSx2.p5.9.m9.1.1.3.2" mathvariant="normal" xref="S1.SSx2.p5.9.m9.1.1.3.2.cmml">ℵ</mi><mn id="S1.SSx2.p5.9.m9.1.1.3.3" xref="S1.SSx2.p5.9.m9.1.1.3.3.cmml">1</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S1.SSx2.p5.9.m9.1b"><apply id="S1.SSx2.p5.9.m9.1.1.cmml" xref="S1.SSx2.p5.9.m9.1.1"><csymbol cd="ambiguous" id="S1.SSx2.p5.9.m9.1.1.1.cmml" xref="S1.SSx2.p5.9.m9.1.1">subscript</csymbol><ci id="S1.SSx2.p5.9.m9.1.1.2.cmml" xref="S1.SSx2.p5.9.m9.1.1.2">𝖬𝖠</ci><apply id="S1.SSx2.p5.9.m9.1.1.3.cmml" xref="S1.SSx2.p5.9.m9.1.1.3"><csymbol cd="ambiguous" id="S1.SSx2.p5.9.m9.1.1.3.1.cmml" xref="S1.SSx2.p5.9.m9.1.1.3">subscript</csymbol><ci id="S1.SSx2.p5.9.m9.1.1.3.2.cmml" xref="S1.SSx2.p5.9.m9.1.1.3.2">ℵ</ci><cn id="S1.SSx2.p5.9.m9.1.1.3.3.cmml" type="integer" xref="S1.SSx2.p5.9.m9.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx2.p5.9.m9.1c">\mathsf{MA}_{\aleph_{1}}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx2.p5.9.m9.1d">sansserif_MA start_POSTSUBSCRIPT roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>, this together with the previous theorem, are roughly the unique limitations if restricted to <math alttext="\aleph_{1}" class="ltx_Math" display="inline" id="S1.SSx2.p5.10.m10.1"><semantics id="S1.SSx2.p5.10.m10.1a"><msub id="S1.SSx2.p5.10.m10.1.1" xref="S1.SSx2.p5.10.m10.1.1.cmml"><mi id="S1.SSx2.p5.10.m10.1.1.2" mathvariant="normal" xref="S1.SSx2.p5.10.m10.1.1.2.cmml">ℵ</mi><mn id="S1.SSx2.p5.10.m10.1.1.3" xref="S1.SSx2.p5.10.m10.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S1.SSx2.p5.10.m10.1b"><apply id="S1.SSx2.p5.10.m10.1.1.cmml" xref="S1.SSx2.p5.10.m10.1.1"><csymbol cd="ambiguous" id="S1.SSx2.p5.10.m10.1.1.1.cmml" xref="S1.SSx2.p5.10.m10.1.1">subscript</csymbol><ci id="S1.SSx2.p5.10.m10.1.1.2.cmml" xref="S1.SSx2.p5.10.m10.1.1.2">ℵ</ci><cn id="S1.SSx2.p5.10.m10.1.1.3.cmml" type="integer" xref="S1.SSx2.p5.10.m10.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx2.p5.10.m10.1c">\aleph_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx2.p5.10.m10.1d">roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-dense lines.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S1.Thmtheorem10"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem10.1.1.1">Theorem 1.10</span></span><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem10.2.2">.</span> </h6> <div class="ltx_para" id="S1.Thmtheorem10.p1"> <p class="ltx_p" id="S1.Thmtheorem10.p1.9"><span class="ltx_text ltx_font_italic" id="S1.Thmtheorem10.p1.9.9">Assume <math alttext="\mathsf{MA}_{\aleph_{1}}" class="ltx_Math" display="inline" id="S1.Thmtheorem10.p1.1.1.m1.1"><semantics id="S1.Thmtheorem10.p1.1.1.m1.1a"><msub id="S1.Thmtheorem10.p1.1.1.m1.1.1" xref="S1.Thmtheorem10.p1.1.1.m1.1.1.cmml"><mi id="S1.Thmtheorem10.p1.1.1.m1.1.1.2" xref="S1.Thmtheorem10.p1.1.1.m1.1.1.2.cmml">𝖬𝖠</mi><msub id="S1.Thmtheorem10.p1.1.1.m1.1.1.3" xref="S1.Thmtheorem10.p1.1.1.m1.1.1.3.cmml"><mi id="S1.Thmtheorem10.p1.1.1.m1.1.1.3.2" mathvariant="normal" xref="S1.Thmtheorem10.p1.1.1.m1.1.1.3.2.cmml">ℵ</mi><mn id="S1.Thmtheorem10.p1.1.1.m1.1.1.3.3" xref="S1.Thmtheorem10.p1.1.1.m1.1.1.3.3.cmml">1</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem10.p1.1.1.m1.1b"><apply id="S1.Thmtheorem10.p1.1.1.m1.1.1.cmml" xref="S1.Thmtheorem10.p1.1.1.m1.1.1"><csymbol cd="ambiguous" id="S1.Thmtheorem10.p1.1.1.m1.1.1.1.cmml" xref="S1.Thmtheorem10.p1.1.1.m1.1.1">subscript</csymbol><ci id="S1.Thmtheorem10.p1.1.1.m1.1.1.2.cmml" xref="S1.Thmtheorem10.p1.1.1.m1.1.1.2">𝖬𝖠</ci><apply id="S1.Thmtheorem10.p1.1.1.m1.1.1.3.cmml" xref="S1.Thmtheorem10.p1.1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S1.Thmtheorem10.p1.1.1.m1.1.1.3.1.cmml" xref="S1.Thmtheorem10.p1.1.1.m1.1.1.3">subscript</csymbol><ci id="S1.Thmtheorem10.p1.1.1.m1.1.1.3.2.cmml" xref="S1.Thmtheorem10.p1.1.1.m1.1.1.3.2">ℵ</ci><cn id="S1.Thmtheorem10.p1.1.1.m1.1.1.3.3.cmml" type="integer" xref="S1.Thmtheorem10.p1.1.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem10.p1.1.1.m1.1c">\mathsf{MA}_{\aleph_{1}}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem10.p1.1.1.m1.1d">sansserif_MA start_POSTSUBSCRIPT roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. Let <math alttext="C^{\prime}\preceq C" class="ltx_Math" display="inline" id="S1.Thmtheorem10.p1.2.2.m2.1"><semantics id="S1.Thmtheorem10.p1.2.2.m2.1a"><mrow id="S1.Thmtheorem10.p1.2.2.m2.1.1" xref="S1.Thmtheorem10.p1.2.2.m2.1.1.cmml"><msup id="S1.Thmtheorem10.p1.2.2.m2.1.1.2" xref="S1.Thmtheorem10.p1.2.2.m2.1.1.2.cmml"><mi id="S1.Thmtheorem10.p1.2.2.m2.1.1.2.2" xref="S1.Thmtheorem10.p1.2.2.m2.1.1.2.2.cmml">C</mi><mo id="S1.Thmtheorem10.p1.2.2.m2.1.1.2.3" xref="S1.Thmtheorem10.p1.2.2.m2.1.1.2.3.cmml">′</mo></msup><mo id="S1.Thmtheorem10.p1.2.2.m2.1.1.1" xref="S1.Thmtheorem10.p1.2.2.m2.1.1.1.cmml">⪯</mo><mi id="S1.Thmtheorem10.p1.2.2.m2.1.1.3" xref="S1.Thmtheorem10.p1.2.2.m2.1.1.3.cmml">C</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem10.p1.2.2.m2.1b"><apply id="S1.Thmtheorem10.p1.2.2.m2.1.1.cmml" xref="S1.Thmtheorem10.p1.2.2.m2.1.1"><csymbol cd="latexml" id="S1.Thmtheorem10.p1.2.2.m2.1.1.1.cmml" xref="S1.Thmtheorem10.p1.2.2.m2.1.1.1">precedes-or-equals</csymbol><apply id="S1.Thmtheorem10.p1.2.2.m2.1.1.2.cmml" xref="S1.Thmtheorem10.p1.2.2.m2.1.1.2"><csymbol cd="ambiguous" id="S1.Thmtheorem10.p1.2.2.m2.1.1.2.1.cmml" xref="S1.Thmtheorem10.p1.2.2.m2.1.1.2">superscript</csymbol><ci id="S1.Thmtheorem10.p1.2.2.m2.1.1.2.2.cmml" xref="S1.Thmtheorem10.p1.2.2.m2.1.1.2.2">𝐶</ci><ci id="S1.Thmtheorem10.p1.2.2.m2.1.1.2.3.cmml" xref="S1.Thmtheorem10.p1.2.2.m2.1.1.2.3">′</ci></apply><ci id="S1.Thmtheorem10.p1.2.2.m2.1.1.3.cmml" xref="S1.Thmtheorem10.p1.2.2.m2.1.1.3">𝐶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem10.p1.2.2.m2.1c">C^{\prime}\preceq C</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem10.p1.2.2.m2.1d">italic_C start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⪯ italic_C</annotation></semantics></math> be <math alttext="\aleph_{1}" class="ltx_Math" display="inline" id="S1.Thmtheorem10.p1.3.3.m3.1"><semantics id="S1.Thmtheorem10.p1.3.3.m3.1a"><msub id="S1.Thmtheorem10.p1.3.3.m3.1.1" xref="S1.Thmtheorem10.p1.3.3.m3.1.1.cmml"><mi id="S1.Thmtheorem10.p1.3.3.m3.1.1.2" mathvariant="normal" xref="S1.Thmtheorem10.p1.3.3.m3.1.1.2.cmml">ℵ</mi><mn id="S1.Thmtheorem10.p1.3.3.m3.1.1.3" xref="S1.Thmtheorem10.p1.3.3.m3.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem10.p1.3.3.m3.1b"><apply id="S1.Thmtheorem10.p1.3.3.m3.1.1.cmml" xref="S1.Thmtheorem10.p1.3.3.m3.1.1"><csymbol cd="ambiguous" id="S1.Thmtheorem10.p1.3.3.m3.1.1.1.cmml" xref="S1.Thmtheorem10.p1.3.3.m3.1.1">subscript</csymbol><ci id="S1.Thmtheorem10.p1.3.3.m3.1.1.2.cmml" xref="S1.Thmtheorem10.p1.3.3.m3.1.1.2">ℵ</ci><cn id="S1.Thmtheorem10.p1.3.3.m3.1.1.3.cmml" type="integer" xref="S1.Thmtheorem10.p1.3.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem10.p1.3.3.m3.1c">\aleph_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem10.p1.3.3.m3.1d">roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-dense Countryman lines with respective decompositions <math alttext="D" class="ltx_Math" display="inline" id="S1.Thmtheorem10.p1.4.4.m4.1"><semantics id="S1.Thmtheorem10.p1.4.4.m4.1a"><mi id="S1.Thmtheorem10.p1.4.4.m4.1.1" xref="S1.Thmtheorem10.p1.4.4.m4.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem10.p1.4.4.m4.1b"><ci id="S1.Thmtheorem10.p1.4.4.m4.1.1.cmml" xref="S1.Thmtheorem10.p1.4.4.m4.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem10.p1.4.4.m4.1c">D</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem10.p1.4.4.m4.1d">italic_D</annotation></semantics></math> and <math alttext="D^{\prime}" class="ltx_Math" display="inline" id="S1.Thmtheorem10.p1.5.5.m5.1"><semantics id="S1.Thmtheorem10.p1.5.5.m5.1a"><msup id="S1.Thmtheorem10.p1.5.5.m5.1.1" xref="S1.Thmtheorem10.p1.5.5.m5.1.1.cmml"><mi id="S1.Thmtheorem10.p1.5.5.m5.1.1.2" xref="S1.Thmtheorem10.p1.5.5.m5.1.1.2.cmml">D</mi><mo id="S1.Thmtheorem10.p1.5.5.m5.1.1.3" xref="S1.Thmtheorem10.p1.5.5.m5.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem10.p1.5.5.m5.1b"><apply id="S1.Thmtheorem10.p1.5.5.m5.1.1.cmml" xref="S1.Thmtheorem10.p1.5.5.m5.1.1"><csymbol cd="ambiguous" id="S1.Thmtheorem10.p1.5.5.m5.1.1.1.cmml" xref="S1.Thmtheorem10.p1.5.5.m5.1.1">superscript</csymbol><ci id="S1.Thmtheorem10.p1.5.5.m5.1.1.2.cmml" xref="S1.Thmtheorem10.p1.5.5.m5.1.1.2">𝐷</ci><ci id="S1.Thmtheorem10.p1.5.5.m5.1.1.3.cmml" xref="S1.Thmtheorem10.p1.5.5.m5.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem10.p1.5.5.m5.1c">D^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem10.p1.5.5.m5.1d">italic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>. If for some club <math alttext="E\subseteq\omega_{1}" class="ltx_Math" display="inline" id="S1.Thmtheorem10.p1.6.6.m6.1"><semantics id="S1.Thmtheorem10.p1.6.6.m6.1a"><mrow id="S1.Thmtheorem10.p1.6.6.m6.1.1" xref="S1.Thmtheorem10.p1.6.6.m6.1.1.cmml"><mi id="S1.Thmtheorem10.p1.6.6.m6.1.1.2" xref="S1.Thmtheorem10.p1.6.6.m6.1.1.2.cmml">E</mi><mo id="S1.Thmtheorem10.p1.6.6.m6.1.1.1" xref="S1.Thmtheorem10.p1.6.6.m6.1.1.1.cmml">⊆</mo><msub id="S1.Thmtheorem10.p1.6.6.m6.1.1.3" xref="S1.Thmtheorem10.p1.6.6.m6.1.1.3.cmml"><mi id="S1.Thmtheorem10.p1.6.6.m6.1.1.3.2" xref="S1.Thmtheorem10.p1.6.6.m6.1.1.3.2.cmml">ω</mi><mn id="S1.Thmtheorem10.p1.6.6.m6.1.1.3.3" xref="S1.Thmtheorem10.p1.6.6.m6.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem10.p1.6.6.m6.1b"><apply id="S1.Thmtheorem10.p1.6.6.m6.1.1.cmml" xref="S1.Thmtheorem10.p1.6.6.m6.1.1"><subset id="S1.Thmtheorem10.p1.6.6.m6.1.1.1.cmml" xref="S1.Thmtheorem10.p1.6.6.m6.1.1.1"></subset><ci id="S1.Thmtheorem10.p1.6.6.m6.1.1.2.cmml" xref="S1.Thmtheorem10.p1.6.6.m6.1.1.2">𝐸</ci><apply id="S1.Thmtheorem10.p1.6.6.m6.1.1.3.cmml" xref="S1.Thmtheorem10.p1.6.6.m6.1.1.3"><csymbol cd="ambiguous" id="S1.Thmtheorem10.p1.6.6.m6.1.1.3.1.cmml" xref="S1.Thmtheorem10.p1.6.6.m6.1.1.3">subscript</csymbol><ci id="S1.Thmtheorem10.p1.6.6.m6.1.1.3.2.cmml" xref="S1.Thmtheorem10.p1.6.6.m6.1.1.3.2">𝜔</ci><cn id="S1.Thmtheorem10.p1.6.6.m6.1.1.3.3.cmml" type="integer" xref="S1.Thmtheorem10.p1.6.6.m6.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem10.p1.6.6.m6.1c">E\subseteq\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem10.p1.6.6.m6.1d">italic_E ⊆ italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\mathscr{L}(C,D)\cap E\subseteq\hat{\mathscr{L}}(C^{\prime},D^{\prime})" class="ltx_Math" display="inline" id="S1.Thmtheorem10.p1.7.7.m7.4"><semantics id="S1.Thmtheorem10.p1.7.7.m7.4a"><mrow id="S1.Thmtheorem10.p1.7.7.m7.4.4" xref="S1.Thmtheorem10.p1.7.7.m7.4.4.cmml"><mrow id="S1.Thmtheorem10.p1.7.7.m7.4.4.4" xref="S1.Thmtheorem10.p1.7.7.m7.4.4.4.cmml"><mrow id="S1.Thmtheorem10.p1.7.7.m7.4.4.4.2" xref="S1.Thmtheorem10.p1.7.7.m7.4.4.4.2.cmml"><mi class="ltx_font_mathscript" id="S1.Thmtheorem10.p1.7.7.m7.4.4.4.2.2" xref="S1.Thmtheorem10.p1.7.7.m7.4.4.4.2.2.cmml">ℒ</mi><mo id="S1.Thmtheorem10.p1.7.7.m7.4.4.4.2.1" xref="S1.Thmtheorem10.p1.7.7.m7.4.4.4.2.1.cmml">⁢</mo><mrow id="S1.Thmtheorem10.p1.7.7.m7.4.4.4.2.3.2" xref="S1.Thmtheorem10.p1.7.7.m7.4.4.4.2.3.1.cmml"><mo id="S1.Thmtheorem10.p1.7.7.m7.4.4.4.2.3.2.1" stretchy="false" xref="S1.Thmtheorem10.p1.7.7.m7.4.4.4.2.3.1.cmml">(</mo><mi id="S1.Thmtheorem10.p1.7.7.m7.1.1" xref="S1.Thmtheorem10.p1.7.7.m7.1.1.cmml">C</mi><mo id="S1.Thmtheorem10.p1.7.7.m7.4.4.4.2.3.2.2" xref="S1.Thmtheorem10.p1.7.7.m7.4.4.4.2.3.1.cmml">,</mo><mi id="S1.Thmtheorem10.p1.7.7.m7.2.2" xref="S1.Thmtheorem10.p1.7.7.m7.2.2.cmml">D</mi><mo id="S1.Thmtheorem10.p1.7.7.m7.4.4.4.2.3.2.3" stretchy="false" xref="S1.Thmtheorem10.p1.7.7.m7.4.4.4.2.3.1.cmml">)</mo></mrow></mrow><mo id="S1.Thmtheorem10.p1.7.7.m7.4.4.4.1" xref="S1.Thmtheorem10.p1.7.7.m7.4.4.4.1.cmml">∩</mo><mi id="S1.Thmtheorem10.p1.7.7.m7.4.4.4.3" xref="S1.Thmtheorem10.p1.7.7.m7.4.4.4.3.cmml">E</mi></mrow><mo id="S1.Thmtheorem10.p1.7.7.m7.4.4.3" xref="S1.Thmtheorem10.p1.7.7.m7.4.4.3.cmml">⊆</mo><mrow id="S1.Thmtheorem10.p1.7.7.m7.4.4.2" xref="S1.Thmtheorem10.p1.7.7.m7.4.4.2.cmml"><mover accent="true" id="S1.Thmtheorem10.p1.7.7.m7.4.4.2.4" xref="S1.Thmtheorem10.p1.7.7.m7.4.4.2.4.cmml"><mi class="ltx_font_mathscript" id="S1.Thmtheorem10.p1.7.7.m7.4.4.2.4.2" xref="S1.Thmtheorem10.p1.7.7.m7.4.4.2.4.2.cmml">ℒ</mi><mo id="S1.Thmtheorem10.p1.7.7.m7.4.4.2.4.1" xref="S1.Thmtheorem10.p1.7.7.m7.4.4.2.4.1.cmml">^</mo></mover><mo id="S1.Thmtheorem10.p1.7.7.m7.4.4.2.3" xref="S1.Thmtheorem10.p1.7.7.m7.4.4.2.3.cmml">⁢</mo><mrow id="S1.Thmtheorem10.p1.7.7.m7.4.4.2.2.2" xref="S1.Thmtheorem10.p1.7.7.m7.4.4.2.2.3.cmml"><mo id="S1.Thmtheorem10.p1.7.7.m7.4.4.2.2.2.3" stretchy="false" xref="S1.Thmtheorem10.p1.7.7.m7.4.4.2.2.3.cmml">(</mo><msup id="S1.Thmtheorem10.p1.7.7.m7.3.3.1.1.1.1" xref="S1.Thmtheorem10.p1.7.7.m7.3.3.1.1.1.1.cmml"><mi id="S1.Thmtheorem10.p1.7.7.m7.3.3.1.1.1.1.2" xref="S1.Thmtheorem10.p1.7.7.m7.3.3.1.1.1.1.2.cmml">C</mi><mo id="S1.Thmtheorem10.p1.7.7.m7.3.3.1.1.1.1.3" xref="S1.Thmtheorem10.p1.7.7.m7.3.3.1.1.1.1.3.cmml">′</mo></msup><mo id="S1.Thmtheorem10.p1.7.7.m7.4.4.2.2.2.4" xref="S1.Thmtheorem10.p1.7.7.m7.4.4.2.2.3.cmml">,</mo><msup id="S1.Thmtheorem10.p1.7.7.m7.4.4.2.2.2.2" xref="S1.Thmtheorem10.p1.7.7.m7.4.4.2.2.2.2.cmml"><mi id="S1.Thmtheorem10.p1.7.7.m7.4.4.2.2.2.2.2" xref="S1.Thmtheorem10.p1.7.7.m7.4.4.2.2.2.2.2.cmml">D</mi><mo id="S1.Thmtheorem10.p1.7.7.m7.4.4.2.2.2.2.3" xref="S1.Thmtheorem10.p1.7.7.m7.4.4.2.2.2.2.3.cmml">′</mo></msup><mo id="S1.Thmtheorem10.p1.7.7.m7.4.4.2.2.2.5" stretchy="false" xref="S1.Thmtheorem10.p1.7.7.m7.4.4.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem10.p1.7.7.m7.4b"><apply id="S1.Thmtheorem10.p1.7.7.m7.4.4.cmml" xref="S1.Thmtheorem10.p1.7.7.m7.4.4"><subset id="S1.Thmtheorem10.p1.7.7.m7.4.4.3.cmml" xref="S1.Thmtheorem10.p1.7.7.m7.4.4.3"></subset><apply id="S1.Thmtheorem10.p1.7.7.m7.4.4.4.cmml" xref="S1.Thmtheorem10.p1.7.7.m7.4.4.4"><intersect id="S1.Thmtheorem10.p1.7.7.m7.4.4.4.1.cmml" xref="S1.Thmtheorem10.p1.7.7.m7.4.4.4.1"></intersect><apply id="S1.Thmtheorem10.p1.7.7.m7.4.4.4.2.cmml" xref="S1.Thmtheorem10.p1.7.7.m7.4.4.4.2"><times id="S1.Thmtheorem10.p1.7.7.m7.4.4.4.2.1.cmml" xref="S1.Thmtheorem10.p1.7.7.m7.4.4.4.2.1"></times><ci id="S1.Thmtheorem10.p1.7.7.m7.4.4.4.2.2.cmml" xref="S1.Thmtheorem10.p1.7.7.m7.4.4.4.2.2">ℒ</ci><interval closure="open" id="S1.Thmtheorem10.p1.7.7.m7.4.4.4.2.3.1.cmml" xref="S1.Thmtheorem10.p1.7.7.m7.4.4.4.2.3.2"><ci id="S1.Thmtheorem10.p1.7.7.m7.1.1.cmml" xref="S1.Thmtheorem10.p1.7.7.m7.1.1">𝐶</ci><ci id="S1.Thmtheorem10.p1.7.7.m7.2.2.cmml" xref="S1.Thmtheorem10.p1.7.7.m7.2.2">𝐷</ci></interval></apply><ci id="S1.Thmtheorem10.p1.7.7.m7.4.4.4.3.cmml" xref="S1.Thmtheorem10.p1.7.7.m7.4.4.4.3">𝐸</ci></apply><apply id="S1.Thmtheorem10.p1.7.7.m7.4.4.2.cmml" xref="S1.Thmtheorem10.p1.7.7.m7.4.4.2"><times id="S1.Thmtheorem10.p1.7.7.m7.4.4.2.3.cmml" xref="S1.Thmtheorem10.p1.7.7.m7.4.4.2.3"></times><apply id="S1.Thmtheorem10.p1.7.7.m7.4.4.2.4.cmml" xref="S1.Thmtheorem10.p1.7.7.m7.4.4.2.4"><ci id="S1.Thmtheorem10.p1.7.7.m7.4.4.2.4.1.cmml" xref="S1.Thmtheorem10.p1.7.7.m7.4.4.2.4.1">^</ci><ci id="S1.Thmtheorem10.p1.7.7.m7.4.4.2.4.2.cmml" xref="S1.Thmtheorem10.p1.7.7.m7.4.4.2.4.2">ℒ</ci></apply><interval closure="open" id="S1.Thmtheorem10.p1.7.7.m7.4.4.2.2.3.cmml" xref="S1.Thmtheorem10.p1.7.7.m7.4.4.2.2.2"><apply id="S1.Thmtheorem10.p1.7.7.m7.3.3.1.1.1.1.cmml" xref="S1.Thmtheorem10.p1.7.7.m7.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S1.Thmtheorem10.p1.7.7.m7.3.3.1.1.1.1.1.cmml" xref="S1.Thmtheorem10.p1.7.7.m7.3.3.1.1.1.1">superscript</csymbol><ci id="S1.Thmtheorem10.p1.7.7.m7.3.3.1.1.1.1.2.cmml" xref="S1.Thmtheorem10.p1.7.7.m7.3.3.1.1.1.1.2">𝐶</ci><ci id="S1.Thmtheorem10.p1.7.7.m7.3.3.1.1.1.1.3.cmml" xref="S1.Thmtheorem10.p1.7.7.m7.3.3.1.1.1.1.3">′</ci></apply><apply id="S1.Thmtheorem10.p1.7.7.m7.4.4.2.2.2.2.cmml" xref="S1.Thmtheorem10.p1.7.7.m7.4.4.2.2.2.2"><csymbol cd="ambiguous" id="S1.Thmtheorem10.p1.7.7.m7.4.4.2.2.2.2.1.cmml" xref="S1.Thmtheorem10.p1.7.7.m7.4.4.2.2.2.2">superscript</csymbol><ci id="S1.Thmtheorem10.p1.7.7.m7.4.4.2.2.2.2.2.cmml" xref="S1.Thmtheorem10.p1.7.7.m7.4.4.2.2.2.2.2">𝐷</ci><ci id="S1.Thmtheorem10.p1.7.7.m7.4.4.2.2.2.2.3.cmml" xref="S1.Thmtheorem10.p1.7.7.m7.4.4.2.2.2.2.3">′</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem10.p1.7.7.m7.4c">\mathscr{L}(C,D)\cap E\subseteq\hat{\mathscr{L}}(C^{\prime},D^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem10.p1.7.7.m7.4d">script_L ( italic_C , italic_D ) ∩ italic_E ⊆ over^ start_ARG script_L end_ARG ( italic_C start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> and <math alttext="\mathscr{R}(C,D)\cap E\subseteq\hat{\mathscr{R}}(C^{\prime},D^{\prime})" class="ltx_Math" display="inline" id="S1.Thmtheorem10.p1.8.8.m8.4"><semantics id="S1.Thmtheorem10.p1.8.8.m8.4a"><mrow id="S1.Thmtheorem10.p1.8.8.m8.4.4" xref="S1.Thmtheorem10.p1.8.8.m8.4.4.cmml"><mrow id="S1.Thmtheorem10.p1.8.8.m8.4.4.4" xref="S1.Thmtheorem10.p1.8.8.m8.4.4.4.cmml"><mrow id="S1.Thmtheorem10.p1.8.8.m8.4.4.4.2" xref="S1.Thmtheorem10.p1.8.8.m8.4.4.4.2.cmml"><mi class="ltx_font_mathscript" id="S1.Thmtheorem10.p1.8.8.m8.4.4.4.2.2" xref="S1.Thmtheorem10.p1.8.8.m8.4.4.4.2.2.cmml">ℛ</mi><mo id="S1.Thmtheorem10.p1.8.8.m8.4.4.4.2.1" xref="S1.Thmtheorem10.p1.8.8.m8.4.4.4.2.1.cmml">⁢</mo><mrow id="S1.Thmtheorem10.p1.8.8.m8.4.4.4.2.3.2" xref="S1.Thmtheorem10.p1.8.8.m8.4.4.4.2.3.1.cmml"><mo id="S1.Thmtheorem10.p1.8.8.m8.4.4.4.2.3.2.1" stretchy="false" xref="S1.Thmtheorem10.p1.8.8.m8.4.4.4.2.3.1.cmml">(</mo><mi id="S1.Thmtheorem10.p1.8.8.m8.1.1" xref="S1.Thmtheorem10.p1.8.8.m8.1.1.cmml">C</mi><mo id="S1.Thmtheorem10.p1.8.8.m8.4.4.4.2.3.2.2" xref="S1.Thmtheorem10.p1.8.8.m8.4.4.4.2.3.1.cmml">,</mo><mi id="S1.Thmtheorem10.p1.8.8.m8.2.2" xref="S1.Thmtheorem10.p1.8.8.m8.2.2.cmml">D</mi><mo id="S1.Thmtheorem10.p1.8.8.m8.4.4.4.2.3.2.3" stretchy="false" xref="S1.Thmtheorem10.p1.8.8.m8.4.4.4.2.3.1.cmml">)</mo></mrow></mrow><mo id="S1.Thmtheorem10.p1.8.8.m8.4.4.4.1" xref="S1.Thmtheorem10.p1.8.8.m8.4.4.4.1.cmml">∩</mo><mi id="S1.Thmtheorem10.p1.8.8.m8.4.4.4.3" xref="S1.Thmtheorem10.p1.8.8.m8.4.4.4.3.cmml">E</mi></mrow><mo id="S1.Thmtheorem10.p1.8.8.m8.4.4.3" xref="S1.Thmtheorem10.p1.8.8.m8.4.4.3.cmml">⊆</mo><mrow id="S1.Thmtheorem10.p1.8.8.m8.4.4.2" xref="S1.Thmtheorem10.p1.8.8.m8.4.4.2.cmml"><mover accent="true" id="S1.Thmtheorem10.p1.8.8.m8.4.4.2.4" xref="S1.Thmtheorem10.p1.8.8.m8.4.4.2.4.cmml"><mi class="ltx_font_mathscript" id="S1.Thmtheorem10.p1.8.8.m8.4.4.2.4.2" xref="S1.Thmtheorem10.p1.8.8.m8.4.4.2.4.2.cmml">ℛ</mi><mo id="S1.Thmtheorem10.p1.8.8.m8.4.4.2.4.1" xref="S1.Thmtheorem10.p1.8.8.m8.4.4.2.4.1.cmml">^</mo></mover><mo id="S1.Thmtheorem10.p1.8.8.m8.4.4.2.3" xref="S1.Thmtheorem10.p1.8.8.m8.4.4.2.3.cmml">⁢</mo><mrow id="S1.Thmtheorem10.p1.8.8.m8.4.4.2.2.2" xref="S1.Thmtheorem10.p1.8.8.m8.4.4.2.2.3.cmml"><mo id="S1.Thmtheorem10.p1.8.8.m8.4.4.2.2.2.3" stretchy="false" xref="S1.Thmtheorem10.p1.8.8.m8.4.4.2.2.3.cmml">(</mo><msup id="S1.Thmtheorem10.p1.8.8.m8.3.3.1.1.1.1" xref="S1.Thmtheorem10.p1.8.8.m8.3.3.1.1.1.1.cmml"><mi id="S1.Thmtheorem10.p1.8.8.m8.3.3.1.1.1.1.2" xref="S1.Thmtheorem10.p1.8.8.m8.3.3.1.1.1.1.2.cmml">C</mi><mo id="S1.Thmtheorem10.p1.8.8.m8.3.3.1.1.1.1.3" xref="S1.Thmtheorem10.p1.8.8.m8.3.3.1.1.1.1.3.cmml">′</mo></msup><mo id="S1.Thmtheorem10.p1.8.8.m8.4.4.2.2.2.4" xref="S1.Thmtheorem10.p1.8.8.m8.4.4.2.2.3.cmml">,</mo><msup id="S1.Thmtheorem10.p1.8.8.m8.4.4.2.2.2.2" xref="S1.Thmtheorem10.p1.8.8.m8.4.4.2.2.2.2.cmml"><mi id="S1.Thmtheorem10.p1.8.8.m8.4.4.2.2.2.2.2" xref="S1.Thmtheorem10.p1.8.8.m8.4.4.2.2.2.2.2.cmml">D</mi><mo id="S1.Thmtheorem10.p1.8.8.m8.4.4.2.2.2.2.3" xref="S1.Thmtheorem10.p1.8.8.m8.4.4.2.2.2.2.3.cmml">′</mo></msup><mo id="S1.Thmtheorem10.p1.8.8.m8.4.4.2.2.2.5" stretchy="false" xref="S1.Thmtheorem10.p1.8.8.m8.4.4.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem10.p1.8.8.m8.4b"><apply id="S1.Thmtheorem10.p1.8.8.m8.4.4.cmml" xref="S1.Thmtheorem10.p1.8.8.m8.4.4"><subset id="S1.Thmtheorem10.p1.8.8.m8.4.4.3.cmml" xref="S1.Thmtheorem10.p1.8.8.m8.4.4.3"></subset><apply id="S1.Thmtheorem10.p1.8.8.m8.4.4.4.cmml" xref="S1.Thmtheorem10.p1.8.8.m8.4.4.4"><intersect id="S1.Thmtheorem10.p1.8.8.m8.4.4.4.1.cmml" xref="S1.Thmtheorem10.p1.8.8.m8.4.4.4.1"></intersect><apply id="S1.Thmtheorem10.p1.8.8.m8.4.4.4.2.cmml" xref="S1.Thmtheorem10.p1.8.8.m8.4.4.4.2"><times id="S1.Thmtheorem10.p1.8.8.m8.4.4.4.2.1.cmml" xref="S1.Thmtheorem10.p1.8.8.m8.4.4.4.2.1"></times><ci id="S1.Thmtheorem10.p1.8.8.m8.4.4.4.2.2.cmml" xref="S1.Thmtheorem10.p1.8.8.m8.4.4.4.2.2">ℛ</ci><interval closure="open" id="S1.Thmtheorem10.p1.8.8.m8.4.4.4.2.3.1.cmml" xref="S1.Thmtheorem10.p1.8.8.m8.4.4.4.2.3.2"><ci id="S1.Thmtheorem10.p1.8.8.m8.1.1.cmml" xref="S1.Thmtheorem10.p1.8.8.m8.1.1">𝐶</ci><ci id="S1.Thmtheorem10.p1.8.8.m8.2.2.cmml" xref="S1.Thmtheorem10.p1.8.8.m8.2.2">𝐷</ci></interval></apply><ci id="S1.Thmtheorem10.p1.8.8.m8.4.4.4.3.cmml" xref="S1.Thmtheorem10.p1.8.8.m8.4.4.4.3">𝐸</ci></apply><apply id="S1.Thmtheorem10.p1.8.8.m8.4.4.2.cmml" xref="S1.Thmtheorem10.p1.8.8.m8.4.4.2"><times id="S1.Thmtheorem10.p1.8.8.m8.4.4.2.3.cmml" xref="S1.Thmtheorem10.p1.8.8.m8.4.4.2.3"></times><apply id="S1.Thmtheorem10.p1.8.8.m8.4.4.2.4.cmml" xref="S1.Thmtheorem10.p1.8.8.m8.4.4.2.4"><ci id="S1.Thmtheorem10.p1.8.8.m8.4.4.2.4.1.cmml" xref="S1.Thmtheorem10.p1.8.8.m8.4.4.2.4.1">^</ci><ci id="S1.Thmtheorem10.p1.8.8.m8.4.4.2.4.2.cmml" xref="S1.Thmtheorem10.p1.8.8.m8.4.4.2.4.2">ℛ</ci></apply><interval closure="open" id="S1.Thmtheorem10.p1.8.8.m8.4.4.2.2.3.cmml" xref="S1.Thmtheorem10.p1.8.8.m8.4.4.2.2.2"><apply id="S1.Thmtheorem10.p1.8.8.m8.3.3.1.1.1.1.cmml" xref="S1.Thmtheorem10.p1.8.8.m8.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S1.Thmtheorem10.p1.8.8.m8.3.3.1.1.1.1.1.cmml" xref="S1.Thmtheorem10.p1.8.8.m8.3.3.1.1.1.1">superscript</csymbol><ci id="S1.Thmtheorem10.p1.8.8.m8.3.3.1.1.1.1.2.cmml" xref="S1.Thmtheorem10.p1.8.8.m8.3.3.1.1.1.1.2">𝐶</ci><ci id="S1.Thmtheorem10.p1.8.8.m8.3.3.1.1.1.1.3.cmml" xref="S1.Thmtheorem10.p1.8.8.m8.3.3.1.1.1.1.3">′</ci></apply><apply id="S1.Thmtheorem10.p1.8.8.m8.4.4.2.2.2.2.cmml" xref="S1.Thmtheorem10.p1.8.8.m8.4.4.2.2.2.2"><csymbol cd="ambiguous" id="S1.Thmtheorem10.p1.8.8.m8.4.4.2.2.2.2.1.cmml" xref="S1.Thmtheorem10.p1.8.8.m8.4.4.2.2.2.2">superscript</csymbol><ci id="S1.Thmtheorem10.p1.8.8.m8.4.4.2.2.2.2.2.cmml" xref="S1.Thmtheorem10.p1.8.8.m8.4.4.2.2.2.2.2">𝐷</ci><ci id="S1.Thmtheorem10.p1.8.8.m8.4.4.2.2.2.2.3.cmml" xref="S1.Thmtheorem10.p1.8.8.m8.4.4.2.2.2.2.3">′</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem10.p1.8.8.m8.4c">\mathscr{R}(C,D)\cap E\subseteq\hat{\mathscr{R}}(C^{\prime},D^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem10.p1.8.8.m8.4d">script_R ( italic_C , italic_D ) ∩ italic_E ⊆ over^ start_ARG script_R end_ARG ( italic_C start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math>, then <math alttext="C\trianglerighteq C^{\prime}" class="ltx_Math" display="inline" id="S1.Thmtheorem10.p1.9.9.m9.1"><semantics id="S1.Thmtheorem10.p1.9.9.m9.1a"><mrow id="S1.Thmtheorem10.p1.9.9.m9.1.1" xref="S1.Thmtheorem10.p1.9.9.m9.1.1.cmml"><mi id="S1.Thmtheorem10.p1.9.9.m9.1.1.2" xref="S1.Thmtheorem10.p1.9.9.m9.1.1.2.cmml">C</mi><mo id="S1.Thmtheorem10.p1.9.9.m9.1.1.1" xref="S1.Thmtheorem10.p1.9.9.m9.1.1.1.cmml">⁢</mo><mi id="S1.Thmtheorem10.p1.9.9.m9.1.1.3" mathvariant="normal" xref="S1.Thmtheorem10.p1.9.9.m9.1.1.3.cmml">⊵</mi><mo id="S1.Thmtheorem10.p1.9.9.m9.1.1.1a" xref="S1.Thmtheorem10.p1.9.9.m9.1.1.1.cmml">⁢</mo><msup id="S1.Thmtheorem10.p1.9.9.m9.1.1.4" xref="S1.Thmtheorem10.p1.9.9.m9.1.1.4.cmml"><mi id="S1.Thmtheorem10.p1.9.9.m9.1.1.4.2" xref="S1.Thmtheorem10.p1.9.9.m9.1.1.4.2.cmml">C</mi><mo id="S1.Thmtheorem10.p1.9.9.m9.1.1.4.3" xref="S1.Thmtheorem10.p1.9.9.m9.1.1.4.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem10.p1.9.9.m9.1b"><apply id="S1.Thmtheorem10.p1.9.9.m9.1.1.cmml" xref="S1.Thmtheorem10.p1.9.9.m9.1.1"><times id="S1.Thmtheorem10.p1.9.9.m9.1.1.1.cmml" xref="S1.Thmtheorem10.p1.9.9.m9.1.1.1"></times><ci id="S1.Thmtheorem10.p1.9.9.m9.1.1.2.cmml" xref="S1.Thmtheorem10.p1.9.9.m9.1.1.2">𝐶</ci><ci id="S1.Thmtheorem10.p1.9.9.m9.1.1.3.cmml" xref="S1.Thmtheorem10.p1.9.9.m9.1.1.3">⊵</ci><apply id="S1.Thmtheorem10.p1.9.9.m9.1.1.4.cmml" xref="S1.Thmtheorem10.p1.9.9.m9.1.1.4"><csymbol cd="ambiguous" id="S1.Thmtheorem10.p1.9.9.m9.1.1.4.1.cmml" xref="S1.Thmtheorem10.p1.9.9.m9.1.1.4">superscript</csymbol><ci id="S1.Thmtheorem10.p1.9.9.m9.1.1.4.2.cmml" xref="S1.Thmtheorem10.p1.9.9.m9.1.1.4.2">𝐶</ci><ci id="S1.Thmtheorem10.p1.9.9.m9.1.1.4.3.cmml" xref="S1.Thmtheorem10.p1.9.9.m9.1.1.4.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem10.p1.9.9.m9.1c">C\trianglerighteq C^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem10.p1.9.9.m9.1d">italic_C ⊵ italic_C start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S1.SSx2.p6"> <p class="ltx_p" id="S1.SSx2.p6.2">To proof this we construct a ccc forcing notion, based in Moore’s forcing in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib16" title="">16</a>]</cite>, which under the correct hypotheses introduces an epimorphism from <math alttext="C" class="ltx_Math" display="inline" id="S1.SSx2.p6.1.m1.1"><semantics id="S1.SSx2.p6.1.m1.1a"><mi id="S1.SSx2.p6.1.m1.1.1" xref="S1.SSx2.p6.1.m1.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S1.SSx2.p6.1.m1.1b"><ci id="S1.SSx2.p6.1.m1.1.1.cmml" xref="S1.SSx2.p6.1.m1.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx2.p6.1.m1.1c">C</annotation><annotation encoding="application/x-llamapun" id="S1.SSx2.p6.1.m1.1d">italic_C</annotation></semantics></math> onto <math alttext="C^{\prime}" class="ltx_Math" display="inline" id="S1.SSx2.p6.2.m2.1"><semantics id="S1.SSx2.p6.2.m2.1a"><msup id="S1.SSx2.p6.2.m2.1.1" xref="S1.SSx2.p6.2.m2.1.1.cmml"><mi id="S1.SSx2.p6.2.m2.1.1.2" xref="S1.SSx2.p6.2.m2.1.1.2.cmml">C</mi><mo id="S1.SSx2.p6.2.m2.1.1.3" xref="S1.SSx2.p6.2.m2.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S1.SSx2.p6.2.m2.1b"><apply id="S1.SSx2.p6.2.m2.1.1.cmml" xref="S1.SSx2.p6.2.m2.1.1"><csymbol cd="ambiguous" id="S1.SSx2.p6.2.m2.1.1.1.cmml" xref="S1.SSx2.p6.2.m2.1.1">superscript</csymbol><ci id="S1.SSx2.p6.2.m2.1.1.2.cmml" xref="S1.SSx2.p6.2.m2.1.1.2">𝐶</ci><ci id="S1.SSx2.p6.2.m2.1.1.3.cmml" xref="S1.SSx2.p6.2.m2.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx2.p6.2.m2.1c">C^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx2.p6.2.m2.1d">italic_C start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S1.SSx2.p7"> <p class="ltx_p" id="S1.SSx2.p7.5">This theorem allows to do three things. First, it gives an alternative proof that any <math alttext="\aleph_{1}" class="ltx_Math" display="inline" id="S1.SSx2.p7.1.m1.1"><semantics id="S1.SSx2.p7.1.m1.1a"><msub id="S1.SSx2.p7.1.m1.1.1" xref="S1.SSx2.p7.1.m1.1.1.cmml"><mi id="S1.SSx2.p7.1.m1.1.1.2" mathvariant="normal" xref="S1.SSx2.p7.1.m1.1.1.2.cmml">ℵ</mi><mn id="S1.SSx2.p7.1.m1.1.1.3" xref="S1.SSx2.p7.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S1.SSx2.p7.1.m1.1b"><apply id="S1.SSx2.p7.1.m1.1.1.cmml" xref="S1.SSx2.p7.1.m1.1.1"><csymbol cd="ambiguous" id="S1.SSx2.p7.1.m1.1.1.1.cmml" xref="S1.SSx2.p7.1.m1.1.1">subscript</csymbol><ci id="S1.SSx2.p7.1.m1.1.1.2.cmml" xref="S1.SSx2.p7.1.m1.1.1.2">ℵ</ci><cn id="S1.SSx2.p7.1.m1.1.1.3.cmml" type="integer" xref="S1.SSx2.p7.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx2.p7.1.m1.1c">\aleph_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx2.p7.1.m1.1d">roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-dense and non stationary Countryman line is strongly surjective under <math alttext="\mathsf{MA}_{\aleph_{1}}" class="ltx_Math" display="inline" id="S1.SSx2.p7.2.m2.1"><semantics id="S1.SSx2.p7.2.m2.1a"><msub id="S1.SSx2.p7.2.m2.1.1" xref="S1.SSx2.p7.2.m2.1.1.cmml"><mi id="S1.SSx2.p7.2.m2.1.1.2" xref="S1.SSx2.p7.2.m2.1.1.2.cmml">𝖬𝖠</mi><msub id="S1.SSx2.p7.2.m2.1.1.3" xref="S1.SSx2.p7.2.m2.1.1.3.cmml"><mi id="S1.SSx2.p7.2.m2.1.1.3.2" mathvariant="normal" xref="S1.SSx2.p7.2.m2.1.1.3.2.cmml">ℵ</mi><mn id="S1.SSx2.p7.2.m2.1.1.3.3" xref="S1.SSx2.p7.2.m2.1.1.3.3.cmml">1</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S1.SSx2.p7.2.m2.1b"><apply id="S1.SSx2.p7.2.m2.1.1.cmml" xref="S1.SSx2.p7.2.m2.1.1"><csymbol cd="ambiguous" id="S1.SSx2.p7.2.m2.1.1.1.cmml" xref="S1.SSx2.p7.2.m2.1.1">subscript</csymbol><ci id="S1.SSx2.p7.2.m2.1.1.2.cmml" xref="S1.SSx2.p7.2.m2.1.1.2">𝖬𝖠</ci><apply id="S1.SSx2.p7.2.m2.1.1.3.cmml" xref="S1.SSx2.p7.2.m2.1.1.3"><csymbol cd="ambiguous" id="S1.SSx2.p7.2.m2.1.1.3.1.cmml" xref="S1.SSx2.p7.2.m2.1.1.3">subscript</csymbol><ci id="S1.SSx2.p7.2.m2.1.1.3.2.cmml" xref="S1.SSx2.p7.2.m2.1.1.3.2">ℵ</ci><cn id="S1.SSx2.p7.2.m2.1.1.3.3.cmml" type="integer" xref="S1.SSx2.p7.2.m2.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx2.p7.2.m2.1c">\mathsf{MA}_{\aleph_{1}}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx2.p7.2.m2.1d">sansserif_MA start_POSTSUBSCRIPT roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. It also allow us to create an infinite <math alttext="\vartriangleleft" class="ltx_Math" display="inline" id="S1.SSx2.p7.3.m3.1"><semantics id="S1.SSx2.p7.3.m3.1a"><mi id="S1.SSx2.p7.3.m3.1.1" mathvariant="normal" xref="S1.SSx2.p7.3.m3.1.1.cmml">⊲</mi><annotation-xml encoding="MathML-Content" id="S1.SSx2.p7.3.m3.1b"><ci id="S1.SSx2.p7.3.m3.1.1.cmml" xref="S1.SSx2.p7.3.m3.1.1">⊲</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx2.p7.3.m3.1c">\vartriangleleft</annotation><annotation encoding="application/x-llamapun" id="S1.SSx2.p7.3.m3.1d">⊲</annotation></semantics></math>-decreasing chain of Countryman lines, thus giving an alternative answer to <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#Thmquestion2" title="Question 2. ‣ Historical and mathematical context ‣ 1. Introduction ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">2</span></a>. Finally, in conjunction with a theorem of Moore, it allow us to proof that the class of Aronszajn lines has a two element <math alttext="\trianglelefteq" class="ltx_Math" display="inline" id="S1.SSx2.p7.4.m4.1"><semantics id="S1.SSx2.p7.4.m4.1a"><mi id="S1.SSx2.p7.4.m4.1.1" mathvariant="normal" xref="S1.SSx2.p7.4.m4.1.1.cmml">⊴</mi><annotation-xml encoding="MathML-Content" id="S1.SSx2.p7.4.m4.1b"><ci id="S1.SSx2.p7.4.m4.1.1.cmml" xref="S1.SSx2.p7.4.m4.1.1">⊴</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx2.p7.4.m4.1c">\trianglelefteq</annotation><annotation encoding="application/x-llamapun" id="S1.SSx2.p7.4.m4.1d">⊴</annotation></semantics></math>-basis under <math alttext="\mathsf{PFA}" class="ltx_Math" display="inline" id="S1.SSx2.p7.5.m5.1"><semantics id="S1.SSx2.p7.5.m5.1a"><mi id="S1.SSx2.p7.5.m5.1.1" xref="S1.SSx2.p7.5.m5.1.1.cmml">𝖯𝖥𝖠</mi><annotation-xml encoding="MathML-Content" id="S1.SSx2.p7.5.m5.1b"><ci id="S1.SSx2.p7.5.m5.1.1.cmml" xref="S1.SSx2.p7.5.m5.1.1">𝖯𝖥𝖠</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx2.p7.5.m5.1c">\mathsf{PFA}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx2.p7.5.m5.1d">sansserif_PFA</annotation></semantics></math>.</p> </div> </section> <section class="ltx_subsection" id="S1.SSx3"> <h3 class="ltx_title ltx_title_subsection">Organization</h3> <div class="ltx_para" id="S1.SSx3.p1"> <p class="ltx_p" id="S1.SSx3.p1.13">In <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S2" title="2. Aronszajn and Countryman lines ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">2</span></a> we develop the mathematical background that is needed later. In <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S3" title="3. Strongly surjective Aronszajn lines ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">3</span></a> we show that under <math alttext="\mathsf{MA}_{\aleph_{1}}" class="ltx_Math" display="inline" id="S1.SSx3.p1.1.m1.1"><semantics id="S1.SSx3.p1.1.m1.1a"><msub id="S1.SSx3.p1.1.m1.1.1" xref="S1.SSx3.p1.1.m1.1.1.cmml"><mi id="S1.SSx3.p1.1.m1.1.1.2" xref="S1.SSx3.p1.1.m1.1.1.2.cmml">𝖬𝖠</mi><msub id="S1.SSx3.p1.1.m1.1.1.3" xref="S1.SSx3.p1.1.m1.1.1.3.cmml"><mi id="S1.SSx3.p1.1.m1.1.1.3.2" mathvariant="normal" xref="S1.SSx3.p1.1.m1.1.1.3.2.cmml">ℵ</mi><mn id="S1.SSx3.p1.1.m1.1.1.3.3" xref="S1.SSx3.p1.1.m1.1.1.3.3.cmml">1</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S1.SSx3.p1.1.m1.1b"><apply id="S1.SSx3.p1.1.m1.1.1.cmml" xref="S1.SSx3.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S1.SSx3.p1.1.m1.1.1.1.cmml" xref="S1.SSx3.p1.1.m1.1.1">subscript</csymbol><ci id="S1.SSx3.p1.1.m1.1.1.2.cmml" xref="S1.SSx3.p1.1.m1.1.1.2">𝖬𝖠</ci><apply id="S1.SSx3.p1.1.m1.1.1.3.cmml" xref="S1.SSx3.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S1.SSx3.p1.1.m1.1.1.3.1.cmml" xref="S1.SSx3.p1.1.m1.1.1.3">subscript</csymbol><ci id="S1.SSx3.p1.1.m1.1.1.3.2.cmml" xref="S1.SSx3.p1.1.m1.1.1.3.2">ℵ</ci><cn id="S1.SSx3.p1.1.m1.1.1.3.3.cmml" type="integer" xref="S1.SSx3.p1.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx3.p1.1.m1.1c">\mathsf{MA}_{\aleph_{1}}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx3.p1.1.m1.1d">sansserif_MA start_POSTSUBSCRIPT roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> there are many strongly surjective Aronszajn lines. In particular we prove that <math alttext="\mathsf{MA}_{\aleph_{1}}" class="ltx_Math" display="inline" id="S1.SSx3.p1.2.m2.1"><semantics id="S1.SSx3.p1.2.m2.1a"><msub id="S1.SSx3.p1.2.m2.1.1" xref="S1.SSx3.p1.2.m2.1.1.cmml"><mi id="S1.SSx3.p1.2.m2.1.1.2" xref="S1.SSx3.p1.2.m2.1.1.2.cmml">𝖬𝖠</mi><msub id="S1.SSx3.p1.2.m2.1.1.3" xref="S1.SSx3.p1.2.m2.1.1.3.cmml"><mi id="S1.SSx3.p1.2.m2.1.1.3.2" mathvariant="normal" xref="S1.SSx3.p1.2.m2.1.1.3.2.cmml">ℵ</mi><mn id="S1.SSx3.p1.2.m2.1.1.3.3" xref="S1.SSx3.p1.2.m2.1.1.3.3.cmml">1</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S1.SSx3.p1.2.m2.1b"><apply id="S1.SSx3.p1.2.m2.1.1.cmml" xref="S1.SSx3.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S1.SSx3.p1.2.m2.1.1.1.cmml" xref="S1.SSx3.p1.2.m2.1.1">subscript</csymbol><ci id="S1.SSx3.p1.2.m2.1.1.2.cmml" xref="S1.SSx3.p1.2.m2.1.1.2">𝖬𝖠</ci><apply id="S1.SSx3.p1.2.m2.1.1.3.cmml" xref="S1.SSx3.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S1.SSx3.p1.2.m2.1.1.3.1.cmml" xref="S1.SSx3.p1.2.m2.1.1.3">subscript</csymbol><ci id="S1.SSx3.p1.2.m2.1.1.3.2.cmml" xref="S1.SSx3.p1.2.m2.1.1.3.2">ℵ</ci><cn id="S1.SSx3.p1.2.m2.1.1.3.3.cmml" type="integer" xref="S1.SSx3.p1.2.m2.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx3.p1.2.m2.1c">\mathsf{MA}_{\aleph_{1}}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx3.p1.2.m2.1d">sansserif_MA start_POSTSUBSCRIPT roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> implies the existence of a strongly surjective Countryman line. In <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S4" title="4. Aronszajn line decompositions ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">4</span></a> we develop the concept of decompositions for Aronszajn lines, and construct Countryman lines with particular configurations of <math alttext="\mathscr{L}" class="ltx_Math" display="inline" id="S1.SSx3.p1.3.m3.1"><semantics id="S1.SSx3.p1.3.m3.1a"><mi class="ltx_font_mathscript" id="S1.SSx3.p1.3.m3.1.1" xref="S1.SSx3.p1.3.m3.1.1.cmml">ℒ</mi><annotation-xml encoding="MathML-Content" id="S1.SSx3.p1.3.m3.1b"><ci id="S1.SSx3.p1.3.m3.1.1.cmml" xref="S1.SSx3.p1.3.m3.1.1">ℒ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx3.p1.3.m3.1c">\mathscr{L}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx3.p1.3.m3.1d">script_L</annotation></semantics></math> and <math alttext="\mathscr{R}" class="ltx_Math" display="inline" id="S1.SSx3.p1.4.m4.1"><semantics id="S1.SSx3.p1.4.m4.1a"><mi class="ltx_font_mathscript" id="S1.SSx3.p1.4.m4.1.1" xref="S1.SSx3.p1.4.m4.1.1.cmml">ℛ</mi><annotation-xml encoding="MathML-Content" id="S1.SSx3.p1.4.m4.1b"><ci id="S1.SSx3.p1.4.m4.1.1.cmml" xref="S1.SSx3.p1.4.m4.1.1">ℛ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx3.p1.4.m4.1c">\mathscr{R}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx3.p1.4.m4.1d">script_R</annotation></semantics></math>. In <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S5" title="5. An infinite antichain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">5</span></a> construct an infinite <math alttext="\trianglelefteq" class="ltx_Math" display="inline" id="S1.SSx3.p1.5.m5.1"><semantics id="S1.SSx3.p1.5.m5.1a"><mi id="S1.SSx3.p1.5.m5.1.1" mathvariant="normal" xref="S1.SSx3.p1.5.m5.1.1.cmml">⊴</mi><annotation-xml encoding="MathML-Content" id="S1.SSx3.p1.5.m5.1b"><ci id="S1.SSx3.p1.5.m5.1.1.cmml" xref="S1.SSx3.p1.5.m5.1.1">⊴</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx3.p1.5.m5.1c">\trianglelefteq</annotation><annotation encoding="application/x-llamapun" id="S1.SSx3.p1.5.m5.1d">⊴</annotation></semantics></math>-antichain of Countryman lines under <math alttext="\mathsf{ZFC}" class="ltx_Math" display="inline" id="S1.SSx3.p1.6.m6.1"><semantics id="S1.SSx3.p1.6.m6.1a"><mi id="S1.SSx3.p1.6.m6.1.1" xref="S1.SSx3.p1.6.m6.1.1.cmml">𝖹𝖥𝖢</mi><annotation-xml encoding="MathML-Content" id="S1.SSx3.p1.6.m6.1b"><ci id="S1.SSx3.p1.6.m6.1.1.cmml" xref="S1.SSx3.p1.6.m6.1.1">𝖹𝖥𝖢</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx3.p1.6.m6.1c">\mathsf{ZFC}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx3.p1.6.m6.1d">sansserif_ZFC</annotation></semantics></math>. In <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6" title="6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">6</span></a> In we construct an infinite <math alttext="\trianglelefteq" class="ltx_Math" display="inline" id="S1.SSx3.p1.7.m7.1"><semantics id="S1.SSx3.p1.7.m7.1a"><mi id="S1.SSx3.p1.7.m7.1.1" mathvariant="normal" xref="S1.SSx3.p1.7.m7.1.1.cmml">⊴</mi><annotation-xml encoding="MathML-Content" id="S1.SSx3.p1.7.m7.1b"><ci id="S1.SSx3.p1.7.m7.1.1.cmml" xref="S1.SSx3.p1.7.m7.1.1">⊴</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx3.p1.7.m7.1c">\trianglelefteq</annotation><annotation encoding="application/x-llamapun" id="S1.SSx3.p1.7.m7.1d">⊴</annotation></semantics></math>-decreasing chain of Countryman lines under <math alttext="\mathsf{MA}_{\aleph_{1}}" class="ltx_Math" display="inline" id="S1.SSx3.p1.8.m8.1"><semantics id="S1.SSx3.p1.8.m8.1a"><msub id="S1.SSx3.p1.8.m8.1.1" xref="S1.SSx3.p1.8.m8.1.1.cmml"><mi id="S1.SSx3.p1.8.m8.1.1.2" xref="S1.SSx3.p1.8.m8.1.1.2.cmml">𝖬𝖠</mi><msub id="S1.SSx3.p1.8.m8.1.1.3" xref="S1.SSx3.p1.8.m8.1.1.3.cmml"><mi id="S1.SSx3.p1.8.m8.1.1.3.2" mathvariant="normal" xref="S1.SSx3.p1.8.m8.1.1.3.2.cmml">ℵ</mi><mn id="S1.SSx3.p1.8.m8.1.1.3.3" xref="S1.SSx3.p1.8.m8.1.1.3.3.cmml">1</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S1.SSx3.p1.8.m8.1b"><apply id="S1.SSx3.p1.8.m8.1.1.cmml" xref="S1.SSx3.p1.8.m8.1.1"><csymbol cd="ambiguous" id="S1.SSx3.p1.8.m8.1.1.1.cmml" xref="S1.SSx3.p1.8.m8.1.1">subscript</csymbol><ci id="S1.SSx3.p1.8.m8.1.1.2.cmml" xref="S1.SSx3.p1.8.m8.1.1.2">𝖬𝖠</ci><apply id="S1.SSx3.p1.8.m8.1.1.3.cmml" xref="S1.SSx3.p1.8.m8.1.1.3"><csymbol cd="ambiguous" id="S1.SSx3.p1.8.m8.1.1.3.1.cmml" xref="S1.SSx3.p1.8.m8.1.1.3">subscript</csymbol><ci id="S1.SSx3.p1.8.m8.1.1.3.2.cmml" xref="S1.SSx3.p1.8.m8.1.1.3.2">ℵ</ci><cn id="S1.SSx3.p1.8.m8.1.1.3.3.cmml" type="integer" xref="S1.SSx3.p1.8.m8.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx3.p1.8.m8.1c">\mathsf{MA}_{\aleph_{1}}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx3.p1.8.m8.1d">sansserif_MA start_POSTSUBSCRIPT roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. It is in this section that we present our main Forcing. In <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S7" title="7. A two element basis for the Aronszajn lines ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">7</span></a> we apply the forcing to show that under <math alttext="\mathsf{PFA}" class="ltx_Math" display="inline" id="S1.SSx3.p1.9.m9.1"><semantics id="S1.SSx3.p1.9.m9.1a"><mi id="S1.SSx3.p1.9.m9.1.1" xref="S1.SSx3.p1.9.m9.1.1.cmml">𝖯𝖥𝖠</mi><annotation-xml encoding="MathML-Content" id="S1.SSx3.p1.9.m9.1b"><ci id="S1.SSx3.p1.9.m9.1.1.cmml" xref="S1.SSx3.p1.9.m9.1.1">𝖯𝖥𝖠</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx3.p1.9.m9.1c">\mathsf{PFA}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx3.p1.9.m9.1d">sansserif_PFA</annotation></semantics></math>, the class of Aronszajn lines has a two element <math alttext="\trianglelefteq" class="ltx_Math" display="inline" id="S1.SSx3.p1.10.m10.1"><semantics id="S1.SSx3.p1.10.m10.1a"><mi id="S1.SSx3.p1.10.m10.1.1" mathvariant="normal" xref="S1.SSx3.p1.10.m10.1.1.cmml">⊴</mi><annotation-xml encoding="MathML-Content" id="S1.SSx3.p1.10.m10.1b"><ci id="S1.SSx3.p1.10.m10.1.1.cmml" xref="S1.SSx3.p1.10.m10.1.1">⊴</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx3.p1.10.m10.1c">\trianglelefteq</annotation><annotation encoding="application/x-llamapun" id="S1.SSx3.p1.10.m10.1d">⊴</annotation></semantics></math>-bassis. We also show that this cannot be extended to all uncountable linear orders by proving that (in <math alttext="\mathsf{ZFC}" class="ltx_Math" display="inline" id="S1.SSx3.p1.11.m11.1"><semantics id="S1.SSx3.p1.11.m11.1a"><mi id="S1.SSx3.p1.11.m11.1.1" xref="S1.SSx3.p1.11.m11.1.1.cmml">𝖹𝖥𝖢</mi><annotation-xml encoding="MathML-Content" id="S1.SSx3.p1.11.m11.1b"><ci id="S1.SSx3.p1.11.m11.1.1.cmml" xref="S1.SSx3.p1.11.m11.1.1">𝖹𝖥𝖢</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx3.p1.11.m11.1c">\mathsf{ZFC}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx3.p1.11.m11.1d">sansserif_ZFC</annotation></semantics></math>) there cannot be a finite <math alttext="\trianglelefteq" class="ltx_Math" display="inline" id="S1.SSx3.p1.12.m12.1"><semantics id="S1.SSx3.p1.12.m12.1a"><mi id="S1.SSx3.p1.12.m12.1.1" mathvariant="normal" xref="S1.SSx3.p1.12.m12.1.1.cmml">⊴</mi><annotation-xml encoding="MathML-Content" id="S1.SSx3.p1.12.m12.1b"><ci id="S1.SSx3.p1.12.m12.1.1.cmml" xref="S1.SSx3.p1.12.m12.1.1">⊴</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx3.p1.12.m12.1c">\trianglelefteq</annotation><annotation encoding="application/x-llamapun" id="S1.SSx3.p1.12.m12.1d">⊴</annotation></semantics></math>-basis for the uncountable real orders. In <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S8" title="8. On a question on Countryman lines ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">8</span></a> we take a detour from the <math alttext="\trianglelefteq" class="ltx_Math" display="inline" id="S1.SSx3.p1.13.m13.1"><semantics id="S1.SSx3.p1.13.m13.1a"><mi id="S1.SSx3.p1.13.m13.1.1" mathvariant="normal" xref="S1.SSx3.p1.13.m13.1.1.cmml">⊴</mi><annotation-xml encoding="MathML-Content" id="S1.SSx3.p1.13.m13.1b"><ci id="S1.SSx3.p1.13.m13.1.1.cmml" xref="S1.SSx3.p1.13.m13.1.1">⊴</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx3.p1.13.m13.1c">\trianglelefteq</annotation><annotation encoding="application/x-llamapun" id="S1.SSx3.p1.13.m13.1d">⊴</annotation></semantics></math> relation to answer a question on Countryman lines that seemed natural to us, and have not seen in the literature.</p> </div> </section> <section class="ltx_subsection" id="S1.SSx4"> <h3 class="ltx_title ltx_title_subsection">Notation and prerequisites</h3> <div class="ltx_para" id="S1.SSx4.p1"> <p class="ltx_p" id="S1.SSx4.p1.3">We assume familiarity with standard terminology and notations of Set Theory. We refer to <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib8" title="">8</a>]</cite> and <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib10" title="">10</a>]</cite> for undefined notions. Particularly relevant is the notion of trees and their interplay with linear orders, the material in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib21" title="">21</a>]</cite> should prove helpful for a deeper understating. Regarding set theoretic hypotheses, we will deal primarily with <math alttext="\mathsf{MA}_{\aleph_{1}}" class="ltx_Math" display="inline" id="S1.SSx4.p1.1.m1.1"><semantics id="S1.SSx4.p1.1.m1.1a"><msub id="S1.SSx4.p1.1.m1.1.1" xref="S1.SSx4.p1.1.m1.1.1.cmml"><mi id="S1.SSx4.p1.1.m1.1.1.2" xref="S1.SSx4.p1.1.m1.1.1.2.cmml">𝖬𝖠</mi><msub id="S1.SSx4.p1.1.m1.1.1.3" xref="S1.SSx4.p1.1.m1.1.1.3.cmml"><mi id="S1.SSx4.p1.1.m1.1.1.3.2" mathvariant="normal" xref="S1.SSx4.p1.1.m1.1.1.3.2.cmml">ℵ</mi><mn id="S1.SSx4.p1.1.m1.1.1.3.3" xref="S1.SSx4.p1.1.m1.1.1.3.3.cmml">1</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S1.SSx4.p1.1.m1.1b"><apply id="S1.SSx4.p1.1.m1.1.1.cmml" xref="S1.SSx4.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S1.SSx4.p1.1.m1.1.1.1.cmml" xref="S1.SSx4.p1.1.m1.1.1">subscript</csymbol><ci id="S1.SSx4.p1.1.m1.1.1.2.cmml" xref="S1.SSx4.p1.1.m1.1.1.2">𝖬𝖠</ci><apply id="S1.SSx4.p1.1.m1.1.1.3.cmml" xref="S1.SSx4.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S1.SSx4.p1.1.m1.1.1.3.1.cmml" xref="S1.SSx4.p1.1.m1.1.1.3">subscript</csymbol><ci id="S1.SSx4.p1.1.m1.1.1.3.2.cmml" xref="S1.SSx4.p1.1.m1.1.1.3.2">ℵ</ci><cn id="S1.SSx4.p1.1.m1.1.1.3.3.cmml" type="integer" xref="S1.SSx4.p1.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx4.p1.1.m1.1c">\mathsf{MA}_{\aleph_{1}}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx4.p1.1.m1.1d">sansserif_MA start_POSTSUBSCRIPT roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\mathsf{PFA}" class="ltx_Math" display="inline" id="S1.SSx4.p1.2.m2.1"><semantics id="S1.SSx4.p1.2.m2.1a"><mi id="S1.SSx4.p1.2.m2.1.1" xref="S1.SSx4.p1.2.m2.1.1.cmml">𝖯𝖥𝖠</mi><annotation-xml encoding="MathML-Content" id="S1.SSx4.p1.2.m2.1b"><ci id="S1.SSx4.p1.2.m2.1.1.cmml" xref="S1.SSx4.p1.2.m2.1.1">𝖯𝖥𝖠</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx4.p1.2.m2.1c">\mathsf{PFA}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx4.p1.2.m2.1d">sansserif_PFA</annotation></semantics></math>, but with the exception of <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6" title="6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">6</span></a> in which knowledge of the statement of <math alttext="\mathsf{MA}_{\aleph_{1}}" class="ltx_Math" display="inline" id="S1.SSx4.p1.3.m3.1"><semantics id="S1.SSx4.p1.3.m3.1a"><msub id="S1.SSx4.p1.3.m3.1.1" xref="S1.SSx4.p1.3.m3.1.1.cmml"><mi id="S1.SSx4.p1.3.m3.1.1.2" xref="S1.SSx4.p1.3.m3.1.1.2.cmml">𝖬𝖠</mi><msub id="S1.SSx4.p1.3.m3.1.1.3" xref="S1.SSx4.p1.3.m3.1.1.3.cmml"><mi id="S1.SSx4.p1.3.m3.1.1.3.2" mathvariant="normal" xref="S1.SSx4.p1.3.m3.1.1.3.2.cmml">ℵ</mi><mn id="S1.SSx4.p1.3.m3.1.1.3.3" xref="S1.SSx4.p1.3.m3.1.1.3.3.cmml">1</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S1.SSx4.p1.3.m3.1b"><apply id="S1.SSx4.p1.3.m3.1.1.cmml" xref="S1.SSx4.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S1.SSx4.p1.3.m3.1.1.1.cmml" xref="S1.SSx4.p1.3.m3.1.1">subscript</csymbol><ci id="S1.SSx4.p1.3.m3.1.1.2.cmml" xref="S1.SSx4.p1.3.m3.1.1.2">𝖬𝖠</ci><apply id="S1.SSx4.p1.3.m3.1.1.3.cmml" xref="S1.SSx4.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S1.SSx4.p1.3.m3.1.1.3.1.cmml" xref="S1.SSx4.p1.3.m3.1.1.3">subscript</csymbol><ci id="S1.SSx4.p1.3.m3.1.1.3.2.cmml" xref="S1.SSx4.p1.3.m3.1.1.3.2">ℵ</ci><cn id="S1.SSx4.p1.3.m3.1.1.3.3.cmml" type="integer" xref="S1.SSx4.p1.3.m3.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx4.p1.3.m3.1c">\mathsf{MA}_{\aleph_{1}}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx4.p1.3.m3.1d">sansserif_MA start_POSTSUBSCRIPT roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> is necessary, these can be used as black boxes in the rest of the text. Knowledge of the specifics of the method of Forcing will be needed in <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6" title="6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Sections</span> <span class="ltx_text ltx_ref_tag">6</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S8" title="8. On a question on Countryman lines ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">8</span></a>.</p> </div> <div class="ltx_para" id="S1.SSx4.p2"> <p class="ltx_p" id="S1.SSx4.p2.9">A <em class="ltx_emph ltx_font_italic" id="S1.SSx4.p2.9.1">sequence</em> is any function with domain an ordinal. We let <math alttext="\sqsubseteq" class="ltx_Math" display="inline" id="S1.SSx4.p2.1.m1.1"><semantics id="S1.SSx4.p2.1.m1.1a"><mo id="S1.SSx4.p2.1.m1.1.1" xref="S1.SSx4.p2.1.m1.1.1.cmml">⊑</mo><annotation-xml encoding="MathML-Content" id="S1.SSx4.p2.1.m1.1b"><csymbol cd="latexml" id="S1.SSx4.p2.1.m1.1.1.cmml" xref="S1.SSx4.p2.1.m1.1.1">square-image-of-or-equals</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx4.p2.1.m1.1c">\sqsubseteq</annotation><annotation encoding="application/x-llamapun" id="S1.SSx4.p2.1.m1.1d">⊑</annotation></semantics></math> denote sequence extension and <math alttext="\sqsubset" class="ltx_Math" display="inline" id="S1.SSx4.p2.2.m2.1"><semantics id="S1.SSx4.p2.2.m2.1a"><mo id="S1.SSx4.p2.2.m2.1.1" xref="S1.SSx4.p2.2.m2.1.1.cmml">⊏</mo><annotation-xml encoding="MathML-Content" id="S1.SSx4.p2.2.m2.1b"><csymbol cd="latexml" id="S1.SSx4.p2.2.m2.1.1.cmml" xref="S1.SSx4.p2.2.m2.1.1">square-image-of</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx4.p2.2.m2.1c">\sqsubset</annotation><annotation encoding="application/x-llamapun" id="S1.SSx4.p2.2.m2.1d">⊏</annotation></semantics></math> proper sequence extension. If <math alttext="S" class="ltx_Math" display="inline" id="S1.SSx4.p2.3.m3.1"><semantics id="S1.SSx4.p2.3.m3.1a"><mi id="S1.SSx4.p2.3.m3.1.1" xref="S1.SSx4.p2.3.m3.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S1.SSx4.p2.3.m3.1b"><ci id="S1.SSx4.p2.3.m3.1.1.cmml" xref="S1.SSx4.p2.3.m3.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx4.p2.3.m3.1c">S</annotation><annotation encoding="application/x-llamapun" id="S1.SSx4.p2.3.m3.1d">italic_S</annotation></semantics></math> is a set of sequences with ranges in a linearly ordered set <math alttext="A" class="ltx_Math" display="inline" id="S1.SSx4.p2.4.m4.1"><semantics id="S1.SSx4.p2.4.m4.1a"><mi id="S1.SSx4.p2.4.m4.1.1" xref="S1.SSx4.p2.4.m4.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S1.SSx4.p2.4.m4.1b"><ci id="S1.SSx4.p2.4.m4.1.1.cmml" xref="S1.SSx4.p2.4.m4.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx4.p2.4.m4.1c">A</annotation><annotation encoding="application/x-llamapun" id="S1.SSx4.p2.4.m4.1d">italic_A</annotation></semantics></math>, then <math alttext="S" class="ltx_Math" display="inline" id="S1.SSx4.p2.5.m5.1"><semantics id="S1.SSx4.p2.5.m5.1a"><mi id="S1.SSx4.p2.5.m5.1.1" xref="S1.SSx4.p2.5.m5.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S1.SSx4.p2.5.m5.1b"><ci id="S1.SSx4.p2.5.m5.1.1.cmml" xref="S1.SSx4.p2.5.m5.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx4.p2.5.m5.1c">S</annotation><annotation encoding="application/x-llamapun" id="S1.SSx4.p2.5.m5.1d">italic_S</annotation></semantics></math> is naturally linearly ordered by the lexicographic ordering, given by <math alttext="t<_{\mathrm{lex}}s" class="ltx_Math" display="inline" id="S1.SSx4.p2.6.m6.1"><semantics id="S1.SSx4.p2.6.m6.1a"><mrow id="S1.SSx4.p2.6.m6.1.1" xref="S1.SSx4.p2.6.m6.1.1.cmml"><mi id="S1.SSx4.p2.6.m6.1.1.2" xref="S1.SSx4.p2.6.m6.1.1.2.cmml">t</mi><msub id="S1.SSx4.p2.6.m6.1.1.1" xref="S1.SSx4.p2.6.m6.1.1.1.cmml"><mo id="S1.SSx4.p2.6.m6.1.1.1.2" xref="S1.SSx4.p2.6.m6.1.1.1.2.cmml"><</mo><mi id="S1.SSx4.p2.6.m6.1.1.1.3" xref="S1.SSx4.p2.6.m6.1.1.1.3.cmml">lex</mi></msub><mi id="S1.SSx4.p2.6.m6.1.1.3" xref="S1.SSx4.p2.6.m6.1.1.3.cmml">s</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.SSx4.p2.6.m6.1b"><apply id="S1.SSx4.p2.6.m6.1.1.cmml" xref="S1.SSx4.p2.6.m6.1.1"><apply id="S1.SSx4.p2.6.m6.1.1.1.cmml" xref="S1.SSx4.p2.6.m6.1.1.1"><csymbol cd="ambiguous" id="S1.SSx4.p2.6.m6.1.1.1.1.cmml" xref="S1.SSx4.p2.6.m6.1.1.1">subscript</csymbol><lt id="S1.SSx4.p2.6.m6.1.1.1.2.cmml" xref="S1.SSx4.p2.6.m6.1.1.1.2"></lt><ci id="S1.SSx4.p2.6.m6.1.1.1.3.cmml" xref="S1.SSx4.p2.6.m6.1.1.1.3">lex</ci></apply><ci id="S1.SSx4.p2.6.m6.1.1.2.cmml" xref="S1.SSx4.p2.6.m6.1.1.2">𝑡</ci><ci id="S1.SSx4.p2.6.m6.1.1.3.cmml" xref="S1.SSx4.p2.6.m6.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx4.p2.6.m6.1c">t<_{\mathrm{lex}}s</annotation><annotation encoding="application/x-llamapun" id="S1.SSx4.p2.6.m6.1d">italic_t < start_POSTSUBSCRIPT roman_lex end_POSTSUBSCRIPT italic_s</annotation></semantics></math> if either <math alttext="t\sqsubset s" class="ltx_Math" display="inline" id="S1.SSx4.p2.7.m7.1"><semantics id="S1.SSx4.p2.7.m7.1a"><mrow id="S1.SSx4.p2.7.m7.1.1" xref="S1.SSx4.p2.7.m7.1.1.cmml"><mi id="S1.SSx4.p2.7.m7.1.1.2" xref="S1.SSx4.p2.7.m7.1.1.2.cmml">t</mi><mo id="S1.SSx4.p2.7.m7.1.1.1" xref="S1.SSx4.p2.7.m7.1.1.1.cmml">⊏</mo><mi id="S1.SSx4.p2.7.m7.1.1.3" xref="S1.SSx4.p2.7.m7.1.1.3.cmml">s</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.SSx4.p2.7.m7.1b"><apply id="S1.SSx4.p2.7.m7.1.1.cmml" xref="S1.SSx4.p2.7.m7.1.1"><csymbol cd="latexml" id="S1.SSx4.p2.7.m7.1.1.1.cmml" xref="S1.SSx4.p2.7.m7.1.1.1">square-image-of</csymbol><ci id="S1.SSx4.p2.7.m7.1.1.2.cmml" xref="S1.SSx4.p2.7.m7.1.1.2">𝑡</ci><ci id="S1.SSx4.p2.7.m7.1.1.3.cmml" xref="S1.SSx4.p2.7.m7.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx4.p2.7.m7.1c">t\sqsubset s</annotation><annotation encoding="application/x-llamapun" id="S1.SSx4.p2.7.m7.1d">italic_t ⊏ italic_s</annotation></semantics></math>, or <math alttext="t(\xi)<_{A}s(\xi)" class="ltx_Math" display="inline" id="S1.SSx4.p2.8.m8.2"><semantics id="S1.SSx4.p2.8.m8.2a"><mrow id="S1.SSx4.p2.8.m8.2.3" xref="S1.SSx4.p2.8.m8.2.3.cmml"><mrow id="S1.SSx4.p2.8.m8.2.3.2" xref="S1.SSx4.p2.8.m8.2.3.2.cmml"><mi id="S1.SSx4.p2.8.m8.2.3.2.2" xref="S1.SSx4.p2.8.m8.2.3.2.2.cmml">t</mi><mo id="S1.SSx4.p2.8.m8.2.3.2.1" xref="S1.SSx4.p2.8.m8.2.3.2.1.cmml">⁢</mo><mrow id="S1.SSx4.p2.8.m8.2.3.2.3.2" xref="S1.SSx4.p2.8.m8.2.3.2.cmml"><mo id="S1.SSx4.p2.8.m8.2.3.2.3.2.1" stretchy="false" xref="S1.SSx4.p2.8.m8.2.3.2.cmml">(</mo><mi id="S1.SSx4.p2.8.m8.1.1" xref="S1.SSx4.p2.8.m8.1.1.cmml">ξ</mi><mo id="S1.SSx4.p2.8.m8.2.3.2.3.2.2" stretchy="false" xref="S1.SSx4.p2.8.m8.2.3.2.cmml">)</mo></mrow></mrow><msub id="S1.SSx4.p2.8.m8.2.3.1" xref="S1.SSx4.p2.8.m8.2.3.1.cmml"><mo id="S1.SSx4.p2.8.m8.2.3.1.2" xref="S1.SSx4.p2.8.m8.2.3.1.2.cmml"><</mo><mi id="S1.SSx4.p2.8.m8.2.3.1.3" xref="S1.SSx4.p2.8.m8.2.3.1.3.cmml">A</mi></msub><mrow id="S1.SSx4.p2.8.m8.2.3.3" xref="S1.SSx4.p2.8.m8.2.3.3.cmml"><mi id="S1.SSx4.p2.8.m8.2.3.3.2" xref="S1.SSx4.p2.8.m8.2.3.3.2.cmml">s</mi><mo id="S1.SSx4.p2.8.m8.2.3.3.1" xref="S1.SSx4.p2.8.m8.2.3.3.1.cmml">⁢</mo><mrow id="S1.SSx4.p2.8.m8.2.3.3.3.2" xref="S1.SSx4.p2.8.m8.2.3.3.cmml"><mo id="S1.SSx4.p2.8.m8.2.3.3.3.2.1" stretchy="false" xref="S1.SSx4.p2.8.m8.2.3.3.cmml">(</mo><mi id="S1.SSx4.p2.8.m8.2.2" xref="S1.SSx4.p2.8.m8.2.2.cmml">ξ</mi><mo id="S1.SSx4.p2.8.m8.2.3.3.3.2.2" stretchy="false" xref="S1.SSx4.p2.8.m8.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SSx4.p2.8.m8.2b"><apply id="S1.SSx4.p2.8.m8.2.3.cmml" xref="S1.SSx4.p2.8.m8.2.3"><apply id="S1.SSx4.p2.8.m8.2.3.1.cmml" xref="S1.SSx4.p2.8.m8.2.3.1"><csymbol cd="ambiguous" id="S1.SSx4.p2.8.m8.2.3.1.1.cmml" xref="S1.SSx4.p2.8.m8.2.3.1">subscript</csymbol><lt id="S1.SSx4.p2.8.m8.2.3.1.2.cmml" xref="S1.SSx4.p2.8.m8.2.3.1.2"></lt><ci id="S1.SSx4.p2.8.m8.2.3.1.3.cmml" xref="S1.SSx4.p2.8.m8.2.3.1.3">𝐴</ci></apply><apply id="S1.SSx4.p2.8.m8.2.3.2.cmml" xref="S1.SSx4.p2.8.m8.2.3.2"><times id="S1.SSx4.p2.8.m8.2.3.2.1.cmml" xref="S1.SSx4.p2.8.m8.2.3.2.1"></times><ci id="S1.SSx4.p2.8.m8.2.3.2.2.cmml" xref="S1.SSx4.p2.8.m8.2.3.2.2">𝑡</ci><ci id="S1.SSx4.p2.8.m8.1.1.cmml" xref="S1.SSx4.p2.8.m8.1.1">𝜉</ci></apply><apply id="S1.SSx4.p2.8.m8.2.3.3.cmml" xref="S1.SSx4.p2.8.m8.2.3.3"><times id="S1.SSx4.p2.8.m8.2.3.3.1.cmml" xref="S1.SSx4.p2.8.m8.2.3.3.1"></times><ci id="S1.SSx4.p2.8.m8.2.3.3.2.cmml" xref="S1.SSx4.p2.8.m8.2.3.3.2">𝑠</ci><ci id="S1.SSx4.p2.8.m8.2.2.cmml" xref="S1.SSx4.p2.8.m8.2.2">𝜉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx4.p2.8.m8.2c">t(\xi)<_{A}s(\xi)</annotation><annotation encoding="application/x-llamapun" id="S1.SSx4.p2.8.m8.2d">italic_t ( italic_ξ ) < start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_s ( italic_ξ )</annotation></semantics></math> where <math alttext="\xi" class="ltx_Math" display="inline" id="S1.SSx4.p2.9.m9.1"><semantics id="S1.SSx4.p2.9.m9.1a"><mi id="S1.SSx4.p2.9.m9.1.1" xref="S1.SSx4.p2.9.m9.1.1.cmml">ξ</mi><annotation-xml encoding="MathML-Content" id="S1.SSx4.p2.9.m9.1b"><ci id="S1.SSx4.p2.9.m9.1.1.cmml" xref="S1.SSx4.p2.9.m9.1.1">𝜉</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx4.p2.9.m9.1c">\xi</annotation><annotation encoding="application/x-llamapun" id="S1.SSx4.p2.9.m9.1d">italic_ξ</annotation></semantics></math> is the first ordinal at which they differ.</p> </div> <div class="ltx_para" id="S1.SSx4.p3"> <p class="ltx_p" id="S1.SSx4.p3.13">When talking about orders we will usually not specify the order relation, and use subscripts to do it as it was done above in the definition of <math alttext="\leq_{\mathrm{lex}}" class="ltx_Math" display="inline" id="S1.SSx4.p3.1.m1.1"><semantics id="S1.SSx4.p3.1.m1.1a"><msub id="S1.SSx4.p3.1.m1.1.1" xref="S1.SSx4.p3.1.m1.1.1.cmml"><mo id="S1.SSx4.p3.1.m1.1.1.2" xref="S1.SSx4.p3.1.m1.1.1.2.cmml">≤</mo><mi id="S1.SSx4.p3.1.m1.1.1.3" xref="S1.SSx4.p3.1.m1.1.1.3.cmml">lex</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SSx4.p3.1.m1.1b"><apply id="S1.SSx4.p3.1.m1.1.1.cmml" xref="S1.SSx4.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S1.SSx4.p3.1.m1.1.1.1.cmml" xref="S1.SSx4.p3.1.m1.1.1">subscript</csymbol><leq id="S1.SSx4.p3.1.m1.1.1.2.cmml" xref="S1.SSx4.p3.1.m1.1.1.2"></leq><ci id="S1.SSx4.p3.1.m1.1.1.3.cmml" xref="S1.SSx4.p3.1.m1.1.1.3">lex</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx4.p3.1.m1.1c">\leq_{\mathrm{lex}}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx4.p3.1.m1.1d">≤ start_POSTSUBSCRIPT roman_lex end_POSTSUBSCRIPT</annotation></semantics></math>. A plain <math alttext="<" class="ltx_Math" display="inline" id="S1.SSx4.p3.2.m2.1"><semantics id="S1.SSx4.p3.2.m2.1a"><mo id="S1.SSx4.p3.2.m2.1.1" xref="S1.SSx4.p3.2.m2.1.1.cmml"><</mo><annotation-xml encoding="MathML-Content" id="S1.SSx4.p3.2.m2.1b"><lt id="S1.SSx4.p3.2.m2.1.1.cmml" xref="S1.SSx4.p3.2.m2.1.1"></lt></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx4.p3.2.m2.1c"><</annotation><annotation encoding="application/x-llamapun" id="S1.SSx4.p3.2.m2.1d"><</annotation></semantics></math> usually refers to the standard order on ordinals. When thinking of <math alttext="A\times B" class="ltx_Math" display="inline" id="S1.SSx4.p3.3.m3.1"><semantics id="S1.SSx4.p3.3.m3.1a"><mrow id="S1.SSx4.p3.3.m3.1.1" xref="S1.SSx4.p3.3.m3.1.1.cmml"><mi id="S1.SSx4.p3.3.m3.1.1.2" xref="S1.SSx4.p3.3.m3.1.1.2.cmml">A</mi><mo id="S1.SSx4.p3.3.m3.1.1.1" lspace="0.222em" rspace="0.222em" xref="S1.SSx4.p3.3.m3.1.1.1.cmml">×</mo><mi id="S1.SSx4.p3.3.m3.1.1.3" xref="S1.SSx4.p3.3.m3.1.1.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.SSx4.p3.3.m3.1b"><apply id="S1.SSx4.p3.3.m3.1.1.cmml" xref="S1.SSx4.p3.3.m3.1.1"><times id="S1.SSx4.p3.3.m3.1.1.1.cmml" xref="S1.SSx4.p3.3.m3.1.1.1"></times><ci id="S1.SSx4.p3.3.m3.1.1.2.cmml" xref="S1.SSx4.p3.3.m3.1.1.2">𝐴</ci><ci id="S1.SSx4.p3.3.m3.1.1.3.cmml" xref="S1.SSx4.p3.3.m3.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx4.p3.3.m3.1c">A\times B</annotation><annotation encoding="application/x-llamapun" id="S1.SSx4.p3.3.m3.1d">italic_A × italic_B</annotation></semantics></math> as a linear order, we give it the lexicographically order, that is <math alttext="A" class="ltx_Math" display="inline" id="S1.SSx4.p3.4.m4.1"><semantics id="S1.SSx4.p3.4.m4.1a"><mi id="S1.SSx4.p3.4.m4.1.1" xref="S1.SSx4.p3.4.m4.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S1.SSx4.p3.4.m4.1b"><ci id="S1.SSx4.p3.4.m4.1.1.cmml" xref="S1.SSx4.p3.4.m4.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx4.p3.4.m4.1c">A</annotation><annotation encoding="application/x-llamapun" id="S1.SSx4.p3.4.m4.1d">italic_A</annotation></semantics></math> copies of <math alttext="B" class="ltx_Math" display="inline" id="S1.SSx4.p3.5.m5.1"><semantics id="S1.SSx4.p3.5.m5.1a"><mi id="S1.SSx4.p3.5.m5.1.1" xref="S1.SSx4.p3.5.m5.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S1.SSx4.p3.5.m5.1b"><ci id="S1.SSx4.p3.5.m5.1.1.cmml" xref="S1.SSx4.p3.5.m5.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx4.p3.5.m5.1c">B</annotation><annotation encoding="application/x-llamapun" id="S1.SSx4.p3.5.m5.1d">italic_B</annotation></semantics></math>. This is somewhat non standard since then <math alttext="\alpha\times\beta\cong\beta\cdot\alpha" class="ltx_Math" display="inline" id="S1.SSx4.p3.6.m6.1"><semantics id="S1.SSx4.p3.6.m6.1a"><mrow id="S1.SSx4.p3.6.m6.1.1" xref="S1.SSx4.p3.6.m6.1.1.cmml"><mrow id="S1.SSx4.p3.6.m6.1.1.2" xref="S1.SSx4.p3.6.m6.1.1.2.cmml"><mi id="S1.SSx4.p3.6.m6.1.1.2.2" xref="S1.SSx4.p3.6.m6.1.1.2.2.cmml">α</mi><mo id="S1.SSx4.p3.6.m6.1.1.2.1" lspace="0.222em" rspace="0.222em" xref="S1.SSx4.p3.6.m6.1.1.2.1.cmml">×</mo><mi id="S1.SSx4.p3.6.m6.1.1.2.3" xref="S1.SSx4.p3.6.m6.1.1.2.3.cmml">β</mi></mrow><mo id="S1.SSx4.p3.6.m6.1.1.1" xref="S1.SSx4.p3.6.m6.1.1.1.cmml">≅</mo><mrow id="S1.SSx4.p3.6.m6.1.1.3" xref="S1.SSx4.p3.6.m6.1.1.3.cmml"><mi id="S1.SSx4.p3.6.m6.1.1.3.2" xref="S1.SSx4.p3.6.m6.1.1.3.2.cmml">β</mi><mo id="S1.SSx4.p3.6.m6.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S1.SSx4.p3.6.m6.1.1.3.1.cmml">⋅</mo><mi id="S1.SSx4.p3.6.m6.1.1.3.3" xref="S1.SSx4.p3.6.m6.1.1.3.3.cmml">α</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SSx4.p3.6.m6.1b"><apply id="S1.SSx4.p3.6.m6.1.1.cmml" xref="S1.SSx4.p3.6.m6.1.1"><approx id="S1.SSx4.p3.6.m6.1.1.1.cmml" xref="S1.SSx4.p3.6.m6.1.1.1"></approx><apply id="S1.SSx4.p3.6.m6.1.1.2.cmml" xref="S1.SSx4.p3.6.m6.1.1.2"><times id="S1.SSx4.p3.6.m6.1.1.2.1.cmml" xref="S1.SSx4.p3.6.m6.1.1.2.1"></times><ci id="S1.SSx4.p3.6.m6.1.1.2.2.cmml" xref="S1.SSx4.p3.6.m6.1.1.2.2">𝛼</ci><ci id="S1.SSx4.p3.6.m6.1.1.2.3.cmml" xref="S1.SSx4.p3.6.m6.1.1.2.3">𝛽</ci></apply><apply id="S1.SSx4.p3.6.m6.1.1.3.cmml" xref="S1.SSx4.p3.6.m6.1.1.3"><ci id="S1.SSx4.p3.6.m6.1.1.3.1.cmml" xref="S1.SSx4.p3.6.m6.1.1.3.1">⋅</ci><ci id="S1.SSx4.p3.6.m6.1.1.3.2.cmml" xref="S1.SSx4.p3.6.m6.1.1.3.2">𝛽</ci><ci id="S1.SSx4.p3.6.m6.1.1.3.3.cmml" xref="S1.SSx4.p3.6.m6.1.1.3.3">𝛼</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx4.p3.6.m6.1c">\alpha\times\beta\cong\beta\cdot\alpha</annotation><annotation encoding="application/x-llamapun" id="S1.SSx4.p3.6.m6.1d">italic_α × italic_β ≅ italic_β ⋅ italic_α</annotation></semantics></math>, when thinking of ordinal multiplication. Also if for all <math alttext="a\in A" class="ltx_Math" display="inline" id="S1.SSx4.p3.7.m7.1"><semantics id="S1.SSx4.p3.7.m7.1a"><mrow id="S1.SSx4.p3.7.m7.1.1" xref="S1.SSx4.p3.7.m7.1.1.cmml"><mi id="S1.SSx4.p3.7.m7.1.1.2" xref="S1.SSx4.p3.7.m7.1.1.2.cmml">a</mi><mo id="S1.SSx4.p3.7.m7.1.1.1" xref="S1.SSx4.p3.7.m7.1.1.1.cmml">∈</mo><mi id="S1.SSx4.p3.7.m7.1.1.3" xref="S1.SSx4.p3.7.m7.1.1.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.SSx4.p3.7.m7.1b"><apply id="S1.SSx4.p3.7.m7.1.1.cmml" xref="S1.SSx4.p3.7.m7.1.1"><in id="S1.SSx4.p3.7.m7.1.1.1.cmml" xref="S1.SSx4.p3.7.m7.1.1.1"></in><ci id="S1.SSx4.p3.7.m7.1.1.2.cmml" xref="S1.SSx4.p3.7.m7.1.1.2">𝑎</ci><ci id="S1.SSx4.p3.7.m7.1.1.3.cmml" xref="S1.SSx4.p3.7.m7.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx4.p3.7.m7.1c">a\in A</annotation><annotation encoding="application/x-llamapun" id="S1.SSx4.p3.7.m7.1d">italic_a ∈ italic_A</annotation></semantics></math>, <math alttext="L_{a}" class="ltx_Math" display="inline" id="S1.SSx4.p3.8.m8.1"><semantics id="S1.SSx4.p3.8.m8.1a"><msub id="S1.SSx4.p3.8.m8.1.1" xref="S1.SSx4.p3.8.m8.1.1.cmml"><mi id="S1.SSx4.p3.8.m8.1.1.2" xref="S1.SSx4.p3.8.m8.1.1.2.cmml">L</mi><mi id="S1.SSx4.p3.8.m8.1.1.3" xref="S1.SSx4.p3.8.m8.1.1.3.cmml">a</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SSx4.p3.8.m8.1b"><apply id="S1.SSx4.p3.8.m8.1.1.cmml" xref="S1.SSx4.p3.8.m8.1.1"><csymbol cd="ambiguous" id="S1.SSx4.p3.8.m8.1.1.1.cmml" xref="S1.SSx4.p3.8.m8.1.1">subscript</csymbol><ci id="S1.SSx4.p3.8.m8.1.1.2.cmml" xref="S1.SSx4.p3.8.m8.1.1.2">𝐿</ci><ci id="S1.SSx4.p3.8.m8.1.1.3.cmml" xref="S1.SSx4.p3.8.m8.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx4.p3.8.m8.1c">L_{a}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx4.p3.8.m8.1d">italic_L start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT</annotation></semantics></math> is a linear order, <math alttext="\sum_{a\in A}L_{a}" class="ltx_Math" display="inline" id="S1.SSx4.p3.9.m9.1"><semantics id="S1.SSx4.p3.9.m9.1a"><mrow id="S1.SSx4.p3.9.m9.1.1" xref="S1.SSx4.p3.9.m9.1.1.cmml"><msub id="S1.SSx4.p3.9.m9.1.1.1" xref="S1.SSx4.p3.9.m9.1.1.1.cmml"><mo id="S1.SSx4.p3.9.m9.1.1.1.2" xref="S1.SSx4.p3.9.m9.1.1.1.2.cmml">∑</mo><mrow id="S1.SSx4.p3.9.m9.1.1.1.3" xref="S1.SSx4.p3.9.m9.1.1.1.3.cmml"><mi id="S1.SSx4.p3.9.m9.1.1.1.3.2" xref="S1.SSx4.p3.9.m9.1.1.1.3.2.cmml">a</mi><mo id="S1.SSx4.p3.9.m9.1.1.1.3.1" xref="S1.SSx4.p3.9.m9.1.1.1.3.1.cmml">∈</mo><mi id="S1.SSx4.p3.9.m9.1.1.1.3.3" xref="S1.SSx4.p3.9.m9.1.1.1.3.3.cmml">A</mi></mrow></msub><msub id="S1.SSx4.p3.9.m9.1.1.2" xref="S1.SSx4.p3.9.m9.1.1.2.cmml"><mi id="S1.SSx4.p3.9.m9.1.1.2.2" xref="S1.SSx4.p3.9.m9.1.1.2.2.cmml">L</mi><mi id="S1.SSx4.p3.9.m9.1.1.2.3" xref="S1.SSx4.p3.9.m9.1.1.2.3.cmml">a</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S1.SSx4.p3.9.m9.1b"><apply id="S1.SSx4.p3.9.m9.1.1.cmml" xref="S1.SSx4.p3.9.m9.1.1"><apply id="S1.SSx4.p3.9.m9.1.1.1.cmml" xref="S1.SSx4.p3.9.m9.1.1.1"><csymbol cd="ambiguous" id="S1.SSx4.p3.9.m9.1.1.1.1.cmml" xref="S1.SSx4.p3.9.m9.1.1.1">subscript</csymbol><sum id="S1.SSx4.p3.9.m9.1.1.1.2.cmml" xref="S1.SSx4.p3.9.m9.1.1.1.2"></sum><apply id="S1.SSx4.p3.9.m9.1.1.1.3.cmml" xref="S1.SSx4.p3.9.m9.1.1.1.3"><in id="S1.SSx4.p3.9.m9.1.1.1.3.1.cmml" xref="S1.SSx4.p3.9.m9.1.1.1.3.1"></in><ci id="S1.SSx4.p3.9.m9.1.1.1.3.2.cmml" xref="S1.SSx4.p3.9.m9.1.1.1.3.2">𝑎</ci><ci id="S1.SSx4.p3.9.m9.1.1.1.3.3.cmml" xref="S1.SSx4.p3.9.m9.1.1.1.3.3">𝐴</ci></apply></apply><apply id="S1.SSx4.p3.9.m9.1.1.2.cmml" xref="S1.SSx4.p3.9.m9.1.1.2"><csymbol cd="ambiguous" id="S1.SSx4.p3.9.m9.1.1.2.1.cmml" xref="S1.SSx4.p3.9.m9.1.1.2">subscript</csymbol><ci id="S1.SSx4.p3.9.m9.1.1.2.2.cmml" xref="S1.SSx4.p3.9.m9.1.1.2.2">𝐿</ci><ci id="S1.SSx4.p3.9.m9.1.1.2.3.cmml" xref="S1.SSx4.p3.9.m9.1.1.2.3">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx4.p3.9.m9.1c">\sum_{a\in A}L_{a}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx4.p3.9.m9.1d">∑ start_POSTSUBSCRIPT italic_a ∈ italic_A end_POSTSUBSCRIPT italic_L start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT</annotation></semantics></math> denotes the order consisting of lifting each <math alttext="a\in A" class="ltx_Math" display="inline" id="S1.SSx4.p3.10.m10.1"><semantics id="S1.SSx4.p3.10.m10.1a"><mrow id="S1.SSx4.p3.10.m10.1.1" xref="S1.SSx4.p3.10.m10.1.1.cmml"><mi id="S1.SSx4.p3.10.m10.1.1.2" xref="S1.SSx4.p3.10.m10.1.1.2.cmml">a</mi><mo id="S1.SSx4.p3.10.m10.1.1.1" xref="S1.SSx4.p3.10.m10.1.1.1.cmml">∈</mo><mi id="S1.SSx4.p3.10.m10.1.1.3" xref="S1.SSx4.p3.10.m10.1.1.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.SSx4.p3.10.m10.1b"><apply id="S1.SSx4.p3.10.m10.1.1.cmml" xref="S1.SSx4.p3.10.m10.1.1"><in id="S1.SSx4.p3.10.m10.1.1.1.cmml" xref="S1.SSx4.p3.10.m10.1.1.1"></in><ci id="S1.SSx4.p3.10.m10.1.1.2.cmml" xref="S1.SSx4.p3.10.m10.1.1.2">𝑎</ci><ci id="S1.SSx4.p3.10.m10.1.1.3.cmml" xref="S1.SSx4.p3.10.m10.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx4.p3.10.m10.1c">a\in A</annotation><annotation encoding="application/x-llamapun" id="S1.SSx4.p3.10.m10.1d">italic_a ∈ italic_A</annotation></semantics></math> to a copy of <math alttext="L_{a}" class="ltx_Math" display="inline" id="S1.SSx4.p3.11.m11.1"><semantics id="S1.SSx4.p3.11.m11.1a"><msub id="S1.SSx4.p3.11.m11.1.1" xref="S1.SSx4.p3.11.m11.1.1.cmml"><mi id="S1.SSx4.p3.11.m11.1.1.2" xref="S1.SSx4.p3.11.m11.1.1.2.cmml">L</mi><mi id="S1.SSx4.p3.11.m11.1.1.3" xref="S1.SSx4.p3.11.m11.1.1.3.cmml">a</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SSx4.p3.11.m11.1b"><apply id="S1.SSx4.p3.11.m11.1.1.cmml" xref="S1.SSx4.p3.11.m11.1.1"><csymbol cd="ambiguous" id="S1.SSx4.p3.11.m11.1.1.1.cmml" xref="S1.SSx4.p3.11.m11.1.1">subscript</csymbol><ci id="S1.SSx4.p3.11.m11.1.1.2.cmml" xref="S1.SSx4.p3.11.m11.1.1.2">𝐿</ci><ci id="S1.SSx4.p3.11.m11.1.1.3.cmml" xref="S1.SSx4.p3.11.m11.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx4.p3.11.m11.1c">L_{a}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx4.p3.11.m11.1d">italic_L start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT</annotation></semantics></math>. Formally this is the lexicographic ordering of <math alttext="\bigcup_{a\in A}\{a\}\times L_{a}" class="ltx_Math" display="inline" id="S1.SSx4.p3.12.m12.1"><semantics id="S1.SSx4.p3.12.m12.1a"><mrow id="S1.SSx4.p3.12.m12.1.2" xref="S1.SSx4.p3.12.m12.1.2.cmml"><msub id="S1.SSx4.p3.12.m12.1.2.1" xref="S1.SSx4.p3.12.m12.1.2.1.cmml"><mo id="S1.SSx4.p3.12.m12.1.2.1.2" xref="S1.SSx4.p3.12.m12.1.2.1.2.cmml">⋃</mo><mrow id="S1.SSx4.p3.12.m12.1.2.1.3" xref="S1.SSx4.p3.12.m12.1.2.1.3.cmml"><mi id="S1.SSx4.p3.12.m12.1.2.1.3.2" xref="S1.SSx4.p3.12.m12.1.2.1.3.2.cmml">a</mi><mo id="S1.SSx4.p3.12.m12.1.2.1.3.1" xref="S1.SSx4.p3.12.m12.1.2.1.3.1.cmml">∈</mo><mi id="S1.SSx4.p3.12.m12.1.2.1.3.3" xref="S1.SSx4.p3.12.m12.1.2.1.3.3.cmml">A</mi></mrow></msub><mrow id="S1.SSx4.p3.12.m12.1.2.2" xref="S1.SSx4.p3.12.m12.1.2.2.cmml"><mrow id="S1.SSx4.p3.12.m12.1.2.2.2.2" xref="S1.SSx4.p3.12.m12.1.2.2.2.1.cmml"><mo id="S1.SSx4.p3.12.m12.1.2.2.2.2.1" lspace="0em" stretchy="false" xref="S1.SSx4.p3.12.m12.1.2.2.2.1.cmml">{</mo><mi id="S1.SSx4.p3.12.m12.1.1" xref="S1.SSx4.p3.12.m12.1.1.cmml">a</mi><mo id="S1.SSx4.p3.12.m12.1.2.2.2.2.2" rspace="0.055em" stretchy="false" xref="S1.SSx4.p3.12.m12.1.2.2.2.1.cmml">}</mo></mrow><mo id="S1.SSx4.p3.12.m12.1.2.2.1" rspace="0.222em" xref="S1.SSx4.p3.12.m12.1.2.2.1.cmml">×</mo><msub id="S1.SSx4.p3.12.m12.1.2.2.3" xref="S1.SSx4.p3.12.m12.1.2.2.3.cmml"><mi id="S1.SSx4.p3.12.m12.1.2.2.3.2" xref="S1.SSx4.p3.12.m12.1.2.2.3.2.cmml">L</mi><mi id="S1.SSx4.p3.12.m12.1.2.2.3.3" xref="S1.SSx4.p3.12.m12.1.2.2.3.3.cmml">a</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SSx4.p3.12.m12.1b"><apply id="S1.SSx4.p3.12.m12.1.2.cmml" xref="S1.SSx4.p3.12.m12.1.2"><apply id="S1.SSx4.p3.12.m12.1.2.1.cmml" xref="S1.SSx4.p3.12.m12.1.2.1"><csymbol cd="ambiguous" id="S1.SSx4.p3.12.m12.1.2.1.1.cmml" xref="S1.SSx4.p3.12.m12.1.2.1">subscript</csymbol><union id="S1.SSx4.p3.12.m12.1.2.1.2.cmml" xref="S1.SSx4.p3.12.m12.1.2.1.2"></union><apply id="S1.SSx4.p3.12.m12.1.2.1.3.cmml" xref="S1.SSx4.p3.12.m12.1.2.1.3"><in id="S1.SSx4.p3.12.m12.1.2.1.3.1.cmml" xref="S1.SSx4.p3.12.m12.1.2.1.3.1"></in><ci id="S1.SSx4.p3.12.m12.1.2.1.3.2.cmml" xref="S1.SSx4.p3.12.m12.1.2.1.3.2">𝑎</ci><ci id="S1.SSx4.p3.12.m12.1.2.1.3.3.cmml" xref="S1.SSx4.p3.12.m12.1.2.1.3.3">𝐴</ci></apply></apply><apply id="S1.SSx4.p3.12.m12.1.2.2.cmml" xref="S1.SSx4.p3.12.m12.1.2.2"><times id="S1.SSx4.p3.12.m12.1.2.2.1.cmml" xref="S1.SSx4.p3.12.m12.1.2.2.1"></times><set id="S1.SSx4.p3.12.m12.1.2.2.2.1.cmml" xref="S1.SSx4.p3.12.m12.1.2.2.2.2"><ci id="S1.SSx4.p3.12.m12.1.1.cmml" xref="S1.SSx4.p3.12.m12.1.1">𝑎</ci></set><apply id="S1.SSx4.p3.12.m12.1.2.2.3.cmml" xref="S1.SSx4.p3.12.m12.1.2.2.3"><csymbol cd="ambiguous" id="S1.SSx4.p3.12.m12.1.2.2.3.1.cmml" xref="S1.SSx4.p3.12.m12.1.2.2.3">subscript</csymbol><ci id="S1.SSx4.p3.12.m12.1.2.2.3.2.cmml" xref="S1.SSx4.p3.12.m12.1.2.2.3.2">𝐿</ci><ci id="S1.SSx4.p3.12.m12.1.2.2.3.3.cmml" xref="S1.SSx4.p3.12.m12.1.2.2.3.3">𝑎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx4.p3.12.m12.1c">\bigcup_{a\in A}\{a\}\times L_{a}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx4.p3.12.m12.1d">⋃ start_POSTSUBSCRIPT italic_a ∈ italic_A end_POSTSUBSCRIPT { italic_a } × italic_L start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT</annotation></semantics></math>. Note that <math alttext="A\times B=\sum_{a\in A}B" class="ltx_Math" display="inline" id="S1.SSx4.p3.13.m13.1"><semantics id="S1.SSx4.p3.13.m13.1a"><mrow id="S1.SSx4.p3.13.m13.1.1" xref="S1.SSx4.p3.13.m13.1.1.cmml"><mrow id="S1.SSx4.p3.13.m13.1.1.2" xref="S1.SSx4.p3.13.m13.1.1.2.cmml"><mi id="S1.SSx4.p3.13.m13.1.1.2.2" xref="S1.SSx4.p3.13.m13.1.1.2.2.cmml">A</mi><mo id="S1.SSx4.p3.13.m13.1.1.2.1" lspace="0.222em" rspace="0.222em" xref="S1.SSx4.p3.13.m13.1.1.2.1.cmml">×</mo><mi id="S1.SSx4.p3.13.m13.1.1.2.3" xref="S1.SSx4.p3.13.m13.1.1.2.3.cmml">B</mi></mrow><mo id="S1.SSx4.p3.13.m13.1.1.1" rspace="0.111em" xref="S1.SSx4.p3.13.m13.1.1.1.cmml">=</mo><mrow id="S1.SSx4.p3.13.m13.1.1.3" xref="S1.SSx4.p3.13.m13.1.1.3.cmml"><msub id="S1.SSx4.p3.13.m13.1.1.3.1" xref="S1.SSx4.p3.13.m13.1.1.3.1.cmml"><mo id="S1.SSx4.p3.13.m13.1.1.3.1.2" xref="S1.SSx4.p3.13.m13.1.1.3.1.2.cmml">∑</mo><mrow id="S1.SSx4.p3.13.m13.1.1.3.1.3" xref="S1.SSx4.p3.13.m13.1.1.3.1.3.cmml"><mi id="S1.SSx4.p3.13.m13.1.1.3.1.3.2" xref="S1.SSx4.p3.13.m13.1.1.3.1.3.2.cmml">a</mi><mo id="S1.SSx4.p3.13.m13.1.1.3.1.3.1" xref="S1.SSx4.p3.13.m13.1.1.3.1.3.1.cmml">∈</mo><mi id="S1.SSx4.p3.13.m13.1.1.3.1.3.3" xref="S1.SSx4.p3.13.m13.1.1.3.1.3.3.cmml">A</mi></mrow></msub><mi id="S1.SSx4.p3.13.m13.1.1.3.2" xref="S1.SSx4.p3.13.m13.1.1.3.2.cmml">B</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SSx4.p3.13.m13.1b"><apply id="S1.SSx4.p3.13.m13.1.1.cmml" xref="S1.SSx4.p3.13.m13.1.1"><eq id="S1.SSx4.p3.13.m13.1.1.1.cmml" xref="S1.SSx4.p3.13.m13.1.1.1"></eq><apply id="S1.SSx4.p3.13.m13.1.1.2.cmml" xref="S1.SSx4.p3.13.m13.1.1.2"><times id="S1.SSx4.p3.13.m13.1.1.2.1.cmml" xref="S1.SSx4.p3.13.m13.1.1.2.1"></times><ci id="S1.SSx4.p3.13.m13.1.1.2.2.cmml" xref="S1.SSx4.p3.13.m13.1.1.2.2">𝐴</ci><ci id="S1.SSx4.p3.13.m13.1.1.2.3.cmml" xref="S1.SSx4.p3.13.m13.1.1.2.3">𝐵</ci></apply><apply id="S1.SSx4.p3.13.m13.1.1.3.cmml" xref="S1.SSx4.p3.13.m13.1.1.3"><apply id="S1.SSx4.p3.13.m13.1.1.3.1.cmml" xref="S1.SSx4.p3.13.m13.1.1.3.1"><csymbol cd="ambiguous" id="S1.SSx4.p3.13.m13.1.1.3.1.1.cmml" xref="S1.SSx4.p3.13.m13.1.1.3.1">subscript</csymbol><sum id="S1.SSx4.p3.13.m13.1.1.3.1.2.cmml" xref="S1.SSx4.p3.13.m13.1.1.3.1.2"></sum><apply id="S1.SSx4.p3.13.m13.1.1.3.1.3.cmml" xref="S1.SSx4.p3.13.m13.1.1.3.1.3"><in id="S1.SSx4.p3.13.m13.1.1.3.1.3.1.cmml" xref="S1.SSx4.p3.13.m13.1.1.3.1.3.1"></in><ci id="S1.SSx4.p3.13.m13.1.1.3.1.3.2.cmml" xref="S1.SSx4.p3.13.m13.1.1.3.1.3.2">𝑎</ci><ci id="S1.SSx4.p3.13.m13.1.1.3.1.3.3.cmml" xref="S1.SSx4.p3.13.m13.1.1.3.1.3.3">𝐴</ci></apply></apply><ci id="S1.SSx4.p3.13.m13.1.1.3.2.cmml" xref="S1.SSx4.p3.13.m13.1.1.3.2">𝐵</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx4.p3.13.m13.1c">A\times B=\sum_{a\in A}B</annotation><annotation encoding="application/x-llamapun" id="S1.SSx4.p3.13.m13.1d">italic_A × italic_B = ∑ start_POSTSUBSCRIPT italic_a ∈ italic_A end_POSTSUBSCRIPT italic_B</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S1.SSx4.p4"> <p class="ltx_p" id="S1.SSx4.p4.18">An <em class="ltx_emph ltx_font_italic" id="S1.SSx4.p4.18.1">interval</em> is any nonempty set <math alttext="I\subseteq A" class="ltx_Math" display="inline" id="S1.SSx4.p4.1.m1.1"><semantics id="S1.SSx4.p4.1.m1.1a"><mrow id="S1.SSx4.p4.1.m1.1.1" xref="S1.SSx4.p4.1.m1.1.1.cmml"><mi id="S1.SSx4.p4.1.m1.1.1.2" xref="S1.SSx4.p4.1.m1.1.1.2.cmml">I</mi><mo id="S1.SSx4.p4.1.m1.1.1.1" xref="S1.SSx4.p4.1.m1.1.1.1.cmml">⊆</mo><mi id="S1.SSx4.p4.1.m1.1.1.3" xref="S1.SSx4.p4.1.m1.1.1.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.SSx4.p4.1.m1.1b"><apply id="S1.SSx4.p4.1.m1.1.1.cmml" xref="S1.SSx4.p4.1.m1.1.1"><subset id="S1.SSx4.p4.1.m1.1.1.1.cmml" xref="S1.SSx4.p4.1.m1.1.1.1"></subset><ci id="S1.SSx4.p4.1.m1.1.1.2.cmml" xref="S1.SSx4.p4.1.m1.1.1.2">𝐼</ci><ci id="S1.SSx4.p4.1.m1.1.1.3.cmml" xref="S1.SSx4.p4.1.m1.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx4.p4.1.m1.1c">I\subseteq A</annotation><annotation encoding="application/x-llamapun" id="S1.SSx4.p4.1.m1.1d">italic_I ⊆ italic_A</annotation></semantics></math> that is <em class="ltx_emph ltx_font_italic" id="S1.SSx4.p4.18.2">convex</em> in the sense that if <math alttext="a<_{A}b" class="ltx_Math" display="inline" id="S1.SSx4.p4.2.m2.1"><semantics id="S1.SSx4.p4.2.m2.1a"><mrow id="S1.SSx4.p4.2.m2.1.1" xref="S1.SSx4.p4.2.m2.1.1.cmml"><mi id="S1.SSx4.p4.2.m2.1.1.2" xref="S1.SSx4.p4.2.m2.1.1.2.cmml">a</mi><msub id="S1.SSx4.p4.2.m2.1.1.1" xref="S1.SSx4.p4.2.m2.1.1.1.cmml"><mo id="S1.SSx4.p4.2.m2.1.1.1.2" xref="S1.SSx4.p4.2.m2.1.1.1.2.cmml"><</mo><mi id="S1.SSx4.p4.2.m2.1.1.1.3" xref="S1.SSx4.p4.2.m2.1.1.1.3.cmml">A</mi></msub><mi id="S1.SSx4.p4.2.m2.1.1.3" xref="S1.SSx4.p4.2.m2.1.1.3.cmml">b</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.SSx4.p4.2.m2.1b"><apply id="S1.SSx4.p4.2.m2.1.1.cmml" xref="S1.SSx4.p4.2.m2.1.1"><apply id="S1.SSx4.p4.2.m2.1.1.1.cmml" xref="S1.SSx4.p4.2.m2.1.1.1"><csymbol cd="ambiguous" id="S1.SSx4.p4.2.m2.1.1.1.1.cmml" xref="S1.SSx4.p4.2.m2.1.1.1">subscript</csymbol><lt id="S1.SSx4.p4.2.m2.1.1.1.2.cmml" xref="S1.SSx4.p4.2.m2.1.1.1.2"></lt><ci id="S1.SSx4.p4.2.m2.1.1.1.3.cmml" xref="S1.SSx4.p4.2.m2.1.1.1.3">𝐴</ci></apply><ci id="S1.SSx4.p4.2.m2.1.1.2.cmml" xref="S1.SSx4.p4.2.m2.1.1.2">𝑎</ci><ci id="S1.SSx4.p4.2.m2.1.1.3.cmml" xref="S1.SSx4.p4.2.m2.1.1.3">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx4.p4.2.m2.1c">a<_{A}b</annotation><annotation encoding="application/x-llamapun" id="S1.SSx4.p4.2.m2.1d">italic_a < start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_b</annotation></semantics></math> are in <math alttext="I" class="ltx_Math" display="inline" id="S1.SSx4.p4.3.m3.1"><semantics id="S1.SSx4.p4.3.m3.1a"><mi id="S1.SSx4.p4.3.m3.1.1" xref="S1.SSx4.p4.3.m3.1.1.cmml">I</mi><annotation-xml encoding="MathML-Content" id="S1.SSx4.p4.3.m3.1b"><ci id="S1.SSx4.p4.3.m3.1.1.cmml" xref="S1.SSx4.p4.3.m3.1.1">𝐼</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx4.p4.3.m3.1c">I</annotation><annotation encoding="application/x-llamapun" id="S1.SSx4.p4.3.m3.1d">italic_I</annotation></semantics></math>, then <math alttext="{[a,b]}_{A}\subseteq I" class="ltx_Math" display="inline" id="S1.SSx4.p4.4.m4.2"><semantics id="S1.SSx4.p4.4.m4.2a"><mrow id="S1.SSx4.p4.4.m4.2.3" xref="S1.SSx4.p4.4.m4.2.3.cmml"><msub id="S1.SSx4.p4.4.m4.2.3.2" xref="S1.SSx4.p4.4.m4.2.3.2.cmml"><mrow id="S1.SSx4.p4.4.m4.2.3.2.2.2" xref="S1.SSx4.p4.4.m4.2.3.2.2.1.cmml"><mo id="S1.SSx4.p4.4.m4.2.3.2.2.2.1" stretchy="false" xref="S1.SSx4.p4.4.m4.2.3.2.2.1.cmml">[</mo><mi id="S1.SSx4.p4.4.m4.1.1" xref="S1.SSx4.p4.4.m4.1.1.cmml">a</mi><mo id="S1.SSx4.p4.4.m4.2.3.2.2.2.2" xref="S1.SSx4.p4.4.m4.2.3.2.2.1.cmml">,</mo><mi id="S1.SSx4.p4.4.m4.2.2" xref="S1.SSx4.p4.4.m4.2.2.cmml">b</mi><mo id="S1.SSx4.p4.4.m4.2.3.2.2.2.3" stretchy="false" xref="S1.SSx4.p4.4.m4.2.3.2.2.1.cmml">]</mo></mrow><mi id="S1.SSx4.p4.4.m4.2.3.2.3" xref="S1.SSx4.p4.4.m4.2.3.2.3.cmml">A</mi></msub><mo id="S1.SSx4.p4.4.m4.2.3.1" xref="S1.SSx4.p4.4.m4.2.3.1.cmml">⊆</mo><mi id="S1.SSx4.p4.4.m4.2.3.3" xref="S1.SSx4.p4.4.m4.2.3.3.cmml">I</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.SSx4.p4.4.m4.2b"><apply id="S1.SSx4.p4.4.m4.2.3.cmml" xref="S1.SSx4.p4.4.m4.2.3"><subset id="S1.SSx4.p4.4.m4.2.3.1.cmml" xref="S1.SSx4.p4.4.m4.2.3.1"></subset><apply id="S1.SSx4.p4.4.m4.2.3.2.cmml" xref="S1.SSx4.p4.4.m4.2.3.2"><csymbol cd="ambiguous" id="S1.SSx4.p4.4.m4.2.3.2.1.cmml" xref="S1.SSx4.p4.4.m4.2.3.2">subscript</csymbol><interval closure="closed" id="S1.SSx4.p4.4.m4.2.3.2.2.1.cmml" xref="S1.SSx4.p4.4.m4.2.3.2.2.2"><ci id="S1.SSx4.p4.4.m4.1.1.cmml" xref="S1.SSx4.p4.4.m4.1.1">𝑎</ci><ci id="S1.SSx4.p4.4.m4.2.2.cmml" xref="S1.SSx4.p4.4.m4.2.2">𝑏</ci></interval><ci id="S1.SSx4.p4.4.m4.2.3.2.3.cmml" xref="S1.SSx4.p4.4.m4.2.3.2.3">𝐴</ci></apply><ci id="S1.SSx4.p4.4.m4.2.3.3.cmml" xref="S1.SSx4.p4.4.m4.2.3.3">𝐼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx4.p4.4.m4.2c">{[a,b]}_{A}\subseteq I</annotation><annotation encoding="application/x-llamapun" id="S1.SSx4.p4.4.m4.2d">[ italic_a , italic_b ] start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⊆ italic_I</annotation></semantics></math>, <math alttext="I" class="ltx_Math" display="inline" id="S1.SSx4.p4.5.m5.1"><semantics id="S1.SSx4.p4.5.m5.1a"><mi id="S1.SSx4.p4.5.m5.1.1" xref="S1.SSx4.p4.5.m5.1.1.cmml">I</mi><annotation-xml encoding="MathML-Content" id="S1.SSx4.p4.5.m5.1b"><ci id="S1.SSx4.p4.5.m5.1.1.cmml" xref="S1.SSx4.p4.5.m5.1.1">𝐼</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx4.p4.5.m5.1c">I</annotation><annotation encoding="application/x-llamapun" id="S1.SSx4.p4.5.m5.1d">italic_I</annotation></semantics></math> is called <em class="ltx_emph ltx_font_italic" id="S1.SSx4.p4.18.3">trivial</em> if <math alttext="|I|=1" class="ltx_Math" display="inline" id="S1.SSx4.p4.6.m6.1"><semantics id="S1.SSx4.p4.6.m6.1a"><mrow id="S1.SSx4.p4.6.m6.1.2" xref="S1.SSx4.p4.6.m6.1.2.cmml"><mrow id="S1.SSx4.p4.6.m6.1.2.2.2" xref="S1.SSx4.p4.6.m6.1.2.2.1.cmml"><mo id="S1.SSx4.p4.6.m6.1.2.2.2.1" stretchy="false" xref="S1.SSx4.p4.6.m6.1.2.2.1.1.cmml">|</mo><mi id="S1.SSx4.p4.6.m6.1.1" xref="S1.SSx4.p4.6.m6.1.1.cmml">I</mi><mo id="S1.SSx4.p4.6.m6.1.2.2.2.2" stretchy="false" xref="S1.SSx4.p4.6.m6.1.2.2.1.1.cmml">|</mo></mrow><mo id="S1.SSx4.p4.6.m6.1.2.1" xref="S1.SSx4.p4.6.m6.1.2.1.cmml">=</mo><mn id="S1.SSx4.p4.6.m6.1.2.3" xref="S1.SSx4.p4.6.m6.1.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.SSx4.p4.6.m6.1b"><apply id="S1.SSx4.p4.6.m6.1.2.cmml" xref="S1.SSx4.p4.6.m6.1.2"><eq id="S1.SSx4.p4.6.m6.1.2.1.cmml" xref="S1.SSx4.p4.6.m6.1.2.1"></eq><apply id="S1.SSx4.p4.6.m6.1.2.2.1.cmml" xref="S1.SSx4.p4.6.m6.1.2.2.2"><abs id="S1.SSx4.p4.6.m6.1.2.2.1.1.cmml" xref="S1.SSx4.p4.6.m6.1.2.2.2.1"></abs><ci id="S1.SSx4.p4.6.m6.1.1.cmml" xref="S1.SSx4.p4.6.m6.1.1">𝐼</ci></apply><cn id="S1.SSx4.p4.6.m6.1.2.3.cmml" type="integer" xref="S1.SSx4.p4.6.m6.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx4.p4.6.m6.1c">|I|=1</annotation><annotation encoding="application/x-llamapun" id="S1.SSx4.p4.6.m6.1d">| italic_I | = 1</annotation></semantics></math>. Examples of intervals are intervals of the form <math alttext="{[a,b]}_{A}" class="ltx_Math" display="inline" id="S1.SSx4.p4.7.m7.2"><semantics id="S1.SSx4.p4.7.m7.2a"><msub id="S1.SSx4.p4.7.m7.2.3" xref="S1.SSx4.p4.7.m7.2.3.cmml"><mrow id="S1.SSx4.p4.7.m7.2.3.2.2" xref="S1.SSx4.p4.7.m7.2.3.2.1.cmml"><mo id="S1.SSx4.p4.7.m7.2.3.2.2.1" stretchy="false" xref="S1.SSx4.p4.7.m7.2.3.2.1.cmml">[</mo><mi id="S1.SSx4.p4.7.m7.1.1" xref="S1.SSx4.p4.7.m7.1.1.cmml">a</mi><mo id="S1.SSx4.p4.7.m7.2.3.2.2.2" xref="S1.SSx4.p4.7.m7.2.3.2.1.cmml">,</mo><mi id="S1.SSx4.p4.7.m7.2.2" xref="S1.SSx4.p4.7.m7.2.2.cmml">b</mi><mo id="S1.SSx4.p4.7.m7.2.3.2.2.3" stretchy="false" xref="S1.SSx4.p4.7.m7.2.3.2.1.cmml">]</mo></mrow><mi id="S1.SSx4.p4.7.m7.2.3.3" xref="S1.SSx4.p4.7.m7.2.3.3.cmml">A</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SSx4.p4.7.m7.2b"><apply id="S1.SSx4.p4.7.m7.2.3.cmml" xref="S1.SSx4.p4.7.m7.2.3"><csymbol cd="ambiguous" id="S1.SSx4.p4.7.m7.2.3.1.cmml" xref="S1.SSx4.p4.7.m7.2.3">subscript</csymbol><interval closure="closed" id="S1.SSx4.p4.7.m7.2.3.2.1.cmml" xref="S1.SSx4.p4.7.m7.2.3.2.2"><ci id="S1.SSx4.p4.7.m7.1.1.cmml" xref="S1.SSx4.p4.7.m7.1.1">𝑎</ci><ci id="S1.SSx4.p4.7.m7.2.2.cmml" xref="S1.SSx4.p4.7.m7.2.2">𝑏</ci></interval><ci id="S1.SSx4.p4.7.m7.2.3.3.cmml" xref="S1.SSx4.p4.7.m7.2.3.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx4.p4.7.m7.2c">{[a,b]}_{A}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx4.p4.7.m7.2d">[ italic_a , italic_b ] start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="{\left]a,b\right[}_{A}" class="ltx_Math" display="inline" id="S1.SSx4.p4.8.m8.2"><semantics id="S1.SSx4.p4.8.m8.2a"><msub id="S1.SSx4.p4.8.m8.2.3" xref="S1.SSx4.p4.8.m8.2.3.cmml"><mrow id="S1.SSx4.p4.8.m8.2.3.2.2" xref="S1.SSx4.p4.8.m8.2.3.2.1.cmml"><mo id="S1.SSx4.p4.8.m8.2.3.2.2.1" rspace="0em" stretchy="true" xref="S1.SSx4.p4.8.m8.2.3.2.1.cmml">]</mo><mi id="S1.SSx4.p4.8.m8.1.1" xref="S1.SSx4.p4.8.m8.1.1.cmml">a</mi><mo id="S1.SSx4.p4.8.m8.2.3.2.2.2" xref="S1.SSx4.p4.8.m8.2.3.2.1.cmml">,</mo><mi id="S1.SSx4.p4.8.m8.2.2" xref="S1.SSx4.p4.8.m8.2.2.cmml">b</mi><mo id="S1.SSx4.p4.8.m8.2.3.2.2.3" lspace="0em" stretchy="true" xref="S1.SSx4.p4.8.m8.2.3.2.1.cmml">[</mo></mrow><mi id="S1.SSx4.p4.8.m8.2.3.3" xref="S1.SSx4.p4.8.m8.2.3.3.cmml">A</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SSx4.p4.8.m8.2b"><apply id="S1.SSx4.p4.8.m8.2.3.cmml" xref="S1.SSx4.p4.8.m8.2.3"><csymbol cd="ambiguous" id="S1.SSx4.p4.8.m8.2.3.1.cmml" xref="S1.SSx4.p4.8.m8.2.3">subscript</csymbol><list id="S1.SSx4.p4.8.m8.2.3.2.1.cmml" xref="S1.SSx4.p4.8.m8.2.3.2.2"><ci id="S1.SSx4.p4.8.m8.1.1.cmml" xref="S1.SSx4.p4.8.m8.1.1">𝑎</ci><ci id="S1.SSx4.p4.8.m8.2.2.cmml" xref="S1.SSx4.p4.8.m8.2.2">𝑏</ci></list><ci id="S1.SSx4.p4.8.m8.2.3.3.cmml" xref="S1.SSx4.p4.8.m8.2.3.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx4.p4.8.m8.2c">{\left]a,b\right[}_{A}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx4.p4.8.m8.2d">] italic_a , italic_b [ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="{\left[a,b\right[}_{A}" class="ltx_Math" display="inline" id="S1.SSx4.p4.9.m9.2"><semantics id="S1.SSx4.p4.9.m9.2a"><msub id="S1.SSx4.p4.9.m9.2.3" xref="S1.SSx4.p4.9.m9.2.3.cmml"><mrow id="S1.SSx4.p4.9.m9.2.3.2.2" xref="S1.SSx4.p4.9.m9.2.3.2.1.cmml"><mo id="S1.SSx4.p4.9.m9.2.3.2.2.1" xref="S1.SSx4.p4.9.m9.2.3.2.1.cmml">[</mo><mi id="S1.SSx4.p4.9.m9.1.1" xref="S1.SSx4.p4.9.m9.1.1.cmml">a</mi><mo id="S1.SSx4.p4.9.m9.2.3.2.2.2" xref="S1.SSx4.p4.9.m9.2.3.2.1.cmml">,</mo><mi id="S1.SSx4.p4.9.m9.2.2" xref="S1.SSx4.p4.9.m9.2.2.cmml">b</mi><mo id="S1.SSx4.p4.9.m9.2.3.2.2.3" lspace="0em" stretchy="true" xref="S1.SSx4.p4.9.m9.2.3.2.1.cmml">[</mo></mrow><mi id="S1.SSx4.p4.9.m9.2.3.3" xref="S1.SSx4.p4.9.m9.2.3.3.cmml">A</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SSx4.p4.9.m9.2b"><apply id="S1.SSx4.p4.9.m9.2.3.cmml" xref="S1.SSx4.p4.9.m9.2.3"><csymbol cd="ambiguous" id="S1.SSx4.p4.9.m9.2.3.1.cmml" xref="S1.SSx4.p4.9.m9.2.3">subscript</csymbol><list id="S1.SSx4.p4.9.m9.2.3.2.1.cmml" xref="S1.SSx4.p4.9.m9.2.3.2.2"><ci id="S1.SSx4.p4.9.m9.1.1.cmml" xref="S1.SSx4.p4.9.m9.1.1">𝑎</ci><ci id="S1.SSx4.p4.9.m9.2.2.cmml" xref="S1.SSx4.p4.9.m9.2.2">𝑏</ci></list><ci id="S1.SSx4.p4.9.m9.2.3.3.cmml" xref="S1.SSx4.p4.9.m9.2.3.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx4.p4.9.m9.2c">{\left[a,b\right[}_{A}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx4.p4.9.m9.2d">[ italic_a , italic_b [ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="{\left]a,b\right]}_{A}" class="ltx_Math" display="inline" id="S1.SSx4.p4.10.m10.2"><semantics id="S1.SSx4.p4.10.m10.2a"><msub id="S1.SSx4.p4.10.m10.2.3" xref="S1.SSx4.p4.10.m10.2.3.cmml"><mrow id="S1.SSx4.p4.10.m10.2.3.2.2" xref="S1.SSx4.p4.10.m10.2.3.2.1.cmml"><mo id="S1.SSx4.p4.10.m10.2.3.2.2.1" rspace="0em" stretchy="true" xref="S1.SSx4.p4.10.m10.2.3.2.1.cmml">]</mo><mi id="S1.SSx4.p4.10.m10.1.1" xref="S1.SSx4.p4.10.m10.1.1.cmml">a</mi><mo id="S1.SSx4.p4.10.m10.2.3.2.2.2" xref="S1.SSx4.p4.10.m10.2.3.2.1.cmml">,</mo><mi id="S1.SSx4.p4.10.m10.2.2" xref="S1.SSx4.p4.10.m10.2.2.cmml">b</mi><mo id="S1.SSx4.p4.10.m10.2.3.2.2.3" xref="S1.SSx4.p4.10.m10.2.3.2.1.cmml">]</mo></mrow><mi id="S1.SSx4.p4.10.m10.2.3.3" xref="S1.SSx4.p4.10.m10.2.3.3.cmml">A</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SSx4.p4.10.m10.2b"><apply id="S1.SSx4.p4.10.m10.2.3.cmml" xref="S1.SSx4.p4.10.m10.2.3"><csymbol cd="ambiguous" id="S1.SSx4.p4.10.m10.2.3.1.cmml" xref="S1.SSx4.p4.10.m10.2.3">subscript</csymbol><list id="S1.SSx4.p4.10.m10.2.3.2.1.cmml" xref="S1.SSx4.p4.10.m10.2.3.2.2"><ci id="S1.SSx4.p4.10.m10.1.1.cmml" xref="S1.SSx4.p4.10.m10.1.1">𝑎</ci><ci id="S1.SSx4.p4.10.m10.2.2.cmml" xref="S1.SSx4.p4.10.m10.2.2">𝑏</ci></list><ci id="S1.SSx4.p4.10.m10.2.3.3.cmml" xref="S1.SSx4.p4.10.m10.2.3.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx4.p4.10.m10.2c">{\left]a,b\right]}_{A}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx4.p4.10.m10.2d">] italic_a , italic_b ] start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT</annotation></semantics></math> which are defined as usual. We usually drop the subscripts if this causes no confusion. At some points we will allow <math alttext="a" class="ltx_Math" display="inline" id="S1.SSx4.p4.11.m11.1"><semantics id="S1.SSx4.p4.11.m11.1a"><mi id="S1.SSx4.p4.11.m11.1.1" xref="S1.SSx4.p4.11.m11.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S1.SSx4.p4.11.m11.1b"><ci id="S1.SSx4.p4.11.m11.1.1.cmml" xref="S1.SSx4.p4.11.m11.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx4.p4.11.m11.1c">a</annotation><annotation encoding="application/x-llamapun" id="S1.SSx4.p4.11.m11.1d">italic_a</annotation></semantics></math> and <math alttext="b" class="ltx_Math" display="inline" id="S1.SSx4.p4.12.m12.1"><semantics id="S1.SSx4.p4.12.m12.1a"><mi id="S1.SSx4.p4.12.m12.1.1" xref="S1.SSx4.p4.12.m12.1.1.cmml">b</mi><annotation-xml encoding="MathML-Content" id="S1.SSx4.p4.12.m12.1b"><ci id="S1.SSx4.p4.12.m12.1.1.cmml" xref="S1.SSx4.p4.12.m12.1.1">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx4.p4.12.m12.1c">b</annotation><annotation encoding="application/x-llamapun" id="S1.SSx4.p4.12.m12.1d">italic_b</annotation></semantics></math> to be <math alttext="+\infty" class="ltx_Math" display="inline" id="S1.SSx4.p4.13.m13.1"><semantics id="S1.SSx4.p4.13.m13.1a"><mrow id="S1.SSx4.p4.13.m13.1.1" xref="S1.SSx4.p4.13.m13.1.1.cmml"><mo id="S1.SSx4.p4.13.m13.1.1a" xref="S1.SSx4.p4.13.m13.1.1.cmml">+</mo><mi id="S1.SSx4.p4.13.m13.1.1.2" mathvariant="normal" xref="S1.SSx4.p4.13.m13.1.1.2.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.SSx4.p4.13.m13.1b"><apply id="S1.SSx4.p4.13.m13.1.1.cmml" xref="S1.SSx4.p4.13.m13.1.1"><plus id="S1.SSx4.p4.13.m13.1.1.1.cmml" xref="S1.SSx4.p4.13.m13.1.1"></plus><infinity id="S1.SSx4.p4.13.m13.1.1.2.cmml" xref="S1.SSx4.p4.13.m13.1.1.2"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx4.p4.13.m13.1c">+\infty</annotation><annotation encoding="application/x-llamapun" id="S1.SSx4.p4.13.m13.1d">+ ∞</annotation></semantics></math>/<math alttext="-\infty" class="ltx_Math" display="inline" id="S1.SSx4.p4.14.m14.1"><semantics id="S1.SSx4.p4.14.m14.1a"><mrow id="S1.SSx4.p4.14.m14.1.1" xref="S1.SSx4.p4.14.m14.1.1.cmml"><mo id="S1.SSx4.p4.14.m14.1.1a" xref="S1.SSx4.p4.14.m14.1.1.cmml">−</mo><mi id="S1.SSx4.p4.14.m14.1.1.2" mathvariant="normal" xref="S1.SSx4.p4.14.m14.1.1.2.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.SSx4.p4.14.m14.1b"><apply id="S1.SSx4.p4.14.m14.1.1.cmml" xref="S1.SSx4.p4.14.m14.1.1"><minus id="S1.SSx4.p4.14.m14.1.1.1.cmml" xref="S1.SSx4.p4.14.m14.1.1"></minus><infinity id="S1.SSx4.p4.14.m14.1.1.2.cmml" xref="S1.SSx4.p4.14.m14.1.1.2"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx4.p4.14.m14.1c">-\infty</annotation><annotation encoding="application/x-llamapun" id="S1.SSx4.p4.14.m14.1d">- ∞</annotation></semantics></math>, giving the obvious interpretation. For example <math alttext="\left[a,+\infty\right[" class="ltx_Math" display="inline" id="S1.SSx4.p4.15.m15.2"><semantics id="S1.SSx4.p4.15.m15.2a"><mrow id="S1.SSx4.p4.15.m15.2.2.1" xref="S1.SSx4.p4.15.m15.2.2.2.cmml"><mo id="S1.SSx4.p4.15.m15.2.2.1.2" xref="S1.SSx4.p4.15.m15.2.2.2.cmml">[</mo><mi id="S1.SSx4.p4.15.m15.1.1" xref="S1.SSx4.p4.15.m15.1.1.cmml">a</mi><mo id="S1.SSx4.p4.15.m15.2.2.1.3" xref="S1.SSx4.p4.15.m15.2.2.2.cmml">,</mo><mrow id="S1.SSx4.p4.15.m15.2.2.1.1" xref="S1.SSx4.p4.15.m15.2.2.1.1.cmml"><mo id="S1.SSx4.p4.15.m15.2.2.1.1a" xref="S1.SSx4.p4.15.m15.2.2.1.1.cmml">+</mo><mi id="S1.SSx4.p4.15.m15.2.2.1.1.2" mathvariant="normal" xref="S1.SSx4.p4.15.m15.2.2.1.1.2.cmml">∞</mi></mrow><mo id="S1.SSx4.p4.15.m15.2.2.1.4" lspace="0em" stretchy="true" xref="S1.SSx4.p4.15.m15.2.2.2.cmml">[</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.SSx4.p4.15.m15.2b"><list id="S1.SSx4.p4.15.m15.2.2.2.cmml" xref="S1.SSx4.p4.15.m15.2.2.1"><ci id="S1.SSx4.p4.15.m15.1.1.cmml" xref="S1.SSx4.p4.15.m15.1.1">𝑎</ci><apply id="S1.SSx4.p4.15.m15.2.2.1.1.cmml" xref="S1.SSx4.p4.15.m15.2.2.1.1"><plus id="S1.SSx4.p4.15.m15.2.2.1.1.1.cmml" xref="S1.SSx4.p4.15.m15.2.2.1.1"></plus><infinity id="S1.SSx4.p4.15.m15.2.2.1.1.2.cmml" xref="S1.SSx4.p4.15.m15.2.2.1.1.2"></infinity></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx4.p4.15.m15.2c">\left[a,+\infty\right[</annotation><annotation encoding="application/x-llamapun" id="S1.SSx4.p4.15.m15.2d">[ italic_a , + ∞ [</annotation></semantics></math> stands for <math alttext="\{b\in A:a<_{A}b\}" class="ltx_Math" display="inline" id="S1.SSx4.p4.16.m16.2"><semantics id="S1.SSx4.p4.16.m16.2a"><mrow id="S1.SSx4.p4.16.m16.2.2.2" xref="S1.SSx4.p4.16.m16.2.2.3.cmml"><mo id="S1.SSx4.p4.16.m16.2.2.2.3" stretchy="false" xref="S1.SSx4.p4.16.m16.2.2.3.1.cmml">{</mo><mrow id="S1.SSx4.p4.16.m16.1.1.1.1" xref="S1.SSx4.p4.16.m16.1.1.1.1.cmml"><mi id="S1.SSx4.p4.16.m16.1.1.1.1.2" xref="S1.SSx4.p4.16.m16.1.1.1.1.2.cmml">b</mi><mo id="S1.SSx4.p4.16.m16.1.1.1.1.1" xref="S1.SSx4.p4.16.m16.1.1.1.1.1.cmml">∈</mo><mi id="S1.SSx4.p4.16.m16.1.1.1.1.3" xref="S1.SSx4.p4.16.m16.1.1.1.1.3.cmml">A</mi></mrow><mo id="S1.SSx4.p4.16.m16.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S1.SSx4.p4.16.m16.2.2.3.1.cmml">:</mo><mrow id="S1.SSx4.p4.16.m16.2.2.2.2" xref="S1.SSx4.p4.16.m16.2.2.2.2.cmml"><mi id="S1.SSx4.p4.16.m16.2.2.2.2.2" xref="S1.SSx4.p4.16.m16.2.2.2.2.2.cmml">a</mi><msub id="S1.SSx4.p4.16.m16.2.2.2.2.1" xref="S1.SSx4.p4.16.m16.2.2.2.2.1.cmml"><mo id="S1.SSx4.p4.16.m16.2.2.2.2.1.2" xref="S1.SSx4.p4.16.m16.2.2.2.2.1.2.cmml"><</mo><mi id="S1.SSx4.p4.16.m16.2.2.2.2.1.3" xref="S1.SSx4.p4.16.m16.2.2.2.2.1.3.cmml">A</mi></msub><mi id="S1.SSx4.p4.16.m16.2.2.2.2.3" xref="S1.SSx4.p4.16.m16.2.2.2.2.3.cmml">b</mi></mrow><mo id="S1.SSx4.p4.16.m16.2.2.2.5" stretchy="false" xref="S1.SSx4.p4.16.m16.2.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.SSx4.p4.16.m16.2b"><apply id="S1.SSx4.p4.16.m16.2.2.3.cmml" xref="S1.SSx4.p4.16.m16.2.2.2"><csymbol cd="latexml" id="S1.SSx4.p4.16.m16.2.2.3.1.cmml" xref="S1.SSx4.p4.16.m16.2.2.2.3">conditional-set</csymbol><apply id="S1.SSx4.p4.16.m16.1.1.1.1.cmml" xref="S1.SSx4.p4.16.m16.1.1.1.1"><in id="S1.SSx4.p4.16.m16.1.1.1.1.1.cmml" xref="S1.SSx4.p4.16.m16.1.1.1.1.1"></in><ci id="S1.SSx4.p4.16.m16.1.1.1.1.2.cmml" xref="S1.SSx4.p4.16.m16.1.1.1.1.2">𝑏</ci><ci id="S1.SSx4.p4.16.m16.1.1.1.1.3.cmml" xref="S1.SSx4.p4.16.m16.1.1.1.1.3">𝐴</ci></apply><apply id="S1.SSx4.p4.16.m16.2.2.2.2.cmml" xref="S1.SSx4.p4.16.m16.2.2.2.2"><apply id="S1.SSx4.p4.16.m16.2.2.2.2.1.cmml" xref="S1.SSx4.p4.16.m16.2.2.2.2.1"><csymbol cd="ambiguous" id="S1.SSx4.p4.16.m16.2.2.2.2.1.1.cmml" xref="S1.SSx4.p4.16.m16.2.2.2.2.1">subscript</csymbol><lt id="S1.SSx4.p4.16.m16.2.2.2.2.1.2.cmml" xref="S1.SSx4.p4.16.m16.2.2.2.2.1.2"></lt><ci id="S1.SSx4.p4.16.m16.2.2.2.2.1.3.cmml" xref="S1.SSx4.p4.16.m16.2.2.2.2.1.3">𝐴</ci></apply><ci id="S1.SSx4.p4.16.m16.2.2.2.2.2.cmml" xref="S1.SSx4.p4.16.m16.2.2.2.2.2">𝑎</ci><ci id="S1.SSx4.p4.16.m16.2.2.2.2.3.cmml" xref="S1.SSx4.p4.16.m16.2.2.2.2.3">𝑏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx4.p4.16.m16.2c">\{b\in A:a<_{A}b\}</annotation><annotation encoding="application/x-llamapun" id="S1.SSx4.p4.16.m16.2d">{ italic_b ∈ italic_A : italic_a < start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_b }</annotation></semantics></math>. Note that these are not necessarily all the intervals of <math alttext="A" class="ltx_Math" display="inline" id="S1.SSx4.p4.17.m17.1"><semantics id="S1.SSx4.p4.17.m17.1a"><mi id="S1.SSx4.p4.17.m17.1.1" xref="S1.SSx4.p4.17.m17.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S1.SSx4.p4.17.m17.1b"><ci id="S1.SSx4.p4.17.m17.1.1.cmml" xref="S1.SSx4.p4.17.m17.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx4.p4.17.m17.1c">A</annotation><annotation encoding="application/x-llamapun" id="S1.SSx4.p4.17.m17.1d">italic_A</annotation></semantics></math>. For example <math alttext="\mathbb{Q}\cap[0,\sqrt{2}]" class="ltx_Math" display="inline" id="S1.SSx4.p4.18.m18.2"><semantics id="S1.SSx4.p4.18.m18.2a"><mrow id="S1.SSx4.p4.18.m18.2.3" xref="S1.SSx4.p4.18.m18.2.3.cmml"><mi id="S1.SSx4.p4.18.m18.2.3.2" xref="S1.SSx4.p4.18.m18.2.3.2.cmml">ℚ</mi><mo id="S1.SSx4.p4.18.m18.2.3.1" xref="S1.SSx4.p4.18.m18.2.3.1.cmml">∩</mo><mrow id="S1.SSx4.p4.18.m18.2.3.3.2" xref="S1.SSx4.p4.18.m18.2.3.3.1.cmml"><mo id="S1.SSx4.p4.18.m18.2.3.3.2.1" stretchy="false" xref="S1.SSx4.p4.18.m18.2.3.3.1.cmml">[</mo><mn id="S1.SSx4.p4.18.m18.1.1" xref="S1.SSx4.p4.18.m18.1.1.cmml">0</mn><mo id="S1.SSx4.p4.18.m18.2.3.3.2.2" xref="S1.SSx4.p4.18.m18.2.3.3.1.cmml">,</mo><msqrt id="S1.SSx4.p4.18.m18.2.2" xref="S1.SSx4.p4.18.m18.2.2.cmml"><mn id="S1.SSx4.p4.18.m18.2.2.2" xref="S1.SSx4.p4.18.m18.2.2.2.cmml">2</mn></msqrt><mo id="S1.SSx4.p4.18.m18.2.3.3.2.3" stretchy="false" xref="S1.SSx4.p4.18.m18.2.3.3.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SSx4.p4.18.m18.2b"><apply id="S1.SSx4.p4.18.m18.2.3.cmml" xref="S1.SSx4.p4.18.m18.2.3"><intersect id="S1.SSx4.p4.18.m18.2.3.1.cmml" xref="S1.SSx4.p4.18.m18.2.3.1"></intersect><ci id="S1.SSx4.p4.18.m18.2.3.2.cmml" xref="S1.SSx4.p4.18.m18.2.3.2">ℚ</ci><interval closure="closed" id="S1.SSx4.p4.18.m18.2.3.3.1.cmml" xref="S1.SSx4.p4.18.m18.2.3.3.2"><cn id="S1.SSx4.p4.18.m18.1.1.cmml" type="integer" xref="S1.SSx4.p4.18.m18.1.1">0</cn><apply id="S1.SSx4.p4.18.m18.2.2.cmml" xref="S1.SSx4.p4.18.m18.2.2"><root id="S1.SSx4.p4.18.m18.2.2a.cmml" xref="S1.SSx4.p4.18.m18.2.2"></root><cn id="S1.SSx4.p4.18.m18.2.2.2.cmml" type="integer" xref="S1.SSx4.p4.18.m18.2.2.2">2</cn></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SSx4.p4.18.m18.2c">\mathbb{Q}\cap[0,\sqrt{2}]</annotation><annotation encoding="application/x-llamapun" id="S1.SSx4.p4.18.m18.2d">blackboard_Q ∩ [ 0 , square-root start_ARG 2 end_ARG ]</annotation></semantics></math> is an interval that cannot be written in that form.</p> </div> </section> </section> <section class="ltx_section" id="S2"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">2. </span>Aronszajn and Countryman lines</h2> <div class="ltx_para" id="S2.p1"> <p class="ltx_p" id="S2.p1.2">In this section we review some known facts about Aronszajn lines, and some of their known properties under axioms such as <math alttext="\mathsf{MA}_{\aleph_{1}}" class="ltx_Math" display="inline" id="S2.p1.1.m1.1"><semantics id="S2.p1.1.m1.1a"><msub id="S2.p1.1.m1.1.1" xref="S2.p1.1.m1.1.1.cmml"><mi id="S2.p1.1.m1.1.1.2" xref="S2.p1.1.m1.1.1.2.cmml">𝖬𝖠</mi><msub id="S2.p1.1.m1.1.1.3" xref="S2.p1.1.m1.1.1.3.cmml"><mi id="S2.p1.1.m1.1.1.3.2" mathvariant="normal" xref="S2.p1.1.m1.1.1.3.2.cmml">ℵ</mi><mn id="S2.p1.1.m1.1.1.3.3" xref="S2.p1.1.m1.1.1.3.3.cmml">1</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S2.p1.1.m1.1b"><apply id="S2.p1.1.m1.1.1.cmml" xref="S2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S2.p1.1.m1.1.1.1.cmml" xref="S2.p1.1.m1.1.1">subscript</csymbol><ci id="S2.p1.1.m1.1.1.2.cmml" xref="S2.p1.1.m1.1.1.2">𝖬𝖠</ci><apply id="S2.p1.1.m1.1.1.3.cmml" xref="S2.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S2.p1.1.m1.1.1.3.1.cmml" xref="S2.p1.1.m1.1.1.3">subscript</csymbol><ci id="S2.p1.1.m1.1.1.3.2.cmml" xref="S2.p1.1.m1.1.1.3.2">ℵ</ci><cn id="S2.p1.1.m1.1.1.3.3.cmml" type="integer" xref="S2.p1.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.1.m1.1c">\mathsf{MA}_{\aleph_{1}}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.1.m1.1d">sansserif_MA start_POSTSUBSCRIPT roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\mathsf{PFA}" class="ltx_Math" display="inline" id="S2.p1.2.m2.1"><semantics id="S2.p1.2.m2.1a"><mi id="S2.p1.2.m2.1.1" xref="S2.p1.2.m2.1.1.cmml">𝖯𝖥𝖠</mi><annotation-xml encoding="MathML-Content" id="S2.p1.2.m2.1b"><ci id="S2.p1.2.m2.1.1.cmml" xref="S2.p1.2.m2.1.1">𝖯𝖥𝖠</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.2.m2.1c">\mathsf{PFA}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.2.m2.1d">sansserif_PFA</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="S2.Thmtheorem1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem1.1.1.1">Definition 2.1</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem1.2.2">.</span> </h6> <div class="ltx_para" id="S2.Thmtheorem1.p1"> <p class="ltx_p" id="S2.Thmtheorem1.p1.5">For a linear order <math alttext="L" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.1.m1.1"><semantics id="S2.Thmtheorem1.p1.1.m1.1a"><mi id="S2.Thmtheorem1.p1.1.m1.1.1" xref="S2.Thmtheorem1.p1.1.m1.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.1.m1.1b"><ci id="S2.Thmtheorem1.p1.1.m1.1.1.cmml" xref="S2.Thmtheorem1.p1.1.m1.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.1.m1.1c">L</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.1.m1.1d">italic_L</annotation></semantics></math> and <math alttext="X\subseteq L" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.2.m2.1"><semantics id="S2.Thmtheorem1.p1.2.m2.1a"><mrow id="S2.Thmtheorem1.p1.2.m2.1.1" xref="S2.Thmtheorem1.p1.2.m2.1.1.cmml"><mi id="S2.Thmtheorem1.p1.2.m2.1.1.2" xref="S2.Thmtheorem1.p1.2.m2.1.1.2.cmml">X</mi><mo id="S2.Thmtheorem1.p1.2.m2.1.1.1" xref="S2.Thmtheorem1.p1.2.m2.1.1.1.cmml">⊆</mo><mi id="S2.Thmtheorem1.p1.2.m2.1.1.3" xref="S2.Thmtheorem1.p1.2.m2.1.1.3.cmml">L</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.2.m2.1b"><apply id="S2.Thmtheorem1.p1.2.m2.1.1.cmml" xref="S2.Thmtheorem1.p1.2.m2.1.1"><subset id="S2.Thmtheorem1.p1.2.m2.1.1.1.cmml" xref="S2.Thmtheorem1.p1.2.m2.1.1.1"></subset><ci id="S2.Thmtheorem1.p1.2.m2.1.1.2.cmml" xref="S2.Thmtheorem1.p1.2.m2.1.1.2">𝑋</ci><ci id="S2.Thmtheorem1.p1.2.m2.1.1.3.cmml" xref="S2.Thmtheorem1.p1.2.m2.1.1.3">𝐿</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.2.m2.1c">X\subseteq L</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.2.m2.1d">italic_X ⊆ italic_L</annotation></semantics></math>, the <em class="ltx_emph ltx_font_italic" id="S2.Thmtheorem1.p1.5.1">complementary intervals</em> of <math alttext="L\setminus X" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.3.m3.1"><semantics id="S2.Thmtheorem1.p1.3.m3.1a"><mrow id="S2.Thmtheorem1.p1.3.m3.1.1" xref="S2.Thmtheorem1.p1.3.m3.1.1.cmml"><mi id="S2.Thmtheorem1.p1.3.m3.1.1.2" xref="S2.Thmtheorem1.p1.3.m3.1.1.2.cmml">L</mi><mo id="S2.Thmtheorem1.p1.3.m3.1.1.1" xref="S2.Thmtheorem1.p1.3.m3.1.1.1.cmml">∖</mo><mi id="S2.Thmtheorem1.p1.3.m3.1.1.3" xref="S2.Thmtheorem1.p1.3.m3.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.3.m3.1b"><apply id="S2.Thmtheorem1.p1.3.m3.1.1.cmml" xref="S2.Thmtheorem1.p1.3.m3.1.1"><setdiff id="S2.Thmtheorem1.p1.3.m3.1.1.1.cmml" xref="S2.Thmtheorem1.p1.3.m3.1.1.1"></setdiff><ci id="S2.Thmtheorem1.p1.3.m3.1.1.2.cmml" xref="S2.Thmtheorem1.p1.3.m3.1.1.2">𝐿</ci><ci id="S2.Thmtheorem1.p1.3.m3.1.1.3.cmml" xref="S2.Thmtheorem1.p1.3.m3.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.3.m3.1c">L\setminus X</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.3.m3.1d">italic_L ∖ italic_X</annotation></semantics></math> are the intervals of <math alttext="L" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.4.m4.1"><semantics id="S2.Thmtheorem1.p1.4.m4.1a"><mi id="S2.Thmtheorem1.p1.4.m4.1.1" xref="S2.Thmtheorem1.p1.4.m4.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.4.m4.1b"><ci id="S2.Thmtheorem1.p1.4.m4.1.1.cmml" xref="S2.Thmtheorem1.p1.4.m4.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.4.m4.1c">L</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.4.m4.1d">italic_L</annotation></semantics></math> which are maximal with respect to not having points in <math alttext="X" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.5.m5.1"><semantics id="S2.Thmtheorem1.p1.5.m5.1a"><mi id="S2.Thmtheorem1.p1.5.m5.1.1" xref="S2.Thmtheorem1.p1.5.m5.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.5.m5.1b"><ci id="S2.Thmtheorem1.p1.5.m5.1.1.cmml" xref="S2.Thmtheorem1.p1.5.m5.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.5.m5.1c">X</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.5.m5.1d">italic_X</annotation></semantics></math>.</p> </div> </div> <div class="ltx_para" id="S2.p2"> <p class="ltx_p" id="S2.p2.4">Note that the complementary intervals of <math alttext="L\setminus X" class="ltx_Math" display="inline" id="S2.p2.1.m1.1"><semantics id="S2.p2.1.m1.1a"><mrow id="S2.p2.1.m1.1.1" xref="S2.p2.1.m1.1.1.cmml"><mi id="S2.p2.1.m1.1.1.2" xref="S2.p2.1.m1.1.1.2.cmml">L</mi><mo id="S2.p2.1.m1.1.1.1" xref="S2.p2.1.m1.1.1.1.cmml">∖</mo><mi id="S2.p2.1.m1.1.1.3" xref="S2.p2.1.m1.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p2.1.m1.1b"><apply id="S2.p2.1.m1.1.1.cmml" xref="S2.p2.1.m1.1.1"><setdiff id="S2.p2.1.m1.1.1.1.cmml" xref="S2.p2.1.m1.1.1.1"></setdiff><ci id="S2.p2.1.m1.1.1.2.cmml" xref="S2.p2.1.m1.1.1.2">𝐿</ci><ci id="S2.p2.1.m1.1.1.3.cmml" xref="S2.p2.1.m1.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.1.m1.1c">L\setminus X</annotation><annotation encoding="application/x-llamapun" id="S2.p2.1.m1.1d">italic_L ∖ italic_X</annotation></semantics></math> always form a partition of <math alttext="L\setminus X" class="ltx_Math" display="inline" id="S2.p2.2.m2.1"><semantics id="S2.p2.2.m2.1a"><mrow id="S2.p2.2.m2.1.1" xref="S2.p2.2.m2.1.1.cmml"><mi id="S2.p2.2.m2.1.1.2" xref="S2.p2.2.m2.1.1.2.cmml">L</mi><mo id="S2.p2.2.m2.1.1.1" xref="S2.p2.2.m2.1.1.1.cmml">∖</mo><mi id="S2.p2.2.m2.1.1.3" xref="S2.p2.2.m2.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p2.2.m2.1b"><apply id="S2.p2.2.m2.1.1.cmml" xref="S2.p2.2.m2.1.1"><setdiff id="S2.p2.2.m2.1.1.1.cmml" xref="S2.p2.2.m2.1.1.1"></setdiff><ci id="S2.p2.2.m2.1.1.2.cmml" xref="S2.p2.2.m2.1.1.2">𝐿</ci><ci id="S2.p2.2.m2.1.1.3.cmml" xref="S2.p2.2.m2.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.2.m2.1c">L\setminus X</annotation><annotation encoding="application/x-llamapun" id="S2.p2.2.m2.1d">italic_L ∖ italic_X</annotation></semantics></math> into intervals, and that if <math alttext="L" class="ltx_Math" display="inline" id="S2.p2.3.m3.1"><semantics id="S2.p2.3.m3.1a"><mi id="S2.p2.3.m3.1.1" xref="S2.p2.3.m3.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S2.p2.3.m3.1b"><ci id="S2.p2.3.m3.1.1.cmml" xref="S2.p2.3.m3.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.3.m3.1c">L</annotation><annotation encoding="application/x-llamapun" id="S2.p2.3.m3.1d">italic_L</annotation></semantics></math> is Aronszajn and <math alttext="X" class="ltx_Math" display="inline" id="S2.p2.4.m4.1"><semantics id="S2.p2.4.m4.1a"><mi id="S2.p2.4.m4.1.1" xref="S2.p2.4.m4.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S2.p2.4.m4.1b"><ci id="S2.p2.4.m4.1.1.cmml" xref="S2.p2.4.m4.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.4.m4.1c">X</annotation><annotation encoding="application/x-llamapun" id="S2.p2.4.m4.1d">italic_X</annotation></semantics></math> is countable, then this partition is countable. Following <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib23" title="">23</a>]</cite> we make the following definition.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="S2.Thmtheorem2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem2.1.1.1">Definition 2.2</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem2.2.2">.</span> </h6> <div class="ltx_para" id="S2.Thmtheorem2.p1"> <p class="ltx_p" id="S2.Thmtheorem2.p1.6">Let <math alttext="A" class="ltx_Math" display="inline" id="S2.Thmtheorem2.p1.1.m1.1"><semantics id="S2.Thmtheorem2.p1.1.m1.1a"><mi id="S2.Thmtheorem2.p1.1.m1.1.1" xref="S2.Thmtheorem2.p1.1.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem2.p1.1.m1.1b"><ci id="S2.Thmtheorem2.p1.1.m1.1.1.cmml" xref="S2.Thmtheorem2.p1.1.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem2.p1.1.m1.1c">A</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem2.p1.1.m1.1d">italic_A</annotation></semantics></math> be an Aronszajn line. A <em class="ltx_emph ltx_font_italic" id="S2.Thmtheorem2.p1.6.1">decomposition for</em> <math alttext="A" class="ltx_Math" display="inline" id="S2.Thmtheorem2.p1.2.m2.1"><semantics id="S2.Thmtheorem2.p1.2.m2.1a"><mi id="S2.Thmtheorem2.p1.2.m2.1.1" xref="S2.Thmtheorem2.p1.2.m2.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem2.p1.2.m2.1b"><ci id="S2.Thmtheorem2.p1.2.m2.1.1.cmml" xref="S2.Thmtheorem2.p1.2.m2.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem2.p1.2.m2.1c">A</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem2.p1.2.m2.1d">italic_A</annotation></semantics></math> is a <math alttext="\subseteq" class="ltx_Math" display="inline" id="S2.Thmtheorem2.p1.3.m3.1"><semantics id="S2.Thmtheorem2.p1.3.m3.1a"><mo id="S2.Thmtheorem2.p1.3.m3.1.1" xref="S2.Thmtheorem2.p1.3.m3.1.1.cmml">⊆</mo><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem2.p1.3.m3.1b"><subset id="S2.Thmtheorem2.p1.3.m3.1.1.cmml" xref="S2.Thmtheorem2.p1.3.m3.1.1"></subset></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem2.p1.3.m3.1c">\subseteq</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem2.p1.3.m3.1d">⊆</annotation></semantics></math>-increasing and continuous (at limits is the union) sequence <math alttext="\langle D_{\xi}:\xi<\omega_{1}\rangle" class="ltx_math_unparsed" display="inline" id="S2.Thmtheorem2.p1.4.m4.1"><semantics id="S2.Thmtheorem2.p1.4.m4.1a"><mrow id="S2.Thmtheorem2.p1.4.m4.1b"><mo id="S2.Thmtheorem2.p1.4.m4.1.1" stretchy="false">⟨</mo><msub id="S2.Thmtheorem2.p1.4.m4.1.2"><mi id="S2.Thmtheorem2.p1.4.m4.1.2.2">D</mi><mi id="S2.Thmtheorem2.p1.4.m4.1.2.3">ξ</mi></msub><mo id="S2.Thmtheorem2.p1.4.m4.1.3" lspace="0.278em" rspace="0.278em">:</mo><mi id="S2.Thmtheorem2.p1.4.m4.1.4">ξ</mi><mo id="S2.Thmtheorem2.p1.4.m4.1.5"><</mo><msub id="S2.Thmtheorem2.p1.4.m4.1.6"><mi id="S2.Thmtheorem2.p1.4.m4.1.6.2">ω</mi><mn id="S2.Thmtheorem2.p1.4.m4.1.6.3">1</mn></msub><mo id="S2.Thmtheorem2.p1.4.m4.1.7" stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex" id="S2.Thmtheorem2.p1.4.m4.1c">\langle D_{\xi}:\xi<\omega_{1}\rangle</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem2.p1.4.m4.1d">⟨ italic_D start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT : italic_ξ < italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⟩</annotation></semantics></math> of countable subsets of <math alttext="A" class="ltx_Math" display="inline" id="S2.Thmtheorem2.p1.5.m5.1"><semantics id="S2.Thmtheorem2.p1.5.m5.1a"><mi id="S2.Thmtheorem2.p1.5.m5.1.1" xref="S2.Thmtheorem2.p1.5.m5.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem2.p1.5.m5.1b"><ci id="S2.Thmtheorem2.p1.5.m5.1.1.cmml" xref="S2.Thmtheorem2.p1.5.m5.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem2.p1.5.m5.1c">A</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem2.p1.5.m5.1d">italic_A</annotation></semantics></math> that covers <math alttext="A" class="ltx_Math" display="inline" id="S2.Thmtheorem2.p1.6.m6.1"><semantics id="S2.Thmtheorem2.p1.6.m6.1a"><mi id="S2.Thmtheorem2.p1.6.m6.1.1" xref="S2.Thmtheorem2.p1.6.m6.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem2.p1.6.m6.1b"><ci id="S2.Thmtheorem2.p1.6.m6.1.1.cmml" xref="S2.Thmtheorem2.p1.6.m6.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem2.p1.6.m6.1c">A</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem2.p1.6.m6.1d">italic_A</annotation></semantics></math>.</p> </div> </div> <div class="ltx_para" id="S2.p3"> <p class="ltx_p" id="S2.p3.1">An important fact is that any Aronszajn line has size <math alttext="\aleph_{1}" class="ltx_Math" display="inline" id="S2.p3.1.m1.1"><semantics id="S2.p3.1.m1.1a"><msub id="S2.p3.1.m1.1.1" xref="S2.p3.1.m1.1.1.cmml"><mi id="S2.p3.1.m1.1.1.2" mathvariant="normal" xref="S2.p3.1.m1.1.1.2.cmml">ℵ</mi><mn id="S2.p3.1.m1.1.1.3" xref="S2.p3.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.p3.1.m1.1b"><apply id="S2.p3.1.m1.1.1.cmml" xref="S2.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S2.p3.1.m1.1.1.1.cmml" xref="S2.p3.1.m1.1.1">subscript</csymbol><ci id="S2.p3.1.m1.1.1.2.cmml" xref="S2.p3.1.m1.1.1.2">ℵ</ci><cn id="S2.p3.1.m1.1.1.3.cmml" type="integer" xref="S2.p3.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.1.m1.1c">\aleph_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.p3.1.m1.1d">roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> (see for example <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib3" title="">3</a>, Corollary 4.3]</cite>), therefore decompositions always exist. The following seems to be folklore.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S2.Thmtheorem3"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem3.1.1.1">Lemma 2.3</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem3.2.2">.</span> </h6> <div class="ltx_para" id="S2.Thmtheorem3.p1"> <p class="ltx_p" id="S2.Thmtheorem3.p1.4">If <math alttext="A" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.1.m1.1"><semantics id="S2.Thmtheorem3.p1.1.m1.1a"><mi id="S2.Thmtheorem3.p1.1.m1.1.1" xref="S2.Thmtheorem3.p1.1.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.1.m1.1b"><ci id="S2.Thmtheorem3.p1.1.m1.1.1.cmml" xref="S2.Thmtheorem3.p1.1.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.1.m1.1c">A</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.1.m1.1d">italic_A</annotation></semantics></math> is Aronszajn and <math alttext="\langle D_{\xi}:\xi<\omega_{1}\rangle" class="ltx_math_unparsed" display="inline" id="S2.Thmtheorem3.p1.2.m2.1"><semantics id="S2.Thmtheorem3.p1.2.m2.1a"><mrow id="S2.Thmtheorem3.p1.2.m2.1b"><mo id="S2.Thmtheorem3.p1.2.m2.1.1" stretchy="false">⟨</mo><msub id="S2.Thmtheorem3.p1.2.m2.1.2"><mi id="S2.Thmtheorem3.p1.2.m2.1.2.2">D</mi><mi id="S2.Thmtheorem3.p1.2.m2.1.2.3">ξ</mi></msub><mo id="S2.Thmtheorem3.p1.2.m2.1.3" lspace="0.278em" rspace="0.278em">:</mo><mi id="S2.Thmtheorem3.p1.2.m2.1.4">ξ</mi><mo id="S2.Thmtheorem3.p1.2.m2.1.5"><</mo><msub id="S2.Thmtheorem3.p1.2.m2.1.6"><mi id="S2.Thmtheorem3.p1.2.m2.1.6.2">ω</mi><mn id="S2.Thmtheorem3.p1.2.m2.1.6.3">1</mn></msub><mo id="S2.Thmtheorem3.p1.2.m2.1.7" stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.2.m2.1c">\langle D_{\xi}:\xi<\omega_{1}\rangle</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.2.m2.1d">⟨ italic_D start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT : italic_ξ < italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⟩</annotation></semantics></math> is a decomposition for <math alttext="A" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.3.m3.1"><semantics id="S2.Thmtheorem3.p1.3.m3.1a"><mi id="S2.Thmtheorem3.p1.3.m3.1.1" xref="S2.Thmtheorem3.p1.3.m3.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.3.m3.1b"><ci id="S2.Thmtheorem3.p1.3.m3.1.1.cmml" xref="S2.Thmtheorem3.p1.3.m3.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.3.m3.1c">A</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.3.m3.1d">italic_A</annotation></semantics></math>, then <math alttext="T:=\bigcup_{\xi<\omega_{1}}\{I:I\text{ is a complementary interval of $A% \setminus D_{\xi}$}\}" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.4.m4.3"><semantics id="S2.Thmtheorem3.p1.4.m4.3a"><mrow id="S2.Thmtheorem3.p1.4.m4.3.3" xref="S2.Thmtheorem3.p1.4.m4.3.3.cmml"><mi id="S2.Thmtheorem3.p1.4.m4.3.3.3" xref="S2.Thmtheorem3.p1.4.m4.3.3.3.cmml">T</mi><mo id="S2.Thmtheorem3.p1.4.m4.3.3.2" lspace="0.278em" rspace="0.111em" xref="S2.Thmtheorem3.p1.4.m4.3.3.2.cmml">:=</mo><mrow id="S2.Thmtheorem3.p1.4.m4.3.3.1" xref="S2.Thmtheorem3.p1.4.m4.3.3.1.cmml"><msub id="S2.Thmtheorem3.p1.4.m4.3.3.1.2" xref="S2.Thmtheorem3.p1.4.m4.3.3.1.2.cmml"><mo id="S2.Thmtheorem3.p1.4.m4.3.3.1.2.2" rspace="0em" xref="S2.Thmtheorem3.p1.4.m4.3.3.1.2.2.cmml">⋃</mo><mrow id="S2.Thmtheorem3.p1.4.m4.3.3.1.2.3" xref="S2.Thmtheorem3.p1.4.m4.3.3.1.2.3.cmml"><mi id="S2.Thmtheorem3.p1.4.m4.3.3.1.2.3.2" xref="S2.Thmtheorem3.p1.4.m4.3.3.1.2.3.2.cmml">ξ</mi><mo id="S2.Thmtheorem3.p1.4.m4.3.3.1.2.3.1" xref="S2.Thmtheorem3.p1.4.m4.3.3.1.2.3.1.cmml"><</mo><msub id="S2.Thmtheorem3.p1.4.m4.3.3.1.2.3.3" xref="S2.Thmtheorem3.p1.4.m4.3.3.1.2.3.3.cmml"><mi id="S2.Thmtheorem3.p1.4.m4.3.3.1.2.3.3.2" xref="S2.Thmtheorem3.p1.4.m4.3.3.1.2.3.3.2.cmml">ω</mi><mn id="S2.Thmtheorem3.p1.4.m4.3.3.1.2.3.3.3" xref="S2.Thmtheorem3.p1.4.m4.3.3.1.2.3.3.3.cmml">1</mn></msub></mrow></msub><mrow id="S2.Thmtheorem3.p1.4.m4.3.3.1.1.1" xref="S2.Thmtheorem3.p1.4.m4.3.3.1.1.2.cmml"><mo id="S2.Thmtheorem3.p1.4.m4.3.3.1.1.1.2" stretchy="false" xref="S2.Thmtheorem3.p1.4.m4.3.3.1.1.2.1.cmml">{</mo><mi id="S2.Thmtheorem3.p1.4.m4.2.2" xref="S2.Thmtheorem3.p1.4.m4.2.2.cmml">I</mi><mo id="S2.Thmtheorem3.p1.4.m4.3.3.1.1.1.3" lspace="0.278em" rspace="0.278em" xref="S2.Thmtheorem3.p1.4.m4.3.3.1.1.2.1.cmml">:</mo><mrow id="S2.Thmtheorem3.p1.4.m4.3.3.1.1.1.1" xref="S2.Thmtheorem3.p1.4.m4.3.3.1.1.1.1.cmml"><mi id="S2.Thmtheorem3.p1.4.m4.3.3.1.1.1.1.2" xref="S2.Thmtheorem3.p1.4.m4.3.3.1.1.1.1.2.cmml">I</mi><mo id="S2.Thmtheorem3.p1.4.m4.3.3.1.1.1.1.1" xref="S2.Thmtheorem3.p1.4.m4.3.3.1.1.1.1.1.cmml">⁢</mo><mrow id="S2.Thmtheorem3.p1.4.m4.1.1.1" xref="S2.Thmtheorem3.p1.4.m4.1.1.1b.cmml"><mtext id="S2.Thmtheorem3.p1.4.m4.1.1.1a" xref="S2.Thmtheorem3.p1.4.m4.1.1.1b.cmml"> is a complementary interval of </mtext><mrow id="S2.Thmtheorem3.p1.4.m4.1.1.1.m1.1.1" xref="S2.Thmtheorem3.p1.4.m4.1.1.1.m1.1.1.cmml"><mi id="S2.Thmtheorem3.p1.4.m4.1.1.1.m1.1.1.2" xref="S2.Thmtheorem3.p1.4.m4.1.1.1.m1.1.1.2.cmml">A</mi><mo id="S2.Thmtheorem3.p1.4.m4.1.1.1.m1.1.1.1" xref="S2.Thmtheorem3.p1.4.m4.1.1.1.m1.1.1.1.cmml">∖</mo><msub id="S2.Thmtheorem3.p1.4.m4.1.1.1.m1.1.1.3" xref="S2.Thmtheorem3.p1.4.m4.1.1.1.m1.1.1.3.cmml"><mi id="S2.Thmtheorem3.p1.4.m4.1.1.1.m1.1.1.3.2" xref="S2.Thmtheorem3.p1.4.m4.1.1.1.m1.1.1.3.2.cmml">D</mi><mi id="S2.Thmtheorem3.p1.4.m4.1.1.1.m1.1.1.3.3" xref="S2.Thmtheorem3.p1.4.m4.1.1.1.m1.1.1.3.3.cmml">ξ</mi></msub></mrow></mrow></mrow><mo id="S2.Thmtheorem3.p1.4.m4.3.3.1.1.1.4" stretchy="false" xref="S2.Thmtheorem3.p1.4.m4.3.3.1.1.2.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.4.m4.3b"><apply id="S2.Thmtheorem3.p1.4.m4.3.3.cmml" xref="S2.Thmtheorem3.p1.4.m4.3.3"><csymbol cd="latexml" id="S2.Thmtheorem3.p1.4.m4.3.3.2.cmml" xref="S2.Thmtheorem3.p1.4.m4.3.3.2">assign</csymbol><ci id="S2.Thmtheorem3.p1.4.m4.3.3.3.cmml" xref="S2.Thmtheorem3.p1.4.m4.3.3.3">𝑇</ci><apply id="S2.Thmtheorem3.p1.4.m4.3.3.1.cmml" xref="S2.Thmtheorem3.p1.4.m4.3.3.1"><apply id="S2.Thmtheorem3.p1.4.m4.3.3.1.2.cmml" xref="S2.Thmtheorem3.p1.4.m4.3.3.1.2"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p1.4.m4.3.3.1.2.1.cmml" xref="S2.Thmtheorem3.p1.4.m4.3.3.1.2">subscript</csymbol><union id="S2.Thmtheorem3.p1.4.m4.3.3.1.2.2.cmml" xref="S2.Thmtheorem3.p1.4.m4.3.3.1.2.2"></union><apply id="S2.Thmtheorem3.p1.4.m4.3.3.1.2.3.cmml" xref="S2.Thmtheorem3.p1.4.m4.3.3.1.2.3"><lt id="S2.Thmtheorem3.p1.4.m4.3.3.1.2.3.1.cmml" xref="S2.Thmtheorem3.p1.4.m4.3.3.1.2.3.1"></lt><ci id="S2.Thmtheorem3.p1.4.m4.3.3.1.2.3.2.cmml" xref="S2.Thmtheorem3.p1.4.m4.3.3.1.2.3.2">𝜉</ci><apply id="S2.Thmtheorem3.p1.4.m4.3.3.1.2.3.3.cmml" xref="S2.Thmtheorem3.p1.4.m4.3.3.1.2.3.3"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p1.4.m4.3.3.1.2.3.3.1.cmml" xref="S2.Thmtheorem3.p1.4.m4.3.3.1.2.3.3">subscript</csymbol><ci id="S2.Thmtheorem3.p1.4.m4.3.3.1.2.3.3.2.cmml" xref="S2.Thmtheorem3.p1.4.m4.3.3.1.2.3.3.2">𝜔</ci><cn id="S2.Thmtheorem3.p1.4.m4.3.3.1.2.3.3.3.cmml" type="integer" xref="S2.Thmtheorem3.p1.4.m4.3.3.1.2.3.3.3">1</cn></apply></apply></apply><apply id="S2.Thmtheorem3.p1.4.m4.3.3.1.1.2.cmml" xref="S2.Thmtheorem3.p1.4.m4.3.3.1.1.1"><csymbol cd="latexml" id="S2.Thmtheorem3.p1.4.m4.3.3.1.1.2.1.cmml" xref="S2.Thmtheorem3.p1.4.m4.3.3.1.1.1.2">conditional-set</csymbol><ci id="S2.Thmtheorem3.p1.4.m4.2.2.cmml" xref="S2.Thmtheorem3.p1.4.m4.2.2">𝐼</ci><apply id="S2.Thmtheorem3.p1.4.m4.3.3.1.1.1.1.cmml" xref="S2.Thmtheorem3.p1.4.m4.3.3.1.1.1.1"><times id="S2.Thmtheorem3.p1.4.m4.3.3.1.1.1.1.1.cmml" xref="S2.Thmtheorem3.p1.4.m4.3.3.1.1.1.1.1"></times><ci id="S2.Thmtheorem3.p1.4.m4.3.3.1.1.1.1.2.cmml" xref="S2.Thmtheorem3.p1.4.m4.3.3.1.1.1.1.2">𝐼</ci><ci id="S2.Thmtheorem3.p1.4.m4.1.1.1b.cmml" xref="S2.Thmtheorem3.p1.4.m4.1.1.1"><mrow id="S2.Thmtheorem3.p1.4.m4.1.1.1.cmml" xref="S2.Thmtheorem3.p1.4.m4.1.1.1"><mtext id="S2.Thmtheorem3.p1.4.m4.1.1.1a.cmml" xref="S2.Thmtheorem3.p1.4.m4.1.1.1"> is a complementary interval of </mtext><mrow id="S2.Thmtheorem3.p1.4.m4.1.1.1.m1.1.1.cmml" xref="S2.Thmtheorem3.p1.4.m4.1.1.1.m1.1.1"><mi id="S2.Thmtheorem3.p1.4.m4.1.1.1.m1.1.1.2.cmml" xref="S2.Thmtheorem3.p1.4.m4.1.1.1.m1.1.1.2">A</mi><mo id="S2.Thmtheorem3.p1.4.m4.1.1.1.m1.1.1.1.cmml" xref="S2.Thmtheorem3.p1.4.m4.1.1.1.m1.1.1.1">∖</mo><msub id="S2.Thmtheorem3.p1.4.m4.1.1.1.m1.1.1.3.cmml" xref="S2.Thmtheorem3.p1.4.m4.1.1.1.m1.1.1.3"><mi id="S2.Thmtheorem3.p1.4.m4.1.1.1.m1.1.1.3.2.cmml" xref="S2.Thmtheorem3.p1.4.m4.1.1.1.m1.1.1.3.2">D</mi><mi id="S2.Thmtheorem3.p1.4.m4.1.1.1.m1.1.1.3.3.cmml" xref="S2.Thmtheorem3.p1.4.m4.1.1.1.m1.1.1.3.3">ξ</mi></msub></mrow></mrow></ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.4.m4.3c">T:=\bigcup_{\xi<\omega_{1}}\{I:I\text{ is a complementary interval of $A% \setminus D_{\xi}$}\}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.4.m4.3d">italic_T := ⋃ start_POSTSUBSCRIPT italic_ξ < italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT { italic_I : italic_I is a complementary interval of italic_A ∖ italic_D start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT }</annotation></semantics></math> ordered by reverse inclusion is an Aronszajn tree.</p> </div> </div> <div class="ltx_para" id="S2.p4"> <p class="ltx_p" id="S2.p4.1">Following <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib16" title="">16</a>]</cite> and <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib1" title="">1</a>]</cite> we make the following definition.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="S2.Thmtheorem4"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem4.1.1.1">Definition 2.4</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem4.2.2">.</span> </h6> <div class="ltx_para" id="S2.Thmtheorem4.p1"> <p class="ltx_p" id="S2.Thmtheorem4.p1.6">An Aronszajn line <math alttext="A" class="ltx_Math" display="inline" id="S2.Thmtheorem4.p1.1.m1.1"><semantics id="S2.Thmtheorem4.p1.1.m1.1a"><mi id="S2.Thmtheorem4.p1.1.m1.1.1" xref="S2.Thmtheorem4.p1.1.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem4.p1.1.m1.1b"><ci id="S2.Thmtheorem4.p1.1.m1.1.1.cmml" xref="S2.Thmtheorem4.p1.1.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem4.p1.1.m1.1c">A</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem4.p1.1.m1.1d">italic_A</annotation></semantics></math> is called <em class="ltx_emph ltx_font_italic" id="S2.Thmtheorem4.p1.6.1">non stationary</em> if it has a decomposition <math alttext="\langle D_{\xi}:\xi<\omega_{1}\rangle" class="ltx_math_unparsed" display="inline" id="S2.Thmtheorem4.p1.2.m2.1"><semantics id="S2.Thmtheorem4.p1.2.m2.1a"><mrow id="S2.Thmtheorem4.p1.2.m2.1b"><mo id="S2.Thmtheorem4.p1.2.m2.1.1" stretchy="false">⟨</mo><msub id="S2.Thmtheorem4.p1.2.m2.1.2"><mi id="S2.Thmtheorem4.p1.2.m2.1.2.2">D</mi><mi id="S2.Thmtheorem4.p1.2.m2.1.2.3">ξ</mi></msub><mo id="S2.Thmtheorem4.p1.2.m2.1.3" lspace="0.278em" rspace="0.278em">:</mo><mi id="S2.Thmtheorem4.p1.2.m2.1.4">ξ</mi><mo id="S2.Thmtheorem4.p1.2.m2.1.5"><</mo><msub id="S2.Thmtheorem4.p1.2.m2.1.6"><mi id="S2.Thmtheorem4.p1.2.m2.1.6.2">ω</mi><mn id="S2.Thmtheorem4.p1.2.m2.1.6.3">1</mn></msub><mo id="S2.Thmtheorem4.p1.2.m2.1.7" stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex" id="S2.Thmtheorem4.p1.2.m2.1c">\langle D_{\xi}:\xi<\omega_{1}\rangle</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem4.p1.2.m2.1d">⟨ italic_D start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT : italic_ξ < italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⟩</annotation></semantics></math> such that for every <math alttext="\xi<\omega_{1}" class="ltx_Math" display="inline" id="S2.Thmtheorem4.p1.3.m3.1"><semantics id="S2.Thmtheorem4.p1.3.m3.1a"><mrow id="S2.Thmtheorem4.p1.3.m3.1.1" xref="S2.Thmtheorem4.p1.3.m3.1.1.cmml"><mi id="S2.Thmtheorem4.p1.3.m3.1.1.2" xref="S2.Thmtheorem4.p1.3.m3.1.1.2.cmml">ξ</mi><mo id="S2.Thmtheorem4.p1.3.m3.1.1.1" xref="S2.Thmtheorem4.p1.3.m3.1.1.1.cmml"><</mo><msub id="S2.Thmtheorem4.p1.3.m3.1.1.3" xref="S2.Thmtheorem4.p1.3.m3.1.1.3.cmml"><mi id="S2.Thmtheorem4.p1.3.m3.1.1.3.2" xref="S2.Thmtheorem4.p1.3.m3.1.1.3.2.cmml">ω</mi><mn id="S2.Thmtheorem4.p1.3.m3.1.1.3.3" xref="S2.Thmtheorem4.p1.3.m3.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem4.p1.3.m3.1b"><apply id="S2.Thmtheorem4.p1.3.m3.1.1.cmml" xref="S2.Thmtheorem4.p1.3.m3.1.1"><lt id="S2.Thmtheorem4.p1.3.m3.1.1.1.cmml" xref="S2.Thmtheorem4.p1.3.m3.1.1.1"></lt><ci id="S2.Thmtheorem4.p1.3.m3.1.1.2.cmml" xref="S2.Thmtheorem4.p1.3.m3.1.1.2">𝜉</ci><apply id="S2.Thmtheorem4.p1.3.m3.1.1.3.cmml" xref="S2.Thmtheorem4.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S2.Thmtheorem4.p1.3.m3.1.1.3.1.cmml" xref="S2.Thmtheorem4.p1.3.m3.1.1.3">subscript</csymbol><ci id="S2.Thmtheorem4.p1.3.m3.1.1.3.2.cmml" xref="S2.Thmtheorem4.p1.3.m3.1.1.3.2">𝜔</ci><cn id="S2.Thmtheorem4.p1.3.m3.1.1.3.3.cmml" type="integer" xref="S2.Thmtheorem4.p1.3.m3.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem4.p1.3.m3.1c">\xi<\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem4.p1.3.m3.1d">italic_ξ < italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, every complementary interval of <math alttext="A\setminus D_{\xi}" class="ltx_Math" display="inline" id="S2.Thmtheorem4.p1.4.m4.1"><semantics id="S2.Thmtheorem4.p1.4.m4.1a"><mrow id="S2.Thmtheorem4.p1.4.m4.1.1" xref="S2.Thmtheorem4.p1.4.m4.1.1.cmml"><mi id="S2.Thmtheorem4.p1.4.m4.1.1.2" xref="S2.Thmtheorem4.p1.4.m4.1.1.2.cmml">A</mi><mo id="S2.Thmtheorem4.p1.4.m4.1.1.1" xref="S2.Thmtheorem4.p1.4.m4.1.1.1.cmml">∖</mo><msub id="S2.Thmtheorem4.p1.4.m4.1.1.3" xref="S2.Thmtheorem4.p1.4.m4.1.1.3.cmml"><mi id="S2.Thmtheorem4.p1.4.m4.1.1.3.2" xref="S2.Thmtheorem4.p1.4.m4.1.1.3.2.cmml">D</mi><mi id="S2.Thmtheorem4.p1.4.m4.1.1.3.3" xref="S2.Thmtheorem4.p1.4.m4.1.1.3.3.cmml">ξ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem4.p1.4.m4.1b"><apply id="S2.Thmtheorem4.p1.4.m4.1.1.cmml" xref="S2.Thmtheorem4.p1.4.m4.1.1"><setdiff id="S2.Thmtheorem4.p1.4.m4.1.1.1.cmml" xref="S2.Thmtheorem4.p1.4.m4.1.1.1"></setdiff><ci id="S2.Thmtheorem4.p1.4.m4.1.1.2.cmml" xref="S2.Thmtheorem4.p1.4.m4.1.1.2">𝐴</ci><apply id="S2.Thmtheorem4.p1.4.m4.1.1.3.cmml" xref="S2.Thmtheorem4.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="S2.Thmtheorem4.p1.4.m4.1.1.3.1.cmml" xref="S2.Thmtheorem4.p1.4.m4.1.1.3">subscript</csymbol><ci id="S2.Thmtheorem4.p1.4.m4.1.1.3.2.cmml" xref="S2.Thmtheorem4.p1.4.m4.1.1.3.2">𝐷</ci><ci id="S2.Thmtheorem4.p1.4.m4.1.1.3.3.cmml" xref="S2.Thmtheorem4.p1.4.m4.1.1.3.3">𝜉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem4.p1.4.m4.1c">A\setminus D_{\xi}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem4.p1.4.m4.1d">italic_A ∖ italic_D start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT</annotation></semantics></math> is without endpoints. <math alttext="A" class="ltx_Math" display="inline" id="S2.Thmtheorem4.p1.5.m5.1"><semantics id="S2.Thmtheorem4.p1.5.m5.1a"><mi id="S2.Thmtheorem4.p1.5.m5.1.1" xref="S2.Thmtheorem4.p1.5.m5.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem4.p1.5.m5.1b"><ci id="S2.Thmtheorem4.p1.5.m5.1.1.cmml" xref="S2.Thmtheorem4.p1.5.m5.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem4.p1.5.m5.1c">A</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem4.p1.5.m5.1d">italic_A</annotation></semantics></math> is called <em class="ltx_emph ltx_font_italic" id="S2.Thmtheorem4.p1.6.2">normal</em> if it is non stationary and <math alttext="\aleph_{1}" class="ltx_Math" display="inline" id="S2.Thmtheorem4.p1.6.m6.1"><semantics id="S2.Thmtheorem4.p1.6.m6.1a"><msub id="S2.Thmtheorem4.p1.6.m6.1.1" xref="S2.Thmtheorem4.p1.6.m6.1.1.cmml"><mi id="S2.Thmtheorem4.p1.6.m6.1.1.2" mathvariant="normal" xref="S2.Thmtheorem4.p1.6.m6.1.1.2.cmml">ℵ</mi><mn id="S2.Thmtheorem4.p1.6.m6.1.1.3" xref="S2.Thmtheorem4.p1.6.m6.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem4.p1.6.m6.1b"><apply id="S2.Thmtheorem4.p1.6.m6.1.1.cmml" xref="S2.Thmtheorem4.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem4.p1.6.m6.1.1.1.cmml" xref="S2.Thmtheorem4.p1.6.m6.1.1">subscript</csymbol><ci id="S2.Thmtheorem4.p1.6.m6.1.1.2.cmml" xref="S2.Thmtheorem4.p1.6.m6.1.1.2">ℵ</ci><cn id="S2.Thmtheorem4.p1.6.m6.1.1.3.cmml" type="integer" xref="S2.Thmtheorem4.p1.6.m6.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem4.p1.6.m6.1c">\aleph_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem4.p1.6.m6.1d">roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-dense.</p> </div> </div> <div class="ltx_para" id="S2.p5"> <p class="ltx_p" id="S2.p5.1">The following, which also seems folklore, will be particularly relevant. We do not know of a proof in the literature, but such a proof can easily be extracted from the material in <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S4" title="4. Aronszajn line decompositions ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">4</span></a>.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S2.Thmtheorem5"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem5.1.1.1">Lemma 2.5</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem5.2.2">.</span> </h6> <div class="ltx_para" id="S2.Thmtheorem5.p1"> <p class="ltx_p" id="S2.Thmtheorem5.p1.1">Every Aronszajn line contains a normal suborder.</p> </div> </div> <div class="ltx_para" id="S2.p6"> <p class="ltx_p" id="S2.p6.1">The following summarizes the most important properties of Countryman lines, which where defined in the introduction. For a proof see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib21" title="">21</a>, Theorem 5.4]</cite>.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S2.Thmtheorem6"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem6.1.1.1">Lemma 2.6</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem6.2.2">.</span> </h6> <div class="ltx_para" id="S2.Thmtheorem6.p1"> <p class="ltx_p" id="S2.Thmtheorem6.p1.1">If <math alttext="C" class="ltx_Math" display="inline" id="S2.Thmtheorem6.p1.1.m1.1"><semantics id="S2.Thmtheorem6.p1.1.m1.1a"><mi id="S2.Thmtheorem6.p1.1.m1.1.1" xref="S2.Thmtheorem6.p1.1.m1.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem6.p1.1.m1.1b"><ci id="S2.Thmtheorem6.p1.1.m1.1.1.cmml" xref="S2.Thmtheorem6.p1.1.m1.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem6.p1.1.m1.1c">C</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem6.p1.1.m1.1d">italic_C</annotation></semantics></math> is a Countryman line then,</p> <ul class="ltx_itemize" id="S2.I1"> <li class="ltx_item" id="S2.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S2.I1.i1.p1"> <p class="ltx_p" id="S2.I1.i1.p1.1"><math alttext="C" class="ltx_Math" display="inline" id="S2.I1.i1.p1.1.m1.1"><semantics id="S2.I1.i1.p1.1.m1.1a"><mi id="S2.I1.i1.p1.1.m1.1.1" xref="S2.I1.i1.p1.1.m1.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S2.I1.i1.p1.1.m1.1b"><ci id="S2.I1.i1.p1.1.m1.1.1.cmml" xref="S2.I1.i1.p1.1.m1.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i1.p1.1.m1.1c">C</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i1.p1.1.m1.1d">italic_C</annotation></semantics></math> is Aronszajn.</p> </div> </li> <li class="ltx_item" id="S2.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S2.I1.i2.p1"> <p class="ltx_p" id="S2.I1.i2.p1.3"><math alttext="C" class="ltx_Math" display="inline" id="S2.I1.i2.p1.1.m1.1"><semantics id="S2.I1.i2.p1.1.m1.1a"><mi id="S2.I1.i2.p1.1.m1.1.1" xref="S2.I1.i2.p1.1.m1.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S2.I1.i2.p1.1.m1.1b"><ci id="S2.I1.i2.p1.1.m1.1.1.cmml" xref="S2.I1.i2.p1.1.m1.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i2.p1.1.m1.1c">C</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i2.p1.1.m1.1d">italic_C</annotation></semantics></math> contains no two uncountable reverse isomorphic suborders, that is, <math alttext="X^{\star}\npreceq C" class="ltx_Math" display="inline" id="S2.I1.i2.p1.2.m2.1"><semantics id="S2.I1.i2.p1.2.m2.1a"><mrow id="S2.I1.i2.p1.2.m2.1.1" xref="S2.I1.i2.p1.2.m2.1.1.cmml"><msup id="S2.I1.i2.p1.2.m2.1.1.2" xref="S2.I1.i2.p1.2.m2.1.1.2.cmml"><mi id="S2.I1.i2.p1.2.m2.1.1.2.2" xref="S2.I1.i2.p1.2.m2.1.1.2.2.cmml">X</mi><mo id="S2.I1.i2.p1.2.m2.1.1.2.3" xref="S2.I1.i2.p1.2.m2.1.1.2.3.cmml">⋆</mo></msup><mo id="S2.I1.i2.p1.2.m2.1.1.1" xref="S2.I1.i2.p1.2.m2.1.1.1.cmml">⋠</mo><mi id="S2.I1.i2.p1.2.m2.1.1.3" xref="S2.I1.i2.p1.2.m2.1.1.3.cmml">C</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.I1.i2.p1.2.m2.1b"><apply id="S2.I1.i2.p1.2.m2.1.1.cmml" xref="S2.I1.i2.p1.2.m2.1.1"><csymbol cd="latexml" id="S2.I1.i2.p1.2.m2.1.1.1.cmml" xref="S2.I1.i2.p1.2.m2.1.1.1">not-precedes-nor-equals</csymbol><apply id="S2.I1.i2.p1.2.m2.1.1.2.cmml" xref="S2.I1.i2.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="S2.I1.i2.p1.2.m2.1.1.2.1.cmml" xref="S2.I1.i2.p1.2.m2.1.1.2">superscript</csymbol><ci id="S2.I1.i2.p1.2.m2.1.1.2.2.cmml" xref="S2.I1.i2.p1.2.m2.1.1.2.2">𝑋</ci><ci id="S2.I1.i2.p1.2.m2.1.1.2.3.cmml" xref="S2.I1.i2.p1.2.m2.1.1.2.3">⋆</ci></apply><ci id="S2.I1.i2.p1.2.m2.1.1.3.cmml" xref="S2.I1.i2.p1.2.m2.1.1.3">𝐶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i2.p1.2.m2.1c">X^{\star}\npreceq C</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i2.p1.2.m2.1d">italic_X start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT ⋠ italic_C</annotation></semantics></math> whenever <math alttext="X\preceq C" class="ltx_Math" display="inline" id="S2.I1.i2.p1.3.m3.1"><semantics id="S2.I1.i2.p1.3.m3.1a"><mrow id="S2.I1.i2.p1.3.m3.1.1" xref="S2.I1.i2.p1.3.m3.1.1.cmml"><mi id="S2.I1.i2.p1.3.m3.1.1.2" xref="S2.I1.i2.p1.3.m3.1.1.2.cmml">X</mi><mo id="S2.I1.i2.p1.3.m3.1.1.1" xref="S2.I1.i2.p1.3.m3.1.1.1.cmml">⪯</mo><mi id="S2.I1.i2.p1.3.m3.1.1.3" xref="S2.I1.i2.p1.3.m3.1.1.3.cmml">C</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.I1.i2.p1.3.m3.1b"><apply id="S2.I1.i2.p1.3.m3.1.1.cmml" xref="S2.I1.i2.p1.3.m3.1.1"><csymbol cd="latexml" id="S2.I1.i2.p1.3.m3.1.1.1.cmml" xref="S2.I1.i2.p1.3.m3.1.1.1">precedes-or-equals</csymbol><ci id="S2.I1.i2.p1.3.m3.1.1.2.cmml" xref="S2.I1.i2.p1.3.m3.1.1.2">𝑋</ci><ci id="S2.I1.i2.p1.3.m3.1.1.3.cmml" xref="S2.I1.i2.p1.3.m3.1.1.3">𝐶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i2.p1.3.m3.1c">X\preceq C</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i2.p1.3.m3.1d">italic_X ⪯ italic_C</annotation></semantics></math> is uncountable.</p> </div> </li> <li class="ltx_item" id="S2.I1.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S2.I1.i3.p1"> <p class="ltx_p" id="S2.I1.i3.p1.2">For every <math alttext="n\geq 1" class="ltx_Math" display="inline" id="S2.I1.i3.p1.1.m1.1"><semantics id="S2.I1.i3.p1.1.m1.1a"><mrow id="S2.I1.i3.p1.1.m1.1.1" xref="S2.I1.i3.p1.1.m1.1.1.cmml"><mi id="S2.I1.i3.p1.1.m1.1.1.2" xref="S2.I1.i3.p1.1.m1.1.1.2.cmml">n</mi><mo id="S2.I1.i3.p1.1.m1.1.1.1" xref="S2.I1.i3.p1.1.m1.1.1.1.cmml">≥</mo><mn id="S2.I1.i3.p1.1.m1.1.1.3" xref="S2.I1.i3.p1.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.I1.i3.p1.1.m1.1b"><apply id="S2.I1.i3.p1.1.m1.1.1.cmml" xref="S2.I1.i3.p1.1.m1.1.1"><geq id="S2.I1.i3.p1.1.m1.1.1.1.cmml" xref="S2.I1.i3.p1.1.m1.1.1.1"></geq><ci id="S2.I1.i3.p1.1.m1.1.1.2.cmml" xref="S2.I1.i3.p1.1.m1.1.1.2">𝑛</ci><cn id="S2.I1.i3.p1.1.m1.1.1.3.cmml" type="integer" xref="S2.I1.i3.p1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i3.p1.1.m1.1c">n\geq 1</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i3.p1.1.m1.1d">italic_n ≥ 1</annotation></semantics></math>, <math alttext="C^{n}" class="ltx_Math" display="inline" id="S2.I1.i3.p1.2.m2.1"><semantics id="S2.I1.i3.p1.2.m2.1a"><msup id="S2.I1.i3.p1.2.m2.1.1" xref="S2.I1.i3.p1.2.m2.1.1.cmml"><mi id="S2.I1.i3.p1.2.m2.1.1.2" xref="S2.I1.i3.p1.2.m2.1.1.2.cmml">C</mi><mi id="S2.I1.i3.p1.2.m2.1.1.3" xref="S2.I1.i3.p1.2.m2.1.1.3.cmml">n</mi></msup><annotation-xml encoding="MathML-Content" id="S2.I1.i3.p1.2.m2.1b"><apply id="S2.I1.i3.p1.2.m2.1.1.cmml" xref="S2.I1.i3.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S2.I1.i3.p1.2.m2.1.1.1.cmml" xref="S2.I1.i3.p1.2.m2.1.1">superscript</csymbol><ci id="S2.I1.i3.p1.2.m2.1.1.2.cmml" xref="S2.I1.i3.p1.2.m2.1.1.2">𝐶</ci><ci id="S2.I1.i3.p1.2.m2.1.1.3.cmml" xref="S2.I1.i3.p1.2.m2.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i3.p1.2.m2.1c">C^{n}</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i3.p1.2.m2.1d">italic_C start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math>, in the product order, is the union of countably many chains.</p> </div> </li> </ul> </div> </div> <div class="ltx_para" id="S2.p7"> <p class="ltx_p" id="S2.p7.1">It is unclear to the author who first proved the following. For a proof see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib23" title="">23</a>, 2.1.12 and 2.1.13]</cite>.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S2.Thmtheorem7"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem7.1.1.1">Theorem 2.7</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem7.2.2">.</span> </h6> <div class="ltx_para" id="S2.Thmtheorem7.p1"> <p class="ltx_p" id="S2.Thmtheorem7.p1.7"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem7.p1.7.7">Assume <math alttext="\mathsf{MA}_{\aleph_{1}}" class="ltx_Math" display="inline" id="S2.Thmtheorem7.p1.1.1.m1.1"><semantics id="S2.Thmtheorem7.p1.1.1.m1.1a"><msub id="S2.Thmtheorem7.p1.1.1.m1.1.1" xref="S2.Thmtheorem7.p1.1.1.m1.1.1.cmml"><mi id="S2.Thmtheorem7.p1.1.1.m1.1.1.2" xref="S2.Thmtheorem7.p1.1.1.m1.1.1.2.cmml">𝖬𝖠</mi><msub id="S2.Thmtheorem7.p1.1.1.m1.1.1.3" xref="S2.Thmtheorem7.p1.1.1.m1.1.1.3.cmml"><mi id="S2.Thmtheorem7.p1.1.1.m1.1.1.3.2" mathvariant="normal" xref="S2.Thmtheorem7.p1.1.1.m1.1.1.3.2.cmml">ℵ</mi><mn id="S2.Thmtheorem7.p1.1.1.m1.1.1.3.3" xref="S2.Thmtheorem7.p1.1.1.m1.1.1.3.3.cmml">1</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem7.p1.1.1.m1.1b"><apply id="S2.Thmtheorem7.p1.1.1.m1.1.1.cmml" xref="S2.Thmtheorem7.p1.1.1.m1.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem7.p1.1.1.m1.1.1.1.cmml" xref="S2.Thmtheorem7.p1.1.1.m1.1.1">subscript</csymbol><ci id="S2.Thmtheorem7.p1.1.1.m1.1.1.2.cmml" xref="S2.Thmtheorem7.p1.1.1.m1.1.1.2">𝖬𝖠</ci><apply id="S2.Thmtheorem7.p1.1.1.m1.1.1.3.cmml" xref="S2.Thmtheorem7.p1.1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S2.Thmtheorem7.p1.1.1.m1.1.1.3.1.cmml" xref="S2.Thmtheorem7.p1.1.1.m1.1.1.3">subscript</csymbol><ci id="S2.Thmtheorem7.p1.1.1.m1.1.1.3.2.cmml" xref="S2.Thmtheorem7.p1.1.1.m1.1.1.3.2">ℵ</ci><cn id="S2.Thmtheorem7.p1.1.1.m1.1.1.3.3.cmml" type="integer" xref="S2.Thmtheorem7.p1.1.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem7.p1.1.1.m1.1c">\mathsf{MA}_{\aleph_{1}}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem7.p1.1.1.m1.1d">sansserif_MA start_POSTSUBSCRIPT roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. The class of Countryman line has exactly two <math alttext="\preceq" class="ltx_Math" display="inline" id="S2.Thmtheorem7.p1.2.2.m2.1"><semantics id="S2.Thmtheorem7.p1.2.2.m2.1a"><mo id="S2.Thmtheorem7.p1.2.2.m2.1.1" xref="S2.Thmtheorem7.p1.2.2.m2.1.1.cmml">⪯</mo><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem7.p1.2.2.m2.1b"><csymbol cd="latexml" id="S2.Thmtheorem7.p1.2.2.m2.1.1.cmml" xref="S2.Thmtheorem7.p1.2.2.m2.1.1">precedes-or-equals</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem7.p1.2.2.m2.1c">\preceq</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem7.p1.2.2.m2.1d">⪯</annotation></semantics></math>-equivalence classes. In particular if <math alttext="C" class="ltx_Math" display="inline" id="S2.Thmtheorem7.p1.3.3.m3.1"><semantics id="S2.Thmtheorem7.p1.3.3.m3.1a"><mi id="S2.Thmtheorem7.p1.3.3.m3.1.1" xref="S2.Thmtheorem7.p1.3.3.m3.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem7.p1.3.3.m3.1b"><ci id="S2.Thmtheorem7.p1.3.3.m3.1.1.cmml" xref="S2.Thmtheorem7.p1.3.3.m3.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem7.p1.3.3.m3.1c">C</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem7.p1.3.3.m3.1d">italic_C</annotation></semantics></math> is any Countryman, then <math alttext="C" class="ltx_Math" display="inline" id="S2.Thmtheorem7.p1.4.4.m4.1"><semantics id="S2.Thmtheorem7.p1.4.4.m4.1a"><mi id="S2.Thmtheorem7.p1.4.4.m4.1.1" xref="S2.Thmtheorem7.p1.4.4.m4.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem7.p1.4.4.m4.1b"><ci id="S2.Thmtheorem7.p1.4.4.m4.1.1.cmml" xref="S2.Thmtheorem7.p1.4.4.m4.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem7.p1.4.4.m4.1c">C</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem7.p1.4.4.m4.1d">italic_C</annotation></semantics></math> is a <math alttext="\preceq" class="ltx_Math" display="inline" id="S2.Thmtheorem7.p1.5.5.m5.1"><semantics id="S2.Thmtheorem7.p1.5.5.m5.1a"><mo id="S2.Thmtheorem7.p1.5.5.m5.1.1" xref="S2.Thmtheorem7.p1.5.5.m5.1.1.cmml">⪯</mo><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem7.p1.5.5.m5.1b"><csymbol cd="latexml" id="S2.Thmtheorem7.p1.5.5.m5.1.1.cmml" xref="S2.Thmtheorem7.p1.5.5.m5.1.1">precedes-or-equals</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem7.p1.5.5.m5.1c">\preceq</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem7.p1.5.5.m5.1d">⪯</annotation></semantics></math>-minimal uncountable order and <math alttext="\{C,C^{\star}\}" class="ltx_Math" display="inline" id="S2.Thmtheorem7.p1.6.6.m6.2"><semantics id="S2.Thmtheorem7.p1.6.6.m6.2a"><mrow id="S2.Thmtheorem7.p1.6.6.m6.2.2.1" xref="S2.Thmtheorem7.p1.6.6.m6.2.2.2.cmml"><mo id="S2.Thmtheorem7.p1.6.6.m6.2.2.1.2" stretchy="false" xref="S2.Thmtheorem7.p1.6.6.m6.2.2.2.cmml">{</mo><mi id="S2.Thmtheorem7.p1.6.6.m6.1.1" xref="S2.Thmtheorem7.p1.6.6.m6.1.1.cmml">C</mi><mo id="S2.Thmtheorem7.p1.6.6.m6.2.2.1.3" xref="S2.Thmtheorem7.p1.6.6.m6.2.2.2.cmml">,</mo><msup id="S2.Thmtheorem7.p1.6.6.m6.2.2.1.1" xref="S2.Thmtheorem7.p1.6.6.m6.2.2.1.1.cmml"><mi id="S2.Thmtheorem7.p1.6.6.m6.2.2.1.1.2" xref="S2.Thmtheorem7.p1.6.6.m6.2.2.1.1.2.cmml">C</mi><mo id="S2.Thmtheorem7.p1.6.6.m6.2.2.1.1.3" xref="S2.Thmtheorem7.p1.6.6.m6.2.2.1.1.3.cmml">⋆</mo></msup><mo id="S2.Thmtheorem7.p1.6.6.m6.2.2.1.4" stretchy="false" xref="S2.Thmtheorem7.p1.6.6.m6.2.2.2.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem7.p1.6.6.m6.2b"><set id="S2.Thmtheorem7.p1.6.6.m6.2.2.2.cmml" xref="S2.Thmtheorem7.p1.6.6.m6.2.2.1"><ci id="S2.Thmtheorem7.p1.6.6.m6.1.1.cmml" xref="S2.Thmtheorem7.p1.6.6.m6.1.1">𝐶</ci><apply id="S2.Thmtheorem7.p1.6.6.m6.2.2.1.1.cmml" xref="S2.Thmtheorem7.p1.6.6.m6.2.2.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem7.p1.6.6.m6.2.2.1.1.1.cmml" xref="S2.Thmtheorem7.p1.6.6.m6.2.2.1.1">superscript</csymbol><ci id="S2.Thmtheorem7.p1.6.6.m6.2.2.1.1.2.cmml" xref="S2.Thmtheorem7.p1.6.6.m6.2.2.1.1.2">𝐶</ci><ci id="S2.Thmtheorem7.p1.6.6.m6.2.2.1.1.3.cmml" xref="S2.Thmtheorem7.p1.6.6.m6.2.2.1.1.3">⋆</ci></apply></set></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem7.p1.6.6.m6.2c">\{C,C^{\star}\}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem7.p1.6.6.m6.2d">{ italic_C , italic_C start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT }</annotation></semantics></math> is a <math alttext="\preceq" class="ltx_Math" display="inline" id="S2.Thmtheorem7.p1.7.7.m7.1"><semantics id="S2.Thmtheorem7.p1.7.7.m7.1a"><mo id="S2.Thmtheorem7.p1.7.7.m7.1.1" xref="S2.Thmtheorem7.p1.7.7.m7.1.1.cmml">⪯</mo><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem7.p1.7.7.m7.1b"><csymbol cd="latexml" id="S2.Thmtheorem7.p1.7.7.m7.1.1.cmml" xref="S2.Thmtheorem7.p1.7.7.m7.1.1">precedes-or-equals</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem7.p1.7.7.m7.1c">\preceq</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem7.p1.7.7.m7.1d">⪯</annotation></semantics></math>-basis for the Countryman lines.</span></p> </div> </div> <div class="ltx_para" id="S2.p8"> <p class="ltx_p" id="S2.p8.1">The following is a strong form of the previous theorem due to Moore (recall <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S2.Thmtheorem5" title="Lemma 2.5. ‣ 2. Aronszajn and Countryman lines ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">2.5</span></a>).</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S2.Thmtheorem8"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem8.1.1.1">Theorem 2.8</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem8.2.2">.</span> </h6> <div class="ltx_para" id="S2.Thmtheorem8.p1"> <p class="ltx_p" id="S2.Thmtheorem8.p1.1"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem8.p1.1.1">(Moore <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib16" title="">16</a>]</cite>) Assume <math alttext="\mathsf{MA}_{\aleph_{1}}" class="ltx_Math" display="inline" id="S2.Thmtheorem8.p1.1.1.m1.1"><semantics id="S2.Thmtheorem8.p1.1.1.m1.1a"><msub id="S2.Thmtheorem8.p1.1.1.m1.1.1" xref="S2.Thmtheorem8.p1.1.1.m1.1.1.cmml"><mi id="S2.Thmtheorem8.p1.1.1.m1.1.1.2" xref="S2.Thmtheorem8.p1.1.1.m1.1.1.2.cmml">𝖬𝖠</mi><msub id="S2.Thmtheorem8.p1.1.1.m1.1.1.3" xref="S2.Thmtheorem8.p1.1.1.m1.1.1.3.cmml"><mi id="S2.Thmtheorem8.p1.1.1.m1.1.1.3.2" mathvariant="normal" xref="S2.Thmtheorem8.p1.1.1.m1.1.1.3.2.cmml">ℵ</mi><mn id="S2.Thmtheorem8.p1.1.1.m1.1.1.3.3" xref="S2.Thmtheorem8.p1.1.1.m1.1.1.3.3.cmml">1</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem8.p1.1.1.m1.1b"><apply id="S2.Thmtheorem8.p1.1.1.m1.1.1.cmml" xref="S2.Thmtheorem8.p1.1.1.m1.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem8.p1.1.1.m1.1.1.1.cmml" xref="S2.Thmtheorem8.p1.1.1.m1.1.1">subscript</csymbol><ci id="S2.Thmtheorem8.p1.1.1.m1.1.1.2.cmml" xref="S2.Thmtheorem8.p1.1.1.m1.1.1.2">𝖬𝖠</ci><apply id="S2.Thmtheorem8.p1.1.1.m1.1.1.3.cmml" xref="S2.Thmtheorem8.p1.1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S2.Thmtheorem8.p1.1.1.m1.1.1.3.1.cmml" xref="S2.Thmtheorem8.p1.1.1.m1.1.1.3">subscript</csymbol><ci id="S2.Thmtheorem8.p1.1.1.m1.1.1.3.2.cmml" xref="S2.Thmtheorem8.p1.1.1.m1.1.1.3.2">ℵ</ci><cn id="S2.Thmtheorem8.p1.1.1.m1.1.1.3.3.cmml" type="integer" xref="S2.Thmtheorem8.p1.1.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem8.p1.1.1.m1.1c">\mathsf{MA}_{\aleph_{1}}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem8.p1.1.1.m1.1d">sansserif_MA start_POSTSUBSCRIPT roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. Any two normal Countryman lines are either isomorphic or reverse isomorphic.</span></p> </div> </div> <div class="ltx_para" id="S2.p9"> <p class="ltx_p" id="S2.p9.2">For the rest of this section we fix a Countryman line <math alttext="C" class="ltx_Math" display="inline" id="S2.p9.1.m1.1"><semantics id="S2.p9.1.m1.1a"><mi id="S2.p9.1.m1.1.1" xref="S2.p9.1.m1.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S2.p9.1.m1.1b"><ci id="S2.p9.1.m1.1.1.cmml" xref="S2.p9.1.m1.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p9.1.m1.1c">C</annotation><annotation encoding="application/x-llamapun" id="S2.p9.1.m1.1d">italic_C</annotation></semantics></math>. Recall the definition of <math alttext="\eta_{C}" class="ltx_Math" display="inline" id="S2.p9.2.m2.1"><semantics id="S2.p9.2.m2.1a"><msub id="S2.p9.2.m2.1.1" xref="S2.p9.2.m2.1.1.cmml"><mi id="S2.p9.2.m2.1.1.2" xref="S2.p9.2.m2.1.1.2.cmml">η</mi><mi id="S2.p9.2.m2.1.1.3" xref="S2.p9.2.m2.1.1.3.cmml">C</mi></msub><annotation-xml encoding="MathML-Content" id="S2.p9.2.m2.1b"><apply id="S2.p9.2.m2.1.1.cmml" xref="S2.p9.2.m2.1.1"><csymbol cd="ambiguous" id="S2.p9.2.m2.1.1.1.cmml" xref="S2.p9.2.m2.1.1">subscript</csymbol><ci id="S2.p9.2.m2.1.1.2.cmml" xref="S2.p9.2.m2.1.1.2">𝜂</ci><ci id="S2.p9.2.m2.1.1.3.cmml" xref="S2.p9.2.m2.1.1.3">𝐶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p9.2.m2.1c">\eta_{C}</annotation><annotation encoding="application/x-llamapun" id="S2.p9.2.m2.1d">italic_η start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT</annotation></semantics></math> from the introduction.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S2.Thmtheorem9"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem9.1.1.1">Theorem 2.9</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem9.2.2">.</span> </h6> <div class="ltx_para" id="S2.Thmtheorem9.p1"> <p class="ltx_p" id="S2.Thmtheorem9.p1.7"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem9.p1.7.7">(Moore <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib16" title="">16</a>]</cite>) Assume <math alttext="\mathsf{PFA}" class="ltx_Math" display="inline" id="S2.Thmtheorem9.p1.1.1.m1.1"><semantics id="S2.Thmtheorem9.p1.1.1.m1.1a"><mi id="S2.Thmtheorem9.p1.1.1.m1.1.1" xref="S2.Thmtheorem9.p1.1.1.m1.1.1.cmml">𝖯𝖥𝖠</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem9.p1.1.1.m1.1b"><ci id="S2.Thmtheorem9.p1.1.1.m1.1.1.cmml" xref="S2.Thmtheorem9.p1.1.1.m1.1.1">𝖯𝖥𝖠</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem9.p1.1.1.m1.1c">\mathsf{PFA}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem9.p1.1.1.m1.1d">sansserif_PFA</annotation></semantics></math>. For every Aronszajn line <math alttext="A" class="ltx_Math" display="inline" id="S2.Thmtheorem9.p1.2.2.m2.1"><semantics id="S2.Thmtheorem9.p1.2.2.m2.1a"><mi id="S2.Thmtheorem9.p1.2.2.m2.1.1" xref="S2.Thmtheorem9.p1.2.2.m2.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem9.p1.2.2.m2.1b"><ci id="S2.Thmtheorem9.p1.2.2.m2.1.1.cmml" xref="S2.Thmtheorem9.p1.2.2.m2.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem9.p1.2.2.m2.1c">A</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem9.p1.2.2.m2.1d">italic_A</annotation></semantics></math>, either <math alttext="\eta_{C}\preceq A" class="ltx_Math" display="inline" id="S2.Thmtheorem9.p1.3.3.m3.1"><semantics id="S2.Thmtheorem9.p1.3.3.m3.1a"><mrow id="S2.Thmtheorem9.p1.3.3.m3.1.1" xref="S2.Thmtheorem9.p1.3.3.m3.1.1.cmml"><msub id="S2.Thmtheorem9.p1.3.3.m3.1.1.2" xref="S2.Thmtheorem9.p1.3.3.m3.1.1.2.cmml"><mi id="S2.Thmtheorem9.p1.3.3.m3.1.1.2.2" xref="S2.Thmtheorem9.p1.3.3.m3.1.1.2.2.cmml">η</mi><mi id="S2.Thmtheorem9.p1.3.3.m3.1.1.2.3" xref="S2.Thmtheorem9.p1.3.3.m3.1.1.2.3.cmml">C</mi></msub><mo id="S2.Thmtheorem9.p1.3.3.m3.1.1.1" xref="S2.Thmtheorem9.p1.3.3.m3.1.1.1.cmml">⪯</mo><mi id="S2.Thmtheorem9.p1.3.3.m3.1.1.3" xref="S2.Thmtheorem9.p1.3.3.m3.1.1.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem9.p1.3.3.m3.1b"><apply id="S2.Thmtheorem9.p1.3.3.m3.1.1.cmml" xref="S2.Thmtheorem9.p1.3.3.m3.1.1"><csymbol cd="latexml" id="S2.Thmtheorem9.p1.3.3.m3.1.1.1.cmml" xref="S2.Thmtheorem9.p1.3.3.m3.1.1.1">precedes-or-equals</csymbol><apply id="S2.Thmtheorem9.p1.3.3.m3.1.1.2.cmml" xref="S2.Thmtheorem9.p1.3.3.m3.1.1.2"><csymbol cd="ambiguous" id="S2.Thmtheorem9.p1.3.3.m3.1.1.2.1.cmml" xref="S2.Thmtheorem9.p1.3.3.m3.1.1.2">subscript</csymbol><ci id="S2.Thmtheorem9.p1.3.3.m3.1.1.2.2.cmml" xref="S2.Thmtheorem9.p1.3.3.m3.1.1.2.2">𝜂</ci><ci id="S2.Thmtheorem9.p1.3.3.m3.1.1.2.3.cmml" xref="S2.Thmtheorem9.p1.3.3.m3.1.1.2.3">𝐶</ci></apply><ci id="S2.Thmtheorem9.p1.3.3.m3.1.1.3.cmml" xref="S2.Thmtheorem9.p1.3.3.m3.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem9.p1.3.3.m3.1c">\eta_{C}\preceq A</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem9.p1.3.3.m3.1d">italic_η start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ⪯ italic_A</annotation></semantics></math>, or <math alttext="A" class="ltx_Math" display="inline" id="S2.Thmtheorem9.p1.4.4.m4.1"><semantics id="S2.Thmtheorem9.p1.4.4.m4.1a"><mi id="S2.Thmtheorem9.p1.4.4.m4.1.1" xref="S2.Thmtheorem9.p1.4.4.m4.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem9.p1.4.4.m4.1b"><ci id="S2.Thmtheorem9.p1.4.4.m4.1.1.cmml" xref="S2.Thmtheorem9.p1.4.4.m4.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem9.p1.4.4.m4.1c">A</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem9.p1.4.4.m4.1d">italic_A</annotation></semantics></math> contains an interval <math alttext="\preceq" class="ltx_Math" display="inline" id="S2.Thmtheorem9.p1.5.5.m5.1"><semantics id="S2.Thmtheorem9.p1.5.5.m5.1a"><mo id="S2.Thmtheorem9.p1.5.5.m5.1.1" xref="S2.Thmtheorem9.p1.5.5.m5.1.1.cmml">⪯</mo><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem9.p1.5.5.m5.1b"><csymbol cd="latexml" id="S2.Thmtheorem9.p1.5.5.m5.1.1.cmml" xref="S2.Thmtheorem9.p1.5.5.m5.1.1">precedes-or-equals</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem9.p1.5.5.m5.1c">\preceq</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem9.p1.5.5.m5.1d">⪯</annotation></semantics></math>-equivalent to <math alttext="C" class="ltx_Math" display="inline" id="S2.Thmtheorem9.p1.6.6.m6.1"><semantics id="S2.Thmtheorem9.p1.6.6.m6.1a"><mi id="S2.Thmtheorem9.p1.6.6.m6.1.1" xref="S2.Thmtheorem9.p1.6.6.m6.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem9.p1.6.6.m6.1b"><ci id="S2.Thmtheorem9.p1.6.6.m6.1.1.cmml" xref="S2.Thmtheorem9.p1.6.6.m6.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem9.p1.6.6.m6.1c">C</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem9.p1.6.6.m6.1d">italic_C</annotation></semantics></math> or to <math alttext="C^{\star}" class="ltx_Math" display="inline" id="S2.Thmtheorem9.p1.7.7.m7.1"><semantics id="S2.Thmtheorem9.p1.7.7.m7.1a"><msup id="S2.Thmtheorem9.p1.7.7.m7.1.1" xref="S2.Thmtheorem9.p1.7.7.m7.1.1.cmml"><mi id="S2.Thmtheorem9.p1.7.7.m7.1.1.2" xref="S2.Thmtheorem9.p1.7.7.m7.1.1.2.cmml">C</mi><mo id="S2.Thmtheorem9.p1.7.7.m7.1.1.3" xref="S2.Thmtheorem9.p1.7.7.m7.1.1.3.cmml">⋆</mo></msup><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem9.p1.7.7.m7.1b"><apply id="S2.Thmtheorem9.p1.7.7.m7.1.1.cmml" xref="S2.Thmtheorem9.p1.7.7.m7.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem9.p1.7.7.m7.1.1.1.cmml" xref="S2.Thmtheorem9.p1.7.7.m7.1.1">superscript</csymbol><ci id="S2.Thmtheorem9.p1.7.7.m7.1.1.2.cmml" xref="S2.Thmtheorem9.p1.7.7.m7.1.1.2">𝐶</ci><ci id="S2.Thmtheorem9.p1.7.7.m7.1.1.3.cmml" xref="S2.Thmtheorem9.p1.7.7.m7.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem9.p1.7.7.m7.1c">C^{\star}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem9.p1.7.7.m7.1d">italic_C start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S2.p10"> <p class="ltx_p" id="S2.p10.4">Making the analogy with the scattered linear orders (linear orders without copies of <math alttext="\mathbb{Q}" class="ltx_Math" display="inline" id="S2.p10.1.m1.1"><semantics id="S2.p10.1.m1.1a"><mi id="S2.p10.1.m1.1.1" xref="S2.p10.1.m1.1.1.cmml">ℚ</mi><annotation-xml encoding="MathML-Content" id="S2.p10.1.m1.1b"><ci id="S2.p10.1.m1.1.1.cmml" xref="S2.p10.1.m1.1.1">ℚ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p10.1.m1.1c">\mathbb{Q}</annotation><annotation encoding="application/x-llamapun" id="S2.p10.1.m1.1d">blackboard_Q</annotation></semantics></math>), Moore calls an Aronszajn line <em class="ltx_emph ltx_font_italic" id="S2.p10.4.1">fragmented</em> if it does not contain a copy of <math alttext="\eta_{C}" class="ltx_Math" display="inline" id="S2.p10.2.m2.1"><semantics id="S2.p10.2.m2.1a"><msub id="S2.p10.2.m2.1.1" xref="S2.p10.2.m2.1.1.cmml"><mi id="S2.p10.2.m2.1.1.2" xref="S2.p10.2.m2.1.1.2.cmml">η</mi><mi id="S2.p10.2.m2.1.1.3" xref="S2.p10.2.m2.1.1.3.cmml">C</mi></msub><annotation-xml encoding="MathML-Content" id="S2.p10.2.m2.1b"><apply id="S2.p10.2.m2.1.1.cmml" xref="S2.p10.2.m2.1.1"><csymbol cd="ambiguous" id="S2.p10.2.m2.1.1.1.cmml" xref="S2.p10.2.m2.1.1">subscript</csymbol><ci id="S2.p10.2.m2.1.1.2.cmml" xref="S2.p10.2.m2.1.1.2">𝜂</ci><ci id="S2.p10.2.m2.1.1.3.cmml" xref="S2.p10.2.m2.1.1.3">𝐶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p10.2.m2.1c">\eta_{C}</annotation><annotation encoding="application/x-llamapun" id="S2.p10.2.m2.1d">italic_η start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT</annotation></semantics></math>. The following definitions are due to Martínez-Ranero <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib14" title="">14</a>]</cite>. Here we assume that <math alttext="C" class="ltx_Math" display="inline" id="S2.p10.3.m3.1"><semantics id="S2.p10.3.m3.1a"><mi id="S2.p10.3.m3.1.1" xref="S2.p10.3.m3.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S2.p10.3.m3.1b"><ci id="S2.p10.3.m3.1.1.cmml" xref="S2.p10.3.m3.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p10.3.m3.1c">C</annotation><annotation encoding="application/x-llamapun" id="S2.p10.3.m3.1d">italic_C</annotation></semantics></math> is <math alttext="\aleph_{1}" class="ltx_Math" display="inline" id="S2.p10.4.m4.1"><semantics id="S2.p10.4.m4.1a"><msub id="S2.p10.4.m4.1.1" xref="S2.p10.4.m4.1.1.cmml"><mi id="S2.p10.4.m4.1.1.2" mathvariant="normal" xref="S2.p10.4.m4.1.1.2.cmml">ℵ</mi><mn id="S2.p10.4.m4.1.1.3" xref="S2.p10.4.m4.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.p10.4.m4.1b"><apply id="S2.p10.4.m4.1.1.cmml" xref="S2.p10.4.m4.1.1"><csymbol cd="ambiguous" id="S2.p10.4.m4.1.1.1.cmml" xref="S2.p10.4.m4.1.1">subscript</csymbol><ci id="S2.p10.4.m4.1.1.2.cmml" xref="S2.p10.4.m4.1.1.2">ℵ</ci><cn id="S2.p10.4.m4.1.1.3.cmml" type="integer" xref="S2.p10.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p10.4.m4.1c">\aleph_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.p10.4.m4.1d">roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-dense.</p> </div> <div class="ltx_para" id="S2.p11"> <p class="ltx_p" id="S2.p11.3">For <math alttext="\alpha<\omega_{1}" class="ltx_Math" display="inline" id="S2.p11.1.m1.1"><semantics id="S2.p11.1.m1.1a"><mrow id="S2.p11.1.m1.1.1" xref="S2.p11.1.m1.1.1.cmml"><mi id="S2.p11.1.m1.1.1.2" xref="S2.p11.1.m1.1.1.2.cmml">α</mi><mo id="S2.p11.1.m1.1.1.1" xref="S2.p11.1.m1.1.1.1.cmml"><</mo><msub id="S2.p11.1.m1.1.1.3" xref="S2.p11.1.m1.1.1.3.cmml"><mi id="S2.p11.1.m1.1.1.3.2" xref="S2.p11.1.m1.1.1.3.2.cmml">ω</mi><mn id="S2.p11.1.m1.1.1.3.3" xref="S2.p11.1.m1.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.p11.1.m1.1b"><apply id="S2.p11.1.m1.1.1.cmml" xref="S2.p11.1.m1.1.1"><lt id="S2.p11.1.m1.1.1.1.cmml" xref="S2.p11.1.m1.1.1.1"></lt><ci id="S2.p11.1.m1.1.1.2.cmml" xref="S2.p11.1.m1.1.1.2">𝛼</ci><apply id="S2.p11.1.m1.1.1.3.cmml" xref="S2.p11.1.m1.1.1.3"><csymbol cd="ambiguous" id="S2.p11.1.m1.1.1.3.1.cmml" xref="S2.p11.1.m1.1.1.3">subscript</csymbol><ci id="S2.p11.1.m1.1.1.3.2.cmml" xref="S2.p11.1.m1.1.1.3.2">𝜔</ci><cn id="S2.p11.1.m1.1.1.3.3.cmml" type="integer" xref="S2.p11.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p11.1.m1.1c">\alpha<\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.p11.1.m1.1d">italic_α < italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> recursively define Aronszajn lines <math alttext="D_{\alpha}^{+}" class="ltx_Math" display="inline" id="S2.p11.2.m2.1"><semantics id="S2.p11.2.m2.1a"><msubsup id="S2.p11.2.m2.1.1" xref="S2.p11.2.m2.1.1.cmml"><mi id="S2.p11.2.m2.1.1.2.2" xref="S2.p11.2.m2.1.1.2.2.cmml">D</mi><mi id="S2.p11.2.m2.1.1.2.3" xref="S2.p11.2.m2.1.1.2.3.cmml">α</mi><mo id="S2.p11.2.m2.1.1.3" xref="S2.p11.2.m2.1.1.3.cmml">+</mo></msubsup><annotation-xml encoding="MathML-Content" id="S2.p11.2.m2.1b"><apply id="S2.p11.2.m2.1.1.cmml" xref="S2.p11.2.m2.1.1"><csymbol cd="ambiguous" id="S2.p11.2.m2.1.1.1.cmml" xref="S2.p11.2.m2.1.1">superscript</csymbol><apply id="S2.p11.2.m2.1.1.2.cmml" xref="S2.p11.2.m2.1.1"><csymbol cd="ambiguous" id="S2.p11.2.m2.1.1.2.1.cmml" xref="S2.p11.2.m2.1.1">subscript</csymbol><ci id="S2.p11.2.m2.1.1.2.2.cmml" xref="S2.p11.2.m2.1.1.2.2">𝐷</ci><ci id="S2.p11.2.m2.1.1.2.3.cmml" xref="S2.p11.2.m2.1.1.2.3">𝛼</ci></apply><plus id="S2.p11.2.m2.1.1.3.cmml" xref="S2.p11.2.m2.1.1.3"></plus></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p11.2.m2.1c">D_{\alpha}^{+}</annotation><annotation encoding="application/x-llamapun" id="S2.p11.2.m2.1d">italic_D start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="D_{\alpha}^{-}" class="ltx_Math" display="inline" id="S2.p11.3.m3.1"><semantics id="S2.p11.3.m3.1a"><msubsup id="S2.p11.3.m3.1.1" xref="S2.p11.3.m3.1.1.cmml"><mi id="S2.p11.3.m3.1.1.2.2" xref="S2.p11.3.m3.1.1.2.2.cmml">D</mi><mi id="S2.p11.3.m3.1.1.2.3" xref="S2.p11.3.m3.1.1.2.3.cmml">α</mi><mo id="S2.p11.3.m3.1.1.3" xref="S2.p11.3.m3.1.1.3.cmml">−</mo></msubsup><annotation-xml encoding="MathML-Content" id="S2.p11.3.m3.1b"><apply id="S2.p11.3.m3.1.1.cmml" xref="S2.p11.3.m3.1.1"><csymbol cd="ambiguous" id="S2.p11.3.m3.1.1.1.cmml" xref="S2.p11.3.m3.1.1">superscript</csymbol><apply id="S2.p11.3.m3.1.1.2.cmml" xref="S2.p11.3.m3.1.1"><csymbol cd="ambiguous" id="S2.p11.3.m3.1.1.2.1.cmml" xref="S2.p11.3.m3.1.1">subscript</csymbol><ci id="S2.p11.3.m3.1.1.2.2.cmml" xref="S2.p11.3.m3.1.1.2.2">𝐷</ci><ci id="S2.p11.3.m3.1.1.2.3.cmml" xref="S2.p11.3.m3.1.1.2.3">𝛼</ci></apply><minus id="S2.p11.3.m3.1.1.3.cmml" xref="S2.p11.3.m3.1.1.3"></minus></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p11.3.m3.1c">D_{\alpha}^{-}</annotation><annotation encoding="application/x-llamapun" id="S2.p11.3.m3.1d">italic_D start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT</annotation></semantics></math> as follows.</p> <ul class="ltx_itemize" id="S2.I2"> <li class="ltx_item" id="S2.I2.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S2.I2.i1.p1"> <p class="ltx_p" id="S2.I2.i1.p1.2"><math alttext="D_{0}^{+}:=C" class="ltx_Math" display="inline" id="S2.I2.i1.p1.1.m1.1"><semantics id="S2.I2.i1.p1.1.m1.1a"><mrow id="S2.I2.i1.p1.1.m1.1.1" xref="S2.I2.i1.p1.1.m1.1.1.cmml"><msubsup id="S2.I2.i1.p1.1.m1.1.1.2" xref="S2.I2.i1.p1.1.m1.1.1.2.cmml"><mi id="S2.I2.i1.p1.1.m1.1.1.2.2.2" xref="S2.I2.i1.p1.1.m1.1.1.2.2.2.cmml">D</mi><mn id="S2.I2.i1.p1.1.m1.1.1.2.2.3" xref="S2.I2.i1.p1.1.m1.1.1.2.2.3.cmml">0</mn><mo id="S2.I2.i1.p1.1.m1.1.1.2.3" xref="S2.I2.i1.p1.1.m1.1.1.2.3.cmml">+</mo></msubsup><mo id="S2.I2.i1.p1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.I2.i1.p1.1.m1.1.1.1.cmml">:=</mo><mi id="S2.I2.i1.p1.1.m1.1.1.3" xref="S2.I2.i1.p1.1.m1.1.1.3.cmml">C</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.I2.i1.p1.1.m1.1b"><apply id="S2.I2.i1.p1.1.m1.1.1.cmml" xref="S2.I2.i1.p1.1.m1.1.1"><csymbol cd="latexml" id="S2.I2.i1.p1.1.m1.1.1.1.cmml" xref="S2.I2.i1.p1.1.m1.1.1.1">assign</csymbol><apply id="S2.I2.i1.p1.1.m1.1.1.2.cmml" xref="S2.I2.i1.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S2.I2.i1.p1.1.m1.1.1.2.1.cmml" xref="S2.I2.i1.p1.1.m1.1.1.2">superscript</csymbol><apply id="S2.I2.i1.p1.1.m1.1.1.2.2.cmml" xref="S2.I2.i1.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S2.I2.i1.p1.1.m1.1.1.2.2.1.cmml" xref="S2.I2.i1.p1.1.m1.1.1.2">subscript</csymbol><ci id="S2.I2.i1.p1.1.m1.1.1.2.2.2.cmml" xref="S2.I2.i1.p1.1.m1.1.1.2.2.2">𝐷</ci><cn id="S2.I2.i1.p1.1.m1.1.1.2.2.3.cmml" type="integer" xref="S2.I2.i1.p1.1.m1.1.1.2.2.3">0</cn></apply><plus id="S2.I2.i1.p1.1.m1.1.1.2.3.cmml" xref="S2.I2.i1.p1.1.m1.1.1.2.3"></plus></apply><ci id="S2.I2.i1.p1.1.m1.1.1.3.cmml" xref="S2.I2.i1.p1.1.m1.1.1.3">𝐶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I2.i1.p1.1.m1.1c">D_{0}^{+}:=C</annotation><annotation encoding="application/x-llamapun" id="S2.I2.i1.p1.1.m1.1d">italic_D start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT := italic_C</annotation></semantics></math> and <math alttext="D_{0}^{-}:=C^{\star}" class="ltx_Math" display="inline" id="S2.I2.i1.p1.2.m2.1"><semantics id="S2.I2.i1.p1.2.m2.1a"><mrow id="S2.I2.i1.p1.2.m2.1.1" xref="S2.I2.i1.p1.2.m2.1.1.cmml"><msubsup id="S2.I2.i1.p1.2.m2.1.1.2" xref="S2.I2.i1.p1.2.m2.1.1.2.cmml"><mi id="S2.I2.i1.p1.2.m2.1.1.2.2.2" xref="S2.I2.i1.p1.2.m2.1.1.2.2.2.cmml">D</mi><mn id="S2.I2.i1.p1.2.m2.1.1.2.2.3" xref="S2.I2.i1.p1.2.m2.1.1.2.2.3.cmml">0</mn><mo id="S2.I2.i1.p1.2.m2.1.1.2.3" xref="S2.I2.i1.p1.2.m2.1.1.2.3.cmml">−</mo></msubsup><mo id="S2.I2.i1.p1.2.m2.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.I2.i1.p1.2.m2.1.1.1.cmml">:=</mo><msup id="S2.I2.i1.p1.2.m2.1.1.3" xref="S2.I2.i1.p1.2.m2.1.1.3.cmml"><mi id="S2.I2.i1.p1.2.m2.1.1.3.2" xref="S2.I2.i1.p1.2.m2.1.1.3.2.cmml">C</mi><mo id="S2.I2.i1.p1.2.m2.1.1.3.3" xref="S2.I2.i1.p1.2.m2.1.1.3.3.cmml">⋆</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.I2.i1.p1.2.m2.1b"><apply id="S2.I2.i1.p1.2.m2.1.1.cmml" xref="S2.I2.i1.p1.2.m2.1.1"><csymbol cd="latexml" id="S2.I2.i1.p1.2.m2.1.1.1.cmml" xref="S2.I2.i1.p1.2.m2.1.1.1">assign</csymbol><apply id="S2.I2.i1.p1.2.m2.1.1.2.cmml" xref="S2.I2.i1.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="S2.I2.i1.p1.2.m2.1.1.2.1.cmml" xref="S2.I2.i1.p1.2.m2.1.1.2">superscript</csymbol><apply id="S2.I2.i1.p1.2.m2.1.1.2.2.cmml" xref="S2.I2.i1.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="S2.I2.i1.p1.2.m2.1.1.2.2.1.cmml" xref="S2.I2.i1.p1.2.m2.1.1.2">subscript</csymbol><ci id="S2.I2.i1.p1.2.m2.1.1.2.2.2.cmml" xref="S2.I2.i1.p1.2.m2.1.1.2.2.2">𝐷</ci><cn id="S2.I2.i1.p1.2.m2.1.1.2.2.3.cmml" type="integer" xref="S2.I2.i1.p1.2.m2.1.1.2.2.3">0</cn></apply><minus id="S2.I2.i1.p1.2.m2.1.1.2.3.cmml" xref="S2.I2.i1.p1.2.m2.1.1.2.3"></minus></apply><apply id="S2.I2.i1.p1.2.m2.1.1.3.cmml" xref="S2.I2.i1.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S2.I2.i1.p1.2.m2.1.1.3.1.cmml" xref="S2.I2.i1.p1.2.m2.1.1.3">superscript</csymbol><ci id="S2.I2.i1.p1.2.m2.1.1.3.2.cmml" xref="S2.I2.i1.p1.2.m2.1.1.3.2">𝐶</ci><ci id="S2.I2.i1.p1.2.m2.1.1.3.3.cmml" xref="S2.I2.i1.p1.2.m2.1.1.3.3">⋆</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I2.i1.p1.2.m2.1c">D_{0}^{-}:=C^{\star}</annotation><annotation encoding="application/x-llamapun" id="S2.I2.i1.p1.2.m2.1d">italic_D start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT := italic_C start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S2.I2.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S2.I2.i2.p1"> <p class="ltx_p" id="S2.I2.i2.p1.2"><math alttext="D_{\alpha+1}^{+}=C\times D_{\alpha}^{-}" class="ltx_Math" display="inline" id="S2.I2.i2.p1.1.m1.1"><semantics id="S2.I2.i2.p1.1.m1.1a"><mrow id="S2.I2.i2.p1.1.m1.1.1" xref="S2.I2.i2.p1.1.m1.1.1.cmml"><msubsup id="S2.I2.i2.p1.1.m1.1.1.2" xref="S2.I2.i2.p1.1.m1.1.1.2.cmml"><mi id="S2.I2.i2.p1.1.m1.1.1.2.2.2" xref="S2.I2.i2.p1.1.m1.1.1.2.2.2.cmml">D</mi><mrow id="S2.I2.i2.p1.1.m1.1.1.2.2.3" xref="S2.I2.i2.p1.1.m1.1.1.2.2.3.cmml"><mi id="S2.I2.i2.p1.1.m1.1.1.2.2.3.2" xref="S2.I2.i2.p1.1.m1.1.1.2.2.3.2.cmml">α</mi><mo id="S2.I2.i2.p1.1.m1.1.1.2.2.3.1" xref="S2.I2.i2.p1.1.m1.1.1.2.2.3.1.cmml">+</mo><mn id="S2.I2.i2.p1.1.m1.1.1.2.2.3.3" xref="S2.I2.i2.p1.1.m1.1.1.2.2.3.3.cmml">1</mn></mrow><mo id="S2.I2.i2.p1.1.m1.1.1.2.3" xref="S2.I2.i2.p1.1.m1.1.1.2.3.cmml">+</mo></msubsup><mo id="S2.I2.i2.p1.1.m1.1.1.1" xref="S2.I2.i2.p1.1.m1.1.1.1.cmml">=</mo><mrow id="S2.I2.i2.p1.1.m1.1.1.3" xref="S2.I2.i2.p1.1.m1.1.1.3.cmml"><mi id="S2.I2.i2.p1.1.m1.1.1.3.2" xref="S2.I2.i2.p1.1.m1.1.1.3.2.cmml">C</mi><mo id="S2.I2.i2.p1.1.m1.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S2.I2.i2.p1.1.m1.1.1.3.1.cmml">×</mo><msubsup id="S2.I2.i2.p1.1.m1.1.1.3.3" xref="S2.I2.i2.p1.1.m1.1.1.3.3.cmml"><mi id="S2.I2.i2.p1.1.m1.1.1.3.3.2.2" xref="S2.I2.i2.p1.1.m1.1.1.3.3.2.2.cmml">D</mi><mi id="S2.I2.i2.p1.1.m1.1.1.3.3.2.3" xref="S2.I2.i2.p1.1.m1.1.1.3.3.2.3.cmml">α</mi><mo id="S2.I2.i2.p1.1.m1.1.1.3.3.3" xref="S2.I2.i2.p1.1.m1.1.1.3.3.3.cmml">−</mo></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.I2.i2.p1.1.m1.1b"><apply id="S2.I2.i2.p1.1.m1.1.1.cmml" xref="S2.I2.i2.p1.1.m1.1.1"><eq id="S2.I2.i2.p1.1.m1.1.1.1.cmml" xref="S2.I2.i2.p1.1.m1.1.1.1"></eq><apply id="S2.I2.i2.p1.1.m1.1.1.2.cmml" xref="S2.I2.i2.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S2.I2.i2.p1.1.m1.1.1.2.1.cmml" xref="S2.I2.i2.p1.1.m1.1.1.2">superscript</csymbol><apply id="S2.I2.i2.p1.1.m1.1.1.2.2.cmml" xref="S2.I2.i2.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S2.I2.i2.p1.1.m1.1.1.2.2.1.cmml" xref="S2.I2.i2.p1.1.m1.1.1.2">subscript</csymbol><ci id="S2.I2.i2.p1.1.m1.1.1.2.2.2.cmml" xref="S2.I2.i2.p1.1.m1.1.1.2.2.2">𝐷</ci><apply id="S2.I2.i2.p1.1.m1.1.1.2.2.3.cmml" xref="S2.I2.i2.p1.1.m1.1.1.2.2.3"><plus id="S2.I2.i2.p1.1.m1.1.1.2.2.3.1.cmml" xref="S2.I2.i2.p1.1.m1.1.1.2.2.3.1"></plus><ci id="S2.I2.i2.p1.1.m1.1.1.2.2.3.2.cmml" xref="S2.I2.i2.p1.1.m1.1.1.2.2.3.2">𝛼</ci><cn id="S2.I2.i2.p1.1.m1.1.1.2.2.3.3.cmml" type="integer" xref="S2.I2.i2.p1.1.m1.1.1.2.2.3.3">1</cn></apply></apply><plus id="S2.I2.i2.p1.1.m1.1.1.2.3.cmml" xref="S2.I2.i2.p1.1.m1.1.1.2.3"></plus></apply><apply id="S2.I2.i2.p1.1.m1.1.1.3.cmml" xref="S2.I2.i2.p1.1.m1.1.1.3"><times id="S2.I2.i2.p1.1.m1.1.1.3.1.cmml" xref="S2.I2.i2.p1.1.m1.1.1.3.1"></times><ci id="S2.I2.i2.p1.1.m1.1.1.3.2.cmml" xref="S2.I2.i2.p1.1.m1.1.1.3.2">𝐶</ci><apply id="S2.I2.i2.p1.1.m1.1.1.3.3.cmml" xref="S2.I2.i2.p1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S2.I2.i2.p1.1.m1.1.1.3.3.1.cmml" xref="S2.I2.i2.p1.1.m1.1.1.3.3">superscript</csymbol><apply id="S2.I2.i2.p1.1.m1.1.1.3.3.2.cmml" xref="S2.I2.i2.p1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S2.I2.i2.p1.1.m1.1.1.3.3.2.1.cmml" xref="S2.I2.i2.p1.1.m1.1.1.3.3">subscript</csymbol><ci id="S2.I2.i2.p1.1.m1.1.1.3.3.2.2.cmml" xref="S2.I2.i2.p1.1.m1.1.1.3.3.2.2">𝐷</ci><ci id="S2.I2.i2.p1.1.m1.1.1.3.3.2.3.cmml" xref="S2.I2.i2.p1.1.m1.1.1.3.3.2.3">𝛼</ci></apply><minus id="S2.I2.i2.p1.1.m1.1.1.3.3.3.cmml" xref="S2.I2.i2.p1.1.m1.1.1.3.3.3"></minus></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I2.i2.p1.1.m1.1c">D_{\alpha+1}^{+}=C\times D_{\alpha}^{-}</annotation><annotation encoding="application/x-llamapun" id="S2.I2.i2.p1.1.m1.1d">italic_D start_POSTSUBSCRIPT italic_α + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT = italic_C × italic_D start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="D_{\alpha+1}^{-}:=C^{\star}\times D_{\alpha}^{+}" class="ltx_Math" display="inline" id="S2.I2.i2.p1.2.m2.1"><semantics id="S2.I2.i2.p1.2.m2.1a"><mrow id="S2.I2.i2.p1.2.m2.1.1" xref="S2.I2.i2.p1.2.m2.1.1.cmml"><msubsup id="S2.I2.i2.p1.2.m2.1.1.2" xref="S2.I2.i2.p1.2.m2.1.1.2.cmml"><mi id="S2.I2.i2.p1.2.m2.1.1.2.2.2" xref="S2.I2.i2.p1.2.m2.1.1.2.2.2.cmml">D</mi><mrow id="S2.I2.i2.p1.2.m2.1.1.2.2.3" xref="S2.I2.i2.p1.2.m2.1.1.2.2.3.cmml"><mi id="S2.I2.i2.p1.2.m2.1.1.2.2.3.2" xref="S2.I2.i2.p1.2.m2.1.1.2.2.3.2.cmml">α</mi><mo id="S2.I2.i2.p1.2.m2.1.1.2.2.3.1" xref="S2.I2.i2.p1.2.m2.1.1.2.2.3.1.cmml">+</mo><mn id="S2.I2.i2.p1.2.m2.1.1.2.2.3.3" xref="S2.I2.i2.p1.2.m2.1.1.2.2.3.3.cmml">1</mn></mrow><mo id="S2.I2.i2.p1.2.m2.1.1.2.3" xref="S2.I2.i2.p1.2.m2.1.1.2.3.cmml">−</mo></msubsup><mo id="S2.I2.i2.p1.2.m2.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.I2.i2.p1.2.m2.1.1.1.cmml">:=</mo><mrow id="S2.I2.i2.p1.2.m2.1.1.3" xref="S2.I2.i2.p1.2.m2.1.1.3.cmml"><msup id="S2.I2.i2.p1.2.m2.1.1.3.2" xref="S2.I2.i2.p1.2.m2.1.1.3.2.cmml"><mi id="S2.I2.i2.p1.2.m2.1.1.3.2.2" xref="S2.I2.i2.p1.2.m2.1.1.3.2.2.cmml">C</mi><mo id="S2.I2.i2.p1.2.m2.1.1.3.2.3" xref="S2.I2.i2.p1.2.m2.1.1.3.2.3.cmml">⋆</mo></msup><mo id="S2.I2.i2.p1.2.m2.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S2.I2.i2.p1.2.m2.1.1.3.1.cmml">×</mo><msubsup id="S2.I2.i2.p1.2.m2.1.1.3.3" xref="S2.I2.i2.p1.2.m2.1.1.3.3.cmml"><mi id="S2.I2.i2.p1.2.m2.1.1.3.3.2.2" xref="S2.I2.i2.p1.2.m2.1.1.3.3.2.2.cmml">D</mi><mi id="S2.I2.i2.p1.2.m2.1.1.3.3.2.3" xref="S2.I2.i2.p1.2.m2.1.1.3.3.2.3.cmml">α</mi><mo id="S2.I2.i2.p1.2.m2.1.1.3.3.3" xref="S2.I2.i2.p1.2.m2.1.1.3.3.3.cmml">+</mo></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.I2.i2.p1.2.m2.1b"><apply id="S2.I2.i2.p1.2.m2.1.1.cmml" xref="S2.I2.i2.p1.2.m2.1.1"><csymbol cd="latexml" id="S2.I2.i2.p1.2.m2.1.1.1.cmml" xref="S2.I2.i2.p1.2.m2.1.1.1">assign</csymbol><apply id="S2.I2.i2.p1.2.m2.1.1.2.cmml" xref="S2.I2.i2.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="S2.I2.i2.p1.2.m2.1.1.2.1.cmml" xref="S2.I2.i2.p1.2.m2.1.1.2">superscript</csymbol><apply id="S2.I2.i2.p1.2.m2.1.1.2.2.cmml" xref="S2.I2.i2.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="S2.I2.i2.p1.2.m2.1.1.2.2.1.cmml" xref="S2.I2.i2.p1.2.m2.1.1.2">subscript</csymbol><ci id="S2.I2.i2.p1.2.m2.1.1.2.2.2.cmml" xref="S2.I2.i2.p1.2.m2.1.1.2.2.2">𝐷</ci><apply id="S2.I2.i2.p1.2.m2.1.1.2.2.3.cmml" xref="S2.I2.i2.p1.2.m2.1.1.2.2.3"><plus id="S2.I2.i2.p1.2.m2.1.1.2.2.3.1.cmml" xref="S2.I2.i2.p1.2.m2.1.1.2.2.3.1"></plus><ci id="S2.I2.i2.p1.2.m2.1.1.2.2.3.2.cmml" xref="S2.I2.i2.p1.2.m2.1.1.2.2.3.2">𝛼</ci><cn id="S2.I2.i2.p1.2.m2.1.1.2.2.3.3.cmml" type="integer" xref="S2.I2.i2.p1.2.m2.1.1.2.2.3.3">1</cn></apply></apply><minus id="S2.I2.i2.p1.2.m2.1.1.2.3.cmml" xref="S2.I2.i2.p1.2.m2.1.1.2.3"></minus></apply><apply id="S2.I2.i2.p1.2.m2.1.1.3.cmml" xref="S2.I2.i2.p1.2.m2.1.1.3"><times id="S2.I2.i2.p1.2.m2.1.1.3.1.cmml" xref="S2.I2.i2.p1.2.m2.1.1.3.1"></times><apply id="S2.I2.i2.p1.2.m2.1.1.3.2.cmml" xref="S2.I2.i2.p1.2.m2.1.1.3.2"><csymbol cd="ambiguous" id="S2.I2.i2.p1.2.m2.1.1.3.2.1.cmml" xref="S2.I2.i2.p1.2.m2.1.1.3.2">superscript</csymbol><ci id="S2.I2.i2.p1.2.m2.1.1.3.2.2.cmml" xref="S2.I2.i2.p1.2.m2.1.1.3.2.2">𝐶</ci><ci id="S2.I2.i2.p1.2.m2.1.1.3.2.3.cmml" xref="S2.I2.i2.p1.2.m2.1.1.3.2.3">⋆</ci></apply><apply id="S2.I2.i2.p1.2.m2.1.1.3.3.cmml" xref="S2.I2.i2.p1.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S2.I2.i2.p1.2.m2.1.1.3.3.1.cmml" xref="S2.I2.i2.p1.2.m2.1.1.3.3">superscript</csymbol><apply id="S2.I2.i2.p1.2.m2.1.1.3.3.2.cmml" xref="S2.I2.i2.p1.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S2.I2.i2.p1.2.m2.1.1.3.3.2.1.cmml" xref="S2.I2.i2.p1.2.m2.1.1.3.3">subscript</csymbol><ci id="S2.I2.i2.p1.2.m2.1.1.3.3.2.2.cmml" xref="S2.I2.i2.p1.2.m2.1.1.3.3.2.2">𝐷</ci><ci id="S2.I2.i2.p1.2.m2.1.1.3.3.2.3.cmml" xref="S2.I2.i2.p1.2.m2.1.1.3.3.2.3">𝛼</ci></apply><plus id="S2.I2.i2.p1.2.m2.1.1.3.3.3.cmml" xref="S2.I2.i2.p1.2.m2.1.1.3.3.3"></plus></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I2.i2.p1.2.m2.1c">D_{\alpha+1}^{-}:=C^{\star}\times D_{\alpha}^{+}</annotation><annotation encoding="application/x-llamapun" id="S2.I2.i2.p1.2.m2.1d">italic_D start_POSTSUBSCRIPT italic_α + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT := italic_C start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT × italic_D start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S2.I2.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S2.I2.i3.p1"> <p class="ltx_p" id="S2.I2.i3.p1.9">If <math alttext="\alpha" class="ltx_Math" display="inline" id="S2.I2.i3.p1.1.m1.1"><semantics id="S2.I2.i3.p1.1.m1.1a"><mi id="S2.I2.i3.p1.1.m1.1.1" xref="S2.I2.i3.p1.1.m1.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S2.I2.i3.p1.1.m1.1b"><ci id="S2.I2.i3.p1.1.m1.1.1.cmml" xref="S2.I2.i3.p1.1.m1.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.I2.i3.p1.1.m1.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S2.I2.i3.p1.1.m1.1d">italic_α</annotation></semantics></math> is limit let <math alttext="D_{\alpha}^{+}:=\sum_{x\in C}A_{x}" class="ltx_Math" display="inline" id="S2.I2.i3.p1.2.m2.1"><semantics id="S2.I2.i3.p1.2.m2.1a"><mrow id="S2.I2.i3.p1.2.m2.1.1" xref="S2.I2.i3.p1.2.m2.1.1.cmml"><msubsup id="S2.I2.i3.p1.2.m2.1.1.2" xref="S2.I2.i3.p1.2.m2.1.1.2.cmml"><mi id="S2.I2.i3.p1.2.m2.1.1.2.2.2" xref="S2.I2.i3.p1.2.m2.1.1.2.2.2.cmml">D</mi><mi id="S2.I2.i3.p1.2.m2.1.1.2.2.3" xref="S2.I2.i3.p1.2.m2.1.1.2.2.3.cmml">α</mi><mo id="S2.I2.i3.p1.2.m2.1.1.2.3" xref="S2.I2.i3.p1.2.m2.1.1.2.3.cmml">+</mo></msubsup><mo id="S2.I2.i3.p1.2.m2.1.1.1" lspace="0.278em" rspace="0.111em" xref="S2.I2.i3.p1.2.m2.1.1.1.cmml">:=</mo><mrow id="S2.I2.i3.p1.2.m2.1.1.3" xref="S2.I2.i3.p1.2.m2.1.1.3.cmml"><msub id="S2.I2.i3.p1.2.m2.1.1.3.1" xref="S2.I2.i3.p1.2.m2.1.1.3.1.cmml"><mo id="S2.I2.i3.p1.2.m2.1.1.3.1.2" xref="S2.I2.i3.p1.2.m2.1.1.3.1.2.cmml">∑</mo><mrow id="S2.I2.i3.p1.2.m2.1.1.3.1.3" xref="S2.I2.i3.p1.2.m2.1.1.3.1.3.cmml"><mi id="S2.I2.i3.p1.2.m2.1.1.3.1.3.2" xref="S2.I2.i3.p1.2.m2.1.1.3.1.3.2.cmml">x</mi><mo id="S2.I2.i3.p1.2.m2.1.1.3.1.3.1" xref="S2.I2.i3.p1.2.m2.1.1.3.1.3.1.cmml">∈</mo><mi id="S2.I2.i3.p1.2.m2.1.1.3.1.3.3" xref="S2.I2.i3.p1.2.m2.1.1.3.1.3.3.cmml">C</mi></mrow></msub><msub id="S2.I2.i3.p1.2.m2.1.1.3.2" xref="S2.I2.i3.p1.2.m2.1.1.3.2.cmml"><mi id="S2.I2.i3.p1.2.m2.1.1.3.2.2" xref="S2.I2.i3.p1.2.m2.1.1.3.2.2.cmml">A</mi><mi id="S2.I2.i3.p1.2.m2.1.1.3.2.3" xref="S2.I2.i3.p1.2.m2.1.1.3.2.3.cmml">x</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.I2.i3.p1.2.m2.1b"><apply id="S2.I2.i3.p1.2.m2.1.1.cmml" xref="S2.I2.i3.p1.2.m2.1.1"><csymbol cd="latexml" id="S2.I2.i3.p1.2.m2.1.1.1.cmml" xref="S2.I2.i3.p1.2.m2.1.1.1">assign</csymbol><apply id="S2.I2.i3.p1.2.m2.1.1.2.cmml" xref="S2.I2.i3.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="S2.I2.i3.p1.2.m2.1.1.2.1.cmml" xref="S2.I2.i3.p1.2.m2.1.1.2">superscript</csymbol><apply id="S2.I2.i3.p1.2.m2.1.1.2.2.cmml" xref="S2.I2.i3.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="S2.I2.i3.p1.2.m2.1.1.2.2.1.cmml" xref="S2.I2.i3.p1.2.m2.1.1.2">subscript</csymbol><ci id="S2.I2.i3.p1.2.m2.1.1.2.2.2.cmml" xref="S2.I2.i3.p1.2.m2.1.1.2.2.2">𝐷</ci><ci id="S2.I2.i3.p1.2.m2.1.1.2.2.3.cmml" xref="S2.I2.i3.p1.2.m2.1.1.2.2.3">𝛼</ci></apply><plus id="S2.I2.i3.p1.2.m2.1.1.2.3.cmml" xref="S2.I2.i3.p1.2.m2.1.1.2.3"></plus></apply><apply id="S2.I2.i3.p1.2.m2.1.1.3.cmml" xref="S2.I2.i3.p1.2.m2.1.1.3"><apply id="S2.I2.i3.p1.2.m2.1.1.3.1.cmml" xref="S2.I2.i3.p1.2.m2.1.1.3.1"><csymbol cd="ambiguous" id="S2.I2.i3.p1.2.m2.1.1.3.1.1.cmml" xref="S2.I2.i3.p1.2.m2.1.1.3.1">subscript</csymbol><sum id="S2.I2.i3.p1.2.m2.1.1.3.1.2.cmml" xref="S2.I2.i3.p1.2.m2.1.1.3.1.2"></sum><apply id="S2.I2.i3.p1.2.m2.1.1.3.1.3.cmml" xref="S2.I2.i3.p1.2.m2.1.1.3.1.3"><in id="S2.I2.i3.p1.2.m2.1.1.3.1.3.1.cmml" xref="S2.I2.i3.p1.2.m2.1.1.3.1.3.1"></in><ci id="S2.I2.i3.p1.2.m2.1.1.3.1.3.2.cmml" xref="S2.I2.i3.p1.2.m2.1.1.3.1.3.2">𝑥</ci><ci id="S2.I2.i3.p1.2.m2.1.1.3.1.3.3.cmml" xref="S2.I2.i3.p1.2.m2.1.1.3.1.3.3">𝐶</ci></apply></apply><apply id="S2.I2.i3.p1.2.m2.1.1.3.2.cmml" xref="S2.I2.i3.p1.2.m2.1.1.3.2"><csymbol cd="ambiguous" id="S2.I2.i3.p1.2.m2.1.1.3.2.1.cmml" xref="S2.I2.i3.p1.2.m2.1.1.3.2">subscript</csymbol><ci id="S2.I2.i3.p1.2.m2.1.1.3.2.2.cmml" xref="S2.I2.i3.p1.2.m2.1.1.3.2.2">𝐴</ci><ci id="S2.I2.i3.p1.2.m2.1.1.3.2.3.cmml" xref="S2.I2.i3.p1.2.m2.1.1.3.2.3">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I2.i3.p1.2.m2.1c">D_{\alpha}^{+}:=\sum_{x\in C}A_{x}</annotation><annotation encoding="application/x-llamapun" id="S2.I2.i3.p1.2.m2.1d">italic_D start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT := ∑ start_POSTSUBSCRIPT italic_x ∈ italic_C end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> such that each <math alttext="A_{x}" class="ltx_Math" display="inline" id="S2.I2.i3.p1.3.m3.1"><semantics id="S2.I2.i3.p1.3.m3.1a"><msub id="S2.I2.i3.p1.3.m3.1.1" xref="S2.I2.i3.p1.3.m3.1.1.cmml"><mi id="S2.I2.i3.p1.3.m3.1.1.2" xref="S2.I2.i3.p1.3.m3.1.1.2.cmml">A</mi><mi id="S2.I2.i3.p1.3.m3.1.1.3" xref="S2.I2.i3.p1.3.m3.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="S2.I2.i3.p1.3.m3.1b"><apply id="S2.I2.i3.p1.3.m3.1.1.cmml" xref="S2.I2.i3.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S2.I2.i3.p1.3.m3.1.1.1.cmml" xref="S2.I2.i3.p1.3.m3.1.1">subscript</csymbol><ci id="S2.I2.i3.p1.3.m3.1.1.2.cmml" xref="S2.I2.i3.p1.3.m3.1.1.2">𝐴</ci><ci id="S2.I2.i3.p1.3.m3.1.1.3.cmml" xref="S2.I2.i3.p1.3.m3.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I2.i3.p1.3.m3.1c">A_{x}</annotation><annotation encoding="application/x-llamapun" id="S2.I2.i3.p1.3.m3.1d">italic_A start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> is <math alttext="D_{\xi}^{-}" class="ltx_Math" display="inline" id="S2.I2.i3.p1.4.m4.1"><semantics id="S2.I2.i3.p1.4.m4.1a"><msubsup id="S2.I2.i3.p1.4.m4.1.1" xref="S2.I2.i3.p1.4.m4.1.1.cmml"><mi id="S2.I2.i3.p1.4.m4.1.1.2.2" xref="S2.I2.i3.p1.4.m4.1.1.2.2.cmml">D</mi><mi id="S2.I2.i3.p1.4.m4.1.1.2.3" xref="S2.I2.i3.p1.4.m4.1.1.2.3.cmml">ξ</mi><mo id="S2.I2.i3.p1.4.m4.1.1.3" xref="S2.I2.i3.p1.4.m4.1.1.3.cmml">−</mo></msubsup><annotation-xml encoding="MathML-Content" id="S2.I2.i3.p1.4.m4.1b"><apply id="S2.I2.i3.p1.4.m4.1.1.cmml" xref="S2.I2.i3.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S2.I2.i3.p1.4.m4.1.1.1.cmml" xref="S2.I2.i3.p1.4.m4.1.1">superscript</csymbol><apply id="S2.I2.i3.p1.4.m4.1.1.2.cmml" xref="S2.I2.i3.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S2.I2.i3.p1.4.m4.1.1.2.1.cmml" xref="S2.I2.i3.p1.4.m4.1.1">subscript</csymbol><ci id="S2.I2.i3.p1.4.m4.1.1.2.2.cmml" xref="S2.I2.i3.p1.4.m4.1.1.2.2">𝐷</ci><ci id="S2.I2.i3.p1.4.m4.1.1.2.3.cmml" xref="S2.I2.i3.p1.4.m4.1.1.2.3">𝜉</ci></apply><minus id="S2.I2.i3.p1.4.m4.1.1.3.cmml" xref="S2.I2.i3.p1.4.m4.1.1.3"></minus></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I2.i3.p1.4.m4.1c">D_{\xi}^{-}</annotation><annotation encoding="application/x-llamapun" id="S2.I2.i3.p1.4.m4.1d">italic_D start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT</annotation></semantics></math> for some <math alttext="\xi<\alpha" class="ltx_Math" display="inline" id="S2.I2.i3.p1.5.m5.1"><semantics id="S2.I2.i3.p1.5.m5.1a"><mrow id="S2.I2.i3.p1.5.m5.1.1" xref="S2.I2.i3.p1.5.m5.1.1.cmml"><mi id="S2.I2.i3.p1.5.m5.1.1.2" xref="S2.I2.i3.p1.5.m5.1.1.2.cmml">ξ</mi><mo id="S2.I2.i3.p1.5.m5.1.1.1" xref="S2.I2.i3.p1.5.m5.1.1.1.cmml"><</mo><mi id="S2.I2.i3.p1.5.m5.1.1.3" xref="S2.I2.i3.p1.5.m5.1.1.3.cmml">α</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.I2.i3.p1.5.m5.1b"><apply id="S2.I2.i3.p1.5.m5.1.1.cmml" xref="S2.I2.i3.p1.5.m5.1.1"><lt id="S2.I2.i3.p1.5.m5.1.1.1.cmml" xref="S2.I2.i3.p1.5.m5.1.1.1"></lt><ci id="S2.I2.i3.p1.5.m5.1.1.2.cmml" xref="S2.I2.i3.p1.5.m5.1.1.2">𝜉</ci><ci id="S2.I2.i3.p1.5.m5.1.1.3.cmml" xref="S2.I2.i3.p1.5.m5.1.1.3">𝛼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I2.i3.p1.5.m5.1c">\xi<\alpha</annotation><annotation encoding="application/x-llamapun" id="S2.I2.i3.p1.5.m5.1d">italic_ξ < italic_α</annotation></semantics></math>, and such that for each <math alttext="\xi<\alpha" class="ltx_Math" display="inline" id="S2.I2.i3.p1.6.m6.1"><semantics id="S2.I2.i3.p1.6.m6.1a"><mrow id="S2.I2.i3.p1.6.m6.1.1" xref="S2.I2.i3.p1.6.m6.1.1.cmml"><mi id="S2.I2.i3.p1.6.m6.1.1.2" xref="S2.I2.i3.p1.6.m6.1.1.2.cmml">ξ</mi><mo id="S2.I2.i3.p1.6.m6.1.1.1" xref="S2.I2.i3.p1.6.m6.1.1.1.cmml"><</mo><mi id="S2.I2.i3.p1.6.m6.1.1.3" xref="S2.I2.i3.p1.6.m6.1.1.3.cmml">α</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.I2.i3.p1.6.m6.1b"><apply id="S2.I2.i3.p1.6.m6.1.1.cmml" xref="S2.I2.i3.p1.6.m6.1.1"><lt id="S2.I2.i3.p1.6.m6.1.1.1.cmml" xref="S2.I2.i3.p1.6.m6.1.1.1"></lt><ci id="S2.I2.i3.p1.6.m6.1.1.2.cmml" xref="S2.I2.i3.p1.6.m6.1.1.2">𝜉</ci><ci id="S2.I2.i3.p1.6.m6.1.1.3.cmml" xref="S2.I2.i3.p1.6.m6.1.1.3">𝛼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I2.i3.p1.6.m6.1c">\xi<\alpha</annotation><annotation encoding="application/x-llamapun" id="S2.I2.i3.p1.6.m6.1d">italic_ξ < italic_α</annotation></semantics></math>, <math alttext="\{x\in C:A_{x}=D_{\xi}^{-}\}" class="ltx_Math" display="inline" id="S2.I2.i3.p1.7.m7.2"><semantics id="S2.I2.i3.p1.7.m7.2a"><mrow id="S2.I2.i3.p1.7.m7.2.2.2" xref="S2.I2.i3.p1.7.m7.2.2.3.cmml"><mo id="S2.I2.i3.p1.7.m7.2.2.2.3" stretchy="false" xref="S2.I2.i3.p1.7.m7.2.2.3.1.cmml">{</mo><mrow id="S2.I2.i3.p1.7.m7.1.1.1.1" xref="S2.I2.i3.p1.7.m7.1.1.1.1.cmml"><mi id="S2.I2.i3.p1.7.m7.1.1.1.1.2" xref="S2.I2.i3.p1.7.m7.1.1.1.1.2.cmml">x</mi><mo id="S2.I2.i3.p1.7.m7.1.1.1.1.1" xref="S2.I2.i3.p1.7.m7.1.1.1.1.1.cmml">∈</mo><mi id="S2.I2.i3.p1.7.m7.1.1.1.1.3" xref="S2.I2.i3.p1.7.m7.1.1.1.1.3.cmml">C</mi></mrow><mo id="S2.I2.i3.p1.7.m7.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S2.I2.i3.p1.7.m7.2.2.3.1.cmml">:</mo><mrow id="S2.I2.i3.p1.7.m7.2.2.2.2" xref="S2.I2.i3.p1.7.m7.2.2.2.2.cmml"><msub id="S2.I2.i3.p1.7.m7.2.2.2.2.2" xref="S2.I2.i3.p1.7.m7.2.2.2.2.2.cmml"><mi id="S2.I2.i3.p1.7.m7.2.2.2.2.2.2" xref="S2.I2.i3.p1.7.m7.2.2.2.2.2.2.cmml">A</mi><mi id="S2.I2.i3.p1.7.m7.2.2.2.2.2.3" xref="S2.I2.i3.p1.7.m7.2.2.2.2.2.3.cmml">x</mi></msub><mo id="S2.I2.i3.p1.7.m7.2.2.2.2.1" xref="S2.I2.i3.p1.7.m7.2.2.2.2.1.cmml">=</mo><msubsup id="S2.I2.i3.p1.7.m7.2.2.2.2.3" xref="S2.I2.i3.p1.7.m7.2.2.2.2.3.cmml"><mi id="S2.I2.i3.p1.7.m7.2.2.2.2.3.2.2" xref="S2.I2.i3.p1.7.m7.2.2.2.2.3.2.2.cmml">D</mi><mi id="S2.I2.i3.p1.7.m7.2.2.2.2.3.2.3" xref="S2.I2.i3.p1.7.m7.2.2.2.2.3.2.3.cmml">ξ</mi><mo id="S2.I2.i3.p1.7.m7.2.2.2.2.3.3" xref="S2.I2.i3.p1.7.m7.2.2.2.2.3.3.cmml">−</mo></msubsup></mrow><mo id="S2.I2.i3.p1.7.m7.2.2.2.5" stretchy="false" xref="S2.I2.i3.p1.7.m7.2.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.I2.i3.p1.7.m7.2b"><apply id="S2.I2.i3.p1.7.m7.2.2.3.cmml" xref="S2.I2.i3.p1.7.m7.2.2.2"><csymbol cd="latexml" id="S2.I2.i3.p1.7.m7.2.2.3.1.cmml" xref="S2.I2.i3.p1.7.m7.2.2.2.3">conditional-set</csymbol><apply id="S2.I2.i3.p1.7.m7.1.1.1.1.cmml" xref="S2.I2.i3.p1.7.m7.1.1.1.1"><in id="S2.I2.i3.p1.7.m7.1.1.1.1.1.cmml" xref="S2.I2.i3.p1.7.m7.1.1.1.1.1"></in><ci id="S2.I2.i3.p1.7.m7.1.1.1.1.2.cmml" xref="S2.I2.i3.p1.7.m7.1.1.1.1.2">𝑥</ci><ci id="S2.I2.i3.p1.7.m7.1.1.1.1.3.cmml" xref="S2.I2.i3.p1.7.m7.1.1.1.1.3">𝐶</ci></apply><apply id="S2.I2.i3.p1.7.m7.2.2.2.2.cmml" xref="S2.I2.i3.p1.7.m7.2.2.2.2"><eq id="S2.I2.i3.p1.7.m7.2.2.2.2.1.cmml" xref="S2.I2.i3.p1.7.m7.2.2.2.2.1"></eq><apply id="S2.I2.i3.p1.7.m7.2.2.2.2.2.cmml" xref="S2.I2.i3.p1.7.m7.2.2.2.2.2"><csymbol cd="ambiguous" id="S2.I2.i3.p1.7.m7.2.2.2.2.2.1.cmml" xref="S2.I2.i3.p1.7.m7.2.2.2.2.2">subscript</csymbol><ci id="S2.I2.i3.p1.7.m7.2.2.2.2.2.2.cmml" xref="S2.I2.i3.p1.7.m7.2.2.2.2.2.2">𝐴</ci><ci id="S2.I2.i3.p1.7.m7.2.2.2.2.2.3.cmml" xref="S2.I2.i3.p1.7.m7.2.2.2.2.2.3">𝑥</ci></apply><apply id="S2.I2.i3.p1.7.m7.2.2.2.2.3.cmml" xref="S2.I2.i3.p1.7.m7.2.2.2.2.3"><csymbol cd="ambiguous" id="S2.I2.i3.p1.7.m7.2.2.2.2.3.1.cmml" xref="S2.I2.i3.p1.7.m7.2.2.2.2.3">superscript</csymbol><apply id="S2.I2.i3.p1.7.m7.2.2.2.2.3.2.cmml" xref="S2.I2.i3.p1.7.m7.2.2.2.2.3"><csymbol cd="ambiguous" id="S2.I2.i3.p1.7.m7.2.2.2.2.3.2.1.cmml" xref="S2.I2.i3.p1.7.m7.2.2.2.2.3">subscript</csymbol><ci id="S2.I2.i3.p1.7.m7.2.2.2.2.3.2.2.cmml" xref="S2.I2.i3.p1.7.m7.2.2.2.2.3.2.2">𝐷</ci><ci id="S2.I2.i3.p1.7.m7.2.2.2.2.3.2.3.cmml" xref="S2.I2.i3.p1.7.m7.2.2.2.2.3.2.3">𝜉</ci></apply><minus id="S2.I2.i3.p1.7.m7.2.2.2.2.3.3.cmml" xref="S2.I2.i3.p1.7.m7.2.2.2.2.3.3"></minus></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I2.i3.p1.7.m7.2c">\{x\in C:A_{x}=D_{\xi}^{-}\}</annotation><annotation encoding="application/x-llamapun" id="S2.I2.i3.p1.7.m7.2d">{ italic_x ∈ italic_C : italic_A start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT = italic_D start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT }</annotation></semantics></math> is dense in <math alttext="C" class="ltx_Math" display="inline" id="S2.I2.i3.p1.8.m8.1"><semantics id="S2.I2.i3.p1.8.m8.1a"><mi id="S2.I2.i3.p1.8.m8.1.1" xref="S2.I2.i3.p1.8.m8.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S2.I2.i3.p1.8.m8.1b"><ci id="S2.I2.i3.p1.8.m8.1.1.cmml" xref="S2.I2.i3.p1.8.m8.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.I2.i3.p1.8.m8.1c">C</annotation><annotation encoding="application/x-llamapun" id="S2.I2.i3.p1.8.m8.1d">italic_C</annotation></semantics></math>. <math alttext="D_{\alpha}^{-}" class="ltx_Math" display="inline" id="S2.I2.i3.p1.9.m9.1"><semantics id="S2.I2.i3.p1.9.m9.1a"><msubsup id="S2.I2.i3.p1.9.m9.1.1" xref="S2.I2.i3.p1.9.m9.1.1.cmml"><mi id="S2.I2.i3.p1.9.m9.1.1.2.2" xref="S2.I2.i3.p1.9.m9.1.1.2.2.cmml">D</mi><mi id="S2.I2.i3.p1.9.m9.1.1.2.3" xref="S2.I2.i3.p1.9.m9.1.1.2.3.cmml">α</mi><mo id="S2.I2.i3.p1.9.m9.1.1.3" xref="S2.I2.i3.p1.9.m9.1.1.3.cmml">−</mo></msubsup><annotation-xml encoding="MathML-Content" id="S2.I2.i3.p1.9.m9.1b"><apply id="S2.I2.i3.p1.9.m9.1.1.cmml" xref="S2.I2.i3.p1.9.m9.1.1"><csymbol cd="ambiguous" id="S2.I2.i3.p1.9.m9.1.1.1.cmml" xref="S2.I2.i3.p1.9.m9.1.1">superscript</csymbol><apply id="S2.I2.i3.p1.9.m9.1.1.2.cmml" xref="S2.I2.i3.p1.9.m9.1.1"><csymbol cd="ambiguous" id="S2.I2.i3.p1.9.m9.1.1.2.1.cmml" xref="S2.I2.i3.p1.9.m9.1.1">subscript</csymbol><ci id="S2.I2.i3.p1.9.m9.1.1.2.2.cmml" xref="S2.I2.i3.p1.9.m9.1.1.2.2">𝐷</ci><ci id="S2.I2.i3.p1.9.m9.1.1.2.3.cmml" xref="S2.I2.i3.p1.9.m9.1.1.2.3">𝛼</ci></apply><minus id="S2.I2.i3.p1.9.m9.1.1.3.cmml" xref="S2.I2.i3.p1.9.m9.1.1.3"></minus></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I2.i3.p1.9.m9.1c">D_{\alpha}^{-}</annotation><annotation encoding="application/x-llamapun" id="S2.I2.i3.p1.9.m9.1d">italic_D start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT</annotation></semantics></math> is defined similarly.</p> </div> </li> </ul> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S2.Thmtheorem10"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem10.1.1.1">Lemma 2.10</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem10.2.2">.</span> </h6> <div class="ltx_para" id="S2.Thmtheorem10.p1"> <p class="ltx_p" id="S2.Thmtheorem10.p1.9">(Martínez-Ranero <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib14" title="">14</a>]</cite>) Assume <math alttext="\mathsf{MA}_{\aleph_{1}}" class="ltx_Math" display="inline" id="S2.Thmtheorem10.p1.1.m1.1"><semantics id="S2.Thmtheorem10.p1.1.m1.1a"><msub id="S2.Thmtheorem10.p1.1.m1.1.1" xref="S2.Thmtheorem10.p1.1.m1.1.1.cmml"><mi id="S2.Thmtheorem10.p1.1.m1.1.1.2" xref="S2.Thmtheorem10.p1.1.m1.1.1.2.cmml">𝖬𝖠</mi><msub id="S2.Thmtheorem10.p1.1.m1.1.1.3" xref="S2.Thmtheorem10.p1.1.m1.1.1.3.cmml"><mi id="S2.Thmtheorem10.p1.1.m1.1.1.3.2" mathvariant="normal" xref="S2.Thmtheorem10.p1.1.m1.1.1.3.2.cmml">ℵ</mi><mn id="S2.Thmtheorem10.p1.1.m1.1.1.3.3" xref="S2.Thmtheorem10.p1.1.m1.1.1.3.3.cmml">1</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem10.p1.1.m1.1b"><apply id="S2.Thmtheorem10.p1.1.m1.1.1.cmml" xref="S2.Thmtheorem10.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem10.p1.1.m1.1.1.1.cmml" xref="S2.Thmtheorem10.p1.1.m1.1.1">subscript</csymbol><ci id="S2.Thmtheorem10.p1.1.m1.1.1.2.cmml" xref="S2.Thmtheorem10.p1.1.m1.1.1.2">𝖬𝖠</ci><apply id="S2.Thmtheorem10.p1.1.m1.1.1.3.cmml" xref="S2.Thmtheorem10.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S2.Thmtheorem10.p1.1.m1.1.1.3.1.cmml" xref="S2.Thmtheorem10.p1.1.m1.1.1.3">subscript</csymbol><ci id="S2.Thmtheorem10.p1.1.m1.1.1.3.2.cmml" xref="S2.Thmtheorem10.p1.1.m1.1.1.3.2">ℵ</ci><cn id="S2.Thmtheorem10.p1.1.m1.1.1.3.3.cmml" type="integer" xref="S2.Thmtheorem10.p1.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem10.p1.1.m1.1c">\mathsf{MA}_{\aleph_{1}}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem10.p1.1.m1.1d">sansserif_MA start_POSTSUBSCRIPT roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. For any fragmented Aronszajn line <math alttext="A" class="ltx_Math" display="inline" id="S2.Thmtheorem10.p1.2.m2.1"><semantics id="S2.Thmtheorem10.p1.2.m2.1a"><mi id="S2.Thmtheorem10.p1.2.m2.1.1" xref="S2.Thmtheorem10.p1.2.m2.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem10.p1.2.m2.1b"><ci id="S2.Thmtheorem10.p1.2.m2.1.1.cmml" xref="S2.Thmtheorem10.p1.2.m2.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem10.p1.2.m2.1c">A</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem10.p1.2.m2.1d">italic_A</annotation></semantics></math>, there is <math alttext="\alpha<\omega_{2}" class="ltx_Math" display="inline" id="S2.Thmtheorem10.p1.3.m3.1"><semantics id="S2.Thmtheorem10.p1.3.m3.1a"><mrow id="S2.Thmtheorem10.p1.3.m3.1.1" xref="S2.Thmtheorem10.p1.3.m3.1.1.cmml"><mi id="S2.Thmtheorem10.p1.3.m3.1.1.2" xref="S2.Thmtheorem10.p1.3.m3.1.1.2.cmml">α</mi><mo id="S2.Thmtheorem10.p1.3.m3.1.1.1" xref="S2.Thmtheorem10.p1.3.m3.1.1.1.cmml"><</mo><msub id="S2.Thmtheorem10.p1.3.m3.1.1.3" xref="S2.Thmtheorem10.p1.3.m3.1.1.3.cmml"><mi id="S2.Thmtheorem10.p1.3.m3.1.1.3.2" xref="S2.Thmtheorem10.p1.3.m3.1.1.3.2.cmml">ω</mi><mn id="S2.Thmtheorem10.p1.3.m3.1.1.3.3" xref="S2.Thmtheorem10.p1.3.m3.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem10.p1.3.m3.1b"><apply id="S2.Thmtheorem10.p1.3.m3.1.1.cmml" xref="S2.Thmtheorem10.p1.3.m3.1.1"><lt id="S2.Thmtheorem10.p1.3.m3.1.1.1.cmml" xref="S2.Thmtheorem10.p1.3.m3.1.1.1"></lt><ci id="S2.Thmtheorem10.p1.3.m3.1.1.2.cmml" xref="S2.Thmtheorem10.p1.3.m3.1.1.2">𝛼</ci><apply id="S2.Thmtheorem10.p1.3.m3.1.1.3.cmml" xref="S2.Thmtheorem10.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S2.Thmtheorem10.p1.3.m3.1.1.3.1.cmml" xref="S2.Thmtheorem10.p1.3.m3.1.1.3">subscript</csymbol><ci id="S2.Thmtheorem10.p1.3.m3.1.1.3.2.cmml" xref="S2.Thmtheorem10.p1.3.m3.1.1.3.2">𝜔</ci><cn id="S2.Thmtheorem10.p1.3.m3.1.1.3.3.cmml" type="integer" xref="S2.Thmtheorem10.p1.3.m3.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem10.p1.3.m3.1c">\alpha<\omega_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem10.p1.3.m3.1d">italic_α < italic_ω start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> such that either <math alttext="A" class="ltx_Math" display="inline" id="S2.Thmtheorem10.p1.4.m4.1"><semantics id="S2.Thmtheorem10.p1.4.m4.1a"><mi id="S2.Thmtheorem10.p1.4.m4.1.1" xref="S2.Thmtheorem10.p1.4.m4.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem10.p1.4.m4.1b"><ci id="S2.Thmtheorem10.p1.4.m4.1.1.cmml" xref="S2.Thmtheorem10.p1.4.m4.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem10.p1.4.m4.1c">A</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem10.p1.4.m4.1d">italic_A</annotation></semantics></math> is <math alttext="\preceq" class="ltx_Math" display="inline" id="S2.Thmtheorem10.p1.5.m5.1"><semantics id="S2.Thmtheorem10.p1.5.m5.1a"><mo id="S2.Thmtheorem10.p1.5.m5.1.1" xref="S2.Thmtheorem10.p1.5.m5.1.1.cmml">⪯</mo><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem10.p1.5.m5.1b"><csymbol cd="latexml" id="S2.Thmtheorem10.p1.5.m5.1.1.cmml" xref="S2.Thmtheorem10.p1.5.m5.1.1">precedes-or-equals</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem10.p1.5.m5.1c">\preceq</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem10.p1.5.m5.1d">⪯</annotation></semantics></math>-equivalent to <math alttext="D_{\alpha}^{-}" class="ltx_Math" display="inline" id="S2.Thmtheorem10.p1.6.m6.1"><semantics id="S2.Thmtheorem10.p1.6.m6.1a"><msubsup id="S2.Thmtheorem10.p1.6.m6.1.1" xref="S2.Thmtheorem10.p1.6.m6.1.1.cmml"><mi id="S2.Thmtheorem10.p1.6.m6.1.1.2.2" xref="S2.Thmtheorem10.p1.6.m6.1.1.2.2.cmml">D</mi><mi id="S2.Thmtheorem10.p1.6.m6.1.1.2.3" xref="S2.Thmtheorem10.p1.6.m6.1.1.2.3.cmml">α</mi><mo id="S2.Thmtheorem10.p1.6.m6.1.1.3" xref="S2.Thmtheorem10.p1.6.m6.1.1.3.cmml">−</mo></msubsup><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem10.p1.6.m6.1b"><apply id="S2.Thmtheorem10.p1.6.m6.1.1.cmml" xref="S2.Thmtheorem10.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem10.p1.6.m6.1.1.1.cmml" xref="S2.Thmtheorem10.p1.6.m6.1.1">superscript</csymbol><apply id="S2.Thmtheorem10.p1.6.m6.1.1.2.cmml" xref="S2.Thmtheorem10.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem10.p1.6.m6.1.1.2.1.cmml" xref="S2.Thmtheorem10.p1.6.m6.1.1">subscript</csymbol><ci id="S2.Thmtheorem10.p1.6.m6.1.1.2.2.cmml" xref="S2.Thmtheorem10.p1.6.m6.1.1.2.2">𝐷</ci><ci id="S2.Thmtheorem10.p1.6.m6.1.1.2.3.cmml" xref="S2.Thmtheorem10.p1.6.m6.1.1.2.3">𝛼</ci></apply><minus id="S2.Thmtheorem10.p1.6.m6.1.1.3.cmml" xref="S2.Thmtheorem10.p1.6.m6.1.1.3"></minus></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem10.p1.6.m6.1c">D_{\alpha}^{-}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem10.p1.6.m6.1d">italic_D start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT</annotation></semantics></math> or <math alttext="D_{\alpha}^{+}" class="ltx_Math" display="inline" id="S2.Thmtheorem10.p1.7.m7.1"><semantics id="S2.Thmtheorem10.p1.7.m7.1a"><msubsup id="S2.Thmtheorem10.p1.7.m7.1.1" xref="S2.Thmtheorem10.p1.7.m7.1.1.cmml"><mi id="S2.Thmtheorem10.p1.7.m7.1.1.2.2" xref="S2.Thmtheorem10.p1.7.m7.1.1.2.2.cmml">D</mi><mi id="S2.Thmtheorem10.p1.7.m7.1.1.2.3" xref="S2.Thmtheorem10.p1.7.m7.1.1.2.3.cmml">α</mi><mo id="S2.Thmtheorem10.p1.7.m7.1.1.3" xref="S2.Thmtheorem10.p1.7.m7.1.1.3.cmml">+</mo></msubsup><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem10.p1.7.m7.1b"><apply id="S2.Thmtheorem10.p1.7.m7.1.1.cmml" xref="S2.Thmtheorem10.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem10.p1.7.m7.1.1.1.cmml" xref="S2.Thmtheorem10.p1.7.m7.1.1">superscript</csymbol><apply id="S2.Thmtheorem10.p1.7.m7.1.1.2.cmml" xref="S2.Thmtheorem10.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem10.p1.7.m7.1.1.2.1.cmml" xref="S2.Thmtheorem10.p1.7.m7.1.1">subscript</csymbol><ci id="S2.Thmtheorem10.p1.7.m7.1.1.2.2.cmml" xref="S2.Thmtheorem10.p1.7.m7.1.1.2.2">𝐷</ci><ci id="S2.Thmtheorem10.p1.7.m7.1.1.2.3.cmml" xref="S2.Thmtheorem10.p1.7.m7.1.1.2.3">𝛼</ci></apply><plus id="S2.Thmtheorem10.p1.7.m7.1.1.3.cmml" xref="S2.Thmtheorem10.p1.7.m7.1.1.3"></plus></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem10.p1.7.m7.1c">D_{\alpha}^{+}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem10.p1.7.m7.1d">italic_D start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math>, or <math alttext="A\prec D_{\alpha}^{-}" class="ltx_Math" display="inline" id="S2.Thmtheorem10.p1.8.m8.1"><semantics id="S2.Thmtheorem10.p1.8.m8.1a"><mrow id="S2.Thmtheorem10.p1.8.m8.1.1" xref="S2.Thmtheorem10.p1.8.m8.1.1.cmml"><mi id="S2.Thmtheorem10.p1.8.m8.1.1.2" xref="S2.Thmtheorem10.p1.8.m8.1.1.2.cmml">A</mi><mo id="S2.Thmtheorem10.p1.8.m8.1.1.1" xref="S2.Thmtheorem10.p1.8.m8.1.1.1.cmml">≺</mo><msubsup id="S2.Thmtheorem10.p1.8.m8.1.1.3" xref="S2.Thmtheorem10.p1.8.m8.1.1.3.cmml"><mi id="S2.Thmtheorem10.p1.8.m8.1.1.3.2.2" xref="S2.Thmtheorem10.p1.8.m8.1.1.3.2.2.cmml">D</mi><mi id="S2.Thmtheorem10.p1.8.m8.1.1.3.2.3" xref="S2.Thmtheorem10.p1.8.m8.1.1.3.2.3.cmml">α</mi><mo id="S2.Thmtheorem10.p1.8.m8.1.1.3.3" xref="S2.Thmtheorem10.p1.8.m8.1.1.3.3.cmml">−</mo></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem10.p1.8.m8.1b"><apply id="S2.Thmtheorem10.p1.8.m8.1.1.cmml" xref="S2.Thmtheorem10.p1.8.m8.1.1"><csymbol cd="latexml" id="S2.Thmtheorem10.p1.8.m8.1.1.1.cmml" xref="S2.Thmtheorem10.p1.8.m8.1.1.1">precedes</csymbol><ci id="S2.Thmtheorem10.p1.8.m8.1.1.2.cmml" xref="S2.Thmtheorem10.p1.8.m8.1.1.2">𝐴</ci><apply id="S2.Thmtheorem10.p1.8.m8.1.1.3.cmml" xref="S2.Thmtheorem10.p1.8.m8.1.1.3"><csymbol cd="ambiguous" id="S2.Thmtheorem10.p1.8.m8.1.1.3.1.cmml" xref="S2.Thmtheorem10.p1.8.m8.1.1.3">superscript</csymbol><apply id="S2.Thmtheorem10.p1.8.m8.1.1.3.2.cmml" xref="S2.Thmtheorem10.p1.8.m8.1.1.3"><csymbol cd="ambiguous" id="S2.Thmtheorem10.p1.8.m8.1.1.3.2.1.cmml" xref="S2.Thmtheorem10.p1.8.m8.1.1.3">subscript</csymbol><ci id="S2.Thmtheorem10.p1.8.m8.1.1.3.2.2.cmml" xref="S2.Thmtheorem10.p1.8.m8.1.1.3.2.2">𝐷</ci><ci id="S2.Thmtheorem10.p1.8.m8.1.1.3.2.3.cmml" xref="S2.Thmtheorem10.p1.8.m8.1.1.3.2.3">𝛼</ci></apply><minus id="S2.Thmtheorem10.p1.8.m8.1.1.3.3.cmml" xref="S2.Thmtheorem10.p1.8.m8.1.1.3.3"></minus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem10.p1.8.m8.1c">A\prec D_{\alpha}^{-}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem10.p1.8.m8.1d">italic_A ≺ italic_D start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="A\prec D_{\alpha}^{+}" class="ltx_Math" display="inline" id="S2.Thmtheorem10.p1.9.m9.1"><semantics id="S2.Thmtheorem10.p1.9.m9.1a"><mrow id="S2.Thmtheorem10.p1.9.m9.1.1" xref="S2.Thmtheorem10.p1.9.m9.1.1.cmml"><mi id="S2.Thmtheorem10.p1.9.m9.1.1.2" xref="S2.Thmtheorem10.p1.9.m9.1.1.2.cmml">A</mi><mo id="S2.Thmtheorem10.p1.9.m9.1.1.1" xref="S2.Thmtheorem10.p1.9.m9.1.1.1.cmml">≺</mo><msubsup id="S2.Thmtheorem10.p1.9.m9.1.1.3" xref="S2.Thmtheorem10.p1.9.m9.1.1.3.cmml"><mi id="S2.Thmtheorem10.p1.9.m9.1.1.3.2.2" xref="S2.Thmtheorem10.p1.9.m9.1.1.3.2.2.cmml">D</mi><mi id="S2.Thmtheorem10.p1.9.m9.1.1.3.2.3" xref="S2.Thmtheorem10.p1.9.m9.1.1.3.2.3.cmml">α</mi><mo id="S2.Thmtheorem10.p1.9.m9.1.1.3.3" xref="S2.Thmtheorem10.p1.9.m9.1.1.3.3.cmml">+</mo></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem10.p1.9.m9.1b"><apply id="S2.Thmtheorem10.p1.9.m9.1.1.cmml" xref="S2.Thmtheorem10.p1.9.m9.1.1"><csymbol cd="latexml" id="S2.Thmtheorem10.p1.9.m9.1.1.1.cmml" xref="S2.Thmtheorem10.p1.9.m9.1.1.1">precedes</csymbol><ci id="S2.Thmtheorem10.p1.9.m9.1.1.2.cmml" xref="S2.Thmtheorem10.p1.9.m9.1.1.2">𝐴</ci><apply id="S2.Thmtheorem10.p1.9.m9.1.1.3.cmml" xref="S2.Thmtheorem10.p1.9.m9.1.1.3"><csymbol cd="ambiguous" id="S2.Thmtheorem10.p1.9.m9.1.1.3.1.cmml" xref="S2.Thmtheorem10.p1.9.m9.1.1.3">superscript</csymbol><apply id="S2.Thmtheorem10.p1.9.m9.1.1.3.2.cmml" xref="S2.Thmtheorem10.p1.9.m9.1.1.3"><csymbol cd="ambiguous" id="S2.Thmtheorem10.p1.9.m9.1.1.3.2.1.cmml" xref="S2.Thmtheorem10.p1.9.m9.1.1.3">subscript</csymbol><ci id="S2.Thmtheorem10.p1.9.m9.1.1.3.2.2.cmml" xref="S2.Thmtheorem10.p1.9.m9.1.1.3.2.2">𝐷</ci><ci id="S2.Thmtheorem10.p1.9.m9.1.1.3.2.3.cmml" xref="S2.Thmtheorem10.p1.9.m9.1.1.3.2.3">𝛼</ci></apply><plus id="S2.Thmtheorem10.p1.9.m9.1.1.3.3.cmml" xref="S2.Thmtheorem10.p1.9.m9.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem10.p1.9.m9.1c">A\prec D_{\alpha}^{+}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem10.p1.9.m9.1d">italic_A ≺ italic_D start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> </div> </section> <section class="ltx_section" id="S3"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">3. </span>Strongly surjective Aronszajn lines</h2> <div class="ltx_para" id="S3.p1"> <p class="ltx_p" id="S3.p1.2">In this chapter we show that under <math alttext="\mathsf{MA}_{\aleph_{1}}" class="ltx_Math" display="inline" id="S3.p1.1.m1.1"><semantics id="S3.p1.1.m1.1a"><msub id="S3.p1.1.m1.1.1" xref="S3.p1.1.m1.1.1.cmml"><mi id="S3.p1.1.m1.1.1.2" xref="S3.p1.1.m1.1.1.2.cmml">𝖬𝖠</mi><msub id="S3.p1.1.m1.1.1.3" xref="S3.p1.1.m1.1.1.3.cmml"><mi id="S3.p1.1.m1.1.1.3.2" mathvariant="normal" xref="S3.p1.1.m1.1.1.3.2.cmml">ℵ</mi><mn id="S3.p1.1.m1.1.1.3.3" xref="S3.p1.1.m1.1.1.3.3.cmml">1</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S3.p1.1.m1.1b"><apply id="S3.p1.1.m1.1.1.cmml" xref="S3.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.p1.1.m1.1.1.1.cmml" xref="S3.p1.1.m1.1.1">subscript</csymbol><ci id="S3.p1.1.m1.1.1.2.cmml" xref="S3.p1.1.m1.1.1.2">𝖬𝖠</ci><apply id="S3.p1.1.m1.1.1.3.cmml" xref="S3.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.p1.1.m1.1.1.3.1.cmml" xref="S3.p1.1.m1.1.1.3">subscript</csymbol><ci id="S3.p1.1.m1.1.1.3.2.cmml" xref="S3.p1.1.m1.1.1.3.2">ℵ</ci><cn id="S3.p1.1.m1.1.1.3.3.cmml" type="integer" xref="S3.p1.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.1.m1.1c">\mathsf{MA}_{\aleph_{1}}</annotation><annotation encoding="application/x-llamapun" id="S3.p1.1.m1.1d">sansserif_MA start_POSTSUBSCRIPT roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> there are many strongly surjective Aronszajn lines; in particular we deduce from Moore’s theorem (<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S2.Thmtheorem8" title="Theorem 2.8. ‣ 2. Aronszajn and Countryman lines ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">2.8</span></a>) that any normal Countryman line is strongly surjective under <math alttext="\mathsf{MA}_{\aleph_{1}}" class="ltx_Math" display="inline" id="S3.p1.2.m2.1"><semantics id="S3.p1.2.m2.1a"><msub id="S3.p1.2.m2.1.1" xref="S3.p1.2.m2.1.1.cmml"><mi id="S3.p1.2.m2.1.1.2" xref="S3.p1.2.m2.1.1.2.cmml">𝖬𝖠</mi><msub id="S3.p1.2.m2.1.1.3" xref="S3.p1.2.m2.1.1.3.cmml"><mi id="S3.p1.2.m2.1.1.3.2" mathvariant="normal" xref="S3.p1.2.m2.1.1.3.2.cmml">ℵ</mi><mn id="S3.p1.2.m2.1.1.3.3" xref="S3.p1.2.m2.1.1.3.3.cmml">1</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S3.p1.2.m2.1b"><apply id="S3.p1.2.m2.1.1.cmml" xref="S3.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.p1.2.m2.1.1.1.cmml" xref="S3.p1.2.m2.1.1">subscript</csymbol><ci id="S3.p1.2.m2.1.1.2.cmml" xref="S3.p1.2.m2.1.1.2">𝖬𝖠</ci><apply id="S3.p1.2.m2.1.1.3.cmml" xref="S3.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.p1.2.m2.1.1.3.1.cmml" xref="S3.p1.2.m2.1.1.3">subscript</csymbol><ci id="S3.p1.2.m2.1.1.3.2.cmml" xref="S3.p1.2.m2.1.1.3.2">ℵ</ci><cn id="S3.p1.2.m2.1.1.3.3.cmml" type="integer" xref="S3.p1.2.m2.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.2.m2.1c">\mathsf{MA}_{\aleph_{1}}</annotation><annotation encoding="application/x-llamapun" id="S3.p1.2.m2.1d">sansserif_MA start_POSTSUBSCRIPT roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. Later, in <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6" title="6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">6</span></a>, we show that this can also be achieved directly by forcing.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S3.Thmtheorem1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem1.1.1.1">Lemma 3.1</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem1.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem1.p1"> <p class="ltx_p" id="S3.Thmtheorem1.p1.3">If <math alttext="A" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.1.m1.1"><semantics id="S3.Thmtheorem1.p1.1.m1.1a"><mi id="S3.Thmtheorem1.p1.1.m1.1.1" xref="S3.Thmtheorem1.p1.1.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.1.m1.1b"><ci id="S3.Thmtheorem1.p1.1.m1.1.1.cmml" xref="S3.Thmtheorem1.p1.1.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.1.m1.1c">A</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.1.m1.1d">italic_A</annotation></semantics></math> is either countable (but nonempty) or Aronszajn, and <math alttext="B" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.2.m2.1"><semantics id="S3.Thmtheorem1.p1.2.m2.1a"><mi id="S3.Thmtheorem1.p1.2.m2.1.1" xref="S3.Thmtheorem1.p1.2.m2.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.2.m2.1b"><ci id="S3.Thmtheorem1.p1.2.m2.1.1.cmml" xref="S3.Thmtheorem1.p1.2.m2.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.2.m2.1c">B</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.2.m2.1d">italic_B</annotation></semantics></math> is any normal Aronszajn line, then <math alttext="A\times B" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.3.m3.1"><semantics id="S3.Thmtheorem1.p1.3.m3.1a"><mrow id="S3.Thmtheorem1.p1.3.m3.1.1" xref="S3.Thmtheorem1.p1.3.m3.1.1.cmml"><mi id="S3.Thmtheorem1.p1.3.m3.1.1.2" xref="S3.Thmtheorem1.p1.3.m3.1.1.2.cmml">A</mi><mo id="S3.Thmtheorem1.p1.3.m3.1.1.1" lspace="0.222em" rspace="0.222em" xref="S3.Thmtheorem1.p1.3.m3.1.1.1.cmml">×</mo><mi id="S3.Thmtheorem1.p1.3.m3.1.1.3" xref="S3.Thmtheorem1.p1.3.m3.1.1.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.3.m3.1b"><apply id="S3.Thmtheorem1.p1.3.m3.1.1.cmml" xref="S3.Thmtheorem1.p1.3.m3.1.1"><times id="S3.Thmtheorem1.p1.3.m3.1.1.1.cmml" xref="S3.Thmtheorem1.p1.3.m3.1.1.1"></times><ci id="S3.Thmtheorem1.p1.3.m3.1.1.2.cmml" xref="S3.Thmtheorem1.p1.3.m3.1.1.2">𝐴</ci><ci id="S3.Thmtheorem1.p1.3.m3.1.1.3.cmml" xref="S3.Thmtheorem1.p1.3.m3.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.3.m3.1c">A\times B</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.3.m3.1d">italic_A × italic_B</annotation></semantics></math> is also a normal Aronszajn line.</p> </div> </div> <div class="ltx_proof" id="S3.2"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S3.1.p1"> <p class="ltx_p" id="S3.1.p1.9">Clearly <math alttext="A\times B" class="ltx_Math" display="inline" id="S3.1.p1.1.m1.1"><semantics id="S3.1.p1.1.m1.1a"><mrow id="S3.1.p1.1.m1.1.1" xref="S3.1.p1.1.m1.1.1.cmml"><mi id="S3.1.p1.1.m1.1.1.2" xref="S3.1.p1.1.m1.1.1.2.cmml">A</mi><mo id="S3.1.p1.1.m1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S3.1.p1.1.m1.1.1.1.cmml">×</mo><mi id="S3.1.p1.1.m1.1.1.3" xref="S3.1.p1.1.m1.1.1.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.1.p1.1.m1.1b"><apply id="S3.1.p1.1.m1.1.1.cmml" xref="S3.1.p1.1.m1.1.1"><times id="S3.1.p1.1.m1.1.1.1.cmml" xref="S3.1.p1.1.m1.1.1.1"></times><ci id="S3.1.p1.1.m1.1.1.2.cmml" xref="S3.1.p1.1.m1.1.1.2">𝐴</ci><ci id="S3.1.p1.1.m1.1.1.3.cmml" xref="S3.1.p1.1.m1.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.1.m1.1c">A\times B</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.1.m1.1d">italic_A × italic_B</annotation></semantics></math> is an <math alttext="\aleph_{1}" class="ltx_Math" display="inline" id="S3.1.p1.2.m2.1"><semantics id="S3.1.p1.2.m2.1a"><msub id="S3.1.p1.2.m2.1.1" xref="S3.1.p1.2.m2.1.1.cmml"><mi id="S3.1.p1.2.m2.1.1.2" mathvariant="normal" xref="S3.1.p1.2.m2.1.1.2.cmml">ℵ</mi><mn id="S3.1.p1.2.m2.1.1.3" xref="S3.1.p1.2.m2.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S3.1.p1.2.m2.1b"><apply id="S3.1.p1.2.m2.1.1.cmml" xref="S3.1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.1.p1.2.m2.1.1.1.cmml" xref="S3.1.p1.2.m2.1.1">subscript</csymbol><ci id="S3.1.p1.2.m2.1.1.2.cmml" xref="S3.1.p1.2.m2.1.1.2">ℵ</ci><cn id="S3.1.p1.2.m2.1.1.3.cmml" type="integer" xref="S3.1.p1.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.2.m2.1c">\aleph_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.2.m2.1d">roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-dense Aronszajn line. It remains to prove that it is also non stationary. Let <math alttext="\langle D_{\xi}:\xi<\omega_{1}\rangle" class="ltx_math_unparsed" display="inline" id="S3.1.p1.3.m3.1"><semantics id="S3.1.p1.3.m3.1a"><mrow id="S3.1.p1.3.m3.1b"><mo id="S3.1.p1.3.m3.1.1" stretchy="false">⟨</mo><msub id="S3.1.p1.3.m3.1.2"><mi id="S3.1.p1.3.m3.1.2.2">D</mi><mi id="S3.1.p1.3.m3.1.2.3">ξ</mi></msub><mo id="S3.1.p1.3.m3.1.3" lspace="0.278em" rspace="0.278em">:</mo><mi id="S3.1.p1.3.m3.1.4">ξ</mi><mo id="S3.1.p1.3.m3.1.5"><</mo><msub id="S3.1.p1.3.m3.1.6"><mi id="S3.1.p1.3.m3.1.6.2">ω</mi><mn id="S3.1.p1.3.m3.1.6.3">1</mn></msub><mo id="S3.1.p1.3.m3.1.7" stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex" id="S3.1.p1.3.m3.1c">\langle D_{\xi}:\xi<\omega_{1}\rangle</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.3.m3.1d">⟨ italic_D start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT : italic_ξ < italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⟩</annotation></semantics></math> witness the non stationarity of <math alttext="B" class="ltx_Math" display="inline" id="S3.1.p1.4.m4.1"><semantics id="S3.1.p1.4.m4.1a"><mi id="S3.1.p1.4.m4.1.1" xref="S3.1.p1.4.m4.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S3.1.p1.4.m4.1b"><ci id="S3.1.p1.4.m4.1.1.cmml" xref="S3.1.p1.4.m4.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.4.m4.1c">B</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.4.m4.1d">italic_B</annotation></semantics></math>, and let <math alttext="\{a_{\xi}:\xi<\omega_{1}\}" class="ltx_Math" display="inline" id="S3.1.p1.5.m5.2"><semantics id="S3.1.p1.5.m5.2a"><mrow id="S3.1.p1.5.m5.2.2.2" xref="S3.1.p1.5.m5.2.2.3.cmml"><mo id="S3.1.p1.5.m5.2.2.2.3" stretchy="false" xref="S3.1.p1.5.m5.2.2.3.1.cmml">{</mo><msub id="S3.1.p1.5.m5.1.1.1.1" xref="S3.1.p1.5.m5.1.1.1.1.cmml"><mi id="S3.1.p1.5.m5.1.1.1.1.2" xref="S3.1.p1.5.m5.1.1.1.1.2.cmml">a</mi><mi id="S3.1.p1.5.m5.1.1.1.1.3" xref="S3.1.p1.5.m5.1.1.1.1.3.cmml">ξ</mi></msub><mo id="S3.1.p1.5.m5.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S3.1.p1.5.m5.2.2.3.1.cmml">:</mo><mrow id="S3.1.p1.5.m5.2.2.2.2" xref="S3.1.p1.5.m5.2.2.2.2.cmml"><mi id="S3.1.p1.5.m5.2.2.2.2.2" xref="S3.1.p1.5.m5.2.2.2.2.2.cmml">ξ</mi><mo id="S3.1.p1.5.m5.2.2.2.2.1" xref="S3.1.p1.5.m5.2.2.2.2.1.cmml"><</mo><msub id="S3.1.p1.5.m5.2.2.2.2.3" xref="S3.1.p1.5.m5.2.2.2.2.3.cmml"><mi id="S3.1.p1.5.m5.2.2.2.2.3.2" xref="S3.1.p1.5.m5.2.2.2.2.3.2.cmml">ω</mi><mn id="S3.1.p1.5.m5.2.2.2.2.3.3" xref="S3.1.p1.5.m5.2.2.2.2.3.3.cmml">1</mn></msub></mrow><mo id="S3.1.p1.5.m5.2.2.2.5" stretchy="false" xref="S3.1.p1.5.m5.2.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.1.p1.5.m5.2b"><apply id="S3.1.p1.5.m5.2.2.3.cmml" xref="S3.1.p1.5.m5.2.2.2"><csymbol cd="latexml" id="S3.1.p1.5.m5.2.2.3.1.cmml" xref="S3.1.p1.5.m5.2.2.2.3">conditional-set</csymbol><apply id="S3.1.p1.5.m5.1.1.1.1.cmml" xref="S3.1.p1.5.m5.1.1.1.1"><csymbol cd="ambiguous" id="S3.1.p1.5.m5.1.1.1.1.1.cmml" xref="S3.1.p1.5.m5.1.1.1.1">subscript</csymbol><ci id="S3.1.p1.5.m5.1.1.1.1.2.cmml" xref="S3.1.p1.5.m5.1.1.1.1.2">𝑎</ci><ci id="S3.1.p1.5.m5.1.1.1.1.3.cmml" xref="S3.1.p1.5.m5.1.1.1.1.3">𝜉</ci></apply><apply id="S3.1.p1.5.m5.2.2.2.2.cmml" xref="S3.1.p1.5.m5.2.2.2.2"><lt id="S3.1.p1.5.m5.2.2.2.2.1.cmml" xref="S3.1.p1.5.m5.2.2.2.2.1"></lt><ci id="S3.1.p1.5.m5.2.2.2.2.2.cmml" xref="S3.1.p1.5.m5.2.2.2.2.2">𝜉</ci><apply id="S3.1.p1.5.m5.2.2.2.2.3.cmml" xref="S3.1.p1.5.m5.2.2.2.2.3"><csymbol cd="ambiguous" id="S3.1.p1.5.m5.2.2.2.2.3.1.cmml" xref="S3.1.p1.5.m5.2.2.2.2.3">subscript</csymbol><ci id="S3.1.p1.5.m5.2.2.2.2.3.2.cmml" xref="S3.1.p1.5.m5.2.2.2.2.3.2">𝜔</ci><cn id="S3.1.p1.5.m5.2.2.2.2.3.3.cmml" type="integer" xref="S3.1.p1.5.m5.2.2.2.2.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.5.m5.2c">\{a_{\xi}:\xi<\omega_{1}\}</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.5.m5.2d">{ italic_a start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT : italic_ξ < italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT }</annotation></semantics></math> enumerate <math alttext="A" class="ltx_Math" display="inline" id="S3.1.p1.6.m6.1"><semantics id="S3.1.p1.6.m6.1a"><mi id="S3.1.p1.6.m6.1.1" xref="S3.1.p1.6.m6.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S3.1.p1.6.m6.1b"><ci id="S3.1.p1.6.m6.1.1.cmml" xref="S3.1.p1.6.m6.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.6.m6.1c">A</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.6.m6.1d">italic_A</annotation></semantics></math> (with repetitions allowed). We claim that <math alttext="\langle U_{\xi}:\xi<\omega_{1}\rangle" class="ltx_math_unparsed" display="inline" id="S3.1.p1.7.m7.1"><semantics id="S3.1.p1.7.m7.1a"><mrow id="S3.1.p1.7.m7.1b"><mo id="S3.1.p1.7.m7.1.1" stretchy="false">⟨</mo><msub id="S3.1.p1.7.m7.1.2"><mi id="S3.1.p1.7.m7.1.2.2">U</mi><mi id="S3.1.p1.7.m7.1.2.3">ξ</mi></msub><mo id="S3.1.p1.7.m7.1.3" lspace="0.278em" rspace="0.278em">:</mo><mi id="S3.1.p1.7.m7.1.4">ξ</mi><mo id="S3.1.p1.7.m7.1.5"><</mo><msub id="S3.1.p1.7.m7.1.6"><mi id="S3.1.p1.7.m7.1.6.2">ω</mi><mn id="S3.1.p1.7.m7.1.6.3">1</mn></msub><mo id="S3.1.p1.7.m7.1.7" stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex" id="S3.1.p1.7.m7.1c">\langle U_{\xi}:\xi<\omega_{1}\rangle</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.7.m7.1d">⟨ italic_U start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT : italic_ξ < italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⟩</annotation></semantics></math> where <math alttext="U_{\xi}:=\{a_{\eta}:\eta<\xi\}\times D_{\xi}" class="ltx_Math" display="inline" id="S3.1.p1.8.m8.2"><semantics id="S3.1.p1.8.m8.2a"><mrow id="S3.1.p1.8.m8.2.2" xref="S3.1.p1.8.m8.2.2.cmml"><msub id="S3.1.p1.8.m8.2.2.4" xref="S3.1.p1.8.m8.2.2.4.cmml"><mi id="S3.1.p1.8.m8.2.2.4.2" xref="S3.1.p1.8.m8.2.2.4.2.cmml">U</mi><mi id="S3.1.p1.8.m8.2.2.4.3" xref="S3.1.p1.8.m8.2.2.4.3.cmml">ξ</mi></msub><mo id="S3.1.p1.8.m8.2.2.3" lspace="0.278em" rspace="0.278em" xref="S3.1.p1.8.m8.2.2.3.cmml">:=</mo><mrow id="S3.1.p1.8.m8.2.2.2" xref="S3.1.p1.8.m8.2.2.2.cmml"><mrow id="S3.1.p1.8.m8.2.2.2.2.2" xref="S3.1.p1.8.m8.2.2.2.2.3.cmml"><mo id="S3.1.p1.8.m8.2.2.2.2.2.3" stretchy="false" xref="S3.1.p1.8.m8.2.2.2.2.3.1.cmml">{</mo><msub id="S3.1.p1.8.m8.1.1.1.1.1.1" xref="S3.1.p1.8.m8.1.1.1.1.1.1.cmml"><mi id="S3.1.p1.8.m8.1.1.1.1.1.1.2" xref="S3.1.p1.8.m8.1.1.1.1.1.1.2.cmml">a</mi><mi id="S3.1.p1.8.m8.1.1.1.1.1.1.3" xref="S3.1.p1.8.m8.1.1.1.1.1.1.3.cmml">η</mi></msub><mo id="S3.1.p1.8.m8.2.2.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S3.1.p1.8.m8.2.2.2.2.3.1.cmml">:</mo><mrow id="S3.1.p1.8.m8.2.2.2.2.2.2" xref="S3.1.p1.8.m8.2.2.2.2.2.2.cmml"><mi id="S3.1.p1.8.m8.2.2.2.2.2.2.2" xref="S3.1.p1.8.m8.2.2.2.2.2.2.2.cmml">η</mi><mo id="S3.1.p1.8.m8.2.2.2.2.2.2.1" xref="S3.1.p1.8.m8.2.2.2.2.2.2.1.cmml"><</mo><mi id="S3.1.p1.8.m8.2.2.2.2.2.2.3" xref="S3.1.p1.8.m8.2.2.2.2.2.2.3.cmml">ξ</mi></mrow><mo id="S3.1.p1.8.m8.2.2.2.2.2.5" rspace="0.055em" stretchy="false" xref="S3.1.p1.8.m8.2.2.2.2.3.1.cmml">}</mo></mrow><mo id="S3.1.p1.8.m8.2.2.2.3" rspace="0.222em" xref="S3.1.p1.8.m8.2.2.2.3.cmml">×</mo><msub id="S3.1.p1.8.m8.2.2.2.4" xref="S3.1.p1.8.m8.2.2.2.4.cmml"><mi id="S3.1.p1.8.m8.2.2.2.4.2" xref="S3.1.p1.8.m8.2.2.2.4.2.cmml">D</mi><mi id="S3.1.p1.8.m8.2.2.2.4.3" xref="S3.1.p1.8.m8.2.2.2.4.3.cmml">ξ</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.1.p1.8.m8.2b"><apply id="S3.1.p1.8.m8.2.2.cmml" xref="S3.1.p1.8.m8.2.2"><csymbol cd="latexml" id="S3.1.p1.8.m8.2.2.3.cmml" xref="S3.1.p1.8.m8.2.2.3">assign</csymbol><apply id="S3.1.p1.8.m8.2.2.4.cmml" xref="S3.1.p1.8.m8.2.2.4"><csymbol cd="ambiguous" id="S3.1.p1.8.m8.2.2.4.1.cmml" xref="S3.1.p1.8.m8.2.2.4">subscript</csymbol><ci id="S3.1.p1.8.m8.2.2.4.2.cmml" xref="S3.1.p1.8.m8.2.2.4.2">𝑈</ci><ci id="S3.1.p1.8.m8.2.2.4.3.cmml" xref="S3.1.p1.8.m8.2.2.4.3">𝜉</ci></apply><apply id="S3.1.p1.8.m8.2.2.2.cmml" xref="S3.1.p1.8.m8.2.2.2"><times id="S3.1.p1.8.m8.2.2.2.3.cmml" xref="S3.1.p1.8.m8.2.2.2.3"></times><apply id="S3.1.p1.8.m8.2.2.2.2.3.cmml" xref="S3.1.p1.8.m8.2.2.2.2.2"><csymbol cd="latexml" id="S3.1.p1.8.m8.2.2.2.2.3.1.cmml" xref="S3.1.p1.8.m8.2.2.2.2.2.3">conditional-set</csymbol><apply id="S3.1.p1.8.m8.1.1.1.1.1.1.cmml" xref="S3.1.p1.8.m8.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.1.p1.8.m8.1.1.1.1.1.1.1.cmml" xref="S3.1.p1.8.m8.1.1.1.1.1.1">subscript</csymbol><ci id="S3.1.p1.8.m8.1.1.1.1.1.1.2.cmml" xref="S3.1.p1.8.m8.1.1.1.1.1.1.2">𝑎</ci><ci id="S3.1.p1.8.m8.1.1.1.1.1.1.3.cmml" xref="S3.1.p1.8.m8.1.1.1.1.1.1.3">𝜂</ci></apply><apply id="S3.1.p1.8.m8.2.2.2.2.2.2.cmml" xref="S3.1.p1.8.m8.2.2.2.2.2.2"><lt id="S3.1.p1.8.m8.2.2.2.2.2.2.1.cmml" xref="S3.1.p1.8.m8.2.2.2.2.2.2.1"></lt><ci id="S3.1.p1.8.m8.2.2.2.2.2.2.2.cmml" xref="S3.1.p1.8.m8.2.2.2.2.2.2.2">𝜂</ci><ci id="S3.1.p1.8.m8.2.2.2.2.2.2.3.cmml" xref="S3.1.p1.8.m8.2.2.2.2.2.2.3">𝜉</ci></apply></apply><apply id="S3.1.p1.8.m8.2.2.2.4.cmml" xref="S3.1.p1.8.m8.2.2.2.4"><csymbol cd="ambiguous" id="S3.1.p1.8.m8.2.2.2.4.1.cmml" xref="S3.1.p1.8.m8.2.2.2.4">subscript</csymbol><ci id="S3.1.p1.8.m8.2.2.2.4.2.cmml" xref="S3.1.p1.8.m8.2.2.2.4.2">𝐷</ci><ci id="S3.1.p1.8.m8.2.2.2.4.3.cmml" xref="S3.1.p1.8.m8.2.2.2.4.3">𝜉</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.8.m8.2c">U_{\xi}:=\{a_{\eta}:\eta<\xi\}\times D_{\xi}</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.8.m8.2d">italic_U start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT := { italic_a start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT : italic_η < italic_ξ } × italic_D start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT</annotation></semantics></math>, witnesses the nonstationarity of <math alttext="A\times B" class="ltx_Math" display="inline" id="S3.1.p1.9.m9.1"><semantics id="S3.1.p1.9.m9.1a"><mrow id="S3.1.p1.9.m9.1.1" xref="S3.1.p1.9.m9.1.1.cmml"><mi id="S3.1.p1.9.m9.1.1.2" xref="S3.1.p1.9.m9.1.1.2.cmml">A</mi><mo id="S3.1.p1.9.m9.1.1.1" lspace="0.222em" rspace="0.222em" xref="S3.1.p1.9.m9.1.1.1.cmml">×</mo><mi id="S3.1.p1.9.m9.1.1.3" xref="S3.1.p1.9.m9.1.1.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.1.p1.9.m9.1b"><apply id="S3.1.p1.9.m9.1.1.cmml" xref="S3.1.p1.9.m9.1.1"><times id="S3.1.p1.9.m9.1.1.1.cmml" xref="S3.1.p1.9.m9.1.1.1"></times><ci id="S3.1.p1.9.m9.1.1.2.cmml" xref="S3.1.p1.9.m9.1.1.2">𝐴</ci><ci id="S3.1.p1.9.m9.1.1.3.cmml" xref="S3.1.p1.9.m9.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.9.m9.1c">A\times B</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.9.m9.1d">italic_A × italic_B</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.2.p2"> <p class="ltx_p" id="S3.2.p2.17">It should be clear that <math alttext="U" class="ltx_Math" display="inline" id="S3.2.p2.1.m1.1"><semantics id="S3.2.p2.1.m1.1a"><mi id="S3.2.p2.1.m1.1.1" xref="S3.2.p2.1.m1.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="S3.2.p2.1.m1.1b"><ci id="S3.2.p2.1.m1.1.1.cmml" xref="S3.2.p2.1.m1.1.1">𝑈</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.2.p2.1.m1.1c">U</annotation><annotation encoding="application/x-llamapun" id="S3.2.p2.1.m1.1d">italic_U</annotation></semantics></math> is increasing, continuous, covers <math alttext="A\times B" class="ltx_Math" display="inline" id="S3.2.p2.2.m2.1"><semantics id="S3.2.p2.2.m2.1a"><mrow id="S3.2.p2.2.m2.1.1" xref="S3.2.p2.2.m2.1.1.cmml"><mi id="S3.2.p2.2.m2.1.1.2" xref="S3.2.p2.2.m2.1.1.2.cmml">A</mi><mo id="S3.2.p2.2.m2.1.1.1" lspace="0.222em" rspace="0.222em" xref="S3.2.p2.2.m2.1.1.1.cmml">×</mo><mi id="S3.2.p2.2.m2.1.1.3" xref="S3.2.p2.2.m2.1.1.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.2.p2.2.m2.1b"><apply id="S3.2.p2.2.m2.1.1.cmml" xref="S3.2.p2.2.m2.1.1"><times id="S3.2.p2.2.m2.1.1.1.cmml" xref="S3.2.p2.2.m2.1.1.1"></times><ci id="S3.2.p2.2.m2.1.1.2.cmml" xref="S3.2.p2.2.m2.1.1.2">𝐴</ci><ci id="S3.2.p2.2.m2.1.1.3.cmml" xref="S3.2.p2.2.m2.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.2.p2.2.m2.1c">A\times B</annotation><annotation encoding="application/x-llamapun" id="S3.2.p2.2.m2.1d">italic_A × italic_B</annotation></semantics></math> and consists of countable subsets of <math alttext="A\times B" class="ltx_Math" display="inline" id="S3.2.p2.3.m3.1"><semantics id="S3.2.p2.3.m3.1a"><mrow id="S3.2.p2.3.m3.1.1" xref="S3.2.p2.3.m3.1.1.cmml"><mi id="S3.2.p2.3.m3.1.1.2" xref="S3.2.p2.3.m3.1.1.2.cmml">A</mi><mo id="S3.2.p2.3.m3.1.1.1" lspace="0.222em" rspace="0.222em" xref="S3.2.p2.3.m3.1.1.1.cmml">×</mo><mi id="S3.2.p2.3.m3.1.1.3" xref="S3.2.p2.3.m3.1.1.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.2.p2.3.m3.1b"><apply id="S3.2.p2.3.m3.1.1.cmml" xref="S3.2.p2.3.m3.1.1"><times id="S3.2.p2.3.m3.1.1.1.cmml" xref="S3.2.p2.3.m3.1.1.1"></times><ci id="S3.2.p2.3.m3.1.1.2.cmml" xref="S3.2.p2.3.m3.1.1.2">𝐴</ci><ci id="S3.2.p2.3.m3.1.1.3.cmml" xref="S3.2.p2.3.m3.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.2.p2.3.m3.1c">A\times B</annotation><annotation encoding="application/x-llamapun" id="S3.2.p2.3.m3.1d">italic_A × italic_B</annotation></semantics></math>. Fix <math alttext="\xi<\omega_{1}" class="ltx_Math" display="inline" id="S3.2.p2.4.m4.1"><semantics id="S3.2.p2.4.m4.1a"><mrow id="S3.2.p2.4.m4.1.1" xref="S3.2.p2.4.m4.1.1.cmml"><mi id="S3.2.p2.4.m4.1.1.2" xref="S3.2.p2.4.m4.1.1.2.cmml">ξ</mi><mo id="S3.2.p2.4.m4.1.1.1" xref="S3.2.p2.4.m4.1.1.1.cmml"><</mo><msub id="S3.2.p2.4.m4.1.1.3" xref="S3.2.p2.4.m4.1.1.3.cmml"><mi id="S3.2.p2.4.m4.1.1.3.2" xref="S3.2.p2.4.m4.1.1.3.2.cmml">ω</mi><mn id="S3.2.p2.4.m4.1.1.3.3" xref="S3.2.p2.4.m4.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.2.p2.4.m4.1b"><apply id="S3.2.p2.4.m4.1.1.cmml" xref="S3.2.p2.4.m4.1.1"><lt id="S3.2.p2.4.m4.1.1.1.cmml" xref="S3.2.p2.4.m4.1.1.1"></lt><ci id="S3.2.p2.4.m4.1.1.2.cmml" xref="S3.2.p2.4.m4.1.1.2">𝜉</ci><apply id="S3.2.p2.4.m4.1.1.3.cmml" xref="S3.2.p2.4.m4.1.1.3"><csymbol cd="ambiguous" id="S3.2.p2.4.m4.1.1.3.1.cmml" xref="S3.2.p2.4.m4.1.1.3">subscript</csymbol><ci id="S3.2.p2.4.m4.1.1.3.2.cmml" xref="S3.2.p2.4.m4.1.1.3.2">𝜔</ci><cn id="S3.2.p2.4.m4.1.1.3.3.cmml" type="integer" xref="S3.2.p2.4.m4.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.2.p2.4.m4.1c">\xi<\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.2.p2.4.m4.1d">italic_ξ < italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="I" class="ltx_Math" display="inline" id="S3.2.p2.5.m5.1"><semantics id="S3.2.p2.5.m5.1a"><mi id="S3.2.p2.5.m5.1.1" xref="S3.2.p2.5.m5.1.1.cmml">I</mi><annotation-xml encoding="MathML-Content" id="S3.2.p2.5.m5.1b"><ci id="S3.2.p2.5.m5.1.1.cmml" xref="S3.2.p2.5.m5.1.1">𝐼</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.2.p2.5.m5.1c">I</annotation><annotation encoding="application/x-llamapun" id="S3.2.p2.5.m5.1d">italic_I</annotation></semantics></math> a complementary interval of <math alttext="(A\times B)\setminus U_{\xi}" class="ltx_Math" display="inline" id="S3.2.p2.6.m6.1"><semantics id="S3.2.p2.6.m6.1a"><mrow id="S3.2.p2.6.m6.1.1" xref="S3.2.p2.6.m6.1.1.cmml"><mrow id="S3.2.p2.6.m6.1.1.1.1" xref="S3.2.p2.6.m6.1.1.1.1.1.cmml"><mo id="S3.2.p2.6.m6.1.1.1.1.2" stretchy="false" xref="S3.2.p2.6.m6.1.1.1.1.1.cmml">(</mo><mrow id="S3.2.p2.6.m6.1.1.1.1.1" xref="S3.2.p2.6.m6.1.1.1.1.1.cmml"><mi id="S3.2.p2.6.m6.1.1.1.1.1.2" xref="S3.2.p2.6.m6.1.1.1.1.1.2.cmml">A</mi><mo id="S3.2.p2.6.m6.1.1.1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S3.2.p2.6.m6.1.1.1.1.1.1.cmml">×</mo><mi id="S3.2.p2.6.m6.1.1.1.1.1.3" xref="S3.2.p2.6.m6.1.1.1.1.1.3.cmml">B</mi></mrow><mo id="S3.2.p2.6.m6.1.1.1.1.3" stretchy="false" xref="S3.2.p2.6.m6.1.1.1.1.1.cmml">)</mo></mrow><mo id="S3.2.p2.6.m6.1.1.2" xref="S3.2.p2.6.m6.1.1.2.cmml">∖</mo><msub id="S3.2.p2.6.m6.1.1.3" xref="S3.2.p2.6.m6.1.1.3.cmml"><mi id="S3.2.p2.6.m6.1.1.3.2" xref="S3.2.p2.6.m6.1.1.3.2.cmml">U</mi><mi id="S3.2.p2.6.m6.1.1.3.3" xref="S3.2.p2.6.m6.1.1.3.3.cmml">ξ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.2.p2.6.m6.1b"><apply id="S3.2.p2.6.m6.1.1.cmml" xref="S3.2.p2.6.m6.1.1"><setdiff id="S3.2.p2.6.m6.1.1.2.cmml" xref="S3.2.p2.6.m6.1.1.2"></setdiff><apply id="S3.2.p2.6.m6.1.1.1.1.1.cmml" xref="S3.2.p2.6.m6.1.1.1.1"><times id="S3.2.p2.6.m6.1.1.1.1.1.1.cmml" xref="S3.2.p2.6.m6.1.1.1.1.1.1"></times><ci id="S3.2.p2.6.m6.1.1.1.1.1.2.cmml" xref="S3.2.p2.6.m6.1.1.1.1.1.2">𝐴</ci><ci id="S3.2.p2.6.m6.1.1.1.1.1.3.cmml" xref="S3.2.p2.6.m6.1.1.1.1.1.3">𝐵</ci></apply><apply id="S3.2.p2.6.m6.1.1.3.cmml" xref="S3.2.p2.6.m6.1.1.3"><csymbol cd="ambiguous" id="S3.2.p2.6.m6.1.1.3.1.cmml" xref="S3.2.p2.6.m6.1.1.3">subscript</csymbol><ci id="S3.2.p2.6.m6.1.1.3.2.cmml" xref="S3.2.p2.6.m6.1.1.3.2">𝑈</ci><ci id="S3.2.p2.6.m6.1.1.3.3.cmml" xref="S3.2.p2.6.m6.1.1.3.3">𝜉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.2.p2.6.m6.1c">(A\times B)\setminus U_{\xi}</annotation><annotation encoding="application/x-llamapun" id="S3.2.p2.6.m6.1d">( italic_A × italic_B ) ∖ italic_U start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT</annotation></semantics></math>, and <math alttext="(a,b)" class="ltx_Math" display="inline" id="S3.2.p2.7.m7.2"><semantics id="S3.2.p2.7.m7.2a"><mrow id="S3.2.p2.7.m7.2.3.2" xref="S3.2.p2.7.m7.2.3.1.cmml"><mo id="S3.2.p2.7.m7.2.3.2.1" stretchy="false" xref="S3.2.p2.7.m7.2.3.1.cmml">(</mo><mi id="S3.2.p2.7.m7.1.1" xref="S3.2.p2.7.m7.1.1.cmml">a</mi><mo id="S3.2.p2.7.m7.2.3.2.2" xref="S3.2.p2.7.m7.2.3.1.cmml">,</mo><mi id="S3.2.p2.7.m7.2.2" xref="S3.2.p2.7.m7.2.2.cmml">b</mi><mo id="S3.2.p2.7.m7.2.3.2.3" stretchy="false" xref="S3.2.p2.7.m7.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.2.p2.7.m7.2b"><interval closure="open" id="S3.2.p2.7.m7.2.3.1.cmml" xref="S3.2.p2.7.m7.2.3.2"><ci id="S3.2.p2.7.m7.1.1.cmml" xref="S3.2.p2.7.m7.1.1">𝑎</ci><ci id="S3.2.p2.7.m7.2.2.cmml" xref="S3.2.p2.7.m7.2.2">𝑏</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.2.p2.7.m7.2c">(a,b)</annotation><annotation encoding="application/x-llamapun" id="S3.2.p2.7.m7.2d">( italic_a , italic_b )</annotation></semantics></math> an element of <math alttext="I" class="ltx_Math" display="inline" id="S3.2.p2.8.m8.1"><semantics id="S3.2.p2.8.m8.1a"><mi id="S3.2.p2.8.m8.1.1" xref="S3.2.p2.8.m8.1.1.cmml">I</mi><annotation-xml encoding="MathML-Content" id="S3.2.p2.8.m8.1b"><ci id="S3.2.p2.8.m8.1.1.cmml" xref="S3.2.p2.8.m8.1.1">𝐼</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.2.p2.8.m8.1c">I</annotation><annotation encoding="application/x-llamapun" id="S3.2.p2.8.m8.1d">italic_I</annotation></semantics></math>. Also let <math alttext="J" class="ltx_Math" display="inline" id="S3.2.p2.9.m9.1"><semantics id="S3.2.p2.9.m9.1a"><mi id="S3.2.p2.9.m9.1.1" xref="S3.2.p2.9.m9.1.1.cmml">J</mi><annotation-xml encoding="MathML-Content" id="S3.2.p2.9.m9.1b"><ci id="S3.2.p2.9.m9.1.1.cmml" xref="S3.2.p2.9.m9.1.1">𝐽</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.2.p2.9.m9.1c">J</annotation><annotation encoding="application/x-llamapun" id="S3.2.p2.9.m9.1d">italic_J</annotation></semantics></math> be the complementary interval of <math alttext="B\setminus D_{\xi}" class="ltx_Math" display="inline" id="S3.2.p2.10.m10.1"><semantics id="S3.2.p2.10.m10.1a"><mrow id="S3.2.p2.10.m10.1.1" xref="S3.2.p2.10.m10.1.1.cmml"><mi id="S3.2.p2.10.m10.1.1.2" xref="S3.2.p2.10.m10.1.1.2.cmml">B</mi><mo id="S3.2.p2.10.m10.1.1.1" xref="S3.2.p2.10.m10.1.1.1.cmml">∖</mo><msub id="S3.2.p2.10.m10.1.1.3" xref="S3.2.p2.10.m10.1.1.3.cmml"><mi id="S3.2.p2.10.m10.1.1.3.2" xref="S3.2.p2.10.m10.1.1.3.2.cmml">D</mi><mi id="S3.2.p2.10.m10.1.1.3.3" xref="S3.2.p2.10.m10.1.1.3.3.cmml">ξ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.2.p2.10.m10.1b"><apply id="S3.2.p2.10.m10.1.1.cmml" xref="S3.2.p2.10.m10.1.1"><setdiff id="S3.2.p2.10.m10.1.1.1.cmml" xref="S3.2.p2.10.m10.1.1.1"></setdiff><ci id="S3.2.p2.10.m10.1.1.2.cmml" xref="S3.2.p2.10.m10.1.1.2">𝐵</ci><apply id="S3.2.p2.10.m10.1.1.3.cmml" xref="S3.2.p2.10.m10.1.1.3"><csymbol cd="ambiguous" id="S3.2.p2.10.m10.1.1.3.1.cmml" xref="S3.2.p2.10.m10.1.1.3">subscript</csymbol><ci id="S3.2.p2.10.m10.1.1.3.2.cmml" xref="S3.2.p2.10.m10.1.1.3.2">𝐷</ci><ci id="S3.2.p2.10.m10.1.1.3.3.cmml" xref="S3.2.p2.10.m10.1.1.3.3">𝜉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.2.p2.10.m10.1c">B\setminus D_{\xi}</annotation><annotation encoding="application/x-llamapun" id="S3.2.p2.10.m10.1d">italic_B ∖ italic_D start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT</annotation></semantics></math> in which <math alttext="b" class="ltx_Math" display="inline" id="S3.2.p2.11.m11.1"><semantics id="S3.2.p2.11.m11.1a"><mi id="S3.2.p2.11.m11.1.1" xref="S3.2.p2.11.m11.1.1.cmml">b</mi><annotation-xml encoding="MathML-Content" id="S3.2.p2.11.m11.1b"><ci id="S3.2.p2.11.m11.1.1.cmml" xref="S3.2.p2.11.m11.1.1">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.2.p2.11.m11.1c">b</annotation><annotation encoding="application/x-llamapun" id="S3.2.p2.11.m11.1d">italic_b</annotation></semantics></math> is. Since <math alttext="J" class="ltx_Math" display="inline" id="S3.2.p2.12.m12.1"><semantics id="S3.2.p2.12.m12.1a"><mi id="S3.2.p2.12.m12.1.1" xref="S3.2.p2.12.m12.1.1.cmml">J</mi><annotation-xml encoding="MathML-Content" id="S3.2.p2.12.m12.1b"><ci id="S3.2.p2.12.m12.1.1.cmml" xref="S3.2.p2.12.m12.1.1">𝐽</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.2.p2.12.m12.1c">J</annotation><annotation encoding="application/x-llamapun" id="S3.2.p2.12.m12.1d">italic_J</annotation></semantics></math> has no endpoints there are <math alttext="b^{\prime},b^{\prime\prime}\in J" class="ltx_Math" display="inline" id="S3.2.p2.13.m13.2"><semantics id="S3.2.p2.13.m13.2a"><mrow id="S3.2.p2.13.m13.2.2" xref="S3.2.p2.13.m13.2.2.cmml"><mrow id="S3.2.p2.13.m13.2.2.2.2" xref="S3.2.p2.13.m13.2.2.2.3.cmml"><msup id="S3.2.p2.13.m13.1.1.1.1.1" xref="S3.2.p2.13.m13.1.1.1.1.1.cmml"><mi id="S3.2.p2.13.m13.1.1.1.1.1.2" xref="S3.2.p2.13.m13.1.1.1.1.1.2.cmml">b</mi><mo id="S3.2.p2.13.m13.1.1.1.1.1.3" xref="S3.2.p2.13.m13.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S3.2.p2.13.m13.2.2.2.2.3" xref="S3.2.p2.13.m13.2.2.2.3.cmml">,</mo><msup id="S3.2.p2.13.m13.2.2.2.2.2" xref="S3.2.p2.13.m13.2.2.2.2.2.cmml"><mi id="S3.2.p2.13.m13.2.2.2.2.2.2" xref="S3.2.p2.13.m13.2.2.2.2.2.2.cmml">b</mi><mo id="S3.2.p2.13.m13.2.2.2.2.2.3" xref="S3.2.p2.13.m13.2.2.2.2.2.3.cmml">′′</mo></msup></mrow><mo id="S3.2.p2.13.m13.2.2.3" xref="S3.2.p2.13.m13.2.2.3.cmml">∈</mo><mi id="S3.2.p2.13.m13.2.2.4" xref="S3.2.p2.13.m13.2.2.4.cmml">J</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.2.p2.13.m13.2b"><apply id="S3.2.p2.13.m13.2.2.cmml" xref="S3.2.p2.13.m13.2.2"><in id="S3.2.p2.13.m13.2.2.3.cmml" xref="S3.2.p2.13.m13.2.2.3"></in><list id="S3.2.p2.13.m13.2.2.2.3.cmml" xref="S3.2.p2.13.m13.2.2.2.2"><apply id="S3.2.p2.13.m13.1.1.1.1.1.cmml" xref="S3.2.p2.13.m13.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.2.p2.13.m13.1.1.1.1.1.1.cmml" xref="S3.2.p2.13.m13.1.1.1.1.1">superscript</csymbol><ci id="S3.2.p2.13.m13.1.1.1.1.1.2.cmml" xref="S3.2.p2.13.m13.1.1.1.1.1.2">𝑏</ci><ci id="S3.2.p2.13.m13.1.1.1.1.1.3.cmml" xref="S3.2.p2.13.m13.1.1.1.1.1.3">′</ci></apply><apply id="S3.2.p2.13.m13.2.2.2.2.2.cmml" xref="S3.2.p2.13.m13.2.2.2.2.2"><csymbol cd="ambiguous" id="S3.2.p2.13.m13.2.2.2.2.2.1.cmml" xref="S3.2.p2.13.m13.2.2.2.2.2">superscript</csymbol><ci id="S3.2.p2.13.m13.2.2.2.2.2.2.cmml" xref="S3.2.p2.13.m13.2.2.2.2.2.2">𝑏</ci><ci id="S3.2.p2.13.m13.2.2.2.2.2.3.cmml" xref="S3.2.p2.13.m13.2.2.2.2.2.3">′′</ci></apply></list><ci id="S3.2.p2.13.m13.2.2.4.cmml" xref="S3.2.p2.13.m13.2.2.4">𝐽</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.2.p2.13.m13.2c">b^{\prime},b^{\prime\prime}\in J</annotation><annotation encoding="application/x-llamapun" id="S3.2.p2.13.m13.2d">italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_b start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ∈ italic_J</annotation></semantics></math> such that <math alttext="b^{\prime}<_{B}b<_{B}b^{\prime\prime}" class="ltx_Math" display="inline" id="S3.2.p2.14.m14.1"><semantics id="S3.2.p2.14.m14.1a"><mrow id="S3.2.p2.14.m14.1.1" xref="S3.2.p2.14.m14.1.1.cmml"><msup id="S3.2.p2.14.m14.1.1.2" xref="S3.2.p2.14.m14.1.1.2.cmml"><mi id="S3.2.p2.14.m14.1.1.2.2" xref="S3.2.p2.14.m14.1.1.2.2.cmml">b</mi><mo id="S3.2.p2.14.m14.1.1.2.3" xref="S3.2.p2.14.m14.1.1.2.3.cmml">′</mo></msup><msub id="S3.2.p2.14.m14.1.1.3" xref="S3.2.p2.14.m14.1.1.3.cmml"><mo id="S3.2.p2.14.m14.1.1.3.2" xref="S3.2.p2.14.m14.1.1.3.2.cmml"><</mo><mi id="S3.2.p2.14.m14.1.1.3.3" xref="S3.2.p2.14.m14.1.1.3.3.cmml">B</mi></msub><mi id="S3.2.p2.14.m14.1.1.4" xref="S3.2.p2.14.m14.1.1.4.cmml">b</mi><msub id="S3.2.p2.14.m14.1.1.5" xref="S3.2.p2.14.m14.1.1.5.cmml"><mo id="S3.2.p2.14.m14.1.1.5.2" xref="S3.2.p2.14.m14.1.1.5.2.cmml"><</mo><mi id="S3.2.p2.14.m14.1.1.5.3" xref="S3.2.p2.14.m14.1.1.5.3.cmml">B</mi></msub><msup id="S3.2.p2.14.m14.1.1.6" xref="S3.2.p2.14.m14.1.1.6.cmml"><mi id="S3.2.p2.14.m14.1.1.6.2" xref="S3.2.p2.14.m14.1.1.6.2.cmml">b</mi><mo id="S3.2.p2.14.m14.1.1.6.3" xref="S3.2.p2.14.m14.1.1.6.3.cmml">′′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.2.p2.14.m14.1b"><apply id="S3.2.p2.14.m14.1.1.cmml" xref="S3.2.p2.14.m14.1.1"><and id="S3.2.p2.14.m14.1.1a.cmml" xref="S3.2.p2.14.m14.1.1"></and><apply id="S3.2.p2.14.m14.1.1b.cmml" xref="S3.2.p2.14.m14.1.1"><apply id="S3.2.p2.14.m14.1.1.3.cmml" xref="S3.2.p2.14.m14.1.1.3"><csymbol cd="ambiguous" id="S3.2.p2.14.m14.1.1.3.1.cmml" xref="S3.2.p2.14.m14.1.1.3">subscript</csymbol><lt id="S3.2.p2.14.m14.1.1.3.2.cmml" xref="S3.2.p2.14.m14.1.1.3.2"></lt><ci id="S3.2.p2.14.m14.1.1.3.3.cmml" xref="S3.2.p2.14.m14.1.1.3.3">𝐵</ci></apply><apply id="S3.2.p2.14.m14.1.1.2.cmml" xref="S3.2.p2.14.m14.1.1.2"><csymbol cd="ambiguous" id="S3.2.p2.14.m14.1.1.2.1.cmml" xref="S3.2.p2.14.m14.1.1.2">superscript</csymbol><ci id="S3.2.p2.14.m14.1.1.2.2.cmml" xref="S3.2.p2.14.m14.1.1.2.2">𝑏</ci><ci id="S3.2.p2.14.m14.1.1.2.3.cmml" xref="S3.2.p2.14.m14.1.1.2.3">′</ci></apply><ci id="S3.2.p2.14.m14.1.1.4.cmml" xref="S3.2.p2.14.m14.1.1.4">𝑏</ci></apply><apply id="S3.2.p2.14.m14.1.1c.cmml" xref="S3.2.p2.14.m14.1.1"><apply id="S3.2.p2.14.m14.1.1.5.cmml" xref="S3.2.p2.14.m14.1.1.5"><csymbol cd="ambiguous" id="S3.2.p2.14.m14.1.1.5.1.cmml" xref="S3.2.p2.14.m14.1.1.5">subscript</csymbol><lt id="S3.2.p2.14.m14.1.1.5.2.cmml" xref="S3.2.p2.14.m14.1.1.5.2"></lt><ci id="S3.2.p2.14.m14.1.1.5.3.cmml" xref="S3.2.p2.14.m14.1.1.5.3">𝐵</ci></apply><share href="https://arxiv.org/html/2503.13728v1#S3.2.p2.14.m14.1.1.4.cmml" id="S3.2.p2.14.m14.1.1d.cmml" xref="S3.2.p2.14.m14.1.1"></share><apply id="S3.2.p2.14.m14.1.1.6.cmml" xref="S3.2.p2.14.m14.1.1.6"><csymbol cd="ambiguous" id="S3.2.p2.14.m14.1.1.6.1.cmml" xref="S3.2.p2.14.m14.1.1.6">superscript</csymbol><ci id="S3.2.p2.14.m14.1.1.6.2.cmml" xref="S3.2.p2.14.m14.1.1.6.2">𝑏</ci><ci id="S3.2.p2.14.m14.1.1.6.3.cmml" xref="S3.2.p2.14.m14.1.1.6.3">′′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.2.p2.14.m14.1c">b^{\prime}<_{B}b<_{B}b^{\prime\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.2.p2.14.m14.1d">italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT < start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_b < start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT</annotation></semantics></math>. But then <math alttext="(a,b^{\prime})<_{A\times B}(a,b)<_{A\times B}(a,b^{\prime\prime})" class="ltx_Math" display="inline" id="S3.2.p2.15.m15.6"><semantics id="S3.2.p2.15.m15.6a"><mrow id="S3.2.p2.15.m15.6.6" xref="S3.2.p2.15.m15.6.6.cmml"><mrow id="S3.2.p2.15.m15.5.5.1.1" xref="S3.2.p2.15.m15.5.5.1.2.cmml"><mo id="S3.2.p2.15.m15.5.5.1.1.2" stretchy="false" xref="S3.2.p2.15.m15.5.5.1.2.cmml">(</mo><mi id="S3.2.p2.15.m15.1.1" xref="S3.2.p2.15.m15.1.1.cmml">a</mi><mo id="S3.2.p2.15.m15.5.5.1.1.3" xref="S3.2.p2.15.m15.5.5.1.2.cmml">,</mo><msup id="S3.2.p2.15.m15.5.5.1.1.1" xref="S3.2.p2.15.m15.5.5.1.1.1.cmml"><mi id="S3.2.p2.15.m15.5.5.1.1.1.2" xref="S3.2.p2.15.m15.5.5.1.1.1.2.cmml">b</mi><mo id="S3.2.p2.15.m15.5.5.1.1.1.3" xref="S3.2.p2.15.m15.5.5.1.1.1.3.cmml">′</mo></msup><mo id="S3.2.p2.15.m15.5.5.1.1.4" stretchy="false" xref="S3.2.p2.15.m15.5.5.1.2.cmml">)</mo></mrow><msub id="S3.2.p2.15.m15.6.6.4" xref="S3.2.p2.15.m15.6.6.4.cmml"><mo id="S3.2.p2.15.m15.6.6.4.2" xref="S3.2.p2.15.m15.6.6.4.2.cmml"><</mo><mrow id="S3.2.p2.15.m15.6.6.4.3" xref="S3.2.p2.15.m15.6.6.4.3.cmml"><mi id="S3.2.p2.15.m15.6.6.4.3.2" xref="S3.2.p2.15.m15.6.6.4.3.2.cmml">A</mi><mo id="S3.2.p2.15.m15.6.6.4.3.1" lspace="0.222em" rspace="0.222em" xref="S3.2.p2.15.m15.6.6.4.3.1.cmml">×</mo><mi id="S3.2.p2.15.m15.6.6.4.3.3" xref="S3.2.p2.15.m15.6.6.4.3.3.cmml">B</mi></mrow></msub><mrow id="S3.2.p2.15.m15.6.6.5.2" xref="S3.2.p2.15.m15.6.6.5.1.cmml"><mo id="S3.2.p2.15.m15.6.6.5.2.1" stretchy="false" xref="S3.2.p2.15.m15.6.6.5.1.cmml">(</mo><mi id="S3.2.p2.15.m15.2.2" xref="S3.2.p2.15.m15.2.2.cmml">a</mi><mo id="S3.2.p2.15.m15.6.6.5.2.2" xref="S3.2.p2.15.m15.6.6.5.1.cmml">,</mo><mi id="S3.2.p2.15.m15.3.3" xref="S3.2.p2.15.m15.3.3.cmml">b</mi><mo id="S3.2.p2.15.m15.6.6.5.2.3" stretchy="false" xref="S3.2.p2.15.m15.6.6.5.1.cmml">)</mo></mrow><msub id="S3.2.p2.15.m15.6.6.6" xref="S3.2.p2.15.m15.6.6.6.cmml"><mo id="S3.2.p2.15.m15.6.6.6.2" xref="S3.2.p2.15.m15.6.6.6.2.cmml"><</mo><mrow id="S3.2.p2.15.m15.6.6.6.3" xref="S3.2.p2.15.m15.6.6.6.3.cmml"><mi id="S3.2.p2.15.m15.6.6.6.3.2" xref="S3.2.p2.15.m15.6.6.6.3.2.cmml">A</mi><mo id="S3.2.p2.15.m15.6.6.6.3.1" lspace="0.222em" rspace="0.222em" xref="S3.2.p2.15.m15.6.6.6.3.1.cmml">×</mo><mi id="S3.2.p2.15.m15.6.6.6.3.3" xref="S3.2.p2.15.m15.6.6.6.3.3.cmml">B</mi></mrow></msub><mrow id="S3.2.p2.15.m15.6.6.2.1" xref="S3.2.p2.15.m15.6.6.2.2.cmml"><mo id="S3.2.p2.15.m15.6.6.2.1.2" stretchy="false" xref="S3.2.p2.15.m15.6.6.2.2.cmml">(</mo><mi id="S3.2.p2.15.m15.4.4" xref="S3.2.p2.15.m15.4.4.cmml">a</mi><mo id="S3.2.p2.15.m15.6.6.2.1.3" xref="S3.2.p2.15.m15.6.6.2.2.cmml">,</mo><msup id="S3.2.p2.15.m15.6.6.2.1.1" xref="S3.2.p2.15.m15.6.6.2.1.1.cmml"><mi id="S3.2.p2.15.m15.6.6.2.1.1.2" xref="S3.2.p2.15.m15.6.6.2.1.1.2.cmml">b</mi><mo id="S3.2.p2.15.m15.6.6.2.1.1.3" xref="S3.2.p2.15.m15.6.6.2.1.1.3.cmml">′′</mo></msup><mo id="S3.2.p2.15.m15.6.6.2.1.4" stretchy="false" xref="S3.2.p2.15.m15.6.6.2.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.2.p2.15.m15.6b"><apply id="S3.2.p2.15.m15.6.6.cmml" xref="S3.2.p2.15.m15.6.6"><and id="S3.2.p2.15.m15.6.6a.cmml" xref="S3.2.p2.15.m15.6.6"></and><apply id="S3.2.p2.15.m15.6.6b.cmml" xref="S3.2.p2.15.m15.6.6"><apply id="S3.2.p2.15.m15.6.6.4.cmml" xref="S3.2.p2.15.m15.6.6.4"><csymbol cd="ambiguous" id="S3.2.p2.15.m15.6.6.4.1.cmml" xref="S3.2.p2.15.m15.6.6.4">subscript</csymbol><lt id="S3.2.p2.15.m15.6.6.4.2.cmml" xref="S3.2.p2.15.m15.6.6.4.2"></lt><apply id="S3.2.p2.15.m15.6.6.4.3.cmml" xref="S3.2.p2.15.m15.6.6.4.3"><times id="S3.2.p2.15.m15.6.6.4.3.1.cmml" xref="S3.2.p2.15.m15.6.6.4.3.1"></times><ci id="S3.2.p2.15.m15.6.6.4.3.2.cmml" xref="S3.2.p2.15.m15.6.6.4.3.2">𝐴</ci><ci id="S3.2.p2.15.m15.6.6.4.3.3.cmml" xref="S3.2.p2.15.m15.6.6.4.3.3">𝐵</ci></apply></apply><interval closure="open" id="S3.2.p2.15.m15.5.5.1.2.cmml" xref="S3.2.p2.15.m15.5.5.1.1"><ci id="S3.2.p2.15.m15.1.1.cmml" xref="S3.2.p2.15.m15.1.1">𝑎</ci><apply id="S3.2.p2.15.m15.5.5.1.1.1.cmml" xref="S3.2.p2.15.m15.5.5.1.1.1"><csymbol cd="ambiguous" id="S3.2.p2.15.m15.5.5.1.1.1.1.cmml" xref="S3.2.p2.15.m15.5.5.1.1.1">superscript</csymbol><ci id="S3.2.p2.15.m15.5.5.1.1.1.2.cmml" xref="S3.2.p2.15.m15.5.5.1.1.1.2">𝑏</ci><ci id="S3.2.p2.15.m15.5.5.1.1.1.3.cmml" xref="S3.2.p2.15.m15.5.5.1.1.1.3">′</ci></apply></interval><interval closure="open" id="S3.2.p2.15.m15.6.6.5.1.cmml" xref="S3.2.p2.15.m15.6.6.5.2"><ci id="S3.2.p2.15.m15.2.2.cmml" xref="S3.2.p2.15.m15.2.2">𝑎</ci><ci id="S3.2.p2.15.m15.3.3.cmml" xref="S3.2.p2.15.m15.3.3">𝑏</ci></interval></apply><apply id="S3.2.p2.15.m15.6.6c.cmml" xref="S3.2.p2.15.m15.6.6"><apply id="S3.2.p2.15.m15.6.6.6.cmml" xref="S3.2.p2.15.m15.6.6.6"><csymbol cd="ambiguous" id="S3.2.p2.15.m15.6.6.6.1.cmml" xref="S3.2.p2.15.m15.6.6.6">subscript</csymbol><lt id="S3.2.p2.15.m15.6.6.6.2.cmml" xref="S3.2.p2.15.m15.6.6.6.2"></lt><apply id="S3.2.p2.15.m15.6.6.6.3.cmml" xref="S3.2.p2.15.m15.6.6.6.3"><times id="S3.2.p2.15.m15.6.6.6.3.1.cmml" xref="S3.2.p2.15.m15.6.6.6.3.1"></times><ci id="S3.2.p2.15.m15.6.6.6.3.2.cmml" xref="S3.2.p2.15.m15.6.6.6.3.2">𝐴</ci><ci id="S3.2.p2.15.m15.6.6.6.3.3.cmml" xref="S3.2.p2.15.m15.6.6.6.3.3">𝐵</ci></apply></apply><share href="https://arxiv.org/html/2503.13728v1#S3.2.p2.15.m15.6.6.5.cmml" id="S3.2.p2.15.m15.6.6d.cmml" xref="S3.2.p2.15.m15.6.6"></share><interval closure="open" id="S3.2.p2.15.m15.6.6.2.2.cmml" xref="S3.2.p2.15.m15.6.6.2.1"><ci id="S3.2.p2.15.m15.4.4.cmml" xref="S3.2.p2.15.m15.4.4">𝑎</ci><apply id="S3.2.p2.15.m15.6.6.2.1.1.cmml" xref="S3.2.p2.15.m15.6.6.2.1.1"><csymbol cd="ambiguous" id="S3.2.p2.15.m15.6.6.2.1.1.1.cmml" xref="S3.2.p2.15.m15.6.6.2.1.1">superscript</csymbol><ci id="S3.2.p2.15.m15.6.6.2.1.1.2.cmml" xref="S3.2.p2.15.m15.6.6.2.1.1.2">𝑏</ci><ci id="S3.2.p2.15.m15.6.6.2.1.1.3.cmml" xref="S3.2.p2.15.m15.6.6.2.1.1.3">′′</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.2.p2.15.m15.6c">(a,b^{\prime})<_{A\times B}(a,b)<_{A\times B}(a,b^{\prime\prime})</annotation><annotation encoding="application/x-llamapun" id="S3.2.p2.15.m15.6d">( italic_a , italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) < start_POSTSUBSCRIPT italic_A × italic_B end_POSTSUBSCRIPT ( italic_a , italic_b ) < start_POSTSUBSCRIPT italic_A × italic_B end_POSTSUBSCRIPT ( italic_a , italic_b start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> and <math alttext="(a,b^{\prime}),(a,b^{\prime\prime})\in I" class="ltx_Math" display="inline" id="S3.2.p2.16.m16.4"><semantics id="S3.2.p2.16.m16.4a"><mrow id="S3.2.p2.16.m16.4.4" xref="S3.2.p2.16.m16.4.4.cmml"><mrow id="S3.2.p2.16.m16.4.4.2.2" xref="S3.2.p2.16.m16.4.4.2.3.cmml"><mrow id="S3.2.p2.16.m16.3.3.1.1.1.1" xref="S3.2.p2.16.m16.3.3.1.1.1.2.cmml"><mo id="S3.2.p2.16.m16.3.3.1.1.1.1.2" stretchy="false" xref="S3.2.p2.16.m16.3.3.1.1.1.2.cmml">(</mo><mi id="S3.2.p2.16.m16.1.1" xref="S3.2.p2.16.m16.1.1.cmml">a</mi><mo id="S3.2.p2.16.m16.3.3.1.1.1.1.3" xref="S3.2.p2.16.m16.3.3.1.1.1.2.cmml">,</mo><msup id="S3.2.p2.16.m16.3.3.1.1.1.1.1" xref="S3.2.p2.16.m16.3.3.1.1.1.1.1.cmml"><mi id="S3.2.p2.16.m16.3.3.1.1.1.1.1.2" xref="S3.2.p2.16.m16.3.3.1.1.1.1.1.2.cmml">b</mi><mo id="S3.2.p2.16.m16.3.3.1.1.1.1.1.3" xref="S3.2.p2.16.m16.3.3.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S3.2.p2.16.m16.3.3.1.1.1.1.4" stretchy="false" xref="S3.2.p2.16.m16.3.3.1.1.1.2.cmml">)</mo></mrow><mo id="S3.2.p2.16.m16.4.4.2.2.3" xref="S3.2.p2.16.m16.4.4.2.3.cmml">,</mo><mrow id="S3.2.p2.16.m16.4.4.2.2.2.1" xref="S3.2.p2.16.m16.4.4.2.2.2.2.cmml"><mo id="S3.2.p2.16.m16.4.4.2.2.2.1.2" stretchy="false" xref="S3.2.p2.16.m16.4.4.2.2.2.2.cmml">(</mo><mi id="S3.2.p2.16.m16.2.2" xref="S3.2.p2.16.m16.2.2.cmml">a</mi><mo id="S3.2.p2.16.m16.4.4.2.2.2.1.3" xref="S3.2.p2.16.m16.4.4.2.2.2.2.cmml">,</mo><msup id="S3.2.p2.16.m16.4.4.2.2.2.1.1" xref="S3.2.p2.16.m16.4.4.2.2.2.1.1.cmml"><mi id="S3.2.p2.16.m16.4.4.2.2.2.1.1.2" xref="S3.2.p2.16.m16.4.4.2.2.2.1.1.2.cmml">b</mi><mo id="S3.2.p2.16.m16.4.4.2.2.2.1.1.3" xref="S3.2.p2.16.m16.4.4.2.2.2.1.1.3.cmml">′′</mo></msup><mo id="S3.2.p2.16.m16.4.4.2.2.2.1.4" stretchy="false" xref="S3.2.p2.16.m16.4.4.2.2.2.2.cmml">)</mo></mrow></mrow><mo id="S3.2.p2.16.m16.4.4.3" xref="S3.2.p2.16.m16.4.4.3.cmml">∈</mo><mi id="S3.2.p2.16.m16.4.4.4" xref="S3.2.p2.16.m16.4.4.4.cmml">I</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.2.p2.16.m16.4b"><apply id="S3.2.p2.16.m16.4.4.cmml" xref="S3.2.p2.16.m16.4.4"><in id="S3.2.p2.16.m16.4.4.3.cmml" xref="S3.2.p2.16.m16.4.4.3"></in><list id="S3.2.p2.16.m16.4.4.2.3.cmml" xref="S3.2.p2.16.m16.4.4.2.2"><interval closure="open" id="S3.2.p2.16.m16.3.3.1.1.1.2.cmml" xref="S3.2.p2.16.m16.3.3.1.1.1.1"><ci id="S3.2.p2.16.m16.1.1.cmml" xref="S3.2.p2.16.m16.1.1">𝑎</ci><apply id="S3.2.p2.16.m16.3.3.1.1.1.1.1.cmml" xref="S3.2.p2.16.m16.3.3.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.2.p2.16.m16.3.3.1.1.1.1.1.1.cmml" xref="S3.2.p2.16.m16.3.3.1.1.1.1.1">superscript</csymbol><ci id="S3.2.p2.16.m16.3.3.1.1.1.1.1.2.cmml" xref="S3.2.p2.16.m16.3.3.1.1.1.1.1.2">𝑏</ci><ci id="S3.2.p2.16.m16.3.3.1.1.1.1.1.3.cmml" xref="S3.2.p2.16.m16.3.3.1.1.1.1.1.3">′</ci></apply></interval><interval closure="open" id="S3.2.p2.16.m16.4.4.2.2.2.2.cmml" xref="S3.2.p2.16.m16.4.4.2.2.2.1"><ci id="S3.2.p2.16.m16.2.2.cmml" xref="S3.2.p2.16.m16.2.2">𝑎</ci><apply id="S3.2.p2.16.m16.4.4.2.2.2.1.1.cmml" xref="S3.2.p2.16.m16.4.4.2.2.2.1.1"><csymbol cd="ambiguous" id="S3.2.p2.16.m16.4.4.2.2.2.1.1.1.cmml" xref="S3.2.p2.16.m16.4.4.2.2.2.1.1">superscript</csymbol><ci id="S3.2.p2.16.m16.4.4.2.2.2.1.1.2.cmml" xref="S3.2.p2.16.m16.4.4.2.2.2.1.1.2">𝑏</ci><ci id="S3.2.p2.16.m16.4.4.2.2.2.1.1.3.cmml" xref="S3.2.p2.16.m16.4.4.2.2.2.1.1.3">′′</ci></apply></interval></list><ci id="S3.2.p2.16.m16.4.4.4.cmml" xref="S3.2.p2.16.m16.4.4.4">𝐼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.2.p2.16.m16.4c">(a,b^{\prime}),(a,b^{\prime\prime})\in I</annotation><annotation encoding="application/x-llamapun" id="S3.2.p2.16.m16.4d">( italic_a , italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) , ( italic_a , italic_b start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) ∈ italic_I</annotation></semantics></math>. We conclude that <math alttext="I" class="ltx_Math" display="inline" id="S3.2.p2.17.m17.1"><semantics id="S3.2.p2.17.m17.1a"><mi id="S3.2.p2.17.m17.1.1" xref="S3.2.p2.17.m17.1.1.cmml">I</mi><annotation-xml encoding="MathML-Content" id="S3.2.p2.17.m17.1b"><ci id="S3.2.p2.17.m17.1.1.cmml" xref="S3.2.p2.17.m17.1.1">𝐼</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.2.p2.17.m17.1c">I</annotation><annotation encoding="application/x-llamapun" id="S3.2.p2.17.m17.1d">italic_I</annotation></semantics></math> has no endpoints either. ∎</p> </div> </div> <div class="ltx_theorem ltx_theorem_corollary" id="S3.Thmtheorem2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem2.1.1.1">Corollary 3.2</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem2.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem2.p1"> <p class="ltx_p" id="S3.Thmtheorem2.p1.5">Assume <math alttext="\mathsf{MA}_{\aleph_{1}}" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.1.m1.1"><semantics id="S3.Thmtheorem2.p1.1.m1.1a"><msub id="S3.Thmtheorem2.p1.1.m1.1.1" xref="S3.Thmtheorem2.p1.1.m1.1.1.cmml"><mi id="S3.Thmtheorem2.p1.1.m1.1.1.2" xref="S3.Thmtheorem2.p1.1.m1.1.1.2.cmml">𝖬𝖠</mi><msub id="S3.Thmtheorem2.p1.1.m1.1.1.3" xref="S3.Thmtheorem2.p1.1.m1.1.1.3.cmml"><mi id="S3.Thmtheorem2.p1.1.m1.1.1.3.2" mathvariant="normal" xref="S3.Thmtheorem2.p1.1.m1.1.1.3.2.cmml">ℵ</mi><mn id="S3.Thmtheorem2.p1.1.m1.1.1.3.3" xref="S3.Thmtheorem2.p1.1.m1.1.1.3.3.cmml">1</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.1.m1.1b"><apply id="S3.Thmtheorem2.p1.1.m1.1.1.cmml" xref="S3.Thmtheorem2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p1.1.m1.1.1.1.cmml" xref="S3.Thmtheorem2.p1.1.m1.1.1">subscript</csymbol><ci id="S3.Thmtheorem2.p1.1.m1.1.1.2.cmml" xref="S3.Thmtheorem2.p1.1.m1.1.1.2">𝖬𝖠</ci><apply id="S3.Thmtheorem2.p1.1.m1.1.1.3.cmml" xref="S3.Thmtheorem2.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p1.1.m1.1.1.3.1.cmml" xref="S3.Thmtheorem2.p1.1.m1.1.1.3">subscript</csymbol><ci id="S3.Thmtheorem2.p1.1.m1.1.1.3.2.cmml" xref="S3.Thmtheorem2.p1.1.m1.1.1.3.2">ℵ</ci><cn id="S3.Thmtheorem2.p1.1.m1.1.1.3.3.cmml" type="integer" xref="S3.Thmtheorem2.p1.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.1.m1.1c">\mathsf{MA}_{\aleph_{1}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.1.m1.1d">sansserif_MA start_POSTSUBSCRIPT roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. If <math alttext="C" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.2.m2.1"><semantics id="S3.Thmtheorem2.p1.2.m2.1a"><mi id="S3.Thmtheorem2.p1.2.m2.1.1" xref="S3.Thmtheorem2.p1.2.m2.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.2.m2.1b"><ci id="S3.Thmtheorem2.p1.2.m2.1.1.cmml" xref="S3.Thmtheorem2.p1.2.m2.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.2.m2.1c">C</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.2.m2.1d">italic_C</annotation></semantics></math> is a normal Countryman line, then <math alttext="A\times C\cong C" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.3.m3.1"><semantics id="S3.Thmtheorem2.p1.3.m3.1a"><mrow id="S3.Thmtheorem2.p1.3.m3.1.1" xref="S3.Thmtheorem2.p1.3.m3.1.1.cmml"><mrow id="S3.Thmtheorem2.p1.3.m3.1.1.2" xref="S3.Thmtheorem2.p1.3.m3.1.1.2.cmml"><mi id="S3.Thmtheorem2.p1.3.m3.1.1.2.2" xref="S3.Thmtheorem2.p1.3.m3.1.1.2.2.cmml">A</mi><mo id="S3.Thmtheorem2.p1.3.m3.1.1.2.1" lspace="0.222em" rspace="0.222em" xref="S3.Thmtheorem2.p1.3.m3.1.1.2.1.cmml">×</mo><mi id="S3.Thmtheorem2.p1.3.m3.1.1.2.3" xref="S3.Thmtheorem2.p1.3.m3.1.1.2.3.cmml">C</mi></mrow><mo id="S3.Thmtheorem2.p1.3.m3.1.1.1" xref="S3.Thmtheorem2.p1.3.m3.1.1.1.cmml">≅</mo><mi id="S3.Thmtheorem2.p1.3.m3.1.1.3" xref="S3.Thmtheorem2.p1.3.m3.1.1.3.cmml">C</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.3.m3.1b"><apply id="S3.Thmtheorem2.p1.3.m3.1.1.cmml" xref="S3.Thmtheorem2.p1.3.m3.1.1"><approx id="S3.Thmtheorem2.p1.3.m3.1.1.1.cmml" xref="S3.Thmtheorem2.p1.3.m3.1.1.1"></approx><apply id="S3.Thmtheorem2.p1.3.m3.1.1.2.cmml" xref="S3.Thmtheorem2.p1.3.m3.1.1.2"><times id="S3.Thmtheorem2.p1.3.m3.1.1.2.1.cmml" xref="S3.Thmtheorem2.p1.3.m3.1.1.2.1"></times><ci id="S3.Thmtheorem2.p1.3.m3.1.1.2.2.cmml" xref="S3.Thmtheorem2.p1.3.m3.1.1.2.2">𝐴</ci><ci id="S3.Thmtheorem2.p1.3.m3.1.1.2.3.cmml" xref="S3.Thmtheorem2.p1.3.m3.1.1.2.3">𝐶</ci></apply><ci id="S3.Thmtheorem2.p1.3.m3.1.1.3.cmml" xref="S3.Thmtheorem2.p1.3.m3.1.1.3">𝐶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.3.m3.1c">A\times C\cong C</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.3.m3.1d">italic_A × italic_C ≅ italic_C</annotation></semantics></math> for every nonempty <math alttext="A\preceq C" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.4.m4.1"><semantics id="S3.Thmtheorem2.p1.4.m4.1a"><mrow id="S3.Thmtheorem2.p1.4.m4.1.1" xref="S3.Thmtheorem2.p1.4.m4.1.1.cmml"><mi id="S3.Thmtheorem2.p1.4.m4.1.1.2" xref="S3.Thmtheorem2.p1.4.m4.1.1.2.cmml">A</mi><mo id="S3.Thmtheorem2.p1.4.m4.1.1.1" xref="S3.Thmtheorem2.p1.4.m4.1.1.1.cmml">⪯</mo><mi id="S3.Thmtheorem2.p1.4.m4.1.1.3" xref="S3.Thmtheorem2.p1.4.m4.1.1.3.cmml">C</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.4.m4.1b"><apply id="S3.Thmtheorem2.p1.4.m4.1.1.cmml" xref="S3.Thmtheorem2.p1.4.m4.1.1"><csymbol cd="latexml" id="S3.Thmtheorem2.p1.4.m4.1.1.1.cmml" xref="S3.Thmtheorem2.p1.4.m4.1.1.1">precedes-or-equals</csymbol><ci id="S3.Thmtheorem2.p1.4.m4.1.1.2.cmml" xref="S3.Thmtheorem2.p1.4.m4.1.1.2">𝐴</ci><ci id="S3.Thmtheorem2.p1.4.m4.1.1.3.cmml" xref="S3.Thmtheorem2.p1.4.m4.1.1.3">𝐶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.4.m4.1c">A\preceq C</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.4.m4.1d">italic_A ⪯ italic_C</annotation></semantics></math>. In particular <math alttext="C" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.5.m5.1"><semantics id="S3.Thmtheorem2.p1.5.m5.1a"><mi id="S3.Thmtheorem2.p1.5.m5.1.1" xref="S3.Thmtheorem2.p1.5.m5.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.5.m5.1b"><ci id="S3.Thmtheorem2.p1.5.m5.1.1.cmml" xref="S3.Thmtheorem2.p1.5.m5.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.5.m5.1c">C</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.5.m5.1d">italic_C</annotation></semantics></math> is strongly surjective.</p> </div> </div> <div class="ltx_proof" id="S3.3"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S3.3.p1"> <p class="ltx_p" id="S3.3.p1.5">If <math alttext="A\preceq C" class="ltx_Math" display="inline" id="S3.3.p1.1.m1.1"><semantics id="S3.3.p1.1.m1.1a"><mrow id="S3.3.p1.1.m1.1.1" xref="S3.3.p1.1.m1.1.1.cmml"><mi id="S3.3.p1.1.m1.1.1.2" xref="S3.3.p1.1.m1.1.1.2.cmml">A</mi><mo id="S3.3.p1.1.m1.1.1.1" xref="S3.3.p1.1.m1.1.1.1.cmml">⪯</mo><mi id="S3.3.p1.1.m1.1.1.3" xref="S3.3.p1.1.m1.1.1.3.cmml">C</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.3.p1.1.m1.1b"><apply id="S3.3.p1.1.m1.1.1.cmml" xref="S3.3.p1.1.m1.1.1"><csymbol cd="latexml" id="S3.3.p1.1.m1.1.1.1.cmml" xref="S3.3.p1.1.m1.1.1.1">precedes-or-equals</csymbol><ci id="S3.3.p1.1.m1.1.1.2.cmml" xref="S3.3.p1.1.m1.1.1.2">𝐴</ci><ci id="S3.3.p1.1.m1.1.1.3.cmml" xref="S3.3.p1.1.m1.1.1.3">𝐶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.3.p1.1.m1.1c">A\preceq C</annotation><annotation encoding="application/x-llamapun" id="S3.3.p1.1.m1.1d">italic_A ⪯ italic_C</annotation></semantics></math> is nonempty, then <math alttext="A\times C" class="ltx_Math" display="inline" id="S3.3.p1.2.m2.1"><semantics id="S3.3.p1.2.m2.1a"><mrow id="S3.3.p1.2.m2.1.1" xref="S3.3.p1.2.m2.1.1.cmml"><mi id="S3.3.p1.2.m2.1.1.2" xref="S3.3.p1.2.m2.1.1.2.cmml">A</mi><mo id="S3.3.p1.2.m2.1.1.1" lspace="0.222em" rspace="0.222em" xref="S3.3.p1.2.m2.1.1.1.cmml">×</mo><mi id="S3.3.p1.2.m2.1.1.3" xref="S3.3.p1.2.m2.1.1.3.cmml">C</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.3.p1.2.m2.1b"><apply id="S3.3.p1.2.m2.1.1.cmml" xref="S3.3.p1.2.m2.1.1"><times id="S3.3.p1.2.m2.1.1.1.cmml" xref="S3.3.p1.2.m2.1.1.1"></times><ci id="S3.3.p1.2.m2.1.1.2.cmml" xref="S3.3.p1.2.m2.1.1.2">𝐴</ci><ci id="S3.3.p1.2.m2.1.1.3.cmml" xref="S3.3.p1.2.m2.1.1.3">𝐶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.3.p1.2.m2.1c">A\times C</annotation><annotation encoding="application/x-llamapun" id="S3.3.p1.2.m2.1d">italic_A × italic_C</annotation></semantics></math> is Countryman and it is normal by <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S3.Thmtheorem1" title="Lemma 3.1. ‣ 3. Strongly surjective Aronszajn lines ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">3.1</span></a>. Then by <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S2.Thmtheorem8" title="Theorem 2.8. ‣ 2. Aronszajn and Countryman lines ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">2.8</span></a>, <math alttext="A\times C" class="ltx_Math" display="inline" id="S3.3.p1.3.m3.1"><semantics id="S3.3.p1.3.m3.1a"><mrow id="S3.3.p1.3.m3.1.1" xref="S3.3.p1.3.m3.1.1.cmml"><mi id="S3.3.p1.3.m3.1.1.2" xref="S3.3.p1.3.m3.1.1.2.cmml">A</mi><mo id="S3.3.p1.3.m3.1.1.1" lspace="0.222em" rspace="0.222em" xref="S3.3.p1.3.m3.1.1.1.cmml">×</mo><mi id="S3.3.p1.3.m3.1.1.3" xref="S3.3.p1.3.m3.1.1.3.cmml">C</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.3.p1.3.m3.1b"><apply id="S3.3.p1.3.m3.1.1.cmml" xref="S3.3.p1.3.m3.1.1"><times id="S3.3.p1.3.m3.1.1.1.cmml" xref="S3.3.p1.3.m3.1.1.1"></times><ci id="S3.3.p1.3.m3.1.1.2.cmml" xref="S3.3.p1.3.m3.1.1.2">𝐴</ci><ci id="S3.3.p1.3.m3.1.1.3.cmml" xref="S3.3.p1.3.m3.1.1.3">𝐶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.3.p1.3.m3.1c">A\times C</annotation><annotation encoding="application/x-llamapun" id="S3.3.p1.3.m3.1d">italic_A × italic_C</annotation></semantics></math> is isomorphic to <math alttext="C" class="ltx_Math" display="inline" id="S3.3.p1.4.m4.1"><semantics id="S3.3.p1.4.m4.1a"><mi id="S3.3.p1.4.m4.1.1" xref="S3.3.p1.4.m4.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S3.3.p1.4.m4.1b"><ci id="S3.3.p1.4.m4.1.1.cmml" xref="S3.3.p1.4.m4.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.3.p1.4.m4.1c">C</annotation><annotation encoding="application/x-llamapun" id="S3.3.p1.4.m4.1d">italic_C</annotation></semantics></math> or to <math alttext="C^{\star}" class="ltx_Math" display="inline" id="S3.3.p1.5.m5.1"><semantics id="S3.3.p1.5.m5.1a"><msup id="S3.3.p1.5.m5.1.1" xref="S3.3.p1.5.m5.1.1.cmml"><mi id="S3.3.p1.5.m5.1.1.2" xref="S3.3.p1.5.m5.1.1.2.cmml">C</mi><mo id="S3.3.p1.5.m5.1.1.3" xref="S3.3.p1.5.m5.1.1.3.cmml">⋆</mo></msup><annotation-xml encoding="MathML-Content" id="S3.3.p1.5.m5.1b"><apply id="S3.3.p1.5.m5.1.1.cmml" xref="S3.3.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S3.3.p1.5.m5.1.1.1.cmml" xref="S3.3.p1.5.m5.1.1">superscript</csymbol><ci id="S3.3.p1.5.m5.1.1.2.cmml" xref="S3.3.p1.5.m5.1.1.2">𝐶</ci><ci id="S3.3.p1.5.m5.1.1.3.cmml" xref="S3.3.p1.5.m5.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.3.p1.5.m5.1c">C^{\star}</annotation><annotation encoding="application/x-llamapun" id="S3.3.p1.5.m5.1d">italic_C start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math>, but the later is impossible by <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S2.Thmtheorem6" title="Lemma 2.6. ‣ 2. Aronszajn and Countryman lines ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">2.6</span></a>. ∎</p> </div> </div> <div class="ltx_para" id="S3.p2"> <p class="ltx_p" id="S3.p2.1">As mentioned in the introduction, this answers positively Soukup questions <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#Thmquestion3" title="Question 3. ‣ Historical and mathematical context ‣ 1. Introduction ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">3</span></a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#Thmquestion4" title="Question 4. ‣ Historical and mathematical context ‣ 1. Introduction ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">4</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#Thmquestion5" title="Question 5. ‣ Historical and mathematical context ‣ 1. Introduction ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">5</span></a>.</p> </div> <div class="ltx_para" id="S3.p3"> <p class="ltx_p" id="S3.p3.5">In view of <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S3.Thmtheorem2" title="Corollary 3.2. ‣ 3. Strongly surjective Aronszajn lines ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Corollary</span> <span class="ltx_text ltx_ref_tag">3.2</span></a>, one could ask if in fact every normal Aronszajn line is strongly surjective under <math alttext="\mathsf{MA}_{\aleph_{1}}" class="ltx_Math" display="inline" id="S3.p3.1.m1.1"><semantics id="S3.p3.1.m1.1a"><msub id="S3.p3.1.m1.1.1" xref="S3.p3.1.m1.1.1.cmml"><mi id="S3.p3.1.m1.1.1.2" xref="S3.p3.1.m1.1.1.2.cmml">𝖬𝖠</mi><msub id="S3.p3.1.m1.1.1.3" xref="S3.p3.1.m1.1.1.3.cmml"><mi id="S3.p3.1.m1.1.1.3.2" mathvariant="normal" xref="S3.p3.1.m1.1.1.3.2.cmml">ℵ</mi><mn id="S3.p3.1.m1.1.1.3.3" xref="S3.p3.1.m1.1.1.3.3.cmml">1</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S3.p3.1.m1.1b"><apply id="S3.p3.1.m1.1.1.cmml" xref="S3.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S3.p3.1.m1.1.1.1.cmml" xref="S3.p3.1.m1.1.1">subscript</csymbol><ci id="S3.p3.1.m1.1.1.2.cmml" xref="S3.p3.1.m1.1.1.2">𝖬𝖠</ci><apply id="S3.p3.1.m1.1.1.3.cmml" xref="S3.p3.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.p3.1.m1.1.1.3.1.cmml" xref="S3.p3.1.m1.1.1.3">subscript</csymbol><ci id="S3.p3.1.m1.1.1.3.2.cmml" xref="S3.p3.1.m1.1.1.3.2">ℵ</ci><cn id="S3.p3.1.m1.1.1.3.3.cmml" type="integer" xref="S3.p3.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.1.m1.1c">\mathsf{MA}_{\aleph_{1}}</annotation><annotation encoding="application/x-llamapun" id="S3.p3.1.m1.1d">sansserif_MA start_POSTSUBSCRIPT roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>, or under stronger assumptions, however we can easily see that this is not the case: if <math alttext="C" class="ltx_Math" display="inline" id="S3.p3.2.m2.1"><semantics id="S3.p3.2.m2.1a"><mi id="S3.p3.2.m2.1.1" xref="S3.p3.2.m2.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S3.p3.2.m2.1b"><ci id="S3.p3.2.m2.1.1.cmml" xref="S3.p3.2.m2.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.2.m2.1c">C</annotation><annotation encoding="application/x-llamapun" id="S3.p3.2.m2.1d">italic_C</annotation></semantics></math> is a normal Countryman line, then <math alttext="C+C^{\star}" class="ltx_Math" display="inline" id="S3.p3.3.m3.1"><semantics id="S3.p3.3.m3.1a"><mrow id="S3.p3.3.m3.1.1" xref="S3.p3.3.m3.1.1.cmml"><mi id="S3.p3.3.m3.1.1.2" xref="S3.p3.3.m3.1.1.2.cmml">C</mi><mo id="S3.p3.3.m3.1.1.1" xref="S3.p3.3.m3.1.1.1.cmml">+</mo><msup id="S3.p3.3.m3.1.1.3" xref="S3.p3.3.m3.1.1.3.cmml"><mi id="S3.p3.3.m3.1.1.3.2" xref="S3.p3.3.m3.1.1.3.2.cmml">C</mi><mo id="S3.p3.3.m3.1.1.3.3" xref="S3.p3.3.m3.1.1.3.3.cmml">⋆</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.p3.3.m3.1b"><apply id="S3.p3.3.m3.1.1.cmml" xref="S3.p3.3.m3.1.1"><plus id="S3.p3.3.m3.1.1.1.cmml" xref="S3.p3.3.m3.1.1.1"></plus><ci id="S3.p3.3.m3.1.1.2.cmml" xref="S3.p3.3.m3.1.1.2">𝐶</ci><apply id="S3.p3.3.m3.1.1.3.cmml" xref="S3.p3.3.m3.1.1.3"><csymbol cd="ambiguous" id="S3.p3.3.m3.1.1.3.1.cmml" xref="S3.p3.3.m3.1.1.3">superscript</csymbol><ci id="S3.p3.3.m3.1.1.3.2.cmml" xref="S3.p3.3.m3.1.1.3.2">𝐶</ci><ci id="S3.p3.3.m3.1.1.3.3.cmml" xref="S3.p3.3.m3.1.1.3.3">⋆</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.3.m3.1c">C+C^{\star}</annotation><annotation encoding="application/x-llamapun" id="S3.p3.3.m3.1d">italic_C + italic_C start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math> is a normal Aronszajn line that does not surject onto <math alttext="C" class="ltx_Math" display="inline" id="S3.p3.4.m4.1"><semantics id="S3.p3.4.m4.1a"><mi id="S3.p3.4.m4.1.1" xref="S3.p3.4.m4.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S3.p3.4.m4.1b"><ci id="S3.p3.4.m4.1.1.cmml" xref="S3.p3.4.m4.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.4.m4.1c">C</annotation><annotation encoding="application/x-llamapun" id="S3.p3.4.m4.1d">italic_C</annotation></semantics></math>. Although normality is not enough, we do have many examples of non Countryman Aronszajn lines that are strongly surjective under <math alttext="\mathsf{MA}_{\aleph_{1}}" class="ltx_Math" display="inline" id="S3.p3.5.m5.1"><semantics id="S3.p3.5.m5.1a"><msub id="S3.p3.5.m5.1.1" xref="S3.p3.5.m5.1.1.cmml"><mi id="S3.p3.5.m5.1.1.2" xref="S3.p3.5.m5.1.1.2.cmml">𝖬𝖠</mi><msub id="S3.p3.5.m5.1.1.3" xref="S3.p3.5.m5.1.1.3.cmml"><mi id="S3.p3.5.m5.1.1.3.2" mathvariant="normal" xref="S3.p3.5.m5.1.1.3.2.cmml">ℵ</mi><mn id="S3.p3.5.m5.1.1.3.3" xref="S3.p3.5.m5.1.1.3.3.cmml">1</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S3.p3.5.m5.1b"><apply id="S3.p3.5.m5.1.1.cmml" xref="S3.p3.5.m5.1.1"><csymbol cd="ambiguous" id="S3.p3.5.m5.1.1.1.cmml" xref="S3.p3.5.m5.1.1">subscript</csymbol><ci id="S3.p3.5.m5.1.1.2.cmml" xref="S3.p3.5.m5.1.1.2">𝖬𝖠</ci><apply id="S3.p3.5.m5.1.1.3.cmml" xref="S3.p3.5.m5.1.1.3"><csymbol cd="ambiguous" id="S3.p3.5.m5.1.1.3.1.cmml" xref="S3.p3.5.m5.1.1.3">subscript</csymbol><ci id="S3.p3.5.m5.1.1.3.2.cmml" xref="S3.p3.5.m5.1.1.3.2">ℵ</ci><cn id="S3.p3.5.m5.1.1.3.3.cmml" type="integer" xref="S3.p3.5.m5.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.5.m5.1c">\mathsf{MA}_{\aleph_{1}}</annotation><annotation encoding="application/x-llamapun" id="S3.p3.5.m5.1d">sansserif_MA start_POSTSUBSCRIPT roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>, as we show in what follows. Recall the following fact from <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib5" title="">5</a>]</cite>.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S3.Thmtheorem3"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem3.1.1.1">Lemma 3.3</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem3.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem3.p1"> <p class="ltx_p" id="S3.Thmtheorem3.p1.1">Every strongly surjective linear order is short<span class="ltx_note ltx_role_footnote" id="footnote2"><sup class="ltx_note_mark">2</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">2</sup><span class="ltx_tag ltx_tag_note">2</span>Recall that <math alttext="A" class="ltx_Math" display="inline" id="footnote2.m1.1"><semantics id="footnote2.m1.1b"><mi id="footnote2.m1.1.1" xref="footnote2.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="footnote2.m1.1c"><ci id="footnote2.m1.1.1.cmml" xref="footnote2.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="footnote2.m1.1d">A</annotation><annotation encoding="application/x-llamapun" id="footnote2.m1.1e">italic_A</annotation></semantics></math> is <em class="ltx_emph ltx_font_italic" id="footnote2.1">short</em> if it does not contain a copies of <math alttext="\omega_{1}" class="ltx_Math" display="inline" id="footnote2.m2.1"><semantics id="footnote2.m2.1b"><msub id="footnote2.m2.1.1" xref="footnote2.m2.1.1.cmml"><mi id="footnote2.m2.1.1.2" xref="footnote2.m2.1.1.2.cmml">ω</mi><mn id="footnote2.m2.1.1.3" xref="footnote2.m2.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="footnote2.m2.1c"><apply id="footnote2.m2.1.1.cmml" xref="footnote2.m2.1.1"><csymbol cd="ambiguous" id="footnote2.m2.1.1.1.cmml" xref="footnote2.m2.1.1">subscript</csymbol><ci id="footnote2.m2.1.1.2.cmml" xref="footnote2.m2.1.1.2">𝜔</ci><cn id="footnote2.m2.1.1.3.cmml" type="integer" xref="footnote2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote2.m2.1d">\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="footnote2.m2.1e">italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> or <math alttext="\omega_{1}^{\star}" class="ltx_Math" display="inline" id="footnote2.m3.1"><semantics id="footnote2.m3.1b"><msubsup id="footnote2.m3.1.1" xref="footnote2.m3.1.1.cmml"><mi id="footnote2.m3.1.1.2.2" xref="footnote2.m3.1.1.2.2.cmml">ω</mi><mn id="footnote2.m3.1.1.2.3" xref="footnote2.m3.1.1.2.3.cmml">1</mn><mo id="footnote2.m3.1.1.3" xref="footnote2.m3.1.1.3.cmml">⋆</mo></msubsup><annotation-xml encoding="MathML-Content" id="footnote2.m3.1c"><apply id="footnote2.m3.1.1.cmml" xref="footnote2.m3.1.1"><csymbol cd="ambiguous" id="footnote2.m3.1.1.1.cmml" xref="footnote2.m3.1.1">superscript</csymbol><apply id="footnote2.m3.1.1.2.cmml" xref="footnote2.m3.1.1"><csymbol cd="ambiguous" id="footnote2.m3.1.1.2.1.cmml" xref="footnote2.m3.1.1">subscript</csymbol><ci id="footnote2.m3.1.1.2.2.cmml" xref="footnote2.m3.1.1.2.2">𝜔</ci><cn id="footnote2.m3.1.1.2.3.cmml" type="integer" xref="footnote2.m3.1.1.2.3">1</cn></apply><ci id="footnote2.m3.1.1.3.cmml" xref="footnote2.m3.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote2.m3.1d">\omega_{1}^{\star}</annotation><annotation encoding="application/x-llamapun" id="footnote2.m3.1e">italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math>.</span></span></span>.</p> </div> </div> <div class="ltx_para" id="S3.p4"> <p class="ltx_p" id="S3.p4.1">In <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib5" title="">5</a>, Corollary 2.14]</cite>, they prove that the product of strongly surjective linear orders is again strongly surjective, but in fact they prove more.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S3.Thmtheorem4"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem4.1.1.1">Lemma 3.4</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem4.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem4.p1"> <p class="ltx_p" id="S3.Thmtheorem4.p1.8">(Camerlo et al. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib5" title="">5</a>]</cite>) If <math alttext="A" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.1.m1.1"><semantics id="S3.Thmtheorem4.p1.1.m1.1a"><mi id="S3.Thmtheorem4.p1.1.m1.1.1" xref="S3.Thmtheorem4.p1.1.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.1.m1.1b"><ci id="S3.Thmtheorem4.p1.1.m1.1.1.cmml" xref="S3.Thmtheorem4.p1.1.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.1.m1.1c">A</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.1.m1.1d">italic_A</annotation></semantics></math> and <math alttext="B" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.2.m2.1"><semantics id="S3.Thmtheorem4.p1.2.m2.1a"><mi id="S3.Thmtheorem4.p1.2.m2.1.1" xref="S3.Thmtheorem4.p1.2.m2.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.2.m2.1b"><ci id="S3.Thmtheorem4.p1.2.m2.1.1.cmml" xref="S3.Thmtheorem4.p1.2.m2.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.2.m2.1c">B</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.2.m2.1d">italic_B</annotation></semantics></math> are short, <math alttext="A^{\prime}\trianglelefteq A" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.3.m3.1"><semantics id="S3.Thmtheorem4.p1.3.m3.1a"><mrow id="S3.Thmtheorem4.p1.3.m3.1.1" xref="S3.Thmtheorem4.p1.3.m3.1.1.cmml"><msup id="S3.Thmtheorem4.p1.3.m3.1.1.2" xref="S3.Thmtheorem4.p1.3.m3.1.1.2.cmml"><mi id="S3.Thmtheorem4.p1.3.m3.1.1.2.2" xref="S3.Thmtheorem4.p1.3.m3.1.1.2.2.cmml">A</mi><mo id="S3.Thmtheorem4.p1.3.m3.1.1.2.3" xref="S3.Thmtheorem4.p1.3.m3.1.1.2.3.cmml">′</mo></msup><mo id="S3.Thmtheorem4.p1.3.m3.1.1.1" xref="S3.Thmtheorem4.p1.3.m3.1.1.1.cmml">⁢</mo><mi id="S3.Thmtheorem4.p1.3.m3.1.1.3" mathvariant="normal" xref="S3.Thmtheorem4.p1.3.m3.1.1.3.cmml">⊴</mi><mo id="S3.Thmtheorem4.p1.3.m3.1.1.1a" xref="S3.Thmtheorem4.p1.3.m3.1.1.1.cmml">⁢</mo><mi id="S3.Thmtheorem4.p1.3.m3.1.1.4" xref="S3.Thmtheorem4.p1.3.m3.1.1.4.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.3.m3.1b"><apply id="S3.Thmtheorem4.p1.3.m3.1.1.cmml" xref="S3.Thmtheorem4.p1.3.m3.1.1"><times id="S3.Thmtheorem4.p1.3.m3.1.1.1.cmml" xref="S3.Thmtheorem4.p1.3.m3.1.1.1"></times><apply id="S3.Thmtheorem4.p1.3.m3.1.1.2.cmml" xref="S3.Thmtheorem4.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p1.3.m3.1.1.2.1.cmml" xref="S3.Thmtheorem4.p1.3.m3.1.1.2">superscript</csymbol><ci id="S3.Thmtheorem4.p1.3.m3.1.1.2.2.cmml" xref="S3.Thmtheorem4.p1.3.m3.1.1.2.2">𝐴</ci><ci id="S3.Thmtheorem4.p1.3.m3.1.1.2.3.cmml" xref="S3.Thmtheorem4.p1.3.m3.1.1.2.3">′</ci></apply><ci id="S3.Thmtheorem4.p1.3.m3.1.1.3.cmml" xref="S3.Thmtheorem4.p1.3.m3.1.1.3">⊴</ci><ci id="S3.Thmtheorem4.p1.3.m3.1.1.4.cmml" xref="S3.Thmtheorem4.p1.3.m3.1.1.4">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.3.m3.1c">A^{\prime}\trianglelefteq A</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.3.m3.1d">italic_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⊴ italic_A</annotation></semantics></math> and <math alttext="B^{\prime}\trianglelefteq B" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.4.m4.1"><semantics id="S3.Thmtheorem4.p1.4.m4.1a"><mrow id="S3.Thmtheorem4.p1.4.m4.1.1" xref="S3.Thmtheorem4.p1.4.m4.1.1.cmml"><msup id="S3.Thmtheorem4.p1.4.m4.1.1.2" xref="S3.Thmtheorem4.p1.4.m4.1.1.2.cmml"><mi id="S3.Thmtheorem4.p1.4.m4.1.1.2.2" xref="S3.Thmtheorem4.p1.4.m4.1.1.2.2.cmml">B</mi><mo id="S3.Thmtheorem4.p1.4.m4.1.1.2.3" xref="S3.Thmtheorem4.p1.4.m4.1.1.2.3.cmml">′</mo></msup><mo id="S3.Thmtheorem4.p1.4.m4.1.1.1" xref="S3.Thmtheorem4.p1.4.m4.1.1.1.cmml">⁢</mo><mi id="S3.Thmtheorem4.p1.4.m4.1.1.3" mathvariant="normal" xref="S3.Thmtheorem4.p1.4.m4.1.1.3.cmml">⊴</mi><mo id="S3.Thmtheorem4.p1.4.m4.1.1.1a" xref="S3.Thmtheorem4.p1.4.m4.1.1.1.cmml">⁢</mo><mi id="S3.Thmtheorem4.p1.4.m4.1.1.4" xref="S3.Thmtheorem4.p1.4.m4.1.1.4.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.4.m4.1b"><apply id="S3.Thmtheorem4.p1.4.m4.1.1.cmml" xref="S3.Thmtheorem4.p1.4.m4.1.1"><times id="S3.Thmtheorem4.p1.4.m4.1.1.1.cmml" xref="S3.Thmtheorem4.p1.4.m4.1.1.1"></times><apply id="S3.Thmtheorem4.p1.4.m4.1.1.2.cmml" xref="S3.Thmtheorem4.p1.4.m4.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p1.4.m4.1.1.2.1.cmml" xref="S3.Thmtheorem4.p1.4.m4.1.1.2">superscript</csymbol><ci id="S3.Thmtheorem4.p1.4.m4.1.1.2.2.cmml" xref="S3.Thmtheorem4.p1.4.m4.1.1.2.2">𝐵</ci><ci id="S3.Thmtheorem4.p1.4.m4.1.1.2.3.cmml" xref="S3.Thmtheorem4.p1.4.m4.1.1.2.3">′</ci></apply><ci id="S3.Thmtheorem4.p1.4.m4.1.1.3.cmml" xref="S3.Thmtheorem4.p1.4.m4.1.1.3">⊴</ci><ci id="S3.Thmtheorem4.p1.4.m4.1.1.4.cmml" xref="S3.Thmtheorem4.p1.4.m4.1.1.4">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.4.m4.1c">B^{\prime}\trianglelefteq B</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.4.m4.1d">italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⊴ italic_B</annotation></semantics></math>, then <math alttext="A^{\prime}\times B^{\prime}\trianglelefteq A\times B" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.5.m5.1"><semantics id="S3.Thmtheorem4.p1.5.m5.1a"><mrow id="S3.Thmtheorem4.p1.5.m5.1.1" xref="S3.Thmtheorem4.p1.5.m5.1.1.cmml"><mrow id="S3.Thmtheorem4.p1.5.m5.1.1.2" xref="S3.Thmtheorem4.p1.5.m5.1.1.2.cmml"><mrow id="S3.Thmtheorem4.p1.5.m5.1.1.2.2" xref="S3.Thmtheorem4.p1.5.m5.1.1.2.2.cmml"><msup id="S3.Thmtheorem4.p1.5.m5.1.1.2.2.2" xref="S3.Thmtheorem4.p1.5.m5.1.1.2.2.2.cmml"><mi id="S3.Thmtheorem4.p1.5.m5.1.1.2.2.2.2" xref="S3.Thmtheorem4.p1.5.m5.1.1.2.2.2.2.cmml">A</mi><mo id="S3.Thmtheorem4.p1.5.m5.1.1.2.2.2.3" xref="S3.Thmtheorem4.p1.5.m5.1.1.2.2.2.3.cmml">′</mo></msup><mo id="S3.Thmtheorem4.p1.5.m5.1.1.2.2.1" lspace="0.222em" rspace="0.222em" xref="S3.Thmtheorem4.p1.5.m5.1.1.2.2.1.cmml">×</mo><msup id="S3.Thmtheorem4.p1.5.m5.1.1.2.2.3" xref="S3.Thmtheorem4.p1.5.m5.1.1.2.2.3.cmml"><mi id="S3.Thmtheorem4.p1.5.m5.1.1.2.2.3.2" xref="S3.Thmtheorem4.p1.5.m5.1.1.2.2.3.2.cmml">B</mi><mo id="S3.Thmtheorem4.p1.5.m5.1.1.2.2.3.3" xref="S3.Thmtheorem4.p1.5.m5.1.1.2.2.3.3.cmml">′</mo></msup></mrow><mo id="S3.Thmtheorem4.p1.5.m5.1.1.2.1" xref="S3.Thmtheorem4.p1.5.m5.1.1.2.1.cmml">⁢</mo><mi id="S3.Thmtheorem4.p1.5.m5.1.1.2.3" mathvariant="normal" xref="S3.Thmtheorem4.p1.5.m5.1.1.2.3.cmml">⊴</mi><mo id="S3.Thmtheorem4.p1.5.m5.1.1.2.1a" xref="S3.Thmtheorem4.p1.5.m5.1.1.2.1.cmml">⁢</mo><mi id="S3.Thmtheorem4.p1.5.m5.1.1.2.4" xref="S3.Thmtheorem4.p1.5.m5.1.1.2.4.cmml">A</mi></mrow><mo id="S3.Thmtheorem4.p1.5.m5.1.1.1" lspace="0.222em" rspace="0.222em" xref="S3.Thmtheorem4.p1.5.m5.1.1.1.cmml">×</mo><mi id="S3.Thmtheorem4.p1.5.m5.1.1.3" xref="S3.Thmtheorem4.p1.5.m5.1.1.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.5.m5.1b"><apply id="S3.Thmtheorem4.p1.5.m5.1.1.cmml" xref="S3.Thmtheorem4.p1.5.m5.1.1"><times id="S3.Thmtheorem4.p1.5.m5.1.1.1.cmml" xref="S3.Thmtheorem4.p1.5.m5.1.1.1"></times><apply id="S3.Thmtheorem4.p1.5.m5.1.1.2.cmml" xref="S3.Thmtheorem4.p1.5.m5.1.1.2"><times id="S3.Thmtheorem4.p1.5.m5.1.1.2.1.cmml" xref="S3.Thmtheorem4.p1.5.m5.1.1.2.1"></times><apply id="S3.Thmtheorem4.p1.5.m5.1.1.2.2.cmml" xref="S3.Thmtheorem4.p1.5.m5.1.1.2.2"><times id="S3.Thmtheorem4.p1.5.m5.1.1.2.2.1.cmml" xref="S3.Thmtheorem4.p1.5.m5.1.1.2.2.1"></times><apply id="S3.Thmtheorem4.p1.5.m5.1.1.2.2.2.cmml" xref="S3.Thmtheorem4.p1.5.m5.1.1.2.2.2"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p1.5.m5.1.1.2.2.2.1.cmml" xref="S3.Thmtheorem4.p1.5.m5.1.1.2.2.2">superscript</csymbol><ci id="S3.Thmtheorem4.p1.5.m5.1.1.2.2.2.2.cmml" xref="S3.Thmtheorem4.p1.5.m5.1.1.2.2.2.2">𝐴</ci><ci id="S3.Thmtheorem4.p1.5.m5.1.1.2.2.2.3.cmml" xref="S3.Thmtheorem4.p1.5.m5.1.1.2.2.2.3">′</ci></apply><apply id="S3.Thmtheorem4.p1.5.m5.1.1.2.2.3.cmml" xref="S3.Thmtheorem4.p1.5.m5.1.1.2.2.3"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p1.5.m5.1.1.2.2.3.1.cmml" xref="S3.Thmtheorem4.p1.5.m5.1.1.2.2.3">superscript</csymbol><ci id="S3.Thmtheorem4.p1.5.m5.1.1.2.2.3.2.cmml" xref="S3.Thmtheorem4.p1.5.m5.1.1.2.2.3.2">𝐵</ci><ci id="S3.Thmtheorem4.p1.5.m5.1.1.2.2.3.3.cmml" xref="S3.Thmtheorem4.p1.5.m5.1.1.2.2.3.3">′</ci></apply></apply><ci id="S3.Thmtheorem4.p1.5.m5.1.1.2.3.cmml" xref="S3.Thmtheorem4.p1.5.m5.1.1.2.3">⊴</ci><ci id="S3.Thmtheorem4.p1.5.m5.1.1.2.4.cmml" xref="S3.Thmtheorem4.p1.5.m5.1.1.2.4">𝐴</ci></apply><ci id="S3.Thmtheorem4.p1.5.m5.1.1.3.cmml" xref="S3.Thmtheorem4.p1.5.m5.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.5.m5.1c">A^{\prime}\times B^{\prime}\trianglelefteq A\times B</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.5.m5.1d">italic_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT × italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⊴ italic_A × italic_B</annotation></semantics></math>. In particular <math alttext="A\times B" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.6.m6.1"><semantics id="S3.Thmtheorem4.p1.6.m6.1a"><mrow id="S3.Thmtheorem4.p1.6.m6.1.1" xref="S3.Thmtheorem4.p1.6.m6.1.1.cmml"><mi id="S3.Thmtheorem4.p1.6.m6.1.1.2" xref="S3.Thmtheorem4.p1.6.m6.1.1.2.cmml">A</mi><mo id="S3.Thmtheorem4.p1.6.m6.1.1.1" lspace="0.222em" rspace="0.222em" xref="S3.Thmtheorem4.p1.6.m6.1.1.1.cmml">×</mo><mi id="S3.Thmtheorem4.p1.6.m6.1.1.3" xref="S3.Thmtheorem4.p1.6.m6.1.1.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.6.m6.1b"><apply id="S3.Thmtheorem4.p1.6.m6.1.1.cmml" xref="S3.Thmtheorem4.p1.6.m6.1.1"><times id="S3.Thmtheorem4.p1.6.m6.1.1.1.cmml" xref="S3.Thmtheorem4.p1.6.m6.1.1.1"></times><ci id="S3.Thmtheorem4.p1.6.m6.1.1.2.cmml" xref="S3.Thmtheorem4.p1.6.m6.1.1.2">𝐴</ci><ci id="S3.Thmtheorem4.p1.6.m6.1.1.3.cmml" xref="S3.Thmtheorem4.p1.6.m6.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.6.m6.1c">A\times B</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.6.m6.1d">italic_A × italic_B</annotation></semantics></math> is strongly surjective whenever <math alttext="A" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.7.m7.1"><semantics id="S3.Thmtheorem4.p1.7.m7.1a"><mi id="S3.Thmtheorem4.p1.7.m7.1.1" xref="S3.Thmtheorem4.p1.7.m7.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.7.m7.1b"><ci id="S3.Thmtheorem4.p1.7.m7.1.1.cmml" xref="S3.Thmtheorem4.p1.7.m7.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.7.m7.1c">A</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.7.m7.1d">italic_A</annotation></semantics></math> and <math alttext="B" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.8.m8.1"><semantics id="S3.Thmtheorem4.p1.8.m8.1a"><mi id="S3.Thmtheorem4.p1.8.m8.1.1" xref="S3.Thmtheorem4.p1.8.m8.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.8.m8.1b"><ci id="S3.Thmtheorem4.p1.8.m8.1.1.cmml" xref="S3.Thmtheorem4.p1.8.m8.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.8.m8.1c">B</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.8.m8.1d">italic_B</annotation></semantics></math> are.</p> </div> </div> <div class="ltx_theorem ltx_theorem_definition" id="S3.Thmtheorem5"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem5.1.1.1">Definition 3.5</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem5.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem5.p1"> <p class="ltx_p" id="S3.Thmtheorem5.p1.5"><math alttext="L" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.1.m1.1"><semantics id="S3.Thmtheorem5.p1.1.m1.1a"><mi id="S3.Thmtheorem5.p1.1.m1.1.1" xref="S3.Thmtheorem5.p1.1.m1.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.1.m1.1b"><ci id="S3.Thmtheorem5.p1.1.m1.1.1.cmml" xref="S3.Thmtheorem5.p1.1.m1.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.1.m1.1c">L</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.1.m1.1d">italic_L</annotation></semantics></math> is said to be <em class="ltx_emph ltx_font_italic" id="S3.Thmtheorem5.p1.5.1">open strongly surjective</em> if for all nonempty <math alttext="A\subseteq L" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.2.m2.1"><semantics id="S3.Thmtheorem5.p1.2.m2.1a"><mrow id="S3.Thmtheorem5.p1.2.m2.1.1" xref="S3.Thmtheorem5.p1.2.m2.1.1.cmml"><mi id="S3.Thmtheorem5.p1.2.m2.1.1.2" xref="S3.Thmtheorem5.p1.2.m2.1.1.2.cmml">A</mi><mo id="S3.Thmtheorem5.p1.2.m2.1.1.1" xref="S3.Thmtheorem5.p1.2.m2.1.1.1.cmml">⊆</mo><mi id="S3.Thmtheorem5.p1.2.m2.1.1.3" xref="S3.Thmtheorem5.p1.2.m2.1.1.3.cmml">L</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.2.m2.1b"><apply id="S3.Thmtheorem5.p1.2.m2.1.1.cmml" xref="S3.Thmtheorem5.p1.2.m2.1.1"><subset id="S3.Thmtheorem5.p1.2.m2.1.1.1.cmml" xref="S3.Thmtheorem5.p1.2.m2.1.1.1"></subset><ci id="S3.Thmtheorem5.p1.2.m2.1.1.2.cmml" xref="S3.Thmtheorem5.p1.2.m2.1.1.2">𝐴</ci><ci id="S3.Thmtheorem5.p1.2.m2.1.1.3.cmml" xref="S3.Thmtheorem5.p1.2.m2.1.1.3">𝐿</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.2.m2.1c">A\subseteq L</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.2.m2.1d">italic_A ⊆ italic_L</annotation></semantics></math> there is an epimorphism <math alttext="f:L\twoheadrightarrow A" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.3.m3.1"><semantics id="S3.Thmtheorem5.p1.3.m3.1a"><mrow id="S3.Thmtheorem5.p1.3.m3.1.1" xref="S3.Thmtheorem5.p1.3.m3.1.1.cmml"><mi id="S3.Thmtheorem5.p1.3.m3.1.1.2" xref="S3.Thmtheorem5.p1.3.m3.1.1.2.cmml">f</mi><mo id="S3.Thmtheorem5.p1.3.m3.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.Thmtheorem5.p1.3.m3.1.1.1.cmml">:</mo><mrow id="S3.Thmtheorem5.p1.3.m3.1.1.3" xref="S3.Thmtheorem5.p1.3.m3.1.1.3.cmml"><mi id="S3.Thmtheorem5.p1.3.m3.1.1.3.2" xref="S3.Thmtheorem5.p1.3.m3.1.1.3.2.cmml">L</mi><mo id="S3.Thmtheorem5.p1.3.m3.1.1.3.1" stretchy="false" xref="S3.Thmtheorem5.p1.3.m3.1.1.3.1.cmml">↠</mo><mi id="S3.Thmtheorem5.p1.3.m3.1.1.3.3" xref="S3.Thmtheorem5.p1.3.m3.1.1.3.3.cmml">A</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.3.m3.1b"><apply id="S3.Thmtheorem5.p1.3.m3.1.1.cmml" xref="S3.Thmtheorem5.p1.3.m3.1.1"><ci id="S3.Thmtheorem5.p1.3.m3.1.1.1.cmml" xref="S3.Thmtheorem5.p1.3.m3.1.1.1">:</ci><ci id="S3.Thmtheorem5.p1.3.m3.1.1.2.cmml" xref="S3.Thmtheorem5.p1.3.m3.1.1.2">𝑓</ci><apply id="S3.Thmtheorem5.p1.3.m3.1.1.3.cmml" xref="S3.Thmtheorem5.p1.3.m3.1.1.3"><ci id="S3.Thmtheorem5.p1.3.m3.1.1.3.1.cmml" xref="S3.Thmtheorem5.p1.3.m3.1.1.3.1">↠</ci><ci id="S3.Thmtheorem5.p1.3.m3.1.1.3.2.cmml" xref="S3.Thmtheorem5.p1.3.m3.1.1.3.2">𝐿</ci><ci id="S3.Thmtheorem5.p1.3.m3.1.1.3.3.cmml" xref="S3.Thmtheorem5.p1.3.m3.1.1.3.3">𝐴</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.3.m3.1c">f:L\twoheadrightarrow A</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.3.m3.1d">italic_f : italic_L ↠ italic_A</annotation></semantics></math> such that for all <math alttext="a\in A" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.4.m4.1"><semantics id="S3.Thmtheorem5.p1.4.m4.1a"><mrow id="S3.Thmtheorem5.p1.4.m4.1.1" xref="S3.Thmtheorem5.p1.4.m4.1.1.cmml"><mi id="S3.Thmtheorem5.p1.4.m4.1.1.2" xref="S3.Thmtheorem5.p1.4.m4.1.1.2.cmml">a</mi><mo id="S3.Thmtheorem5.p1.4.m4.1.1.1" xref="S3.Thmtheorem5.p1.4.m4.1.1.1.cmml">∈</mo><mi id="S3.Thmtheorem5.p1.4.m4.1.1.3" xref="S3.Thmtheorem5.p1.4.m4.1.1.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.4.m4.1b"><apply id="S3.Thmtheorem5.p1.4.m4.1.1.cmml" xref="S3.Thmtheorem5.p1.4.m4.1.1"><in id="S3.Thmtheorem5.p1.4.m4.1.1.1.cmml" xref="S3.Thmtheorem5.p1.4.m4.1.1.1"></in><ci id="S3.Thmtheorem5.p1.4.m4.1.1.2.cmml" xref="S3.Thmtheorem5.p1.4.m4.1.1.2">𝑎</ci><ci id="S3.Thmtheorem5.p1.4.m4.1.1.3.cmml" xref="S3.Thmtheorem5.p1.4.m4.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.4.m4.1c">a\in A</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.4.m4.1d">italic_a ∈ italic_A</annotation></semantics></math>, <math alttext="f^{-1}(a)" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.5.m5.1"><semantics id="S3.Thmtheorem5.p1.5.m5.1a"><mrow id="S3.Thmtheorem5.p1.5.m5.1.2" xref="S3.Thmtheorem5.p1.5.m5.1.2.cmml"><msup id="S3.Thmtheorem5.p1.5.m5.1.2.2" xref="S3.Thmtheorem5.p1.5.m5.1.2.2.cmml"><mi id="S3.Thmtheorem5.p1.5.m5.1.2.2.2" xref="S3.Thmtheorem5.p1.5.m5.1.2.2.2.cmml">f</mi><mrow id="S3.Thmtheorem5.p1.5.m5.1.2.2.3" xref="S3.Thmtheorem5.p1.5.m5.1.2.2.3.cmml"><mo id="S3.Thmtheorem5.p1.5.m5.1.2.2.3a" xref="S3.Thmtheorem5.p1.5.m5.1.2.2.3.cmml">−</mo><mn id="S3.Thmtheorem5.p1.5.m5.1.2.2.3.2" xref="S3.Thmtheorem5.p1.5.m5.1.2.2.3.2.cmml">1</mn></mrow></msup><mo id="S3.Thmtheorem5.p1.5.m5.1.2.1" xref="S3.Thmtheorem5.p1.5.m5.1.2.1.cmml">⁢</mo><mrow id="S3.Thmtheorem5.p1.5.m5.1.2.3.2" xref="S3.Thmtheorem5.p1.5.m5.1.2.cmml"><mo id="S3.Thmtheorem5.p1.5.m5.1.2.3.2.1" stretchy="false" xref="S3.Thmtheorem5.p1.5.m5.1.2.cmml">(</mo><mi id="S3.Thmtheorem5.p1.5.m5.1.1" xref="S3.Thmtheorem5.p1.5.m5.1.1.cmml">a</mi><mo id="S3.Thmtheorem5.p1.5.m5.1.2.3.2.2" stretchy="false" xref="S3.Thmtheorem5.p1.5.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.5.m5.1b"><apply id="S3.Thmtheorem5.p1.5.m5.1.2.cmml" xref="S3.Thmtheorem5.p1.5.m5.1.2"><times id="S3.Thmtheorem5.p1.5.m5.1.2.1.cmml" xref="S3.Thmtheorem5.p1.5.m5.1.2.1"></times><apply id="S3.Thmtheorem5.p1.5.m5.1.2.2.cmml" xref="S3.Thmtheorem5.p1.5.m5.1.2.2"><csymbol cd="ambiguous" id="S3.Thmtheorem5.p1.5.m5.1.2.2.1.cmml" xref="S3.Thmtheorem5.p1.5.m5.1.2.2">superscript</csymbol><ci id="S3.Thmtheorem5.p1.5.m5.1.2.2.2.cmml" xref="S3.Thmtheorem5.p1.5.m5.1.2.2.2">𝑓</ci><apply id="S3.Thmtheorem5.p1.5.m5.1.2.2.3.cmml" xref="S3.Thmtheorem5.p1.5.m5.1.2.2.3"><minus id="S3.Thmtheorem5.p1.5.m5.1.2.2.3.1.cmml" xref="S3.Thmtheorem5.p1.5.m5.1.2.2.3"></minus><cn id="S3.Thmtheorem5.p1.5.m5.1.2.2.3.2.cmml" type="integer" xref="S3.Thmtheorem5.p1.5.m5.1.2.2.3.2">1</cn></apply></apply><ci id="S3.Thmtheorem5.p1.5.m5.1.1.cmml" xref="S3.Thmtheorem5.p1.5.m5.1.1">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.5.m5.1c">f^{-1}(a)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.5.m5.1d">italic_f start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_a )</annotation></semantics></math> has no endpoints.</p> </div> </div> <div class="ltx_theorem ltx_theorem_definition" id="S3.Thmtheorem6"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem6.1.1.1">Definition 3.6</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem6.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem6.p1"> <p class="ltx_p" id="S3.Thmtheorem6.p1.7">Let <math alttext="I" class="ltx_Math" display="inline" id="S3.Thmtheorem6.p1.1.m1.1"><semantics id="S3.Thmtheorem6.p1.1.m1.1a"><mi id="S3.Thmtheorem6.p1.1.m1.1.1" xref="S3.Thmtheorem6.p1.1.m1.1.1.cmml">I</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem6.p1.1.m1.1b"><ci id="S3.Thmtheorem6.p1.1.m1.1.1.cmml" xref="S3.Thmtheorem6.p1.1.m1.1.1">𝐼</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.1.m1.1c">I</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.1.m1.1d">italic_I</annotation></semantics></math> be any dense linear order, an <em class="ltx_emph ltx_font_italic" id="S3.Thmtheorem6.p1.2.1"><math alttext="I" class="ltx_Math" display="inline" id="S3.Thmtheorem6.p1.2.1.m1.1"><semantics id="S3.Thmtheorem6.p1.2.1.m1.1a"><mi id="S3.Thmtheorem6.p1.2.1.m1.1.1" xref="S3.Thmtheorem6.p1.2.1.m1.1.1.cmml">I</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem6.p1.2.1.m1.1b"><ci id="S3.Thmtheorem6.p1.2.1.m1.1.1.cmml" xref="S3.Thmtheorem6.p1.2.1.m1.1.1">𝐼</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.2.1.m1.1c">I</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.2.1.m1.1d">italic_I</annotation></semantics></math>-mixed sum</em> is any order of the form <math alttext="\sum_{i\in I}A_{i}" class="ltx_Math" display="inline" id="S3.Thmtheorem6.p1.3.m2.1"><semantics id="S3.Thmtheorem6.p1.3.m2.1a"><mrow id="S3.Thmtheorem6.p1.3.m2.1.1" xref="S3.Thmtheorem6.p1.3.m2.1.1.cmml"><msub id="S3.Thmtheorem6.p1.3.m2.1.1.1" xref="S3.Thmtheorem6.p1.3.m2.1.1.1.cmml"><mo id="S3.Thmtheorem6.p1.3.m2.1.1.1.2" xref="S3.Thmtheorem6.p1.3.m2.1.1.1.2.cmml">∑</mo><mrow id="S3.Thmtheorem6.p1.3.m2.1.1.1.3" xref="S3.Thmtheorem6.p1.3.m2.1.1.1.3.cmml"><mi id="S3.Thmtheorem6.p1.3.m2.1.1.1.3.2" xref="S3.Thmtheorem6.p1.3.m2.1.1.1.3.2.cmml">i</mi><mo id="S3.Thmtheorem6.p1.3.m2.1.1.1.3.1" xref="S3.Thmtheorem6.p1.3.m2.1.1.1.3.1.cmml">∈</mo><mi id="S3.Thmtheorem6.p1.3.m2.1.1.1.3.3" xref="S3.Thmtheorem6.p1.3.m2.1.1.1.3.3.cmml">I</mi></mrow></msub><msub id="S3.Thmtheorem6.p1.3.m2.1.1.2" xref="S3.Thmtheorem6.p1.3.m2.1.1.2.cmml"><mi id="S3.Thmtheorem6.p1.3.m2.1.1.2.2" xref="S3.Thmtheorem6.p1.3.m2.1.1.2.2.cmml">A</mi><mi id="S3.Thmtheorem6.p1.3.m2.1.1.2.3" xref="S3.Thmtheorem6.p1.3.m2.1.1.2.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem6.p1.3.m2.1b"><apply id="S3.Thmtheorem6.p1.3.m2.1.1.cmml" xref="S3.Thmtheorem6.p1.3.m2.1.1"><apply id="S3.Thmtheorem6.p1.3.m2.1.1.1.cmml" xref="S3.Thmtheorem6.p1.3.m2.1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem6.p1.3.m2.1.1.1.1.cmml" xref="S3.Thmtheorem6.p1.3.m2.1.1.1">subscript</csymbol><sum id="S3.Thmtheorem6.p1.3.m2.1.1.1.2.cmml" xref="S3.Thmtheorem6.p1.3.m2.1.1.1.2"></sum><apply id="S3.Thmtheorem6.p1.3.m2.1.1.1.3.cmml" xref="S3.Thmtheorem6.p1.3.m2.1.1.1.3"><in id="S3.Thmtheorem6.p1.3.m2.1.1.1.3.1.cmml" xref="S3.Thmtheorem6.p1.3.m2.1.1.1.3.1"></in><ci id="S3.Thmtheorem6.p1.3.m2.1.1.1.3.2.cmml" xref="S3.Thmtheorem6.p1.3.m2.1.1.1.3.2">𝑖</ci><ci id="S3.Thmtheorem6.p1.3.m2.1.1.1.3.3.cmml" xref="S3.Thmtheorem6.p1.3.m2.1.1.1.3.3">𝐼</ci></apply></apply><apply id="S3.Thmtheorem6.p1.3.m2.1.1.2.cmml" xref="S3.Thmtheorem6.p1.3.m2.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem6.p1.3.m2.1.1.2.1.cmml" xref="S3.Thmtheorem6.p1.3.m2.1.1.2">subscript</csymbol><ci id="S3.Thmtheorem6.p1.3.m2.1.1.2.2.cmml" xref="S3.Thmtheorem6.p1.3.m2.1.1.2.2">𝐴</ci><ci id="S3.Thmtheorem6.p1.3.m2.1.1.2.3.cmml" xref="S3.Thmtheorem6.p1.3.m2.1.1.2.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.3.m2.1c">\sum_{i\in I}A_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.3.m2.1d">∑ start_POSTSUBSCRIPT italic_i ∈ italic_I end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> where each <math alttext="A_{i}" class="ltx_Math" display="inline" id="S3.Thmtheorem6.p1.4.m3.1"><semantics id="S3.Thmtheorem6.p1.4.m3.1a"><msub id="S3.Thmtheorem6.p1.4.m3.1.1" xref="S3.Thmtheorem6.p1.4.m3.1.1.cmml"><mi id="S3.Thmtheorem6.p1.4.m3.1.1.2" xref="S3.Thmtheorem6.p1.4.m3.1.1.2.cmml">A</mi><mi id="S3.Thmtheorem6.p1.4.m3.1.1.3" xref="S3.Thmtheorem6.p1.4.m3.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem6.p1.4.m3.1b"><apply id="S3.Thmtheorem6.p1.4.m3.1.1.cmml" xref="S3.Thmtheorem6.p1.4.m3.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem6.p1.4.m3.1.1.1.cmml" xref="S3.Thmtheorem6.p1.4.m3.1.1">subscript</csymbol><ci id="S3.Thmtheorem6.p1.4.m3.1.1.2.cmml" xref="S3.Thmtheorem6.p1.4.m3.1.1.2">𝐴</ci><ci id="S3.Thmtheorem6.p1.4.m3.1.1.3.cmml" xref="S3.Thmtheorem6.p1.4.m3.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.4.m3.1c">A_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.4.m3.1d">italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> is nonempty and such that for all <math alttext="i\in I" class="ltx_Math" display="inline" id="S3.Thmtheorem6.p1.5.m4.1"><semantics id="S3.Thmtheorem6.p1.5.m4.1a"><mrow id="S3.Thmtheorem6.p1.5.m4.1.1" xref="S3.Thmtheorem6.p1.5.m4.1.1.cmml"><mi id="S3.Thmtheorem6.p1.5.m4.1.1.2" xref="S3.Thmtheorem6.p1.5.m4.1.1.2.cmml">i</mi><mo id="S3.Thmtheorem6.p1.5.m4.1.1.1" xref="S3.Thmtheorem6.p1.5.m4.1.1.1.cmml">∈</mo><mi id="S3.Thmtheorem6.p1.5.m4.1.1.3" xref="S3.Thmtheorem6.p1.5.m4.1.1.3.cmml">I</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem6.p1.5.m4.1b"><apply id="S3.Thmtheorem6.p1.5.m4.1.1.cmml" xref="S3.Thmtheorem6.p1.5.m4.1.1"><in id="S3.Thmtheorem6.p1.5.m4.1.1.1.cmml" xref="S3.Thmtheorem6.p1.5.m4.1.1.1"></in><ci id="S3.Thmtheorem6.p1.5.m4.1.1.2.cmml" xref="S3.Thmtheorem6.p1.5.m4.1.1.2">𝑖</ci><ci id="S3.Thmtheorem6.p1.5.m4.1.1.3.cmml" xref="S3.Thmtheorem6.p1.5.m4.1.1.3">𝐼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.5.m4.1c">i\in I</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.5.m4.1d">italic_i ∈ italic_I</annotation></semantics></math>, <math alttext="\{j\in I:A_{i}=A_{j}\}" class="ltx_Math" display="inline" id="S3.Thmtheorem6.p1.6.m5.2"><semantics id="S3.Thmtheorem6.p1.6.m5.2a"><mrow id="S3.Thmtheorem6.p1.6.m5.2.2.2" xref="S3.Thmtheorem6.p1.6.m5.2.2.3.cmml"><mo id="S3.Thmtheorem6.p1.6.m5.2.2.2.3" stretchy="false" xref="S3.Thmtheorem6.p1.6.m5.2.2.3.1.cmml">{</mo><mrow id="S3.Thmtheorem6.p1.6.m5.1.1.1.1" xref="S3.Thmtheorem6.p1.6.m5.1.1.1.1.cmml"><mi id="S3.Thmtheorem6.p1.6.m5.1.1.1.1.2" xref="S3.Thmtheorem6.p1.6.m5.1.1.1.1.2.cmml">j</mi><mo id="S3.Thmtheorem6.p1.6.m5.1.1.1.1.1" xref="S3.Thmtheorem6.p1.6.m5.1.1.1.1.1.cmml">∈</mo><mi id="S3.Thmtheorem6.p1.6.m5.1.1.1.1.3" xref="S3.Thmtheorem6.p1.6.m5.1.1.1.1.3.cmml">I</mi></mrow><mo id="S3.Thmtheorem6.p1.6.m5.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S3.Thmtheorem6.p1.6.m5.2.2.3.1.cmml">:</mo><mrow id="S3.Thmtheorem6.p1.6.m5.2.2.2.2" xref="S3.Thmtheorem6.p1.6.m5.2.2.2.2.cmml"><msub id="S3.Thmtheorem6.p1.6.m5.2.2.2.2.2" xref="S3.Thmtheorem6.p1.6.m5.2.2.2.2.2.cmml"><mi id="S3.Thmtheorem6.p1.6.m5.2.2.2.2.2.2" xref="S3.Thmtheorem6.p1.6.m5.2.2.2.2.2.2.cmml">A</mi><mi id="S3.Thmtheorem6.p1.6.m5.2.2.2.2.2.3" xref="S3.Thmtheorem6.p1.6.m5.2.2.2.2.2.3.cmml">i</mi></msub><mo id="S3.Thmtheorem6.p1.6.m5.2.2.2.2.1" xref="S3.Thmtheorem6.p1.6.m5.2.2.2.2.1.cmml">=</mo><msub id="S3.Thmtheorem6.p1.6.m5.2.2.2.2.3" xref="S3.Thmtheorem6.p1.6.m5.2.2.2.2.3.cmml"><mi id="S3.Thmtheorem6.p1.6.m5.2.2.2.2.3.2" xref="S3.Thmtheorem6.p1.6.m5.2.2.2.2.3.2.cmml">A</mi><mi id="S3.Thmtheorem6.p1.6.m5.2.2.2.2.3.3" xref="S3.Thmtheorem6.p1.6.m5.2.2.2.2.3.3.cmml">j</mi></msub></mrow><mo id="S3.Thmtheorem6.p1.6.m5.2.2.2.5" stretchy="false" xref="S3.Thmtheorem6.p1.6.m5.2.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem6.p1.6.m5.2b"><apply id="S3.Thmtheorem6.p1.6.m5.2.2.3.cmml" xref="S3.Thmtheorem6.p1.6.m5.2.2.2"><csymbol cd="latexml" id="S3.Thmtheorem6.p1.6.m5.2.2.3.1.cmml" xref="S3.Thmtheorem6.p1.6.m5.2.2.2.3">conditional-set</csymbol><apply id="S3.Thmtheorem6.p1.6.m5.1.1.1.1.cmml" xref="S3.Thmtheorem6.p1.6.m5.1.1.1.1"><in id="S3.Thmtheorem6.p1.6.m5.1.1.1.1.1.cmml" xref="S3.Thmtheorem6.p1.6.m5.1.1.1.1.1"></in><ci id="S3.Thmtheorem6.p1.6.m5.1.1.1.1.2.cmml" xref="S3.Thmtheorem6.p1.6.m5.1.1.1.1.2">𝑗</ci><ci id="S3.Thmtheorem6.p1.6.m5.1.1.1.1.3.cmml" xref="S3.Thmtheorem6.p1.6.m5.1.1.1.1.3">𝐼</ci></apply><apply id="S3.Thmtheorem6.p1.6.m5.2.2.2.2.cmml" xref="S3.Thmtheorem6.p1.6.m5.2.2.2.2"><eq id="S3.Thmtheorem6.p1.6.m5.2.2.2.2.1.cmml" xref="S3.Thmtheorem6.p1.6.m5.2.2.2.2.1"></eq><apply id="S3.Thmtheorem6.p1.6.m5.2.2.2.2.2.cmml" xref="S3.Thmtheorem6.p1.6.m5.2.2.2.2.2"><csymbol cd="ambiguous" id="S3.Thmtheorem6.p1.6.m5.2.2.2.2.2.1.cmml" xref="S3.Thmtheorem6.p1.6.m5.2.2.2.2.2">subscript</csymbol><ci id="S3.Thmtheorem6.p1.6.m5.2.2.2.2.2.2.cmml" xref="S3.Thmtheorem6.p1.6.m5.2.2.2.2.2.2">𝐴</ci><ci id="S3.Thmtheorem6.p1.6.m5.2.2.2.2.2.3.cmml" xref="S3.Thmtheorem6.p1.6.m5.2.2.2.2.2.3">𝑖</ci></apply><apply id="S3.Thmtheorem6.p1.6.m5.2.2.2.2.3.cmml" xref="S3.Thmtheorem6.p1.6.m5.2.2.2.2.3"><csymbol cd="ambiguous" id="S3.Thmtheorem6.p1.6.m5.2.2.2.2.3.1.cmml" xref="S3.Thmtheorem6.p1.6.m5.2.2.2.2.3">subscript</csymbol><ci id="S3.Thmtheorem6.p1.6.m5.2.2.2.2.3.2.cmml" xref="S3.Thmtheorem6.p1.6.m5.2.2.2.2.3.2">𝐴</ci><ci id="S3.Thmtheorem6.p1.6.m5.2.2.2.2.3.3.cmml" xref="S3.Thmtheorem6.p1.6.m5.2.2.2.2.3.3">𝑗</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.6.m5.2c">\{j\in I:A_{i}=A_{j}\}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.6.m5.2d">{ italic_j ∈ italic_I : italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_A start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT }</annotation></semantics></math> is dense in <math alttext="I" class="ltx_Math" display="inline" id="S3.Thmtheorem6.p1.7.m6.1"><semantics id="S3.Thmtheorem6.p1.7.m6.1a"><mi id="S3.Thmtheorem6.p1.7.m6.1.1" xref="S3.Thmtheorem6.p1.7.m6.1.1.cmml">I</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem6.p1.7.m6.1b"><ci id="S3.Thmtheorem6.p1.7.m6.1.1.cmml" xref="S3.Thmtheorem6.p1.7.m6.1.1">𝐼</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.7.m6.1c">I</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.7.m6.1d">italic_I</annotation></semantics></math>.</p> </div> </div> <div class="ltx_para" id="S3.p5"> <p class="ltx_p" id="S3.p5.1">The following lemma is already present in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib19" title="">19</a>]</cite> for the case <math alttext="I=\mathbb{Q}" class="ltx_Math" display="inline" id="S3.p5.1.m1.1"><semantics id="S3.p5.1.m1.1a"><mrow id="S3.p5.1.m1.1.1" xref="S3.p5.1.m1.1.1.cmml"><mi id="S3.p5.1.m1.1.1.2" xref="S3.p5.1.m1.1.1.2.cmml">I</mi><mo id="S3.p5.1.m1.1.1.1" xref="S3.p5.1.m1.1.1.1.cmml">=</mo><mi id="S3.p5.1.m1.1.1.3" xref="S3.p5.1.m1.1.1.3.cmml">ℚ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.1.m1.1b"><apply id="S3.p5.1.m1.1.1.cmml" xref="S3.p5.1.m1.1.1"><eq id="S3.p5.1.m1.1.1.1.cmml" xref="S3.p5.1.m1.1.1.1"></eq><ci id="S3.p5.1.m1.1.1.2.cmml" xref="S3.p5.1.m1.1.1.2">𝐼</ci><ci id="S3.p5.1.m1.1.1.3.cmml" xref="S3.p5.1.m1.1.1.3">ℚ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.1.m1.1c">I=\mathbb{Q}</annotation><annotation encoding="application/x-llamapun" id="S3.p5.1.m1.1d">italic_I = blackboard_Q</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S3.Thmtheorem7"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem7.1.1.1">Lemma 3.7</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem7.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem7.p1"> <p class="ltx_p" id="S3.Thmtheorem7.p1.2">If <math alttext="I" class="ltx_Math" display="inline" id="S3.Thmtheorem7.p1.1.m1.1"><semantics id="S3.Thmtheorem7.p1.1.m1.1a"><mi id="S3.Thmtheorem7.p1.1.m1.1.1" xref="S3.Thmtheorem7.p1.1.m1.1.1.cmml">I</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem7.p1.1.m1.1b"><ci id="S3.Thmtheorem7.p1.1.m1.1.1.cmml" xref="S3.Thmtheorem7.p1.1.m1.1.1">𝐼</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem7.p1.1.m1.1c">I</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem7.p1.1.m1.1d">italic_I</annotation></semantics></math> is an open strongly surjective dense order without endpoints, then any <math alttext="I" class="ltx_Math" display="inline" id="S3.Thmtheorem7.p1.2.m2.1"><semantics id="S3.Thmtheorem7.p1.2.m2.1a"><mi id="S3.Thmtheorem7.p1.2.m2.1.1" xref="S3.Thmtheorem7.p1.2.m2.1.1.cmml">I</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem7.p1.2.m2.1b"><ci id="S3.Thmtheorem7.p1.2.m2.1.1.cmml" xref="S3.Thmtheorem7.p1.2.m2.1.1">𝐼</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem7.p1.2.m2.1c">I</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem7.p1.2.m2.1d">italic_I</annotation></semantics></math>-mixed sum of nonempty strongly surjective orders is again strongly surjective.</p> </div> </div> <div class="ltx_proof" id="S3.7"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S3.4.p1"> <p class="ltx_p" id="S3.4.p1.15">Let <math alttext="A=\sum_{i\in I}A_{i}" class="ltx_Math" display="inline" id="S3.4.p1.1.m1.1"><semantics id="S3.4.p1.1.m1.1a"><mrow id="S3.4.p1.1.m1.1.1" xref="S3.4.p1.1.m1.1.1.cmml"><mi id="S3.4.p1.1.m1.1.1.2" xref="S3.4.p1.1.m1.1.1.2.cmml">A</mi><mo id="S3.4.p1.1.m1.1.1.1" rspace="0.111em" xref="S3.4.p1.1.m1.1.1.1.cmml">=</mo><mrow id="S3.4.p1.1.m1.1.1.3" xref="S3.4.p1.1.m1.1.1.3.cmml"><msub id="S3.4.p1.1.m1.1.1.3.1" xref="S3.4.p1.1.m1.1.1.3.1.cmml"><mo id="S3.4.p1.1.m1.1.1.3.1.2" xref="S3.4.p1.1.m1.1.1.3.1.2.cmml">∑</mo><mrow id="S3.4.p1.1.m1.1.1.3.1.3" xref="S3.4.p1.1.m1.1.1.3.1.3.cmml"><mi id="S3.4.p1.1.m1.1.1.3.1.3.2" xref="S3.4.p1.1.m1.1.1.3.1.3.2.cmml">i</mi><mo id="S3.4.p1.1.m1.1.1.3.1.3.1" xref="S3.4.p1.1.m1.1.1.3.1.3.1.cmml">∈</mo><mi id="S3.4.p1.1.m1.1.1.3.1.3.3" xref="S3.4.p1.1.m1.1.1.3.1.3.3.cmml">I</mi></mrow></msub><msub id="S3.4.p1.1.m1.1.1.3.2" xref="S3.4.p1.1.m1.1.1.3.2.cmml"><mi id="S3.4.p1.1.m1.1.1.3.2.2" xref="S3.4.p1.1.m1.1.1.3.2.2.cmml">A</mi><mi id="S3.4.p1.1.m1.1.1.3.2.3" xref="S3.4.p1.1.m1.1.1.3.2.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.4.p1.1.m1.1b"><apply id="S3.4.p1.1.m1.1.1.cmml" xref="S3.4.p1.1.m1.1.1"><eq id="S3.4.p1.1.m1.1.1.1.cmml" xref="S3.4.p1.1.m1.1.1.1"></eq><ci id="S3.4.p1.1.m1.1.1.2.cmml" xref="S3.4.p1.1.m1.1.1.2">𝐴</ci><apply id="S3.4.p1.1.m1.1.1.3.cmml" xref="S3.4.p1.1.m1.1.1.3"><apply id="S3.4.p1.1.m1.1.1.3.1.cmml" xref="S3.4.p1.1.m1.1.1.3.1"><csymbol cd="ambiguous" id="S3.4.p1.1.m1.1.1.3.1.1.cmml" xref="S3.4.p1.1.m1.1.1.3.1">subscript</csymbol><sum id="S3.4.p1.1.m1.1.1.3.1.2.cmml" xref="S3.4.p1.1.m1.1.1.3.1.2"></sum><apply id="S3.4.p1.1.m1.1.1.3.1.3.cmml" xref="S3.4.p1.1.m1.1.1.3.1.3"><in id="S3.4.p1.1.m1.1.1.3.1.3.1.cmml" xref="S3.4.p1.1.m1.1.1.3.1.3.1"></in><ci id="S3.4.p1.1.m1.1.1.3.1.3.2.cmml" xref="S3.4.p1.1.m1.1.1.3.1.3.2">𝑖</ci><ci id="S3.4.p1.1.m1.1.1.3.1.3.3.cmml" xref="S3.4.p1.1.m1.1.1.3.1.3.3">𝐼</ci></apply></apply><apply id="S3.4.p1.1.m1.1.1.3.2.cmml" xref="S3.4.p1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S3.4.p1.1.m1.1.1.3.2.1.cmml" xref="S3.4.p1.1.m1.1.1.3.2">subscript</csymbol><ci id="S3.4.p1.1.m1.1.1.3.2.2.cmml" xref="S3.4.p1.1.m1.1.1.3.2.2">𝐴</ci><ci id="S3.4.p1.1.m1.1.1.3.2.3.cmml" xref="S3.4.p1.1.m1.1.1.3.2.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p1.1.m1.1c">A=\sum_{i\in I}A_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.4.p1.1.m1.1d">italic_A = ∑ start_POSTSUBSCRIPT italic_i ∈ italic_I end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> be the mixed sum and let <math alttext="B\subseteq A" class="ltx_Math" display="inline" id="S3.4.p1.2.m2.1"><semantics id="S3.4.p1.2.m2.1a"><mrow id="S3.4.p1.2.m2.1.1" xref="S3.4.p1.2.m2.1.1.cmml"><mi id="S3.4.p1.2.m2.1.1.2" xref="S3.4.p1.2.m2.1.1.2.cmml">B</mi><mo id="S3.4.p1.2.m2.1.1.1" xref="S3.4.p1.2.m2.1.1.1.cmml">⊆</mo><mi id="S3.4.p1.2.m2.1.1.3" xref="S3.4.p1.2.m2.1.1.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.4.p1.2.m2.1b"><apply id="S3.4.p1.2.m2.1.1.cmml" xref="S3.4.p1.2.m2.1.1"><subset id="S3.4.p1.2.m2.1.1.1.cmml" xref="S3.4.p1.2.m2.1.1.1"></subset><ci id="S3.4.p1.2.m2.1.1.2.cmml" xref="S3.4.p1.2.m2.1.1.2">𝐵</ci><ci id="S3.4.p1.2.m2.1.1.3.cmml" xref="S3.4.p1.2.m2.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p1.2.m2.1c">B\subseteq A</annotation><annotation encoding="application/x-llamapun" id="S3.4.p1.2.m2.1d">italic_B ⊆ italic_A</annotation></semantics></math> be nonempty, we show that <math alttext="A\trianglerighteq B" class="ltx_Math" display="inline" id="S3.4.p1.3.m3.1"><semantics id="S3.4.p1.3.m3.1a"><mrow id="S3.4.p1.3.m3.1.1" xref="S3.4.p1.3.m3.1.1.cmml"><mi id="S3.4.p1.3.m3.1.1.2" xref="S3.4.p1.3.m3.1.1.2.cmml">A</mi><mo id="S3.4.p1.3.m3.1.1.1" xref="S3.4.p1.3.m3.1.1.1.cmml">⁢</mo><mi id="S3.4.p1.3.m3.1.1.3" mathvariant="normal" xref="S3.4.p1.3.m3.1.1.3.cmml">⊵</mi><mo id="S3.4.p1.3.m3.1.1.1a" xref="S3.4.p1.3.m3.1.1.1.cmml">⁢</mo><mi id="S3.4.p1.3.m3.1.1.4" xref="S3.4.p1.3.m3.1.1.4.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.4.p1.3.m3.1b"><apply id="S3.4.p1.3.m3.1.1.cmml" xref="S3.4.p1.3.m3.1.1"><times id="S3.4.p1.3.m3.1.1.1.cmml" xref="S3.4.p1.3.m3.1.1.1"></times><ci id="S3.4.p1.3.m3.1.1.2.cmml" xref="S3.4.p1.3.m3.1.1.2">𝐴</ci><ci id="S3.4.p1.3.m3.1.1.3.cmml" xref="S3.4.p1.3.m3.1.1.3">⊵</ci><ci id="S3.4.p1.3.m3.1.1.4.cmml" xref="S3.4.p1.3.m3.1.1.4">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p1.3.m3.1c">A\trianglerighteq B</annotation><annotation encoding="application/x-llamapun" id="S3.4.p1.3.m3.1d">italic_A ⊵ italic_B</annotation></semantics></math>. First note that <math alttext="B" class="ltx_Math" display="inline" id="S3.4.p1.4.m4.1"><semantics id="S3.4.p1.4.m4.1a"><mi id="S3.4.p1.4.m4.1.1" xref="S3.4.p1.4.m4.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S3.4.p1.4.m4.1b"><ci id="S3.4.p1.4.m4.1.1.cmml" xref="S3.4.p1.4.m4.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p1.4.m4.1c">B</annotation><annotation encoding="application/x-llamapun" id="S3.4.p1.4.m4.1d">italic_B</annotation></semantics></math> can be written as <math alttext="\sum_{i\in I^{\prime}}A_{i}^{\prime}" class="ltx_Math" display="inline" id="S3.4.p1.5.m5.1"><semantics id="S3.4.p1.5.m5.1a"><mrow id="S3.4.p1.5.m5.1.1" xref="S3.4.p1.5.m5.1.1.cmml"><msub id="S3.4.p1.5.m5.1.1.1" xref="S3.4.p1.5.m5.1.1.1.cmml"><mo id="S3.4.p1.5.m5.1.1.1.2" xref="S3.4.p1.5.m5.1.1.1.2.cmml">∑</mo><mrow id="S3.4.p1.5.m5.1.1.1.3" xref="S3.4.p1.5.m5.1.1.1.3.cmml"><mi id="S3.4.p1.5.m5.1.1.1.3.2" xref="S3.4.p1.5.m5.1.1.1.3.2.cmml">i</mi><mo id="S3.4.p1.5.m5.1.1.1.3.1" xref="S3.4.p1.5.m5.1.1.1.3.1.cmml">∈</mo><msup id="S3.4.p1.5.m5.1.1.1.3.3" xref="S3.4.p1.5.m5.1.1.1.3.3.cmml"><mi id="S3.4.p1.5.m5.1.1.1.3.3.2" xref="S3.4.p1.5.m5.1.1.1.3.3.2.cmml">I</mi><mo id="S3.4.p1.5.m5.1.1.1.3.3.3" xref="S3.4.p1.5.m5.1.1.1.3.3.3.cmml">′</mo></msup></mrow></msub><msubsup id="S3.4.p1.5.m5.1.1.2" xref="S3.4.p1.5.m5.1.1.2.cmml"><mi id="S3.4.p1.5.m5.1.1.2.2.2" xref="S3.4.p1.5.m5.1.1.2.2.2.cmml">A</mi><mi id="S3.4.p1.5.m5.1.1.2.2.3" xref="S3.4.p1.5.m5.1.1.2.2.3.cmml">i</mi><mo id="S3.4.p1.5.m5.1.1.2.3" xref="S3.4.p1.5.m5.1.1.2.3.cmml">′</mo></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S3.4.p1.5.m5.1b"><apply id="S3.4.p1.5.m5.1.1.cmml" xref="S3.4.p1.5.m5.1.1"><apply id="S3.4.p1.5.m5.1.1.1.cmml" xref="S3.4.p1.5.m5.1.1.1"><csymbol cd="ambiguous" id="S3.4.p1.5.m5.1.1.1.1.cmml" xref="S3.4.p1.5.m5.1.1.1">subscript</csymbol><sum id="S3.4.p1.5.m5.1.1.1.2.cmml" xref="S3.4.p1.5.m5.1.1.1.2"></sum><apply id="S3.4.p1.5.m5.1.1.1.3.cmml" xref="S3.4.p1.5.m5.1.1.1.3"><in id="S3.4.p1.5.m5.1.1.1.3.1.cmml" xref="S3.4.p1.5.m5.1.1.1.3.1"></in><ci id="S3.4.p1.5.m5.1.1.1.3.2.cmml" xref="S3.4.p1.5.m5.1.1.1.3.2">𝑖</ci><apply id="S3.4.p1.5.m5.1.1.1.3.3.cmml" xref="S3.4.p1.5.m5.1.1.1.3.3"><csymbol cd="ambiguous" id="S3.4.p1.5.m5.1.1.1.3.3.1.cmml" xref="S3.4.p1.5.m5.1.1.1.3.3">superscript</csymbol><ci id="S3.4.p1.5.m5.1.1.1.3.3.2.cmml" xref="S3.4.p1.5.m5.1.1.1.3.3.2">𝐼</ci><ci id="S3.4.p1.5.m5.1.1.1.3.3.3.cmml" xref="S3.4.p1.5.m5.1.1.1.3.3.3">′</ci></apply></apply></apply><apply id="S3.4.p1.5.m5.1.1.2.cmml" xref="S3.4.p1.5.m5.1.1.2"><csymbol cd="ambiguous" id="S3.4.p1.5.m5.1.1.2.1.cmml" xref="S3.4.p1.5.m5.1.1.2">superscript</csymbol><apply id="S3.4.p1.5.m5.1.1.2.2.cmml" xref="S3.4.p1.5.m5.1.1.2"><csymbol cd="ambiguous" id="S3.4.p1.5.m5.1.1.2.2.1.cmml" xref="S3.4.p1.5.m5.1.1.2">subscript</csymbol><ci id="S3.4.p1.5.m5.1.1.2.2.2.cmml" xref="S3.4.p1.5.m5.1.1.2.2.2">𝐴</ci><ci id="S3.4.p1.5.m5.1.1.2.2.3.cmml" xref="S3.4.p1.5.m5.1.1.2.2.3">𝑖</ci></apply><ci id="S3.4.p1.5.m5.1.1.2.3.cmml" xref="S3.4.p1.5.m5.1.1.2.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p1.5.m5.1c">\sum_{i\in I^{\prime}}A_{i}^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.4.p1.5.m5.1d">∑ start_POSTSUBSCRIPT italic_i ∈ italic_I start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> where <math alttext="I^{\prime}\subseteq I" class="ltx_Math" display="inline" id="S3.4.p1.6.m6.1"><semantics id="S3.4.p1.6.m6.1a"><mrow id="S3.4.p1.6.m6.1.1" xref="S3.4.p1.6.m6.1.1.cmml"><msup id="S3.4.p1.6.m6.1.1.2" xref="S3.4.p1.6.m6.1.1.2.cmml"><mi id="S3.4.p1.6.m6.1.1.2.2" xref="S3.4.p1.6.m6.1.1.2.2.cmml">I</mi><mo id="S3.4.p1.6.m6.1.1.2.3" xref="S3.4.p1.6.m6.1.1.2.3.cmml">′</mo></msup><mo id="S3.4.p1.6.m6.1.1.1" xref="S3.4.p1.6.m6.1.1.1.cmml">⊆</mo><mi id="S3.4.p1.6.m6.1.1.3" xref="S3.4.p1.6.m6.1.1.3.cmml">I</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.4.p1.6.m6.1b"><apply id="S3.4.p1.6.m6.1.1.cmml" xref="S3.4.p1.6.m6.1.1"><subset id="S3.4.p1.6.m6.1.1.1.cmml" xref="S3.4.p1.6.m6.1.1.1"></subset><apply id="S3.4.p1.6.m6.1.1.2.cmml" xref="S3.4.p1.6.m6.1.1.2"><csymbol cd="ambiguous" id="S3.4.p1.6.m6.1.1.2.1.cmml" xref="S3.4.p1.6.m6.1.1.2">superscript</csymbol><ci id="S3.4.p1.6.m6.1.1.2.2.cmml" xref="S3.4.p1.6.m6.1.1.2.2">𝐼</ci><ci id="S3.4.p1.6.m6.1.1.2.3.cmml" xref="S3.4.p1.6.m6.1.1.2.3">′</ci></apply><ci id="S3.4.p1.6.m6.1.1.3.cmml" xref="S3.4.p1.6.m6.1.1.3">𝐼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p1.6.m6.1c">I^{\prime}\subseteq I</annotation><annotation encoding="application/x-llamapun" id="S3.4.p1.6.m6.1d">italic_I start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⊆ italic_I</annotation></semantics></math> and <math alttext="A_{i}^{\prime}\subseteq A_{i}" class="ltx_Math" display="inline" id="S3.4.p1.7.m7.1"><semantics id="S3.4.p1.7.m7.1a"><mrow id="S3.4.p1.7.m7.1.1" xref="S3.4.p1.7.m7.1.1.cmml"><msubsup id="S3.4.p1.7.m7.1.1.2" xref="S3.4.p1.7.m7.1.1.2.cmml"><mi id="S3.4.p1.7.m7.1.1.2.2.2" xref="S3.4.p1.7.m7.1.1.2.2.2.cmml">A</mi><mi id="S3.4.p1.7.m7.1.1.2.2.3" xref="S3.4.p1.7.m7.1.1.2.2.3.cmml">i</mi><mo id="S3.4.p1.7.m7.1.1.2.3" xref="S3.4.p1.7.m7.1.1.2.3.cmml">′</mo></msubsup><mo id="S3.4.p1.7.m7.1.1.1" xref="S3.4.p1.7.m7.1.1.1.cmml">⊆</mo><msub id="S3.4.p1.7.m7.1.1.3" xref="S3.4.p1.7.m7.1.1.3.cmml"><mi id="S3.4.p1.7.m7.1.1.3.2" xref="S3.4.p1.7.m7.1.1.3.2.cmml">A</mi><mi id="S3.4.p1.7.m7.1.1.3.3" xref="S3.4.p1.7.m7.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.4.p1.7.m7.1b"><apply id="S3.4.p1.7.m7.1.1.cmml" xref="S3.4.p1.7.m7.1.1"><subset id="S3.4.p1.7.m7.1.1.1.cmml" xref="S3.4.p1.7.m7.1.1.1"></subset><apply id="S3.4.p1.7.m7.1.1.2.cmml" xref="S3.4.p1.7.m7.1.1.2"><csymbol cd="ambiguous" id="S3.4.p1.7.m7.1.1.2.1.cmml" xref="S3.4.p1.7.m7.1.1.2">superscript</csymbol><apply id="S3.4.p1.7.m7.1.1.2.2.cmml" xref="S3.4.p1.7.m7.1.1.2"><csymbol cd="ambiguous" id="S3.4.p1.7.m7.1.1.2.2.1.cmml" xref="S3.4.p1.7.m7.1.1.2">subscript</csymbol><ci id="S3.4.p1.7.m7.1.1.2.2.2.cmml" xref="S3.4.p1.7.m7.1.1.2.2.2">𝐴</ci><ci id="S3.4.p1.7.m7.1.1.2.2.3.cmml" xref="S3.4.p1.7.m7.1.1.2.2.3">𝑖</ci></apply><ci id="S3.4.p1.7.m7.1.1.2.3.cmml" xref="S3.4.p1.7.m7.1.1.2.3">′</ci></apply><apply id="S3.4.p1.7.m7.1.1.3.cmml" xref="S3.4.p1.7.m7.1.1.3"><csymbol cd="ambiguous" id="S3.4.p1.7.m7.1.1.3.1.cmml" xref="S3.4.p1.7.m7.1.1.3">subscript</csymbol><ci id="S3.4.p1.7.m7.1.1.3.2.cmml" xref="S3.4.p1.7.m7.1.1.3.2">𝐴</ci><ci id="S3.4.p1.7.m7.1.1.3.3.cmml" xref="S3.4.p1.7.m7.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p1.7.m7.1c">A_{i}^{\prime}\subseteq A_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.4.p1.7.m7.1d">italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⊆ italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> are all nonempty. Since each <math alttext="A_{i}" class="ltx_Math" display="inline" id="S3.4.p1.8.m8.1"><semantics id="S3.4.p1.8.m8.1a"><msub id="S3.4.p1.8.m8.1.1" xref="S3.4.p1.8.m8.1.1.cmml"><mi id="S3.4.p1.8.m8.1.1.2" xref="S3.4.p1.8.m8.1.1.2.cmml">A</mi><mi id="S3.4.p1.8.m8.1.1.3" xref="S3.4.p1.8.m8.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.4.p1.8.m8.1b"><apply id="S3.4.p1.8.m8.1.1.cmml" xref="S3.4.p1.8.m8.1.1"><csymbol cd="ambiguous" id="S3.4.p1.8.m8.1.1.1.cmml" xref="S3.4.p1.8.m8.1.1">subscript</csymbol><ci id="S3.4.p1.8.m8.1.1.2.cmml" xref="S3.4.p1.8.m8.1.1.2">𝐴</ci><ci id="S3.4.p1.8.m8.1.1.3.cmml" xref="S3.4.p1.8.m8.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p1.8.m8.1c">A_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.4.p1.8.m8.1d">italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> is strongly surjective, <math alttext="A\trianglerighteq B" class="ltx_Math" display="inline" id="S3.4.p1.9.m9.1"><semantics id="S3.4.p1.9.m9.1a"><mrow id="S3.4.p1.9.m9.1.1" xref="S3.4.p1.9.m9.1.1.cmml"><mi id="S3.4.p1.9.m9.1.1.2" xref="S3.4.p1.9.m9.1.1.2.cmml">A</mi><mo id="S3.4.p1.9.m9.1.1.1" xref="S3.4.p1.9.m9.1.1.1.cmml">⁢</mo><mi id="S3.4.p1.9.m9.1.1.3" mathvariant="normal" xref="S3.4.p1.9.m9.1.1.3.cmml">⊵</mi><mo id="S3.4.p1.9.m9.1.1.1a" xref="S3.4.p1.9.m9.1.1.1.cmml">⁢</mo><mi id="S3.4.p1.9.m9.1.1.4" xref="S3.4.p1.9.m9.1.1.4.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.4.p1.9.m9.1b"><apply id="S3.4.p1.9.m9.1.1.cmml" xref="S3.4.p1.9.m9.1.1"><times id="S3.4.p1.9.m9.1.1.1.cmml" xref="S3.4.p1.9.m9.1.1.1"></times><ci id="S3.4.p1.9.m9.1.1.2.cmml" xref="S3.4.p1.9.m9.1.1.2">𝐴</ci><ci id="S3.4.p1.9.m9.1.1.3.cmml" xref="S3.4.p1.9.m9.1.1.3">⊵</ci><ci id="S3.4.p1.9.m9.1.1.4.cmml" xref="S3.4.p1.9.m9.1.1.4">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p1.9.m9.1c">A\trianglerighteq B</annotation><annotation encoding="application/x-llamapun" id="S3.4.p1.9.m9.1d">italic_A ⊵ italic_B</annotation></semantics></math> follows from <math alttext="A\trianglerighteq\sum_{i\in I^{\prime}}A_{i}" class="ltx_Math" display="inline" id="S3.4.p1.10.m10.1"><semantics id="S3.4.p1.10.m10.1a"><mrow id="S3.4.p1.10.m10.1.1" xref="S3.4.p1.10.m10.1.1.cmml"><mi id="S3.4.p1.10.m10.1.1.2" xref="S3.4.p1.10.m10.1.1.2.cmml">A</mi><mo id="S3.4.p1.10.m10.1.1.1" xref="S3.4.p1.10.m10.1.1.1.cmml">⁢</mo><mi id="S3.4.p1.10.m10.1.1.3" mathvariant="normal" xref="S3.4.p1.10.m10.1.1.3.cmml">⊵</mi><mo id="S3.4.p1.10.m10.1.1.1a" xref="S3.4.p1.10.m10.1.1.1.cmml">⁢</mo><mrow id="S3.4.p1.10.m10.1.1.4" xref="S3.4.p1.10.m10.1.1.4.cmml"><msub id="S3.4.p1.10.m10.1.1.4.1" xref="S3.4.p1.10.m10.1.1.4.1.cmml"><mo id="S3.4.p1.10.m10.1.1.4.1.2" xref="S3.4.p1.10.m10.1.1.4.1.2.cmml">∑</mo><mrow id="S3.4.p1.10.m10.1.1.4.1.3" xref="S3.4.p1.10.m10.1.1.4.1.3.cmml"><mi id="S3.4.p1.10.m10.1.1.4.1.3.2" xref="S3.4.p1.10.m10.1.1.4.1.3.2.cmml">i</mi><mo id="S3.4.p1.10.m10.1.1.4.1.3.1" xref="S3.4.p1.10.m10.1.1.4.1.3.1.cmml">∈</mo><msup id="S3.4.p1.10.m10.1.1.4.1.3.3" xref="S3.4.p1.10.m10.1.1.4.1.3.3.cmml"><mi id="S3.4.p1.10.m10.1.1.4.1.3.3.2" xref="S3.4.p1.10.m10.1.1.4.1.3.3.2.cmml">I</mi><mo id="S3.4.p1.10.m10.1.1.4.1.3.3.3" xref="S3.4.p1.10.m10.1.1.4.1.3.3.3.cmml">′</mo></msup></mrow></msub><msub id="S3.4.p1.10.m10.1.1.4.2" xref="S3.4.p1.10.m10.1.1.4.2.cmml"><mi id="S3.4.p1.10.m10.1.1.4.2.2" xref="S3.4.p1.10.m10.1.1.4.2.2.cmml">A</mi><mi id="S3.4.p1.10.m10.1.1.4.2.3" xref="S3.4.p1.10.m10.1.1.4.2.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.4.p1.10.m10.1b"><apply id="S3.4.p1.10.m10.1.1.cmml" xref="S3.4.p1.10.m10.1.1"><times id="S3.4.p1.10.m10.1.1.1.cmml" xref="S3.4.p1.10.m10.1.1.1"></times><ci id="S3.4.p1.10.m10.1.1.2.cmml" xref="S3.4.p1.10.m10.1.1.2">𝐴</ci><ci id="S3.4.p1.10.m10.1.1.3.cmml" xref="S3.4.p1.10.m10.1.1.3">⊵</ci><apply id="S3.4.p1.10.m10.1.1.4.cmml" xref="S3.4.p1.10.m10.1.1.4"><apply id="S3.4.p1.10.m10.1.1.4.1.cmml" xref="S3.4.p1.10.m10.1.1.4.1"><csymbol cd="ambiguous" id="S3.4.p1.10.m10.1.1.4.1.1.cmml" xref="S3.4.p1.10.m10.1.1.4.1">subscript</csymbol><sum id="S3.4.p1.10.m10.1.1.4.1.2.cmml" xref="S3.4.p1.10.m10.1.1.4.1.2"></sum><apply id="S3.4.p1.10.m10.1.1.4.1.3.cmml" xref="S3.4.p1.10.m10.1.1.4.1.3"><in id="S3.4.p1.10.m10.1.1.4.1.3.1.cmml" xref="S3.4.p1.10.m10.1.1.4.1.3.1"></in><ci id="S3.4.p1.10.m10.1.1.4.1.3.2.cmml" xref="S3.4.p1.10.m10.1.1.4.1.3.2">𝑖</ci><apply id="S3.4.p1.10.m10.1.1.4.1.3.3.cmml" xref="S3.4.p1.10.m10.1.1.4.1.3.3"><csymbol cd="ambiguous" id="S3.4.p1.10.m10.1.1.4.1.3.3.1.cmml" xref="S3.4.p1.10.m10.1.1.4.1.3.3">superscript</csymbol><ci id="S3.4.p1.10.m10.1.1.4.1.3.3.2.cmml" xref="S3.4.p1.10.m10.1.1.4.1.3.3.2">𝐼</ci><ci id="S3.4.p1.10.m10.1.1.4.1.3.3.3.cmml" xref="S3.4.p1.10.m10.1.1.4.1.3.3.3">′</ci></apply></apply></apply><apply id="S3.4.p1.10.m10.1.1.4.2.cmml" xref="S3.4.p1.10.m10.1.1.4.2"><csymbol cd="ambiguous" id="S3.4.p1.10.m10.1.1.4.2.1.cmml" xref="S3.4.p1.10.m10.1.1.4.2">subscript</csymbol><ci id="S3.4.p1.10.m10.1.1.4.2.2.cmml" xref="S3.4.p1.10.m10.1.1.4.2.2">𝐴</ci><ci id="S3.4.p1.10.m10.1.1.4.2.3.cmml" xref="S3.4.p1.10.m10.1.1.4.2.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p1.10.m10.1c">A\trianglerighteq\sum_{i\in I^{\prime}}A_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.4.p1.10.m10.1d">italic_A ⊵ ∑ start_POSTSUBSCRIPT italic_i ∈ italic_I start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. Let <math alttext="f:I\twoheadrightarrow I^{\prime}" class="ltx_Math" display="inline" id="S3.4.p1.11.m11.1"><semantics id="S3.4.p1.11.m11.1a"><mrow id="S3.4.p1.11.m11.1.1" xref="S3.4.p1.11.m11.1.1.cmml"><mi id="S3.4.p1.11.m11.1.1.2" xref="S3.4.p1.11.m11.1.1.2.cmml">f</mi><mo id="S3.4.p1.11.m11.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.4.p1.11.m11.1.1.1.cmml">:</mo><mrow id="S3.4.p1.11.m11.1.1.3" xref="S3.4.p1.11.m11.1.1.3.cmml"><mi id="S3.4.p1.11.m11.1.1.3.2" xref="S3.4.p1.11.m11.1.1.3.2.cmml">I</mi><mo id="S3.4.p1.11.m11.1.1.3.1" stretchy="false" xref="S3.4.p1.11.m11.1.1.3.1.cmml">↠</mo><msup id="S3.4.p1.11.m11.1.1.3.3" xref="S3.4.p1.11.m11.1.1.3.3.cmml"><mi id="S3.4.p1.11.m11.1.1.3.3.2" xref="S3.4.p1.11.m11.1.1.3.3.2.cmml">I</mi><mo id="S3.4.p1.11.m11.1.1.3.3.3" xref="S3.4.p1.11.m11.1.1.3.3.3.cmml">′</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.4.p1.11.m11.1b"><apply id="S3.4.p1.11.m11.1.1.cmml" xref="S3.4.p1.11.m11.1.1"><ci id="S3.4.p1.11.m11.1.1.1.cmml" xref="S3.4.p1.11.m11.1.1.1">:</ci><ci id="S3.4.p1.11.m11.1.1.2.cmml" xref="S3.4.p1.11.m11.1.1.2">𝑓</ci><apply id="S3.4.p1.11.m11.1.1.3.cmml" xref="S3.4.p1.11.m11.1.1.3"><ci id="S3.4.p1.11.m11.1.1.3.1.cmml" xref="S3.4.p1.11.m11.1.1.3.1">↠</ci><ci id="S3.4.p1.11.m11.1.1.3.2.cmml" xref="S3.4.p1.11.m11.1.1.3.2">𝐼</ci><apply id="S3.4.p1.11.m11.1.1.3.3.cmml" xref="S3.4.p1.11.m11.1.1.3.3"><csymbol cd="ambiguous" id="S3.4.p1.11.m11.1.1.3.3.1.cmml" xref="S3.4.p1.11.m11.1.1.3.3">superscript</csymbol><ci id="S3.4.p1.11.m11.1.1.3.3.2.cmml" xref="S3.4.p1.11.m11.1.1.3.3.2">𝐼</ci><ci id="S3.4.p1.11.m11.1.1.3.3.3.cmml" xref="S3.4.p1.11.m11.1.1.3.3.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p1.11.m11.1c">f:I\twoheadrightarrow I^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.4.p1.11.m11.1d">italic_f : italic_I ↠ italic_I start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> be an epimorphism with open preimages and for each <math alttext="i\in I^{\prime}" class="ltx_Math" display="inline" id="S3.4.p1.12.m12.1"><semantics id="S3.4.p1.12.m12.1a"><mrow id="S3.4.p1.12.m12.1.1" xref="S3.4.p1.12.m12.1.1.cmml"><mi id="S3.4.p1.12.m12.1.1.2" xref="S3.4.p1.12.m12.1.1.2.cmml">i</mi><mo id="S3.4.p1.12.m12.1.1.1" xref="S3.4.p1.12.m12.1.1.1.cmml">∈</mo><msup id="S3.4.p1.12.m12.1.1.3" xref="S3.4.p1.12.m12.1.1.3.cmml"><mi id="S3.4.p1.12.m12.1.1.3.2" xref="S3.4.p1.12.m12.1.1.3.2.cmml">I</mi><mo id="S3.4.p1.12.m12.1.1.3.3" xref="S3.4.p1.12.m12.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.4.p1.12.m12.1b"><apply id="S3.4.p1.12.m12.1.1.cmml" xref="S3.4.p1.12.m12.1.1"><in id="S3.4.p1.12.m12.1.1.1.cmml" xref="S3.4.p1.12.m12.1.1.1"></in><ci id="S3.4.p1.12.m12.1.1.2.cmml" xref="S3.4.p1.12.m12.1.1.2">𝑖</ci><apply id="S3.4.p1.12.m12.1.1.3.cmml" xref="S3.4.p1.12.m12.1.1.3"><csymbol cd="ambiguous" id="S3.4.p1.12.m12.1.1.3.1.cmml" xref="S3.4.p1.12.m12.1.1.3">superscript</csymbol><ci id="S3.4.p1.12.m12.1.1.3.2.cmml" xref="S3.4.p1.12.m12.1.1.3.2">𝐼</ci><ci id="S3.4.p1.12.m12.1.1.3.3.cmml" xref="S3.4.p1.12.m12.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p1.12.m12.1c">i\in I^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.4.p1.12.m12.1d">italic_i ∈ italic_I start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, let <math alttext="I_{i}:=f^{-1}(i)" class="ltx_Math" display="inline" id="S3.4.p1.13.m13.1"><semantics id="S3.4.p1.13.m13.1a"><mrow id="S3.4.p1.13.m13.1.2" xref="S3.4.p1.13.m13.1.2.cmml"><msub id="S3.4.p1.13.m13.1.2.2" xref="S3.4.p1.13.m13.1.2.2.cmml"><mi id="S3.4.p1.13.m13.1.2.2.2" xref="S3.4.p1.13.m13.1.2.2.2.cmml">I</mi><mi id="S3.4.p1.13.m13.1.2.2.3" xref="S3.4.p1.13.m13.1.2.2.3.cmml">i</mi></msub><mo id="S3.4.p1.13.m13.1.2.1" lspace="0.278em" rspace="0.278em" xref="S3.4.p1.13.m13.1.2.1.cmml">:=</mo><mrow id="S3.4.p1.13.m13.1.2.3" xref="S3.4.p1.13.m13.1.2.3.cmml"><msup id="S3.4.p1.13.m13.1.2.3.2" xref="S3.4.p1.13.m13.1.2.3.2.cmml"><mi id="S3.4.p1.13.m13.1.2.3.2.2" xref="S3.4.p1.13.m13.1.2.3.2.2.cmml">f</mi><mrow id="S3.4.p1.13.m13.1.2.3.2.3" xref="S3.4.p1.13.m13.1.2.3.2.3.cmml"><mo id="S3.4.p1.13.m13.1.2.3.2.3a" xref="S3.4.p1.13.m13.1.2.3.2.3.cmml">−</mo><mn id="S3.4.p1.13.m13.1.2.3.2.3.2" xref="S3.4.p1.13.m13.1.2.3.2.3.2.cmml">1</mn></mrow></msup><mo id="S3.4.p1.13.m13.1.2.3.1" xref="S3.4.p1.13.m13.1.2.3.1.cmml">⁢</mo><mrow id="S3.4.p1.13.m13.1.2.3.3.2" xref="S3.4.p1.13.m13.1.2.3.cmml"><mo id="S3.4.p1.13.m13.1.2.3.3.2.1" stretchy="false" xref="S3.4.p1.13.m13.1.2.3.cmml">(</mo><mi id="S3.4.p1.13.m13.1.1" xref="S3.4.p1.13.m13.1.1.cmml">i</mi><mo id="S3.4.p1.13.m13.1.2.3.3.2.2" stretchy="false" xref="S3.4.p1.13.m13.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.4.p1.13.m13.1b"><apply id="S3.4.p1.13.m13.1.2.cmml" xref="S3.4.p1.13.m13.1.2"><csymbol cd="latexml" id="S3.4.p1.13.m13.1.2.1.cmml" xref="S3.4.p1.13.m13.1.2.1">assign</csymbol><apply id="S3.4.p1.13.m13.1.2.2.cmml" xref="S3.4.p1.13.m13.1.2.2"><csymbol cd="ambiguous" id="S3.4.p1.13.m13.1.2.2.1.cmml" xref="S3.4.p1.13.m13.1.2.2">subscript</csymbol><ci id="S3.4.p1.13.m13.1.2.2.2.cmml" xref="S3.4.p1.13.m13.1.2.2.2">𝐼</ci><ci id="S3.4.p1.13.m13.1.2.2.3.cmml" xref="S3.4.p1.13.m13.1.2.2.3">𝑖</ci></apply><apply id="S3.4.p1.13.m13.1.2.3.cmml" xref="S3.4.p1.13.m13.1.2.3"><times id="S3.4.p1.13.m13.1.2.3.1.cmml" xref="S3.4.p1.13.m13.1.2.3.1"></times><apply id="S3.4.p1.13.m13.1.2.3.2.cmml" xref="S3.4.p1.13.m13.1.2.3.2"><csymbol cd="ambiguous" id="S3.4.p1.13.m13.1.2.3.2.1.cmml" xref="S3.4.p1.13.m13.1.2.3.2">superscript</csymbol><ci id="S3.4.p1.13.m13.1.2.3.2.2.cmml" xref="S3.4.p1.13.m13.1.2.3.2.2">𝑓</ci><apply id="S3.4.p1.13.m13.1.2.3.2.3.cmml" xref="S3.4.p1.13.m13.1.2.3.2.3"><minus id="S3.4.p1.13.m13.1.2.3.2.3.1.cmml" xref="S3.4.p1.13.m13.1.2.3.2.3"></minus><cn id="S3.4.p1.13.m13.1.2.3.2.3.2.cmml" type="integer" xref="S3.4.p1.13.m13.1.2.3.2.3.2">1</cn></apply></apply><ci id="S3.4.p1.13.m13.1.1.cmml" xref="S3.4.p1.13.m13.1.1">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p1.13.m13.1c">I_{i}:=f^{-1}(i)</annotation><annotation encoding="application/x-llamapun" id="S3.4.p1.13.m13.1d">italic_I start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT := italic_f start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_i )</annotation></semantics></math>. It is enough to show that for each <math alttext="i\in I^{\prime}" class="ltx_Math" display="inline" id="S3.4.p1.14.m14.1"><semantics id="S3.4.p1.14.m14.1a"><mrow id="S3.4.p1.14.m14.1.1" xref="S3.4.p1.14.m14.1.1.cmml"><mi id="S3.4.p1.14.m14.1.1.2" xref="S3.4.p1.14.m14.1.1.2.cmml">i</mi><mo id="S3.4.p1.14.m14.1.1.1" xref="S3.4.p1.14.m14.1.1.1.cmml">∈</mo><msup id="S3.4.p1.14.m14.1.1.3" xref="S3.4.p1.14.m14.1.1.3.cmml"><mi id="S3.4.p1.14.m14.1.1.3.2" xref="S3.4.p1.14.m14.1.1.3.2.cmml">I</mi><mo id="S3.4.p1.14.m14.1.1.3.3" xref="S3.4.p1.14.m14.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.4.p1.14.m14.1b"><apply id="S3.4.p1.14.m14.1.1.cmml" xref="S3.4.p1.14.m14.1.1"><in id="S3.4.p1.14.m14.1.1.1.cmml" xref="S3.4.p1.14.m14.1.1.1"></in><ci id="S3.4.p1.14.m14.1.1.2.cmml" xref="S3.4.p1.14.m14.1.1.2">𝑖</ci><apply id="S3.4.p1.14.m14.1.1.3.cmml" xref="S3.4.p1.14.m14.1.1.3"><csymbol cd="ambiguous" id="S3.4.p1.14.m14.1.1.3.1.cmml" xref="S3.4.p1.14.m14.1.1.3">superscript</csymbol><ci id="S3.4.p1.14.m14.1.1.3.2.cmml" xref="S3.4.p1.14.m14.1.1.3.2">𝐼</ci><ci id="S3.4.p1.14.m14.1.1.3.3.cmml" xref="S3.4.p1.14.m14.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p1.14.m14.1c">i\in I^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.4.p1.14.m14.1d">italic_i ∈ italic_I start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="\sum_{j\in I_{i}}A_{j}\trianglerighteq A_{i}" class="ltx_Math" display="inline" id="S3.4.p1.15.m15.1"><semantics id="S3.4.p1.15.m15.1a"><mrow id="S3.4.p1.15.m15.1.1" xref="S3.4.p1.15.m15.1.1.cmml"><msub id="S3.4.p1.15.m15.1.1.1" xref="S3.4.p1.15.m15.1.1.1.cmml"><mo id="S3.4.p1.15.m15.1.1.1.2" xref="S3.4.p1.15.m15.1.1.1.2.cmml">∑</mo><mrow id="S3.4.p1.15.m15.1.1.1.3" xref="S3.4.p1.15.m15.1.1.1.3.cmml"><mi id="S3.4.p1.15.m15.1.1.1.3.2" xref="S3.4.p1.15.m15.1.1.1.3.2.cmml">j</mi><mo id="S3.4.p1.15.m15.1.1.1.3.1" xref="S3.4.p1.15.m15.1.1.1.3.1.cmml">∈</mo><msub id="S3.4.p1.15.m15.1.1.1.3.3" xref="S3.4.p1.15.m15.1.1.1.3.3.cmml"><mi id="S3.4.p1.15.m15.1.1.1.3.3.2" xref="S3.4.p1.15.m15.1.1.1.3.3.2.cmml">I</mi><mi id="S3.4.p1.15.m15.1.1.1.3.3.3" xref="S3.4.p1.15.m15.1.1.1.3.3.3.cmml">i</mi></msub></mrow></msub><mrow id="S3.4.p1.15.m15.1.1.2" xref="S3.4.p1.15.m15.1.1.2.cmml"><msub id="S3.4.p1.15.m15.1.1.2.2" xref="S3.4.p1.15.m15.1.1.2.2.cmml"><mi id="S3.4.p1.15.m15.1.1.2.2.2" xref="S3.4.p1.15.m15.1.1.2.2.2.cmml">A</mi><mi id="S3.4.p1.15.m15.1.1.2.2.3" xref="S3.4.p1.15.m15.1.1.2.2.3.cmml">j</mi></msub><mo id="S3.4.p1.15.m15.1.1.2.1" xref="S3.4.p1.15.m15.1.1.2.1.cmml">⁢</mo><mi id="S3.4.p1.15.m15.1.1.2.3" mathvariant="normal" xref="S3.4.p1.15.m15.1.1.2.3.cmml">⊵</mi><mo id="S3.4.p1.15.m15.1.1.2.1a" xref="S3.4.p1.15.m15.1.1.2.1.cmml">⁢</mo><msub id="S3.4.p1.15.m15.1.1.2.4" xref="S3.4.p1.15.m15.1.1.2.4.cmml"><mi id="S3.4.p1.15.m15.1.1.2.4.2" xref="S3.4.p1.15.m15.1.1.2.4.2.cmml">A</mi><mi id="S3.4.p1.15.m15.1.1.2.4.3" xref="S3.4.p1.15.m15.1.1.2.4.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.4.p1.15.m15.1b"><apply id="S3.4.p1.15.m15.1.1.cmml" xref="S3.4.p1.15.m15.1.1"><apply id="S3.4.p1.15.m15.1.1.1.cmml" xref="S3.4.p1.15.m15.1.1.1"><csymbol cd="ambiguous" id="S3.4.p1.15.m15.1.1.1.1.cmml" xref="S3.4.p1.15.m15.1.1.1">subscript</csymbol><sum id="S3.4.p1.15.m15.1.1.1.2.cmml" xref="S3.4.p1.15.m15.1.1.1.2"></sum><apply id="S3.4.p1.15.m15.1.1.1.3.cmml" xref="S3.4.p1.15.m15.1.1.1.3"><in id="S3.4.p1.15.m15.1.1.1.3.1.cmml" xref="S3.4.p1.15.m15.1.1.1.3.1"></in><ci id="S3.4.p1.15.m15.1.1.1.3.2.cmml" xref="S3.4.p1.15.m15.1.1.1.3.2">𝑗</ci><apply id="S3.4.p1.15.m15.1.1.1.3.3.cmml" xref="S3.4.p1.15.m15.1.1.1.3.3"><csymbol cd="ambiguous" id="S3.4.p1.15.m15.1.1.1.3.3.1.cmml" xref="S3.4.p1.15.m15.1.1.1.3.3">subscript</csymbol><ci id="S3.4.p1.15.m15.1.1.1.3.3.2.cmml" xref="S3.4.p1.15.m15.1.1.1.3.3.2">𝐼</ci><ci id="S3.4.p1.15.m15.1.1.1.3.3.3.cmml" xref="S3.4.p1.15.m15.1.1.1.3.3.3">𝑖</ci></apply></apply></apply><apply id="S3.4.p1.15.m15.1.1.2.cmml" xref="S3.4.p1.15.m15.1.1.2"><times id="S3.4.p1.15.m15.1.1.2.1.cmml" xref="S3.4.p1.15.m15.1.1.2.1"></times><apply id="S3.4.p1.15.m15.1.1.2.2.cmml" xref="S3.4.p1.15.m15.1.1.2.2"><csymbol cd="ambiguous" id="S3.4.p1.15.m15.1.1.2.2.1.cmml" xref="S3.4.p1.15.m15.1.1.2.2">subscript</csymbol><ci id="S3.4.p1.15.m15.1.1.2.2.2.cmml" xref="S3.4.p1.15.m15.1.1.2.2.2">𝐴</ci><ci id="S3.4.p1.15.m15.1.1.2.2.3.cmml" xref="S3.4.p1.15.m15.1.1.2.2.3">𝑗</ci></apply><ci id="S3.4.p1.15.m15.1.1.2.3.cmml" xref="S3.4.p1.15.m15.1.1.2.3">⊵</ci><apply id="S3.4.p1.15.m15.1.1.2.4.cmml" xref="S3.4.p1.15.m15.1.1.2.4"><csymbol cd="ambiguous" id="S3.4.p1.15.m15.1.1.2.4.1.cmml" xref="S3.4.p1.15.m15.1.1.2.4">subscript</csymbol><ci id="S3.4.p1.15.m15.1.1.2.4.2.cmml" xref="S3.4.p1.15.m15.1.1.2.4.2">𝐴</ci><ci id="S3.4.p1.15.m15.1.1.2.4.3.cmml" xref="S3.4.p1.15.m15.1.1.2.4.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p1.15.m15.1c">\sum_{j\in I_{i}}A_{j}\trianglerighteq A_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.4.p1.15.m15.1d">∑ start_POSTSUBSCRIPT italic_j ∈ italic_I start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ⊵ italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.5.p2"> <p class="ltx_p" id="S3.5.p2.10">Fix <math alttext="i\in I^{\prime}" class="ltx_Math" display="inline" id="S3.5.p2.1.m1.1"><semantics id="S3.5.p2.1.m1.1a"><mrow id="S3.5.p2.1.m1.1.1" xref="S3.5.p2.1.m1.1.1.cmml"><mi id="S3.5.p2.1.m1.1.1.2" xref="S3.5.p2.1.m1.1.1.2.cmml">i</mi><mo id="S3.5.p2.1.m1.1.1.1" xref="S3.5.p2.1.m1.1.1.1.cmml">∈</mo><msup id="S3.5.p2.1.m1.1.1.3" xref="S3.5.p2.1.m1.1.1.3.cmml"><mi id="S3.5.p2.1.m1.1.1.3.2" xref="S3.5.p2.1.m1.1.1.3.2.cmml">I</mi><mo id="S3.5.p2.1.m1.1.1.3.3" xref="S3.5.p2.1.m1.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.5.p2.1.m1.1b"><apply id="S3.5.p2.1.m1.1.1.cmml" xref="S3.5.p2.1.m1.1.1"><in id="S3.5.p2.1.m1.1.1.1.cmml" xref="S3.5.p2.1.m1.1.1.1"></in><ci id="S3.5.p2.1.m1.1.1.2.cmml" xref="S3.5.p2.1.m1.1.1.2">𝑖</ci><apply id="S3.5.p2.1.m1.1.1.3.cmml" xref="S3.5.p2.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.5.p2.1.m1.1.1.3.1.cmml" xref="S3.5.p2.1.m1.1.1.3">superscript</csymbol><ci id="S3.5.p2.1.m1.1.1.3.2.cmml" xref="S3.5.p2.1.m1.1.1.3.2">𝐼</ci><ci id="S3.5.p2.1.m1.1.1.3.3.cmml" xref="S3.5.p2.1.m1.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.5.p2.1.m1.1c">i\in I^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.5.p2.1.m1.1d">italic_i ∈ italic_I start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>. Using the fact that <math alttext="I_{i}" class="ltx_Math" display="inline" id="S3.5.p2.2.m2.1"><semantics id="S3.5.p2.2.m2.1a"><msub id="S3.5.p2.2.m2.1.1" xref="S3.5.p2.2.m2.1.1.cmml"><mi id="S3.5.p2.2.m2.1.1.2" xref="S3.5.p2.2.m2.1.1.2.cmml">I</mi><mi id="S3.5.p2.2.m2.1.1.3" xref="S3.5.p2.2.m2.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.5.p2.2.m2.1b"><apply id="S3.5.p2.2.m2.1.1.cmml" xref="S3.5.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S3.5.p2.2.m2.1.1.1.cmml" xref="S3.5.p2.2.m2.1.1">subscript</csymbol><ci id="S3.5.p2.2.m2.1.1.2.cmml" xref="S3.5.p2.2.m2.1.1.2">𝐼</ci><ci id="S3.5.p2.2.m2.1.1.3.cmml" xref="S3.5.p2.2.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.5.p2.2.m2.1c">I_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.5.p2.2.m2.1d">italic_I start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> has countable cofinality and coinitiality, plus the density of <math alttext="\{j:A_{j}=A_{i}\}" class="ltx_Math" display="inline" id="S3.5.p2.3.m3.2"><semantics id="S3.5.p2.3.m3.2a"><mrow id="S3.5.p2.3.m3.2.2.1" xref="S3.5.p2.3.m3.2.2.2.cmml"><mo id="S3.5.p2.3.m3.2.2.1.2" stretchy="false" xref="S3.5.p2.3.m3.2.2.2.1.cmml">{</mo><mi id="S3.5.p2.3.m3.1.1" xref="S3.5.p2.3.m3.1.1.cmml">j</mi><mo id="S3.5.p2.3.m3.2.2.1.3" lspace="0.278em" rspace="0.278em" xref="S3.5.p2.3.m3.2.2.2.1.cmml">:</mo><mrow id="S3.5.p2.3.m3.2.2.1.1" xref="S3.5.p2.3.m3.2.2.1.1.cmml"><msub id="S3.5.p2.3.m3.2.2.1.1.2" xref="S3.5.p2.3.m3.2.2.1.1.2.cmml"><mi id="S3.5.p2.3.m3.2.2.1.1.2.2" xref="S3.5.p2.3.m3.2.2.1.1.2.2.cmml">A</mi><mi id="S3.5.p2.3.m3.2.2.1.1.2.3" xref="S3.5.p2.3.m3.2.2.1.1.2.3.cmml">j</mi></msub><mo id="S3.5.p2.3.m3.2.2.1.1.1" xref="S3.5.p2.3.m3.2.2.1.1.1.cmml">=</mo><msub id="S3.5.p2.3.m3.2.2.1.1.3" xref="S3.5.p2.3.m3.2.2.1.1.3.cmml"><mi id="S3.5.p2.3.m3.2.2.1.1.3.2" xref="S3.5.p2.3.m3.2.2.1.1.3.2.cmml">A</mi><mi id="S3.5.p2.3.m3.2.2.1.1.3.3" xref="S3.5.p2.3.m3.2.2.1.1.3.3.cmml">i</mi></msub></mrow><mo id="S3.5.p2.3.m3.2.2.1.4" stretchy="false" xref="S3.5.p2.3.m3.2.2.2.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.5.p2.3.m3.2b"><apply id="S3.5.p2.3.m3.2.2.2.cmml" xref="S3.5.p2.3.m3.2.2.1"><csymbol cd="latexml" id="S3.5.p2.3.m3.2.2.2.1.cmml" xref="S3.5.p2.3.m3.2.2.1.2">conditional-set</csymbol><ci id="S3.5.p2.3.m3.1.1.cmml" xref="S3.5.p2.3.m3.1.1">𝑗</ci><apply id="S3.5.p2.3.m3.2.2.1.1.cmml" xref="S3.5.p2.3.m3.2.2.1.1"><eq id="S3.5.p2.3.m3.2.2.1.1.1.cmml" xref="S3.5.p2.3.m3.2.2.1.1.1"></eq><apply id="S3.5.p2.3.m3.2.2.1.1.2.cmml" xref="S3.5.p2.3.m3.2.2.1.1.2"><csymbol cd="ambiguous" id="S3.5.p2.3.m3.2.2.1.1.2.1.cmml" xref="S3.5.p2.3.m3.2.2.1.1.2">subscript</csymbol><ci id="S3.5.p2.3.m3.2.2.1.1.2.2.cmml" xref="S3.5.p2.3.m3.2.2.1.1.2.2">𝐴</ci><ci id="S3.5.p2.3.m3.2.2.1.1.2.3.cmml" xref="S3.5.p2.3.m3.2.2.1.1.2.3">𝑗</ci></apply><apply id="S3.5.p2.3.m3.2.2.1.1.3.cmml" xref="S3.5.p2.3.m3.2.2.1.1.3"><csymbol cd="ambiguous" id="S3.5.p2.3.m3.2.2.1.1.3.1.cmml" xref="S3.5.p2.3.m3.2.2.1.1.3">subscript</csymbol><ci id="S3.5.p2.3.m3.2.2.1.1.3.2.cmml" xref="S3.5.p2.3.m3.2.2.1.1.3.2">𝐴</ci><ci id="S3.5.p2.3.m3.2.2.1.1.3.3.cmml" xref="S3.5.p2.3.m3.2.2.1.1.3.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.5.p2.3.m3.2c">\{j:A_{j}=A_{i}\}</annotation><annotation encoding="application/x-llamapun" id="S3.5.p2.3.m3.2d">{ italic_j : italic_A start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT }</annotation></semantics></math>, pick <math alttext="\{j_{z}:z\in\mathbb{Z}\}" class="ltx_Math" display="inline" id="S3.5.p2.4.m4.2"><semantics id="S3.5.p2.4.m4.2a"><mrow id="S3.5.p2.4.m4.2.2.2" xref="S3.5.p2.4.m4.2.2.3.cmml"><mo id="S3.5.p2.4.m4.2.2.2.3" stretchy="false" xref="S3.5.p2.4.m4.2.2.3.1.cmml">{</mo><msub id="S3.5.p2.4.m4.1.1.1.1" xref="S3.5.p2.4.m4.1.1.1.1.cmml"><mi id="S3.5.p2.4.m4.1.1.1.1.2" xref="S3.5.p2.4.m4.1.1.1.1.2.cmml">j</mi><mi id="S3.5.p2.4.m4.1.1.1.1.3" xref="S3.5.p2.4.m4.1.1.1.1.3.cmml">z</mi></msub><mo id="S3.5.p2.4.m4.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S3.5.p2.4.m4.2.2.3.1.cmml">:</mo><mrow id="S3.5.p2.4.m4.2.2.2.2" xref="S3.5.p2.4.m4.2.2.2.2.cmml"><mi id="S3.5.p2.4.m4.2.2.2.2.2" xref="S3.5.p2.4.m4.2.2.2.2.2.cmml">z</mi><mo id="S3.5.p2.4.m4.2.2.2.2.1" xref="S3.5.p2.4.m4.2.2.2.2.1.cmml">∈</mo><mi id="S3.5.p2.4.m4.2.2.2.2.3" xref="S3.5.p2.4.m4.2.2.2.2.3.cmml">ℤ</mi></mrow><mo id="S3.5.p2.4.m4.2.2.2.5" stretchy="false" xref="S3.5.p2.4.m4.2.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.5.p2.4.m4.2b"><apply id="S3.5.p2.4.m4.2.2.3.cmml" xref="S3.5.p2.4.m4.2.2.2"><csymbol cd="latexml" id="S3.5.p2.4.m4.2.2.3.1.cmml" xref="S3.5.p2.4.m4.2.2.2.3">conditional-set</csymbol><apply id="S3.5.p2.4.m4.1.1.1.1.cmml" xref="S3.5.p2.4.m4.1.1.1.1"><csymbol cd="ambiguous" id="S3.5.p2.4.m4.1.1.1.1.1.cmml" xref="S3.5.p2.4.m4.1.1.1.1">subscript</csymbol><ci id="S3.5.p2.4.m4.1.1.1.1.2.cmml" xref="S3.5.p2.4.m4.1.1.1.1.2">𝑗</ci><ci id="S3.5.p2.4.m4.1.1.1.1.3.cmml" xref="S3.5.p2.4.m4.1.1.1.1.3">𝑧</ci></apply><apply id="S3.5.p2.4.m4.2.2.2.2.cmml" xref="S3.5.p2.4.m4.2.2.2.2"><in id="S3.5.p2.4.m4.2.2.2.2.1.cmml" xref="S3.5.p2.4.m4.2.2.2.2.1"></in><ci id="S3.5.p2.4.m4.2.2.2.2.2.cmml" xref="S3.5.p2.4.m4.2.2.2.2.2">𝑧</ci><ci id="S3.5.p2.4.m4.2.2.2.2.3.cmml" xref="S3.5.p2.4.m4.2.2.2.2.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.5.p2.4.m4.2c">\{j_{z}:z\in\mathbb{Z}\}</annotation><annotation encoding="application/x-llamapun" id="S3.5.p2.4.m4.2d">{ italic_j start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT : italic_z ∈ blackboard_Z }</annotation></semantics></math> a copy of <math alttext="\mathbb{Z}" class="ltx_Math" display="inline" id="S3.5.p2.5.m5.1"><semantics id="S3.5.p2.5.m5.1a"><mi id="S3.5.p2.5.m5.1.1" xref="S3.5.p2.5.m5.1.1.cmml">ℤ</mi><annotation-xml encoding="MathML-Content" id="S3.5.p2.5.m5.1b"><ci id="S3.5.p2.5.m5.1.1.cmml" xref="S3.5.p2.5.m5.1.1">ℤ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.5.p2.5.m5.1c">\mathbb{Z}</annotation><annotation encoding="application/x-llamapun" id="S3.5.p2.5.m5.1d">blackboard_Z</annotation></semantics></math> coinitial and cofinal in <math alttext="I_{i}" class="ltx_Math" display="inline" id="S3.5.p2.6.m6.1"><semantics id="S3.5.p2.6.m6.1a"><msub id="S3.5.p2.6.m6.1.1" xref="S3.5.p2.6.m6.1.1.cmml"><mi id="S3.5.p2.6.m6.1.1.2" xref="S3.5.p2.6.m6.1.1.2.cmml">I</mi><mi id="S3.5.p2.6.m6.1.1.3" xref="S3.5.p2.6.m6.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.5.p2.6.m6.1b"><apply id="S3.5.p2.6.m6.1.1.cmml" xref="S3.5.p2.6.m6.1.1"><csymbol cd="ambiguous" id="S3.5.p2.6.m6.1.1.1.cmml" xref="S3.5.p2.6.m6.1.1">subscript</csymbol><ci id="S3.5.p2.6.m6.1.1.2.cmml" xref="S3.5.p2.6.m6.1.1.2">𝐼</ci><ci id="S3.5.p2.6.m6.1.1.3.cmml" xref="S3.5.p2.6.m6.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.5.p2.6.m6.1c">I_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.5.p2.6.m6.1d">italic_I start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> such that for all <math alttext="z" class="ltx_Math" display="inline" id="S3.5.p2.7.m7.1"><semantics id="S3.5.p2.7.m7.1a"><mi id="S3.5.p2.7.m7.1.1" xref="S3.5.p2.7.m7.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S3.5.p2.7.m7.1b"><ci id="S3.5.p2.7.m7.1.1.cmml" xref="S3.5.p2.7.m7.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.5.p2.7.m7.1c">z</annotation><annotation encoding="application/x-llamapun" id="S3.5.p2.7.m7.1d">italic_z</annotation></semantics></math>, <math alttext="A_{j_{z}}=A_{i}" class="ltx_Math" display="inline" id="S3.5.p2.8.m8.1"><semantics id="S3.5.p2.8.m8.1a"><mrow id="S3.5.p2.8.m8.1.1" xref="S3.5.p2.8.m8.1.1.cmml"><msub id="S3.5.p2.8.m8.1.1.2" xref="S3.5.p2.8.m8.1.1.2.cmml"><mi id="S3.5.p2.8.m8.1.1.2.2" xref="S3.5.p2.8.m8.1.1.2.2.cmml">A</mi><msub id="S3.5.p2.8.m8.1.1.2.3" xref="S3.5.p2.8.m8.1.1.2.3.cmml"><mi id="S3.5.p2.8.m8.1.1.2.3.2" xref="S3.5.p2.8.m8.1.1.2.3.2.cmml">j</mi><mi id="S3.5.p2.8.m8.1.1.2.3.3" xref="S3.5.p2.8.m8.1.1.2.3.3.cmml">z</mi></msub></msub><mo id="S3.5.p2.8.m8.1.1.1" xref="S3.5.p2.8.m8.1.1.1.cmml">=</mo><msub id="S3.5.p2.8.m8.1.1.3" xref="S3.5.p2.8.m8.1.1.3.cmml"><mi id="S3.5.p2.8.m8.1.1.3.2" xref="S3.5.p2.8.m8.1.1.3.2.cmml">A</mi><mi id="S3.5.p2.8.m8.1.1.3.3" xref="S3.5.p2.8.m8.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.5.p2.8.m8.1b"><apply id="S3.5.p2.8.m8.1.1.cmml" xref="S3.5.p2.8.m8.1.1"><eq id="S3.5.p2.8.m8.1.1.1.cmml" xref="S3.5.p2.8.m8.1.1.1"></eq><apply id="S3.5.p2.8.m8.1.1.2.cmml" xref="S3.5.p2.8.m8.1.1.2"><csymbol cd="ambiguous" id="S3.5.p2.8.m8.1.1.2.1.cmml" xref="S3.5.p2.8.m8.1.1.2">subscript</csymbol><ci id="S3.5.p2.8.m8.1.1.2.2.cmml" xref="S3.5.p2.8.m8.1.1.2.2">𝐴</ci><apply id="S3.5.p2.8.m8.1.1.2.3.cmml" xref="S3.5.p2.8.m8.1.1.2.3"><csymbol cd="ambiguous" id="S3.5.p2.8.m8.1.1.2.3.1.cmml" xref="S3.5.p2.8.m8.1.1.2.3">subscript</csymbol><ci id="S3.5.p2.8.m8.1.1.2.3.2.cmml" xref="S3.5.p2.8.m8.1.1.2.3.2">𝑗</ci><ci id="S3.5.p2.8.m8.1.1.2.3.3.cmml" xref="S3.5.p2.8.m8.1.1.2.3.3">𝑧</ci></apply></apply><apply id="S3.5.p2.8.m8.1.1.3.cmml" xref="S3.5.p2.8.m8.1.1.3"><csymbol cd="ambiguous" id="S3.5.p2.8.m8.1.1.3.1.cmml" xref="S3.5.p2.8.m8.1.1.3">subscript</csymbol><ci id="S3.5.p2.8.m8.1.1.3.2.cmml" xref="S3.5.p2.8.m8.1.1.3.2">𝐴</ci><ci id="S3.5.p2.8.m8.1.1.3.3.cmml" xref="S3.5.p2.8.m8.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.5.p2.8.m8.1c">A_{j_{z}}=A_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.5.p2.8.m8.1d">italic_A start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT end_POSTSUBSCRIPT = italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. From this one easily sees that <math alttext="\sum_{j\in I_{i}}A_{j}\trianglerighteq\cdots+1+A_{i}+1+A_{i}+1\cdots" class="ltx_Math" display="inline" id="S3.5.p2.9.m9.1"><semantics id="S3.5.p2.9.m9.1a"><mrow id="S3.5.p2.9.m9.1.1" xref="S3.5.p2.9.m9.1.1.cmml"><mrow id="S3.5.p2.9.m9.1.1.2" xref="S3.5.p2.9.m9.1.1.2.cmml"><msub id="S3.5.p2.9.m9.1.1.2.1" xref="S3.5.p2.9.m9.1.1.2.1.cmml"><mo id="S3.5.p2.9.m9.1.1.2.1.2" xref="S3.5.p2.9.m9.1.1.2.1.2.cmml">∑</mo><mrow id="S3.5.p2.9.m9.1.1.2.1.3" xref="S3.5.p2.9.m9.1.1.2.1.3.cmml"><mi id="S3.5.p2.9.m9.1.1.2.1.3.2" xref="S3.5.p2.9.m9.1.1.2.1.3.2.cmml">j</mi><mo id="S3.5.p2.9.m9.1.1.2.1.3.1" xref="S3.5.p2.9.m9.1.1.2.1.3.1.cmml">∈</mo><msub id="S3.5.p2.9.m9.1.1.2.1.3.3" xref="S3.5.p2.9.m9.1.1.2.1.3.3.cmml"><mi id="S3.5.p2.9.m9.1.1.2.1.3.3.2" xref="S3.5.p2.9.m9.1.1.2.1.3.3.2.cmml">I</mi><mi id="S3.5.p2.9.m9.1.1.2.1.3.3.3" xref="S3.5.p2.9.m9.1.1.2.1.3.3.3.cmml">i</mi></msub></mrow></msub><mrow id="S3.5.p2.9.m9.1.1.2.2" xref="S3.5.p2.9.m9.1.1.2.2.cmml"><msub id="S3.5.p2.9.m9.1.1.2.2.2" xref="S3.5.p2.9.m9.1.1.2.2.2.cmml"><mi id="S3.5.p2.9.m9.1.1.2.2.2.2" xref="S3.5.p2.9.m9.1.1.2.2.2.2.cmml">A</mi><mi id="S3.5.p2.9.m9.1.1.2.2.2.3" xref="S3.5.p2.9.m9.1.1.2.2.2.3.cmml">j</mi></msub><mo id="S3.5.p2.9.m9.1.1.2.2.1" xref="S3.5.p2.9.m9.1.1.2.2.1.cmml">⁢</mo><mi id="S3.5.p2.9.m9.1.1.2.2.3" mathvariant="normal" xref="S3.5.p2.9.m9.1.1.2.2.3.cmml">⊵</mi><mo id="S3.5.p2.9.m9.1.1.2.2.1a" xref="S3.5.p2.9.m9.1.1.2.2.1.cmml">⁢</mo><mi id="S3.5.p2.9.m9.1.1.2.2.4" mathvariant="normal" xref="S3.5.p2.9.m9.1.1.2.2.4.cmml">⋯</mi></mrow></mrow><mo id="S3.5.p2.9.m9.1.1.1" xref="S3.5.p2.9.m9.1.1.1.cmml">+</mo><mn id="S3.5.p2.9.m9.1.1.3" xref="S3.5.p2.9.m9.1.1.3.cmml">1</mn><mo id="S3.5.p2.9.m9.1.1.1a" xref="S3.5.p2.9.m9.1.1.1.cmml">+</mo><msub id="S3.5.p2.9.m9.1.1.4" xref="S3.5.p2.9.m9.1.1.4.cmml"><mi id="S3.5.p2.9.m9.1.1.4.2" xref="S3.5.p2.9.m9.1.1.4.2.cmml">A</mi><mi id="S3.5.p2.9.m9.1.1.4.3" xref="S3.5.p2.9.m9.1.1.4.3.cmml">i</mi></msub><mo id="S3.5.p2.9.m9.1.1.1b" xref="S3.5.p2.9.m9.1.1.1.cmml">+</mo><mn id="S3.5.p2.9.m9.1.1.5" xref="S3.5.p2.9.m9.1.1.5.cmml">1</mn><mo id="S3.5.p2.9.m9.1.1.1c" xref="S3.5.p2.9.m9.1.1.1.cmml">+</mo><msub id="S3.5.p2.9.m9.1.1.6" xref="S3.5.p2.9.m9.1.1.6.cmml"><mi id="S3.5.p2.9.m9.1.1.6.2" xref="S3.5.p2.9.m9.1.1.6.2.cmml">A</mi><mi id="S3.5.p2.9.m9.1.1.6.3" xref="S3.5.p2.9.m9.1.1.6.3.cmml">i</mi></msub><mo id="S3.5.p2.9.m9.1.1.1d" xref="S3.5.p2.9.m9.1.1.1.cmml">+</mo><mrow id="S3.5.p2.9.m9.1.1.7" xref="S3.5.p2.9.m9.1.1.7.cmml"><mn id="S3.5.p2.9.m9.1.1.7.2" xref="S3.5.p2.9.m9.1.1.7.2.cmml">1</mn><mo id="S3.5.p2.9.m9.1.1.7.1" xref="S3.5.p2.9.m9.1.1.7.1.cmml">⁢</mo><mi id="S3.5.p2.9.m9.1.1.7.3" mathvariant="normal" xref="S3.5.p2.9.m9.1.1.7.3.cmml">⋯</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.5.p2.9.m9.1b"><apply id="S3.5.p2.9.m9.1.1.cmml" xref="S3.5.p2.9.m9.1.1"><plus id="S3.5.p2.9.m9.1.1.1.cmml" xref="S3.5.p2.9.m9.1.1.1"></plus><apply id="S3.5.p2.9.m9.1.1.2.cmml" xref="S3.5.p2.9.m9.1.1.2"><apply id="S3.5.p2.9.m9.1.1.2.1.cmml" xref="S3.5.p2.9.m9.1.1.2.1"><csymbol cd="ambiguous" id="S3.5.p2.9.m9.1.1.2.1.1.cmml" xref="S3.5.p2.9.m9.1.1.2.1">subscript</csymbol><sum id="S3.5.p2.9.m9.1.1.2.1.2.cmml" xref="S3.5.p2.9.m9.1.1.2.1.2"></sum><apply id="S3.5.p2.9.m9.1.1.2.1.3.cmml" xref="S3.5.p2.9.m9.1.1.2.1.3"><in id="S3.5.p2.9.m9.1.1.2.1.3.1.cmml" xref="S3.5.p2.9.m9.1.1.2.1.3.1"></in><ci id="S3.5.p2.9.m9.1.1.2.1.3.2.cmml" xref="S3.5.p2.9.m9.1.1.2.1.3.2">𝑗</ci><apply id="S3.5.p2.9.m9.1.1.2.1.3.3.cmml" xref="S3.5.p2.9.m9.1.1.2.1.3.3"><csymbol cd="ambiguous" id="S3.5.p2.9.m9.1.1.2.1.3.3.1.cmml" xref="S3.5.p2.9.m9.1.1.2.1.3.3">subscript</csymbol><ci id="S3.5.p2.9.m9.1.1.2.1.3.3.2.cmml" xref="S3.5.p2.9.m9.1.1.2.1.3.3.2">𝐼</ci><ci id="S3.5.p2.9.m9.1.1.2.1.3.3.3.cmml" xref="S3.5.p2.9.m9.1.1.2.1.3.3.3">𝑖</ci></apply></apply></apply><apply id="S3.5.p2.9.m9.1.1.2.2.cmml" xref="S3.5.p2.9.m9.1.1.2.2"><times id="S3.5.p2.9.m9.1.1.2.2.1.cmml" xref="S3.5.p2.9.m9.1.1.2.2.1"></times><apply id="S3.5.p2.9.m9.1.1.2.2.2.cmml" xref="S3.5.p2.9.m9.1.1.2.2.2"><csymbol cd="ambiguous" id="S3.5.p2.9.m9.1.1.2.2.2.1.cmml" xref="S3.5.p2.9.m9.1.1.2.2.2">subscript</csymbol><ci id="S3.5.p2.9.m9.1.1.2.2.2.2.cmml" xref="S3.5.p2.9.m9.1.1.2.2.2.2">𝐴</ci><ci id="S3.5.p2.9.m9.1.1.2.2.2.3.cmml" xref="S3.5.p2.9.m9.1.1.2.2.2.3">𝑗</ci></apply><ci id="S3.5.p2.9.m9.1.1.2.2.3.cmml" xref="S3.5.p2.9.m9.1.1.2.2.3">⊵</ci><ci id="S3.5.p2.9.m9.1.1.2.2.4.cmml" xref="S3.5.p2.9.m9.1.1.2.2.4">⋯</ci></apply></apply><cn id="S3.5.p2.9.m9.1.1.3.cmml" type="integer" xref="S3.5.p2.9.m9.1.1.3">1</cn><apply id="S3.5.p2.9.m9.1.1.4.cmml" xref="S3.5.p2.9.m9.1.1.4"><csymbol cd="ambiguous" id="S3.5.p2.9.m9.1.1.4.1.cmml" xref="S3.5.p2.9.m9.1.1.4">subscript</csymbol><ci id="S3.5.p2.9.m9.1.1.4.2.cmml" xref="S3.5.p2.9.m9.1.1.4.2">𝐴</ci><ci id="S3.5.p2.9.m9.1.1.4.3.cmml" xref="S3.5.p2.9.m9.1.1.4.3">𝑖</ci></apply><cn id="S3.5.p2.9.m9.1.1.5.cmml" type="integer" xref="S3.5.p2.9.m9.1.1.5">1</cn><apply id="S3.5.p2.9.m9.1.1.6.cmml" xref="S3.5.p2.9.m9.1.1.6"><csymbol cd="ambiguous" id="S3.5.p2.9.m9.1.1.6.1.cmml" xref="S3.5.p2.9.m9.1.1.6">subscript</csymbol><ci id="S3.5.p2.9.m9.1.1.6.2.cmml" xref="S3.5.p2.9.m9.1.1.6.2">𝐴</ci><ci id="S3.5.p2.9.m9.1.1.6.3.cmml" xref="S3.5.p2.9.m9.1.1.6.3">𝑖</ci></apply><apply id="S3.5.p2.9.m9.1.1.7.cmml" xref="S3.5.p2.9.m9.1.1.7"><times id="S3.5.p2.9.m9.1.1.7.1.cmml" xref="S3.5.p2.9.m9.1.1.7.1"></times><cn id="S3.5.p2.9.m9.1.1.7.2.cmml" type="integer" xref="S3.5.p2.9.m9.1.1.7.2">1</cn><ci id="S3.5.p2.9.m9.1.1.7.3.cmml" xref="S3.5.p2.9.m9.1.1.7.3">⋯</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.5.p2.9.m9.1c">\sum_{j\in I_{i}}A_{j}\trianglerighteq\cdots+1+A_{i}+1+A_{i}+1\cdots</annotation><annotation encoding="application/x-llamapun" id="S3.5.p2.9.m9.1d">∑ start_POSTSUBSCRIPT italic_j ∈ italic_I start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ⊵ ⋯ + 1 + italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + 1 + italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + 1 ⋯</annotation></semantics></math>, therefore it is enough to prove that <math alttext="\cdots+1+A_{i}+1+A_{i}+1\cdots\trianglerighteq A_{i}" class="ltx_Math" display="inline" id="S3.5.p2.10.m10.1"><semantics id="S3.5.p2.10.m10.1a"><mrow id="S3.5.p2.10.m10.1.1" xref="S3.5.p2.10.m10.1.1.cmml"><mi id="S3.5.p2.10.m10.1.1.2" mathvariant="normal" xref="S3.5.p2.10.m10.1.1.2.cmml">⋯</mi><mo id="S3.5.p2.10.m10.1.1.1" xref="S3.5.p2.10.m10.1.1.1.cmml">+</mo><mn id="S3.5.p2.10.m10.1.1.3" xref="S3.5.p2.10.m10.1.1.3.cmml">1</mn><mo id="S3.5.p2.10.m10.1.1.1a" xref="S3.5.p2.10.m10.1.1.1.cmml">+</mo><msub id="S3.5.p2.10.m10.1.1.4" xref="S3.5.p2.10.m10.1.1.4.cmml"><mi id="S3.5.p2.10.m10.1.1.4.2" xref="S3.5.p2.10.m10.1.1.4.2.cmml">A</mi><mi id="S3.5.p2.10.m10.1.1.4.3" xref="S3.5.p2.10.m10.1.1.4.3.cmml">i</mi></msub><mo id="S3.5.p2.10.m10.1.1.1b" xref="S3.5.p2.10.m10.1.1.1.cmml">+</mo><mn id="S3.5.p2.10.m10.1.1.5" xref="S3.5.p2.10.m10.1.1.5.cmml">1</mn><mo id="S3.5.p2.10.m10.1.1.1c" xref="S3.5.p2.10.m10.1.1.1.cmml">+</mo><msub id="S3.5.p2.10.m10.1.1.6" xref="S3.5.p2.10.m10.1.1.6.cmml"><mi id="S3.5.p2.10.m10.1.1.6.2" xref="S3.5.p2.10.m10.1.1.6.2.cmml">A</mi><mi id="S3.5.p2.10.m10.1.1.6.3" xref="S3.5.p2.10.m10.1.1.6.3.cmml">i</mi></msub><mo id="S3.5.p2.10.m10.1.1.1d" xref="S3.5.p2.10.m10.1.1.1.cmml">+</mo><mrow id="S3.5.p2.10.m10.1.1.7" xref="S3.5.p2.10.m10.1.1.7.cmml"><mn id="S3.5.p2.10.m10.1.1.7.2" xref="S3.5.p2.10.m10.1.1.7.2.cmml">1</mn><mo id="S3.5.p2.10.m10.1.1.7.1" xref="S3.5.p2.10.m10.1.1.7.1.cmml">⁢</mo><mi id="S3.5.p2.10.m10.1.1.7.3" mathvariant="normal" xref="S3.5.p2.10.m10.1.1.7.3.cmml">⋯</mi><mo id="S3.5.p2.10.m10.1.1.7.1a" xref="S3.5.p2.10.m10.1.1.7.1.cmml">⁢</mo><mi id="S3.5.p2.10.m10.1.1.7.4" mathvariant="normal" xref="S3.5.p2.10.m10.1.1.7.4.cmml">⊵</mi><mo id="S3.5.p2.10.m10.1.1.7.1b" xref="S3.5.p2.10.m10.1.1.7.1.cmml">⁢</mo><msub id="S3.5.p2.10.m10.1.1.7.5" xref="S3.5.p2.10.m10.1.1.7.5.cmml"><mi id="S3.5.p2.10.m10.1.1.7.5.2" xref="S3.5.p2.10.m10.1.1.7.5.2.cmml">A</mi><mi id="S3.5.p2.10.m10.1.1.7.5.3" xref="S3.5.p2.10.m10.1.1.7.5.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.5.p2.10.m10.1b"><apply id="S3.5.p2.10.m10.1.1.cmml" xref="S3.5.p2.10.m10.1.1"><plus id="S3.5.p2.10.m10.1.1.1.cmml" xref="S3.5.p2.10.m10.1.1.1"></plus><ci id="S3.5.p2.10.m10.1.1.2.cmml" xref="S3.5.p2.10.m10.1.1.2">⋯</ci><cn id="S3.5.p2.10.m10.1.1.3.cmml" type="integer" xref="S3.5.p2.10.m10.1.1.3">1</cn><apply id="S3.5.p2.10.m10.1.1.4.cmml" xref="S3.5.p2.10.m10.1.1.4"><csymbol cd="ambiguous" id="S3.5.p2.10.m10.1.1.4.1.cmml" xref="S3.5.p2.10.m10.1.1.4">subscript</csymbol><ci id="S3.5.p2.10.m10.1.1.4.2.cmml" xref="S3.5.p2.10.m10.1.1.4.2">𝐴</ci><ci id="S3.5.p2.10.m10.1.1.4.3.cmml" xref="S3.5.p2.10.m10.1.1.4.3">𝑖</ci></apply><cn id="S3.5.p2.10.m10.1.1.5.cmml" type="integer" xref="S3.5.p2.10.m10.1.1.5">1</cn><apply id="S3.5.p2.10.m10.1.1.6.cmml" xref="S3.5.p2.10.m10.1.1.6"><csymbol cd="ambiguous" id="S3.5.p2.10.m10.1.1.6.1.cmml" xref="S3.5.p2.10.m10.1.1.6">subscript</csymbol><ci id="S3.5.p2.10.m10.1.1.6.2.cmml" xref="S3.5.p2.10.m10.1.1.6.2">𝐴</ci><ci id="S3.5.p2.10.m10.1.1.6.3.cmml" xref="S3.5.p2.10.m10.1.1.6.3">𝑖</ci></apply><apply id="S3.5.p2.10.m10.1.1.7.cmml" xref="S3.5.p2.10.m10.1.1.7"><times id="S3.5.p2.10.m10.1.1.7.1.cmml" xref="S3.5.p2.10.m10.1.1.7.1"></times><cn id="S3.5.p2.10.m10.1.1.7.2.cmml" type="integer" xref="S3.5.p2.10.m10.1.1.7.2">1</cn><ci id="S3.5.p2.10.m10.1.1.7.3.cmml" xref="S3.5.p2.10.m10.1.1.7.3">⋯</ci><ci id="S3.5.p2.10.m10.1.1.7.4.cmml" xref="S3.5.p2.10.m10.1.1.7.4">⊵</ci><apply id="S3.5.p2.10.m10.1.1.7.5.cmml" xref="S3.5.p2.10.m10.1.1.7.5"><csymbol cd="ambiguous" id="S3.5.p2.10.m10.1.1.7.5.1.cmml" xref="S3.5.p2.10.m10.1.1.7.5">subscript</csymbol><ci id="S3.5.p2.10.m10.1.1.7.5.2.cmml" xref="S3.5.p2.10.m10.1.1.7.5.2">𝐴</ci><ci id="S3.5.p2.10.m10.1.1.7.5.3.cmml" xref="S3.5.p2.10.m10.1.1.7.5.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.5.p2.10.m10.1c">\cdots+1+A_{i}+1+A_{i}+1\cdots\trianglerighteq A_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.5.p2.10.m10.1d">⋯ + 1 + italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + 1 + italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + 1 ⋯ ⊵ italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.6.p3"> <p class="ltx_p" id="S3.6.p3.8">This is done by cases depending on the cofinality and coinitiality of <math alttext="A_{i}" class="ltx_Math" display="inline" id="S3.6.p3.1.m1.1"><semantics id="S3.6.p3.1.m1.1a"><msub id="S3.6.p3.1.m1.1.1" xref="S3.6.p3.1.m1.1.1.cmml"><mi id="S3.6.p3.1.m1.1.1.2" xref="S3.6.p3.1.m1.1.1.2.cmml">A</mi><mi id="S3.6.p3.1.m1.1.1.3" xref="S3.6.p3.1.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.6.p3.1.m1.1b"><apply id="S3.6.p3.1.m1.1.1.cmml" xref="S3.6.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S3.6.p3.1.m1.1.1.1.cmml" xref="S3.6.p3.1.m1.1.1">subscript</csymbol><ci id="S3.6.p3.1.m1.1.1.2.cmml" xref="S3.6.p3.1.m1.1.1.2">𝐴</ci><ci id="S3.6.p3.1.m1.1.1.3.cmml" xref="S3.6.p3.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.6.p3.1.m1.1c">A_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.6.p3.1.m1.1d">italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. If both the cofinality and the coinitiality of <math alttext="A_{i}" class="ltx_Math" display="inline" id="S3.6.p3.2.m2.1"><semantics id="S3.6.p3.2.m2.1a"><msub id="S3.6.p3.2.m2.1.1" xref="S3.6.p3.2.m2.1.1.cmml"><mi id="S3.6.p3.2.m2.1.1.2" xref="S3.6.p3.2.m2.1.1.2.cmml">A</mi><mi id="S3.6.p3.2.m2.1.1.3" xref="S3.6.p3.2.m2.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.6.p3.2.m2.1b"><apply id="S3.6.p3.2.m2.1.1.cmml" xref="S3.6.p3.2.m2.1.1"><csymbol cd="ambiguous" id="S3.6.p3.2.m2.1.1.1.cmml" xref="S3.6.p3.2.m2.1.1">subscript</csymbol><ci id="S3.6.p3.2.m2.1.1.2.cmml" xref="S3.6.p3.2.m2.1.1.2">𝐴</ci><ci id="S3.6.p3.2.m2.1.1.3.cmml" xref="S3.6.p3.2.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.6.p3.2.m2.1c">A_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.6.p3.2.m2.1d">italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> are <math alttext="1" class="ltx_Math" display="inline" id="S3.6.p3.3.m3.1"><semantics id="S3.6.p3.3.m3.1a"><mn id="S3.6.p3.3.m3.1.1" xref="S3.6.p3.3.m3.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S3.6.p3.3.m3.1b"><cn id="S3.6.p3.3.m3.1.1.cmml" type="integer" xref="S3.6.p3.3.m3.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.6.p3.3.m3.1c">1</annotation><annotation encoding="application/x-llamapun" id="S3.6.p3.3.m3.1d">1</annotation></semantics></math> this is trivial. We do the case when the coinitiality is <math alttext="1" class="ltx_Math" display="inline" id="S3.6.p3.4.m4.1"><semantics id="S3.6.p3.4.m4.1a"><mn id="S3.6.p3.4.m4.1.1" xref="S3.6.p3.4.m4.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S3.6.p3.4.m4.1b"><cn id="S3.6.p3.4.m4.1.1.cmml" type="integer" xref="S3.6.p3.4.m4.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.6.p3.4.m4.1c">1</annotation><annotation encoding="application/x-llamapun" id="S3.6.p3.4.m4.1d">1</annotation></semantics></math> and the cofinality is <math alttext="\omega" class="ltx_Math" display="inline" id="S3.6.p3.5.m5.1"><semantics id="S3.6.p3.5.m5.1a"><mi id="S3.6.p3.5.m5.1.1" xref="S3.6.p3.5.m5.1.1.cmml">ω</mi><annotation-xml encoding="MathML-Content" id="S3.6.p3.5.m5.1b"><ci id="S3.6.p3.5.m5.1.1.cmml" xref="S3.6.p3.5.m5.1.1">𝜔</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.6.p3.5.m5.1c">\omega</annotation><annotation encoding="application/x-llamapun" id="S3.6.p3.5.m5.1d">italic_ω</annotation></semantics></math>, the case when <math alttext="(\operatorname{coi}(A_{i}),\operatorname{cof}(A_{i}))=(\omega^{\star},1)" class="ltx_Math" display="inline" id="S3.6.p3.6.m6.6"><semantics id="S3.6.p3.6.m6.6a"><mrow id="S3.6.p3.6.m6.6.6" xref="S3.6.p3.6.m6.6.6.cmml"><mrow id="S3.6.p3.6.m6.5.5.2.2" xref="S3.6.p3.6.m6.5.5.2.3.cmml"><mo id="S3.6.p3.6.m6.5.5.2.2.3" stretchy="false" xref="S3.6.p3.6.m6.5.5.2.3.cmml">(</mo><mrow id="S3.6.p3.6.m6.4.4.1.1.1.1" xref="S3.6.p3.6.m6.4.4.1.1.1.2.cmml"><mi id="S3.6.p3.6.m6.1.1" xref="S3.6.p3.6.m6.1.1.cmml">coi</mi><mo id="S3.6.p3.6.m6.4.4.1.1.1.1a" xref="S3.6.p3.6.m6.4.4.1.1.1.2.cmml">⁡</mo><mrow id="S3.6.p3.6.m6.4.4.1.1.1.1.1" xref="S3.6.p3.6.m6.4.4.1.1.1.2.cmml"><mo id="S3.6.p3.6.m6.4.4.1.1.1.1.1.2" stretchy="false" xref="S3.6.p3.6.m6.4.4.1.1.1.2.cmml">(</mo><msub id="S3.6.p3.6.m6.4.4.1.1.1.1.1.1" xref="S3.6.p3.6.m6.4.4.1.1.1.1.1.1.cmml"><mi id="S3.6.p3.6.m6.4.4.1.1.1.1.1.1.2" xref="S3.6.p3.6.m6.4.4.1.1.1.1.1.1.2.cmml">A</mi><mi id="S3.6.p3.6.m6.4.4.1.1.1.1.1.1.3" xref="S3.6.p3.6.m6.4.4.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S3.6.p3.6.m6.4.4.1.1.1.1.1.3" stretchy="false" xref="S3.6.p3.6.m6.4.4.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S3.6.p3.6.m6.5.5.2.2.4" xref="S3.6.p3.6.m6.5.5.2.3.cmml">,</mo><mrow id="S3.6.p3.6.m6.5.5.2.2.2.1" xref="S3.6.p3.6.m6.5.5.2.2.2.2.cmml"><mi id="S3.6.p3.6.m6.2.2" xref="S3.6.p3.6.m6.2.2.cmml">cof</mi><mo id="S3.6.p3.6.m6.5.5.2.2.2.1a" xref="S3.6.p3.6.m6.5.5.2.2.2.2.cmml">⁡</mo><mrow id="S3.6.p3.6.m6.5.5.2.2.2.1.1" xref="S3.6.p3.6.m6.5.5.2.2.2.2.cmml"><mo id="S3.6.p3.6.m6.5.5.2.2.2.1.1.2" stretchy="false" xref="S3.6.p3.6.m6.5.5.2.2.2.2.cmml">(</mo><msub id="S3.6.p3.6.m6.5.5.2.2.2.1.1.1" xref="S3.6.p3.6.m6.5.5.2.2.2.1.1.1.cmml"><mi id="S3.6.p3.6.m6.5.5.2.2.2.1.1.1.2" xref="S3.6.p3.6.m6.5.5.2.2.2.1.1.1.2.cmml">A</mi><mi id="S3.6.p3.6.m6.5.5.2.2.2.1.1.1.3" xref="S3.6.p3.6.m6.5.5.2.2.2.1.1.1.3.cmml">i</mi></msub><mo id="S3.6.p3.6.m6.5.5.2.2.2.1.1.3" stretchy="false" xref="S3.6.p3.6.m6.5.5.2.2.2.2.cmml">)</mo></mrow></mrow><mo id="S3.6.p3.6.m6.5.5.2.2.5" stretchy="false" xref="S3.6.p3.6.m6.5.5.2.3.cmml">)</mo></mrow><mo id="S3.6.p3.6.m6.6.6.4" xref="S3.6.p3.6.m6.6.6.4.cmml">=</mo><mrow id="S3.6.p3.6.m6.6.6.3.1" xref="S3.6.p3.6.m6.6.6.3.2.cmml"><mo id="S3.6.p3.6.m6.6.6.3.1.2" stretchy="false" xref="S3.6.p3.6.m6.6.6.3.2.cmml">(</mo><msup id="S3.6.p3.6.m6.6.6.3.1.1" xref="S3.6.p3.6.m6.6.6.3.1.1.cmml"><mi id="S3.6.p3.6.m6.6.6.3.1.1.2" xref="S3.6.p3.6.m6.6.6.3.1.1.2.cmml">ω</mi><mo id="S3.6.p3.6.m6.6.6.3.1.1.3" xref="S3.6.p3.6.m6.6.6.3.1.1.3.cmml">⋆</mo></msup><mo id="S3.6.p3.6.m6.6.6.3.1.3" xref="S3.6.p3.6.m6.6.6.3.2.cmml">,</mo><mn id="S3.6.p3.6.m6.3.3" xref="S3.6.p3.6.m6.3.3.cmml">1</mn><mo id="S3.6.p3.6.m6.6.6.3.1.4" stretchy="false" xref="S3.6.p3.6.m6.6.6.3.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.6.p3.6.m6.6b"><apply id="S3.6.p3.6.m6.6.6.cmml" xref="S3.6.p3.6.m6.6.6"><eq id="S3.6.p3.6.m6.6.6.4.cmml" xref="S3.6.p3.6.m6.6.6.4"></eq><interval closure="open" id="S3.6.p3.6.m6.5.5.2.3.cmml" xref="S3.6.p3.6.m6.5.5.2.2"><apply id="S3.6.p3.6.m6.4.4.1.1.1.2.cmml" xref="S3.6.p3.6.m6.4.4.1.1.1.1"><ci id="S3.6.p3.6.m6.1.1.cmml" xref="S3.6.p3.6.m6.1.1">coi</ci><apply id="S3.6.p3.6.m6.4.4.1.1.1.1.1.1.cmml" xref="S3.6.p3.6.m6.4.4.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.6.p3.6.m6.4.4.1.1.1.1.1.1.1.cmml" xref="S3.6.p3.6.m6.4.4.1.1.1.1.1.1">subscript</csymbol><ci id="S3.6.p3.6.m6.4.4.1.1.1.1.1.1.2.cmml" xref="S3.6.p3.6.m6.4.4.1.1.1.1.1.1.2">𝐴</ci><ci id="S3.6.p3.6.m6.4.4.1.1.1.1.1.1.3.cmml" xref="S3.6.p3.6.m6.4.4.1.1.1.1.1.1.3">𝑖</ci></apply></apply><apply id="S3.6.p3.6.m6.5.5.2.2.2.2.cmml" xref="S3.6.p3.6.m6.5.5.2.2.2.1"><ci id="S3.6.p3.6.m6.2.2.cmml" xref="S3.6.p3.6.m6.2.2">cof</ci><apply id="S3.6.p3.6.m6.5.5.2.2.2.1.1.1.cmml" xref="S3.6.p3.6.m6.5.5.2.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.6.p3.6.m6.5.5.2.2.2.1.1.1.1.cmml" xref="S3.6.p3.6.m6.5.5.2.2.2.1.1.1">subscript</csymbol><ci id="S3.6.p3.6.m6.5.5.2.2.2.1.1.1.2.cmml" xref="S3.6.p3.6.m6.5.5.2.2.2.1.1.1.2">𝐴</ci><ci id="S3.6.p3.6.m6.5.5.2.2.2.1.1.1.3.cmml" xref="S3.6.p3.6.m6.5.5.2.2.2.1.1.1.3">𝑖</ci></apply></apply></interval><interval closure="open" id="S3.6.p3.6.m6.6.6.3.2.cmml" xref="S3.6.p3.6.m6.6.6.3.1"><apply id="S3.6.p3.6.m6.6.6.3.1.1.cmml" xref="S3.6.p3.6.m6.6.6.3.1.1"><csymbol cd="ambiguous" id="S3.6.p3.6.m6.6.6.3.1.1.1.cmml" xref="S3.6.p3.6.m6.6.6.3.1.1">superscript</csymbol><ci id="S3.6.p3.6.m6.6.6.3.1.1.2.cmml" xref="S3.6.p3.6.m6.6.6.3.1.1.2">𝜔</ci><ci id="S3.6.p3.6.m6.6.6.3.1.1.3.cmml" xref="S3.6.p3.6.m6.6.6.3.1.1.3">⋆</ci></apply><cn id="S3.6.p3.6.m6.3.3.cmml" type="integer" xref="S3.6.p3.6.m6.3.3">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.6.p3.6.m6.6c">(\operatorname{coi}(A_{i}),\operatorname{cof}(A_{i}))=(\omega^{\star},1)</annotation><annotation encoding="application/x-llamapun" id="S3.6.p3.6.m6.6d">( roman_coi ( italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) , roman_cof ( italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ) = ( italic_ω start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT , 1 )</annotation></semantics></math> is symmetric, and when it is <math alttext="(\omega^{\star},\omega)" class="ltx_Math" display="inline" id="S3.6.p3.7.m7.2"><semantics id="S3.6.p3.7.m7.2a"><mrow id="S3.6.p3.7.m7.2.2.1" xref="S3.6.p3.7.m7.2.2.2.cmml"><mo id="S3.6.p3.7.m7.2.2.1.2" stretchy="false" xref="S3.6.p3.7.m7.2.2.2.cmml">(</mo><msup id="S3.6.p3.7.m7.2.2.1.1" xref="S3.6.p3.7.m7.2.2.1.1.cmml"><mi id="S3.6.p3.7.m7.2.2.1.1.2" xref="S3.6.p3.7.m7.2.2.1.1.2.cmml">ω</mi><mo id="S3.6.p3.7.m7.2.2.1.1.3" xref="S3.6.p3.7.m7.2.2.1.1.3.cmml">⋆</mo></msup><mo id="S3.6.p3.7.m7.2.2.1.3" xref="S3.6.p3.7.m7.2.2.2.cmml">,</mo><mi id="S3.6.p3.7.m7.1.1" xref="S3.6.p3.7.m7.1.1.cmml">ω</mi><mo id="S3.6.p3.7.m7.2.2.1.4" stretchy="false" xref="S3.6.p3.7.m7.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.6.p3.7.m7.2b"><interval closure="open" id="S3.6.p3.7.m7.2.2.2.cmml" xref="S3.6.p3.7.m7.2.2.1"><apply id="S3.6.p3.7.m7.2.2.1.1.cmml" xref="S3.6.p3.7.m7.2.2.1.1"><csymbol cd="ambiguous" id="S3.6.p3.7.m7.2.2.1.1.1.cmml" xref="S3.6.p3.7.m7.2.2.1.1">superscript</csymbol><ci id="S3.6.p3.7.m7.2.2.1.1.2.cmml" xref="S3.6.p3.7.m7.2.2.1.1.2">𝜔</ci><ci id="S3.6.p3.7.m7.2.2.1.1.3.cmml" xref="S3.6.p3.7.m7.2.2.1.1.3">⋆</ci></apply><ci id="S3.6.p3.7.m7.1.1.cmml" xref="S3.6.p3.7.m7.1.1">𝜔</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.6.p3.7.m7.2c">(\omega^{\star},\omega)</annotation><annotation encoding="application/x-llamapun" id="S3.6.p3.7.m7.2d">( italic_ω start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT , italic_ω )</annotation></semantics></math> is very similar. Recall that <math alttext="A_{i}" class="ltx_Math" display="inline" id="S3.6.p3.8.m8.1"><semantics id="S3.6.p3.8.m8.1a"><msub id="S3.6.p3.8.m8.1.1" xref="S3.6.p3.8.m8.1.1.cmml"><mi id="S3.6.p3.8.m8.1.1.2" xref="S3.6.p3.8.m8.1.1.2.cmml">A</mi><mi id="S3.6.p3.8.m8.1.1.3" xref="S3.6.p3.8.m8.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.6.p3.8.m8.1b"><apply id="S3.6.p3.8.m8.1.1.cmml" xref="S3.6.p3.8.m8.1.1"><csymbol cd="ambiguous" id="S3.6.p3.8.m8.1.1.1.cmml" xref="S3.6.p3.8.m8.1.1">subscript</csymbol><ci id="S3.6.p3.8.m8.1.1.2.cmml" xref="S3.6.p3.8.m8.1.1.2">𝐴</ci><ci id="S3.6.p3.8.m8.1.1.3.cmml" xref="S3.6.p3.8.m8.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.6.p3.8.m8.1c">A_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.6.p3.8.m8.1d">italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, begin strongly surjective, is short, and thus these are all possible cases.</p> </div> <div class="ltx_para" id="S3.7.p4"> <p class="ltx_p" id="S3.7.p4.5">Let <math alttext="\langle a_{n}:n<\omega\rangle" class="ltx_math_unparsed" display="inline" id="S3.7.p4.1.m1.1"><semantics id="S3.7.p4.1.m1.1a"><mrow id="S3.7.p4.1.m1.1b"><mo id="S3.7.p4.1.m1.1.2" stretchy="false">⟨</mo><msub id="S3.7.p4.1.m1.1.3"><mi id="S3.7.p4.1.m1.1.3.2">a</mi><mi id="S3.7.p4.1.m1.1.3.3">n</mi></msub><mo id="S3.7.p4.1.m1.1.4" lspace="0.278em" rspace="0.278em">:</mo><mi id="S3.7.p4.1.m1.1.5">n</mi><mo id="S3.7.p4.1.m1.1.6"><</mo><mi id="S3.7.p4.1.m1.1.1">ω</mi><mo id="S3.7.p4.1.m1.1.7" stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex" id="S3.7.p4.1.m1.1c">\langle a_{n}:n<\omega\rangle</annotation><annotation encoding="application/x-llamapun" id="S3.7.p4.1.m1.1d">⟨ italic_a start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT : italic_n < italic_ω ⟩</annotation></semantics></math> be increasing and cofinal in <math alttext="A" class="ltx_Math" display="inline" id="S3.7.p4.2.m2.1"><semantics id="S3.7.p4.2.m2.1a"><mi id="S3.7.p4.2.m2.1.1" xref="S3.7.p4.2.m2.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S3.7.p4.2.m2.1b"><ci id="S3.7.p4.2.m2.1.1.cmml" xref="S3.7.p4.2.m2.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.7.p4.2.m2.1c">A</annotation><annotation encoding="application/x-llamapun" id="S3.7.p4.2.m2.1d">italic_A</annotation></semantics></math>, and such that <math alttext="a_{0}" class="ltx_Math" display="inline" id="S3.7.p4.3.m3.1"><semantics id="S3.7.p4.3.m3.1a"><msub id="S3.7.p4.3.m3.1.1" xref="S3.7.p4.3.m3.1.1.cmml"><mi id="S3.7.p4.3.m3.1.1.2" xref="S3.7.p4.3.m3.1.1.2.cmml">a</mi><mn id="S3.7.p4.3.m3.1.1.3" xref="S3.7.p4.3.m3.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S3.7.p4.3.m3.1b"><apply id="S3.7.p4.3.m3.1.1.cmml" xref="S3.7.p4.3.m3.1.1"><csymbol cd="ambiguous" id="S3.7.p4.3.m3.1.1.1.cmml" xref="S3.7.p4.3.m3.1.1">subscript</csymbol><ci id="S3.7.p4.3.m3.1.1.2.cmml" xref="S3.7.p4.3.m3.1.1.2">𝑎</ci><cn id="S3.7.p4.3.m3.1.1.3.cmml" type="integer" xref="S3.7.p4.3.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.7.p4.3.m3.1c">a_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.7.p4.3.m3.1d">italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> is the left endpoint of <math alttext="A" class="ltx_Math" display="inline" id="S3.7.p4.4.m4.1"><semantics id="S3.7.p4.4.m4.1a"><mi id="S3.7.p4.4.m4.1.1" xref="S3.7.p4.4.m4.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S3.7.p4.4.m4.1b"><ci id="S3.7.p4.4.m4.1.1.cmml" xref="S3.7.p4.4.m4.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.7.p4.4.m4.1c">A</annotation><annotation encoding="application/x-llamapun" id="S3.7.p4.4.m4.1d">italic_A</annotation></semantics></math>. Note that formally we think of <math alttext="\cdots+1+A_{i}+1+A_{i}+1\cdots" class="ltx_Math" display="inline" id="S3.7.p4.5.m5.1"><semantics id="S3.7.p4.5.m5.1a"><mrow id="S3.7.p4.5.m5.1.1" xref="S3.7.p4.5.m5.1.1.cmml"><mi id="S3.7.p4.5.m5.1.1.2" mathvariant="normal" xref="S3.7.p4.5.m5.1.1.2.cmml">⋯</mi><mo id="S3.7.p4.5.m5.1.1.1" xref="S3.7.p4.5.m5.1.1.1.cmml">+</mo><mn id="S3.7.p4.5.m5.1.1.3" xref="S3.7.p4.5.m5.1.1.3.cmml">1</mn><mo id="S3.7.p4.5.m5.1.1.1a" xref="S3.7.p4.5.m5.1.1.1.cmml">+</mo><msub id="S3.7.p4.5.m5.1.1.4" xref="S3.7.p4.5.m5.1.1.4.cmml"><mi id="S3.7.p4.5.m5.1.1.4.2" xref="S3.7.p4.5.m5.1.1.4.2.cmml">A</mi><mi id="S3.7.p4.5.m5.1.1.4.3" xref="S3.7.p4.5.m5.1.1.4.3.cmml">i</mi></msub><mo id="S3.7.p4.5.m5.1.1.1b" xref="S3.7.p4.5.m5.1.1.1.cmml">+</mo><mn id="S3.7.p4.5.m5.1.1.5" xref="S3.7.p4.5.m5.1.1.5.cmml">1</mn><mo id="S3.7.p4.5.m5.1.1.1c" xref="S3.7.p4.5.m5.1.1.1.cmml">+</mo><msub id="S3.7.p4.5.m5.1.1.6" xref="S3.7.p4.5.m5.1.1.6.cmml"><mi id="S3.7.p4.5.m5.1.1.6.2" xref="S3.7.p4.5.m5.1.1.6.2.cmml">A</mi><mi id="S3.7.p4.5.m5.1.1.6.3" xref="S3.7.p4.5.m5.1.1.6.3.cmml">i</mi></msub><mo id="S3.7.p4.5.m5.1.1.1d" xref="S3.7.p4.5.m5.1.1.1.cmml">+</mo><mrow id="S3.7.p4.5.m5.1.1.7" xref="S3.7.p4.5.m5.1.1.7.cmml"><mn id="S3.7.p4.5.m5.1.1.7.2" xref="S3.7.p4.5.m5.1.1.7.2.cmml">1</mn><mo id="S3.7.p4.5.m5.1.1.7.1" xref="S3.7.p4.5.m5.1.1.7.1.cmml">⁢</mo><mi id="S3.7.p4.5.m5.1.1.7.3" mathvariant="normal" xref="S3.7.p4.5.m5.1.1.7.3.cmml">⋯</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.7.p4.5.m5.1b"><apply id="S3.7.p4.5.m5.1.1.cmml" xref="S3.7.p4.5.m5.1.1"><plus id="S3.7.p4.5.m5.1.1.1.cmml" xref="S3.7.p4.5.m5.1.1.1"></plus><ci id="S3.7.p4.5.m5.1.1.2.cmml" xref="S3.7.p4.5.m5.1.1.2">⋯</ci><cn id="S3.7.p4.5.m5.1.1.3.cmml" type="integer" xref="S3.7.p4.5.m5.1.1.3">1</cn><apply id="S3.7.p4.5.m5.1.1.4.cmml" xref="S3.7.p4.5.m5.1.1.4"><csymbol cd="ambiguous" id="S3.7.p4.5.m5.1.1.4.1.cmml" xref="S3.7.p4.5.m5.1.1.4">subscript</csymbol><ci id="S3.7.p4.5.m5.1.1.4.2.cmml" xref="S3.7.p4.5.m5.1.1.4.2">𝐴</ci><ci id="S3.7.p4.5.m5.1.1.4.3.cmml" xref="S3.7.p4.5.m5.1.1.4.3">𝑖</ci></apply><cn id="S3.7.p4.5.m5.1.1.5.cmml" type="integer" xref="S3.7.p4.5.m5.1.1.5">1</cn><apply id="S3.7.p4.5.m5.1.1.6.cmml" xref="S3.7.p4.5.m5.1.1.6"><csymbol cd="ambiguous" id="S3.7.p4.5.m5.1.1.6.1.cmml" xref="S3.7.p4.5.m5.1.1.6">subscript</csymbol><ci id="S3.7.p4.5.m5.1.1.6.2.cmml" xref="S3.7.p4.5.m5.1.1.6.2">𝐴</ci><ci id="S3.7.p4.5.m5.1.1.6.3.cmml" xref="S3.7.p4.5.m5.1.1.6.3">𝑖</ci></apply><apply id="S3.7.p4.5.m5.1.1.7.cmml" xref="S3.7.p4.5.m5.1.1.7"><times id="S3.7.p4.5.m5.1.1.7.1.cmml" xref="S3.7.p4.5.m5.1.1.7.1"></times><cn id="S3.7.p4.5.m5.1.1.7.2.cmml" type="integer" xref="S3.7.p4.5.m5.1.1.7.2">1</cn><ci id="S3.7.p4.5.m5.1.1.7.3.cmml" xref="S3.7.p4.5.m5.1.1.7.3">⋯</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.7.p4.5.m5.1c">\cdots+1+A_{i}+1+A_{i}+1\cdots</annotation><annotation encoding="application/x-llamapun" id="S3.7.p4.5.m5.1d">⋯ + 1 + italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + 1 + italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + 1 ⋯</annotation></semantics></math> as</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\bigcup_{n\in\mathbb{Z}}\{2n\}\times 1\cup\bigcup_{n\in\mathbb{Z}}\{2n+1\}% \times A_{i}" class="ltx_Math" display="block" id="S3.Ex1.m1.2"><semantics id="S3.Ex1.m1.2a"><mrow id="S3.Ex1.m1.2.2" xref="S3.Ex1.m1.2.2.cmml"><mrow id="S3.Ex1.m1.1.1.1" xref="S3.Ex1.m1.1.1.1.cmml"><munder id="S3.Ex1.m1.1.1.1.2" xref="S3.Ex1.m1.1.1.1.2.cmml"><mo id="S3.Ex1.m1.1.1.1.2.2" movablelimits="false" xref="S3.Ex1.m1.1.1.1.2.2.cmml">⋃</mo><mrow id="S3.Ex1.m1.1.1.1.2.3" xref="S3.Ex1.m1.1.1.1.2.3.cmml"><mi id="S3.Ex1.m1.1.1.1.2.3.2" xref="S3.Ex1.m1.1.1.1.2.3.2.cmml">n</mi><mo id="S3.Ex1.m1.1.1.1.2.3.1" xref="S3.Ex1.m1.1.1.1.2.3.1.cmml">∈</mo><mi id="S3.Ex1.m1.1.1.1.2.3.3" xref="S3.Ex1.m1.1.1.1.2.3.3.cmml">ℤ</mi></mrow></munder><mrow id="S3.Ex1.m1.1.1.1.1" xref="S3.Ex1.m1.1.1.1.1.cmml"><mrow id="S3.Ex1.m1.1.1.1.1.1.1" xref="S3.Ex1.m1.1.1.1.1.1.2.cmml"><mo id="S3.Ex1.m1.1.1.1.1.1.1.2" lspace="0em" stretchy="false" xref="S3.Ex1.m1.1.1.1.1.1.2.cmml">{</mo><mrow id="S3.Ex1.m1.1.1.1.1.1.1.1" xref="S3.Ex1.m1.1.1.1.1.1.1.1.cmml"><mn id="S3.Ex1.m1.1.1.1.1.1.1.1.2" xref="S3.Ex1.m1.1.1.1.1.1.1.1.2.cmml">2</mn><mo id="S3.Ex1.m1.1.1.1.1.1.1.1.1" xref="S3.Ex1.m1.1.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S3.Ex1.m1.1.1.1.1.1.1.1.3" xref="S3.Ex1.m1.1.1.1.1.1.1.1.3.cmml">n</mi></mrow><mo id="S3.Ex1.m1.1.1.1.1.1.1.3" rspace="0.055em" stretchy="false" xref="S3.Ex1.m1.1.1.1.1.1.2.cmml">}</mo></mrow><mo id="S3.Ex1.m1.1.1.1.1.2" rspace="0.222em" xref="S3.Ex1.m1.1.1.1.1.2.cmml">×</mo><mn id="S3.Ex1.m1.1.1.1.1.3" xref="S3.Ex1.m1.1.1.1.1.3.cmml">1</mn></mrow></mrow><mo id="S3.Ex1.m1.2.2.3" rspace="0.055em" xref="S3.Ex1.m1.2.2.3.cmml">∪</mo><mrow id="S3.Ex1.m1.2.2.2" xref="S3.Ex1.m1.2.2.2.cmml"><munder id="S3.Ex1.m1.2.2.2.2" xref="S3.Ex1.m1.2.2.2.2.cmml"><mo id="S3.Ex1.m1.2.2.2.2.2" movablelimits="false" rspace="0em" xref="S3.Ex1.m1.2.2.2.2.2.cmml">⋃</mo><mrow id="S3.Ex1.m1.2.2.2.2.3" xref="S3.Ex1.m1.2.2.2.2.3.cmml"><mi id="S3.Ex1.m1.2.2.2.2.3.2" xref="S3.Ex1.m1.2.2.2.2.3.2.cmml">n</mi><mo id="S3.Ex1.m1.2.2.2.2.3.1" xref="S3.Ex1.m1.2.2.2.2.3.1.cmml">∈</mo><mi id="S3.Ex1.m1.2.2.2.2.3.3" xref="S3.Ex1.m1.2.2.2.2.3.3.cmml">ℤ</mi></mrow></munder><mrow id="S3.Ex1.m1.2.2.2.1" xref="S3.Ex1.m1.2.2.2.1.cmml"><mrow id="S3.Ex1.m1.2.2.2.1.1.1" xref="S3.Ex1.m1.2.2.2.1.1.2.cmml"><mo id="S3.Ex1.m1.2.2.2.1.1.1.2" stretchy="false" xref="S3.Ex1.m1.2.2.2.1.1.2.cmml">{</mo><mrow id="S3.Ex1.m1.2.2.2.1.1.1.1" xref="S3.Ex1.m1.2.2.2.1.1.1.1.cmml"><mrow id="S3.Ex1.m1.2.2.2.1.1.1.1.2" xref="S3.Ex1.m1.2.2.2.1.1.1.1.2.cmml"><mn id="S3.Ex1.m1.2.2.2.1.1.1.1.2.2" xref="S3.Ex1.m1.2.2.2.1.1.1.1.2.2.cmml">2</mn><mo id="S3.Ex1.m1.2.2.2.1.1.1.1.2.1" xref="S3.Ex1.m1.2.2.2.1.1.1.1.2.1.cmml">⁢</mo><mi id="S3.Ex1.m1.2.2.2.1.1.1.1.2.3" xref="S3.Ex1.m1.2.2.2.1.1.1.1.2.3.cmml">n</mi></mrow><mo id="S3.Ex1.m1.2.2.2.1.1.1.1.1" xref="S3.Ex1.m1.2.2.2.1.1.1.1.1.cmml">+</mo><mn id="S3.Ex1.m1.2.2.2.1.1.1.1.3" xref="S3.Ex1.m1.2.2.2.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.Ex1.m1.2.2.2.1.1.1.3" rspace="0.055em" stretchy="false" xref="S3.Ex1.m1.2.2.2.1.1.2.cmml">}</mo></mrow><mo id="S3.Ex1.m1.2.2.2.1.2" rspace="0.222em" xref="S3.Ex1.m1.2.2.2.1.2.cmml">×</mo><msub id="S3.Ex1.m1.2.2.2.1.3" xref="S3.Ex1.m1.2.2.2.1.3.cmml"><mi id="S3.Ex1.m1.2.2.2.1.3.2" xref="S3.Ex1.m1.2.2.2.1.3.2.cmml">A</mi><mi id="S3.Ex1.m1.2.2.2.1.3.3" xref="S3.Ex1.m1.2.2.2.1.3.3.cmml">i</mi></msub></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex1.m1.2b"><apply id="S3.Ex1.m1.2.2.cmml" xref="S3.Ex1.m1.2.2"><union id="S3.Ex1.m1.2.2.3.cmml" xref="S3.Ex1.m1.2.2.3"></union><apply id="S3.Ex1.m1.1.1.1.cmml" xref="S3.Ex1.m1.1.1.1"><apply id="S3.Ex1.m1.1.1.1.2.cmml" xref="S3.Ex1.m1.1.1.1.2"><csymbol cd="ambiguous" id="S3.Ex1.m1.1.1.1.2.1.cmml" xref="S3.Ex1.m1.1.1.1.2">subscript</csymbol><union id="S3.Ex1.m1.1.1.1.2.2.cmml" xref="S3.Ex1.m1.1.1.1.2.2"></union><apply id="S3.Ex1.m1.1.1.1.2.3.cmml" xref="S3.Ex1.m1.1.1.1.2.3"><in id="S3.Ex1.m1.1.1.1.2.3.1.cmml" xref="S3.Ex1.m1.1.1.1.2.3.1"></in><ci id="S3.Ex1.m1.1.1.1.2.3.2.cmml" xref="S3.Ex1.m1.1.1.1.2.3.2">𝑛</ci><ci id="S3.Ex1.m1.1.1.1.2.3.3.cmml" xref="S3.Ex1.m1.1.1.1.2.3.3">ℤ</ci></apply></apply><apply id="S3.Ex1.m1.1.1.1.1.cmml" xref="S3.Ex1.m1.1.1.1.1"><times id="S3.Ex1.m1.1.1.1.1.2.cmml" xref="S3.Ex1.m1.1.1.1.1.2"></times><set id="S3.Ex1.m1.1.1.1.1.1.2.cmml" xref="S3.Ex1.m1.1.1.1.1.1.1"><apply id="S3.Ex1.m1.1.1.1.1.1.1.1.cmml" xref="S3.Ex1.m1.1.1.1.1.1.1.1"><times id="S3.Ex1.m1.1.1.1.1.1.1.1.1.cmml" xref="S3.Ex1.m1.1.1.1.1.1.1.1.1"></times><cn id="S3.Ex1.m1.1.1.1.1.1.1.1.2.cmml" type="integer" xref="S3.Ex1.m1.1.1.1.1.1.1.1.2">2</cn><ci id="S3.Ex1.m1.1.1.1.1.1.1.1.3.cmml" xref="S3.Ex1.m1.1.1.1.1.1.1.1.3">𝑛</ci></apply></set><cn id="S3.Ex1.m1.1.1.1.1.3.cmml" type="integer" xref="S3.Ex1.m1.1.1.1.1.3">1</cn></apply></apply><apply id="S3.Ex1.m1.2.2.2.cmml" xref="S3.Ex1.m1.2.2.2"><apply id="S3.Ex1.m1.2.2.2.2.cmml" xref="S3.Ex1.m1.2.2.2.2"><csymbol cd="ambiguous" id="S3.Ex1.m1.2.2.2.2.1.cmml" xref="S3.Ex1.m1.2.2.2.2">subscript</csymbol><union id="S3.Ex1.m1.2.2.2.2.2.cmml" xref="S3.Ex1.m1.2.2.2.2.2"></union><apply id="S3.Ex1.m1.2.2.2.2.3.cmml" xref="S3.Ex1.m1.2.2.2.2.3"><in id="S3.Ex1.m1.2.2.2.2.3.1.cmml" xref="S3.Ex1.m1.2.2.2.2.3.1"></in><ci id="S3.Ex1.m1.2.2.2.2.3.2.cmml" xref="S3.Ex1.m1.2.2.2.2.3.2">𝑛</ci><ci id="S3.Ex1.m1.2.2.2.2.3.3.cmml" xref="S3.Ex1.m1.2.2.2.2.3.3">ℤ</ci></apply></apply><apply id="S3.Ex1.m1.2.2.2.1.cmml" xref="S3.Ex1.m1.2.2.2.1"><times id="S3.Ex1.m1.2.2.2.1.2.cmml" xref="S3.Ex1.m1.2.2.2.1.2"></times><set id="S3.Ex1.m1.2.2.2.1.1.2.cmml" xref="S3.Ex1.m1.2.2.2.1.1.1"><apply id="S3.Ex1.m1.2.2.2.1.1.1.1.cmml" xref="S3.Ex1.m1.2.2.2.1.1.1.1"><plus id="S3.Ex1.m1.2.2.2.1.1.1.1.1.cmml" xref="S3.Ex1.m1.2.2.2.1.1.1.1.1"></plus><apply id="S3.Ex1.m1.2.2.2.1.1.1.1.2.cmml" xref="S3.Ex1.m1.2.2.2.1.1.1.1.2"><times id="S3.Ex1.m1.2.2.2.1.1.1.1.2.1.cmml" xref="S3.Ex1.m1.2.2.2.1.1.1.1.2.1"></times><cn id="S3.Ex1.m1.2.2.2.1.1.1.1.2.2.cmml" type="integer" xref="S3.Ex1.m1.2.2.2.1.1.1.1.2.2">2</cn><ci id="S3.Ex1.m1.2.2.2.1.1.1.1.2.3.cmml" xref="S3.Ex1.m1.2.2.2.1.1.1.1.2.3">𝑛</ci></apply><cn id="S3.Ex1.m1.2.2.2.1.1.1.1.3.cmml" type="integer" xref="S3.Ex1.m1.2.2.2.1.1.1.1.3">1</cn></apply></set><apply id="S3.Ex1.m1.2.2.2.1.3.cmml" xref="S3.Ex1.m1.2.2.2.1.3"><csymbol cd="ambiguous" id="S3.Ex1.m1.2.2.2.1.3.1.cmml" xref="S3.Ex1.m1.2.2.2.1.3">subscript</csymbol><ci id="S3.Ex1.m1.2.2.2.1.3.2.cmml" xref="S3.Ex1.m1.2.2.2.1.3.2">𝐴</ci><ci id="S3.Ex1.m1.2.2.2.1.3.3.cmml" xref="S3.Ex1.m1.2.2.2.1.3.3">𝑖</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex1.m1.2c">\bigcup_{n\in\mathbb{Z}}\{2n\}\times 1\cup\bigcup_{n\in\mathbb{Z}}\{2n+1\}% \times A_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.Ex1.m1.2d">⋃ start_POSTSUBSCRIPT italic_n ∈ blackboard_Z end_POSTSUBSCRIPT { 2 italic_n } × 1 ∪ ⋃ start_POSTSUBSCRIPT italic_n ∈ blackboard_Z end_POSTSUBSCRIPT { 2 italic_n + 1 } × italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.7.p4.7">with the lexicographical ordering. Then</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex2"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="f(n,x):=\begin{cases}a_{0}&\text{if $n<0$,}\\ a_{k}&\text{if $0\leq n=2k$,}\\ \max_{A_{i}}(a_{k},\min_{A_{i}}(x,a_{k+1}))&\text{if $0\leq n=2k+1$.}\end{cases}" class="ltx_Math" display="block" id="S3.Ex2.m1.8"><semantics id="S3.Ex2.m1.8a"><mrow id="S3.Ex2.m1.8.9" xref="S3.Ex2.m1.8.9.cmml"><mrow id="S3.Ex2.m1.8.9.2" xref="S3.Ex2.m1.8.9.2.cmml"><mi id="S3.Ex2.m1.8.9.2.2" xref="S3.Ex2.m1.8.9.2.2.cmml">f</mi><mo id="S3.Ex2.m1.8.9.2.1" xref="S3.Ex2.m1.8.9.2.1.cmml">⁢</mo><mrow id="S3.Ex2.m1.8.9.2.3.2" xref="S3.Ex2.m1.8.9.2.3.1.cmml"><mo id="S3.Ex2.m1.8.9.2.3.2.1" stretchy="false" xref="S3.Ex2.m1.8.9.2.3.1.cmml">(</mo><mi id="S3.Ex2.m1.7.7" xref="S3.Ex2.m1.7.7.cmml">n</mi><mo id="S3.Ex2.m1.8.9.2.3.2.2" xref="S3.Ex2.m1.8.9.2.3.1.cmml">,</mo><mi id="S3.Ex2.m1.8.8" xref="S3.Ex2.m1.8.8.cmml">x</mi><mo id="S3.Ex2.m1.8.9.2.3.2.3" rspace="0.278em" stretchy="false" xref="S3.Ex2.m1.8.9.2.3.1.cmml">)</mo></mrow></mrow><mo id="S3.Ex2.m1.8.9.1" rspace="0.278em" xref="S3.Ex2.m1.8.9.1.cmml">:=</mo><mrow id="S3.Ex2.m1.6.6" xref="S3.Ex2.m1.8.9.3.1.cmml"><mo id="S3.Ex2.m1.6.6.7" xref="S3.Ex2.m1.8.9.3.1.1.cmml">{</mo><mtable columnspacing="5pt" displaystyle="true" id="S3.Ex2.m1.6.6.6" rowspacing="0pt" xref="S3.Ex2.m1.8.9.3.1.cmml"><mtr id="S3.Ex2.m1.6.6.6a" xref="S3.Ex2.m1.8.9.3.1.cmml"><mtd class="ltx_align_left" columnalign="left" id="S3.Ex2.m1.6.6.6b" xref="S3.Ex2.m1.8.9.3.1.cmml"><msub id="S3.Ex2.m1.4.4.4.4.2.1" xref="S3.Ex2.m1.4.4.4.4.2.1.cmml"><mi id="S3.Ex2.m1.4.4.4.4.2.1.2" xref="S3.Ex2.m1.4.4.4.4.2.1.2.cmml">a</mi><mn id="S3.Ex2.m1.4.4.4.4.2.1.3" xref="S3.Ex2.m1.4.4.4.4.2.1.3.cmml">0</mn></msub></mtd><mtd class="ltx_align_left" columnalign="left" id="S3.Ex2.m1.6.6.6c" xref="S3.Ex2.m1.8.9.3.1.cmml"><mrow id="S3.Ex2.m1.1.1.1.1.1.1" xref="S3.Ex2.m1.1.1.1.1.1.1c.cmml"><mtext id="S3.Ex2.m1.1.1.1.1.1.1a" xref="S3.Ex2.m1.1.1.1.1.1.1c.cmml">if </mtext><mrow id="S3.Ex2.m1.1.1.1.1.1.1.1.1.m1.1.1" xref="S3.Ex2.m1.1.1.1.1.1.1.1.1.m1.1.1.cmml"><mi id="S3.Ex2.m1.1.1.1.1.1.1.1.1.m1.1.1.2" xref="S3.Ex2.m1.1.1.1.1.1.1.1.1.m1.1.1.2.cmml">n</mi><mo id="S3.Ex2.m1.1.1.1.1.1.1.1.1.m1.1.1.1" xref="S3.Ex2.m1.1.1.1.1.1.1.1.1.m1.1.1.1.cmml"><</mo><mn id="S3.Ex2.m1.1.1.1.1.1.1.1.1.m1.1.1.3" xref="S3.Ex2.m1.1.1.1.1.1.1.1.1.m1.1.1.3.cmml">0</mn></mrow><mtext id="S3.Ex2.m1.1.1.1.1.1.1b" xref="S3.Ex2.m1.1.1.1.1.1.1c.cmml">,</mtext></mrow></mtd></mtr><mtr id="S3.Ex2.m1.6.6.6d" xref="S3.Ex2.m1.8.9.3.1.cmml"><mtd class="ltx_align_left" columnalign="left" id="S3.Ex2.m1.6.6.6e" xref="S3.Ex2.m1.8.9.3.1.cmml"><msub id="S3.Ex2.m1.5.5.5.5.2.1" xref="S3.Ex2.m1.5.5.5.5.2.1.cmml"><mi id="S3.Ex2.m1.5.5.5.5.2.1.2" xref="S3.Ex2.m1.5.5.5.5.2.1.2.cmml">a</mi><mi id="S3.Ex2.m1.5.5.5.5.2.1.3" xref="S3.Ex2.m1.5.5.5.5.2.1.3.cmml">k</mi></msub></mtd><mtd class="ltx_align_left" columnalign="left" id="S3.Ex2.m1.6.6.6f" xref="S3.Ex2.m1.8.9.3.1.cmml"><mrow id="S3.Ex2.m1.2.2.2.2.1.1" xref="S3.Ex2.m1.2.2.2.2.1.1c.cmml"><mtext id="S3.Ex2.m1.2.2.2.2.1.1a" xref="S3.Ex2.m1.2.2.2.2.1.1c.cmml">if </mtext><mrow id="S3.Ex2.m1.2.2.2.2.1.1.1.1.m1.1.1" xref="S3.Ex2.m1.2.2.2.2.1.1.1.1.m1.1.1.cmml"><mn id="S3.Ex2.m1.2.2.2.2.1.1.1.1.m1.1.1.2" xref="S3.Ex2.m1.2.2.2.2.1.1.1.1.m1.1.1.2.cmml">0</mn><mo id="S3.Ex2.m1.2.2.2.2.1.1.1.1.m1.1.1.3" xref="S3.Ex2.m1.2.2.2.2.1.1.1.1.m1.1.1.3.cmml">≤</mo><mi id="S3.Ex2.m1.2.2.2.2.1.1.1.1.m1.1.1.4" xref="S3.Ex2.m1.2.2.2.2.1.1.1.1.m1.1.1.4.cmml">n</mi><mo id="S3.Ex2.m1.2.2.2.2.1.1.1.1.m1.1.1.5" xref="S3.Ex2.m1.2.2.2.2.1.1.1.1.m1.1.1.5.cmml">=</mo><mrow id="S3.Ex2.m1.2.2.2.2.1.1.1.1.m1.1.1.6" xref="S3.Ex2.m1.2.2.2.2.1.1.1.1.m1.1.1.6.cmml"><mn id="S3.Ex2.m1.2.2.2.2.1.1.1.1.m1.1.1.6.2" xref="S3.Ex2.m1.2.2.2.2.1.1.1.1.m1.1.1.6.2.cmml">2</mn><mo id="S3.Ex2.m1.2.2.2.2.1.1.1.1.m1.1.1.6.1" xref="S3.Ex2.m1.2.2.2.2.1.1.1.1.m1.1.1.6.1.cmml">⁢</mo><mi id="S3.Ex2.m1.2.2.2.2.1.1.1.1.m1.1.1.6.3" xref="S3.Ex2.m1.2.2.2.2.1.1.1.1.m1.1.1.6.3.cmml">k</mi></mrow></mrow><mtext id="S3.Ex2.m1.2.2.2.2.1.1b" xref="S3.Ex2.m1.2.2.2.2.1.1c.cmml">,</mtext></mrow></mtd></mtr><mtr id="S3.Ex2.m1.6.6.6g" xref="S3.Ex2.m1.8.9.3.1.cmml"><mtd class="ltx_align_left" columnalign="left" id="S3.Ex2.m1.6.6.6h" xref="S3.Ex2.m1.8.9.3.1.cmml"><mrow id="S3.Ex2.m1.6.6.6.6.2.1.4" xref="S3.Ex2.m1.6.6.6.6.2.1.5.cmml"><msub id="S3.Ex2.m1.6.6.6.6.2.1.2.1" xref="S3.Ex2.m1.6.6.6.6.2.1.2.1.cmml"><mi id="S3.Ex2.m1.6.6.6.6.2.1.2.1.2" xref="S3.Ex2.m1.6.6.6.6.2.1.2.1.2.cmml">max</mi><msub id="S3.Ex2.m1.6.6.6.6.2.1.2.1.3" xref="S3.Ex2.m1.6.6.6.6.2.1.2.1.3.cmml"><mi id="S3.Ex2.m1.6.6.6.6.2.1.2.1.3.2" xref="S3.Ex2.m1.6.6.6.6.2.1.2.1.3.2.cmml">A</mi><mi id="S3.Ex2.m1.6.6.6.6.2.1.2.1.3.3" xref="S3.Ex2.m1.6.6.6.6.2.1.2.1.3.3.cmml">i</mi></msub></msub><mo id="S3.Ex2.m1.6.6.6.6.2.1.4a" xref="S3.Ex2.m1.6.6.6.6.2.1.5.cmml">⁡</mo><mrow id="S3.Ex2.m1.6.6.6.6.2.1.4.3" xref="S3.Ex2.m1.6.6.6.6.2.1.5.cmml"><mo id="S3.Ex2.m1.6.6.6.6.2.1.4.3.3" stretchy="false" xref="S3.Ex2.m1.6.6.6.6.2.1.5.cmml">(</mo><msub id="S3.Ex2.m1.6.6.6.6.2.1.3.2.1" xref="S3.Ex2.m1.6.6.6.6.2.1.3.2.1.cmml"><mi id="S3.Ex2.m1.6.6.6.6.2.1.3.2.1.2" xref="S3.Ex2.m1.6.6.6.6.2.1.3.2.1.2.cmml">a</mi><mi id="S3.Ex2.m1.6.6.6.6.2.1.3.2.1.3" xref="S3.Ex2.m1.6.6.6.6.2.1.3.2.1.3.cmml">k</mi></msub><mo id="S3.Ex2.m1.6.6.6.6.2.1.4.3.4" xref="S3.Ex2.m1.6.6.6.6.2.1.5.cmml">,</mo><mrow id="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.2" xref="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.3.cmml"><msub id="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.1.1" xref="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.1.1.cmml"><mi id="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.1.1.2" xref="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.1.1.2.cmml">min</mi><msub id="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.1.1.3" xref="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.1.1.3.cmml"><mi id="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.1.1.3.2" xref="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.1.1.3.2.cmml">A</mi><mi id="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.1.1.3.3" xref="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.1.1.3.3.cmml">i</mi></msub></msub><mo id="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.2a" xref="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.3.cmml">⁡</mo><mrow id="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.2.2" xref="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.3.cmml"><mo id="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.2.2.2" stretchy="false" xref="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.3.cmml">(</mo><mi id="S3.Ex2.m1.6.6.6.6.2.1.1" xref="S3.Ex2.m1.6.6.6.6.2.1.1.cmml">x</mi><mo id="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.2.2.3" xref="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.3.cmml">,</mo><msub id="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.2.2.1" xref="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.2.2.1.cmml"><mi id="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.2.2.1.2" xref="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.2.2.1.2.cmml">a</mi><mrow id="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.2.2.1.3" xref="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.2.2.1.3.cmml"><mi id="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.2.2.1.3.2" xref="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.2.2.1.3.2.cmml">k</mi><mo id="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.2.2.1.3.1" xref="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.2.2.1.3.1.cmml">+</mo><mn id="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.2.2.1.3.3" xref="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.2.2.1.3.3.cmml">1</mn></mrow></msub><mo id="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.2.2.4" stretchy="false" xref="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.3.cmml">)</mo></mrow></mrow><mo id="S3.Ex2.m1.6.6.6.6.2.1.4.3.5" stretchy="false" xref="S3.Ex2.m1.6.6.6.6.2.1.5.cmml">)</mo></mrow></mrow></mtd><mtd class="ltx_align_left" columnalign="left" id="S3.Ex2.m1.6.6.6i" xref="S3.Ex2.m1.8.9.3.1.cmml"><mrow id="S3.Ex2.m1.3.3.3.3.1.1" xref="S3.Ex2.m1.3.3.3.3.1.1c.cmml"><mtext id="S3.Ex2.m1.3.3.3.3.1.1a" xref="S3.Ex2.m1.3.3.3.3.1.1c.cmml">if </mtext><mrow id="S3.Ex2.m1.3.3.3.3.1.1.1.1.m1.1.1" xref="S3.Ex2.m1.3.3.3.3.1.1.1.1.m1.1.1.cmml"><mn id="S3.Ex2.m1.3.3.3.3.1.1.1.1.m1.1.1.2" xref="S3.Ex2.m1.3.3.3.3.1.1.1.1.m1.1.1.2.cmml">0</mn><mo id="S3.Ex2.m1.3.3.3.3.1.1.1.1.m1.1.1.3" xref="S3.Ex2.m1.3.3.3.3.1.1.1.1.m1.1.1.3.cmml">≤</mo><mi id="S3.Ex2.m1.3.3.3.3.1.1.1.1.m1.1.1.4" xref="S3.Ex2.m1.3.3.3.3.1.1.1.1.m1.1.1.4.cmml">n</mi><mo id="S3.Ex2.m1.3.3.3.3.1.1.1.1.m1.1.1.5" xref="S3.Ex2.m1.3.3.3.3.1.1.1.1.m1.1.1.5.cmml">=</mo><mrow id="S3.Ex2.m1.3.3.3.3.1.1.1.1.m1.1.1.6" xref="S3.Ex2.m1.3.3.3.3.1.1.1.1.m1.1.1.6.cmml"><mrow id="S3.Ex2.m1.3.3.3.3.1.1.1.1.m1.1.1.6.2" xref="S3.Ex2.m1.3.3.3.3.1.1.1.1.m1.1.1.6.2.cmml"><mn id="S3.Ex2.m1.3.3.3.3.1.1.1.1.m1.1.1.6.2.2" xref="S3.Ex2.m1.3.3.3.3.1.1.1.1.m1.1.1.6.2.2.cmml">2</mn><mo id="S3.Ex2.m1.3.3.3.3.1.1.1.1.m1.1.1.6.2.1" xref="S3.Ex2.m1.3.3.3.3.1.1.1.1.m1.1.1.6.2.1.cmml">⁢</mo><mi id="S3.Ex2.m1.3.3.3.3.1.1.1.1.m1.1.1.6.2.3" xref="S3.Ex2.m1.3.3.3.3.1.1.1.1.m1.1.1.6.2.3.cmml">k</mi></mrow><mo id="S3.Ex2.m1.3.3.3.3.1.1.1.1.m1.1.1.6.1" xref="S3.Ex2.m1.3.3.3.3.1.1.1.1.m1.1.1.6.1.cmml">+</mo><mn id="S3.Ex2.m1.3.3.3.3.1.1.1.1.m1.1.1.6.3" xref="S3.Ex2.m1.3.3.3.3.1.1.1.1.m1.1.1.6.3.cmml">1</mn></mrow></mrow><mtext id="S3.Ex2.m1.3.3.3.3.1.1b" xref="S3.Ex2.m1.3.3.3.3.1.1c.cmml">.</mtext></mrow></mtd></mtr></mtable></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex2.m1.8b"><apply id="S3.Ex2.m1.8.9.cmml" xref="S3.Ex2.m1.8.9"><csymbol cd="latexml" id="S3.Ex2.m1.8.9.1.cmml" xref="S3.Ex2.m1.8.9.1">assign</csymbol><apply id="S3.Ex2.m1.8.9.2.cmml" xref="S3.Ex2.m1.8.9.2"><times id="S3.Ex2.m1.8.9.2.1.cmml" xref="S3.Ex2.m1.8.9.2.1"></times><ci id="S3.Ex2.m1.8.9.2.2.cmml" xref="S3.Ex2.m1.8.9.2.2">𝑓</ci><interval closure="open" id="S3.Ex2.m1.8.9.2.3.1.cmml" xref="S3.Ex2.m1.8.9.2.3.2"><ci id="S3.Ex2.m1.7.7.cmml" xref="S3.Ex2.m1.7.7">𝑛</ci><ci id="S3.Ex2.m1.8.8.cmml" xref="S3.Ex2.m1.8.8">𝑥</ci></interval></apply><apply id="S3.Ex2.m1.8.9.3.1.cmml" xref="S3.Ex2.m1.6.6"><csymbol cd="latexml" id="S3.Ex2.m1.8.9.3.1.1.cmml" xref="S3.Ex2.m1.6.6.7">cases</csymbol><apply id="S3.Ex2.m1.4.4.4.4.2.1.cmml" xref="S3.Ex2.m1.4.4.4.4.2.1"><csymbol cd="ambiguous" id="S3.Ex2.m1.4.4.4.4.2.1.1.cmml" xref="S3.Ex2.m1.4.4.4.4.2.1">subscript</csymbol><ci id="S3.Ex2.m1.4.4.4.4.2.1.2.cmml" xref="S3.Ex2.m1.4.4.4.4.2.1.2">𝑎</ci><cn id="S3.Ex2.m1.4.4.4.4.2.1.3.cmml" type="integer" xref="S3.Ex2.m1.4.4.4.4.2.1.3">0</cn></apply><ci id="S3.Ex2.m1.1.1.1.1.1.1c.cmml" xref="S3.Ex2.m1.1.1.1.1.1.1"><mrow id="S3.Ex2.m1.1.1.1.1.1.1.cmml" xref="S3.Ex2.m1.1.1.1.1.1.1"><mtext id="S3.Ex2.m1.1.1.1.1.1.1a.cmml" xref="S3.Ex2.m1.1.1.1.1.1.1">if </mtext><mrow id="S3.Ex2.m1.1.1.1.1.1.1.1.1.m1.1.1.cmml" xref="S3.Ex2.m1.1.1.1.1.1.1.1.1.m1.1.1"><mi id="S3.Ex2.m1.1.1.1.1.1.1.1.1.m1.1.1.2.cmml" xref="S3.Ex2.m1.1.1.1.1.1.1.1.1.m1.1.1.2">n</mi><mo id="S3.Ex2.m1.1.1.1.1.1.1.1.1.m1.1.1.1.cmml" xref="S3.Ex2.m1.1.1.1.1.1.1.1.1.m1.1.1.1"><</mo><mn id="S3.Ex2.m1.1.1.1.1.1.1.1.1.m1.1.1.3.cmml" xref="S3.Ex2.m1.1.1.1.1.1.1.1.1.m1.1.1.3">0</mn></mrow><mtext id="S3.Ex2.m1.1.1.1.1.1.1b.cmml" xref="S3.Ex2.m1.1.1.1.1.1.1">,</mtext></mrow></ci><apply id="S3.Ex2.m1.5.5.5.5.2.1.cmml" xref="S3.Ex2.m1.5.5.5.5.2.1"><csymbol cd="ambiguous" id="S3.Ex2.m1.5.5.5.5.2.1.1.cmml" xref="S3.Ex2.m1.5.5.5.5.2.1">subscript</csymbol><ci id="S3.Ex2.m1.5.5.5.5.2.1.2.cmml" xref="S3.Ex2.m1.5.5.5.5.2.1.2">𝑎</ci><ci id="S3.Ex2.m1.5.5.5.5.2.1.3.cmml" xref="S3.Ex2.m1.5.5.5.5.2.1.3">𝑘</ci></apply><ci id="S3.Ex2.m1.2.2.2.2.1.1c.cmml" xref="S3.Ex2.m1.2.2.2.2.1.1"><mrow id="S3.Ex2.m1.2.2.2.2.1.1.cmml" xref="S3.Ex2.m1.2.2.2.2.1.1"><mtext id="S3.Ex2.m1.2.2.2.2.1.1a.cmml" xref="S3.Ex2.m1.2.2.2.2.1.1">if </mtext><mrow id="S3.Ex2.m1.2.2.2.2.1.1.1.1.m1.1.1.cmml" xref="S3.Ex2.m1.2.2.2.2.1.1.1.1.m1.1.1"><mn id="S3.Ex2.m1.2.2.2.2.1.1.1.1.m1.1.1.2.cmml" xref="S3.Ex2.m1.2.2.2.2.1.1.1.1.m1.1.1.2">0</mn><mo id="S3.Ex2.m1.2.2.2.2.1.1.1.1.m1.1.1.3.cmml" xref="S3.Ex2.m1.2.2.2.2.1.1.1.1.m1.1.1.3">≤</mo><mi id="S3.Ex2.m1.2.2.2.2.1.1.1.1.m1.1.1.4.cmml" xref="S3.Ex2.m1.2.2.2.2.1.1.1.1.m1.1.1.4">n</mi><mo id="S3.Ex2.m1.2.2.2.2.1.1.1.1.m1.1.1.5.cmml" xref="S3.Ex2.m1.2.2.2.2.1.1.1.1.m1.1.1.5">=</mo><mrow id="S3.Ex2.m1.2.2.2.2.1.1.1.1.m1.1.1.6.cmml" xref="S3.Ex2.m1.2.2.2.2.1.1.1.1.m1.1.1.6"><mn id="S3.Ex2.m1.2.2.2.2.1.1.1.1.m1.1.1.6.2.cmml" xref="S3.Ex2.m1.2.2.2.2.1.1.1.1.m1.1.1.6.2">2</mn><mo id="S3.Ex2.m1.2.2.2.2.1.1.1.1.m1.1.1.6.1.cmml" xref="S3.Ex2.m1.2.2.2.2.1.1.1.1.m1.1.1.6.1">⁢</mo><mi id="S3.Ex2.m1.2.2.2.2.1.1.1.1.m1.1.1.6.3.cmml" xref="S3.Ex2.m1.2.2.2.2.1.1.1.1.m1.1.1.6.3">k</mi></mrow></mrow><mtext id="S3.Ex2.m1.2.2.2.2.1.1b.cmml" xref="S3.Ex2.m1.2.2.2.2.1.1">,</mtext></mrow></ci><apply id="S3.Ex2.m1.6.6.6.6.2.1.5.cmml" xref="S3.Ex2.m1.6.6.6.6.2.1.4"><apply id="S3.Ex2.m1.6.6.6.6.2.1.2.1.cmml" xref="S3.Ex2.m1.6.6.6.6.2.1.2.1"><csymbol cd="ambiguous" id="S3.Ex2.m1.6.6.6.6.2.1.2.1.1.cmml" xref="S3.Ex2.m1.6.6.6.6.2.1.2.1">subscript</csymbol><max id="S3.Ex2.m1.6.6.6.6.2.1.2.1.2.cmml" xref="S3.Ex2.m1.6.6.6.6.2.1.2.1.2"></max><apply id="S3.Ex2.m1.6.6.6.6.2.1.2.1.3.cmml" xref="S3.Ex2.m1.6.6.6.6.2.1.2.1.3"><csymbol cd="ambiguous" id="S3.Ex2.m1.6.6.6.6.2.1.2.1.3.1.cmml" xref="S3.Ex2.m1.6.6.6.6.2.1.2.1.3">subscript</csymbol><ci id="S3.Ex2.m1.6.6.6.6.2.1.2.1.3.2.cmml" xref="S3.Ex2.m1.6.6.6.6.2.1.2.1.3.2">𝐴</ci><ci id="S3.Ex2.m1.6.6.6.6.2.1.2.1.3.3.cmml" xref="S3.Ex2.m1.6.6.6.6.2.1.2.1.3.3">𝑖</ci></apply></apply><apply id="S3.Ex2.m1.6.6.6.6.2.1.3.2.1.cmml" xref="S3.Ex2.m1.6.6.6.6.2.1.3.2.1"><csymbol cd="ambiguous" id="S3.Ex2.m1.6.6.6.6.2.1.3.2.1.1.cmml" xref="S3.Ex2.m1.6.6.6.6.2.1.3.2.1">subscript</csymbol><ci id="S3.Ex2.m1.6.6.6.6.2.1.3.2.1.2.cmml" xref="S3.Ex2.m1.6.6.6.6.2.1.3.2.1.2">𝑎</ci><ci id="S3.Ex2.m1.6.6.6.6.2.1.3.2.1.3.cmml" xref="S3.Ex2.m1.6.6.6.6.2.1.3.2.1.3">𝑘</ci></apply><apply id="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.3.cmml" xref="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.2"><apply id="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.1.1.cmml" xref="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.1.1"><csymbol cd="ambiguous" id="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.1.1.1.cmml" xref="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.1.1">subscript</csymbol><min id="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.1.1.2.cmml" xref="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.1.1.2"></min><apply id="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.1.1.3.cmml" xref="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.1.1.3"><csymbol cd="ambiguous" id="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.1.1.3.1.cmml" xref="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.1.1.3">subscript</csymbol><ci id="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.1.1.3.2.cmml" xref="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.1.1.3.2">𝐴</ci><ci id="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.1.1.3.3.cmml" xref="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.1.1.3.3">𝑖</ci></apply></apply><ci id="S3.Ex2.m1.6.6.6.6.2.1.1.cmml" xref="S3.Ex2.m1.6.6.6.6.2.1.1">𝑥</ci><apply id="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.2.2.1.cmml" xref="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.2.2.1"><csymbol cd="ambiguous" id="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.2.2.1.1.cmml" xref="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.2.2.1">subscript</csymbol><ci id="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.2.2.1.2.cmml" xref="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.2.2.1.2">𝑎</ci><apply id="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.2.2.1.3.cmml" xref="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.2.2.1.3"><plus id="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.2.2.1.3.1.cmml" xref="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.2.2.1.3.1"></plus><ci id="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.2.2.1.3.2.cmml" xref="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.2.2.1.3.2">𝑘</ci><cn id="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.2.2.1.3.3.cmml" type="integer" xref="S3.Ex2.m1.6.6.6.6.2.1.4.3.2.2.2.1.3.3">1</cn></apply></apply></apply></apply><ci id="S3.Ex2.m1.3.3.3.3.1.1c.cmml" xref="S3.Ex2.m1.3.3.3.3.1.1"><mrow id="S3.Ex2.m1.3.3.3.3.1.1.cmml" xref="S3.Ex2.m1.3.3.3.3.1.1"><mtext id="S3.Ex2.m1.3.3.3.3.1.1a.cmml" xref="S3.Ex2.m1.3.3.3.3.1.1">if </mtext><mrow id="S3.Ex2.m1.3.3.3.3.1.1.1.1.m1.1.1.cmml" xref="S3.Ex2.m1.3.3.3.3.1.1.1.1.m1.1.1"><mn id="S3.Ex2.m1.3.3.3.3.1.1.1.1.m1.1.1.2.cmml" xref="S3.Ex2.m1.3.3.3.3.1.1.1.1.m1.1.1.2">0</mn><mo id="S3.Ex2.m1.3.3.3.3.1.1.1.1.m1.1.1.3.cmml" xref="S3.Ex2.m1.3.3.3.3.1.1.1.1.m1.1.1.3">≤</mo><mi id="S3.Ex2.m1.3.3.3.3.1.1.1.1.m1.1.1.4.cmml" xref="S3.Ex2.m1.3.3.3.3.1.1.1.1.m1.1.1.4">n</mi><mo id="S3.Ex2.m1.3.3.3.3.1.1.1.1.m1.1.1.5.cmml" xref="S3.Ex2.m1.3.3.3.3.1.1.1.1.m1.1.1.5">=</mo><mrow id="S3.Ex2.m1.3.3.3.3.1.1.1.1.m1.1.1.6.cmml" xref="S3.Ex2.m1.3.3.3.3.1.1.1.1.m1.1.1.6"><mrow id="S3.Ex2.m1.3.3.3.3.1.1.1.1.m1.1.1.6.2.cmml" xref="S3.Ex2.m1.3.3.3.3.1.1.1.1.m1.1.1.6.2"><mn id="S3.Ex2.m1.3.3.3.3.1.1.1.1.m1.1.1.6.2.2.cmml" xref="S3.Ex2.m1.3.3.3.3.1.1.1.1.m1.1.1.6.2.2">2</mn><mo id="S3.Ex2.m1.3.3.3.3.1.1.1.1.m1.1.1.6.2.1.cmml" xref="S3.Ex2.m1.3.3.3.3.1.1.1.1.m1.1.1.6.2.1">⁢</mo><mi id="S3.Ex2.m1.3.3.3.3.1.1.1.1.m1.1.1.6.2.3.cmml" xref="S3.Ex2.m1.3.3.3.3.1.1.1.1.m1.1.1.6.2.3">k</mi></mrow><mo id="S3.Ex2.m1.3.3.3.3.1.1.1.1.m1.1.1.6.1.cmml" xref="S3.Ex2.m1.3.3.3.3.1.1.1.1.m1.1.1.6.1">+</mo><mn id="S3.Ex2.m1.3.3.3.3.1.1.1.1.m1.1.1.6.3.cmml" xref="S3.Ex2.m1.3.3.3.3.1.1.1.1.m1.1.1.6.3">1</mn></mrow></mrow><mtext id="S3.Ex2.m1.3.3.3.3.1.1b.cmml" xref="S3.Ex2.m1.3.3.3.3.1.1">.</mtext></mrow></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex2.m1.8c">f(n,x):=\begin{cases}a_{0}&\text{if $n<0$,}\\ a_{k}&\text{if $0\leq n=2k$,}\\ \max_{A_{i}}(a_{k},\min_{A_{i}}(x,a_{k+1}))&\text{if $0\leq n=2k+1$.}\end{cases}</annotation><annotation encoding="application/x-llamapun" id="S3.Ex2.m1.8d">italic_f ( italic_n , italic_x ) := { start_ROW start_CELL italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_CELL start_CELL if italic_n < 0 , end_CELL end_ROW start_ROW start_CELL italic_a start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_CELL start_CELL if 0 ≤ italic_n = 2 italic_k , end_CELL end_ROW start_ROW start_CELL roman_max start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , roman_min start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x , italic_a start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT ) ) end_CELL start_CELL if 0 ≤ italic_n = 2 italic_k + 1 . end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.7.p4.6">witnesses the fact that <math alttext="\cdots+1+A_{i}+1+A_{i}+1\cdots\trianglerighteq A_{i}" class="ltx_Math" display="inline" id="S3.7.p4.6.m1.1"><semantics id="S3.7.p4.6.m1.1a"><mrow id="S3.7.p4.6.m1.1.1" xref="S3.7.p4.6.m1.1.1.cmml"><mi id="S3.7.p4.6.m1.1.1.2" mathvariant="normal" xref="S3.7.p4.6.m1.1.1.2.cmml">⋯</mi><mo id="S3.7.p4.6.m1.1.1.1" xref="S3.7.p4.6.m1.1.1.1.cmml">+</mo><mn id="S3.7.p4.6.m1.1.1.3" xref="S3.7.p4.6.m1.1.1.3.cmml">1</mn><mo id="S3.7.p4.6.m1.1.1.1a" xref="S3.7.p4.6.m1.1.1.1.cmml">+</mo><msub id="S3.7.p4.6.m1.1.1.4" xref="S3.7.p4.6.m1.1.1.4.cmml"><mi id="S3.7.p4.6.m1.1.1.4.2" xref="S3.7.p4.6.m1.1.1.4.2.cmml">A</mi><mi id="S3.7.p4.6.m1.1.1.4.3" xref="S3.7.p4.6.m1.1.1.4.3.cmml">i</mi></msub><mo id="S3.7.p4.6.m1.1.1.1b" xref="S3.7.p4.6.m1.1.1.1.cmml">+</mo><mn id="S3.7.p4.6.m1.1.1.5" xref="S3.7.p4.6.m1.1.1.5.cmml">1</mn><mo id="S3.7.p4.6.m1.1.1.1c" xref="S3.7.p4.6.m1.1.1.1.cmml">+</mo><msub id="S3.7.p4.6.m1.1.1.6" xref="S3.7.p4.6.m1.1.1.6.cmml"><mi id="S3.7.p4.6.m1.1.1.6.2" xref="S3.7.p4.6.m1.1.1.6.2.cmml">A</mi><mi id="S3.7.p4.6.m1.1.1.6.3" xref="S3.7.p4.6.m1.1.1.6.3.cmml">i</mi></msub><mo id="S3.7.p4.6.m1.1.1.1d" xref="S3.7.p4.6.m1.1.1.1.cmml">+</mo><mrow id="S3.7.p4.6.m1.1.1.7" xref="S3.7.p4.6.m1.1.1.7.cmml"><mn id="S3.7.p4.6.m1.1.1.7.2" xref="S3.7.p4.6.m1.1.1.7.2.cmml">1</mn><mo id="S3.7.p4.6.m1.1.1.7.1" xref="S3.7.p4.6.m1.1.1.7.1.cmml">⁢</mo><mi id="S3.7.p4.6.m1.1.1.7.3" mathvariant="normal" xref="S3.7.p4.6.m1.1.1.7.3.cmml">⋯</mi><mo id="S3.7.p4.6.m1.1.1.7.1a" xref="S3.7.p4.6.m1.1.1.7.1.cmml">⁢</mo><mi id="S3.7.p4.6.m1.1.1.7.4" mathvariant="normal" xref="S3.7.p4.6.m1.1.1.7.4.cmml">⊵</mi><mo id="S3.7.p4.6.m1.1.1.7.1b" xref="S3.7.p4.6.m1.1.1.7.1.cmml">⁢</mo><msub id="S3.7.p4.6.m1.1.1.7.5" xref="S3.7.p4.6.m1.1.1.7.5.cmml"><mi id="S3.7.p4.6.m1.1.1.7.5.2" xref="S3.7.p4.6.m1.1.1.7.5.2.cmml">A</mi><mi id="S3.7.p4.6.m1.1.1.7.5.3" xref="S3.7.p4.6.m1.1.1.7.5.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.7.p4.6.m1.1b"><apply id="S3.7.p4.6.m1.1.1.cmml" xref="S3.7.p4.6.m1.1.1"><plus id="S3.7.p4.6.m1.1.1.1.cmml" xref="S3.7.p4.6.m1.1.1.1"></plus><ci id="S3.7.p4.6.m1.1.1.2.cmml" xref="S3.7.p4.6.m1.1.1.2">⋯</ci><cn id="S3.7.p4.6.m1.1.1.3.cmml" type="integer" xref="S3.7.p4.6.m1.1.1.3">1</cn><apply id="S3.7.p4.6.m1.1.1.4.cmml" xref="S3.7.p4.6.m1.1.1.4"><csymbol cd="ambiguous" id="S3.7.p4.6.m1.1.1.4.1.cmml" xref="S3.7.p4.6.m1.1.1.4">subscript</csymbol><ci id="S3.7.p4.6.m1.1.1.4.2.cmml" xref="S3.7.p4.6.m1.1.1.4.2">𝐴</ci><ci id="S3.7.p4.6.m1.1.1.4.3.cmml" xref="S3.7.p4.6.m1.1.1.4.3">𝑖</ci></apply><cn id="S3.7.p4.6.m1.1.1.5.cmml" type="integer" xref="S3.7.p4.6.m1.1.1.5">1</cn><apply id="S3.7.p4.6.m1.1.1.6.cmml" xref="S3.7.p4.6.m1.1.1.6"><csymbol cd="ambiguous" id="S3.7.p4.6.m1.1.1.6.1.cmml" xref="S3.7.p4.6.m1.1.1.6">subscript</csymbol><ci id="S3.7.p4.6.m1.1.1.6.2.cmml" xref="S3.7.p4.6.m1.1.1.6.2">𝐴</ci><ci id="S3.7.p4.6.m1.1.1.6.3.cmml" xref="S3.7.p4.6.m1.1.1.6.3">𝑖</ci></apply><apply id="S3.7.p4.6.m1.1.1.7.cmml" xref="S3.7.p4.6.m1.1.1.7"><times id="S3.7.p4.6.m1.1.1.7.1.cmml" xref="S3.7.p4.6.m1.1.1.7.1"></times><cn id="S3.7.p4.6.m1.1.1.7.2.cmml" type="integer" xref="S3.7.p4.6.m1.1.1.7.2">1</cn><ci id="S3.7.p4.6.m1.1.1.7.3.cmml" xref="S3.7.p4.6.m1.1.1.7.3">⋯</ci><ci id="S3.7.p4.6.m1.1.1.7.4.cmml" xref="S3.7.p4.6.m1.1.1.7.4">⊵</ci><apply id="S3.7.p4.6.m1.1.1.7.5.cmml" xref="S3.7.p4.6.m1.1.1.7.5"><csymbol cd="ambiguous" id="S3.7.p4.6.m1.1.1.7.5.1.cmml" xref="S3.7.p4.6.m1.1.1.7.5">subscript</csymbol><ci id="S3.7.p4.6.m1.1.1.7.5.2.cmml" xref="S3.7.p4.6.m1.1.1.7.5.2">𝐴</ci><ci id="S3.7.p4.6.m1.1.1.7.5.3.cmml" xref="S3.7.p4.6.m1.1.1.7.5.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.7.p4.6.m1.1c">\cdots+1+A_{i}+1+A_{i}+1\cdots\trianglerighteq A_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.7.p4.6.m1.1d">⋯ + 1 + italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + 1 + italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + 1 ⋯ ⊵ italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> ∎</p> </div> </div> <div class="ltx_theorem ltx_theorem_corollary" id="S3.Thmtheorem8"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem8.1.1.1">Corollary 3.8</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem8.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem8.p1"> <p class="ltx_p" id="S3.Thmtheorem8.p1.5">Assume <math alttext="\mathsf{MA}_{\aleph_{1}}" class="ltx_Math" display="inline" id="S3.Thmtheorem8.p1.1.m1.1"><semantics id="S3.Thmtheorem8.p1.1.m1.1a"><msub id="S3.Thmtheorem8.p1.1.m1.1.1" xref="S3.Thmtheorem8.p1.1.m1.1.1.cmml"><mi id="S3.Thmtheorem8.p1.1.m1.1.1.2" xref="S3.Thmtheorem8.p1.1.m1.1.1.2.cmml">𝖬𝖠</mi><msub id="S3.Thmtheorem8.p1.1.m1.1.1.3" xref="S3.Thmtheorem8.p1.1.m1.1.1.3.cmml"><mi id="S3.Thmtheorem8.p1.1.m1.1.1.3.2" mathvariant="normal" xref="S3.Thmtheorem8.p1.1.m1.1.1.3.2.cmml">ℵ</mi><mn id="S3.Thmtheorem8.p1.1.m1.1.1.3.3" xref="S3.Thmtheorem8.p1.1.m1.1.1.3.3.cmml">1</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem8.p1.1.m1.1b"><apply id="S3.Thmtheorem8.p1.1.m1.1.1.cmml" xref="S3.Thmtheorem8.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem8.p1.1.m1.1.1.1.cmml" xref="S3.Thmtheorem8.p1.1.m1.1.1">subscript</csymbol><ci id="S3.Thmtheorem8.p1.1.m1.1.1.2.cmml" xref="S3.Thmtheorem8.p1.1.m1.1.1.2">𝖬𝖠</ci><apply id="S3.Thmtheorem8.p1.1.m1.1.1.3.cmml" xref="S3.Thmtheorem8.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem8.p1.1.m1.1.1.3.1.cmml" xref="S3.Thmtheorem8.p1.1.m1.1.1.3">subscript</csymbol><ci id="S3.Thmtheorem8.p1.1.m1.1.1.3.2.cmml" xref="S3.Thmtheorem8.p1.1.m1.1.1.3.2">ℵ</ci><cn id="S3.Thmtheorem8.p1.1.m1.1.1.3.3.cmml" type="integer" xref="S3.Thmtheorem8.p1.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem8.p1.1.m1.1c">\mathsf{MA}_{\aleph_{1}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem8.p1.1.m1.1d">sansserif_MA start_POSTSUBSCRIPT roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. For every family <math alttext="\cal{F}" class="ltx_Math" display="inline" id="S3.Thmtheorem8.p1.2.m2.1"><semantics id="S3.Thmtheorem8.p1.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheorem8.p1.2.m2.1.1" xref="S3.Thmtheorem8.p1.2.m2.1.1.cmml">ℱ</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem8.p1.2.m2.1b"><ci id="S3.Thmtheorem8.p1.2.m2.1.1.cmml" xref="S3.Thmtheorem8.p1.2.m2.1.1">ℱ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem8.p1.2.m2.1c">\cal{F}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem8.p1.2.m2.1d">caligraphic_F</annotation></semantics></math> of at most <math alttext="\omega_{1}" class="ltx_Math" display="inline" id="S3.Thmtheorem8.p1.3.m3.1"><semantics id="S3.Thmtheorem8.p1.3.m3.1a"><msub id="S3.Thmtheorem8.p1.3.m3.1.1" xref="S3.Thmtheorem8.p1.3.m3.1.1.cmml"><mi id="S3.Thmtheorem8.p1.3.m3.1.1.2" xref="S3.Thmtheorem8.p1.3.m3.1.1.2.cmml">ω</mi><mn id="S3.Thmtheorem8.p1.3.m3.1.1.3" xref="S3.Thmtheorem8.p1.3.m3.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem8.p1.3.m3.1b"><apply id="S3.Thmtheorem8.p1.3.m3.1.1.cmml" xref="S3.Thmtheorem8.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem8.p1.3.m3.1.1.1.cmml" xref="S3.Thmtheorem8.p1.3.m3.1.1">subscript</csymbol><ci id="S3.Thmtheorem8.p1.3.m3.1.1.2.cmml" xref="S3.Thmtheorem8.p1.3.m3.1.1.2">𝜔</ci><cn id="S3.Thmtheorem8.p1.3.m3.1.1.3.cmml" type="integer" xref="S3.Thmtheorem8.p1.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem8.p1.3.m3.1c">\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem8.p1.3.m3.1d">italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> strongly surjective orders, there is a strongly surjective order of size <math alttext="\sup\{\omega_{1},\sup\{|A|:A\in\cal{F}\}\}" class="ltx_math_unparsed" display="inline" id="S3.Thmtheorem8.p1.4.m4.2"><semantics id="S3.Thmtheorem8.p1.4.m4.2a"><mrow id="S3.Thmtheorem8.p1.4.m4.2b"><mo id="S3.Thmtheorem8.p1.4.m4.2.3" rspace="0em">sup</mo><mrow id="S3.Thmtheorem8.p1.4.m4.2.4"><mo id="S3.Thmtheorem8.p1.4.m4.2.4.1" stretchy="false">{</mo><msub id="S3.Thmtheorem8.p1.4.m4.2.4.2"><mi id="S3.Thmtheorem8.p1.4.m4.2.4.2.2">ω</mi><mn id="S3.Thmtheorem8.p1.4.m4.2.4.2.3">1</mn></msub><mo id="S3.Thmtheorem8.p1.4.m4.2.4.3">,</mo><mo id="S3.Thmtheorem8.p1.4.m4.1.1" lspace="0em" rspace="0em">sup</mo><mrow id="S3.Thmtheorem8.p1.4.m4.2.4.4"><mo id="S3.Thmtheorem8.p1.4.m4.2.4.4.1" stretchy="false">{</mo><mo fence="false" id="S3.Thmtheorem8.p1.4.m4.2.4.4.2" rspace="0.167em" stretchy="false">|</mo><mi id="S3.Thmtheorem8.p1.4.m4.2.2">A</mi><mo fence="false" id="S3.Thmtheorem8.p1.4.m4.2.4.4.3" rspace="0.167em" stretchy="false">|</mo><mo id="S3.Thmtheorem8.p1.4.m4.2.4.4.4" rspace="0.278em">:</mo><mi id="S3.Thmtheorem8.p1.4.m4.2.4.4.5">A</mi><mo id="S3.Thmtheorem8.p1.4.m4.2.4.4.6">∈</mo><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheorem8.p1.4.m4.2.4.4.7">ℱ</mi><mo id="S3.Thmtheorem8.p1.4.m4.2.4.4.8" stretchy="false">}</mo></mrow><mo id="S3.Thmtheorem8.p1.4.m4.2.4.5" stretchy="false">}</mo></mrow></mrow><annotation encoding="application/x-tex" id="S3.Thmtheorem8.p1.4.m4.2c">\sup\{\omega_{1},\sup\{|A|:A\in\cal{F}\}\}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem8.p1.4.m4.2d">roman_sup { italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , roman_sup { | italic_A | : italic_A ∈ caligraphic_F } }</annotation></semantics></math> that contains a copy of every order in <math alttext="\cal{F}" class="ltx_Math" display="inline" id="S3.Thmtheorem8.p1.5.m5.1"><semantics id="S3.Thmtheorem8.p1.5.m5.1a"><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheorem8.p1.5.m5.1.1" xref="S3.Thmtheorem8.p1.5.m5.1.1.cmml">ℱ</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem8.p1.5.m5.1b"><ci id="S3.Thmtheorem8.p1.5.m5.1.1.cmml" xref="S3.Thmtheorem8.p1.5.m5.1.1">ℱ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem8.p1.5.m5.1c">\cal{F}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem8.p1.5.m5.1d">caligraphic_F</annotation></semantics></math>.</p> </div> </div> <div class="ltx_proof" id="S3.8"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S3.8.p1"> <p class="ltx_p" id="S3.8.p1.4">Simply note that <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S3.Thmtheorem2" title="Corollary 3.2. ‣ 3. Strongly surjective Aronszajn lines ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Corollary</span> <span class="ltx_text ltx_ref_tag">3.2</span></a> implies that every normal Countryman line is in fact open strongly surjective under <math alttext="\mathsf{MA}_{\aleph_{1}}" class="ltx_Math" display="inline" id="S3.8.p1.1.m1.1"><semantics id="S3.8.p1.1.m1.1a"><msub id="S3.8.p1.1.m1.1.1" xref="S3.8.p1.1.m1.1.1.cmml"><mi id="S3.8.p1.1.m1.1.1.2" xref="S3.8.p1.1.m1.1.1.2.cmml">𝖬𝖠</mi><msub id="S3.8.p1.1.m1.1.1.3" xref="S3.8.p1.1.m1.1.1.3.cmml"><mi id="S3.8.p1.1.m1.1.1.3.2" mathvariant="normal" xref="S3.8.p1.1.m1.1.1.3.2.cmml">ℵ</mi><mn id="S3.8.p1.1.m1.1.1.3.3" xref="S3.8.p1.1.m1.1.1.3.3.cmml">1</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S3.8.p1.1.m1.1b"><apply id="S3.8.p1.1.m1.1.1.cmml" xref="S3.8.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.8.p1.1.m1.1.1.1.cmml" xref="S3.8.p1.1.m1.1.1">subscript</csymbol><ci id="S3.8.p1.1.m1.1.1.2.cmml" xref="S3.8.p1.1.m1.1.1.2">𝖬𝖠</ci><apply id="S3.8.p1.1.m1.1.1.3.cmml" xref="S3.8.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.8.p1.1.m1.1.1.3.1.cmml" xref="S3.8.p1.1.m1.1.1.3">subscript</csymbol><ci id="S3.8.p1.1.m1.1.1.3.2.cmml" xref="S3.8.p1.1.m1.1.1.3.2">ℵ</ci><cn id="S3.8.p1.1.m1.1.1.3.3.cmml" type="integer" xref="S3.8.p1.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.8.p1.1.m1.1c">\mathsf{MA}_{\aleph_{1}}</annotation><annotation encoding="application/x-llamapun" id="S3.8.p1.1.m1.1d">sansserif_MA start_POSTSUBSCRIPT roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. Then if <math alttext="C" class="ltx_Math" display="inline" id="S3.8.p1.2.m2.1"><semantics id="S3.8.p1.2.m2.1a"><mi id="S3.8.p1.2.m2.1.1" xref="S3.8.p1.2.m2.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S3.8.p1.2.m2.1b"><ci id="S3.8.p1.2.m2.1.1.cmml" xref="S3.8.p1.2.m2.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.8.p1.2.m2.1c">C</annotation><annotation encoding="application/x-llamapun" id="S3.8.p1.2.m2.1d">italic_C</annotation></semantics></math> is a normal Countryman line, any <math alttext="C" class="ltx_Math" display="inline" id="S3.8.p1.3.m3.1"><semantics id="S3.8.p1.3.m3.1a"><mi id="S3.8.p1.3.m3.1.1" xref="S3.8.p1.3.m3.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S3.8.p1.3.m3.1b"><ci id="S3.8.p1.3.m3.1.1.cmml" xref="S3.8.p1.3.m3.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.8.p1.3.m3.1c">C</annotation><annotation encoding="application/x-llamapun" id="S3.8.p1.3.m3.1d">italic_C</annotation></semantics></math>-mixed sum of <math alttext="\cal{F}" class="ltx_Math" display="inline" id="S3.8.p1.4.m4.1"><semantics id="S3.8.p1.4.m4.1a"><mi class="ltx_font_mathcaligraphic" id="S3.8.p1.4.m4.1.1" xref="S3.8.p1.4.m4.1.1.cmml">ℱ</mi><annotation-xml encoding="MathML-Content" id="S3.8.p1.4.m4.1b"><ci id="S3.8.p1.4.m4.1.1.cmml" xref="S3.8.p1.4.m4.1.1">ℱ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.8.p1.4.m4.1c">\cal{F}</annotation><annotation encoding="application/x-llamapun" id="S3.8.p1.4.m4.1d">caligraphic_F</annotation></semantics></math> works. ∎</p> </div> </div> <div class="ltx_para" id="S3.p6"> <p class="ltx_p" id="S3.p6.4">Note that as in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib19" title="">19</a>]</cite>, using <math alttext="\mathbb{Q}" class="ltx_Math" display="inline" id="S3.p6.1.m1.1"><semantics id="S3.p6.1.m1.1a"><mi id="S3.p6.1.m1.1.1" xref="S3.p6.1.m1.1.1.cmml">ℚ</mi><annotation-xml encoding="MathML-Content" id="S3.p6.1.m1.1b"><ci id="S3.p6.1.m1.1.1.cmml" xref="S3.p6.1.m1.1.1">ℚ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.1.m1.1c">\mathbb{Q}</annotation><annotation encoding="application/x-llamapun" id="S3.p6.1.m1.1d">blackboard_Q</annotation></semantics></math> instead of <math alttext="C" class="ltx_Math" display="inline" id="S3.p6.2.m2.1"><semantics id="S3.p6.2.m2.1a"><mi id="S3.p6.2.m2.1.1" xref="S3.p6.2.m2.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S3.p6.2.m2.1b"><ci id="S3.p6.2.m2.1.1.cmml" xref="S3.p6.2.m2.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.2.m2.1c">C</annotation><annotation encoding="application/x-llamapun" id="S3.p6.2.m2.1d">italic_C</annotation></semantics></math> gives a <math alttext="\mathsf{ZFC}" class="ltx_Math" display="inline" id="S3.p6.3.m3.1"><semantics id="S3.p6.3.m3.1a"><mi id="S3.p6.3.m3.1.1" xref="S3.p6.3.m3.1.1.cmml">𝖹𝖥𝖢</mi><annotation-xml encoding="MathML-Content" id="S3.p6.3.m3.1b"><ci id="S3.p6.3.m3.1.1.cmml" xref="S3.p6.3.m3.1.1">𝖹𝖥𝖢</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.3.m3.1c">\mathsf{ZFC}</annotation><annotation encoding="application/x-llamapun" id="S3.p6.3.m3.1d">sansserif_ZFC</annotation></semantics></math> proof of the previous corollary when <math alttext="|F|\leq\aleph_{0}" class="ltx_Math" display="inline" id="S3.p6.4.m4.1"><semantics id="S3.p6.4.m4.1a"><mrow id="S3.p6.4.m4.1.2" xref="S3.p6.4.m4.1.2.cmml"><mrow id="S3.p6.4.m4.1.2.2.2" xref="S3.p6.4.m4.1.2.2.1.cmml"><mo id="S3.p6.4.m4.1.2.2.2.1" stretchy="false" xref="S3.p6.4.m4.1.2.2.1.1.cmml">|</mo><mi id="S3.p6.4.m4.1.1" xref="S3.p6.4.m4.1.1.cmml">F</mi><mo id="S3.p6.4.m4.1.2.2.2.2" stretchy="false" xref="S3.p6.4.m4.1.2.2.1.1.cmml">|</mo></mrow><mo id="S3.p6.4.m4.1.2.1" xref="S3.p6.4.m4.1.2.1.cmml">≤</mo><msub id="S3.p6.4.m4.1.2.3" xref="S3.p6.4.m4.1.2.3.cmml"><mi id="S3.p6.4.m4.1.2.3.2" mathvariant="normal" xref="S3.p6.4.m4.1.2.3.2.cmml">ℵ</mi><mn id="S3.p6.4.m4.1.2.3.3" xref="S3.p6.4.m4.1.2.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.p6.4.m4.1b"><apply id="S3.p6.4.m4.1.2.cmml" xref="S3.p6.4.m4.1.2"><leq id="S3.p6.4.m4.1.2.1.cmml" xref="S3.p6.4.m4.1.2.1"></leq><apply id="S3.p6.4.m4.1.2.2.1.cmml" xref="S3.p6.4.m4.1.2.2.2"><abs id="S3.p6.4.m4.1.2.2.1.1.cmml" xref="S3.p6.4.m4.1.2.2.2.1"></abs><ci id="S3.p6.4.m4.1.1.cmml" xref="S3.p6.4.m4.1.1">𝐹</ci></apply><apply id="S3.p6.4.m4.1.2.3.cmml" xref="S3.p6.4.m4.1.2.3"><csymbol cd="ambiguous" id="S3.p6.4.m4.1.2.3.1.cmml" xref="S3.p6.4.m4.1.2.3">subscript</csymbol><ci id="S3.p6.4.m4.1.2.3.2.cmml" xref="S3.p6.4.m4.1.2.3.2">ℵ</ci><cn id="S3.p6.4.m4.1.2.3.3.cmml" type="integer" xref="S3.p6.4.m4.1.2.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.4.m4.1c">|F|\leq\aleph_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.p6.4.m4.1d">| italic_F | ≤ roman_ℵ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.p7"> <p class="ltx_p" id="S3.p7.4">Recall the definition of <math alttext="D_{\alpha}^{+}" class="ltx_Math" display="inline" id="S3.p7.1.m1.1"><semantics id="S3.p7.1.m1.1a"><msubsup id="S3.p7.1.m1.1.1" xref="S3.p7.1.m1.1.1.cmml"><mi id="S3.p7.1.m1.1.1.2.2" xref="S3.p7.1.m1.1.1.2.2.cmml">D</mi><mi id="S3.p7.1.m1.1.1.2.3" xref="S3.p7.1.m1.1.1.2.3.cmml">α</mi><mo id="S3.p7.1.m1.1.1.3" xref="S3.p7.1.m1.1.1.3.cmml">+</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.p7.1.m1.1b"><apply id="S3.p7.1.m1.1.1.cmml" xref="S3.p7.1.m1.1.1"><csymbol cd="ambiguous" id="S3.p7.1.m1.1.1.1.cmml" xref="S3.p7.1.m1.1.1">superscript</csymbol><apply id="S3.p7.1.m1.1.1.2.cmml" xref="S3.p7.1.m1.1.1"><csymbol cd="ambiguous" id="S3.p7.1.m1.1.1.2.1.cmml" xref="S3.p7.1.m1.1.1">subscript</csymbol><ci id="S3.p7.1.m1.1.1.2.2.cmml" xref="S3.p7.1.m1.1.1.2.2">𝐷</ci><ci id="S3.p7.1.m1.1.1.2.3.cmml" xref="S3.p7.1.m1.1.1.2.3">𝛼</ci></apply><plus id="S3.p7.1.m1.1.1.3.cmml" xref="S3.p7.1.m1.1.1.3"></plus></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p7.1.m1.1c">D_{\alpha}^{+}</annotation><annotation encoding="application/x-llamapun" id="S3.p7.1.m1.1d">italic_D start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="D_{\alpha}^{-}" class="ltx_Math" display="inline" id="S3.p7.2.m2.1"><semantics id="S3.p7.2.m2.1a"><msubsup id="S3.p7.2.m2.1.1" xref="S3.p7.2.m2.1.1.cmml"><mi id="S3.p7.2.m2.1.1.2.2" xref="S3.p7.2.m2.1.1.2.2.cmml">D</mi><mi id="S3.p7.2.m2.1.1.2.3" xref="S3.p7.2.m2.1.1.2.3.cmml">α</mi><mo id="S3.p7.2.m2.1.1.3" xref="S3.p7.2.m2.1.1.3.cmml">−</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.p7.2.m2.1b"><apply id="S3.p7.2.m2.1.1.cmml" xref="S3.p7.2.m2.1.1"><csymbol cd="ambiguous" id="S3.p7.2.m2.1.1.1.cmml" xref="S3.p7.2.m2.1.1">superscript</csymbol><apply id="S3.p7.2.m2.1.1.2.cmml" xref="S3.p7.2.m2.1.1"><csymbol cd="ambiguous" id="S3.p7.2.m2.1.1.2.1.cmml" xref="S3.p7.2.m2.1.1">subscript</csymbol><ci id="S3.p7.2.m2.1.1.2.2.cmml" xref="S3.p7.2.m2.1.1.2.2">𝐷</ci><ci id="S3.p7.2.m2.1.1.2.3.cmml" xref="S3.p7.2.m2.1.1.2.3">𝛼</ci></apply><minus id="S3.p7.2.m2.1.1.3.cmml" xref="S3.p7.2.m2.1.1.3"></minus></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p7.2.m2.1c">D_{\alpha}^{-}</annotation><annotation encoding="application/x-llamapun" id="S3.p7.2.m2.1d">italic_D start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT</annotation></semantics></math> from <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S2.Thmtheorem10" title="Lemma 2.10. ‣ 2. Aronszajn and Countryman lines ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">2.10</span></a>. One easily sees that <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S3.Thmtheorem7" title="Lemma 3.7. ‣ 3. Strongly surjective Aronszajn lines ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">3.7</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S3.Thmtheorem4" title="Lemma 3.4. ‣ 3. Strongly surjective Aronszajn lines ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">3.4</span></a> imply the following, which shows that under <math alttext="\mathsf{MA}_{\aleph_{1}}" class="ltx_Math" display="inline" id="S3.p7.3.m3.1"><semantics id="S3.p7.3.m3.1a"><msub id="S3.p7.3.m3.1.1" xref="S3.p7.3.m3.1.1.cmml"><mi id="S3.p7.3.m3.1.1.2" xref="S3.p7.3.m3.1.1.2.cmml">𝖬𝖠</mi><msub id="S3.p7.3.m3.1.1.3" xref="S3.p7.3.m3.1.1.3.cmml"><mi id="S3.p7.3.m3.1.1.3.2" mathvariant="normal" xref="S3.p7.3.m3.1.1.3.2.cmml">ℵ</mi><mn id="S3.p7.3.m3.1.1.3.3" xref="S3.p7.3.m3.1.1.3.3.cmml">1</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S3.p7.3.m3.1b"><apply id="S3.p7.3.m3.1.1.cmml" xref="S3.p7.3.m3.1.1"><csymbol cd="ambiguous" id="S3.p7.3.m3.1.1.1.cmml" xref="S3.p7.3.m3.1.1">subscript</csymbol><ci id="S3.p7.3.m3.1.1.2.cmml" xref="S3.p7.3.m3.1.1.2">𝖬𝖠</ci><apply id="S3.p7.3.m3.1.1.3.cmml" xref="S3.p7.3.m3.1.1.3"><csymbol cd="ambiguous" id="S3.p7.3.m3.1.1.3.1.cmml" xref="S3.p7.3.m3.1.1.3">subscript</csymbol><ci id="S3.p7.3.m3.1.1.3.2.cmml" xref="S3.p7.3.m3.1.1.3.2">ℵ</ci><cn id="S3.p7.3.m3.1.1.3.3.cmml" type="integer" xref="S3.p7.3.m3.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p7.3.m3.1c">\mathsf{MA}_{\aleph_{1}}</annotation><annotation encoding="application/x-llamapun" id="S3.p7.3.m3.1d">sansserif_MA start_POSTSUBSCRIPT roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> there are many really different (non <math alttext="\preceq" class="ltx_Math" display="inline" id="S3.p7.4.m4.1"><semantics id="S3.p7.4.m4.1a"><mo id="S3.p7.4.m4.1.1" xref="S3.p7.4.m4.1.1.cmml">⪯</mo><annotation-xml encoding="MathML-Content" id="S3.p7.4.m4.1b"><csymbol cd="latexml" id="S3.p7.4.m4.1.1.cmml" xref="S3.p7.4.m4.1.1">precedes-or-equals</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.p7.4.m4.1c">\preceq</annotation><annotation encoding="application/x-llamapun" id="S3.p7.4.m4.1d">⪯</annotation></semantics></math>-equivalent) Aronszajn lines.</p> </div> <div class="ltx_theorem ltx_theorem_corollary" id="S3.Thmtheorem9"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem9.1.1.1">Corollary 3.9</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem9.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem9.p1"> <p class="ltx_p" id="S3.Thmtheorem9.p1.4">Assume <math alttext="(\mathsf{MA}_{\aleph_{1}})" class="ltx_Math" display="inline" id="S3.Thmtheorem9.p1.1.m1.1"><semantics id="S3.Thmtheorem9.p1.1.m1.1a"><mrow id="S3.Thmtheorem9.p1.1.m1.1.1.1" xref="S3.Thmtheorem9.p1.1.m1.1.1.1.1.cmml"><mo id="S3.Thmtheorem9.p1.1.m1.1.1.1.2" stretchy="false" xref="S3.Thmtheorem9.p1.1.m1.1.1.1.1.cmml">(</mo><msub id="S3.Thmtheorem9.p1.1.m1.1.1.1.1" xref="S3.Thmtheorem9.p1.1.m1.1.1.1.1.cmml"><mi id="S3.Thmtheorem9.p1.1.m1.1.1.1.1.2" xref="S3.Thmtheorem9.p1.1.m1.1.1.1.1.2.cmml">𝖬𝖠</mi><msub id="S3.Thmtheorem9.p1.1.m1.1.1.1.1.3" xref="S3.Thmtheorem9.p1.1.m1.1.1.1.1.3.cmml"><mi id="S3.Thmtheorem9.p1.1.m1.1.1.1.1.3.2" mathvariant="normal" xref="S3.Thmtheorem9.p1.1.m1.1.1.1.1.3.2.cmml">ℵ</mi><mn id="S3.Thmtheorem9.p1.1.m1.1.1.1.1.3.3" xref="S3.Thmtheorem9.p1.1.m1.1.1.1.1.3.3.cmml">1</mn></msub></msub><mo id="S3.Thmtheorem9.p1.1.m1.1.1.1.3" stretchy="false" xref="S3.Thmtheorem9.p1.1.m1.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem9.p1.1.m1.1b"><apply id="S3.Thmtheorem9.p1.1.m1.1.1.1.1.cmml" xref="S3.Thmtheorem9.p1.1.m1.1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem9.p1.1.m1.1.1.1.1.1.cmml" xref="S3.Thmtheorem9.p1.1.m1.1.1.1">subscript</csymbol><ci id="S3.Thmtheorem9.p1.1.m1.1.1.1.1.2.cmml" xref="S3.Thmtheorem9.p1.1.m1.1.1.1.1.2">𝖬𝖠</ci><apply id="S3.Thmtheorem9.p1.1.m1.1.1.1.1.3.cmml" xref="S3.Thmtheorem9.p1.1.m1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem9.p1.1.m1.1.1.1.1.3.1.cmml" xref="S3.Thmtheorem9.p1.1.m1.1.1.1.1.3">subscript</csymbol><ci id="S3.Thmtheorem9.p1.1.m1.1.1.1.1.3.2.cmml" xref="S3.Thmtheorem9.p1.1.m1.1.1.1.1.3.2">ℵ</ci><cn id="S3.Thmtheorem9.p1.1.m1.1.1.1.1.3.3.cmml" type="integer" xref="S3.Thmtheorem9.p1.1.m1.1.1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem9.p1.1.m1.1c">(\mathsf{MA}_{\aleph_{1}})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem9.p1.1.m1.1d">( sansserif_MA start_POSTSUBSCRIPT roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT )</annotation></semantics></math>. If the Countryman line used to build them is normal, then for every <math alttext="\alpha<\omega_{2}" class="ltx_Math" display="inline" id="S3.Thmtheorem9.p1.2.m2.1"><semantics id="S3.Thmtheorem9.p1.2.m2.1a"><mrow id="S3.Thmtheorem9.p1.2.m2.1.1" xref="S3.Thmtheorem9.p1.2.m2.1.1.cmml"><mi id="S3.Thmtheorem9.p1.2.m2.1.1.2" xref="S3.Thmtheorem9.p1.2.m2.1.1.2.cmml">α</mi><mo id="S3.Thmtheorem9.p1.2.m2.1.1.1" xref="S3.Thmtheorem9.p1.2.m2.1.1.1.cmml"><</mo><msub id="S3.Thmtheorem9.p1.2.m2.1.1.3" xref="S3.Thmtheorem9.p1.2.m2.1.1.3.cmml"><mi id="S3.Thmtheorem9.p1.2.m2.1.1.3.2" xref="S3.Thmtheorem9.p1.2.m2.1.1.3.2.cmml">ω</mi><mn id="S3.Thmtheorem9.p1.2.m2.1.1.3.3" xref="S3.Thmtheorem9.p1.2.m2.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem9.p1.2.m2.1b"><apply id="S3.Thmtheorem9.p1.2.m2.1.1.cmml" xref="S3.Thmtheorem9.p1.2.m2.1.1"><lt id="S3.Thmtheorem9.p1.2.m2.1.1.1.cmml" xref="S3.Thmtheorem9.p1.2.m2.1.1.1"></lt><ci id="S3.Thmtheorem9.p1.2.m2.1.1.2.cmml" xref="S3.Thmtheorem9.p1.2.m2.1.1.2">𝛼</ci><apply id="S3.Thmtheorem9.p1.2.m2.1.1.3.cmml" xref="S3.Thmtheorem9.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem9.p1.2.m2.1.1.3.1.cmml" xref="S3.Thmtheorem9.p1.2.m2.1.1.3">subscript</csymbol><ci id="S3.Thmtheorem9.p1.2.m2.1.1.3.2.cmml" xref="S3.Thmtheorem9.p1.2.m2.1.1.3.2">𝜔</ci><cn id="S3.Thmtheorem9.p1.2.m2.1.1.3.3.cmml" type="integer" xref="S3.Thmtheorem9.p1.2.m2.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem9.p1.2.m2.1c">\alpha<\omega_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem9.p1.2.m2.1d">italic_α < italic_ω start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="D_{\alpha}^{+}" class="ltx_Math" display="inline" id="S3.Thmtheorem9.p1.3.m3.1"><semantics id="S3.Thmtheorem9.p1.3.m3.1a"><msubsup id="S3.Thmtheorem9.p1.3.m3.1.1" xref="S3.Thmtheorem9.p1.3.m3.1.1.cmml"><mi id="S3.Thmtheorem9.p1.3.m3.1.1.2.2" xref="S3.Thmtheorem9.p1.3.m3.1.1.2.2.cmml">D</mi><mi id="S3.Thmtheorem9.p1.3.m3.1.1.2.3" xref="S3.Thmtheorem9.p1.3.m3.1.1.2.3.cmml">α</mi><mo id="S3.Thmtheorem9.p1.3.m3.1.1.3" xref="S3.Thmtheorem9.p1.3.m3.1.1.3.cmml">+</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem9.p1.3.m3.1b"><apply id="S3.Thmtheorem9.p1.3.m3.1.1.cmml" xref="S3.Thmtheorem9.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem9.p1.3.m3.1.1.1.cmml" xref="S3.Thmtheorem9.p1.3.m3.1.1">superscript</csymbol><apply id="S3.Thmtheorem9.p1.3.m3.1.1.2.cmml" xref="S3.Thmtheorem9.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem9.p1.3.m3.1.1.2.1.cmml" xref="S3.Thmtheorem9.p1.3.m3.1.1">subscript</csymbol><ci id="S3.Thmtheorem9.p1.3.m3.1.1.2.2.cmml" xref="S3.Thmtheorem9.p1.3.m3.1.1.2.2">𝐷</ci><ci id="S3.Thmtheorem9.p1.3.m3.1.1.2.3.cmml" xref="S3.Thmtheorem9.p1.3.m3.1.1.2.3">𝛼</ci></apply><plus id="S3.Thmtheorem9.p1.3.m3.1.1.3.cmml" xref="S3.Thmtheorem9.p1.3.m3.1.1.3"></plus></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem9.p1.3.m3.1c">D_{\alpha}^{+}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem9.p1.3.m3.1d">italic_D start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="D_{\alpha}^{-}" class="ltx_Math" display="inline" id="S3.Thmtheorem9.p1.4.m4.1"><semantics id="S3.Thmtheorem9.p1.4.m4.1a"><msubsup id="S3.Thmtheorem9.p1.4.m4.1.1" xref="S3.Thmtheorem9.p1.4.m4.1.1.cmml"><mi id="S3.Thmtheorem9.p1.4.m4.1.1.2.2" xref="S3.Thmtheorem9.p1.4.m4.1.1.2.2.cmml">D</mi><mi id="S3.Thmtheorem9.p1.4.m4.1.1.2.3" xref="S3.Thmtheorem9.p1.4.m4.1.1.2.3.cmml">α</mi><mo id="S3.Thmtheorem9.p1.4.m4.1.1.3" xref="S3.Thmtheorem9.p1.4.m4.1.1.3.cmml">−</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem9.p1.4.m4.1b"><apply id="S3.Thmtheorem9.p1.4.m4.1.1.cmml" xref="S3.Thmtheorem9.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem9.p1.4.m4.1.1.1.cmml" xref="S3.Thmtheorem9.p1.4.m4.1.1">superscript</csymbol><apply id="S3.Thmtheorem9.p1.4.m4.1.1.2.cmml" xref="S3.Thmtheorem9.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem9.p1.4.m4.1.1.2.1.cmml" xref="S3.Thmtheorem9.p1.4.m4.1.1">subscript</csymbol><ci id="S3.Thmtheorem9.p1.4.m4.1.1.2.2.cmml" xref="S3.Thmtheorem9.p1.4.m4.1.1.2.2">𝐷</ci><ci id="S3.Thmtheorem9.p1.4.m4.1.1.2.3.cmml" xref="S3.Thmtheorem9.p1.4.m4.1.1.2.3">𝛼</ci></apply><minus id="S3.Thmtheorem9.p1.4.m4.1.1.3.cmml" xref="S3.Thmtheorem9.p1.4.m4.1.1.3"></minus></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem9.p1.4.m4.1c">D_{\alpha}^{-}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem9.p1.4.m4.1d">italic_D start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT</annotation></semantics></math> are strongly surjective.</p> </div> </div> <div class="ltx_para" id="S3.p8"> <p class="ltx_p" id="S3.p8.1">For the rest of this section fix <math alttext="C" class="ltx_Math" display="inline" id="S3.p8.1.m1.1"><semantics id="S3.p8.1.m1.1a"><mi id="S3.p8.1.m1.1.1" xref="S3.p8.1.m1.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S3.p8.1.m1.1b"><ci id="S3.p8.1.m1.1.1.cmml" xref="S3.p8.1.m1.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p8.1.m1.1c">C</annotation><annotation encoding="application/x-llamapun" id="S3.p8.1.m1.1d">italic_C</annotation></semantics></math> a normal Countryman line. While we do not have an answer to <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#Thmquestion6" title="Question 6. ‣ Historical and mathematical context ‣ 1. Introduction ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">6</span></a>, we do have the following.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S3.Thmtheorem10"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem10.1.1.1">Lemma 3.10</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem10.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem10.p1"> <p class="ltx_p" id="S3.Thmtheorem10.p1.3">Assume <math alttext="\mathsf{PFA}" class="ltx_Math" display="inline" id="S3.Thmtheorem10.p1.1.m1.1"><semantics id="S3.Thmtheorem10.p1.1.m1.1a"><mi id="S3.Thmtheorem10.p1.1.m1.1.1" xref="S3.Thmtheorem10.p1.1.m1.1.1.cmml">𝖯𝖥𝖠</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem10.p1.1.m1.1b"><ci id="S3.Thmtheorem10.p1.1.m1.1.1.cmml" xref="S3.Thmtheorem10.p1.1.m1.1.1">𝖯𝖥𝖠</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem10.p1.1.m1.1c">\mathsf{PFA}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem10.p1.1.m1.1d">sansserif_PFA</annotation></semantics></math>. For every fragmented Aronszajn line <math alttext="A" class="ltx_Math" display="inline" id="S3.Thmtheorem10.p1.2.m2.1"><semantics id="S3.Thmtheorem10.p1.2.m2.1a"><mi id="S3.Thmtheorem10.p1.2.m2.1.1" xref="S3.Thmtheorem10.p1.2.m2.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem10.p1.2.m2.1b"><ci id="S3.Thmtheorem10.p1.2.m2.1.1.cmml" xref="S3.Thmtheorem10.p1.2.m2.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem10.p1.2.m2.1c">A</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem10.p1.2.m2.1d">italic_A</annotation></semantics></math>, <math alttext="\eta_{C}\trianglerighteq A" class="ltx_Math" display="inline" id="S3.Thmtheorem10.p1.3.m3.1"><semantics id="S3.Thmtheorem10.p1.3.m3.1a"><mrow id="S3.Thmtheorem10.p1.3.m3.1.1" xref="S3.Thmtheorem10.p1.3.m3.1.1.cmml"><msub id="S3.Thmtheorem10.p1.3.m3.1.1.2" xref="S3.Thmtheorem10.p1.3.m3.1.1.2.cmml"><mi id="S3.Thmtheorem10.p1.3.m3.1.1.2.2" xref="S3.Thmtheorem10.p1.3.m3.1.1.2.2.cmml">η</mi><mi id="S3.Thmtheorem10.p1.3.m3.1.1.2.3" xref="S3.Thmtheorem10.p1.3.m3.1.1.2.3.cmml">C</mi></msub><mo id="S3.Thmtheorem10.p1.3.m3.1.1.1" xref="S3.Thmtheorem10.p1.3.m3.1.1.1.cmml">⁢</mo><mi id="S3.Thmtheorem10.p1.3.m3.1.1.3" mathvariant="normal" xref="S3.Thmtheorem10.p1.3.m3.1.1.3.cmml">⊵</mi><mo id="S3.Thmtheorem10.p1.3.m3.1.1.1a" xref="S3.Thmtheorem10.p1.3.m3.1.1.1.cmml">⁢</mo><mi id="S3.Thmtheorem10.p1.3.m3.1.1.4" xref="S3.Thmtheorem10.p1.3.m3.1.1.4.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem10.p1.3.m3.1b"><apply id="S3.Thmtheorem10.p1.3.m3.1.1.cmml" xref="S3.Thmtheorem10.p1.3.m3.1.1"><times id="S3.Thmtheorem10.p1.3.m3.1.1.1.cmml" xref="S3.Thmtheorem10.p1.3.m3.1.1.1"></times><apply id="S3.Thmtheorem10.p1.3.m3.1.1.2.cmml" xref="S3.Thmtheorem10.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem10.p1.3.m3.1.1.2.1.cmml" xref="S3.Thmtheorem10.p1.3.m3.1.1.2">subscript</csymbol><ci id="S3.Thmtheorem10.p1.3.m3.1.1.2.2.cmml" xref="S3.Thmtheorem10.p1.3.m3.1.1.2.2">𝜂</ci><ci id="S3.Thmtheorem10.p1.3.m3.1.1.2.3.cmml" xref="S3.Thmtheorem10.p1.3.m3.1.1.2.3">𝐶</ci></apply><ci id="S3.Thmtheorem10.p1.3.m3.1.1.3.cmml" xref="S3.Thmtheorem10.p1.3.m3.1.1.3">⊵</ci><ci id="S3.Thmtheorem10.p1.3.m3.1.1.4.cmml" xref="S3.Thmtheorem10.p1.3.m3.1.1.4">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem10.p1.3.m3.1c">\eta_{C}\trianglerighteq A</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem10.p1.3.m3.1d">italic_η start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ⊵ italic_A</annotation></semantics></math>.</p> </div> </div> <div class="ltx_proof" id="S3.12"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S3.9.p1"> <p class="ltx_p" id="S3.9.p1.5">By <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S2.Thmtheorem10" title="Lemma 2.10. ‣ 2. Aronszajn and Countryman lines ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">2.10</span></a> we know that for any fragmented Aronszajn line there is a least <math alttext="\alpha" class="ltx_Math" display="inline" id="S3.9.p1.1.m1.1"><semantics id="S3.9.p1.1.m1.1a"><mi id="S3.9.p1.1.m1.1.1" xref="S3.9.p1.1.m1.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S3.9.p1.1.m1.1b"><ci id="S3.9.p1.1.m1.1.1.cmml" xref="S3.9.p1.1.m1.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.9.p1.1.m1.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.9.p1.1.m1.1d">italic_α</annotation></semantics></math> such that <math alttext="A\preceq D_{\alpha}^{-}" class="ltx_Math" display="inline" id="S3.9.p1.2.m2.1"><semantics id="S3.9.p1.2.m2.1a"><mrow id="S3.9.p1.2.m2.1.1" xref="S3.9.p1.2.m2.1.1.cmml"><mi id="S3.9.p1.2.m2.1.1.2" xref="S3.9.p1.2.m2.1.1.2.cmml">A</mi><mo id="S3.9.p1.2.m2.1.1.1" xref="S3.9.p1.2.m2.1.1.1.cmml">⪯</mo><msubsup id="S3.9.p1.2.m2.1.1.3" xref="S3.9.p1.2.m2.1.1.3.cmml"><mi id="S3.9.p1.2.m2.1.1.3.2.2" xref="S3.9.p1.2.m2.1.1.3.2.2.cmml">D</mi><mi id="S3.9.p1.2.m2.1.1.3.2.3" xref="S3.9.p1.2.m2.1.1.3.2.3.cmml">α</mi><mo id="S3.9.p1.2.m2.1.1.3.3" xref="S3.9.p1.2.m2.1.1.3.3.cmml">−</mo></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S3.9.p1.2.m2.1b"><apply id="S3.9.p1.2.m2.1.1.cmml" xref="S3.9.p1.2.m2.1.1"><csymbol cd="latexml" id="S3.9.p1.2.m2.1.1.1.cmml" xref="S3.9.p1.2.m2.1.1.1">precedes-or-equals</csymbol><ci id="S3.9.p1.2.m2.1.1.2.cmml" xref="S3.9.p1.2.m2.1.1.2">𝐴</ci><apply id="S3.9.p1.2.m2.1.1.3.cmml" xref="S3.9.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.9.p1.2.m2.1.1.3.1.cmml" xref="S3.9.p1.2.m2.1.1.3">superscript</csymbol><apply id="S3.9.p1.2.m2.1.1.3.2.cmml" xref="S3.9.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.9.p1.2.m2.1.1.3.2.1.cmml" xref="S3.9.p1.2.m2.1.1.3">subscript</csymbol><ci id="S3.9.p1.2.m2.1.1.3.2.2.cmml" xref="S3.9.p1.2.m2.1.1.3.2.2">𝐷</ci><ci id="S3.9.p1.2.m2.1.1.3.2.3.cmml" xref="S3.9.p1.2.m2.1.1.3.2.3">𝛼</ci></apply><minus id="S3.9.p1.2.m2.1.1.3.3.cmml" xref="S3.9.p1.2.m2.1.1.3.3"></minus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.9.p1.2.m2.1c">A\preceq D_{\alpha}^{-}</annotation><annotation encoding="application/x-llamapun" id="S3.9.p1.2.m2.1d">italic_A ⪯ italic_D start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT</annotation></semantics></math> or <math alttext="A\preceq D_{\alpha}^{+}" class="ltx_Math" display="inline" id="S3.9.p1.3.m3.1"><semantics id="S3.9.p1.3.m3.1a"><mrow id="S3.9.p1.3.m3.1.1" xref="S3.9.p1.3.m3.1.1.cmml"><mi id="S3.9.p1.3.m3.1.1.2" xref="S3.9.p1.3.m3.1.1.2.cmml">A</mi><mo id="S3.9.p1.3.m3.1.1.1" xref="S3.9.p1.3.m3.1.1.1.cmml">⪯</mo><msubsup id="S3.9.p1.3.m3.1.1.3" xref="S3.9.p1.3.m3.1.1.3.cmml"><mi id="S3.9.p1.3.m3.1.1.3.2.2" xref="S3.9.p1.3.m3.1.1.3.2.2.cmml">D</mi><mi id="S3.9.p1.3.m3.1.1.3.2.3" xref="S3.9.p1.3.m3.1.1.3.2.3.cmml">α</mi><mo id="S3.9.p1.3.m3.1.1.3.3" xref="S3.9.p1.3.m3.1.1.3.3.cmml">+</mo></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S3.9.p1.3.m3.1b"><apply id="S3.9.p1.3.m3.1.1.cmml" xref="S3.9.p1.3.m3.1.1"><csymbol cd="latexml" id="S3.9.p1.3.m3.1.1.1.cmml" xref="S3.9.p1.3.m3.1.1.1">precedes-or-equals</csymbol><ci id="S3.9.p1.3.m3.1.1.2.cmml" xref="S3.9.p1.3.m3.1.1.2">𝐴</ci><apply id="S3.9.p1.3.m3.1.1.3.cmml" xref="S3.9.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S3.9.p1.3.m3.1.1.3.1.cmml" xref="S3.9.p1.3.m3.1.1.3">superscript</csymbol><apply id="S3.9.p1.3.m3.1.1.3.2.cmml" xref="S3.9.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S3.9.p1.3.m3.1.1.3.2.1.cmml" xref="S3.9.p1.3.m3.1.1.3">subscript</csymbol><ci id="S3.9.p1.3.m3.1.1.3.2.2.cmml" xref="S3.9.p1.3.m3.1.1.3.2.2">𝐷</ci><ci id="S3.9.p1.3.m3.1.1.3.2.3.cmml" xref="S3.9.p1.3.m3.1.1.3.2.3">𝛼</ci></apply><plus id="S3.9.p1.3.m3.1.1.3.3.cmml" xref="S3.9.p1.3.m3.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.9.p1.3.m3.1c">A\preceq D_{\alpha}^{+}</annotation><annotation encoding="application/x-llamapun" id="S3.9.p1.3.m3.1d">italic_A ⪯ italic_D start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math>. Thus by <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S3.Thmtheorem9" title="Corollary 3.9. ‣ 3. Strongly surjective Aronszajn lines ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Corollary</span> <span class="ltx_text ltx_ref_tag">3.9</span></a>, it is enough to prove that <math alttext="\eta_{C}\trianglerighteq D_{\alpha}^{+},D_{\alpha}^{-}" class="ltx_Math" display="inline" id="S3.9.p1.4.m4.2"><semantics id="S3.9.p1.4.m4.2a"><mrow id="S3.9.p1.4.m4.2.2.2" xref="S3.9.p1.4.m4.2.2.3.cmml"><mrow id="S3.9.p1.4.m4.1.1.1.1" xref="S3.9.p1.4.m4.1.1.1.1.cmml"><msub id="S3.9.p1.4.m4.1.1.1.1.2" xref="S3.9.p1.4.m4.1.1.1.1.2.cmml"><mi id="S3.9.p1.4.m4.1.1.1.1.2.2" xref="S3.9.p1.4.m4.1.1.1.1.2.2.cmml">η</mi><mi id="S3.9.p1.4.m4.1.1.1.1.2.3" xref="S3.9.p1.4.m4.1.1.1.1.2.3.cmml">C</mi></msub><mo id="S3.9.p1.4.m4.1.1.1.1.1" xref="S3.9.p1.4.m4.1.1.1.1.1.cmml">⁢</mo><mi id="S3.9.p1.4.m4.1.1.1.1.3" mathvariant="normal" xref="S3.9.p1.4.m4.1.1.1.1.3.cmml">⊵</mi><mo id="S3.9.p1.4.m4.1.1.1.1.1a" xref="S3.9.p1.4.m4.1.1.1.1.1.cmml">⁢</mo><msubsup id="S3.9.p1.4.m4.1.1.1.1.4" xref="S3.9.p1.4.m4.1.1.1.1.4.cmml"><mi id="S3.9.p1.4.m4.1.1.1.1.4.2.2" xref="S3.9.p1.4.m4.1.1.1.1.4.2.2.cmml">D</mi><mi id="S3.9.p1.4.m4.1.1.1.1.4.2.3" xref="S3.9.p1.4.m4.1.1.1.1.4.2.3.cmml">α</mi><mo id="S3.9.p1.4.m4.1.1.1.1.4.3" xref="S3.9.p1.4.m4.1.1.1.1.4.3.cmml">+</mo></msubsup></mrow><mo id="S3.9.p1.4.m4.2.2.2.3" xref="S3.9.p1.4.m4.2.2.3.cmml">,</mo><msubsup id="S3.9.p1.4.m4.2.2.2.2" xref="S3.9.p1.4.m4.2.2.2.2.cmml"><mi id="S3.9.p1.4.m4.2.2.2.2.2.2" xref="S3.9.p1.4.m4.2.2.2.2.2.2.cmml">D</mi><mi id="S3.9.p1.4.m4.2.2.2.2.2.3" xref="S3.9.p1.4.m4.2.2.2.2.2.3.cmml">α</mi><mo id="S3.9.p1.4.m4.2.2.2.2.3" xref="S3.9.p1.4.m4.2.2.2.2.3.cmml">−</mo></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S3.9.p1.4.m4.2b"><list id="S3.9.p1.4.m4.2.2.3.cmml" xref="S3.9.p1.4.m4.2.2.2"><apply id="S3.9.p1.4.m4.1.1.1.1.cmml" xref="S3.9.p1.4.m4.1.1.1.1"><times id="S3.9.p1.4.m4.1.1.1.1.1.cmml" xref="S3.9.p1.4.m4.1.1.1.1.1"></times><apply id="S3.9.p1.4.m4.1.1.1.1.2.cmml" xref="S3.9.p1.4.m4.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.9.p1.4.m4.1.1.1.1.2.1.cmml" xref="S3.9.p1.4.m4.1.1.1.1.2">subscript</csymbol><ci id="S3.9.p1.4.m4.1.1.1.1.2.2.cmml" xref="S3.9.p1.4.m4.1.1.1.1.2.2">𝜂</ci><ci id="S3.9.p1.4.m4.1.1.1.1.2.3.cmml" xref="S3.9.p1.4.m4.1.1.1.1.2.3">𝐶</ci></apply><ci id="S3.9.p1.4.m4.1.1.1.1.3.cmml" xref="S3.9.p1.4.m4.1.1.1.1.3">⊵</ci><apply id="S3.9.p1.4.m4.1.1.1.1.4.cmml" xref="S3.9.p1.4.m4.1.1.1.1.4"><csymbol cd="ambiguous" id="S3.9.p1.4.m4.1.1.1.1.4.1.cmml" xref="S3.9.p1.4.m4.1.1.1.1.4">superscript</csymbol><apply id="S3.9.p1.4.m4.1.1.1.1.4.2.cmml" xref="S3.9.p1.4.m4.1.1.1.1.4"><csymbol cd="ambiguous" id="S3.9.p1.4.m4.1.1.1.1.4.2.1.cmml" xref="S3.9.p1.4.m4.1.1.1.1.4">subscript</csymbol><ci id="S3.9.p1.4.m4.1.1.1.1.4.2.2.cmml" xref="S3.9.p1.4.m4.1.1.1.1.4.2.2">𝐷</ci><ci id="S3.9.p1.4.m4.1.1.1.1.4.2.3.cmml" xref="S3.9.p1.4.m4.1.1.1.1.4.2.3">𝛼</ci></apply><plus id="S3.9.p1.4.m4.1.1.1.1.4.3.cmml" xref="S3.9.p1.4.m4.1.1.1.1.4.3"></plus></apply></apply><apply id="S3.9.p1.4.m4.2.2.2.2.cmml" xref="S3.9.p1.4.m4.2.2.2.2"><csymbol cd="ambiguous" id="S3.9.p1.4.m4.2.2.2.2.1.cmml" xref="S3.9.p1.4.m4.2.2.2.2">superscript</csymbol><apply id="S3.9.p1.4.m4.2.2.2.2.2.cmml" xref="S3.9.p1.4.m4.2.2.2.2"><csymbol cd="ambiguous" id="S3.9.p1.4.m4.2.2.2.2.2.1.cmml" xref="S3.9.p1.4.m4.2.2.2.2">subscript</csymbol><ci id="S3.9.p1.4.m4.2.2.2.2.2.2.cmml" xref="S3.9.p1.4.m4.2.2.2.2.2.2">𝐷</ci><ci id="S3.9.p1.4.m4.2.2.2.2.2.3.cmml" xref="S3.9.p1.4.m4.2.2.2.2.2.3">𝛼</ci></apply><minus id="S3.9.p1.4.m4.2.2.2.2.3.cmml" xref="S3.9.p1.4.m4.2.2.2.2.3"></minus></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.9.p1.4.m4.2c">\eta_{C}\trianglerighteq D_{\alpha}^{+},D_{\alpha}^{-}</annotation><annotation encoding="application/x-llamapun" id="S3.9.p1.4.m4.2d">italic_η start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ⊵ italic_D start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT , italic_D start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT</annotation></semantics></math> for every <math alttext="\alpha<\omega_{1}" class="ltx_Math" display="inline" id="S3.9.p1.5.m5.1"><semantics id="S3.9.p1.5.m5.1a"><mrow id="S3.9.p1.5.m5.1.1" xref="S3.9.p1.5.m5.1.1.cmml"><mi id="S3.9.p1.5.m5.1.1.2" xref="S3.9.p1.5.m5.1.1.2.cmml">α</mi><mo id="S3.9.p1.5.m5.1.1.1" xref="S3.9.p1.5.m5.1.1.1.cmml"><</mo><msub id="S3.9.p1.5.m5.1.1.3" xref="S3.9.p1.5.m5.1.1.3.cmml"><mi id="S3.9.p1.5.m5.1.1.3.2" xref="S3.9.p1.5.m5.1.1.3.2.cmml">ω</mi><mn id="S3.9.p1.5.m5.1.1.3.3" xref="S3.9.p1.5.m5.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.9.p1.5.m5.1b"><apply id="S3.9.p1.5.m5.1.1.cmml" xref="S3.9.p1.5.m5.1.1"><lt id="S3.9.p1.5.m5.1.1.1.cmml" xref="S3.9.p1.5.m5.1.1.1"></lt><ci id="S3.9.p1.5.m5.1.1.2.cmml" xref="S3.9.p1.5.m5.1.1.2">𝛼</ci><apply id="S3.9.p1.5.m5.1.1.3.cmml" xref="S3.9.p1.5.m5.1.1.3"><csymbol cd="ambiguous" id="S3.9.p1.5.m5.1.1.3.1.cmml" xref="S3.9.p1.5.m5.1.1.3">subscript</csymbol><ci id="S3.9.p1.5.m5.1.1.3.2.cmml" xref="S3.9.p1.5.m5.1.1.3.2">𝜔</ci><cn id="S3.9.p1.5.m5.1.1.3.3.cmml" type="integer" xref="S3.9.p1.5.m5.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.9.p1.5.m5.1c">\alpha<\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.9.p1.5.m5.1d">italic_α < italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.10.p2"> <p class="ltx_p" id="S3.10.p2.9">We do it by induction on <math alttext="\alpha" class="ltx_Math" display="inline" id="S3.10.p2.1.m1.1"><semantics id="S3.10.p2.1.m1.1a"><mi id="S3.10.p2.1.m1.1.1" xref="S3.10.p2.1.m1.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S3.10.p2.1.m1.1b"><ci id="S3.10.p2.1.m1.1.1.cmml" xref="S3.10.p2.1.m1.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.10.p2.1.m1.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.10.p2.1.m1.1d">italic_α</annotation></semantics></math>. For <math alttext="\alpha=0" class="ltx_Math" display="inline" id="S3.10.p2.2.m2.1"><semantics id="S3.10.p2.2.m2.1a"><mrow id="S3.10.p2.2.m2.1.1" xref="S3.10.p2.2.m2.1.1.cmml"><mi id="S3.10.p2.2.m2.1.1.2" xref="S3.10.p2.2.m2.1.1.2.cmml">α</mi><mo id="S3.10.p2.2.m2.1.1.1" xref="S3.10.p2.2.m2.1.1.1.cmml">=</mo><mn id="S3.10.p2.2.m2.1.1.3" xref="S3.10.p2.2.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.10.p2.2.m2.1b"><apply id="S3.10.p2.2.m2.1.1.cmml" xref="S3.10.p2.2.m2.1.1"><eq id="S3.10.p2.2.m2.1.1.1.cmml" xref="S3.10.p2.2.m2.1.1.1"></eq><ci id="S3.10.p2.2.m2.1.1.2.cmml" xref="S3.10.p2.2.m2.1.1.2">𝛼</ci><cn id="S3.10.p2.2.m2.1.1.3.cmml" type="integer" xref="S3.10.p2.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.10.p2.2.m2.1c">\alpha=0</annotation><annotation encoding="application/x-llamapun" id="S3.10.p2.2.m2.1d">italic_α = 0</annotation></semantics></math> we need to show that <math alttext="C,C^{\star}\trianglelefteq\eta_{C}" class="ltx_Math" display="inline" id="S3.10.p2.3.m3.2"><semantics id="S3.10.p2.3.m3.2a"><mrow id="S3.10.p2.3.m3.2.2.1" xref="S3.10.p2.3.m3.2.2.2.cmml"><mi id="S3.10.p2.3.m3.1.1" xref="S3.10.p2.3.m3.1.1.cmml">C</mi><mo id="S3.10.p2.3.m3.2.2.1.2" xref="S3.10.p2.3.m3.2.2.2.cmml">,</mo><mrow id="S3.10.p2.3.m3.2.2.1.1" xref="S3.10.p2.3.m3.2.2.1.1.cmml"><msup id="S3.10.p2.3.m3.2.2.1.1.2" xref="S3.10.p2.3.m3.2.2.1.1.2.cmml"><mi id="S3.10.p2.3.m3.2.2.1.1.2.2" xref="S3.10.p2.3.m3.2.2.1.1.2.2.cmml">C</mi><mo id="S3.10.p2.3.m3.2.2.1.1.2.3" xref="S3.10.p2.3.m3.2.2.1.1.2.3.cmml">⋆</mo></msup><mo id="S3.10.p2.3.m3.2.2.1.1.1" xref="S3.10.p2.3.m3.2.2.1.1.1.cmml">⁢</mo><mi id="S3.10.p2.3.m3.2.2.1.1.3" mathvariant="normal" xref="S3.10.p2.3.m3.2.2.1.1.3.cmml">⊴</mi><mo id="S3.10.p2.3.m3.2.2.1.1.1a" xref="S3.10.p2.3.m3.2.2.1.1.1.cmml">⁢</mo><msub id="S3.10.p2.3.m3.2.2.1.1.4" xref="S3.10.p2.3.m3.2.2.1.1.4.cmml"><mi id="S3.10.p2.3.m3.2.2.1.1.4.2" xref="S3.10.p2.3.m3.2.2.1.1.4.2.cmml">η</mi><mi id="S3.10.p2.3.m3.2.2.1.1.4.3" xref="S3.10.p2.3.m3.2.2.1.1.4.3.cmml">C</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.10.p2.3.m3.2b"><list id="S3.10.p2.3.m3.2.2.2.cmml" xref="S3.10.p2.3.m3.2.2.1"><ci id="S3.10.p2.3.m3.1.1.cmml" xref="S3.10.p2.3.m3.1.1">𝐶</ci><apply id="S3.10.p2.3.m3.2.2.1.1.cmml" xref="S3.10.p2.3.m3.2.2.1.1"><times id="S3.10.p2.3.m3.2.2.1.1.1.cmml" xref="S3.10.p2.3.m3.2.2.1.1.1"></times><apply id="S3.10.p2.3.m3.2.2.1.1.2.cmml" xref="S3.10.p2.3.m3.2.2.1.1.2"><csymbol cd="ambiguous" id="S3.10.p2.3.m3.2.2.1.1.2.1.cmml" xref="S3.10.p2.3.m3.2.2.1.1.2">superscript</csymbol><ci id="S3.10.p2.3.m3.2.2.1.1.2.2.cmml" xref="S3.10.p2.3.m3.2.2.1.1.2.2">𝐶</ci><ci id="S3.10.p2.3.m3.2.2.1.1.2.3.cmml" xref="S3.10.p2.3.m3.2.2.1.1.2.3">⋆</ci></apply><ci id="S3.10.p2.3.m3.2.2.1.1.3.cmml" xref="S3.10.p2.3.m3.2.2.1.1.3">⊴</ci><apply id="S3.10.p2.3.m3.2.2.1.1.4.cmml" xref="S3.10.p2.3.m3.2.2.1.1.4"><csymbol cd="ambiguous" id="S3.10.p2.3.m3.2.2.1.1.4.1.cmml" xref="S3.10.p2.3.m3.2.2.1.1.4">subscript</csymbol><ci id="S3.10.p2.3.m3.2.2.1.1.4.2.cmml" xref="S3.10.p2.3.m3.2.2.1.1.4.2">𝜂</ci><ci id="S3.10.p2.3.m3.2.2.1.1.4.3.cmml" xref="S3.10.p2.3.m3.2.2.1.1.4.3">𝐶</ci></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.10.p2.3.m3.2c">C,C^{\star}\trianglelefteq\eta_{C}</annotation><annotation encoding="application/x-llamapun" id="S3.10.p2.3.m3.2d">italic_C , italic_C start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT ⊴ italic_η start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT</annotation></semantics></math>. We do it for <math alttext="C" class="ltx_Math" display="inline" id="S3.10.p2.4.m4.1"><semantics id="S3.10.p2.4.m4.1a"><mi id="S3.10.p2.4.m4.1.1" xref="S3.10.p2.4.m4.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S3.10.p2.4.m4.1b"><ci id="S3.10.p2.4.m4.1.1.cmml" xref="S3.10.p2.4.m4.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.10.p2.4.m4.1c">C</annotation><annotation encoding="application/x-llamapun" id="S3.10.p2.4.m4.1d">italic_C</annotation></semantics></math>, the other case is identical. Note that <math alttext="\eta_{C}\trianglerighteq{(C^{\star}+1+C)}^{2}\trianglerighteq\mathbb{Z}\times(% 1+C)\cong C" class="ltx_Math" display="inline" id="S3.10.p2.5.m5.2"><semantics id="S3.10.p2.5.m5.2a"><mrow id="S3.10.p2.5.m5.2.2" xref="S3.10.p2.5.m5.2.2.cmml"><mrow id="S3.10.p2.5.m5.2.2.2" xref="S3.10.p2.5.m5.2.2.2.cmml"><mrow id="S3.10.p2.5.m5.1.1.1.1" xref="S3.10.p2.5.m5.1.1.1.1.cmml"><msub id="S3.10.p2.5.m5.1.1.1.1.3" xref="S3.10.p2.5.m5.1.1.1.1.3.cmml"><mi id="S3.10.p2.5.m5.1.1.1.1.3.2" xref="S3.10.p2.5.m5.1.1.1.1.3.2.cmml">η</mi><mi id="S3.10.p2.5.m5.1.1.1.1.3.3" xref="S3.10.p2.5.m5.1.1.1.1.3.3.cmml">C</mi></msub><mo id="S3.10.p2.5.m5.1.1.1.1.2" xref="S3.10.p2.5.m5.1.1.1.1.2.cmml">⁢</mo><mi id="S3.10.p2.5.m5.1.1.1.1.4" mathvariant="normal" xref="S3.10.p2.5.m5.1.1.1.1.4.cmml">⊵</mi><mo id="S3.10.p2.5.m5.1.1.1.1.2a" xref="S3.10.p2.5.m5.1.1.1.1.2.cmml">⁢</mo><msup id="S3.10.p2.5.m5.1.1.1.1.1" xref="S3.10.p2.5.m5.1.1.1.1.1.cmml"><mrow id="S3.10.p2.5.m5.1.1.1.1.1.1.1" xref="S3.10.p2.5.m5.1.1.1.1.1.1.1.1.cmml"><mo id="S3.10.p2.5.m5.1.1.1.1.1.1.1.2" stretchy="false" xref="S3.10.p2.5.m5.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.10.p2.5.m5.1.1.1.1.1.1.1.1" xref="S3.10.p2.5.m5.1.1.1.1.1.1.1.1.cmml"><msup id="S3.10.p2.5.m5.1.1.1.1.1.1.1.1.2" xref="S3.10.p2.5.m5.1.1.1.1.1.1.1.1.2.cmml"><mi id="S3.10.p2.5.m5.1.1.1.1.1.1.1.1.2.2" xref="S3.10.p2.5.m5.1.1.1.1.1.1.1.1.2.2.cmml">C</mi><mo id="S3.10.p2.5.m5.1.1.1.1.1.1.1.1.2.3" xref="S3.10.p2.5.m5.1.1.1.1.1.1.1.1.2.3.cmml">⋆</mo></msup><mo id="S3.10.p2.5.m5.1.1.1.1.1.1.1.1.1" xref="S3.10.p2.5.m5.1.1.1.1.1.1.1.1.1.cmml">+</mo><mn id="S3.10.p2.5.m5.1.1.1.1.1.1.1.1.3" xref="S3.10.p2.5.m5.1.1.1.1.1.1.1.1.3.cmml">1</mn><mo id="S3.10.p2.5.m5.1.1.1.1.1.1.1.1.1a" xref="S3.10.p2.5.m5.1.1.1.1.1.1.1.1.1.cmml">+</mo><mi id="S3.10.p2.5.m5.1.1.1.1.1.1.1.1.4" xref="S3.10.p2.5.m5.1.1.1.1.1.1.1.1.4.cmml">C</mi></mrow><mo id="S3.10.p2.5.m5.1.1.1.1.1.1.1.3" stretchy="false" xref="S3.10.p2.5.m5.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mn id="S3.10.p2.5.m5.1.1.1.1.1.3" xref="S3.10.p2.5.m5.1.1.1.1.1.3.cmml">2</mn></msup><mo id="S3.10.p2.5.m5.1.1.1.1.2b" xref="S3.10.p2.5.m5.1.1.1.1.2.cmml">⁢</mo><mi id="S3.10.p2.5.m5.1.1.1.1.5" mathvariant="normal" xref="S3.10.p2.5.m5.1.1.1.1.5.cmml">⊵</mi><mo id="S3.10.p2.5.m5.1.1.1.1.2c" xref="S3.10.p2.5.m5.1.1.1.1.2.cmml">⁢</mo><mi id="S3.10.p2.5.m5.1.1.1.1.6" xref="S3.10.p2.5.m5.1.1.1.1.6.cmml">ℤ</mi></mrow><mo id="S3.10.p2.5.m5.2.2.2.3" lspace="0.222em" rspace="0.222em" xref="S3.10.p2.5.m5.2.2.2.3.cmml">×</mo><mrow id="S3.10.p2.5.m5.2.2.2.2.1" xref="S3.10.p2.5.m5.2.2.2.2.1.1.cmml"><mo id="S3.10.p2.5.m5.2.2.2.2.1.2" stretchy="false" xref="S3.10.p2.5.m5.2.2.2.2.1.1.cmml">(</mo><mrow id="S3.10.p2.5.m5.2.2.2.2.1.1" xref="S3.10.p2.5.m5.2.2.2.2.1.1.cmml"><mn id="S3.10.p2.5.m5.2.2.2.2.1.1.2" xref="S3.10.p2.5.m5.2.2.2.2.1.1.2.cmml">1</mn><mo id="S3.10.p2.5.m5.2.2.2.2.1.1.1" xref="S3.10.p2.5.m5.2.2.2.2.1.1.1.cmml">+</mo><mi id="S3.10.p2.5.m5.2.2.2.2.1.1.3" xref="S3.10.p2.5.m5.2.2.2.2.1.1.3.cmml">C</mi></mrow><mo id="S3.10.p2.5.m5.2.2.2.2.1.3" stretchy="false" xref="S3.10.p2.5.m5.2.2.2.2.1.1.cmml">)</mo></mrow></mrow><mo id="S3.10.p2.5.m5.2.2.3" xref="S3.10.p2.5.m5.2.2.3.cmml">≅</mo><mi id="S3.10.p2.5.m5.2.2.4" xref="S3.10.p2.5.m5.2.2.4.cmml">C</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.10.p2.5.m5.2b"><apply id="S3.10.p2.5.m5.2.2.cmml" xref="S3.10.p2.5.m5.2.2"><approx id="S3.10.p2.5.m5.2.2.3.cmml" xref="S3.10.p2.5.m5.2.2.3"></approx><apply id="S3.10.p2.5.m5.2.2.2.cmml" xref="S3.10.p2.5.m5.2.2.2"><times id="S3.10.p2.5.m5.2.2.2.3.cmml" xref="S3.10.p2.5.m5.2.2.2.3"></times><apply id="S3.10.p2.5.m5.1.1.1.1.cmml" xref="S3.10.p2.5.m5.1.1.1.1"><times id="S3.10.p2.5.m5.1.1.1.1.2.cmml" xref="S3.10.p2.5.m5.1.1.1.1.2"></times><apply id="S3.10.p2.5.m5.1.1.1.1.3.cmml" xref="S3.10.p2.5.m5.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.10.p2.5.m5.1.1.1.1.3.1.cmml" xref="S3.10.p2.5.m5.1.1.1.1.3">subscript</csymbol><ci id="S3.10.p2.5.m5.1.1.1.1.3.2.cmml" xref="S3.10.p2.5.m5.1.1.1.1.3.2">𝜂</ci><ci id="S3.10.p2.5.m5.1.1.1.1.3.3.cmml" xref="S3.10.p2.5.m5.1.1.1.1.3.3">𝐶</ci></apply><ci id="S3.10.p2.5.m5.1.1.1.1.4.cmml" xref="S3.10.p2.5.m5.1.1.1.1.4">⊵</ci><apply id="S3.10.p2.5.m5.1.1.1.1.1.cmml" xref="S3.10.p2.5.m5.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.10.p2.5.m5.1.1.1.1.1.2.cmml" xref="S3.10.p2.5.m5.1.1.1.1.1">superscript</csymbol><apply id="S3.10.p2.5.m5.1.1.1.1.1.1.1.1.cmml" xref="S3.10.p2.5.m5.1.1.1.1.1.1.1"><plus id="S3.10.p2.5.m5.1.1.1.1.1.1.1.1.1.cmml" xref="S3.10.p2.5.m5.1.1.1.1.1.1.1.1.1"></plus><apply id="S3.10.p2.5.m5.1.1.1.1.1.1.1.1.2.cmml" xref="S3.10.p2.5.m5.1.1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.10.p2.5.m5.1.1.1.1.1.1.1.1.2.1.cmml" xref="S3.10.p2.5.m5.1.1.1.1.1.1.1.1.2">superscript</csymbol><ci id="S3.10.p2.5.m5.1.1.1.1.1.1.1.1.2.2.cmml" xref="S3.10.p2.5.m5.1.1.1.1.1.1.1.1.2.2">𝐶</ci><ci id="S3.10.p2.5.m5.1.1.1.1.1.1.1.1.2.3.cmml" xref="S3.10.p2.5.m5.1.1.1.1.1.1.1.1.2.3">⋆</ci></apply><cn id="S3.10.p2.5.m5.1.1.1.1.1.1.1.1.3.cmml" type="integer" xref="S3.10.p2.5.m5.1.1.1.1.1.1.1.1.3">1</cn><ci id="S3.10.p2.5.m5.1.1.1.1.1.1.1.1.4.cmml" xref="S3.10.p2.5.m5.1.1.1.1.1.1.1.1.4">𝐶</ci></apply><cn id="S3.10.p2.5.m5.1.1.1.1.1.3.cmml" type="integer" xref="S3.10.p2.5.m5.1.1.1.1.1.3">2</cn></apply><ci id="S3.10.p2.5.m5.1.1.1.1.5.cmml" xref="S3.10.p2.5.m5.1.1.1.1.5">⊵</ci><ci id="S3.10.p2.5.m5.1.1.1.1.6.cmml" xref="S3.10.p2.5.m5.1.1.1.1.6">ℤ</ci></apply><apply id="S3.10.p2.5.m5.2.2.2.2.1.1.cmml" xref="S3.10.p2.5.m5.2.2.2.2.1"><plus id="S3.10.p2.5.m5.2.2.2.2.1.1.1.cmml" xref="S3.10.p2.5.m5.2.2.2.2.1.1.1"></plus><cn id="S3.10.p2.5.m5.2.2.2.2.1.1.2.cmml" type="integer" xref="S3.10.p2.5.m5.2.2.2.2.1.1.2">1</cn><ci id="S3.10.p2.5.m5.2.2.2.2.1.1.3.cmml" xref="S3.10.p2.5.m5.2.2.2.2.1.1.3">𝐶</ci></apply></apply><ci id="S3.10.p2.5.m5.2.2.4.cmml" xref="S3.10.p2.5.m5.2.2.4">𝐶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.10.p2.5.m5.2c">\eta_{C}\trianglerighteq{(C^{\star}+1+C)}^{2}\trianglerighteq\mathbb{Z}\times(% 1+C)\cong C</annotation><annotation encoding="application/x-llamapun" id="S3.10.p2.5.m5.2d">italic_η start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ⊵ ( italic_C start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT + 1 + italic_C ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ⊵ blackboard_Z × ( 1 + italic_C ) ≅ italic_C</annotation></semantics></math>, where the first inequality is witnessed by <math alttext="\langle x_{n}:n<\omega\rangle\mapsto(x_{0},x_{1})" class="ltx_math_unparsed" display="inline" id="S3.10.p2.6.m6.1"><semantics id="S3.10.p2.6.m6.1a"><mrow id="S3.10.p2.6.m6.1b"><mrow id="S3.10.p2.6.m6.1.2"><mo id="S3.10.p2.6.m6.1.2.1" stretchy="false">⟨</mo><msub id="S3.10.p2.6.m6.1.2.2"><mi id="S3.10.p2.6.m6.1.2.2.2">x</mi><mi id="S3.10.p2.6.m6.1.2.2.3">n</mi></msub><mo id="S3.10.p2.6.m6.1.2.3" lspace="0.278em" rspace="0.278em">:</mo><mi id="S3.10.p2.6.m6.1.2.4">n</mi><mo id="S3.10.p2.6.m6.1.2.5"><</mo><mi id="S3.10.p2.6.m6.1.1">ω</mi><mo id="S3.10.p2.6.m6.1.2.6" stretchy="false">⟩</mo></mrow><mo id="S3.10.p2.6.m6.1.3" stretchy="false">↦</mo><mrow id="S3.10.p2.6.m6.1.4"><mo id="S3.10.p2.6.m6.1.4.1" stretchy="false">(</mo><msub id="S3.10.p2.6.m6.1.4.2"><mi id="S3.10.p2.6.m6.1.4.2.2">x</mi><mn id="S3.10.p2.6.m6.1.4.2.3">0</mn></msub><mo id="S3.10.p2.6.m6.1.4.3">,</mo><msub id="S3.10.p2.6.m6.1.4.4"><mi id="S3.10.p2.6.m6.1.4.4.2">x</mi><mn id="S3.10.p2.6.m6.1.4.4.3">1</mn></msub><mo id="S3.10.p2.6.m6.1.4.5" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="S3.10.p2.6.m6.1c">\langle x_{n}:n<\omega\rangle\mapsto(x_{0},x_{1})</annotation><annotation encoding="application/x-llamapun" id="S3.10.p2.6.m6.1d">⟨ italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT : italic_n < italic_ω ⟩ ↦ ( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math>, the second follows from the strong surjectivity of <math alttext="C" class="ltx_Math" display="inline" id="S3.10.p2.7.m7.1"><semantics id="S3.10.p2.7.m7.1a"><mi id="S3.10.p2.7.m7.1.1" xref="S3.10.p2.7.m7.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S3.10.p2.7.m7.1b"><ci id="S3.10.p2.7.m7.1.1.cmml" xref="S3.10.p2.7.m7.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.10.p2.7.m7.1c">C</annotation><annotation encoding="application/x-llamapun" id="S3.10.p2.7.m7.1d">italic_C</annotation></semantics></math> and <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S3.Thmtheorem4" title="Lemma 3.4. ‣ 3. Strongly surjective Aronszajn lines ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">3.4</span></a>, and the isomorphism follows from <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S2.Thmtheorem8" title="Theorem 2.8. ‣ 2. Aronszajn and Countryman lines ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">2.8</span></a>. Also a direct epimorphism from <math alttext="\mathbb{Z}\times(1+C)" class="ltx_Math" display="inline" id="S3.10.p2.8.m8.1"><semantics id="S3.10.p2.8.m8.1a"><mrow id="S3.10.p2.8.m8.1.1" xref="S3.10.p2.8.m8.1.1.cmml"><mi id="S3.10.p2.8.m8.1.1.3" xref="S3.10.p2.8.m8.1.1.3.cmml">ℤ</mi><mo id="S3.10.p2.8.m8.1.1.2" lspace="0.222em" rspace="0.222em" xref="S3.10.p2.8.m8.1.1.2.cmml">×</mo><mrow id="S3.10.p2.8.m8.1.1.1.1" xref="S3.10.p2.8.m8.1.1.1.1.1.cmml"><mo id="S3.10.p2.8.m8.1.1.1.1.2" stretchy="false" xref="S3.10.p2.8.m8.1.1.1.1.1.cmml">(</mo><mrow id="S3.10.p2.8.m8.1.1.1.1.1" xref="S3.10.p2.8.m8.1.1.1.1.1.cmml"><mn id="S3.10.p2.8.m8.1.1.1.1.1.2" xref="S3.10.p2.8.m8.1.1.1.1.1.2.cmml">1</mn><mo id="S3.10.p2.8.m8.1.1.1.1.1.1" xref="S3.10.p2.8.m8.1.1.1.1.1.1.cmml">+</mo><mi id="S3.10.p2.8.m8.1.1.1.1.1.3" xref="S3.10.p2.8.m8.1.1.1.1.1.3.cmml">C</mi></mrow><mo id="S3.10.p2.8.m8.1.1.1.1.3" stretchy="false" xref="S3.10.p2.8.m8.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.10.p2.8.m8.1b"><apply id="S3.10.p2.8.m8.1.1.cmml" xref="S3.10.p2.8.m8.1.1"><times id="S3.10.p2.8.m8.1.1.2.cmml" xref="S3.10.p2.8.m8.1.1.2"></times><ci id="S3.10.p2.8.m8.1.1.3.cmml" xref="S3.10.p2.8.m8.1.1.3">ℤ</ci><apply id="S3.10.p2.8.m8.1.1.1.1.1.cmml" xref="S3.10.p2.8.m8.1.1.1.1"><plus id="S3.10.p2.8.m8.1.1.1.1.1.1.cmml" xref="S3.10.p2.8.m8.1.1.1.1.1.1"></plus><cn id="S3.10.p2.8.m8.1.1.1.1.1.2.cmml" type="integer" xref="S3.10.p2.8.m8.1.1.1.1.1.2">1</cn><ci id="S3.10.p2.8.m8.1.1.1.1.1.3.cmml" xref="S3.10.p2.8.m8.1.1.1.1.1.3">𝐶</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.10.p2.8.m8.1c">\mathbb{Z}\times(1+C)</annotation><annotation encoding="application/x-llamapun" id="S3.10.p2.8.m8.1d">blackboard_Z × ( 1 + italic_C )</annotation></semantics></math> onto <math alttext="C" class="ltx_Math" display="inline" id="S3.10.p2.9.m9.1"><semantics id="S3.10.p2.9.m9.1a"><mi id="S3.10.p2.9.m9.1.1" xref="S3.10.p2.9.m9.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S3.10.p2.9.m9.1b"><ci id="S3.10.p2.9.m9.1.1.cmml" xref="S3.10.p2.9.m9.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.10.p2.9.m9.1c">C</annotation><annotation encoding="application/x-llamapun" id="S3.10.p2.9.m9.1d">italic_C</annotation></semantics></math> is not hard to construct.</p> </div> <div class="ltx_para" id="S3.11.p3"> <p class="ltx_p" id="S3.11.p3.9">If <math alttext="\alpha>0" class="ltx_Math" display="inline" id="S3.11.p3.1.m1.1"><semantics id="S3.11.p3.1.m1.1a"><mrow id="S3.11.p3.1.m1.1.1" xref="S3.11.p3.1.m1.1.1.cmml"><mi id="S3.11.p3.1.m1.1.1.2" xref="S3.11.p3.1.m1.1.1.2.cmml">α</mi><mo id="S3.11.p3.1.m1.1.1.1" xref="S3.11.p3.1.m1.1.1.1.cmml">></mo><mn id="S3.11.p3.1.m1.1.1.3" xref="S3.11.p3.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.11.p3.1.m1.1b"><apply id="S3.11.p3.1.m1.1.1.cmml" xref="S3.11.p3.1.m1.1.1"><gt id="S3.11.p3.1.m1.1.1.1.cmml" xref="S3.11.p3.1.m1.1.1.1"></gt><ci id="S3.11.p3.1.m1.1.1.2.cmml" xref="S3.11.p3.1.m1.1.1.2">𝛼</ci><cn id="S3.11.p3.1.m1.1.1.3.cmml" type="integer" xref="S3.11.p3.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.11.p3.1.m1.1c">\alpha>0</annotation><annotation encoding="application/x-llamapun" id="S3.11.p3.1.m1.1d">italic_α > 0</annotation></semantics></math>, then regardless of whether <math alttext="\alpha" class="ltx_Math" display="inline" id="S3.11.p3.2.m2.1"><semantics id="S3.11.p3.2.m2.1a"><mi id="S3.11.p3.2.m2.1.1" xref="S3.11.p3.2.m2.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S3.11.p3.2.m2.1b"><ci id="S3.11.p3.2.m2.1.1.cmml" xref="S3.11.p3.2.m2.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.11.p3.2.m2.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.11.p3.2.m2.1d">italic_α</annotation></semantics></math> is limit or successor and whether <math alttext="i=+" class="ltx_Math" display="inline" id="S3.11.p3.3.m3.1"><semantics id="S3.11.p3.3.m3.1a"><mrow id="S3.11.p3.3.m3.1.1" xref="S3.11.p3.3.m3.1.1.cmml"><mi id="S3.11.p3.3.m3.1.1.2" xref="S3.11.p3.3.m3.1.1.2.cmml">i</mi><mo id="S3.11.p3.3.m3.1.1.1" rspace="0em" xref="S3.11.p3.3.m3.1.1.1.cmml">=</mo><mo id="S3.11.p3.3.m3.1.1.3" lspace="0em" xref="S3.11.p3.3.m3.1.1.3.cmml">+</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.11.p3.3.m3.1b"><apply id="S3.11.p3.3.m3.1.1.cmml" xref="S3.11.p3.3.m3.1.1"><eq id="S3.11.p3.3.m3.1.1.1.cmml" xref="S3.11.p3.3.m3.1.1.1"></eq><ci id="S3.11.p3.3.m3.1.1.2.cmml" xref="S3.11.p3.3.m3.1.1.2">𝑖</ci><plus id="S3.11.p3.3.m3.1.1.3.cmml" xref="S3.11.p3.3.m3.1.1.3"></plus></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.11.p3.3.m3.1c">i=+</annotation><annotation encoding="application/x-llamapun" id="S3.11.p3.3.m3.1d">italic_i = +</annotation></semantics></math> or <math alttext="i=-" class="ltx_Math" display="inline" id="S3.11.p3.4.m4.1"><semantics id="S3.11.p3.4.m4.1a"><mrow id="S3.11.p3.4.m4.1.1" xref="S3.11.p3.4.m4.1.1.cmml"><mi id="S3.11.p3.4.m4.1.1.2" xref="S3.11.p3.4.m4.1.1.2.cmml">i</mi><mo id="S3.11.p3.4.m4.1.1.1" rspace="0em" xref="S3.11.p3.4.m4.1.1.1.cmml">=</mo><mo id="S3.11.p3.4.m4.1.1.3" lspace="0em" xref="S3.11.p3.4.m4.1.1.3.cmml">−</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.11.p3.4.m4.1b"><apply id="S3.11.p3.4.m4.1.1.cmml" xref="S3.11.p3.4.m4.1.1"><eq id="S3.11.p3.4.m4.1.1.1.cmml" xref="S3.11.p3.4.m4.1.1.1"></eq><ci id="S3.11.p3.4.m4.1.1.2.cmml" xref="S3.11.p3.4.m4.1.1.2">𝑖</ci><minus id="S3.11.p3.4.m4.1.1.3.cmml" xref="S3.11.p3.4.m4.1.1.3"></minus></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.11.p3.4.m4.1c">i=-</annotation><annotation encoding="application/x-llamapun" id="S3.11.p3.4.m4.1d">italic_i = -</annotation></semantics></math>, <math alttext="D_{\alpha}^{i}" class="ltx_Math" display="inline" id="S3.11.p3.5.m5.1"><semantics id="S3.11.p3.5.m5.1a"><msubsup id="S3.11.p3.5.m5.1.1" xref="S3.11.p3.5.m5.1.1.cmml"><mi id="S3.11.p3.5.m5.1.1.2.2" xref="S3.11.p3.5.m5.1.1.2.2.cmml">D</mi><mi id="S3.11.p3.5.m5.1.1.2.3" xref="S3.11.p3.5.m5.1.1.2.3.cmml">α</mi><mi id="S3.11.p3.5.m5.1.1.3" xref="S3.11.p3.5.m5.1.1.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.11.p3.5.m5.1b"><apply id="S3.11.p3.5.m5.1.1.cmml" xref="S3.11.p3.5.m5.1.1"><csymbol cd="ambiguous" id="S3.11.p3.5.m5.1.1.1.cmml" xref="S3.11.p3.5.m5.1.1">superscript</csymbol><apply id="S3.11.p3.5.m5.1.1.2.cmml" xref="S3.11.p3.5.m5.1.1"><csymbol cd="ambiguous" id="S3.11.p3.5.m5.1.1.2.1.cmml" xref="S3.11.p3.5.m5.1.1">subscript</csymbol><ci id="S3.11.p3.5.m5.1.1.2.2.cmml" xref="S3.11.p3.5.m5.1.1.2.2">𝐷</ci><ci id="S3.11.p3.5.m5.1.1.2.3.cmml" xref="S3.11.p3.5.m5.1.1.2.3">𝛼</ci></apply><ci id="S3.11.p3.5.m5.1.1.3.cmml" xref="S3.11.p3.5.m5.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.11.p3.5.m5.1c">D_{\alpha}^{i}</annotation><annotation encoding="application/x-llamapun" id="S3.11.p3.5.m5.1d">italic_D start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> can be written as <math alttext="\sum_{x\in C}A_{x}" class="ltx_Math" display="inline" id="S3.11.p3.6.m6.1"><semantics id="S3.11.p3.6.m6.1a"><mrow id="S3.11.p3.6.m6.1.1" xref="S3.11.p3.6.m6.1.1.cmml"><msub id="S3.11.p3.6.m6.1.1.1" xref="S3.11.p3.6.m6.1.1.1.cmml"><mo id="S3.11.p3.6.m6.1.1.1.2" xref="S3.11.p3.6.m6.1.1.1.2.cmml">∑</mo><mrow id="S3.11.p3.6.m6.1.1.1.3" xref="S3.11.p3.6.m6.1.1.1.3.cmml"><mi id="S3.11.p3.6.m6.1.1.1.3.2" xref="S3.11.p3.6.m6.1.1.1.3.2.cmml">x</mi><mo id="S3.11.p3.6.m6.1.1.1.3.1" xref="S3.11.p3.6.m6.1.1.1.3.1.cmml">∈</mo><mi id="S3.11.p3.6.m6.1.1.1.3.3" xref="S3.11.p3.6.m6.1.1.1.3.3.cmml">C</mi></mrow></msub><msub id="S3.11.p3.6.m6.1.1.2" xref="S3.11.p3.6.m6.1.1.2.cmml"><mi id="S3.11.p3.6.m6.1.1.2.2" xref="S3.11.p3.6.m6.1.1.2.2.cmml">A</mi><mi id="S3.11.p3.6.m6.1.1.2.3" xref="S3.11.p3.6.m6.1.1.2.3.cmml">x</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.11.p3.6.m6.1b"><apply id="S3.11.p3.6.m6.1.1.cmml" xref="S3.11.p3.6.m6.1.1"><apply id="S3.11.p3.6.m6.1.1.1.cmml" xref="S3.11.p3.6.m6.1.1.1"><csymbol cd="ambiguous" id="S3.11.p3.6.m6.1.1.1.1.cmml" xref="S3.11.p3.6.m6.1.1.1">subscript</csymbol><sum id="S3.11.p3.6.m6.1.1.1.2.cmml" xref="S3.11.p3.6.m6.1.1.1.2"></sum><apply id="S3.11.p3.6.m6.1.1.1.3.cmml" xref="S3.11.p3.6.m6.1.1.1.3"><in id="S3.11.p3.6.m6.1.1.1.3.1.cmml" xref="S3.11.p3.6.m6.1.1.1.3.1"></in><ci id="S3.11.p3.6.m6.1.1.1.3.2.cmml" xref="S3.11.p3.6.m6.1.1.1.3.2">𝑥</ci><ci id="S3.11.p3.6.m6.1.1.1.3.3.cmml" xref="S3.11.p3.6.m6.1.1.1.3.3">𝐶</ci></apply></apply><apply id="S3.11.p3.6.m6.1.1.2.cmml" xref="S3.11.p3.6.m6.1.1.2"><csymbol cd="ambiguous" id="S3.11.p3.6.m6.1.1.2.1.cmml" xref="S3.11.p3.6.m6.1.1.2">subscript</csymbol><ci id="S3.11.p3.6.m6.1.1.2.2.cmml" xref="S3.11.p3.6.m6.1.1.2.2">𝐴</ci><ci id="S3.11.p3.6.m6.1.1.2.3.cmml" xref="S3.11.p3.6.m6.1.1.2.3">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.11.p3.6.m6.1c">\sum_{x\in C}A_{x}</annotation><annotation encoding="application/x-llamapun" id="S3.11.p3.6.m6.1d">∑ start_POSTSUBSCRIPT italic_x ∈ italic_C end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> where each <math alttext="A_{x}" class="ltx_Math" display="inline" id="S3.11.p3.7.m7.1"><semantics id="S3.11.p3.7.m7.1a"><msub id="S3.11.p3.7.m7.1.1" xref="S3.11.p3.7.m7.1.1.cmml"><mi id="S3.11.p3.7.m7.1.1.2" xref="S3.11.p3.7.m7.1.1.2.cmml">A</mi><mi id="S3.11.p3.7.m7.1.1.3" xref="S3.11.p3.7.m7.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="S3.11.p3.7.m7.1b"><apply id="S3.11.p3.7.m7.1.1.cmml" xref="S3.11.p3.7.m7.1.1"><csymbol cd="ambiguous" id="S3.11.p3.7.m7.1.1.1.cmml" xref="S3.11.p3.7.m7.1.1">subscript</csymbol><ci id="S3.11.p3.7.m7.1.1.2.cmml" xref="S3.11.p3.7.m7.1.1.2">𝐴</ci><ci id="S3.11.p3.7.m7.1.1.3.cmml" xref="S3.11.p3.7.m7.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.11.p3.7.m7.1c">A_{x}</annotation><annotation encoding="application/x-llamapun" id="S3.11.p3.7.m7.1d">italic_A start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> is in <math alttext="\{D_{\xi}^{x}:\xi<\alpha,x\in\{-,+\}\}" class="ltx_Math" display="inline" id="S3.11.p3.8.m8.6"><semantics id="S3.11.p3.8.m8.6a"><mrow id="S3.11.p3.8.m8.6.6.2" xref="S3.11.p3.8.m8.6.6.3.cmml"><mo id="S3.11.p3.8.m8.6.6.2.3" stretchy="false" xref="S3.11.p3.8.m8.6.6.3.1.cmml">{</mo><msubsup id="S3.11.p3.8.m8.5.5.1.1" xref="S3.11.p3.8.m8.5.5.1.1.cmml"><mi id="S3.11.p3.8.m8.5.5.1.1.2.2" xref="S3.11.p3.8.m8.5.5.1.1.2.2.cmml">D</mi><mi id="S3.11.p3.8.m8.5.5.1.1.2.3" xref="S3.11.p3.8.m8.5.5.1.1.2.3.cmml">ξ</mi><mi id="S3.11.p3.8.m8.5.5.1.1.3" xref="S3.11.p3.8.m8.5.5.1.1.3.cmml">x</mi></msubsup><mo id="S3.11.p3.8.m8.6.6.2.4" lspace="0.278em" rspace="0.278em" xref="S3.11.p3.8.m8.6.6.3.1.cmml">:</mo><mrow id="S3.11.p3.8.m8.6.6.2.2.2" xref="S3.11.p3.8.m8.6.6.2.2.3.cmml"><mrow id="S3.11.p3.8.m8.6.6.2.2.1.1" xref="S3.11.p3.8.m8.6.6.2.2.1.1.cmml"><mi id="S3.11.p3.8.m8.6.6.2.2.1.1.2" xref="S3.11.p3.8.m8.6.6.2.2.1.1.2.cmml">ξ</mi><mo id="S3.11.p3.8.m8.6.6.2.2.1.1.1" xref="S3.11.p3.8.m8.6.6.2.2.1.1.1.cmml"><</mo><mi id="S3.11.p3.8.m8.3.3" xref="S3.11.p3.8.m8.3.3.cmml">α</mi></mrow><mo id="S3.11.p3.8.m8.6.6.2.2.2.3" xref="S3.11.p3.8.m8.6.6.2.2.3a.cmml">,</mo><mrow id="S3.11.p3.8.m8.6.6.2.2.2.2" xref="S3.11.p3.8.m8.6.6.2.2.2.2.cmml"><mi id="S3.11.p3.8.m8.4.4" xref="S3.11.p3.8.m8.4.4.cmml">x</mi><mo id="S3.11.p3.8.m8.6.6.2.2.2.2.1" xref="S3.11.p3.8.m8.6.6.2.2.2.2.1.cmml">∈</mo><mrow id="S3.11.p3.8.m8.6.6.2.2.2.2.2.2" xref="S3.11.p3.8.m8.6.6.2.2.2.2.2.1.cmml"><mo id="S3.11.p3.8.m8.6.6.2.2.2.2.2.2.1" stretchy="false" xref="S3.11.p3.8.m8.6.6.2.2.2.2.2.1.cmml">{</mo><mo id="S3.11.p3.8.m8.1.1" lspace="0em" rspace="0em" xref="S3.11.p3.8.m8.1.1.cmml">−</mo><mo id="S3.11.p3.8.m8.6.6.2.2.2.2.2.2.2" rspace="0em" xref="S3.11.p3.8.m8.6.6.2.2.2.2.2.1.cmml">,</mo><mo id="S3.11.p3.8.m8.2.2" lspace="0em" rspace="0em" xref="S3.11.p3.8.m8.2.2.cmml">+</mo><mo id="S3.11.p3.8.m8.6.6.2.2.2.2.2.2.3" stretchy="false" xref="S3.11.p3.8.m8.6.6.2.2.2.2.2.1.cmml">}</mo></mrow></mrow></mrow><mo id="S3.11.p3.8.m8.6.6.2.5" stretchy="false" xref="S3.11.p3.8.m8.6.6.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.11.p3.8.m8.6b"><apply id="S3.11.p3.8.m8.6.6.3.cmml" xref="S3.11.p3.8.m8.6.6.2"><csymbol cd="latexml" id="S3.11.p3.8.m8.6.6.3.1.cmml" xref="S3.11.p3.8.m8.6.6.2.3">conditional-set</csymbol><apply id="S3.11.p3.8.m8.5.5.1.1.cmml" xref="S3.11.p3.8.m8.5.5.1.1"><csymbol cd="ambiguous" id="S3.11.p3.8.m8.5.5.1.1.1.cmml" xref="S3.11.p3.8.m8.5.5.1.1">superscript</csymbol><apply id="S3.11.p3.8.m8.5.5.1.1.2.cmml" xref="S3.11.p3.8.m8.5.5.1.1"><csymbol cd="ambiguous" id="S3.11.p3.8.m8.5.5.1.1.2.1.cmml" xref="S3.11.p3.8.m8.5.5.1.1">subscript</csymbol><ci id="S3.11.p3.8.m8.5.5.1.1.2.2.cmml" xref="S3.11.p3.8.m8.5.5.1.1.2.2">𝐷</ci><ci id="S3.11.p3.8.m8.5.5.1.1.2.3.cmml" xref="S3.11.p3.8.m8.5.5.1.1.2.3">𝜉</ci></apply><ci id="S3.11.p3.8.m8.5.5.1.1.3.cmml" xref="S3.11.p3.8.m8.5.5.1.1.3">𝑥</ci></apply><apply id="S3.11.p3.8.m8.6.6.2.2.3.cmml" xref="S3.11.p3.8.m8.6.6.2.2.2"><csymbol cd="ambiguous" id="S3.11.p3.8.m8.6.6.2.2.3a.cmml" xref="S3.11.p3.8.m8.6.6.2.2.2.3">formulae-sequence</csymbol><apply id="S3.11.p3.8.m8.6.6.2.2.1.1.cmml" xref="S3.11.p3.8.m8.6.6.2.2.1.1"><lt id="S3.11.p3.8.m8.6.6.2.2.1.1.1.cmml" xref="S3.11.p3.8.m8.6.6.2.2.1.1.1"></lt><ci id="S3.11.p3.8.m8.6.6.2.2.1.1.2.cmml" xref="S3.11.p3.8.m8.6.6.2.2.1.1.2">𝜉</ci><ci id="S3.11.p3.8.m8.3.3.cmml" xref="S3.11.p3.8.m8.3.3">𝛼</ci></apply><apply id="S3.11.p3.8.m8.6.6.2.2.2.2.cmml" xref="S3.11.p3.8.m8.6.6.2.2.2.2"><in id="S3.11.p3.8.m8.6.6.2.2.2.2.1.cmml" xref="S3.11.p3.8.m8.6.6.2.2.2.2.1"></in><ci id="S3.11.p3.8.m8.4.4.cmml" xref="S3.11.p3.8.m8.4.4">𝑥</ci><set id="S3.11.p3.8.m8.6.6.2.2.2.2.2.1.cmml" xref="S3.11.p3.8.m8.6.6.2.2.2.2.2.2"><minus id="S3.11.p3.8.m8.1.1.cmml" xref="S3.11.p3.8.m8.1.1"></minus><plus id="S3.11.p3.8.m8.2.2.cmml" xref="S3.11.p3.8.m8.2.2"></plus></set></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.11.p3.8.m8.6c">\{D_{\xi}^{x}:\xi<\alpha,x\in\{-,+\}\}</annotation><annotation encoding="application/x-llamapun" id="S3.11.p3.8.m8.6d">{ italic_D start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT : italic_ξ < italic_α , italic_x ∈ { - , + } }</annotation></semantics></math>. Also note that <math alttext="{(C^{\star}+1+C)}^{2}\times\eta_{C}\cong\eta_{C}" class="ltx_Math" display="inline" id="S3.11.p3.9.m9.1"><semantics id="S3.11.p3.9.m9.1a"><mrow id="S3.11.p3.9.m9.1.1" xref="S3.11.p3.9.m9.1.1.cmml"><mrow id="S3.11.p3.9.m9.1.1.1" xref="S3.11.p3.9.m9.1.1.1.cmml"><msup id="S3.11.p3.9.m9.1.1.1.1" xref="S3.11.p3.9.m9.1.1.1.1.cmml"><mrow id="S3.11.p3.9.m9.1.1.1.1.1.1" xref="S3.11.p3.9.m9.1.1.1.1.1.1.1.cmml"><mo id="S3.11.p3.9.m9.1.1.1.1.1.1.2" stretchy="false" xref="S3.11.p3.9.m9.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.11.p3.9.m9.1.1.1.1.1.1.1" xref="S3.11.p3.9.m9.1.1.1.1.1.1.1.cmml"><msup id="S3.11.p3.9.m9.1.1.1.1.1.1.1.2" xref="S3.11.p3.9.m9.1.1.1.1.1.1.1.2.cmml"><mi id="S3.11.p3.9.m9.1.1.1.1.1.1.1.2.2" xref="S3.11.p3.9.m9.1.1.1.1.1.1.1.2.2.cmml">C</mi><mo id="S3.11.p3.9.m9.1.1.1.1.1.1.1.2.3" xref="S3.11.p3.9.m9.1.1.1.1.1.1.1.2.3.cmml">⋆</mo></msup><mo id="S3.11.p3.9.m9.1.1.1.1.1.1.1.1" xref="S3.11.p3.9.m9.1.1.1.1.1.1.1.1.cmml">+</mo><mn id="S3.11.p3.9.m9.1.1.1.1.1.1.1.3" xref="S3.11.p3.9.m9.1.1.1.1.1.1.1.3.cmml">1</mn><mo id="S3.11.p3.9.m9.1.1.1.1.1.1.1.1a" xref="S3.11.p3.9.m9.1.1.1.1.1.1.1.1.cmml">+</mo><mi id="S3.11.p3.9.m9.1.1.1.1.1.1.1.4" xref="S3.11.p3.9.m9.1.1.1.1.1.1.1.4.cmml">C</mi></mrow><mo id="S3.11.p3.9.m9.1.1.1.1.1.1.3" stretchy="false" xref="S3.11.p3.9.m9.1.1.1.1.1.1.1.cmml">)</mo></mrow><mn id="S3.11.p3.9.m9.1.1.1.1.3" xref="S3.11.p3.9.m9.1.1.1.1.3.cmml">2</mn></msup><mo id="S3.11.p3.9.m9.1.1.1.2" lspace="0.222em" rspace="0.222em" xref="S3.11.p3.9.m9.1.1.1.2.cmml">×</mo><msub id="S3.11.p3.9.m9.1.1.1.3" xref="S3.11.p3.9.m9.1.1.1.3.cmml"><mi id="S3.11.p3.9.m9.1.1.1.3.2" xref="S3.11.p3.9.m9.1.1.1.3.2.cmml">η</mi><mi id="S3.11.p3.9.m9.1.1.1.3.3" xref="S3.11.p3.9.m9.1.1.1.3.3.cmml">C</mi></msub></mrow><mo id="S3.11.p3.9.m9.1.1.2" xref="S3.11.p3.9.m9.1.1.2.cmml">≅</mo><msub id="S3.11.p3.9.m9.1.1.3" xref="S3.11.p3.9.m9.1.1.3.cmml"><mi id="S3.11.p3.9.m9.1.1.3.2" xref="S3.11.p3.9.m9.1.1.3.2.cmml">η</mi><mi id="S3.11.p3.9.m9.1.1.3.3" xref="S3.11.p3.9.m9.1.1.3.3.cmml">C</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.11.p3.9.m9.1b"><apply id="S3.11.p3.9.m9.1.1.cmml" xref="S3.11.p3.9.m9.1.1"><approx id="S3.11.p3.9.m9.1.1.2.cmml" xref="S3.11.p3.9.m9.1.1.2"></approx><apply id="S3.11.p3.9.m9.1.1.1.cmml" xref="S3.11.p3.9.m9.1.1.1"><times id="S3.11.p3.9.m9.1.1.1.2.cmml" xref="S3.11.p3.9.m9.1.1.1.2"></times><apply id="S3.11.p3.9.m9.1.1.1.1.cmml" xref="S3.11.p3.9.m9.1.1.1.1"><csymbol cd="ambiguous" id="S3.11.p3.9.m9.1.1.1.1.2.cmml" xref="S3.11.p3.9.m9.1.1.1.1">superscript</csymbol><apply id="S3.11.p3.9.m9.1.1.1.1.1.1.1.cmml" xref="S3.11.p3.9.m9.1.1.1.1.1.1"><plus id="S3.11.p3.9.m9.1.1.1.1.1.1.1.1.cmml" xref="S3.11.p3.9.m9.1.1.1.1.1.1.1.1"></plus><apply id="S3.11.p3.9.m9.1.1.1.1.1.1.1.2.cmml" xref="S3.11.p3.9.m9.1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.11.p3.9.m9.1.1.1.1.1.1.1.2.1.cmml" xref="S3.11.p3.9.m9.1.1.1.1.1.1.1.2">superscript</csymbol><ci id="S3.11.p3.9.m9.1.1.1.1.1.1.1.2.2.cmml" xref="S3.11.p3.9.m9.1.1.1.1.1.1.1.2.2">𝐶</ci><ci id="S3.11.p3.9.m9.1.1.1.1.1.1.1.2.3.cmml" xref="S3.11.p3.9.m9.1.1.1.1.1.1.1.2.3">⋆</ci></apply><cn id="S3.11.p3.9.m9.1.1.1.1.1.1.1.3.cmml" type="integer" xref="S3.11.p3.9.m9.1.1.1.1.1.1.1.3">1</cn><ci id="S3.11.p3.9.m9.1.1.1.1.1.1.1.4.cmml" xref="S3.11.p3.9.m9.1.1.1.1.1.1.1.4">𝐶</ci></apply><cn id="S3.11.p3.9.m9.1.1.1.1.3.cmml" type="integer" xref="S3.11.p3.9.m9.1.1.1.1.3">2</cn></apply><apply id="S3.11.p3.9.m9.1.1.1.3.cmml" xref="S3.11.p3.9.m9.1.1.1.3"><csymbol cd="ambiguous" id="S3.11.p3.9.m9.1.1.1.3.1.cmml" xref="S3.11.p3.9.m9.1.1.1.3">subscript</csymbol><ci id="S3.11.p3.9.m9.1.1.1.3.2.cmml" xref="S3.11.p3.9.m9.1.1.1.3.2">𝜂</ci><ci id="S3.11.p3.9.m9.1.1.1.3.3.cmml" xref="S3.11.p3.9.m9.1.1.1.3.3">𝐶</ci></apply></apply><apply id="S3.11.p3.9.m9.1.1.3.cmml" xref="S3.11.p3.9.m9.1.1.3"><csymbol cd="ambiguous" id="S3.11.p3.9.m9.1.1.3.1.cmml" xref="S3.11.p3.9.m9.1.1.3">subscript</csymbol><ci id="S3.11.p3.9.m9.1.1.3.2.cmml" xref="S3.11.p3.9.m9.1.1.3.2">𝜂</ci><ci id="S3.11.p3.9.m9.1.1.3.3.cmml" xref="S3.11.p3.9.m9.1.1.3.3">𝐶</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.11.p3.9.m9.1c">{(C^{\star}+1+C)}^{2}\times\eta_{C}\cong\eta_{C}</annotation><annotation encoding="application/x-llamapun" id="S3.11.p3.9.m9.1d">( italic_C start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT + 1 + italic_C ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT × italic_η start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ≅ italic_η start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT</annotation></semantics></math>. Thus it is enough to prove that</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex3"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="{(C^{\star}+1+C)}^{2}\times\eta_{C}\trianglerighteq\sum_{x\in C}A_{x}." class="ltx_Math" display="block" id="S3.Ex3.m1.1"><semantics id="S3.Ex3.m1.1a"><mrow id="S3.Ex3.m1.1.1.1" xref="S3.Ex3.m1.1.1.1.1.cmml"><mrow id="S3.Ex3.m1.1.1.1.1" xref="S3.Ex3.m1.1.1.1.1.cmml"><mrow id="S3.Ex3.m1.1.1.1.1.1" xref="S3.Ex3.m1.1.1.1.1.1.cmml"><msup id="S3.Ex3.m1.1.1.1.1.1.1" xref="S3.Ex3.m1.1.1.1.1.1.1.cmml"><mrow id="S3.Ex3.m1.1.1.1.1.1.1.1.1" xref="S3.Ex3.m1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S3.Ex3.m1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S3.Ex3.m1.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.Ex3.m1.1.1.1.1.1.1.1.1.1" xref="S3.Ex3.m1.1.1.1.1.1.1.1.1.1.cmml"><msup id="S3.Ex3.m1.1.1.1.1.1.1.1.1.1.2" xref="S3.Ex3.m1.1.1.1.1.1.1.1.1.1.2.cmml"><mi id="S3.Ex3.m1.1.1.1.1.1.1.1.1.1.2.2" xref="S3.Ex3.m1.1.1.1.1.1.1.1.1.1.2.2.cmml">C</mi><mo id="S3.Ex3.m1.1.1.1.1.1.1.1.1.1.2.3" xref="S3.Ex3.m1.1.1.1.1.1.1.1.1.1.2.3.cmml">⋆</mo></msup><mo id="S3.Ex3.m1.1.1.1.1.1.1.1.1.1.1" xref="S3.Ex3.m1.1.1.1.1.1.1.1.1.1.1.cmml">+</mo><mn id="S3.Ex3.m1.1.1.1.1.1.1.1.1.1.3" xref="S3.Ex3.m1.1.1.1.1.1.1.1.1.1.3.cmml">1</mn><mo id="S3.Ex3.m1.1.1.1.1.1.1.1.1.1.1a" xref="S3.Ex3.m1.1.1.1.1.1.1.1.1.1.1.cmml">+</mo><mi id="S3.Ex3.m1.1.1.1.1.1.1.1.1.1.4" xref="S3.Ex3.m1.1.1.1.1.1.1.1.1.1.4.cmml">C</mi></mrow><mo id="S3.Ex3.m1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S3.Ex3.m1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mn id="S3.Ex3.m1.1.1.1.1.1.1.3" xref="S3.Ex3.m1.1.1.1.1.1.1.3.cmml">2</mn></msup><mo id="S3.Ex3.m1.1.1.1.1.1.2" lspace="0.222em" rspace="0.222em" xref="S3.Ex3.m1.1.1.1.1.1.2.cmml">×</mo><msub id="S3.Ex3.m1.1.1.1.1.1.3" xref="S3.Ex3.m1.1.1.1.1.1.3.cmml"><mi id="S3.Ex3.m1.1.1.1.1.1.3.2" xref="S3.Ex3.m1.1.1.1.1.1.3.2.cmml">η</mi><mi id="S3.Ex3.m1.1.1.1.1.1.3.3" xref="S3.Ex3.m1.1.1.1.1.1.3.3.cmml">C</mi></msub></mrow><mo id="S3.Ex3.m1.1.1.1.1.2" xref="S3.Ex3.m1.1.1.1.1.2.cmml">⁢</mo><mi id="S3.Ex3.m1.1.1.1.1.3" mathvariant="normal" xref="S3.Ex3.m1.1.1.1.1.3.cmml">⊵</mi><mo id="S3.Ex3.m1.1.1.1.1.2a" xref="S3.Ex3.m1.1.1.1.1.2.cmml">⁢</mo><mrow id="S3.Ex3.m1.1.1.1.1.4" xref="S3.Ex3.m1.1.1.1.1.4.cmml"><munder id="S3.Ex3.m1.1.1.1.1.4.1" xref="S3.Ex3.m1.1.1.1.1.4.1.cmml"><mo id="S3.Ex3.m1.1.1.1.1.4.1.2" movablelimits="false" xref="S3.Ex3.m1.1.1.1.1.4.1.2.cmml">∑</mo><mrow id="S3.Ex3.m1.1.1.1.1.4.1.3" xref="S3.Ex3.m1.1.1.1.1.4.1.3.cmml"><mi id="S3.Ex3.m1.1.1.1.1.4.1.3.2" xref="S3.Ex3.m1.1.1.1.1.4.1.3.2.cmml">x</mi><mo id="S3.Ex3.m1.1.1.1.1.4.1.3.1" xref="S3.Ex3.m1.1.1.1.1.4.1.3.1.cmml">∈</mo><mi id="S3.Ex3.m1.1.1.1.1.4.1.3.3" xref="S3.Ex3.m1.1.1.1.1.4.1.3.3.cmml">C</mi></mrow></munder><msub id="S3.Ex3.m1.1.1.1.1.4.2" xref="S3.Ex3.m1.1.1.1.1.4.2.cmml"><mi id="S3.Ex3.m1.1.1.1.1.4.2.2" xref="S3.Ex3.m1.1.1.1.1.4.2.2.cmml">A</mi><mi id="S3.Ex3.m1.1.1.1.1.4.2.3" xref="S3.Ex3.m1.1.1.1.1.4.2.3.cmml">x</mi></msub></mrow></mrow><mo id="S3.Ex3.m1.1.1.1.2" lspace="0em" xref="S3.Ex3.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex3.m1.1b"><apply id="S3.Ex3.m1.1.1.1.1.cmml" xref="S3.Ex3.m1.1.1.1"><times id="S3.Ex3.m1.1.1.1.1.2.cmml" xref="S3.Ex3.m1.1.1.1.1.2"></times><apply id="S3.Ex3.m1.1.1.1.1.1.cmml" xref="S3.Ex3.m1.1.1.1.1.1"><times id="S3.Ex3.m1.1.1.1.1.1.2.cmml" xref="S3.Ex3.m1.1.1.1.1.1.2"></times><apply id="S3.Ex3.m1.1.1.1.1.1.1.cmml" xref="S3.Ex3.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.Ex3.m1.1.1.1.1.1.1.2.cmml" xref="S3.Ex3.m1.1.1.1.1.1.1">superscript</csymbol><apply id="S3.Ex3.m1.1.1.1.1.1.1.1.1.1.cmml" xref="S3.Ex3.m1.1.1.1.1.1.1.1.1"><plus id="S3.Ex3.m1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S3.Ex3.m1.1.1.1.1.1.1.1.1.1.1"></plus><apply id="S3.Ex3.m1.1.1.1.1.1.1.1.1.1.2.cmml" xref="S3.Ex3.m1.1.1.1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.Ex3.m1.1.1.1.1.1.1.1.1.1.2.1.cmml" xref="S3.Ex3.m1.1.1.1.1.1.1.1.1.1.2">superscript</csymbol><ci id="S3.Ex3.m1.1.1.1.1.1.1.1.1.1.2.2.cmml" xref="S3.Ex3.m1.1.1.1.1.1.1.1.1.1.2.2">𝐶</ci><ci id="S3.Ex3.m1.1.1.1.1.1.1.1.1.1.2.3.cmml" xref="S3.Ex3.m1.1.1.1.1.1.1.1.1.1.2.3">⋆</ci></apply><cn id="S3.Ex3.m1.1.1.1.1.1.1.1.1.1.3.cmml" type="integer" xref="S3.Ex3.m1.1.1.1.1.1.1.1.1.1.3">1</cn><ci id="S3.Ex3.m1.1.1.1.1.1.1.1.1.1.4.cmml" xref="S3.Ex3.m1.1.1.1.1.1.1.1.1.1.4">𝐶</ci></apply><cn id="S3.Ex3.m1.1.1.1.1.1.1.3.cmml" type="integer" xref="S3.Ex3.m1.1.1.1.1.1.1.3">2</cn></apply><apply id="S3.Ex3.m1.1.1.1.1.1.3.cmml" xref="S3.Ex3.m1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.Ex3.m1.1.1.1.1.1.3.1.cmml" xref="S3.Ex3.m1.1.1.1.1.1.3">subscript</csymbol><ci id="S3.Ex3.m1.1.1.1.1.1.3.2.cmml" xref="S3.Ex3.m1.1.1.1.1.1.3.2">𝜂</ci><ci id="S3.Ex3.m1.1.1.1.1.1.3.3.cmml" xref="S3.Ex3.m1.1.1.1.1.1.3.3">𝐶</ci></apply></apply><ci id="S3.Ex3.m1.1.1.1.1.3.cmml" xref="S3.Ex3.m1.1.1.1.1.3">⊵</ci><apply id="S3.Ex3.m1.1.1.1.1.4.cmml" xref="S3.Ex3.m1.1.1.1.1.4"><apply id="S3.Ex3.m1.1.1.1.1.4.1.cmml" xref="S3.Ex3.m1.1.1.1.1.4.1"><csymbol cd="ambiguous" id="S3.Ex3.m1.1.1.1.1.4.1.1.cmml" xref="S3.Ex3.m1.1.1.1.1.4.1">subscript</csymbol><sum id="S3.Ex3.m1.1.1.1.1.4.1.2.cmml" xref="S3.Ex3.m1.1.1.1.1.4.1.2"></sum><apply id="S3.Ex3.m1.1.1.1.1.4.1.3.cmml" xref="S3.Ex3.m1.1.1.1.1.4.1.3"><in id="S3.Ex3.m1.1.1.1.1.4.1.3.1.cmml" xref="S3.Ex3.m1.1.1.1.1.4.1.3.1"></in><ci id="S3.Ex3.m1.1.1.1.1.4.1.3.2.cmml" xref="S3.Ex3.m1.1.1.1.1.4.1.3.2">𝑥</ci><ci id="S3.Ex3.m1.1.1.1.1.4.1.3.3.cmml" xref="S3.Ex3.m1.1.1.1.1.4.1.3.3">𝐶</ci></apply></apply><apply id="S3.Ex3.m1.1.1.1.1.4.2.cmml" xref="S3.Ex3.m1.1.1.1.1.4.2"><csymbol cd="ambiguous" id="S3.Ex3.m1.1.1.1.1.4.2.1.cmml" xref="S3.Ex3.m1.1.1.1.1.4.2">subscript</csymbol><ci id="S3.Ex3.m1.1.1.1.1.4.2.2.cmml" xref="S3.Ex3.m1.1.1.1.1.4.2.2">𝐴</ci><ci id="S3.Ex3.m1.1.1.1.1.4.2.3.cmml" xref="S3.Ex3.m1.1.1.1.1.4.2.3">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex3.m1.1c">{(C^{\star}+1+C)}^{2}\times\eta_{C}\trianglerighteq\sum_{x\in C}A_{x}.</annotation><annotation encoding="application/x-llamapun" id="S3.Ex3.m1.1d">( italic_C start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT + 1 + italic_C ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT × italic_η start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ⊵ ∑ start_POSTSUBSCRIPT italic_x ∈ italic_C end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S3.12.p4"> <p class="ltx_p" id="S3.12.p4.5">Let <math alttext="f:{(C^{\star}+1+C)}^{2}\twoheadrightarrow C" class="ltx_Math" display="inline" id="S3.12.p4.1.m1.1"><semantics id="S3.12.p4.1.m1.1a"><mrow id="S3.12.p4.1.m1.1.1" xref="S3.12.p4.1.m1.1.1.cmml"><mi id="S3.12.p4.1.m1.1.1.3" xref="S3.12.p4.1.m1.1.1.3.cmml">f</mi><mo id="S3.12.p4.1.m1.1.1.2" lspace="0.278em" rspace="0.278em" xref="S3.12.p4.1.m1.1.1.2.cmml">:</mo><mrow id="S3.12.p4.1.m1.1.1.1" xref="S3.12.p4.1.m1.1.1.1.cmml"><msup id="S3.12.p4.1.m1.1.1.1.1" xref="S3.12.p4.1.m1.1.1.1.1.cmml"><mrow id="S3.12.p4.1.m1.1.1.1.1.1.1" xref="S3.12.p4.1.m1.1.1.1.1.1.1.1.cmml"><mo id="S3.12.p4.1.m1.1.1.1.1.1.1.2" stretchy="false" xref="S3.12.p4.1.m1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.12.p4.1.m1.1.1.1.1.1.1.1" xref="S3.12.p4.1.m1.1.1.1.1.1.1.1.cmml"><msup id="S3.12.p4.1.m1.1.1.1.1.1.1.1.2" xref="S3.12.p4.1.m1.1.1.1.1.1.1.1.2.cmml"><mi id="S3.12.p4.1.m1.1.1.1.1.1.1.1.2.2" xref="S3.12.p4.1.m1.1.1.1.1.1.1.1.2.2.cmml">C</mi><mo id="S3.12.p4.1.m1.1.1.1.1.1.1.1.2.3" xref="S3.12.p4.1.m1.1.1.1.1.1.1.1.2.3.cmml">⋆</mo></msup><mo id="S3.12.p4.1.m1.1.1.1.1.1.1.1.1" xref="S3.12.p4.1.m1.1.1.1.1.1.1.1.1.cmml">+</mo><mn id="S3.12.p4.1.m1.1.1.1.1.1.1.1.3" xref="S3.12.p4.1.m1.1.1.1.1.1.1.1.3.cmml">1</mn><mo id="S3.12.p4.1.m1.1.1.1.1.1.1.1.1a" xref="S3.12.p4.1.m1.1.1.1.1.1.1.1.1.cmml">+</mo><mi id="S3.12.p4.1.m1.1.1.1.1.1.1.1.4" xref="S3.12.p4.1.m1.1.1.1.1.1.1.1.4.cmml">C</mi></mrow><mo id="S3.12.p4.1.m1.1.1.1.1.1.1.3" stretchy="false" xref="S3.12.p4.1.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mn id="S3.12.p4.1.m1.1.1.1.1.3" xref="S3.12.p4.1.m1.1.1.1.1.3.cmml">2</mn></msup><mo id="S3.12.p4.1.m1.1.1.1.2" stretchy="false" xref="S3.12.p4.1.m1.1.1.1.2.cmml">↠</mo><mi id="S3.12.p4.1.m1.1.1.1.3" xref="S3.12.p4.1.m1.1.1.1.3.cmml">C</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.12.p4.1.m1.1b"><apply id="S3.12.p4.1.m1.1.1.cmml" xref="S3.12.p4.1.m1.1.1"><ci id="S3.12.p4.1.m1.1.1.2.cmml" xref="S3.12.p4.1.m1.1.1.2">:</ci><ci id="S3.12.p4.1.m1.1.1.3.cmml" xref="S3.12.p4.1.m1.1.1.3">𝑓</ci><apply id="S3.12.p4.1.m1.1.1.1.cmml" xref="S3.12.p4.1.m1.1.1.1"><ci id="S3.12.p4.1.m1.1.1.1.2.cmml" xref="S3.12.p4.1.m1.1.1.1.2">↠</ci><apply id="S3.12.p4.1.m1.1.1.1.1.cmml" xref="S3.12.p4.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="S3.12.p4.1.m1.1.1.1.1.2.cmml" xref="S3.12.p4.1.m1.1.1.1.1">superscript</csymbol><apply id="S3.12.p4.1.m1.1.1.1.1.1.1.1.cmml" xref="S3.12.p4.1.m1.1.1.1.1.1.1"><plus id="S3.12.p4.1.m1.1.1.1.1.1.1.1.1.cmml" xref="S3.12.p4.1.m1.1.1.1.1.1.1.1.1"></plus><apply id="S3.12.p4.1.m1.1.1.1.1.1.1.1.2.cmml" xref="S3.12.p4.1.m1.1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.12.p4.1.m1.1.1.1.1.1.1.1.2.1.cmml" xref="S3.12.p4.1.m1.1.1.1.1.1.1.1.2">superscript</csymbol><ci id="S3.12.p4.1.m1.1.1.1.1.1.1.1.2.2.cmml" xref="S3.12.p4.1.m1.1.1.1.1.1.1.1.2.2">𝐶</ci><ci id="S3.12.p4.1.m1.1.1.1.1.1.1.1.2.3.cmml" xref="S3.12.p4.1.m1.1.1.1.1.1.1.1.2.3">⋆</ci></apply><cn id="S3.12.p4.1.m1.1.1.1.1.1.1.1.3.cmml" type="integer" xref="S3.12.p4.1.m1.1.1.1.1.1.1.1.3">1</cn><ci id="S3.12.p4.1.m1.1.1.1.1.1.1.1.4.cmml" xref="S3.12.p4.1.m1.1.1.1.1.1.1.1.4">𝐶</ci></apply><cn id="S3.12.p4.1.m1.1.1.1.1.3.cmml" type="integer" xref="S3.12.p4.1.m1.1.1.1.1.3">2</cn></apply><ci id="S3.12.p4.1.m1.1.1.1.3.cmml" xref="S3.12.p4.1.m1.1.1.1.3">𝐶</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.12.p4.1.m1.1c">f:{(C^{\star}+1+C)}^{2}\twoheadrightarrow C</annotation><annotation encoding="application/x-llamapun" id="S3.12.p4.1.m1.1d">italic_f : ( italic_C start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT + 1 + italic_C ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ↠ italic_C</annotation></semantics></math> be an epimorphism, and for each <math alttext="x\in C" class="ltx_Math" display="inline" id="S3.12.p4.2.m2.1"><semantics id="S3.12.p4.2.m2.1a"><mrow id="S3.12.p4.2.m2.1.1" xref="S3.12.p4.2.m2.1.1.cmml"><mi id="S3.12.p4.2.m2.1.1.2" xref="S3.12.p4.2.m2.1.1.2.cmml">x</mi><mo id="S3.12.p4.2.m2.1.1.1" xref="S3.12.p4.2.m2.1.1.1.cmml">∈</mo><mi id="S3.12.p4.2.m2.1.1.3" xref="S3.12.p4.2.m2.1.1.3.cmml">C</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.12.p4.2.m2.1b"><apply id="S3.12.p4.2.m2.1.1.cmml" xref="S3.12.p4.2.m2.1.1"><in id="S3.12.p4.2.m2.1.1.1.cmml" xref="S3.12.p4.2.m2.1.1.1"></in><ci id="S3.12.p4.2.m2.1.1.2.cmml" xref="S3.12.p4.2.m2.1.1.2">𝑥</ci><ci id="S3.12.p4.2.m2.1.1.3.cmml" xref="S3.12.p4.2.m2.1.1.3">𝐶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.12.p4.2.m2.1c">x\in C</annotation><annotation encoding="application/x-llamapun" id="S3.12.p4.2.m2.1d">italic_x ∈ italic_C</annotation></semantics></math>, let <math alttext="I_{x}=f^{-1}(x)" class="ltx_Math" display="inline" id="S3.12.p4.3.m3.1"><semantics id="S3.12.p4.3.m3.1a"><mrow id="S3.12.p4.3.m3.1.2" xref="S3.12.p4.3.m3.1.2.cmml"><msub id="S3.12.p4.3.m3.1.2.2" xref="S3.12.p4.3.m3.1.2.2.cmml"><mi id="S3.12.p4.3.m3.1.2.2.2" xref="S3.12.p4.3.m3.1.2.2.2.cmml">I</mi><mi id="S3.12.p4.3.m3.1.2.2.3" xref="S3.12.p4.3.m3.1.2.2.3.cmml">x</mi></msub><mo id="S3.12.p4.3.m3.1.2.1" xref="S3.12.p4.3.m3.1.2.1.cmml">=</mo><mrow id="S3.12.p4.3.m3.1.2.3" xref="S3.12.p4.3.m3.1.2.3.cmml"><msup id="S3.12.p4.3.m3.1.2.3.2" xref="S3.12.p4.3.m3.1.2.3.2.cmml"><mi id="S3.12.p4.3.m3.1.2.3.2.2" xref="S3.12.p4.3.m3.1.2.3.2.2.cmml">f</mi><mrow id="S3.12.p4.3.m3.1.2.3.2.3" xref="S3.12.p4.3.m3.1.2.3.2.3.cmml"><mo id="S3.12.p4.3.m3.1.2.3.2.3a" xref="S3.12.p4.3.m3.1.2.3.2.3.cmml">−</mo><mn id="S3.12.p4.3.m3.1.2.3.2.3.2" xref="S3.12.p4.3.m3.1.2.3.2.3.2.cmml">1</mn></mrow></msup><mo id="S3.12.p4.3.m3.1.2.3.1" xref="S3.12.p4.3.m3.1.2.3.1.cmml">⁢</mo><mrow id="S3.12.p4.3.m3.1.2.3.3.2" xref="S3.12.p4.3.m3.1.2.3.cmml"><mo id="S3.12.p4.3.m3.1.2.3.3.2.1" stretchy="false" xref="S3.12.p4.3.m3.1.2.3.cmml">(</mo><mi id="S3.12.p4.3.m3.1.1" xref="S3.12.p4.3.m3.1.1.cmml">x</mi><mo id="S3.12.p4.3.m3.1.2.3.3.2.2" stretchy="false" xref="S3.12.p4.3.m3.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.12.p4.3.m3.1b"><apply id="S3.12.p4.3.m3.1.2.cmml" xref="S3.12.p4.3.m3.1.2"><eq id="S3.12.p4.3.m3.1.2.1.cmml" xref="S3.12.p4.3.m3.1.2.1"></eq><apply id="S3.12.p4.3.m3.1.2.2.cmml" xref="S3.12.p4.3.m3.1.2.2"><csymbol cd="ambiguous" id="S3.12.p4.3.m3.1.2.2.1.cmml" xref="S3.12.p4.3.m3.1.2.2">subscript</csymbol><ci id="S3.12.p4.3.m3.1.2.2.2.cmml" xref="S3.12.p4.3.m3.1.2.2.2">𝐼</ci><ci id="S3.12.p4.3.m3.1.2.2.3.cmml" xref="S3.12.p4.3.m3.1.2.2.3">𝑥</ci></apply><apply id="S3.12.p4.3.m3.1.2.3.cmml" xref="S3.12.p4.3.m3.1.2.3"><times id="S3.12.p4.3.m3.1.2.3.1.cmml" xref="S3.12.p4.3.m3.1.2.3.1"></times><apply id="S3.12.p4.3.m3.1.2.3.2.cmml" xref="S3.12.p4.3.m3.1.2.3.2"><csymbol cd="ambiguous" id="S3.12.p4.3.m3.1.2.3.2.1.cmml" xref="S3.12.p4.3.m3.1.2.3.2">superscript</csymbol><ci id="S3.12.p4.3.m3.1.2.3.2.2.cmml" xref="S3.12.p4.3.m3.1.2.3.2.2">𝑓</ci><apply id="S3.12.p4.3.m3.1.2.3.2.3.cmml" xref="S3.12.p4.3.m3.1.2.3.2.3"><minus id="S3.12.p4.3.m3.1.2.3.2.3.1.cmml" xref="S3.12.p4.3.m3.1.2.3.2.3"></minus><cn id="S3.12.p4.3.m3.1.2.3.2.3.2.cmml" type="integer" xref="S3.12.p4.3.m3.1.2.3.2.3.2">1</cn></apply></apply><ci id="S3.12.p4.3.m3.1.1.cmml" xref="S3.12.p4.3.m3.1.1">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.12.p4.3.m3.1c">I_{x}=f^{-1}(x)</annotation><annotation encoding="application/x-llamapun" id="S3.12.p4.3.m3.1d">italic_I start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT = italic_f start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_x )</annotation></semantics></math>. Now from <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S3.Thmtheorem4" title="Lemma 3.4. ‣ 3. Strongly surjective Aronszajn lines ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">3.4</span></a>, we know that <math alttext="I_{x}\times\eta_{C}\trianglerighteq\eta_{C}" class="ltx_Math" display="inline" id="S3.12.p4.4.m4.1"><semantics id="S3.12.p4.4.m4.1a"><mrow id="S3.12.p4.4.m4.1.1" xref="S3.12.p4.4.m4.1.1.cmml"><mrow id="S3.12.p4.4.m4.1.1.2" xref="S3.12.p4.4.m4.1.1.2.cmml"><msub id="S3.12.p4.4.m4.1.1.2.2" xref="S3.12.p4.4.m4.1.1.2.2.cmml"><mi id="S3.12.p4.4.m4.1.1.2.2.2" xref="S3.12.p4.4.m4.1.1.2.2.2.cmml">I</mi><mi id="S3.12.p4.4.m4.1.1.2.2.3" xref="S3.12.p4.4.m4.1.1.2.2.3.cmml">x</mi></msub><mo id="S3.12.p4.4.m4.1.1.2.1" lspace="0.222em" rspace="0.222em" xref="S3.12.p4.4.m4.1.1.2.1.cmml">×</mo><msub id="S3.12.p4.4.m4.1.1.2.3" xref="S3.12.p4.4.m4.1.1.2.3.cmml"><mi id="S3.12.p4.4.m4.1.1.2.3.2" xref="S3.12.p4.4.m4.1.1.2.3.2.cmml">η</mi><mi id="S3.12.p4.4.m4.1.1.2.3.3" xref="S3.12.p4.4.m4.1.1.2.3.3.cmml">C</mi></msub></mrow><mo id="S3.12.p4.4.m4.1.1.1" xref="S3.12.p4.4.m4.1.1.1.cmml">⁢</mo><mi id="S3.12.p4.4.m4.1.1.3" mathvariant="normal" xref="S3.12.p4.4.m4.1.1.3.cmml">⊵</mi><mo id="S3.12.p4.4.m4.1.1.1a" xref="S3.12.p4.4.m4.1.1.1.cmml">⁢</mo><msub id="S3.12.p4.4.m4.1.1.4" xref="S3.12.p4.4.m4.1.1.4.cmml"><mi id="S3.12.p4.4.m4.1.1.4.2" xref="S3.12.p4.4.m4.1.1.4.2.cmml">η</mi><mi id="S3.12.p4.4.m4.1.1.4.3" xref="S3.12.p4.4.m4.1.1.4.3.cmml">C</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.12.p4.4.m4.1b"><apply id="S3.12.p4.4.m4.1.1.cmml" xref="S3.12.p4.4.m4.1.1"><times id="S3.12.p4.4.m4.1.1.1.cmml" xref="S3.12.p4.4.m4.1.1.1"></times><apply id="S3.12.p4.4.m4.1.1.2.cmml" xref="S3.12.p4.4.m4.1.1.2"><times id="S3.12.p4.4.m4.1.1.2.1.cmml" xref="S3.12.p4.4.m4.1.1.2.1"></times><apply id="S3.12.p4.4.m4.1.1.2.2.cmml" xref="S3.12.p4.4.m4.1.1.2.2"><csymbol cd="ambiguous" id="S3.12.p4.4.m4.1.1.2.2.1.cmml" xref="S3.12.p4.4.m4.1.1.2.2">subscript</csymbol><ci id="S3.12.p4.4.m4.1.1.2.2.2.cmml" xref="S3.12.p4.4.m4.1.1.2.2.2">𝐼</ci><ci id="S3.12.p4.4.m4.1.1.2.2.3.cmml" xref="S3.12.p4.4.m4.1.1.2.2.3">𝑥</ci></apply><apply id="S3.12.p4.4.m4.1.1.2.3.cmml" xref="S3.12.p4.4.m4.1.1.2.3"><csymbol cd="ambiguous" id="S3.12.p4.4.m4.1.1.2.3.1.cmml" xref="S3.12.p4.4.m4.1.1.2.3">subscript</csymbol><ci id="S3.12.p4.4.m4.1.1.2.3.2.cmml" xref="S3.12.p4.4.m4.1.1.2.3.2">𝜂</ci><ci id="S3.12.p4.4.m4.1.1.2.3.3.cmml" xref="S3.12.p4.4.m4.1.1.2.3.3">𝐶</ci></apply></apply><ci id="S3.12.p4.4.m4.1.1.3.cmml" xref="S3.12.p4.4.m4.1.1.3">⊵</ci><apply id="S3.12.p4.4.m4.1.1.4.cmml" xref="S3.12.p4.4.m4.1.1.4"><csymbol cd="ambiguous" id="S3.12.p4.4.m4.1.1.4.1.cmml" xref="S3.12.p4.4.m4.1.1.4">subscript</csymbol><ci id="S3.12.p4.4.m4.1.1.4.2.cmml" xref="S3.12.p4.4.m4.1.1.4.2">𝜂</ci><ci id="S3.12.p4.4.m4.1.1.4.3.cmml" xref="S3.12.p4.4.m4.1.1.4.3">𝐶</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.12.p4.4.m4.1c">I_{x}\times\eta_{C}\trianglerighteq\eta_{C}</annotation><annotation encoding="application/x-llamapun" id="S3.12.p4.4.m4.1d">italic_I start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT × italic_η start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ⊵ italic_η start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT</annotation></semantics></math>, and by induction we obtain <math alttext="\eta_{C}\trianglerighteq A_{x}" class="ltx_Math" display="inline" id="S3.12.p4.5.m5.1"><semantics id="S3.12.p4.5.m5.1a"><mrow id="S3.12.p4.5.m5.1.1" xref="S3.12.p4.5.m5.1.1.cmml"><msub id="S3.12.p4.5.m5.1.1.2" xref="S3.12.p4.5.m5.1.1.2.cmml"><mi id="S3.12.p4.5.m5.1.1.2.2" xref="S3.12.p4.5.m5.1.1.2.2.cmml">η</mi><mi id="S3.12.p4.5.m5.1.1.2.3" xref="S3.12.p4.5.m5.1.1.2.3.cmml">C</mi></msub><mo id="S3.12.p4.5.m5.1.1.1" xref="S3.12.p4.5.m5.1.1.1.cmml">⁢</mo><mi id="S3.12.p4.5.m5.1.1.3" mathvariant="normal" xref="S3.12.p4.5.m5.1.1.3.cmml">⊵</mi><mo id="S3.12.p4.5.m5.1.1.1a" xref="S3.12.p4.5.m5.1.1.1.cmml">⁢</mo><msub id="S3.12.p4.5.m5.1.1.4" xref="S3.12.p4.5.m5.1.1.4.cmml"><mi id="S3.12.p4.5.m5.1.1.4.2" xref="S3.12.p4.5.m5.1.1.4.2.cmml">A</mi><mi id="S3.12.p4.5.m5.1.1.4.3" xref="S3.12.p4.5.m5.1.1.4.3.cmml">x</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.12.p4.5.m5.1b"><apply id="S3.12.p4.5.m5.1.1.cmml" xref="S3.12.p4.5.m5.1.1"><times id="S3.12.p4.5.m5.1.1.1.cmml" xref="S3.12.p4.5.m5.1.1.1"></times><apply id="S3.12.p4.5.m5.1.1.2.cmml" xref="S3.12.p4.5.m5.1.1.2"><csymbol cd="ambiguous" id="S3.12.p4.5.m5.1.1.2.1.cmml" xref="S3.12.p4.5.m5.1.1.2">subscript</csymbol><ci id="S3.12.p4.5.m5.1.1.2.2.cmml" xref="S3.12.p4.5.m5.1.1.2.2">𝜂</ci><ci id="S3.12.p4.5.m5.1.1.2.3.cmml" xref="S3.12.p4.5.m5.1.1.2.3">𝐶</ci></apply><ci id="S3.12.p4.5.m5.1.1.3.cmml" xref="S3.12.p4.5.m5.1.1.3">⊵</ci><apply id="S3.12.p4.5.m5.1.1.4.cmml" xref="S3.12.p4.5.m5.1.1.4"><csymbol cd="ambiguous" id="S3.12.p4.5.m5.1.1.4.1.cmml" xref="S3.12.p4.5.m5.1.1.4">subscript</csymbol><ci id="S3.12.p4.5.m5.1.1.4.2.cmml" xref="S3.12.p4.5.m5.1.1.4.2">𝐴</ci><ci id="S3.12.p4.5.m5.1.1.4.3.cmml" xref="S3.12.p4.5.m5.1.1.4.3">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.12.p4.5.m5.1c">\eta_{C}\trianglerighteq A_{x}</annotation><annotation encoding="application/x-llamapun" id="S3.12.p4.5.m5.1d">italic_η start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ⊵ italic_A start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math>. Finally,</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex4"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="{(C^{\star}+1+C)}^{2}\times\eta_{C}=\sum_{x\in C}f^{-1}(x)\times\eta_{C}% \trianglerighteq\sum_{x\in C}A_{x}=D_{\alpha}^{i}." class="ltx_Math" display="block" id="S3.Ex4.m1.2"><semantics id="S3.Ex4.m1.2a"><mrow id="S3.Ex4.m1.2.2.1" xref="S3.Ex4.m1.2.2.1.1.cmml"><mrow id="S3.Ex4.m1.2.2.1.1" xref="S3.Ex4.m1.2.2.1.1.cmml"><mrow id="S3.Ex4.m1.2.2.1.1.1" xref="S3.Ex4.m1.2.2.1.1.1.cmml"><msup id="S3.Ex4.m1.2.2.1.1.1.1" xref="S3.Ex4.m1.2.2.1.1.1.1.cmml"><mrow id="S3.Ex4.m1.2.2.1.1.1.1.1.1" xref="S3.Ex4.m1.2.2.1.1.1.1.1.1.1.cmml"><mo id="S3.Ex4.m1.2.2.1.1.1.1.1.1.2" stretchy="false" xref="S3.Ex4.m1.2.2.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.Ex4.m1.2.2.1.1.1.1.1.1.1" xref="S3.Ex4.m1.2.2.1.1.1.1.1.1.1.cmml"><msup id="S3.Ex4.m1.2.2.1.1.1.1.1.1.1.2" xref="S3.Ex4.m1.2.2.1.1.1.1.1.1.1.2.cmml"><mi id="S3.Ex4.m1.2.2.1.1.1.1.1.1.1.2.2" xref="S3.Ex4.m1.2.2.1.1.1.1.1.1.1.2.2.cmml">C</mi><mo id="S3.Ex4.m1.2.2.1.1.1.1.1.1.1.2.3" xref="S3.Ex4.m1.2.2.1.1.1.1.1.1.1.2.3.cmml">⋆</mo></msup><mo id="S3.Ex4.m1.2.2.1.1.1.1.1.1.1.1" xref="S3.Ex4.m1.2.2.1.1.1.1.1.1.1.1.cmml">+</mo><mn id="S3.Ex4.m1.2.2.1.1.1.1.1.1.1.3" xref="S3.Ex4.m1.2.2.1.1.1.1.1.1.1.3.cmml">1</mn><mo id="S3.Ex4.m1.2.2.1.1.1.1.1.1.1.1a" xref="S3.Ex4.m1.2.2.1.1.1.1.1.1.1.1.cmml">+</mo><mi id="S3.Ex4.m1.2.2.1.1.1.1.1.1.1.4" xref="S3.Ex4.m1.2.2.1.1.1.1.1.1.1.4.cmml">C</mi></mrow><mo id="S3.Ex4.m1.2.2.1.1.1.1.1.1.3" stretchy="false" xref="S3.Ex4.m1.2.2.1.1.1.1.1.1.1.cmml">)</mo></mrow><mn id="S3.Ex4.m1.2.2.1.1.1.1.3" xref="S3.Ex4.m1.2.2.1.1.1.1.3.cmml">2</mn></msup><mo id="S3.Ex4.m1.2.2.1.1.1.2" lspace="0.222em" rspace="0.222em" xref="S3.Ex4.m1.2.2.1.1.1.2.cmml">×</mo><msub id="S3.Ex4.m1.2.2.1.1.1.3" xref="S3.Ex4.m1.2.2.1.1.1.3.cmml"><mi id="S3.Ex4.m1.2.2.1.1.1.3.2" xref="S3.Ex4.m1.2.2.1.1.1.3.2.cmml">η</mi><mi id="S3.Ex4.m1.2.2.1.1.1.3.3" xref="S3.Ex4.m1.2.2.1.1.1.3.3.cmml">C</mi></msub></mrow><mo id="S3.Ex4.m1.2.2.1.1.3" rspace="0.111em" xref="S3.Ex4.m1.2.2.1.1.3.cmml">=</mo><mrow id="S3.Ex4.m1.2.2.1.1.4" xref="S3.Ex4.m1.2.2.1.1.4.cmml"><munder id="S3.Ex4.m1.2.2.1.1.4.1" xref="S3.Ex4.m1.2.2.1.1.4.1.cmml"><mo id="S3.Ex4.m1.2.2.1.1.4.1.2" movablelimits="false" xref="S3.Ex4.m1.2.2.1.1.4.1.2.cmml">∑</mo><mrow id="S3.Ex4.m1.2.2.1.1.4.1.3" xref="S3.Ex4.m1.2.2.1.1.4.1.3.cmml"><mi id="S3.Ex4.m1.2.2.1.1.4.1.3.2" xref="S3.Ex4.m1.2.2.1.1.4.1.3.2.cmml">x</mi><mo id="S3.Ex4.m1.2.2.1.1.4.1.3.1" xref="S3.Ex4.m1.2.2.1.1.4.1.3.1.cmml">∈</mo><mi id="S3.Ex4.m1.2.2.1.1.4.1.3.3" xref="S3.Ex4.m1.2.2.1.1.4.1.3.3.cmml">C</mi></mrow></munder><mrow id="S3.Ex4.m1.2.2.1.1.4.2" xref="S3.Ex4.m1.2.2.1.1.4.2.cmml"><mrow id="S3.Ex4.m1.2.2.1.1.4.2.2" xref="S3.Ex4.m1.2.2.1.1.4.2.2.cmml"><mrow id="S3.Ex4.m1.2.2.1.1.4.2.2.2" xref="S3.Ex4.m1.2.2.1.1.4.2.2.2.cmml"><msup id="S3.Ex4.m1.2.2.1.1.4.2.2.2.2" xref="S3.Ex4.m1.2.2.1.1.4.2.2.2.2.cmml"><mi id="S3.Ex4.m1.2.2.1.1.4.2.2.2.2.2" xref="S3.Ex4.m1.2.2.1.1.4.2.2.2.2.2.cmml">f</mi><mrow id="S3.Ex4.m1.2.2.1.1.4.2.2.2.2.3" xref="S3.Ex4.m1.2.2.1.1.4.2.2.2.2.3.cmml"><mo id="S3.Ex4.m1.2.2.1.1.4.2.2.2.2.3a" xref="S3.Ex4.m1.2.2.1.1.4.2.2.2.2.3.cmml">−</mo><mn id="S3.Ex4.m1.2.2.1.1.4.2.2.2.2.3.2" xref="S3.Ex4.m1.2.2.1.1.4.2.2.2.2.3.2.cmml">1</mn></mrow></msup><mo id="S3.Ex4.m1.2.2.1.1.4.2.2.2.1" xref="S3.Ex4.m1.2.2.1.1.4.2.2.2.1.cmml">⁢</mo><mrow id="S3.Ex4.m1.2.2.1.1.4.2.2.2.3.2" xref="S3.Ex4.m1.2.2.1.1.4.2.2.2.cmml"><mo id="S3.Ex4.m1.2.2.1.1.4.2.2.2.3.2.1" stretchy="false" xref="S3.Ex4.m1.2.2.1.1.4.2.2.2.cmml">(</mo><mi id="S3.Ex4.m1.1.1" xref="S3.Ex4.m1.1.1.cmml">x</mi><mo id="S3.Ex4.m1.2.2.1.1.4.2.2.2.3.2.2" rspace="0.055em" stretchy="false" xref="S3.Ex4.m1.2.2.1.1.4.2.2.2.cmml">)</mo></mrow></mrow><mo id="S3.Ex4.m1.2.2.1.1.4.2.2.1" rspace="0.222em" xref="S3.Ex4.m1.2.2.1.1.4.2.2.1.cmml">×</mo><msub id="S3.Ex4.m1.2.2.1.1.4.2.2.3" xref="S3.Ex4.m1.2.2.1.1.4.2.2.3.cmml"><mi id="S3.Ex4.m1.2.2.1.1.4.2.2.3.2" xref="S3.Ex4.m1.2.2.1.1.4.2.2.3.2.cmml">η</mi><mi id="S3.Ex4.m1.2.2.1.1.4.2.2.3.3" xref="S3.Ex4.m1.2.2.1.1.4.2.2.3.3.cmml">C</mi></msub></mrow><mo id="S3.Ex4.m1.2.2.1.1.4.2.1" xref="S3.Ex4.m1.2.2.1.1.4.2.1.cmml">⁢</mo><mi id="S3.Ex4.m1.2.2.1.1.4.2.3" mathvariant="normal" xref="S3.Ex4.m1.2.2.1.1.4.2.3.cmml">⊵</mi><mo id="S3.Ex4.m1.2.2.1.1.4.2.1a" xref="S3.Ex4.m1.2.2.1.1.4.2.1.cmml">⁢</mo><mrow id="S3.Ex4.m1.2.2.1.1.4.2.4" xref="S3.Ex4.m1.2.2.1.1.4.2.4.cmml"><munder id="S3.Ex4.m1.2.2.1.1.4.2.4.1" xref="S3.Ex4.m1.2.2.1.1.4.2.4.1.cmml"><mo id="S3.Ex4.m1.2.2.1.1.4.2.4.1.2" movablelimits="false" xref="S3.Ex4.m1.2.2.1.1.4.2.4.1.2.cmml">∑</mo><mrow id="S3.Ex4.m1.2.2.1.1.4.2.4.1.3" xref="S3.Ex4.m1.2.2.1.1.4.2.4.1.3.cmml"><mi id="S3.Ex4.m1.2.2.1.1.4.2.4.1.3.2" xref="S3.Ex4.m1.2.2.1.1.4.2.4.1.3.2.cmml">x</mi><mo id="S3.Ex4.m1.2.2.1.1.4.2.4.1.3.1" xref="S3.Ex4.m1.2.2.1.1.4.2.4.1.3.1.cmml">∈</mo><mi id="S3.Ex4.m1.2.2.1.1.4.2.4.1.3.3" xref="S3.Ex4.m1.2.2.1.1.4.2.4.1.3.3.cmml">C</mi></mrow></munder><msub id="S3.Ex4.m1.2.2.1.1.4.2.4.2" xref="S3.Ex4.m1.2.2.1.1.4.2.4.2.cmml"><mi id="S3.Ex4.m1.2.2.1.1.4.2.4.2.2" xref="S3.Ex4.m1.2.2.1.1.4.2.4.2.2.cmml">A</mi><mi id="S3.Ex4.m1.2.2.1.1.4.2.4.2.3" xref="S3.Ex4.m1.2.2.1.1.4.2.4.2.3.cmml">x</mi></msub></mrow></mrow></mrow><mo id="S3.Ex4.m1.2.2.1.1.5" xref="S3.Ex4.m1.2.2.1.1.5.cmml">=</mo><msubsup id="S3.Ex4.m1.2.2.1.1.6" xref="S3.Ex4.m1.2.2.1.1.6.cmml"><mi id="S3.Ex4.m1.2.2.1.1.6.2.2" xref="S3.Ex4.m1.2.2.1.1.6.2.2.cmml">D</mi><mi id="S3.Ex4.m1.2.2.1.1.6.2.3" xref="S3.Ex4.m1.2.2.1.1.6.2.3.cmml">α</mi><mi id="S3.Ex4.m1.2.2.1.1.6.3" xref="S3.Ex4.m1.2.2.1.1.6.3.cmml">i</mi></msubsup></mrow><mo id="S3.Ex4.m1.2.2.1.2" lspace="0em" xref="S3.Ex4.m1.2.2.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex4.m1.2b"><apply id="S3.Ex4.m1.2.2.1.1.cmml" xref="S3.Ex4.m1.2.2.1"><and id="S3.Ex4.m1.2.2.1.1a.cmml" xref="S3.Ex4.m1.2.2.1"></and><apply id="S3.Ex4.m1.2.2.1.1b.cmml" xref="S3.Ex4.m1.2.2.1"><eq id="S3.Ex4.m1.2.2.1.1.3.cmml" xref="S3.Ex4.m1.2.2.1.1.3"></eq><apply id="S3.Ex4.m1.2.2.1.1.1.cmml" xref="S3.Ex4.m1.2.2.1.1.1"><times id="S3.Ex4.m1.2.2.1.1.1.2.cmml" xref="S3.Ex4.m1.2.2.1.1.1.2"></times><apply id="S3.Ex4.m1.2.2.1.1.1.1.cmml" xref="S3.Ex4.m1.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S3.Ex4.m1.2.2.1.1.1.1.2.cmml" xref="S3.Ex4.m1.2.2.1.1.1.1">superscript</csymbol><apply id="S3.Ex4.m1.2.2.1.1.1.1.1.1.1.cmml" xref="S3.Ex4.m1.2.2.1.1.1.1.1.1"><plus id="S3.Ex4.m1.2.2.1.1.1.1.1.1.1.1.cmml" xref="S3.Ex4.m1.2.2.1.1.1.1.1.1.1.1"></plus><apply id="S3.Ex4.m1.2.2.1.1.1.1.1.1.1.2.cmml" xref="S3.Ex4.m1.2.2.1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.Ex4.m1.2.2.1.1.1.1.1.1.1.2.1.cmml" xref="S3.Ex4.m1.2.2.1.1.1.1.1.1.1.2">superscript</csymbol><ci id="S3.Ex4.m1.2.2.1.1.1.1.1.1.1.2.2.cmml" xref="S3.Ex4.m1.2.2.1.1.1.1.1.1.1.2.2">𝐶</ci><ci id="S3.Ex4.m1.2.2.1.1.1.1.1.1.1.2.3.cmml" xref="S3.Ex4.m1.2.2.1.1.1.1.1.1.1.2.3">⋆</ci></apply><cn id="S3.Ex4.m1.2.2.1.1.1.1.1.1.1.3.cmml" type="integer" xref="S3.Ex4.m1.2.2.1.1.1.1.1.1.1.3">1</cn><ci id="S3.Ex4.m1.2.2.1.1.1.1.1.1.1.4.cmml" xref="S3.Ex4.m1.2.2.1.1.1.1.1.1.1.4">𝐶</ci></apply><cn id="S3.Ex4.m1.2.2.1.1.1.1.3.cmml" type="integer" xref="S3.Ex4.m1.2.2.1.1.1.1.3">2</cn></apply><apply id="S3.Ex4.m1.2.2.1.1.1.3.cmml" xref="S3.Ex4.m1.2.2.1.1.1.3"><csymbol cd="ambiguous" id="S3.Ex4.m1.2.2.1.1.1.3.1.cmml" xref="S3.Ex4.m1.2.2.1.1.1.3">subscript</csymbol><ci id="S3.Ex4.m1.2.2.1.1.1.3.2.cmml" xref="S3.Ex4.m1.2.2.1.1.1.3.2">𝜂</ci><ci id="S3.Ex4.m1.2.2.1.1.1.3.3.cmml" xref="S3.Ex4.m1.2.2.1.1.1.3.3">𝐶</ci></apply></apply><apply id="S3.Ex4.m1.2.2.1.1.4.cmml" xref="S3.Ex4.m1.2.2.1.1.4"><apply id="S3.Ex4.m1.2.2.1.1.4.1.cmml" xref="S3.Ex4.m1.2.2.1.1.4.1"><csymbol cd="ambiguous" id="S3.Ex4.m1.2.2.1.1.4.1.1.cmml" xref="S3.Ex4.m1.2.2.1.1.4.1">subscript</csymbol><sum id="S3.Ex4.m1.2.2.1.1.4.1.2.cmml" xref="S3.Ex4.m1.2.2.1.1.4.1.2"></sum><apply id="S3.Ex4.m1.2.2.1.1.4.1.3.cmml" xref="S3.Ex4.m1.2.2.1.1.4.1.3"><in id="S3.Ex4.m1.2.2.1.1.4.1.3.1.cmml" xref="S3.Ex4.m1.2.2.1.1.4.1.3.1"></in><ci id="S3.Ex4.m1.2.2.1.1.4.1.3.2.cmml" xref="S3.Ex4.m1.2.2.1.1.4.1.3.2">𝑥</ci><ci id="S3.Ex4.m1.2.2.1.1.4.1.3.3.cmml" xref="S3.Ex4.m1.2.2.1.1.4.1.3.3">𝐶</ci></apply></apply><apply id="S3.Ex4.m1.2.2.1.1.4.2.cmml" xref="S3.Ex4.m1.2.2.1.1.4.2"><times id="S3.Ex4.m1.2.2.1.1.4.2.1.cmml" xref="S3.Ex4.m1.2.2.1.1.4.2.1"></times><apply id="S3.Ex4.m1.2.2.1.1.4.2.2.cmml" xref="S3.Ex4.m1.2.2.1.1.4.2.2"><times id="S3.Ex4.m1.2.2.1.1.4.2.2.1.cmml" xref="S3.Ex4.m1.2.2.1.1.4.2.2.1"></times><apply id="S3.Ex4.m1.2.2.1.1.4.2.2.2.cmml" xref="S3.Ex4.m1.2.2.1.1.4.2.2.2"><times id="S3.Ex4.m1.2.2.1.1.4.2.2.2.1.cmml" xref="S3.Ex4.m1.2.2.1.1.4.2.2.2.1"></times><apply id="S3.Ex4.m1.2.2.1.1.4.2.2.2.2.cmml" xref="S3.Ex4.m1.2.2.1.1.4.2.2.2.2"><csymbol cd="ambiguous" id="S3.Ex4.m1.2.2.1.1.4.2.2.2.2.1.cmml" xref="S3.Ex4.m1.2.2.1.1.4.2.2.2.2">superscript</csymbol><ci id="S3.Ex4.m1.2.2.1.1.4.2.2.2.2.2.cmml" xref="S3.Ex4.m1.2.2.1.1.4.2.2.2.2.2">𝑓</ci><apply id="S3.Ex4.m1.2.2.1.1.4.2.2.2.2.3.cmml" xref="S3.Ex4.m1.2.2.1.1.4.2.2.2.2.3"><minus id="S3.Ex4.m1.2.2.1.1.4.2.2.2.2.3.1.cmml" xref="S3.Ex4.m1.2.2.1.1.4.2.2.2.2.3"></minus><cn id="S3.Ex4.m1.2.2.1.1.4.2.2.2.2.3.2.cmml" type="integer" xref="S3.Ex4.m1.2.2.1.1.4.2.2.2.2.3.2">1</cn></apply></apply><ci id="S3.Ex4.m1.1.1.cmml" xref="S3.Ex4.m1.1.1">𝑥</ci></apply><apply id="S3.Ex4.m1.2.2.1.1.4.2.2.3.cmml" xref="S3.Ex4.m1.2.2.1.1.4.2.2.3"><csymbol cd="ambiguous" id="S3.Ex4.m1.2.2.1.1.4.2.2.3.1.cmml" xref="S3.Ex4.m1.2.2.1.1.4.2.2.3">subscript</csymbol><ci id="S3.Ex4.m1.2.2.1.1.4.2.2.3.2.cmml" xref="S3.Ex4.m1.2.2.1.1.4.2.2.3.2">𝜂</ci><ci id="S3.Ex4.m1.2.2.1.1.4.2.2.3.3.cmml" xref="S3.Ex4.m1.2.2.1.1.4.2.2.3.3">𝐶</ci></apply></apply><ci id="S3.Ex4.m1.2.2.1.1.4.2.3.cmml" xref="S3.Ex4.m1.2.2.1.1.4.2.3">⊵</ci><apply id="S3.Ex4.m1.2.2.1.1.4.2.4.cmml" xref="S3.Ex4.m1.2.2.1.1.4.2.4"><apply id="S3.Ex4.m1.2.2.1.1.4.2.4.1.cmml" xref="S3.Ex4.m1.2.2.1.1.4.2.4.1"><csymbol cd="ambiguous" id="S3.Ex4.m1.2.2.1.1.4.2.4.1.1.cmml" xref="S3.Ex4.m1.2.2.1.1.4.2.4.1">subscript</csymbol><sum id="S3.Ex4.m1.2.2.1.1.4.2.4.1.2.cmml" xref="S3.Ex4.m1.2.2.1.1.4.2.4.1.2"></sum><apply id="S3.Ex4.m1.2.2.1.1.4.2.4.1.3.cmml" xref="S3.Ex4.m1.2.2.1.1.4.2.4.1.3"><in id="S3.Ex4.m1.2.2.1.1.4.2.4.1.3.1.cmml" xref="S3.Ex4.m1.2.2.1.1.4.2.4.1.3.1"></in><ci id="S3.Ex4.m1.2.2.1.1.4.2.4.1.3.2.cmml" xref="S3.Ex4.m1.2.2.1.1.4.2.4.1.3.2">𝑥</ci><ci id="S3.Ex4.m1.2.2.1.1.4.2.4.1.3.3.cmml" xref="S3.Ex4.m1.2.2.1.1.4.2.4.1.3.3">𝐶</ci></apply></apply><apply id="S3.Ex4.m1.2.2.1.1.4.2.4.2.cmml" xref="S3.Ex4.m1.2.2.1.1.4.2.4.2"><csymbol cd="ambiguous" id="S3.Ex4.m1.2.2.1.1.4.2.4.2.1.cmml" xref="S3.Ex4.m1.2.2.1.1.4.2.4.2">subscript</csymbol><ci id="S3.Ex4.m1.2.2.1.1.4.2.4.2.2.cmml" xref="S3.Ex4.m1.2.2.1.1.4.2.4.2.2">𝐴</ci><ci id="S3.Ex4.m1.2.2.1.1.4.2.4.2.3.cmml" xref="S3.Ex4.m1.2.2.1.1.4.2.4.2.3">𝑥</ci></apply></apply></apply></apply></apply><apply id="S3.Ex4.m1.2.2.1.1c.cmml" xref="S3.Ex4.m1.2.2.1"><eq id="S3.Ex4.m1.2.2.1.1.5.cmml" xref="S3.Ex4.m1.2.2.1.1.5"></eq><share href="https://arxiv.org/html/2503.13728v1#S3.Ex4.m1.2.2.1.1.4.cmml" id="S3.Ex4.m1.2.2.1.1d.cmml" xref="S3.Ex4.m1.2.2.1"></share><apply id="S3.Ex4.m1.2.2.1.1.6.cmml" xref="S3.Ex4.m1.2.2.1.1.6"><csymbol cd="ambiguous" id="S3.Ex4.m1.2.2.1.1.6.1.cmml" xref="S3.Ex4.m1.2.2.1.1.6">superscript</csymbol><apply id="S3.Ex4.m1.2.2.1.1.6.2.cmml" xref="S3.Ex4.m1.2.2.1.1.6"><csymbol cd="ambiguous" id="S3.Ex4.m1.2.2.1.1.6.2.1.cmml" xref="S3.Ex4.m1.2.2.1.1.6">subscript</csymbol><ci id="S3.Ex4.m1.2.2.1.1.6.2.2.cmml" xref="S3.Ex4.m1.2.2.1.1.6.2.2">𝐷</ci><ci id="S3.Ex4.m1.2.2.1.1.6.2.3.cmml" xref="S3.Ex4.m1.2.2.1.1.6.2.3">𝛼</ci></apply><ci id="S3.Ex4.m1.2.2.1.1.6.3.cmml" xref="S3.Ex4.m1.2.2.1.1.6.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex4.m1.2c">{(C^{\star}+1+C)}^{2}\times\eta_{C}=\sum_{x\in C}f^{-1}(x)\times\eta_{C}% \trianglerighteq\sum_{x\in C}A_{x}=D_{\alpha}^{i}.</annotation><annotation encoding="application/x-llamapun" id="S3.Ex4.m1.2d">( italic_C start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT + 1 + italic_C ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT × italic_η start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_x ∈ italic_C end_POSTSUBSCRIPT italic_f start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_x ) × italic_η start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ⊵ ∑ start_POSTSUBSCRIPT italic_x ∈ italic_C end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT = italic_D start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.12.p4.6">∎</p> </div> </div> </section> <section class="ltx_section" id="S4"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">4. </span>Aronszajn line decompositions</h2> <div class="ltx_para" id="S4.p1"> <p class="ltx_p" id="S4.p1.1">In this section we develop the notion of decompositions in Aronszajn lines (recall <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S2.Thmtheorem2" title="Definition 2.2. ‣ 2. Aronszajn and Countryman lines ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Definition</span> <span class="ltx_text ltx_ref_tag">2.2</span></a>). In particular we construct several Aronszajn lines with decompositions that achieve specific properties that will be used to study the <math alttext="\trianglelefteq" class="ltx_Math" display="inline" id="S4.p1.1.m1.1"><semantics id="S4.p1.1.m1.1a"><mi id="S4.p1.1.m1.1.1" mathvariant="normal" xref="S4.p1.1.m1.1.1.cmml">⊴</mi><annotation-xml encoding="MathML-Content" id="S4.p1.1.m1.1b"><ci id="S4.p1.1.m1.1.1.cmml" xref="S4.p1.1.m1.1.1">⊴</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.1.m1.1c">\trianglelefteq</annotation><annotation encoding="application/x-llamapun" id="S4.p1.1.m1.1d">⊴</annotation></semantics></math> relation in <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S5" title="5. An infinite antichain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Sections</span> <span class="ltx_text ltx_ref_tag">5</span></a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6" title="6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">6</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S7" title="7. A two element basis for the Aronszajn lines ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">7</span></a>.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S4.Thmtheorem1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem1.1.1.1">Lemma 4.1</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem1.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem1.p1"> <p class="ltx_p" id="S4.Thmtheorem1.p1.7">Let <math alttext="A" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.1.m1.1"><semantics id="S4.Thmtheorem1.p1.1.m1.1a"><mi id="S4.Thmtheorem1.p1.1.m1.1.1" xref="S4.Thmtheorem1.p1.1.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.1.m1.1b"><ci id="S4.Thmtheorem1.p1.1.m1.1.1.cmml" xref="S4.Thmtheorem1.p1.1.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.1.m1.1c">A</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.1.m1.1d">italic_A</annotation></semantics></math> be an Aronszajn line, and <math alttext="D" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.2.m2.1"><semantics id="S4.Thmtheorem1.p1.2.m2.1a"><mi id="S4.Thmtheorem1.p1.2.m2.1.1" xref="S4.Thmtheorem1.p1.2.m2.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.2.m2.1b"><ci id="S4.Thmtheorem1.p1.2.m2.1.1.cmml" xref="S4.Thmtheorem1.p1.2.m2.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.2.m2.1c">D</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.2.m2.1d">italic_D</annotation></semantics></math>, <math alttext="D^{\prime}" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.3.m3.1"><semantics id="S4.Thmtheorem1.p1.3.m3.1a"><msup id="S4.Thmtheorem1.p1.3.m3.1.1" xref="S4.Thmtheorem1.p1.3.m3.1.1.cmml"><mi id="S4.Thmtheorem1.p1.3.m3.1.1.2" xref="S4.Thmtheorem1.p1.3.m3.1.1.2.cmml">D</mi><mo id="S4.Thmtheorem1.p1.3.m3.1.1.3" xref="S4.Thmtheorem1.p1.3.m3.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.3.m3.1b"><apply id="S4.Thmtheorem1.p1.3.m3.1.1.cmml" xref="S4.Thmtheorem1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem1.p1.3.m3.1.1.1.cmml" xref="S4.Thmtheorem1.p1.3.m3.1.1">superscript</csymbol><ci id="S4.Thmtheorem1.p1.3.m3.1.1.2.cmml" xref="S4.Thmtheorem1.p1.3.m3.1.1.2">𝐷</ci><ci id="S4.Thmtheorem1.p1.3.m3.1.1.3.cmml" xref="S4.Thmtheorem1.p1.3.m3.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.3.m3.1c">D^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.3.m3.1d">italic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> be two decompositions for <math alttext="A" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.4.m4.1"><semantics id="S4.Thmtheorem1.p1.4.m4.1a"><mi id="S4.Thmtheorem1.p1.4.m4.1.1" xref="S4.Thmtheorem1.p1.4.m4.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.4.m4.1b"><ci id="S4.Thmtheorem1.p1.4.m4.1.1.cmml" xref="S4.Thmtheorem1.p1.4.m4.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.4.m4.1c">A</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.4.m4.1d">italic_A</annotation></semantics></math>. Then there is a club <math alttext="E\subseteq\omega_{1}" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.5.m5.1"><semantics id="S4.Thmtheorem1.p1.5.m5.1a"><mrow id="S4.Thmtheorem1.p1.5.m5.1.1" xref="S4.Thmtheorem1.p1.5.m5.1.1.cmml"><mi id="S4.Thmtheorem1.p1.5.m5.1.1.2" xref="S4.Thmtheorem1.p1.5.m5.1.1.2.cmml">E</mi><mo id="S4.Thmtheorem1.p1.5.m5.1.1.1" xref="S4.Thmtheorem1.p1.5.m5.1.1.1.cmml">⊆</mo><msub id="S4.Thmtheorem1.p1.5.m5.1.1.3" xref="S4.Thmtheorem1.p1.5.m5.1.1.3.cmml"><mi id="S4.Thmtheorem1.p1.5.m5.1.1.3.2" xref="S4.Thmtheorem1.p1.5.m5.1.1.3.2.cmml">ω</mi><mn id="S4.Thmtheorem1.p1.5.m5.1.1.3.3" xref="S4.Thmtheorem1.p1.5.m5.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.5.m5.1b"><apply id="S4.Thmtheorem1.p1.5.m5.1.1.cmml" xref="S4.Thmtheorem1.p1.5.m5.1.1"><subset id="S4.Thmtheorem1.p1.5.m5.1.1.1.cmml" xref="S4.Thmtheorem1.p1.5.m5.1.1.1"></subset><ci id="S4.Thmtheorem1.p1.5.m5.1.1.2.cmml" xref="S4.Thmtheorem1.p1.5.m5.1.1.2">𝐸</ci><apply id="S4.Thmtheorem1.p1.5.m5.1.1.3.cmml" xref="S4.Thmtheorem1.p1.5.m5.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem1.p1.5.m5.1.1.3.1.cmml" xref="S4.Thmtheorem1.p1.5.m5.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem1.p1.5.m5.1.1.3.2.cmml" xref="S4.Thmtheorem1.p1.5.m5.1.1.3.2">𝜔</ci><cn id="S4.Thmtheorem1.p1.5.m5.1.1.3.3.cmml" type="integer" xref="S4.Thmtheorem1.p1.5.m5.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.5.m5.1c">E\subseteq\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.5.m5.1d">italic_E ⊆ italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> such that for all <math alttext="\nu\in E" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.6.m6.1"><semantics id="S4.Thmtheorem1.p1.6.m6.1a"><mrow id="S4.Thmtheorem1.p1.6.m6.1.1" xref="S4.Thmtheorem1.p1.6.m6.1.1.cmml"><mi id="S4.Thmtheorem1.p1.6.m6.1.1.2" xref="S4.Thmtheorem1.p1.6.m6.1.1.2.cmml">ν</mi><mo id="S4.Thmtheorem1.p1.6.m6.1.1.1" xref="S4.Thmtheorem1.p1.6.m6.1.1.1.cmml">∈</mo><mi id="S4.Thmtheorem1.p1.6.m6.1.1.3" xref="S4.Thmtheorem1.p1.6.m6.1.1.3.cmml">E</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.6.m6.1b"><apply id="S4.Thmtheorem1.p1.6.m6.1.1.cmml" xref="S4.Thmtheorem1.p1.6.m6.1.1"><in id="S4.Thmtheorem1.p1.6.m6.1.1.1.cmml" xref="S4.Thmtheorem1.p1.6.m6.1.1.1"></in><ci id="S4.Thmtheorem1.p1.6.m6.1.1.2.cmml" xref="S4.Thmtheorem1.p1.6.m6.1.1.2">𝜈</ci><ci id="S4.Thmtheorem1.p1.6.m6.1.1.3.cmml" xref="S4.Thmtheorem1.p1.6.m6.1.1.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.6.m6.1c">\nu\in E</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.6.m6.1d">italic_ν ∈ italic_E</annotation></semantics></math>, <math alttext="D_{\nu}=D^{\prime}_{\nu}" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.7.m7.1"><semantics id="S4.Thmtheorem1.p1.7.m7.1a"><mrow id="S4.Thmtheorem1.p1.7.m7.1.1" xref="S4.Thmtheorem1.p1.7.m7.1.1.cmml"><msub id="S4.Thmtheorem1.p1.7.m7.1.1.2" xref="S4.Thmtheorem1.p1.7.m7.1.1.2.cmml"><mi id="S4.Thmtheorem1.p1.7.m7.1.1.2.2" xref="S4.Thmtheorem1.p1.7.m7.1.1.2.2.cmml">D</mi><mi id="S4.Thmtheorem1.p1.7.m7.1.1.2.3" xref="S4.Thmtheorem1.p1.7.m7.1.1.2.3.cmml">ν</mi></msub><mo id="S4.Thmtheorem1.p1.7.m7.1.1.1" xref="S4.Thmtheorem1.p1.7.m7.1.1.1.cmml">=</mo><msubsup id="S4.Thmtheorem1.p1.7.m7.1.1.3" xref="S4.Thmtheorem1.p1.7.m7.1.1.3.cmml"><mi id="S4.Thmtheorem1.p1.7.m7.1.1.3.2.2" xref="S4.Thmtheorem1.p1.7.m7.1.1.3.2.2.cmml">D</mi><mi id="S4.Thmtheorem1.p1.7.m7.1.1.3.3" xref="S4.Thmtheorem1.p1.7.m7.1.1.3.3.cmml">ν</mi><mo id="S4.Thmtheorem1.p1.7.m7.1.1.3.2.3" xref="S4.Thmtheorem1.p1.7.m7.1.1.3.2.3.cmml">′</mo></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.7.m7.1b"><apply id="S4.Thmtheorem1.p1.7.m7.1.1.cmml" xref="S4.Thmtheorem1.p1.7.m7.1.1"><eq id="S4.Thmtheorem1.p1.7.m7.1.1.1.cmml" xref="S4.Thmtheorem1.p1.7.m7.1.1.1"></eq><apply id="S4.Thmtheorem1.p1.7.m7.1.1.2.cmml" xref="S4.Thmtheorem1.p1.7.m7.1.1.2"><csymbol cd="ambiguous" id="S4.Thmtheorem1.p1.7.m7.1.1.2.1.cmml" xref="S4.Thmtheorem1.p1.7.m7.1.1.2">subscript</csymbol><ci id="S4.Thmtheorem1.p1.7.m7.1.1.2.2.cmml" xref="S4.Thmtheorem1.p1.7.m7.1.1.2.2">𝐷</ci><ci id="S4.Thmtheorem1.p1.7.m7.1.1.2.3.cmml" xref="S4.Thmtheorem1.p1.7.m7.1.1.2.3">𝜈</ci></apply><apply id="S4.Thmtheorem1.p1.7.m7.1.1.3.cmml" xref="S4.Thmtheorem1.p1.7.m7.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem1.p1.7.m7.1.1.3.1.cmml" xref="S4.Thmtheorem1.p1.7.m7.1.1.3">subscript</csymbol><apply id="S4.Thmtheorem1.p1.7.m7.1.1.3.2.cmml" xref="S4.Thmtheorem1.p1.7.m7.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem1.p1.7.m7.1.1.3.2.1.cmml" xref="S4.Thmtheorem1.p1.7.m7.1.1.3">superscript</csymbol><ci id="S4.Thmtheorem1.p1.7.m7.1.1.3.2.2.cmml" xref="S4.Thmtheorem1.p1.7.m7.1.1.3.2.2">𝐷</ci><ci id="S4.Thmtheorem1.p1.7.m7.1.1.3.2.3.cmml" xref="S4.Thmtheorem1.p1.7.m7.1.1.3.2.3">′</ci></apply><ci id="S4.Thmtheorem1.p1.7.m7.1.1.3.3.cmml" xref="S4.Thmtheorem1.p1.7.m7.1.1.3.3">𝜈</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.7.m7.1c">D_{\nu}=D^{\prime}_{\nu}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.7.m7.1d">italic_D start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT = italic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> </div> <div class="ltx_proof" id="S4.2"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S4.1.p1"> <p class="ltx_p" id="S4.1.p1.5">Assume that <math alttext="A" class="ltx_Math" display="inline" id="S4.1.p1.1.m1.1"><semantics id="S4.1.p1.1.m1.1a"><mi id="S4.1.p1.1.m1.1.1" xref="S4.1.p1.1.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S4.1.p1.1.m1.1b"><ci id="S4.1.p1.1.m1.1.1.cmml" xref="S4.1.p1.1.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.1.p1.1.m1.1c">A</annotation><annotation encoding="application/x-llamapun" id="S4.1.p1.1.m1.1d">italic_A</annotation></semantics></math> is <math alttext="\omega_{1}" class="ltx_Math" display="inline" id="S4.1.p1.2.m2.1"><semantics id="S4.1.p1.2.m2.1a"><msub id="S4.1.p1.2.m2.1.1" xref="S4.1.p1.2.m2.1.1.cmml"><mi id="S4.1.p1.2.m2.1.1.2" xref="S4.1.p1.2.m2.1.1.2.cmml">ω</mi><mn id="S4.1.p1.2.m2.1.1.3" xref="S4.1.p1.2.m2.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.1.p1.2.m2.1b"><apply id="S4.1.p1.2.m2.1.1.cmml" xref="S4.1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S4.1.p1.2.m2.1.1.1.cmml" xref="S4.1.p1.2.m2.1.1">subscript</csymbol><ci id="S4.1.p1.2.m2.1.1.2.cmml" xref="S4.1.p1.2.m2.1.1.2">𝜔</ci><cn id="S4.1.p1.2.m2.1.1.3.cmml" type="integer" xref="S4.1.p1.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.1.p1.2.m2.1c">\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.1.p1.2.m2.1d">italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> as a set and let <math alttext="E\subseteq\omega_{1}" class="ltx_Math" display="inline" id="S4.1.p1.3.m3.1"><semantics id="S4.1.p1.3.m3.1a"><mrow id="S4.1.p1.3.m3.1.1" xref="S4.1.p1.3.m3.1.1.cmml"><mi id="S4.1.p1.3.m3.1.1.2" xref="S4.1.p1.3.m3.1.1.2.cmml">E</mi><mo id="S4.1.p1.3.m3.1.1.1" xref="S4.1.p1.3.m3.1.1.1.cmml">⊆</mo><msub id="S4.1.p1.3.m3.1.1.3" xref="S4.1.p1.3.m3.1.1.3.cmml"><mi id="S4.1.p1.3.m3.1.1.3.2" xref="S4.1.p1.3.m3.1.1.3.2.cmml">ω</mi><mn id="S4.1.p1.3.m3.1.1.3.3" xref="S4.1.p1.3.m3.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.1.p1.3.m3.1b"><apply id="S4.1.p1.3.m3.1.1.cmml" xref="S4.1.p1.3.m3.1.1"><subset id="S4.1.p1.3.m3.1.1.1.cmml" xref="S4.1.p1.3.m3.1.1.1"></subset><ci id="S4.1.p1.3.m3.1.1.2.cmml" xref="S4.1.p1.3.m3.1.1.2">𝐸</ci><apply id="S4.1.p1.3.m3.1.1.3.cmml" xref="S4.1.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S4.1.p1.3.m3.1.1.3.1.cmml" xref="S4.1.p1.3.m3.1.1.3">subscript</csymbol><ci id="S4.1.p1.3.m3.1.1.3.2.cmml" xref="S4.1.p1.3.m3.1.1.3.2">𝜔</ci><cn id="S4.1.p1.3.m3.1.1.3.3.cmml" type="integer" xref="S4.1.p1.3.m3.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.1.p1.3.m3.1c">E\subseteq\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.1.p1.3.m3.1d">italic_E ⊆ italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> be a club of limit ordinals closed under <math alttext="f" class="ltx_Math" display="inline" id="S4.1.p1.4.m4.1"><semantics id="S4.1.p1.4.m4.1a"><mi id="S4.1.p1.4.m4.1.1" xref="S4.1.p1.4.m4.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S4.1.p1.4.m4.1b"><ci id="S4.1.p1.4.m4.1.1.cmml" xref="S4.1.p1.4.m4.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.1.p1.4.m4.1c">f</annotation><annotation encoding="application/x-llamapun" id="S4.1.p1.4.m4.1d">italic_f</annotation></semantics></math> and <math alttext="g" class="ltx_Math" display="inline" id="S4.1.p1.5.m5.1"><semantics id="S4.1.p1.5.m5.1a"><mi id="S4.1.p1.5.m5.1.1" xref="S4.1.p1.5.m5.1.1.cmml">g</mi><annotation-xml encoding="MathML-Content" id="S4.1.p1.5.m5.1b"><ci id="S4.1.p1.5.m5.1.1.cmml" xref="S4.1.p1.5.m5.1.1">𝑔</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.1.p1.5.m5.1c">g</annotation><annotation encoding="application/x-llamapun" id="S4.1.p1.5.m5.1d">italic_g</annotation></semantics></math> where</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex5"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="f(\xi):=\min\{\alpha:\xi\in D_{\alpha}\}\text{ and }g(\xi):=\sup\{\eta+1:\eta% \in D_{\xi}\}." class="ltx_Math" display="block" id="S4.Ex5.m1.4"><semantics id="S4.Ex5.m1.4a"><mrow id="S4.Ex5.m1.4.4.1" xref="S4.Ex5.m1.4.4.1.1.cmml"><mrow id="S4.Ex5.m1.4.4.1.1" xref="S4.Ex5.m1.4.4.1.1.cmml"><mrow id="S4.Ex5.m1.4.4.1.1.5" xref="S4.Ex5.m1.4.4.1.1.5.cmml"><mi id="S4.Ex5.m1.4.4.1.1.5.2" xref="S4.Ex5.m1.4.4.1.1.5.2.cmml">f</mi><mo id="S4.Ex5.m1.4.4.1.1.5.1" xref="S4.Ex5.m1.4.4.1.1.5.1.cmml">⁢</mo><mrow id="S4.Ex5.m1.4.4.1.1.5.3.2" xref="S4.Ex5.m1.4.4.1.1.5.cmml"><mo id="S4.Ex5.m1.4.4.1.1.5.3.2.1" stretchy="false" xref="S4.Ex5.m1.4.4.1.1.5.cmml">(</mo><mi id="S4.Ex5.m1.1.1" xref="S4.Ex5.m1.1.1.cmml">ξ</mi><mo id="S4.Ex5.m1.4.4.1.1.5.3.2.2" rspace="0.278em" stretchy="false" xref="S4.Ex5.m1.4.4.1.1.5.cmml">)</mo></mrow></mrow><mo id="S4.Ex5.m1.4.4.1.1.6" rspace="0.278em" xref="S4.Ex5.m1.4.4.1.1.6.cmml">:=</mo><mrow id="S4.Ex5.m1.4.4.1.1.1" xref="S4.Ex5.m1.4.4.1.1.1.cmml"><mrow id="S4.Ex5.m1.4.4.1.1.1.1.1" xref="S4.Ex5.m1.4.4.1.1.1.1.2.cmml"><mi id="S4.Ex5.m1.2.2" xref="S4.Ex5.m1.2.2.cmml">min</mi><mo id="S4.Ex5.m1.4.4.1.1.1.1.1a" xref="S4.Ex5.m1.4.4.1.1.1.1.2.cmml">⁡</mo><mrow id="S4.Ex5.m1.4.4.1.1.1.1.1.1" xref="S4.Ex5.m1.4.4.1.1.1.1.2.cmml"><mo id="S4.Ex5.m1.4.4.1.1.1.1.1.1.2" stretchy="false" xref="S4.Ex5.m1.4.4.1.1.1.1.2.cmml">{</mo><mrow id="S4.Ex5.m1.4.4.1.1.1.1.1.1.1" xref="S4.Ex5.m1.4.4.1.1.1.1.1.1.1.cmml"><mi id="S4.Ex5.m1.4.4.1.1.1.1.1.1.1.2" xref="S4.Ex5.m1.4.4.1.1.1.1.1.1.1.2.cmml">α</mi><mo id="S4.Ex5.m1.4.4.1.1.1.1.1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S4.Ex5.m1.4.4.1.1.1.1.1.1.1.1.cmml">:</mo><mrow id="S4.Ex5.m1.4.4.1.1.1.1.1.1.1.3" xref="S4.Ex5.m1.4.4.1.1.1.1.1.1.1.3.cmml"><mi id="S4.Ex5.m1.4.4.1.1.1.1.1.1.1.3.2" xref="S4.Ex5.m1.4.4.1.1.1.1.1.1.1.3.2.cmml">ξ</mi><mo id="S4.Ex5.m1.4.4.1.1.1.1.1.1.1.3.1" xref="S4.Ex5.m1.4.4.1.1.1.1.1.1.1.3.1.cmml">∈</mo><msub id="S4.Ex5.m1.4.4.1.1.1.1.1.1.1.3.3" xref="S4.Ex5.m1.4.4.1.1.1.1.1.1.1.3.3.cmml"><mi id="S4.Ex5.m1.4.4.1.1.1.1.1.1.1.3.3.2" xref="S4.Ex5.m1.4.4.1.1.1.1.1.1.1.3.3.2.cmml">D</mi><mi id="S4.Ex5.m1.4.4.1.1.1.1.1.1.1.3.3.3" xref="S4.Ex5.m1.4.4.1.1.1.1.1.1.1.3.3.3.cmml">α</mi></msub></mrow></mrow><mo id="S4.Ex5.m1.4.4.1.1.1.1.1.1.3" stretchy="false" xref="S4.Ex5.m1.4.4.1.1.1.1.2.cmml">}</mo></mrow></mrow><mo id="S4.Ex5.m1.4.4.1.1.1.2" xref="S4.Ex5.m1.4.4.1.1.1.2.cmml">⁢</mo><mtext id="S4.Ex5.m1.4.4.1.1.1.3" xref="S4.Ex5.m1.4.4.1.1.1.3a.cmml"> and </mtext><mo id="S4.Ex5.m1.4.4.1.1.1.2a" xref="S4.Ex5.m1.4.4.1.1.1.2.cmml">⁢</mo><mi id="S4.Ex5.m1.4.4.1.1.1.4" xref="S4.Ex5.m1.4.4.1.1.1.4.cmml">g</mi><mo id="S4.Ex5.m1.4.4.1.1.1.2b" xref="S4.Ex5.m1.4.4.1.1.1.2.cmml">⁢</mo><mrow id="S4.Ex5.m1.4.4.1.1.1.5.2" xref="S4.Ex5.m1.4.4.1.1.1.cmml"><mo id="S4.Ex5.m1.4.4.1.1.1.5.2.1" stretchy="false" xref="S4.Ex5.m1.4.4.1.1.1.cmml">(</mo><mi id="S4.Ex5.m1.3.3" xref="S4.Ex5.m1.3.3.cmml">ξ</mi><mo id="S4.Ex5.m1.4.4.1.1.1.5.2.2" rspace="0.278em" stretchy="false" xref="S4.Ex5.m1.4.4.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex5.m1.4.4.1.1.7" xref="S4.Ex5.m1.4.4.1.1.7.cmml">:=</mo><mrow id="S4.Ex5.m1.4.4.1.1.3" xref="S4.Ex5.m1.4.4.1.1.3.cmml"><mo id="S4.Ex5.m1.4.4.1.1.3.3" lspace="0.111em" movablelimits="false" rspace="0em" xref="S4.Ex5.m1.4.4.1.1.3.3.cmml">sup</mo><mrow id="S4.Ex5.m1.4.4.1.1.3.2.2" xref="S4.Ex5.m1.4.4.1.1.3.2.3.cmml"><mo id="S4.Ex5.m1.4.4.1.1.3.2.2.3" stretchy="false" xref="S4.Ex5.m1.4.4.1.1.3.2.3.1.cmml">{</mo><mrow id="S4.Ex5.m1.4.4.1.1.2.1.1.1" xref="S4.Ex5.m1.4.4.1.1.2.1.1.1.cmml"><mi id="S4.Ex5.m1.4.4.1.1.2.1.1.1.2" xref="S4.Ex5.m1.4.4.1.1.2.1.1.1.2.cmml">η</mi><mo id="S4.Ex5.m1.4.4.1.1.2.1.1.1.1" xref="S4.Ex5.m1.4.4.1.1.2.1.1.1.1.cmml">+</mo><mn id="S4.Ex5.m1.4.4.1.1.2.1.1.1.3" xref="S4.Ex5.m1.4.4.1.1.2.1.1.1.3.cmml">1</mn></mrow><mo id="S4.Ex5.m1.4.4.1.1.3.2.2.4" lspace="0.278em" rspace="0.278em" xref="S4.Ex5.m1.4.4.1.1.3.2.3.1.cmml">:</mo><mrow id="S4.Ex5.m1.4.4.1.1.3.2.2.2" xref="S4.Ex5.m1.4.4.1.1.3.2.2.2.cmml"><mi id="S4.Ex5.m1.4.4.1.1.3.2.2.2.2" xref="S4.Ex5.m1.4.4.1.1.3.2.2.2.2.cmml">η</mi><mo id="S4.Ex5.m1.4.4.1.1.3.2.2.2.1" xref="S4.Ex5.m1.4.4.1.1.3.2.2.2.1.cmml">∈</mo><msub id="S4.Ex5.m1.4.4.1.1.3.2.2.2.3" xref="S4.Ex5.m1.4.4.1.1.3.2.2.2.3.cmml"><mi id="S4.Ex5.m1.4.4.1.1.3.2.2.2.3.2" xref="S4.Ex5.m1.4.4.1.1.3.2.2.2.3.2.cmml">D</mi><mi id="S4.Ex5.m1.4.4.1.1.3.2.2.2.3.3" xref="S4.Ex5.m1.4.4.1.1.3.2.2.2.3.3.cmml">ξ</mi></msub></mrow><mo id="S4.Ex5.m1.4.4.1.1.3.2.2.5" stretchy="false" xref="S4.Ex5.m1.4.4.1.1.3.2.3.1.cmml">}</mo></mrow></mrow></mrow><mo id="S4.Ex5.m1.4.4.1.2" lspace="0em" xref="S4.Ex5.m1.4.4.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex5.m1.4b"><apply id="S4.Ex5.m1.4.4.1.1.cmml" xref="S4.Ex5.m1.4.4.1"><and id="S4.Ex5.m1.4.4.1.1a.cmml" xref="S4.Ex5.m1.4.4.1"></and><apply id="S4.Ex5.m1.4.4.1.1b.cmml" xref="S4.Ex5.m1.4.4.1"><csymbol cd="latexml" id="S4.Ex5.m1.4.4.1.1.6.cmml" xref="S4.Ex5.m1.4.4.1.1.6">assign</csymbol><apply id="S4.Ex5.m1.4.4.1.1.5.cmml" xref="S4.Ex5.m1.4.4.1.1.5"><times id="S4.Ex5.m1.4.4.1.1.5.1.cmml" xref="S4.Ex5.m1.4.4.1.1.5.1"></times><ci id="S4.Ex5.m1.4.4.1.1.5.2.cmml" xref="S4.Ex5.m1.4.4.1.1.5.2">𝑓</ci><ci id="S4.Ex5.m1.1.1.cmml" xref="S4.Ex5.m1.1.1">𝜉</ci></apply><apply id="S4.Ex5.m1.4.4.1.1.1.cmml" xref="S4.Ex5.m1.4.4.1.1.1"><times id="S4.Ex5.m1.4.4.1.1.1.2.cmml" xref="S4.Ex5.m1.4.4.1.1.1.2"></times><apply id="S4.Ex5.m1.4.4.1.1.1.1.2.cmml" xref="S4.Ex5.m1.4.4.1.1.1.1.1"><min id="S4.Ex5.m1.2.2.cmml" xref="S4.Ex5.m1.2.2"></min><apply id="S4.Ex5.m1.4.4.1.1.1.1.1.1.1.cmml" xref="S4.Ex5.m1.4.4.1.1.1.1.1.1.1"><ci id="S4.Ex5.m1.4.4.1.1.1.1.1.1.1.1.cmml" xref="S4.Ex5.m1.4.4.1.1.1.1.1.1.1.1">:</ci><ci id="S4.Ex5.m1.4.4.1.1.1.1.1.1.1.2.cmml" xref="S4.Ex5.m1.4.4.1.1.1.1.1.1.1.2">𝛼</ci><apply id="S4.Ex5.m1.4.4.1.1.1.1.1.1.1.3.cmml" xref="S4.Ex5.m1.4.4.1.1.1.1.1.1.1.3"><in id="S4.Ex5.m1.4.4.1.1.1.1.1.1.1.3.1.cmml" xref="S4.Ex5.m1.4.4.1.1.1.1.1.1.1.3.1"></in><ci id="S4.Ex5.m1.4.4.1.1.1.1.1.1.1.3.2.cmml" xref="S4.Ex5.m1.4.4.1.1.1.1.1.1.1.3.2">𝜉</ci><apply id="S4.Ex5.m1.4.4.1.1.1.1.1.1.1.3.3.cmml" xref="S4.Ex5.m1.4.4.1.1.1.1.1.1.1.3.3"><csymbol cd="ambiguous" id="S4.Ex5.m1.4.4.1.1.1.1.1.1.1.3.3.1.cmml" xref="S4.Ex5.m1.4.4.1.1.1.1.1.1.1.3.3">subscript</csymbol><ci id="S4.Ex5.m1.4.4.1.1.1.1.1.1.1.3.3.2.cmml" xref="S4.Ex5.m1.4.4.1.1.1.1.1.1.1.3.3.2">𝐷</ci><ci id="S4.Ex5.m1.4.4.1.1.1.1.1.1.1.3.3.3.cmml" xref="S4.Ex5.m1.4.4.1.1.1.1.1.1.1.3.3.3">𝛼</ci></apply></apply></apply></apply><ci id="S4.Ex5.m1.4.4.1.1.1.3a.cmml" xref="S4.Ex5.m1.4.4.1.1.1.3"><mtext id="S4.Ex5.m1.4.4.1.1.1.3.cmml" xref="S4.Ex5.m1.4.4.1.1.1.3"> and </mtext></ci><ci id="S4.Ex5.m1.4.4.1.1.1.4.cmml" xref="S4.Ex5.m1.4.4.1.1.1.4">𝑔</ci><ci id="S4.Ex5.m1.3.3.cmml" xref="S4.Ex5.m1.3.3">𝜉</ci></apply></apply><apply id="S4.Ex5.m1.4.4.1.1c.cmml" xref="S4.Ex5.m1.4.4.1"><csymbol cd="latexml" id="S4.Ex5.m1.4.4.1.1.7.cmml" xref="S4.Ex5.m1.4.4.1.1.7">assign</csymbol><share href="https://arxiv.org/html/2503.13728v1#S4.Ex5.m1.4.4.1.1.1.cmml" id="S4.Ex5.m1.4.4.1.1d.cmml" xref="S4.Ex5.m1.4.4.1"></share><apply id="S4.Ex5.m1.4.4.1.1.3.cmml" xref="S4.Ex5.m1.4.4.1.1.3"><csymbol cd="latexml" id="S4.Ex5.m1.4.4.1.1.3.3.cmml" xref="S4.Ex5.m1.4.4.1.1.3.3">supremum</csymbol><apply id="S4.Ex5.m1.4.4.1.1.3.2.3.cmml" xref="S4.Ex5.m1.4.4.1.1.3.2.2"><csymbol cd="latexml" id="S4.Ex5.m1.4.4.1.1.3.2.3.1.cmml" xref="S4.Ex5.m1.4.4.1.1.3.2.2.3">conditional-set</csymbol><apply id="S4.Ex5.m1.4.4.1.1.2.1.1.1.cmml" xref="S4.Ex5.m1.4.4.1.1.2.1.1.1"><plus id="S4.Ex5.m1.4.4.1.1.2.1.1.1.1.cmml" xref="S4.Ex5.m1.4.4.1.1.2.1.1.1.1"></plus><ci id="S4.Ex5.m1.4.4.1.1.2.1.1.1.2.cmml" xref="S4.Ex5.m1.4.4.1.1.2.1.1.1.2">𝜂</ci><cn id="S4.Ex5.m1.4.4.1.1.2.1.1.1.3.cmml" type="integer" xref="S4.Ex5.m1.4.4.1.1.2.1.1.1.3">1</cn></apply><apply id="S4.Ex5.m1.4.4.1.1.3.2.2.2.cmml" xref="S4.Ex5.m1.4.4.1.1.3.2.2.2"><in id="S4.Ex5.m1.4.4.1.1.3.2.2.2.1.cmml" xref="S4.Ex5.m1.4.4.1.1.3.2.2.2.1"></in><ci id="S4.Ex5.m1.4.4.1.1.3.2.2.2.2.cmml" xref="S4.Ex5.m1.4.4.1.1.3.2.2.2.2">𝜂</ci><apply id="S4.Ex5.m1.4.4.1.1.3.2.2.2.3.cmml" xref="S4.Ex5.m1.4.4.1.1.3.2.2.2.3"><csymbol cd="ambiguous" id="S4.Ex5.m1.4.4.1.1.3.2.2.2.3.1.cmml" xref="S4.Ex5.m1.4.4.1.1.3.2.2.2.3">subscript</csymbol><ci id="S4.Ex5.m1.4.4.1.1.3.2.2.2.3.2.cmml" xref="S4.Ex5.m1.4.4.1.1.3.2.2.2.3.2">𝐷</ci><ci id="S4.Ex5.m1.4.4.1.1.3.2.2.2.3.3.cmml" xref="S4.Ex5.m1.4.4.1.1.3.2.2.2.3.3">𝜉</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex5.m1.4c">f(\xi):=\min\{\alpha:\xi\in D_{\alpha}\}\text{ and }g(\xi):=\sup\{\eta+1:\eta% \in D_{\xi}\}.</annotation><annotation encoding="application/x-llamapun" id="S4.Ex5.m1.4d">italic_f ( italic_ξ ) := roman_min { italic_α : italic_ξ ∈ italic_D start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT } and italic_g ( italic_ξ ) := roman_sup { italic_η + 1 : italic_η ∈ italic_D start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT } .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.1.p1.17">We show that if <math alttext="\nu\in E" class="ltx_Math" display="inline" id="S4.1.p1.6.m1.1"><semantics id="S4.1.p1.6.m1.1a"><mrow id="S4.1.p1.6.m1.1.1" xref="S4.1.p1.6.m1.1.1.cmml"><mi id="S4.1.p1.6.m1.1.1.2" xref="S4.1.p1.6.m1.1.1.2.cmml">ν</mi><mo id="S4.1.p1.6.m1.1.1.1" xref="S4.1.p1.6.m1.1.1.1.cmml">∈</mo><mi id="S4.1.p1.6.m1.1.1.3" xref="S4.1.p1.6.m1.1.1.3.cmml">E</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.1.p1.6.m1.1b"><apply id="S4.1.p1.6.m1.1.1.cmml" xref="S4.1.p1.6.m1.1.1"><in id="S4.1.p1.6.m1.1.1.1.cmml" xref="S4.1.p1.6.m1.1.1.1"></in><ci id="S4.1.p1.6.m1.1.1.2.cmml" xref="S4.1.p1.6.m1.1.1.2">𝜈</ci><ci id="S4.1.p1.6.m1.1.1.3.cmml" xref="S4.1.p1.6.m1.1.1.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.1.p1.6.m1.1c">\nu\in E</annotation><annotation encoding="application/x-llamapun" id="S4.1.p1.6.m1.1d">italic_ν ∈ italic_E</annotation></semantics></math>, then <math alttext="D_{\nu}=\nu" class="ltx_Math" display="inline" id="S4.1.p1.7.m2.1"><semantics id="S4.1.p1.7.m2.1a"><mrow id="S4.1.p1.7.m2.1.1" xref="S4.1.p1.7.m2.1.1.cmml"><msub id="S4.1.p1.7.m2.1.1.2" xref="S4.1.p1.7.m2.1.1.2.cmml"><mi id="S4.1.p1.7.m2.1.1.2.2" xref="S4.1.p1.7.m2.1.1.2.2.cmml">D</mi><mi id="S4.1.p1.7.m2.1.1.2.3" xref="S4.1.p1.7.m2.1.1.2.3.cmml">ν</mi></msub><mo id="S4.1.p1.7.m2.1.1.1" xref="S4.1.p1.7.m2.1.1.1.cmml">=</mo><mi id="S4.1.p1.7.m2.1.1.3" xref="S4.1.p1.7.m2.1.1.3.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.1.p1.7.m2.1b"><apply id="S4.1.p1.7.m2.1.1.cmml" xref="S4.1.p1.7.m2.1.1"><eq id="S4.1.p1.7.m2.1.1.1.cmml" xref="S4.1.p1.7.m2.1.1.1"></eq><apply id="S4.1.p1.7.m2.1.1.2.cmml" xref="S4.1.p1.7.m2.1.1.2"><csymbol cd="ambiguous" id="S4.1.p1.7.m2.1.1.2.1.cmml" xref="S4.1.p1.7.m2.1.1.2">subscript</csymbol><ci id="S4.1.p1.7.m2.1.1.2.2.cmml" xref="S4.1.p1.7.m2.1.1.2.2">𝐷</ci><ci id="S4.1.p1.7.m2.1.1.2.3.cmml" xref="S4.1.p1.7.m2.1.1.2.3">𝜈</ci></apply><ci id="S4.1.p1.7.m2.1.1.3.cmml" xref="S4.1.p1.7.m2.1.1.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.1.p1.7.m2.1c">D_{\nu}=\nu</annotation><annotation encoding="application/x-llamapun" id="S4.1.p1.7.m2.1d">italic_D start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT = italic_ν</annotation></semantics></math>. First observe that for all <math alttext="\xi<\nu" class="ltx_Math" display="inline" id="S4.1.p1.8.m3.1"><semantics id="S4.1.p1.8.m3.1a"><mrow id="S4.1.p1.8.m3.1.1" xref="S4.1.p1.8.m3.1.1.cmml"><mi id="S4.1.p1.8.m3.1.1.2" xref="S4.1.p1.8.m3.1.1.2.cmml">ξ</mi><mo id="S4.1.p1.8.m3.1.1.1" xref="S4.1.p1.8.m3.1.1.1.cmml"><</mo><mi id="S4.1.p1.8.m3.1.1.3" xref="S4.1.p1.8.m3.1.1.3.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.1.p1.8.m3.1b"><apply id="S4.1.p1.8.m3.1.1.cmml" xref="S4.1.p1.8.m3.1.1"><lt id="S4.1.p1.8.m3.1.1.1.cmml" xref="S4.1.p1.8.m3.1.1.1"></lt><ci id="S4.1.p1.8.m3.1.1.2.cmml" xref="S4.1.p1.8.m3.1.1.2">𝜉</ci><ci id="S4.1.p1.8.m3.1.1.3.cmml" xref="S4.1.p1.8.m3.1.1.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.1.p1.8.m3.1c">\xi<\nu</annotation><annotation encoding="application/x-llamapun" id="S4.1.p1.8.m3.1d">italic_ξ < italic_ν</annotation></semantics></math>, <math alttext="D_{\xi}\subseteq g(\xi)<\nu" class="ltx_Math" display="inline" id="S4.1.p1.9.m4.1"><semantics id="S4.1.p1.9.m4.1a"><mrow id="S4.1.p1.9.m4.1.2" xref="S4.1.p1.9.m4.1.2.cmml"><msub id="S4.1.p1.9.m4.1.2.2" xref="S4.1.p1.9.m4.1.2.2.cmml"><mi id="S4.1.p1.9.m4.1.2.2.2" xref="S4.1.p1.9.m4.1.2.2.2.cmml">D</mi><mi id="S4.1.p1.9.m4.1.2.2.3" xref="S4.1.p1.9.m4.1.2.2.3.cmml">ξ</mi></msub><mo id="S4.1.p1.9.m4.1.2.3" xref="S4.1.p1.9.m4.1.2.3.cmml">⊆</mo><mrow id="S4.1.p1.9.m4.1.2.4" xref="S4.1.p1.9.m4.1.2.4.cmml"><mi id="S4.1.p1.9.m4.1.2.4.2" xref="S4.1.p1.9.m4.1.2.4.2.cmml">g</mi><mo id="S4.1.p1.9.m4.1.2.4.1" xref="S4.1.p1.9.m4.1.2.4.1.cmml">⁢</mo><mrow id="S4.1.p1.9.m4.1.2.4.3.2" xref="S4.1.p1.9.m4.1.2.4.cmml"><mo id="S4.1.p1.9.m4.1.2.4.3.2.1" stretchy="false" xref="S4.1.p1.9.m4.1.2.4.cmml">(</mo><mi id="S4.1.p1.9.m4.1.1" xref="S4.1.p1.9.m4.1.1.cmml">ξ</mi><mo id="S4.1.p1.9.m4.1.2.4.3.2.2" stretchy="false" xref="S4.1.p1.9.m4.1.2.4.cmml">)</mo></mrow></mrow><mo id="S4.1.p1.9.m4.1.2.5" xref="S4.1.p1.9.m4.1.2.5.cmml"><</mo><mi id="S4.1.p1.9.m4.1.2.6" xref="S4.1.p1.9.m4.1.2.6.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.1.p1.9.m4.1b"><apply id="S4.1.p1.9.m4.1.2.cmml" xref="S4.1.p1.9.m4.1.2"><and id="S4.1.p1.9.m4.1.2a.cmml" xref="S4.1.p1.9.m4.1.2"></and><apply id="S4.1.p1.9.m4.1.2b.cmml" xref="S4.1.p1.9.m4.1.2"><subset id="S4.1.p1.9.m4.1.2.3.cmml" xref="S4.1.p1.9.m4.1.2.3"></subset><apply id="S4.1.p1.9.m4.1.2.2.cmml" xref="S4.1.p1.9.m4.1.2.2"><csymbol cd="ambiguous" id="S4.1.p1.9.m4.1.2.2.1.cmml" xref="S4.1.p1.9.m4.1.2.2">subscript</csymbol><ci id="S4.1.p1.9.m4.1.2.2.2.cmml" xref="S4.1.p1.9.m4.1.2.2.2">𝐷</ci><ci id="S4.1.p1.9.m4.1.2.2.3.cmml" xref="S4.1.p1.9.m4.1.2.2.3">𝜉</ci></apply><apply id="S4.1.p1.9.m4.1.2.4.cmml" xref="S4.1.p1.9.m4.1.2.4"><times id="S4.1.p1.9.m4.1.2.4.1.cmml" xref="S4.1.p1.9.m4.1.2.4.1"></times><ci id="S4.1.p1.9.m4.1.2.4.2.cmml" xref="S4.1.p1.9.m4.1.2.4.2">𝑔</ci><ci id="S4.1.p1.9.m4.1.1.cmml" xref="S4.1.p1.9.m4.1.1">𝜉</ci></apply></apply><apply id="S4.1.p1.9.m4.1.2c.cmml" xref="S4.1.p1.9.m4.1.2"><lt id="S4.1.p1.9.m4.1.2.5.cmml" xref="S4.1.p1.9.m4.1.2.5"></lt><share href="https://arxiv.org/html/2503.13728v1#S4.1.p1.9.m4.1.2.4.cmml" id="S4.1.p1.9.m4.1.2d.cmml" xref="S4.1.p1.9.m4.1.2"></share><ci id="S4.1.p1.9.m4.1.2.6.cmml" xref="S4.1.p1.9.m4.1.2.6">𝜈</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.1.p1.9.m4.1c">D_{\xi}\subseteq g(\xi)<\nu</annotation><annotation encoding="application/x-llamapun" id="S4.1.p1.9.m4.1d">italic_D start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT ⊆ italic_g ( italic_ξ ) < italic_ν</annotation></semantics></math>, and since decompositions are the union at limit ordinals, we conclude that <math alttext="D_{\nu}\subseteq\nu" class="ltx_Math" display="inline" id="S4.1.p1.10.m5.1"><semantics id="S4.1.p1.10.m5.1a"><mrow id="S4.1.p1.10.m5.1.1" xref="S4.1.p1.10.m5.1.1.cmml"><msub id="S4.1.p1.10.m5.1.1.2" xref="S4.1.p1.10.m5.1.1.2.cmml"><mi id="S4.1.p1.10.m5.1.1.2.2" xref="S4.1.p1.10.m5.1.1.2.2.cmml">D</mi><mi id="S4.1.p1.10.m5.1.1.2.3" xref="S4.1.p1.10.m5.1.1.2.3.cmml">ν</mi></msub><mo id="S4.1.p1.10.m5.1.1.1" xref="S4.1.p1.10.m5.1.1.1.cmml">⊆</mo><mi id="S4.1.p1.10.m5.1.1.3" xref="S4.1.p1.10.m5.1.1.3.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.1.p1.10.m5.1b"><apply id="S4.1.p1.10.m5.1.1.cmml" xref="S4.1.p1.10.m5.1.1"><subset id="S4.1.p1.10.m5.1.1.1.cmml" xref="S4.1.p1.10.m5.1.1.1"></subset><apply id="S4.1.p1.10.m5.1.1.2.cmml" xref="S4.1.p1.10.m5.1.1.2"><csymbol cd="ambiguous" id="S4.1.p1.10.m5.1.1.2.1.cmml" xref="S4.1.p1.10.m5.1.1.2">subscript</csymbol><ci id="S4.1.p1.10.m5.1.1.2.2.cmml" xref="S4.1.p1.10.m5.1.1.2.2">𝐷</ci><ci id="S4.1.p1.10.m5.1.1.2.3.cmml" xref="S4.1.p1.10.m5.1.1.2.3">𝜈</ci></apply><ci id="S4.1.p1.10.m5.1.1.3.cmml" xref="S4.1.p1.10.m5.1.1.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.1.p1.10.m5.1c">D_{\nu}\subseteq\nu</annotation><annotation encoding="application/x-llamapun" id="S4.1.p1.10.m5.1d">italic_D start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT ⊆ italic_ν</annotation></semantics></math>. Suppose that for some <math alttext="\xi" class="ltx_Math" display="inline" id="S4.1.p1.11.m6.1"><semantics id="S4.1.p1.11.m6.1a"><mi id="S4.1.p1.11.m6.1.1" xref="S4.1.p1.11.m6.1.1.cmml">ξ</mi><annotation-xml encoding="MathML-Content" id="S4.1.p1.11.m6.1b"><ci id="S4.1.p1.11.m6.1.1.cmml" xref="S4.1.p1.11.m6.1.1">𝜉</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.1.p1.11.m6.1c">\xi</annotation><annotation encoding="application/x-llamapun" id="S4.1.p1.11.m6.1d">italic_ξ</annotation></semantics></math>, <math alttext="\xi\in\nu\setminus D_{\nu}" class="ltx_Math" display="inline" id="S4.1.p1.12.m7.1"><semantics id="S4.1.p1.12.m7.1a"><mrow id="S4.1.p1.12.m7.1.1" xref="S4.1.p1.12.m7.1.1.cmml"><mi id="S4.1.p1.12.m7.1.1.2" xref="S4.1.p1.12.m7.1.1.2.cmml">ξ</mi><mo id="S4.1.p1.12.m7.1.1.1" xref="S4.1.p1.12.m7.1.1.1.cmml">∈</mo><mrow id="S4.1.p1.12.m7.1.1.3" xref="S4.1.p1.12.m7.1.1.3.cmml"><mi id="S4.1.p1.12.m7.1.1.3.2" xref="S4.1.p1.12.m7.1.1.3.2.cmml">ν</mi><mo id="S4.1.p1.12.m7.1.1.3.1" xref="S4.1.p1.12.m7.1.1.3.1.cmml">∖</mo><msub id="S4.1.p1.12.m7.1.1.3.3" xref="S4.1.p1.12.m7.1.1.3.3.cmml"><mi id="S4.1.p1.12.m7.1.1.3.3.2" xref="S4.1.p1.12.m7.1.1.3.3.2.cmml">D</mi><mi id="S4.1.p1.12.m7.1.1.3.3.3" xref="S4.1.p1.12.m7.1.1.3.3.3.cmml">ν</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.1.p1.12.m7.1b"><apply id="S4.1.p1.12.m7.1.1.cmml" xref="S4.1.p1.12.m7.1.1"><in id="S4.1.p1.12.m7.1.1.1.cmml" xref="S4.1.p1.12.m7.1.1.1"></in><ci id="S4.1.p1.12.m7.1.1.2.cmml" xref="S4.1.p1.12.m7.1.1.2">𝜉</ci><apply id="S4.1.p1.12.m7.1.1.3.cmml" xref="S4.1.p1.12.m7.1.1.3"><setdiff id="S4.1.p1.12.m7.1.1.3.1.cmml" xref="S4.1.p1.12.m7.1.1.3.1"></setdiff><ci id="S4.1.p1.12.m7.1.1.3.2.cmml" xref="S4.1.p1.12.m7.1.1.3.2">𝜈</ci><apply id="S4.1.p1.12.m7.1.1.3.3.cmml" xref="S4.1.p1.12.m7.1.1.3.3"><csymbol cd="ambiguous" id="S4.1.p1.12.m7.1.1.3.3.1.cmml" xref="S4.1.p1.12.m7.1.1.3.3">subscript</csymbol><ci id="S4.1.p1.12.m7.1.1.3.3.2.cmml" xref="S4.1.p1.12.m7.1.1.3.3.2">𝐷</ci><ci id="S4.1.p1.12.m7.1.1.3.3.3.cmml" xref="S4.1.p1.12.m7.1.1.3.3.3">𝜈</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.1.p1.12.m7.1c">\xi\in\nu\setminus D_{\nu}</annotation><annotation encoding="application/x-llamapun" id="S4.1.p1.12.m7.1d">italic_ξ ∈ italic_ν ∖ italic_D start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT</annotation></semantics></math>. Then <math alttext="f(\xi)<\nu" class="ltx_Math" display="inline" id="S4.1.p1.13.m8.1"><semantics id="S4.1.p1.13.m8.1a"><mrow id="S4.1.p1.13.m8.1.2" xref="S4.1.p1.13.m8.1.2.cmml"><mrow id="S4.1.p1.13.m8.1.2.2" xref="S4.1.p1.13.m8.1.2.2.cmml"><mi id="S4.1.p1.13.m8.1.2.2.2" xref="S4.1.p1.13.m8.1.2.2.2.cmml">f</mi><mo id="S4.1.p1.13.m8.1.2.2.1" xref="S4.1.p1.13.m8.1.2.2.1.cmml">⁢</mo><mrow id="S4.1.p1.13.m8.1.2.2.3.2" xref="S4.1.p1.13.m8.1.2.2.cmml"><mo id="S4.1.p1.13.m8.1.2.2.3.2.1" stretchy="false" xref="S4.1.p1.13.m8.1.2.2.cmml">(</mo><mi id="S4.1.p1.13.m8.1.1" xref="S4.1.p1.13.m8.1.1.cmml">ξ</mi><mo id="S4.1.p1.13.m8.1.2.2.3.2.2" stretchy="false" xref="S4.1.p1.13.m8.1.2.2.cmml">)</mo></mrow></mrow><mo id="S4.1.p1.13.m8.1.2.1" xref="S4.1.p1.13.m8.1.2.1.cmml"><</mo><mi id="S4.1.p1.13.m8.1.2.3" xref="S4.1.p1.13.m8.1.2.3.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.1.p1.13.m8.1b"><apply id="S4.1.p1.13.m8.1.2.cmml" xref="S4.1.p1.13.m8.1.2"><lt id="S4.1.p1.13.m8.1.2.1.cmml" xref="S4.1.p1.13.m8.1.2.1"></lt><apply id="S4.1.p1.13.m8.1.2.2.cmml" xref="S4.1.p1.13.m8.1.2.2"><times id="S4.1.p1.13.m8.1.2.2.1.cmml" xref="S4.1.p1.13.m8.1.2.2.1"></times><ci id="S4.1.p1.13.m8.1.2.2.2.cmml" xref="S4.1.p1.13.m8.1.2.2.2">𝑓</ci><ci id="S4.1.p1.13.m8.1.1.cmml" xref="S4.1.p1.13.m8.1.1">𝜉</ci></apply><ci id="S4.1.p1.13.m8.1.2.3.cmml" xref="S4.1.p1.13.m8.1.2.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.1.p1.13.m8.1c">f(\xi)<\nu</annotation><annotation encoding="application/x-llamapun" id="S4.1.p1.13.m8.1d">italic_f ( italic_ξ ) < italic_ν</annotation></semantics></math>, which implies that for some <math alttext="\alpha<\nu" class="ltx_Math" display="inline" id="S4.1.p1.14.m9.1"><semantics id="S4.1.p1.14.m9.1a"><mrow id="S4.1.p1.14.m9.1.1" xref="S4.1.p1.14.m9.1.1.cmml"><mi id="S4.1.p1.14.m9.1.1.2" xref="S4.1.p1.14.m9.1.1.2.cmml">α</mi><mo id="S4.1.p1.14.m9.1.1.1" xref="S4.1.p1.14.m9.1.1.1.cmml"><</mo><mi id="S4.1.p1.14.m9.1.1.3" xref="S4.1.p1.14.m9.1.1.3.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.1.p1.14.m9.1b"><apply id="S4.1.p1.14.m9.1.1.cmml" xref="S4.1.p1.14.m9.1.1"><lt id="S4.1.p1.14.m9.1.1.1.cmml" xref="S4.1.p1.14.m9.1.1.1"></lt><ci id="S4.1.p1.14.m9.1.1.2.cmml" xref="S4.1.p1.14.m9.1.1.2">𝛼</ci><ci id="S4.1.p1.14.m9.1.1.3.cmml" xref="S4.1.p1.14.m9.1.1.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.1.p1.14.m9.1c">\alpha<\nu</annotation><annotation encoding="application/x-llamapun" id="S4.1.p1.14.m9.1d">italic_α < italic_ν</annotation></semantics></math>, <math alttext="\xi\in D_{\alpha}" class="ltx_Math" display="inline" id="S4.1.p1.15.m10.1"><semantics id="S4.1.p1.15.m10.1a"><mrow id="S4.1.p1.15.m10.1.1" xref="S4.1.p1.15.m10.1.1.cmml"><mi id="S4.1.p1.15.m10.1.1.2" xref="S4.1.p1.15.m10.1.1.2.cmml">ξ</mi><mo id="S4.1.p1.15.m10.1.1.1" xref="S4.1.p1.15.m10.1.1.1.cmml">∈</mo><msub id="S4.1.p1.15.m10.1.1.3" xref="S4.1.p1.15.m10.1.1.3.cmml"><mi id="S4.1.p1.15.m10.1.1.3.2" xref="S4.1.p1.15.m10.1.1.3.2.cmml">D</mi><mi id="S4.1.p1.15.m10.1.1.3.3" xref="S4.1.p1.15.m10.1.1.3.3.cmml">α</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.1.p1.15.m10.1b"><apply id="S4.1.p1.15.m10.1.1.cmml" xref="S4.1.p1.15.m10.1.1"><in id="S4.1.p1.15.m10.1.1.1.cmml" xref="S4.1.p1.15.m10.1.1.1"></in><ci id="S4.1.p1.15.m10.1.1.2.cmml" xref="S4.1.p1.15.m10.1.1.2">𝜉</ci><apply id="S4.1.p1.15.m10.1.1.3.cmml" xref="S4.1.p1.15.m10.1.1.3"><csymbol cd="ambiguous" id="S4.1.p1.15.m10.1.1.3.1.cmml" xref="S4.1.p1.15.m10.1.1.3">subscript</csymbol><ci id="S4.1.p1.15.m10.1.1.3.2.cmml" xref="S4.1.p1.15.m10.1.1.3.2">𝐷</ci><ci id="S4.1.p1.15.m10.1.1.3.3.cmml" xref="S4.1.p1.15.m10.1.1.3.3">𝛼</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.1.p1.15.m10.1c">\xi\in D_{\alpha}</annotation><annotation encoding="application/x-llamapun" id="S4.1.p1.15.m10.1d">italic_ξ ∈ italic_D start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT</annotation></semantics></math>. But then since <math alttext="D" class="ltx_Math" display="inline" id="S4.1.p1.16.m11.1"><semantics id="S4.1.p1.16.m11.1a"><mi id="S4.1.p1.16.m11.1.1" xref="S4.1.p1.16.m11.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S4.1.p1.16.m11.1b"><ci id="S4.1.p1.16.m11.1.1.cmml" xref="S4.1.p1.16.m11.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.1.p1.16.m11.1c">D</annotation><annotation encoding="application/x-llamapun" id="S4.1.p1.16.m11.1d">italic_D</annotation></semantics></math> is increasing, <math alttext="\xi\in D_{\nu}" class="ltx_Math" display="inline" id="S4.1.p1.17.m12.1"><semantics id="S4.1.p1.17.m12.1a"><mrow id="S4.1.p1.17.m12.1.1" xref="S4.1.p1.17.m12.1.1.cmml"><mi id="S4.1.p1.17.m12.1.1.2" xref="S4.1.p1.17.m12.1.1.2.cmml">ξ</mi><mo id="S4.1.p1.17.m12.1.1.1" xref="S4.1.p1.17.m12.1.1.1.cmml">∈</mo><msub id="S4.1.p1.17.m12.1.1.3" xref="S4.1.p1.17.m12.1.1.3.cmml"><mi id="S4.1.p1.17.m12.1.1.3.2" xref="S4.1.p1.17.m12.1.1.3.2.cmml">D</mi><mi id="S4.1.p1.17.m12.1.1.3.3" xref="S4.1.p1.17.m12.1.1.3.3.cmml">ν</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.1.p1.17.m12.1b"><apply id="S4.1.p1.17.m12.1.1.cmml" xref="S4.1.p1.17.m12.1.1"><in id="S4.1.p1.17.m12.1.1.1.cmml" xref="S4.1.p1.17.m12.1.1.1"></in><ci id="S4.1.p1.17.m12.1.1.2.cmml" xref="S4.1.p1.17.m12.1.1.2">𝜉</ci><apply id="S4.1.p1.17.m12.1.1.3.cmml" xref="S4.1.p1.17.m12.1.1.3"><csymbol cd="ambiguous" id="S4.1.p1.17.m12.1.1.3.1.cmml" xref="S4.1.p1.17.m12.1.1.3">subscript</csymbol><ci id="S4.1.p1.17.m12.1.1.3.2.cmml" xref="S4.1.p1.17.m12.1.1.3.2">𝐷</ci><ci id="S4.1.p1.17.m12.1.1.3.3.cmml" xref="S4.1.p1.17.m12.1.1.3.3">𝜈</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.1.p1.17.m12.1c">\xi\in D_{\nu}</annotation><annotation encoding="application/x-llamapun" id="S4.1.p1.17.m12.1d">italic_ξ ∈ italic_D start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT</annotation></semantics></math>, which is a contradiction.</p> </div> <div class="ltx_para" id="S4.2.p2"> <p class="ltx_p" id="S4.2.p2.4">Using an analogous argument for <math alttext="D^{\prime}" class="ltx_Math" display="inline" id="S4.2.p2.1.m1.1"><semantics id="S4.2.p2.1.m1.1a"><msup id="S4.2.p2.1.m1.1.1" xref="S4.2.p2.1.m1.1.1.cmml"><mi id="S4.2.p2.1.m1.1.1.2" xref="S4.2.p2.1.m1.1.1.2.cmml">D</mi><mo id="S4.2.p2.1.m1.1.1.3" xref="S4.2.p2.1.m1.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.2.p2.1.m1.1b"><apply id="S4.2.p2.1.m1.1.1.cmml" xref="S4.2.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S4.2.p2.1.m1.1.1.1.cmml" xref="S4.2.p2.1.m1.1.1">superscript</csymbol><ci id="S4.2.p2.1.m1.1.1.2.cmml" xref="S4.2.p2.1.m1.1.1.2">𝐷</ci><ci id="S4.2.p2.1.m1.1.1.3.cmml" xref="S4.2.p2.1.m1.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.2.p2.1.m1.1c">D^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.2.p2.1.m1.1d">italic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, and intersecting the resulting clubs, we get a club <math alttext="E" class="ltx_Math" display="inline" id="S4.2.p2.2.m2.1"><semantics id="S4.2.p2.2.m2.1a"><mi id="S4.2.p2.2.m2.1.1" xref="S4.2.p2.2.m2.1.1.cmml">E</mi><annotation-xml encoding="MathML-Content" id="S4.2.p2.2.m2.1b"><ci id="S4.2.p2.2.m2.1.1.cmml" xref="S4.2.p2.2.m2.1.1">𝐸</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.2.p2.2.m2.1c">E</annotation><annotation encoding="application/x-llamapun" id="S4.2.p2.2.m2.1d">italic_E</annotation></semantics></math> such that for all <math alttext="\nu\in E" class="ltx_Math" display="inline" id="S4.2.p2.3.m3.1"><semantics id="S4.2.p2.3.m3.1a"><mrow id="S4.2.p2.3.m3.1.1" xref="S4.2.p2.3.m3.1.1.cmml"><mi id="S4.2.p2.3.m3.1.1.2" xref="S4.2.p2.3.m3.1.1.2.cmml">ν</mi><mo id="S4.2.p2.3.m3.1.1.1" xref="S4.2.p2.3.m3.1.1.1.cmml">∈</mo><mi id="S4.2.p2.3.m3.1.1.3" xref="S4.2.p2.3.m3.1.1.3.cmml">E</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.2.p2.3.m3.1b"><apply id="S4.2.p2.3.m3.1.1.cmml" xref="S4.2.p2.3.m3.1.1"><in id="S4.2.p2.3.m3.1.1.1.cmml" xref="S4.2.p2.3.m3.1.1.1"></in><ci id="S4.2.p2.3.m3.1.1.2.cmml" xref="S4.2.p2.3.m3.1.1.2">𝜈</ci><ci id="S4.2.p2.3.m3.1.1.3.cmml" xref="S4.2.p2.3.m3.1.1.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.2.p2.3.m3.1c">\nu\in E</annotation><annotation encoding="application/x-llamapun" id="S4.2.p2.3.m3.1d">italic_ν ∈ italic_E</annotation></semantics></math>, <math alttext="D_{\nu}=\nu=D^{\prime}_{\nu}" class="ltx_Math" display="inline" id="S4.2.p2.4.m4.1"><semantics id="S4.2.p2.4.m4.1a"><mrow id="S4.2.p2.4.m4.1.1" xref="S4.2.p2.4.m4.1.1.cmml"><msub id="S4.2.p2.4.m4.1.1.2" xref="S4.2.p2.4.m4.1.1.2.cmml"><mi id="S4.2.p2.4.m4.1.1.2.2" xref="S4.2.p2.4.m4.1.1.2.2.cmml">D</mi><mi id="S4.2.p2.4.m4.1.1.2.3" xref="S4.2.p2.4.m4.1.1.2.3.cmml">ν</mi></msub><mo id="S4.2.p2.4.m4.1.1.3" xref="S4.2.p2.4.m4.1.1.3.cmml">=</mo><mi id="S4.2.p2.4.m4.1.1.4" xref="S4.2.p2.4.m4.1.1.4.cmml">ν</mi><mo id="S4.2.p2.4.m4.1.1.5" xref="S4.2.p2.4.m4.1.1.5.cmml">=</mo><msubsup id="S4.2.p2.4.m4.1.1.6" xref="S4.2.p2.4.m4.1.1.6.cmml"><mi id="S4.2.p2.4.m4.1.1.6.2.2" xref="S4.2.p2.4.m4.1.1.6.2.2.cmml">D</mi><mi id="S4.2.p2.4.m4.1.1.6.3" xref="S4.2.p2.4.m4.1.1.6.3.cmml">ν</mi><mo id="S4.2.p2.4.m4.1.1.6.2.3" xref="S4.2.p2.4.m4.1.1.6.2.3.cmml">′</mo></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S4.2.p2.4.m4.1b"><apply id="S4.2.p2.4.m4.1.1.cmml" xref="S4.2.p2.4.m4.1.1"><and id="S4.2.p2.4.m4.1.1a.cmml" xref="S4.2.p2.4.m4.1.1"></and><apply id="S4.2.p2.4.m4.1.1b.cmml" xref="S4.2.p2.4.m4.1.1"><eq id="S4.2.p2.4.m4.1.1.3.cmml" xref="S4.2.p2.4.m4.1.1.3"></eq><apply id="S4.2.p2.4.m4.1.1.2.cmml" xref="S4.2.p2.4.m4.1.1.2"><csymbol cd="ambiguous" id="S4.2.p2.4.m4.1.1.2.1.cmml" xref="S4.2.p2.4.m4.1.1.2">subscript</csymbol><ci id="S4.2.p2.4.m4.1.1.2.2.cmml" xref="S4.2.p2.4.m4.1.1.2.2">𝐷</ci><ci id="S4.2.p2.4.m4.1.1.2.3.cmml" xref="S4.2.p2.4.m4.1.1.2.3">𝜈</ci></apply><ci id="S4.2.p2.4.m4.1.1.4.cmml" xref="S4.2.p2.4.m4.1.1.4">𝜈</ci></apply><apply id="S4.2.p2.4.m4.1.1c.cmml" xref="S4.2.p2.4.m4.1.1"><eq id="S4.2.p2.4.m4.1.1.5.cmml" xref="S4.2.p2.4.m4.1.1.5"></eq><share href="https://arxiv.org/html/2503.13728v1#S4.2.p2.4.m4.1.1.4.cmml" id="S4.2.p2.4.m4.1.1d.cmml" xref="S4.2.p2.4.m4.1.1"></share><apply id="S4.2.p2.4.m4.1.1.6.cmml" xref="S4.2.p2.4.m4.1.1.6"><csymbol cd="ambiguous" id="S4.2.p2.4.m4.1.1.6.1.cmml" xref="S4.2.p2.4.m4.1.1.6">subscript</csymbol><apply id="S4.2.p2.4.m4.1.1.6.2.cmml" xref="S4.2.p2.4.m4.1.1.6"><csymbol cd="ambiguous" id="S4.2.p2.4.m4.1.1.6.2.1.cmml" xref="S4.2.p2.4.m4.1.1.6">superscript</csymbol><ci id="S4.2.p2.4.m4.1.1.6.2.2.cmml" xref="S4.2.p2.4.m4.1.1.6.2.2">𝐷</ci><ci id="S4.2.p2.4.m4.1.1.6.2.3.cmml" xref="S4.2.p2.4.m4.1.1.6.2.3">′</ci></apply><ci id="S4.2.p2.4.m4.1.1.6.3.cmml" xref="S4.2.p2.4.m4.1.1.6.3">𝜈</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.2.p2.4.m4.1c">D_{\nu}=\nu=D^{\prime}_{\nu}</annotation><annotation encoding="application/x-llamapun" id="S4.2.p2.4.m4.1d">italic_D start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT = italic_ν = italic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT</annotation></semantics></math>. ∎</p> </div> </div> <div class="ltx_para" id="S4.p2"> <p class="ltx_p" id="S4.p2.13">Let us fix an Aronszajn line <math alttext="A" class="ltx_Math" display="inline" id="S4.p2.1.m1.1"><semantics id="S4.p2.1.m1.1a"><mi id="S4.p2.1.m1.1.1" xref="S4.p2.1.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S4.p2.1.m1.1b"><ci id="S4.p2.1.m1.1.1.cmml" xref="S4.p2.1.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.1.m1.1c">A</annotation><annotation encoding="application/x-llamapun" id="S4.p2.1.m1.1d">italic_A</annotation></semantics></math>. If <math alttext="D=\langle D_{\xi}:\xi<\omega_{1}\rangle" class="ltx_math_unparsed" display="inline" id="S4.p2.2.m2.1"><semantics id="S4.p2.2.m2.1a"><mrow id="S4.p2.2.m2.1b"><mi id="S4.p2.2.m2.1.1">D</mi><mo id="S4.p2.2.m2.1.2">=</mo><mrow id="S4.p2.2.m2.1.3"><mo id="S4.p2.2.m2.1.3.1" stretchy="false">⟨</mo><msub id="S4.p2.2.m2.1.3.2"><mi id="S4.p2.2.m2.1.3.2.2">D</mi><mi id="S4.p2.2.m2.1.3.2.3">ξ</mi></msub><mo id="S4.p2.2.m2.1.3.3" lspace="0.278em" rspace="0.278em">:</mo><mi id="S4.p2.2.m2.1.3.4">ξ</mi><mo id="S4.p2.2.m2.1.3.5"><</mo><msub id="S4.p2.2.m2.1.3.6"><mi id="S4.p2.2.m2.1.3.6.2">ω</mi><mn id="S4.p2.2.m2.1.3.6.3">1</mn></msub><mo id="S4.p2.2.m2.1.3.7" stretchy="false">⟩</mo></mrow></mrow><annotation encoding="application/x-tex" id="S4.p2.2.m2.1c">D=\langle D_{\xi}:\xi<\omega_{1}\rangle</annotation><annotation encoding="application/x-llamapun" id="S4.p2.2.m2.1d">italic_D = ⟨ italic_D start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT : italic_ξ < italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⟩</annotation></semantics></math> is a decomposition for <math alttext="A" class="ltx_Math" display="inline" id="S4.p2.3.m3.1"><semantics id="S4.p2.3.m3.1a"><mi id="S4.p2.3.m3.1.1" xref="S4.p2.3.m3.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S4.p2.3.m3.1b"><ci id="S4.p2.3.m3.1.1.cmml" xref="S4.p2.3.m3.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.3.m3.1c">A</annotation><annotation encoding="application/x-llamapun" id="S4.p2.3.m3.1d">italic_A</annotation></semantics></math>, let <math alttext="\mathscr{L}(A,D)" class="ltx_Math" display="inline" id="S4.p2.4.m4.2"><semantics id="S4.p2.4.m4.2a"><mrow id="S4.p2.4.m4.2.3" xref="S4.p2.4.m4.2.3.cmml"><mi class="ltx_font_mathscript" id="S4.p2.4.m4.2.3.2" xref="S4.p2.4.m4.2.3.2.cmml">ℒ</mi><mo id="S4.p2.4.m4.2.3.1" xref="S4.p2.4.m4.2.3.1.cmml">⁢</mo><mrow id="S4.p2.4.m4.2.3.3.2" xref="S4.p2.4.m4.2.3.3.1.cmml"><mo id="S4.p2.4.m4.2.3.3.2.1" stretchy="false" xref="S4.p2.4.m4.2.3.3.1.cmml">(</mo><mi id="S4.p2.4.m4.1.1" xref="S4.p2.4.m4.1.1.cmml">A</mi><mo id="S4.p2.4.m4.2.3.3.2.2" xref="S4.p2.4.m4.2.3.3.1.cmml">,</mo><mi id="S4.p2.4.m4.2.2" xref="S4.p2.4.m4.2.2.cmml">D</mi><mo id="S4.p2.4.m4.2.3.3.2.3" stretchy="false" xref="S4.p2.4.m4.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.4.m4.2b"><apply id="S4.p2.4.m4.2.3.cmml" xref="S4.p2.4.m4.2.3"><times id="S4.p2.4.m4.2.3.1.cmml" xref="S4.p2.4.m4.2.3.1"></times><ci id="S4.p2.4.m4.2.3.2.cmml" xref="S4.p2.4.m4.2.3.2">ℒ</ci><interval closure="open" id="S4.p2.4.m4.2.3.3.1.cmml" xref="S4.p2.4.m4.2.3.3.2"><ci id="S4.p2.4.m4.1.1.cmml" xref="S4.p2.4.m4.1.1">𝐴</ci><ci id="S4.p2.4.m4.2.2.cmml" xref="S4.p2.4.m4.2.2">𝐷</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.4.m4.2c">\mathscr{L}(A,D)</annotation><annotation encoding="application/x-llamapun" id="S4.p2.4.m4.2d">script_L ( italic_A , italic_D )</annotation></semantics></math> denote the <math alttext="\xi<\omega_{1}" class="ltx_Math" display="inline" id="S4.p2.5.m5.1"><semantics id="S4.p2.5.m5.1a"><mrow id="S4.p2.5.m5.1.1" xref="S4.p2.5.m5.1.1.cmml"><mi id="S4.p2.5.m5.1.1.2" xref="S4.p2.5.m5.1.1.2.cmml">ξ</mi><mo id="S4.p2.5.m5.1.1.1" xref="S4.p2.5.m5.1.1.1.cmml"><</mo><msub id="S4.p2.5.m5.1.1.3" xref="S4.p2.5.m5.1.1.3.cmml"><mi id="S4.p2.5.m5.1.1.3.2" xref="S4.p2.5.m5.1.1.3.2.cmml">ω</mi><mn id="S4.p2.5.m5.1.1.3.3" xref="S4.p2.5.m5.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.5.m5.1b"><apply id="S4.p2.5.m5.1.1.cmml" xref="S4.p2.5.m5.1.1"><lt id="S4.p2.5.m5.1.1.1.cmml" xref="S4.p2.5.m5.1.1.1"></lt><ci id="S4.p2.5.m5.1.1.2.cmml" xref="S4.p2.5.m5.1.1.2">𝜉</ci><apply id="S4.p2.5.m5.1.1.3.cmml" xref="S4.p2.5.m5.1.1.3"><csymbol cd="ambiguous" id="S4.p2.5.m5.1.1.3.1.cmml" xref="S4.p2.5.m5.1.1.3">subscript</csymbol><ci id="S4.p2.5.m5.1.1.3.2.cmml" xref="S4.p2.5.m5.1.1.3.2">𝜔</ci><cn id="S4.p2.5.m5.1.1.3.3.cmml" type="integer" xref="S4.p2.5.m5.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.5.m5.1c">\xi<\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.5.m5.1d">italic_ξ < italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> such that some complementary interval of <math alttext="A\setminus D_{\xi}" class="ltx_Math" display="inline" id="S4.p2.6.m6.1"><semantics id="S4.p2.6.m6.1a"><mrow id="S4.p2.6.m6.1.1" xref="S4.p2.6.m6.1.1.cmml"><mi id="S4.p2.6.m6.1.1.2" xref="S4.p2.6.m6.1.1.2.cmml">A</mi><mo id="S4.p2.6.m6.1.1.1" xref="S4.p2.6.m6.1.1.1.cmml">∖</mo><msub id="S4.p2.6.m6.1.1.3" xref="S4.p2.6.m6.1.1.3.cmml"><mi id="S4.p2.6.m6.1.1.3.2" xref="S4.p2.6.m6.1.1.3.2.cmml">D</mi><mi id="S4.p2.6.m6.1.1.3.3" xref="S4.p2.6.m6.1.1.3.3.cmml">ξ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.6.m6.1b"><apply id="S4.p2.6.m6.1.1.cmml" xref="S4.p2.6.m6.1.1"><setdiff id="S4.p2.6.m6.1.1.1.cmml" xref="S4.p2.6.m6.1.1.1"></setdiff><ci id="S4.p2.6.m6.1.1.2.cmml" xref="S4.p2.6.m6.1.1.2">𝐴</ci><apply id="S4.p2.6.m6.1.1.3.cmml" xref="S4.p2.6.m6.1.1.3"><csymbol cd="ambiguous" id="S4.p2.6.m6.1.1.3.1.cmml" xref="S4.p2.6.m6.1.1.3">subscript</csymbol><ci id="S4.p2.6.m6.1.1.3.2.cmml" xref="S4.p2.6.m6.1.1.3.2">𝐷</ci><ci id="S4.p2.6.m6.1.1.3.3.cmml" xref="S4.p2.6.m6.1.1.3.3">𝜉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.6.m6.1c">A\setminus D_{\xi}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.6.m6.1d">italic_A ∖ italic_D start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT</annotation></semantics></math> has a left endpoint. And let <math alttext="\hat{\mathscr{L}}(A,D)" class="ltx_Math" display="inline" id="S4.p2.7.m7.2"><semantics id="S4.p2.7.m7.2a"><mrow id="S4.p2.7.m7.2.3" xref="S4.p2.7.m7.2.3.cmml"><mover accent="true" id="S4.p2.7.m7.2.3.2" xref="S4.p2.7.m7.2.3.2.cmml"><mi class="ltx_font_mathscript" id="S4.p2.7.m7.2.3.2.2" xref="S4.p2.7.m7.2.3.2.2.cmml">ℒ</mi><mo id="S4.p2.7.m7.2.3.2.1" xref="S4.p2.7.m7.2.3.2.1.cmml">^</mo></mover><mo id="S4.p2.7.m7.2.3.1" xref="S4.p2.7.m7.2.3.1.cmml">⁢</mo><mrow id="S4.p2.7.m7.2.3.3.2" xref="S4.p2.7.m7.2.3.3.1.cmml"><mo id="S4.p2.7.m7.2.3.3.2.1" stretchy="false" xref="S4.p2.7.m7.2.3.3.1.cmml">(</mo><mi id="S4.p2.7.m7.1.1" xref="S4.p2.7.m7.1.1.cmml">A</mi><mo id="S4.p2.7.m7.2.3.3.2.2" xref="S4.p2.7.m7.2.3.3.1.cmml">,</mo><mi id="S4.p2.7.m7.2.2" xref="S4.p2.7.m7.2.2.cmml">D</mi><mo id="S4.p2.7.m7.2.3.3.2.3" stretchy="false" xref="S4.p2.7.m7.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.7.m7.2b"><apply id="S4.p2.7.m7.2.3.cmml" xref="S4.p2.7.m7.2.3"><times id="S4.p2.7.m7.2.3.1.cmml" xref="S4.p2.7.m7.2.3.1"></times><apply id="S4.p2.7.m7.2.3.2.cmml" xref="S4.p2.7.m7.2.3.2"><ci id="S4.p2.7.m7.2.3.2.1.cmml" xref="S4.p2.7.m7.2.3.2.1">^</ci><ci id="S4.p2.7.m7.2.3.2.2.cmml" xref="S4.p2.7.m7.2.3.2.2">ℒ</ci></apply><interval closure="open" id="S4.p2.7.m7.2.3.3.1.cmml" xref="S4.p2.7.m7.2.3.3.2"><ci id="S4.p2.7.m7.1.1.cmml" xref="S4.p2.7.m7.1.1">𝐴</ci><ci id="S4.p2.7.m7.2.2.cmml" xref="S4.p2.7.m7.2.2">𝐷</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.7.m7.2c">\hat{\mathscr{L}}(A,D)</annotation><annotation encoding="application/x-llamapun" id="S4.p2.7.m7.2d">over^ start_ARG script_L end_ARG ( italic_A , italic_D )</annotation></semantics></math> denote the set of <math alttext="\xi<\omega_{1}" class="ltx_Math" display="inline" id="S4.p2.8.m8.1"><semantics id="S4.p2.8.m8.1a"><mrow id="S4.p2.8.m8.1.1" xref="S4.p2.8.m8.1.1.cmml"><mi id="S4.p2.8.m8.1.1.2" xref="S4.p2.8.m8.1.1.2.cmml">ξ</mi><mo id="S4.p2.8.m8.1.1.1" xref="S4.p2.8.m8.1.1.1.cmml"><</mo><msub id="S4.p2.8.m8.1.1.3" xref="S4.p2.8.m8.1.1.3.cmml"><mi id="S4.p2.8.m8.1.1.3.2" xref="S4.p2.8.m8.1.1.3.2.cmml">ω</mi><mn id="S4.p2.8.m8.1.1.3.3" xref="S4.p2.8.m8.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.8.m8.1b"><apply id="S4.p2.8.m8.1.1.cmml" xref="S4.p2.8.m8.1.1"><lt id="S4.p2.8.m8.1.1.1.cmml" xref="S4.p2.8.m8.1.1.1"></lt><ci id="S4.p2.8.m8.1.1.2.cmml" xref="S4.p2.8.m8.1.1.2">𝜉</ci><apply id="S4.p2.8.m8.1.1.3.cmml" xref="S4.p2.8.m8.1.1.3"><csymbol cd="ambiguous" id="S4.p2.8.m8.1.1.3.1.cmml" xref="S4.p2.8.m8.1.1.3">subscript</csymbol><ci id="S4.p2.8.m8.1.1.3.2.cmml" xref="S4.p2.8.m8.1.1.3.2">𝜔</ci><cn id="S4.p2.8.m8.1.1.3.3.cmml" type="integer" xref="S4.p2.8.m8.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.8.m8.1c">\xi<\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.8.m8.1d">italic_ξ < italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> such that every complementary interval of <math alttext="A\setminus D_{\xi}" class="ltx_Math" display="inline" id="S4.p2.9.m9.1"><semantics id="S4.p2.9.m9.1a"><mrow id="S4.p2.9.m9.1.1" xref="S4.p2.9.m9.1.1.cmml"><mi id="S4.p2.9.m9.1.1.2" xref="S4.p2.9.m9.1.1.2.cmml">A</mi><mo id="S4.p2.9.m9.1.1.1" xref="S4.p2.9.m9.1.1.1.cmml">∖</mo><msub id="S4.p2.9.m9.1.1.3" xref="S4.p2.9.m9.1.1.3.cmml"><mi id="S4.p2.9.m9.1.1.3.2" xref="S4.p2.9.m9.1.1.3.2.cmml">D</mi><mi id="S4.p2.9.m9.1.1.3.3" xref="S4.p2.9.m9.1.1.3.3.cmml">ξ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.9.m9.1b"><apply id="S4.p2.9.m9.1.1.cmml" xref="S4.p2.9.m9.1.1"><setdiff id="S4.p2.9.m9.1.1.1.cmml" xref="S4.p2.9.m9.1.1.1"></setdiff><ci id="S4.p2.9.m9.1.1.2.cmml" xref="S4.p2.9.m9.1.1.2">𝐴</ci><apply id="S4.p2.9.m9.1.1.3.cmml" xref="S4.p2.9.m9.1.1.3"><csymbol cd="ambiguous" id="S4.p2.9.m9.1.1.3.1.cmml" xref="S4.p2.9.m9.1.1.3">subscript</csymbol><ci id="S4.p2.9.m9.1.1.3.2.cmml" xref="S4.p2.9.m9.1.1.3.2">𝐷</ci><ci id="S4.p2.9.m9.1.1.3.3.cmml" xref="S4.p2.9.m9.1.1.3.3">𝜉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.9.m9.1c">A\setminus D_{\xi}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.9.m9.1d">italic_A ∖ italic_D start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT</annotation></semantics></math> has a left endpoint. Let <math alttext="\mathscr{R}(A,D)" class="ltx_Math" display="inline" id="S4.p2.10.m10.2"><semantics id="S4.p2.10.m10.2a"><mrow id="S4.p2.10.m10.2.3" xref="S4.p2.10.m10.2.3.cmml"><mi class="ltx_font_mathscript" id="S4.p2.10.m10.2.3.2" xref="S4.p2.10.m10.2.3.2.cmml">ℛ</mi><mo id="S4.p2.10.m10.2.3.1" xref="S4.p2.10.m10.2.3.1.cmml">⁢</mo><mrow id="S4.p2.10.m10.2.3.3.2" xref="S4.p2.10.m10.2.3.3.1.cmml"><mo id="S4.p2.10.m10.2.3.3.2.1" stretchy="false" xref="S4.p2.10.m10.2.3.3.1.cmml">(</mo><mi id="S4.p2.10.m10.1.1" xref="S4.p2.10.m10.1.1.cmml">A</mi><mo id="S4.p2.10.m10.2.3.3.2.2" xref="S4.p2.10.m10.2.3.3.1.cmml">,</mo><mi id="S4.p2.10.m10.2.2" xref="S4.p2.10.m10.2.2.cmml">D</mi><mo id="S4.p2.10.m10.2.3.3.2.3" stretchy="false" xref="S4.p2.10.m10.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.10.m10.2b"><apply id="S4.p2.10.m10.2.3.cmml" xref="S4.p2.10.m10.2.3"><times id="S4.p2.10.m10.2.3.1.cmml" xref="S4.p2.10.m10.2.3.1"></times><ci id="S4.p2.10.m10.2.3.2.cmml" xref="S4.p2.10.m10.2.3.2">ℛ</ci><interval closure="open" id="S4.p2.10.m10.2.3.3.1.cmml" xref="S4.p2.10.m10.2.3.3.2"><ci id="S4.p2.10.m10.1.1.cmml" xref="S4.p2.10.m10.1.1">𝐴</ci><ci id="S4.p2.10.m10.2.2.cmml" xref="S4.p2.10.m10.2.2">𝐷</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.10.m10.2c">\mathscr{R}(A,D)</annotation><annotation encoding="application/x-llamapun" id="S4.p2.10.m10.2d">script_R ( italic_A , italic_D )</annotation></semantics></math> and <math alttext="\hat{\mathscr{R}}(A,D)" class="ltx_Math" display="inline" id="S4.p2.11.m11.2"><semantics id="S4.p2.11.m11.2a"><mrow id="S4.p2.11.m11.2.3" xref="S4.p2.11.m11.2.3.cmml"><mover accent="true" id="S4.p2.11.m11.2.3.2" xref="S4.p2.11.m11.2.3.2.cmml"><mi class="ltx_font_mathscript" id="S4.p2.11.m11.2.3.2.2" xref="S4.p2.11.m11.2.3.2.2.cmml">ℛ</mi><mo id="S4.p2.11.m11.2.3.2.1" xref="S4.p2.11.m11.2.3.2.1.cmml">^</mo></mover><mo id="S4.p2.11.m11.2.3.1" xref="S4.p2.11.m11.2.3.1.cmml">⁢</mo><mrow id="S4.p2.11.m11.2.3.3.2" xref="S4.p2.11.m11.2.3.3.1.cmml"><mo id="S4.p2.11.m11.2.3.3.2.1" stretchy="false" xref="S4.p2.11.m11.2.3.3.1.cmml">(</mo><mi id="S4.p2.11.m11.1.1" xref="S4.p2.11.m11.1.1.cmml">A</mi><mo id="S4.p2.11.m11.2.3.3.2.2" xref="S4.p2.11.m11.2.3.3.1.cmml">,</mo><mi id="S4.p2.11.m11.2.2" xref="S4.p2.11.m11.2.2.cmml">D</mi><mo id="S4.p2.11.m11.2.3.3.2.3" stretchy="false" xref="S4.p2.11.m11.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.11.m11.2b"><apply id="S4.p2.11.m11.2.3.cmml" xref="S4.p2.11.m11.2.3"><times id="S4.p2.11.m11.2.3.1.cmml" xref="S4.p2.11.m11.2.3.1"></times><apply id="S4.p2.11.m11.2.3.2.cmml" xref="S4.p2.11.m11.2.3.2"><ci id="S4.p2.11.m11.2.3.2.1.cmml" xref="S4.p2.11.m11.2.3.2.1">^</ci><ci id="S4.p2.11.m11.2.3.2.2.cmml" xref="S4.p2.11.m11.2.3.2.2">ℛ</ci></apply><interval closure="open" id="S4.p2.11.m11.2.3.3.1.cmml" xref="S4.p2.11.m11.2.3.3.2"><ci id="S4.p2.11.m11.1.1.cmml" xref="S4.p2.11.m11.1.1">𝐴</ci><ci id="S4.p2.11.m11.2.2.cmml" xref="S4.p2.11.m11.2.2">𝐷</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.11.m11.2c">\hat{\mathscr{R}}(A,D)</annotation><annotation encoding="application/x-llamapun" id="S4.p2.11.m11.2d">over^ start_ARG script_R end_ARG ( italic_A , italic_D )</annotation></semantics></math> be analogous replacing left by right. Recall <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S2.Thmtheorem4" title="Definition 2.4. ‣ 2. Aronszajn and Countryman lines ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Definition</span> <span class="ltx_text ltx_ref_tag">2.4</span></a>. With this notation one easily sees that the definition of non stationary in Aronszajn lines, says exactly that for some decomposition <math alttext="D" class="ltx_Math" display="inline" id="S4.p2.12.m12.1"><semantics id="S4.p2.12.m12.1a"><mi id="S4.p2.12.m12.1.1" xref="S4.p2.12.m12.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S4.p2.12.m12.1b"><ci id="S4.p2.12.m12.1.1.cmml" xref="S4.p2.12.m12.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.12.m12.1c">D</annotation><annotation encoding="application/x-llamapun" id="S4.p2.12.m12.1d">italic_D</annotation></semantics></math>, <math alttext="\mathscr{L}(A,D)=\varnothing=\mathscr{R}(A,D)" class="ltx_Math" display="inline" id="S4.p2.13.m13.4"><semantics id="S4.p2.13.m13.4a"><mrow id="S4.p2.13.m13.4.5" xref="S4.p2.13.m13.4.5.cmml"><mrow id="S4.p2.13.m13.4.5.2" xref="S4.p2.13.m13.4.5.2.cmml"><mi class="ltx_font_mathscript" id="S4.p2.13.m13.4.5.2.2" xref="S4.p2.13.m13.4.5.2.2.cmml">ℒ</mi><mo id="S4.p2.13.m13.4.5.2.1" xref="S4.p2.13.m13.4.5.2.1.cmml">⁢</mo><mrow id="S4.p2.13.m13.4.5.2.3.2" xref="S4.p2.13.m13.4.5.2.3.1.cmml"><mo id="S4.p2.13.m13.4.5.2.3.2.1" stretchy="false" xref="S4.p2.13.m13.4.5.2.3.1.cmml">(</mo><mi id="S4.p2.13.m13.1.1" xref="S4.p2.13.m13.1.1.cmml">A</mi><mo id="S4.p2.13.m13.4.5.2.3.2.2" xref="S4.p2.13.m13.4.5.2.3.1.cmml">,</mo><mi id="S4.p2.13.m13.2.2" xref="S4.p2.13.m13.2.2.cmml">D</mi><mo id="S4.p2.13.m13.4.5.2.3.2.3" stretchy="false" xref="S4.p2.13.m13.4.5.2.3.1.cmml">)</mo></mrow></mrow><mo id="S4.p2.13.m13.4.5.3" xref="S4.p2.13.m13.4.5.3.cmml">=</mo><mi id="S4.p2.13.m13.4.5.4" mathvariant="normal" xref="S4.p2.13.m13.4.5.4.cmml">∅</mi><mo id="S4.p2.13.m13.4.5.5" xref="S4.p2.13.m13.4.5.5.cmml">=</mo><mrow id="S4.p2.13.m13.4.5.6" xref="S4.p2.13.m13.4.5.6.cmml"><mi class="ltx_font_mathscript" id="S4.p2.13.m13.4.5.6.2" xref="S4.p2.13.m13.4.5.6.2.cmml">ℛ</mi><mo id="S4.p2.13.m13.4.5.6.1" xref="S4.p2.13.m13.4.5.6.1.cmml">⁢</mo><mrow id="S4.p2.13.m13.4.5.6.3.2" xref="S4.p2.13.m13.4.5.6.3.1.cmml"><mo id="S4.p2.13.m13.4.5.6.3.2.1" stretchy="false" xref="S4.p2.13.m13.4.5.6.3.1.cmml">(</mo><mi id="S4.p2.13.m13.3.3" xref="S4.p2.13.m13.3.3.cmml">A</mi><mo id="S4.p2.13.m13.4.5.6.3.2.2" xref="S4.p2.13.m13.4.5.6.3.1.cmml">,</mo><mi id="S4.p2.13.m13.4.4" xref="S4.p2.13.m13.4.4.cmml">D</mi><mo id="S4.p2.13.m13.4.5.6.3.2.3" stretchy="false" xref="S4.p2.13.m13.4.5.6.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.13.m13.4b"><apply id="S4.p2.13.m13.4.5.cmml" xref="S4.p2.13.m13.4.5"><and id="S4.p2.13.m13.4.5a.cmml" xref="S4.p2.13.m13.4.5"></and><apply id="S4.p2.13.m13.4.5b.cmml" xref="S4.p2.13.m13.4.5"><eq id="S4.p2.13.m13.4.5.3.cmml" xref="S4.p2.13.m13.4.5.3"></eq><apply id="S4.p2.13.m13.4.5.2.cmml" xref="S4.p2.13.m13.4.5.2"><times id="S4.p2.13.m13.4.5.2.1.cmml" xref="S4.p2.13.m13.4.5.2.1"></times><ci id="S4.p2.13.m13.4.5.2.2.cmml" xref="S4.p2.13.m13.4.5.2.2">ℒ</ci><interval closure="open" id="S4.p2.13.m13.4.5.2.3.1.cmml" xref="S4.p2.13.m13.4.5.2.3.2"><ci id="S4.p2.13.m13.1.1.cmml" xref="S4.p2.13.m13.1.1">𝐴</ci><ci id="S4.p2.13.m13.2.2.cmml" xref="S4.p2.13.m13.2.2">𝐷</ci></interval></apply><emptyset id="S4.p2.13.m13.4.5.4.cmml" xref="S4.p2.13.m13.4.5.4"></emptyset></apply><apply id="S4.p2.13.m13.4.5c.cmml" xref="S4.p2.13.m13.4.5"><eq id="S4.p2.13.m13.4.5.5.cmml" xref="S4.p2.13.m13.4.5.5"></eq><share href="https://arxiv.org/html/2503.13728v1#S4.p2.13.m13.4.5.4.cmml" id="S4.p2.13.m13.4.5d.cmml" xref="S4.p2.13.m13.4.5"></share><apply id="S4.p2.13.m13.4.5.6.cmml" xref="S4.p2.13.m13.4.5.6"><times id="S4.p2.13.m13.4.5.6.1.cmml" xref="S4.p2.13.m13.4.5.6.1"></times><ci id="S4.p2.13.m13.4.5.6.2.cmml" xref="S4.p2.13.m13.4.5.6.2">ℛ</ci><interval closure="open" id="S4.p2.13.m13.4.5.6.3.1.cmml" xref="S4.p2.13.m13.4.5.6.3.2"><ci id="S4.p2.13.m13.3.3.cmml" xref="S4.p2.13.m13.3.3">𝐴</ci><ci id="S4.p2.13.m13.4.4.cmml" xref="S4.p2.13.m13.4.4">𝐷</ci></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.13.m13.4c">\mathscr{L}(A,D)=\varnothing=\mathscr{R}(A,D)</annotation><annotation encoding="application/x-llamapun" id="S4.p2.13.m13.4d">script_L ( italic_A , italic_D ) = ∅ = script_R ( italic_A , italic_D )</annotation></semantics></math>. This, together with <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S4.Thmtheorem1" title="Lemma 4.1. ‣ 4. Aronszajn line decompositions ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">4.1</span></a>, easily implies the following, which also explains the choice of naming in that definition.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="S4.Thmtheorem2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem2.1.1.1">Proposition 4.2</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem2.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem2.p1"> <p class="ltx_p" id="S4.Thmtheorem2.p1.5"><math alttext="A" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.1.m1.1"><semantics id="S4.Thmtheorem2.p1.1.m1.1a"><mi id="S4.Thmtheorem2.p1.1.m1.1.1" xref="S4.Thmtheorem2.p1.1.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.1.m1.1b"><ci id="S4.Thmtheorem2.p1.1.m1.1.1.cmml" xref="S4.Thmtheorem2.p1.1.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.1.m1.1c">A</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.1.m1.1d">italic_A</annotation></semantics></math> is non stationary iff for every decomposition <math alttext="D" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.2.m2.1"><semantics id="S4.Thmtheorem2.p1.2.m2.1a"><mi id="S4.Thmtheorem2.p1.2.m2.1.1" xref="S4.Thmtheorem2.p1.2.m2.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.2.m2.1b"><ci id="S4.Thmtheorem2.p1.2.m2.1.1.cmml" xref="S4.Thmtheorem2.p1.2.m2.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.2.m2.1c">D</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.2.m2.1d">italic_D</annotation></semantics></math>, <math alttext="\mathscr{L}(A,D)" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.3.m3.2"><semantics id="S4.Thmtheorem2.p1.3.m3.2a"><mrow id="S4.Thmtheorem2.p1.3.m3.2.3" xref="S4.Thmtheorem2.p1.3.m3.2.3.cmml"><mi class="ltx_font_mathscript" id="S4.Thmtheorem2.p1.3.m3.2.3.2" xref="S4.Thmtheorem2.p1.3.m3.2.3.2.cmml">ℒ</mi><mo id="S4.Thmtheorem2.p1.3.m3.2.3.1" xref="S4.Thmtheorem2.p1.3.m3.2.3.1.cmml">⁢</mo><mrow id="S4.Thmtheorem2.p1.3.m3.2.3.3.2" xref="S4.Thmtheorem2.p1.3.m3.2.3.3.1.cmml"><mo id="S4.Thmtheorem2.p1.3.m3.2.3.3.2.1" stretchy="false" xref="S4.Thmtheorem2.p1.3.m3.2.3.3.1.cmml">(</mo><mi id="S4.Thmtheorem2.p1.3.m3.1.1" xref="S4.Thmtheorem2.p1.3.m3.1.1.cmml">A</mi><mo id="S4.Thmtheorem2.p1.3.m3.2.3.3.2.2" xref="S4.Thmtheorem2.p1.3.m3.2.3.3.1.cmml">,</mo><mi id="S4.Thmtheorem2.p1.3.m3.2.2" xref="S4.Thmtheorem2.p1.3.m3.2.2.cmml">D</mi><mo id="S4.Thmtheorem2.p1.3.m3.2.3.3.2.3" stretchy="false" xref="S4.Thmtheorem2.p1.3.m3.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.3.m3.2b"><apply id="S4.Thmtheorem2.p1.3.m3.2.3.cmml" xref="S4.Thmtheorem2.p1.3.m3.2.3"><times id="S4.Thmtheorem2.p1.3.m3.2.3.1.cmml" xref="S4.Thmtheorem2.p1.3.m3.2.3.1"></times><ci id="S4.Thmtheorem2.p1.3.m3.2.3.2.cmml" xref="S4.Thmtheorem2.p1.3.m3.2.3.2">ℒ</ci><interval closure="open" id="S4.Thmtheorem2.p1.3.m3.2.3.3.1.cmml" xref="S4.Thmtheorem2.p1.3.m3.2.3.3.2"><ci id="S4.Thmtheorem2.p1.3.m3.1.1.cmml" xref="S4.Thmtheorem2.p1.3.m3.1.1">𝐴</ci><ci id="S4.Thmtheorem2.p1.3.m3.2.2.cmml" xref="S4.Thmtheorem2.p1.3.m3.2.2">𝐷</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.3.m3.2c">\mathscr{L}(A,D)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.3.m3.2d">script_L ( italic_A , italic_D )</annotation></semantics></math> and <math alttext="\mathscr{R}(A,D)" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.4.m4.2"><semantics id="S4.Thmtheorem2.p1.4.m4.2a"><mrow id="S4.Thmtheorem2.p1.4.m4.2.3" xref="S4.Thmtheorem2.p1.4.m4.2.3.cmml"><mi class="ltx_font_mathscript" id="S4.Thmtheorem2.p1.4.m4.2.3.2" xref="S4.Thmtheorem2.p1.4.m4.2.3.2.cmml">ℛ</mi><mo id="S4.Thmtheorem2.p1.4.m4.2.3.1" xref="S4.Thmtheorem2.p1.4.m4.2.3.1.cmml">⁢</mo><mrow id="S4.Thmtheorem2.p1.4.m4.2.3.3.2" xref="S4.Thmtheorem2.p1.4.m4.2.3.3.1.cmml"><mo id="S4.Thmtheorem2.p1.4.m4.2.3.3.2.1" stretchy="false" xref="S4.Thmtheorem2.p1.4.m4.2.3.3.1.cmml">(</mo><mi id="S4.Thmtheorem2.p1.4.m4.1.1" xref="S4.Thmtheorem2.p1.4.m4.1.1.cmml">A</mi><mo id="S4.Thmtheorem2.p1.4.m4.2.3.3.2.2" xref="S4.Thmtheorem2.p1.4.m4.2.3.3.1.cmml">,</mo><mi id="S4.Thmtheorem2.p1.4.m4.2.2" xref="S4.Thmtheorem2.p1.4.m4.2.2.cmml">D</mi><mo id="S4.Thmtheorem2.p1.4.m4.2.3.3.2.3" stretchy="false" xref="S4.Thmtheorem2.p1.4.m4.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.4.m4.2b"><apply id="S4.Thmtheorem2.p1.4.m4.2.3.cmml" xref="S4.Thmtheorem2.p1.4.m4.2.3"><times id="S4.Thmtheorem2.p1.4.m4.2.3.1.cmml" xref="S4.Thmtheorem2.p1.4.m4.2.3.1"></times><ci id="S4.Thmtheorem2.p1.4.m4.2.3.2.cmml" xref="S4.Thmtheorem2.p1.4.m4.2.3.2">ℛ</ci><interval closure="open" id="S4.Thmtheorem2.p1.4.m4.2.3.3.1.cmml" xref="S4.Thmtheorem2.p1.4.m4.2.3.3.2"><ci id="S4.Thmtheorem2.p1.4.m4.1.1.cmml" xref="S4.Thmtheorem2.p1.4.m4.1.1">𝐴</ci><ci id="S4.Thmtheorem2.p1.4.m4.2.2.cmml" xref="S4.Thmtheorem2.p1.4.m4.2.2">𝐷</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.4.m4.2c">\mathscr{R}(A,D)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.4.m4.2d">script_R ( italic_A , italic_D )</annotation></semantics></math> are non stationary subsets of <math alttext="\omega_{1}" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.5.m5.1"><semantics id="S4.Thmtheorem2.p1.5.m5.1a"><msub id="S4.Thmtheorem2.p1.5.m5.1.1" xref="S4.Thmtheorem2.p1.5.m5.1.1.cmml"><mi id="S4.Thmtheorem2.p1.5.m5.1.1.2" xref="S4.Thmtheorem2.p1.5.m5.1.1.2.cmml">ω</mi><mn id="S4.Thmtheorem2.p1.5.m5.1.1.3" xref="S4.Thmtheorem2.p1.5.m5.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.5.m5.1b"><apply id="S4.Thmtheorem2.p1.5.m5.1.1.cmml" xref="S4.Thmtheorem2.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.5.m5.1.1.1.cmml" xref="S4.Thmtheorem2.p1.5.m5.1.1">subscript</csymbol><ci id="S4.Thmtheorem2.p1.5.m5.1.1.2.cmml" xref="S4.Thmtheorem2.p1.5.m5.1.1.2">𝜔</ci><cn id="S4.Thmtheorem2.p1.5.m5.1.1.3.cmml" type="integer" xref="S4.Thmtheorem2.p1.5.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.5.m5.1c">\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.5.m5.1d">italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> </div> <div class="ltx_para" id="S4.p3"> <p class="ltx_p" id="S4.p3.2">We dedicate the rest of this section to construct Aronszajn lines with specific configurations of <math alttext="\mathscr{L}" class="ltx_Math" display="inline" id="S4.p3.1.m1.1"><semantics id="S4.p3.1.m1.1a"><mi class="ltx_font_mathscript" id="S4.p3.1.m1.1.1" xref="S4.p3.1.m1.1.1.cmml">ℒ</mi><annotation-xml encoding="MathML-Content" id="S4.p3.1.m1.1b"><ci id="S4.p3.1.m1.1.1.cmml" xref="S4.p3.1.m1.1.1">ℒ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.1.m1.1c">\mathscr{L}</annotation><annotation encoding="application/x-llamapun" id="S4.p3.1.m1.1d">script_L</annotation></semantics></math> and <math alttext="\mathscr{R}" class="ltx_Math" display="inline" id="S4.p3.2.m2.1"><semantics id="S4.p3.2.m2.1a"><mi class="ltx_font_mathscript" id="S4.p3.2.m2.1.1" xref="S4.p3.2.m2.1.1.cmml">ℛ</mi><annotation-xml encoding="MathML-Content" id="S4.p3.2.m2.1b"><ci id="S4.p3.2.m2.1.1.cmml" xref="S4.p3.2.m2.1.1">ℛ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.2.m2.1c">\mathscr{R}</annotation><annotation encoding="application/x-llamapun" id="S4.p3.2.m2.1d">script_R</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.p4"> <p class="ltx_p" id="S4.p4.10">Recall that for functions with the same domain, <math alttext="f=^{*}g" class="ltx_Math" display="inline" id="S4.p4.1.m1.1"><semantics id="S4.p4.1.m1.1a"><mrow id="S4.p4.1.m1.1.1" xref="S4.p4.1.m1.1.1.cmml"><mi id="S4.p4.1.m1.1.1.2" xref="S4.p4.1.m1.1.1.2.cmml">f</mi><msup id="S4.p4.1.m1.1.1.1" xref="S4.p4.1.m1.1.1.1.cmml"><mo id="S4.p4.1.m1.1.1.1.2" xref="S4.p4.1.m1.1.1.1.2.cmml">=</mo><mo id="S4.p4.1.m1.1.1.1.3" xref="S4.p4.1.m1.1.1.1.3.cmml">∗</mo></msup><mi id="S4.p4.1.m1.1.1.3" xref="S4.p4.1.m1.1.1.3.cmml">g</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.p4.1.m1.1b"><apply id="S4.p4.1.m1.1.1.cmml" xref="S4.p4.1.m1.1.1"><apply id="S4.p4.1.m1.1.1.1.cmml" xref="S4.p4.1.m1.1.1.1"><csymbol cd="ambiguous" id="S4.p4.1.m1.1.1.1.1.cmml" xref="S4.p4.1.m1.1.1.1">superscript</csymbol><eq id="S4.p4.1.m1.1.1.1.2.cmml" xref="S4.p4.1.m1.1.1.1.2"></eq><times id="S4.p4.1.m1.1.1.1.3.cmml" xref="S4.p4.1.m1.1.1.1.3"></times></apply><ci id="S4.p4.1.m1.1.1.2.cmml" xref="S4.p4.1.m1.1.1.2">𝑓</ci><ci id="S4.p4.1.m1.1.1.3.cmml" xref="S4.p4.1.m1.1.1.3">𝑔</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p4.1.m1.1c">f=^{*}g</annotation><annotation encoding="application/x-llamapun" id="S4.p4.1.m1.1d">italic_f = start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_g</annotation></semantics></math> denotes that <math alttext="\{x\in\operatorname{dom}(f):f(x)\neq g(x)\}" class="ltx_Math" display="inline" id="S4.p4.2.m2.6"><semantics id="S4.p4.2.m2.6a"><mrow id="S4.p4.2.m2.6.6.2" xref="S4.p4.2.m2.6.6.3.cmml"><mo id="S4.p4.2.m2.6.6.2.3" stretchy="false" xref="S4.p4.2.m2.6.6.3.1.cmml">{</mo><mrow id="S4.p4.2.m2.5.5.1.1" xref="S4.p4.2.m2.5.5.1.1.cmml"><mi id="S4.p4.2.m2.5.5.1.1.2" xref="S4.p4.2.m2.5.5.1.1.2.cmml">x</mi><mo id="S4.p4.2.m2.5.5.1.1.1" xref="S4.p4.2.m2.5.5.1.1.1.cmml">∈</mo><mrow id="S4.p4.2.m2.5.5.1.1.3.2" xref="S4.p4.2.m2.5.5.1.1.3.1.cmml"><mi id="S4.p4.2.m2.1.1" xref="S4.p4.2.m2.1.1.cmml">dom</mi><mo id="S4.p4.2.m2.5.5.1.1.3.2a" xref="S4.p4.2.m2.5.5.1.1.3.1.cmml">⁡</mo><mrow id="S4.p4.2.m2.5.5.1.1.3.2.1" xref="S4.p4.2.m2.5.5.1.1.3.1.cmml"><mo id="S4.p4.2.m2.5.5.1.1.3.2.1.1" stretchy="false" xref="S4.p4.2.m2.5.5.1.1.3.1.cmml">(</mo><mi id="S4.p4.2.m2.2.2" xref="S4.p4.2.m2.2.2.cmml">f</mi><mo id="S4.p4.2.m2.5.5.1.1.3.2.1.2" rspace="0.278em" stretchy="false" xref="S4.p4.2.m2.5.5.1.1.3.1.cmml">)</mo></mrow></mrow></mrow><mo id="S4.p4.2.m2.6.6.2.4" rspace="0.278em" xref="S4.p4.2.m2.6.6.3.1.cmml">:</mo><mrow id="S4.p4.2.m2.6.6.2.2" xref="S4.p4.2.m2.6.6.2.2.cmml"><mrow id="S4.p4.2.m2.6.6.2.2.2" xref="S4.p4.2.m2.6.6.2.2.2.cmml"><mi id="S4.p4.2.m2.6.6.2.2.2.2" xref="S4.p4.2.m2.6.6.2.2.2.2.cmml">f</mi><mo id="S4.p4.2.m2.6.6.2.2.2.1" xref="S4.p4.2.m2.6.6.2.2.2.1.cmml">⁢</mo><mrow id="S4.p4.2.m2.6.6.2.2.2.3.2" xref="S4.p4.2.m2.6.6.2.2.2.cmml"><mo id="S4.p4.2.m2.6.6.2.2.2.3.2.1" stretchy="false" xref="S4.p4.2.m2.6.6.2.2.2.cmml">(</mo><mi id="S4.p4.2.m2.3.3" xref="S4.p4.2.m2.3.3.cmml">x</mi><mo id="S4.p4.2.m2.6.6.2.2.2.3.2.2" stretchy="false" xref="S4.p4.2.m2.6.6.2.2.2.cmml">)</mo></mrow></mrow><mo id="S4.p4.2.m2.6.6.2.2.1" xref="S4.p4.2.m2.6.6.2.2.1.cmml">≠</mo><mrow id="S4.p4.2.m2.6.6.2.2.3" xref="S4.p4.2.m2.6.6.2.2.3.cmml"><mi id="S4.p4.2.m2.6.6.2.2.3.2" xref="S4.p4.2.m2.6.6.2.2.3.2.cmml">g</mi><mo id="S4.p4.2.m2.6.6.2.2.3.1" xref="S4.p4.2.m2.6.6.2.2.3.1.cmml">⁢</mo><mrow id="S4.p4.2.m2.6.6.2.2.3.3.2" xref="S4.p4.2.m2.6.6.2.2.3.cmml"><mo id="S4.p4.2.m2.6.6.2.2.3.3.2.1" stretchy="false" xref="S4.p4.2.m2.6.6.2.2.3.cmml">(</mo><mi id="S4.p4.2.m2.4.4" xref="S4.p4.2.m2.4.4.cmml">x</mi><mo id="S4.p4.2.m2.6.6.2.2.3.3.2.2" stretchy="false" xref="S4.p4.2.m2.6.6.2.2.3.cmml">)</mo></mrow></mrow></mrow><mo id="S4.p4.2.m2.6.6.2.5" stretchy="false" xref="S4.p4.2.m2.6.6.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.p4.2.m2.6b"><apply id="S4.p4.2.m2.6.6.3.cmml" xref="S4.p4.2.m2.6.6.2"><csymbol cd="latexml" id="S4.p4.2.m2.6.6.3.1.cmml" xref="S4.p4.2.m2.6.6.2.3">conditional-set</csymbol><apply id="S4.p4.2.m2.5.5.1.1.cmml" xref="S4.p4.2.m2.5.5.1.1"><in id="S4.p4.2.m2.5.5.1.1.1.cmml" xref="S4.p4.2.m2.5.5.1.1.1"></in><ci id="S4.p4.2.m2.5.5.1.1.2.cmml" xref="S4.p4.2.m2.5.5.1.1.2">𝑥</ci><apply id="S4.p4.2.m2.5.5.1.1.3.1.cmml" xref="S4.p4.2.m2.5.5.1.1.3.2"><ci id="S4.p4.2.m2.1.1.cmml" xref="S4.p4.2.m2.1.1">dom</ci><ci id="S4.p4.2.m2.2.2.cmml" xref="S4.p4.2.m2.2.2">𝑓</ci></apply></apply><apply id="S4.p4.2.m2.6.6.2.2.cmml" xref="S4.p4.2.m2.6.6.2.2"><neq id="S4.p4.2.m2.6.6.2.2.1.cmml" xref="S4.p4.2.m2.6.6.2.2.1"></neq><apply id="S4.p4.2.m2.6.6.2.2.2.cmml" xref="S4.p4.2.m2.6.6.2.2.2"><times id="S4.p4.2.m2.6.6.2.2.2.1.cmml" xref="S4.p4.2.m2.6.6.2.2.2.1"></times><ci id="S4.p4.2.m2.6.6.2.2.2.2.cmml" xref="S4.p4.2.m2.6.6.2.2.2.2">𝑓</ci><ci id="S4.p4.2.m2.3.3.cmml" xref="S4.p4.2.m2.3.3">𝑥</ci></apply><apply id="S4.p4.2.m2.6.6.2.2.3.cmml" xref="S4.p4.2.m2.6.6.2.2.3"><times id="S4.p4.2.m2.6.6.2.2.3.1.cmml" xref="S4.p4.2.m2.6.6.2.2.3.1"></times><ci id="S4.p4.2.m2.6.6.2.2.3.2.cmml" xref="S4.p4.2.m2.6.6.2.2.3.2">𝑔</ci><ci id="S4.p4.2.m2.4.4.cmml" xref="S4.p4.2.m2.4.4">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p4.2.m2.6c">\{x\in\operatorname{dom}(f):f(x)\neq g(x)\}</annotation><annotation encoding="application/x-llamapun" id="S4.p4.2.m2.6d">{ italic_x ∈ roman_dom ( italic_f ) : italic_f ( italic_x ) ≠ italic_g ( italic_x ) }</annotation></semantics></math> is finite. We also say that a function <math alttext="f" class="ltx_Math" display="inline" id="S4.p4.3.m3.1"><semantics id="S4.p4.3.m3.1a"><mi id="S4.p4.3.m3.1.1" xref="S4.p4.3.m3.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S4.p4.3.m3.1b"><ci id="S4.p4.3.m3.1.1.cmml" xref="S4.p4.3.m3.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p4.3.m3.1c">f</annotation><annotation encoding="application/x-llamapun" id="S4.p4.3.m3.1d">italic_f</annotation></semantics></math> is <em class="ltx_emph ltx_font_italic" id="S4.p4.10.1">finite-to-one</em> if <math alttext="f^{-1}(y)" class="ltx_Math" display="inline" id="S4.p4.4.m4.1"><semantics id="S4.p4.4.m4.1a"><mrow id="S4.p4.4.m4.1.2" xref="S4.p4.4.m4.1.2.cmml"><msup id="S4.p4.4.m4.1.2.2" xref="S4.p4.4.m4.1.2.2.cmml"><mi id="S4.p4.4.m4.1.2.2.2" xref="S4.p4.4.m4.1.2.2.2.cmml">f</mi><mrow id="S4.p4.4.m4.1.2.2.3" xref="S4.p4.4.m4.1.2.2.3.cmml"><mo id="S4.p4.4.m4.1.2.2.3a" xref="S4.p4.4.m4.1.2.2.3.cmml">−</mo><mn id="S4.p4.4.m4.1.2.2.3.2" xref="S4.p4.4.m4.1.2.2.3.2.cmml">1</mn></mrow></msup><mo id="S4.p4.4.m4.1.2.1" xref="S4.p4.4.m4.1.2.1.cmml">⁢</mo><mrow id="S4.p4.4.m4.1.2.3.2" xref="S4.p4.4.m4.1.2.cmml"><mo id="S4.p4.4.m4.1.2.3.2.1" stretchy="false" xref="S4.p4.4.m4.1.2.cmml">(</mo><mi id="S4.p4.4.m4.1.1" xref="S4.p4.4.m4.1.1.cmml">y</mi><mo id="S4.p4.4.m4.1.2.3.2.2" stretchy="false" xref="S4.p4.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p4.4.m4.1b"><apply id="S4.p4.4.m4.1.2.cmml" xref="S4.p4.4.m4.1.2"><times id="S4.p4.4.m4.1.2.1.cmml" xref="S4.p4.4.m4.1.2.1"></times><apply id="S4.p4.4.m4.1.2.2.cmml" xref="S4.p4.4.m4.1.2.2"><csymbol cd="ambiguous" id="S4.p4.4.m4.1.2.2.1.cmml" xref="S4.p4.4.m4.1.2.2">superscript</csymbol><ci id="S4.p4.4.m4.1.2.2.2.cmml" xref="S4.p4.4.m4.1.2.2.2">𝑓</ci><apply id="S4.p4.4.m4.1.2.2.3.cmml" xref="S4.p4.4.m4.1.2.2.3"><minus id="S4.p4.4.m4.1.2.2.3.1.cmml" xref="S4.p4.4.m4.1.2.2.3"></minus><cn id="S4.p4.4.m4.1.2.2.3.2.cmml" type="integer" xref="S4.p4.4.m4.1.2.2.3.2">1</cn></apply></apply><ci id="S4.p4.4.m4.1.1.cmml" xref="S4.p4.4.m4.1.1">𝑦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p4.4.m4.1c">f^{-1}(y)</annotation><annotation encoding="application/x-llamapun" id="S4.p4.4.m4.1d">italic_f start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_y )</annotation></semantics></math> is finite for every <math alttext="y\in\operatorname{ran}(f)" class="ltx_Math" display="inline" id="S4.p4.5.m5.2"><semantics id="S4.p4.5.m5.2a"><mrow id="S4.p4.5.m5.2.3" xref="S4.p4.5.m5.2.3.cmml"><mi id="S4.p4.5.m5.2.3.2" xref="S4.p4.5.m5.2.3.2.cmml">y</mi><mo id="S4.p4.5.m5.2.3.1" xref="S4.p4.5.m5.2.3.1.cmml">∈</mo><mrow id="S4.p4.5.m5.2.3.3.2" xref="S4.p4.5.m5.2.3.3.1.cmml"><mi id="S4.p4.5.m5.1.1" xref="S4.p4.5.m5.1.1.cmml">ran</mi><mo id="S4.p4.5.m5.2.3.3.2a" xref="S4.p4.5.m5.2.3.3.1.cmml">⁡</mo><mrow id="S4.p4.5.m5.2.3.3.2.1" xref="S4.p4.5.m5.2.3.3.1.cmml"><mo id="S4.p4.5.m5.2.3.3.2.1.1" stretchy="false" xref="S4.p4.5.m5.2.3.3.1.cmml">(</mo><mi id="S4.p4.5.m5.2.2" xref="S4.p4.5.m5.2.2.cmml">f</mi><mo id="S4.p4.5.m5.2.3.3.2.1.2" stretchy="false" xref="S4.p4.5.m5.2.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p4.5.m5.2b"><apply id="S4.p4.5.m5.2.3.cmml" xref="S4.p4.5.m5.2.3"><in id="S4.p4.5.m5.2.3.1.cmml" xref="S4.p4.5.m5.2.3.1"></in><ci id="S4.p4.5.m5.2.3.2.cmml" xref="S4.p4.5.m5.2.3.2">𝑦</ci><apply id="S4.p4.5.m5.2.3.3.1.cmml" xref="S4.p4.5.m5.2.3.3.2"><ci id="S4.p4.5.m5.1.1.cmml" xref="S4.p4.5.m5.1.1">ran</ci><ci id="S4.p4.5.m5.2.2.cmml" xref="S4.p4.5.m5.2.2">𝑓</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p4.5.m5.2c">y\in\operatorname{ran}(f)</annotation><annotation encoding="application/x-llamapun" id="S4.p4.5.m5.2d">italic_y ∈ roman_ran ( italic_f )</annotation></semantics></math>. If <math alttext="T" class="ltx_Math" display="inline" id="S4.p4.6.m6.1"><semantics id="S4.p4.6.m6.1a"><mi id="S4.p4.6.m6.1.1" xref="S4.p4.6.m6.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S4.p4.6.m6.1b"><ci id="S4.p4.6.m6.1.1.cmml" xref="S4.p4.6.m6.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p4.6.m6.1c">T</annotation><annotation encoding="application/x-llamapun" id="S4.p4.6.m6.1d">italic_T</annotation></semantics></math> is a set of sequences (i.e., functions with domain an ordinal), <math alttext="T" class="ltx_Math" display="inline" id="S4.p4.7.m7.1"><semantics id="S4.p4.7.m7.1a"><mi id="S4.p4.7.m7.1.1" xref="S4.p4.7.m7.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S4.p4.7.m7.1b"><ci id="S4.p4.7.m7.1.1.cmml" xref="S4.p4.7.m7.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p4.7.m7.1c">T</annotation><annotation encoding="application/x-llamapun" id="S4.p4.7.m7.1d">italic_T</annotation></semantics></math> is called <em class="ltx_emph ltx_font_italic" id="S4.p4.10.2">coherent</em> if for all <math alttext="t,s\in T" class="ltx_Math" display="inline" id="S4.p4.8.m8.2"><semantics id="S4.p4.8.m8.2a"><mrow id="S4.p4.8.m8.2.3" xref="S4.p4.8.m8.2.3.cmml"><mrow id="S4.p4.8.m8.2.3.2.2" xref="S4.p4.8.m8.2.3.2.1.cmml"><mi id="S4.p4.8.m8.1.1" xref="S4.p4.8.m8.1.1.cmml">t</mi><mo id="S4.p4.8.m8.2.3.2.2.1" xref="S4.p4.8.m8.2.3.2.1.cmml">,</mo><mi id="S4.p4.8.m8.2.2" xref="S4.p4.8.m8.2.2.cmml">s</mi></mrow><mo id="S4.p4.8.m8.2.3.1" xref="S4.p4.8.m8.2.3.1.cmml">∈</mo><mi id="S4.p4.8.m8.2.3.3" xref="S4.p4.8.m8.2.3.3.cmml">T</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.p4.8.m8.2b"><apply id="S4.p4.8.m8.2.3.cmml" xref="S4.p4.8.m8.2.3"><in id="S4.p4.8.m8.2.3.1.cmml" xref="S4.p4.8.m8.2.3.1"></in><list id="S4.p4.8.m8.2.3.2.1.cmml" xref="S4.p4.8.m8.2.3.2.2"><ci id="S4.p4.8.m8.1.1.cmml" xref="S4.p4.8.m8.1.1">𝑡</ci><ci id="S4.p4.8.m8.2.2.cmml" xref="S4.p4.8.m8.2.2">𝑠</ci></list><ci id="S4.p4.8.m8.2.3.3.cmml" xref="S4.p4.8.m8.2.3.3">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p4.8.m8.2c">t,s\in T</annotation><annotation encoding="application/x-llamapun" id="S4.p4.8.m8.2d">italic_t , italic_s ∈ italic_T</annotation></semantics></math> with <math alttext="\operatorname{dom}(t)\leq\operatorname{dom}(s)" class="ltx_Math" display="inline" id="S4.p4.9.m9.4"><semantics id="S4.p4.9.m9.4a"><mrow id="S4.p4.9.m9.4.5" xref="S4.p4.9.m9.4.5.cmml"><mrow id="S4.p4.9.m9.4.5.2.2" xref="S4.p4.9.m9.4.5.2.1.cmml"><mi id="S4.p4.9.m9.1.1" xref="S4.p4.9.m9.1.1.cmml">dom</mi><mo id="S4.p4.9.m9.4.5.2.2a" xref="S4.p4.9.m9.4.5.2.1.cmml">⁡</mo><mrow id="S4.p4.9.m9.4.5.2.2.1" xref="S4.p4.9.m9.4.5.2.1.cmml"><mo id="S4.p4.9.m9.4.5.2.2.1.1" stretchy="false" xref="S4.p4.9.m9.4.5.2.1.cmml">(</mo><mi id="S4.p4.9.m9.2.2" xref="S4.p4.9.m9.2.2.cmml">t</mi><mo id="S4.p4.9.m9.4.5.2.2.1.2" stretchy="false" xref="S4.p4.9.m9.4.5.2.1.cmml">)</mo></mrow></mrow><mo id="S4.p4.9.m9.4.5.1" xref="S4.p4.9.m9.4.5.1.cmml">≤</mo><mrow id="S4.p4.9.m9.4.5.3.2" xref="S4.p4.9.m9.4.5.3.1.cmml"><mi id="S4.p4.9.m9.3.3" xref="S4.p4.9.m9.3.3.cmml">dom</mi><mo id="S4.p4.9.m9.4.5.3.2a" xref="S4.p4.9.m9.4.5.3.1.cmml">⁡</mo><mrow id="S4.p4.9.m9.4.5.3.2.1" xref="S4.p4.9.m9.4.5.3.1.cmml"><mo id="S4.p4.9.m9.4.5.3.2.1.1" stretchy="false" xref="S4.p4.9.m9.4.5.3.1.cmml">(</mo><mi id="S4.p4.9.m9.4.4" xref="S4.p4.9.m9.4.4.cmml">s</mi><mo id="S4.p4.9.m9.4.5.3.2.1.2" stretchy="false" xref="S4.p4.9.m9.4.5.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p4.9.m9.4b"><apply id="S4.p4.9.m9.4.5.cmml" xref="S4.p4.9.m9.4.5"><leq id="S4.p4.9.m9.4.5.1.cmml" xref="S4.p4.9.m9.4.5.1"></leq><apply id="S4.p4.9.m9.4.5.2.1.cmml" xref="S4.p4.9.m9.4.5.2.2"><ci id="S4.p4.9.m9.1.1.cmml" xref="S4.p4.9.m9.1.1">dom</ci><ci id="S4.p4.9.m9.2.2.cmml" xref="S4.p4.9.m9.2.2">𝑡</ci></apply><apply id="S4.p4.9.m9.4.5.3.1.cmml" xref="S4.p4.9.m9.4.5.3.2"><ci id="S4.p4.9.m9.3.3.cmml" xref="S4.p4.9.m9.3.3">dom</ci><ci id="S4.p4.9.m9.4.4.cmml" xref="S4.p4.9.m9.4.4">𝑠</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p4.9.m9.4c">\operatorname{dom}(t)\leq\operatorname{dom}(s)</annotation><annotation encoding="application/x-llamapun" id="S4.p4.9.m9.4d">roman_dom ( italic_t ) ≤ roman_dom ( italic_s )</annotation></semantics></math>, <math alttext="t=^{*}s|_{\operatorname{dom}(t)}" class="ltx_Math" display="inline" id="S4.p4.10.m10.3"><semantics id="S4.p4.10.m10.3a"><mrow id="S4.p4.10.m10.3.4" xref="S4.p4.10.m10.3.4.cmml"><mi id="S4.p4.10.m10.3.4.2" xref="S4.p4.10.m10.3.4.2.cmml">t</mi><msup id="S4.p4.10.m10.3.4.1" xref="S4.p4.10.m10.3.4.1.cmml"><mo id="S4.p4.10.m10.3.4.1.2" xref="S4.p4.10.m10.3.4.1.2.cmml">=</mo><mo id="S4.p4.10.m10.3.4.1.3" xref="S4.p4.10.m10.3.4.1.3.cmml">∗</mo></msup><msub id="S4.p4.10.m10.3.4.3.2" xref="S4.p4.10.m10.3.4.3.1.cmml"><mrow id="S4.p4.10.m10.3.4.3.2.2" xref="S4.p4.10.m10.3.4.3.1.cmml"><mi id="S4.p4.10.m10.3.3" xref="S4.p4.10.m10.3.3.cmml">s</mi><mo id="S4.p4.10.m10.3.4.3.2.2.1" stretchy="false" xref="S4.p4.10.m10.3.4.3.1.1.cmml">|</mo></mrow><mrow id="S4.p4.10.m10.2.2.2.4" xref="S4.p4.10.m10.2.2.2.3.cmml"><mi id="S4.p4.10.m10.1.1.1.1" xref="S4.p4.10.m10.1.1.1.1.cmml">dom</mi><mo id="S4.p4.10.m10.2.2.2.4a" xref="S4.p4.10.m10.2.2.2.3.cmml">⁡</mo><mrow id="S4.p4.10.m10.2.2.2.4.1" xref="S4.p4.10.m10.2.2.2.3.cmml"><mo id="S4.p4.10.m10.2.2.2.4.1.1" stretchy="false" xref="S4.p4.10.m10.2.2.2.3.cmml">(</mo><mi id="S4.p4.10.m10.2.2.2.2" xref="S4.p4.10.m10.2.2.2.2.cmml">t</mi><mo id="S4.p4.10.m10.2.2.2.4.1.2" stretchy="false" xref="S4.p4.10.m10.2.2.2.3.cmml">)</mo></mrow></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.p4.10.m10.3b"><apply id="S4.p4.10.m10.3.4.cmml" xref="S4.p4.10.m10.3.4"><apply id="S4.p4.10.m10.3.4.1.cmml" xref="S4.p4.10.m10.3.4.1"><csymbol cd="ambiguous" id="S4.p4.10.m10.3.4.1.1.cmml" xref="S4.p4.10.m10.3.4.1">superscript</csymbol><eq id="S4.p4.10.m10.3.4.1.2.cmml" xref="S4.p4.10.m10.3.4.1.2"></eq><times id="S4.p4.10.m10.3.4.1.3.cmml" xref="S4.p4.10.m10.3.4.1.3"></times></apply><ci id="S4.p4.10.m10.3.4.2.cmml" xref="S4.p4.10.m10.3.4.2">𝑡</ci><apply id="S4.p4.10.m10.3.4.3.1.cmml" xref="S4.p4.10.m10.3.4.3.2"><csymbol cd="latexml" id="S4.p4.10.m10.3.4.3.1.1.cmml" xref="S4.p4.10.m10.3.4.3.2.2.1">evaluated-at</csymbol><ci id="S4.p4.10.m10.3.3.cmml" xref="S4.p4.10.m10.3.3">𝑠</ci><apply id="S4.p4.10.m10.2.2.2.3.cmml" xref="S4.p4.10.m10.2.2.2.4"><ci id="S4.p4.10.m10.1.1.1.1.cmml" xref="S4.p4.10.m10.1.1.1.1">dom</ci><ci id="S4.p4.10.m10.2.2.2.2.cmml" xref="S4.p4.10.m10.2.2.2.2">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p4.10.m10.3c">t=^{*}s|_{\operatorname{dom}(t)}</annotation><annotation encoding="application/x-llamapun" id="S4.p4.10.m10.3d">italic_t = start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_s | start_POSTSUBSCRIPT roman_dom ( italic_t ) end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.p5"> <p class="ltx_p" id="S4.p5.4">Let <math alttext="\langle f_{\alpha}:\alpha<\omega_{1}\rangle" class="ltx_math_unparsed" display="inline" id="S4.p5.1.m1.1"><semantics id="S4.p5.1.m1.1a"><mrow id="S4.p5.1.m1.1b"><mo id="S4.p5.1.m1.1.1" stretchy="false">⟨</mo><msub id="S4.p5.1.m1.1.2"><mi id="S4.p5.1.m1.1.2.2">f</mi><mi id="S4.p5.1.m1.1.2.3">α</mi></msub><mo id="S4.p5.1.m1.1.3" lspace="0.278em" rspace="0.278em">:</mo><mi id="S4.p5.1.m1.1.4">α</mi><mo id="S4.p5.1.m1.1.5"><</mo><msub id="S4.p5.1.m1.1.6"><mi id="S4.p5.1.m1.1.6.2">ω</mi><mn id="S4.p5.1.m1.1.6.3">1</mn></msub><mo id="S4.p5.1.m1.1.7" stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex" id="S4.p5.1.m1.1c">\langle f_{\alpha}:\alpha<\omega_{1}\rangle</annotation><annotation encoding="application/x-llamapun" id="S4.p5.1.m1.1d">⟨ italic_f start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT : italic_α < italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⟩</annotation></semantics></math> be a coherent sequence of finite-one-functions <math alttext="f_{\alpha}:\alpha\to\omega" class="ltx_Math" display="inline" id="S4.p5.2.m2.1"><semantics id="S4.p5.2.m2.1a"><mrow id="S4.p5.2.m2.1.1" xref="S4.p5.2.m2.1.1.cmml"><msub id="S4.p5.2.m2.1.1.2" xref="S4.p5.2.m2.1.1.2.cmml"><mi id="S4.p5.2.m2.1.1.2.2" xref="S4.p5.2.m2.1.1.2.2.cmml">f</mi><mi id="S4.p5.2.m2.1.1.2.3" xref="S4.p5.2.m2.1.1.2.3.cmml">α</mi></msub><mo id="S4.p5.2.m2.1.1.1" lspace="0.278em" rspace="0.278em" xref="S4.p5.2.m2.1.1.1.cmml">:</mo><mrow id="S4.p5.2.m2.1.1.3" xref="S4.p5.2.m2.1.1.3.cmml"><mi id="S4.p5.2.m2.1.1.3.2" xref="S4.p5.2.m2.1.1.3.2.cmml">α</mi><mo id="S4.p5.2.m2.1.1.3.1" stretchy="false" xref="S4.p5.2.m2.1.1.3.1.cmml">→</mo><mi id="S4.p5.2.m2.1.1.3.3" xref="S4.p5.2.m2.1.1.3.3.cmml">ω</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p5.2.m2.1b"><apply id="S4.p5.2.m2.1.1.cmml" xref="S4.p5.2.m2.1.1"><ci id="S4.p5.2.m2.1.1.1.cmml" xref="S4.p5.2.m2.1.1.1">:</ci><apply id="S4.p5.2.m2.1.1.2.cmml" xref="S4.p5.2.m2.1.1.2"><csymbol cd="ambiguous" id="S4.p5.2.m2.1.1.2.1.cmml" xref="S4.p5.2.m2.1.1.2">subscript</csymbol><ci id="S4.p5.2.m2.1.1.2.2.cmml" xref="S4.p5.2.m2.1.1.2.2">𝑓</ci><ci id="S4.p5.2.m2.1.1.2.3.cmml" xref="S4.p5.2.m2.1.1.2.3">𝛼</ci></apply><apply id="S4.p5.2.m2.1.1.3.cmml" xref="S4.p5.2.m2.1.1.3"><ci id="S4.p5.2.m2.1.1.3.1.cmml" xref="S4.p5.2.m2.1.1.3.1">→</ci><ci id="S4.p5.2.m2.1.1.3.2.cmml" xref="S4.p5.2.m2.1.1.3.2">𝛼</ci><ci id="S4.p5.2.m2.1.1.3.3.cmml" xref="S4.p5.2.m2.1.1.3.3">𝜔</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p5.2.m2.1c">f_{\alpha}:\alpha\to\omega</annotation><annotation encoding="application/x-llamapun" id="S4.p5.2.m2.1d">italic_f start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT : italic_α → italic_ω</annotation></semantics></math>. It is well known that such a sequence exists in <math alttext="\mathsf{ZFC}" class="ltx_Math" display="inline" id="S4.p5.3.m3.1"><semantics id="S4.p5.3.m3.1a"><mi id="S4.p5.3.m3.1.1" xref="S4.p5.3.m3.1.1.cmml">𝖹𝖥𝖢</mi><annotation-xml encoding="MathML-Content" id="S4.p5.3.m3.1b"><ci id="S4.p5.3.m3.1.1.cmml" xref="S4.p5.3.m3.1.1">𝖹𝖥𝖢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p5.3.m3.1c">\mathsf{ZFC}</annotation><annotation encoding="application/x-llamapun" id="S4.p5.3.m3.1d">sansserif_ZFC</annotation></semantics></math>, and that <math alttext="T:=\{t\in\alpha^{\omega}:\alpha<\omega_{1},t=^{*}f_{\alpha}\}" class="ltx_Math" display="inline" id="S4.p5.4.m4.3"><semantics id="S4.p5.4.m4.3a"><mrow id="S4.p5.4.m4.3.3" xref="S4.p5.4.m4.3.3.cmml"><mi id="S4.p5.4.m4.3.3.4" xref="S4.p5.4.m4.3.3.4.cmml">T</mi><mo id="S4.p5.4.m4.3.3.3" lspace="0.278em" rspace="0.278em" xref="S4.p5.4.m4.3.3.3.cmml">:=</mo><mrow id="S4.p5.4.m4.3.3.2.2" xref="S4.p5.4.m4.3.3.2.3.cmml"><mo id="S4.p5.4.m4.3.3.2.2.3" stretchy="false" xref="S4.p5.4.m4.3.3.2.3.1.cmml">{</mo><mrow id="S4.p5.4.m4.2.2.1.1.1" xref="S4.p5.4.m4.2.2.1.1.1.cmml"><mi id="S4.p5.4.m4.2.2.1.1.1.2" xref="S4.p5.4.m4.2.2.1.1.1.2.cmml">t</mi><mo id="S4.p5.4.m4.2.2.1.1.1.1" xref="S4.p5.4.m4.2.2.1.1.1.1.cmml">∈</mo><msup id="S4.p5.4.m4.2.2.1.1.1.3" xref="S4.p5.4.m4.2.2.1.1.1.3.cmml"><mi id="S4.p5.4.m4.2.2.1.1.1.3.2" xref="S4.p5.4.m4.2.2.1.1.1.3.2.cmml">α</mi><mi id="S4.p5.4.m4.2.2.1.1.1.3.3" xref="S4.p5.4.m4.2.2.1.1.1.3.3.cmml">ω</mi></msup></mrow><mo id="S4.p5.4.m4.3.3.2.2.4" lspace="0.278em" rspace="0.278em" xref="S4.p5.4.m4.3.3.2.3.1.cmml">:</mo><mrow id="S4.p5.4.m4.3.3.2.2.2.2" xref="S4.p5.4.m4.3.3.2.2.2.3.cmml"><mrow id="S4.p5.4.m4.3.3.2.2.2.1.1" xref="S4.p5.4.m4.3.3.2.2.2.1.1.cmml"><mi id="S4.p5.4.m4.3.3.2.2.2.1.1.2" xref="S4.p5.4.m4.3.3.2.2.2.1.1.2.cmml">α</mi><mo id="S4.p5.4.m4.3.3.2.2.2.1.1.1" xref="S4.p5.4.m4.3.3.2.2.2.1.1.1.cmml"><</mo><msub id="S4.p5.4.m4.3.3.2.2.2.1.1.3" xref="S4.p5.4.m4.3.3.2.2.2.1.1.3.cmml"><mi id="S4.p5.4.m4.3.3.2.2.2.1.1.3.2" xref="S4.p5.4.m4.3.3.2.2.2.1.1.3.2.cmml">ω</mi><mn id="S4.p5.4.m4.3.3.2.2.2.1.1.3.3" xref="S4.p5.4.m4.3.3.2.2.2.1.1.3.3.cmml">1</mn></msub></mrow><mo id="S4.p5.4.m4.3.3.2.2.2.2.3" xref="S4.p5.4.m4.3.3.2.2.2.3a.cmml">,</mo><mrow id="S4.p5.4.m4.3.3.2.2.2.2.2" xref="S4.p5.4.m4.3.3.2.2.2.2.2.cmml"><mi id="S4.p5.4.m4.1.1" xref="S4.p5.4.m4.1.1.cmml">t</mi><msup id="S4.p5.4.m4.3.3.2.2.2.2.2.1" xref="S4.p5.4.m4.3.3.2.2.2.2.2.1.cmml"><mo id="S4.p5.4.m4.3.3.2.2.2.2.2.1.2" xref="S4.p5.4.m4.3.3.2.2.2.2.2.1.2.cmml">=</mo><mo id="S4.p5.4.m4.3.3.2.2.2.2.2.1.3" xref="S4.p5.4.m4.3.3.2.2.2.2.2.1.3.cmml">∗</mo></msup><msub id="S4.p5.4.m4.3.3.2.2.2.2.2.2" xref="S4.p5.4.m4.3.3.2.2.2.2.2.2.cmml"><mi id="S4.p5.4.m4.3.3.2.2.2.2.2.2.2" xref="S4.p5.4.m4.3.3.2.2.2.2.2.2.2.cmml">f</mi><mi id="S4.p5.4.m4.3.3.2.2.2.2.2.2.3" xref="S4.p5.4.m4.3.3.2.2.2.2.2.2.3.cmml">α</mi></msub></mrow></mrow><mo id="S4.p5.4.m4.3.3.2.2.5" stretchy="false" xref="S4.p5.4.m4.3.3.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p5.4.m4.3b"><apply id="S4.p5.4.m4.3.3.cmml" xref="S4.p5.4.m4.3.3"><csymbol cd="latexml" id="S4.p5.4.m4.3.3.3.cmml" xref="S4.p5.4.m4.3.3.3">assign</csymbol><ci id="S4.p5.4.m4.3.3.4.cmml" xref="S4.p5.4.m4.3.3.4">𝑇</ci><apply id="S4.p5.4.m4.3.3.2.3.cmml" xref="S4.p5.4.m4.3.3.2.2"><csymbol cd="latexml" id="S4.p5.4.m4.3.3.2.3.1.cmml" xref="S4.p5.4.m4.3.3.2.2.3">conditional-set</csymbol><apply id="S4.p5.4.m4.2.2.1.1.1.cmml" xref="S4.p5.4.m4.2.2.1.1.1"><in id="S4.p5.4.m4.2.2.1.1.1.1.cmml" xref="S4.p5.4.m4.2.2.1.1.1.1"></in><ci id="S4.p5.4.m4.2.2.1.1.1.2.cmml" xref="S4.p5.4.m4.2.2.1.1.1.2">𝑡</ci><apply id="S4.p5.4.m4.2.2.1.1.1.3.cmml" xref="S4.p5.4.m4.2.2.1.1.1.3"><csymbol cd="ambiguous" id="S4.p5.4.m4.2.2.1.1.1.3.1.cmml" xref="S4.p5.4.m4.2.2.1.1.1.3">superscript</csymbol><ci id="S4.p5.4.m4.2.2.1.1.1.3.2.cmml" xref="S4.p5.4.m4.2.2.1.1.1.3.2">𝛼</ci><ci id="S4.p5.4.m4.2.2.1.1.1.3.3.cmml" xref="S4.p5.4.m4.2.2.1.1.1.3.3">𝜔</ci></apply></apply><apply id="S4.p5.4.m4.3.3.2.2.2.3.cmml" xref="S4.p5.4.m4.3.3.2.2.2.2"><csymbol cd="ambiguous" id="S4.p5.4.m4.3.3.2.2.2.3a.cmml" xref="S4.p5.4.m4.3.3.2.2.2.2.3">formulae-sequence</csymbol><apply id="S4.p5.4.m4.3.3.2.2.2.1.1.cmml" xref="S4.p5.4.m4.3.3.2.2.2.1.1"><lt id="S4.p5.4.m4.3.3.2.2.2.1.1.1.cmml" xref="S4.p5.4.m4.3.3.2.2.2.1.1.1"></lt><ci id="S4.p5.4.m4.3.3.2.2.2.1.1.2.cmml" xref="S4.p5.4.m4.3.3.2.2.2.1.1.2">𝛼</ci><apply id="S4.p5.4.m4.3.3.2.2.2.1.1.3.cmml" xref="S4.p5.4.m4.3.3.2.2.2.1.1.3"><csymbol cd="ambiguous" id="S4.p5.4.m4.3.3.2.2.2.1.1.3.1.cmml" xref="S4.p5.4.m4.3.3.2.2.2.1.1.3">subscript</csymbol><ci id="S4.p5.4.m4.3.3.2.2.2.1.1.3.2.cmml" xref="S4.p5.4.m4.3.3.2.2.2.1.1.3.2">𝜔</ci><cn id="S4.p5.4.m4.3.3.2.2.2.1.1.3.3.cmml" type="integer" xref="S4.p5.4.m4.3.3.2.2.2.1.1.3.3">1</cn></apply></apply><apply id="S4.p5.4.m4.3.3.2.2.2.2.2.cmml" xref="S4.p5.4.m4.3.3.2.2.2.2.2"><apply id="S4.p5.4.m4.3.3.2.2.2.2.2.1.cmml" xref="S4.p5.4.m4.3.3.2.2.2.2.2.1"><csymbol cd="ambiguous" id="S4.p5.4.m4.3.3.2.2.2.2.2.1.1.cmml" xref="S4.p5.4.m4.3.3.2.2.2.2.2.1">superscript</csymbol><eq id="S4.p5.4.m4.3.3.2.2.2.2.2.1.2.cmml" xref="S4.p5.4.m4.3.3.2.2.2.2.2.1.2"></eq><times id="S4.p5.4.m4.3.3.2.2.2.2.2.1.3.cmml" xref="S4.p5.4.m4.3.3.2.2.2.2.2.1.3"></times></apply><ci id="S4.p5.4.m4.1.1.cmml" xref="S4.p5.4.m4.1.1">𝑡</ci><apply id="S4.p5.4.m4.3.3.2.2.2.2.2.2.cmml" xref="S4.p5.4.m4.3.3.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.p5.4.m4.3.3.2.2.2.2.2.2.1.cmml" xref="S4.p5.4.m4.3.3.2.2.2.2.2.2">subscript</csymbol><ci id="S4.p5.4.m4.3.3.2.2.2.2.2.2.2.cmml" xref="S4.p5.4.m4.3.3.2.2.2.2.2.2.2">𝑓</ci><ci id="S4.p5.4.m4.3.3.2.2.2.2.2.2.3.cmml" xref="S4.p5.4.m4.3.3.2.2.2.2.2.2.3">𝛼</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p5.4.m4.3c">T:=\{t\in\alpha^{\omega}:\alpha<\omega_{1},t=^{*}f_{\alpha}\}</annotation><annotation encoding="application/x-llamapun" id="S4.p5.4.m4.3d">italic_T := { italic_t ∈ italic_α start_POSTSUPERSCRIPT italic_ω end_POSTSUPERSCRIPT : italic_α < italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_t = start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_f start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT }</annotation></semantics></math> is an Aronszajn tree. The following is a theorem of Todorcevic (see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib22" title="">22</a>]</cite>). For a somewhat simpler proof see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib9" title="">9</a>]</cite>.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S4.Thmtheorem3"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem3.1.1.1">Lemma 4.3</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem3.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem3.p1"> <p class="ltx_p" id="S4.Thmtheorem3.p1.1"><math alttext="T" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.1.m1.1"><semantics id="S4.Thmtheorem3.p1.1.m1.1a"><mi id="S4.Thmtheorem3.p1.1.m1.1.1" xref="S4.Thmtheorem3.p1.1.m1.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.1.m1.1b"><ci id="S4.Thmtheorem3.p1.1.m1.1.1.cmml" xref="S4.Thmtheorem3.p1.1.m1.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.1.m1.1c">T</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.1.m1.1d">italic_T</annotation></semantics></math> is Countryman with the natural lexicographic ordering.</p> </div> </div> <div class="ltx_para" id="S4.p6"> <p class="ltx_p" id="S4.p6.9">Let <math alttext="\Lambda" class="ltx_Math" display="inline" id="S4.p6.1.m1.1"><semantics id="S4.p6.1.m1.1a"><mi id="S4.p6.1.m1.1.1" mathvariant="normal" xref="S4.p6.1.m1.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S4.p6.1.m1.1b"><ci id="S4.p6.1.m1.1.1.cmml" xref="S4.p6.1.m1.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p6.1.m1.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S4.p6.1.m1.1d">roman_Λ</annotation></semantics></math> denote the set of countable limit ordinals, and <math alttext="\Lambda^{\prime}" class="ltx_Math" display="inline" id="S4.p6.2.m2.1"><semantics id="S4.p6.2.m2.1a"><msup id="S4.p6.2.m2.1.1" xref="S4.p6.2.m2.1.1.cmml"><mi id="S4.p6.2.m2.1.1.2" mathvariant="normal" xref="S4.p6.2.m2.1.1.2.cmml">Λ</mi><mo id="S4.p6.2.m2.1.1.3" xref="S4.p6.2.m2.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.p6.2.m2.1b"><apply id="S4.p6.2.m2.1.1.cmml" xref="S4.p6.2.m2.1.1"><csymbol cd="ambiguous" id="S4.p6.2.m2.1.1.1.cmml" xref="S4.p6.2.m2.1.1">superscript</csymbol><ci id="S4.p6.2.m2.1.1.2.cmml" xref="S4.p6.2.m2.1.1.2">Λ</ci><ci id="S4.p6.2.m2.1.1.3.cmml" xref="S4.p6.2.m2.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p6.2.m2.1c">\Lambda^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.p6.2.m2.1d">roman_Λ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> those that are limit of elements in <math alttext="\Lambda" class="ltx_Math" display="inline" id="S4.p6.3.m3.1"><semantics id="S4.p6.3.m3.1a"><mi id="S4.p6.3.m3.1.1" mathvariant="normal" xref="S4.p6.3.m3.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S4.p6.3.m3.1b"><ci id="S4.p6.3.m3.1.1.cmml" xref="S4.p6.3.m3.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p6.3.m3.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S4.p6.3.m3.1d">roman_Λ</annotation></semantics></math>. Both these sets are club. Define <math alttext="A^{+}:=A_{0}\cup A_{1}" class="ltx_Math" display="inline" id="S4.p6.4.m4.1"><semantics id="S4.p6.4.m4.1a"><mrow id="S4.p6.4.m4.1.1" xref="S4.p6.4.m4.1.1.cmml"><msup id="S4.p6.4.m4.1.1.2" xref="S4.p6.4.m4.1.1.2.cmml"><mi id="S4.p6.4.m4.1.1.2.2" xref="S4.p6.4.m4.1.1.2.2.cmml">A</mi><mo id="S4.p6.4.m4.1.1.2.3" xref="S4.p6.4.m4.1.1.2.3.cmml">+</mo></msup><mo id="S4.p6.4.m4.1.1.1" lspace="0.278em" rspace="0.278em" xref="S4.p6.4.m4.1.1.1.cmml">:=</mo><mrow id="S4.p6.4.m4.1.1.3" xref="S4.p6.4.m4.1.1.3.cmml"><msub id="S4.p6.4.m4.1.1.3.2" xref="S4.p6.4.m4.1.1.3.2.cmml"><mi id="S4.p6.4.m4.1.1.3.2.2" xref="S4.p6.4.m4.1.1.3.2.2.cmml">A</mi><mn id="S4.p6.4.m4.1.1.3.2.3" xref="S4.p6.4.m4.1.1.3.2.3.cmml">0</mn></msub><mo id="S4.p6.4.m4.1.1.3.1" xref="S4.p6.4.m4.1.1.3.1.cmml">∪</mo><msub id="S4.p6.4.m4.1.1.3.3" xref="S4.p6.4.m4.1.1.3.3.cmml"><mi id="S4.p6.4.m4.1.1.3.3.2" xref="S4.p6.4.m4.1.1.3.3.2.cmml">A</mi><mn id="S4.p6.4.m4.1.1.3.3.3" xref="S4.p6.4.m4.1.1.3.3.3.cmml">1</mn></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p6.4.m4.1b"><apply id="S4.p6.4.m4.1.1.cmml" xref="S4.p6.4.m4.1.1"><csymbol cd="latexml" id="S4.p6.4.m4.1.1.1.cmml" xref="S4.p6.4.m4.1.1.1">assign</csymbol><apply id="S4.p6.4.m4.1.1.2.cmml" xref="S4.p6.4.m4.1.1.2"><csymbol cd="ambiguous" id="S4.p6.4.m4.1.1.2.1.cmml" xref="S4.p6.4.m4.1.1.2">superscript</csymbol><ci id="S4.p6.4.m4.1.1.2.2.cmml" xref="S4.p6.4.m4.1.1.2.2">𝐴</ci><plus id="S4.p6.4.m4.1.1.2.3.cmml" xref="S4.p6.4.m4.1.1.2.3"></plus></apply><apply id="S4.p6.4.m4.1.1.3.cmml" xref="S4.p6.4.m4.1.1.3"><union id="S4.p6.4.m4.1.1.3.1.cmml" xref="S4.p6.4.m4.1.1.3.1"></union><apply id="S4.p6.4.m4.1.1.3.2.cmml" xref="S4.p6.4.m4.1.1.3.2"><csymbol cd="ambiguous" id="S4.p6.4.m4.1.1.3.2.1.cmml" xref="S4.p6.4.m4.1.1.3.2">subscript</csymbol><ci id="S4.p6.4.m4.1.1.3.2.2.cmml" xref="S4.p6.4.m4.1.1.3.2.2">𝐴</ci><cn id="S4.p6.4.m4.1.1.3.2.3.cmml" type="integer" xref="S4.p6.4.m4.1.1.3.2.3">0</cn></apply><apply id="S4.p6.4.m4.1.1.3.3.cmml" xref="S4.p6.4.m4.1.1.3.3"><csymbol cd="ambiguous" id="S4.p6.4.m4.1.1.3.3.1.cmml" xref="S4.p6.4.m4.1.1.3.3">subscript</csymbol><ci id="S4.p6.4.m4.1.1.3.3.2.cmml" xref="S4.p6.4.m4.1.1.3.3.2">𝐴</ci><cn id="S4.p6.4.m4.1.1.3.3.3.cmml" type="integer" xref="S4.p6.4.m4.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p6.4.m4.1c">A^{+}:=A_{0}\cup A_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.p6.4.m4.1d">italic_A start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT := italic_A start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∪ italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> where <math alttext="A_{0}:=\{t\in T:\operatorname{dom}(t)\in\Lambda\}" class="ltx_Math" display="inline" id="S4.p6.5.m5.4"><semantics id="S4.p6.5.m5.4a"><mrow id="S4.p6.5.m5.4.4" xref="S4.p6.5.m5.4.4.cmml"><msub id="S4.p6.5.m5.4.4.4" xref="S4.p6.5.m5.4.4.4.cmml"><mi id="S4.p6.5.m5.4.4.4.2" xref="S4.p6.5.m5.4.4.4.2.cmml">A</mi><mn id="S4.p6.5.m5.4.4.4.3" xref="S4.p6.5.m5.4.4.4.3.cmml">0</mn></msub><mo id="S4.p6.5.m5.4.4.3" lspace="0.278em" rspace="0.278em" xref="S4.p6.5.m5.4.4.3.cmml">:=</mo><mrow id="S4.p6.5.m5.4.4.2.2" xref="S4.p6.5.m5.4.4.2.3.cmml"><mo id="S4.p6.5.m5.4.4.2.2.3" stretchy="false" xref="S4.p6.5.m5.4.4.2.3.1.cmml">{</mo><mrow id="S4.p6.5.m5.3.3.1.1.1" xref="S4.p6.5.m5.3.3.1.1.1.cmml"><mi id="S4.p6.5.m5.3.3.1.1.1.2" xref="S4.p6.5.m5.3.3.1.1.1.2.cmml">t</mi><mo id="S4.p6.5.m5.3.3.1.1.1.1" xref="S4.p6.5.m5.3.3.1.1.1.1.cmml">∈</mo><mi id="S4.p6.5.m5.3.3.1.1.1.3" xref="S4.p6.5.m5.3.3.1.1.1.3.cmml">T</mi></mrow><mo id="S4.p6.5.m5.4.4.2.2.4" lspace="0.278em" rspace="0.278em" xref="S4.p6.5.m5.4.4.2.3.1.cmml">:</mo><mrow id="S4.p6.5.m5.4.4.2.2.2" xref="S4.p6.5.m5.4.4.2.2.2.cmml"><mrow id="S4.p6.5.m5.4.4.2.2.2.2.2" xref="S4.p6.5.m5.4.4.2.2.2.2.1.cmml"><mi id="S4.p6.5.m5.1.1" xref="S4.p6.5.m5.1.1.cmml">dom</mi><mo id="S4.p6.5.m5.4.4.2.2.2.2.2a" xref="S4.p6.5.m5.4.4.2.2.2.2.1.cmml">⁡</mo><mrow id="S4.p6.5.m5.4.4.2.2.2.2.2.1" xref="S4.p6.5.m5.4.4.2.2.2.2.1.cmml"><mo id="S4.p6.5.m5.4.4.2.2.2.2.2.1.1" stretchy="false" xref="S4.p6.5.m5.4.4.2.2.2.2.1.cmml">(</mo><mi id="S4.p6.5.m5.2.2" xref="S4.p6.5.m5.2.2.cmml">t</mi><mo id="S4.p6.5.m5.4.4.2.2.2.2.2.1.2" stretchy="false" xref="S4.p6.5.m5.4.4.2.2.2.2.1.cmml">)</mo></mrow></mrow><mo id="S4.p6.5.m5.4.4.2.2.2.1" xref="S4.p6.5.m5.4.4.2.2.2.1.cmml">∈</mo><mi id="S4.p6.5.m5.4.4.2.2.2.3" mathvariant="normal" xref="S4.p6.5.m5.4.4.2.2.2.3.cmml">Λ</mi></mrow><mo id="S4.p6.5.m5.4.4.2.2.5" stretchy="false" xref="S4.p6.5.m5.4.4.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p6.5.m5.4b"><apply id="S4.p6.5.m5.4.4.cmml" xref="S4.p6.5.m5.4.4"><csymbol cd="latexml" id="S4.p6.5.m5.4.4.3.cmml" xref="S4.p6.5.m5.4.4.3">assign</csymbol><apply id="S4.p6.5.m5.4.4.4.cmml" xref="S4.p6.5.m5.4.4.4"><csymbol cd="ambiguous" id="S4.p6.5.m5.4.4.4.1.cmml" xref="S4.p6.5.m5.4.4.4">subscript</csymbol><ci id="S4.p6.5.m5.4.4.4.2.cmml" xref="S4.p6.5.m5.4.4.4.2">𝐴</ci><cn id="S4.p6.5.m5.4.4.4.3.cmml" type="integer" xref="S4.p6.5.m5.4.4.4.3">0</cn></apply><apply id="S4.p6.5.m5.4.4.2.3.cmml" xref="S4.p6.5.m5.4.4.2.2"><csymbol cd="latexml" id="S4.p6.5.m5.4.4.2.3.1.cmml" xref="S4.p6.5.m5.4.4.2.2.3">conditional-set</csymbol><apply id="S4.p6.5.m5.3.3.1.1.1.cmml" xref="S4.p6.5.m5.3.3.1.1.1"><in id="S4.p6.5.m5.3.3.1.1.1.1.cmml" xref="S4.p6.5.m5.3.3.1.1.1.1"></in><ci id="S4.p6.5.m5.3.3.1.1.1.2.cmml" xref="S4.p6.5.m5.3.3.1.1.1.2">𝑡</ci><ci id="S4.p6.5.m5.3.3.1.1.1.3.cmml" xref="S4.p6.5.m5.3.3.1.1.1.3">𝑇</ci></apply><apply id="S4.p6.5.m5.4.4.2.2.2.cmml" xref="S4.p6.5.m5.4.4.2.2.2"><in id="S4.p6.5.m5.4.4.2.2.2.1.cmml" xref="S4.p6.5.m5.4.4.2.2.2.1"></in><apply id="S4.p6.5.m5.4.4.2.2.2.2.1.cmml" xref="S4.p6.5.m5.4.4.2.2.2.2.2"><ci id="S4.p6.5.m5.1.1.cmml" xref="S4.p6.5.m5.1.1">dom</ci><ci id="S4.p6.5.m5.2.2.cmml" xref="S4.p6.5.m5.2.2">𝑡</ci></apply><ci id="S4.p6.5.m5.4.4.2.2.2.3.cmml" xref="S4.p6.5.m5.4.4.2.2.2.3">Λ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p6.5.m5.4c">A_{0}:=\{t\in T:\operatorname{dom}(t)\in\Lambda\}</annotation><annotation encoding="application/x-llamapun" id="S4.p6.5.m5.4d">italic_A start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT := { italic_t ∈ italic_T : roman_dom ( italic_t ) ∈ roman_Λ }</annotation></semantics></math> and <math alttext="A_{1}:=\{t^{\frown}\langle\omega\rangle:t\in A_{0}\}" class="ltx_Math" display="inline" id="S4.p6.6.m6.3"><semantics id="S4.p6.6.m6.3a"><mrow id="S4.p6.6.m6.3.3" xref="S4.p6.6.m6.3.3.cmml"><msub id="S4.p6.6.m6.3.3.4" xref="S4.p6.6.m6.3.3.4.cmml"><mi id="S4.p6.6.m6.3.3.4.2" xref="S4.p6.6.m6.3.3.4.2.cmml">A</mi><mn id="S4.p6.6.m6.3.3.4.3" xref="S4.p6.6.m6.3.3.4.3.cmml">1</mn></msub><mo id="S4.p6.6.m6.3.3.3" lspace="0.278em" rspace="0.278em" xref="S4.p6.6.m6.3.3.3.cmml">:=</mo><mrow id="S4.p6.6.m6.3.3.2.2" xref="S4.p6.6.m6.3.3.2.3.cmml"><mo id="S4.p6.6.m6.3.3.2.2.3" stretchy="false" xref="S4.p6.6.m6.3.3.2.3.1.cmml">{</mo><mrow id="S4.p6.6.m6.2.2.1.1.1" xref="S4.p6.6.m6.2.2.1.1.1.cmml"><msup id="S4.p6.6.m6.2.2.1.1.1.2" xref="S4.p6.6.m6.2.2.1.1.1.2.cmml"><mi id="S4.p6.6.m6.2.2.1.1.1.2.2" xref="S4.p6.6.m6.2.2.1.1.1.2.2.cmml">t</mi><mo id="S4.p6.6.m6.2.2.1.1.1.2.3" xref="S4.p6.6.m6.2.2.1.1.1.2.3.cmml">⌢</mo></msup><mo id="S4.p6.6.m6.2.2.1.1.1.1" xref="S4.p6.6.m6.2.2.1.1.1.1.cmml">⁢</mo><mrow id="S4.p6.6.m6.2.2.1.1.1.3.2" xref="S4.p6.6.m6.2.2.1.1.1.3.1.cmml"><mo id="S4.p6.6.m6.2.2.1.1.1.3.2.1" stretchy="false" xref="S4.p6.6.m6.2.2.1.1.1.3.1.1.cmml">⟨</mo><mi id="S4.p6.6.m6.1.1" xref="S4.p6.6.m6.1.1.cmml">ω</mi><mo id="S4.p6.6.m6.2.2.1.1.1.3.2.2" rspace="0.278em" stretchy="false" xref="S4.p6.6.m6.2.2.1.1.1.3.1.1.cmml">⟩</mo></mrow></mrow><mo id="S4.p6.6.m6.3.3.2.2.4" rspace="0.278em" xref="S4.p6.6.m6.3.3.2.3.1.cmml">:</mo><mrow id="S4.p6.6.m6.3.3.2.2.2" xref="S4.p6.6.m6.3.3.2.2.2.cmml"><mi id="S4.p6.6.m6.3.3.2.2.2.2" xref="S4.p6.6.m6.3.3.2.2.2.2.cmml">t</mi><mo id="S4.p6.6.m6.3.3.2.2.2.1" xref="S4.p6.6.m6.3.3.2.2.2.1.cmml">∈</mo><msub id="S4.p6.6.m6.3.3.2.2.2.3" xref="S4.p6.6.m6.3.3.2.2.2.3.cmml"><mi id="S4.p6.6.m6.3.3.2.2.2.3.2" xref="S4.p6.6.m6.3.3.2.2.2.3.2.cmml">A</mi><mn id="S4.p6.6.m6.3.3.2.2.2.3.3" xref="S4.p6.6.m6.3.3.2.2.2.3.3.cmml">0</mn></msub></mrow><mo id="S4.p6.6.m6.3.3.2.2.5" stretchy="false" xref="S4.p6.6.m6.3.3.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p6.6.m6.3b"><apply id="S4.p6.6.m6.3.3.cmml" xref="S4.p6.6.m6.3.3"><csymbol cd="latexml" id="S4.p6.6.m6.3.3.3.cmml" xref="S4.p6.6.m6.3.3.3">assign</csymbol><apply id="S4.p6.6.m6.3.3.4.cmml" xref="S4.p6.6.m6.3.3.4"><csymbol cd="ambiguous" id="S4.p6.6.m6.3.3.4.1.cmml" xref="S4.p6.6.m6.3.3.4">subscript</csymbol><ci id="S4.p6.6.m6.3.3.4.2.cmml" xref="S4.p6.6.m6.3.3.4.2">𝐴</ci><cn id="S4.p6.6.m6.3.3.4.3.cmml" type="integer" xref="S4.p6.6.m6.3.3.4.3">1</cn></apply><apply id="S4.p6.6.m6.3.3.2.3.cmml" xref="S4.p6.6.m6.3.3.2.2"><csymbol cd="latexml" id="S4.p6.6.m6.3.3.2.3.1.cmml" xref="S4.p6.6.m6.3.3.2.2.3">conditional-set</csymbol><apply id="S4.p6.6.m6.2.2.1.1.1.cmml" xref="S4.p6.6.m6.2.2.1.1.1"><times id="S4.p6.6.m6.2.2.1.1.1.1.cmml" xref="S4.p6.6.m6.2.2.1.1.1.1"></times><apply id="S4.p6.6.m6.2.2.1.1.1.2.cmml" xref="S4.p6.6.m6.2.2.1.1.1.2"><csymbol cd="ambiguous" id="S4.p6.6.m6.2.2.1.1.1.2.1.cmml" xref="S4.p6.6.m6.2.2.1.1.1.2">superscript</csymbol><ci id="S4.p6.6.m6.2.2.1.1.1.2.2.cmml" xref="S4.p6.6.m6.2.2.1.1.1.2.2">𝑡</ci><ci id="S4.p6.6.m6.2.2.1.1.1.2.3.cmml" xref="S4.p6.6.m6.2.2.1.1.1.2.3">⌢</ci></apply><apply id="S4.p6.6.m6.2.2.1.1.1.3.1.cmml" xref="S4.p6.6.m6.2.2.1.1.1.3.2"><csymbol cd="latexml" id="S4.p6.6.m6.2.2.1.1.1.3.1.1.cmml" xref="S4.p6.6.m6.2.2.1.1.1.3.2.1">delimited-⟨⟩</csymbol><ci id="S4.p6.6.m6.1.1.cmml" xref="S4.p6.6.m6.1.1">𝜔</ci></apply></apply><apply id="S4.p6.6.m6.3.3.2.2.2.cmml" xref="S4.p6.6.m6.3.3.2.2.2"><in id="S4.p6.6.m6.3.3.2.2.2.1.cmml" xref="S4.p6.6.m6.3.3.2.2.2.1"></in><ci id="S4.p6.6.m6.3.3.2.2.2.2.cmml" xref="S4.p6.6.m6.3.3.2.2.2.2">𝑡</ci><apply id="S4.p6.6.m6.3.3.2.2.2.3.cmml" xref="S4.p6.6.m6.3.3.2.2.2.3"><csymbol cd="ambiguous" id="S4.p6.6.m6.3.3.2.2.2.3.1.cmml" xref="S4.p6.6.m6.3.3.2.2.2.3">subscript</csymbol><ci id="S4.p6.6.m6.3.3.2.2.2.3.2.cmml" xref="S4.p6.6.m6.3.3.2.2.2.3.2">𝐴</ci><cn id="S4.p6.6.m6.3.3.2.2.2.3.3.cmml" type="integer" xref="S4.p6.6.m6.3.3.2.2.2.3.3">0</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p6.6.m6.3c">A_{1}:=\{t^{\frown}\langle\omega\rangle:t\in A_{0}\}</annotation><annotation encoding="application/x-llamapun" id="S4.p6.6.m6.3d">italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT := { italic_t start_POSTSUPERSCRIPT ⌢ end_POSTSUPERSCRIPT ⟨ italic_ω ⟩ : italic_t ∈ italic_A start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT }</annotation></semantics></math>. We understand <math alttext="A^{+}" class="ltx_Math" display="inline" id="S4.p6.7.m7.1"><semantics id="S4.p6.7.m7.1a"><msup id="S4.p6.7.m7.1.1" xref="S4.p6.7.m7.1.1.cmml"><mi id="S4.p6.7.m7.1.1.2" xref="S4.p6.7.m7.1.1.2.cmml">A</mi><mo id="S4.p6.7.m7.1.1.3" xref="S4.p6.7.m7.1.1.3.cmml">+</mo></msup><annotation-xml encoding="MathML-Content" id="S4.p6.7.m7.1b"><apply id="S4.p6.7.m7.1.1.cmml" xref="S4.p6.7.m7.1.1"><csymbol cd="ambiguous" id="S4.p6.7.m7.1.1.1.cmml" xref="S4.p6.7.m7.1.1">superscript</csymbol><ci id="S4.p6.7.m7.1.1.2.cmml" xref="S4.p6.7.m7.1.1.2">𝐴</ci><plus id="S4.p6.7.m7.1.1.3.cmml" xref="S4.p6.7.m7.1.1.3"></plus></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p6.7.m7.1c">A^{+}</annotation><annotation encoding="application/x-llamapun" id="S4.p6.7.m7.1d">italic_A start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> as a linear order with the lexicographic ordering inherited from <math alttext="T" class="ltx_Math" display="inline" id="S4.p6.8.m8.1"><semantics id="S4.p6.8.m8.1a"><mi id="S4.p6.8.m8.1.1" xref="S4.p6.8.m8.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S4.p6.8.m8.1b"><ci id="S4.p6.8.m8.1.1.cmml" xref="S4.p6.8.m8.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p6.8.m8.1c">T</annotation><annotation encoding="application/x-llamapun" id="S4.p6.8.m8.1d">italic_T</annotation></semantics></math>, so <math alttext="A^{+}" class="ltx_Math" display="inline" id="S4.p6.9.m9.1"><semantics id="S4.p6.9.m9.1a"><msup id="S4.p6.9.m9.1.1" xref="S4.p6.9.m9.1.1.cmml"><mi id="S4.p6.9.m9.1.1.2" xref="S4.p6.9.m9.1.1.2.cmml">A</mi><mo id="S4.p6.9.m9.1.1.3" xref="S4.p6.9.m9.1.1.3.cmml">+</mo></msup><annotation-xml encoding="MathML-Content" id="S4.p6.9.m9.1b"><apply id="S4.p6.9.m9.1.1.cmml" xref="S4.p6.9.m9.1.1"><csymbol cd="ambiguous" id="S4.p6.9.m9.1.1.1.cmml" xref="S4.p6.9.m9.1.1">superscript</csymbol><ci id="S4.p6.9.m9.1.1.2.cmml" xref="S4.p6.9.m9.1.1.2">𝐴</ci><plus id="S4.p6.9.m9.1.1.3.cmml" xref="S4.p6.9.m9.1.1.3"></plus></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p6.9.m9.1c">A^{+}</annotation><annotation encoding="application/x-llamapun" id="S4.p6.9.m9.1d">italic_A start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> is an Aronszajn line.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S4.Thmtheorem4"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem4.1.1.1">Lemma 4.4</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem4.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem4.p1"> <p class="ltx_p" id="S4.Thmtheorem4.p1.2"><math alttext="A^{+}" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.1.m1.1"><semantics id="S4.Thmtheorem4.p1.1.m1.1a"><msup id="S4.Thmtheorem4.p1.1.m1.1.1" xref="S4.Thmtheorem4.p1.1.m1.1.1.cmml"><mi id="S4.Thmtheorem4.p1.1.m1.1.1.2" xref="S4.Thmtheorem4.p1.1.m1.1.1.2.cmml">A</mi><mo id="S4.Thmtheorem4.p1.1.m1.1.1.3" xref="S4.Thmtheorem4.p1.1.m1.1.1.3.cmml">+</mo></msup><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.1.m1.1b"><apply id="S4.Thmtheorem4.p1.1.m1.1.1.cmml" xref="S4.Thmtheorem4.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem4.p1.1.m1.1.1.1.cmml" xref="S4.Thmtheorem4.p1.1.m1.1.1">superscript</csymbol><ci id="S4.Thmtheorem4.p1.1.m1.1.1.2.cmml" xref="S4.Thmtheorem4.p1.1.m1.1.1.2">𝐴</ci><plus id="S4.Thmtheorem4.p1.1.m1.1.1.3.cmml" xref="S4.Thmtheorem4.p1.1.m1.1.1.3"></plus></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.1.m1.1c">A^{+}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.1.m1.1d">italic_A start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> is <math alttext="\aleph_{1}" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.2.m2.1"><semantics id="S4.Thmtheorem4.p1.2.m2.1a"><msub id="S4.Thmtheorem4.p1.2.m2.1.1" xref="S4.Thmtheorem4.p1.2.m2.1.1.cmml"><mi id="S4.Thmtheorem4.p1.2.m2.1.1.2" mathvariant="normal" xref="S4.Thmtheorem4.p1.2.m2.1.1.2.cmml">ℵ</mi><mn id="S4.Thmtheorem4.p1.2.m2.1.1.3" xref="S4.Thmtheorem4.p1.2.m2.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.2.m2.1b"><apply id="S4.Thmtheorem4.p1.2.m2.1.1.cmml" xref="S4.Thmtheorem4.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem4.p1.2.m2.1.1.1.cmml" xref="S4.Thmtheorem4.p1.2.m2.1.1">subscript</csymbol><ci id="S4.Thmtheorem4.p1.2.m2.1.1.2.cmml" xref="S4.Thmtheorem4.p1.2.m2.1.1.2">ℵ</ci><cn id="S4.Thmtheorem4.p1.2.m2.1.1.3.cmml" type="integer" xref="S4.Thmtheorem4.p1.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.2.m2.1c">\aleph_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.2.m2.1d">roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-dense.</p> </div> </div> <div class="ltx_proof" id="S4.3"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S4.3.p1"> <p class="ltx_p" id="S4.3.p1.20">We show that if <math alttext="t<_{\mathrm{lex}}s" class="ltx_Math" display="inline" id="S4.3.p1.1.m1.1"><semantics id="S4.3.p1.1.m1.1a"><mrow id="S4.3.p1.1.m1.1.1" xref="S4.3.p1.1.m1.1.1.cmml"><mi id="S4.3.p1.1.m1.1.1.2" xref="S4.3.p1.1.m1.1.1.2.cmml">t</mi><msub id="S4.3.p1.1.m1.1.1.1" xref="S4.3.p1.1.m1.1.1.1.cmml"><mo id="S4.3.p1.1.m1.1.1.1.2" xref="S4.3.p1.1.m1.1.1.1.2.cmml"><</mo><mi id="S4.3.p1.1.m1.1.1.1.3" xref="S4.3.p1.1.m1.1.1.1.3.cmml">lex</mi></msub><mi id="S4.3.p1.1.m1.1.1.3" xref="S4.3.p1.1.m1.1.1.3.cmml">s</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.3.p1.1.m1.1b"><apply id="S4.3.p1.1.m1.1.1.cmml" xref="S4.3.p1.1.m1.1.1"><apply id="S4.3.p1.1.m1.1.1.1.cmml" xref="S4.3.p1.1.m1.1.1.1"><csymbol cd="ambiguous" id="S4.3.p1.1.m1.1.1.1.1.cmml" xref="S4.3.p1.1.m1.1.1.1">subscript</csymbol><lt id="S4.3.p1.1.m1.1.1.1.2.cmml" xref="S4.3.p1.1.m1.1.1.1.2"></lt><ci id="S4.3.p1.1.m1.1.1.1.3.cmml" xref="S4.3.p1.1.m1.1.1.1.3">lex</ci></apply><ci id="S4.3.p1.1.m1.1.1.2.cmml" xref="S4.3.p1.1.m1.1.1.2">𝑡</ci><ci id="S4.3.p1.1.m1.1.1.3.cmml" xref="S4.3.p1.1.m1.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.3.p1.1.m1.1c">t<_{\mathrm{lex}}s</annotation><annotation encoding="application/x-llamapun" id="S4.3.p1.1.m1.1d">italic_t < start_POSTSUBSCRIPT roman_lex end_POSTSUBSCRIPT italic_s</annotation></semantics></math> are in <math alttext="A^{+}" class="ltx_Math" display="inline" id="S4.3.p1.2.m2.1"><semantics id="S4.3.p1.2.m2.1a"><msup id="S4.3.p1.2.m2.1.1" xref="S4.3.p1.2.m2.1.1.cmml"><mi id="S4.3.p1.2.m2.1.1.2" xref="S4.3.p1.2.m2.1.1.2.cmml">A</mi><mo id="S4.3.p1.2.m2.1.1.3" xref="S4.3.p1.2.m2.1.1.3.cmml">+</mo></msup><annotation-xml encoding="MathML-Content" id="S4.3.p1.2.m2.1b"><apply id="S4.3.p1.2.m2.1.1.cmml" xref="S4.3.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S4.3.p1.2.m2.1.1.1.cmml" xref="S4.3.p1.2.m2.1.1">superscript</csymbol><ci id="S4.3.p1.2.m2.1.1.2.cmml" xref="S4.3.p1.2.m2.1.1.2">𝐴</ci><plus id="S4.3.p1.2.m2.1.1.3.cmml" xref="S4.3.p1.2.m2.1.1.3"></plus></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.3.p1.2.m2.1c">A^{+}</annotation><annotation encoding="application/x-llamapun" id="S4.3.p1.2.m2.1d">italic_A start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math>, then <math alttext="[t,s]_{\mathrm{lex}}\cap A^{+}" class="ltx_Math" display="inline" id="S4.3.p1.3.m3.2"><semantics id="S4.3.p1.3.m3.2a"><mrow id="S4.3.p1.3.m3.2.3" xref="S4.3.p1.3.m3.2.3.cmml"><msub id="S4.3.p1.3.m3.2.3.2" xref="S4.3.p1.3.m3.2.3.2.cmml"><mrow id="S4.3.p1.3.m3.2.3.2.2.2" xref="S4.3.p1.3.m3.2.3.2.2.1.cmml"><mo id="S4.3.p1.3.m3.2.3.2.2.2.1" stretchy="false" xref="S4.3.p1.3.m3.2.3.2.2.1.cmml">[</mo><mi id="S4.3.p1.3.m3.1.1" xref="S4.3.p1.3.m3.1.1.cmml">t</mi><mo id="S4.3.p1.3.m3.2.3.2.2.2.2" xref="S4.3.p1.3.m3.2.3.2.2.1.cmml">,</mo><mi id="S4.3.p1.3.m3.2.2" xref="S4.3.p1.3.m3.2.2.cmml">s</mi><mo id="S4.3.p1.3.m3.2.3.2.2.2.3" stretchy="false" xref="S4.3.p1.3.m3.2.3.2.2.1.cmml">]</mo></mrow><mi id="S4.3.p1.3.m3.2.3.2.3" xref="S4.3.p1.3.m3.2.3.2.3.cmml">lex</mi></msub><mo id="S4.3.p1.3.m3.2.3.1" xref="S4.3.p1.3.m3.2.3.1.cmml">∩</mo><msup id="S4.3.p1.3.m3.2.3.3" xref="S4.3.p1.3.m3.2.3.3.cmml"><mi id="S4.3.p1.3.m3.2.3.3.2" xref="S4.3.p1.3.m3.2.3.3.2.cmml">A</mi><mo id="S4.3.p1.3.m3.2.3.3.3" xref="S4.3.p1.3.m3.2.3.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.3.p1.3.m3.2b"><apply id="S4.3.p1.3.m3.2.3.cmml" xref="S4.3.p1.3.m3.2.3"><intersect id="S4.3.p1.3.m3.2.3.1.cmml" xref="S4.3.p1.3.m3.2.3.1"></intersect><apply id="S4.3.p1.3.m3.2.3.2.cmml" xref="S4.3.p1.3.m3.2.3.2"><csymbol cd="ambiguous" id="S4.3.p1.3.m3.2.3.2.1.cmml" xref="S4.3.p1.3.m3.2.3.2">subscript</csymbol><interval closure="closed" id="S4.3.p1.3.m3.2.3.2.2.1.cmml" xref="S4.3.p1.3.m3.2.3.2.2.2"><ci id="S4.3.p1.3.m3.1.1.cmml" xref="S4.3.p1.3.m3.1.1">𝑡</ci><ci id="S4.3.p1.3.m3.2.2.cmml" xref="S4.3.p1.3.m3.2.2">𝑠</ci></interval><ci id="S4.3.p1.3.m3.2.3.2.3.cmml" xref="S4.3.p1.3.m3.2.3.2.3">lex</ci></apply><apply id="S4.3.p1.3.m3.2.3.3.cmml" xref="S4.3.p1.3.m3.2.3.3"><csymbol cd="ambiguous" id="S4.3.p1.3.m3.2.3.3.1.cmml" xref="S4.3.p1.3.m3.2.3.3">superscript</csymbol><ci id="S4.3.p1.3.m3.2.3.3.2.cmml" xref="S4.3.p1.3.m3.2.3.3.2">𝐴</ci><plus id="S4.3.p1.3.m3.2.3.3.3.cmml" xref="S4.3.p1.3.m3.2.3.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.3.p1.3.m3.2c">[t,s]_{\mathrm{lex}}\cap A^{+}</annotation><annotation encoding="application/x-llamapun" id="S4.3.p1.3.m3.2d">[ italic_t , italic_s ] start_POSTSUBSCRIPT roman_lex end_POSTSUBSCRIPT ∩ italic_A start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> is uncountable. The proof that <math alttext="A^{+}" class="ltx_Math" display="inline" id="S4.3.p1.4.m4.1"><semantics id="S4.3.p1.4.m4.1a"><msup id="S4.3.p1.4.m4.1.1" xref="S4.3.p1.4.m4.1.1.cmml"><mi id="S4.3.p1.4.m4.1.1.2" xref="S4.3.p1.4.m4.1.1.2.cmml">A</mi><mo id="S4.3.p1.4.m4.1.1.3" xref="S4.3.p1.4.m4.1.1.3.cmml">+</mo></msup><annotation-xml encoding="MathML-Content" id="S4.3.p1.4.m4.1b"><apply id="S4.3.p1.4.m4.1.1.cmml" xref="S4.3.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S4.3.p1.4.m4.1.1.1.cmml" xref="S4.3.p1.4.m4.1.1">superscript</csymbol><ci id="S4.3.p1.4.m4.1.1.2.cmml" xref="S4.3.p1.4.m4.1.1.2">𝐴</ci><plus id="S4.3.p1.4.m4.1.1.3.cmml" xref="S4.3.p1.4.m4.1.1.3"></plus></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.3.p1.4.m4.1c">A^{+}</annotation><annotation encoding="application/x-llamapun" id="S4.3.p1.4.m4.1d">italic_A start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> has no endpoints is similar. First observe that it is enough to find <math alttext="u\in T" class="ltx_Math" display="inline" id="S4.3.p1.5.m5.1"><semantics id="S4.3.p1.5.m5.1a"><mrow id="S4.3.p1.5.m5.1.1" xref="S4.3.p1.5.m5.1.1.cmml"><mi id="S4.3.p1.5.m5.1.1.2" xref="S4.3.p1.5.m5.1.1.2.cmml">u</mi><mo id="S4.3.p1.5.m5.1.1.1" xref="S4.3.p1.5.m5.1.1.1.cmml">∈</mo><mi id="S4.3.p1.5.m5.1.1.3" xref="S4.3.p1.5.m5.1.1.3.cmml">T</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.3.p1.5.m5.1b"><apply id="S4.3.p1.5.m5.1.1.cmml" xref="S4.3.p1.5.m5.1.1"><in id="S4.3.p1.5.m5.1.1.1.cmml" xref="S4.3.p1.5.m5.1.1.1"></in><ci id="S4.3.p1.5.m5.1.1.2.cmml" xref="S4.3.p1.5.m5.1.1.2">𝑢</ci><ci id="S4.3.p1.5.m5.1.1.3.cmml" xref="S4.3.p1.5.m5.1.1.3">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.3.p1.5.m5.1c">u\in T</annotation><annotation encoding="application/x-llamapun" id="S4.3.p1.5.m5.1d">italic_u ∈ italic_T</annotation></semantics></math> such that <math alttext="t<_{\mathrm{lex}}u<_{\mathrm{lex}}s" class="ltx_Math" display="inline" id="S4.3.p1.6.m6.1"><semantics id="S4.3.p1.6.m6.1a"><mrow id="S4.3.p1.6.m6.1.1" xref="S4.3.p1.6.m6.1.1.cmml"><mi id="S4.3.p1.6.m6.1.1.2" xref="S4.3.p1.6.m6.1.1.2.cmml">t</mi><msub id="S4.3.p1.6.m6.1.1.3" xref="S4.3.p1.6.m6.1.1.3.cmml"><mo id="S4.3.p1.6.m6.1.1.3.2" xref="S4.3.p1.6.m6.1.1.3.2.cmml"><</mo><mi id="S4.3.p1.6.m6.1.1.3.3" xref="S4.3.p1.6.m6.1.1.3.3.cmml">lex</mi></msub><mi id="S4.3.p1.6.m6.1.1.4" xref="S4.3.p1.6.m6.1.1.4.cmml">u</mi><msub id="S4.3.p1.6.m6.1.1.5" xref="S4.3.p1.6.m6.1.1.5.cmml"><mo id="S4.3.p1.6.m6.1.1.5.2" xref="S4.3.p1.6.m6.1.1.5.2.cmml"><</mo><mi id="S4.3.p1.6.m6.1.1.5.3" xref="S4.3.p1.6.m6.1.1.5.3.cmml">lex</mi></msub><mi id="S4.3.p1.6.m6.1.1.6" xref="S4.3.p1.6.m6.1.1.6.cmml">s</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.3.p1.6.m6.1b"><apply id="S4.3.p1.6.m6.1.1.cmml" xref="S4.3.p1.6.m6.1.1"><and id="S4.3.p1.6.m6.1.1a.cmml" xref="S4.3.p1.6.m6.1.1"></and><apply id="S4.3.p1.6.m6.1.1b.cmml" xref="S4.3.p1.6.m6.1.1"><apply id="S4.3.p1.6.m6.1.1.3.cmml" xref="S4.3.p1.6.m6.1.1.3"><csymbol cd="ambiguous" id="S4.3.p1.6.m6.1.1.3.1.cmml" xref="S4.3.p1.6.m6.1.1.3">subscript</csymbol><lt id="S4.3.p1.6.m6.1.1.3.2.cmml" xref="S4.3.p1.6.m6.1.1.3.2"></lt><ci id="S4.3.p1.6.m6.1.1.3.3.cmml" xref="S4.3.p1.6.m6.1.1.3.3">lex</ci></apply><ci id="S4.3.p1.6.m6.1.1.2.cmml" xref="S4.3.p1.6.m6.1.1.2">𝑡</ci><ci id="S4.3.p1.6.m6.1.1.4.cmml" xref="S4.3.p1.6.m6.1.1.4">𝑢</ci></apply><apply id="S4.3.p1.6.m6.1.1c.cmml" xref="S4.3.p1.6.m6.1.1"><apply id="S4.3.p1.6.m6.1.1.5.cmml" xref="S4.3.p1.6.m6.1.1.5"><csymbol cd="ambiguous" id="S4.3.p1.6.m6.1.1.5.1.cmml" xref="S4.3.p1.6.m6.1.1.5">subscript</csymbol><lt id="S4.3.p1.6.m6.1.1.5.2.cmml" xref="S4.3.p1.6.m6.1.1.5.2"></lt><ci id="S4.3.p1.6.m6.1.1.5.3.cmml" xref="S4.3.p1.6.m6.1.1.5.3">lex</ci></apply><share href="https://arxiv.org/html/2503.13728v1#S4.3.p1.6.m6.1.1.4.cmml" id="S4.3.p1.6.m6.1.1d.cmml" xref="S4.3.p1.6.m6.1.1"></share><ci id="S4.3.p1.6.m6.1.1.6.cmml" xref="S4.3.p1.6.m6.1.1.6">𝑠</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.3.p1.6.m6.1c">t<_{\mathrm{lex}}u<_{\mathrm{lex}}s</annotation><annotation encoding="application/x-llamapun" id="S4.3.p1.6.m6.1d">italic_t < start_POSTSUBSCRIPT roman_lex end_POSTSUBSCRIPT italic_u < start_POSTSUBSCRIPT roman_lex end_POSTSUBSCRIPT italic_s</annotation></semantics></math> and is <math alttext="T" class="ltx_Math" display="inline" id="S4.3.p1.7.m7.1"><semantics id="S4.3.p1.7.m7.1a"><mi id="S4.3.p1.7.m7.1.1" xref="S4.3.p1.7.m7.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S4.3.p1.7.m7.1b"><ci id="S4.3.p1.7.m7.1.1.cmml" xref="S4.3.p1.7.m7.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.3.p1.7.m7.1c">T</annotation><annotation encoding="application/x-llamapun" id="S4.3.p1.7.m7.1d">italic_T</annotation></semantics></math>-incomparable with <math alttext="s" class="ltx_Math" display="inline" id="S4.3.p1.8.m8.1"><semantics id="S4.3.p1.8.m8.1a"><mi id="S4.3.p1.8.m8.1.1" xref="S4.3.p1.8.m8.1.1.cmml">s</mi><annotation-xml encoding="MathML-Content" id="S4.3.p1.8.m8.1b"><ci id="S4.3.p1.8.m8.1.1.cmml" xref="S4.3.p1.8.m8.1.1">𝑠</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.3.p1.8.m8.1c">s</annotation><annotation encoding="application/x-llamapun" id="S4.3.p1.8.m8.1d">italic_s</annotation></semantics></math>, because then <math alttext="u\uparrow\cap A^{+}" class="ltx_Math" display="inline" id="S4.3.p1.9.m9.1"><semantics id="S4.3.p1.9.m9.1a"><mrow id="S4.3.p1.9.m9.1.1" xref="S4.3.p1.9.m9.1.1.cmml"><mi id="S4.3.p1.9.m9.1.1.2" xref="S4.3.p1.9.m9.1.1.2.cmml">u</mi><mo id="S4.3.p1.9.m9.1.1.1" rspace="0.1389em" stretchy="false" xref="S4.3.p1.9.m9.1.1.1.cmml">↑</mo><mrow id="S4.3.p1.9.m9.1.1.3" xref="S4.3.p1.9.m9.1.1.3.cmml"><mo id="S4.3.p1.9.m9.1.1.3a" lspace="0.1389em" rspace="0em" xref="S4.3.p1.9.m9.1.1.3.cmml">∩</mo><msup id="S4.3.p1.9.m9.1.1.3.2" xref="S4.3.p1.9.m9.1.1.3.2.cmml"><mi id="S4.3.p1.9.m9.1.1.3.2.2" xref="S4.3.p1.9.m9.1.1.3.2.2.cmml">A</mi><mo id="S4.3.p1.9.m9.1.1.3.2.3" xref="S4.3.p1.9.m9.1.1.3.2.3.cmml">+</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.3.p1.9.m9.1b"><apply id="S4.3.p1.9.m9.1.1.cmml" xref="S4.3.p1.9.m9.1.1"><ci id="S4.3.p1.9.m9.1.1.1.cmml" xref="S4.3.p1.9.m9.1.1.1">↑</ci><ci id="S4.3.p1.9.m9.1.1.2.cmml" xref="S4.3.p1.9.m9.1.1.2">𝑢</ci><apply id="S4.3.p1.9.m9.1.1.3.cmml" xref="S4.3.p1.9.m9.1.1.3"><intersect id="S4.3.p1.9.m9.1.1.3.1.cmml" xref="S4.3.p1.9.m9.1.1.3"></intersect><apply id="S4.3.p1.9.m9.1.1.3.2.cmml" xref="S4.3.p1.9.m9.1.1.3.2"><csymbol cd="ambiguous" id="S4.3.p1.9.m9.1.1.3.2.1.cmml" xref="S4.3.p1.9.m9.1.1.3.2">superscript</csymbol><ci id="S4.3.p1.9.m9.1.1.3.2.2.cmml" xref="S4.3.p1.9.m9.1.1.3.2.2">𝐴</ci><plus id="S4.3.p1.9.m9.1.1.3.2.3.cmml" xref="S4.3.p1.9.m9.1.1.3.2.3"></plus></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.3.p1.9.m9.1c">u\uparrow\cap A^{+}</annotation><annotation encoding="application/x-llamapun" id="S4.3.p1.9.m9.1d">italic_u ↑ ∩ italic_A start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_note ltx_role_footnote" id="footnote3"><sup class="ltx_note_mark">3</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">3</sup><span class="ltx_tag ltx_tag_note">3</span> <math alttext="u\uparrow" class="ltx_Math" display="inline" id="footnote3.m1.1"><semantics id="footnote3.m1.1b"><mrow id="footnote3.m1.1.1" xref="footnote3.m1.1.1.cmml"><mi id="footnote3.m1.1.1.2" xref="footnote3.m1.1.1.2.cmml">u</mi><mo id="footnote3.m1.1.1.1" stretchy="false" xref="footnote3.m1.1.1.1.cmml">↑</mo><mi id="footnote3.m1.1.1.3" xref="footnote3.m1.1.1.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="footnote3.m1.1c"><apply id="footnote3.m1.1.1.cmml" xref="footnote3.m1.1.1"><ci id="footnote3.m1.1.1.1.cmml" xref="footnote3.m1.1.1.1">↑</ci><ci id="footnote3.m1.1.1.2.cmml" xref="footnote3.m1.1.1.2">𝑢</ci><csymbol cd="latexml" id="footnote3.m1.1.1.3.cmml" xref="footnote3.m1.1.1.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote3.m1.1d">u\uparrow</annotation><annotation encoding="application/x-llamapun" id="footnote3.m1.1e">italic_u ↑</annotation></semantics></math> stands for <math alttext="\{t\in T:u\leq_{T}t\}" class="ltx_Math" display="inline" id="footnote3.m2.2"><semantics id="footnote3.m2.2b"><mrow id="footnote3.m2.2.2.2" xref="footnote3.m2.2.2.3.cmml"><mo id="footnote3.m2.2.2.2.3" stretchy="false" xref="footnote3.m2.2.2.3.1.cmml">{</mo><mrow id="footnote3.m2.1.1.1.1" xref="footnote3.m2.1.1.1.1.cmml"><mi id="footnote3.m2.1.1.1.1.2" xref="footnote3.m2.1.1.1.1.2.cmml">t</mi><mo id="footnote3.m2.1.1.1.1.1" xref="footnote3.m2.1.1.1.1.1.cmml">∈</mo><mi id="footnote3.m2.1.1.1.1.3" xref="footnote3.m2.1.1.1.1.3.cmml">T</mi></mrow><mo id="footnote3.m2.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="footnote3.m2.2.2.3.1.cmml">:</mo><mrow id="footnote3.m2.2.2.2.2" xref="footnote3.m2.2.2.2.2.cmml"><mi id="footnote3.m2.2.2.2.2.2" xref="footnote3.m2.2.2.2.2.2.cmml">u</mi><msub id="footnote3.m2.2.2.2.2.1" xref="footnote3.m2.2.2.2.2.1.cmml"><mo id="footnote3.m2.2.2.2.2.1.2" xref="footnote3.m2.2.2.2.2.1.2.cmml">≤</mo><mi id="footnote3.m2.2.2.2.2.1.3" xref="footnote3.m2.2.2.2.2.1.3.cmml">T</mi></msub><mi id="footnote3.m2.2.2.2.2.3" xref="footnote3.m2.2.2.2.2.3.cmml">t</mi></mrow><mo id="footnote3.m2.2.2.2.5" stretchy="false" xref="footnote3.m2.2.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="footnote3.m2.2c"><apply id="footnote3.m2.2.2.3.cmml" xref="footnote3.m2.2.2.2"><csymbol cd="latexml" id="footnote3.m2.2.2.3.1.cmml" xref="footnote3.m2.2.2.2.3">conditional-set</csymbol><apply id="footnote3.m2.1.1.1.1.cmml" xref="footnote3.m2.1.1.1.1"><in id="footnote3.m2.1.1.1.1.1.cmml" xref="footnote3.m2.1.1.1.1.1"></in><ci id="footnote3.m2.1.1.1.1.2.cmml" xref="footnote3.m2.1.1.1.1.2">𝑡</ci><ci id="footnote3.m2.1.1.1.1.3.cmml" xref="footnote3.m2.1.1.1.1.3">𝑇</ci></apply><apply id="footnote3.m2.2.2.2.2.cmml" xref="footnote3.m2.2.2.2.2"><apply id="footnote3.m2.2.2.2.2.1.cmml" xref="footnote3.m2.2.2.2.2.1"><csymbol cd="ambiguous" id="footnote3.m2.2.2.2.2.1.1.cmml" xref="footnote3.m2.2.2.2.2.1">subscript</csymbol><leq id="footnote3.m2.2.2.2.2.1.2.cmml" xref="footnote3.m2.2.2.2.2.1.2"></leq><ci id="footnote3.m2.2.2.2.2.1.3.cmml" xref="footnote3.m2.2.2.2.2.1.3">𝑇</ci></apply><ci id="footnote3.m2.2.2.2.2.2.cmml" xref="footnote3.m2.2.2.2.2.2">𝑢</ci><ci id="footnote3.m2.2.2.2.2.3.cmml" xref="footnote3.m2.2.2.2.2.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote3.m2.2d">\{t\in T:u\leq_{T}t\}</annotation><annotation encoding="application/x-llamapun" id="footnote3.m2.2e">{ italic_t ∈ italic_T : italic_u ≤ start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT italic_t }</annotation></semantics></math></span></span></span>. is uncountable and contained in <math alttext="[t,s]_{\mathrm{lex}}" class="ltx_Math" display="inline" id="S4.3.p1.10.m10.2"><semantics id="S4.3.p1.10.m10.2a"><msub id="S4.3.p1.10.m10.2.3" xref="S4.3.p1.10.m10.2.3.cmml"><mrow id="S4.3.p1.10.m10.2.3.2.2" xref="S4.3.p1.10.m10.2.3.2.1.cmml"><mo id="S4.3.p1.10.m10.2.3.2.2.1" stretchy="false" xref="S4.3.p1.10.m10.2.3.2.1.cmml">[</mo><mi id="S4.3.p1.10.m10.1.1" xref="S4.3.p1.10.m10.1.1.cmml">t</mi><mo id="S4.3.p1.10.m10.2.3.2.2.2" xref="S4.3.p1.10.m10.2.3.2.1.cmml">,</mo><mi id="S4.3.p1.10.m10.2.2" xref="S4.3.p1.10.m10.2.2.cmml">s</mi><mo id="S4.3.p1.10.m10.2.3.2.2.3" stretchy="false" xref="S4.3.p1.10.m10.2.3.2.1.cmml">]</mo></mrow><mi id="S4.3.p1.10.m10.2.3.3" xref="S4.3.p1.10.m10.2.3.3.cmml">lex</mi></msub><annotation-xml encoding="MathML-Content" id="S4.3.p1.10.m10.2b"><apply id="S4.3.p1.10.m10.2.3.cmml" xref="S4.3.p1.10.m10.2.3"><csymbol cd="ambiguous" id="S4.3.p1.10.m10.2.3.1.cmml" xref="S4.3.p1.10.m10.2.3">subscript</csymbol><interval closure="closed" id="S4.3.p1.10.m10.2.3.2.1.cmml" xref="S4.3.p1.10.m10.2.3.2.2"><ci id="S4.3.p1.10.m10.1.1.cmml" xref="S4.3.p1.10.m10.1.1">𝑡</ci><ci id="S4.3.p1.10.m10.2.2.cmml" xref="S4.3.p1.10.m10.2.2">𝑠</ci></interval><ci id="S4.3.p1.10.m10.2.3.3.cmml" xref="S4.3.p1.10.m10.2.3.3">lex</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.3.p1.10.m10.2c">[t,s]_{\mathrm{lex}}</annotation><annotation encoding="application/x-llamapun" id="S4.3.p1.10.m10.2d">[ italic_t , italic_s ] start_POSTSUBSCRIPT roman_lex end_POSTSUBSCRIPT</annotation></semantics></math>. This is clear if <math alttext="t\not\sqsubset s" class="ltx_Math" display="inline" id="S4.3.p1.11.m11.1"><semantics id="S4.3.p1.11.m11.1a"><mrow id="S4.3.p1.11.m11.1.1" xref="S4.3.p1.11.m11.1.1.cmml"><mi id="S4.3.p1.11.m11.1.1.2" xref="S4.3.p1.11.m11.1.1.2.cmml">t</mi><mo id="S4.3.p1.11.m11.1.1.1" xref="S4.3.p1.11.m11.1.1.1.cmml">⊏̸</mo><mi id="S4.3.p1.11.m11.1.1.3" xref="S4.3.p1.11.m11.1.1.3.cmml">s</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.3.p1.11.m11.1b"><apply id="S4.3.p1.11.m11.1.1.cmml" xref="S4.3.p1.11.m11.1.1"><csymbol cd="latexml" id="S4.3.p1.11.m11.1.1.1.cmml" xref="S4.3.p1.11.m11.1.1.1">not-square-image-of</csymbol><ci id="S4.3.p1.11.m11.1.1.2.cmml" xref="S4.3.p1.11.m11.1.1.2">𝑡</ci><ci id="S4.3.p1.11.m11.1.1.3.cmml" xref="S4.3.p1.11.m11.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.3.p1.11.m11.1c">t\not\sqsubset s</annotation><annotation encoding="application/x-llamapun" id="S4.3.p1.11.m11.1d">italic_t ⊏̸ italic_s</annotation></semantics></math> (simply take <math alttext="u:=t)" class="ltx_math_unparsed" display="inline" id="S4.3.p1.12.m12.1"><semantics id="S4.3.p1.12.m12.1a"><mrow id="S4.3.p1.12.m12.1b"><mi id="S4.3.p1.12.m12.1.1">u</mi><mo id="S4.3.p1.12.m12.1.2" lspace="0.278em" rspace="0.278em">:=</mo><mi id="S4.3.p1.12.m12.1.3">t</mi><mo id="S4.3.p1.12.m12.1.4" stretchy="false">)</mo></mrow><annotation encoding="application/x-tex" id="S4.3.p1.12.m12.1c">u:=t)</annotation><annotation encoding="application/x-llamapun" id="S4.3.p1.12.m12.1d">italic_u := italic_t )</annotation></semantics></math>, thus assume that <math alttext="t\sqsubset s" class="ltx_Math" display="inline" id="S4.3.p1.13.m13.1"><semantics id="S4.3.p1.13.m13.1a"><mrow id="S4.3.p1.13.m13.1.1" xref="S4.3.p1.13.m13.1.1.cmml"><mi id="S4.3.p1.13.m13.1.1.2" xref="S4.3.p1.13.m13.1.1.2.cmml">t</mi><mo id="S4.3.p1.13.m13.1.1.1" xref="S4.3.p1.13.m13.1.1.1.cmml">⊏</mo><mi id="S4.3.p1.13.m13.1.1.3" xref="S4.3.p1.13.m13.1.1.3.cmml">s</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.3.p1.13.m13.1b"><apply id="S4.3.p1.13.m13.1.1.cmml" xref="S4.3.p1.13.m13.1.1"><csymbol cd="latexml" id="S4.3.p1.13.m13.1.1.1.cmml" xref="S4.3.p1.13.m13.1.1.1">square-image-of</csymbol><ci id="S4.3.p1.13.m13.1.1.2.cmml" xref="S4.3.p1.13.m13.1.1.2">𝑡</ci><ci id="S4.3.p1.13.m13.1.1.3.cmml" xref="S4.3.p1.13.m13.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.3.p1.13.m13.1c">t\sqsubset s</annotation><annotation encoding="application/x-llamapun" id="S4.3.p1.13.m13.1d">italic_t ⊏ italic_s</annotation></semantics></math>. Note then that <math alttext="t\notin A_{1}" class="ltx_Math" display="inline" id="S4.3.p1.14.m14.1"><semantics id="S4.3.p1.14.m14.1a"><mrow id="S4.3.p1.14.m14.1.1" xref="S4.3.p1.14.m14.1.1.cmml"><mi id="S4.3.p1.14.m14.1.1.2" xref="S4.3.p1.14.m14.1.1.2.cmml">t</mi><mo id="S4.3.p1.14.m14.1.1.1" xref="S4.3.p1.14.m14.1.1.1.cmml">∉</mo><msub id="S4.3.p1.14.m14.1.1.3" xref="S4.3.p1.14.m14.1.1.3.cmml"><mi id="S4.3.p1.14.m14.1.1.3.2" xref="S4.3.p1.14.m14.1.1.3.2.cmml">A</mi><mn id="S4.3.p1.14.m14.1.1.3.3" xref="S4.3.p1.14.m14.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.3.p1.14.m14.1b"><apply id="S4.3.p1.14.m14.1.1.cmml" xref="S4.3.p1.14.m14.1.1"><notin id="S4.3.p1.14.m14.1.1.1.cmml" xref="S4.3.p1.14.m14.1.1.1"></notin><ci id="S4.3.p1.14.m14.1.1.2.cmml" xref="S4.3.p1.14.m14.1.1.2">𝑡</ci><apply id="S4.3.p1.14.m14.1.1.3.cmml" xref="S4.3.p1.14.m14.1.1.3"><csymbol cd="ambiguous" id="S4.3.p1.14.m14.1.1.3.1.cmml" xref="S4.3.p1.14.m14.1.1.3">subscript</csymbol><ci id="S4.3.p1.14.m14.1.1.3.2.cmml" xref="S4.3.p1.14.m14.1.1.3.2">𝐴</ci><cn id="S4.3.p1.14.m14.1.1.3.3.cmml" type="integer" xref="S4.3.p1.14.m14.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.3.p1.14.m14.1c">t\notin A_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.3.p1.14.m14.1d">italic_t ∉ italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, because those have no extensions, so <math alttext="t\in A_{0}" class="ltx_Math" display="inline" id="S4.3.p1.15.m15.1"><semantics id="S4.3.p1.15.m15.1a"><mrow id="S4.3.p1.15.m15.1.1" xref="S4.3.p1.15.m15.1.1.cmml"><mi id="S4.3.p1.15.m15.1.1.2" xref="S4.3.p1.15.m15.1.1.2.cmml">t</mi><mo id="S4.3.p1.15.m15.1.1.1" xref="S4.3.p1.15.m15.1.1.1.cmml">∈</mo><msub id="S4.3.p1.15.m15.1.1.3" xref="S4.3.p1.15.m15.1.1.3.cmml"><mi id="S4.3.p1.15.m15.1.1.3.2" xref="S4.3.p1.15.m15.1.1.3.2.cmml">A</mi><mn id="S4.3.p1.15.m15.1.1.3.3" xref="S4.3.p1.15.m15.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.3.p1.15.m15.1b"><apply id="S4.3.p1.15.m15.1.1.cmml" xref="S4.3.p1.15.m15.1.1"><in id="S4.3.p1.15.m15.1.1.1.cmml" xref="S4.3.p1.15.m15.1.1.1"></in><ci id="S4.3.p1.15.m15.1.1.2.cmml" xref="S4.3.p1.15.m15.1.1.2">𝑡</ci><apply id="S4.3.p1.15.m15.1.1.3.cmml" xref="S4.3.p1.15.m15.1.1.3"><csymbol cd="ambiguous" id="S4.3.p1.15.m15.1.1.3.1.cmml" xref="S4.3.p1.15.m15.1.1.3">subscript</csymbol><ci id="S4.3.p1.15.m15.1.1.3.2.cmml" xref="S4.3.p1.15.m15.1.1.3.2">𝐴</ci><cn id="S4.3.p1.15.m15.1.1.3.3.cmml" type="integer" xref="S4.3.p1.15.m15.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.3.p1.15.m15.1c">t\in A_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.3.p1.15.m15.1d">italic_t ∈ italic_A start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>. Now observe that <math alttext="\operatorname{dom}(t)+\omega\leq\operatorname{dom}(s)" class="ltx_Math" display="inline" id="S4.3.p1.16.m16.4"><semantics id="S4.3.p1.16.m16.4a"><mrow id="S4.3.p1.16.m16.4.5" xref="S4.3.p1.16.m16.4.5.cmml"><mrow id="S4.3.p1.16.m16.4.5.2" xref="S4.3.p1.16.m16.4.5.2.cmml"><mrow id="S4.3.p1.16.m16.4.5.2.2.2" xref="S4.3.p1.16.m16.4.5.2.2.1.cmml"><mi id="S4.3.p1.16.m16.1.1" xref="S4.3.p1.16.m16.1.1.cmml">dom</mi><mo id="S4.3.p1.16.m16.4.5.2.2.2a" xref="S4.3.p1.16.m16.4.5.2.2.1.cmml">⁡</mo><mrow id="S4.3.p1.16.m16.4.5.2.2.2.1" xref="S4.3.p1.16.m16.4.5.2.2.1.cmml"><mo id="S4.3.p1.16.m16.4.5.2.2.2.1.1" stretchy="false" xref="S4.3.p1.16.m16.4.5.2.2.1.cmml">(</mo><mi id="S4.3.p1.16.m16.2.2" xref="S4.3.p1.16.m16.2.2.cmml">t</mi><mo id="S4.3.p1.16.m16.4.5.2.2.2.1.2" stretchy="false" xref="S4.3.p1.16.m16.4.5.2.2.1.cmml">)</mo></mrow></mrow><mo id="S4.3.p1.16.m16.4.5.2.1" xref="S4.3.p1.16.m16.4.5.2.1.cmml">+</mo><mi id="S4.3.p1.16.m16.4.5.2.3" xref="S4.3.p1.16.m16.4.5.2.3.cmml">ω</mi></mrow><mo id="S4.3.p1.16.m16.4.5.1" xref="S4.3.p1.16.m16.4.5.1.cmml">≤</mo><mrow id="S4.3.p1.16.m16.4.5.3.2" xref="S4.3.p1.16.m16.4.5.3.1.cmml"><mi id="S4.3.p1.16.m16.3.3" xref="S4.3.p1.16.m16.3.3.cmml">dom</mi><mo id="S4.3.p1.16.m16.4.5.3.2a" xref="S4.3.p1.16.m16.4.5.3.1.cmml">⁡</mo><mrow id="S4.3.p1.16.m16.4.5.3.2.1" xref="S4.3.p1.16.m16.4.5.3.1.cmml"><mo id="S4.3.p1.16.m16.4.5.3.2.1.1" stretchy="false" xref="S4.3.p1.16.m16.4.5.3.1.cmml">(</mo><mi id="S4.3.p1.16.m16.4.4" xref="S4.3.p1.16.m16.4.4.cmml">s</mi><mo id="S4.3.p1.16.m16.4.5.3.2.1.2" stretchy="false" xref="S4.3.p1.16.m16.4.5.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.3.p1.16.m16.4b"><apply id="S4.3.p1.16.m16.4.5.cmml" xref="S4.3.p1.16.m16.4.5"><leq id="S4.3.p1.16.m16.4.5.1.cmml" xref="S4.3.p1.16.m16.4.5.1"></leq><apply id="S4.3.p1.16.m16.4.5.2.cmml" xref="S4.3.p1.16.m16.4.5.2"><plus id="S4.3.p1.16.m16.4.5.2.1.cmml" xref="S4.3.p1.16.m16.4.5.2.1"></plus><apply id="S4.3.p1.16.m16.4.5.2.2.1.cmml" xref="S4.3.p1.16.m16.4.5.2.2.2"><ci id="S4.3.p1.16.m16.1.1.cmml" xref="S4.3.p1.16.m16.1.1">dom</ci><ci id="S4.3.p1.16.m16.2.2.cmml" xref="S4.3.p1.16.m16.2.2">𝑡</ci></apply><ci id="S4.3.p1.16.m16.4.5.2.3.cmml" xref="S4.3.p1.16.m16.4.5.2.3">𝜔</ci></apply><apply id="S4.3.p1.16.m16.4.5.3.1.cmml" xref="S4.3.p1.16.m16.4.5.3.2"><ci id="S4.3.p1.16.m16.3.3.cmml" xref="S4.3.p1.16.m16.3.3">dom</ci><ci id="S4.3.p1.16.m16.4.4.cmml" xref="S4.3.p1.16.m16.4.4">𝑠</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.3.p1.16.m16.4c">\operatorname{dom}(t)+\omega\leq\operatorname{dom}(s)</annotation><annotation encoding="application/x-llamapun" id="S4.3.p1.16.m16.4d">roman_dom ( italic_t ) + italic_ω ≤ roman_dom ( italic_s )</annotation></semantics></math>, and since <math alttext="s" class="ltx_Math" display="inline" id="S4.3.p1.17.m17.1"><semantics id="S4.3.p1.17.m17.1a"><mi id="S4.3.p1.17.m17.1.1" xref="S4.3.p1.17.m17.1.1.cmml">s</mi><annotation-xml encoding="MathML-Content" id="S4.3.p1.17.m17.1b"><ci id="S4.3.p1.17.m17.1.1.cmml" xref="S4.3.p1.17.m17.1.1">𝑠</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.3.p1.17.m17.1c">s</annotation><annotation encoding="application/x-llamapun" id="S4.3.p1.17.m17.1d">italic_s</annotation></semantics></math> is finite-to-one, for some <math alttext="n<\omega" class="ltx_Math" display="inline" id="S4.3.p1.18.m18.1"><semantics id="S4.3.p1.18.m18.1a"><mrow id="S4.3.p1.18.m18.1.1" xref="S4.3.p1.18.m18.1.1.cmml"><mi id="S4.3.p1.18.m18.1.1.2" xref="S4.3.p1.18.m18.1.1.2.cmml">n</mi><mo id="S4.3.p1.18.m18.1.1.1" xref="S4.3.p1.18.m18.1.1.1.cmml"><</mo><mi id="S4.3.p1.18.m18.1.1.3" xref="S4.3.p1.18.m18.1.1.3.cmml">ω</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.3.p1.18.m18.1b"><apply id="S4.3.p1.18.m18.1.1.cmml" xref="S4.3.p1.18.m18.1.1"><lt id="S4.3.p1.18.m18.1.1.1.cmml" xref="S4.3.p1.18.m18.1.1.1"></lt><ci id="S4.3.p1.18.m18.1.1.2.cmml" xref="S4.3.p1.18.m18.1.1.2">𝑛</ci><ci id="S4.3.p1.18.m18.1.1.3.cmml" xref="S4.3.p1.18.m18.1.1.3">𝜔</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.3.p1.18.m18.1c">n<\omega</annotation><annotation encoding="application/x-llamapun" id="S4.3.p1.18.m18.1d">italic_n < italic_ω</annotation></semantics></math>, <math alttext="s(\operatorname{dom}(t)+n+1)\neq 0" class="ltx_Math" display="inline" id="S4.3.p1.19.m19.3"><semantics id="S4.3.p1.19.m19.3a"><mrow id="S4.3.p1.19.m19.3.3" xref="S4.3.p1.19.m19.3.3.cmml"><mrow id="S4.3.p1.19.m19.3.3.1" xref="S4.3.p1.19.m19.3.3.1.cmml"><mi id="S4.3.p1.19.m19.3.3.1.3" xref="S4.3.p1.19.m19.3.3.1.3.cmml">s</mi><mo id="S4.3.p1.19.m19.3.3.1.2" xref="S4.3.p1.19.m19.3.3.1.2.cmml">⁢</mo><mrow id="S4.3.p1.19.m19.3.3.1.1.1" xref="S4.3.p1.19.m19.3.3.1.1.1.1.cmml"><mo id="S4.3.p1.19.m19.3.3.1.1.1.2" stretchy="false" xref="S4.3.p1.19.m19.3.3.1.1.1.1.cmml">(</mo><mrow id="S4.3.p1.19.m19.3.3.1.1.1.1" xref="S4.3.p1.19.m19.3.3.1.1.1.1.cmml"><mrow id="S4.3.p1.19.m19.3.3.1.1.1.1.2.2" xref="S4.3.p1.19.m19.3.3.1.1.1.1.2.1.cmml"><mi id="S4.3.p1.19.m19.1.1" xref="S4.3.p1.19.m19.1.1.cmml">dom</mi><mo id="S4.3.p1.19.m19.3.3.1.1.1.1.2.2a" xref="S4.3.p1.19.m19.3.3.1.1.1.1.2.1.cmml">⁡</mo><mrow id="S4.3.p1.19.m19.3.3.1.1.1.1.2.2.1" xref="S4.3.p1.19.m19.3.3.1.1.1.1.2.1.cmml"><mo id="S4.3.p1.19.m19.3.3.1.1.1.1.2.2.1.1" stretchy="false" xref="S4.3.p1.19.m19.3.3.1.1.1.1.2.1.cmml">(</mo><mi id="S4.3.p1.19.m19.2.2" xref="S4.3.p1.19.m19.2.2.cmml">t</mi><mo id="S4.3.p1.19.m19.3.3.1.1.1.1.2.2.1.2" stretchy="false" xref="S4.3.p1.19.m19.3.3.1.1.1.1.2.1.cmml">)</mo></mrow></mrow><mo id="S4.3.p1.19.m19.3.3.1.1.1.1.1" xref="S4.3.p1.19.m19.3.3.1.1.1.1.1.cmml">+</mo><mi id="S4.3.p1.19.m19.3.3.1.1.1.1.3" xref="S4.3.p1.19.m19.3.3.1.1.1.1.3.cmml">n</mi><mo id="S4.3.p1.19.m19.3.3.1.1.1.1.1a" xref="S4.3.p1.19.m19.3.3.1.1.1.1.1.cmml">+</mo><mn id="S4.3.p1.19.m19.3.3.1.1.1.1.4" xref="S4.3.p1.19.m19.3.3.1.1.1.1.4.cmml">1</mn></mrow><mo id="S4.3.p1.19.m19.3.3.1.1.1.3" stretchy="false" xref="S4.3.p1.19.m19.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.3.p1.19.m19.3.3.2" xref="S4.3.p1.19.m19.3.3.2.cmml">≠</mo><mn id="S4.3.p1.19.m19.3.3.3" xref="S4.3.p1.19.m19.3.3.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.3.p1.19.m19.3b"><apply id="S4.3.p1.19.m19.3.3.cmml" xref="S4.3.p1.19.m19.3.3"><neq id="S4.3.p1.19.m19.3.3.2.cmml" xref="S4.3.p1.19.m19.3.3.2"></neq><apply id="S4.3.p1.19.m19.3.3.1.cmml" xref="S4.3.p1.19.m19.3.3.1"><times id="S4.3.p1.19.m19.3.3.1.2.cmml" xref="S4.3.p1.19.m19.3.3.1.2"></times><ci id="S4.3.p1.19.m19.3.3.1.3.cmml" xref="S4.3.p1.19.m19.3.3.1.3">𝑠</ci><apply id="S4.3.p1.19.m19.3.3.1.1.1.1.cmml" xref="S4.3.p1.19.m19.3.3.1.1.1"><plus id="S4.3.p1.19.m19.3.3.1.1.1.1.1.cmml" xref="S4.3.p1.19.m19.3.3.1.1.1.1.1"></plus><apply id="S4.3.p1.19.m19.3.3.1.1.1.1.2.1.cmml" xref="S4.3.p1.19.m19.3.3.1.1.1.1.2.2"><ci id="S4.3.p1.19.m19.1.1.cmml" xref="S4.3.p1.19.m19.1.1">dom</ci><ci id="S4.3.p1.19.m19.2.2.cmml" xref="S4.3.p1.19.m19.2.2">𝑡</ci></apply><ci id="S4.3.p1.19.m19.3.3.1.1.1.1.3.cmml" xref="S4.3.p1.19.m19.3.3.1.1.1.1.3">𝑛</ci><cn id="S4.3.p1.19.m19.3.3.1.1.1.1.4.cmml" type="integer" xref="S4.3.p1.19.m19.3.3.1.1.1.1.4">1</cn></apply></apply><cn id="S4.3.p1.19.m19.3.3.3.cmml" type="integer" xref="S4.3.p1.19.m19.3.3.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.3.p1.19.m19.3c">s(\operatorname{dom}(t)+n+1)\neq 0</annotation><annotation encoding="application/x-llamapun" id="S4.3.p1.19.m19.3d">italic_s ( roman_dom ( italic_t ) + italic_n + 1 ) ≠ 0</annotation></semantics></math>. Then <math alttext="u:=(s|_{\operatorname{dom}(t)+n})^{\frown}\langle 0\rangle" class="ltx_Math" display="inline" id="S4.3.p1.20.m20.5"><semantics id="S4.3.p1.20.m20.5a"><mrow id="S4.3.p1.20.m20.5.5" xref="S4.3.p1.20.m20.5.5.cmml"><mi id="S4.3.p1.20.m20.5.5.3" xref="S4.3.p1.20.m20.5.5.3.cmml">u</mi><mo id="S4.3.p1.20.m20.5.5.2" lspace="0.278em" rspace="0.278em" xref="S4.3.p1.20.m20.5.5.2.cmml">:=</mo><mrow id="S4.3.p1.20.m20.5.5.1" xref="S4.3.p1.20.m20.5.5.1.cmml"><msup id="S4.3.p1.20.m20.5.5.1.1" xref="S4.3.p1.20.m20.5.5.1.1.cmml"><mrow id="S4.3.p1.20.m20.5.5.1.1.1.1" xref="S4.3.p1.20.m20.5.5.1.1.cmml"><mo id="S4.3.p1.20.m20.5.5.1.1.1.1.2" stretchy="false" xref="S4.3.p1.20.m20.5.5.1.1.cmml">(</mo><msub id="S4.3.p1.20.m20.5.5.1.1.1.1.1.2" xref="S4.3.p1.20.m20.5.5.1.1.1.1.1.1.cmml"><mrow id="S4.3.p1.20.m20.5.5.1.1.1.1.1.2.2" xref="S4.3.p1.20.m20.5.5.1.1.1.1.1.1.cmml"><mi id="S4.3.p1.20.m20.3.3" xref="S4.3.p1.20.m20.3.3.cmml">s</mi><mo id="S4.3.p1.20.m20.5.5.1.1.1.1.1.2.2.1" stretchy="false" xref="S4.3.p1.20.m20.5.5.1.1.1.1.1.1.1.cmml">|</mo></mrow><mrow id="S4.3.p1.20.m20.2.2.2" xref="S4.3.p1.20.m20.2.2.2.cmml"><mrow id="S4.3.p1.20.m20.2.2.2.4.2" xref="S4.3.p1.20.m20.2.2.2.4.1.cmml"><mi id="S4.3.p1.20.m20.1.1.1.1" xref="S4.3.p1.20.m20.1.1.1.1.cmml">dom</mi><mo id="S4.3.p1.20.m20.2.2.2.4.2a" xref="S4.3.p1.20.m20.2.2.2.4.1.cmml">⁡</mo><mrow id="S4.3.p1.20.m20.2.2.2.4.2.1" xref="S4.3.p1.20.m20.2.2.2.4.1.cmml"><mo id="S4.3.p1.20.m20.2.2.2.4.2.1.1" stretchy="false" xref="S4.3.p1.20.m20.2.2.2.4.1.cmml">(</mo><mi id="S4.3.p1.20.m20.2.2.2.2" xref="S4.3.p1.20.m20.2.2.2.2.cmml">t</mi><mo id="S4.3.p1.20.m20.2.2.2.4.2.1.2" stretchy="false" xref="S4.3.p1.20.m20.2.2.2.4.1.cmml">)</mo></mrow></mrow><mo id="S4.3.p1.20.m20.2.2.2.3" xref="S4.3.p1.20.m20.2.2.2.3.cmml">+</mo><mi id="S4.3.p1.20.m20.2.2.2.5" xref="S4.3.p1.20.m20.2.2.2.5.cmml">n</mi></mrow></msub><mo id="S4.3.p1.20.m20.5.5.1.1.1.1.3" stretchy="false" xref="S4.3.p1.20.m20.5.5.1.1.cmml">)</mo></mrow><mo id="S4.3.p1.20.m20.5.5.1.1.3" xref="S4.3.p1.20.m20.5.5.1.1.3.cmml">⌢</mo></msup><mo id="S4.3.p1.20.m20.5.5.1.2" xref="S4.3.p1.20.m20.5.5.1.2.cmml">⁢</mo><mrow id="S4.3.p1.20.m20.5.5.1.3.2" xref="S4.3.p1.20.m20.5.5.1.3.1.cmml"><mo id="S4.3.p1.20.m20.5.5.1.3.2.1" stretchy="false" xref="S4.3.p1.20.m20.5.5.1.3.1.1.cmml">⟨</mo><mn id="S4.3.p1.20.m20.4.4" xref="S4.3.p1.20.m20.4.4.cmml">0</mn><mo id="S4.3.p1.20.m20.5.5.1.3.2.2" stretchy="false" xref="S4.3.p1.20.m20.5.5.1.3.1.1.cmml">⟩</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.3.p1.20.m20.5b"><apply id="S4.3.p1.20.m20.5.5.cmml" xref="S4.3.p1.20.m20.5.5"><csymbol cd="latexml" id="S4.3.p1.20.m20.5.5.2.cmml" xref="S4.3.p1.20.m20.5.5.2">assign</csymbol><ci id="S4.3.p1.20.m20.5.5.3.cmml" xref="S4.3.p1.20.m20.5.5.3">𝑢</ci><apply id="S4.3.p1.20.m20.5.5.1.cmml" xref="S4.3.p1.20.m20.5.5.1"><times id="S4.3.p1.20.m20.5.5.1.2.cmml" xref="S4.3.p1.20.m20.5.5.1.2"></times><apply id="S4.3.p1.20.m20.5.5.1.1.cmml" xref="S4.3.p1.20.m20.5.5.1.1"><csymbol cd="ambiguous" id="S4.3.p1.20.m20.5.5.1.1.2.cmml" xref="S4.3.p1.20.m20.5.5.1.1">superscript</csymbol><apply id="S4.3.p1.20.m20.5.5.1.1.1.1.1.1.cmml" xref="S4.3.p1.20.m20.5.5.1.1.1.1.1.2"><csymbol cd="latexml" id="S4.3.p1.20.m20.5.5.1.1.1.1.1.1.1.cmml" xref="S4.3.p1.20.m20.5.5.1.1.1.1.1.2.2.1">evaluated-at</csymbol><ci id="S4.3.p1.20.m20.3.3.cmml" xref="S4.3.p1.20.m20.3.3">𝑠</ci><apply id="S4.3.p1.20.m20.2.2.2.cmml" xref="S4.3.p1.20.m20.2.2.2"><plus id="S4.3.p1.20.m20.2.2.2.3.cmml" xref="S4.3.p1.20.m20.2.2.2.3"></plus><apply id="S4.3.p1.20.m20.2.2.2.4.1.cmml" xref="S4.3.p1.20.m20.2.2.2.4.2"><ci id="S4.3.p1.20.m20.1.1.1.1.cmml" xref="S4.3.p1.20.m20.1.1.1.1">dom</ci><ci id="S4.3.p1.20.m20.2.2.2.2.cmml" xref="S4.3.p1.20.m20.2.2.2.2">𝑡</ci></apply><ci id="S4.3.p1.20.m20.2.2.2.5.cmml" xref="S4.3.p1.20.m20.2.2.2.5">𝑛</ci></apply></apply><ci id="S4.3.p1.20.m20.5.5.1.1.3.cmml" xref="S4.3.p1.20.m20.5.5.1.1.3">⌢</ci></apply><apply id="S4.3.p1.20.m20.5.5.1.3.1.cmml" xref="S4.3.p1.20.m20.5.5.1.3.2"><csymbol cd="latexml" id="S4.3.p1.20.m20.5.5.1.3.1.1.cmml" xref="S4.3.p1.20.m20.5.5.1.3.2.1">delimited-⟨⟩</csymbol><cn id="S4.3.p1.20.m20.4.4.cmml" type="integer" xref="S4.3.p1.20.m20.4.4">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.3.p1.20.m20.5c">u:=(s|_{\operatorname{dom}(t)+n})^{\frown}\langle 0\rangle</annotation><annotation encoding="application/x-llamapun" id="S4.3.p1.20.m20.5d">italic_u := ( italic_s | start_POSTSUBSCRIPT roman_dom ( italic_t ) + italic_n end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT ⌢ end_POSTSUPERSCRIPT ⟨ 0 ⟩</annotation></semantics></math> is as wanted. ∎</p> </div> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S4.Thmtheorem5"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem5.1.1.1">Theorem 4.5</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem5.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem5.p1"> <p class="ltx_p" id="S4.Thmtheorem5.p1.5"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem5.p1.5.5">For every <math alttext="X,Y\subseteq\omega_{1}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p1.1.1.m1.2"><semantics id="S4.Thmtheorem5.p1.1.1.m1.2a"><mrow id="S4.Thmtheorem5.p1.1.1.m1.2.3" xref="S4.Thmtheorem5.p1.1.1.m1.2.3.cmml"><mrow id="S4.Thmtheorem5.p1.1.1.m1.2.3.2.2" xref="S4.Thmtheorem5.p1.1.1.m1.2.3.2.1.cmml"><mi id="S4.Thmtheorem5.p1.1.1.m1.1.1" xref="S4.Thmtheorem5.p1.1.1.m1.1.1.cmml">X</mi><mo id="S4.Thmtheorem5.p1.1.1.m1.2.3.2.2.1" xref="S4.Thmtheorem5.p1.1.1.m1.2.3.2.1.cmml">,</mo><mi id="S4.Thmtheorem5.p1.1.1.m1.2.2" xref="S4.Thmtheorem5.p1.1.1.m1.2.2.cmml">Y</mi></mrow><mo id="S4.Thmtheorem5.p1.1.1.m1.2.3.1" xref="S4.Thmtheorem5.p1.1.1.m1.2.3.1.cmml">⊆</mo><msub id="S4.Thmtheorem5.p1.1.1.m1.2.3.3" xref="S4.Thmtheorem5.p1.1.1.m1.2.3.3.cmml"><mi id="S4.Thmtheorem5.p1.1.1.m1.2.3.3.2" xref="S4.Thmtheorem5.p1.1.1.m1.2.3.3.2.cmml">ω</mi><mn id="S4.Thmtheorem5.p1.1.1.m1.2.3.3.3" xref="S4.Thmtheorem5.p1.1.1.m1.2.3.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p1.1.1.m1.2b"><apply id="S4.Thmtheorem5.p1.1.1.m1.2.3.cmml" xref="S4.Thmtheorem5.p1.1.1.m1.2.3"><subset id="S4.Thmtheorem5.p1.1.1.m1.2.3.1.cmml" xref="S4.Thmtheorem5.p1.1.1.m1.2.3.1"></subset><list id="S4.Thmtheorem5.p1.1.1.m1.2.3.2.1.cmml" xref="S4.Thmtheorem5.p1.1.1.m1.2.3.2.2"><ci id="S4.Thmtheorem5.p1.1.1.m1.1.1.cmml" xref="S4.Thmtheorem5.p1.1.1.m1.1.1">𝑋</ci><ci id="S4.Thmtheorem5.p1.1.1.m1.2.2.cmml" xref="S4.Thmtheorem5.p1.1.1.m1.2.2">𝑌</ci></list><apply id="S4.Thmtheorem5.p1.1.1.m1.2.3.3.cmml" xref="S4.Thmtheorem5.p1.1.1.m1.2.3.3"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p1.1.1.m1.2.3.3.1.cmml" xref="S4.Thmtheorem5.p1.1.1.m1.2.3.3">subscript</csymbol><ci id="S4.Thmtheorem5.p1.1.1.m1.2.3.3.2.cmml" xref="S4.Thmtheorem5.p1.1.1.m1.2.3.3.2">𝜔</ci><cn id="S4.Thmtheorem5.p1.1.1.m1.2.3.3.3.cmml" type="integer" xref="S4.Thmtheorem5.p1.1.1.m1.2.3.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p1.1.1.m1.2c">X,Y\subseteq\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p1.1.1.m1.2d">italic_X , italic_Y ⊆ italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> there is an <math alttext="\aleph_{1}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p1.2.2.m2.1"><semantics id="S4.Thmtheorem5.p1.2.2.m2.1a"><msub id="S4.Thmtheorem5.p1.2.2.m2.1.1" xref="S4.Thmtheorem5.p1.2.2.m2.1.1.cmml"><mi id="S4.Thmtheorem5.p1.2.2.m2.1.1.2" mathvariant="normal" xref="S4.Thmtheorem5.p1.2.2.m2.1.1.2.cmml">ℵ</mi><mn id="S4.Thmtheorem5.p1.2.2.m2.1.1.3" xref="S4.Thmtheorem5.p1.2.2.m2.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p1.2.2.m2.1b"><apply id="S4.Thmtheorem5.p1.2.2.m2.1.1.cmml" xref="S4.Thmtheorem5.p1.2.2.m2.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p1.2.2.m2.1.1.1.cmml" xref="S4.Thmtheorem5.p1.2.2.m2.1.1">subscript</csymbol><ci id="S4.Thmtheorem5.p1.2.2.m2.1.1.2.cmml" xref="S4.Thmtheorem5.p1.2.2.m2.1.1.2">ℵ</ci><cn id="S4.Thmtheorem5.p1.2.2.m2.1.1.3.cmml" type="integer" xref="S4.Thmtheorem5.p1.2.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p1.2.2.m2.1c">\aleph_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p1.2.2.m2.1d">roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-dense Aronszajn line <math alttext="A" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p1.3.3.m3.1"><semantics id="S4.Thmtheorem5.p1.3.3.m3.1a"><mi id="S4.Thmtheorem5.p1.3.3.m3.1.1" xref="S4.Thmtheorem5.p1.3.3.m3.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p1.3.3.m3.1b"><ci id="S4.Thmtheorem5.p1.3.3.m3.1.1.cmml" xref="S4.Thmtheorem5.p1.3.3.m3.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p1.3.3.m3.1c">A</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p1.3.3.m3.1d">italic_A</annotation></semantics></math>, and decomposition <math alttext="D" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p1.4.4.m4.1"><semantics id="S4.Thmtheorem5.p1.4.4.m4.1a"><mi id="S4.Thmtheorem5.p1.4.4.m4.1.1" xref="S4.Thmtheorem5.p1.4.4.m4.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p1.4.4.m4.1b"><ci id="S4.Thmtheorem5.p1.4.4.m4.1.1.cmml" xref="S4.Thmtheorem5.p1.4.4.m4.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p1.4.4.m4.1c">D</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p1.4.4.m4.1d">italic_D</annotation></semantics></math> for <math alttext="A" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p1.5.5.m5.1"><semantics id="S4.Thmtheorem5.p1.5.5.m5.1a"><mi id="S4.Thmtheorem5.p1.5.5.m5.1.1" xref="S4.Thmtheorem5.p1.5.5.m5.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p1.5.5.m5.1b"><ci id="S4.Thmtheorem5.p1.5.5.m5.1.1.cmml" xref="S4.Thmtheorem5.p1.5.5.m5.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p1.5.5.m5.1c">A</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p1.5.5.m5.1d">italic_A</annotation></semantics></math>, such that,</span></p> <ul class="ltx_itemize" id="S4.I1"> <li class="ltx_item" id="S4.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S4.I1.i1.p1"> <p class="ltx_p" id="S4.I1.i1.p1.1"><math alttext="\hat{\mathscr{L}}(A,D)=\mathscr{L}(A,D)=X" class="ltx_Math" display="inline" id="S4.I1.i1.p1.1.m1.4"><semantics id="S4.I1.i1.p1.1.m1.4a"><mrow id="S4.I1.i1.p1.1.m1.4.5" xref="S4.I1.i1.p1.1.m1.4.5.cmml"><mrow id="S4.I1.i1.p1.1.m1.4.5.2" xref="S4.I1.i1.p1.1.m1.4.5.2.cmml"><mover accent="true" id="S4.I1.i1.p1.1.m1.4.5.2.2" xref="S4.I1.i1.p1.1.m1.4.5.2.2.cmml"><mi class="ltx_font_mathscript" id="S4.I1.i1.p1.1.m1.4.5.2.2.2" xref="S4.I1.i1.p1.1.m1.4.5.2.2.2.cmml">ℒ</mi><mo id="S4.I1.i1.p1.1.m1.4.5.2.2.1" xref="S4.I1.i1.p1.1.m1.4.5.2.2.1.cmml">^</mo></mover><mo id="S4.I1.i1.p1.1.m1.4.5.2.1" xref="S4.I1.i1.p1.1.m1.4.5.2.1.cmml">⁢</mo><mrow id="S4.I1.i1.p1.1.m1.4.5.2.3.2" xref="S4.I1.i1.p1.1.m1.4.5.2.3.1.cmml"><mo id="S4.I1.i1.p1.1.m1.4.5.2.3.2.1" stretchy="false" xref="S4.I1.i1.p1.1.m1.4.5.2.3.1.cmml">(</mo><mi id="S4.I1.i1.p1.1.m1.1.1" xref="S4.I1.i1.p1.1.m1.1.1.cmml">A</mi><mo id="S4.I1.i1.p1.1.m1.4.5.2.3.2.2" xref="S4.I1.i1.p1.1.m1.4.5.2.3.1.cmml">,</mo><mi id="S4.I1.i1.p1.1.m1.2.2" xref="S4.I1.i1.p1.1.m1.2.2.cmml">D</mi><mo id="S4.I1.i1.p1.1.m1.4.5.2.3.2.3" stretchy="false" xref="S4.I1.i1.p1.1.m1.4.5.2.3.1.cmml">)</mo></mrow></mrow><mo id="S4.I1.i1.p1.1.m1.4.5.3" xref="S4.I1.i1.p1.1.m1.4.5.3.cmml">=</mo><mrow id="S4.I1.i1.p1.1.m1.4.5.4" xref="S4.I1.i1.p1.1.m1.4.5.4.cmml"><mi class="ltx_font_mathscript" id="S4.I1.i1.p1.1.m1.4.5.4.2" xref="S4.I1.i1.p1.1.m1.4.5.4.2.cmml">ℒ</mi><mo id="S4.I1.i1.p1.1.m1.4.5.4.1" xref="S4.I1.i1.p1.1.m1.4.5.4.1.cmml">⁢</mo><mrow id="S4.I1.i1.p1.1.m1.4.5.4.3.2" xref="S4.I1.i1.p1.1.m1.4.5.4.3.1.cmml"><mo id="S4.I1.i1.p1.1.m1.4.5.4.3.2.1" stretchy="false" xref="S4.I1.i1.p1.1.m1.4.5.4.3.1.cmml">(</mo><mi id="S4.I1.i1.p1.1.m1.3.3" xref="S4.I1.i1.p1.1.m1.3.3.cmml">A</mi><mo id="S4.I1.i1.p1.1.m1.4.5.4.3.2.2" xref="S4.I1.i1.p1.1.m1.4.5.4.3.1.cmml">,</mo><mi id="S4.I1.i1.p1.1.m1.4.4" xref="S4.I1.i1.p1.1.m1.4.4.cmml">D</mi><mo id="S4.I1.i1.p1.1.m1.4.5.4.3.2.3" stretchy="false" xref="S4.I1.i1.p1.1.m1.4.5.4.3.1.cmml">)</mo></mrow></mrow><mo id="S4.I1.i1.p1.1.m1.4.5.5" xref="S4.I1.i1.p1.1.m1.4.5.5.cmml">=</mo><mi id="S4.I1.i1.p1.1.m1.4.5.6" xref="S4.I1.i1.p1.1.m1.4.5.6.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p1.1.m1.4b"><apply id="S4.I1.i1.p1.1.m1.4.5.cmml" xref="S4.I1.i1.p1.1.m1.4.5"><and id="S4.I1.i1.p1.1.m1.4.5a.cmml" xref="S4.I1.i1.p1.1.m1.4.5"></and><apply id="S4.I1.i1.p1.1.m1.4.5b.cmml" xref="S4.I1.i1.p1.1.m1.4.5"><eq id="S4.I1.i1.p1.1.m1.4.5.3.cmml" xref="S4.I1.i1.p1.1.m1.4.5.3"></eq><apply id="S4.I1.i1.p1.1.m1.4.5.2.cmml" xref="S4.I1.i1.p1.1.m1.4.5.2"><times id="S4.I1.i1.p1.1.m1.4.5.2.1.cmml" xref="S4.I1.i1.p1.1.m1.4.5.2.1"></times><apply id="S4.I1.i1.p1.1.m1.4.5.2.2.cmml" xref="S4.I1.i1.p1.1.m1.4.5.2.2"><ci id="S4.I1.i1.p1.1.m1.4.5.2.2.1.cmml" xref="S4.I1.i1.p1.1.m1.4.5.2.2.1">^</ci><ci id="S4.I1.i1.p1.1.m1.4.5.2.2.2.cmml" xref="S4.I1.i1.p1.1.m1.4.5.2.2.2">ℒ</ci></apply><interval closure="open" id="S4.I1.i1.p1.1.m1.4.5.2.3.1.cmml" xref="S4.I1.i1.p1.1.m1.4.5.2.3.2"><ci id="S4.I1.i1.p1.1.m1.1.1.cmml" xref="S4.I1.i1.p1.1.m1.1.1">𝐴</ci><ci id="S4.I1.i1.p1.1.m1.2.2.cmml" xref="S4.I1.i1.p1.1.m1.2.2">𝐷</ci></interval></apply><apply id="S4.I1.i1.p1.1.m1.4.5.4.cmml" xref="S4.I1.i1.p1.1.m1.4.5.4"><times id="S4.I1.i1.p1.1.m1.4.5.4.1.cmml" xref="S4.I1.i1.p1.1.m1.4.5.4.1"></times><ci id="S4.I1.i1.p1.1.m1.4.5.4.2.cmml" xref="S4.I1.i1.p1.1.m1.4.5.4.2">ℒ</ci><interval closure="open" id="S4.I1.i1.p1.1.m1.4.5.4.3.1.cmml" xref="S4.I1.i1.p1.1.m1.4.5.4.3.2"><ci id="S4.I1.i1.p1.1.m1.3.3.cmml" xref="S4.I1.i1.p1.1.m1.3.3">𝐴</ci><ci id="S4.I1.i1.p1.1.m1.4.4.cmml" xref="S4.I1.i1.p1.1.m1.4.4">𝐷</ci></interval></apply></apply><apply id="S4.I1.i1.p1.1.m1.4.5c.cmml" xref="S4.I1.i1.p1.1.m1.4.5"><eq id="S4.I1.i1.p1.1.m1.4.5.5.cmml" xref="S4.I1.i1.p1.1.m1.4.5.5"></eq><share href="https://arxiv.org/html/2503.13728v1#S4.I1.i1.p1.1.m1.4.5.4.cmml" id="S4.I1.i1.p1.1.m1.4.5d.cmml" xref="S4.I1.i1.p1.1.m1.4.5"></share><ci id="S4.I1.i1.p1.1.m1.4.5.6.cmml" xref="S4.I1.i1.p1.1.m1.4.5.6">𝑋</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p1.1.m1.4c">\hat{\mathscr{L}}(A,D)=\mathscr{L}(A,D)=X</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p1.1.m1.4d">over^ start_ARG script_L end_ARG ( italic_A , italic_D ) = script_L ( italic_A , italic_D ) = italic_X</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I1.i1.p1.1.1">,</span></p> </div> </li> <li class="ltx_item" id="S4.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S4.I1.i2.p1"> <p class="ltx_p" id="S4.I1.i2.p1.1"><math alttext="\hat{\mathscr{R}}(A,D)=\mathscr{R}(A,D)=Y" class="ltx_Math" display="inline" id="S4.I1.i2.p1.1.m1.4"><semantics id="S4.I1.i2.p1.1.m1.4a"><mrow id="S4.I1.i2.p1.1.m1.4.5" xref="S4.I1.i2.p1.1.m1.4.5.cmml"><mrow id="S4.I1.i2.p1.1.m1.4.5.2" xref="S4.I1.i2.p1.1.m1.4.5.2.cmml"><mover accent="true" id="S4.I1.i2.p1.1.m1.4.5.2.2" xref="S4.I1.i2.p1.1.m1.4.5.2.2.cmml"><mi class="ltx_font_mathscript" id="S4.I1.i2.p1.1.m1.4.5.2.2.2" xref="S4.I1.i2.p1.1.m1.4.5.2.2.2.cmml">ℛ</mi><mo id="S4.I1.i2.p1.1.m1.4.5.2.2.1" xref="S4.I1.i2.p1.1.m1.4.5.2.2.1.cmml">^</mo></mover><mo id="S4.I1.i2.p1.1.m1.4.5.2.1" xref="S4.I1.i2.p1.1.m1.4.5.2.1.cmml">⁢</mo><mrow id="S4.I1.i2.p1.1.m1.4.5.2.3.2" xref="S4.I1.i2.p1.1.m1.4.5.2.3.1.cmml"><mo id="S4.I1.i2.p1.1.m1.4.5.2.3.2.1" stretchy="false" xref="S4.I1.i2.p1.1.m1.4.5.2.3.1.cmml">(</mo><mi id="S4.I1.i2.p1.1.m1.1.1" xref="S4.I1.i2.p1.1.m1.1.1.cmml">A</mi><mo id="S4.I1.i2.p1.1.m1.4.5.2.3.2.2" xref="S4.I1.i2.p1.1.m1.4.5.2.3.1.cmml">,</mo><mi id="S4.I1.i2.p1.1.m1.2.2" xref="S4.I1.i2.p1.1.m1.2.2.cmml">D</mi><mo id="S4.I1.i2.p1.1.m1.4.5.2.3.2.3" stretchy="false" xref="S4.I1.i2.p1.1.m1.4.5.2.3.1.cmml">)</mo></mrow></mrow><mo id="S4.I1.i2.p1.1.m1.4.5.3" xref="S4.I1.i2.p1.1.m1.4.5.3.cmml">=</mo><mrow id="S4.I1.i2.p1.1.m1.4.5.4" xref="S4.I1.i2.p1.1.m1.4.5.4.cmml"><mi class="ltx_font_mathscript" id="S4.I1.i2.p1.1.m1.4.5.4.2" xref="S4.I1.i2.p1.1.m1.4.5.4.2.cmml">ℛ</mi><mo id="S4.I1.i2.p1.1.m1.4.5.4.1" xref="S4.I1.i2.p1.1.m1.4.5.4.1.cmml">⁢</mo><mrow id="S4.I1.i2.p1.1.m1.4.5.4.3.2" xref="S4.I1.i2.p1.1.m1.4.5.4.3.1.cmml"><mo id="S4.I1.i2.p1.1.m1.4.5.4.3.2.1" stretchy="false" xref="S4.I1.i2.p1.1.m1.4.5.4.3.1.cmml">(</mo><mi id="S4.I1.i2.p1.1.m1.3.3" xref="S4.I1.i2.p1.1.m1.3.3.cmml">A</mi><mo id="S4.I1.i2.p1.1.m1.4.5.4.3.2.2" xref="S4.I1.i2.p1.1.m1.4.5.4.3.1.cmml">,</mo><mi id="S4.I1.i2.p1.1.m1.4.4" xref="S4.I1.i2.p1.1.m1.4.4.cmml">D</mi><mo id="S4.I1.i2.p1.1.m1.4.5.4.3.2.3" stretchy="false" xref="S4.I1.i2.p1.1.m1.4.5.4.3.1.cmml">)</mo></mrow></mrow><mo id="S4.I1.i2.p1.1.m1.4.5.5" xref="S4.I1.i2.p1.1.m1.4.5.5.cmml">=</mo><mi id="S4.I1.i2.p1.1.m1.4.5.6" xref="S4.I1.i2.p1.1.m1.4.5.6.cmml">Y</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i2.p1.1.m1.4b"><apply id="S4.I1.i2.p1.1.m1.4.5.cmml" xref="S4.I1.i2.p1.1.m1.4.5"><and id="S4.I1.i2.p1.1.m1.4.5a.cmml" xref="S4.I1.i2.p1.1.m1.4.5"></and><apply id="S4.I1.i2.p1.1.m1.4.5b.cmml" xref="S4.I1.i2.p1.1.m1.4.5"><eq id="S4.I1.i2.p1.1.m1.4.5.3.cmml" xref="S4.I1.i2.p1.1.m1.4.5.3"></eq><apply id="S4.I1.i2.p1.1.m1.4.5.2.cmml" xref="S4.I1.i2.p1.1.m1.4.5.2"><times id="S4.I1.i2.p1.1.m1.4.5.2.1.cmml" xref="S4.I1.i2.p1.1.m1.4.5.2.1"></times><apply id="S4.I1.i2.p1.1.m1.4.5.2.2.cmml" xref="S4.I1.i2.p1.1.m1.4.5.2.2"><ci id="S4.I1.i2.p1.1.m1.4.5.2.2.1.cmml" xref="S4.I1.i2.p1.1.m1.4.5.2.2.1">^</ci><ci id="S4.I1.i2.p1.1.m1.4.5.2.2.2.cmml" xref="S4.I1.i2.p1.1.m1.4.5.2.2.2">ℛ</ci></apply><interval closure="open" id="S4.I1.i2.p1.1.m1.4.5.2.3.1.cmml" xref="S4.I1.i2.p1.1.m1.4.5.2.3.2"><ci id="S4.I1.i2.p1.1.m1.1.1.cmml" xref="S4.I1.i2.p1.1.m1.1.1">𝐴</ci><ci id="S4.I1.i2.p1.1.m1.2.2.cmml" xref="S4.I1.i2.p1.1.m1.2.2">𝐷</ci></interval></apply><apply id="S4.I1.i2.p1.1.m1.4.5.4.cmml" xref="S4.I1.i2.p1.1.m1.4.5.4"><times id="S4.I1.i2.p1.1.m1.4.5.4.1.cmml" xref="S4.I1.i2.p1.1.m1.4.5.4.1"></times><ci id="S4.I1.i2.p1.1.m1.4.5.4.2.cmml" xref="S4.I1.i2.p1.1.m1.4.5.4.2">ℛ</ci><interval closure="open" id="S4.I1.i2.p1.1.m1.4.5.4.3.1.cmml" xref="S4.I1.i2.p1.1.m1.4.5.4.3.2"><ci id="S4.I1.i2.p1.1.m1.3.3.cmml" xref="S4.I1.i2.p1.1.m1.3.3">𝐴</ci><ci id="S4.I1.i2.p1.1.m1.4.4.cmml" xref="S4.I1.i2.p1.1.m1.4.4">𝐷</ci></interval></apply></apply><apply id="S4.I1.i2.p1.1.m1.4.5c.cmml" xref="S4.I1.i2.p1.1.m1.4.5"><eq id="S4.I1.i2.p1.1.m1.4.5.5.cmml" xref="S4.I1.i2.p1.1.m1.4.5.5"></eq><share href="https://arxiv.org/html/2503.13728v1#S4.I1.i2.p1.1.m1.4.5.4.cmml" id="S4.I1.i2.p1.1.m1.4.5d.cmml" xref="S4.I1.i2.p1.1.m1.4.5"></share><ci id="S4.I1.i2.p1.1.m1.4.5.6.cmml" xref="S4.I1.i2.p1.1.m1.4.5.6">𝑌</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i2.p1.1.m1.4c">\hat{\mathscr{R}}(A,D)=\mathscr{R}(A,D)=Y</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i2.p1.1.m1.4d">over^ start_ARG script_R end_ARG ( italic_A , italic_D ) = script_R ( italic_A , italic_D ) = italic_Y</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I1.i2.p1.1.1"> and,</span></p> </div> </li> <li class="ltx_item" id="S4.I1.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S4.I1.i3.p1"> <p class="ltx_p" id="S4.I1.i3.p1.2"><span class="ltx_text ltx_font_italic" id="S4.I1.i3.p1.2.1">For every </span><math alttext="\xi<\omega_{1}" class="ltx_Math" display="inline" id="S4.I1.i3.p1.1.m1.1"><semantics id="S4.I1.i3.p1.1.m1.1a"><mrow id="S4.I1.i3.p1.1.m1.1.1" xref="S4.I1.i3.p1.1.m1.1.1.cmml"><mi id="S4.I1.i3.p1.1.m1.1.1.2" xref="S4.I1.i3.p1.1.m1.1.1.2.cmml">ξ</mi><mo id="S4.I1.i3.p1.1.m1.1.1.1" xref="S4.I1.i3.p1.1.m1.1.1.1.cmml"><</mo><msub id="S4.I1.i3.p1.1.m1.1.1.3" xref="S4.I1.i3.p1.1.m1.1.1.3.cmml"><mi id="S4.I1.i3.p1.1.m1.1.1.3.2" xref="S4.I1.i3.p1.1.m1.1.1.3.2.cmml">ω</mi><mn id="S4.I1.i3.p1.1.m1.1.1.3.3" xref="S4.I1.i3.p1.1.m1.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i3.p1.1.m1.1b"><apply id="S4.I1.i3.p1.1.m1.1.1.cmml" xref="S4.I1.i3.p1.1.m1.1.1"><lt id="S4.I1.i3.p1.1.m1.1.1.1.cmml" xref="S4.I1.i3.p1.1.m1.1.1.1"></lt><ci id="S4.I1.i3.p1.1.m1.1.1.2.cmml" xref="S4.I1.i3.p1.1.m1.1.1.2">𝜉</ci><apply id="S4.I1.i3.p1.1.m1.1.1.3.cmml" xref="S4.I1.i3.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.I1.i3.p1.1.m1.1.1.3.1.cmml" xref="S4.I1.i3.p1.1.m1.1.1.3">subscript</csymbol><ci id="S4.I1.i3.p1.1.m1.1.1.3.2.cmml" xref="S4.I1.i3.p1.1.m1.1.1.3.2">𝜔</ci><cn id="S4.I1.i3.p1.1.m1.1.1.3.3.cmml" type="integer" xref="S4.I1.i3.p1.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i3.p1.1.m1.1c">\xi<\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i3.p1.1.m1.1d">italic_ξ < italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I1.i3.p1.2.2">, no complementary interval of </span><math alttext="A\setminus D_{\xi}" class="ltx_Math" display="inline" id="S4.I1.i3.p1.2.m2.1"><semantics id="S4.I1.i3.p1.2.m2.1a"><mrow id="S4.I1.i3.p1.2.m2.1.1" xref="S4.I1.i3.p1.2.m2.1.1.cmml"><mi id="S4.I1.i3.p1.2.m2.1.1.2" xref="S4.I1.i3.p1.2.m2.1.1.2.cmml">A</mi><mo id="S4.I1.i3.p1.2.m2.1.1.1" xref="S4.I1.i3.p1.2.m2.1.1.1.cmml">∖</mo><msub id="S4.I1.i3.p1.2.m2.1.1.3" xref="S4.I1.i3.p1.2.m2.1.1.3.cmml"><mi id="S4.I1.i3.p1.2.m2.1.1.3.2" xref="S4.I1.i3.p1.2.m2.1.1.3.2.cmml">D</mi><mi id="S4.I1.i3.p1.2.m2.1.1.3.3" xref="S4.I1.i3.p1.2.m2.1.1.3.3.cmml">ξ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i3.p1.2.m2.1b"><apply id="S4.I1.i3.p1.2.m2.1.1.cmml" xref="S4.I1.i3.p1.2.m2.1.1"><setdiff id="S4.I1.i3.p1.2.m2.1.1.1.cmml" xref="S4.I1.i3.p1.2.m2.1.1.1"></setdiff><ci id="S4.I1.i3.p1.2.m2.1.1.2.cmml" xref="S4.I1.i3.p1.2.m2.1.1.2">𝐴</ci><apply id="S4.I1.i3.p1.2.m2.1.1.3.cmml" xref="S4.I1.i3.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S4.I1.i3.p1.2.m2.1.1.3.1.cmml" xref="S4.I1.i3.p1.2.m2.1.1.3">subscript</csymbol><ci id="S4.I1.i3.p1.2.m2.1.1.3.2.cmml" xref="S4.I1.i3.p1.2.m2.1.1.3.2">𝐷</ci><ci id="S4.I1.i3.p1.2.m2.1.1.3.3.cmml" xref="S4.I1.i3.p1.2.m2.1.1.3.3">𝜉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i3.p1.2.m2.1c">A\setminus D_{\xi}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i3.p1.2.m2.1d">italic_A ∖ italic_D start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I1.i3.p1.2.3"> is singleton.</span></p> </div> </li> </ul> </div> </div> <div class="ltx_proof" id="S4.6"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S4.4.p1"> <p class="ltx_p" id="S4.4.p1.6">Fix <math alttext="X" class="ltx_Math" display="inline" id="S4.4.p1.1.m1.1"><semantics id="S4.4.p1.1.m1.1a"><mi id="S4.4.p1.1.m1.1.1" xref="S4.4.p1.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S4.4.p1.1.m1.1b"><ci id="S4.4.p1.1.m1.1.1.cmml" xref="S4.4.p1.1.m1.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.4.p1.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S4.4.p1.1.m1.1d">italic_X</annotation></semantics></math> and <math alttext="Y" class="ltx_Math" display="inline" id="S4.4.p1.2.m2.1"><semantics id="S4.4.p1.2.m2.1a"><mi id="S4.4.p1.2.m2.1.1" xref="S4.4.p1.2.m2.1.1.cmml">Y</mi><annotation-xml encoding="MathML-Content" id="S4.4.p1.2.m2.1b"><ci id="S4.4.p1.2.m2.1.1.cmml" xref="S4.4.p1.2.m2.1.1">𝑌</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.4.p1.2.m2.1c">Y</annotation><annotation encoding="application/x-llamapun" id="S4.4.p1.2.m2.1d">italic_Y</annotation></semantics></math> and let <math alttext="\langle\lambda_{\xi}:\xi<\omega_{1}\rangle" class="ltx_math_unparsed" display="inline" id="S4.4.p1.3.m3.1"><semantics id="S4.4.p1.3.m3.1a"><mrow id="S4.4.p1.3.m3.1b"><mo id="S4.4.p1.3.m3.1.1" stretchy="false">⟨</mo><msub id="S4.4.p1.3.m3.1.2"><mi id="S4.4.p1.3.m3.1.2.2">λ</mi><mi id="S4.4.p1.3.m3.1.2.3">ξ</mi></msub><mo id="S4.4.p1.3.m3.1.3" lspace="0.278em" rspace="0.278em">:</mo><mi id="S4.4.p1.3.m3.1.4">ξ</mi><mo id="S4.4.p1.3.m3.1.5"><</mo><msub id="S4.4.p1.3.m3.1.6"><mi id="S4.4.p1.3.m3.1.6.2">ω</mi><mn id="S4.4.p1.3.m3.1.6.3">1</mn></msub><mo id="S4.4.p1.3.m3.1.7" stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex" id="S4.4.p1.3.m3.1c">\langle\lambda_{\xi}:\xi<\omega_{1}\rangle</annotation><annotation encoding="application/x-llamapun" id="S4.4.p1.3.m3.1d">⟨ italic_λ start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT : italic_ξ < italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⟩</annotation></semantics></math> be the increasing enumeration of <math alttext="\Lambda^{\prime}" class="ltx_Math" display="inline" id="S4.4.p1.4.m4.1"><semantics id="S4.4.p1.4.m4.1a"><msup id="S4.4.p1.4.m4.1.1" xref="S4.4.p1.4.m4.1.1.cmml"><mi id="S4.4.p1.4.m4.1.1.2" mathvariant="normal" xref="S4.4.p1.4.m4.1.1.2.cmml">Λ</mi><mo id="S4.4.p1.4.m4.1.1.3" xref="S4.4.p1.4.m4.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.4.p1.4.m4.1b"><apply id="S4.4.p1.4.m4.1.1.cmml" xref="S4.4.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S4.4.p1.4.m4.1.1.1.cmml" xref="S4.4.p1.4.m4.1.1">superscript</csymbol><ci id="S4.4.p1.4.m4.1.1.2.cmml" xref="S4.4.p1.4.m4.1.1.2">Λ</ci><ci id="S4.4.p1.4.m4.1.1.3.cmml" xref="S4.4.p1.4.m4.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.4.p1.4.m4.1c">\Lambda^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.4.p1.4.m4.1d">roman_Λ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>. We will construct <math alttext="A" class="ltx_Math" display="inline" id="S4.4.p1.5.m5.1"><semantics id="S4.4.p1.5.m5.1a"><mi id="S4.4.p1.5.m5.1.1" xref="S4.4.p1.5.m5.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S4.4.p1.5.m5.1b"><ci id="S4.4.p1.5.m5.1.1.cmml" xref="S4.4.p1.5.m5.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.4.p1.5.m5.1c">A</annotation><annotation encoding="application/x-llamapun" id="S4.4.p1.5.m5.1d">italic_A</annotation></semantics></math> as a subtree of <math alttext="A^{+}" class="ltx_Math" display="inline" id="S4.4.p1.6.m6.1"><semantics id="S4.4.p1.6.m6.1a"><msup id="S4.4.p1.6.m6.1.1" xref="S4.4.p1.6.m6.1.1.cmml"><mi id="S4.4.p1.6.m6.1.1.2" xref="S4.4.p1.6.m6.1.1.2.cmml">A</mi><mo id="S4.4.p1.6.m6.1.1.3" xref="S4.4.p1.6.m6.1.1.3.cmml">+</mo></msup><annotation-xml encoding="MathML-Content" id="S4.4.p1.6.m6.1b"><apply id="S4.4.p1.6.m6.1.1.cmml" xref="S4.4.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S4.4.p1.6.m6.1.1.1.cmml" xref="S4.4.p1.6.m6.1.1">superscript</csymbol><ci id="S4.4.p1.6.m6.1.1.2.cmml" xref="S4.4.p1.6.m6.1.1.2">𝐴</ci><plus id="S4.4.p1.6.m6.1.1.3.cmml" xref="S4.4.p1.6.m6.1.1.3"></plus></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.4.p1.6.m6.1c">A^{+}</annotation><annotation encoding="application/x-llamapun" id="S4.4.p1.6.m6.1d">italic_A start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> by removing the unwanted endpoints. For this let</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex6"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="A:=\{t\in A_{0}:\operatorname{dom}(t)=\lambda_{\xi},\xi\in X\}\cup\{t^{\frown}% \langle\omega\rangle\in A_{1}:\operatorname{dom}(t)=\lambda_{\xi},\xi\in Y\}." class="ltx_Math" display="block" id="S4.Ex6.m1.6"><semantics id="S4.Ex6.m1.6a"><mrow id="S4.Ex6.m1.6.6.1" xref="S4.Ex6.m1.6.6.1.1.cmml"><mrow id="S4.Ex6.m1.6.6.1.1" xref="S4.Ex6.m1.6.6.1.1.cmml"><mi id="S4.Ex6.m1.6.6.1.1.6" xref="S4.Ex6.m1.6.6.1.1.6.cmml">A</mi><mo id="S4.Ex6.m1.6.6.1.1.5" lspace="0.278em" rspace="0.278em" xref="S4.Ex6.m1.6.6.1.1.5.cmml">:=</mo><mrow id="S4.Ex6.m1.6.6.1.1.4" xref="S4.Ex6.m1.6.6.1.1.4.cmml"><mrow id="S4.Ex6.m1.6.6.1.1.2.2.2" xref="S4.Ex6.m1.6.6.1.1.2.2.3.cmml"><mo id="S4.Ex6.m1.6.6.1.1.2.2.2.3" stretchy="false" xref="S4.Ex6.m1.6.6.1.1.2.2.3.1.cmml">{</mo><mrow id="S4.Ex6.m1.6.6.1.1.1.1.1.1" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.cmml"><mi id="S4.Ex6.m1.6.6.1.1.1.1.1.1.2" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.2.cmml">t</mi><mo id="S4.Ex6.m1.6.6.1.1.1.1.1.1.1" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.cmml">∈</mo><msub id="S4.Ex6.m1.6.6.1.1.1.1.1.1.3" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.3.cmml"><mi id="S4.Ex6.m1.6.6.1.1.1.1.1.1.3.2" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.3.2.cmml">A</mi><mn id="S4.Ex6.m1.6.6.1.1.1.1.1.1.3.3" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.3.3.cmml">0</mn></msub></mrow><mo id="S4.Ex6.m1.6.6.1.1.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S4.Ex6.m1.6.6.1.1.2.2.3.1.cmml">:</mo><mrow id="S4.Ex6.m1.6.6.1.1.2.2.2.2.2" xref="S4.Ex6.m1.6.6.1.1.2.2.2.2.3.cmml"><mrow id="S4.Ex6.m1.6.6.1.1.2.2.2.2.1.1" xref="S4.Ex6.m1.6.6.1.1.2.2.2.2.1.1.cmml"><mrow id="S4.Ex6.m1.6.6.1.1.2.2.2.2.1.1.2.2" xref="S4.Ex6.m1.6.6.1.1.2.2.2.2.1.1.2.1.cmml"><mi id="S4.Ex6.m1.1.1" xref="S4.Ex6.m1.1.1.cmml">dom</mi><mo id="S4.Ex6.m1.6.6.1.1.2.2.2.2.1.1.2.2a" xref="S4.Ex6.m1.6.6.1.1.2.2.2.2.1.1.2.1.cmml">⁡</mo><mrow id="S4.Ex6.m1.6.6.1.1.2.2.2.2.1.1.2.2.1" xref="S4.Ex6.m1.6.6.1.1.2.2.2.2.1.1.2.1.cmml"><mo id="S4.Ex6.m1.6.6.1.1.2.2.2.2.1.1.2.2.1.1" stretchy="false" xref="S4.Ex6.m1.6.6.1.1.2.2.2.2.1.1.2.1.cmml">(</mo><mi id="S4.Ex6.m1.2.2" xref="S4.Ex6.m1.2.2.cmml">t</mi><mo id="S4.Ex6.m1.6.6.1.1.2.2.2.2.1.1.2.2.1.2" stretchy="false" xref="S4.Ex6.m1.6.6.1.1.2.2.2.2.1.1.2.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex6.m1.6.6.1.1.2.2.2.2.1.1.1" xref="S4.Ex6.m1.6.6.1.1.2.2.2.2.1.1.1.cmml">=</mo><msub id="S4.Ex6.m1.6.6.1.1.2.2.2.2.1.1.3" xref="S4.Ex6.m1.6.6.1.1.2.2.2.2.1.1.3.cmml"><mi id="S4.Ex6.m1.6.6.1.1.2.2.2.2.1.1.3.2" xref="S4.Ex6.m1.6.6.1.1.2.2.2.2.1.1.3.2.cmml">λ</mi><mi id="S4.Ex6.m1.6.6.1.1.2.2.2.2.1.1.3.3" xref="S4.Ex6.m1.6.6.1.1.2.2.2.2.1.1.3.3.cmml">ξ</mi></msub></mrow><mo id="S4.Ex6.m1.6.6.1.1.2.2.2.2.2.3" xref="S4.Ex6.m1.6.6.1.1.2.2.2.2.3a.cmml">,</mo><mrow id="S4.Ex6.m1.6.6.1.1.2.2.2.2.2.2" xref="S4.Ex6.m1.6.6.1.1.2.2.2.2.2.2.cmml"><mi id="S4.Ex6.m1.6.6.1.1.2.2.2.2.2.2.2" xref="S4.Ex6.m1.6.6.1.1.2.2.2.2.2.2.2.cmml">ξ</mi><mo id="S4.Ex6.m1.6.6.1.1.2.2.2.2.2.2.1" xref="S4.Ex6.m1.6.6.1.1.2.2.2.2.2.2.1.cmml">∈</mo><mi id="S4.Ex6.m1.6.6.1.1.2.2.2.2.2.2.3" xref="S4.Ex6.m1.6.6.1.1.2.2.2.2.2.2.3.cmml">X</mi></mrow></mrow><mo id="S4.Ex6.m1.6.6.1.1.2.2.2.5" stretchy="false" xref="S4.Ex6.m1.6.6.1.1.2.2.3.1.cmml">}</mo></mrow><mo id="S4.Ex6.m1.6.6.1.1.4.5" xref="S4.Ex6.m1.6.6.1.1.4.5.cmml">∪</mo><mrow id="S4.Ex6.m1.6.6.1.1.4.4.2" xref="S4.Ex6.m1.6.6.1.1.4.4.3.cmml"><mo id="S4.Ex6.m1.6.6.1.1.4.4.2.3" stretchy="false" xref="S4.Ex6.m1.6.6.1.1.4.4.3.1.cmml">{</mo><mrow id="S4.Ex6.m1.6.6.1.1.3.3.1.1" xref="S4.Ex6.m1.6.6.1.1.3.3.1.1.cmml"><mrow id="S4.Ex6.m1.6.6.1.1.3.3.1.1.2" xref="S4.Ex6.m1.6.6.1.1.3.3.1.1.2.cmml"><msup id="S4.Ex6.m1.6.6.1.1.3.3.1.1.2.2" xref="S4.Ex6.m1.6.6.1.1.3.3.1.1.2.2.cmml"><mi id="S4.Ex6.m1.6.6.1.1.3.3.1.1.2.2.2" xref="S4.Ex6.m1.6.6.1.1.3.3.1.1.2.2.2.cmml">t</mi><mo id="S4.Ex6.m1.6.6.1.1.3.3.1.1.2.2.3" xref="S4.Ex6.m1.6.6.1.1.3.3.1.1.2.2.3.cmml">⌢</mo></msup><mo id="S4.Ex6.m1.6.6.1.1.3.3.1.1.2.1" xref="S4.Ex6.m1.6.6.1.1.3.3.1.1.2.1.cmml">⁢</mo><mrow id="S4.Ex6.m1.6.6.1.1.3.3.1.1.2.3.2" xref="S4.Ex6.m1.6.6.1.1.3.3.1.1.2.3.1.cmml"><mo id="S4.Ex6.m1.6.6.1.1.3.3.1.1.2.3.2.1" stretchy="false" xref="S4.Ex6.m1.6.6.1.1.3.3.1.1.2.3.1.1.cmml">⟨</mo><mi id="S4.Ex6.m1.3.3" xref="S4.Ex6.m1.3.3.cmml">ω</mi><mo id="S4.Ex6.m1.6.6.1.1.3.3.1.1.2.3.2.2" stretchy="false" xref="S4.Ex6.m1.6.6.1.1.3.3.1.1.2.3.1.1.cmml">⟩</mo></mrow></mrow><mo id="S4.Ex6.m1.6.6.1.1.3.3.1.1.1" xref="S4.Ex6.m1.6.6.1.1.3.3.1.1.1.cmml">∈</mo><msub id="S4.Ex6.m1.6.6.1.1.3.3.1.1.3" xref="S4.Ex6.m1.6.6.1.1.3.3.1.1.3.cmml"><mi id="S4.Ex6.m1.6.6.1.1.3.3.1.1.3.2" xref="S4.Ex6.m1.6.6.1.1.3.3.1.1.3.2.cmml">A</mi><mn id="S4.Ex6.m1.6.6.1.1.3.3.1.1.3.3" xref="S4.Ex6.m1.6.6.1.1.3.3.1.1.3.3.cmml">1</mn></msub></mrow><mo id="S4.Ex6.m1.6.6.1.1.4.4.2.4" lspace="0.278em" rspace="0.278em" xref="S4.Ex6.m1.6.6.1.1.4.4.3.1.cmml">:</mo><mrow id="S4.Ex6.m1.6.6.1.1.4.4.2.2.2" xref="S4.Ex6.m1.6.6.1.1.4.4.2.2.3.cmml"><mrow id="S4.Ex6.m1.6.6.1.1.4.4.2.2.1.1" xref="S4.Ex6.m1.6.6.1.1.4.4.2.2.1.1.cmml"><mrow id="S4.Ex6.m1.6.6.1.1.4.4.2.2.1.1.2.2" xref="S4.Ex6.m1.6.6.1.1.4.4.2.2.1.1.2.1.cmml"><mi id="S4.Ex6.m1.4.4" xref="S4.Ex6.m1.4.4.cmml">dom</mi><mo id="S4.Ex6.m1.6.6.1.1.4.4.2.2.1.1.2.2a" xref="S4.Ex6.m1.6.6.1.1.4.4.2.2.1.1.2.1.cmml">⁡</mo><mrow id="S4.Ex6.m1.6.6.1.1.4.4.2.2.1.1.2.2.1" xref="S4.Ex6.m1.6.6.1.1.4.4.2.2.1.1.2.1.cmml"><mo id="S4.Ex6.m1.6.6.1.1.4.4.2.2.1.1.2.2.1.1" stretchy="false" xref="S4.Ex6.m1.6.6.1.1.4.4.2.2.1.1.2.1.cmml">(</mo><mi id="S4.Ex6.m1.5.5" xref="S4.Ex6.m1.5.5.cmml">t</mi><mo id="S4.Ex6.m1.6.6.1.1.4.4.2.2.1.1.2.2.1.2" stretchy="false" xref="S4.Ex6.m1.6.6.1.1.4.4.2.2.1.1.2.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex6.m1.6.6.1.1.4.4.2.2.1.1.1" xref="S4.Ex6.m1.6.6.1.1.4.4.2.2.1.1.1.cmml">=</mo><msub id="S4.Ex6.m1.6.6.1.1.4.4.2.2.1.1.3" xref="S4.Ex6.m1.6.6.1.1.4.4.2.2.1.1.3.cmml"><mi id="S4.Ex6.m1.6.6.1.1.4.4.2.2.1.1.3.2" xref="S4.Ex6.m1.6.6.1.1.4.4.2.2.1.1.3.2.cmml">λ</mi><mi id="S4.Ex6.m1.6.6.1.1.4.4.2.2.1.1.3.3" xref="S4.Ex6.m1.6.6.1.1.4.4.2.2.1.1.3.3.cmml">ξ</mi></msub></mrow><mo id="S4.Ex6.m1.6.6.1.1.4.4.2.2.2.3" xref="S4.Ex6.m1.6.6.1.1.4.4.2.2.3a.cmml">,</mo><mrow id="S4.Ex6.m1.6.6.1.1.4.4.2.2.2.2" xref="S4.Ex6.m1.6.6.1.1.4.4.2.2.2.2.cmml"><mi id="S4.Ex6.m1.6.6.1.1.4.4.2.2.2.2.2" xref="S4.Ex6.m1.6.6.1.1.4.4.2.2.2.2.2.cmml">ξ</mi><mo id="S4.Ex6.m1.6.6.1.1.4.4.2.2.2.2.1" xref="S4.Ex6.m1.6.6.1.1.4.4.2.2.2.2.1.cmml">∈</mo><mi id="S4.Ex6.m1.6.6.1.1.4.4.2.2.2.2.3" xref="S4.Ex6.m1.6.6.1.1.4.4.2.2.2.2.3.cmml">Y</mi></mrow></mrow><mo id="S4.Ex6.m1.6.6.1.1.4.4.2.5" stretchy="false" xref="S4.Ex6.m1.6.6.1.1.4.4.3.1.cmml">}</mo></mrow></mrow></mrow><mo id="S4.Ex6.m1.6.6.1.2" lspace="0em" xref="S4.Ex6.m1.6.6.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex6.m1.6b"><apply id="S4.Ex6.m1.6.6.1.1.cmml" xref="S4.Ex6.m1.6.6.1"><csymbol cd="latexml" id="S4.Ex6.m1.6.6.1.1.5.cmml" xref="S4.Ex6.m1.6.6.1.1.5">assign</csymbol><ci id="S4.Ex6.m1.6.6.1.1.6.cmml" xref="S4.Ex6.m1.6.6.1.1.6">𝐴</ci><apply id="S4.Ex6.m1.6.6.1.1.4.cmml" xref="S4.Ex6.m1.6.6.1.1.4"><union id="S4.Ex6.m1.6.6.1.1.4.5.cmml" xref="S4.Ex6.m1.6.6.1.1.4.5"></union><apply id="S4.Ex6.m1.6.6.1.1.2.2.3.cmml" xref="S4.Ex6.m1.6.6.1.1.2.2.2"><csymbol cd="latexml" id="S4.Ex6.m1.6.6.1.1.2.2.3.1.cmml" xref="S4.Ex6.m1.6.6.1.1.2.2.2.3">conditional-set</csymbol><apply id="S4.Ex6.m1.6.6.1.1.1.1.1.1.cmml" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1"><in id="S4.Ex6.m1.6.6.1.1.1.1.1.1.1.cmml" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.1"></in><ci id="S4.Ex6.m1.6.6.1.1.1.1.1.1.2.cmml" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.2">𝑡</ci><apply id="S4.Ex6.m1.6.6.1.1.1.1.1.1.3.cmml" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.Ex6.m1.6.6.1.1.1.1.1.1.3.1.cmml" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.3">subscript</csymbol><ci id="S4.Ex6.m1.6.6.1.1.1.1.1.1.3.2.cmml" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.3.2">𝐴</ci><cn id="S4.Ex6.m1.6.6.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S4.Ex6.m1.6.6.1.1.1.1.1.1.3.3">0</cn></apply></apply><apply id="S4.Ex6.m1.6.6.1.1.2.2.2.2.3.cmml" xref="S4.Ex6.m1.6.6.1.1.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.Ex6.m1.6.6.1.1.2.2.2.2.3a.cmml" xref="S4.Ex6.m1.6.6.1.1.2.2.2.2.2.3">formulae-sequence</csymbol><apply id="S4.Ex6.m1.6.6.1.1.2.2.2.2.1.1.cmml" xref="S4.Ex6.m1.6.6.1.1.2.2.2.2.1.1"><eq id="S4.Ex6.m1.6.6.1.1.2.2.2.2.1.1.1.cmml" xref="S4.Ex6.m1.6.6.1.1.2.2.2.2.1.1.1"></eq><apply id="S4.Ex6.m1.6.6.1.1.2.2.2.2.1.1.2.1.cmml" xref="S4.Ex6.m1.6.6.1.1.2.2.2.2.1.1.2.2"><ci id="S4.Ex6.m1.1.1.cmml" xref="S4.Ex6.m1.1.1">dom</ci><ci id="S4.Ex6.m1.2.2.cmml" xref="S4.Ex6.m1.2.2">𝑡</ci></apply><apply id="S4.Ex6.m1.6.6.1.1.2.2.2.2.1.1.3.cmml" xref="S4.Ex6.m1.6.6.1.1.2.2.2.2.1.1.3"><csymbol cd="ambiguous" id="S4.Ex6.m1.6.6.1.1.2.2.2.2.1.1.3.1.cmml" xref="S4.Ex6.m1.6.6.1.1.2.2.2.2.1.1.3">subscript</csymbol><ci id="S4.Ex6.m1.6.6.1.1.2.2.2.2.1.1.3.2.cmml" xref="S4.Ex6.m1.6.6.1.1.2.2.2.2.1.1.3.2">𝜆</ci><ci id="S4.Ex6.m1.6.6.1.1.2.2.2.2.1.1.3.3.cmml" xref="S4.Ex6.m1.6.6.1.1.2.2.2.2.1.1.3.3">𝜉</ci></apply></apply><apply id="S4.Ex6.m1.6.6.1.1.2.2.2.2.2.2.cmml" xref="S4.Ex6.m1.6.6.1.1.2.2.2.2.2.2"><in id="S4.Ex6.m1.6.6.1.1.2.2.2.2.2.2.1.cmml" xref="S4.Ex6.m1.6.6.1.1.2.2.2.2.2.2.1"></in><ci id="S4.Ex6.m1.6.6.1.1.2.2.2.2.2.2.2.cmml" xref="S4.Ex6.m1.6.6.1.1.2.2.2.2.2.2.2">𝜉</ci><ci id="S4.Ex6.m1.6.6.1.1.2.2.2.2.2.2.3.cmml" xref="S4.Ex6.m1.6.6.1.1.2.2.2.2.2.2.3">𝑋</ci></apply></apply></apply><apply id="S4.Ex6.m1.6.6.1.1.4.4.3.cmml" xref="S4.Ex6.m1.6.6.1.1.4.4.2"><csymbol cd="latexml" id="S4.Ex6.m1.6.6.1.1.4.4.3.1.cmml" xref="S4.Ex6.m1.6.6.1.1.4.4.2.3">conditional-set</csymbol><apply id="S4.Ex6.m1.6.6.1.1.3.3.1.1.cmml" xref="S4.Ex6.m1.6.6.1.1.3.3.1.1"><in id="S4.Ex6.m1.6.6.1.1.3.3.1.1.1.cmml" xref="S4.Ex6.m1.6.6.1.1.3.3.1.1.1"></in><apply id="S4.Ex6.m1.6.6.1.1.3.3.1.1.2.cmml" xref="S4.Ex6.m1.6.6.1.1.3.3.1.1.2"><times id="S4.Ex6.m1.6.6.1.1.3.3.1.1.2.1.cmml" xref="S4.Ex6.m1.6.6.1.1.3.3.1.1.2.1"></times><apply id="S4.Ex6.m1.6.6.1.1.3.3.1.1.2.2.cmml" xref="S4.Ex6.m1.6.6.1.1.3.3.1.1.2.2"><csymbol cd="ambiguous" id="S4.Ex6.m1.6.6.1.1.3.3.1.1.2.2.1.cmml" xref="S4.Ex6.m1.6.6.1.1.3.3.1.1.2.2">superscript</csymbol><ci id="S4.Ex6.m1.6.6.1.1.3.3.1.1.2.2.2.cmml" xref="S4.Ex6.m1.6.6.1.1.3.3.1.1.2.2.2">𝑡</ci><ci id="S4.Ex6.m1.6.6.1.1.3.3.1.1.2.2.3.cmml" xref="S4.Ex6.m1.6.6.1.1.3.3.1.1.2.2.3">⌢</ci></apply><apply id="S4.Ex6.m1.6.6.1.1.3.3.1.1.2.3.1.cmml" xref="S4.Ex6.m1.6.6.1.1.3.3.1.1.2.3.2"><csymbol cd="latexml" id="S4.Ex6.m1.6.6.1.1.3.3.1.1.2.3.1.1.cmml" xref="S4.Ex6.m1.6.6.1.1.3.3.1.1.2.3.2.1">delimited-⟨⟩</csymbol><ci id="S4.Ex6.m1.3.3.cmml" xref="S4.Ex6.m1.3.3">𝜔</ci></apply></apply><apply id="S4.Ex6.m1.6.6.1.1.3.3.1.1.3.cmml" xref="S4.Ex6.m1.6.6.1.1.3.3.1.1.3"><csymbol cd="ambiguous" id="S4.Ex6.m1.6.6.1.1.3.3.1.1.3.1.cmml" xref="S4.Ex6.m1.6.6.1.1.3.3.1.1.3">subscript</csymbol><ci id="S4.Ex6.m1.6.6.1.1.3.3.1.1.3.2.cmml" xref="S4.Ex6.m1.6.6.1.1.3.3.1.1.3.2">𝐴</ci><cn id="S4.Ex6.m1.6.6.1.1.3.3.1.1.3.3.cmml" type="integer" xref="S4.Ex6.m1.6.6.1.1.3.3.1.1.3.3">1</cn></apply></apply><apply id="S4.Ex6.m1.6.6.1.1.4.4.2.2.3.cmml" xref="S4.Ex6.m1.6.6.1.1.4.4.2.2.2"><csymbol cd="ambiguous" id="S4.Ex6.m1.6.6.1.1.4.4.2.2.3a.cmml" xref="S4.Ex6.m1.6.6.1.1.4.4.2.2.2.3">formulae-sequence</csymbol><apply id="S4.Ex6.m1.6.6.1.1.4.4.2.2.1.1.cmml" xref="S4.Ex6.m1.6.6.1.1.4.4.2.2.1.1"><eq id="S4.Ex6.m1.6.6.1.1.4.4.2.2.1.1.1.cmml" xref="S4.Ex6.m1.6.6.1.1.4.4.2.2.1.1.1"></eq><apply id="S4.Ex6.m1.6.6.1.1.4.4.2.2.1.1.2.1.cmml" xref="S4.Ex6.m1.6.6.1.1.4.4.2.2.1.1.2.2"><ci id="S4.Ex6.m1.4.4.cmml" xref="S4.Ex6.m1.4.4">dom</ci><ci id="S4.Ex6.m1.5.5.cmml" xref="S4.Ex6.m1.5.5">𝑡</ci></apply><apply id="S4.Ex6.m1.6.6.1.1.4.4.2.2.1.1.3.cmml" xref="S4.Ex6.m1.6.6.1.1.4.4.2.2.1.1.3"><csymbol cd="ambiguous" id="S4.Ex6.m1.6.6.1.1.4.4.2.2.1.1.3.1.cmml" xref="S4.Ex6.m1.6.6.1.1.4.4.2.2.1.1.3">subscript</csymbol><ci id="S4.Ex6.m1.6.6.1.1.4.4.2.2.1.1.3.2.cmml" xref="S4.Ex6.m1.6.6.1.1.4.4.2.2.1.1.3.2">𝜆</ci><ci id="S4.Ex6.m1.6.6.1.1.4.4.2.2.1.1.3.3.cmml" xref="S4.Ex6.m1.6.6.1.1.4.4.2.2.1.1.3.3">𝜉</ci></apply></apply><apply id="S4.Ex6.m1.6.6.1.1.4.4.2.2.2.2.cmml" xref="S4.Ex6.m1.6.6.1.1.4.4.2.2.2.2"><in id="S4.Ex6.m1.6.6.1.1.4.4.2.2.2.2.1.cmml" xref="S4.Ex6.m1.6.6.1.1.4.4.2.2.2.2.1"></in><ci id="S4.Ex6.m1.6.6.1.1.4.4.2.2.2.2.2.cmml" xref="S4.Ex6.m1.6.6.1.1.4.4.2.2.2.2.2">𝜉</ci><ci id="S4.Ex6.m1.6.6.1.1.4.4.2.2.2.2.3.cmml" xref="S4.Ex6.m1.6.6.1.1.4.4.2.2.2.2.3">𝑌</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex6.m1.6c">A:=\{t\in A_{0}:\operatorname{dom}(t)=\lambda_{\xi},\xi\in X\}\cup\{t^{\frown}% \langle\omega\rangle\in A_{1}:\operatorname{dom}(t)=\lambda_{\xi},\xi\in Y\}.</annotation><annotation encoding="application/x-llamapun" id="S4.Ex6.m1.6d">italic_A := { italic_t ∈ italic_A start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT : roman_dom ( italic_t ) = italic_λ start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT , italic_ξ ∈ italic_X } ∪ { italic_t start_POSTSUPERSCRIPT ⌢ end_POSTSUPERSCRIPT ⟨ italic_ω ⟩ ∈ italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT : roman_dom ( italic_t ) = italic_λ start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT , italic_ξ ∈ italic_Y } .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.4.p1.11">The <math alttext="\aleph_{1}" class="ltx_Math" display="inline" id="S4.4.p1.7.m1.1"><semantics id="S4.4.p1.7.m1.1a"><msub id="S4.4.p1.7.m1.1.1" xref="S4.4.p1.7.m1.1.1.cmml"><mi id="S4.4.p1.7.m1.1.1.2" mathvariant="normal" xref="S4.4.p1.7.m1.1.1.2.cmml">ℵ</mi><mn id="S4.4.p1.7.m1.1.1.3" xref="S4.4.p1.7.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.4.p1.7.m1.1b"><apply id="S4.4.p1.7.m1.1.1.cmml" xref="S4.4.p1.7.m1.1.1"><csymbol cd="ambiguous" id="S4.4.p1.7.m1.1.1.1.cmml" xref="S4.4.p1.7.m1.1.1">subscript</csymbol><ci id="S4.4.p1.7.m1.1.1.2.cmml" xref="S4.4.p1.7.m1.1.1.2">ℵ</ci><cn id="S4.4.p1.7.m1.1.1.3.cmml" type="integer" xref="S4.4.p1.7.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.4.p1.7.m1.1c">\aleph_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.4.p1.7.m1.1d">roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-density of <math alttext="A" class="ltx_Math" display="inline" id="S4.4.p1.8.m2.1"><semantics id="S4.4.p1.8.m2.1a"><mi id="S4.4.p1.8.m2.1.1" xref="S4.4.p1.8.m2.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S4.4.p1.8.m2.1b"><ci id="S4.4.p1.8.m2.1.1.cmml" xref="S4.4.p1.8.m2.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.4.p1.8.m2.1c">A</annotation><annotation encoding="application/x-llamapun" id="S4.4.p1.8.m2.1d">italic_A</annotation></semantics></math> follows from the fact that in the proof of <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S4.Thmtheorem4" title="Lemma 4.4. ‣ 4. Aronszajn line decompositions ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">4.4</span></a>, the <math alttext="u" class="ltx_Math" display="inline" id="S4.4.p1.9.m3.1"><semantics id="S4.4.p1.9.m3.1a"><mi id="S4.4.p1.9.m3.1.1" xref="S4.4.p1.9.m3.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.4.p1.9.m3.1b"><ci id="S4.4.p1.9.m3.1.1.cmml" xref="S4.4.p1.9.m3.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.4.p1.9.m3.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.4.p1.9.m3.1d">italic_u</annotation></semantics></math> satisfies that <math alttext="u\uparrow\cap A" class="ltx_Math" display="inline" id="S4.4.p1.10.m4.1"><semantics id="S4.4.p1.10.m4.1a"><mrow id="S4.4.p1.10.m4.1.1" xref="S4.4.p1.10.m4.1.1.cmml"><mi id="S4.4.p1.10.m4.1.1.2" xref="S4.4.p1.10.m4.1.1.2.cmml">u</mi><mo id="S4.4.p1.10.m4.1.1.1" rspace="0.1389em" stretchy="false" xref="S4.4.p1.10.m4.1.1.1.cmml">↑</mo><mrow id="S4.4.p1.10.m4.1.1.3" xref="S4.4.p1.10.m4.1.1.3.cmml"><mo id="S4.4.p1.10.m4.1.1.3a" lspace="0.1389em" rspace="0em" xref="S4.4.p1.10.m4.1.1.3.cmml">∩</mo><mi id="S4.4.p1.10.m4.1.1.3.2" xref="S4.4.p1.10.m4.1.1.3.2.cmml">A</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.4.p1.10.m4.1b"><apply id="S4.4.p1.10.m4.1.1.cmml" xref="S4.4.p1.10.m4.1.1"><ci id="S4.4.p1.10.m4.1.1.1.cmml" xref="S4.4.p1.10.m4.1.1.1">↑</ci><ci id="S4.4.p1.10.m4.1.1.2.cmml" xref="S4.4.p1.10.m4.1.1.2">𝑢</ci><apply id="S4.4.p1.10.m4.1.1.3.cmml" xref="S4.4.p1.10.m4.1.1.3"><intersect id="S4.4.p1.10.m4.1.1.3.1.cmml" xref="S4.4.p1.10.m4.1.1.3"></intersect><ci id="S4.4.p1.10.m4.1.1.3.2.cmml" xref="S4.4.p1.10.m4.1.1.3.2">𝐴</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.4.p1.10.m4.1c">u\uparrow\cap A</annotation><annotation encoding="application/x-llamapun" id="S4.4.p1.10.m4.1d">italic_u ↑ ∩ italic_A</annotation></semantics></math> is also uncountable. To see this note that above every countable ordinal, there are uncountable many in <math alttext="\Lambda\setminus\Lambda^{\prime}" class="ltx_Math" display="inline" id="S4.4.p1.11.m5.1"><semantics id="S4.4.p1.11.m5.1a"><mrow id="S4.4.p1.11.m5.1.1" xref="S4.4.p1.11.m5.1.1.cmml"><mi id="S4.4.p1.11.m5.1.1.2" mathvariant="normal" xref="S4.4.p1.11.m5.1.1.2.cmml">Λ</mi><mo id="S4.4.p1.11.m5.1.1.1" xref="S4.4.p1.11.m5.1.1.1.cmml">∖</mo><msup id="S4.4.p1.11.m5.1.1.3" xref="S4.4.p1.11.m5.1.1.3.cmml"><mi id="S4.4.p1.11.m5.1.1.3.2" mathvariant="normal" xref="S4.4.p1.11.m5.1.1.3.2.cmml">Λ</mi><mo id="S4.4.p1.11.m5.1.1.3.3" xref="S4.4.p1.11.m5.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.4.p1.11.m5.1b"><apply id="S4.4.p1.11.m5.1.1.cmml" xref="S4.4.p1.11.m5.1.1"><setdiff id="S4.4.p1.11.m5.1.1.1.cmml" xref="S4.4.p1.11.m5.1.1.1"></setdiff><ci id="S4.4.p1.11.m5.1.1.2.cmml" xref="S4.4.p1.11.m5.1.1.2">Λ</ci><apply id="S4.4.p1.11.m5.1.1.3.cmml" xref="S4.4.p1.11.m5.1.1.3"><csymbol cd="ambiguous" id="S4.4.p1.11.m5.1.1.3.1.cmml" xref="S4.4.p1.11.m5.1.1.3">superscript</csymbol><ci id="S4.4.p1.11.m5.1.1.3.2.cmml" xref="S4.4.p1.11.m5.1.1.3.2">Λ</ci><ci id="S4.4.p1.11.m5.1.1.3.3.cmml" xref="S4.4.p1.11.m5.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.4.p1.11.m5.1c">\Lambda\setminus\Lambda^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.4.p1.11.m5.1d">roman_Λ ∖ roman_Λ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.5.p2"> <p class="ltx_p" id="S4.5.p2.13">For <math alttext="\xi<\omega_{1}" class="ltx_Math" display="inline" id="S4.5.p2.1.m1.1"><semantics id="S4.5.p2.1.m1.1a"><mrow id="S4.5.p2.1.m1.1.1" xref="S4.5.p2.1.m1.1.1.cmml"><mi id="S4.5.p2.1.m1.1.1.2" xref="S4.5.p2.1.m1.1.1.2.cmml">ξ</mi><mo id="S4.5.p2.1.m1.1.1.1" xref="S4.5.p2.1.m1.1.1.1.cmml"><</mo><msub id="S4.5.p2.1.m1.1.1.3" xref="S4.5.p2.1.m1.1.1.3.cmml"><mi id="S4.5.p2.1.m1.1.1.3.2" xref="S4.5.p2.1.m1.1.1.3.2.cmml">ω</mi><mn id="S4.5.p2.1.m1.1.1.3.3" xref="S4.5.p2.1.m1.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.5.p2.1.m1.1b"><apply id="S4.5.p2.1.m1.1.1.cmml" xref="S4.5.p2.1.m1.1.1"><lt id="S4.5.p2.1.m1.1.1.1.cmml" xref="S4.5.p2.1.m1.1.1.1"></lt><ci id="S4.5.p2.1.m1.1.1.2.cmml" xref="S4.5.p2.1.m1.1.1.2">𝜉</ci><apply id="S4.5.p2.1.m1.1.1.3.cmml" xref="S4.5.p2.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.5.p2.1.m1.1.1.3.1.cmml" xref="S4.5.p2.1.m1.1.1.3">subscript</csymbol><ci id="S4.5.p2.1.m1.1.1.3.2.cmml" xref="S4.5.p2.1.m1.1.1.3.2">𝜔</ci><cn id="S4.5.p2.1.m1.1.1.3.3.cmml" type="integer" xref="S4.5.p2.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p2.1.m1.1c">\xi<\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.5.p2.1.m1.1d">italic_ξ < italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, let <math alttext="D_{\xi}:=\{t\in A:\operatorname{dom}(t)<\lambda_{\xi}\}" class="ltx_Math" display="inline" id="S4.5.p2.2.m2.4"><semantics id="S4.5.p2.2.m2.4a"><mrow id="S4.5.p2.2.m2.4.4" xref="S4.5.p2.2.m2.4.4.cmml"><msub id="S4.5.p2.2.m2.4.4.4" xref="S4.5.p2.2.m2.4.4.4.cmml"><mi id="S4.5.p2.2.m2.4.4.4.2" xref="S4.5.p2.2.m2.4.4.4.2.cmml">D</mi><mi id="S4.5.p2.2.m2.4.4.4.3" xref="S4.5.p2.2.m2.4.4.4.3.cmml">ξ</mi></msub><mo id="S4.5.p2.2.m2.4.4.3" lspace="0.278em" rspace="0.278em" xref="S4.5.p2.2.m2.4.4.3.cmml">:=</mo><mrow id="S4.5.p2.2.m2.4.4.2.2" xref="S4.5.p2.2.m2.4.4.2.3.cmml"><mo id="S4.5.p2.2.m2.4.4.2.2.3" stretchy="false" xref="S4.5.p2.2.m2.4.4.2.3.1.cmml">{</mo><mrow id="S4.5.p2.2.m2.3.3.1.1.1" xref="S4.5.p2.2.m2.3.3.1.1.1.cmml"><mi id="S4.5.p2.2.m2.3.3.1.1.1.2" xref="S4.5.p2.2.m2.3.3.1.1.1.2.cmml">t</mi><mo id="S4.5.p2.2.m2.3.3.1.1.1.1" xref="S4.5.p2.2.m2.3.3.1.1.1.1.cmml">∈</mo><mi id="S4.5.p2.2.m2.3.3.1.1.1.3" xref="S4.5.p2.2.m2.3.3.1.1.1.3.cmml">A</mi></mrow><mo id="S4.5.p2.2.m2.4.4.2.2.4" lspace="0.278em" rspace="0.278em" xref="S4.5.p2.2.m2.4.4.2.3.1.cmml">:</mo><mrow id="S4.5.p2.2.m2.4.4.2.2.2" xref="S4.5.p2.2.m2.4.4.2.2.2.cmml"><mrow id="S4.5.p2.2.m2.4.4.2.2.2.2.2" xref="S4.5.p2.2.m2.4.4.2.2.2.2.1.cmml"><mi id="S4.5.p2.2.m2.1.1" xref="S4.5.p2.2.m2.1.1.cmml">dom</mi><mo id="S4.5.p2.2.m2.4.4.2.2.2.2.2a" xref="S4.5.p2.2.m2.4.4.2.2.2.2.1.cmml">⁡</mo><mrow id="S4.5.p2.2.m2.4.4.2.2.2.2.2.1" xref="S4.5.p2.2.m2.4.4.2.2.2.2.1.cmml"><mo id="S4.5.p2.2.m2.4.4.2.2.2.2.2.1.1" stretchy="false" xref="S4.5.p2.2.m2.4.4.2.2.2.2.1.cmml">(</mo><mi id="S4.5.p2.2.m2.2.2" xref="S4.5.p2.2.m2.2.2.cmml">t</mi><mo id="S4.5.p2.2.m2.4.4.2.2.2.2.2.1.2" stretchy="false" xref="S4.5.p2.2.m2.4.4.2.2.2.2.1.cmml">)</mo></mrow></mrow><mo id="S4.5.p2.2.m2.4.4.2.2.2.1" xref="S4.5.p2.2.m2.4.4.2.2.2.1.cmml"><</mo><msub id="S4.5.p2.2.m2.4.4.2.2.2.3" xref="S4.5.p2.2.m2.4.4.2.2.2.3.cmml"><mi id="S4.5.p2.2.m2.4.4.2.2.2.3.2" xref="S4.5.p2.2.m2.4.4.2.2.2.3.2.cmml">λ</mi><mi id="S4.5.p2.2.m2.4.4.2.2.2.3.3" xref="S4.5.p2.2.m2.4.4.2.2.2.3.3.cmml">ξ</mi></msub></mrow><mo id="S4.5.p2.2.m2.4.4.2.2.5" stretchy="false" xref="S4.5.p2.2.m2.4.4.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.5.p2.2.m2.4b"><apply id="S4.5.p2.2.m2.4.4.cmml" xref="S4.5.p2.2.m2.4.4"><csymbol cd="latexml" id="S4.5.p2.2.m2.4.4.3.cmml" xref="S4.5.p2.2.m2.4.4.3">assign</csymbol><apply id="S4.5.p2.2.m2.4.4.4.cmml" xref="S4.5.p2.2.m2.4.4.4"><csymbol cd="ambiguous" id="S4.5.p2.2.m2.4.4.4.1.cmml" xref="S4.5.p2.2.m2.4.4.4">subscript</csymbol><ci id="S4.5.p2.2.m2.4.4.4.2.cmml" xref="S4.5.p2.2.m2.4.4.4.2">𝐷</ci><ci id="S4.5.p2.2.m2.4.4.4.3.cmml" xref="S4.5.p2.2.m2.4.4.4.3">𝜉</ci></apply><apply id="S4.5.p2.2.m2.4.4.2.3.cmml" xref="S4.5.p2.2.m2.4.4.2.2"><csymbol cd="latexml" id="S4.5.p2.2.m2.4.4.2.3.1.cmml" xref="S4.5.p2.2.m2.4.4.2.2.3">conditional-set</csymbol><apply id="S4.5.p2.2.m2.3.3.1.1.1.cmml" xref="S4.5.p2.2.m2.3.3.1.1.1"><in id="S4.5.p2.2.m2.3.3.1.1.1.1.cmml" xref="S4.5.p2.2.m2.3.3.1.1.1.1"></in><ci id="S4.5.p2.2.m2.3.3.1.1.1.2.cmml" xref="S4.5.p2.2.m2.3.3.1.1.1.2">𝑡</ci><ci id="S4.5.p2.2.m2.3.3.1.1.1.3.cmml" xref="S4.5.p2.2.m2.3.3.1.1.1.3">𝐴</ci></apply><apply id="S4.5.p2.2.m2.4.4.2.2.2.cmml" xref="S4.5.p2.2.m2.4.4.2.2.2"><lt id="S4.5.p2.2.m2.4.4.2.2.2.1.cmml" xref="S4.5.p2.2.m2.4.4.2.2.2.1"></lt><apply id="S4.5.p2.2.m2.4.4.2.2.2.2.1.cmml" xref="S4.5.p2.2.m2.4.4.2.2.2.2.2"><ci id="S4.5.p2.2.m2.1.1.cmml" xref="S4.5.p2.2.m2.1.1">dom</ci><ci id="S4.5.p2.2.m2.2.2.cmml" xref="S4.5.p2.2.m2.2.2">𝑡</ci></apply><apply id="S4.5.p2.2.m2.4.4.2.2.2.3.cmml" xref="S4.5.p2.2.m2.4.4.2.2.2.3"><csymbol cd="ambiguous" id="S4.5.p2.2.m2.4.4.2.2.2.3.1.cmml" xref="S4.5.p2.2.m2.4.4.2.2.2.3">subscript</csymbol><ci id="S4.5.p2.2.m2.4.4.2.2.2.3.2.cmml" xref="S4.5.p2.2.m2.4.4.2.2.2.3.2">𝜆</ci><ci id="S4.5.p2.2.m2.4.4.2.2.2.3.3.cmml" xref="S4.5.p2.2.m2.4.4.2.2.2.3.3">𝜉</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p2.2.m2.4c">D_{\xi}:=\{t\in A:\operatorname{dom}(t)<\lambda_{\xi}\}</annotation><annotation encoding="application/x-llamapun" id="S4.5.p2.2.m2.4d">italic_D start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT := { italic_t ∈ italic_A : roman_dom ( italic_t ) < italic_λ start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT }</annotation></semantics></math>. Clearly <math alttext="D:=\langle D_{\xi}:\xi<\omega_{1}\rangle" class="ltx_math_unparsed" display="inline" id="S4.5.p2.3.m3.1"><semantics id="S4.5.p2.3.m3.1a"><mrow id="S4.5.p2.3.m3.1b"><mi id="S4.5.p2.3.m3.1.1">D</mi><mo id="S4.5.p2.3.m3.1.2" lspace="0.278em" rspace="0.278em">:=</mo><mrow id="S4.5.p2.3.m3.1.3"><mo id="S4.5.p2.3.m3.1.3.1" stretchy="false">⟨</mo><msub id="S4.5.p2.3.m3.1.3.2"><mi id="S4.5.p2.3.m3.1.3.2.2">D</mi><mi id="S4.5.p2.3.m3.1.3.2.3">ξ</mi></msub><mo id="S4.5.p2.3.m3.1.3.3" lspace="0.278em" rspace="0.278em">:</mo><mi id="S4.5.p2.3.m3.1.3.4">ξ</mi><mo id="S4.5.p2.3.m3.1.3.5"><</mo><msub id="S4.5.p2.3.m3.1.3.6"><mi id="S4.5.p2.3.m3.1.3.6.2">ω</mi><mn id="S4.5.p2.3.m3.1.3.6.3">1</mn></msub><mo id="S4.5.p2.3.m3.1.3.7" stretchy="false">⟩</mo></mrow></mrow><annotation encoding="application/x-tex" id="S4.5.p2.3.m3.1c">D:=\langle D_{\xi}:\xi<\omega_{1}\rangle</annotation><annotation encoding="application/x-llamapun" id="S4.5.p2.3.m3.1d">italic_D := ⟨ italic_D start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT : italic_ξ < italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⟩</annotation></semantics></math> is a decomposition for <math alttext="A" class="ltx_Math" display="inline" id="S4.5.p2.4.m4.1"><semantics id="S4.5.p2.4.m4.1a"><mi id="S4.5.p2.4.m4.1.1" xref="S4.5.p2.4.m4.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S4.5.p2.4.m4.1b"><ci id="S4.5.p2.4.m4.1.1.cmml" xref="S4.5.p2.4.m4.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p2.4.m4.1c">A</annotation><annotation encoding="application/x-llamapun" id="S4.5.p2.4.m4.1d">italic_A</annotation></semantics></math>. We claim that for every <math alttext="\xi<\omega_{1}" class="ltx_Math" display="inline" id="S4.5.p2.5.m5.1"><semantics id="S4.5.p2.5.m5.1a"><mrow id="S4.5.p2.5.m5.1.1" xref="S4.5.p2.5.m5.1.1.cmml"><mi id="S4.5.p2.5.m5.1.1.2" xref="S4.5.p2.5.m5.1.1.2.cmml">ξ</mi><mo id="S4.5.p2.5.m5.1.1.1" xref="S4.5.p2.5.m5.1.1.1.cmml"><</mo><msub id="S4.5.p2.5.m5.1.1.3" xref="S4.5.p2.5.m5.1.1.3.cmml"><mi id="S4.5.p2.5.m5.1.1.3.2" xref="S4.5.p2.5.m5.1.1.3.2.cmml">ω</mi><mn id="S4.5.p2.5.m5.1.1.3.3" xref="S4.5.p2.5.m5.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.5.p2.5.m5.1b"><apply id="S4.5.p2.5.m5.1.1.cmml" xref="S4.5.p2.5.m5.1.1"><lt id="S4.5.p2.5.m5.1.1.1.cmml" xref="S4.5.p2.5.m5.1.1.1"></lt><ci id="S4.5.p2.5.m5.1.1.2.cmml" xref="S4.5.p2.5.m5.1.1.2">𝜉</ci><apply id="S4.5.p2.5.m5.1.1.3.cmml" xref="S4.5.p2.5.m5.1.1.3"><csymbol cd="ambiguous" id="S4.5.p2.5.m5.1.1.3.1.cmml" xref="S4.5.p2.5.m5.1.1.3">subscript</csymbol><ci id="S4.5.p2.5.m5.1.1.3.2.cmml" xref="S4.5.p2.5.m5.1.1.3.2">𝜔</ci><cn id="S4.5.p2.5.m5.1.1.3.3.cmml" type="integer" xref="S4.5.p2.5.m5.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p2.5.m5.1c">\xi<\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.5.p2.5.m5.1d">italic_ξ < italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, the complementary intervals of <math alttext="A\setminus D_{\xi}" class="ltx_Math" display="inline" id="S4.5.p2.6.m6.1"><semantics id="S4.5.p2.6.m6.1a"><mrow id="S4.5.p2.6.m6.1.1" xref="S4.5.p2.6.m6.1.1.cmml"><mi id="S4.5.p2.6.m6.1.1.2" xref="S4.5.p2.6.m6.1.1.2.cmml">A</mi><mo id="S4.5.p2.6.m6.1.1.1" xref="S4.5.p2.6.m6.1.1.1.cmml">∖</mo><msub id="S4.5.p2.6.m6.1.1.3" xref="S4.5.p2.6.m6.1.1.3.cmml"><mi id="S4.5.p2.6.m6.1.1.3.2" xref="S4.5.p2.6.m6.1.1.3.2.cmml">D</mi><mi id="S4.5.p2.6.m6.1.1.3.3" xref="S4.5.p2.6.m6.1.1.3.3.cmml">ξ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.5.p2.6.m6.1b"><apply id="S4.5.p2.6.m6.1.1.cmml" xref="S4.5.p2.6.m6.1.1"><setdiff id="S4.5.p2.6.m6.1.1.1.cmml" xref="S4.5.p2.6.m6.1.1.1"></setdiff><ci id="S4.5.p2.6.m6.1.1.2.cmml" xref="S4.5.p2.6.m6.1.1.2">𝐴</ci><apply id="S4.5.p2.6.m6.1.1.3.cmml" xref="S4.5.p2.6.m6.1.1.3"><csymbol cd="ambiguous" id="S4.5.p2.6.m6.1.1.3.1.cmml" xref="S4.5.p2.6.m6.1.1.3">subscript</csymbol><ci id="S4.5.p2.6.m6.1.1.3.2.cmml" xref="S4.5.p2.6.m6.1.1.3.2">𝐷</ci><ci id="S4.5.p2.6.m6.1.1.3.3.cmml" xref="S4.5.p2.6.m6.1.1.3.3">𝜉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p2.6.m6.1c">A\setminus D_{\xi}</annotation><annotation encoding="application/x-llamapun" id="S4.5.p2.6.m6.1d">italic_A ∖ italic_D start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT</annotation></semantics></math> are all of the form <math alttext="\{t\in A:t_{0}\sqsubseteq t\}" class="ltx_Math" display="inline" id="S4.5.p2.7.m7.2"><semantics id="S4.5.p2.7.m7.2a"><mrow id="S4.5.p2.7.m7.2.2.2" xref="S4.5.p2.7.m7.2.2.3.cmml"><mo id="S4.5.p2.7.m7.2.2.2.3" stretchy="false" xref="S4.5.p2.7.m7.2.2.3.1.cmml">{</mo><mrow id="S4.5.p2.7.m7.1.1.1.1" xref="S4.5.p2.7.m7.1.1.1.1.cmml"><mi id="S4.5.p2.7.m7.1.1.1.1.2" xref="S4.5.p2.7.m7.1.1.1.1.2.cmml">t</mi><mo id="S4.5.p2.7.m7.1.1.1.1.1" xref="S4.5.p2.7.m7.1.1.1.1.1.cmml">∈</mo><mi id="S4.5.p2.7.m7.1.1.1.1.3" xref="S4.5.p2.7.m7.1.1.1.1.3.cmml">A</mi></mrow><mo id="S4.5.p2.7.m7.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S4.5.p2.7.m7.2.2.3.1.cmml">:</mo><mrow id="S4.5.p2.7.m7.2.2.2.2" xref="S4.5.p2.7.m7.2.2.2.2.cmml"><msub id="S4.5.p2.7.m7.2.2.2.2.2" xref="S4.5.p2.7.m7.2.2.2.2.2.cmml"><mi id="S4.5.p2.7.m7.2.2.2.2.2.2" xref="S4.5.p2.7.m7.2.2.2.2.2.2.cmml">t</mi><mn id="S4.5.p2.7.m7.2.2.2.2.2.3" xref="S4.5.p2.7.m7.2.2.2.2.2.3.cmml">0</mn></msub><mo id="S4.5.p2.7.m7.2.2.2.2.1" xref="S4.5.p2.7.m7.2.2.2.2.1.cmml">⊑</mo><mi id="S4.5.p2.7.m7.2.2.2.2.3" xref="S4.5.p2.7.m7.2.2.2.2.3.cmml">t</mi></mrow><mo id="S4.5.p2.7.m7.2.2.2.5" stretchy="false" xref="S4.5.p2.7.m7.2.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.5.p2.7.m7.2b"><apply id="S4.5.p2.7.m7.2.2.3.cmml" xref="S4.5.p2.7.m7.2.2.2"><csymbol cd="latexml" id="S4.5.p2.7.m7.2.2.3.1.cmml" xref="S4.5.p2.7.m7.2.2.2.3">conditional-set</csymbol><apply id="S4.5.p2.7.m7.1.1.1.1.cmml" xref="S4.5.p2.7.m7.1.1.1.1"><in id="S4.5.p2.7.m7.1.1.1.1.1.cmml" xref="S4.5.p2.7.m7.1.1.1.1.1"></in><ci id="S4.5.p2.7.m7.1.1.1.1.2.cmml" xref="S4.5.p2.7.m7.1.1.1.1.2">𝑡</ci><ci id="S4.5.p2.7.m7.1.1.1.1.3.cmml" xref="S4.5.p2.7.m7.1.1.1.1.3">𝐴</ci></apply><apply id="S4.5.p2.7.m7.2.2.2.2.cmml" xref="S4.5.p2.7.m7.2.2.2.2"><csymbol cd="latexml" id="S4.5.p2.7.m7.2.2.2.2.1.cmml" xref="S4.5.p2.7.m7.2.2.2.2.1">square-image-of-or-equals</csymbol><apply id="S4.5.p2.7.m7.2.2.2.2.2.cmml" xref="S4.5.p2.7.m7.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.5.p2.7.m7.2.2.2.2.2.1.cmml" xref="S4.5.p2.7.m7.2.2.2.2.2">subscript</csymbol><ci id="S4.5.p2.7.m7.2.2.2.2.2.2.cmml" xref="S4.5.p2.7.m7.2.2.2.2.2.2">𝑡</ci><cn id="S4.5.p2.7.m7.2.2.2.2.2.3.cmml" type="integer" xref="S4.5.p2.7.m7.2.2.2.2.2.3">0</cn></apply><ci id="S4.5.p2.7.m7.2.2.2.2.3.cmml" xref="S4.5.p2.7.m7.2.2.2.2.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p2.7.m7.2c">\{t\in A:t_{0}\sqsubseteq t\}</annotation><annotation encoding="application/x-llamapun" id="S4.5.p2.7.m7.2d">{ italic_t ∈ italic_A : italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⊑ italic_t }</annotation></semantics></math> for some <math alttext="t_{0}\in A_{1}" class="ltx_Math" display="inline" id="S4.5.p2.8.m8.1"><semantics id="S4.5.p2.8.m8.1a"><mrow id="S4.5.p2.8.m8.1.1" xref="S4.5.p2.8.m8.1.1.cmml"><msub id="S4.5.p2.8.m8.1.1.2" xref="S4.5.p2.8.m8.1.1.2.cmml"><mi id="S4.5.p2.8.m8.1.1.2.2" xref="S4.5.p2.8.m8.1.1.2.2.cmml">t</mi><mn id="S4.5.p2.8.m8.1.1.2.3" xref="S4.5.p2.8.m8.1.1.2.3.cmml">0</mn></msub><mo id="S4.5.p2.8.m8.1.1.1" xref="S4.5.p2.8.m8.1.1.1.cmml">∈</mo><msub id="S4.5.p2.8.m8.1.1.3" xref="S4.5.p2.8.m8.1.1.3.cmml"><mi id="S4.5.p2.8.m8.1.1.3.2" xref="S4.5.p2.8.m8.1.1.3.2.cmml">A</mi><mn id="S4.5.p2.8.m8.1.1.3.3" xref="S4.5.p2.8.m8.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.5.p2.8.m8.1b"><apply id="S4.5.p2.8.m8.1.1.cmml" xref="S4.5.p2.8.m8.1.1"><in id="S4.5.p2.8.m8.1.1.1.cmml" xref="S4.5.p2.8.m8.1.1.1"></in><apply id="S4.5.p2.8.m8.1.1.2.cmml" xref="S4.5.p2.8.m8.1.1.2"><csymbol cd="ambiguous" id="S4.5.p2.8.m8.1.1.2.1.cmml" xref="S4.5.p2.8.m8.1.1.2">subscript</csymbol><ci id="S4.5.p2.8.m8.1.1.2.2.cmml" xref="S4.5.p2.8.m8.1.1.2.2">𝑡</ci><cn id="S4.5.p2.8.m8.1.1.2.3.cmml" type="integer" xref="S4.5.p2.8.m8.1.1.2.3">0</cn></apply><apply id="S4.5.p2.8.m8.1.1.3.cmml" xref="S4.5.p2.8.m8.1.1.3"><csymbol cd="ambiguous" id="S4.5.p2.8.m8.1.1.3.1.cmml" xref="S4.5.p2.8.m8.1.1.3">subscript</csymbol><ci id="S4.5.p2.8.m8.1.1.3.2.cmml" xref="S4.5.p2.8.m8.1.1.3.2">𝐴</ci><cn id="S4.5.p2.8.m8.1.1.3.3.cmml" type="integer" xref="S4.5.p2.8.m8.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p2.8.m8.1c">t_{0}\in A_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.5.p2.8.m8.1d">italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> with <math alttext="\operatorname{dom}(t_{0})=\lambda_{\xi}" class="ltx_Math" display="inline" id="S4.5.p2.9.m9.2"><semantics id="S4.5.p2.9.m9.2a"><mrow id="S4.5.p2.9.m9.2.2" xref="S4.5.p2.9.m9.2.2.cmml"><mrow id="S4.5.p2.9.m9.2.2.1.1" xref="S4.5.p2.9.m9.2.2.1.2.cmml"><mi id="S4.5.p2.9.m9.1.1" xref="S4.5.p2.9.m9.1.1.cmml">dom</mi><mo id="S4.5.p2.9.m9.2.2.1.1a" xref="S4.5.p2.9.m9.2.2.1.2.cmml">⁡</mo><mrow id="S4.5.p2.9.m9.2.2.1.1.1" xref="S4.5.p2.9.m9.2.2.1.2.cmml"><mo id="S4.5.p2.9.m9.2.2.1.1.1.2" stretchy="false" xref="S4.5.p2.9.m9.2.2.1.2.cmml">(</mo><msub id="S4.5.p2.9.m9.2.2.1.1.1.1" xref="S4.5.p2.9.m9.2.2.1.1.1.1.cmml"><mi id="S4.5.p2.9.m9.2.2.1.1.1.1.2" xref="S4.5.p2.9.m9.2.2.1.1.1.1.2.cmml">t</mi><mn id="S4.5.p2.9.m9.2.2.1.1.1.1.3" xref="S4.5.p2.9.m9.2.2.1.1.1.1.3.cmml">0</mn></msub><mo id="S4.5.p2.9.m9.2.2.1.1.1.3" stretchy="false" xref="S4.5.p2.9.m9.2.2.1.2.cmml">)</mo></mrow></mrow><mo id="S4.5.p2.9.m9.2.2.2" xref="S4.5.p2.9.m9.2.2.2.cmml">=</mo><msub id="S4.5.p2.9.m9.2.2.3" xref="S4.5.p2.9.m9.2.2.3.cmml"><mi id="S4.5.p2.9.m9.2.2.3.2" xref="S4.5.p2.9.m9.2.2.3.2.cmml">λ</mi><mi id="S4.5.p2.9.m9.2.2.3.3" xref="S4.5.p2.9.m9.2.2.3.3.cmml">ξ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.5.p2.9.m9.2b"><apply id="S4.5.p2.9.m9.2.2.cmml" xref="S4.5.p2.9.m9.2.2"><eq id="S4.5.p2.9.m9.2.2.2.cmml" xref="S4.5.p2.9.m9.2.2.2"></eq><apply id="S4.5.p2.9.m9.2.2.1.2.cmml" xref="S4.5.p2.9.m9.2.2.1.1"><ci id="S4.5.p2.9.m9.1.1.cmml" xref="S4.5.p2.9.m9.1.1">dom</ci><apply id="S4.5.p2.9.m9.2.2.1.1.1.1.cmml" xref="S4.5.p2.9.m9.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S4.5.p2.9.m9.2.2.1.1.1.1.1.cmml" xref="S4.5.p2.9.m9.2.2.1.1.1.1">subscript</csymbol><ci id="S4.5.p2.9.m9.2.2.1.1.1.1.2.cmml" xref="S4.5.p2.9.m9.2.2.1.1.1.1.2">𝑡</ci><cn id="S4.5.p2.9.m9.2.2.1.1.1.1.3.cmml" type="integer" xref="S4.5.p2.9.m9.2.2.1.1.1.1.3">0</cn></apply></apply><apply id="S4.5.p2.9.m9.2.2.3.cmml" xref="S4.5.p2.9.m9.2.2.3"><csymbol cd="ambiguous" id="S4.5.p2.9.m9.2.2.3.1.cmml" xref="S4.5.p2.9.m9.2.2.3">subscript</csymbol><ci id="S4.5.p2.9.m9.2.2.3.2.cmml" xref="S4.5.p2.9.m9.2.2.3.2">𝜆</ci><ci id="S4.5.p2.9.m9.2.2.3.3.cmml" xref="S4.5.p2.9.m9.2.2.3.3">𝜉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p2.9.m9.2c">\operatorname{dom}(t_{0})=\lambda_{\xi}</annotation><annotation encoding="application/x-llamapun" id="S4.5.p2.9.m9.2d">roman_dom ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) = italic_λ start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT</annotation></semantics></math>. Before doing this note that this implies the desired result, since this interval has a left endpoint iff <math alttext="t_{0}\in A" class="ltx_Math" display="inline" id="S4.5.p2.10.m10.1"><semantics id="S4.5.p2.10.m10.1a"><mrow id="S4.5.p2.10.m10.1.1" xref="S4.5.p2.10.m10.1.1.cmml"><msub id="S4.5.p2.10.m10.1.1.2" xref="S4.5.p2.10.m10.1.1.2.cmml"><mi id="S4.5.p2.10.m10.1.1.2.2" xref="S4.5.p2.10.m10.1.1.2.2.cmml">t</mi><mn id="S4.5.p2.10.m10.1.1.2.3" xref="S4.5.p2.10.m10.1.1.2.3.cmml">0</mn></msub><mo id="S4.5.p2.10.m10.1.1.1" xref="S4.5.p2.10.m10.1.1.1.cmml">∈</mo><mi id="S4.5.p2.10.m10.1.1.3" xref="S4.5.p2.10.m10.1.1.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.5.p2.10.m10.1b"><apply id="S4.5.p2.10.m10.1.1.cmml" xref="S4.5.p2.10.m10.1.1"><in id="S4.5.p2.10.m10.1.1.1.cmml" xref="S4.5.p2.10.m10.1.1.1"></in><apply id="S4.5.p2.10.m10.1.1.2.cmml" xref="S4.5.p2.10.m10.1.1.2"><csymbol cd="ambiguous" id="S4.5.p2.10.m10.1.1.2.1.cmml" xref="S4.5.p2.10.m10.1.1.2">subscript</csymbol><ci id="S4.5.p2.10.m10.1.1.2.2.cmml" xref="S4.5.p2.10.m10.1.1.2.2">𝑡</ci><cn id="S4.5.p2.10.m10.1.1.2.3.cmml" type="integer" xref="S4.5.p2.10.m10.1.1.2.3">0</cn></apply><ci id="S4.5.p2.10.m10.1.1.3.cmml" xref="S4.5.p2.10.m10.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p2.10.m10.1c">t_{0}\in A</annotation><annotation encoding="application/x-llamapun" id="S4.5.p2.10.m10.1d">italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_A</annotation></semantics></math> iff <math alttext="\xi\in X" class="ltx_Math" display="inline" id="S4.5.p2.11.m11.1"><semantics id="S4.5.p2.11.m11.1a"><mrow id="S4.5.p2.11.m11.1.1" xref="S4.5.p2.11.m11.1.1.cmml"><mi id="S4.5.p2.11.m11.1.1.2" xref="S4.5.p2.11.m11.1.1.2.cmml">ξ</mi><mo id="S4.5.p2.11.m11.1.1.1" xref="S4.5.p2.11.m11.1.1.1.cmml">∈</mo><mi id="S4.5.p2.11.m11.1.1.3" xref="S4.5.p2.11.m11.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.5.p2.11.m11.1b"><apply id="S4.5.p2.11.m11.1.1.cmml" xref="S4.5.p2.11.m11.1.1"><in id="S4.5.p2.11.m11.1.1.1.cmml" xref="S4.5.p2.11.m11.1.1.1"></in><ci id="S4.5.p2.11.m11.1.1.2.cmml" xref="S4.5.p2.11.m11.1.1.2">𝜉</ci><ci id="S4.5.p2.11.m11.1.1.3.cmml" xref="S4.5.p2.11.m11.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p2.11.m11.1c">\xi\in X</annotation><annotation encoding="application/x-llamapun" id="S4.5.p2.11.m11.1d">italic_ξ ∈ italic_X</annotation></semantics></math>, and a right one iff <math alttext="{t_{0}}^{\frown}\langle\omega\rangle\in A" class="ltx_Math" display="inline" id="S4.5.p2.12.m12.1"><semantics id="S4.5.p2.12.m12.1a"><mrow id="S4.5.p2.12.m12.1.2" xref="S4.5.p2.12.m12.1.2.cmml"><mrow id="S4.5.p2.12.m12.1.2.2" xref="S4.5.p2.12.m12.1.2.2.cmml"><mmultiscripts id="S4.5.p2.12.m12.1.2.2.2" xref="S4.5.p2.12.m12.1.2.2.2.cmml"><mi id="S4.5.p2.12.m12.1.2.2.2.2.2" xref="S4.5.p2.12.m12.1.2.2.2.2.2.cmml">t</mi><mn id="S4.5.p2.12.m12.1.2.2.2.2.3" xref="S4.5.p2.12.m12.1.2.2.2.2.3.cmml">0</mn><mrow id="S4.5.p2.12.m12.1.2.2.2a" xref="S4.5.p2.12.m12.1.2.2.2.cmml"></mrow><mrow id="S4.5.p2.12.m12.1.2.2.2b" xref="S4.5.p2.12.m12.1.2.2.2.cmml"></mrow><mo id="S4.5.p2.12.m12.1.2.2.2.3" xref="S4.5.p2.12.m12.1.2.2.2.3.cmml">⌢</mo></mmultiscripts><mo id="S4.5.p2.12.m12.1.2.2.1" xref="S4.5.p2.12.m12.1.2.2.1.cmml">⁢</mo><mrow id="S4.5.p2.12.m12.1.2.2.3.2" xref="S4.5.p2.12.m12.1.2.2.3.1.cmml"><mo id="S4.5.p2.12.m12.1.2.2.3.2.1" stretchy="false" xref="S4.5.p2.12.m12.1.2.2.3.1.1.cmml">⟨</mo><mi id="S4.5.p2.12.m12.1.1" xref="S4.5.p2.12.m12.1.1.cmml">ω</mi><mo id="S4.5.p2.12.m12.1.2.2.3.2.2" stretchy="false" xref="S4.5.p2.12.m12.1.2.2.3.1.1.cmml">⟩</mo></mrow></mrow><mo id="S4.5.p2.12.m12.1.2.1" xref="S4.5.p2.12.m12.1.2.1.cmml">∈</mo><mi id="S4.5.p2.12.m12.1.2.3" xref="S4.5.p2.12.m12.1.2.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.5.p2.12.m12.1b"><apply id="S4.5.p2.12.m12.1.2.cmml" xref="S4.5.p2.12.m12.1.2"><in id="S4.5.p2.12.m12.1.2.1.cmml" xref="S4.5.p2.12.m12.1.2.1"></in><apply id="S4.5.p2.12.m12.1.2.2.cmml" xref="S4.5.p2.12.m12.1.2.2"><times id="S4.5.p2.12.m12.1.2.2.1.cmml" xref="S4.5.p2.12.m12.1.2.2.1"></times><apply id="S4.5.p2.12.m12.1.2.2.2.cmml" xref="S4.5.p2.12.m12.1.2.2.2"><csymbol cd="ambiguous" id="S4.5.p2.12.m12.1.2.2.2.1.cmml" xref="S4.5.p2.12.m12.1.2.2.2">superscript</csymbol><apply id="S4.5.p2.12.m12.1.2.2.2.2.cmml" xref="S4.5.p2.12.m12.1.2.2.2"><csymbol cd="ambiguous" id="S4.5.p2.12.m12.1.2.2.2.2.1.cmml" xref="S4.5.p2.12.m12.1.2.2.2">subscript</csymbol><ci id="S4.5.p2.12.m12.1.2.2.2.2.2.cmml" xref="S4.5.p2.12.m12.1.2.2.2.2.2">𝑡</ci><cn id="S4.5.p2.12.m12.1.2.2.2.2.3.cmml" type="integer" xref="S4.5.p2.12.m12.1.2.2.2.2.3">0</cn></apply><ci id="S4.5.p2.12.m12.1.2.2.2.3.cmml" xref="S4.5.p2.12.m12.1.2.2.2.3">⌢</ci></apply><apply id="S4.5.p2.12.m12.1.2.2.3.1.cmml" xref="S4.5.p2.12.m12.1.2.2.3.2"><csymbol cd="latexml" id="S4.5.p2.12.m12.1.2.2.3.1.1.cmml" xref="S4.5.p2.12.m12.1.2.2.3.2.1">delimited-⟨⟩</csymbol><ci id="S4.5.p2.12.m12.1.1.cmml" xref="S4.5.p2.12.m12.1.1">𝜔</ci></apply></apply><ci id="S4.5.p2.12.m12.1.2.3.cmml" xref="S4.5.p2.12.m12.1.2.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p2.12.m12.1c">{t_{0}}^{\frown}\langle\omega\rangle\in A</annotation><annotation encoding="application/x-llamapun" id="S4.5.p2.12.m12.1d">italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⌢ end_POSTSUPERSCRIPT ⟨ italic_ω ⟩ ∈ italic_A</annotation></semantics></math> iff <math alttext="\xi\in Y" class="ltx_Math" display="inline" id="S4.5.p2.13.m13.1"><semantics id="S4.5.p2.13.m13.1a"><mrow id="S4.5.p2.13.m13.1.1" xref="S4.5.p2.13.m13.1.1.cmml"><mi id="S4.5.p2.13.m13.1.1.2" xref="S4.5.p2.13.m13.1.1.2.cmml">ξ</mi><mo id="S4.5.p2.13.m13.1.1.1" xref="S4.5.p2.13.m13.1.1.1.cmml">∈</mo><mi id="S4.5.p2.13.m13.1.1.3" xref="S4.5.p2.13.m13.1.1.3.cmml">Y</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.5.p2.13.m13.1b"><apply id="S4.5.p2.13.m13.1.1.cmml" xref="S4.5.p2.13.m13.1.1"><in id="S4.5.p2.13.m13.1.1.1.cmml" xref="S4.5.p2.13.m13.1.1.1"></in><ci id="S4.5.p2.13.m13.1.1.2.cmml" xref="S4.5.p2.13.m13.1.1.2">𝜉</ci><ci id="S4.5.p2.13.m13.1.1.3.cmml" xref="S4.5.p2.13.m13.1.1.3">𝑌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p2.13.m13.1c">\xi\in Y</annotation><annotation encoding="application/x-llamapun" id="S4.5.p2.13.m13.1d">italic_ξ ∈ italic_Y</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.6.p3"> <p class="ltx_p" id="S4.6.p3.19">Fix <math alttext="\xi<\omega_{1}" class="ltx_Math" display="inline" id="S4.6.p3.1.m1.1"><semantics id="S4.6.p3.1.m1.1a"><mrow id="S4.6.p3.1.m1.1.1" xref="S4.6.p3.1.m1.1.1.cmml"><mi id="S4.6.p3.1.m1.1.1.2" xref="S4.6.p3.1.m1.1.1.2.cmml">ξ</mi><mo id="S4.6.p3.1.m1.1.1.1" xref="S4.6.p3.1.m1.1.1.1.cmml"><</mo><msub id="S4.6.p3.1.m1.1.1.3" xref="S4.6.p3.1.m1.1.1.3.cmml"><mi id="S4.6.p3.1.m1.1.1.3.2" xref="S4.6.p3.1.m1.1.1.3.2.cmml">ω</mi><mn id="S4.6.p3.1.m1.1.1.3.3" xref="S4.6.p3.1.m1.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.6.p3.1.m1.1b"><apply id="S4.6.p3.1.m1.1.1.cmml" xref="S4.6.p3.1.m1.1.1"><lt id="S4.6.p3.1.m1.1.1.1.cmml" xref="S4.6.p3.1.m1.1.1.1"></lt><ci id="S4.6.p3.1.m1.1.1.2.cmml" xref="S4.6.p3.1.m1.1.1.2">𝜉</ci><apply id="S4.6.p3.1.m1.1.1.3.cmml" xref="S4.6.p3.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.6.p3.1.m1.1.1.3.1.cmml" xref="S4.6.p3.1.m1.1.1.3">subscript</csymbol><ci id="S4.6.p3.1.m1.1.1.3.2.cmml" xref="S4.6.p3.1.m1.1.1.3.2">𝜔</ci><cn id="S4.6.p3.1.m1.1.1.3.3.cmml" type="integer" xref="S4.6.p3.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.6.p3.1.m1.1c">\xi<\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.6.p3.1.m1.1d">italic_ξ < italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="t<_{\mathrm{lex}}s" class="ltx_Math" display="inline" id="S4.6.p3.2.m2.1"><semantics id="S4.6.p3.2.m2.1a"><mrow id="S4.6.p3.2.m2.1.1" xref="S4.6.p3.2.m2.1.1.cmml"><mi id="S4.6.p3.2.m2.1.1.2" xref="S4.6.p3.2.m2.1.1.2.cmml">t</mi><msub id="S4.6.p3.2.m2.1.1.1" xref="S4.6.p3.2.m2.1.1.1.cmml"><mo id="S4.6.p3.2.m2.1.1.1.2" xref="S4.6.p3.2.m2.1.1.1.2.cmml"><</mo><mi id="S4.6.p3.2.m2.1.1.1.3" xref="S4.6.p3.2.m2.1.1.1.3.cmml">lex</mi></msub><mi id="S4.6.p3.2.m2.1.1.3" xref="S4.6.p3.2.m2.1.1.3.cmml">s</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.6.p3.2.m2.1b"><apply id="S4.6.p3.2.m2.1.1.cmml" xref="S4.6.p3.2.m2.1.1"><apply id="S4.6.p3.2.m2.1.1.1.cmml" xref="S4.6.p3.2.m2.1.1.1"><csymbol cd="ambiguous" id="S4.6.p3.2.m2.1.1.1.1.cmml" xref="S4.6.p3.2.m2.1.1.1">subscript</csymbol><lt id="S4.6.p3.2.m2.1.1.1.2.cmml" xref="S4.6.p3.2.m2.1.1.1.2"></lt><ci id="S4.6.p3.2.m2.1.1.1.3.cmml" xref="S4.6.p3.2.m2.1.1.1.3">lex</ci></apply><ci id="S4.6.p3.2.m2.1.1.2.cmml" xref="S4.6.p3.2.m2.1.1.2">𝑡</ci><ci id="S4.6.p3.2.m2.1.1.3.cmml" xref="S4.6.p3.2.m2.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.6.p3.2.m2.1c">t<_{\mathrm{lex}}s</annotation><annotation encoding="application/x-llamapun" id="S4.6.p3.2.m2.1d">italic_t < start_POSTSUBSCRIPT roman_lex end_POSTSUBSCRIPT italic_s</annotation></semantics></math> in <math alttext="A\setminus D_{\xi}" class="ltx_Math" display="inline" id="S4.6.p3.3.m3.1"><semantics id="S4.6.p3.3.m3.1a"><mrow id="S4.6.p3.3.m3.1.1" xref="S4.6.p3.3.m3.1.1.cmml"><mi id="S4.6.p3.3.m3.1.1.2" xref="S4.6.p3.3.m3.1.1.2.cmml">A</mi><mo id="S4.6.p3.3.m3.1.1.1" xref="S4.6.p3.3.m3.1.1.1.cmml">∖</mo><msub id="S4.6.p3.3.m3.1.1.3" xref="S4.6.p3.3.m3.1.1.3.cmml"><mi id="S4.6.p3.3.m3.1.1.3.2" xref="S4.6.p3.3.m3.1.1.3.2.cmml">D</mi><mi id="S4.6.p3.3.m3.1.1.3.3" xref="S4.6.p3.3.m3.1.1.3.3.cmml">ξ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.6.p3.3.m3.1b"><apply id="S4.6.p3.3.m3.1.1.cmml" xref="S4.6.p3.3.m3.1.1"><setdiff id="S4.6.p3.3.m3.1.1.1.cmml" xref="S4.6.p3.3.m3.1.1.1"></setdiff><ci id="S4.6.p3.3.m3.1.1.2.cmml" xref="S4.6.p3.3.m3.1.1.2">𝐴</ci><apply id="S4.6.p3.3.m3.1.1.3.cmml" xref="S4.6.p3.3.m3.1.1.3"><csymbol cd="ambiguous" id="S4.6.p3.3.m3.1.1.3.1.cmml" xref="S4.6.p3.3.m3.1.1.3">subscript</csymbol><ci id="S4.6.p3.3.m3.1.1.3.2.cmml" xref="S4.6.p3.3.m3.1.1.3.2">𝐷</ci><ci id="S4.6.p3.3.m3.1.1.3.3.cmml" xref="S4.6.p3.3.m3.1.1.3.3">𝜉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.6.p3.3.m3.1c">A\setminus D_{\xi}</annotation><annotation encoding="application/x-llamapun" id="S4.6.p3.3.m3.1d">italic_A ∖ italic_D start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT</annotation></semantics></math>. Also let <math alttext="t_{0}:=t|_{\lambda_{\xi}}" class="ltx_Math" display="inline" id="S4.6.p3.4.m4.2"><semantics id="S4.6.p3.4.m4.2a"><mrow id="S4.6.p3.4.m4.2.3" xref="S4.6.p3.4.m4.2.3.cmml"><msub id="S4.6.p3.4.m4.2.3.2" xref="S4.6.p3.4.m4.2.3.2.cmml"><mi id="S4.6.p3.4.m4.2.3.2.2" xref="S4.6.p3.4.m4.2.3.2.2.cmml">t</mi><mn id="S4.6.p3.4.m4.2.3.2.3" xref="S4.6.p3.4.m4.2.3.2.3.cmml">0</mn></msub><mo id="S4.6.p3.4.m4.2.3.1" lspace="0.278em" rspace="0.278em" xref="S4.6.p3.4.m4.2.3.1.cmml">:=</mo><msub id="S4.6.p3.4.m4.2.3.3.2" xref="S4.6.p3.4.m4.2.3.3.1.cmml"><mrow id="S4.6.p3.4.m4.2.3.3.2.2" xref="S4.6.p3.4.m4.2.3.3.1.cmml"><mi id="S4.6.p3.4.m4.1.1" xref="S4.6.p3.4.m4.1.1.cmml">t</mi><mo id="S4.6.p3.4.m4.2.3.3.2.2.1" stretchy="false" xref="S4.6.p3.4.m4.2.3.3.1.1.cmml">|</mo></mrow><msub id="S4.6.p3.4.m4.2.2.1" xref="S4.6.p3.4.m4.2.2.1.cmml"><mi id="S4.6.p3.4.m4.2.2.1.2" xref="S4.6.p3.4.m4.2.2.1.2.cmml">λ</mi><mi id="S4.6.p3.4.m4.2.2.1.3" xref="S4.6.p3.4.m4.2.2.1.3.cmml">ξ</mi></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.6.p3.4.m4.2b"><apply id="S4.6.p3.4.m4.2.3.cmml" xref="S4.6.p3.4.m4.2.3"><csymbol cd="latexml" id="S4.6.p3.4.m4.2.3.1.cmml" xref="S4.6.p3.4.m4.2.3.1">assign</csymbol><apply id="S4.6.p3.4.m4.2.3.2.cmml" xref="S4.6.p3.4.m4.2.3.2"><csymbol cd="ambiguous" id="S4.6.p3.4.m4.2.3.2.1.cmml" xref="S4.6.p3.4.m4.2.3.2">subscript</csymbol><ci id="S4.6.p3.4.m4.2.3.2.2.cmml" xref="S4.6.p3.4.m4.2.3.2.2">𝑡</ci><cn id="S4.6.p3.4.m4.2.3.2.3.cmml" type="integer" xref="S4.6.p3.4.m4.2.3.2.3">0</cn></apply><apply id="S4.6.p3.4.m4.2.3.3.1.cmml" xref="S4.6.p3.4.m4.2.3.3.2"><csymbol cd="latexml" id="S4.6.p3.4.m4.2.3.3.1.1.cmml" xref="S4.6.p3.4.m4.2.3.3.2.2.1">evaluated-at</csymbol><ci id="S4.6.p3.4.m4.1.1.cmml" xref="S4.6.p3.4.m4.1.1">𝑡</ci><apply id="S4.6.p3.4.m4.2.2.1.cmml" xref="S4.6.p3.4.m4.2.2.1"><csymbol cd="ambiguous" id="S4.6.p3.4.m4.2.2.1.1.cmml" xref="S4.6.p3.4.m4.2.2.1">subscript</csymbol><ci id="S4.6.p3.4.m4.2.2.1.2.cmml" xref="S4.6.p3.4.m4.2.2.1.2">𝜆</ci><ci id="S4.6.p3.4.m4.2.2.1.3.cmml" xref="S4.6.p3.4.m4.2.2.1.3">𝜉</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.6.p3.4.m4.2c">t_{0}:=t|_{\lambda_{\xi}}</annotation><annotation encoding="application/x-llamapun" id="S4.6.p3.4.m4.2d">italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT := italic_t | start_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="t_{1}:=s|_{\lambda_{\xi}}" class="ltx_Math" display="inline" id="S4.6.p3.5.m5.2"><semantics id="S4.6.p3.5.m5.2a"><mrow id="S4.6.p3.5.m5.2.3" xref="S4.6.p3.5.m5.2.3.cmml"><msub id="S4.6.p3.5.m5.2.3.2" xref="S4.6.p3.5.m5.2.3.2.cmml"><mi id="S4.6.p3.5.m5.2.3.2.2" xref="S4.6.p3.5.m5.2.3.2.2.cmml">t</mi><mn id="S4.6.p3.5.m5.2.3.2.3" xref="S4.6.p3.5.m5.2.3.2.3.cmml">1</mn></msub><mo id="S4.6.p3.5.m5.2.3.1" lspace="0.278em" rspace="0.278em" xref="S4.6.p3.5.m5.2.3.1.cmml">:=</mo><msub id="S4.6.p3.5.m5.2.3.3.2" xref="S4.6.p3.5.m5.2.3.3.1.cmml"><mrow id="S4.6.p3.5.m5.2.3.3.2.2" xref="S4.6.p3.5.m5.2.3.3.1.cmml"><mi id="S4.6.p3.5.m5.1.1" xref="S4.6.p3.5.m5.1.1.cmml">s</mi><mo id="S4.6.p3.5.m5.2.3.3.2.2.1" stretchy="false" xref="S4.6.p3.5.m5.2.3.3.1.1.cmml">|</mo></mrow><msub id="S4.6.p3.5.m5.2.2.1" xref="S4.6.p3.5.m5.2.2.1.cmml"><mi id="S4.6.p3.5.m5.2.2.1.2" xref="S4.6.p3.5.m5.2.2.1.2.cmml">λ</mi><mi id="S4.6.p3.5.m5.2.2.1.3" xref="S4.6.p3.5.m5.2.2.1.3.cmml">ξ</mi></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.6.p3.5.m5.2b"><apply id="S4.6.p3.5.m5.2.3.cmml" xref="S4.6.p3.5.m5.2.3"><csymbol cd="latexml" id="S4.6.p3.5.m5.2.3.1.cmml" xref="S4.6.p3.5.m5.2.3.1">assign</csymbol><apply id="S4.6.p3.5.m5.2.3.2.cmml" xref="S4.6.p3.5.m5.2.3.2"><csymbol cd="ambiguous" id="S4.6.p3.5.m5.2.3.2.1.cmml" xref="S4.6.p3.5.m5.2.3.2">subscript</csymbol><ci id="S4.6.p3.5.m5.2.3.2.2.cmml" xref="S4.6.p3.5.m5.2.3.2.2">𝑡</ci><cn id="S4.6.p3.5.m5.2.3.2.3.cmml" type="integer" xref="S4.6.p3.5.m5.2.3.2.3">1</cn></apply><apply id="S4.6.p3.5.m5.2.3.3.1.cmml" xref="S4.6.p3.5.m5.2.3.3.2"><csymbol cd="latexml" id="S4.6.p3.5.m5.2.3.3.1.1.cmml" xref="S4.6.p3.5.m5.2.3.3.2.2.1">evaluated-at</csymbol><ci id="S4.6.p3.5.m5.1.1.cmml" xref="S4.6.p3.5.m5.1.1">𝑠</ci><apply id="S4.6.p3.5.m5.2.2.1.cmml" xref="S4.6.p3.5.m5.2.2.1"><csymbol cd="ambiguous" id="S4.6.p3.5.m5.2.2.1.1.cmml" xref="S4.6.p3.5.m5.2.2.1">subscript</csymbol><ci id="S4.6.p3.5.m5.2.2.1.2.cmml" xref="S4.6.p3.5.m5.2.2.1.2">𝜆</ci><ci id="S4.6.p3.5.m5.2.2.1.3.cmml" xref="S4.6.p3.5.m5.2.2.1.3">𝜉</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.6.p3.5.m5.2c">t_{1}:=s|_{\lambda_{\xi}}</annotation><annotation encoding="application/x-llamapun" id="S4.6.p3.5.m5.2d">italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT := italic_s | start_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. It is enough to prove that if <math alttext="t_{0}\neq t_{1}" class="ltx_Math" display="inline" id="S4.6.p3.6.m6.1"><semantics id="S4.6.p3.6.m6.1a"><mrow id="S4.6.p3.6.m6.1.1" xref="S4.6.p3.6.m6.1.1.cmml"><msub id="S4.6.p3.6.m6.1.1.2" xref="S4.6.p3.6.m6.1.1.2.cmml"><mi id="S4.6.p3.6.m6.1.1.2.2" xref="S4.6.p3.6.m6.1.1.2.2.cmml">t</mi><mn id="S4.6.p3.6.m6.1.1.2.3" xref="S4.6.p3.6.m6.1.1.2.3.cmml">0</mn></msub><mo id="S4.6.p3.6.m6.1.1.1" xref="S4.6.p3.6.m6.1.1.1.cmml">≠</mo><msub id="S4.6.p3.6.m6.1.1.3" xref="S4.6.p3.6.m6.1.1.3.cmml"><mi id="S4.6.p3.6.m6.1.1.3.2" xref="S4.6.p3.6.m6.1.1.3.2.cmml">t</mi><mn id="S4.6.p3.6.m6.1.1.3.3" xref="S4.6.p3.6.m6.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.6.p3.6.m6.1b"><apply id="S4.6.p3.6.m6.1.1.cmml" xref="S4.6.p3.6.m6.1.1"><neq id="S4.6.p3.6.m6.1.1.1.cmml" xref="S4.6.p3.6.m6.1.1.1"></neq><apply id="S4.6.p3.6.m6.1.1.2.cmml" xref="S4.6.p3.6.m6.1.1.2"><csymbol cd="ambiguous" id="S4.6.p3.6.m6.1.1.2.1.cmml" xref="S4.6.p3.6.m6.1.1.2">subscript</csymbol><ci id="S4.6.p3.6.m6.1.1.2.2.cmml" xref="S4.6.p3.6.m6.1.1.2.2">𝑡</ci><cn id="S4.6.p3.6.m6.1.1.2.3.cmml" type="integer" xref="S4.6.p3.6.m6.1.1.2.3">0</cn></apply><apply id="S4.6.p3.6.m6.1.1.3.cmml" xref="S4.6.p3.6.m6.1.1.3"><csymbol cd="ambiguous" id="S4.6.p3.6.m6.1.1.3.1.cmml" xref="S4.6.p3.6.m6.1.1.3">subscript</csymbol><ci id="S4.6.p3.6.m6.1.1.3.2.cmml" xref="S4.6.p3.6.m6.1.1.3.2">𝑡</ci><cn id="S4.6.p3.6.m6.1.1.3.3.cmml" type="integer" xref="S4.6.p3.6.m6.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.6.p3.6.m6.1c">t_{0}\neq t_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.6.p3.6.m6.1d">italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≠ italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, then they are not in the same complementary interval of <math alttext="A\setminus D_{\xi}" class="ltx_Math" display="inline" id="S4.6.p3.7.m7.1"><semantics id="S4.6.p3.7.m7.1a"><mrow id="S4.6.p3.7.m7.1.1" xref="S4.6.p3.7.m7.1.1.cmml"><mi id="S4.6.p3.7.m7.1.1.2" xref="S4.6.p3.7.m7.1.1.2.cmml">A</mi><mo id="S4.6.p3.7.m7.1.1.1" xref="S4.6.p3.7.m7.1.1.1.cmml">∖</mo><msub id="S4.6.p3.7.m7.1.1.3" xref="S4.6.p3.7.m7.1.1.3.cmml"><mi id="S4.6.p3.7.m7.1.1.3.2" xref="S4.6.p3.7.m7.1.1.3.2.cmml">D</mi><mi id="S4.6.p3.7.m7.1.1.3.3" xref="S4.6.p3.7.m7.1.1.3.3.cmml">ξ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.6.p3.7.m7.1b"><apply id="S4.6.p3.7.m7.1.1.cmml" xref="S4.6.p3.7.m7.1.1"><setdiff id="S4.6.p3.7.m7.1.1.1.cmml" xref="S4.6.p3.7.m7.1.1.1"></setdiff><ci id="S4.6.p3.7.m7.1.1.2.cmml" xref="S4.6.p3.7.m7.1.1.2">𝐴</ci><apply id="S4.6.p3.7.m7.1.1.3.cmml" xref="S4.6.p3.7.m7.1.1.3"><csymbol cd="ambiguous" id="S4.6.p3.7.m7.1.1.3.1.cmml" xref="S4.6.p3.7.m7.1.1.3">subscript</csymbol><ci id="S4.6.p3.7.m7.1.1.3.2.cmml" xref="S4.6.p3.7.m7.1.1.3.2">𝐷</ci><ci id="S4.6.p3.7.m7.1.1.3.3.cmml" xref="S4.6.p3.7.m7.1.1.3.3">𝜉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.6.p3.7.m7.1c">A\setminus D_{\xi}</annotation><annotation encoding="application/x-llamapun" id="S4.6.p3.7.m7.1d">italic_A ∖ italic_D start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT</annotation></semantics></math>. Assume that <math alttext="t_{0}\neq t_{1}" class="ltx_Math" display="inline" id="S4.6.p3.8.m8.1"><semantics id="S4.6.p3.8.m8.1a"><mrow id="S4.6.p3.8.m8.1.1" xref="S4.6.p3.8.m8.1.1.cmml"><msub id="S4.6.p3.8.m8.1.1.2" xref="S4.6.p3.8.m8.1.1.2.cmml"><mi id="S4.6.p3.8.m8.1.1.2.2" xref="S4.6.p3.8.m8.1.1.2.2.cmml">t</mi><mn id="S4.6.p3.8.m8.1.1.2.3" xref="S4.6.p3.8.m8.1.1.2.3.cmml">0</mn></msub><mo id="S4.6.p3.8.m8.1.1.1" xref="S4.6.p3.8.m8.1.1.1.cmml">≠</mo><msub id="S4.6.p3.8.m8.1.1.3" xref="S4.6.p3.8.m8.1.1.3.cmml"><mi id="S4.6.p3.8.m8.1.1.3.2" xref="S4.6.p3.8.m8.1.1.3.2.cmml">t</mi><mn id="S4.6.p3.8.m8.1.1.3.3" xref="S4.6.p3.8.m8.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.6.p3.8.m8.1b"><apply id="S4.6.p3.8.m8.1.1.cmml" xref="S4.6.p3.8.m8.1.1"><neq id="S4.6.p3.8.m8.1.1.1.cmml" xref="S4.6.p3.8.m8.1.1.1"></neq><apply id="S4.6.p3.8.m8.1.1.2.cmml" xref="S4.6.p3.8.m8.1.1.2"><csymbol cd="ambiguous" id="S4.6.p3.8.m8.1.1.2.1.cmml" xref="S4.6.p3.8.m8.1.1.2">subscript</csymbol><ci id="S4.6.p3.8.m8.1.1.2.2.cmml" xref="S4.6.p3.8.m8.1.1.2.2">𝑡</ci><cn id="S4.6.p3.8.m8.1.1.2.3.cmml" type="integer" xref="S4.6.p3.8.m8.1.1.2.3">0</cn></apply><apply id="S4.6.p3.8.m8.1.1.3.cmml" xref="S4.6.p3.8.m8.1.1.3"><csymbol cd="ambiguous" id="S4.6.p3.8.m8.1.1.3.1.cmml" xref="S4.6.p3.8.m8.1.1.3">subscript</csymbol><ci id="S4.6.p3.8.m8.1.1.3.2.cmml" xref="S4.6.p3.8.m8.1.1.3.2">𝑡</ci><cn id="S4.6.p3.8.m8.1.1.3.3.cmml" type="integer" xref="S4.6.p3.8.m8.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.6.p3.8.m8.1c">t_{0}\neq t_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.6.p3.8.m8.1d">italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≠ italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, so that in fact <math alttext="t_{0}<_{\mathrm{lex}}t_{1}" class="ltx_Math" display="inline" id="S4.6.p3.9.m9.1"><semantics id="S4.6.p3.9.m9.1a"><mrow id="S4.6.p3.9.m9.1.1" xref="S4.6.p3.9.m9.1.1.cmml"><msub id="S4.6.p3.9.m9.1.1.2" xref="S4.6.p3.9.m9.1.1.2.cmml"><mi id="S4.6.p3.9.m9.1.1.2.2" xref="S4.6.p3.9.m9.1.1.2.2.cmml">t</mi><mn id="S4.6.p3.9.m9.1.1.2.3" xref="S4.6.p3.9.m9.1.1.2.3.cmml">0</mn></msub><msub id="S4.6.p3.9.m9.1.1.1" xref="S4.6.p3.9.m9.1.1.1.cmml"><mo id="S4.6.p3.9.m9.1.1.1.2" xref="S4.6.p3.9.m9.1.1.1.2.cmml"><</mo><mi id="S4.6.p3.9.m9.1.1.1.3" xref="S4.6.p3.9.m9.1.1.1.3.cmml">lex</mi></msub><msub id="S4.6.p3.9.m9.1.1.3" xref="S4.6.p3.9.m9.1.1.3.cmml"><mi id="S4.6.p3.9.m9.1.1.3.2" xref="S4.6.p3.9.m9.1.1.3.2.cmml">t</mi><mn id="S4.6.p3.9.m9.1.1.3.3" xref="S4.6.p3.9.m9.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.6.p3.9.m9.1b"><apply id="S4.6.p3.9.m9.1.1.cmml" xref="S4.6.p3.9.m9.1.1"><apply id="S4.6.p3.9.m9.1.1.1.cmml" xref="S4.6.p3.9.m9.1.1.1"><csymbol cd="ambiguous" id="S4.6.p3.9.m9.1.1.1.1.cmml" xref="S4.6.p3.9.m9.1.1.1">subscript</csymbol><lt id="S4.6.p3.9.m9.1.1.1.2.cmml" xref="S4.6.p3.9.m9.1.1.1.2"></lt><ci id="S4.6.p3.9.m9.1.1.1.3.cmml" xref="S4.6.p3.9.m9.1.1.1.3">lex</ci></apply><apply id="S4.6.p3.9.m9.1.1.2.cmml" xref="S4.6.p3.9.m9.1.1.2"><csymbol cd="ambiguous" id="S4.6.p3.9.m9.1.1.2.1.cmml" xref="S4.6.p3.9.m9.1.1.2">subscript</csymbol><ci id="S4.6.p3.9.m9.1.1.2.2.cmml" xref="S4.6.p3.9.m9.1.1.2.2">𝑡</ci><cn id="S4.6.p3.9.m9.1.1.2.3.cmml" type="integer" xref="S4.6.p3.9.m9.1.1.2.3">0</cn></apply><apply id="S4.6.p3.9.m9.1.1.3.cmml" xref="S4.6.p3.9.m9.1.1.3"><csymbol cd="ambiguous" id="S4.6.p3.9.m9.1.1.3.1.cmml" xref="S4.6.p3.9.m9.1.1.3">subscript</csymbol><ci id="S4.6.p3.9.m9.1.1.3.2.cmml" xref="S4.6.p3.9.m9.1.1.3.2">𝑡</ci><cn id="S4.6.p3.9.m9.1.1.3.3.cmml" type="integer" xref="S4.6.p3.9.m9.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.6.p3.9.m9.1c">t_{0}<_{\mathrm{lex}}t_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.6.p3.9.m9.1d">italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT < start_POSTSUBSCRIPT roman_lex end_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>. Let <math alttext="\delta<\lambda_{\xi}" class="ltx_Math" display="inline" id="S4.6.p3.10.m10.1"><semantics id="S4.6.p3.10.m10.1a"><mrow id="S4.6.p3.10.m10.1.1" xref="S4.6.p3.10.m10.1.1.cmml"><mi id="S4.6.p3.10.m10.1.1.2" xref="S4.6.p3.10.m10.1.1.2.cmml">δ</mi><mo id="S4.6.p3.10.m10.1.1.1" xref="S4.6.p3.10.m10.1.1.1.cmml"><</mo><msub id="S4.6.p3.10.m10.1.1.3" xref="S4.6.p3.10.m10.1.1.3.cmml"><mi id="S4.6.p3.10.m10.1.1.3.2" xref="S4.6.p3.10.m10.1.1.3.2.cmml">λ</mi><mi id="S4.6.p3.10.m10.1.1.3.3" xref="S4.6.p3.10.m10.1.1.3.3.cmml">ξ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.6.p3.10.m10.1b"><apply id="S4.6.p3.10.m10.1.1.cmml" xref="S4.6.p3.10.m10.1.1"><lt id="S4.6.p3.10.m10.1.1.1.cmml" xref="S4.6.p3.10.m10.1.1.1"></lt><ci id="S4.6.p3.10.m10.1.1.2.cmml" xref="S4.6.p3.10.m10.1.1.2">𝛿</ci><apply id="S4.6.p3.10.m10.1.1.3.cmml" xref="S4.6.p3.10.m10.1.1.3"><csymbol cd="ambiguous" id="S4.6.p3.10.m10.1.1.3.1.cmml" xref="S4.6.p3.10.m10.1.1.3">subscript</csymbol><ci id="S4.6.p3.10.m10.1.1.3.2.cmml" xref="S4.6.p3.10.m10.1.1.3.2">𝜆</ci><ci id="S4.6.p3.10.m10.1.1.3.3.cmml" xref="S4.6.p3.10.m10.1.1.3.3">𝜉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.6.p3.10.m10.1c">\delta<\lambda_{\xi}</annotation><annotation encoding="application/x-llamapun" id="S4.6.p3.10.m10.1d">italic_δ < italic_λ start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT</annotation></semantics></math> be the least such that <math alttext="t_{0}(\delta)\neq t_{1}(\delta)" class="ltx_Math" display="inline" id="S4.6.p3.11.m11.2"><semantics id="S4.6.p3.11.m11.2a"><mrow id="S4.6.p3.11.m11.2.3" xref="S4.6.p3.11.m11.2.3.cmml"><mrow id="S4.6.p3.11.m11.2.3.2" xref="S4.6.p3.11.m11.2.3.2.cmml"><msub id="S4.6.p3.11.m11.2.3.2.2" xref="S4.6.p3.11.m11.2.3.2.2.cmml"><mi id="S4.6.p3.11.m11.2.3.2.2.2" xref="S4.6.p3.11.m11.2.3.2.2.2.cmml">t</mi><mn id="S4.6.p3.11.m11.2.3.2.2.3" xref="S4.6.p3.11.m11.2.3.2.2.3.cmml">0</mn></msub><mo id="S4.6.p3.11.m11.2.3.2.1" xref="S4.6.p3.11.m11.2.3.2.1.cmml">⁢</mo><mrow id="S4.6.p3.11.m11.2.3.2.3.2" xref="S4.6.p3.11.m11.2.3.2.cmml"><mo id="S4.6.p3.11.m11.2.3.2.3.2.1" stretchy="false" xref="S4.6.p3.11.m11.2.3.2.cmml">(</mo><mi id="S4.6.p3.11.m11.1.1" xref="S4.6.p3.11.m11.1.1.cmml">δ</mi><mo id="S4.6.p3.11.m11.2.3.2.3.2.2" stretchy="false" xref="S4.6.p3.11.m11.2.3.2.cmml">)</mo></mrow></mrow><mo id="S4.6.p3.11.m11.2.3.1" xref="S4.6.p3.11.m11.2.3.1.cmml">≠</mo><mrow id="S4.6.p3.11.m11.2.3.3" xref="S4.6.p3.11.m11.2.3.3.cmml"><msub id="S4.6.p3.11.m11.2.3.3.2" xref="S4.6.p3.11.m11.2.3.3.2.cmml"><mi id="S4.6.p3.11.m11.2.3.3.2.2" xref="S4.6.p3.11.m11.2.3.3.2.2.cmml">t</mi><mn id="S4.6.p3.11.m11.2.3.3.2.3" xref="S4.6.p3.11.m11.2.3.3.2.3.cmml">1</mn></msub><mo id="S4.6.p3.11.m11.2.3.3.1" xref="S4.6.p3.11.m11.2.3.3.1.cmml">⁢</mo><mrow id="S4.6.p3.11.m11.2.3.3.3.2" xref="S4.6.p3.11.m11.2.3.3.cmml"><mo id="S4.6.p3.11.m11.2.3.3.3.2.1" stretchy="false" xref="S4.6.p3.11.m11.2.3.3.cmml">(</mo><mi id="S4.6.p3.11.m11.2.2" xref="S4.6.p3.11.m11.2.2.cmml">δ</mi><mo id="S4.6.p3.11.m11.2.3.3.3.2.2" stretchy="false" xref="S4.6.p3.11.m11.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.6.p3.11.m11.2b"><apply id="S4.6.p3.11.m11.2.3.cmml" xref="S4.6.p3.11.m11.2.3"><neq id="S4.6.p3.11.m11.2.3.1.cmml" xref="S4.6.p3.11.m11.2.3.1"></neq><apply id="S4.6.p3.11.m11.2.3.2.cmml" xref="S4.6.p3.11.m11.2.3.2"><times id="S4.6.p3.11.m11.2.3.2.1.cmml" xref="S4.6.p3.11.m11.2.3.2.1"></times><apply id="S4.6.p3.11.m11.2.3.2.2.cmml" xref="S4.6.p3.11.m11.2.3.2.2"><csymbol cd="ambiguous" id="S4.6.p3.11.m11.2.3.2.2.1.cmml" xref="S4.6.p3.11.m11.2.3.2.2">subscript</csymbol><ci id="S4.6.p3.11.m11.2.3.2.2.2.cmml" xref="S4.6.p3.11.m11.2.3.2.2.2">𝑡</ci><cn id="S4.6.p3.11.m11.2.3.2.2.3.cmml" type="integer" xref="S4.6.p3.11.m11.2.3.2.2.3">0</cn></apply><ci id="S4.6.p3.11.m11.1.1.cmml" xref="S4.6.p3.11.m11.1.1">𝛿</ci></apply><apply id="S4.6.p3.11.m11.2.3.3.cmml" xref="S4.6.p3.11.m11.2.3.3"><times id="S4.6.p3.11.m11.2.3.3.1.cmml" xref="S4.6.p3.11.m11.2.3.3.1"></times><apply id="S4.6.p3.11.m11.2.3.3.2.cmml" xref="S4.6.p3.11.m11.2.3.3.2"><csymbol cd="ambiguous" id="S4.6.p3.11.m11.2.3.3.2.1.cmml" xref="S4.6.p3.11.m11.2.3.3.2">subscript</csymbol><ci id="S4.6.p3.11.m11.2.3.3.2.2.cmml" xref="S4.6.p3.11.m11.2.3.3.2.2">𝑡</ci><cn id="S4.6.p3.11.m11.2.3.3.2.3.cmml" type="integer" xref="S4.6.p3.11.m11.2.3.3.2.3">1</cn></apply><ci id="S4.6.p3.11.m11.2.2.cmml" xref="S4.6.p3.11.m11.2.2">𝛿</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.6.p3.11.m11.2c">t_{0}(\delta)\neq t_{1}(\delta)</annotation><annotation encoding="application/x-llamapun" id="S4.6.p3.11.m11.2d">italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_δ ) ≠ italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_δ )</annotation></semantics></math>. Now <math alttext="u:=t_{0}|_{\delta+\omega}" class="ltx_Math" display="inline" id="S4.6.p3.12.m12.2"><semantics id="S4.6.p3.12.m12.2a"><mrow id="S4.6.p3.12.m12.2.2" xref="S4.6.p3.12.m12.2.2.cmml"><mi id="S4.6.p3.12.m12.2.2.3" xref="S4.6.p3.12.m12.2.2.3.cmml">u</mi><mo id="S4.6.p3.12.m12.2.2.2" lspace="0.278em" rspace="0.278em" xref="S4.6.p3.12.m12.2.2.2.cmml">:=</mo><msub id="S4.6.p3.12.m12.2.2.1.1" xref="S4.6.p3.12.m12.2.2.1.2.cmml"><mrow id="S4.6.p3.12.m12.2.2.1.1.1" xref="S4.6.p3.12.m12.2.2.1.2.cmml"><msub id="S4.6.p3.12.m12.2.2.1.1.1.1" xref="S4.6.p3.12.m12.2.2.1.1.1.1.cmml"><mi id="S4.6.p3.12.m12.2.2.1.1.1.1.2" xref="S4.6.p3.12.m12.2.2.1.1.1.1.2.cmml">t</mi><mn id="S4.6.p3.12.m12.2.2.1.1.1.1.3" xref="S4.6.p3.12.m12.2.2.1.1.1.1.3.cmml">0</mn></msub><mo id="S4.6.p3.12.m12.2.2.1.1.1.2" stretchy="false" xref="S4.6.p3.12.m12.2.2.1.2.1.cmml">|</mo></mrow><mrow id="S4.6.p3.12.m12.1.1.1" xref="S4.6.p3.12.m12.1.1.1.cmml"><mi id="S4.6.p3.12.m12.1.1.1.2" xref="S4.6.p3.12.m12.1.1.1.2.cmml">δ</mi><mo id="S4.6.p3.12.m12.1.1.1.1" xref="S4.6.p3.12.m12.1.1.1.1.cmml">+</mo><mi id="S4.6.p3.12.m12.1.1.1.3" xref="S4.6.p3.12.m12.1.1.1.3.cmml">ω</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.6.p3.12.m12.2b"><apply id="S4.6.p3.12.m12.2.2.cmml" xref="S4.6.p3.12.m12.2.2"><csymbol cd="latexml" id="S4.6.p3.12.m12.2.2.2.cmml" xref="S4.6.p3.12.m12.2.2.2">assign</csymbol><ci id="S4.6.p3.12.m12.2.2.3.cmml" xref="S4.6.p3.12.m12.2.2.3">𝑢</ci><apply id="S4.6.p3.12.m12.2.2.1.2.cmml" xref="S4.6.p3.12.m12.2.2.1.1"><csymbol cd="latexml" id="S4.6.p3.12.m12.2.2.1.2.1.cmml" xref="S4.6.p3.12.m12.2.2.1.1.1.2">evaluated-at</csymbol><apply id="S4.6.p3.12.m12.2.2.1.1.1.1.cmml" xref="S4.6.p3.12.m12.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S4.6.p3.12.m12.2.2.1.1.1.1.1.cmml" xref="S4.6.p3.12.m12.2.2.1.1.1.1">subscript</csymbol><ci id="S4.6.p3.12.m12.2.2.1.1.1.1.2.cmml" xref="S4.6.p3.12.m12.2.2.1.1.1.1.2">𝑡</ci><cn id="S4.6.p3.12.m12.2.2.1.1.1.1.3.cmml" type="integer" xref="S4.6.p3.12.m12.2.2.1.1.1.1.3">0</cn></apply><apply id="S4.6.p3.12.m12.1.1.1.cmml" xref="S4.6.p3.12.m12.1.1.1"><plus id="S4.6.p3.12.m12.1.1.1.1.cmml" xref="S4.6.p3.12.m12.1.1.1.1"></plus><ci id="S4.6.p3.12.m12.1.1.1.2.cmml" xref="S4.6.p3.12.m12.1.1.1.2">𝛿</ci><ci id="S4.6.p3.12.m12.1.1.1.3.cmml" xref="S4.6.p3.12.m12.1.1.1.3">𝜔</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.6.p3.12.m12.2c">u:=t_{0}|_{\delta+\omega}</annotation><annotation encoding="application/x-llamapun" id="S4.6.p3.12.m12.2d">italic_u := italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT | start_POSTSUBSCRIPT italic_δ + italic_ω end_POSTSUBSCRIPT</annotation></semantics></math> satisfies that <math alttext="t<_{\mathrm{lex}}u<_{\mathrm{lex}}s" class="ltx_Math" display="inline" id="S4.6.p3.13.m13.1"><semantics id="S4.6.p3.13.m13.1a"><mrow id="S4.6.p3.13.m13.1.1" xref="S4.6.p3.13.m13.1.1.cmml"><mi id="S4.6.p3.13.m13.1.1.2" xref="S4.6.p3.13.m13.1.1.2.cmml">t</mi><msub id="S4.6.p3.13.m13.1.1.3" xref="S4.6.p3.13.m13.1.1.3.cmml"><mo id="S4.6.p3.13.m13.1.1.3.2" xref="S4.6.p3.13.m13.1.1.3.2.cmml"><</mo><mi id="S4.6.p3.13.m13.1.1.3.3" xref="S4.6.p3.13.m13.1.1.3.3.cmml">lex</mi></msub><mi id="S4.6.p3.13.m13.1.1.4" xref="S4.6.p3.13.m13.1.1.4.cmml">u</mi><msub id="S4.6.p3.13.m13.1.1.5" xref="S4.6.p3.13.m13.1.1.5.cmml"><mo id="S4.6.p3.13.m13.1.1.5.2" xref="S4.6.p3.13.m13.1.1.5.2.cmml"><</mo><mi id="S4.6.p3.13.m13.1.1.5.3" xref="S4.6.p3.13.m13.1.1.5.3.cmml">lex</mi></msub><mi id="S4.6.p3.13.m13.1.1.6" xref="S4.6.p3.13.m13.1.1.6.cmml">s</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.6.p3.13.m13.1b"><apply id="S4.6.p3.13.m13.1.1.cmml" xref="S4.6.p3.13.m13.1.1"><and id="S4.6.p3.13.m13.1.1a.cmml" xref="S4.6.p3.13.m13.1.1"></and><apply id="S4.6.p3.13.m13.1.1b.cmml" xref="S4.6.p3.13.m13.1.1"><apply id="S4.6.p3.13.m13.1.1.3.cmml" xref="S4.6.p3.13.m13.1.1.3"><csymbol cd="ambiguous" id="S4.6.p3.13.m13.1.1.3.1.cmml" xref="S4.6.p3.13.m13.1.1.3">subscript</csymbol><lt id="S4.6.p3.13.m13.1.1.3.2.cmml" xref="S4.6.p3.13.m13.1.1.3.2"></lt><ci id="S4.6.p3.13.m13.1.1.3.3.cmml" xref="S4.6.p3.13.m13.1.1.3.3">lex</ci></apply><ci id="S4.6.p3.13.m13.1.1.2.cmml" xref="S4.6.p3.13.m13.1.1.2">𝑡</ci><ci id="S4.6.p3.13.m13.1.1.4.cmml" xref="S4.6.p3.13.m13.1.1.4">𝑢</ci></apply><apply id="S4.6.p3.13.m13.1.1c.cmml" xref="S4.6.p3.13.m13.1.1"><apply id="S4.6.p3.13.m13.1.1.5.cmml" xref="S4.6.p3.13.m13.1.1.5"><csymbol cd="ambiguous" id="S4.6.p3.13.m13.1.1.5.1.cmml" xref="S4.6.p3.13.m13.1.1.5">subscript</csymbol><lt id="S4.6.p3.13.m13.1.1.5.2.cmml" xref="S4.6.p3.13.m13.1.1.5.2"></lt><ci id="S4.6.p3.13.m13.1.1.5.3.cmml" xref="S4.6.p3.13.m13.1.1.5.3">lex</ci></apply><share href="https://arxiv.org/html/2503.13728v1#S4.6.p3.13.m13.1.1.4.cmml" id="S4.6.p3.13.m13.1.1d.cmml" xref="S4.6.p3.13.m13.1.1"></share><ci id="S4.6.p3.13.m13.1.1.6.cmml" xref="S4.6.p3.13.m13.1.1.6">𝑠</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.6.p3.13.m13.1c">t<_{\mathrm{lex}}u<_{\mathrm{lex}}s</annotation><annotation encoding="application/x-llamapun" id="S4.6.p3.13.m13.1d">italic_t < start_POSTSUBSCRIPT roman_lex end_POSTSUBSCRIPT italic_u < start_POSTSUBSCRIPT roman_lex end_POSTSUBSCRIPT italic_s</annotation></semantics></math>, <math alttext="\operatorname{dom}(u)<\lambda_{\xi}" class="ltx_Math" display="inline" id="S4.6.p3.14.m14.2"><semantics id="S4.6.p3.14.m14.2a"><mrow id="S4.6.p3.14.m14.2.3" xref="S4.6.p3.14.m14.2.3.cmml"><mrow id="S4.6.p3.14.m14.2.3.2.2" xref="S4.6.p3.14.m14.2.3.2.1.cmml"><mi id="S4.6.p3.14.m14.1.1" xref="S4.6.p3.14.m14.1.1.cmml">dom</mi><mo id="S4.6.p3.14.m14.2.3.2.2a" xref="S4.6.p3.14.m14.2.3.2.1.cmml">⁡</mo><mrow id="S4.6.p3.14.m14.2.3.2.2.1" xref="S4.6.p3.14.m14.2.3.2.1.cmml"><mo id="S4.6.p3.14.m14.2.3.2.2.1.1" stretchy="false" xref="S4.6.p3.14.m14.2.3.2.1.cmml">(</mo><mi id="S4.6.p3.14.m14.2.2" xref="S4.6.p3.14.m14.2.2.cmml">u</mi><mo id="S4.6.p3.14.m14.2.3.2.2.1.2" stretchy="false" xref="S4.6.p3.14.m14.2.3.2.1.cmml">)</mo></mrow></mrow><mo id="S4.6.p3.14.m14.2.3.1" xref="S4.6.p3.14.m14.2.3.1.cmml"><</mo><msub id="S4.6.p3.14.m14.2.3.3" xref="S4.6.p3.14.m14.2.3.3.cmml"><mi id="S4.6.p3.14.m14.2.3.3.2" xref="S4.6.p3.14.m14.2.3.3.2.cmml">λ</mi><mi id="S4.6.p3.14.m14.2.3.3.3" xref="S4.6.p3.14.m14.2.3.3.3.cmml">ξ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.6.p3.14.m14.2b"><apply id="S4.6.p3.14.m14.2.3.cmml" xref="S4.6.p3.14.m14.2.3"><lt id="S4.6.p3.14.m14.2.3.1.cmml" xref="S4.6.p3.14.m14.2.3.1"></lt><apply id="S4.6.p3.14.m14.2.3.2.1.cmml" xref="S4.6.p3.14.m14.2.3.2.2"><ci id="S4.6.p3.14.m14.1.1.cmml" xref="S4.6.p3.14.m14.1.1">dom</ci><ci id="S4.6.p3.14.m14.2.2.cmml" xref="S4.6.p3.14.m14.2.2">𝑢</ci></apply><apply id="S4.6.p3.14.m14.2.3.3.cmml" xref="S4.6.p3.14.m14.2.3.3"><csymbol cd="ambiguous" id="S4.6.p3.14.m14.2.3.3.1.cmml" xref="S4.6.p3.14.m14.2.3.3">subscript</csymbol><ci id="S4.6.p3.14.m14.2.3.3.2.cmml" xref="S4.6.p3.14.m14.2.3.3.2">𝜆</ci><ci id="S4.6.p3.14.m14.2.3.3.3.cmml" xref="S4.6.p3.14.m14.2.3.3.3">𝜉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.6.p3.14.m14.2c">\operatorname{dom}(u)<\lambda_{\xi}</annotation><annotation encoding="application/x-llamapun" id="S4.6.p3.14.m14.2d">roman_dom ( italic_u ) < italic_λ start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\operatorname{dom}(u)\notin\Lambda^{\prime}" class="ltx_Math" display="inline" id="S4.6.p3.15.m15.2"><semantics id="S4.6.p3.15.m15.2a"><mrow id="S4.6.p3.15.m15.2.3" xref="S4.6.p3.15.m15.2.3.cmml"><mrow id="S4.6.p3.15.m15.2.3.2.2" xref="S4.6.p3.15.m15.2.3.2.1.cmml"><mi id="S4.6.p3.15.m15.1.1" xref="S4.6.p3.15.m15.1.1.cmml">dom</mi><mo id="S4.6.p3.15.m15.2.3.2.2a" xref="S4.6.p3.15.m15.2.3.2.1.cmml">⁡</mo><mrow id="S4.6.p3.15.m15.2.3.2.2.1" xref="S4.6.p3.15.m15.2.3.2.1.cmml"><mo id="S4.6.p3.15.m15.2.3.2.2.1.1" stretchy="false" xref="S4.6.p3.15.m15.2.3.2.1.cmml">(</mo><mi id="S4.6.p3.15.m15.2.2" xref="S4.6.p3.15.m15.2.2.cmml">u</mi><mo id="S4.6.p3.15.m15.2.3.2.2.1.2" stretchy="false" xref="S4.6.p3.15.m15.2.3.2.1.cmml">)</mo></mrow></mrow><mo id="S4.6.p3.15.m15.2.3.1" xref="S4.6.p3.15.m15.2.3.1.cmml">∉</mo><msup id="S4.6.p3.15.m15.2.3.3" xref="S4.6.p3.15.m15.2.3.3.cmml"><mi id="S4.6.p3.15.m15.2.3.3.2" mathvariant="normal" xref="S4.6.p3.15.m15.2.3.3.2.cmml">Λ</mi><mo id="S4.6.p3.15.m15.2.3.3.3" xref="S4.6.p3.15.m15.2.3.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.6.p3.15.m15.2b"><apply id="S4.6.p3.15.m15.2.3.cmml" xref="S4.6.p3.15.m15.2.3"><notin id="S4.6.p3.15.m15.2.3.1.cmml" xref="S4.6.p3.15.m15.2.3.1"></notin><apply id="S4.6.p3.15.m15.2.3.2.1.cmml" xref="S4.6.p3.15.m15.2.3.2.2"><ci id="S4.6.p3.15.m15.1.1.cmml" xref="S4.6.p3.15.m15.1.1">dom</ci><ci id="S4.6.p3.15.m15.2.2.cmml" xref="S4.6.p3.15.m15.2.2">𝑢</ci></apply><apply id="S4.6.p3.15.m15.2.3.3.cmml" xref="S4.6.p3.15.m15.2.3.3"><csymbol cd="ambiguous" id="S4.6.p3.15.m15.2.3.3.1.cmml" xref="S4.6.p3.15.m15.2.3.3">superscript</csymbol><ci id="S4.6.p3.15.m15.2.3.3.2.cmml" xref="S4.6.p3.15.m15.2.3.3.2">Λ</ci><ci id="S4.6.p3.15.m15.2.3.3.3.cmml" xref="S4.6.p3.15.m15.2.3.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.6.p3.15.m15.2c">\operatorname{dom}(u)\notin\Lambda^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.6.p3.15.m15.2d">roman_dom ( italic_u ) ∉ roman_Λ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, so <math alttext="u\in A" class="ltx_Math" display="inline" id="S4.6.p3.16.m16.1"><semantics id="S4.6.p3.16.m16.1a"><mrow id="S4.6.p3.16.m16.1.1" xref="S4.6.p3.16.m16.1.1.cmml"><mi id="S4.6.p3.16.m16.1.1.2" xref="S4.6.p3.16.m16.1.1.2.cmml">u</mi><mo id="S4.6.p3.16.m16.1.1.1" xref="S4.6.p3.16.m16.1.1.1.cmml">∈</mo><mi id="S4.6.p3.16.m16.1.1.3" xref="S4.6.p3.16.m16.1.1.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.6.p3.16.m16.1b"><apply id="S4.6.p3.16.m16.1.1.cmml" xref="S4.6.p3.16.m16.1.1"><in id="S4.6.p3.16.m16.1.1.1.cmml" xref="S4.6.p3.16.m16.1.1.1"></in><ci id="S4.6.p3.16.m16.1.1.2.cmml" xref="S4.6.p3.16.m16.1.1.2">𝑢</ci><ci id="S4.6.p3.16.m16.1.1.3.cmml" xref="S4.6.p3.16.m16.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.6.p3.16.m16.1c">u\in A</annotation><annotation encoding="application/x-llamapun" id="S4.6.p3.16.m16.1d">italic_u ∈ italic_A</annotation></semantics></math> witnesses that <math alttext="t_{0}" class="ltx_Math" display="inline" id="S4.6.p3.17.m17.1"><semantics id="S4.6.p3.17.m17.1a"><msub id="S4.6.p3.17.m17.1.1" xref="S4.6.p3.17.m17.1.1.cmml"><mi id="S4.6.p3.17.m17.1.1.2" xref="S4.6.p3.17.m17.1.1.2.cmml">t</mi><mn id="S4.6.p3.17.m17.1.1.3" xref="S4.6.p3.17.m17.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S4.6.p3.17.m17.1b"><apply id="S4.6.p3.17.m17.1.1.cmml" xref="S4.6.p3.17.m17.1.1"><csymbol cd="ambiguous" id="S4.6.p3.17.m17.1.1.1.cmml" xref="S4.6.p3.17.m17.1.1">subscript</csymbol><ci id="S4.6.p3.17.m17.1.1.2.cmml" xref="S4.6.p3.17.m17.1.1.2">𝑡</ci><cn id="S4.6.p3.17.m17.1.1.3.cmml" type="integer" xref="S4.6.p3.17.m17.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.6.p3.17.m17.1c">t_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.6.p3.17.m17.1d">italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="t_{1}" class="ltx_Math" display="inline" id="S4.6.p3.18.m18.1"><semantics id="S4.6.p3.18.m18.1a"><msub id="S4.6.p3.18.m18.1.1" xref="S4.6.p3.18.m18.1.1.cmml"><mi id="S4.6.p3.18.m18.1.1.2" xref="S4.6.p3.18.m18.1.1.2.cmml">t</mi><mn id="S4.6.p3.18.m18.1.1.3" xref="S4.6.p3.18.m18.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.6.p3.18.m18.1b"><apply id="S4.6.p3.18.m18.1.1.cmml" xref="S4.6.p3.18.m18.1.1"><csymbol cd="ambiguous" id="S4.6.p3.18.m18.1.1.1.cmml" xref="S4.6.p3.18.m18.1.1">subscript</csymbol><ci id="S4.6.p3.18.m18.1.1.2.cmml" xref="S4.6.p3.18.m18.1.1.2">𝑡</ci><cn id="S4.6.p3.18.m18.1.1.3.cmml" type="integer" xref="S4.6.p3.18.m18.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.6.p3.18.m18.1c">t_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.6.p3.18.m18.1d">italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> are in distinct complementary intervals of <math alttext="A\setminus D_{\xi}" class="ltx_Math" display="inline" id="S4.6.p3.19.m19.1"><semantics id="S4.6.p3.19.m19.1a"><mrow id="S4.6.p3.19.m19.1.1" xref="S4.6.p3.19.m19.1.1.cmml"><mi id="S4.6.p3.19.m19.1.1.2" xref="S4.6.p3.19.m19.1.1.2.cmml">A</mi><mo id="S4.6.p3.19.m19.1.1.1" xref="S4.6.p3.19.m19.1.1.1.cmml">∖</mo><msub id="S4.6.p3.19.m19.1.1.3" xref="S4.6.p3.19.m19.1.1.3.cmml"><mi id="S4.6.p3.19.m19.1.1.3.2" xref="S4.6.p3.19.m19.1.1.3.2.cmml">D</mi><mi id="S4.6.p3.19.m19.1.1.3.3" xref="S4.6.p3.19.m19.1.1.3.3.cmml">ξ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.6.p3.19.m19.1b"><apply id="S4.6.p3.19.m19.1.1.cmml" xref="S4.6.p3.19.m19.1.1"><setdiff id="S4.6.p3.19.m19.1.1.1.cmml" xref="S4.6.p3.19.m19.1.1.1"></setdiff><ci id="S4.6.p3.19.m19.1.1.2.cmml" xref="S4.6.p3.19.m19.1.1.2">𝐴</ci><apply id="S4.6.p3.19.m19.1.1.3.cmml" xref="S4.6.p3.19.m19.1.1.3"><csymbol cd="ambiguous" id="S4.6.p3.19.m19.1.1.3.1.cmml" xref="S4.6.p3.19.m19.1.1.3">subscript</csymbol><ci id="S4.6.p3.19.m19.1.1.3.2.cmml" xref="S4.6.p3.19.m19.1.1.3.2">𝐷</ci><ci id="S4.6.p3.19.m19.1.1.3.3.cmml" xref="S4.6.p3.19.m19.1.1.3.3">𝜉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.6.p3.19.m19.1c">A\setminus D_{\xi}</annotation><annotation encoding="application/x-llamapun" id="S4.6.p3.19.m19.1d">italic_A ∖ italic_D start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT</annotation></semantics></math>. ∎</p> </div> </div> <div class="ltx_theorem ltx_theorem_corollary" id="S4.Thmtheorem6"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem6.1.1.1">Corollary 4.6</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem6.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem6.p1"> <p class="ltx_p" id="S4.Thmtheorem6.p1.2">If <math alttext="Y=\varnothing" class="ltx_Math" display="inline" id="S4.Thmtheorem6.p1.1.m1.1"><semantics id="S4.Thmtheorem6.p1.1.m1.1a"><mrow id="S4.Thmtheorem6.p1.1.m1.1.1" xref="S4.Thmtheorem6.p1.1.m1.1.1.cmml"><mi id="S4.Thmtheorem6.p1.1.m1.1.1.2" xref="S4.Thmtheorem6.p1.1.m1.1.1.2.cmml">Y</mi><mo id="S4.Thmtheorem6.p1.1.m1.1.1.1" xref="S4.Thmtheorem6.p1.1.m1.1.1.1.cmml">=</mo><mi id="S4.Thmtheorem6.p1.1.m1.1.1.3" mathvariant="normal" xref="S4.Thmtheorem6.p1.1.m1.1.1.3.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem6.p1.1.m1.1b"><apply id="S4.Thmtheorem6.p1.1.m1.1.1.cmml" xref="S4.Thmtheorem6.p1.1.m1.1.1"><eq id="S4.Thmtheorem6.p1.1.m1.1.1.1.cmml" xref="S4.Thmtheorem6.p1.1.m1.1.1.1"></eq><ci id="S4.Thmtheorem6.p1.1.m1.1.1.2.cmml" xref="S4.Thmtheorem6.p1.1.m1.1.1.2">𝑌</ci><emptyset id="S4.Thmtheorem6.p1.1.m1.1.1.3.cmml" xref="S4.Thmtheorem6.p1.1.m1.1.1.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem6.p1.1.m1.1c">Y=\varnothing</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem6.p1.1.m1.1d">italic_Y = ∅</annotation></semantics></math>, then we can require <math alttext="A" class="ltx_Math" display="inline" id="S4.Thmtheorem6.p1.2.m2.1"><semantics id="S4.Thmtheorem6.p1.2.m2.1a"><mi id="S4.Thmtheorem6.p1.2.m2.1.1" xref="S4.Thmtheorem6.p1.2.m2.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem6.p1.2.m2.1b"><ci id="S4.Thmtheorem6.p1.2.m2.1.1.cmml" xref="S4.Thmtheorem6.p1.2.m2.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem6.p1.2.m2.1c">A</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem6.p1.2.m2.1d">italic_A</annotation></semantics></math> in the conclusion of <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S4.Thmtheorem5" title="Theorem 4.5. ‣ 4. Aronszajn line decompositions ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">4.5</span></a> to be Countryman.</p> </div> </div> <div class="ltx_proof" id="S4.7"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S4.7.p1"> <p class="ltx_p" id="S4.7.p1.2">Simply note that in the proof of <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S4.Thmtheorem5" title="Theorem 4.5. ‣ 4. Aronszajn line decompositions ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">4.5</span></a> if <math alttext="Y=0" class="ltx_Math" display="inline" id="S4.7.p1.1.m1.1"><semantics id="S4.7.p1.1.m1.1a"><mrow id="S4.7.p1.1.m1.1.1" xref="S4.7.p1.1.m1.1.1.cmml"><mi id="S4.7.p1.1.m1.1.1.2" xref="S4.7.p1.1.m1.1.1.2.cmml">Y</mi><mo id="S4.7.p1.1.m1.1.1.1" xref="S4.7.p1.1.m1.1.1.1.cmml">=</mo><mn id="S4.7.p1.1.m1.1.1.3" xref="S4.7.p1.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.7.p1.1.m1.1b"><apply id="S4.7.p1.1.m1.1.1.cmml" xref="S4.7.p1.1.m1.1.1"><eq id="S4.7.p1.1.m1.1.1.1.cmml" xref="S4.7.p1.1.m1.1.1.1"></eq><ci id="S4.7.p1.1.m1.1.1.2.cmml" xref="S4.7.p1.1.m1.1.1.2">𝑌</ci><cn id="S4.7.p1.1.m1.1.1.3.cmml" type="integer" xref="S4.7.p1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.7.p1.1.m1.1c">Y=0</annotation><annotation encoding="application/x-llamapun" id="S4.7.p1.1.m1.1d">italic_Y = 0</annotation></semantics></math> then <math alttext="A\subseteq T" class="ltx_Math" display="inline" id="S4.7.p1.2.m2.1"><semantics id="S4.7.p1.2.m2.1a"><mrow id="S4.7.p1.2.m2.1.1" xref="S4.7.p1.2.m2.1.1.cmml"><mi id="S4.7.p1.2.m2.1.1.2" xref="S4.7.p1.2.m2.1.1.2.cmml">A</mi><mo id="S4.7.p1.2.m2.1.1.1" xref="S4.7.p1.2.m2.1.1.1.cmml">⊆</mo><mi id="S4.7.p1.2.m2.1.1.3" xref="S4.7.p1.2.m2.1.1.3.cmml">T</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.7.p1.2.m2.1b"><apply id="S4.7.p1.2.m2.1.1.cmml" xref="S4.7.p1.2.m2.1.1"><subset id="S4.7.p1.2.m2.1.1.1.cmml" xref="S4.7.p1.2.m2.1.1.1"></subset><ci id="S4.7.p1.2.m2.1.1.2.cmml" xref="S4.7.p1.2.m2.1.1.2">𝐴</ci><ci id="S4.7.p1.2.m2.1.1.3.cmml" xref="S4.7.p1.2.m2.1.1.3">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.7.p1.2.m2.1c">A\subseteq T</annotation><annotation encoding="application/x-llamapun" id="S4.7.p1.2.m2.1d">italic_A ⊆ italic_T</annotation></semantics></math>. So this follows from <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S4.Thmtheorem3" title="Lemma 4.3. ‣ 4. Aronszajn line decompositions ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">4.3</span></a>. ∎</p> </div> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S4.Thmtheorem7"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem7.1.1.1">Lemma 4.7</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem7.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem7.p1"> <p class="ltx_p" id="S4.Thmtheorem7.p1.2"><math alttext="A^{+}" class="ltx_Math" display="inline" id="S4.Thmtheorem7.p1.1.m1.1"><semantics id="S4.Thmtheorem7.p1.1.m1.1a"><msup id="S4.Thmtheorem7.p1.1.m1.1.1" xref="S4.Thmtheorem7.p1.1.m1.1.1.cmml"><mi id="S4.Thmtheorem7.p1.1.m1.1.1.2" xref="S4.Thmtheorem7.p1.1.m1.1.1.2.cmml">A</mi><mo id="S4.Thmtheorem7.p1.1.m1.1.1.3" xref="S4.Thmtheorem7.p1.1.m1.1.1.3.cmml">+</mo></msup><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem7.p1.1.m1.1b"><apply id="S4.Thmtheorem7.p1.1.m1.1.1.cmml" xref="S4.Thmtheorem7.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem7.p1.1.m1.1.1.1.cmml" xref="S4.Thmtheorem7.p1.1.m1.1.1">superscript</csymbol><ci id="S4.Thmtheorem7.p1.1.m1.1.1.2.cmml" xref="S4.Thmtheorem7.p1.1.m1.1.1.2">𝐴</ci><plus id="S4.Thmtheorem7.p1.1.m1.1.1.3.cmml" xref="S4.Thmtheorem7.p1.1.m1.1.1.3"></plus></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem7.p1.1.m1.1c">A^{+}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem7.p1.1.m1.1d">italic_A start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> is Countryman iff <math alttext="T" class="ltx_Math" display="inline" id="S4.Thmtheorem7.p1.2.m2.1"><semantics id="S4.Thmtheorem7.p1.2.m2.1a"><mi id="S4.Thmtheorem7.p1.2.m2.1.1" xref="S4.Thmtheorem7.p1.2.m2.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem7.p1.2.m2.1b"><ci id="S4.Thmtheorem7.p1.2.m2.1.1.cmml" xref="S4.Thmtheorem7.p1.2.m2.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem7.p1.2.m2.1c">T</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem7.p1.2.m2.1d">italic_T</annotation></semantics></math> is special<span class="ltx_note ltx_role_footnote" id="footnote4"><sup class="ltx_note_mark">4</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">4</sup><span class="ltx_tag ltx_tag_note">4</span>A tree <math alttext="T" class="ltx_Math" display="inline" id="footnote4.m1.1"><semantics id="footnote4.m1.1b"><mi id="footnote4.m1.1.1" xref="footnote4.m1.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="footnote4.m1.1c"><ci id="footnote4.m1.1.1.cmml" xref="footnote4.m1.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="footnote4.m1.1d">T</annotation><annotation encoding="application/x-llamapun" id="footnote4.m1.1e">italic_T</annotation></semantics></math> is called <em class="ltx_emph ltx_font_italic" id="footnote4.1">special</em> if it is the union of countably many antichains. Equivalently, if there is <math alttext="f:T\to\omega" class="ltx_Math" display="inline" id="footnote4.m2.1"><semantics id="footnote4.m2.1b"><mrow id="footnote4.m2.1.1" xref="footnote4.m2.1.1.cmml"><mi id="footnote4.m2.1.1.2" xref="footnote4.m2.1.1.2.cmml">f</mi><mo id="footnote4.m2.1.1.1" lspace="0.278em" rspace="0.278em" xref="footnote4.m2.1.1.1.cmml">:</mo><mrow id="footnote4.m2.1.1.3" xref="footnote4.m2.1.1.3.cmml"><mi id="footnote4.m2.1.1.3.2" xref="footnote4.m2.1.1.3.2.cmml">T</mi><mo id="footnote4.m2.1.1.3.1" stretchy="false" xref="footnote4.m2.1.1.3.1.cmml">→</mo><mi id="footnote4.m2.1.1.3.3" xref="footnote4.m2.1.1.3.3.cmml">ω</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="footnote4.m2.1c"><apply id="footnote4.m2.1.1.cmml" xref="footnote4.m2.1.1"><ci id="footnote4.m2.1.1.1.cmml" xref="footnote4.m2.1.1.1">:</ci><ci id="footnote4.m2.1.1.2.cmml" xref="footnote4.m2.1.1.2">𝑓</ci><apply id="footnote4.m2.1.1.3.cmml" xref="footnote4.m2.1.1.3"><ci id="footnote4.m2.1.1.3.1.cmml" xref="footnote4.m2.1.1.3.1">→</ci><ci id="footnote4.m2.1.1.3.2.cmml" xref="footnote4.m2.1.1.3.2">𝑇</ci><ci id="footnote4.m2.1.1.3.3.cmml" xref="footnote4.m2.1.1.3.3">𝜔</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote4.m2.1d">f:T\to\omega</annotation><annotation encoding="application/x-llamapun" id="footnote4.m2.1e">italic_f : italic_T → italic_ω</annotation></semantics></math> such that <math alttext="f(t)\neq f(s)" class="ltx_Math" display="inline" id="footnote4.m3.2"><semantics id="footnote4.m3.2b"><mrow id="footnote4.m3.2.3" xref="footnote4.m3.2.3.cmml"><mrow id="footnote4.m3.2.3.2" xref="footnote4.m3.2.3.2.cmml"><mi id="footnote4.m3.2.3.2.2" xref="footnote4.m3.2.3.2.2.cmml">f</mi><mo id="footnote4.m3.2.3.2.1" xref="footnote4.m3.2.3.2.1.cmml">⁢</mo><mrow id="footnote4.m3.2.3.2.3.2" xref="footnote4.m3.2.3.2.cmml"><mo id="footnote4.m3.2.3.2.3.2.1" stretchy="false" xref="footnote4.m3.2.3.2.cmml">(</mo><mi id="footnote4.m3.1.1" xref="footnote4.m3.1.1.cmml">t</mi><mo id="footnote4.m3.2.3.2.3.2.2" stretchy="false" xref="footnote4.m3.2.3.2.cmml">)</mo></mrow></mrow><mo id="footnote4.m3.2.3.1" xref="footnote4.m3.2.3.1.cmml">≠</mo><mrow id="footnote4.m3.2.3.3" xref="footnote4.m3.2.3.3.cmml"><mi id="footnote4.m3.2.3.3.2" xref="footnote4.m3.2.3.3.2.cmml">f</mi><mo id="footnote4.m3.2.3.3.1" xref="footnote4.m3.2.3.3.1.cmml">⁢</mo><mrow id="footnote4.m3.2.3.3.3.2" xref="footnote4.m3.2.3.3.cmml"><mo id="footnote4.m3.2.3.3.3.2.1" stretchy="false" xref="footnote4.m3.2.3.3.cmml">(</mo><mi id="footnote4.m3.2.2" xref="footnote4.m3.2.2.cmml">s</mi><mo id="footnote4.m3.2.3.3.3.2.2" stretchy="false" xref="footnote4.m3.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="footnote4.m3.2c"><apply id="footnote4.m3.2.3.cmml" xref="footnote4.m3.2.3"><neq id="footnote4.m3.2.3.1.cmml" xref="footnote4.m3.2.3.1"></neq><apply id="footnote4.m3.2.3.2.cmml" xref="footnote4.m3.2.3.2"><times id="footnote4.m3.2.3.2.1.cmml" xref="footnote4.m3.2.3.2.1"></times><ci id="footnote4.m3.2.3.2.2.cmml" xref="footnote4.m3.2.3.2.2">𝑓</ci><ci id="footnote4.m3.1.1.cmml" xref="footnote4.m3.1.1">𝑡</ci></apply><apply id="footnote4.m3.2.3.3.cmml" xref="footnote4.m3.2.3.3"><times id="footnote4.m3.2.3.3.1.cmml" xref="footnote4.m3.2.3.3.1"></times><ci id="footnote4.m3.2.3.3.2.cmml" xref="footnote4.m3.2.3.3.2">𝑓</ci><ci id="footnote4.m3.2.2.cmml" xref="footnote4.m3.2.2">𝑠</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote4.m3.2d">f(t)\neq f(s)</annotation><annotation encoding="application/x-llamapun" id="footnote4.m3.2e">italic_f ( italic_t ) ≠ italic_f ( italic_s )</annotation></semantics></math> whenever <math alttext="t<_{T}s" class="ltx_Math" display="inline" id="footnote4.m4.1"><semantics id="footnote4.m4.1b"><mrow id="footnote4.m4.1.1" xref="footnote4.m4.1.1.cmml"><mi id="footnote4.m4.1.1.2" xref="footnote4.m4.1.1.2.cmml">t</mi><msub id="footnote4.m4.1.1.1" xref="footnote4.m4.1.1.1.cmml"><mo id="footnote4.m4.1.1.1.2" xref="footnote4.m4.1.1.1.2.cmml"><</mo><mi id="footnote4.m4.1.1.1.3" xref="footnote4.m4.1.1.1.3.cmml">T</mi></msub><mi id="footnote4.m4.1.1.3" xref="footnote4.m4.1.1.3.cmml">s</mi></mrow><annotation-xml encoding="MathML-Content" id="footnote4.m4.1c"><apply id="footnote4.m4.1.1.cmml" xref="footnote4.m4.1.1"><apply id="footnote4.m4.1.1.1.cmml" xref="footnote4.m4.1.1.1"><csymbol cd="ambiguous" id="footnote4.m4.1.1.1.1.cmml" xref="footnote4.m4.1.1.1">subscript</csymbol><lt id="footnote4.m4.1.1.1.2.cmml" xref="footnote4.m4.1.1.1.2"></lt><ci id="footnote4.m4.1.1.1.3.cmml" xref="footnote4.m4.1.1.1.3">𝑇</ci></apply><ci id="footnote4.m4.1.1.2.cmml" xref="footnote4.m4.1.1.2">𝑡</ci><ci id="footnote4.m4.1.1.3.cmml" xref="footnote4.m4.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote4.m4.1d">t<_{T}s</annotation><annotation encoding="application/x-llamapun" id="footnote4.m4.1e">italic_t < start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT italic_s</annotation></semantics></math>. </span></span></span>.</p> </div> </div> <div class="ltx_proof" id="S4.12"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S4.8.p1"> <p class="ltx_p" id="S4.8.p1.10"><math alttext="(\rightarrow)" class="ltx_Math" display="inline" id="S4.8.p1.1.m1.1"><semantics id="S4.8.p1.1.m1.1a"><mrow id="S4.8.p1.1.m1.1.2.2"><mo id="S4.8.p1.1.m1.1.2.2.1" stretchy="false">(</mo><mo id="S4.8.p1.1.m1.1.1" lspace="0em" rspace="0em" stretchy="false" xref="S4.8.p1.1.m1.1.1.cmml">→</mo><mo id="S4.8.p1.1.m1.1.2.2.2" stretchy="false">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.8.p1.1.m1.1b"><ci id="S4.8.p1.1.m1.1.1.cmml" xref="S4.8.p1.1.m1.1.1">→</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.8.p1.1.m1.1c">(\rightarrow)</annotation><annotation encoding="application/x-llamapun" id="S4.8.p1.1.m1.1d">( → )</annotation></semantics></math>. Let <math alttext="T_{s}" class="ltx_Math" display="inline" id="S4.8.p1.2.m2.1"><semantics id="S4.8.p1.2.m2.1a"><msub id="S4.8.p1.2.m2.1.1" xref="S4.8.p1.2.m2.1.1.cmml"><mi id="S4.8.p1.2.m2.1.1.2" xref="S4.8.p1.2.m2.1.1.2.cmml">T</mi><mi id="S4.8.p1.2.m2.1.1.3" xref="S4.8.p1.2.m2.1.1.3.cmml">s</mi></msub><annotation-xml encoding="MathML-Content" id="S4.8.p1.2.m2.1b"><apply id="S4.8.p1.2.m2.1.1.cmml" xref="S4.8.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S4.8.p1.2.m2.1.1.1.cmml" xref="S4.8.p1.2.m2.1.1">subscript</csymbol><ci id="S4.8.p1.2.m2.1.1.2.cmml" xref="S4.8.p1.2.m2.1.1.2">𝑇</ci><ci id="S4.8.p1.2.m2.1.1.3.cmml" xref="S4.8.p1.2.m2.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.8.p1.2.m2.1c">T_{s}</annotation><annotation encoding="application/x-llamapun" id="S4.8.p1.2.m2.1d">italic_T start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math> be the set of <math alttext="t\in T" class="ltx_Math" display="inline" id="S4.8.p1.3.m3.1"><semantics id="S4.8.p1.3.m3.1a"><mrow id="S4.8.p1.3.m3.1.1" xref="S4.8.p1.3.m3.1.1.cmml"><mi id="S4.8.p1.3.m3.1.1.2" xref="S4.8.p1.3.m3.1.1.2.cmml">t</mi><mo id="S4.8.p1.3.m3.1.1.1" xref="S4.8.p1.3.m3.1.1.1.cmml">∈</mo><mi id="S4.8.p1.3.m3.1.1.3" xref="S4.8.p1.3.m3.1.1.3.cmml">T</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.8.p1.3.m3.1b"><apply id="S4.8.p1.3.m3.1.1.cmml" xref="S4.8.p1.3.m3.1.1"><in id="S4.8.p1.3.m3.1.1.1.cmml" xref="S4.8.p1.3.m3.1.1.1"></in><ci id="S4.8.p1.3.m3.1.1.2.cmml" xref="S4.8.p1.3.m3.1.1.2">𝑡</ci><ci id="S4.8.p1.3.m3.1.1.3.cmml" xref="S4.8.p1.3.m3.1.1.3">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.8.p1.3.m3.1c">t\in T</annotation><annotation encoding="application/x-llamapun" id="S4.8.p1.3.m3.1d">italic_t ∈ italic_T</annotation></semantics></math> that have successor length, and for <math alttext="t\in T_{s}" class="ltx_Math" display="inline" id="S4.8.p1.4.m4.1"><semantics id="S4.8.p1.4.m4.1a"><mrow id="S4.8.p1.4.m4.1.1" xref="S4.8.p1.4.m4.1.1.cmml"><mi id="S4.8.p1.4.m4.1.1.2" xref="S4.8.p1.4.m4.1.1.2.cmml">t</mi><mo id="S4.8.p1.4.m4.1.1.1" xref="S4.8.p1.4.m4.1.1.1.cmml">∈</mo><msub id="S4.8.p1.4.m4.1.1.3" xref="S4.8.p1.4.m4.1.1.3.cmml"><mi id="S4.8.p1.4.m4.1.1.3.2" xref="S4.8.p1.4.m4.1.1.3.2.cmml">T</mi><mi id="S4.8.p1.4.m4.1.1.3.3" xref="S4.8.p1.4.m4.1.1.3.3.cmml">s</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.8.p1.4.m4.1b"><apply id="S4.8.p1.4.m4.1.1.cmml" xref="S4.8.p1.4.m4.1.1"><in id="S4.8.p1.4.m4.1.1.1.cmml" xref="S4.8.p1.4.m4.1.1.1"></in><ci id="S4.8.p1.4.m4.1.1.2.cmml" xref="S4.8.p1.4.m4.1.1.2">𝑡</ci><apply id="S4.8.p1.4.m4.1.1.3.cmml" xref="S4.8.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="S4.8.p1.4.m4.1.1.3.1.cmml" xref="S4.8.p1.4.m4.1.1.3">subscript</csymbol><ci id="S4.8.p1.4.m4.1.1.3.2.cmml" xref="S4.8.p1.4.m4.1.1.3.2">𝑇</ci><ci id="S4.8.p1.4.m4.1.1.3.3.cmml" xref="S4.8.p1.4.m4.1.1.3.3">𝑠</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.8.p1.4.m4.1c">t\in T_{s}</annotation><annotation encoding="application/x-llamapun" id="S4.8.p1.4.m4.1d">italic_t ∈ italic_T start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math> let <math alttext="l(t):=t(\alpha)" class="ltx_Math" display="inline" id="S4.8.p1.5.m5.2"><semantics id="S4.8.p1.5.m5.2a"><mrow id="S4.8.p1.5.m5.2.3" xref="S4.8.p1.5.m5.2.3.cmml"><mrow id="S4.8.p1.5.m5.2.3.2" xref="S4.8.p1.5.m5.2.3.2.cmml"><mi id="S4.8.p1.5.m5.2.3.2.2" xref="S4.8.p1.5.m5.2.3.2.2.cmml">l</mi><mo id="S4.8.p1.5.m5.2.3.2.1" xref="S4.8.p1.5.m5.2.3.2.1.cmml">⁢</mo><mrow id="S4.8.p1.5.m5.2.3.2.3.2" xref="S4.8.p1.5.m5.2.3.2.cmml"><mo id="S4.8.p1.5.m5.2.3.2.3.2.1" stretchy="false" xref="S4.8.p1.5.m5.2.3.2.cmml">(</mo><mi id="S4.8.p1.5.m5.1.1" xref="S4.8.p1.5.m5.1.1.cmml">t</mi><mo id="S4.8.p1.5.m5.2.3.2.3.2.2" rspace="0.278em" stretchy="false" xref="S4.8.p1.5.m5.2.3.2.cmml">)</mo></mrow></mrow><mo id="S4.8.p1.5.m5.2.3.1" rspace="0.278em" xref="S4.8.p1.5.m5.2.3.1.cmml">:=</mo><mrow id="S4.8.p1.5.m5.2.3.3" xref="S4.8.p1.5.m5.2.3.3.cmml"><mi id="S4.8.p1.5.m5.2.3.3.2" xref="S4.8.p1.5.m5.2.3.3.2.cmml">t</mi><mo id="S4.8.p1.5.m5.2.3.3.1" xref="S4.8.p1.5.m5.2.3.3.1.cmml">⁢</mo><mrow id="S4.8.p1.5.m5.2.3.3.3.2" xref="S4.8.p1.5.m5.2.3.3.cmml"><mo id="S4.8.p1.5.m5.2.3.3.3.2.1" stretchy="false" xref="S4.8.p1.5.m5.2.3.3.cmml">(</mo><mi id="S4.8.p1.5.m5.2.2" xref="S4.8.p1.5.m5.2.2.cmml">α</mi><mo id="S4.8.p1.5.m5.2.3.3.3.2.2" stretchy="false" xref="S4.8.p1.5.m5.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.8.p1.5.m5.2b"><apply id="S4.8.p1.5.m5.2.3.cmml" xref="S4.8.p1.5.m5.2.3"><csymbol cd="latexml" id="S4.8.p1.5.m5.2.3.1.cmml" xref="S4.8.p1.5.m5.2.3.1">assign</csymbol><apply id="S4.8.p1.5.m5.2.3.2.cmml" xref="S4.8.p1.5.m5.2.3.2"><times id="S4.8.p1.5.m5.2.3.2.1.cmml" xref="S4.8.p1.5.m5.2.3.2.1"></times><ci id="S4.8.p1.5.m5.2.3.2.2.cmml" xref="S4.8.p1.5.m5.2.3.2.2">𝑙</ci><ci id="S4.8.p1.5.m5.1.1.cmml" xref="S4.8.p1.5.m5.1.1">𝑡</ci></apply><apply id="S4.8.p1.5.m5.2.3.3.cmml" xref="S4.8.p1.5.m5.2.3.3"><times id="S4.8.p1.5.m5.2.3.3.1.cmml" xref="S4.8.p1.5.m5.2.3.3.1"></times><ci id="S4.8.p1.5.m5.2.3.3.2.cmml" xref="S4.8.p1.5.m5.2.3.3.2">𝑡</ci><ci id="S4.8.p1.5.m5.2.2.cmml" xref="S4.8.p1.5.m5.2.2">𝛼</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.8.p1.5.m5.2c">l(t):=t(\alpha)</annotation><annotation encoding="application/x-llamapun" id="S4.8.p1.5.m5.2d">italic_l ( italic_t ) := italic_t ( italic_α )</annotation></semantics></math> where <math alttext="\operatorname{dom}(t)=\alpha+1" class="ltx_Math" display="inline" id="S4.8.p1.6.m6.2"><semantics id="S4.8.p1.6.m6.2a"><mrow id="S4.8.p1.6.m6.2.3" xref="S4.8.p1.6.m6.2.3.cmml"><mrow id="S4.8.p1.6.m6.2.3.2.2" xref="S4.8.p1.6.m6.2.3.2.1.cmml"><mi id="S4.8.p1.6.m6.1.1" xref="S4.8.p1.6.m6.1.1.cmml">dom</mi><mo id="S4.8.p1.6.m6.2.3.2.2a" xref="S4.8.p1.6.m6.2.3.2.1.cmml">⁡</mo><mrow id="S4.8.p1.6.m6.2.3.2.2.1" xref="S4.8.p1.6.m6.2.3.2.1.cmml"><mo id="S4.8.p1.6.m6.2.3.2.2.1.1" stretchy="false" xref="S4.8.p1.6.m6.2.3.2.1.cmml">(</mo><mi id="S4.8.p1.6.m6.2.2" xref="S4.8.p1.6.m6.2.2.cmml">t</mi><mo id="S4.8.p1.6.m6.2.3.2.2.1.2" stretchy="false" xref="S4.8.p1.6.m6.2.3.2.1.cmml">)</mo></mrow></mrow><mo id="S4.8.p1.6.m6.2.3.1" xref="S4.8.p1.6.m6.2.3.1.cmml">=</mo><mrow id="S4.8.p1.6.m6.2.3.3" xref="S4.8.p1.6.m6.2.3.3.cmml"><mi id="S4.8.p1.6.m6.2.3.3.2" xref="S4.8.p1.6.m6.2.3.3.2.cmml">α</mi><mo id="S4.8.p1.6.m6.2.3.3.1" xref="S4.8.p1.6.m6.2.3.3.1.cmml">+</mo><mn id="S4.8.p1.6.m6.2.3.3.3" xref="S4.8.p1.6.m6.2.3.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.8.p1.6.m6.2b"><apply id="S4.8.p1.6.m6.2.3.cmml" xref="S4.8.p1.6.m6.2.3"><eq id="S4.8.p1.6.m6.2.3.1.cmml" xref="S4.8.p1.6.m6.2.3.1"></eq><apply id="S4.8.p1.6.m6.2.3.2.1.cmml" xref="S4.8.p1.6.m6.2.3.2.2"><ci id="S4.8.p1.6.m6.1.1.cmml" xref="S4.8.p1.6.m6.1.1">dom</ci><ci id="S4.8.p1.6.m6.2.2.cmml" xref="S4.8.p1.6.m6.2.2">𝑡</ci></apply><apply id="S4.8.p1.6.m6.2.3.3.cmml" xref="S4.8.p1.6.m6.2.3.3"><plus id="S4.8.p1.6.m6.2.3.3.1.cmml" xref="S4.8.p1.6.m6.2.3.3.1"></plus><ci id="S4.8.p1.6.m6.2.3.3.2.cmml" xref="S4.8.p1.6.m6.2.3.3.2">𝛼</ci><cn id="S4.8.p1.6.m6.2.3.3.3.cmml" type="integer" xref="S4.8.p1.6.m6.2.3.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.8.p1.6.m6.2c">\operatorname{dom}(t)=\alpha+1</annotation><annotation encoding="application/x-llamapun" id="S4.8.p1.6.m6.2d">roman_dom ( italic_t ) = italic_α + 1</annotation></semantics></math>. Since <math alttext="T" class="ltx_Math" display="inline" id="S4.8.p1.7.m7.1"><semantics id="S4.8.p1.7.m7.1a"><mi id="S4.8.p1.7.m7.1.1" xref="S4.8.p1.7.m7.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S4.8.p1.7.m7.1b"><ci id="S4.8.p1.7.m7.1.1.cmml" xref="S4.8.p1.7.m7.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.8.p1.7.m7.1c">T</annotation><annotation encoding="application/x-llamapun" id="S4.8.p1.7.m7.1d">italic_T</annotation></semantics></math> consists of finite-to-one functions, one sees that <math alttext="t\mapsto(l(t),|\{\xi<\operatorname{dom}(t):t(\xi)=l(t)\}|)" class="ltx_Math" display="inline" id="S4.8.p1.8.m8.7"><semantics id="S4.8.p1.8.m8.7a"><mrow id="S4.8.p1.8.m8.7.7" xref="S4.8.p1.8.m8.7.7.cmml"><mi id="S4.8.p1.8.m8.7.7.4" xref="S4.8.p1.8.m8.7.7.4.cmml">t</mi><mo id="S4.8.p1.8.m8.7.7.3" stretchy="false" xref="S4.8.p1.8.m8.7.7.3.cmml">↦</mo><mrow id="S4.8.p1.8.m8.7.7.2.2" xref="S4.8.p1.8.m8.7.7.2.3.cmml"><mo id="S4.8.p1.8.m8.7.7.2.2.3" stretchy="false" xref="S4.8.p1.8.m8.7.7.2.3.cmml">(</mo><mrow id="S4.8.p1.8.m8.6.6.1.1.1" xref="S4.8.p1.8.m8.6.6.1.1.1.cmml"><mi id="S4.8.p1.8.m8.6.6.1.1.1.2" xref="S4.8.p1.8.m8.6.6.1.1.1.2.cmml">l</mi><mo id="S4.8.p1.8.m8.6.6.1.1.1.1" xref="S4.8.p1.8.m8.6.6.1.1.1.1.cmml">⁢</mo><mrow id="S4.8.p1.8.m8.6.6.1.1.1.3.2" xref="S4.8.p1.8.m8.6.6.1.1.1.cmml"><mo id="S4.8.p1.8.m8.6.6.1.1.1.3.2.1" stretchy="false" xref="S4.8.p1.8.m8.6.6.1.1.1.cmml">(</mo><mi id="S4.8.p1.8.m8.1.1" xref="S4.8.p1.8.m8.1.1.cmml">t</mi><mo id="S4.8.p1.8.m8.6.6.1.1.1.3.2.2" stretchy="false" xref="S4.8.p1.8.m8.6.6.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.8.p1.8.m8.7.7.2.2.4" xref="S4.8.p1.8.m8.7.7.2.3.cmml">,</mo><mrow id="S4.8.p1.8.m8.7.7.2.2.2.1" xref="S4.8.p1.8.m8.7.7.2.2.2.2.cmml"><mo id="S4.8.p1.8.m8.7.7.2.2.2.1.2" stretchy="false" xref="S4.8.p1.8.m8.7.7.2.2.2.2.1.cmml">|</mo><mrow id="S4.8.p1.8.m8.7.7.2.2.2.1.1.2" xref="S4.8.p1.8.m8.7.7.2.2.2.1.1.3.cmml"><mo id="S4.8.p1.8.m8.7.7.2.2.2.1.1.2.3" stretchy="false" xref="S4.8.p1.8.m8.7.7.2.2.2.1.1.3.1.cmml">{</mo><mrow id="S4.8.p1.8.m8.7.7.2.2.2.1.1.1.1" xref="S4.8.p1.8.m8.7.7.2.2.2.1.1.1.1.cmml"><mi id="S4.8.p1.8.m8.7.7.2.2.2.1.1.1.1.2" xref="S4.8.p1.8.m8.7.7.2.2.2.1.1.1.1.2.cmml">ξ</mi><mo id="S4.8.p1.8.m8.7.7.2.2.2.1.1.1.1.1" xref="S4.8.p1.8.m8.7.7.2.2.2.1.1.1.1.1.cmml"><</mo><mrow id="S4.8.p1.8.m8.7.7.2.2.2.1.1.1.1.3.2" xref="S4.8.p1.8.m8.7.7.2.2.2.1.1.1.1.3.1.cmml"><mi id="S4.8.p1.8.m8.2.2" xref="S4.8.p1.8.m8.2.2.cmml">dom</mi><mo id="S4.8.p1.8.m8.7.7.2.2.2.1.1.1.1.3.2a" xref="S4.8.p1.8.m8.7.7.2.2.2.1.1.1.1.3.1.cmml">⁡</mo><mrow id="S4.8.p1.8.m8.7.7.2.2.2.1.1.1.1.3.2.1" xref="S4.8.p1.8.m8.7.7.2.2.2.1.1.1.1.3.1.cmml"><mo id="S4.8.p1.8.m8.7.7.2.2.2.1.1.1.1.3.2.1.1" stretchy="false" xref="S4.8.p1.8.m8.7.7.2.2.2.1.1.1.1.3.1.cmml">(</mo><mi id="S4.8.p1.8.m8.3.3" xref="S4.8.p1.8.m8.3.3.cmml">t</mi><mo id="S4.8.p1.8.m8.7.7.2.2.2.1.1.1.1.3.2.1.2" rspace="0.278em" stretchy="false" xref="S4.8.p1.8.m8.7.7.2.2.2.1.1.1.1.3.1.cmml">)</mo></mrow></mrow></mrow><mo id="S4.8.p1.8.m8.7.7.2.2.2.1.1.2.4" rspace="0.278em" xref="S4.8.p1.8.m8.7.7.2.2.2.1.1.3.1.cmml">:</mo><mrow id="S4.8.p1.8.m8.7.7.2.2.2.1.1.2.2" xref="S4.8.p1.8.m8.7.7.2.2.2.1.1.2.2.cmml"><mrow id="S4.8.p1.8.m8.7.7.2.2.2.1.1.2.2.2" xref="S4.8.p1.8.m8.7.7.2.2.2.1.1.2.2.2.cmml"><mi id="S4.8.p1.8.m8.7.7.2.2.2.1.1.2.2.2.2" xref="S4.8.p1.8.m8.7.7.2.2.2.1.1.2.2.2.2.cmml">t</mi><mo id="S4.8.p1.8.m8.7.7.2.2.2.1.1.2.2.2.1" xref="S4.8.p1.8.m8.7.7.2.2.2.1.1.2.2.2.1.cmml">⁢</mo><mrow id="S4.8.p1.8.m8.7.7.2.2.2.1.1.2.2.2.3.2" xref="S4.8.p1.8.m8.7.7.2.2.2.1.1.2.2.2.cmml"><mo id="S4.8.p1.8.m8.7.7.2.2.2.1.1.2.2.2.3.2.1" stretchy="false" xref="S4.8.p1.8.m8.7.7.2.2.2.1.1.2.2.2.cmml">(</mo><mi id="S4.8.p1.8.m8.4.4" xref="S4.8.p1.8.m8.4.4.cmml">ξ</mi><mo id="S4.8.p1.8.m8.7.7.2.2.2.1.1.2.2.2.3.2.2" stretchy="false" xref="S4.8.p1.8.m8.7.7.2.2.2.1.1.2.2.2.cmml">)</mo></mrow></mrow><mo id="S4.8.p1.8.m8.7.7.2.2.2.1.1.2.2.1" xref="S4.8.p1.8.m8.7.7.2.2.2.1.1.2.2.1.cmml">=</mo><mrow id="S4.8.p1.8.m8.7.7.2.2.2.1.1.2.2.3" xref="S4.8.p1.8.m8.7.7.2.2.2.1.1.2.2.3.cmml"><mi id="S4.8.p1.8.m8.7.7.2.2.2.1.1.2.2.3.2" xref="S4.8.p1.8.m8.7.7.2.2.2.1.1.2.2.3.2.cmml">l</mi><mo id="S4.8.p1.8.m8.7.7.2.2.2.1.1.2.2.3.1" xref="S4.8.p1.8.m8.7.7.2.2.2.1.1.2.2.3.1.cmml">⁢</mo><mrow id="S4.8.p1.8.m8.7.7.2.2.2.1.1.2.2.3.3.2" xref="S4.8.p1.8.m8.7.7.2.2.2.1.1.2.2.3.cmml"><mo id="S4.8.p1.8.m8.7.7.2.2.2.1.1.2.2.3.3.2.1" stretchy="false" xref="S4.8.p1.8.m8.7.7.2.2.2.1.1.2.2.3.cmml">(</mo><mi id="S4.8.p1.8.m8.5.5" xref="S4.8.p1.8.m8.5.5.cmml">t</mi><mo id="S4.8.p1.8.m8.7.7.2.2.2.1.1.2.2.3.3.2.2" stretchy="false" xref="S4.8.p1.8.m8.7.7.2.2.2.1.1.2.2.3.cmml">)</mo></mrow></mrow></mrow><mo id="S4.8.p1.8.m8.7.7.2.2.2.1.1.2.5" stretchy="false" xref="S4.8.p1.8.m8.7.7.2.2.2.1.1.3.1.cmml">}</mo></mrow><mo id="S4.8.p1.8.m8.7.7.2.2.2.1.3" stretchy="false" xref="S4.8.p1.8.m8.7.7.2.2.2.2.1.cmml">|</mo></mrow><mo id="S4.8.p1.8.m8.7.7.2.2.5" stretchy="false" xref="S4.8.p1.8.m8.7.7.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.8.p1.8.m8.7b"><apply id="S4.8.p1.8.m8.7.7.cmml" xref="S4.8.p1.8.m8.7.7"><csymbol cd="latexml" id="S4.8.p1.8.m8.7.7.3.cmml" xref="S4.8.p1.8.m8.7.7.3">maps-to</csymbol><ci id="S4.8.p1.8.m8.7.7.4.cmml" xref="S4.8.p1.8.m8.7.7.4">𝑡</ci><interval closure="open" id="S4.8.p1.8.m8.7.7.2.3.cmml" xref="S4.8.p1.8.m8.7.7.2.2"><apply id="S4.8.p1.8.m8.6.6.1.1.1.cmml" xref="S4.8.p1.8.m8.6.6.1.1.1"><times id="S4.8.p1.8.m8.6.6.1.1.1.1.cmml" xref="S4.8.p1.8.m8.6.6.1.1.1.1"></times><ci id="S4.8.p1.8.m8.6.6.1.1.1.2.cmml" xref="S4.8.p1.8.m8.6.6.1.1.1.2">𝑙</ci><ci id="S4.8.p1.8.m8.1.1.cmml" xref="S4.8.p1.8.m8.1.1">𝑡</ci></apply><apply id="S4.8.p1.8.m8.7.7.2.2.2.2.cmml" xref="S4.8.p1.8.m8.7.7.2.2.2.1"><abs id="S4.8.p1.8.m8.7.7.2.2.2.2.1.cmml" xref="S4.8.p1.8.m8.7.7.2.2.2.1.2"></abs><apply id="S4.8.p1.8.m8.7.7.2.2.2.1.1.3.cmml" xref="S4.8.p1.8.m8.7.7.2.2.2.1.1.2"><csymbol cd="latexml" id="S4.8.p1.8.m8.7.7.2.2.2.1.1.3.1.cmml" xref="S4.8.p1.8.m8.7.7.2.2.2.1.1.2.3">conditional-set</csymbol><apply id="S4.8.p1.8.m8.7.7.2.2.2.1.1.1.1.cmml" xref="S4.8.p1.8.m8.7.7.2.2.2.1.1.1.1"><lt id="S4.8.p1.8.m8.7.7.2.2.2.1.1.1.1.1.cmml" xref="S4.8.p1.8.m8.7.7.2.2.2.1.1.1.1.1"></lt><ci id="S4.8.p1.8.m8.7.7.2.2.2.1.1.1.1.2.cmml" xref="S4.8.p1.8.m8.7.7.2.2.2.1.1.1.1.2">𝜉</ci><apply id="S4.8.p1.8.m8.7.7.2.2.2.1.1.1.1.3.1.cmml" xref="S4.8.p1.8.m8.7.7.2.2.2.1.1.1.1.3.2"><ci id="S4.8.p1.8.m8.2.2.cmml" xref="S4.8.p1.8.m8.2.2">dom</ci><ci id="S4.8.p1.8.m8.3.3.cmml" xref="S4.8.p1.8.m8.3.3">𝑡</ci></apply></apply><apply id="S4.8.p1.8.m8.7.7.2.2.2.1.1.2.2.cmml" xref="S4.8.p1.8.m8.7.7.2.2.2.1.1.2.2"><eq id="S4.8.p1.8.m8.7.7.2.2.2.1.1.2.2.1.cmml" xref="S4.8.p1.8.m8.7.7.2.2.2.1.1.2.2.1"></eq><apply id="S4.8.p1.8.m8.7.7.2.2.2.1.1.2.2.2.cmml" xref="S4.8.p1.8.m8.7.7.2.2.2.1.1.2.2.2"><times id="S4.8.p1.8.m8.7.7.2.2.2.1.1.2.2.2.1.cmml" xref="S4.8.p1.8.m8.7.7.2.2.2.1.1.2.2.2.1"></times><ci id="S4.8.p1.8.m8.7.7.2.2.2.1.1.2.2.2.2.cmml" xref="S4.8.p1.8.m8.7.7.2.2.2.1.1.2.2.2.2">𝑡</ci><ci id="S4.8.p1.8.m8.4.4.cmml" xref="S4.8.p1.8.m8.4.4">𝜉</ci></apply><apply id="S4.8.p1.8.m8.7.7.2.2.2.1.1.2.2.3.cmml" xref="S4.8.p1.8.m8.7.7.2.2.2.1.1.2.2.3"><times id="S4.8.p1.8.m8.7.7.2.2.2.1.1.2.2.3.1.cmml" xref="S4.8.p1.8.m8.7.7.2.2.2.1.1.2.2.3.1"></times><ci id="S4.8.p1.8.m8.7.7.2.2.2.1.1.2.2.3.2.cmml" xref="S4.8.p1.8.m8.7.7.2.2.2.1.1.2.2.3.2">𝑙</ci><ci id="S4.8.p1.8.m8.5.5.cmml" xref="S4.8.p1.8.m8.5.5">𝑡</ci></apply></apply></apply></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.8.p1.8.m8.7c">t\mapsto(l(t),|\{\xi<\operatorname{dom}(t):t(\xi)=l(t)\}|)</annotation><annotation encoding="application/x-llamapun" id="S4.8.p1.8.m8.7d">italic_t ↦ ( italic_l ( italic_t ) , | { italic_ξ < roman_dom ( italic_t ) : italic_t ( italic_ξ ) = italic_l ( italic_t ) } | )</annotation></semantics></math> witnesses that <math alttext="T_{s}" class="ltx_Math" display="inline" id="S4.8.p1.9.m9.1"><semantics id="S4.8.p1.9.m9.1a"><msub id="S4.8.p1.9.m9.1.1" xref="S4.8.p1.9.m9.1.1.cmml"><mi id="S4.8.p1.9.m9.1.1.2" xref="S4.8.p1.9.m9.1.1.2.cmml">T</mi><mi id="S4.8.p1.9.m9.1.1.3" xref="S4.8.p1.9.m9.1.1.3.cmml">s</mi></msub><annotation-xml encoding="MathML-Content" id="S4.8.p1.9.m9.1b"><apply id="S4.8.p1.9.m9.1.1.cmml" xref="S4.8.p1.9.m9.1.1"><csymbol cd="ambiguous" id="S4.8.p1.9.m9.1.1.1.cmml" xref="S4.8.p1.9.m9.1.1">subscript</csymbol><ci id="S4.8.p1.9.m9.1.1.2.cmml" xref="S4.8.p1.9.m9.1.1.2">𝑇</ci><ci id="S4.8.p1.9.m9.1.1.3.cmml" xref="S4.8.p1.9.m9.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.8.p1.9.m9.1c">T_{s}</annotation><annotation encoding="application/x-llamapun" id="S4.8.p1.9.m9.1d">italic_T start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math> is special. Thus it is enough to prove that <math alttext="A_{0}" class="ltx_Math" display="inline" id="S4.8.p1.10.m10.1"><semantics id="S4.8.p1.10.m10.1a"><msub id="S4.8.p1.10.m10.1.1" xref="S4.8.p1.10.m10.1.1.cmml"><mi id="S4.8.p1.10.m10.1.1.2" xref="S4.8.p1.10.m10.1.1.2.cmml">A</mi><mn id="S4.8.p1.10.m10.1.1.3" xref="S4.8.p1.10.m10.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S4.8.p1.10.m10.1b"><apply id="S4.8.p1.10.m10.1.1.cmml" xref="S4.8.p1.10.m10.1.1"><csymbol cd="ambiguous" id="S4.8.p1.10.m10.1.1.1.cmml" xref="S4.8.p1.10.m10.1.1">subscript</csymbol><ci id="S4.8.p1.10.m10.1.1.2.cmml" xref="S4.8.p1.10.m10.1.1.2">𝐴</ci><cn id="S4.8.p1.10.m10.1.1.3.cmml" type="integer" xref="S4.8.p1.10.m10.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.8.p1.10.m10.1c">A_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.8.p1.10.m10.1d">italic_A start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> is special.</p> </div> <div class="ltx_para" id="S4.9.p2"> <p class="ltx_p" id="S4.9.p2.9">Let <math alttext="c:A^{+}\times A^{+}\to\omega" class="ltx_Math" display="inline" id="S4.9.p2.1.m1.1"><semantics id="S4.9.p2.1.m1.1a"><mrow id="S4.9.p2.1.m1.1.1" xref="S4.9.p2.1.m1.1.1.cmml"><mi id="S4.9.p2.1.m1.1.1.2" xref="S4.9.p2.1.m1.1.1.2.cmml">c</mi><mo id="S4.9.p2.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S4.9.p2.1.m1.1.1.1.cmml">:</mo><mrow id="S4.9.p2.1.m1.1.1.3" xref="S4.9.p2.1.m1.1.1.3.cmml"><mrow id="S4.9.p2.1.m1.1.1.3.2" xref="S4.9.p2.1.m1.1.1.3.2.cmml"><msup id="S4.9.p2.1.m1.1.1.3.2.2" xref="S4.9.p2.1.m1.1.1.3.2.2.cmml"><mi id="S4.9.p2.1.m1.1.1.3.2.2.2" xref="S4.9.p2.1.m1.1.1.3.2.2.2.cmml">A</mi><mo id="S4.9.p2.1.m1.1.1.3.2.2.3" xref="S4.9.p2.1.m1.1.1.3.2.2.3.cmml">+</mo></msup><mo id="S4.9.p2.1.m1.1.1.3.2.1" lspace="0.222em" rspace="0.222em" xref="S4.9.p2.1.m1.1.1.3.2.1.cmml">×</mo><msup id="S4.9.p2.1.m1.1.1.3.2.3" xref="S4.9.p2.1.m1.1.1.3.2.3.cmml"><mi id="S4.9.p2.1.m1.1.1.3.2.3.2" xref="S4.9.p2.1.m1.1.1.3.2.3.2.cmml">A</mi><mo id="S4.9.p2.1.m1.1.1.3.2.3.3" xref="S4.9.p2.1.m1.1.1.3.2.3.3.cmml">+</mo></msup></mrow><mo id="S4.9.p2.1.m1.1.1.3.1" stretchy="false" xref="S4.9.p2.1.m1.1.1.3.1.cmml">→</mo><mi id="S4.9.p2.1.m1.1.1.3.3" xref="S4.9.p2.1.m1.1.1.3.3.cmml">ω</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.9.p2.1.m1.1b"><apply id="S4.9.p2.1.m1.1.1.cmml" xref="S4.9.p2.1.m1.1.1"><ci id="S4.9.p2.1.m1.1.1.1.cmml" xref="S4.9.p2.1.m1.1.1.1">:</ci><ci id="S4.9.p2.1.m1.1.1.2.cmml" xref="S4.9.p2.1.m1.1.1.2">𝑐</ci><apply id="S4.9.p2.1.m1.1.1.3.cmml" xref="S4.9.p2.1.m1.1.1.3"><ci id="S4.9.p2.1.m1.1.1.3.1.cmml" xref="S4.9.p2.1.m1.1.1.3.1">→</ci><apply id="S4.9.p2.1.m1.1.1.3.2.cmml" xref="S4.9.p2.1.m1.1.1.3.2"><times id="S4.9.p2.1.m1.1.1.3.2.1.cmml" xref="S4.9.p2.1.m1.1.1.3.2.1"></times><apply id="S4.9.p2.1.m1.1.1.3.2.2.cmml" xref="S4.9.p2.1.m1.1.1.3.2.2"><csymbol cd="ambiguous" id="S4.9.p2.1.m1.1.1.3.2.2.1.cmml" xref="S4.9.p2.1.m1.1.1.3.2.2">superscript</csymbol><ci id="S4.9.p2.1.m1.1.1.3.2.2.2.cmml" xref="S4.9.p2.1.m1.1.1.3.2.2.2">𝐴</ci><plus id="S4.9.p2.1.m1.1.1.3.2.2.3.cmml" xref="S4.9.p2.1.m1.1.1.3.2.2.3"></plus></apply><apply id="S4.9.p2.1.m1.1.1.3.2.3.cmml" xref="S4.9.p2.1.m1.1.1.3.2.3"><csymbol cd="ambiguous" id="S4.9.p2.1.m1.1.1.3.2.3.1.cmml" xref="S4.9.p2.1.m1.1.1.3.2.3">superscript</csymbol><ci id="S4.9.p2.1.m1.1.1.3.2.3.2.cmml" xref="S4.9.p2.1.m1.1.1.3.2.3.2">𝐴</ci><plus id="S4.9.p2.1.m1.1.1.3.2.3.3.cmml" xref="S4.9.p2.1.m1.1.1.3.2.3.3"></plus></apply></apply><ci id="S4.9.p2.1.m1.1.1.3.3.cmml" xref="S4.9.p2.1.m1.1.1.3.3">𝜔</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.9.p2.1.m1.1c">c:A^{+}\times A^{+}\to\omega</annotation><annotation encoding="application/x-llamapun" id="S4.9.p2.1.m1.1d">italic_c : italic_A start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT × italic_A start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT → italic_ω</annotation></semantics></math> witness that <math alttext="A^{+}" class="ltx_Math" display="inline" id="S4.9.p2.2.m2.1"><semantics id="S4.9.p2.2.m2.1a"><msup id="S4.9.p2.2.m2.1.1" xref="S4.9.p2.2.m2.1.1.cmml"><mi id="S4.9.p2.2.m2.1.1.2" xref="S4.9.p2.2.m2.1.1.2.cmml">A</mi><mo id="S4.9.p2.2.m2.1.1.3" xref="S4.9.p2.2.m2.1.1.3.cmml">+</mo></msup><annotation-xml encoding="MathML-Content" id="S4.9.p2.2.m2.1b"><apply id="S4.9.p2.2.m2.1.1.cmml" xref="S4.9.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S4.9.p2.2.m2.1.1.1.cmml" xref="S4.9.p2.2.m2.1.1">superscript</csymbol><ci id="S4.9.p2.2.m2.1.1.2.cmml" xref="S4.9.p2.2.m2.1.1.2">𝐴</ci><plus id="S4.9.p2.2.m2.1.1.3.cmml" xref="S4.9.p2.2.m2.1.1.3"></plus></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.9.p2.2.m2.1c">A^{+}</annotation><annotation encoding="application/x-llamapun" id="S4.9.p2.2.m2.1d">italic_A start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> is Countryman, i.e., for every <math alttext="n<\omega" class="ltx_Math" display="inline" id="S4.9.p2.3.m3.1"><semantics id="S4.9.p2.3.m3.1a"><mrow id="S4.9.p2.3.m3.1.1" xref="S4.9.p2.3.m3.1.1.cmml"><mi id="S4.9.p2.3.m3.1.1.2" xref="S4.9.p2.3.m3.1.1.2.cmml">n</mi><mo id="S4.9.p2.3.m3.1.1.1" xref="S4.9.p2.3.m3.1.1.1.cmml"><</mo><mi id="S4.9.p2.3.m3.1.1.3" xref="S4.9.p2.3.m3.1.1.3.cmml">ω</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.9.p2.3.m3.1b"><apply id="S4.9.p2.3.m3.1.1.cmml" xref="S4.9.p2.3.m3.1.1"><lt id="S4.9.p2.3.m3.1.1.1.cmml" xref="S4.9.p2.3.m3.1.1.1"></lt><ci id="S4.9.p2.3.m3.1.1.2.cmml" xref="S4.9.p2.3.m3.1.1.2">𝑛</ci><ci id="S4.9.p2.3.m3.1.1.3.cmml" xref="S4.9.p2.3.m3.1.1.3">𝜔</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.9.p2.3.m3.1c">n<\omega</annotation><annotation encoding="application/x-llamapun" id="S4.9.p2.3.m3.1d">italic_n < italic_ω</annotation></semantics></math>, <math alttext="f^{-1}(n)" class="ltx_Math" display="inline" id="S4.9.p2.4.m4.1"><semantics id="S4.9.p2.4.m4.1a"><mrow id="S4.9.p2.4.m4.1.2" xref="S4.9.p2.4.m4.1.2.cmml"><msup id="S4.9.p2.4.m4.1.2.2" xref="S4.9.p2.4.m4.1.2.2.cmml"><mi id="S4.9.p2.4.m4.1.2.2.2" xref="S4.9.p2.4.m4.1.2.2.2.cmml">f</mi><mrow id="S4.9.p2.4.m4.1.2.2.3" xref="S4.9.p2.4.m4.1.2.2.3.cmml"><mo id="S4.9.p2.4.m4.1.2.2.3a" xref="S4.9.p2.4.m4.1.2.2.3.cmml">−</mo><mn id="S4.9.p2.4.m4.1.2.2.3.2" xref="S4.9.p2.4.m4.1.2.2.3.2.cmml">1</mn></mrow></msup><mo id="S4.9.p2.4.m4.1.2.1" xref="S4.9.p2.4.m4.1.2.1.cmml">⁢</mo><mrow id="S4.9.p2.4.m4.1.2.3.2" xref="S4.9.p2.4.m4.1.2.cmml"><mo id="S4.9.p2.4.m4.1.2.3.2.1" stretchy="false" xref="S4.9.p2.4.m4.1.2.cmml">(</mo><mi id="S4.9.p2.4.m4.1.1" xref="S4.9.p2.4.m4.1.1.cmml">n</mi><mo id="S4.9.p2.4.m4.1.2.3.2.2" stretchy="false" xref="S4.9.p2.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.9.p2.4.m4.1b"><apply id="S4.9.p2.4.m4.1.2.cmml" xref="S4.9.p2.4.m4.1.2"><times id="S4.9.p2.4.m4.1.2.1.cmml" xref="S4.9.p2.4.m4.1.2.1"></times><apply id="S4.9.p2.4.m4.1.2.2.cmml" xref="S4.9.p2.4.m4.1.2.2"><csymbol cd="ambiguous" id="S4.9.p2.4.m4.1.2.2.1.cmml" xref="S4.9.p2.4.m4.1.2.2">superscript</csymbol><ci id="S4.9.p2.4.m4.1.2.2.2.cmml" xref="S4.9.p2.4.m4.1.2.2.2">𝑓</ci><apply id="S4.9.p2.4.m4.1.2.2.3.cmml" xref="S4.9.p2.4.m4.1.2.2.3"><minus id="S4.9.p2.4.m4.1.2.2.3.1.cmml" xref="S4.9.p2.4.m4.1.2.2.3"></minus><cn id="S4.9.p2.4.m4.1.2.2.3.2.cmml" type="integer" xref="S4.9.p2.4.m4.1.2.2.3.2">1</cn></apply></apply><ci id="S4.9.p2.4.m4.1.1.cmml" xref="S4.9.p2.4.m4.1.1">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.9.p2.4.m4.1c">f^{-1}(n)</annotation><annotation encoding="application/x-llamapun" id="S4.9.p2.4.m4.1d">italic_f start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_n )</annotation></semantics></math> is a chain in the product order of <math alttext="A^{+}\times A^{+}" class="ltx_Math" display="inline" id="S4.9.p2.5.m5.1"><semantics id="S4.9.p2.5.m5.1a"><mrow id="S4.9.p2.5.m5.1.1" xref="S4.9.p2.5.m5.1.1.cmml"><msup id="S4.9.p2.5.m5.1.1.2" xref="S4.9.p2.5.m5.1.1.2.cmml"><mi id="S4.9.p2.5.m5.1.1.2.2" xref="S4.9.p2.5.m5.1.1.2.2.cmml">A</mi><mo id="S4.9.p2.5.m5.1.1.2.3" xref="S4.9.p2.5.m5.1.1.2.3.cmml">+</mo></msup><mo id="S4.9.p2.5.m5.1.1.1" lspace="0.222em" rspace="0.222em" xref="S4.9.p2.5.m5.1.1.1.cmml">×</mo><msup id="S4.9.p2.5.m5.1.1.3" xref="S4.9.p2.5.m5.1.1.3.cmml"><mi id="S4.9.p2.5.m5.1.1.3.2" xref="S4.9.p2.5.m5.1.1.3.2.cmml">A</mi><mo id="S4.9.p2.5.m5.1.1.3.3" xref="S4.9.p2.5.m5.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.9.p2.5.m5.1b"><apply id="S4.9.p2.5.m5.1.1.cmml" xref="S4.9.p2.5.m5.1.1"><times id="S4.9.p2.5.m5.1.1.1.cmml" xref="S4.9.p2.5.m5.1.1.1"></times><apply id="S4.9.p2.5.m5.1.1.2.cmml" xref="S4.9.p2.5.m5.1.1.2"><csymbol cd="ambiguous" id="S4.9.p2.5.m5.1.1.2.1.cmml" xref="S4.9.p2.5.m5.1.1.2">superscript</csymbol><ci id="S4.9.p2.5.m5.1.1.2.2.cmml" xref="S4.9.p2.5.m5.1.1.2.2">𝐴</ci><plus id="S4.9.p2.5.m5.1.1.2.3.cmml" xref="S4.9.p2.5.m5.1.1.2.3"></plus></apply><apply id="S4.9.p2.5.m5.1.1.3.cmml" xref="S4.9.p2.5.m5.1.1.3"><csymbol cd="ambiguous" id="S4.9.p2.5.m5.1.1.3.1.cmml" xref="S4.9.p2.5.m5.1.1.3">superscript</csymbol><ci id="S4.9.p2.5.m5.1.1.3.2.cmml" xref="S4.9.p2.5.m5.1.1.3.2">𝐴</ci><plus id="S4.9.p2.5.m5.1.1.3.3.cmml" xref="S4.9.p2.5.m5.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.9.p2.5.m5.1c">A^{+}\times A^{+}</annotation><annotation encoding="application/x-llamapun" id="S4.9.p2.5.m5.1d">italic_A start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT × italic_A start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math>. We define <math alttext="f:A_{0}\to\omega" class="ltx_Math" display="inline" id="S4.9.p2.6.m6.1"><semantics id="S4.9.p2.6.m6.1a"><mrow id="S4.9.p2.6.m6.1.1" xref="S4.9.p2.6.m6.1.1.cmml"><mi id="S4.9.p2.6.m6.1.1.2" xref="S4.9.p2.6.m6.1.1.2.cmml">f</mi><mo id="S4.9.p2.6.m6.1.1.1" lspace="0.278em" rspace="0.278em" xref="S4.9.p2.6.m6.1.1.1.cmml">:</mo><mrow id="S4.9.p2.6.m6.1.1.3" xref="S4.9.p2.6.m6.1.1.3.cmml"><msub id="S4.9.p2.6.m6.1.1.3.2" xref="S4.9.p2.6.m6.1.1.3.2.cmml"><mi id="S4.9.p2.6.m6.1.1.3.2.2" xref="S4.9.p2.6.m6.1.1.3.2.2.cmml">A</mi><mn id="S4.9.p2.6.m6.1.1.3.2.3" xref="S4.9.p2.6.m6.1.1.3.2.3.cmml">0</mn></msub><mo id="S4.9.p2.6.m6.1.1.3.1" stretchy="false" xref="S4.9.p2.6.m6.1.1.3.1.cmml">→</mo><mi id="S4.9.p2.6.m6.1.1.3.3" xref="S4.9.p2.6.m6.1.1.3.3.cmml">ω</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.9.p2.6.m6.1b"><apply id="S4.9.p2.6.m6.1.1.cmml" xref="S4.9.p2.6.m6.1.1"><ci id="S4.9.p2.6.m6.1.1.1.cmml" xref="S4.9.p2.6.m6.1.1.1">:</ci><ci id="S4.9.p2.6.m6.1.1.2.cmml" xref="S4.9.p2.6.m6.1.1.2">𝑓</ci><apply id="S4.9.p2.6.m6.1.1.3.cmml" xref="S4.9.p2.6.m6.1.1.3"><ci id="S4.9.p2.6.m6.1.1.3.1.cmml" xref="S4.9.p2.6.m6.1.1.3.1">→</ci><apply id="S4.9.p2.6.m6.1.1.3.2.cmml" xref="S4.9.p2.6.m6.1.1.3.2"><csymbol cd="ambiguous" id="S4.9.p2.6.m6.1.1.3.2.1.cmml" xref="S4.9.p2.6.m6.1.1.3.2">subscript</csymbol><ci id="S4.9.p2.6.m6.1.1.3.2.2.cmml" xref="S4.9.p2.6.m6.1.1.3.2.2">𝐴</ci><cn id="S4.9.p2.6.m6.1.1.3.2.3.cmml" type="integer" xref="S4.9.p2.6.m6.1.1.3.2.3">0</cn></apply><ci id="S4.9.p2.6.m6.1.1.3.3.cmml" xref="S4.9.p2.6.m6.1.1.3.3">𝜔</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.9.p2.6.m6.1c">f:A_{0}\to\omega</annotation><annotation encoding="application/x-llamapun" id="S4.9.p2.6.m6.1d">italic_f : italic_A start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT → italic_ω</annotation></semantics></math> by letting <math alttext="f(t):=c(t^{\frown}\langle\omega\rangle,t)" class="ltx_Math" display="inline" id="S4.9.p2.7.m7.4"><semantics id="S4.9.p2.7.m7.4a"><mrow id="S4.9.p2.7.m7.4.4" xref="S4.9.p2.7.m7.4.4.cmml"><mrow id="S4.9.p2.7.m7.4.4.3" xref="S4.9.p2.7.m7.4.4.3.cmml"><mi id="S4.9.p2.7.m7.4.4.3.2" xref="S4.9.p2.7.m7.4.4.3.2.cmml">f</mi><mo id="S4.9.p2.7.m7.4.4.3.1" xref="S4.9.p2.7.m7.4.4.3.1.cmml">⁢</mo><mrow id="S4.9.p2.7.m7.4.4.3.3.2" xref="S4.9.p2.7.m7.4.4.3.cmml"><mo id="S4.9.p2.7.m7.4.4.3.3.2.1" stretchy="false" xref="S4.9.p2.7.m7.4.4.3.cmml">(</mo><mi id="S4.9.p2.7.m7.1.1" xref="S4.9.p2.7.m7.1.1.cmml">t</mi><mo id="S4.9.p2.7.m7.4.4.3.3.2.2" rspace="0.278em" stretchy="false" xref="S4.9.p2.7.m7.4.4.3.cmml">)</mo></mrow></mrow><mo id="S4.9.p2.7.m7.4.4.2" rspace="0.278em" xref="S4.9.p2.7.m7.4.4.2.cmml">:=</mo><mrow id="S4.9.p2.7.m7.4.4.1" xref="S4.9.p2.7.m7.4.4.1.cmml"><mi id="S4.9.p2.7.m7.4.4.1.3" xref="S4.9.p2.7.m7.4.4.1.3.cmml">c</mi><mo id="S4.9.p2.7.m7.4.4.1.2" xref="S4.9.p2.7.m7.4.4.1.2.cmml">⁢</mo><mrow id="S4.9.p2.7.m7.4.4.1.1.1" xref="S4.9.p2.7.m7.4.4.1.1.2.cmml"><mo id="S4.9.p2.7.m7.4.4.1.1.1.2" stretchy="false" xref="S4.9.p2.7.m7.4.4.1.1.2.cmml">(</mo><mrow id="S4.9.p2.7.m7.4.4.1.1.1.1" xref="S4.9.p2.7.m7.4.4.1.1.1.1.cmml"><msup id="S4.9.p2.7.m7.4.4.1.1.1.1.2" xref="S4.9.p2.7.m7.4.4.1.1.1.1.2.cmml"><mi id="S4.9.p2.7.m7.4.4.1.1.1.1.2.2" xref="S4.9.p2.7.m7.4.4.1.1.1.1.2.2.cmml">t</mi><mo id="S4.9.p2.7.m7.4.4.1.1.1.1.2.3" xref="S4.9.p2.7.m7.4.4.1.1.1.1.2.3.cmml">⌢</mo></msup><mo id="S4.9.p2.7.m7.4.4.1.1.1.1.1" xref="S4.9.p2.7.m7.4.4.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.9.p2.7.m7.4.4.1.1.1.1.3.2" xref="S4.9.p2.7.m7.4.4.1.1.1.1.3.1.cmml"><mo id="S4.9.p2.7.m7.4.4.1.1.1.1.3.2.1" stretchy="false" xref="S4.9.p2.7.m7.4.4.1.1.1.1.3.1.1.cmml">⟨</mo><mi id="S4.9.p2.7.m7.2.2" xref="S4.9.p2.7.m7.2.2.cmml">ω</mi><mo id="S4.9.p2.7.m7.4.4.1.1.1.1.3.2.2" stretchy="false" xref="S4.9.p2.7.m7.4.4.1.1.1.1.3.1.1.cmml">⟩</mo></mrow></mrow><mo id="S4.9.p2.7.m7.4.4.1.1.1.3" xref="S4.9.p2.7.m7.4.4.1.1.2.cmml">,</mo><mi id="S4.9.p2.7.m7.3.3" xref="S4.9.p2.7.m7.3.3.cmml">t</mi><mo id="S4.9.p2.7.m7.4.4.1.1.1.4" stretchy="false" xref="S4.9.p2.7.m7.4.4.1.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.9.p2.7.m7.4b"><apply id="S4.9.p2.7.m7.4.4.cmml" xref="S4.9.p2.7.m7.4.4"><csymbol cd="latexml" id="S4.9.p2.7.m7.4.4.2.cmml" xref="S4.9.p2.7.m7.4.4.2">assign</csymbol><apply id="S4.9.p2.7.m7.4.4.3.cmml" xref="S4.9.p2.7.m7.4.4.3"><times id="S4.9.p2.7.m7.4.4.3.1.cmml" xref="S4.9.p2.7.m7.4.4.3.1"></times><ci id="S4.9.p2.7.m7.4.4.3.2.cmml" xref="S4.9.p2.7.m7.4.4.3.2">𝑓</ci><ci id="S4.9.p2.7.m7.1.1.cmml" xref="S4.9.p2.7.m7.1.1">𝑡</ci></apply><apply id="S4.9.p2.7.m7.4.4.1.cmml" xref="S4.9.p2.7.m7.4.4.1"><times id="S4.9.p2.7.m7.4.4.1.2.cmml" xref="S4.9.p2.7.m7.4.4.1.2"></times><ci id="S4.9.p2.7.m7.4.4.1.3.cmml" xref="S4.9.p2.7.m7.4.4.1.3">𝑐</ci><interval closure="open" id="S4.9.p2.7.m7.4.4.1.1.2.cmml" xref="S4.9.p2.7.m7.4.4.1.1.1"><apply id="S4.9.p2.7.m7.4.4.1.1.1.1.cmml" xref="S4.9.p2.7.m7.4.4.1.1.1.1"><times id="S4.9.p2.7.m7.4.4.1.1.1.1.1.cmml" xref="S4.9.p2.7.m7.4.4.1.1.1.1.1"></times><apply id="S4.9.p2.7.m7.4.4.1.1.1.1.2.cmml" xref="S4.9.p2.7.m7.4.4.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.9.p2.7.m7.4.4.1.1.1.1.2.1.cmml" xref="S4.9.p2.7.m7.4.4.1.1.1.1.2">superscript</csymbol><ci id="S4.9.p2.7.m7.4.4.1.1.1.1.2.2.cmml" xref="S4.9.p2.7.m7.4.4.1.1.1.1.2.2">𝑡</ci><ci id="S4.9.p2.7.m7.4.4.1.1.1.1.2.3.cmml" xref="S4.9.p2.7.m7.4.4.1.1.1.1.2.3">⌢</ci></apply><apply id="S4.9.p2.7.m7.4.4.1.1.1.1.3.1.cmml" xref="S4.9.p2.7.m7.4.4.1.1.1.1.3.2"><csymbol cd="latexml" id="S4.9.p2.7.m7.4.4.1.1.1.1.3.1.1.cmml" xref="S4.9.p2.7.m7.4.4.1.1.1.1.3.2.1">delimited-⟨⟩</csymbol><ci id="S4.9.p2.7.m7.2.2.cmml" xref="S4.9.p2.7.m7.2.2">𝜔</ci></apply></apply><ci id="S4.9.p2.7.m7.3.3.cmml" xref="S4.9.p2.7.m7.3.3">𝑡</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.9.p2.7.m7.4c">f(t):=c(t^{\frown}\langle\omega\rangle,t)</annotation><annotation encoding="application/x-llamapun" id="S4.9.p2.7.m7.4d">italic_f ( italic_t ) := italic_c ( italic_t start_POSTSUPERSCRIPT ⌢ end_POSTSUPERSCRIPT ⟨ italic_ω ⟩ , italic_t )</annotation></semantics></math>, and prove that for <math alttext="t\sqsubset s" class="ltx_Math" display="inline" id="S4.9.p2.8.m8.1"><semantics id="S4.9.p2.8.m8.1a"><mrow id="S4.9.p2.8.m8.1.1" xref="S4.9.p2.8.m8.1.1.cmml"><mi id="S4.9.p2.8.m8.1.1.2" xref="S4.9.p2.8.m8.1.1.2.cmml">t</mi><mo id="S4.9.p2.8.m8.1.1.1" xref="S4.9.p2.8.m8.1.1.1.cmml">⊏</mo><mi id="S4.9.p2.8.m8.1.1.3" xref="S4.9.p2.8.m8.1.1.3.cmml">s</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.9.p2.8.m8.1b"><apply id="S4.9.p2.8.m8.1.1.cmml" xref="S4.9.p2.8.m8.1.1"><csymbol cd="latexml" id="S4.9.p2.8.m8.1.1.1.cmml" xref="S4.9.p2.8.m8.1.1.1">square-image-of</csymbol><ci id="S4.9.p2.8.m8.1.1.2.cmml" xref="S4.9.p2.8.m8.1.1.2">𝑡</ci><ci id="S4.9.p2.8.m8.1.1.3.cmml" xref="S4.9.p2.8.m8.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.9.p2.8.m8.1c">t\sqsubset s</annotation><annotation encoding="application/x-llamapun" id="S4.9.p2.8.m8.1d">italic_t ⊏ italic_s</annotation></semantics></math>, <math alttext="f(t)\neq f(s)" class="ltx_Math" display="inline" id="S4.9.p2.9.m9.2"><semantics id="S4.9.p2.9.m9.2a"><mrow id="S4.9.p2.9.m9.2.3" xref="S4.9.p2.9.m9.2.3.cmml"><mrow id="S4.9.p2.9.m9.2.3.2" xref="S4.9.p2.9.m9.2.3.2.cmml"><mi id="S4.9.p2.9.m9.2.3.2.2" xref="S4.9.p2.9.m9.2.3.2.2.cmml">f</mi><mo id="S4.9.p2.9.m9.2.3.2.1" xref="S4.9.p2.9.m9.2.3.2.1.cmml">⁢</mo><mrow id="S4.9.p2.9.m9.2.3.2.3.2" xref="S4.9.p2.9.m9.2.3.2.cmml"><mo id="S4.9.p2.9.m9.2.3.2.3.2.1" stretchy="false" xref="S4.9.p2.9.m9.2.3.2.cmml">(</mo><mi id="S4.9.p2.9.m9.1.1" xref="S4.9.p2.9.m9.1.1.cmml">t</mi><mo id="S4.9.p2.9.m9.2.3.2.3.2.2" stretchy="false" xref="S4.9.p2.9.m9.2.3.2.cmml">)</mo></mrow></mrow><mo id="S4.9.p2.9.m9.2.3.1" xref="S4.9.p2.9.m9.2.3.1.cmml">≠</mo><mrow id="S4.9.p2.9.m9.2.3.3" xref="S4.9.p2.9.m9.2.3.3.cmml"><mi id="S4.9.p2.9.m9.2.3.3.2" xref="S4.9.p2.9.m9.2.3.3.2.cmml">f</mi><mo id="S4.9.p2.9.m9.2.3.3.1" xref="S4.9.p2.9.m9.2.3.3.1.cmml">⁢</mo><mrow id="S4.9.p2.9.m9.2.3.3.3.2" xref="S4.9.p2.9.m9.2.3.3.cmml"><mo id="S4.9.p2.9.m9.2.3.3.3.2.1" stretchy="false" xref="S4.9.p2.9.m9.2.3.3.cmml">(</mo><mi id="S4.9.p2.9.m9.2.2" xref="S4.9.p2.9.m9.2.2.cmml">s</mi><mo id="S4.9.p2.9.m9.2.3.3.3.2.2" stretchy="false" xref="S4.9.p2.9.m9.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.9.p2.9.m9.2b"><apply id="S4.9.p2.9.m9.2.3.cmml" xref="S4.9.p2.9.m9.2.3"><neq id="S4.9.p2.9.m9.2.3.1.cmml" xref="S4.9.p2.9.m9.2.3.1"></neq><apply id="S4.9.p2.9.m9.2.3.2.cmml" xref="S4.9.p2.9.m9.2.3.2"><times id="S4.9.p2.9.m9.2.3.2.1.cmml" xref="S4.9.p2.9.m9.2.3.2.1"></times><ci id="S4.9.p2.9.m9.2.3.2.2.cmml" xref="S4.9.p2.9.m9.2.3.2.2">𝑓</ci><ci id="S4.9.p2.9.m9.1.1.cmml" xref="S4.9.p2.9.m9.1.1">𝑡</ci></apply><apply id="S4.9.p2.9.m9.2.3.3.cmml" xref="S4.9.p2.9.m9.2.3.3"><times id="S4.9.p2.9.m9.2.3.3.1.cmml" xref="S4.9.p2.9.m9.2.3.3.1"></times><ci id="S4.9.p2.9.m9.2.3.3.2.cmml" xref="S4.9.p2.9.m9.2.3.3.2">𝑓</ci><ci id="S4.9.p2.9.m9.2.2.cmml" xref="S4.9.p2.9.m9.2.2">𝑠</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.9.p2.9.m9.2c">f(t)\neq f(s)</annotation><annotation encoding="application/x-llamapun" id="S4.9.p2.9.m9.2d">italic_f ( italic_t ) ≠ italic_f ( italic_s )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.10.p3"> <p class="ltx_p" id="S4.10.p3.6">Assume <math alttext="t\sqsubset s" class="ltx_Math" display="inline" id="S4.10.p3.1.m1.1"><semantics id="S4.10.p3.1.m1.1a"><mrow id="S4.10.p3.1.m1.1.1" xref="S4.10.p3.1.m1.1.1.cmml"><mi id="S4.10.p3.1.m1.1.1.2" xref="S4.10.p3.1.m1.1.1.2.cmml">t</mi><mo id="S4.10.p3.1.m1.1.1.1" xref="S4.10.p3.1.m1.1.1.1.cmml">⊏</mo><mi id="S4.10.p3.1.m1.1.1.3" xref="S4.10.p3.1.m1.1.1.3.cmml">s</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.10.p3.1.m1.1b"><apply id="S4.10.p3.1.m1.1.1.cmml" xref="S4.10.p3.1.m1.1.1"><csymbol cd="latexml" id="S4.10.p3.1.m1.1.1.1.cmml" xref="S4.10.p3.1.m1.1.1.1">square-image-of</csymbol><ci id="S4.10.p3.1.m1.1.1.2.cmml" xref="S4.10.p3.1.m1.1.1.2">𝑡</ci><ci id="S4.10.p3.1.m1.1.1.3.cmml" xref="S4.10.p3.1.m1.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.10.p3.1.m1.1c">t\sqsubset s</annotation><annotation encoding="application/x-llamapun" id="S4.10.p3.1.m1.1d">italic_t ⊏ italic_s</annotation></semantics></math> and <math alttext="f(t)=f(s)" class="ltx_Math" display="inline" id="S4.10.p3.2.m2.2"><semantics id="S4.10.p3.2.m2.2a"><mrow id="S4.10.p3.2.m2.2.3" xref="S4.10.p3.2.m2.2.3.cmml"><mrow id="S4.10.p3.2.m2.2.3.2" xref="S4.10.p3.2.m2.2.3.2.cmml"><mi id="S4.10.p3.2.m2.2.3.2.2" xref="S4.10.p3.2.m2.2.3.2.2.cmml">f</mi><mo id="S4.10.p3.2.m2.2.3.2.1" xref="S4.10.p3.2.m2.2.3.2.1.cmml">⁢</mo><mrow id="S4.10.p3.2.m2.2.3.2.3.2" xref="S4.10.p3.2.m2.2.3.2.cmml"><mo id="S4.10.p3.2.m2.2.3.2.3.2.1" stretchy="false" xref="S4.10.p3.2.m2.2.3.2.cmml">(</mo><mi id="S4.10.p3.2.m2.1.1" xref="S4.10.p3.2.m2.1.1.cmml">t</mi><mo id="S4.10.p3.2.m2.2.3.2.3.2.2" stretchy="false" xref="S4.10.p3.2.m2.2.3.2.cmml">)</mo></mrow></mrow><mo id="S4.10.p3.2.m2.2.3.1" xref="S4.10.p3.2.m2.2.3.1.cmml">=</mo><mrow id="S4.10.p3.2.m2.2.3.3" xref="S4.10.p3.2.m2.2.3.3.cmml"><mi id="S4.10.p3.2.m2.2.3.3.2" xref="S4.10.p3.2.m2.2.3.3.2.cmml">f</mi><mo id="S4.10.p3.2.m2.2.3.3.1" xref="S4.10.p3.2.m2.2.3.3.1.cmml">⁢</mo><mrow id="S4.10.p3.2.m2.2.3.3.3.2" xref="S4.10.p3.2.m2.2.3.3.cmml"><mo id="S4.10.p3.2.m2.2.3.3.3.2.1" stretchy="false" xref="S4.10.p3.2.m2.2.3.3.cmml">(</mo><mi id="S4.10.p3.2.m2.2.2" xref="S4.10.p3.2.m2.2.2.cmml">s</mi><mo id="S4.10.p3.2.m2.2.3.3.3.2.2" stretchy="false" xref="S4.10.p3.2.m2.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.10.p3.2.m2.2b"><apply id="S4.10.p3.2.m2.2.3.cmml" xref="S4.10.p3.2.m2.2.3"><eq id="S4.10.p3.2.m2.2.3.1.cmml" xref="S4.10.p3.2.m2.2.3.1"></eq><apply id="S4.10.p3.2.m2.2.3.2.cmml" xref="S4.10.p3.2.m2.2.3.2"><times id="S4.10.p3.2.m2.2.3.2.1.cmml" xref="S4.10.p3.2.m2.2.3.2.1"></times><ci id="S4.10.p3.2.m2.2.3.2.2.cmml" xref="S4.10.p3.2.m2.2.3.2.2">𝑓</ci><ci id="S4.10.p3.2.m2.1.1.cmml" xref="S4.10.p3.2.m2.1.1">𝑡</ci></apply><apply id="S4.10.p3.2.m2.2.3.3.cmml" xref="S4.10.p3.2.m2.2.3.3"><times id="S4.10.p3.2.m2.2.3.3.1.cmml" xref="S4.10.p3.2.m2.2.3.3.1"></times><ci id="S4.10.p3.2.m2.2.3.3.2.cmml" xref="S4.10.p3.2.m2.2.3.3.2">𝑓</ci><ci id="S4.10.p3.2.m2.2.2.cmml" xref="S4.10.p3.2.m2.2.2">𝑠</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.10.p3.2.m2.2c">f(t)=f(s)</annotation><annotation encoding="application/x-llamapun" id="S4.10.p3.2.m2.2d">italic_f ( italic_t ) = italic_f ( italic_s )</annotation></semantics></math>. From the first it follows that <math alttext="t^{\frown}\langle\omega\rangle>_{\mathrm{lex}}s^{\frown}\langle\omega\rangle" class="ltx_Math" display="inline" id="S4.10.p3.3.m3.2"><semantics id="S4.10.p3.3.m3.2a"><mrow id="S4.10.p3.3.m3.2.3" xref="S4.10.p3.3.m3.2.3.cmml"><mrow id="S4.10.p3.3.m3.2.3.2" xref="S4.10.p3.3.m3.2.3.2.cmml"><msup id="S4.10.p3.3.m3.2.3.2.2" xref="S4.10.p3.3.m3.2.3.2.2.cmml"><mi id="S4.10.p3.3.m3.2.3.2.2.2" xref="S4.10.p3.3.m3.2.3.2.2.2.cmml">t</mi><mo id="S4.10.p3.3.m3.2.3.2.2.3" xref="S4.10.p3.3.m3.2.3.2.2.3.cmml">⌢</mo></msup><mo id="S4.10.p3.3.m3.2.3.2.1" xref="S4.10.p3.3.m3.2.3.2.1.cmml">⁢</mo><mrow id="S4.10.p3.3.m3.2.3.2.3.2" xref="S4.10.p3.3.m3.2.3.2.3.1.cmml"><mo id="S4.10.p3.3.m3.2.3.2.3.2.1" stretchy="false" xref="S4.10.p3.3.m3.2.3.2.3.1.1.cmml">⟨</mo><mi id="S4.10.p3.3.m3.1.1" xref="S4.10.p3.3.m3.1.1.cmml">ω</mi><mo id="S4.10.p3.3.m3.2.3.2.3.2.2" stretchy="false" xref="S4.10.p3.3.m3.2.3.2.3.1.1.cmml">⟩</mo></mrow></mrow><msub id="S4.10.p3.3.m3.2.3.1" xref="S4.10.p3.3.m3.2.3.1.cmml"><mo id="S4.10.p3.3.m3.2.3.1.2" xref="S4.10.p3.3.m3.2.3.1.2.cmml">></mo><mi id="S4.10.p3.3.m3.2.3.1.3" xref="S4.10.p3.3.m3.2.3.1.3.cmml">lex</mi></msub><mrow id="S4.10.p3.3.m3.2.3.3" xref="S4.10.p3.3.m3.2.3.3.cmml"><msup id="S4.10.p3.3.m3.2.3.3.2" xref="S4.10.p3.3.m3.2.3.3.2.cmml"><mi id="S4.10.p3.3.m3.2.3.3.2.2" xref="S4.10.p3.3.m3.2.3.3.2.2.cmml">s</mi><mo id="S4.10.p3.3.m3.2.3.3.2.3" xref="S4.10.p3.3.m3.2.3.3.2.3.cmml">⌢</mo></msup><mo id="S4.10.p3.3.m3.2.3.3.1" xref="S4.10.p3.3.m3.2.3.3.1.cmml">⁢</mo><mrow id="S4.10.p3.3.m3.2.3.3.3.2" xref="S4.10.p3.3.m3.2.3.3.3.1.cmml"><mo id="S4.10.p3.3.m3.2.3.3.3.2.1" stretchy="false" xref="S4.10.p3.3.m3.2.3.3.3.1.1.cmml">⟨</mo><mi id="S4.10.p3.3.m3.2.2" xref="S4.10.p3.3.m3.2.2.cmml">ω</mi><mo id="S4.10.p3.3.m3.2.3.3.3.2.2" stretchy="false" xref="S4.10.p3.3.m3.2.3.3.3.1.1.cmml">⟩</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.10.p3.3.m3.2b"><apply id="S4.10.p3.3.m3.2.3.cmml" xref="S4.10.p3.3.m3.2.3"><apply id="S4.10.p3.3.m3.2.3.1.cmml" xref="S4.10.p3.3.m3.2.3.1"><csymbol cd="ambiguous" id="S4.10.p3.3.m3.2.3.1.1.cmml" xref="S4.10.p3.3.m3.2.3.1">subscript</csymbol><gt id="S4.10.p3.3.m3.2.3.1.2.cmml" xref="S4.10.p3.3.m3.2.3.1.2"></gt><ci id="S4.10.p3.3.m3.2.3.1.3.cmml" xref="S4.10.p3.3.m3.2.3.1.3">lex</ci></apply><apply id="S4.10.p3.3.m3.2.3.2.cmml" xref="S4.10.p3.3.m3.2.3.2"><times id="S4.10.p3.3.m3.2.3.2.1.cmml" xref="S4.10.p3.3.m3.2.3.2.1"></times><apply id="S4.10.p3.3.m3.2.3.2.2.cmml" xref="S4.10.p3.3.m3.2.3.2.2"><csymbol cd="ambiguous" id="S4.10.p3.3.m3.2.3.2.2.1.cmml" xref="S4.10.p3.3.m3.2.3.2.2">superscript</csymbol><ci id="S4.10.p3.3.m3.2.3.2.2.2.cmml" xref="S4.10.p3.3.m3.2.3.2.2.2">𝑡</ci><ci id="S4.10.p3.3.m3.2.3.2.2.3.cmml" xref="S4.10.p3.3.m3.2.3.2.2.3">⌢</ci></apply><apply id="S4.10.p3.3.m3.2.3.2.3.1.cmml" xref="S4.10.p3.3.m3.2.3.2.3.2"><csymbol cd="latexml" id="S4.10.p3.3.m3.2.3.2.3.1.1.cmml" xref="S4.10.p3.3.m3.2.3.2.3.2.1">delimited-⟨⟩</csymbol><ci id="S4.10.p3.3.m3.1.1.cmml" xref="S4.10.p3.3.m3.1.1">𝜔</ci></apply></apply><apply id="S4.10.p3.3.m3.2.3.3.cmml" xref="S4.10.p3.3.m3.2.3.3"><times id="S4.10.p3.3.m3.2.3.3.1.cmml" xref="S4.10.p3.3.m3.2.3.3.1"></times><apply id="S4.10.p3.3.m3.2.3.3.2.cmml" xref="S4.10.p3.3.m3.2.3.3.2"><csymbol cd="ambiguous" id="S4.10.p3.3.m3.2.3.3.2.1.cmml" xref="S4.10.p3.3.m3.2.3.3.2">superscript</csymbol><ci id="S4.10.p3.3.m3.2.3.3.2.2.cmml" xref="S4.10.p3.3.m3.2.3.3.2.2">𝑠</ci><ci id="S4.10.p3.3.m3.2.3.3.2.3.cmml" xref="S4.10.p3.3.m3.2.3.3.2.3">⌢</ci></apply><apply id="S4.10.p3.3.m3.2.3.3.3.1.cmml" xref="S4.10.p3.3.m3.2.3.3.3.2"><csymbol cd="latexml" id="S4.10.p3.3.m3.2.3.3.3.1.1.cmml" xref="S4.10.p3.3.m3.2.3.3.3.2.1">delimited-⟨⟩</csymbol><ci id="S4.10.p3.3.m3.2.2.cmml" xref="S4.10.p3.3.m3.2.2">𝜔</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.10.p3.3.m3.2c">t^{\frown}\langle\omega\rangle>_{\mathrm{lex}}s^{\frown}\langle\omega\rangle</annotation><annotation encoding="application/x-llamapun" id="S4.10.p3.3.m3.2d">italic_t start_POSTSUPERSCRIPT ⌢ end_POSTSUPERSCRIPT ⟨ italic_ω ⟩ > start_POSTSUBSCRIPT roman_lex end_POSTSUBSCRIPT italic_s start_POSTSUPERSCRIPT ⌢ end_POSTSUPERSCRIPT ⟨ italic_ω ⟩</annotation></semantics></math>, and from the second that <math alttext="c(t^{\frown}\langle\omega\rangle,t)=c(s^{\frown}\langle\omega\rangle,s)" class="ltx_Math" display="inline" id="S4.10.p3.4.m4.6"><semantics id="S4.10.p3.4.m4.6a"><mrow id="S4.10.p3.4.m4.6.6" xref="S4.10.p3.4.m4.6.6.cmml"><mrow id="S4.10.p3.4.m4.5.5.1" xref="S4.10.p3.4.m4.5.5.1.cmml"><mi id="S4.10.p3.4.m4.5.5.1.3" xref="S4.10.p3.4.m4.5.5.1.3.cmml">c</mi><mo id="S4.10.p3.4.m4.5.5.1.2" xref="S4.10.p3.4.m4.5.5.1.2.cmml">⁢</mo><mrow id="S4.10.p3.4.m4.5.5.1.1.1" xref="S4.10.p3.4.m4.5.5.1.1.2.cmml"><mo id="S4.10.p3.4.m4.5.5.1.1.1.2" stretchy="false" xref="S4.10.p3.4.m4.5.5.1.1.2.cmml">(</mo><mrow id="S4.10.p3.4.m4.5.5.1.1.1.1" xref="S4.10.p3.4.m4.5.5.1.1.1.1.cmml"><msup id="S4.10.p3.4.m4.5.5.1.1.1.1.2" xref="S4.10.p3.4.m4.5.5.1.1.1.1.2.cmml"><mi id="S4.10.p3.4.m4.5.5.1.1.1.1.2.2" xref="S4.10.p3.4.m4.5.5.1.1.1.1.2.2.cmml">t</mi><mo id="S4.10.p3.4.m4.5.5.1.1.1.1.2.3" xref="S4.10.p3.4.m4.5.5.1.1.1.1.2.3.cmml">⌢</mo></msup><mo id="S4.10.p3.4.m4.5.5.1.1.1.1.1" xref="S4.10.p3.4.m4.5.5.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.10.p3.4.m4.5.5.1.1.1.1.3.2" xref="S4.10.p3.4.m4.5.5.1.1.1.1.3.1.cmml"><mo id="S4.10.p3.4.m4.5.5.1.1.1.1.3.2.1" stretchy="false" xref="S4.10.p3.4.m4.5.5.1.1.1.1.3.1.1.cmml">⟨</mo><mi id="S4.10.p3.4.m4.1.1" xref="S4.10.p3.4.m4.1.1.cmml">ω</mi><mo id="S4.10.p3.4.m4.5.5.1.1.1.1.3.2.2" stretchy="false" xref="S4.10.p3.4.m4.5.5.1.1.1.1.3.1.1.cmml">⟩</mo></mrow></mrow><mo id="S4.10.p3.4.m4.5.5.1.1.1.3" xref="S4.10.p3.4.m4.5.5.1.1.2.cmml">,</mo><mi id="S4.10.p3.4.m4.2.2" xref="S4.10.p3.4.m4.2.2.cmml">t</mi><mo id="S4.10.p3.4.m4.5.5.1.1.1.4" stretchy="false" xref="S4.10.p3.4.m4.5.5.1.1.2.cmml">)</mo></mrow></mrow><mo id="S4.10.p3.4.m4.6.6.3" xref="S4.10.p3.4.m4.6.6.3.cmml">=</mo><mrow id="S4.10.p3.4.m4.6.6.2" xref="S4.10.p3.4.m4.6.6.2.cmml"><mi id="S4.10.p3.4.m4.6.6.2.3" xref="S4.10.p3.4.m4.6.6.2.3.cmml">c</mi><mo id="S4.10.p3.4.m4.6.6.2.2" xref="S4.10.p3.4.m4.6.6.2.2.cmml">⁢</mo><mrow id="S4.10.p3.4.m4.6.6.2.1.1" xref="S4.10.p3.4.m4.6.6.2.1.2.cmml"><mo id="S4.10.p3.4.m4.6.6.2.1.1.2" stretchy="false" xref="S4.10.p3.4.m4.6.6.2.1.2.cmml">(</mo><mrow id="S4.10.p3.4.m4.6.6.2.1.1.1" xref="S4.10.p3.4.m4.6.6.2.1.1.1.cmml"><msup id="S4.10.p3.4.m4.6.6.2.1.1.1.2" xref="S4.10.p3.4.m4.6.6.2.1.1.1.2.cmml"><mi id="S4.10.p3.4.m4.6.6.2.1.1.1.2.2" xref="S4.10.p3.4.m4.6.6.2.1.1.1.2.2.cmml">s</mi><mo id="S4.10.p3.4.m4.6.6.2.1.1.1.2.3" xref="S4.10.p3.4.m4.6.6.2.1.1.1.2.3.cmml">⌢</mo></msup><mo id="S4.10.p3.4.m4.6.6.2.1.1.1.1" xref="S4.10.p3.4.m4.6.6.2.1.1.1.1.cmml">⁢</mo><mrow id="S4.10.p3.4.m4.6.6.2.1.1.1.3.2" xref="S4.10.p3.4.m4.6.6.2.1.1.1.3.1.cmml"><mo id="S4.10.p3.4.m4.6.6.2.1.1.1.3.2.1" stretchy="false" xref="S4.10.p3.4.m4.6.6.2.1.1.1.3.1.1.cmml">⟨</mo><mi id="S4.10.p3.4.m4.3.3" xref="S4.10.p3.4.m4.3.3.cmml">ω</mi><mo id="S4.10.p3.4.m4.6.6.2.1.1.1.3.2.2" stretchy="false" xref="S4.10.p3.4.m4.6.6.2.1.1.1.3.1.1.cmml">⟩</mo></mrow></mrow><mo id="S4.10.p3.4.m4.6.6.2.1.1.3" xref="S4.10.p3.4.m4.6.6.2.1.2.cmml">,</mo><mi id="S4.10.p3.4.m4.4.4" xref="S4.10.p3.4.m4.4.4.cmml">s</mi><mo id="S4.10.p3.4.m4.6.6.2.1.1.4" stretchy="false" xref="S4.10.p3.4.m4.6.6.2.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.10.p3.4.m4.6b"><apply id="S4.10.p3.4.m4.6.6.cmml" xref="S4.10.p3.4.m4.6.6"><eq id="S4.10.p3.4.m4.6.6.3.cmml" xref="S4.10.p3.4.m4.6.6.3"></eq><apply id="S4.10.p3.4.m4.5.5.1.cmml" xref="S4.10.p3.4.m4.5.5.1"><times id="S4.10.p3.4.m4.5.5.1.2.cmml" xref="S4.10.p3.4.m4.5.5.1.2"></times><ci id="S4.10.p3.4.m4.5.5.1.3.cmml" xref="S4.10.p3.4.m4.5.5.1.3">𝑐</ci><interval closure="open" id="S4.10.p3.4.m4.5.5.1.1.2.cmml" xref="S4.10.p3.4.m4.5.5.1.1.1"><apply id="S4.10.p3.4.m4.5.5.1.1.1.1.cmml" xref="S4.10.p3.4.m4.5.5.1.1.1.1"><times id="S4.10.p3.4.m4.5.5.1.1.1.1.1.cmml" xref="S4.10.p3.4.m4.5.5.1.1.1.1.1"></times><apply id="S4.10.p3.4.m4.5.5.1.1.1.1.2.cmml" xref="S4.10.p3.4.m4.5.5.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.10.p3.4.m4.5.5.1.1.1.1.2.1.cmml" xref="S4.10.p3.4.m4.5.5.1.1.1.1.2">superscript</csymbol><ci id="S4.10.p3.4.m4.5.5.1.1.1.1.2.2.cmml" xref="S4.10.p3.4.m4.5.5.1.1.1.1.2.2">𝑡</ci><ci id="S4.10.p3.4.m4.5.5.1.1.1.1.2.3.cmml" xref="S4.10.p3.4.m4.5.5.1.1.1.1.2.3">⌢</ci></apply><apply id="S4.10.p3.4.m4.5.5.1.1.1.1.3.1.cmml" xref="S4.10.p3.4.m4.5.5.1.1.1.1.3.2"><csymbol cd="latexml" id="S4.10.p3.4.m4.5.5.1.1.1.1.3.1.1.cmml" xref="S4.10.p3.4.m4.5.5.1.1.1.1.3.2.1">delimited-⟨⟩</csymbol><ci id="S4.10.p3.4.m4.1.1.cmml" xref="S4.10.p3.4.m4.1.1">𝜔</ci></apply></apply><ci id="S4.10.p3.4.m4.2.2.cmml" xref="S4.10.p3.4.m4.2.2">𝑡</ci></interval></apply><apply id="S4.10.p3.4.m4.6.6.2.cmml" xref="S4.10.p3.4.m4.6.6.2"><times id="S4.10.p3.4.m4.6.6.2.2.cmml" xref="S4.10.p3.4.m4.6.6.2.2"></times><ci id="S4.10.p3.4.m4.6.6.2.3.cmml" xref="S4.10.p3.4.m4.6.6.2.3">𝑐</ci><interval closure="open" id="S4.10.p3.4.m4.6.6.2.1.2.cmml" xref="S4.10.p3.4.m4.6.6.2.1.1"><apply id="S4.10.p3.4.m4.6.6.2.1.1.1.cmml" xref="S4.10.p3.4.m4.6.6.2.1.1.1"><times id="S4.10.p3.4.m4.6.6.2.1.1.1.1.cmml" xref="S4.10.p3.4.m4.6.6.2.1.1.1.1"></times><apply id="S4.10.p3.4.m4.6.6.2.1.1.1.2.cmml" xref="S4.10.p3.4.m4.6.6.2.1.1.1.2"><csymbol cd="ambiguous" id="S4.10.p3.4.m4.6.6.2.1.1.1.2.1.cmml" xref="S4.10.p3.4.m4.6.6.2.1.1.1.2">superscript</csymbol><ci id="S4.10.p3.4.m4.6.6.2.1.1.1.2.2.cmml" xref="S4.10.p3.4.m4.6.6.2.1.1.1.2.2">𝑠</ci><ci id="S4.10.p3.4.m4.6.6.2.1.1.1.2.3.cmml" xref="S4.10.p3.4.m4.6.6.2.1.1.1.2.3">⌢</ci></apply><apply id="S4.10.p3.4.m4.6.6.2.1.1.1.3.1.cmml" xref="S4.10.p3.4.m4.6.6.2.1.1.1.3.2"><csymbol cd="latexml" id="S4.10.p3.4.m4.6.6.2.1.1.1.3.1.1.cmml" xref="S4.10.p3.4.m4.6.6.2.1.1.1.3.2.1">delimited-⟨⟩</csymbol><ci id="S4.10.p3.4.m4.3.3.cmml" xref="S4.10.p3.4.m4.3.3">𝜔</ci></apply></apply><ci id="S4.10.p3.4.m4.4.4.cmml" xref="S4.10.p3.4.m4.4.4">𝑠</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.10.p3.4.m4.6c">c(t^{\frown}\langle\omega\rangle,t)=c(s^{\frown}\langle\omega\rangle,s)</annotation><annotation encoding="application/x-llamapun" id="S4.10.p3.4.m4.6d">italic_c ( italic_t start_POSTSUPERSCRIPT ⌢ end_POSTSUPERSCRIPT ⟨ italic_ω ⟩ , italic_t ) = italic_c ( italic_s start_POSTSUPERSCRIPT ⌢ end_POSTSUPERSCRIPT ⟨ italic_ω ⟩ , italic_s )</annotation></semantics></math>. Summing up we get that <math alttext="t\geq_{\mathrm{lex}}s" class="ltx_Math" display="inline" id="S4.10.p3.5.m5.1"><semantics id="S4.10.p3.5.m5.1a"><mrow id="S4.10.p3.5.m5.1.1" xref="S4.10.p3.5.m5.1.1.cmml"><mi id="S4.10.p3.5.m5.1.1.2" xref="S4.10.p3.5.m5.1.1.2.cmml">t</mi><msub id="S4.10.p3.5.m5.1.1.1" xref="S4.10.p3.5.m5.1.1.1.cmml"><mo id="S4.10.p3.5.m5.1.1.1.2" xref="S4.10.p3.5.m5.1.1.1.2.cmml">≥</mo><mi id="S4.10.p3.5.m5.1.1.1.3" xref="S4.10.p3.5.m5.1.1.1.3.cmml">lex</mi></msub><mi id="S4.10.p3.5.m5.1.1.3" xref="S4.10.p3.5.m5.1.1.3.cmml">s</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.10.p3.5.m5.1b"><apply id="S4.10.p3.5.m5.1.1.cmml" xref="S4.10.p3.5.m5.1.1"><apply id="S4.10.p3.5.m5.1.1.1.cmml" xref="S4.10.p3.5.m5.1.1.1"><csymbol cd="ambiguous" id="S4.10.p3.5.m5.1.1.1.1.cmml" xref="S4.10.p3.5.m5.1.1.1">subscript</csymbol><geq id="S4.10.p3.5.m5.1.1.1.2.cmml" xref="S4.10.p3.5.m5.1.1.1.2"></geq><ci id="S4.10.p3.5.m5.1.1.1.3.cmml" xref="S4.10.p3.5.m5.1.1.1.3">lex</ci></apply><ci id="S4.10.p3.5.m5.1.1.2.cmml" xref="S4.10.p3.5.m5.1.1.2">𝑡</ci><ci id="S4.10.p3.5.m5.1.1.3.cmml" xref="S4.10.p3.5.m5.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.10.p3.5.m5.1c">t\geq_{\mathrm{lex}}s</annotation><annotation encoding="application/x-llamapun" id="S4.10.p3.5.m5.1d">italic_t ≥ start_POSTSUBSCRIPT roman_lex end_POSTSUBSCRIPT italic_s</annotation></semantics></math>, which contradicts that <math alttext="t\sqsubset s" class="ltx_Math" display="inline" id="S4.10.p3.6.m6.1"><semantics id="S4.10.p3.6.m6.1a"><mrow id="S4.10.p3.6.m6.1.1" xref="S4.10.p3.6.m6.1.1.cmml"><mi id="S4.10.p3.6.m6.1.1.2" xref="S4.10.p3.6.m6.1.1.2.cmml">t</mi><mo id="S4.10.p3.6.m6.1.1.1" xref="S4.10.p3.6.m6.1.1.1.cmml">⊏</mo><mi id="S4.10.p3.6.m6.1.1.3" xref="S4.10.p3.6.m6.1.1.3.cmml">s</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.10.p3.6.m6.1b"><apply id="S4.10.p3.6.m6.1.1.cmml" xref="S4.10.p3.6.m6.1.1"><csymbol cd="latexml" id="S4.10.p3.6.m6.1.1.1.cmml" xref="S4.10.p3.6.m6.1.1.1">square-image-of</csymbol><ci id="S4.10.p3.6.m6.1.1.2.cmml" xref="S4.10.p3.6.m6.1.1.2">𝑡</ci><ci id="S4.10.p3.6.m6.1.1.3.cmml" xref="S4.10.p3.6.m6.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.10.p3.6.m6.1c">t\sqsubset s</annotation><annotation encoding="application/x-llamapun" id="S4.10.p3.6.m6.1d">italic_t ⊏ italic_s</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.11.p4"> <p class="ltx_p" id="S4.11.p4.9">(<math alttext="\leftarrow" class="ltx_Math" display="inline" id="S4.11.p4.1.m1.1"><semantics id="S4.11.p4.1.m1.1a"><mo id="S4.11.p4.1.m1.1.1" stretchy="false" xref="S4.11.p4.1.m1.1.1.cmml">←</mo><annotation-xml encoding="MathML-Content" id="S4.11.p4.1.m1.1b"><ci id="S4.11.p4.1.m1.1.1.cmml" xref="S4.11.p4.1.m1.1.1">←</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.11.p4.1.m1.1c">\leftarrow</annotation><annotation encoding="application/x-llamapun" id="S4.11.p4.1.m1.1d">←</annotation></semantics></math>). As mentioned <math alttext="(T,<_{\mathrm{lex}})" class="ltx_Math" display="inline" id="S4.11.p4.2.m2.2"><semantics id="S4.11.p4.2.m2.2a"><mrow id="S4.11.p4.2.m2.2.2.1" xref="S4.11.p4.2.m2.2.2.2.cmml"><mo id="S4.11.p4.2.m2.2.2.1.2" stretchy="false" xref="S4.11.p4.2.m2.2.2.2.cmml">(</mo><mi id="S4.11.p4.2.m2.1.1" xref="S4.11.p4.2.m2.1.1.cmml">T</mi><mo id="S4.11.p4.2.m2.2.2.1.3" xref="S4.11.p4.2.m2.2.2.2.cmml">,</mo><msub id="S4.11.p4.2.m2.2.2.1.1" xref="S4.11.p4.2.m2.2.2.1.1.cmml"><mo id="S4.11.p4.2.m2.2.2.1.1.2" lspace="0em" rspace="0em" xref="S4.11.p4.2.m2.2.2.1.1.2.cmml"><</mo><mi id="S4.11.p4.2.m2.2.2.1.1.3" xref="S4.11.p4.2.m2.2.2.1.1.3.cmml">lex</mi></msub><mo id="S4.11.p4.2.m2.2.2.1.4" stretchy="false" xref="S4.11.p4.2.m2.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.11.p4.2.m2.2b"><interval closure="open" id="S4.11.p4.2.m2.2.2.2.cmml" xref="S4.11.p4.2.m2.2.2.1"><ci id="S4.11.p4.2.m2.1.1.cmml" xref="S4.11.p4.2.m2.1.1">𝑇</ci><apply id="S4.11.p4.2.m2.2.2.1.1.cmml" xref="S4.11.p4.2.m2.2.2.1.1"><csymbol cd="ambiguous" id="S4.11.p4.2.m2.2.2.1.1.1.cmml" xref="S4.11.p4.2.m2.2.2.1.1">subscript</csymbol><lt id="S4.11.p4.2.m2.2.2.1.1.2.cmml" xref="S4.11.p4.2.m2.2.2.1.1.2"></lt><ci id="S4.11.p4.2.m2.2.2.1.1.3.cmml" xref="S4.11.p4.2.m2.2.2.1.1.3">lex</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.11.p4.2.m2.2c">(T,<_{\mathrm{lex}})</annotation><annotation encoding="application/x-llamapun" id="S4.11.p4.2.m2.2d">( italic_T , < start_POSTSUBSCRIPT roman_lex end_POSTSUBSCRIPT )</annotation></semantics></math> is always Countryman, so let <math alttext="c:T\times T\to\omega" class="ltx_Math" display="inline" id="S4.11.p4.3.m3.1"><semantics id="S4.11.p4.3.m3.1a"><mrow id="S4.11.p4.3.m3.1.1" xref="S4.11.p4.3.m3.1.1.cmml"><mi id="S4.11.p4.3.m3.1.1.2" xref="S4.11.p4.3.m3.1.1.2.cmml">c</mi><mo id="S4.11.p4.3.m3.1.1.1" lspace="0.278em" rspace="0.278em" xref="S4.11.p4.3.m3.1.1.1.cmml">:</mo><mrow id="S4.11.p4.3.m3.1.1.3" xref="S4.11.p4.3.m3.1.1.3.cmml"><mrow id="S4.11.p4.3.m3.1.1.3.2" xref="S4.11.p4.3.m3.1.1.3.2.cmml"><mi id="S4.11.p4.3.m3.1.1.3.2.2" xref="S4.11.p4.3.m3.1.1.3.2.2.cmml">T</mi><mo id="S4.11.p4.3.m3.1.1.3.2.1" lspace="0.222em" rspace="0.222em" xref="S4.11.p4.3.m3.1.1.3.2.1.cmml">×</mo><mi id="S4.11.p4.3.m3.1.1.3.2.3" xref="S4.11.p4.3.m3.1.1.3.2.3.cmml">T</mi></mrow><mo id="S4.11.p4.3.m3.1.1.3.1" stretchy="false" xref="S4.11.p4.3.m3.1.1.3.1.cmml">→</mo><mi id="S4.11.p4.3.m3.1.1.3.3" xref="S4.11.p4.3.m3.1.1.3.3.cmml">ω</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.11.p4.3.m3.1b"><apply id="S4.11.p4.3.m3.1.1.cmml" xref="S4.11.p4.3.m3.1.1"><ci id="S4.11.p4.3.m3.1.1.1.cmml" xref="S4.11.p4.3.m3.1.1.1">:</ci><ci id="S4.11.p4.3.m3.1.1.2.cmml" xref="S4.11.p4.3.m3.1.1.2">𝑐</ci><apply id="S4.11.p4.3.m3.1.1.3.cmml" xref="S4.11.p4.3.m3.1.1.3"><ci id="S4.11.p4.3.m3.1.1.3.1.cmml" xref="S4.11.p4.3.m3.1.1.3.1">→</ci><apply id="S4.11.p4.3.m3.1.1.3.2.cmml" xref="S4.11.p4.3.m3.1.1.3.2"><times id="S4.11.p4.3.m3.1.1.3.2.1.cmml" xref="S4.11.p4.3.m3.1.1.3.2.1"></times><ci id="S4.11.p4.3.m3.1.1.3.2.2.cmml" xref="S4.11.p4.3.m3.1.1.3.2.2">𝑇</ci><ci id="S4.11.p4.3.m3.1.1.3.2.3.cmml" xref="S4.11.p4.3.m3.1.1.3.2.3">𝑇</ci></apply><ci id="S4.11.p4.3.m3.1.1.3.3.cmml" xref="S4.11.p4.3.m3.1.1.3.3">𝜔</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.11.p4.3.m3.1c">c:T\times T\to\omega</annotation><annotation encoding="application/x-llamapun" id="S4.11.p4.3.m3.1d">italic_c : italic_T × italic_T → italic_ω</annotation></semantics></math> witness this, i.e., that for all <math alttext="t,s,t^{\prime},s^{\prime}\in T" class="ltx_Math" display="inline" id="S4.11.p4.4.m4.4"><semantics id="S4.11.p4.4.m4.4a"><mrow id="S4.11.p4.4.m4.4.4" xref="S4.11.p4.4.m4.4.4.cmml"><mrow id="S4.11.p4.4.m4.4.4.2.2" xref="S4.11.p4.4.m4.4.4.2.3.cmml"><mi id="S4.11.p4.4.m4.1.1" xref="S4.11.p4.4.m4.1.1.cmml">t</mi><mo id="S4.11.p4.4.m4.4.4.2.2.3" xref="S4.11.p4.4.m4.4.4.2.3.cmml">,</mo><mi id="S4.11.p4.4.m4.2.2" xref="S4.11.p4.4.m4.2.2.cmml">s</mi><mo id="S4.11.p4.4.m4.4.4.2.2.4" xref="S4.11.p4.4.m4.4.4.2.3.cmml">,</mo><msup id="S4.11.p4.4.m4.3.3.1.1.1" xref="S4.11.p4.4.m4.3.3.1.1.1.cmml"><mi id="S4.11.p4.4.m4.3.3.1.1.1.2" xref="S4.11.p4.4.m4.3.3.1.1.1.2.cmml">t</mi><mo id="S4.11.p4.4.m4.3.3.1.1.1.3" xref="S4.11.p4.4.m4.3.3.1.1.1.3.cmml">′</mo></msup><mo id="S4.11.p4.4.m4.4.4.2.2.5" xref="S4.11.p4.4.m4.4.4.2.3.cmml">,</mo><msup id="S4.11.p4.4.m4.4.4.2.2.2" xref="S4.11.p4.4.m4.4.4.2.2.2.cmml"><mi id="S4.11.p4.4.m4.4.4.2.2.2.2" xref="S4.11.p4.4.m4.4.4.2.2.2.2.cmml">s</mi><mo id="S4.11.p4.4.m4.4.4.2.2.2.3" xref="S4.11.p4.4.m4.4.4.2.2.2.3.cmml">′</mo></msup></mrow><mo id="S4.11.p4.4.m4.4.4.3" xref="S4.11.p4.4.m4.4.4.3.cmml">∈</mo><mi id="S4.11.p4.4.m4.4.4.4" xref="S4.11.p4.4.m4.4.4.4.cmml">T</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.11.p4.4.m4.4b"><apply id="S4.11.p4.4.m4.4.4.cmml" xref="S4.11.p4.4.m4.4.4"><in id="S4.11.p4.4.m4.4.4.3.cmml" xref="S4.11.p4.4.m4.4.4.3"></in><list id="S4.11.p4.4.m4.4.4.2.3.cmml" xref="S4.11.p4.4.m4.4.4.2.2"><ci id="S4.11.p4.4.m4.1.1.cmml" xref="S4.11.p4.4.m4.1.1">𝑡</ci><ci id="S4.11.p4.4.m4.2.2.cmml" xref="S4.11.p4.4.m4.2.2">𝑠</ci><apply id="S4.11.p4.4.m4.3.3.1.1.1.cmml" xref="S4.11.p4.4.m4.3.3.1.1.1"><csymbol cd="ambiguous" id="S4.11.p4.4.m4.3.3.1.1.1.1.cmml" xref="S4.11.p4.4.m4.3.3.1.1.1">superscript</csymbol><ci id="S4.11.p4.4.m4.3.3.1.1.1.2.cmml" xref="S4.11.p4.4.m4.3.3.1.1.1.2">𝑡</ci><ci id="S4.11.p4.4.m4.3.3.1.1.1.3.cmml" xref="S4.11.p4.4.m4.3.3.1.1.1.3">′</ci></apply><apply id="S4.11.p4.4.m4.4.4.2.2.2.cmml" xref="S4.11.p4.4.m4.4.4.2.2.2"><csymbol cd="ambiguous" id="S4.11.p4.4.m4.4.4.2.2.2.1.cmml" xref="S4.11.p4.4.m4.4.4.2.2.2">superscript</csymbol><ci id="S4.11.p4.4.m4.4.4.2.2.2.2.cmml" xref="S4.11.p4.4.m4.4.4.2.2.2.2">𝑠</ci><ci id="S4.11.p4.4.m4.4.4.2.2.2.3.cmml" xref="S4.11.p4.4.m4.4.4.2.2.2.3">′</ci></apply></list><ci id="S4.11.p4.4.m4.4.4.4.cmml" xref="S4.11.p4.4.m4.4.4.4">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.11.p4.4.m4.4c">t,s,t^{\prime},s^{\prime}\in T</annotation><annotation encoding="application/x-llamapun" id="S4.11.p4.4.m4.4d">italic_t , italic_s , italic_t start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_T</annotation></semantics></math>, if <math alttext="c(t,s)=c(t^{\prime},s^{\prime})" class="ltx_Math" display="inline" id="S4.11.p4.5.m5.4"><semantics id="S4.11.p4.5.m5.4a"><mrow id="S4.11.p4.5.m5.4.4" xref="S4.11.p4.5.m5.4.4.cmml"><mrow id="S4.11.p4.5.m5.4.4.4" xref="S4.11.p4.5.m5.4.4.4.cmml"><mi id="S4.11.p4.5.m5.4.4.4.2" xref="S4.11.p4.5.m5.4.4.4.2.cmml">c</mi><mo id="S4.11.p4.5.m5.4.4.4.1" xref="S4.11.p4.5.m5.4.4.4.1.cmml">⁢</mo><mrow id="S4.11.p4.5.m5.4.4.4.3.2" xref="S4.11.p4.5.m5.4.4.4.3.1.cmml"><mo id="S4.11.p4.5.m5.4.4.4.3.2.1" stretchy="false" xref="S4.11.p4.5.m5.4.4.4.3.1.cmml">(</mo><mi id="S4.11.p4.5.m5.1.1" xref="S4.11.p4.5.m5.1.1.cmml">t</mi><mo id="S4.11.p4.5.m5.4.4.4.3.2.2" xref="S4.11.p4.5.m5.4.4.4.3.1.cmml">,</mo><mi id="S4.11.p4.5.m5.2.2" xref="S4.11.p4.5.m5.2.2.cmml">s</mi><mo id="S4.11.p4.5.m5.4.4.4.3.2.3" stretchy="false" xref="S4.11.p4.5.m5.4.4.4.3.1.cmml">)</mo></mrow></mrow><mo id="S4.11.p4.5.m5.4.4.3" xref="S4.11.p4.5.m5.4.4.3.cmml">=</mo><mrow id="S4.11.p4.5.m5.4.4.2" xref="S4.11.p4.5.m5.4.4.2.cmml"><mi id="S4.11.p4.5.m5.4.4.2.4" xref="S4.11.p4.5.m5.4.4.2.4.cmml">c</mi><mo id="S4.11.p4.5.m5.4.4.2.3" xref="S4.11.p4.5.m5.4.4.2.3.cmml">⁢</mo><mrow id="S4.11.p4.5.m5.4.4.2.2.2" xref="S4.11.p4.5.m5.4.4.2.2.3.cmml"><mo id="S4.11.p4.5.m5.4.4.2.2.2.3" stretchy="false" xref="S4.11.p4.5.m5.4.4.2.2.3.cmml">(</mo><msup id="S4.11.p4.5.m5.3.3.1.1.1.1" xref="S4.11.p4.5.m5.3.3.1.1.1.1.cmml"><mi id="S4.11.p4.5.m5.3.3.1.1.1.1.2" xref="S4.11.p4.5.m5.3.3.1.1.1.1.2.cmml">t</mi><mo id="S4.11.p4.5.m5.3.3.1.1.1.1.3" xref="S4.11.p4.5.m5.3.3.1.1.1.1.3.cmml">′</mo></msup><mo id="S4.11.p4.5.m5.4.4.2.2.2.4" xref="S4.11.p4.5.m5.4.4.2.2.3.cmml">,</mo><msup id="S4.11.p4.5.m5.4.4.2.2.2.2" xref="S4.11.p4.5.m5.4.4.2.2.2.2.cmml"><mi id="S4.11.p4.5.m5.4.4.2.2.2.2.2" xref="S4.11.p4.5.m5.4.4.2.2.2.2.2.cmml">s</mi><mo id="S4.11.p4.5.m5.4.4.2.2.2.2.3" xref="S4.11.p4.5.m5.4.4.2.2.2.2.3.cmml">′</mo></msup><mo id="S4.11.p4.5.m5.4.4.2.2.2.5" stretchy="false" xref="S4.11.p4.5.m5.4.4.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.11.p4.5.m5.4b"><apply id="S4.11.p4.5.m5.4.4.cmml" xref="S4.11.p4.5.m5.4.4"><eq id="S4.11.p4.5.m5.4.4.3.cmml" xref="S4.11.p4.5.m5.4.4.3"></eq><apply id="S4.11.p4.5.m5.4.4.4.cmml" xref="S4.11.p4.5.m5.4.4.4"><times id="S4.11.p4.5.m5.4.4.4.1.cmml" xref="S4.11.p4.5.m5.4.4.4.1"></times><ci id="S4.11.p4.5.m5.4.4.4.2.cmml" xref="S4.11.p4.5.m5.4.4.4.2">𝑐</ci><interval closure="open" id="S4.11.p4.5.m5.4.4.4.3.1.cmml" xref="S4.11.p4.5.m5.4.4.4.3.2"><ci id="S4.11.p4.5.m5.1.1.cmml" xref="S4.11.p4.5.m5.1.1">𝑡</ci><ci id="S4.11.p4.5.m5.2.2.cmml" xref="S4.11.p4.5.m5.2.2">𝑠</ci></interval></apply><apply id="S4.11.p4.5.m5.4.4.2.cmml" xref="S4.11.p4.5.m5.4.4.2"><times id="S4.11.p4.5.m5.4.4.2.3.cmml" xref="S4.11.p4.5.m5.4.4.2.3"></times><ci id="S4.11.p4.5.m5.4.4.2.4.cmml" xref="S4.11.p4.5.m5.4.4.2.4">𝑐</ci><interval closure="open" id="S4.11.p4.5.m5.4.4.2.2.3.cmml" xref="S4.11.p4.5.m5.4.4.2.2.2"><apply id="S4.11.p4.5.m5.3.3.1.1.1.1.cmml" xref="S4.11.p4.5.m5.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S4.11.p4.5.m5.3.3.1.1.1.1.1.cmml" xref="S4.11.p4.5.m5.3.3.1.1.1.1">superscript</csymbol><ci id="S4.11.p4.5.m5.3.3.1.1.1.1.2.cmml" xref="S4.11.p4.5.m5.3.3.1.1.1.1.2">𝑡</ci><ci id="S4.11.p4.5.m5.3.3.1.1.1.1.3.cmml" xref="S4.11.p4.5.m5.3.3.1.1.1.1.3">′</ci></apply><apply id="S4.11.p4.5.m5.4.4.2.2.2.2.cmml" xref="S4.11.p4.5.m5.4.4.2.2.2.2"><csymbol cd="ambiguous" id="S4.11.p4.5.m5.4.4.2.2.2.2.1.cmml" xref="S4.11.p4.5.m5.4.4.2.2.2.2">superscript</csymbol><ci id="S4.11.p4.5.m5.4.4.2.2.2.2.2.cmml" xref="S4.11.p4.5.m5.4.4.2.2.2.2.2">𝑠</ci><ci id="S4.11.p4.5.m5.4.4.2.2.2.2.3.cmml" xref="S4.11.p4.5.m5.4.4.2.2.2.2.3">′</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.11.p4.5.m5.4c">c(t,s)=c(t^{\prime},s^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S4.11.p4.5.m5.4d">italic_c ( italic_t , italic_s ) = italic_c ( italic_t start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math>, then <math alttext="t<_{\mathrm{lex}}t^{\prime}\rightarrow s\leq_{\mathrm{lex}}s^{\prime}" class="ltx_Math" display="inline" id="S4.11.p4.6.m6.1"><semantics id="S4.11.p4.6.m6.1a"><mrow id="S4.11.p4.6.m6.1.1" xref="S4.11.p4.6.m6.1.1.cmml"><mi id="S4.11.p4.6.m6.1.1.2" xref="S4.11.p4.6.m6.1.1.2.cmml">t</mi><msub id="S4.11.p4.6.m6.1.1.3" xref="S4.11.p4.6.m6.1.1.3.cmml"><mo id="S4.11.p4.6.m6.1.1.3.2" xref="S4.11.p4.6.m6.1.1.3.2.cmml"><</mo><mi id="S4.11.p4.6.m6.1.1.3.3" xref="S4.11.p4.6.m6.1.1.3.3.cmml">lex</mi></msub><msup id="S4.11.p4.6.m6.1.1.4" xref="S4.11.p4.6.m6.1.1.4.cmml"><mi id="S4.11.p4.6.m6.1.1.4.2" xref="S4.11.p4.6.m6.1.1.4.2.cmml">t</mi><mo id="S4.11.p4.6.m6.1.1.4.3" xref="S4.11.p4.6.m6.1.1.4.3.cmml">′</mo></msup><mo id="S4.11.p4.6.m6.1.1.5" stretchy="false" xref="S4.11.p4.6.m6.1.1.5.cmml">→</mo><mi id="S4.11.p4.6.m6.1.1.6" xref="S4.11.p4.6.m6.1.1.6.cmml">s</mi><msub id="S4.11.p4.6.m6.1.1.7" xref="S4.11.p4.6.m6.1.1.7.cmml"><mo id="S4.11.p4.6.m6.1.1.7.2" xref="S4.11.p4.6.m6.1.1.7.2.cmml">≤</mo><mi id="S4.11.p4.6.m6.1.1.7.3" xref="S4.11.p4.6.m6.1.1.7.3.cmml">lex</mi></msub><msup id="S4.11.p4.6.m6.1.1.8" xref="S4.11.p4.6.m6.1.1.8.cmml"><mi id="S4.11.p4.6.m6.1.1.8.2" xref="S4.11.p4.6.m6.1.1.8.2.cmml">s</mi><mo id="S4.11.p4.6.m6.1.1.8.3" xref="S4.11.p4.6.m6.1.1.8.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.11.p4.6.m6.1b"><apply id="S4.11.p4.6.m6.1.1.cmml" xref="S4.11.p4.6.m6.1.1"><and id="S4.11.p4.6.m6.1.1a.cmml" xref="S4.11.p4.6.m6.1.1"></and><apply id="S4.11.p4.6.m6.1.1b.cmml" xref="S4.11.p4.6.m6.1.1"><apply id="S4.11.p4.6.m6.1.1.3.cmml" xref="S4.11.p4.6.m6.1.1.3"><csymbol cd="ambiguous" id="S4.11.p4.6.m6.1.1.3.1.cmml" xref="S4.11.p4.6.m6.1.1.3">subscript</csymbol><lt id="S4.11.p4.6.m6.1.1.3.2.cmml" xref="S4.11.p4.6.m6.1.1.3.2"></lt><ci id="S4.11.p4.6.m6.1.1.3.3.cmml" xref="S4.11.p4.6.m6.1.1.3.3">lex</ci></apply><ci id="S4.11.p4.6.m6.1.1.2.cmml" xref="S4.11.p4.6.m6.1.1.2">𝑡</ci><apply id="S4.11.p4.6.m6.1.1.4.cmml" xref="S4.11.p4.6.m6.1.1.4"><csymbol cd="ambiguous" id="S4.11.p4.6.m6.1.1.4.1.cmml" xref="S4.11.p4.6.m6.1.1.4">superscript</csymbol><ci id="S4.11.p4.6.m6.1.1.4.2.cmml" xref="S4.11.p4.6.m6.1.1.4.2">𝑡</ci><ci id="S4.11.p4.6.m6.1.1.4.3.cmml" xref="S4.11.p4.6.m6.1.1.4.3">′</ci></apply></apply><apply id="S4.11.p4.6.m6.1.1c.cmml" xref="S4.11.p4.6.m6.1.1"><ci id="S4.11.p4.6.m6.1.1.5.cmml" xref="S4.11.p4.6.m6.1.1.5">→</ci><share href="https://arxiv.org/html/2503.13728v1#S4.11.p4.6.m6.1.1.4.cmml" id="S4.11.p4.6.m6.1.1d.cmml" xref="S4.11.p4.6.m6.1.1"></share><ci id="S4.11.p4.6.m6.1.1.6.cmml" xref="S4.11.p4.6.m6.1.1.6">𝑠</ci></apply><apply id="S4.11.p4.6.m6.1.1e.cmml" xref="S4.11.p4.6.m6.1.1"><apply id="S4.11.p4.6.m6.1.1.7.cmml" xref="S4.11.p4.6.m6.1.1.7"><csymbol cd="ambiguous" id="S4.11.p4.6.m6.1.1.7.1.cmml" xref="S4.11.p4.6.m6.1.1.7">subscript</csymbol><leq id="S4.11.p4.6.m6.1.1.7.2.cmml" xref="S4.11.p4.6.m6.1.1.7.2"></leq><ci id="S4.11.p4.6.m6.1.1.7.3.cmml" xref="S4.11.p4.6.m6.1.1.7.3">lex</ci></apply><share href="https://arxiv.org/html/2503.13728v1#S4.11.p4.6.m6.1.1.6.cmml" id="S4.11.p4.6.m6.1.1f.cmml" xref="S4.11.p4.6.m6.1.1"></share><apply id="S4.11.p4.6.m6.1.1.8.cmml" xref="S4.11.p4.6.m6.1.1.8"><csymbol cd="ambiguous" id="S4.11.p4.6.m6.1.1.8.1.cmml" xref="S4.11.p4.6.m6.1.1.8">superscript</csymbol><ci id="S4.11.p4.6.m6.1.1.8.2.cmml" xref="S4.11.p4.6.m6.1.1.8.2">𝑠</ci><ci id="S4.11.p4.6.m6.1.1.8.3.cmml" xref="S4.11.p4.6.m6.1.1.8.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.11.p4.6.m6.1c">t<_{\mathrm{lex}}t^{\prime}\rightarrow s\leq_{\mathrm{lex}}s^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.11.p4.6.m6.1d">italic_t < start_POSTSUBSCRIPT roman_lex end_POSTSUBSCRIPT italic_t start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT → italic_s ≤ start_POSTSUBSCRIPT roman_lex end_POSTSUBSCRIPT italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>. It is enough to define a similar mapping <math alttext="\tilde{c}" class="ltx_Math" display="inline" id="S4.11.p4.7.m7.1"><semantics id="S4.11.p4.7.m7.1a"><mover accent="true" id="S4.11.p4.7.m7.1.1" xref="S4.11.p4.7.m7.1.1.cmml"><mi id="S4.11.p4.7.m7.1.1.2" xref="S4.11.p4.7.m7.1.1.2.cmml">c</mi><mo id="S4.11.p4.7.m7.1.1.1" xref="S4.11.p4.7.m7.1.1.1.cmml">~</mo></mover><annotation-xml encoding="MathML-Content" id="S4.11.p4.7.m7.1b"><apply id="S4.11.p4.7.m7.1.1.cmml" xref="S4.11.p4.7.m7.1.1"><ci id="S4.11.p4.7.m7.1.1.1.cmml" xref="S4.11.p4.7.m7.1.1.1">~</ci><ci id="S4.11.p4.7.m7.1.1.2.cmml" xref="S4.11.p4.7.m7.1.1.2">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.11.p4.7.m7.1c">\tilde{c}</annotation><annotation encoding="application/x-llamapun" id="S4.11.p4.7.m7.1d">over~ start_ARG italic_c end_ARG</annotation></semantics></math> for <math alttext="A^{+}\times A^{+}" class="ltx_Math" display="inline" id="S4.11.p4.8.m8.1"><semantics id="S4.11.p4.8.m8.1a"><mrow id="S4.11.p4.8.m8.1.1" xref="S4.11.p4.8.m8.1.1.cmml"><msup id="S4.11.p4.8.m8.1.1.2" xref="S4.11.p4.8.m8.1.1.2.cmml"><mi id="S4.11.p4.8.m8.1.1.2.2" xref="S4.11.p4.8.m8.1.1.2.2.cmml">A</mi><mo id="S4.11.p4.8.m8.1.1.2.3" xref="S4.11.p4.8.m8.1.1.2.3.cmml">+</mo></msup><mo id="S4.11.p4.8.m8.1.1.1" lspace="0.222em" rspace="0.222em" xref="S4.11.p4.8.m8.1.1.1.cmml">×</mo><msup id="S4.11.p4.8.m8.1.1.3" xref="S4.11.p4.8.m8.1.1.3.cmml"><mi id="S4.11.p4.8.m8.1.1.3.2" xref="S4.11.p4.8.m8.1.1.3.2.cmml">A</mi><mo id="S4.11.p4.8.m8.1.1.3.3" xref="S4.11.p4.8.m8.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.11.p4.8.m8.1b"><apply id="S4.11.p4.8.m8.1.1.cmml" xref="S4.11.p4.8.m8.1.1"><times id="S4.11.p4.8.m8.1.1.1.cmml" xref="S4.11.p4.8.m8.1.1.1"></times><apply id="S4.11.p4.8.m8.1.1.2.cmml" xref="S4.11.p4.8.m8.1.1.2"><csymbol cd="ambiguous" id="S4.11.p4.8.m8.1.1.2.1.cmml" xref="S4.11.p4.8.m8.1.1.2">superscript</csymbol><ci id="S4.11.p4.8.m8.1.1.2.2.cmml" xref="S4.11.p4.8.m8.1.1.2.2">𝐴</ci><plus id="S4.11.p4.8.m8.1.1.2.3.cmml" xref="S4.11.p4.8.m8.1.1.2.3"></plus></apply><apply id="S4.11.p4.8.m8.1.1.3.cmml" xref="S4.11.p4.8.m8.1.1.3"><csymbol cd="ambiguous" id="S4.11.p4.8.m8.1.1.3.1.cmml" xref="S4.11.p4.8.m8.1.1.3">superscript</csymbol><ci id="S4.11.p4.8.m8.1.1.3.2.cmml" xref="S4.11.p4.8.m8.1.1.3.2">𝐴</ci><plus id="S4.11.p4.8.m8.1.1.3.3.cmml" xref="S4.11.p4.8.m8.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.11.p4.8.m8.1c">A^{+}\times A^{+}</annotation><annotation encoding="application/x-llamapun" id="S4.11.p4.8.m8.1d">italic_A start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT × italic_A start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math>. For convenience our mapping will have range in <math alttext="\omega^{<\omega}" class="ltx_Math" display="inline" id="S4.11.p4.9.m9.1"><semantics id="S4.11.p4.9.m9.1a"><msup id="S4.11.p4.9.m9.1.1" xref="S4.11.p4.9.m9.1.1.cmml"><mi id="S4.11.p4.9.m9.1.1.2" xref="S4.11.p4.9.m9.1.1.2.cmml">ω</mi><mrow id="S4.11.p4.9.m9.1.1.3" xref="S4.11.p4.9.m9.1.1.3.cmml"><mi id="S4.11.p4.9.m9.1.1.3.2" xref="S4.11.p4.9.m9.1.1.3.2.cmml"></mi><mo id="S4.11.p4.9.m9.1.1.3.1" xref="S4.11.p4.9.m9.1.1.3.1.cmml"><</mo><mi id="S4.11.p4.9.m9.1.1.3.3" xref="S4.11.p4.9.m9.1.1.3.3.cmml">ω</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="S4.11.p4.9.m9.1b"><apply id="S4.11.p4.9.m9.1.1.cmml" xref="S4.11.p4.9.m9.1.1"><csymbol cd="ambiguous" id="S4.11.p4.9.m9.1.1.1.cmml" xref="S4.11.p4.9.m9.1.1">superscript</csymbol><ci id="S4.11.p4.9.m9.1.1.2.cmml" xref="S4.11.p4.9.m9.1.1.2">𝜔</ci><apply id="S4.11.p4.9.m9.1.1.3.cmml" xref="S4.11.p4.9.m9.1.1.3"><lt id="S4.11.p4.9.m9.1.1.3.1.cmml" xref="S4.11.p4.9.m9.1.1.3.1"></lt><csymbol cd="latexml" id="S4.11.p4.9.m9.1.1.3.2.cmml" xref="S4.11.p4.9.m9.1.1.3.2">absent</csymbol><ci id="S4.11.p4.9.m9.1.1.3.3.cmml" xref="S4.11.p4.9.m9.1.1.3.3">𝜔</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.11.p4.9.m9.1c">\omega^{<\omega}</annotation><annotation encoding="application/x-llamapun" id="S4.11.p4.9.m9.1d">italic_ω start_POSTSUPERSCRIPT < italic_ω end_POSTSUPERSCRIPT</annotation></semantics></math>, which is enough since this set is countable.</p> </div> <div class="ltx_para" id="S4.12.p5"> <p class="ltx_p" id="S4.12.p5.3">Let <math alttext="f:T\to\omega" class="ltx_Math" display="inline" id="S4.12.p5.1.m1.1"><semantics id="S4.12.p5.1.m1.1a"><mrow id="S4.12.p5.1.m1.1.1" xref="S4.12.p5.1.m1.1.1.cmml"><mi id="S4.12.p5.1.m1.1.1.2" xref="S4.12.p5.1.m1.1.1.2.cmml">f</mi><mo id="S4.12.p5.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S4.12.p5.1.m1.1.1.1.cmml">:</mo><mrow id="S4.12.p5.1.m1.1.1.3" xref="S4.12.p5.1.m1.1.1.3.cmml"><mi id="S4.12.p5.1.m1.1.1.3.2" xref="S4.12.p5.1.m1.1.1.3.2.cmml">T</mi><mo id="S4.12.p5.1.m1.1.1.3.1" stretchy="false" xref="S4.12.p5.1.m1.1.1.3.1.cmml">→</mo><mi id="S4.12.p5.1.m1.1.1.3.3" xref="S4.12.p5.1.m1.1.1.3.3.cmml">ω</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.12.p5.1.m1.1b"><apply id="S4.12.p5.1.m1.1.1.cmml" xref="S4.12.p5.1.m1.1.1"><ci id="S4.12.p5.1.m1.1.1.1.cmml" xref="S4.12.p5.1.m1.1.1.1">:</ci><ci id="S4.12.p5.1.m1.1.1.2.cmml" xref="S4.12.p5.1.m1.1.1.2">𝑓</ci><apply id="S4.12.p5.1.m1.1.1.3.cmml" xref="S4.12.p5.1.m1.1.1.3"><ci id="S4.12.p5.1.m1.1.1.3.1.cmml" xref="S4.12.p5.1.m1.1.1.3.1">→</ci><ci id="S4.12.p5.1.m1.1.1.3.2.cmml" xref="S4.12.p5.1.m1.1.1.3.2">𝑇</ci><ci id="S4.12.p5.1.m1.1.1.3.3.cmml" xref="S4.12.p5.1.m1.1.1.3.3">𝜔</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.12.p5.1.m1.1c">f:T\to\omega</annotation><annotation encoding="application/x-llamapun" id="S4.12.p5.1.m1.1d">italic_f : italic_T → italic_ω</annotation></semantics></math> witness that <math alttext="T" class="ltx_Math" display="inline" id="S4.12.p5.2.m2.1"><semantics id="S4.12.p5.2.m2.1a"><mi id="S4.12.p5.2.m2.1.1" xref="S4.12.p5.2.m2.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S4.12.p5.2.m2.1b"><ci id="S4.12.p5.2.m2.1.1.cmml" xref="S4.12.p5.2.m2.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.12.p5.2.m2.1c">T</annotation><annotation encoding="application/x-llamapun" id="S4.12.p5.2.m2.1d">italic_T</annotation></semantics></math> is special. We define <math alttext="\tilde{c}" class="ltx_Math" display="inline" id="S4.12.p5.3.m3.1"><semantics id="S4.12.p5.3.m3.1a"><mover accent="true" id="S4.12.p5.3.m3.1.1" xref="S4.12.p5.3.m3.1.1.cmml"><mi id="S4.12.p5.3.m3.1.1.2" xref="S4.12.p5.3.m3.1.1.2.cmml">c</mi><mo id="S4.12.p5.3.m3.1.1.1" xref="S4.12.p5.3.m3.1.1.1.cmml">~</mo></mover><annotation-xml encoding="MathML-Content" id="S4.12.p5.3.m3.1b"><apply id="S4.12.p5.3.m3.1.1.cmml" xref="S4.12.p5.3.m3.1.1"><ci id="S4.12.p5.3.m3.1.1.1.cmml" xref="S4.12.p5.3.m3.1.1.1">~</ci><ci id="S4.12.p5.3.m3.1.1.2.cmml" xref="S4.12.p5.3.m3.1.1.2">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.12.p5.3.m3.1c">\tilde{c}</annotation><annotation encoding="application/x-llamapun" id="S4.12.p5.3.m3.1d">over~ start_ARG italic_c end_ARG</annotation></semantics></math> as follows:</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S9.EGx1"> <tbody id="S4.Ex7"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\tilde{c}(t,s)" class="ltx_Math" display="inline" id="S4.Ex7.m1.2"><semantics id="S4.Ex7.m1.2a"><mrow id="S4.Ex7.m1.2.3" xref="S4.Ex7.m1.2.3.cmml"><mover accent="true" id="S4.Ex7.m1.2.3.2" xref="S4.Ex7.m1.2.3.2.cmml"><mi id="S4.Ex7.m1.2.3.2.2" xref="S4.Ex7.m1.2.3.2.2.cmml">c</mi><mo id="S4.Ex7.m1.2.3.2.1" xref="S4.Ex7.m1.2.3.2.1.cmml">~</mo></mover><mo id="S4.Ex7.m1.2.3.1" xref="S4.Ex7.m1.2.3.1.cmml">⁢</mo><mrow id="S4.Ex7.m1.2.3.3.2" xref="S4.Ex7.m1.2.3.3.1.cmml"><mo id="S4.Ex7.m1.2.3.3.2.1" stretchy="false" xref="S4.Ex7.m1.2.3.3.1.cmml">(</mo><mi id="S4.Ex7.m1.1.1" xref="S4.Ex7.m1.1.1.cmml">t</mi><mo id="S4.Ex7.m1.2.3.3.2.2" xref="S4.Ex7.m1.2.3.3.1.cmml">,</mo><mi id="S4.Ex7.m1.2.2" xref="S4.Ex7.m1.2.2.cmml">s</mi><mo id="S4.Ex7.m1.2.3.3.2.3" stretchy="false" xref="S4.Ex7.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex7.m1.2b"><apply id="S4.Ex7.m1.2.3.cmml" xref="S4.Ex7.m1.2.3"><times id="S4.Ex7.m1.2.3.1.cmml" xref="S4.Ex7.m1.2.3.1"></times><apply id="S4.Ex7.m1.2.3.2.cmml" xref="S4.Ex7.m1.2.3.2"><ci id="S4.Ex7.m1.2.3.2.1.cmml" xref="S4.Ex7.m1.2.3.2.1">~</ci><ci id="S4.Ex7.m1.2.3.2.2.cmml" xref="S4.Ex7.m1.2.3.2.2">𝑐</ci></apply><interval closure="open" id="S4.Ex7.m1.2.3.3.1.cmml" xref="S4.Ex7.m1.2.3.3.2"><ci id="S4.Ex7.m1.1.1.cmml" xref="S4.Ex7.m1.1.1">𝑡</ci><ci id="S4.Ex7.m1.2.2.cmml" xref="S4.Ex7.m1.2.2">𝑠</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex7.m1.2c">\displaystyle\tilde{c}(t,s)</annotation><annotation encoding="application/x-llamapun" id="S4.Ex7.m1.2d">over~ start_ARG italic_c end_ARG ( italic_t , italic_s )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle:=(0,0,f(t),f(s),c(t,s))," class="ltx_Math" display="inline" id="S4.Ex7.m2.7"><semantics id="S4.Ex7.m2.7a"><mrow id="S4.Ex7.m2.7.7.1" xref="S4.Ex7.m2.7.7.1.1.cmml"><mrow id="S4.Ex7.m2.7.7.1.1" xref="S4.Ex7.m2.7.7.1.1.cmml"><mi id="S4.Ex7.m2.7.7.1.1.5" xref="S4.Ex7.m2.7.7.1.1.5.cmml"></mi><mo id="S4.Ex7.m2.7.7.1.1.4" lspace="0.278em" rspace="0.278em" xref="S4.Ex7.m2.7.7.1.1.4.cmml">:=</mo><mrow id="S4.Ex7.m2.7.7.1.1.3.3" xref="S4.Ex7.m2.7.7.1.1.3.4.cmml"><mo id="S4.Ex7.m2.7.7.1.1.3.3.4" stretchy="false" xref="S4.Ex7.m2.7.7.1.1.3.4.cmml">(</mo><mn id="S4.Ex7.m2.5.5" xref="S4.Ex7.m2.5.5.cmml">0</mn><mo id="S4.Ex7.m2.7.7.1.1.3.3.5" xref="S4.Ex7.m2.7.7.1.1.3.4.cmml">,</mo><mn id="S4.Ex7.m2.6.6" xref="S4.Ex7.m2.6.6.cmml">0</mn><mo id="S4.Ex7.m2.7.7.1.1.3.3.6" xref="S4.Ex7.m2.7.7.1.1.3.4.cmml">,</mo><mrow id="S4.Ex7.m2.7.7.1.1.1.1.1" xref="S4.Ex7.m2.7.7.1.1.1.1.1.cmml"><mi id="S4.Ex7.m2.7.7.1.1.1.1.1.2" xref="S4.Ex7.m2.7.7.1.1.1.1.1.2.cmml">f</mi><mo id="S4.Ex7.m2.7.7.1.1.1.1.1.1" xref="S4.Ex7.m2.7.7.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.Ex7.m2.7.7.1.1.1.1.1.3.2" xref="S4.Ex7.m2.7.7.1.1.1.1.1.cmml"><mo id="S4.Ex7.m2.7.7.1.1.1.1.1.3.2.1" stretchy="false" xref="S4.Ex7.m2.7.7.1.1.1.1.1.cmml">(</mo><mi id="S4.Ex7.m2.1.1" xref="S4.Ex7.m2.1.1.cmml">t</mi><mo id="S4.Ex7.m2.7.7.1.1.1.1.1.3.2.2" stretchy="false" xref="S4.Ex7.m2.7.7.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex7.m2.7.7.1.1.3.3.7" xref="S4.Ex7.m2.7.7.1.1.3.4.cmml">,</mo><mrow id="S4.Ex7.m2.7.7.1.1.2.2.2" xref="S4.Ex7.m2.7.7.1.1.2.2.2.cmml"><mi id="S4.Ex7.m2.7.7.1.1.2.2.2.2" xref="S4.Ex7.m2.7.7.1.1.2.2.2.2.cmml">f</mi><mo id="S4.Ex7.m2.7.7.1.1.2.2.2.1" xref="S4.Ex7.m2.7.7.1.1.2.2.2.1.cmml">⁢</mo><mrow id="S4.Ex7.m2.7.7.1.1.2.2.2.3.2" xref="S4.Ex7.m2.7.7.1.1.2.2.2.cmml"><mo id="S4.Ex7.m2.7.7.1.1.2.2.2.3.2.1" stretchy="false" xref="S4.Ex7.m2.7.7.1.1.2.2.2.cmml">(</mo><mi id="S4.Ex7.m2.2.2" xref="S4.Ex7.m2.2.2.cmml">s</mi><mo id="S4.Ex7.m2.7.7.1.1.2.2.2.3.2.2" stretchy="false" xref="S4.Ex7.m2.7.7.1.1.2.2.2.cmml">)</mo></mrow></mrow><mo id="S4.Ex7.m2.7.7.1.1.3.3.8" xref="S4.Ex7.m2.7.7.1.1.3.4.cmml">,</mo><mrow id="S4.Ex7.m2.7.7.1.1.3.3.3" xref="S4.Ex7.m2.7.7.1.1.3.3.3.cmml"><mi id="S4.Ex7.m2.7.7.1.1.3.3.3.2" xref="S4.Ex7.m2.7.7.1.1.3.3.3.2.cmml">c</mi><mo id="S4.Ex7.m2.7.7.1.1.3.3.3.1" xref="S4.Ex7.m2.7.7.1.1.3.3.3.1.cmml">⁢</mo><mrow id="S4.Ex7.m2.7.7.1.1.3.3.3.3.2" xref="S4.Ex7.m2.7.7.1.1.3.3.3.3.1.cmml"><mo id="S4.Ex7.m2.7.7.1.1.3.3.3.3.2.1" stretchy="false" xref="S4.Ex7.m2.7.7.1.1.3.3.3.3.1.cmml">(</mo><mi id="S4.Ex7.m2.3.3" xref="S4.Ex7.m2.3.3.cmml">t</mi><mo id="S4.Ex7.m2.7.7.1.1.3.3.3.3.2.2" xref="S4.Ex7.m2.7.7.1.1.3.3.3.3.1.cmml">,</mo><mi id="S4.Ex7.m2.4.4" xref="S4.Ex7.m2.4.4.cmml">s</mi><mo id="S4.Ex7.m2.7.7.1.1.3.3.3.3.2.3" stretchy="false" xref="S4.Ex7.m2.7.7.1.1.3.3.3.3.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex7.m2.7.7.1.1.3.3.9" stretchy="false" xref="S4.Ex7.m2.7.7.1.1.3.4.cmml">)</mo></mrow></mrow><mo id="S4.Ex7.m2.7.7.1.2" xref="S4.Ex7.m2.7.7.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex7.m2.7b"><apply id="S4.Ex7.m2.7.7.1.1.cmml" xref="S4.Ex7.m2.7.7.1"><csymbol cd="latexml" id="S4.Ex7.m2.7.7.1.1.4.cmml" xref="S4.Ex7.m2.7.7.1.1.4">assign</csymbol><csymbol cd="latexml" id="S4.Ex7.m2.7.7.1.1.5.cmml" xref="S4.Ex7.m2.7.7.1.1.5">absent</csymbol><vector id="S4.Ex7.m2.7.7.1.1.3.4.cmml" xref="S4.Ex7.m2.7.7.1.1.3.3"><cn id="S4.Ex7.m2.5.5.cmml" type="integer" xref="S4.Ex7.m2.5.5">0</cn><cn id="S4.Ex7.m2.6.6.cmml" type="integer" xref="S4.Ex7.m2.6.6">0</cn><apply id="S4.Ex7.m2.7.7.1.1.1.1.1.cmml" xref="S4.Ex7.m2.7.7.1.1.1.1.1"><times id="S4.Ex7.m2.7.7.1.1.1.1.1.1.cmml" xref="S4.Ex7.m2.7.7.1.1.1.1.1.1"></times><ci id="S4.Ex7.m2.7.7.1.1.1.1.1.2.cmml" xref="S4.Ex7.m2.7.7.1.1.1.1.1.2">𝑓</ci><ci id="S4.Ex7.m2.1.1.cmml" xref="S4.Ex7.m2.1.1">𝑡</ci></apply><apply id="S4.Ex7.m2.7.7.1.1.2.2.2.cmml" xref="S4.Ex7.m2.7.7.1.1.2.2.2"><times id="S4.Ex7.m2.7.7.1.1.2.2.2.1.cmml" xref="S4.Ex7.m2.7.7.1.1.2.2.2.1"></times><ci id="S4.Ex7.m2.7.7.1.1.2.2.2.2.cmml" xref="S4.Ex7.m2.7.7.1.1.2.2.2.2">𝑓</ci><ci id="S4.Ex7.m2.2.2.cmml" xref="S4.Ex7.m2.2.2">𝑠</ci></apply><apply id="S4.Ex7.m2.7.7.1.1.3.3.3.cmml" xref="S4.Ex7.m2.7.7.1.1.3.3.3"><times id="S4.Ex7.m2.7.7.1.1.3.3.3.1.cmml" xref="S4.Ex7.m2.7.7.1.1.3.3.3.1"></times><ci id="S4.Ex7.m2.7.7.1.1.3.3.3.2.cmml" xref="S4.Ex7.m2.7.7.1.1.3.3.3.2">𝑐</ci><interval closure="open" id="S4.Ex7.m2.7.7.1.1.3.3.3.3.1.cmml" xref="S4.Ex7.m2.7.7.1.1.3.3.3.3.2"><ci id="S4.Ex7.m2.3.3.cmml" xref="S4.Ex7.m2.3.3">𝑡</ci><ci id="S4.Ex7.m2.4.4.cmml" xref="S4.Ex7.m2.4.4">𝑠</ci></interval></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex7.m2.7c">\displaystyle:=(0,0,f(t),f(s),c(t,s)),</annotation><annotation encoding="application/x-llamapun" id="S4.Ex7.m2.7d">:= ( 0 , 0 , italic_f ( italic_t ) , italic_f ( italic_s ) , italic_c ( italic_t , italic_s ) ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S4.Ex8"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\tilde{c}(t,s^{\frown}\langle\omega\rangle)" class="ltx_Math" display="inline" id="S4.Ex8.m1.3"><semantics id="S4.Ex8.m1.3a"><mrow id="S4.Ex8.m1.3.3" xref="S4.Ex8.m1.3.3.cmml"><mover accent="true" id="S4.Ex8.m1.3.3.3" xref="S4.Ex8.m1.3.3.3.cmml"><mi id="S4.Ex8.m1.3.3.3.2" xref="S4.Ex8.m1.3.3.3.2.cmml">c</mi><mo id="S4.Ex8.m1.3.3.3.1" xref="S4.Ex8.m1.3.3.3.1.cmml">~</mo></mover><mo id="S4.Ex8.m1.3.3.2" xref="S4.Ex8.m1.3.3.2.cmml">⁢</mo><mrow id="S4.Ex8.m1.3.3.1.1" xref="S4.Ex8.m1.3.3.1.2.cmml"><mo id="S4.Ex8.m1.3.3.1.1.2" stretchy="false" xref="S4.Ex8.m1.3.3.1.2.cmml">(</mo><mi id="S4.Ex8.m1.2.2" xref="S4.Ex8.m1.2.2.cmml">t</mi><mo id="S4.Ex8.m1.3.3.1.1.3" xref="S4.Ex8.m1.3.3.1.2.cmml">,</mo><mrow id="S4.Ex8.m1.3.3.1.1.1" xref="S4.Ex8.m1.3.3.1.1.1.cmml"><msup id="S4.Ex8.m1.3.3.1.1.1.2" xref="S4.Ex8.m1.3.3.1.1.1.2.cmml"><mi id="S4.Ex8.m1.3.3.1.1.1.2.2" xref="S4.Ex8.m1.3.3.1.1.1.2.2.cmml">s</mi><mo id="S4.Ex8.m1.3.3.1.1.1.2.3" xref="S4.Ex8.m1.3.3.1.1.1.2.3.cmml">⌢</mo></msup><mo id="S4.Ex8.m1.3.3.1.1.1.1" xref="S4.Ex8.m1.3.3.1.1.1.1.cmml">⁢</mo><mrow id="S4.Ex8.m1.3.3.1.1.1.3.2" xref="S4.Ex8.m1.3.3.1.1.1.3.1.cmml"><mo id="S4.Ex8.m1.3.3.1.1.1.3.2.1" stretchy="false" xref="S4.Ex8.m1.3.3.1.1.1.3.1.1.cmml">⟨</mo><mi id="S4.Ex8.m1.1.1" xref="S4.Ex8.m1.1.1.cmml">ω</mi><mo id="S4.Ex8.m1.3.3.1.1.1.3.2.2" stretchy="false" xref="S4.Ex8.m1.3.3.1.1.1.3.1.1.cmml">⟩</mo></mrow></mrow><mo id="S4.Ex8.m1.3.3.1.1.4" stretchy="false" xref="S4.Ex8.m1.3.3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex8.m1.3b"><apply id="S4.Ex8.m1.3.3.cmml" xref="S4.Ex8.m1.3.3"><times id="S4.Ex8.m1.3.3.2.cmml" xref="S4.Ex8.m1.3.3.2"></times><apply id="S4.Ex8.m1.3.3.3.cmml" xref="S4.Ex8.m1.3.3.3"><ci id="S4.Ex8.m1.3.3.3.1.cmml" xref="S4.Ex8.m1.3.3.3.1">~</ci><ci id="S4.Ex8.m1.3.3.3.2.cmml" xref="S4.Ex8.m1.3.3.3.2">𝑐</ci></apply><interval closure="open" id="S4.Ex8.m1.3.3.1.2.cmml" xref="S4.Ex8.m1.3.3.1.1"><ci id="S4.Ex8.m1.2.2.cmml" xref="S4.Ex8.m1.2.2">𝑡</ci><apply id="S4.Ex8.m1.3.3.1.1.1.cmml" xref="S4.Ex8.m1.3.3.1.1.1"><times id="S4.Ex8.m1.3.3.1.1.1.1.cmml" xref="S4.Ex8.m1.3.3.1.1.1.1"></times><apply id="S4.Ex8.m1.3.3.1.1.1.2.cmml" xref="S4.Ex8.m1.3.3.1.1.1.2"><csymbol cd="ambiguous" id="S4.Ex8.m1.3.3.1.1.1.2.1.cmml" xref="S4.Ex8.m1.3.3.1.1.1.2">superscript</csymbol><ci id="S4.Ex8.m1.3.3.1.1.1.2.2.cmml" xref="S4.Ex8.m1.3.3.1.1.1.2.2">𝑠</ci><ci id="S4.Ex8.m1.3.3.1.1.1.2.3.cmml" xref="S4.Ex8.m1.3.3.1.1.1.2.3">⌢</ci></apply><apply id="S4.Ex8.m1.3.3.1.1.1.3.1.cmml" xref="S4.Ex8.m1.3.3.1.1.1.3.2"><csymbol cd="latexml" id="S4.Ex8.m1.3.3.1.1.1.3.1.1.cmml" xref="S4.Ex8.m1.3.3.1.1.1.3.2.1">delimited-⟨⟩</csymbol><ci id="S4.Ex8.m1.1.1.cmml" xref="S4.Ex8.m1.1.1">𝜔</ci></apply></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex8.m1.3c">\displaystyle\tilde{c}(t,s^{\frown}\langle\omega\rangle)</annotation><annotation encoding="application/x-llamapun" id="S4.Ex8.m1.3d">over~ start_ARG italic_c end_ARG ( italic_t , italic_s start_POSTSUPERSCRIPT ⌢ end_POSTSUPERSCRIPT ⟨ italic_ω ⟩ )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle:=(0,1,f(t),f(s),c(t,s))," class="ltx_Math" display="inline" id="S4.Ex8.m2.7"><semantics id="S4.Ex8.m2.7a"><mrow id="S4.Ex8.m2.7.7.1" xref="S4.Ex8.m2.7.7.1.1.cmml"><mrow id="S4.Ex8.m2.7.7.1.1" xref="S4.Ex8.m2.7.7.1.1.cmml"><mi id="S4.Ex8.m2.7.7.1.1.5" xref="S4.Ex8.m2.7.7.1.1.5.cmml"></mi><mo id="S4.Ex8.m2.7.7.1.1.4" lspace="0.278em" rspace="0.278em" xref="S4.Ex8.m2.7.7.1.1.4.cmml">:=</mo><mrow id="S4.Ex8.m2.7.7.1.1.3.3" xref="S4.Ex8.m2.7.7.1.1.3.4.cmml"><mo id="S4.Ex8.m2.7.7.1.1.3.3.4" stretchy="false" xref="S4.Ex8.m2.7.7.1.1.3.4.cmml">(</mo><mn id="S4.Ex8.m2.5.5" xref="S4.Ex8.m2.5.5.cmml">0</mn><mo id="S4.Ex8.m2.7.7.1.1.3.3.5" xref="S4.Ex8.m2.7.7.1.1.3.4.cmml">,</mo><mn id="S4.Ex8.m2.6.6" xref="S4.Ex8.m2.6.6.cmml">1</mn><mo id="S4.Ex8.m2.7.7.1.1.3.3.6" xref="S4.Ex8.m2.7.7.1.1.3.4.cmml">,</mo><mrow id="S4.Ex8.m2.7.7.1.1.1.1.1" xref="S4.Ex8.m2.7.7.1.1.1.1.1.cmml"><mi id="S4.Ex8.m2.7.7.1.1.1.1.1.2" xref="S4.Ex8.m2.7.7.1.1.1.1.1.2.cmml">f</mi><mo id="S4.Ex8.m2.7.7.1.1.1.1.1.1" xref="S4.Ex8.m2.7.7.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.Ex8.m2.7.7.1.1.1.1.1.3.2" xref="S4.Ex8.m2.7.7.1.1.1.1.1.cmml"><mo id="S4.Ex8.m2.7.7.1.1.1.1.1.3.2.1" stretchy="false" xref="S4.Ex8.m2.7.7.1.1.1.1.1.cmml">(</mo><mi id="S4.Ex8.m2.1.1" xref="S4.Ex8.m2.1.1.cmml">t</mi><mo id="S4.Ex8.m2.7.7.1.1.1.1.1.3.2.2" stretchy="false" xref="S4.Ex8.m2.7.7.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex8.m2.7.7.1.1.3.3.7" xref="S4.Ex8.m2.7.7.1.1.3.4.cmml">,</mo><mrow id="S4.Ex8.m2.7.7.1.1.2.2.2" xref="S4.Ex8.m2.7.7.1.1.2.2.2.cmml"><mi id="S4.Ex8.m2.7.7.1.1.2.2.2.2" xref="S4.Ex8.m2.7.7.1.1.2.2.2.2.cmml">f</mi><mo id="S4.Ex8.m2.7.7.1.1.2.2.2.1" xref="S4.Ex8.m2.7.7.1.1.2.2.2.1.cmml">⁢</mo><mrow id="S4.Ex8.m2.7.7.1.1.2.2.2.3.2" xref="S4.Ex8.m2.7.7.1.1.2.2.2.cmml"><mo id="S4.Ex8.m2.7.7.1.1.2.2.2.3.2.1" stretchy="false" xref="S4.Ex8.m2.7.7.1.1.2.2.2.cmml">(</mo><mi id="S4.Ex8.m2.2.2" xref="S4.Ex8.m2.2.2.cmml">s</mi><mo id="S4.Ex8.m2.7.7.1.1.2.2.2.3.2.2" stretchy="false" xref="S4.Ex8.m2.7.7.1.1.2.2.2.cmml">)</mo></mrow></mrow><mo id="S4.Ex8.m2.7.7.1.1.3.3.8" xref="S4.Ex8.m2.7.7.1.1.3.4.cmml">,</mo><mrow id="S4.Ex8.m2.7.7.1.1.3.3.3" xref="S4.Ex8.m2.7.7.1.1.3.3.3.cmml"><mi id="S4.Ex8.m2.7.7.1.1.3.3.3.2" xref="S4.Ex8.m2.7.7.1.1.3.3.3.2.cmml">c</mi><mo id="S4.Ex8.m2.7.7.1.1.3.3.3.1" xref="S4.Ex8.m2.7.7.1.1.3.3.3.1.cmml">⁢</mo><mrow id="S4.Ex8.m2.7.7.1.1.3.3.3.3.2" xref="S4.Ex8.m2.7.7.1.1.3.3.3.3.1.cmml"><mo id="S4.Ex8.m2.7.7.1.1.3.3.3.3.2.1" stretchy="false" xref="S4.Ex8.m2.7.7.1.1.3.3.3.3.1.cmml">(</mo><mi id="S4.Ex8.m2.3.3" xref="S4.Ex8.m2.3.3.cmml">t</mi><mo id="S4.Ex8.m2.7.7.1.1.3.3.3.3.2.2" xref="S4.Ex8.m2.7.7.1.1.3.3.3.3.1.cmml">,</mo><mi id="S4.Ex8.m2.4.4" xref="S4.Ex8.m2.4.4.cmml">s</mi><mo id="S4.Ex8.m2.7.7.1.1.3.3.3.3.2.3" stretchy="false" xref="S4.Ex8.m2.7.7.1.1.3.3.3.3.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex8.m2.7.7.1.1.3.3.9" stretchy="false" xref="S4.Ex8.m2.7.7.1.1.3.4.cmml">)</mo></mrow></mrow><mo id="S4.Ex8.m2.7.7.1.2" xref="S4.Ex8.m2.7.7.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex8.m2.7b"><apply id="S4.Ex8.m2.7.7.1.1.cmml" xref="S4.Ex8.m2.7.7.1"><csymbol cd="latexml" id="S4.Ex8.m2.7.7.1.1.4.cmml" xref="S4.Ex8.m2.7.7.1.1.4">assign</csymbol><csymbol cd="latexml" id="S4.Ex8.m2.7.7.1.1.5.cmml" xref="S4.Ex8.m2.7.7.1.1.5">absent</csymbol><vector id="S4.Ex8.m2.7.7.1.1.3.4.cmml" xref="S4.Ex8.m2.7.7.1.1.3.3"><cn id="S4.Ex8.m2.5.5.cmml" type="integer" xref="S4.Ex8.m2.5.5">0</cn><cn id="S4.Ex8.m2.6.6.cmml" type="integer" xref="S4.Ex8.m2.6.6">1</cn><apply id="S4.Ex8.m2.7.7.1.1.1.1.1.cmml" xref="S4.Ex8.m2.7.7.1.1.1.1.1"><times id="S4.Ex8.m2.7.7.1.1.1.1.1.1.cmml" xref="S4.Ex8.m2.7.7.1.1.1.1.1.1"></times><ci id="S4.Ex8.m2.7.7.1.1.1.1.1.2.cmml" xref="S4.Ex8.m2.7.7.1.1.1.1.1.2">𝑓</ci><ci id="S4.Ex8.m2.1.1.cmml" xref="S4.Ex8.m2.1.1">𝑡</ci></apply><apply id="S4.Ex8.m2.7.7.1.1.2.2.2.cmml" xref="S4.Ex8.m2.7.7.1.1.2.2.2"><times id="S4.Ex8.m2.7.7.1.1.2.2.2.1.cmml" xref="S4.Ex8.m2.7.7.1.1.2.2.2.1"></times><ci id="S4.Ex8.m2.7.7.1.1.2.2.2.2.cmml" xref="S4.Ex8.m2.7.7.1.1.2.2.2.2">𝑓</ci><ci id="S4.Ex8.m2.2.2.cmml" xref="S4.Ex8.m2.2.2">𝑠</ci></apply><apply id="S4.Ex8.m2.7.7.1.1.3.3.3.cmml" xref="S4.Ex8.m2.7.7.1.1.3.3.3"><times id="S4.Ex8.m2.7.7.1.1.3.3.3.1.cmml" xref="S4.Ex8.m2.7.7.1.1.3.3.3.1"></times><ci id="S4.Ex8.m2.7.7.1.1.3.3.3.2.cmml" xref="S4.Ex8.m2.7.7.1.1.3.3.3.2">𝑐</ci><interval closure="open" id="S4.Ex8.m2.7.7.1.1.3.3.3.3.1.cmml" xref="S4.Ex8.m2.7.7.1.1.3.3.3.3.2"><ci id="S4.Ex8.m2.3.3.cmml" xref="S4.Ex8.m2.3.3">𝑡</ci><ci id="S4.Ex8.m2.4.4.cmml" xref="S4.Ex8.m2.4.4">𝑠</ci></interval></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex8.m2.7c">\displaystyle:=(0,1,f(t),f(s),c(t,s)),</annotation><annotation encoding="application/x-llamapun" id="S4.Ex8.m2.7d">:= ( 0 , 1 , italic_f ( italic_t ) , italic_f ( italic_s ) , italic_c ( italic_t , italic_s ) ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S4.Ex9"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\tilde{c}(t^{\frown}\langle\omega\rangle,s)" class="ltx_Math" display="inline" id="S4.Ex9.m1.3"><semantics id="S4.Ex9.m1.3a"><mrow id="S4.Ex9.m1.3.3" xref="S4.Ex9.m1.3.3.cmml"><mover accent="true" id="S4.Ex9.m1.3.3.3" xref="S4.Ex9.m1.3.3.3.cmml"><mi id="S4.Ex9.m1.3.3.3.2" xref="S4.Ex9.m1.3.3.3.2.cmml">c</mi><mo id="S4.Ex9.m1.3.3.3.1" xref="S4.Ex9.m1.3.3.3.1.cmml">~</mo></mover><mo id="S4.Ex9.m1.3.3.2" xref="S4.Ex9.m1.3.3.2.cmml">⁢</mo><mrow id="S4.Ex9.m1.3.3.1.1" xref="S4.Ex9.m1.3.3.1.2.cmml"><mo id="S4.Ex9.m1.3.3.1.1.2" stretchy="false" xref="S4.Ex9.m1.3.3.1.2.cmml">(</mo><mrow id="S4.Ex9.m1.3.3.1.1.1" xref="S4.Ex9.m1.3.3.1.1.1.cmml"><msup id="S4.Ex9.m1.3.3.1.1.1.2" xref="S4.Ex9.m1.3.3.1.1.1.2.cmml"><mi id="S4.Ex9.m1.3.3.1.1.1.2.2" xref="S4.Ex9.m1.3.3.1.1.1.2.2.cmml">t</mi><mo id="S4.Ex9.m1.3.3.1.1.1.2.3" xref="S4.Ex9.m1.3.3.1.1.1.2.3.cmml">⌢</mo></msup><mo id="S4.Ex9.m1.3.3.1.1.1.1" xref="S4.Ex9.m1.3.3.1.1.1.1.cmml">⁢</mo><mrow id="S4.Ex9.m1.3.3.1.1.1.3.2" xref="S4.Ex9.m1.3.3.1.1.1.3.1.cmml"><mo id="S4.Ex9.m1.3.3.1.1.1.3.2.1" stretchy="false" xref="S4.Ex9.m1.3.3.1.1.1.3.1.1.cmml">⟨</mo><mi id="S4.Ex9.m1.1.1" xref="S4.Ex9.m1.1.1.cmml">ω</mi><mo id="S4.Ex9.m1.3.3.1.1.1.3.2.2" stretchy="false" xref="S4.Ex9.m1.3.3.1.1.1.3.1.1.cmml">⟩</mo></mrow></mrow><mo id="S4.Ex9.m1.3.3.1.1.3" xref="S4.Ex9.m1.3.3.1.2.cmml">,</mo><mi id="S4.Ex9.m1.2.2" xref="S4.Ex9.m1.2.2.cmml">s</mi><mo id="S4.Ex9.m1.3.3.1.1.4" stretchy="false" xref="S4.Ex9.m1.3.3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex9.m1.3b"><apply id="S4.Ex9.m1.3.3.cmml" xref="S4.Ex9.m1.3.3"><times id="S4.Ex9.m1.3.3.2.cmml" xref="S4.Ex9.m1.3.3.2"></times><apply id="S4.Ex9.m1.3.3.3.cmml" xref="S4.Ex9.m1.3.3.3"><ci id="S4.Ex9.m1.3.3.3.1.cmml" xref="S4.Ex9.m1.3.3.3.1">~</ci><ci id="S4.Ex9.m1.3.3.3.2.cmml" xref="S4.Ex9.m1.3.3.3.2">𝑐</ci></apply><interval closure="open" id="S4.Ex9.m1.3.3.1.2.cmml" xref="S4.Ex9.m1.3.3.1.1"><apply id="S4.Ex9.m1.3.3.1.1.1.cmml" xref="S4.Ex9.m1.3.3.1.1.1"><times id="S4.Ex9.m1.3.3.1.1.1.1.cmml" xref="S4.Ex9.m1.3.3.1.1.1.1"></times><apply id="S4.Ex9.m1.3.3.1.1.1.2.cmml" xref="S4.Ex9.m1.3.3.1.1.1.2"><csymbol cd="ambiguous" id="S4.Ex9.m1.3.3.1.1.1.2.1.cmml" xref="S4.Ex9.m1.3.3.1.1.1.2">superscript</csymbol><ci id="S4.Ex9.m1.3.3.1.1.1.2.2.cmml" xref="S4.Ex9.m1.3.3.1.1.1.2.2">𝑡</ci><ci id="S4.Ex9.m1.3.3.1.1.1.2.3.cmml" xref="S4.Ex9.m1.3.3.1.1.1.2.3">⌢</ci></apply><apply id="S4.Ex9.m1.3.3.1.1.1.3.1.cmml" xref="S4.Ex9.m1.3.3.1.1.1.3.2"><csymbol cd="latexml" id="S4.Ex9.m1.3.3.1.1.1.3.1.1.cmml" xref="S4.Ex9.m1.3.3.1.1.1.3.2.1">delimited-⟨⟩</csymbol><ci id="S4.Ex9.m1.1.1.cmml" xref="S4.Ex9.m1.1.1">𝜔</ci></apply></apply><ci id="S4.Ex9.m1.2.2.cmml" xref="S4.Ex9.m1.2.2">𝑠</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex9.m1.3c">\displaystyle\tilde{c}(t^{\frown}\langle\omega\rangle,s)</annotation><annotation encoding="application/x-llamapun" id="S4.Ex9.m1.3d">over~ start_ARG italic_c end_ARG ( italic_t start_POSTSUPERSCRIPT ⌢ end_POSTSUPERSCRIPT ⟨ italic_ω ⟩ , italic_s )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle:=(1,0,f(t),f(s),c(t,s))," class="ltx_Math" display="inline" id="S4.Ex9.m2.7"><semantics id="S4.Ex9.m2.7a"><mrow id="S4.Ex9.m2.7.7.1" xref="S4.Ex9.m2.7.7.1.1.cmml"><mrow id="S4.Ex9.m2.7.7.1.1" xref="S4.Ex9.m2.7.7.1.1.cmml"><mi id="S4.Ex9.m2.7.7.1.1.5" xref="S4.Ex9.m2.7.7.1.1.5.cmml"></mi><mo id="S4.Ex9.m2.7.7.1.1.4" lspace="0.278em" rspace="0.278em" xref="S4.Ex9.m2.7.7.1.1.4.cmml">:=</mo><mrow id="S4.Ex9.m2.7.7.1.1.3.3" xref="S4.Ex9.m2.7.7.1.1.3.4.cmml"><mo id="S4.Ex9.m2.7.7.1.1.3.3.4" stretchy="false" xref="S4.Ex9.m2.7.7.1.1.3.4.cmml">(</mo><mn id="S4.Ex9.m2.5.5" xref="S4.Ex9.m2.5.5.cmml">1</mn><mo id="S4.Ex9.m2.7.7.1.1.3.3.5" xref="S4.Ex9.m2.7.7.1.1.3.4.cmml">,</mo><mn id="S4.Ex9.m2.6.6" xref="S4.Ex9.m2.6.6.cmml">0</mn><mo id="S4.Ex9.m2.7.7.1.1.3.3.6" xref="S4.Ex9.m2.7.7.1.1.3.4.cmml">,</mo><mrow id="S4.Ex9.m2.7.7.1.1.1.1.1" xref="S4.Ex9.m2.7.7.1.1.1.1.1.cmml"><mi id="S4.Ex9.m2.7.7.1.1.1.1.1.2" xref="S4.Ex9.m2.7.7.1.1.1.1.1.2.cmml">f</mi><mo id="S4.Ex9.m2.7.7.1.1.1.1.1.1" xref="S4.Ex9.m2.7.7.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.Ex9.m2.7.7.1.1.1.1.1.3.2" xref="S4.Ex9.m2.7.7.1.1.1.1.1.cmml"><mo id="S4.Ex9.m2.7.7.1.1.1.1.1.3.2.1" stretchy="false" xref="S4.Ex9.m2.7.7.1.1.1.1.1.cmml">(</mo><mi id="S4.Ex9.m2.1.1" xref="S4.Ex9.m2.1.1.cmml">t</mi><mo id="S4.Ex9.m2.7.7.1.1.1.1.1.3.2.2" stretchy="false" xref="S4.Ex9.m2.7.7.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex9.m2.7.7.1.1.3.3.7" xref="S4.Ex9.m2.7.7.1.1.3.4.cmml">,</mo><mrow id="S4.Ex9.m2.7.7.1.1.2.2.2" xref="S4.Ex9.m2.7.7.1.1.2.2.2.cmml"><mi id="S4.Ex9.m2.7.7.1.1.2.2.2.2" xref="S4.Ex9.m2.7.7.1.1.2.2.2.2.cmml">f</mi><mo id="S4.Ex9.m2.7.7.1.1.2.2.2.1" xref="S4.Ex9.m2.7.7.1.1.2.2.2.1.cmml">⁢</mo><mrow id="S4.Ex9.m2.7.7.1.1.2.2.2.3.2" xref="S4.Ex9.m2.7.7.1.1.2.2.2.cmml"><mo id="S4.Ex9.m2.7.7.1.1.2.2.2.3.2.1" stretchy="false" xref="S4.Ex9.m2.7.7.1.1.2.2.2.cmml">(</mo><mi id="S4.Ex9.m2.2.2" xref="S4.Ex9.m2.2.2.cmml">s</mi><mo id="S4.Ex9.m2.7.7.1.1.2.2.2.3.2.2" stretchy="false" xref="S4.Ex9.m2.7.7.1.1.2.2.2.cmml">)</mo></mrow></mrow><mo id="S4.Ex9.m2.7.7.1.1.3.3.8" xref="S4.Ex9.m2.7.7.1.1.3.4.cmml">,</mo><mrow id="S4.Ex9.m2.7.7.1.1.3.3.3" xref="S4.Ex9.m2.7.7.1.1.3.3.3.cmml"><mi id="S4.Ex9.m2.7.7.1.1.3.3.3.2" xref="S4.Ex9.m2.7.7.1.1.3.3.3.2.cmml">c</mi><mo id="S4.Ex9.m2.7.7.1.1.3.3.3.1" xref="S4.Ex9.m2.7.7.1.1.3.3.3.1.cmml">⁢</mo><mrow id="S4.Ex9.m2.7.7.1.1.3.3.3.3.2" xref="S4.Ex9.m2.7.7.1.1.3.3.3.3.1.cmml"><mo id="S4.Ex9.m2.7.7.1.1.3.3.3.3.2.1" stretchy="false" xref="S4.Ex9.m2.7.7.1.1.3.3.3.3.1.cmml">(</mo><mi id="S4.Ex9.m2.3.3" xref="S4.Ex9.m2.3.3.cmml">t</mi><mo id="S4.Ex9.m2.7.7.1.1.3.3.3.3.2.2" xref="S4.Ex9.m2.7.7.1.1.3.3.3.3.1.cmml">,</mo><mi id="S4.Ex9.m2.4.4" xref="S4.Ex9.m2.4.4.cmml">s</mi><mo id="S4.Ex9.m2.7.7.1.1.3.3.3.3.2.3" stretchy="false" xref="S4.Ex9.m2.7.7.1.1.3.3.3.3.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex9.m2.7.7.1.1.3.3.9" stretchy="false" xref="S4.Ex9.m2.7.7.1.1.3.4.cmml">)</mo></mrow></mrow><mo id="S4.Ex9.m2.7.7.1.2" xref="S4.Ex9.m2.7.7.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex9.m2.7b"><apply id="S4.Ex9.m2.7.7.1.1.cmml" xref="S4.Ex9.m2.7.7.1"><csymbol cd="latexml" id="S4.Ex9.m2.7.7.1.1.4.cmml" xref="S4.Ex9.m2.7.7.1.1.4">assign</csymbol><csymbol cd="latexml" id="S4.Ex9.m2.7.7.1.1.5.cmml" xref="S4.Ex9.m2.7.7.1.1.5">absent</csymbol><vector id="S4.Ex9.m2.7.7.1.1.3.4.cmml" xref="S4.Ex9.m2.7.7.1.1.3.3"><cn id="S4.Ex9.m2.5.5.cmml" type="integer" xref="S4.Ex9.m2.5.5">1</cn><cn id="S4.Ex9.m2.6.6.cmml" type="integer" xref="S4.Ex9.m2.6.6">0</cn><apply id="S4.Ex9.m2.7.7.1.1.1.1.1.cmml" xref="S4.Ex9.m2.7.7.1.1.1.1.1"><times id="S4.Ex9.m2.7.7.1.1.1.1.1.1.cmml" xref="S4.Ex9.m2.7.7.1.1.1.1.1.1"></times><ci id="S4.Ex9.m2.7.7.1.1.1.1.1.2.cmml" xref="S4.Ex9.m2.7.7.1.1.1.1.1.2">𝑓</ci><ci id="S4.Ex9.m2.1.1.cmml" xref="S4.Ex9.m2.1.1">𝑡</ci></apply><apply id="S4.Ex9.m2.7.7.1.1.2.2.2.cmml" xref="S4.Ex9.m2.7.7.1.1.2.2.2"><times id="S4.Ex9.m2.7.7.1.1.2.2.2.1.cmml" xref="S4.Ex9.m2.7.7.1.1.2.2.2.1"></times><ci id="S4.Ex9.m2.7.7.1.1.2.2.2.2.cmml" xref="S4.Ex9.m2.7.7.1.1.2.2.2.2">𝑓</ci><ci id="S4.Ex9.m2.2.2.cmml" xref="S4.Ex9.m2.2.2">𝑠</ci></apply><apply id="S4.Ex9.m2.7.7.1.1.3.3.3.cmml" xref="S4.Ex9.m2.7.7.1.1.3.3.3"><times id="S4.Ex9.m2.7.7.1.1.3.3.3.1.cmml" xref="S4.Ex9.m2.7.7.1.1.3.3.3.1"></times><ci id="S4.Ex9.m2.7.7.1.1.3.3.3.2.cmml" xref="S4.Ex9.m2.7.7.1.1.3.3.3.2">𝑐</ci><interval closure="open" id="S4.Ex9.m2.7.7.1.1.3.3.3.3.1.cmml" xref="S4.Ex9.m2.7.7.1.1.3.3.3.3.2"><ci id="S4.Ex9.m2.3.3.cmml" xref="S4.Ex9.m2.3.3">𝑡</ci><ci id="S4.Ex9.m2.4.4.cmml" xref="S4.Ex9.m2.4.4">𝑠</ci></interval></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex9.m2.7c">\displaystyle:=(1,0,f(t),f(s),c(t,s)),</annotation><annotation encoding="application/x-llamapun" id="S4.Ex9.m2.7d">:= ( 1 , 0 , italic_f ( italic_t ) , italic_f ( italic_s ) , italic_c ( italic_t , italic_s ) ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S4.Ex10"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\tilde{c}(t^{\frown}\langle\omega\rangle,s^{\frown}\langle\omega\rangle)" class="ltx_Math" display="inline" id="S4.Ex10.m1.4"><semantics id="S4.Ex10.m1.4a"><mrow id="S4.Ex10.m1.4.4" xref="S4.Ex10.m1.4.4.cmml"><mover accent="true" id="S4.Ex10.m1.4.4.4" xref="S4.Ex10.m1.4.4.4.cmml"><mi id="S4.Ex10.m1.4.4.4.2" xref="S4.Ex10.m1.4.4.4.2.cmml">c</mi><mo id="S4.Ex10.m1.4.4.4.1" xref="S4.Ex10.m1.4.4.4.1.cmml">~</mo></mover><mo id="S4.Ex10.m1.4.4.3" xref="S4.Ex10.m1.4.4.3.cmml">⁢</mo><mrow id="S4.Ex10.m1.4.4.2.2" xref="S4.Ex10.m1.4.4.2.3.cmml"><mo id="S4.Ex10.m1.4.4.2.2.3" stretchy="false" xref="S4.Ex10.m1.4.4.2.3.cmml">(</mo><mrow id="S4.Ex10.m1.3.3.1.1.1" xref="S4.Ex10.m1.3.3.1.1.1.cmml"><msup id="S4.Ex10.m1.3.3.1.1.1.2" xref="S4.Ex10.m1.3.3.1.1.1.2.cmml"><mi id="S4.Ex10.m1.3.3.1.1.1.2.2" xref="S4.Ex10.m1.3.3.1.1.1.2.2.cmml">t</mi><mo id="S4.Ex10.m1.3.3.1.1.1.2.3" xref="S4.Ex10.m1.3.3.1.1.1.2.3.cmml">⌢</mo></msup><mo id="S4.Ex10.m1.3.3.1.1.1.1" xref="S4.Ex10.m1.3.3.1.1.1.1.cmml">⁢</mo><mrow id="S4.Ex10.m1.3.3.1.1.1.3.2" xref="S4.Ex10.m1.3.3.1.1.1.3.1.cmml"><mo id="S4.Ex10.m1.3.3.1.1.1.3.2.1" stretchy="false" xref="S4.Ex10.m1.3.3.1.1.1.3.1.1.cmml">⟨</mo><mi id="S4.Ex10.m1.1.1" xref="S4.Ex10.m1.1.1.cmml">ω</mi><mo id="S4.Ex10.m1.3.3.1.1.1.3.2.2" stretchy="false" xref="S4.Ex10.m1.3.3.1.1.1.3.1.1.cmml">⟩</mo></mrow></mrow><mo id="S4.Ex10.m1.4.4.2.2.4" xref="S4.Ex10.m1.4.4.2.3.cmml">,</mo><mrow id="S4.Ex10.m1.4.4.2.2.2" xref="S4.Ex10.m1.4.4.2.2.2.cmml"><msup id="S4.Ex10.m1.4.4.2.2.2.2" xref="S4.Ex10.m1.4.4.2.2.2.2.cmml"><mi id="S4.Ex10.m1.4.4.2.2.2.2.2" xref="S4.Ex10.m1.4.4.2.2.2.2.2.cmml">s</mi><mo id="S4.Ex10.m1.4.4.2.2.2.2.3" xref="S4.Ex10.m1.4.4.2.2.2.2.3.cmml">⌢</mo></msup><mo id="S4.Ex10.m1.4.4.2.2.2.1" xref="S4.Ex10.m1.4.4.2.2.2.1.cmml">⁢</mo><mrow id="S4.Ex10.m1.4.4.2.2.2.3.2" xref="S4.Ex10.m1.4.4.2.2.2.3.1.cmml"><mo id="S4.Ex10.m1.4.4.2.2.2.3.2.1" stretchy="false" xref="S4.Ex10.m1.4.4.2.2.2.3.1.1.cmml">⟨</mo><mi id="S4.Ex10.m1.2.2" xref="S4.Ex10.m1.2.2.cmml">ω</mi><mo id="S4.Ex10.m1.4.4.2.2.2.3.2.2" stretchy="false" xref="S4.Ex10.m1.4.4.2.2.2.3.1.1.cmml">⟩</mo></mrow></mrow><mo id="S4.Ex10.m1.4.4.2.2.5" stretchy="false" xref="S4.Ex10.m1.4.4.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex10.m1.4b"><apply id="S4.Ex10.m1.4.4.cmml" xref="S4.Ex10.m1.4.4"><times id="S4.Ex10.m1.4.4.3.cmml" xref="S4.Ex10.m1.4.4.3"></times><apply id="S4.Ex10.m1.4.4.4.cmml" xref="S4.Ex10.m1.4.4.4"><ci id="S4.Ex10.m1.4.4.4.1.cmml" xref="S4.Ex10.m1.4.4.4.1">~</ci><ci id="S4.Ex10.m1.4.4.4.2.cmml" xref="S4.Ex10.m1.4.4.4.2">𝑐</ci></apply><interval closure="open" id="S4.Ex10.m1.4.4.2.3.cmml" xref="S4.Ex10.m1.4.4.2.2"><apply id="S4.Ex10.m1.3.3.1.1.1.cmml" xref="S4.Ex10.m1.3.3.1.1.1"><times id="S4.Ex10.m1.3.3.1.1.1.1.cmml" xref="S4.Ex10.m1.3.3.1.1.1.1"></times><apply id="S4.Ex10.m1.3.3.1.1.1.2.cmml" xref="S4.Ex10.m1.3.3.1.1.1.2"><csymbol cd="ambiguous" id="S4.Ex10.m1.3.3.1.1.1.2.1.cmml" xref="S4.Ex10.m1.3.3.1.1.1.2">superscript</csymbol><ci id="S4.Ex10.m1.3.3.1.1.1.2.2.cmml" xref="S4.Ex10.m1.3.3.1.1.1.2.2">𝑡</ci><ci id="S4.Ex10.m1.3.3.1.1.1.2.3.cmml" xref="S4.Ex10.m1.3.3.1.1.1.2.3">⌢</ci></apply><apply id="S4.Ex10.m1.3.3.1.1.1.3.1.cmml" xref="S4.Ex10.m1.3.3.1.1.1.3.2"><csymbol cd="latexml" id="S4.Ex10.m1.3.3.1.1.1.3.1.1.cmml" xref="S4.Ex10.m1.3.3.1.1.1.3.2.1">delimited-⟨⟩</csymbol><ci id="S4.Ex10.m1.1.1.cmml" xref="S4.Ex10.m1.1.1">𝜔</ci></apply></apply><apply id="S4.Ex10.m1.4.4.2.2.2.cmml" xref="S4.Ex10.m1.4.4.2.2.2"><times id="S4.Ex10.m1.4.4.2.2.2.1.cmml" xref="S4.Ex10.m1.4.4.2.2.2.1"></times><apply id="S4.Ex10.m1.4.4.2.2.2.2.cmml" xref="S4.Ex10.m1.4.4.2.2.2.2"><csymbol cd="ambiguous" id="S4.Ex10.m1.4.4.2.2.2.2.1.cmml" xref="S4.Ex10.m1.4.4.2.2.2.2">superscript</csymbol><ci id="S4.Ex10.m1.4.4.2.2.2.2.2.cmml" xref="S4.Ex10.m1.4.4.2.2.2.2.2">𝑠</ci><ci id="S4.Ex10.m1.4.4.2.2.2.2.3.cmml" xref="S4.Ex10.m1.4.4.2.2.2.2.3">⌢</ci></apply><apply id="S4.Ex10.m1.4.4.2.2.2.3.1.cmml" xref="S4.Ex10.m1.4.4.2.2.2.3.2"><csymbol cd="latexml" id="S4.Ex10.m1.4.4.2.2.2.3.1.1.cmml" xref="S4.Ex10.m1.4.4.2.2.2.3.2.1">delimited-⟨⟩</csymbol><ci id="S4.Ex10.m1.2.2.cmml" xref="S4.Ex10.m1.2.2">𝜔</ci></apply></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex10.m1.4c">\displaystyle\tilde{c}(t^{\frown}\langle\omega\rangle,s^{\frown}\langle\omega\rangle)</annotation><annotation encoding="application/x-llamapun" id="S4.Ex10.m1.4d">over~ start_ARG italic_c end_ARG ( italic_t start_POSTSUPERSCRIPT ⌢ end_POSTSUPERSCRIPT ⟨ italic_ω ⟩ , italic_s start_POSTSUPERSCRIPT ⌢ end_POSTSUPERSCRIPT ⟨ italic_ω ⟩ )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle:=(1,1,f(t),f(s),c(t,s))." class="ltx_Math" display="inline" id="S4.Ex10.m2.7"><semantics id="S4.Ex10.m2.7a"><mrow id="S4.Ex10.m2.7.7.1" xref="S4.Ex10.m2.7.7.1.1.cmml"><mrow id="S4.Ex10.m2.7.7.1.1" xref="S4.Ex10.m2.7.7.1.1.cmml"><mi id="S4.Ex10.m2.7.7.1.1.5" xref="S4.Ex10.m2.7.7.1.1.5.cmml"></mi><mo id="S4.Ex10.m2.7.7.1.1.4" lspace="0.278em" rspace="0.278em" xref="S4.Ex10.m2.7.7.1.1.4.cmml">:=</mo><mrow id="S4.Ex10.m2.7.7.1.1.3.3" xref="S4.Ex10.m2.7.7.1.1.3.4.cmml"><mo id="S4.Ex10.m2.7.7.1.1.3.3.4" stretchy="false" xref="S4.Ex10.m2.7.7.1.1.3.4.cmml">(</mo><mn id="S4.Ex10.m2.5.5" xref="S4.Ex10.m2.5.5.cmml">1</mn><mo id="S4.Ex10.m2.7.7.1.1.3.3.5" xref="S4.Ex10.m2.7.7.1.1.3.4.cmml">,</mo><mn id="S4.Ex10.m2.6.6" xref="S4.Ex10.m2.6.6.cmml">1</mn><mo id="S4.Ex10.m2.7.7.1.1.3.3.6" xref="S4.Ex10.m2.7.7.1.1.3.4.cmml">,</mo><mrow id="S4.Ex10.m2.7.7.1.1.1.1.1" xref="S4.Ex10.m2.7.7.1.1.1.1.1.cmml"><mi id="S4.Ex10.m2.7.7.1.1.1.1.1.2" xref="S4.Ex10.m2.7.7.1.1.1.1.1.2.cmml">f</mi><mo id="S4.Ex10.m2.7.7.1.1.1.1.1.1" xref="S4.Ex10.m2.7.7.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.Ex10.m2.7.7.1.1.1.1.1.3.2" xref="S4.Ex10.m2.7.7.1.1.1.1.1.cmml"><mo id="S4.Ex10.m2.7.7.1.1.1.1.1.3.2.1" stretchy="false" xref="S4.Ex10.m2.7.7.1.1.1.1.1.cmml">(</mo><mi id="S4.Ex10.m2.1.1" xref="S4.Ex10.m2.1.1.cmml">t</mi><mo id="S4.Ex10.m2.7.7.1.1.1.1.1.3.2.2" stretchy="false" xref="S4.Ex10.m2.7.7.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex10.m2.7.7.1.1.3.3.7" xref="S4.Ex10.m2.7.7.1.1.3.4.cmml">,</mo><mrow id="S4.Ex10.m2.7.7.1.1.2.2.2" xref="S4.Ex10.m2.7.7.1.1.2.2.2.cmml"><mi id="S4.Ex10.m2.7.7.1.1.2.2.2.2" xref="S4.Ex10.m2.7.7.1.1.2.2.2.2.cmml">f</mi><mo id="S4.Ex10.m2.7.7.1.1.2.2.2.1" xref="S4.Ex10.m2.7.7.1.1.2.2.2.1.cmml">⁢</mo><mrow id="S4.Ex10.m2.7.7.1.1.2.2.2.3.2" xref="S4.Ex10.m2.7.7.1.1.2.2.2.cmml"><mo id="S4.Ex10.m2.7.7.1.1.2.2.2.3.2.1" stretchy="false" xref="S4.Ex10.m2.7.7.1.1.2.2.2.cmml">(</mo><mi id="S4.Ex10.m2.2.2" xref="S4.Ex10.m2.2.2.cmml">s</mi><mo id="S4.Ex10.m2.7.7.1.1.2.2.2.3.2.2" stretchy="false" xref="S4.Ex10.m2.7.7.1.1.2.2.2.cmml">)</mo></mrow></mrow><mo id="S4.Ex10.m2.7.7.1.1.3.3.8" xref="S4.Ex10.m2.7.7.1.1.3.4.cmml">,</mo><mrow id="S4.Ex10.m2.7.7.1.1.3.3.3" xref="S4.Ex10.m2.7.7.1.1.3.3.3.cmml"><mi id="S4.Ex10.m2.7.7.1.1.3.3.3.2" xref="S4.Ex10.m2.7.7.1.1.3.3.3.2.cmml">c</mi><mo id="S4.Ex10.m2.7.7.1.1.3.3.3.1" xref="S4.Ex10.m2.7.7.1.1.3.3.3.1.cmml">⁢</mo><mrow id="S4.Ex10.m2.7.7.1.1.3.3.3.3.2" xref="S4.Ex10.m2.7.7.1.1.3.3.3.3.1.cmml"><mo id="S4.Ex10.m2.7.7.1.1.3.3.3.3.2.1" stretchy="false" xref="S4.Ex10.m2.7.7.1.1.3.3.3.3.1.cmml">(</mo><mi id="S4.Ex10.m2.3.3" xref="S4.Ex10.m2.3.3.cmml">t</mi><mo id="S4.Ex10.m2.7.7.1.1.3.3.3.3.2.2" xref="S4.Ex10.m2.7.7.1.1.3.3.3.3.1.cmml">,</mo><mi id="S4.Ex10.m2.4.4" xref="S4.Ex10.m2.4.4.cmml">s</mi><mo id="S4.Ex10.m2.7.7.1.1.3.3.3.3.2.3" stretchy="false" xref="S4.Ex10.m2.7.7.1.1.3.3.3.3.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex10.m2.7.7.1.1.3.3.9" stretchy="false" xref="S4.Ex10.m2.7.7.1.1.3.4.cmml">)</mo></mrow></mrow><mo id="S4.Ex10.m2.7.7.1.2" lspace="0em" xref="S4.Ex10.m2.7.7.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex10.m2.7b"><apply id="S4.Ex10.m2.7.7.1.1.cmml" xref="S4.Ex10.m2.7.7.1"><csymbol cd="latexml" id="S4.Ex10.m2.7.7.1.1.4.cmml" xref="S4.Ex10.m2.7.7.1.1.4">assign</csymbol><csymbol cd="latexml" id="S4.Ex10.m2.7.7.1.1.5.cmml" xref="S4.Ex10.m2.7.7.1.1.5">absent</csymbol><vector id="S4.Ex10.m2.7.7.1.1.3.4.cmml" xref="S4.Ex10.m2.7.7.1.1.3.3"><cn id="S4.Ex10.m2.5.5.cmml" type="integer" xref="S4.Ex10.m2.5.5">1</cn><cn id="S4.Ex10.m2.6.6.cmml" type="integer" xref="S4.Ex10.m2.6.6">1</cn><apply id="S4.Ex10.m2.7.7.1.1.1.1.1.cmml" xref="S4.Ex10.m2.7.7.1.1.1.1.1"><times id="S4.Ex10.m2.7.7.1.1.1.1.1.1.cmml" xref="S4.Ex10.m2.7.7.1.1.1.1.1.1"></times><ci id="S4.Ex10.m2.7.7.1.1.1.1.1.2.cmml" xref="S4.Ex10.m2.7.7.1.1.1.1.1.2">𝑓</ci><ci id="S4.Ex10.m2.1.1.cmml" xref="S4.Ex10.m2.1.1">𝑡</ci></apply><apply id="S4.Ex10.m2.7.7.1.1.2.2.2.cmml" xref="S4.Ex10.m2.7.7.1.1.2.2.2"><times id="S4.Ex10.m2.7.7.1.1.2.2.2.1.cmml" xref="S4.Ex10.m2.7.7.1.1.2.2.2.1"></times><ci id="S4.Ex10.m2.7.7.1.1.2.2.2.2.cmml" xref="S4.Ex10.m2.7.7.1.1.2.2.2.2">𝑓</ci><ci id="S4.Ex10.m2.2.2.cmml" xref="S4.Ex10.m2.2.2">𝑠</ci></apply><apply id="S4.Ex10.m2.7.7.1.1.3.3.3.cmml" xref="S4.Ex10.m2.7.7.1.1.3.3.3"><times id="S4.Ex10.m2.7.7.1.1.3.3.3.1.cmml" xref="S4.Ex10.m2.7.7.1.1.3.3.3.1"></times><ci id="S4.Ex10.m2.7.7.1.1.3.3.3.2.cmml" xref="S4.Ex10.m2.7.7.1.1.3.3.3.2">𝑐</ci><interval closure="open" id="S4.Ex10.m2.7.7.1.1.3.3.3.3.1.cmml" xref="S4.Ex10.m2.7.7.1.1.3.3.3.3.2"><ci id="S4.Ex10.m2.3.3.cmml" xref="S4.Ex10.m2.3.3">𝑡</ci><ci id="S4.Ex10.m2.4.4.cmml" xref="S4.Ex10.m2.4.4">𝑠</ci></interval></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex10.m2.7c">\displaystyle:=(1,1,f(t),f(s),c(t,s)).</annotation><annotation encoding="application/x-llamapun" id="S4.Ex10.m2.7d">:= ( 1 , 1 , italic_f ( italic_t ) , italic_f ( italic_s ) , italic_c ( italic_t , italic_s ) ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.12.p5.11">The idea of using the antichains, is that while for <math alttext="t,t^{\prime}\in A_{0}" class="ltx_Math" display="inline" id="S4.12.p5.4.m1.2"><semantics id="S4.12.p5.4.m1.2a"><mrow id="S4.12.p5.4.m1.2.2" xref="S4.12.p5.4.m1.2.2.cmml"><mrow id="S4.12.p5.4.m1.2.2.1.1" xref="S4.12.p5.4.m1.2.2.1.2.cmml"><mi id="S4.12.p5.4.m1.1.1" xref="S4.12.p5.4.m1.1.1.cmml">t</mi><mo id="S4.12.p5.4.m1.2.2.1.1.2" xref="S4.12.p5.4.m1.2.2.1.2.cmml">,</mo><msup id="S4.12.p5.4.m1.2.2.1.1.1" xref="S4.12.p5.4.m1.2.2.1.1.1.cmml"><mi id="S4.12.p5.4.m1.2.2.1.1.1.2" xref="S4.12.p5.4.m1.2.2.1.1.1.2.cmml">t</mi><mo id="S4.12.p5.4.m1.2.2.1.1.1.3" xref="S4.12.p5.4.m1.2.2.1.1.1.3.cmml">′</mo></msup></mrow><mo id="S4.12.p5.4.m1.2.2.2" xref="S4.12.p5.4.m1.2.2.2.cmml">∈</mo><msub id="S4.12.p5.4.m1.2.2.3" xref="S4.12.p5.4.m1.2.2.3.cmml"><mi id="S4.12.p5.4.m1.2.2.3.2" xref="S4.12.p5.4.m1.2.2.3.2.cmml">A</mi><mn id="S4.12.p5.4.m1.2.2.3.3" xref="S4.12.p5.4.m1.2.2.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.12.p5.4.m1.2b"><apply id="S4.12.p5.4.m1.2.2.cmml" xref="S4.12.p5.4.m1.2.2"><in id="S4.12.p5.4.m1.2.2.2.cmml" xref="S4.12.p5.4.m1.2.2.2"></in><list id="S4.12.p5.4.m1.2.2.1.2.cmml" xref="S4.12.p5.4.m1.2.2.1.1"><ci id="S4.12.p5.4.m1.1.1.cmml" xref="S4.12.p5.4.m1.1.1">𝑡</ci><apply id="S4.12.p5.4.m1.2.2.1.1.1.cmml" xref="S4.12.p5.4.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.12.p5.4.m1.2.2.1.1.1.1.cmml" xref="S4.12.p5.4.m1.2.2.1.1.1">superscript</csymbol><ci id="S4.12.p5.4.m1.2.2.1.1.1.2.cmml" xref="S4.12.p5.4.m1.2.2.1.1.1.2">𝑡</ci><ci id="S4.12.p5.4.m1.2.2.1.1.1.3.cmml" xref="S4.12.p5.4.m1.2.2.1.1.1.3">′</ci></apply></list><apply id="S4.12.p5.4.m1.2.2.3.cmml" xref="S4.12.p5.4.m1.2.2.3"><csymbol cd="ambiguous" id="S4.12.p5.4.m1.2.2.3.1.cmml" xref="S4.12.p5.4.m1.2.2.3">subscript</csymbol><ci id="S4.12.p5.4.m1.2.2.3.2.cmml" xref="S4.12.p5.4.m1.2.2.3.2">𝐴</ci><cn id="S4.12.p5.4.m1.2.2.3.3.cmml" type="integer" xref="S4.12.p5.4.m1.2.2.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.12.p5.4.m1.2c">t,t^{\prime}\in A_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.12.p5.4.m1.2d">italic_t , italic_t start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_A start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> <math alttext="t^{\frown}\langle\omega\rangle<_{\mathrm{lex}}t^{\prime\frown}\langle\omega\rangle" class="ltx_Math" display="inline" id="S4.12.p5.5.m2.4"><semantics id="S4.12.p5.5.m2.4a"><mrow id="S4.12.p5.5.m2.4.5" xref="S4.12.p5.5.m2.4.5.cmml"><mrow id="S4.12.p5.5.m2.4.5.2" xref="S4.12.p5.5.m2.4.5.2.cmml"><msup id="S4.12.p5.5.m2.4.5.2.2" xref="S4.12.p5.5.m2.4.5.2.2.cmml"><mi id="S4.12.p5.5.m2.4.5.2.2.2" xref="S4.12.p5.5.m2.4.5.2.2.2.cmml">t</mi><mo id="S4.12.p5.5.m2.4.5.2.2.3" xref="S4.12.p5.5.m2.4.5.2.2.3.cmml">⌢</mo></msup><mo id="S4.12.p5.5.m2.4.5.2.1" xref="S4.12.p5.5.m2.4.5.2.1.cmml">⁢</mo><mrow id="S4.12.p5.5.m2.4.5.2.3.2" xref="S4.12.p5.5.m2.4.5.2.3.1.cmml"><mo id="S4.12.p5.5.m2.4.5.2.3.2.1" stretchy="false" xref="S4.12.p5.5.m2.4.5.2.3.1.1.cmml">⟨</mo><mi id="S4.12.p5.5.m2.3.3" xref="S4.12.p5.5.m2.3.3.cmml">ω</mi><mo id="S4.12.p5.5.m2.4.5.2.3.2.2" stretchy="false" xref="S4.12.p5.5.m2.4.5.2.3.1.1.cmml">⟩</mo></mrow></mrow><msub id="S4.12.p5.5.m2.4.5.1" xref="S4.12.p5.5.m2.4.5.1.cmml"><mo id="S4.12.p5.5.m2.4.5.1.2" xref="S4.12.p5.5.m2.4.5.1.2.cmml"><</mo><mi id="S4.12.p5.5.m2.4.5.1.3" xref="S4.12.p5.5.m2.4.5.1.3.cmml">lex</mi></msub><mrow id="S4.12.p5.5.m2.4.5.3" xref="S4.12.p5.5.m2.4.5.3.cmml"><msup id="S4.12.p5.5.m2.4.5.3.2" xref="S4.12.p5.5.m2.4.5.3.2.cmml"><mi id="S4.12.p5.5.m2.4.5.3.2.2" xref="S4.12.p5.5.m2.4.5.3.2.2.cmml">t</mi><mrow id="S4.12.p5.5.m2.2.2.2.2" xref="S4.12.p5.5.m2.2.2.2.3.cmml"><mo id="S4.12.p5.5.m2.2.2.2.2.1" mathsize="142%" xref="S4.12.p5.5.m2.2.2.2.2.1.cmml">′</mo><mo id="S4.12.p5.5.m2.2.2.2.2.2" lspace="0.278em" xref="S4.12.p5.5.m2.2.2.2.3.cmml">⁣</mo><mo id="S4.12.p5.5.m2.1.1.1.1" xref="S4.12.p5.5.m2.1.1.1.1.cmml">⌢</mo></mrow></msup><mo id="S4.12.p5.5.m2.4.5.3.1" xref="S4.12.p5.5.m2.4.5.3.1.cmml">⁢</mo><mrow id="S4.12.p5.5.m2.4.5.3.3.2" xref="S4.12.p5.5.m2.4.5.3.3.1.cmml"><mo id="S4.12.p5.5.m2.4.5.3.3.2.1" stretchy="false" xref="S4.12.p5.5.m2.4.5.3.3.1.1.cmml">⟨</mo><mi id="S4.12.p5.5.m2.4.4" xref="S4.12.p5.5.m2.4.4.cmml">ω</mi><mo id="S4.12.p5.5.m2.4.5.3.3.2.2" stretchy="false" xref="S4.12.p5.5.m2.4.5.3.3.1.1.cmml">⟩</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.12.p5.5.m2.4b"><apply id="S4.12.p5.5.m2.4.5.cmml" xref="S4.12.p5.5.m2.4.5"><apply id="S4.12.p5.5.m2.4.5.1.cmml" xref="S4.12.p5.5.m2.4.5.1"><csymbol cd="ambiguous" id="S4.12.p5.5.m2.4.5.1.1.cmml" xref="S4.12.p5.5.m2.4.5.1">subscript</csymbol><lt id="S4.12.p5.5.m2.4.5.1.2.cmml" xref="S4.12.p5.5.m2.4.5.1.2"></lt><ci id="S4.12.p5.5.m2.4.5.1.3.cmml" xref="S4.12.p5.5.m2.4.5.1.3">lex</ci></apply><apply id="S4.12.p5.5.m2.4.5.2.cmml" xref="S4.12.p5.5.m2.4.5.2"><times id="S4.12.p5.5.m2.4.5.2.1.cmml" xref="S4.12.p5.5.m2.4.5.2.1"></times><apply id="S4.12.p5.5.m2.4.5.2.2.cmml" xref="S4.12.p5.5.m2.4.5.2.2"><csymbol cd="ambiguous" id="S4.12.p5.5.m2.4.5.2.2.1.cmml" xref="S4.12.p5.5.m2.4.5.2.2">superscript</csymbol><ci id="S4.12.p5.5.m2.4.5.2.2.2.cmml" xref="S4.12.p5.5.m2.4.5.2.2.2">𝑡</ci><ci id="S4.12.p5.5.m2.4.5.2.2.3.cmml" xref="S4.12.p5.5.m2.4.5.2.2.3">⌢</ci></apply><apply id="S4.12.p5.5.m2.4.5.2.3.1.cmml" xref="S4.12.p5.5.m2.4.5.2.3.2"><csymbol cd="latexml" id="S4.12.p5.5.m2.4.5.2.3.1.1.cmml" xref="S4.12.p5.5.m2.4.5.2.3.2.1">delimited-⟨⟩</csymbol><ci id="S4.12.p5.5.m2.3.3.cmml" xref="S4.12.p5.5.m2.3.3">𝜔</ci></apply></apply><apply id="S4.12.p5.5.m2.4.5.3.cmml" xref="S4.12.p5.5.m2.4.5.3"><times id="S4.12.p5.5.m2.4.5.3.1.cmml" xref="S4.12.p5.5.m2.4.5.3.1"></times><apply id="S4.12.p5.5.m2.4.5.3.2.cmml" xref="S4.12.p5.5.m2.4.5.3.2"><csymbol cd="ambiguous" id="S4.12.p5.5.m2.4.5.3.2.1.cmml" xref="S4.12.p5.5.m2.4.5.3.2">superscript</csymbol><ci id="S4.12.p5.5.m2.4.5.3.2.2.cmml" xref="S4.12.p5.5.m2.4.5.3.2.2">𝑡</ci><list id="S4.12.p5.5.m2.2.2.2.3.cmml" xref="S4.12.p5.5.m2.2.2.2.2"><ci id="S4.12.p5.5.m2.2.2.2.2.1.cmml" xref="S4.12.p5.5.m2.2.2.2.2.1">′</ci><ci id="S4.12.p5.5.m2.1.1.1.1.cmml" xref="S4.12.p5.5.m2.1.1.1.1">⌢</ci></list></apply><apply id="S4.12.p5.5.m2.4.5.3.3.1.cmml" xref="S4.12.p5.5.m2.4.5.3.3.2"><csymbol cd="latexml" id="S4.12.p5.5.m2.4.5.3.3.1.1.cmml" xref="S4.12.p5.5.m2.4.5.3.3.2.1">delimited-⟨⟩</csymbol><ci id="S4.12.p5.5.m2.4.4.cmml" xref="S4.12.p5.5.m2.4.4">𝜔</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.12.p5.5.m2.4c">t^{\frown}\langle\omega\rangle<_{\mathrm{lex}}t^{\prime\frown}\langle\omega\rangle</annotation><annotation encoding="application/x-llamapun" id="S4.12.p5.5.m2.4d">italic_t start_POSTSUPERSCRIPT ⌢ end_POSTSUPERSCRIPT ⟨ italic_ω ⟩ < start_POSTSUBSCRIPT roman_lex end_POSTSUBSCRIPT italic_t start_POSTSUPERSCRIPT ′ ⌢ end_POSTSUPERSCRIPT ⟨ italic_ω ⟩</annotation></semantics></math> does not necessarily follow from <math alttext="t<_{\mathrm{lex}}t^{\prime}" class="ltx_Math" display="inline" id="S4.12.p5.6.m3.1"><semantics id="S4.12.p5.6.m3.1a"><mrow id="S4.12.p5.6.m3.1.1" xref="S4.12.p5.6.m3.1.1.cmml"><mi id="S4.12.p5.6.m3.1.1.2" xref="S4.12.p5.6.m3.1.1.2.cmml">t</mi><msub id="S4.12.p5.6.m3.1.1.1" xref="S4.12.p5.6.m3.1.1.1.cmml"><mo id="S4.12.p5.6.m3.1.1.1.2" xref="S4.12.p5.6.m3.1.1.1.2.cmml"><</mo><mi id="S4.12.p5.6.m3.1.1.1.3" xref="S4.12.p5.6.m3.1.1.1.3.cmml">lex</mi></msub><msup id="S4.12.p5.6.m3.1.1.3" xref="S4.12.p5.6.m3.1.1.3.cmml"><mi id="S4.12.p5.6.m3.1.1.3.2" xref="S4.12.p5.6.m3.1.1.3.2.cmml">t</mi><mo id="S4.12.p5.6.m3.1.1.3.3" xref="S4.12.p5.6.m3.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.12.p5.6.m3.1b"><apply id="S4.12.p5.6.m3.1.1.cmml" xref="S4.12.p5.6.m3.1.1"><apply id="S4.12.p5.6.m3.1.1.1.cmml" xref="S4.12.p5.6.m3.1.1.1"><csymbol cd="ambiguous" id="S4.12.p5.6.m3.1.1.1.1.cmml" xref="S4.12.p5.6.m3.1.1.1">subscript</csymbol><lt id="S4.12.p5.6.m3.1.1.1.2.cmml" xref="S4.12.p5.6.m3.1.1.1.2"></lt><ci id="S4.12.p5.6.m3.1.1.1.3.cmml" xref="S4.12.p5.6.m3.1.1.1.3">lex</ci></apply><ci id="S4.12.p5.6.m3.1.1.2.cmml" xref="S4.12.p5.6.m3.1.1.2">𝑡</ci><apply id="S4.12.p5.6.m3.1.1.3.cmml" xref="S4.12.p5.6.m3.1.1.3"><csymbol cd="ambiguous" id="S4.12.p5.6.m3.1.1.3.1.cmml" xref="S4.12.p5.6.m3.1.1.3">superscript</csymbol><ci id="S4.12.p5.6.m3.1.1.3.2.cmml" xref="S4.12.p5.6.m3.1.1.3.2">𝑡</ci><ci id="S4.12.p5.6.m3.1.1.3.3.cmml" xref="S4.12.p5.6.m3.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.12.p5.6.m3.1c">t<_{\mathrm{lex}}t^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.12.p5.6.m3.1d">italic_t < start_POSTSUBSCRIPT roman_lex end_POSTSUBSCRIPT italic_t start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, it does follow if we also have that <math alttext="t" class="ltx_Math" display="inline" id="S4.12.p5.7.m4.1"><semantics id="S4.12.p5.7.m4.1a"><mi id="S4.12.p5.7.m4.1.1" xref="S4.12.p5.7.m4.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S4.12.p5.7.m4.1b"><ci id="S4.12.p5.7.m4.1.1.cmml" xref="S4.12.p5.7.m4.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.12.p5.7.m4.1c">t</annotation><annotation encoding="application/x-llamapun" id="S4.12.p5.7.m4.1d">italic_t</annotation></semantics></math> and <math alttext="t^{\prime}" class="ltx_Math" display="inline" id="S4.12.p5.8.m5.1"><semantics id="S4.12.p5.8.m5.1a"><msup id="S4.12.p5.8.m5.1.1" xref="S4.12.p5.8.m5.1.1.cmml"><mi id="S4.12.p5.8.m5.1.1.2" xref="S4.12.p5.8.m5.1.1.2.cmml">t</mi><mo id="S4.12.p5.8.m5.1.1.3" xref="S4.12.p5.8.m5.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.12.p5.8.m5.1b"><apply id="S4.12.p5.8.m5.1.1.cmml" xref="S4.12.p5.8.m5.1.1"><csymbol cd="ambiguous" id="S4.12.p5.8.m5.1.1.1.cmml" xref="S4.12.p5.8.m5.1.1">superscript</csymbol><ci id="S4.12.p5.8.m5.1.1.2.cmml" xref="S4.12.p5.8.m5.1.1.2">𝑡</ci><ci id="S4.12.p5.8.m5.1.1.3.cmml" xref="S4.12.p5.8.m5.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.12.p5.8.m5.1c">t^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.12.p5.8.m5.1d">italic_t start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> are <math alttext="T" class="ltx_Math" display="inline" id="S4.12.p5.9.m6.1"><semantics id="S4.12.p5.9.m6.1a"><mi id="S4.12.p5.9.m6.1.1" xref="S4.12.p5.9.m6.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S4.12.p5.9.m6.1b"><ci id="S4.12.p5.9.m6.1.1.cmml" xref="S4.12.p5.9.m6.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.12.p5.9.m6.1c">T</annotation><annotation encoding="application/x-llamapun" id="S4.12.p5.9.m6.1d">italic_T</annotation></semantics></math>-incomparable. Thus we have that if <math alttext="f(t)=f(t^{\prime})" class="ltx_Math" display="inline" id="S4.12.p5.10.m7.2"><semantics id="S4.12.p5.10.m7.2a"><mrow id="S4.12.p5.10.m7.2.2" xref="S4.12.p5.10.m7.2.2.cmml"><mrow id="S4.12.p5.10.m7.2.2.3" xref="S4.12.p5.10.m7.2.2.3.cmml"><mi id="S4.12.p5.10.m7.2.2.3.2" xref="S4.12.p5.10.m7.2.2.3.2.cmml">f</mi><mo id="S4.12.p5.10.m7.2.2.3.1" xref="S4.12.p5.10.m7.2.2.3.1.cmml">⁢</mo><mrow id="S4.12.p5.10.m7.2.2.3.3.2" xref="S4.12.p5.10.m7.2.2.3.cmml"><mo id="S4.12.p5.10.m7.2.2.3.3.2.1" stretchy="false" xref="S4.12.p5.10.m7.2.2.3.cmml">(</mo><mi id="S4.12.p5.10.m7.1.1" xref="S4.12.p5.10.m7.1.1.cmml">t</mi><mo id="S4.12.p5.10.m7.2.2.3.3.2.2" stretchy="false" xref="S4.12.p5.10.m7.2.2.3.cmml">)</mo></mrow></mrow><mo id="S4.12.p5.10.m7.2.2.2" xref="S4.12.p5.10.m7.2.2.2.cmml">=</mo><mrow id="S4.12.p5.10.m7.2.2.1" xref="S4.12.p5.10.m7.2.2.1.cmml"><mi id="S4.12.p5.10.m7.2.2.1.3" xref="S4.12.p5.10.m7.2.2.1.3.cmml">f</mi><mo id="S4.12.p5.10.m7.2.2.1.2" xref="S4.12.p5.10.m7.2.2.1.2.cmml">⁢</mo><mrow id="S4.12.p5.10.m7.2.2.1.1.1" xref="S4.12.p5.10.m7.2.2.1.1.1.1.cmml"><mo id="S4.12.p5.10.m7.2.2.1.1.1.2" stretchy="false" xref="S4.12.p5.10.m7.2.2.1.1.1.1.cmml">(</mo><msup id="S4.12.p5.10.m7.2.2.1.1.1.1" xref="S4.12.p5.10.m7.2.2.1.1.1.1.cmml"><mi id="S4.12.p5.10.m7.2.2.1.1.1.1.2" xref="S4.12.p5.10.m7.2.2.1.1.1.1.2.cmml">t</mi><mo id="S4.12.p5.10.m7.2.2.1.1.1.1.3" xref="S4.12.p5.10.m7.2.2.1.1.1.1.3.cmml">′</mo></msup><mo id="S4.12.p5.10.m7.2.2.1.1.1.3" stretchy="false" xref="S4.12.p5.10.m7.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.12.p5.10.m7.2b"><apply id="S4.12.p5.10.m7.2.2.cmml" xref="S4.12.p5.10.m7.2.2"><eq id="S4.12.p5.10.m7.2.2.2.cmml" xref="S4.12.p5.10.m7.2.2.2"></eq><apply id="S4.12.p5.10.m7.2.2.3.cmml" xref="S4.12.p5.10.m7.2.2.3"><times id="S4.12.p5.10.m7.2.2.3.1.cmml" xref="S4.12.p5.10.m7.2.2.3.1"></times><ci id="S4.12.p5.10.m7.2.2.3.2.cmml" xref="S4.12.p5.10.m7.2.2.3.2">𝑓</ci><ci id="S4.12.p5.10.m7.1.1.cmml" xref="S4.12.p5.10.m7.1.1">𝑡</ci></apply><apply id="S4.12.p5.10.m7.2.2.1.cmml" xref="S4.12.p5.10.m7.2.2.1"><times id="S4.12.p5.10.m7.2.2.1.2.cmml" xref="S4.12.p5.10.m7.2.2.1.2"></times><ci id="S4.12.p5.10.m7.2.2.1.3.cmml" xref="S4.12.p5.10.m7.2.2.1.3">𝑓</ci><apply id="S4.12.p5.10.m7.2.2.1.1.1.1.cmml" xref="S4.12.p5.10.m7.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.12.p5.10.m7.2.2.1.1.1.1.1.cmml" xref="S4.12.p5.10.m7.2.2.1.1.1">superscript</csymbol><ci id="S4.12.p5.10.m7.2.2.1.1.1.1.2.cmml" xref="S4.12.p5.10.m7.2.2.1.1.1.1.2">𝑡</ci><ci id="S4.12.p5.10.m7.2.2.1.1.1.1.3.cmml" xref="S4.12.p5.10.m7.2.2.1.1.1.1.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.12.p5.10.m7.2c">f(t)=f(t^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S4.12.p5.10.m7.2d">italic_f ( italic_t ) = italic_f ( italic_t start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> and <math alttext="t\neq t^{\prime}" class="ltx_Math" display="inline" id="S4.12.p5.11.m8.1"><semantics id="S4.12.p5.11.m8.1a"><mrow id="S4.12.p5.11.m8.1.1" xref="S4.12.p5.11.m8.1.1.cmml"><mi id="S4.12.p5.11.m8.1.1.2" xref="S4.12.p5.11.m8.1.1.2.cmml">t</mi><mo id="S4.12.p5.11.m8.1.1.1" xref="S4.12.p5.11.m8.1.1.1.cmml">≠</mo><msup id="S4.12.p5.11.m8.1.1.3" xref="S4.12.p5.11.m8.1.1.3.cmml"><mi id="S4.12.p5.11.m8.1.1.3.2" xref="S4.12.p5.11.m8.1.1.3.2.cmml">t</mi><mo id="S4.12.p5.11.m8.1.1.3.3" xref="S4.12.p5.11.m8.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.12.p5.11.m8.1b"><apply id="S4.12.p5.11.m8.1.1.cmml" xref="S4.12.p5.11.m8.1.1"><neq id="S4.12.p5.11.m8.1.1.1.cmml" xref="S4.12.p5.11.m8.1.1.1"></neq><ci id="S4.12.p5.11.m8.1.1.2.cmml" xref="S4.12.p5.11.m8.1.1.2">𝑡</ci><apply id="S4.12.p5.11.m8.1.1.3.cmml" xref="S4.12.p5.11.m8.1.1.3"><csymbol cd="ambiguous" id="S4.12.p5.11.m8.1.1.3.1.cmml" xref="S4.12.p5.11.m8.1.1.3">superscript</csymbol><ci id="S4.12.p5.11.m8.1.1.3.2.cmml" xref="S4.12.p5.11.m8.1.1.3.2">𝑡</ci><ci id="S4.12.p5.11.m8.1.1.3.3.cmml" xref="S4.12.p5.11.m8.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.12.p5.11.m8.1c">t\neq t^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.12.p5.11.m8.1d">italic_t ≠ italic_t start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, then</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex11"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="t<_{\mathrm{lex}}t^{\prime}\leftrightarrow t^{\frown}\langle\omega\rangle<_{% \mathrm{lex}}t^{\prime}\leftrightarrow t<_{\mathrm{lex}}t^{\prime\frown}% \langle\omega\rangle\leftrightarrow t^{\frown}\langle\omega\rangle<_{\mathrm{% lex}}t^{\prime\frown}\langle\omega\rangle" class="ltx_Math" display="block" id="S4.Ex11.m1.8"><semantics id="S4.Ex11.m1.8a"><mrow id="S4.Ex11.m1.8.9" xref="S4.Ex11.m1.8.9.cmml"><mrow id="S4.Ex11.m1.8.9.2" xref="S4.Ex11.m1.8.9.2.cmml"><mi id="S4.Ex11.m1.8.9.2.2" xref="S4.Ex11.m1.8.9.2.2.cmml">t</mi><msub id="S4.Ex11.m1.8.9.2.1" xref="S4.Ex11.m1.8.9.2.1.cmml"><mo id="S4.Ex11.m1.8.9.2.1.2" xref="S4.Ex11.m1.8.9.2.1.2.cmml"><</mo><mi id="S4.Ex11.m1.8.9.2.1.3" xref="S4.Ex11.m1.8.9.2.1.3.cmml">lex</mi></msub><msup id="S4.Ex11.m1.8.9.2.3" xref="S4.Ex11.m1.8.9.2.3.cmml"><mi id="S4.Ex11.m1.8.9.2.3.2" xref="S4.Ex11.m1.8.9.2.3.2.cmml">t</mi><mo id="S4.Ex11.m1.8.9.2.3.3" xref="S4.Ex11.m1.8.9.2.3.3.cmml">′</mo></msup></mrow><mo id="S4.Ex11.m1.8.9.3" stretchy="false" xref="S4.Ex11.m1.8.9.3.cmml">↔</mo><mrow id="S4.Ex11.m1.8.9.4" xref="S4.Ex11.m1.8.9.4.cmml"><mrow id="S4.Ex11.m1.8.9.4.2" xref="S4.Ex11.m1.8.9.4.2.cmml"><msup id="S4.Ex11.m1.8.9.4.2.2" xref="S4.Ex11.m1.8.9.4.2.2.cmml"><mi id="S4.Ex11.m1.8.9.4.2.2.2" xref="S4.Ex11.m1.8.9.4.2.2.2.cmml">t</mi><mo id="S4.Ex11.m1.8.9.4.2.2.3" xref="S4.Ex11.m1.8.9.4.2.2.3.cmml">⌢</mo></msup><mo id="S4.Ex11.m1.8.9.4.2.1" xref="S4.Ex11.m1.8.9.4.2.1.cmml">⁢</mo><mrow id="S4.Ex11.m1.8.9.4.2.3.2" xref="S4.Ex11.m1.8.9.4.2.3.1.cmml"><mo id="S4.Ex11.m1.8.9.4.2.3.2.1" stretchy="false" xref="S4.Ex11.m1.8.9.4.2.3.1.1.cmml">⟨</mo><mi id="S4.Ex11.m1.5.5" xref="S4.Ex11.m1.5.5.cmml">ω</mi><mo id="S4.Ex11.m1.8.9.4.2.3.2.2" stretchy="false" xref="S4.Ex11.m1.8.9.4.2.3.1.1.cmml">⟩</mo></mrow></mrow><msub id="S4.Ex11.m1.8.9.4.1" xref="S4.Ex11.m1.8.9.4.1.cmml"><mo id="S4.Ex11.m1.8.9.4.1.2" xref="S4.Ex11.m1.8.9.4.1.2.cmml"><</mo><mi id="S4.Ex11.m1.8.9.4.1.3" xref="S4.Ex11.m1.8.9.4.1.3.cmml">lex</mi></msub><msup id="S4.Ex11.m1.8.9.4.3" xref="S4.Ex11.m1.8.9.4.3.cmml"><mi id="S4.Ex11.m1.8.9.4.3.2" xref="S4.Ex11.m1.8.9.4.3.2.cmml">t</mi><mo id="S4.Ex11.m1.8.9.4.3.3" xref="S4.Ex11.m1.8.9.4.3.3.cmml">′</mo></msup></mrow><mo id="S4.Ex11.m1.8.9.5" stretchy="false" xref="S4.Ex11.m1.8.9.5.cmml">↔</mo><mrow id="S4.Ex11.m1.8.9.6" xref="S4.Ex11.m1.8.9.6.cmml"><mi id="S4.Ex11.m1.8.9.6.2" xref="S4.Ex11.m1.8.9.6.2.cmml">t</mi><msub id="S4.Ex11.m1.8.9.6.1" xref="S4.Ex11.m1.8.9.6.1.cmml"><mo id="S4.Ex11.m1.8.9.6.1.2" xref="S4.Ex11.m1.8.9.6.1.2.cmml"><</mo><mi id="S4.Ex11.m1.8.9.6.1.3" xref="S4.Ex11.m1.8.9.6.1.3.cmml">lex</mi></msub><mrow id="S4.Ex11.m1.8.9.6.3" xref="S4.Ex11.m1.8.9.6.3.cmml"><msup id="S4.Ex11.m1.8.9.6.3.2" xref="S4.Ex11.m1.8.9.6.3.2.cmml"><mi id="S4.Ex11.m1.8.9.6.3.2.2" xref="S4.Ex11.m1.8.9.6.3.2.2.cmml">t</mi><mrow id="S4.Ex11.m1.2.2.2.2" xref="S4.Ex11.m1.2.2.2.3.cmml"><mo id="S4.Ex11.m1.2.2.2.2.1" mathsize="142%" xref="S4.Ex11.m1.2.2.2.2.1.cmml">′</mo><mo id="S4.Ex11.m1.2.2.2.2.2" lspace="0.278em" xref="S4.Ex11.m1.2.2.2.3.cmml">⁣</mo><mo id="S4.Ex11.m1.1.1.1.1" xref="S4.Ex11.m1.1.1.1.1.cmml">⌢</mo></mrow></msup><mo id="S4.Ex11.m1.8.9.6.3.1" xref="S4.Ex11.m1.8.9.6.3.1.cmml">⁢</mo><mrow id="S4.Ex11.m1.8.9.6.3.3.2" xref="S4.Ex11.m1.8.9.6.3.3.1.cmml"><mo id="S4.Ex11.m1.8.9.6.3.3.2.1" stretchy="false" xref="S4.Ex11.m1.8.9.6.3.3.1.1.cmml">⟨</mo><mi id="S4.Ex11.m1.6.6" xref="S4.Ex11.m1.6.6.cmml">ω</mi><mo id="S4.Ex11.m1.8.9.6.3.3.2.2" stretchy="false" xref="S4.Ex11.m1.8.9.6.3.3.1.1.cmml">⟩</mo></mrow></mrow></mrow><mo id="S4.Ex11.m1.8.9.7" stretchy="false" xref="S4.Ex11.m1.8.9.7.cmml">↔</mo><mrow id="S4.Ex11.m1.8.9.8" xref="S4.Ex11.m1.8.9.8.cmml"><mrow id="S4.Ex11.m1.8.9.8.2" xref="S4.Ex11.m1.8.9.8.2.cmml"><msup id="S4.Ex11.m1.8.9.8.2.2" xref="S4.Ex11.m1.8.9.8.2.2.cmml"><mi id="S4.Ex11.m1.8.9.8.2.2.2" xref="S4.Ex11.m1.8.9.8.2.2.2.cmml">t</mi><mo id="S4.Ex11.m1.8.9.8.2.2.3" xref="S4.Ex11.m1.8.9.8.2.2.3.cmml">⌢</mo></msup><mo id="S4.Ex11.m1.8.9.8.2.1" xref="S4.Ex11.m1.8.9.8.2.1.cmml">⁢</mo><mrow id="S4.Ex11.m1.8.9.8.2.3.2" xref="S4.Ex11.m1.8.9.8.2.3.1.cmml"><mo id="S4.Ex11.m1.8.9.8.2.3.2.1" stretchy="false" xref="S4.Ex11.m1.8.9.8.2.3.1.1.cmml">⟨</mo><mi id="S4.Ex11.m1.7.7" xref="S4.Ex11.m1.7.7.cmml">ω</mi><mo id="S4.Ex11.m1.8.9.8.2.3.2.2" stretchy="false" xref="S4.Ex11.m1.8.9.8.2.3.1.1.cmml">⟩</mo></mrow></mrow><msub id="S4.Ex11.m1.8.9.8.1" xref="S4.Ex11.m1.8.9.8.1.cmml"><mo id="S4.Ex11.m1.8.9.8.1.2" xref="S4.Ex11.m1.8.9.8.1.2.cmml"><</mo><mi id="S4.Ex11.m1.8.9.8.1.3" xref="S4.Ex11.m1.8.9.8.1.3.cmml">lex</mi></msub><mrow id="S4.Ex11.m1.8.9.8.3" xref="S4.Ex11.m1.8.9.8.3.cmml"><msup id="S4.Ex11.m1.8.9.8.3.2" xref="S4.Ex11.m1.8.9.8.3.2.cmml"><mi id="S4.Ex11.m1.8.9.8.3.2.2" xref="S4.Ex11.m1.8.9.8.3.2.2.cmml">t</mi><mrow id="S4.Ex11.m1.4.4.2.2" xref="S4.Ex11.m1.4.4.2.3.cmml"><mo id="S4.Ex11.m1.4.4.2.2.1" mathsize="142%" xref="S4.Ex11.m1.4.4.2.2.1.cmml">′</mo><mo id="S4.Ex11.m1.4.4.2.2.2" lspace="0.278em" xref="S4.Ex11.m1.4.4.2.3.cmml">⁣</mo><mo id="S4.Ex11.m1.3.3.1.1" xref="S4.Ex11.m1.3.3.1.1.cmml">⌢</mo></mrow></msup><mo id="S4.Ex11.m1.8.9.8.3.1" xref="S4.Ex11.m1.8.9.8.3.1.cmml">⁢</mo><mrow id="S4.Ex11.m1.8.9.8.3.3.2" xref="S4.Ex11.m1.8.9.8.3.3.1.cmml"><mo id="S4.Ex11.m1.8.9.8.3.3.2.1" stretchy="false" xref="S4.Ex11.m1.8.9.8.3.3.1.1.cmml">⟨</mo><mi id="S4.Ex11.m1.8.8" xref="S4.Ex11.m1.8.8.cmml">ω</mi><mo id="S4.Ex11.m1.8.9.8.3.3.2.2" stretchy="false" xref="S4.Ex11.m1.8.9.8.3.3.1.1.cmml">⟩</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex11.m1.8b"><apply id="S4.Ex11.m1.8.9.cmml" xref="S4.Ex11.m1.8.9"><and id="S4.Ex11.m1.8.9a.cmml" xref="S4.Ex11.m1.8.9"></and><apply id="S4.Ex11.m1.8.9b.cmml" xref="S4.Ex11.m1.8.9"><ci id="S4.Ex11.m1.8.9.3.cmml" xref="S4.Ex11.m1.8.9.3">↔</ci><apply id="S4.Ex11.m1.8.9.2.cmml" xref="S4.Ex11.m1.8.9.2"><apply id="S4.Ex11.m1.8.9.2.1.cmml" xref="S4.Ex11.m1.8.9.2.1"><csymbol cd="ambiguous" id="S4.Ex11.m1.8.9.2.1.1.cmml" xref="S4.Ex11.m1.8.9.2.1">subscript</csymbol><lt id="S4.Ex11.m1.8.9.2.1.2.cmml" xref="S4.Ex11.m1.8.9.2.1.2"></lt><ci id="S4.Ex11.m1.8.9.2.1.3.cmml" xref="S4.Ex11.m1.8.9.2.1.3">lex</ci></apply><ci id="S4.Ex11.m1.8.9.2.2.cmml" xref="S4.Ex11.m1.8.9.2.2">𝑡</ci><apply id="S4.Ex11.m1.8.9.2.3.cmml" xref="S4.Ex11.m1.8.9.2.3"><csymbol cd="ambiguous" id="S4.Ex11.m1.8.9.2.3.1.cmml" xref="S4.Ex11.m1.8.9.2.3">superscript</csymbol><ci id="S4.Ex11.m1.8.9.2.3.2.cmml" xref="S4.Ex11.m1.8.9.2.3.2">𝑡</ci><ci id="S4.Ex11.m1.8.9.2.3.3.cmml" xref="S4.Ex11.m1.8.9.2.3.3">′</ci></apply></apply><apply id="S4.Ex11.m1.8.9.4.cmml" xref="S4.Ex11.m1.8.9.4"><apply id="S4.Ex11.m1.8.9.4.1.cmml" xref="S4.Ex11.m1.8.9.4.1"><csymbol cd="ambiguous" id="S4.Ex11.m1.8.9.4.1.1.cmml" xref="S4.Ex11.m1.8.9.4.1">subscript</csymbol><lt id="S4.Ex11.m1.8.9.4.1.2.cmml" xref="S4.Ex11.m1.8.9.4.1.2"></lt><ci id="S4.Ex11.m1.8.9.4.1.3.cmml" xref="S4.Ex11.m1.8.9.4.1.3">lex</ci></apply><apply id="S4.Ex11.m1.8.9.4.2.cmml" xref="S4.Ex11.m1.8.9.4.2"><times id="S4.Ex11.m1.8.9.4.2.1.cmml" xref="S4.Ex11.m1.8.9.4.2.1"></times><apply id="S4.Ex11.m1.8.9.4.2.2.cmml" xref="S4.Ex11.m1.8.9.4.2.2"><csymbol cd="ambiguous" id="S4.Ex11.m1.8.9.4.2.2.1.cmml" xref="S4.Ex11.m1.8.9.4.2.2">superscript</csymbol><ci id="S4.Ex11.m1.8.9.4.2.2.2.cmml" xref="S4.Ex11.m1.8.9.4.2.2.2">𝑡</ci><ci id="S4.Ex11.m1.8.9.4.2.2.3.cmml" xref="S4.Ex11.m1.8.9.4.2.2.3">⌢</ci></apply><apply id="S4.Ex11.m1.8.9.4.2.3.1.cmml" xref="S4.Ex11.m1.8.9.4.2.3.2"><csymbol cd="latexml" id="S4.Ex11.m1.8.9.4.2.3.1.1.cmml" xref="S4.Ex11.m1.8.9.4.2.3.2.1">delimited-⟨⟩</csymbol><ci id="S4.Ex11.m1.5.5.cmml" xref="S4.Ex11.m1.5.5">𝜔</ci></apply></apply><apply id="S4.Ex11.m1.8.9.4.3.cmml" xref="S4.Ex11.m1.8.9.4.3"><csymbol cd="ambiguous" id="S4.Ex11.m1.8.9.4.3.1.cmml" xref="S4.Ex11.m1.8.9.4.3">superscript</csymbol><ci id="S4.Ex11.m1.8.9.4.3.2.cmml" xref="S4.Ex11.m1.8.9.4.3.2">𝑡</ci><ci id="S4.Ex11.m1.8.9.4.3.3.cmml" xref="S4.Ex11.m1.8.9.4.3.3">′</ci></apply></apply></apply><apply id="S4.Ex11.m1.8.9c.cmml" xref="S4.Ex11.m1.8.9"><ci id="S4.Ex11.m1.8.9.5.cmml" xref="S4.Ex11.m1.8.9.5">↔</ci><share href="https://arxiv.org/html/2503.13728v1#S4.Ex11.m1.8.9.4.cmml" id="S4.Ex11.m1.8.9d.cmml" xref="S4.Ex11.m1.8.9"></share><apply id="S4.Ex11.m1.8.9.6.cmml" xref="S4.Ex11.m1.8.9.6"><apply id="S4.Ex11.m1.8.9.6.1.cmml" xref="S4.Ex11.m1.8.9.6.1"><csymbol cd="ambiguous" id="S4.Ex11.m1.8.9.6.1.1.cmml" xref="S4.Ex11.m1.8.9.6.1">subscript</csymbol><lt id="S4.Ex11.m1.8.9.6.1.2.cmml" xref="S4.Ex11.m1.8.9.6.1.2"></lt><ci id="S4.Ex11.m1.8.9.6.1.3.cmml" xref="S4.Ex11.m1.8.9.6.1.3">lex</ci></apply><ci id="S4.Ex11.m1.8.9.6.2.cmml" xref="S4.Ex11.m1.8.9.6.2">𝑡</ci><apply id="S4.Ex11.m1.8.9.6.3.cmml" xref="S4.Ex11.m1.8.9.6.3"><times id="S4.Ex11.m1.8.9.6.3.1.cmml" xref="S4.Ex11.m1.8.9.6.3.1"></times><apply id="S4.Ex11.m1.8.9.6.3.2.cmml" xref="S4.Ex11.m1.8.9.6.3.2"><csymbol cd="ambiguous" id="S4.Ex11.m1.8.9.6.3.2.1.cmml" xref="S4.Ex11.m1.8.9.6.3.2">superscript</csymbol><ci id="S4.Ex11.m1.8.9.6.3.2.2.cmml" xref="S4.Ex11.m1.8.9.6.3.2.2">𝑡</ci><list id="S4.Ex11.m1.2.2.2.3.cmml" xref="S4.Ex11.m1.2.2.2.2"><ci id="S4.Ex11.m1.2.2.2.2.1.cmml" xref="S4.Ex11.m1.2.2.2.2.1">′</ci><ci id="S4.Ex11.m1.1.1.1.1.cmml" xref="S4.Ex11.m1.1.1.1.1">⌢</ci></list></apply><apply id="S4.Ex11.m1.8.9.6.3.3.1.cmml" xref="S4.Ex11.m1.8.9.6.3.3.2"><csymbol cd="latexml" id="S4.Ex11.m1.8.9.6.3.3.1.1.cmml" xref="S4.Ex11.m1.8.9.6.3.3.2.1">delimited-⟨⟩</csymbol><ci id="S4.Ex11.m1.6.6.cmml" xref="S4.Ex11.m1.6.6">𝜔</ci></apply></apply></apply></apply><apply id="S4.Ex11.m1.8.9e.cmml" xref="S4.Ex11.m1.8.9"><ci id="S4.Ex11.m1.8.9.7.cmml" xref="S4.Ex11.m1.8.9.7">↔</ci><share href="https://arxiv.org/html/2503.13728v1#S4.Ex11.m1.8.9.6.cmml" id="S4.Ex11.m1.8.9f.cmml" xref="S4.Ex11.m1.8.9"></share><apply id="S4.Ex11.m1.8.9.8.cmml" xref="S4.Ex11.m1.8.9.8"><apply id="S4.Ex11.m1.8.9.8.1.cmml" xref="S4.Ex11.m1.8.9.8.1"><csymbol cd="ambiguous" id="S4.Ex11.m1.8.9.8.1.1.cmml" xref="S4.Ex11.m1.8.9.8.1">subscript</csymbol><lt id="S4.Ex11.m1.8.9.8.1.2.cmml" xref="S4.Ex11.m1.8.9.8.1.2"></lt><ci id="S4.Ex11.m1.8.9.8.1.3.cmml" xref="S4.Ex11.m1.8.9.8.1.3">lex</ci></apply><apply id="S4.Ex11.m1.8.9.8.2.cmml" xref="S4.Ex11.m1.8.9.8.2"><times id="S4.Ex11.m1.8.9.8.2.1.cmml" xref="S4.Ex11.m1.8.9.8.2.1"></times><apply id="S4.Ex11.m1.8.9.8.2.2.cmml" xref="S4.Ex11.m1.8.9.8.2.2"><csymbol cd="ambiguous" id="S4.Ex11.m1.8.9.8.2.2.1.cmml" xref="S4.Ex11.m1.8.9.8.2.2">superscript</csymbol><ci id="S4.Ex11.m1.8.9.8.2.2.2.cmml" xref="S4.Ex11.m1.8.9.8.2.2.2">𝑡</ci><ci id="S4.Ex11.m1.8.9.8.2.2.3.cmml" xref="S4.Ex11.m1.8.9.8.2.2.3">⌢</ci></apply><apply id="S4.Ex11.m1.8.9.8.2.3.1.cmml" xref="S4.Ex11.m1.8.9.8.2.3.2"><csymbol cd="latexml" id="S4.Ex11.m1.8.9.8.2.3.1.1.cmml" xref="S4.Ex11.m1.8.9.8.2.3.2.1">delimited-⟨⟩</csymbol><ci id="S4.Ex11.m1.7.7.cmml" xref="S4.Ex11.m1.7.7">𝜔</ci></apply></apply><apply id="S4.Ex11.m1.8.9.8.3.cmml" xref="S4.Ex11.m1.8.9.8.3"><times id="S4.Ex11.m1.8.9.8.3.1.cmml" xref="S4.Ex11.m1.8.9.8.3.1"></times><apply id="S4.Ex11.m1.8.9.8.3.2.cmml" xref="S4.Ex11.m1.8.9.8.3.2"><csymbol cd="ambiguous" id="S4.Ex11.m1.8.9.8.3.2.1.cmml" xref="S4.Ex11.m1.8.9.8.3.2">superscript</csymbol><ci id="S4.Ex11.m1.8.9.8.3.2.2.cmml" xref="S4.Ex11.m1.8.9.8.3.2.2">𝑡</ci><list id="S4.Ex11.m1.4.4.2.3.cmml" xref="S4.Ex11.m1.4.4.2.2"><ci id="S4.Ex11.m1.4.4.2.2.1.cmml" xref="S4.Ex11.m1.4.4.2.2.1">′</ci><ci id="S4.Ex11.m1.3.3.1.1.cmml" xref="S4.Ex11.m1.3.3.1.1">⌢</ci></list></apply><apply id="S4.Ex11.m1.8.9.8.3.3.1.cmml" xref="S4.Ex11.m1.8.9.8.3.3.2"><csymbol cd="latexml" id="S4.Ex11.m1.8.9.8.3.3.1.1.cmml" xref="S4.Ex11.m1.8.9.8.3.3.2.1">delimited-⟨⟩</csymbol><ci id="S4.Ex11.m1.8.8.cmml" xref="S4.Ex11.m1.8.8">𝜔</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex11.m1.8c">t<_{\mathrm{lex}}t^{\prime}\leftrightarrow t^{\frown}\langle\omega\rangle<_{% \mathrm{lex}}t^{\prime}\leftrightarrow t<_{\mathrm{lex}}t^{\prime\frown}% \langle\omega\rangle\leftrightarrow t^{\frown}\langle\omega\rangle<_{\mathrm{% lex}}t^{\prime\frown}\langle\omega\rangle</annotation><annotation encoding="application/x-llamapun" id="S4.Ex11.m1.8d">italic_t < start_POSTSUBSCRIPT roman_lex end_POSTSUBSCRIPT italic_t start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ↔ italic_t start_POSTSUPERSCRIPT ⌢ end_POSTSUPERSCRIPT ⟨ italic_ω ⟩ < start_POSTSUBSCRIPT roman_lex end_POSTSUBSCRIPT italic_t start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ↔ italic_t < start_POSTSUBSCRIPT roman_lex end_POSTSUBSCRIPT italic_t start_POSTSUPERSCRIPT ′ ⌢ end_POSTSUPERSCRIPT ⟨ italic_ω ⟩ ↔ italic_t start_POSTSUPERSCRIPT ⌢ end_POSTSUPERSCRIPT ⟨ italic_ω ⟩ < start_POSTSUBSCRIPT roman_lex end_POSTSUBSCRIPT italic_t start_POSTSUPERSCRIPT ′ ⌢ end_POSTSUPERSCRIPT ⟨ italic_ω ⟩</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.12.p5.16">This in turn implies that if <math alttext="t,t^{\prime},s,s^{\prime}\in A^{+}" class="ltx_Math" display="inline" id="S4.12.p5.12.m1.4"><semantics id="S4.12.p5.12.m1.4a"><mrow id="S4.12.p5.12.m1.4.4" xref="S4.12.p5.12.m1.4.4.cmml"><mrow id="S4.12.p5.12.m1.4.4.2.2" xref="S4.12.p5.12.m1.4.4.2.3.cmml"><mi id="S4.12.p5.12.m1.1.1" xref="S4.12.p5.12.m1.1.1.cmml">t</mi><mo id="S4.12.p5.12.m1.4.4.2.2.3" xref="S4.12.p5.12.m1.4.4.2.3.cmml">,</mo><msup id="S4.12.p5.12.m1.3.3.1.1.1" xref="S4.12.p5.12.m1.3.3.1.1.1.cmml"><mi id="S4.12.p5.12.m1.3.3.1.1.1.2" xref="S4.12.p5.12.m1.3.3.1.1.1.2.cmml">t</mi><mo id="S4.12.p5.12.m1.3.3.1.1.1.3" xref="S4.12.p5.12.m1.3.3.1.1.1.3.cmml">′</mo></msup><mo id="S4.12.p5.12.m1.4.4.2.2.4" xref="S4.12.p5.12.m1.4.4.2.3.cmml">,</mo><mi id="S4.12.p5.12.m1.2.2" xref="S4.12.p5.12.m1.2.2.cmml">s</mi><mo id="S4.12.p5.12.m1.4.4.2.2.5" xref="S4.12.p5.12.m1.4.4.2.3.cmml">,</mo><msup id="S4.12.p5.12.m1.4.4.2.2.2" xref="S4.12.p5.12.m1.4.4.2.2.2.cmml"><mi id="S4.12.p5.12.m1.4.4.2.2.2.2" xref="S4.12.p5.12.m1.4.4.2.2.2.2.cmml">s</mi><mo id="S4.12.p5.12.m1.4.4.2.2.2.3" xref="S4.12.p5.12.m1.4.4.2.2.2.3.cmml">′</mo></msup></mrow><mo id="S4.12.p5.12.m1.4.4.3" xref="S4.12.p5.12.m1.4.4.3.cmml">∈</mo><msup id="S4.12.p5.12.m1.4.4.4" xref="S4.12.p5.12.m1.4.4.4.cmml"><mi id="S4.12.p5.12.m1.4.4.4.2" xref="S4.12.p5.12.m1.4.4.4.2.cmml">A</mi><mo id="S4.12.p5.12.m1.4.4.4.3" xref="S4.12.p5.12.m1.4.4.4.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.12.p5.12.m1.4b"><apply id="S4.12.p5.12.m1.4.4.cmml" xref="S4.12.p5.12.m1.4.4"><in id="S4.12.p5.12.m1.4.4.3.cmml" xref="S4.12.p5.12.m1.4.4.3"></in><list id="S4.12.p5.12.m1.4.4.2.3.cmml" xref="S4.12.p5.12.m1.4.4.2.2"><ci id="S4.12.p5.12.m1.1.1.cmml" xref="S4.12.p5.12.m1.1.1">𝑡</ci><apply id="S4.12.p5.12.m1.3.3.1.1.1.cmml" xref="S4.12.p5.12.m1.3.3.1.1.1"><csymbol cd="ambiguous" id="S4.12.p5.12.m1.3.3.1.1.1.1.cmml" xref="S4.12.p5.12.m1.3.3.1.1.1">superscript</csymbol><ci id="S4.12.p5.12.m1.3.3.1.1.1.2.cmml" xref="S4.12.p5.12.m1.3.3.1.1.1.2">𝑡</ci><ci id="S4.12.p5.12.m1.3.3.1.1.1.3.cmml" xref="S4.12.p5.12.m1.3.3.1.1.1.3">′</ci></apply><ci id="S4.12.p5.12.m1.2.2.cmml" xref="S4.12.p5.12.m1.2.2">𝑠</ci><apply id="S4.12.p5.12.m1.4.4.2.2.2.cmml" xref="S4.12.p5.12.m1.4.4.2.2.2"><csymbol cd="ambiguous" id="S4.12.p5.12.m1.4.4.2.2.2.1.cmml" xref="S4.12.p5.12.m1.4.4.2.2.2">superscript</csymbol><ci id="S4.12.p5.12.m1.4.4.2.2.2.2.cmml" xref="S4.12.p5.12.m1.4.4.2.2.2.2">𝑠</ci><ci id="S4.12.p5.12.m1.4.4.2.2.2.3.cmml" xref="S4.12.p5.12.m1.4.4.2.2.2.3">′</ci></apply></list><apply id="S4.12.p5.12.m1.4.4.4.cmml" xref="S4.12.p5.12.m1.4.4.4"><csymbol cd="ambiguous" id="S4.12.p5.12.m1.4.4.4.1.cmml" xref="S4.12.p5.12.m1.4.4.4">superscript</csymbol><ci id="S4.12.p5.12.m1.4.4.4.2.cmml" xref="S4.12.p5.12.m1.4.4.4.2">𝐴</ci><plus id="S4.12.p5.12.m1.4.4.4.3.cmml" xref="S4.12.p5.12.m1.4.4.4.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.12.p5.12.m1.4c">t,t^{\prime},s,s^{\prime}\in A^{+}</annotation><annotation encoding="application/x-llamapun" id="S4.12.p5.12.m1.4d">italic_t , italic_t start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_s , italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_A start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="t\neq t^{\prime}" class="ltx_Math" display="inline" id="S4.12.p5.13.m2.1"><semantics id="S4.12.p5.13.m2.1a"><mrow id="S4.12.p5.13.m2.1.1" xref="S4.12.p5.13.m2.1.1.cmml"><mi id="S4.12.p5.13.m2.1.1.2" xref="S4.12.p5.13.m2.1.1.2.cmml">t</mi><mo id="S4.12.p5.13.m2.1.1.1" xref="S4.12.p5.13.m2.1.1.1.cmml">≠</mo><msup id="S4.12.p5.13.m2.1.1.3" xref="S4.12.p5.13.m2.1.1.3.cmml"><mi id="S4.12.p5.13.m2.1.1.3.2" xref="S4.12.p5.13.m2.1.1.3.2.cmml">t</mi><mo id="S4.12.p5.13.m2.1.1.3.3" xref="S4.12.p5.13.m2.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.12.p5.13.m2.1b"><apply id="S4.12.p5.13.m2.1.1.cmml" xref="S4.12.p5.13.m2.1.1"><neq id="S4.12.p5.13.m2.1.1.1.cmml" xref="S4.12.p5.13.m2.1.1.1"></neq><ci id="S4.12.p5.13.m2.1.1.2.cmml" xref="S4.12.p5.13.m2.1.1.2">𝑡</ci><apply id="S4.12.p5.13.m2.1.1.3.cmml" xref="S4.12.p5.13.m2.1.1.3"><csymbol cd="ambiguous" id="S4.12.p5.13.m2.1.1.3.1.cmml" xref="S4.12.p5.13.m2.1.1.3">superscript</csymbol><ci id="S4.12.p5.13.m2.1.1.3.2.cmml" xref="S4.12.p5.13.m2.1.1.3.2">𝑡</ci><ci id="S4.12.p5.13.m2.1.1.3.3.cmml" xref="S4.12.p5.13.m2.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.12.p5.13.m2.1c">t\neq t^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.12.p5.13.m2.1d">italic_t ≠ italic_t start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="s\neq s^{\prime}" class="ltx_Math" display="inline" id="S4.12.p5.14.m3.1"><semantics id="S4.12.p5.14.m3.1a"><mrow id="S4.12.p5.14.m3.1.1" xref="S4.12.p5.14.m3.1.1.cmml"><mi id="S4.12.p5.14.m3.1.1.2" xref="S4.12.p5.14.m3.1.1.2.cmml">s</mi><mo id="S4.12.p5.14.m3.1.1.1" xref="S4.12.p5.14.m3.1.1.1.cmml">≠</mo><msup id="S4.12.p5.14.m3.1.1.3" xref="S4.12.p5.14.m3.1.1.3.cmml"><mi id="S4.12.p5.14.m3.1.1.3.2" xref="S4.12.p5.14.m3.1.1.3.2.cmml">s</mi><mo id="S4.12.p5.14.m3.1.1.3.3" xref="S4.12.p5.14.m3.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.12.p5.14.m3.1b"><apply id="S4.12.p5.14.m3.1.1.cmml" xref="S4.12.p5.14.m3.1.1"><neq id="S4.12.p5.14.m3.1.1.1.cmml" xref="S4.12.p5.14.m3.1.1.1"></neq><ci id="S4.12.p5.14.m3.1.1.2.cmml" xref="S4.12.p5.14.m3.1.1.2">𝑠</ci><apply id="S4.12.p5.14.m3.1.1.3.cmml" xref="S4.12.p5.14.m3.1.1.3"><csymbol cd="ambiguous" id="S4.12.p5.14.m3.1.1.3.1.cmml" xref="S4.12.p5.14.m3.1.1.3">superscript</csymbol><ci id="S4.12.p5.14.m3.1.1.3.2.cmml" xref="S4.12.p5.14.m3.1.1.3.2">𝑠</ci><ci id="S4.12.p5.14.m3.1.1.3.3.cmml" xref="S4.12.p5.14.m3.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.12.p5.14.m3.1c">s\neq s^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.12.p5.14.m3.1d">italic_s ≠ italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\tilde{c}(t,s)=\tilde{c}(t^{\prime},s^{\prime})" class="ltx_Math" display="inline" id="S4.12.p5.15.m4.4"><semantics id="S4.12.p5.15.m4.4a"><mrow id="S4.12.p5.15.m4.4.4" xref="S4.12.p5.15.m4.4.4.cmml"><mrow id="S4.12.p5.15.m4.4.4.4" xref="S4.12.p5.15.m4.4.4.4.cmml"><mover accent="true" id="S4.12.p5.15.m4.4.4.4.2" xref="S4.12.p5.15.m4.4.4.4.2.cmml"><mi id="S4.12.p5.15.m4.4.4.4.2.2" xref="S4.12.p5.15.m4.4.4.4.2.2.cmml">c</mi><mo id="S4.12.p5.15.m4.4.4.4.2.1" xref="S4.12.p5.15.m4.4.4.4.2.1.cmml">~</mo></mover><mo id="S4.12.p5.15.m4.4.4.4.1" xref="S4.12.p5.15.m4.4.4.4.1.cmml">⁢</mo><mrow id="S4.12.p5.15.m4.4.4.4.3.2" xref="S4.12.p5.15.m4.4.4.4.3.1.cmml"><mo id="S4.12.p5.15.m4.4.4.4.3.2.1" stretchy="false" xref="S4.12.p5.15.m4.4.4.4.3.1.cmml">(</mo><mi id="S4.12.p5.15.m4.1.1" xref="S4.12.p5.15.m4.1.1.cmml">t</mi><mo id="S4.12.p5.15.m4.4.4.4.3.2.2" xref="S4.12.p5.15.m4.4.4.4.3.1.cmml">,</mo><mi id="S4.12.p5.15.m4.2.2" xref="S4.12.p5.15.m4.2.2.cmml">s</mi><mo id="S4.12.p5.15.m4.4.4.4.3.2.3" stretchy="false" xref="S4.12.p5.15.m4.4.4.4.3.1.cmml">)</mo></mrow></mrow><mo id="S4.12.p5.15.m4.4.4.3" xref="S4.12.p5.15.m4.4.4.3.cmml">=</mo><mrow id="S4.12.p5.15.m4.4.4.2" xref="S4.12.p5.15.m4.4.4.2.cmml"><mover accent="true" id="S4.12.p5.15.m4.4.4.2.4" xref="S4.12.p5.15.m4.4.4.2.4.cmml"><mi id="S4.12.p5.15.m4.4.4.2.4.2" xref="S4.12.p5.15.m4.4.4.2.4.2.cmml">c</mi><mo id="S4.12.p5.15.m4.4.4.2.4.1" xref="S4.12.p5.15.m4.4.4.2.4.1.cmml">~</mo></mover><mo id="S4.12.p5.15.m4.4.4.2.3" xref="S4.12.p5.15.m4.4.4.2.3.cmml">⁢</mo><mrow id="S4.12.p5.15.m4.4.4.2.2.2" xref="S4.12.p5.15.m4.4.4.2.2.3.cmml"><mo id="S4.12.p5.15.m4.4.4.2.2.2.3" stretchy="false" xref="S4.12.p5.15.m4.4.4.2.2.3.cmml">(</mo><msup id="S4.12.p5.15.m4.3.3.1.1.1.1" xref="S4.12.p5.15.m4.3.3.1.1.1.1.cmml"><mi id="S4.12.p5.15.m4.3.3.1.1.1.1.2" xref="S4.12.p5.15.m4.3.3.1.1.1.1.2.cmml">t</mi><mo id="S4.12.p5.15.m4.3.3.1.1.1.1.3" xref="S4.12.p5.15.m4.3.3.1.1.1.1.3.cmml">′</mo></msup><mo id="S4.12.p5.15.m4.4.4.2.2.2.4" xref="S4.12.p5.15.m4.4.4.2.2.3.cmml">,</mo><msup id="S4.12.p5.15.m4.4.4.2.2.2.2" xref="S4.12.p5.15.m4.4.4.2.2.2.2.cmml"><mi id="S4.12.p5.15.m4.4.4.2.2.2.2.2" xref="S4.12.p5.15.m4.4.4.2.2.2.2.2.cmml">s</mi><mo id="S4.12.p5.15.m4.4.4.2.2.2.2.3" xref="S4.12.p5.15.m4.4.4.2.2.2.2.3.cmml">′</mo></msup><mo id="S4.12.p5.15.m4.4.4.2.2.2.5" stretchy="false" xref="S4.12.p5.15.m4.4.4.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.12.p5.15.m4.4b"><apply id="S4.12.p5.15.m4.4.4.cmml" xref="S4.12.p5.15.m4.4.4"><eq id="S4.12.p5.15.m4.4.4.3.cmml" xref="S4.12.p5.15.m4.4.4.3"></eq><apply id="S4.12.p5.15.m4.4.4.4.cmml" xref="S4.12.p5.15.m4.4.4.4"><times id="S4.12.p5.15.m4.4.4.4.1.cmml" xref="S4.12.p5.15.m4.4.4.4.1"></times><apply id="S4.12.p5.15.m4.4.4.4.2.cmml" xref="S4.12.p5.15.m4.4.4.4.2"><ci id="S4.12.p5.15.m4.4.4.4.2.1.cmml" xref="S4.12.p5.15.m4.4.4.4.2.1">~</ci><ci id="S4.12.p5.15.m4.4.4.4.2.2.cmml" xref="S4.12.p5.15.m4.4.4.4.2.2">𝑐</ci></apply><interval closure="open" id="S4.12.p5.15.m4.4.4.4.3.1.cmml" xref="S4.12.p5.15.m4.4.4.4.3.2"><ci id="S4.12.p5.15.m4.1.1.cmml" xref="S4.12.p5.15.m4.1.1">𝑡</ci><ci id="S4.12.p5.15.m4.2.2.cmml" xref="S4.12.p5.15.m4.2.2">𝑠</ci></interval></apply><apply id="S4.12.p5.15.m4.4.4.2.cmml" xref="S4.12.p5.15.m4.4.4.2"><times id="S4.12.p5.15.m4.4.4.2.3.cmml" xref="S4.12.p5.15.m4.4.4.2.3"></times><apply id="S4.12.p5.15.m4.4.4.2.4.cmml" xref="S4.12.p5.15.m4.4.4.2.4"><ci id="S4.12.p5.15.m4.4.4.2.4.1.cmml" xref="S4.12.p5.15.m4.4.4.2.4.1">~</ci><ci id="S4.12.p5.15.m4.4.4.2.4.2.cmml" xref="S4.12.p5.15.m4.4.4.2.4.2">𝑐</ci></apply><interval closure="open" id="S4.12.p5.15.m4.4.4.2.2.3.cmml" xref="S4.12.p5.15.m4.4.4.2.2.2"><apply id="S4.12.p5.15.m4.3.3.1.1.1.1.cmml" xref="S4.12.p5.15.m4.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S4.12.p5.15.m4.3.3.1.1.1.1.1.cmml" xref="S4.12.p5.15.m4.3.3.1.1.1.1">superscript</csymbol><ci id="S4.12.p5.15.m4.3.3.1.1.1.1.2.cmml" xref="S4.12.p5.15.m4.3.3.1.1.1.1.2">𝑡</ci><ci id="S4.12.p5.15.m4.3.3.1.1.1.1.3.cmml" xref="S4.12.p5.15.m4.3.3.1.1.1.1.3">′</ci></apply><apply id="S4.12.p5.15.m4.4.4.2.2.2.2.cmml" xref="S4.12.p5.15.m4.4.4.2.2.2.2"><csymbol cd="ambiguous" id="S4.12.p5.15.m4.4.4.2.2.2.2.1.cmml" xref="S4.12.p5.15.m4.4.4.2.2.2.2">superscript</csymbol><ci id="S4.12.p5.15.m4.4.4.2.2.2.2.2.cmml" xref="S4.12.p5.15.m4.4.4.2.2.2.2.2">𝑠</ci><ci id="S4.12.p5.15.m4.4.4.2.2.2.2.3.cmml" xref="S4.12.p5.15.m4.4.4.2.2.2.2.3">′</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.12.p5.15.m4.4c">\tilde{c}(t,s)=\tilde{c}(t^{\prime},s^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S4.12.p5.15.m4.4d">over~ start_ARG italic_c end_ARG ( italic_t , italic_s ) = over~ start_ARG italic_c end_ARG ( italic_t start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math>, then <math alttext="t<_{\mathrm{lex}}t^{\prime}\rightarrow s<_{\mathrm{lex}}s^{\prime}" class="ltx_Math" display="inline" id="S4.12.p5.16.m5.1"><semantics id="S4.12.p5.16.m5.1a"><mrow id="S4.12.p5.16.m5.1.1" xref="S4.12.p5.16.m5.1.1.cmml"><mi id="S4.12.p5.16.m5.1.1.2" xref="S4.12.p5.16.m5.1.1.2.cmml">t</mi><msub id="S4.12.p5.16.m5.1.1.3" xref="S4.12.p5.16.m5.1.1.3.cmml"><mo id="S4.12.p5.16.m5.1.1.3.2" xref="S4.12.p5.16.m5.1.1.3.2.cmml"><</mo><mi id="S4.12.p5.16.m5.1.1.3.3" xref="S4.12.p5.16.m5.1.1.3.3.cmml">lex</mi></msub><msup id="S4.12.p5.16.m5.1.1.4" xref="S4.12.p5.16.m5.1.1.4.cmml"><mi id="S4.12.p5.16.m5.1.1.4.2" xref="S4.12.p5.16.m5.1.1.4.2.cmml">t</mi><mo id="S4.12.p5.16.m5.1.1.4.3" xref="S4.12.p5.16.m5.1.1.4.3.cmml">′</mo></msup><mo id="S4.12.p5.16.m5.1.1.5" stretchy="false" xref="S4.12.p5.16.m5.1.1.5.cmml">→</mo><mi id="S4.12.p5.16.m5.1.1.6" xref="S4.12.p5.16.m5.1.1.6.cmml">s</mi><msub id="S4.12.p5.16.m5.1.1.7" xref="S4.12.p5.16.m5.1.1.7.cmml"><mo id="S4.12.p5.16.m5.1.1.7.2" xref="S4.12.p5.16.m5.1.1.7.2.cmml"><</mo><mi id="S4.12.p5.16.m5.1.1.7.3" xref="S4.12.p5.16.m5.1.1.7.3.cmml">lex</mi></msub><msup id="S4.12.p5.16.m5.1.1.8" xref="S4.12.p5.16.m5.1.1.8.cmml"><mi id="S4.12.p5.16.m5.1.1.8.2" xref="S4.12.p5.16.m5.1.1.8.2.cmml">s</mi><mo id="S4.12.p5.16.m5.1.1.8.3" xref="S4.12.p5.16.m5.1.1.8.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.12.p5.16.m5.1b"><apply id="S4.12.p5.16.m5.1.1.cmml" xref="S4.12.p5.16.m5.1.1"><and id="S4.12.p5.16.m5.1.1a.cmml" xref="S4.12.p5.16.m5.1.1"></and><apply id="S4.12.p5.16.m5.1.1b.cmml" xref="S4.12.p5.16.m5.1.1"><apply id="S4.12.p5.16.m5.1.1.3.cmml" xref="S4.12.p5.16.m5.1.1.3"><csymbol cd="ambiguous" id="S4.12.p5.16.m5.1.1.3.1.cmml" xref="S4.12.p5.16.m5.1.1.3">subscript</csymbol><lt id="S4.12.p5.16.m5.1.1.3.2.cmml" xref="S4.12.p5.16.m5.1.1.3.2"></lt><ci id="S4.12.p5.16.m5.1.1.3.3.cmml" xref="S4.12.p5.16.m5.1.1.3.3">lex</ci></apply><ci id="S4.12.p5.16.m5.1.1.2.cmml" xref="S4.12.p5.16.m5.1.1.2">𝑡</ci><apply id="S4.12.p5.16.m5.1.1.4.cmml" xref="S4.12.p5.16.m5.1.1.4"><csymbol cd="ambiguous" id="S4.12.p5.16.m5.1.1.4.1.cmml" xref="S4.12.p5.16.m5.1.1.4">superscript</csymbol><ci id="S4.12.p5.16.m5.1.1.4.2.cmml" xref="S4.12.p5.16.m5.1.1.4.2">𝑡</ci><ci id="S4.12.p5.16.m5.1.1.4.3.cmml" xref="S4.12.p5.16.m5.1.1.4.3">′</ci></apply></apply><apply id="S4.12.p5.16.m5.1.1c.cmml" xref="S4.12.p5.16.m5.1.1"><ci id="S4.12.p5.16.m5.1.1.5.cmml" xref="S4.12.p5.16.m5.1.1.5">→</ci><share href="https://arxiv.org/html/2503.13728v1#S4.12.p5.16.m5.1.1.4.cmml" id="S4.12.p5.16.m5.1.1d.cmml" xref="S4.12.p5.16.m5.1.1"></share><ci id="S4.12.p5.16.m5.1.1.6.cmml" xref="S4.12.p5.16.m5.1.1.6">𝑠</ci></apply><apply id="S4.12.p5.16.m5.1.1e.cmml" xref="S4.12.p5.16.m5.1.1"><apply id="S4.12.p5.16.m5.1.1.7.cmml" xref="S4.12.p5.16.m5.1.1.7"><csymbol cd="ambiguous" id="S4.12.p5.16.m5.1.1.7.1.cmml" xref="S4.12.p5.16.m5.1.1.7">subscript</csymbol><lt id="S4.12.p5.16.m5.1.1.7.2.cmml" xref="S4.12.p5.16.m5.1.1.7.2"></lt><ci id="S4.12.p5.16.m5.1.1.7.3.cmml" xref="S4.12.p5.16.m5.1.1.7.3">lex</ci></apply><share href="https://arxiv.org/html/2503.13728v1#S4.12.p5.16.m5.1.1.6.cmml" id="S4.12.p5.16.m5.1.1f.cmml" xref="S4.12.p5.16.m5.1.1"></share><apply id="S4.12.p5.16.m5.1.1.8.cmml" xref="S4.12.p5.16.m5.1.1.8"><csymbol cd="ambiguous" id="S4.12.p5.16.m5.1.1.8.1.cmml" xref="S4.12.p5.16.m5.1.1.8">superscript</csymbol><ci id="S4.12.p5.16.m5.1.1.8.2.cmml" xref="S4.12.p5.16.m5.1.1.8.2">𝑠</ci><ci id="S4.12.p5.16.m5.1.1.8.3.cmml" xref="S4.12.p5.16.m5.1.1.8.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.12.p5.16.m5.1c">t<_{\mathrm{lex}}t^{\prime}\rightarrow s<_{\mathrm{lex}}s^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.12.p5.16.m5.1d">italic_t < start_POSTSUBSCRIPT roman_lex end_POSTSUBSCRIPT italic_t start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT → italic_s < start_POSTSUBSCRIPT roman_lex end_POSTSUBSCRIPT italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>. Which finishes the proof. ∎</p> </div> </div> <div class="ltx_para" id="S4.p7"> <p class="ltx_p" id="S4.p7.1">By recalling that under <math alttext="\mathsf{MA}_{\aleph_{1}}" class="ltx_Math" display="inline" id="S4.p7.1.m1.1"><semantics id="S4.p7.1.m1.1a"><msub id="S4.p7.1.m1.1.1" xref="S4.p7.1.m1.1.1.cmml"><mi id="S4.p7.1.m1.1.1.2" xref="S4.p7.1.m1.1.1.2.cmml">𝖬𝖠</mi><msub id="S4.p7.1.m1.1.1.3" xref="S4.p7.1.m1.1.1.3.cmml"><mi id="S4.p7.1.m1.1.1.3.2" mathvariant="normal" xref="S4.p7.1.m1.1.1.3.2.cmml">ℵ</mi><mn id="S4.p7.1.m1.1.1.3.3" xref="S4.p7.1.m1.1.1.3.3.cmml">1</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S4.p7.1.m1.1b"><apply id="S4.p7.1.m1.1.1.cmml" xref="S4.p7.1.m1.1.1"><csymbol cd="ambiguous" id="S4.p7.1.m1.1.1.1.cmml" xref="S4.p7.1.m1.1.1">subscript</csymbol><ci id="S4.p7.1.m1.1.1.2.cmml" xref="S4.p7.1.m1.1.1.2">𝖬𝖠</ci><apply id="S4.p7.1.m1.1.1.3.cmml" xref="S4.p7.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.p7.1.m1.1.1.3.1.cmml" xref="S4.p7.1.m1.1.1.3">subscript</csymbol><ci id="S4.p7.1.m1.1.1.3.2.cmml" xref="S4.p7.1.m1.1.1.3.2">ℵ</ci><cn id="S4.p7.1.m1.1.1.3.3.cmml" type="integer" xref="S4.p7.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p7.1.m1.1c">\mathsf{MA}_{\aleph_{1}}</annotation><annotation encoding="application/x-llamapun" id="S4.p7.1.m1.1d">sansserif_MA start_POSTSUBSCRIPT roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> every tree is special, we immediately get the following.</p> </div> <div class="ltx_theorem ltx_theorem_corollary" id="S4.Thmtheorem8"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem8.1.1.1">Corollary 4.8</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem8.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem8.p1"> <p class="ltx_p" id="S4.Thmtheorem8.p1.3">Assume <math alttext="\mathsf{MA}_{\aleph_{1}}" class="ltx_Math" display="inline" id="S4.Thmtheorem8.p1.1.m1.1"><semantics id="S4.Thmtheorem8.p1.1.m1.1a"><msub id="S4.Thmtheorem8.p1.1.m1.1.1" xref="S4.Thmtheorem8.p1.1.m1.1.1.cmml"><mi id="S4.Thmtheorem8.p1.1.m1.1.1.2" xref="S4.Thmtheorem8.p1.1.m1.1.1.2.cmml">𝖬𝖠</mi><msub id="S4.Thmtheorem8.p1.1.m1.1.1.3" xref="S4.Thmtheorem8.p1.1.m1.1.1.3.cmml"><mi id="S4.Thmtheorem8.p1.1.m1.1.1.3.2" mathvariant="normal" xref="S4.Thmtheorem8.p1.1.m1.1.1.3.2.cmml">ℵ</mi><mn id="S4.Thmtheorem8.p1.1.m1.1.1.3.3" xref="S4.Thmtheorem8.p1.1.m1.1.1.3.3.cmml">1</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem8.p1.1.m1.1b"><apply id="S4.Thmtheorem8.p1.1.m1.1.1.cmml" xref="S4.Thmtheorem8.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem8.p1.1.m1.1.1.1.cmml" xref="S4.Thmtheorem8.p1.1.m1.1.1">subscript</csymbol><ci id="S4.Thmtheorem8.p1.1.m1.1.1.2.cmml" xref="S4.Thmtheorem8.p1.1.m1.1.1.2">𝖬𝖠</ci><apply id="S4.Thmtheorem8.p1.1.m1.1.1.3.cmml" xref="S4.Thmtheorem8.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem8.p1.1.m1.1.1.3.1.cmml" xref="S4.Thmtheorem8.p1.1.m1.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem8.p1.1.m1.1.1.3.2.cmml" xref="S4.Thmtheorem8.p1.1.m1.1.1.3.2">ℵ</ci><cn id="S4.Thmtheorem8.p1.1.m1.1.1.3.3.cmml" type="integer" xref="S4.Thmtheorem8.p1.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem8.p1.1.m1.1c">\mathsf{MA}_{\aleph_{1}}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem8.p1.1.m1.1d">sansserif_MA start_POSTSUBSCRIPT roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. For every <math alttext="X,Y\subseteq\omega_{1}" class="ltx_Math" display="inline" id="S4.Thmtheorem8.p1.2.m2.2"><semantics id="S4.Thmtheorem8.p1.2.m2.2a"><mrow id="S4.Thmtheorem8.p1.2.m2.2.3" xref="S4.Thmtheorem8.p1.2.m2.2.3.cmml"><mrow id="S4.Thmtheorem8.p1.2.m2.2.3.2.2" xref="S4.Thmtheorem8.p1.2.m2.2.3.2.1.cmml"><mi id="S4.Thmtheorem8.p1.2.m2.1.1" xref="S4.Thmtheorem8.p1.2.m2.1.1.cmml">X</mi><mo id="S4.Thmtheorem8.p1.2.m2.2.3.2.2.1" xref="S4.Thmtheorem8.p1.2.m2.2.3.2.1.cmml">,</mo><mi id="S4.Thmtheorem8.p1.2.m2.2.2" xref="S4.Thmtheorem8.p1.2.m2.2.2.cmml">Y</mi></mrow><mo id="S4.Thmtheorem8.p1.2.m2.2.3.1" xref="S4.Thmtheorem8.p1.2.m2.2.3.1.cmml">⊆</mo><msub id="S4.Thmtheorem8.p1.2.m2.2.3.3" xref="S4.Thmtheorem8.p1.2.m2.2.3.3.cmml"><mi id="S4.Thmtheorem8.p1.2.m2.2.3.3.2" xref="S4.Thmtheorem8.p1.2.m2.2.3.3.2.cmml">ω</mi><mn id="S4.Thmtheorem8.p1.2.m2.2.3.3.3" xref="S4.Thmtheorem8.p1.2.m2.2.3.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem8.p1.2.m2.2b"><apply id="S4.Thmtheorem8.p1.2.m2.2.3.cmml" xref="S4.Thmtheorem8.p1.2.m2.2.3"><subset id="S4.Thmtheorem8.p1.2.m2.2.3.1.cmml" xref="S4.Thmtheorem8.p1.2.m2.2.3.1"></subset><list id="S4.Thmtheorem8.p1.2.m2.2.3.2.1.cmml" xref="S4.Thmtheorem8.p1.2.m2.2.3.2.2"><ci id="S4.Thmtheorem8.p1.2.m2.1.1.cmml" xref="S4.Thmtheorem8.p1.2.m2.1.1">𝑋</ci><ci id="S4.Thmtheorem8.p1.2.m2.2.2.cmml" xref="S4.Thmtheorem8.p1.2.m2.2.2">𝑌</ci></list><apply id="S4.Thmtheorem8.p1.2.m2.2.3.3.cmml" xref="S4.Thmtheorem8.p1.2.m2.2.3.3"><csymbol cd="ambiguous" id="S4.Thmtheorem8.p1.2.m2.2.3.3.1.cmml" xref="S4.Thmtheorem8.p1.2.m2.2.3.3">subscript</csymbol><ci id="S4.Thmtheorem8.p1.2.m2.2.3.3.2.cmml" xref="S4.Thmtheorem8.p1.2.m2.2.3.3.2">𝜔</ci><cn id="S4.Thmtheorem8.p1.2.m2.2.3.3.3.cmml" type="integer" xref="S4.Thmtheorem8.p1.2.m2.2.3.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem8.p1.2.m2.2c">X,Y\subseteq\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem8.p1.2.m2.2d">italic_X , italic_Y ⊆ italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> we can require <math alttext="A" class="ltx_Math" display="inline" id="S4.Thmtheorem8.p1.3.m3.1"><semantics id="S4.Thmtheorem8.p1.3.m3.1a"><mi id="S4.Thmtheorem8.p1.3.m3.1.1" xref="S4.Thmtheorem8.p1.3.m3.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem8.p1.3.m3.1b"><ci id="S4.Thmtheorem8.p1.3.m3.1.1.cmml" xref="S4.Thmtheorem8.p1.3.m3.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem8.p1.3.m3.1c">A</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem8.p1.3.m3.1d">italic_A</annotation></semantics></math> in the conclusion of <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S4.Thmtheorem5" title="Theorem 4.5. ‣ 4. Aronszajn line decompositions ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">4.5</span></a> to be Countryman.</p> </div> </div> <div class="ltx_theorem ltx_theorem_remark" id="S4.Thmtheorem9"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem9.1.1.1">Remark 4.9</span></span><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem9.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem9.p1"> <p class="ltx_p" id="S4.Thmtheorem9.p1.8">At the end of Section 4.5 in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib23" title="">23</a>]</cite> it is claimed that a simple modification of <math alttext="T^{*}(\rho_{1})" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p1.1.m1.1"><semantics id="S4.Thmtheorem9.p1.1.m1.1a"><mrow id="S4.Thmtheorem9.p1.1.m1.1.1" xref="S4.Thmtheorem9.p1.1.m1.1.1.cmml"><msup id="S4.Thmtheorem9.p1.1.m1.1.1.3" xref="S4.Thmtheorem9.p1.1.m1.1.1.3.cmml"><mi id="S4.Thmtheorem9.p1.1.m1.1.1.3.2" xref="S4.Thmtheorem9.p1.1.m1.1.1.3.2.cmml">T</mi><mo id="S4.Thmtheorem9.p1.1.m1.1.1.3.3" xref="S4.Thmtheorem9.p1.1.m1.1.1.3.3.cmml">∗</mo></msup><mo id="S4.Thmtheorem9.p1.1.m1.1.1.2" xref="S4.Thmtheorem9.p1.1.m1.1.1.2.cmml">⁢</mo><mrow id="S4.Thmtheorem9.p1.1.m1.1.1.1.1" xref="S4.Thmtheorem9.p1.1.m1.1.1.1.1.1.cmml"><mo id="S4.Thmtheorem9.p1.1.m1.1.1.1.1.2" stretchy="false" xref="S4.Thmtheorem9.p1.1.m1.1.1.1.1.1.cmml">(</mo><msub id="S4.Thmtheorem9.p1.1.m1.1.1.1.1.1" xref="S4.Thmtheorem9.p1.1.m1.1.1.1.1.1.cmml"><mi id="S4.Thmtheorem9.p1.1.m1.1.1.1.1.1.2" xref="S4.Thmtheorem9.p1.1.m1.1.1.1.1.1.2.cmml">ρ</mi><mn id="S4.Thmtheorem9.p1.1.m1.1.1.1.1.1.3" xref="S4.Thmtheorem9.p1.1.m1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.Thmtheorem9.p1.1.m1.1.1.1.1.3" stretchy="false" xref="S4.Thmtheorem9.p1.1.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p1.1.m1.1b"><apply id="S4.Thmtheorem9.p1.1.m1.1.1.cmml" xref="S4.Thmtheorem9.p1.1.m1.1.1"><times id="S4.Thmtheorem9.p1.1.m1.1.1.2.cmml" xref="S4.Thmtheorem9.p1.1.m1.1.1.2"></times><apply id="S4.Thmtheorem9.p1.1.m1.1.1.3.cmml" xref="S4.Thmtheorem9.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p1.1.m1.1.1.3.1.cmml" xref="S4.Thmtheorem9.p1.1.m1.1.1.3">superscript</csymbol><ci id="S4.Thmtheorem9.p1.1.m1.1.1.3.2.cmml" xref="S4.Thmtheorem9.p1.1.m1.1.1.3.2">𝑇</ci><times id="S4.Thmtheorem9.p1.1.m1.1.1.3.3.cmml" xref="S4.Thmtheorem9.p1.1.m1.1.1.3.3"></times></apply><apply id="S4.Thmtheorem9.p1.1.m1.1.1.1.1.1.cmml" xref="S4.Thmtheorem9.p1.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p1.1.m1.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem9.p1.1.m1.1.1.1.1">subscript</csymbol><ci id="S4.Thmtheorem9.p1.1.m1.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem9.p1.1.m1.1.1.1.1.1.2">𝜌</ci><cn id="S4.Thmtheorem9.p1.1.m1.1.1.1.1.1.3.cmml" type="integer" xref="S4.Thmtheorem9.p1.1.m1.1.1.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p1.1.m1.1c">T^{*}(\rho_{1})</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p1.1.m1.1d">italic_T start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math> gives, in <math alttext="\mathsf{ZFC}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p1.2.m2.1"><semantics id="S4.Thmtheorem9.p1.2.m2.1a"><mi id="S4.Thmtheorem9.p1.2.m2.1.1" xref="S4.Thmtheorem9.p1.2.m2.1.1.cmml">𝖹𝖥𝖢</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p1.2.m2.1b"><ci id="S4.Thmtheorem9.p1.2.m2.1.1.cmml" xref="S4.Thmtheorem9.p1.2.m2.1.1">𝖹𝖥𝖢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p1.2.m2.1c">\mathsf{ZFC}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p1.2.m2.1d">sansserif_ZFC</annotation></semantics></math>, a Countryman line and decomposition <math alttext="D" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p1.3.m3.1"><semantics id="S4.Thmtheorem9.p1.3.m3.1a"><mi id="S4.Thmtheorem9.p1.3.m3.1.1" xref="S4.Thmtheorem9.p1.3.m3.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p1.3.m3.1b"><ci id="S4.Thmtheorem9.p1.3.m3.1.1.cmml" xref="S4.Thmtheorem9.p1.3.m3.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p1.3.m3.1c">D</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p1.3.m3.1d">italic_D</annotation></semantics></math> such that <math alttext="\hat{\mathscr{L}}(C,D)\cap\hat{\mathscr{R}}(C,D)" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p1.4.m4.4"><semantics id="S4.Thmtheorem9.p1.4.m4.4a"><mrow id="S4.Thmtheorem9.p1.4.m4.4.5" xref="S4.Thmtheorem9.p1.4.m4.4.5.cmml"><mrow id="S4.Thmtheorem9.p1.4.m4.4.5.2" xref="S4.Thmtheorem9.p1.4.m4.4.5.2.cmml"><mover accent="true" id="S4.Thmtheorem9.p1.4.m4.4.5.2.2" xref="S4.Thmtheorem9.p1.4.m4.4.5.2.2.cmml"><mi class="ltx_font_mathscript" id="S4.Thmtheorem9.p1.4.m4.4.5.2.2.2" xref="S4.Thmtheorem9.p1.4.m4.4.5.2.2.2.cmml">ℒ</mi><mo id="S4.Thmtheorem9.p1.4.m4.4.5.2.2.1" xref="S4.Thmtheorem9.p1.4.m4.4.5.2.2.1.cmml">^</mo></mover><mo id="S4.Thmtheorem9.p1.4.m4.4.5.2.1" xref="S4.Thmtheorem9.p1.4.m4.4.5.2.1.cmml">⁢</mo><mrow id="S4.Thmtheorem9.p1.4.m4.4.5.2.3.2" xref="S4.Thmtheorem9.p1.4.m4.4.5.2.3.1.cmml"><mo id="S4.Thmtheorem9.p1.4.m4.4.5.2.3.2.1" stretchy="false" xref="S4.Thmtheorem9.p1.4.m4.4.5.2.3.1.cmml">(</mo><mi id="S4.Thmtheorem9.p1.4.m4.1.1" xref="S4.Thmtheorem9.p1.4.m4.1.1.cmml">C</mi><mo id="S4.Thmtheorem9.p1.4.m4.4.5.2.3.2.2" xref="S4.Thmtheorem9.p1.4.m4.4.5.2.3.1.cmml">,</mo><mi id="S4.Thmtheorem9.p1.4.m4.2.2" xref="S4.Thmtheorem9.p1.4.m4.2.2.cmml">D</mi><mo id="S4.Thmtheorem9.p1.4.m4.4.5.2.3.2.3" stretchy="false" xref="S4.Thmtheorem9.p1.4.m4.4.5.2.3.1.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem9.p1.4.m4.4.5.1" xref="S4.Thmtheorem9.p1.4.m4.4.5.1.cmml">∩</mo><mrow id="S4.Thmtheorem9.p1.4.m4.4.5.3" xref="S4.Thmtheorem9.p1.4.m4.4.5.3.cmml"><mover accent="true" id="S4.Thmtheorem9.p1.4.m4.4.5.3.2" xref="S4.Thmtheorem9.p1.4.m4.4.5.3.2.cmml"><mi class="ltx_font_mathscript" id="S4.Thmtheorem9.p1.4.m4.4.5.3.2.2" xref="S4.Thmtheorem9.p1.4.m4.4.5.3.2.2.cmml">ℛ</mi><mo id="S4.Thmtheorem9.p1.4.m4.4.5.3.2.1" xref="S4.Thmtheorem9.p1.4.m4.4.5.3.2.1.cmml">^</mo></mover><mo id="S4.Thmtheorem9.p1.4.m4.4.5.3.1" xref="S4.Thmtheorem9.p1.4.m4.4.5.3.1.cmml">⁢</mo><mrow id="S4.Thmtheorem9.p1.4.m4.4.5.3.3.2" xref="S4.Thmtheorem9.p1.4.m4.4.5.3.3.1.cmml"><mo id="S4.Thmtheorem9.p1.4.m4.4.5.3.3.2.1" stretchy="false" xref="S4.Thmtheorem9.p1.4.m4.4.5.3.3.1.cmml">(</mo><mi id="S4.Thmtheorem9.p1.4.m4.3.3" xref="S4.Thmtheorem9.p1.4.m4.3.3.cmml">C</mi><mo id="S4.Thmtheorem9.p1.4.m4.4.5.3.3.2.2" xref="S4.Thmtheorem9.p1.4.m4.4.5.3.3.1.cmml">,</mo><mi id="S4.Thmtheorem9.p1.4.m4.4.4" xref="S4.Thmtheorem9.p1.4.m4.4.4.cmml">D</mi><mo id="S4.Thmtheorem9.p1.4.m4.4.5.3.3.2.3" stretchy="false" xref="S4.Thmtheorem9.p1.4.m4.4.5.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p1.4.m4.4b"><apply id="S4.Thmtheorem9.p1.4.m4.4.5.cmml" xref="S4.Thmtheorem9.p1.4.m4.4.5"><intersect id="S4.Thmtheorem9.p1.4.m4.4.5.1.cmml" xref="S4.Thmtheorem9.p1.4.m4.4.5.1"></intersect><apply id="S4.Thmtheorem9.p1.4.m4.4.5.2.cmml" xref="S4.Thmtheorem9.p1.4.m4.4.5.2"><times id="S4.Thmtheorem9.p1.4.m4.4.5.2.1.cmml" xref="S4.Thmtheorem9.p1.4.m4.4.5.2.1"></times><apply id="S4.Thmtheorem9.p1.4.m4.4.5.2.2.cmml" xref="S4.Thmtheorem9.p1.4.m4.4.5.2.2"><ci id="S4.Thmtheorem9.p1.4.m4.4.5.2.2.1.cmml" xref="S4.Thmtheorem9.p1.4.m4.4.5.2.2.1">^</ci><ci id="S4.Thmtheorem9.p1.4.m4.4.5.2.2.2.cmml" xref="S4.Thmtheorem9.p1.4.m4.4.5.2.2.2">ℒ</ci></apply><interval closure="open" id="S4.Thmtheorem9.p1.4.m4.4.5.2.3.1.cmml" xref="S4.Thmtheorem9.p1.4.m4.4.5.2.3.2"><ci id="S4.Thmtheorem9.p1.4.m4.1.1.cmml" xref="S4.Thmtheorem9.p1.4.m4.1.1">𝐶</ci><ci id="S4.Thmtheorem9.p1.4.m4.2.2.cmml" xref="S4.Thmtheorem9.p1.4.m4.2.2">𝐷</ci></interval></apply><apply id="S4.Thmtheorem9.p1.4.m4.4.5.3.cmml" xref="S4.Thmtheorem9.p1.4.m4.4.5.3"><times id="S4.Thmtheorem9.p1.4.m4.4.5.3.1.cmml" xref="S4.Thmtheorem9.p1.4.m4.4.5.3.1"></times><apply id="S4.Thmtheorem9.p1.4.m4.4.5.3.2.cmml" xref="S4.Thmtheorem9.p1.4.m4.4.5.3.2"><ci id="S4.Thmtheorem9.p1.4.m4.4.5.3.2.1.cmml" xref="S4.Thmtheorem9.p1.4.m4.4.5.3.2.1">^</ci><ci id="S4.Thmtheorem9.p1.4.m4.4.5.3.2.2.cmml" xref="S4.Thmtheorem9.p1.4.m4.4.5.3.2.2">ℛ</ci></apply><interval closure="open" id="S4.Thmtheorem9.p1.4.m4.4.5.3.3.1.cmml" xref="S4.Thmtheorem9.p1.4.m4.4.5.3.3.2"><ci id="S4.Thmtheorem9.p1.4.m4.3.3.cmml" xref="S4.Thmtheorem9.p1.4.m4.3.3">𝐶</ci><ci id="S4.Thmtheorem9.p1.4.m4.4.4.cmml" xref="S4.Thmtheorem9.p1.4.m4.4.4">𝐷</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p1.4.m4.4c">\hat{\mathscr{L}}(C,D)\cap\hat{\mathscr{R}}(C,D)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p1.4.m4.4d">over^ start_ARG script_L end_ARG ( italic_C , italic_D ) ∩ over^ start_ARG script_R end_ARG ( italic_C , italic_D )</annotation></semantics></math> contains a club. The construction as stated does not seem to make sense, and in view of <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S4.Thmtheorem7" title="Lemma 4.7. ‣ 4. Aronszajn line decompositions ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">4.7</span></a> and the fact that consistently <math alttext="T^{*}(\rho_{1})" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p1.5.m5.1"><semantics id="S4.Thmtheorem9.p1.5.m5.1a"><mrow id="S4.Thmtheorem9.p1.5.m5.1.1" xref="S4.Thmtheorem9.p1.5.m5.1.1.cmml"><msup id="S4.Thmtheorem9.p1.5.m5.1.1.3" xref="S4.Thmtheorem9.p1.5.m5.1.1.3.cmml"><mi id="S4.Thmtheorem9.p1.5.m5.1.1.3.2" xref="S4.Thmtheorem9.p1.5.m5.1.1.3.2.cmml">T</mi><mo id="S4.Thmtheorem9.p1.5.m5.1.1.3.3" xref="S4.Thmtheorem9.p1.5.m5.1.1.3.3.cmml">∗</mo></msup><mo id="S4.Thmtheorem9.p1.5.m5.1.1.2" xref="S4.Thmtheorem9.p1.5.m5.1.1.2.cmml">⁢</mo><mrow id="S4.Thmtheorem9.p1.5.m5.1.1.1.1" xref="S4.Thmtheorem9.p1.5.m5.1.1.1.1.1.cmml"><mo id="S4.Thmtheorem9.p1.5.m5.1.1.1.1.2" stretchy="false" xref="S4.Thmtheorem9.p1.5.m5.1.1.1.1.1.cmml">(</mo><msub id="S4.Thmtheorem9.p1.5.m5.1.1.1.1.1" xref="S4.Thmtheorem9.p1.5.m5.1.1.1.1.1.cmml"><mi id="S4.Thmtheorem9.p1.5.m5.1.1.1.1.1.2" xref="S4.Thmtheorem9.p1.5.m5.1.1.1.1.1.2.cmml">ρ</mi><mn id="S4.Thmtheorem9.p1.5.m5.1.1.1.1.1.3" xref="S4.Thmtheorem9.p1.5.m5.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.Thmtheorem9.p1.5.m5.1.1.1.1.3" stretchy="false" xref="S4.Thmtheorem9.p1.5.m5.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p1.5.m5.1b"><apply id="S4.Thmtheorem9.p1.5.m5.1.1.cmml" xref="S4.Thmtheorem9.p1.5.m5.1.1"><times id="S4.Thmtheorem9.p1.5.m5.1.1.2.cmml" xref="S4.Thmtheorem9.p1.5.m5.1.1.2"></times><apply id="S4.Thmtheorem9.p1.5.m5.1.1.3.cmml" xref="S4.Thmtheorem9.p1.5.m5.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p1.5.m5.1.1.3.1.cmml" xref="S4.Thmtheorem9.p1.5.m5.1.1.3">superscript</csymbol><ci id="S4.Thmtheorem9.p1.5.m5.1.1.3.2.cmml" xref="S4.Thmtheorem9.p1.5.m5.1.1.3.2">𝑇</ci><times id="S4.Thmtheorem9.p1.5.m5.1.1.3.3.cmml" xref="S4.Thmtheorem9.p1.5.m5.1.1.3.3"></times></apply><apply id="S4.Thmtheorem9.p1.5.m5.1.1.1.1.1.cmml" xref="S4.Thmtheorem9.p1.5.m5.1.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p1.5.m5.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem9.p1.5.m5.1.1.1.1">subscript</csymbol><ci id="S4.Thmtheorem9.p1.5.m5.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem9.p1.5.m5.1.1.1.1.1.2">𝜌</ci><cn id="S4.Thmtheorem9.p1.5.m5.1.1.1.1.1.3.cmml" type="integer" xref="S4.Thmtheorem9.p1.5.m5.1.1.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p1.5.m5.1c">T^{*}(\rho_{1})</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p1.5.m5.1d">italic_T start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math> is not special (see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib23" title="">23</a>, 2.2.16]</cite>), it is not clear what was intended. However, if one is willing to do more work, it is possible to construct <math alttext="A^{+}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p1.6.m6.1"><semantics id="S4.Thmtheorem9.p1.6.m6.1a"><msup id="S4.Thmtheorem9.p1.6.m6.1.1" xref="S4.Thmtheorem9.p1.6.m6.1.1.cmml"><mi id="S4.Thmtheorem9.p1.6.m6.1.1.2" xref="S4.Thmtheorem9.p1.6.m6.1.1.2.cmml">A</mi><mo id="S4.Thmtheorem9.p1.6.m6.1.1.3" xref="S4.Thmtheorem9.p1.6.m6.1.1.3.cmml">+</mo></msup><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p1.6.m6.1b"><apply id="S4.Thmtheorem9.p1.6.m6.1.1.cmml" xref="S4.Thmtheorem9.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p1.6.m6.1.1.1.cmml" xref="S4.Thmtheorem9.p1.6.m6.1.1">superscript</csymbol><ci id="S4.Thmtheorem9.p1.6.m6.1.1.2.cmml" xref="S4.Thmtheorem9.p1.6.m6.1.1.2">𝐴</ci><plus id="S4.Thmtheorem9.p1.6.m6.1.1.3.cmml" xref="S4.Thmtheorem9.p1.6.m6.1.1.3"></plus></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p1.6.m6.1c">A^{+}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p1.6.m6.1d">italic_A start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> as to be Countryman in <math alttext="\mathsf{ZFC}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p1.7.m7.1"><semantics id="S4.Thmtheorem9.p1.7.m7.1a"><mi id="S4.Thmtheorem9.p1.7.m7.1.1" xref="S4.Thmtheorem9.p1.7.m7.1.1.cmml">𝖹𝖥𝖢</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p1.7.m7.1b"><ci id="S4.Thmtheorem9.p1.7.m7.1.1.cmml" xref="S4.Thmtheorem9.p1.7.m7.1.1">𝖹𝖥𝖢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p1.7.m7.1c">\mathsf{ZFC}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p1.7.m7.1d">sansserif_ZFC</annotation></semantics></math>, thus removing the need of <math alttext="\mathsf{MA}_{\aleph_{1}}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p1.8.m8.1"><semantics id="S4.Thmtheorem9.p1.8.m8.1a"><msub id="S4.Thmtheorem9.p1.8.m8.1.1" xref="S4.Thmtheorem9.p1.8.m8.1.1.cmml"><mi id="S4.Thmtheorem9.p1.8.m8.1.1.2" xref="S4.Thmtheorem9.p1.8.m8.1.1.2.cmml">𝖬𝖠</mi><msub id="S4.Thmtheorem9.p1.8.m8.1.1.3" xref="S4.Thmtheorem9.p1.8.m8.1.1.3.cmml"><mi id="S4.Thmtheorem9.p1.8.m8.1.1.3.2" mathvariant="normal" xref="S4.Thmtheorem9.p1.8.m8.1.1.3.2.cmml">ℵ</mi><mn id="S4.Thmtheorem9.p1.8.m8.1.1.3.3" xref="S4.Thmtheorem9.p1.8.m8.1.1.3.3.cmml">1</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p1.8.m8.1b"><apply id="S4.Thmtheorem9.p1.8.m8.1.1.cmml" xref="S4.Thmtheorem9.p1.8.m8.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p1.8.m8.1.1.1.cmml" xref="S4.Thmtheorem9.p1.8.m8.1.1">subscript</csymbol><ci id="S4.Thmtheorem9.p1.8.m8.1.1.2.cmml" xref="S4.Thmtheorem9.p1.8.m8.1.1.2">𝖬𝖠</ci><apply id="S4.Thmtheorem9.p1.8.m8.1.1.3.cmml" xref="S4.Thmtheorem9.p1.8.m8.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p1.8.m8.1.1.3.1.cmml" xref="S4.Thmtheorem9.p1.8.m8.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem9.p1.8.m8.1.1.3.2.cmml" xref="S4.Thmtheorem9.p1.8.m8.1.1.3.2">ℵ</ci><cn id="S4.Thmtheorem9.p1.8.m8.1.1.3.3.cmml" type="integer" xref="S4.Thmtheorem9.p1.8.m8.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p1.8.m8.1c">\mathsf{MA}_{\aleph_{1}}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p1.8.m8.1d">sansserif_MA start_POSTSUBSCRIPT roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> in <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S4.Thmtheorem8" title="Corollary 4.8. ‣ 4. Aronszajn line decompositions ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Corollary</span> <span class="ltx_text ltx_ref_tag">4.8</span></a>.</p> </div> <div class="ltx_para" id="S4.Thmtheorem9.p2"> <p class="ltx_p" id="S4.Thmtheorem9.p2.26">We provide a sketch for the interested reader. We use the definitions and notations of <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib6" title="">6</a>, Section 3]</cite>. Let <math alttext="U\subseteq\mathbb{S}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p2.1.m1.1"><semantics id="S4.Thmtheorem9.p2.1.m1.1a"><mrow id="S4.Thmtheorem9.p2.1.m1.1.1" xref="S4.Thmtheorem9.p2.1.m1.1.1.cmml"><mi id="S4.Thmtheorem9.p2.1.m1.1.1.2" xref="S4.Thmtheorem9.p2.1.m1.1.1.2.cmml">U</mi><mo id="S4.Thmtheorem9.p2.1.m1.1.1.1" xref="S4.Thmtheorem9.p2.1.m1.1.1.1.cmml">⊆</mo><mi id="S4.Thmtheorem9.p2.1.m1.1.1.3" xref="S4.Thmtheorem9.p2.1.m1.1.1.3.cmml">𝕊</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p2.1.m1.1b"><apply id="S4.Thmtheorem9.p2.1.m1.1.1.cmml" xref="S4.Thmtheorem9.p2.1.m1.1.1"><subset id="S4.Thmtheorem9.p2.1.m1.1.1.1.cmml" xref="S4.Thmtheorem9.p2.1.m1.1.1.1"></subset><ci id="S4.Thmtheorem9.p2.1.m1.1.1.2.cmml" xref="S4.Thmtheorem9.p2.1.m1.1.1.2">𝑈</ci><ci id="S4.Thmtheorem9.p2.1.m1.1.1.3.cmml" xref="S4.Thmtheorem9.p2.1.m1.1.1.3">𝕊</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p2.1.m1.1c">U\subseteq\mathbb{S}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p2.1.m1.1d">italic_U ⊆ blackboard_S</annotation></semantics></math> be <math alttext="\varrho_{2}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p2.2.m2.1"><semantics id="S4.Thmtheorem9.p2.2.m2.1a"><msub id="S4.Thmtheorem9.p2.2.m2.1.1" xref="S4.Thmtheorem9.p2.2.m2.1.1.cmml"><mi id="S4.Thmtheorem9.p2.2.m2.1.1.2" xref="S4.Thmtheorem9.p2.2.m2.1.1.2.cmml">ϱ</mi><mn id="S4.Thmtheorem9.p2.2.m2.1.1.3" xref="S4.Thmtheorem9.p2.2.m2.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p2.2.m2.1b"><apply id="S4.Thmtheorem9.p2.2.m2.1.1.cmml" xref="S4.Thmtheorem9.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p2.2.m2.1.1.1.cmml" xref="S4.Thmtheorem9.p2.2.m2.1.1">subscript</csymbol><ci id="S4.Thmtheorem9.p2.2.m2.1.1.2.cmml" xref="S4.Thmtheorem9.p2.2.m2.1.1.2">italic-ϱ</ci><cn id="S4.Thmtheorem9.p2.2.m2.1.1.3.cmml" type="integer" xref="S4.Thmtheorem9.p2.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p2.2.m2.1c">\varrho_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p2.2.m2.1d">italic_ϱ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>-coherent and full. That such a tree exists in <math alttext="\mathsf{ZFC}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p2.3.m3.1"><semantics id="S4.Thmtheorem9.p2.3.m3.1a"><mi id="S4.Thmtheorem9.p2.3.m3.1.1" xref="S4.Thmtheorem9.p2.3.m3.1.1.cmml">𝖹𝖥𝖢</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p2.3.m3.1b"><ci id="S4.Thmtheorem9.p2.3.m3.1.1.cmml" xref="S4.Thmtheorem9.p2.3.m3.1.1">𝖹𝖥𝖢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p2.3.m3.1c">\mathsf{ZFC}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p2.3.m3.1d">sansserif_ZFC</annotation></semantics></math> is pointed out after Lemma 3.15 in that paper. Also <math alttext="(U,<_{\mathrm{lex}})" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p2.4.m4.2"><semantics id="S4.Thmtheorem9.p2.4.m4.2a"><mrow id="S4.Thmtheorem9.p2.4.m4.2.2.1" xref="S4.Thmtheorem9.p2.4.m4.2.2.2.cmml"><mo id="S4.Thmtheorem9.p2.4.m4.2.2.1.2" stretchy="false" xref="S4.Thmtheorem9.p2.4.m4.2.2.2.cmml">(</mo><mi id="S4.Thmtheorem9.p2.4.m4.1.1" xref="S4.Thmtheorem9.p2.4.m4.1.1.cmml">U</mi><mo id="S4.Thmtheorem9.p2.4.m4.2.2.1.3" xref="S4.Thmtheorem9.p2.4.m4.2.2.2.cmml">,</mo><msub id="S4.Thmtheorem9.p2.4.m4.2.2.1.1" xref="S4.Thmtheorem9.p2.4.m4.2.2.1.1.cmml"><mo id="S4.Thmtheorem9.p2.4.m4.2.2.1.1.2" lspace="0em" rspace="0em" xref="S4.Thmtheorem9.p2.4.m4.2.2.1.1.2.cmml"><</mo><mi id="S4.Thmtheorem9.p2.4.m4.2.2.1.1.3" xref="S4.Thmtheorem9.p2.4.m4.2.2.1.1.3.cmml">lex</mi></msub><mo id="S4.Thmtheorem9.p2.4.m4.2.2.1.4" stretchy="false" xref="S4.Thmtheorem9.p2.4.m4.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p2.4.m4.2b"><interval closure="open" id="S4.Thmtheorem9.p2.4.m4.2.2.2.cmml" xref="S4.Thmtheorem9.p2.4.m4.2.2.1"><ci id="S4.Thmtheorem9.p2.4.m4.1.1.cmml" xref="S4.Thmtheorem9.p2.4.m4.1.1">𝑈</ci><apply id="S4.Thmtheorem9.p2.4.m4.2.2.1.1.cmml" xref="S4.Thmtheorem9.p2.4.m4.2.2.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p2.4.m4.2.2.1.1.1.cmml" xref="S4.Thmtheorem9.p2.4.m4.2.2.1.1">subscript</csymbol><lt id="S4.Thmtheorem9.p2.4.m4.2.2.1.1.2.cmml" xref="S4.Thmtheorem9.p2.4.m4.2.2.1.1.2"></lt><ci id="S4.Thmtheorem9.p2.4.m4.2.2.1.1.3.cmml" xref="S4.Thmtheorem9.p2.4.m4.2.2.1.1.3">lex</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p2.4.m4.2c">(U,<_{\mathrm{lex}})</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p2.4.m4.2d">( italic_U , < start_POSTSUBSCRIPT roman_lex end_POSTSUBSCRIPT )</annotation></semantics></math> is Countryman by Theorem 3.5 there, and it is easily seen that <math alttext="U" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p2.5.m5.1"><semantics id="S4.Thmtheorem9.p2.5.m5.1a"><mi id="S4.Thmtheorem9.p2.5.m5.1.1" xref="S4.Thmtheorem9.p2.5.m5.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p2.5.m5.1b"><ci id="S4.Thmtheorem9.p2.5.m5.1.1.cmml" xref="S4.Thmtheorem9.p2.5.m5.1.1">𝑈</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p2.5.m5.1c">U</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p2.5.m5.1d">italic_U</annotation></semantics></math> is also special. Now let <math alttext="T" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p2.6.m6.1"><semantics id="S4.Thmtheorem9.p2.6.m6.1a"><mi id="S4.Thmtheorem9.p2.6.m6.1.1" xref="S4.Thmtheorem9.p2.6.m6.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p2.6.m6.1b"><ci id="S4.Thmtheorem9.p2.6.m6.1.1.cmml" xref="S4.Thmtheorem9.p2.6.m6.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p2.6.m6.1c">T</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p2.6.m6.1d">italic_T</annotation></semantics></math> be the result of adding to <math alttext="U" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p2.7.m7.1"><semantics id="S4.Thmtheorem9.p2.7.m7.1a"><mi id="S4.Thmtheorem9.p2.7.m7.1.1" xref="S4.Thmtheorem9.p2.7.m7.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p2.7.m7.1b"><ci id="S4.Thmtheorem9.p2.7.m7.1.1.cmml" xref="S4.Thmtheorem9.p2.7.m7.1.1">𝑈</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p2.7.m7.1c">U</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p2.7.m7.1d">italic_U</annotation></semantics></math> the initial segments of limit length of elements in <math alttext="U" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p2.8.m8.1"><semantics id="S4.Thmtheorem9.p2.8.m8.1a"><mi id="S4.Thmtheorem9.p2.8.m8.1.1" xref="S4.Thmtheorem9.p2.8.m8.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p2.8.m8.1b"><ci id="S4.Thmtheorem9.p2.8.m8.1.1.cmml" xref="S4.Thmtheorem9.p2.8.m8.1.1">𝑈</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p2.8.m8.1c">U</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p2.8.m8.1d">italic_U</annotation></semantics></math>. Note then that the <math alttext="\varrho_{2}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p2.9.m9.1"><semantics id="S4.Thmtheorem9.p2.9.m9.1a"><msub id="S4.Thmtheorem9.p2.9.m9.1.1" xref="S4.Thmtheorem9.p2.9.m9.1.1.cmml"><mi id="S4.Thmtheorem9.p2.9.m9.1.1.2" xref="S4.Thmtheorem9.p2.9.m9.1.1.2.cmml">ϱ</mi><mn id="S4.Thmtheorem9.p2.9.m9.1.1.3" xref="S4.Thmtheorem9.p2.9.m9.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p2.9.m9.1b"><apply id="S4.Thmtheorem9.p2.9.m9.1.1.cmml" xref="S4.Thmtheorem9.p2.9.m9.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p2.9.m9.1.1.1.cmml" xref="S4.Thmtheorem9.p2.9.m9.1.1">subscript</csymbol><ci id="S4.Thmtheorem9.p2.9.m9.1.1.2.cmml" xref="S4.Thmtheorem9.p2.9.m9.1.1.2">italic-ϱ</ci><cn id="S4.Thmtheorem9.p2.9.m9.1.1.3.cmml" type="integer" xref="S4.Thmtheorem9.p2.9.m9.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p2.9.m9.1c">\varrho_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p2.9.m9.1d">italic_ϱ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>-coherence of <math alttext="U" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p2.10.m10.1"><semantics id="S4.Thmtheorem9.p2.10.m10.1a"><mi id="S4.Thmtheorem9.p2.10.m10.1.1" xref="S4.Thmtheorem9.p2.10.m10.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p2.10.m10.1b"><ci id="S4.Thmtheorem9.p2.10.m10.1.1.cmml" xref="S4.Thmtheorem9.p2.10.m10.1.1">𝑈</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p2.10.m10.1c">U</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p2.10.m10.1d">italic_U</annotation></semantics></math> implies that if <math alttext="t\in T\setminus U" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p2.11.m11.1"><semantics id="S4.Thmtheorem9.p2.11.m11.1a"><mrow id="S4.Thmtheorem9.p2.11.m11.1.1" xref="S4.Thmtheorem9.p2.11.m11.1.1.cmml"><mi id="S4.Thmtheorem9.p2.11.m11.1.1.2" xref="S4.Thmtheorem9.p2.11.m11.1.1.2.cmml">t</mi><mo id="S4.Thmtheorem9.p2.11.m11.1.1.1" xref="S4.Thmtheorem9.p2.11.m11.1.1.1.cmml">∈</mo><mrow id="S4.Thmtheorem9.p2.11.m11.1.1.3" xref="S4.Thmtheorem9.p2.11.m11.1.1.3.cmml"><mi id="S4.Thmtheorem9.p2.11.m11.1.1.3.2" xref="S4.Thmtheorem9.p2.11.m11.1.1.3.2.cmml">T</mi><mo id="S4.Thmtheorem9.p2.11.m11.1.1.3.1" xref="S4.Thmtheorem9.p2.11.m11.1.1.3.1.cmml">∖</mo><mi id="S4.Thmtheorem9.p2.11.m11.1.1.3.3" xref="S4.Thmtheorem9.p2.11.m11.1.1.3.3.cmml">U</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p2.11.m11.1b"><apply id="S4.Thmtheorem9.p2.11.m11.1.1.cmml" xref="S4.Thmtheorem9.p2.11.m11.1.1"><in id="S4.Thmtheorem9.p2.11.m11.1.1.1.cmml" xref="S4.Thmtheorem9.p2.11.m11.1.1.1"></in><ci id="S4.Thmtheorem9.p2.11.m11.1.1.2.cmml" xref="S4.Thmtheorem9.p2.11.m11.1.1.2">𝑡</ci><apply id="S4.Thmtheorem9.p2.11.m11.1.1.3.cmml" xref="S4.Thmtheorem9.p2.11.m11.1.1.3"><setdiff id="S4.Thmtheorem9.p2.11.m11.1.1.3.1.cmml" xref="S4.Thmtheorem9.p2.11.m11.1.1.3.1"></setdiff><ci id="S4.Thmtheorem9.p2.11.m11.1.1.3.2.cmml" xref="S4.Thmtheorem9.p2.11.m11.1.1.3.2">𝑇</ci><ci id="S4.Thmtheorem9.p2.11.m11.1.1.3.3.cmml" xref="S4.Thmtheorem9.p2.11.m11.1.1.3.3">𝑈</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p2.11.m11.1c">t\in T\setminus U</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p2.11.m11.1d">italic_t ∈ italic_T ∖ italic_U</annotation></semantics></math>, then there is a unique <math alttext="n(t)\in\omega" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p2.12.m12.1"><semantics id="S4.Thmtheorem9.p2.12.m12.1a"><mrow id="S4.Thmtheorem9.p2.12.m12.1.2" xref="S4.Thmtheorem9.p2.12.m12.1.2.cmml"><mrow id="S4.Thmtheorem9.p2.12.m12.1.2.2" xref="S4.Thmtheorem9.p2.12.m12.1.2.2.cmml"><mi id="S4.Thmtheorem9.p2.12.m12.1.2.2.2" xref="S4.Thmtheorem9.p2.12.m12.1.2.2.2.cmml">n</mi><mo id="S4.Thmtheorem9.p2.12.m12.1.2.2.1" xref="S4.Thmtheorem9.p2.12.m12.1.2.2.1.cmml">⁢</mo><mrow id="S4.Thmtheorem9.p2.12.m12.1.2.2.3.2" xref="S4.Thmtheorem9.p2.12.m12.1.2.2.cmml"><mo id="S4.Thmtheorem9.p2.12.m12.1.2.2.3.2.1" stretchy="false" xref="S4.Thmtheorem9.p2.12.m12.1.2.2.cmml">(</mo><mi id="S4.Thmtheorem9.p2.12.m12.1.1" xref="S4.Thmtheorem9.p2.12.m12.1.1.cmml">t</mi><mo id="S4.Thmtheorem9.p2.12.m12.1.2.2.3.2.2" stretchy="false" xref="S4.Thmtheorem9.p2.12.m12.1.2.2.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem9.p2.12.m12.1.2.1" xref="S4.Thmtheorem9.p2.12.m12.1.2.1.cmml">∈</mo><mi id="S4.Thmtheorem9.p2.12.m12.1.2.3" xref="S4.Thmtheorem9.p2.12.m12.1.2.3.cmml">ω</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p2.12.m12.1b"><apply id="S4.Thmtheorem9.p2.12.m12.1.2.cmml" xref="S4.Thmtheorem9.p2.12.m12.1.2"><in id="S4.Thmtheorem9.p2.12.m12.1.2.1.cmml" xref="S4.Thmtheorem9.p2.12.m12.1.2.1"></in><apply id="S4.Thmtheorem9.p2.12.m12.1.2.2.cmml" xref="S4.Thmtheorem9.p2.12.m12.1.2.2"><times id="S4.Thmtheorem9.p2.12.m12.1.2.2.1.cmml" xref="S4.Thmtheorem9.p2.12.m12.1.2.2.1"></times><ci id="S4.Thmtheorem9.p2.12.m12.1.2.2.2.cmml" xref="S4.Thmtheorem9.p2.12.m12.1.2.2.2">𝑛</ci><ci id="S4.Thmtheorem9.p2.12.m12.1.1.cmml" xref="S4.Thmtheorem9.p2.12.m12.1.1">𝑡</ci></apply><ci id="S4.Thmtheorem9.p2.12.m12.1.2.3.cmml" xref="S4.Thmtheorem9.p2.12.m12.1.2.3">𝜔</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p2.12.m12.1c">n(t)\in\omega</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p2.12.m12.1d">italic_n ( italic_t ) ∈ italic_ω</annotation></semantics></math> such that <math alttext="t^{\frown}\langle n(t)\rangle\in U" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p2.13.m13.2"><semantics id="S4.Thmtheorem9.p2.13.m13.2a"><mrow id="S4.Thmtheorem9.p2.13.m13.2.2" xref="S4.Thmtheorem9.p2.13.m13.2.2.cmml"><mrow id="S4.Thmtheorem9.p2.13.m13.2.2.1" xref="S4.Thmtheorem9.p2.13.m13.2.2.1.cmml"><msup id="S4.Thmtheorem9.p2.13.m13.2.2.1.3" xref="S4.Thmtheorem9.p2.13.m13.2.2.1.3.cmml"><mi id="S4.Thmtheorem9.p2.13.m13.2.2.1.3.2" xref="S4.Thmtheorem9.p2.13.m13.2.2.1.3.2.cmml">t</mi><mo id="S4.Thmtheorem9.p2.13.m13.2.2.1.3.3" xref="S4.Thmtheorem9.p2.13.m13.2.2.1.3.3.cmml">⌢</mo></msup><mo id="S4.Thmtheorem9.p2.13.m13.2.2.1.2" xref="S4.Thmtheorem9.p2.13.m13.2.2.1.2.cmml">⁢</mo><mrow id="S4.Thmtheorem9.p2.13.m13.2.2.1.1.1" xref="S4.Thmtheorem9.p2.13.m13.2.2.1.1.2.cmml"><mo id="S4.Thmtheorem9.p2.13.m13.2.2.1.1.1.2" stretchy="false" xref="S4.Thmtheorem9.p2.13.m13.2.2.1.1.2.1.cmml">⟨</mo><mrow id="S4.Thmtheorem9.p2.13.m13.2.2.1.1.1.1" xref="S4.Thmtheorem9.p2.13.m13.2.2.1.1.1.1.cmml"><mi id="S4.Thmtheorem9.p2.13.m13.2.2.1.1.1.1.2" xref="S4.Thmtheorem9.p2.13.m13.2.2.1.1.1.1.2.cmml">n</mi><mo id="S4.Thmtheorem9.p2.13.m13.2.2.1.1.1.1.1" xref="S4.Thmtheorem9.p2.13.m13.2.2.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.Thmtheorem9.p2.13.m13.2.2.1.1.1.1.3.2" xref="S4.Thmtheorem9.p2.13.m13.2.2.1.1.1.1.cmml"><mo id="S4.Thmtheorem9.p2.13.m13.2.2.1.1.1.1.3.2.1" stretchy="false" xref="S4.Thmtheorem9.p2.13.m13.2.2.1.1.1.1.cmml">(</mo><mi id="S4.Thmtheorem9.p2.13.m13.1.1" xref="S4.Thmtheorem9.p2.13.m13.1.1.cmml">t</mi><mo id="S4.Thmtheorem9.p2.13.m13.2.2.1.1.1.1.3.2.2" stretchy="false" xref="S4.Thmtheorem9.p2.13.m13.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem9.p2.13.m13.2.2.1.1.1.3" stretchy="false" xref="S4.Thmtheorem9.p2.13.m13.2.2.1.1.2.1.cmml">⟩</mo></mrow></mrow><mo id="S4.Thmtheorem9.p2.13.m13.2.2.2" xref="S4.Thmtheorem9.p2.13.m13.2.2.2.cmml">∈</mo><mi id="S4.Thmtheorem9.p2.13.m13.2.2.3" xref="S4.Thmtheorem9.p2.13.m13.2.2.3.cmml">U</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p2.13.m13.2b"><apply id="S4.Thmtheorem9.p2.13.m13.2.2.cmml" xref="S4.Thmtheorem9.p2.13.m13.2.2"><in id="S4.Thmtheorem9.p2.13.m13.2.2.2.cmml" xref="S4.Thmtheorem9.p2.13.m13.2.2.2"></in><apply id="S4.Thmtheorem9.p2.13.m13.2.2.1.cmml" xref="S4.Thmtheorem9.p2.13.m13.2.2.1"><times id="S4.Thmtheorem9.p2.13.m13.2.2.1.2.cmml" xref="S4.Thmtheorem9.p2.13.m13.2.2.1.2"></times><apply id="S4.Thmtheorem9.p2.13.m13.2.2.1.3.cmml" xref="S4.Thmtheorem9.p2.13.m13.2.2.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p2.13.m13.2.2.1.3.1.cmml" xref="S4.Thmtheorem9.p2.13.m13.2.2.1.3">superscript</csymbol><ci id="S4.Thmtheorem9.p2.13.m13.2.2.1.3.2.cmml" xref="S4.Thmtheorem9.p2.13.m13.2.2.1.3.2">𝑡</ci><ci id="S4.Thmtheorem9.p2.13.m13.2.2.1.3.3.cmml" xref="S4.Thmtheorem9.p2.13.m13.2.2.1.3.3">⌢</ci></apply><apply id="S4.Thmtheorem9.p2.13.m13.2.2.1.1.2.cmml" xref="S4.Thmtheorem9.p2.13.m13.2.2.1.1.1"><csymbol cd="latexml" id="S4.Thmtheorem9.p2.13.m13.2.2.1.1.2.1.cmml" xref="S4.Thmtheorem9.p2.13.m13.2.2.1.1.1.2">delimited-⟨⟩</csymbol><apply id="S4.Thmtheorem9.p2.13.m13.2.2.1.1.1.1.cmml" xref="S4.Thmtheorem9.p2.13.m13.2.2.1.1.1.1"><times id="S4.Thmtheorem9.p2.13.m13.2.2.1.1.1.1.1.cmml" xref="S4.Thmtheorem9.p2.13.m13.2.2.1.1.1.1.1"></times><ci id="S4.Thmtheorem9.p2.13.m13.2.2.1.1.1.1.2.cmml" xref="S4.Thmtheorem9.p2.13.m13.2.2.1.1.1.1.2">𝑛</ci><ci id="S4.Thmtheorem9.p2.13.m13.1.1.cmml" xref="S4.Thmtheorem9.p2.13.m13.1.1">𝑡</ci></apply></apply></apply><ci id="S4.Thmtheorem9.p2.13.m13.2.2.3.cmml" xref="S4.Thmtheorem9.p2.13.m13.2.2.3">𝑈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p2.13.m13.2c">t^{\frown}\langle n(t)\rangle\in U</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p2.13.m13.2d">italic_t start_POSTSUPERSCRIPT ⌢ end_POSTSUPERSCRIPT ⟨ italic_n ( italic_t ) ⟩ ∈ italic_U</annotation></semantics></math>. Let <math alttext="f:T\to U" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p2.14.m14.1"><semantics id="S4.Thmtheorem9.p2.14.m14.1a"><mrow id="S4.Thmtheorem9.p2.14.m14.1.1" xref="S4.Thmtheorem9.p2.14.m14.1.1.cmml"><mi id="S4.Thmtheorem9.p2.14.m14.1.1.2" xref="S4.Thmtheorem9.p2.14.m14.1.1.2.cmml">f</mi><mo id="S4.Thmtheorem9.p2.14.m14.1.1.1" lspace="0.278em" rspace="0.278em" xref="S4.Thmtheorem9.p2.14.m14.1.1.1.cmml">:</mo><mrow id="S4.Thmtheorem9.p2.14.m14.1.1.3" xref="S4.Thmtheorem9.p2.14.m14.1.1.3.cmml"><mi id="S4.Thmtheorem9.p2.14.m14.1.1.3.2" xref="S4.Thmtheorem9.p2.14.m14.1.1.3.2.cmml">T</mi><mo id="S4.Thmtheorem9.p2.14.m14.1.1.3.1" stretchy="false" xref="S4.Thmtheorem9.p2.14.m14.1.1.3.1.cmml">→</mo><mi id="S4.Thmtheorem9.p2.14.m14.1.1.3.3" xref="S4.Thmtheorem9.p2.14.m14.1.1.3.3.cmml">U</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p2.14.m14.1b"><apply id="S4.Thmtheorem9.p2.14.m14.1.1.cmml" xref="S4.Thmtheorem9.p2.14.m14.1.1"><ci id="S4.Thmtheorem9.p2.14.m14.1.1.1.cmml" xref="S4.Thmtheorem9.p2.14.m14.1.1.1">:</ci><ci id="S4.Thmtheorem9.p2.14.m14.1.1.2.cmml" xref="S4.Thmtheorem9.p2.14.m14.1.1.2">𝑓</ci><apply id="S4.Thmtheorem9.p2.14.m14.1.1.3.cmml" xref="S4.Thmtheorem9.p2.14.m14.1.1.3"><ci id="S4.Thmtheorem9.p2.14.m14.1.1.3.1.cmml" xref="S4.Thmtheorem9.p2.14.m14.1.1.3.1">→</ci><ci id="S4.Thmtheorem9.p2.14.m14.1.1.3.2.cmml" xref="S4.Thmtheorem9.p2.14.m14.1.1.3.2">𝑇</ci><ci id="S4.Thmtheorem9.p2.14.m14.1.1.3.3.cmml" xref="S4.Thmtheorem9.p2.14.m14.1.1.3.3">𝑈</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p2.14.m14.1c">f:T\to U</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p2.14.m14.1d">italic_f : italic_T → italic_U</annotation></semantics></math> be defined by letting <math alttext="f(t):=t^{\frown}\langle n(t)\rangle" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p2.15.m15.3"><semantics id="S4.Thmtheorem9.p2.15.m15.3a"><mrow id="S4.Thmtheorem9.p2.15.m15.3.3" xref="S4.Thmtheorem9.p2.15.m15.3.3.cmml"><mrow id="S4.Thmtheorem9.p2.15.m15.3.3.3" xref="S4.Thmtheorem9.p2.15.m15.3.3.3.cmml"><mi id="S4.Thmtheorem9.p2.15.m15.3.3.3.2" xref="S4.Thmtheorem9.p2.15.m15.3.3.3.2.cmml">f</mi><mo id="S4.Thmtheorem9.p2.15.m15.3.3.3.1" xref="S4.Thmtheorem9.p2.15.m15.3.3.3.1.cmml">⁢</mo><mrow id="S4.Thmtheorem9.p2.15.m15.3.3.3.3.2" xref="S4.Thmtheorem9.p2.15.m15.3.3.3.cmml"><mo id="S4.Thmtheorem9.p2.15.m15.3.3.3.3.2.1" stretchy="false" xref="S4.Thmtheorem9.p2.15.m15.3.3.3.cmml">(</mo><mi id="S4.Thmtheorem9.p2.15.m15.1.1" xref="S4.Thmtheorem9.p2.15.m15.1.1.cmml">t</mi><mo id="S4.Thmtheorem9.p2.15.m15.3.3.3.3.2.2" rspace="0.278em" stretchy="false" xref="S4.Thmtheorem9.p2.15.m15.3.3.3.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem9.p2.15.m15.3.3.2" rspace="0.278em" xref="S4.Thmtheorem9.p2.15.m15.3.3.2.cmml">:=</mo><mrow id="S4.Thmtheorem9.p2.15.m15.3.3.1" xref="S4.Thmtheorem9.p2.15.m15.3.3.1.cmml"><msup id="S4.Thmtheorem9.p2.15.m15.3.3.1.3" xref="S4.Thmtheorem9.p2.15.m15.3.3.1.3.cmml"><mi id="S4.Thmtheorem9.p2.15.m15.3.3.1.3.2" xref="S4.Thmtheorem9.p2.15.m15.3.3.1.3.2.cmml">t</mi><mo id="S4.Thmtheorem9.p2.15.m15.3.3.1.3.3" xref="S4.Thmtheorem9.p2.15.m15.3.3.1.3.3.cmml">⌢</mo></msup><mo id="S4.Thmtheorem9.p2.15.m15.3.3.1.2" xref="S4.Thmtheorem9.p2.15.m15.3.3.1.2.cmml">⁢</mo><mrow id="S4.Thmtheorem9.p2.15.m15.3.3.1.1.1" xref="S4.Thmtheorem9.p2.15.m15.3.3.1.1.2.cmml"><mo id="S4.Thmtheorem9.p2.15.m15.3.3.1.1.1.2" stretchy="false" xref="S4.Thmtheorem9.p2.15.m15.3.3.1.1.2.1.cmml">⟨</mo><mrow id="S4.Thmtheorem9.p2.15.m15.3.3.1.1.1.1" xref="S4.Thmtheorem9.p2.15.m15.3.3.1.1.1.1.cmml"><mi id="S4.Thmtheorem9.p2.15.m15.3.3.1.1.1.1.2" xref="S4.Thmtheorem9.p2.15.m15.3.3.1.1.1.1.2.cmml">n</mi><mo id="S4.Thmtheorem9.p2.15.m15.3.3.1.1.1.1.1" xref="S4.Thmtheorem9.p2.15.m15.3.3.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.Thmtheorem9.p2.15.m15.3.3.1.1.1.1.3.2" xref="S4.Thmtheorem9.p2.15.m15.3.3.1.1.1.1.cmml"><mo id="S4.Thmtheorem9.p2.15.m15.3.3.1.1.1.1.3.2.1" stretchy="false" xref="S4.Thmtheorem9.p2.15.m15.3.3.1.1.1.1.cmml">(</mo><mi id="S4.Thmtheorem9.p2.15.m15.2.2" xref="S4.Thmtheorem9.p2.15.m15.2.2.cmml">t</mi><mo id="S4.Thmtheorem9.p2.15.m15.3.3.1.1.1.1.3.2.2" stretchy="false" xref="S4.Thmtheorem9.p2.15.m15.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem9.p2.15.m15.3.3.1.1.1.3" stretchy="false" xref="S4.Thmtheorem9.p2.15.m15.3.3.1.1.2.1.cmml">⟩</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p2.15.m15.3b"><apply id="S4.Thmtheorem9.p2.15.m15.3.3.cmml" xref="S4.Thmtheorem9.p2.15.m15.3.3"><csymbol cd="latexml" id="S4.Thmtheorem9.p2.15.m15.3.3.2.cmml" xref="S4.Thmtheorem9.p2.15.m15.3.3.2">assign</csymbol><apply id="S4.Thmtheorem9.p2.15.m15.3.3.3.cmml" xref="S4.Thmtheorem9.p2.15.m15.3.3.3"><times id="S4.Thmtheorem9.p2.15.m15.3.3.3.1.cmml" xref="S4.Thmtheorem9.p2.15.m15.3.3.3.1"></times><ci id="S4.Thmtheorem9.p2.15.m15.3.3.3.2.cmml" xref="S4.Thmtheorem9.p2.15.m15.3.3.3.2">𝑓</ci><ci id="S4.Thmtheorem9.p2.15.m15.1.1.cmml" xref="S4.Thmtheorem9.p2.15.m15.1.1">𝑡</ci></apply><apply id="S4.Thmtheorem9.p2.15.m15.3.3.1.cmml" xref="S4.Thmtheorem9.p2.15.m15.3.3.1"><times id="S4.Thmtheorem9.p2.15.m15.3.3.1.2.cmml" xref="S4.Thmtheorem9.p2.15.m15.3.3.1.2"></times><apply id="S4.Thmtheorem9.p2.15.m15.3.3.1.3.cmml" xref="S4.Thmtheorem9.p2.15.m15.3.3.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p2.15.m15.3.3.1.3.1.cmml" xref="S4.Thmtheorem9.p2.15.m15.3.3.1.3">superscript</csymbol><ci id="S4.Thmtheorem9.p2.15.m15.3.3.1.3.2.cmml" xref="S4.Thmtheorem9.p2.15.m15.3.3.1.3.2">𝑡</ci><ci id="S4.Thmtheorem9.p2.15.m15.3.3.1.3.3.cmml" xref="S4.Thmtheorem9.p2.15.m15.3.3.1.3.3">⌢</ci></apply><apply id="S4.Thmtheorem9.p2.15.m15.3.3.1.1.2.cmml" xref="S4.Thmtheorem9.p2.15.m15.3.3.1.1.1"><csymbol cd="latexml" id="S4.Thmtheorem9.p2.15.m15.3.3.1.1.2.1.cmml" xref="S4.Thmtheorem9.p2.15.m15.3.3.1.1.1.2">delimited-⟨⟩</csymbol><apply id="S4.Thmtheorem9.p2.15.m15.3.3.1.1.1.1.cmml" xref="S4.Thmtheorem9.p2.15.m15.3.3.1.1.1.1"><times id="S4.Thmtheorem9.p2.15.m15.3.3.1.1.1.1.1.cmml" xref="S4.Thmtheorem9.p2.15.m15.3.3.1.1.1.1.1"></times><ci id="S4.Thmtheorem9.p2.15.m15.3.3.1.1.1.1.2.cmml" xref="S4.Thmtheorem9.p2.15.m15.3.3.1.1.1.1.2">𝑛</ci><ci id="S4.Thmtheorem9.p2.15.m15.2.2.cmml" xref="S4.Thmtheorem9.p2.15.m15.2.2">𝑡</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p2.15.m15.3c">f(t):=t^{\frown}\langle n(t)\rangle</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p2.15.m15.3d">italic_f ( italic_t ) := italic_t start_POSTSUPERSCRIPT ⌢ end_POSTSUPERSCRIPT ⟨ italic_n ( italic_t ) ⟩</annotation></semantics></math> if <math alttext="t\in T\setminus U" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p2.16.m16.1"><semantics id="S4.Thmtheorem9.p2.16.m16.1a"><mrow id="S4.Thmtheorem9.p2.16.m16.1.1" xref="S4.Thmtheorem9.p2.16.m16.1.1.cmml"><mi id="S4.Thmtheorem9.p2.16.m16.1.1.2" xref="S4.Thmtheorem9.p2.16.m16.1.1.2.cmml">t</mi><mo id="S4.Thmtheorem9.p2.16.m16.1.1.1" xref="S4.Thmtheorem9.p2.16.m16.1.1.1.cmml">∈</mo><mrow id="S4.Thmtheorem9.p2.16.m16.1.1.3" xref="S4.Thmtheorem9.p2.16.m16.1.1.3.cmml"><mi id="S4.Thmtheorem9.p2.16.m16.1.1.3.2" xref="S4.Thmtheorem9.p2.16.m16.1.1.3.2.cmml">T</mi><mo id="S4.Thmtheorem9.p2.16.m16.1.1.3.1" xref="S4.Thmtheorem9.p2.16.m16.1.1.3.1.cmml">∖</mo><mi id="S4.Thmtheorem9.p2.16.m16.1.1.3.3" xref="S4.Thmtheorem9.p2.16.m16.1.1.3.3.cmml">U</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p2.16.m16.1b"><apply id="S4.Thmtheorem9.p2.16.m16.1.1.cmml" xref="S4.Thmtheorem9.p2.16.m16.1.1"><in id="S4.Thmtheorem9.p2.16.m16.1.1.1.cmml" xref="S4.Thmtheorem9.p2.16.m16.1.1.1"></in><ci id="S4.Thmtheorem9.p2.16.m16.1.1.2.cmml" xref="S4.Thmtheorem9.p2.16.m16.1.1.2">𝑡</ci><apply id="S4.Thmtheorem9.p2.16.m16.1.1.3.cmml" xref="S4.Thmtheorem9.p2.16.m16.1.1.3"><setdiff id="S4.Thmtheorem9.p2.16.m16.1.1.3.1.cmml" xref="S4.Thmtheorem9.p2.16.m16.1.1.3.1"></setdiff><ci id="S4.Thmtheorem9.p2.16.m16.1.1.3.2.cmml" xref="S4.Thmtheorem9.p2.16.m16.1.1.3.2">𝑇</ci><ci id="S4.Thmtheorem9.p2.16.m16.1.1.3.3.cmml" xref="S4.Thmtheorem9.p2.16.m16.1.1.3.3">𝑈</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p2.16.m16.1c">t\in T\setminus U</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p2.16.m16.1d">italic_t ∈ italic_T ∖ italic_U</annotation></semantics></math> and <math alttext="f(t):=t^{\frown}\langle 0\rangle" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p2.17.m17.2"><semantics id="S4.Thmtheorem9.p2.17.m17.2a"><mrow id="S4.Thmtheorem9.p2.17.m17.2.3" xref="S4.Thmtheorem9.p2.17.m17.2.3.cmml"><mrow id="S4.Thmtheorem9.p2.17.m17.2.3.2" xref="S4.Thmtheorem9.p2.17.m17.2.3.2.cmml"><mi id="S4.Thmtheorem9.p2.17.m17.2.3.2.2" xref="S4.Thmtheorem9.p2.17.m17.2.3.2.2.cmml">f</mi><mo id="S4.Thmtheorem9.p2.17.m17.2.3.2.1" xref="S4.Thmtheorem9.p2.17.m17.2.3.2.1.cmml">⁢</mo><mrow id="S4.Thmtheorem9.p2.17.m17.2.3.2.3.2" xref="S4.Thmtheorem9.p2.17.m17.2.3.2.cmml"><mo id="S4.Thmtheorem9.p2.17.m17.2.3.2.3.2.1" stretchy="false" xref="S4.Thmtheorem9.p2.17.m17.2.3.2.cmml">(</mo><mi id="S4.Thmtheorem9.p2.17.m17.1.1" xref="S4.Thmtheorem9.p2.17.m17.1.1.cmml">t</mi><mo id="S4.Thmtheorem9.p2.17.m17.2.3.2.3.2.2" rspace="0.278em" stretchy="false" xref="S4.Thmtheorem9.p2.17.m17.2.3.2.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem9.p2.17.m17.2.3.1" rspace="0.278em" xref="S4.Thmtheorem9.p2.17.m17.2.3.1.cmml">:=</mo><mrow id="S4.Thmtheorem9.p2.17.m17.2.3.3" xref="S4.Thmtheorem9.p2.17.m17.2.3.3.cmml"><msup id="S4.Thmtheorem9.p2.17.m17.2.3.3.2" xref="S4.Thmtheorem9.p2.17.m17.2.3.3.2.cmml"><mi id="S4.Thmtheorem9.p2.17.m17.2.3.3.2.2" xref="S4.Thmtheorem9.p2.17.m17.2.3.3.2.2.cmml">t</mi><mo id="S4.Thmtheorem9.p2.17.m17.2.3.3.2.3" xref="S4.Thmtheorem9.p2.17.m17.2.3.3.2.3.cmml">⌢</mo></msup><mo id="S4.Thmtheorem9.p2.17.m17.2.3.3.1" xref="S4.Thmtheorem9.p2.17.m17.2.3.3.1.cmml">⁢</mo><mrow id="S4.Thmtheorem9.p2.17.m17.2.3.3.3.2" xref="S4.Thmtheorem9.p2.17.m17.2.3.3.3.1.cmml"><mo id="S4.Thmtheorem9.p2.17.m17.2.3.3.3.2.1" stretchy="false" xref="S4.Thmtheorem9.p2.17.m17.2.3.3.3.1.1.cmml">⟨</mo><mn id="S4.Thmtheorem9.p2.17.m17.2.2" xref="S4.Thmtheorem9.p2.17.m17.2.2.cmml">0</mn><mo id="S4.Thmtheorem9.p2.17.m17.2.3.3.3.2.2" stretchy="false" xref="S4.Thmtheorem9.p2.17.m17.2.3.3.3.1.1.cmml">⟩</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p2.17.m17.2b"><apply id="S4.Thmtheorem9.p2.17.m17.2.3.cmml" xref="S4.Thmtheorem9.p2.17.m17.2.3"><csymbol cd="latexml" id="S4.Thmtheorem9.p2.17.m17.2.3.1.cmml" xref="S4.Thmtheorem9.p2.17.m17.2.3.1">assign</csymbol><apply id="S4.Thmtheorem9.p2.17.m17.2.3.2.cmml" xref="S4.Thmtheorem9.p2.17.m17.2.3.2"><times id="S4.Thmtheorem9.p2.17.m17.2.3.2.1.cmml" xref="S4.Thmtheorem9.p2.17.m17.2.3.2.1"></times><ci id="S4.Thmtheorem9.p2.17.m17.2.3.2.2.cmml" xref="S4.Thmtheorem9.p2.17.m17.2.3.2.2">𝑓</ci><ci id="S4.Thmtheorem9.p2.17.m17.1.1.cmml" xref="S4.Thmtheorem9.p2.17.m17.1.1">𝑡</ci></apply><apply id="S4.Thmtheorem9.p2.17.m17.2.3.3.cmml" xref="S4.Thmtheorem9.p2.17.m17.2.3.3"><times id="S4.Thmtheorem9.p2.17.m17.2.3.3.1.cmml" xref="S4.Thmtheorem9.p2.17.m17.2.3.3.1"></times><apply id="S4.Thmtheorem9.p2.17.m17.2.3.3.2.cmml" xref="S4.Thmtheorem9.p2.17.m17.2.3.3.2"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p2.17.m17.2.3.3.2.1.cmml" xref="S4.Thmtheorem9.p2.17.m17.2.3.3.2">superscript</csymbol><ci id="S4.Thmtheorem9.p2.17.m17.2.3.3.2.2.cmml" xref="S4.Thmtheorem9.p2.17.m17.2.3.3.2.2">𝑡</ci><ci id="S4.Thmtheorem9.p2.17.m17.2.3.3.2.3.cmml" xref="S4.Thmtheorem9.p2.17.m17.2.3.3.2.3">⌢</ci></apply><apply id="S4.Thmtheorem9.p2.17.m17.2.3.3.3.1.cmml" xref="S4.Thmtheorem9.p2.17.m17.2.3.3.3.2"><csymbol cd="latexml" id="S4.Thmtheorem9.p2.17.m17.2.3.3.3.1.1.cmml" xref="S4.Thmtheorem9.p2.17.m17.2.3.3.3.2.1">delimited-⟨⟩</csymbol><cn id="S4.Thmtheorem9.p2.17.m17.2.2.cmml" type="integer" xref="S4.Thmtheorem9.p2.17.m17.2.2">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p2.17.m17.2c">f(t):=t^{\frown}\langle 0\rangle</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p2.17.m17.2d">italic_f ( italic_t ) := italic_t start_POSTSUPERSCRIPT ⌢ end_POSTSUPERSCRIPT ⟨ 0 ⟩</annotation></semantics></math> otherwise. Then <math alttext="f" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p2.18.m18.1"><semantics id="S4.Thmtheorem9.p2.18.m18.1a"><mi id="S4.Thmtheorem9.p2.18.m18.1.1" xref="S4.Thmtheorem9.p2.18.m18.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p2.18.m18.1b"><ci id="S4.Thmtheorem9.p2.18.m18.1.1.cmml" xref="S4.Thmtheorem9.p2.18.m18.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p2.18.m18.1c">f</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p2.18.m18.1d">italic_f</annotation></semantics></math> witnesses that <math alttext="(T,<_{\mathrm{lex}})\preceq(U,<_{\mathrm{lex}})" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p2.19.m19.4"><semantics id="S4.Thmtheorem9.p2.19.m19.4a"><mrow id="S4.Thmtheorem9.p2.19.m19.4.4" xref="S4.Thmtheorem9.p2.19.m19.4.4.cmml"><mrow id="S4.Thmtheorem9.p2.19.m19.3.3.1.1" xref="S4.Thmtheorem9.p2.19.m19.3.3.1.2.cmml"><mo id="S4.Thmtheorem9.p2.19.m19.3.3.1.1.2" stretchy="false" xref="S4.Thmtheorem9.p2.19.m19.3.3.1.2.cmml">(</mo><mi id="S4.Thmtheorem9.p2.19.m19.1.1" xref="S4.Thmtheorem9.p2.19.m19.1.1.cmml">T</mi><mo id="S4.Thmtheorem9.p2.19.m19.3.3.1.1.3" xref="S4.Thmtheorem9.p2.19.m19.3.3.1.2.cmml">,</mo><msub id="S4.Thmtheorem9.p2.19.m19.3.3.1.1.1" xref="S4.Thmtheorem9.p2.19.m19.3.3.1.1.1.cmml"><mo id="S4.Thmtheorem9.p2.19.m19.3.3.1.1.1.2" lspace="0em" rspace="0em" xref="S4.Thmtheorem9.p2.19.m19.3.3.1.1.1.2.cmml"><</mo><mi id="S4.Thmtheorem9.p2.19.m19.3.3.1.1.1.3" xref="S4.Thmtheorem9.p2.19.m19.3.3.1.1.1.3.cmml">lex</mi></msub><mo id="S4.Thmtheorem9.p2.19.m19.3.3.1.1.4" stretchy="false" xref="S4.Thmtheorem9.p2.19.m19.3.3.1.2.cmml">)</mo></mrow><mo id="S4.Thmtheorem9.p2.19.m19.4.4.3" xref="S4.Thmtheorem9.p2.19.m19.4.4.3.cmml">⪯</mo><mrow id="S4.Thmtheorem9.p2.19.m19.4.4.2.1" xref="S4.Thmtheorem9.p2.19.m19.4.4.2.2.cmml"><mo id="S4.Thmtheorem9.p2.19.m19.4.4.2.1.2" stretchy="false" xref="S4.Thmtheorem9.p2.19.m19.4.4.2.2.cmml">(</mo><mi id="S4.Thmtheorem9.p2.19.m19.2.2" xref="S4.Thmtheorem9.p2.19.m19.2.2.cmml">U</mi><mo id="S4.Thmtheorem9.p2.19.m19.4.4.2.1.3" xref="S4.Thmtheorem9.p2.19.m19.4.4.2.2.cmml">,</mo><msub id="S4.Thmtheorem9.p2.19.m19.4.4.2.1.1" xref="S4.Thmtheorem9.p2.19.m19.4.4.2.1.1.cmml"><mo id="S4.Thmtheorem9.p2.19.m19.4.4.2.1.1.2" lspace="0em" rspace="0em" xref="S4.Thmtheorem9.p2.19.m19.4.4.2.1.1.2.cmml"><</mo><mi id="S4.Thmtheorem9.p2.19.m19.4.4.2.1.1.3" xref="S4.Thmtheorem9.p2.19.m19.4.4.2.1.1.3.cmml">lex</mi></msub><mo id="S4.Thmtheorem9.p2.19.m19.4.4.2.1.4" stretchy="false" xref="S4.Thmtheorem9.p2.19.m19.4.4.2.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p2.19.m19.4b"><apply id="S4.Thmtheorem9.p2.19.m19.4.4.cmml" xref="S4.Thmtheorem9.p2.19.m19.4.4"><csymbol cd="latexml" id="S4.Thmtheorem9.p2.19.m19.4.4.3.cmml" xref="S4.Thmtheorem9.p2.19.m19.4.4.3">precedes-or-equals</csymbol><interval closure="open" id="S4.Thmtheorem9.p2.19.m19.3.3.1.2.cmml" xref="S4.Thmtheorem9.p2.19.m19.3.3.1.1"><ci id="S4.Thmtheorem9.p2.19.m19.1.1.cmml" xref="S4.Thmtheorem9.p2.19.m19.1.1">𝑇</ci><apply id="S4.Thmtheorem9.p2.19.m19.3.3.1.1.1.cmml" xref="S4.Thmtheorem9.p2.19.m19.3.3.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p2.19.m19.3.3.1.1.1.1.cmml" xref="S4.Thmtheorem9.p2.19.m19.3.3.1.1.1">subscript</csymbol><lt id="S4.Thmtheorem9.p2.19.m19.3.3.1.1.1.2.cmml" xref="S4.Thmtheorem9.p2.19.m19.3.3.1.1.1.2"></lt><ci id="S4.Thmtheorem9.p2.19.m19.3.3.1.1.1.3.cmml" xref="S4.Thmtheorem9.p2.19.m19.3.3.1.1.1.3">lex</ci></apply></interval><interval closure="open" id="S4.Thmtheorem9.p2.19.m19.4.4.2.2.cmml" xref="S4.Thmtheorem9.p2.19.m19.4.4.2.1"><ci id="S4.Thmtheorem9.p2.19.m19.2.2.cmml" xref="S4.Thmtheorem9.p2.19.m19.2.2">𝑈</ci><apply id="S4.Thmtheorem9.p2.19.m19.4.4.2.1.1.cmml" xref="S4.Thmtheorem9.p2.19.m19.4.4.2.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p2.19.m19.4.4.2.1.1.1.cmml" xref="S4.Thmtheorem9.p2.19.m19.4.4.2.1.1">subscript</csymbol><lt id="S4.Thmtheorem9.p2.19.m19.4.4.2.1.1.2.cmml" xref="S4.Thmtheorem9.p2.19.m19.4.4.2.1.1.2"></lt><ci id="S4.Thmtheorem9.p2.19.m19.4.4.2.1.1.3.cmml" xref="S4.Thmtheorem9.p2.19.m19.4.4.2.1.1.3">lex</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p2.19.m19.4c">(T,<_{\mathrm{lex}})\preceq(U,<_{\mathrm{lex}})</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p2.19.m19.4d">( italic_T , < start_POSTSUBSCRIPT roman_lex end_POSTSUBSCRIPT ) ⪯ ( italic_U , < start_POSTSUBSCRIPT roman_lex end_POSTSUBSCRIPT )</annotation></semantics></math>, so <math alttext="(T,<_{\mathrm{lex}})" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p2.20.m20.2"><semantics id="S4.Thmtheorem9.p2.20.m20.2a"><mrow id="S4.Thmtheorem9.p2.20.m20.2.2.1" xref="S4.Thmtheorem9.p2.20.m20.2.2.2.cmml"><mo id="S4.Thmtheorem9.p2.20.m20.2.2.1.2" stretchy="false" xref="S4.Thmtheorem9.p2.20.m20.2.2.2.cmml">(</mo><mi id="S4.Thmtheorem9.p2.20.m20.1.1" xref="S4.Thmtheorem9.p2.20.m20.1.1.cmml">T</mi><mo id="S4.Thmtheorem9.p2.20.m20.2.2.1.3" xref="S4.Thmtheorem9.p2.20.m20.2.2.2.cmml">,</mo><msub id="S4.Thmtheorem9.p2.20.m20.2.2.1.1" xref="S4.Thmtheorem9.p2.20.m20.2.2.1.1.cmml"><mo id="S4.Thmtheorem9.p2.20.m20.2.2.1.1.2" lspace="0em" rspace="0em" xref="S4.Thmtheorem9.p2.20.m20.2.2.1.1.2.cmml"><</mo><mi id="S4.Thmtheorem9.p2.20.m20.2.2.1.1.3" xref="S4.Thmtheorem9.p2.20.m20.2.2.1.1.3.cmml">lex</mi></msub><mo id="S4.Thmtheorem9.p2.20.m20.2.2.1.4" stretchy="false" xref="S4.Thmtheorem9.p2.20.m20.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p2.20.m20.2b"><interval closure="open" id="S4.Thmtheorem9.p2.20.m20.2.2.2.cmml" xref="S4.Thmtheorem9.p2.20.m20.2.2.1"><ci id="S4.Thmtheorem9.p2.20.m20.1.1.cmml" xref="S4.Thmtheorem9.p2.20.m20.1.1">𝑇</ci><apply id="S4.Thmtheorem9.p2.20.m20.2.2.1.1.cmml" xref="S4.Thmtheorem9.p2.20.m20.2.2.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p2.20.m20.2.2.1.1.1.cmml" xref="S4.Thmtheorem9.p2.20.m20.2.2.1.1">subscript</csymbol><lt id="S4.Thmtheorem9.p2.20.m20.2.2.1.1.2.cmml" xref="S4.Thmtheorem9.p2.20.m20.2.2.1.1.2"></lt><ci id="S4.Thmtheorem9.p2.20.m20.2.2.1.1.3.cmml" xref="S4.Thmtheorem9.p2.20.m20.2.2.1.1.3">lex</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p2.20.m20.2c">(T,<_{\mathrm{lex}})</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p2.20.m20.2d">( italic_T , < start_POSTSUBSCRIPT roman_lex end_POSTSUBSCRIPT )</annotation></semantics></math> is again Countryman. And <math alttext="f" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p2.21.m21.1"><semantics id="S4.Thmtheorem9.p2.21.m21.1a"><mi id="S4.Thmtheorem9.p2.21.m21.1.1" xref="S4.Thmtheorem9.p2.21.m21.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p2.21.m21.1b"><ci id="S4.Thmtheorem9.p2.21.m21.1.1.cmml" xref="S4.Thmtheorem9.p2.21.m21.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p2.21.m21.1c">f</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p2.21.m21.1d">italic_f</annotation></semantics></math> is a tree embedding of <math alttext="T\setminus U" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p2.22.m22.1"><semantics id="S4.Thmtheorem9.p2.22.m22.1a"><mrow id="S4.Thmtheorem9.p2.22.m22.1.1" xref="S4.Thmtheorem9.p2.22.m22.1.1.cmml"><mi id="S4.Thmtheorem9.p2.22.m22.1.1.2" xref="S4.Thmtheorem9.p2.22.m22.1.1.2.cmml">T</mi><mo id="S4.Thmtheorem9.p2.22.m22.1.1.1" xref="S4.Thmtheorem9.p2.22.m22.1.1.1.cmml">∖</mo><mi id="S4.Thmtheorem9.p2.22.m22.1.1.3" xref="S4.Thmtheorem9.p2.22.m22.1.1.3.cmml">U</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p2.22.m22.1b"><apply id="S4.Thmtheorem9.p2.22.m22.1.1.cmml" xref="S4.Thmtheorem9.p2.22.m22.1.1"><setdiff id="S4.Thmtheorem9.p2.22.m22.1.1.1.cmml" xref="S4.Thmtheorem9.p2.22.m22.1.1.1"></setdiff><ci id="S4.Thmtheorem9.p2.22.m22.1.1.2.cmml" xref="S4.Thmtheorem9.p2.22.m22.1.1.2">𝑇</ci><ci id="S4.Thmtheorem9.p2.22.m22.1.1.3.cmml" xref="S4.Thmtheorem9.p2.22.m22.1.1.3">𝑈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p2.22.m22.1c">T\setminus U</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p2.22.m22.1d">italic_T ∖ italic_U</annotation></semantics></math> into <math alttext="U" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p2.23.m23.1"><semantics id="S4.Thmtheorem9.p2.23.m23.1a"><mi id="S4.Thmtheorem9.p2.23.m23.1.1" xref="S4.Thmtheorem9.p2.23.m23.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p2.23.m23.1b"><ci id="S4.Thmtheorem9.p2.23.m23.1.1.cmml" xref="S4.Thmtheorem9.p2.23.m23.1.1">𝑈</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p2.23.m23.1c">U</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p2.23.m23.1d">italic_U</annotation></semantics></math>, so <math alttext="T" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p2.24.m24.1"><semantics id="S4.Thmtheorem9.p2.24.m24.1a"><mi id="S4.Thmtheorem9.p2.24.m24.1.1" xref="S4.Thmtheorem9.p2.24.m24.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p2.24.m24.1b"><ci id="S4.Thmtheorem9.p2.24.m24.1.1.cmml" xref="S4.Thmtheorem9.p2.24.m24.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p2.24.m24.1c">T</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p2.24.m24.1d">italic_T</annotation></semantics></math> is again special. Then if one defines <math alttext="A^{+}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p2.25.m25.1"><semantics id="S4.Thmtheorem9.p2.25.m25.1a"><msup id="S4.Thmtheorem9.p2.25.m25.1.1" xref="S4.Thmtheorem9.p2.25.m25.1.1.cmml"><mi id="S4.Thmtheorem9.p2.25.m25.1.1.2" xref="S4.Thmtheorem9.p2.25.m25.1.1.2.cmml">A</mi><mo id="S4.Thmtheorem9.p2.25.m25.1.1.3" xref="S4.Thmtheorem9.p2.25.m25.1.1.3.cmml">+</mo></msup><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p2.25.m25.1b"><apply id="S4.Thmtheorem9.p2.25.m25.1.1.cmml" xref="S4.Thmtheorem9.p2.25.m25.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p2.25.m25.1.1.1.cmml" xref="S4.Thmtheorem9.p2.25.m25.1.1">superscript</csymbol><ci id="S4.Thmtheorem9.p2.25.m25.1.1.2.cmml" xref="S4.Thmtheorem9.p2.25.m25.1.1.2">𝐴</ci><plus id="S4.Thmtheorem9.p2.25.m25.1.1.3.cmml" xref="S4.Thmtheorem9.p2.25.m25.1.1.3"></plus></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p2.25.m25.1c">A^{+}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p2.25.m25.1d">italic_A start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> from this <math alttext="T" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p2.26.m26.1"><semantics id="S4.Thmtheorem9.p2.26.m26.1a"><mi id="S4.Thmtheorem9.p2.26.m26.1.1" xref="S4.Thmtheorem9.p2.26.m26.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p2.26.m26.1b"><ci id="S4.Thmtheorem9.p2.26.m26.1.1.cmml" xref="S4.Thmtheorem9.p2.26.m26.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p2.26.m26.1c">T</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p2.26.m26.1d">italic_T</annotation></semantics></math> exactly as we did for our tree, one can replicate <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S4.Thmtheorem4" title="Lemma 4.4. ‣ 4. Aronszajn line decompositions ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">4.4</span></a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S4.Thmtheorem5" title="Theorem 4.5. ‣ 4. Aronszajn line decompositions ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">4.5</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S4.Thmtheorem7" title="Lemma 4.7. ‣ 4. Aronszajn line decompositions ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">4.7</span></a> verbatim.</p> </div> <div class="ltx_para" id="S4.Thmtheorem9.p3"> <p class="ltx_p" id="S4.Thmtheorem9.p3.3">Another option would be to construct the <math alttext="C" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p3.1.m1.1"><semantics id="S4.Thmtheorem9.p3.1.m1.1a"><mi id="S4.Thmtheorem9.p3.1.m1.1.1" xref="S4.Thmtheorem9.p3.1.m1.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p3.1.m1.1b"><ci id="S4.Thmtheorem9.p3.1.m1.1.1.cmml" xref="S4.Thmtheorem9.p3.1.m1.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p3.1.m1.1c">C</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p3.1.m1.1d">italic_C</annotation></semantics></math>-sequence as to ensure that <math alttext="T^{*}(\rho_{1})" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p3.2.m2.1"><semantics id="S4.Thmtheorem9.p3.2.m2.1a"><mrow id="S4.Thmtheorem9.p3.2.m2.1.1" xref="S4.Thmtheorem9.p3.2.m2.1.1.cmml"><msup id="S4.Thmtheorem9.p3.2.m2.1.1.3" xref="S4.Thmtheorem9.p3.2.m2.1.1.3.cmml"><mi id="S4.Thmtheorem9.p3.2.m2.1.1.3.2" xref="S4.Thmtheorem9.p3.2.m2.1.1.3.2.cmml">T</mi><mo id="S4.Thmtheorem9.p3.2.m2.1.1.3.3" xref="S4.Thmtheorem9.p3.2.m2.1.1.3.3.cmml">∗</mo></msup><mo id="S4.Thmtheorem9.p3.2.m2.1.1.2" xref="S4.Thmtheorem9.p3.2.m2.1.1.2.cmml">⁢</mo><mrow id="S4.Thmtheorem9.p3.2.m2.1.1.1.1" xref="S4.Thmtheorem9.p3.2.m2.1.1.1.1.1.cmml"><mo id="S4.Thmtheorem9.p3.2.m2.1.1.1.1.2" stretchy="false" xref="S4.Thmtheorem9.p3.2.m2.1.1.1.1.1.cmml">(</mo><msub id="S4.Thmtheorem9.p3.2.m2.1.1.1.1.1" xref="S4.Thmtheorem9.p3.2.m2.1.1.1.1.1.cmml"><mi id="S4.Thmtheorem9.p3.2.m2.1.1.1.1.1.2" xref="S4.Thmtheorem9.p3.2.m2.1.1.1.1.1.2.cmml">ρ</mi><mn id="S4.Thmtheorem9.p3.2.m2.1.1.1.1.1.3" xref="S4.Thmtheorem9.p3.2.m2.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.Thmtheorem9.p3.2.m2.1.1.1.1.3" stretchy="false" xref="S4.Thmtheorem9.p3.2.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p3.2.m2.1b"><apply id="S4.Thmtheorem9.p3.2.m2.1.1.cmml" xref="S4.Thmtheorem9.p3.2.m2.1.1"><times id="S4.Thmtheorem9.p3.2.m2.1.1.2.cmml" xref="S4.Thmtheorem9.p3.2.m2.1.1.2"></times><apply id="S4.Thmtheorem9.p3.2.m2.1.1.3.cmml" xref="S4.Thmtheorem9.p3.2.m2.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p3.2.m2.1.1.3.1.cmml" xref="S4.Thmtheorem9.p3.2.m2.1.1.3">superscript</csymbol><ci id="S4.Thmtheorem9.p3.2.m2.1.1.3.2.cmml" xref="S4.Thmtheorem9.p3.2.m2.1.1.3.2">𝑇</ci><times id="S4.Thmtheorem9.p3.2.m2.1.1.3.3.cmml" xref="S4.Thmtheorem9.p3.2.m2.1.1.3.3"></times></apply><apply id="S4.Thmtheorem9.p3.2.m2.1.1.1.1.1.cmml" xref="S4.Thmtheorem9.p3.2.m2.1.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p3.2.m2.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem9.p3.2.m2.1.1.1.1">subscript</csymbol><ci id="S4.Thmtheorem9.p3.2.m2.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem9.p3.2.m2.1.1.1.1.1.2">𝜌</ci><cn id="S4.Thmtheorem9.p3.2.m2.1.1.1.1.1.3.cmml" type="integer" xref="S4.Thmtheorem9.p3.2.m2.1.1.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p3.2.m2.1c">T^{*}(\rho_{1})</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p3.2.m2.1d">italic_T start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math> is special. We do not know if this is possible in <math alttext="\mathsf{ZFC}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p3.3.m3.1"><semantics id="S4.Thmtheorem9.p3.3.m3.1a"><mi id="S4.Thmtheorem9.p3.3.m3.1.1" xref="S4.Thmtheorem9.p3.3.m3.1.1.cmml">𝖹𝖥𝖢</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p3.3.m3.1b"><ci id="S4.Thmtheorem9.p3.3.m3.1.1.cmml" xref="S4.Thmtheorem9.p3.3.m3.1.1">𝖹𝖥𝖢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p3.3.m3.1c">\mathsf{ZFC}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p3.3.m3.1d">sansserif_ZFC</annotation></semantics></math>, see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib23" title="">23</a>, Question 2.2.18]</cite>.</p> </div> </div> </section> <section class="ltx_section" id="S5"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">5. </span>An infinite antichain</h2> <div class="ltx_para" id="S5.p1"> <p class="ltx_p" id="S5.p1.4">In this section we show that already in <math alttext="\mathsf{ZFC}" class="ltx_Math" display="inline" id="S5.p1.1.m1.1"><semantics id="S5.p1.1.m1.1a"><mi id="S5.p1.1.m1.1.1" xref="S5.p1.1.m1.1.1.cmml">𝖹𝖥𝖢</mi><annotation-xml encoding="MathML-Content" id="S5.p1.1.m1.1b"><ci id="S5.p1.1.m1.1.1.cmml" xref="S5.p1.1.m1.1.1">𝖹𝖥𝖢</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.1.m1.1c">\mathsf{ZFC}</annotation><annotation encoding="application/x-llamapun" id="S5.p1.1.m1.1d">sansserif_ZFC</annotation></semantics></math> there is a <math alttext="\trianglelefteq" class="ltx_Math" display="inline" id="S5.p1.2.m2.1"><semantics id="S5.p1.2.m2.1a"><mi id="S5.p1.2.m2.1.1" mathvariant="normal" xref="S5.p1.2.m2.1.1.cmml">⊴</mi><annotation-xml encoding="MathML-Content" id="S5.p1.2.m2.1b"><ci id="S5.p1.2.m2.1.1.cmml" xref="S5.p1.2.m2.1.1">⊴</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.2.m2.1c">\trianglelefteq</annotation><annotation encoding="application/x-llamapun" id="S5.p1.2.m2.1d">⊴</annotation></semantics></math>-antichain of <math alttext="\aleph_{1}" class="ltx_Math" display="inline" id="S5.p1.3.m3.1"><semantics id="S5.p1.3.m3.1a"><msub id="S5.p1.3.m3.1.1" xref="S5.p1.3.m3.1.1.cmml"><mi id="S5.p1.3.m3.1.1.2" mathvariant="normal" xref="S5.p1.3.m3.1.1.2.cmml">ℵ</mi><mn id="S5.p1.3.m3.1.1.3" xref="S5.p1.3.m3.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S5.p1.3.m3.1b"><apply id="S5.p1.3.m3.1.1.cmml" xref="S5.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S5.p1.3.m3.1.1.1.cmml" xref="S5.p1.3.m3.1.1">subscript</csymbol><ci id="S5.p1.3.m3.1.1.2.cmml" xref="S5.p1.3.m3.1.1.2">ℵ</ci><cn id="S5.p1.3.m3.1.1.3.cmml" type="integer" xref="S5.p1.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.3.m3.1c">\aleph_{1}</annotation><annotation encoding="application/x-llamapun" id="S5.p1.3.m3.1d">roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-dense Aronszajn lines of cardinality <math alttext="2^{\aleph_{1}}" class="ltx_Math" display="inline" id="S5.p1.4.m4.1"><semantics id="S5.p1.4.m4.1a"><msup id="S5.p1.4.m4.1.1" xref="S5.p1.4.m4.1.1.cmml"><mn id="S5.p1.4.m4.1.1.2" xref="S5.p1.4.m4.1.1.2.cmml">2</mn><msub id="S5.p1.4.m4.1.1.3" xref="S5.p1.4.m4.1.1.3.cmml"><mi id="S5.p1.4.m4.1.1.3.2" mathvariant="normal" xref="S5.p1.4.m4.1.1.3.2.cmml">ℵ</mi><mn id="S5.p1.4.m4.1.1.3.3" xref="S5.p1.4.m4.1.1.3.3.cmml">1</mn></msub></msup><annotation-xml encoding="MathML-Content" id="S5.p1.4.m4.1b"><apply id="S5.p1.4.m4.1.1.cmml" xref="S5.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S5.p1.4.m4.1.1.1.cmml" xref="S5.p1.4.m4.1.1">superscript</csymbol><cn id="S5.p1.4.m4.1.1.2.cmml" type="integer" xref="S5.p1.4.m4.1.1.2">2</cn><apply id="S5.p1.4.m4.1.1.3.cmml" xref="S5.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="S5.p1.4.m4.1.1.3.1.cmml" xref="S5.p1.4.m4.1.1.3">subscript</csymbol><ci id="S5.p1.4.m4.1.1.3.2.cmml" xref="S5.p1.4.m4.1.1.3.2">ℵ</ci><cn id="S5.p1.4.m4.1.1.3.3.cmml" type="integer" xref="S5.p1.4.m4.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.4.m4.1c">2^{\aleph_{1}}</annotation><annotation encoding="application/x-llamapun" id="S5.p1.4.m4.1d">2 start_POSTSUPERSCRIPT roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math>, giving a negative answer to <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#Thmquestion2" title="Question 2. ‣ Historical and mathematical context ‣ 1. Introduction ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">2</span></a>, even if we restrict to the class of Countryman lines.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="S5.Thmtheorem1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem1.1.1.1">Definition 5.1</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem1.2.2">.</span> </h6> <div class="ltx_para" id="S5.Thmtheorem1.p1"> <p class="ltx_p" id="S5.Thmtheorem1.p1.7">Let <math alttext="A" class="ltx_Math" display="inline" id="S5.Thmtheorem1.p1.1.m1.1"><semantics id="S5.Thmtheorem1.p1.1.m1.1a"><mi id="S5.Thmtheorem1.p1.1.m1.1.1" xref="S5.Thmtheorem1.p1.1.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem1.p1.1.m1.1b"><ci id="S5.Thmtheorem1.p1.1.m1.1.1.cmml" xref="S5.Thmtheorem1.p1.1.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem1.p1.1.m1.1c">A</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem1.p1.1.m1.1d">italic_A</annotation></semantics></math> be an Aronszajn line and <math alttext="B\subseteq A" class="ltx_Math" display="inline" id="S5.Thmtheorem1.p1.2.m2.1"><semantics id="S5.Thmtheorem1.p1.2.m2.1a"><mrow id="S5.Thmtheorem1.p1.2.m2.1.1" xref="S5.Thmtheorem1.p1.2.m2.1.1.cmml"><mi id="S5.Thmtheorem1.p1.2.m2.1.1.2" xref="S5.Thmtheorem1.p1.2.m2.1.1.2.cmml">B</mi><mo id="S5.Thmtheorem1.p1.2.m2.1.1.1" xref="S5.Thmtheorem1.p1.2.m2.1.1.1.cmml">⊆</mo><mi id="S5.Thmtheorem1.p1.2.m2.1.1.3" xref="S5.Thmtheorem1.p1.2.m2.1.1.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem1.p1.2.m2.1b"><apply id="S5.Thmtheorem1.p1.2.m2.1.1.cmml" xref="S5.Thmtheorem1.p1.2.m2.1.1"><subset id="S5.Thmtheorem1.p1.2.m2.1.1.1.cmml" xref="S5.Thmtheorem1.p1.2.m2.1.1.1"></subset><ci id="S5.Thmtheorem1.p1.2.m2.1.1.2.cmml" xref="S5.Thmtheorem1.p1.2.m2.1.1.2">𝐵</ci><ci id="S5.Thmtheorem1.p1.2.m2.1.1.3.cmml" xref="S5.Thmtheorem1.p1.2.m2.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem1.p1.2.m2.1c">B\subseteq A</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem1.p1.2.m2.1d">italic_B ⊆ italic_A</annotation></semantics></math> be any subset. We say that <math alttext="B" class="ltx_Math" display="inline" id="S5.Thmtheorem1.p1.3.m3.1"><semantics id="S5.Thmtheorem1.p1.3.m3.1a"><mi id="S5.Thmtheorem1.p1.3.m3.1.1" xref="S5.Thmtheorem1.p1.3.m3.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem1.p1.3.m3.1b"><ci id="S5.Thmtheorem1.p1.3.m3.1.1.cmml" xref="S5.Thmtheorem1.p1.3.m3.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem1.p1.3.m3.1c">B</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem1.p1.3.m3.1d">italic_B</annotation></semantics></math> <em class="ltx_emph ltx_font_italic" id="S5.Thmtheorem1.p1.7.1">approximates</em> <math alttext="a\in A" class="ltx_Math" display="inline" id="S5.Thmtheorem1.p1.4.m4.1"><semantics id="S5.Thmtheorem1.p1.4.m4.1a"><mrow id="S5.Thmtheorem1.p1.4.m4.1.1" xref="S5.Thmtheorem1.p1.4.m4.1.1.cmml"><mi id="S5.Thmtheorem1.p1.4.m4.1.1.2" xref="S5.Thmtheorem1.p1.4.m4.1.1.2.cmml">a</mi><mo id="S5.Thmtheorem1.p1.4.m4.1.1.1" xref="S5.Thmtheorem1.p1.4.m4.1.1.1.cmml">∈</mo><mi id="S5.Thmtheorem1.p1.4.m4.1.1.3" xref="S5.Thmtheorem1.p1.4.m4.1.1.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem1.p1.4.m4.1b"><apply id="S5.Thmtheorem1.p1.4.m4.1.1.cmml" xref="S5.Thmtheorem1.p1.4.m4.1.1"><in id="S5.Thmtheorem1.p1.4.m4.1.1.1.cmml" xref="S5.Thmtheorem1.p1.4.m4.1.1.1"></in><ci id="S5.Thmtheorem1.p1.4.m4.1.1.2.cmml" xref="S5.Thmtheorem1.p1.4.m4.1.1.2">𝑎</ci><ci id="S5.Thmtheorem1.p1.4.m4.1.1.3.cmml" xref="S5.Thmtheorem1.p1.4.m4.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem1.p1.4.m4.1c">a\in A</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem1.p1.4.m4.1d">italic_a ∈ italic_A</annotation></semantics></math> if <math alttext="B" class="ltx_Math" display="inline" id="S5.Thmtheorem1.p1.5.m5.1"><semantics id="S5.Thmtheorem1.p1.5.m5.1a"><mi id="S5.Thmtheorem1.p1.5.m5.1.1" xref="S5.Thmtheorem1.p1.5.m5.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem1.p1.5.m5.1b"><ci id="S5.Thmtheorem1.p1.5.m5.1.1.cmml" xref="S5.Thmtheorem1.p1.5.m5.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem1.p1.5.m5.1c">B</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem1.p1.5.m5.1d">italic_B</annotation></semantics></math> is cofinal in <math alttext="\left]-\infty,a\right[" class="ltx_Math" display="inline" id="S5.Thmtheorem1.p1.6.m6.2"><semantics id="S5.Thmtheorem1.p1.6.m6.2a"><mrow id="S5.Thmtheorem1.p1.6.m6.2.2.1" xref="S5.Thmtheorem1.p1.6.m6.2.2.2.cmml"><mo id="S5.Thmtheorem1.p1.6.m6.2.2.1.2" rspace="0em" stretchy="true" xref="S5.Thmtheorem1.p1.6.m6.2.2.2.cmml">]</mo><mrow id="S5.Thmtheorem1.p1.6.m6.2.2.1.1" xref="S5.Thmtheorem1.p1.6.m6.2.2.1.1.cmml"><mo id="S5.Thmtheorem1.p1.6.m6.2.2.1.1a" xref="S5.Thmtheorem1.p1.6.m6.2.2.1.1.cmml">−</mo><mi id="S5.Thmtheorem1.p1.6.m6.2.2.1.1.2" mathvariant="normal" xref="S5.Thmtheorem1.p1.6.m6.2.2.1.1.2.cmml">∞</mi></mrow><mo id="S5.Thmtheorem1.p1.6.m6.2.2.1.3" xref="S5.Thmtheorem1.p1.6.m6.2.2.2.cmml">,</mo><mi id="S5.Thmtheorem1.p1.6.m6.1.1" xref="S5.Thmtheorem1.p1.6.m6.1.1.cmml">a</mi><mo id="S5.Thmtheorem1.p1.6.m6.2.2.1.4" lspace="0em" stretchy="true" xref="S5.Thmtheorem1.p1.6.m6.2.2.2.cmml">[</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem1.p1.6.m6.2b"><list id="S5.Thmtheorem1.p1.6.m6.2.2.2.cmml" xref="S5.Thmtheorem1.p1.6.m6.2.2.1"><apply id="S5.Thmtheorem1.p1.6.m6.2.2.1.1.cmml" xref="S5.Thmtheorem1.p1.6.m6.2.2.1.1"><minus id="S5.Thmtheorem1.p1.6.m6.2.2.1.1.1.cmml" xref="S5.Thmtheorem1.p1.6.m6.2.2.1.1"></minus><infinity id="S5.Thmtheorem1.p1.6.m6.2.2.1.1.2.cmml" xref="S5.Thmtheorem1.p1.6.m6.2.2.1.1.2"></infinity></apply><ci id="S5.Thmtheorem1.p1.6.m6.1.1.cmml" xref="S5.Thmtheorem1.p1.6.m6.1.1">𝑎</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem1.p1.6.m6.2c">\left]-\infty,a\right[</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem1.p1.6.m6.2d">] - ∞ , italic_a [</annotation></semantics></math> and coinitial in <math alttext="\left]a,+\infty\right[" class="ltx_Math" display="inline" id="S5.Thmtheorem1.p1.7.m7.2"><semantics id="S5.Thmtheorem1.p1.7.m7.2a"><mrow id="S5.Thmtheorem1.p1.7.m7.2.2.1" xref="S5.Thmtheorem1.p1.7.m7.2.2.2.cmml"><mo id="S5.Thmtheorem1.p1.7.m7.2.2.1.2" rspace="0em" stretchy="true" xref="S5.Thmtheorem1.p1.7.m7.2.2.2.cmml">]</mo><mi id="S5.Thmtheorem1.p1.7.m7.1.1" xref="S5.Thmtheorem1.p1.7.m7.1.1.cmml">a</mi><mo id="S5.Thmtheorem1.p1.7.m7.2.2.1.3" xref="S5.Thmtheorem1.p1.7.m7.2.2.2.cmml">,</mo><mrow id="S5.Thmtheorem1.p1.7.m7.2.2.1.1" xref="S5.Thmtheorem1.p1.7.m7.2.2.1.1.cmml"><mo id="S5.Thmtheorem1.p1.7.m7.2.2.1.1a" xref="S5.Thmtheorem1.p1.7.m7.2.2.1.1.cmml">+</mo><mi id="S5.Thmtheorem1.p1.7.m7.2.2.1.1.2" mathvariant="normal" xref="S5.Thmtheorem1.p1.7.m7.2.2.1.1.2.cmml">∞</mi></mrow><mo id="S5.Thmtheorem1.p1.7.m7.2.2.1.4" lspace="0em" stretchy="true" xref="S5.Thmtheorem1.p1.7.m7.2.2.2.cmml">[</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem1.p1.7.m7.2b"><list id="S5.Thmtheorem1.p1.7.m7.2.2.2.cmml" xref="S5.Thmtheorem1.p1.7.m7.2.2.1"><ci id="S5.Thmtheorem1.p1.7.m7.1.1.cmml" xref="S5.Thmtheorem1.p1.7.m7.1.1">𝑎</ci><apply id="S5.Thmtheorem1.p1.7.m7.2.2.1.1.cmml" xref="S5.Thmtheorem1.p1.7.m7.2.2.1.1"><plus id="S5.Thmtheorem1.p1.7.m7.2.2.1.1.1.cmml" xref="S5.Thmtheorem1.p1.7.m7.2.2.1.1"></plus><infinity id="S5.Thmtheorem1.p1.7.m7.2.2.1.1.2.cmml" xref="S5.Thmtheorem1.p1.7.m7.2.2.1.1.2"></infinity></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem1.p1.7.m7.2c">\left]a,+\infty\right[</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem1.p1.7.m7.2d">] italic_a , + ∞ [</annotation></semantics></math>.</p> </div> </div> <div class="ltx_para" id="S5.p2"> <p class="ltx_p" id="S5.p2.8">Note that if <math alttext="B" class="ltx_Math" display="inline" id="S5.p2.1.m1.1"><semantics id="S5.p2.1.m1.1a"><mi id="S5.p2.1.m1.1.1" xref="S5.p2.1.m1.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S5.p2.1.m1.1b"><ci id="S5.p2.1.m1.1.1.cmml" xref="S5.p2.1.m1.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.1.m1.1c">B</annotation><annotation encoding="application/x-llamapun" id="S5.p2.1.m1.1d">italic_B</annotation></semantics></math> approximates <math alttext="a\in A" class="ltx_Math" display="inline" id="S5.p2.2.m2.1"><semantics id="S5.p2.2.m2.1a"><mrow id="S5.p2.2.m2.1.1" xref="S5.p2.2.m2.1.1.cmml"><mi id="S5.p2.2.m2.1.1.2" xref="S5.p2.2.m2.1.1.2.cmml">a</mi><mo id="S5.p2.2.m2.1.1.1" xref="S5.p2.2.m2.1.1.1.cmml">∈</mo><mi id="S5.p2.2.m2.1.1.3" xref="S5.p2.2.m2.1.1.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.p2.2.m2.1b"><apply id="S5.p2.2.m2.1.1.cmml" xref="S5.p2.2.m2.1.1"><in id="S5.p2.2.m2.1.1.1.cmml" xref="S5.p2.2.m2.1.1.1"></in><ci id="S5.p2.2.m2.1.1.2.cmml" xref="S5.p2.2.m2.1.1.2">𝑎</ci><ci id="S5.p2.2.m2.1.1.3.cmml" xref="S5.p2.2.m2.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.2.m2.1c">a\in A</annotation><annotation encoding="application/x-llamapun" id="S5.p2.2.m2.1d">italic_a ∈ italic_A</annotation></semantics></math>, and <math alttext="a" class="ltx_Math" display="inline" id="S5.p2.3.m3.1"><semantics id="S5.p2.3.m3.1a"><mi id="S5.p2.3.m3.1.1" xref="S5.p2.3.m3.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S5.p2.3.m3.1b"><ci id="S5.p2.3.m3.1.1.cmml" xref="S5.p2.3.m3.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.3.m3.1c">a</annotation><annotation encoding="application/x-llamapun" id="S5.p2.3.m3.1d">italic_a</annotation></semantics></math> is not the right endpoint of <math alttext="A" class="ltx_Math" display="inline" id="S5.p2.4.m4.1"><semantics id="S5.p2.4.m4.1a"><mi id="S5.p2.4.m4.1.1" xref="S5.p2.4.m4.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S5.p2.4.m4.1b"><ci id="S5.p2.4.m4.1.1.cmml" xref="S5.p2.4.m4.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.4.m4.1c">A</annotation><annotation encoding="application/x-llamapun" id="S5.p2.4.m4.1d">italic_A</annotation></semantics></math>, then for every <math alttext="a^{\prime}>_{A}a" class="ltx_Math" display="inline" id="S5.p2.5.m5.1"><semantics id="S5.p2.5.m5.1a"><mrow id="S5.p2.5.m5.1.1" xref="S5.p2.5.m5.1.1.cmml"><msup id="S5.p2.5.m5.1.1.2" xref="S5.p2.5.m5.1.1.2.cmml"><mi id="S5.p2.5.m5.1.1.2.2" xref="S5.p2.5.m5.1.1.2.2.cmml">a</mi><mo id="S5.p2.5.m5.1.1.2.3" xref="S5.p2.5.m5.1.1.2.3.cmml">′</mo></msup><msub id="S5.p2.5.m5.1.1.1" xref="S5.p2.5.m5.1.1.1.cmml"><mo id="S5.p2.5.m5.1.1.1.2" xref="S5.p2.5.m5.1.1.1.2.cmml">></mo><mi id="S5.p2.5.m5.1.1.1.3" xref="S5.p2.5.m5.1.1.1.3.cmml">A</mi></msub><mi id="S5.p2.5.m5.1.1.3" xref="S5.p2.5.m5.1.1.3.cmml">a</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.p2.5.m5.1b"><apply id="S5.p2.5.m5.1.1.cmml" xref="S5.p2.5.m5.1.1"><apply id="S5.p2.5.m5.1.1.1.cmml" xref="S5.p2.5.m5.1.1.1"><csymbol cd="ambiguous" id="S5.p2.5.m5.1.1.1.1.cmml" xref="S5.p2.5.m5.1.1.1">subscript</csymbol><gt id="S5.p2.5.m5.1.1.1.2.cmml" xref="S5.p2.5.m5.1.1.1.2"></gt><ci id="S5.p2.5.m5.1.1.1.3.cmml" xref="S5.p2.5.m5.1.1.1.3">𝐴</ci></apply><apply id="S5.p2.5.m5.1.1.2.cmml" xref="S5.p2.5.m5.1.1.2"><csymbol cd="ambiguous" id="S5.p2.5.m5.1.1.2.1.cmml" xref="S5.p2.5.m5.1.1.2">superscript</csymbol><ci id="S5.p2.5.m5.1.1.2.2.cmml" xref="S5.p2.5.m5.1.1.2.2">𝑎</ci><ci id="S5.p2.5.m5.1.1.2.3.cmml" xref="S5.p2.5.m5.1.1.2.3">′</ci></apply><ci id="S5.p2.5.m5.1.1.3.cmml" xref="S5.p2.5.m5.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.5.m5.1c">a^{\prime}>_{A}a</annotation><annotation encoding="application/x-llamapun" id="S5.p2.5.m5.1d">italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT > start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_a</annotation></semantics></math> there is <math alttext="b\in B" class="ltx_Math" display="inline" id="S5.p2.6.m6.1"><semantics id="S5.p2.6.m6.1a"><mrow id="S5.p2.6.m6.1.1" xref="S5.p2.6.m6.1.1.cmml"><mi id="S5.p2.6.m6.1.1.2" xref="S5.p2.6.m6.1.1.2.cmml">b</mi><mo id="S5.p2.6.m6.1.1.1" xref="S5.p2.6.m6.1.1.1.cmml">∈</mo><mi id="S5.p2.6.m6.1.1.3" xref="S5.p2.6.m6.1.1.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.p2.6.m6.1b"><apply id="S5.p2.6.m6.1.1.cmml" xref="S5.p2.6.m6.1.1"><in id="S5.p2.6.m6.1.1.1.cmml" xref="S5.p2.6.m6.1.1.1"></in><ci id="S5.p2.6.m6.1.1.2.cmml" xref="S5.p2.6.m6.1.1.2">𝑏</ci><ci id="S5.p2.6.m6.1.1.3.cmml" xref="S5.p2.6.m6.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.6.m6.1c">b\in B</annotation><annotation encoding="application/x-llamapun" id="S5.p2.6.m6.1d">italic_b ∈ italic_B</annotation></semantics></math> such that <math alttext="a<_{A}b\leq_{A}a^{\prime}" class="ltx_Math" display="inline" id="S5.p2.7.m7.1"><semantics id="S5.p2.7.m7.1a"><mrow id="S5.p2.7.m7.1.1" xref="S5.p2.7.m7.1.1.cmml"><mi id="S5.p2.7.m7.1.1.2" xref="S5.p2.7.m7.1.1.2.cmml">a</mi><msub id="S5.p2.7.m7.1.1.3" xref="S5.p2.7.m7.1.1.3.cmml"><mo id="S5.p2.7.m7.1.1.3.2" xref="S5.p2.7.m7.1.1.3.2.cmml"><</mo><mi id="S5.p2.7.m7.1.1.3.3" xref="S5.p2.7.m7.1.1.3.3.cmml">A</mi></msub><mi id="S5.p2.7.m7.1.1.4" xref="S5.p2.7.m7.1.1.4.cmml">b</mi><msub id="S5.p2.7.m7.1.1.5" xref="S5.p2.7.m7.1.1.5.cmml"><mo id="S5.p2.7.m7.1.1.5.2" xref="S5.p2.7.m7.1.1.5.2.cmml">≤</mo><mi id="S5.p2.7.m7.1.1.5.3" xref="S5.p2.7.m7.1.1.5.3.cmml">A</mi></msub><msup id="S5.p2.7.m7.1.1.6" xref="S5.p2.7.m7.1.1.6.cmml"><mi id="S5.p2.7.m7.1.1.6.2" xref="S5.p2.7.m7.1.1.6.2.cmml">a</mi><mo id="S5.p2.7.m7.1.1.6.3" xref="S5.p2.7.m7.1.1.6.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.p2.7.m7.1b"><apply id="S5.p2.7.m7.1.1.cmml" xref="S5.p2.7.m7.1.1"><and id="S5.p2.7.m7.1.1a.cmml" xref="S5.p2.7.m7.1.1"></and><apply id="S5.p2.7.m7.1.1b.cmml" xref="S5.p2.7.m7.1.1"><apply id="S5.p2.7.m7.1.1.3.cmml" xref="S5.p2.7.m7.1.1.3"><csymbol cd="ambiguous" id="S5.p2.7.m7.1.1.3.1.cmml" xref="S5.p2.7.m7.1.1.3">subscript</csymbol><lt id="S5.p2.7.m7.1.1.3.2.cmml" xref="S5.p2.7.m7.1.1.3.2"></lt><ci id="S5.p2.7.m7.1.1.3.3.cmml" xref="S5.p2.7.m7.1.1.3.3">𝐴</ci></apply><ci id="S5.p2.7.m7.1.1.2.cmml" xref="S5.p2.7.m7.1.1.2">𝑎</ci><ci id="S5.p2.7.m7.1.1.4.cmml" xref="S5.p2.7.m7.1.1.4">𝑏</ci></apply><apply id="S5.p2.7.m7.1.1c.cmml" xref="S5.p2.7.m7.1.1"><apply id="S5.p2.7.m7.1.1.5.cmml" xref="S5.p2.7.m7.1.1.5"><csymbol cd="ambiguous" id="S5.p2.7.m7.1.1.5.1.cmml" xref="S5.p2.7.m7.1.1.5">subscript</csymbol><leq id="S5.p2.7.m7.1.1.5.2.cmml" xref="S5.p2.7.m7.1.1.5.2"></leq><ci id="S5.p2.7.m7.1.1.5.3.cmml" xref="S5.p2.7.m7.1.1.5.3">𝐴</ci></apply><share href="https://arxiv.org/html/2503.13728v1#S5.p2.7.m7.1.1.4.cmml" id="S5.p2.7.m7.1.1d.cmml" xref="S5.p2.7.m7.1.1"></share><apply id="S5.p2.7.m7.1.1.6.cmml" xref="S5.p2.7.m7.1.1.6"><csymbol cd="ambiguous" id="S5.p2.7.m7.1.1.6.1.cmml" xref="S5.p2.7.m7.1.1.6">superscript</csymbol><ci id="S5.p2.7.m7.1.1.6.2.cmml" xref="S5.p2.7.m7.1.1.6.2">𝑎</ci><ci id="S5.p2.7.m7.1.1.6.3.cmml" xref="S5.p2.7.m7.1.1.6.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.7.m7.1c">a<_{A}b\leq_{A}a^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S5.p2.7.m7.1d">italic_a < start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_b ≤ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, and if <math alttext="a^{\prime}\notin B" class="ltx_Math" display="inline" id="S5.p2.8.m8.1"><semantics id="S5.p2.8.m8.1a"><mrow id="S5.p2.8.m8.1.1" xref="S5.p2.8.m8.1.1.cmml"><msup id="S5.p2.8.m8.1.1.2" xref="S5.p2.8.m8.1.1.2.cmml"><mi id="S5.p2.8.m8.1.1.2.2" xref="S5.p2.8.m8.1.1.2.2.cmml">a</mi><mo id="S5.p2.8.m8.1.1.2.3" xref="S5.p2.8.m8.1.1.2.3.cmml">′</mo></msup><mo id="S5.p2.8.m8.1.1.1" xref="S5.p2.8.m8.1.1.1.cmml">∉</mo><mi id="S5.p2.8.m8.1.1.3" xref="S5.p2.8.m8.1.1.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.p2.8.m8.1b"><apply id="S5.p2.8.m8.1.1.cmml" xref="S5.p2.8.m8.1.1"><notin id="S5.p2.8.m8.1.1.1.cmml" xref="S5.p2.8.m8.1.1.1"></notin><apply id="S5.p2.8.m8.1.1.2.cmml" xref="S5.p2.8.m8.1.1.2"><csymbol cd="ambiguous" id="S5.p2.8.m8.1.1.2.1.cmml" xref="S5.p2.8.m8.1.1.2">superscript</csymbol><ci id="S5.p2.8.m8.1.1.2.2.cmml" xref="S5.p2.8.m8.1.1.2.2">𝑎</ci><ci id="S5.p2.8.m8.1.1.2.3.cmml" xref="S5.p2.8.m8.1.1.2.3">′</ci></apply><ci id="S5.p2.8.m8.1.1.3.cmml" xref="S5.p2.8.m8.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.8.m8.1c">a^{\prime}\notin B</annotation><annotation encoding="application/x-llamapun" id="S5.p2.8.m8.1d">italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∉ italic_B</annotation></semantics></math>, then the second inequality is strict.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S5.Thmtheorem2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem2.1.1.1">Lemma 5.2</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem2.2.2">.</span> </h6> <div class="ltx_para" id="S5.Thmtheorem2.p1"> <p class="ltx_p" id="S5.Thmtheorem2.p1.10">Let <math alttext="A" class="ltx_Math" display="inline" id="S5.Thmtheorem2.p1.1.m1.1"><semantics id="S5.Thmtheorem2.p1.1.m1.1a"><mi id="S5.Thmtheorem2.p1.1.m1.1.1" xref="S5.Thmtheorem2.p1.1.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem2.p1.1.m1.1b"><ci id="S5.Thmtheorem2.p1.1.m1.1.1.cmml" xref="S5.Thmtheorem2.p1.1.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem2.p1.1.m1.1c">A</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem2.p1.1.m1.1d">italic_A</annotation></semantics></math> and <math alttext="X" class="ltx_Math" display="inline" id="S5.Thmtheorem2.p1.2.m2.1"><semantics id="S5.Thmtheorem2.p1.2.m2.1a"><mi id="S5.Thmtheorem2.p1.2.m2.1.1" xref="S5.Thmtheorem2.p1.2.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem2.p1.2.m2.1b"><ci id="S5.Thmtheorem2.p1.2.m2.1.1.cmml" xref="S5.Thmtheorem2.p1.2.m2.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem2.p1.2.m2.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem2.p1.2.m2.1d">italic_X</annotation></semantics></math> be Aronszajn lines. If for some (equivalently any) pair of decompositions <math alttext="D^{A}" class="ltx_Math" display="inline" id="S5.Thmtheorem2.p1.3.m3.1"><semantics id="S5.Thmtheorem2.p1.3.m3.1a"><msup id="S5.Thmtheorem2.p1.3.m3.1.1" xref="S5.Thmtheorem2.p1.3.m3.1.1.cmml"><mi id="S5.Thmtheorem2.p1.3.m3.1.1.2" xref="S5.Thmtheorem2.p1.3.m3.1.1.2.cmml">D</mi><mi id="S5.Thmtheorem2.p1.3.m3.1.1.3" xref="S5.Thmtheorem2.p1.3.m3.1.1.3.cmml">A</mi></msup><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem2.p1.3.m3.1b"><apply id="S5.Thmtheorem2.p1.3.m3.1.1.cmml" xref="S5.Thmtheorem2.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem2.p1.3.m3.1.1.1.cmml" xref="S5.Thmtheorem2.p1.3.m3.1.1">superscript</csymbol><ci id="S5.Thmtheorem2.p1.3.m3.1.1.2.cmml" xref="S5.Thmtheorem2.p1.3.m3.1.1.2">𝐷</ci><ci id="S5.Thmtheorem2.p1.3.m3.1.1.3.cmml" xref="S5.Thmtheorem2.p1.3.m3.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem2.p1.3.m3.1c">D^{A}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem2.p1.3.m3.1d">italic_D start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="D^{X}" class="ltx_Math" display="inline" id="S5.Thmtheorem2.p1.4.m4.1"><semantics id="S5.Thmtheorem2.p1.4.m4.1a"><msup id="S5.Thmtheorem2.p1.4.m4.1.1" xref="S5.Thmtheorem2.p1.4.m4.1.1.cmml"><mi id="S5.Thmtheorem2.p1.4.m4.1.1.2" xref="S5.Thmtheorem2.p1.4.m4.1.1.2.cmml">D</mi><mi id="S5.Thmtheorem2.p1.4.m4.1.1.3" xref="S5.Thmtheorem2.p1.4.m4.1.1.3.cmml">X</mi></msup><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem2.p1.4.m4.1b"><apply id="S5.Thmtheorem2.p1.4.m4.1.1.cmml" xref="S5.Thmtheorem2.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem2.p1.4.m4.1.1.1.cmml" xref="S5.Thmtheorem2.p1.4.m4.1.1">superscript</csymbol><ci id="S5.Thmtheorem2.p1.4.m4.1.1.2.cmml" xref="S5.Thmtheorem2.p1.4.m4.1.1.2">𝐷</ci><ci id="S5.Thmtheorem2.p1.4.m4.1.1.3.cmml" xref="S5.Thmtheorem2.p1.4.m4.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem2.p1.4.m4.1c">D^{X}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem2.p1.4.m4.1d">italic_D start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT</annotation></semantics></math> for <math alttext="A" class="ltx_Math" display="inline" id="S5.Thmtheorem2.p1.5.m5.1"><semantics id="S5.Thmtheorem2.p1.5.m5.1a"><mi id="S5.Thmtheorem2.p1.5.m5.1.1" xref="S5.Thmtheorem2.p1.5.m5.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem2.p1.5.m5.1b"><ci id="S5.Thmtheorem2.p1.5.m5.1.1.cmml" xref="S5.Thmtheorem2.p1.5.m5.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem2.p1.5.m5.1c">A</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem2.p1.5.m5.1d">italic_A</annotation></semantics></math> and <math alttext="X" class="ltx_Math" display="inline" id="S5.Thmtheorem2.p1.6.m6.1"><semantics id="S5.Thmtheorem2.p1.6.m6.1a"><mi id="S5.Thmtheorem2.p1.6.m6.1.1" xref="S5.Thmtheorem2.p1.6.m6.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem2.p1.6.m6.1b"><ci id="S5.Thmtheorem2.p1.6.m6.1.1.cmml" xref="S5.Thmtheorem2.p1.6.m6.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem2.p1.6.m6.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem2.p1.6.m6.1d">italic_X</annotation></semantics></math> respectively, <math alttext="\hat{\mathscr{L}}(A,D^{A})\setminus\hat{\mathscr{L}}(X,D^{X})" class="ltx_Math" display="inline" id="S5.Thmtheorem2.p1.7.m7.4"><semantics id="S5.Thmtheorem2.p1.7.m7.4a"><mrow id="S5.Thmtheorem2.p1.7.m7.4.4" xref="S5.Thmtheorem2.p1.7.m7.4.4.cmml"><mrow id="S5.Thmtheorem2.p1.7.m7.3.3.1" xref="S5.Thmtheorem2.p1.7.m7.3.3.1.cmml"><mover accent="true" id="S5.Thmtheorem2.p1.7.m7.3.3.1.3" xref="S5.Thmtheorem2.p1.7.m7.3.3.1.3.cmml"><mi class="ltx_font_mathscript" id="S5.Thmtheorem2.p1.7.m7.3.3.1.3.2" xref="S5.Thmtheorem2.p1.7.m7.3.3.1.3.2.cmml">ℒ</mi><mo id="S5.Thmtheorem2.p1.7.m7.3.3.1.3.1" xref="S5.Thmtheorem2.p1.7.m7.3.3.1.3.1.cmml">^</mo></mover><mo id="S5.Thmtheorem2.p1.7.m7.3.3.1.2" xref="S5.Thmtheorem2.p1.7.m7.3.3.1.2.cmml">⁢</mo><mrow id="S5.Thmtheorem2.p1.7.m7.3.3.1.1.1" xref="S5.Thmtheorem2.p1.7.m7.3.3.1.1.2.cmml"><mo id="S5.Thmtheorem2.p1.7.m7.3.3.1.1.1.2" stretchy="false" xref="S5.Thmtheorem2.p1.7.m7.3.3.1.1.2.cmml">(</mo><mi id="S5.Thmtheorem2.p1.7.m7.1.1" xref="S5.Thmtheorem2.p1.7.m7.1.1.cmml">A</mi><mo id="S5.Thmtheorem2.p1.7.m7.3.3.1.1.1.3" xref="S5.Thmtheorem2.p1.7.m7.3.3.1.1.2.cmml">,</mo><msup id="S5.Thmtheorem2.p1.7.m7.3.3.1.1.1.1" xref="S5.Thmtheorem2.p1.7.m7.3.3.1.1.1.1.cmml"><mi id="S5.Thmtheorem2.p1.7.m7.3.3.1.1.1.1.2" xref="S5.Thmtheorem2.p1.7.m7.3.3.1.1.1.1.2.cmml">D</mi><mi id="S5.Thmtheorem2.p1.7.m7.3.3.1.1.1.1.3" xref="S5.Thmtheorem2.p1.7.m7.3.3.1.1.1.1.3.cmml">A</mi></msup><mo id="S5.Thmtheorem2.p1.7.m7.3.3.1.1.1.4" stretchy="false" xref="S5.Thmtheorem2.p1.7.m7.3.3.1.1.2.cmml">)</mo></mrow></mrow><mo id="S5.Thmtheorem2.p1.7.m7.4.4.3" xref="S5.Thmtheorem2.p1.7.m7.4.4.3.cmml">∖</mo><mrow id="S5.Thmtheorem2.p1.7.m7.4.4.2" xref="S5.Thmtheorem2.p1.7.m7.4.4.2.cmml"><mover accent="true" id="S5.Thmtheorem2.p1.7.m7.4.4.2.3" xref="S5.Thmtheorem2.p1.7.m7.4.4.2.3.cmml"><mi class="ltx_font_mathscript" id="S5.Thmtheorem2.p1.7.m7.4.4.2.3.2" xref="S5.Thmtheorem2.p1.7.m7.4.4.2.3.2.cmml">ℒ</mi><mo id="S5.Thmtheorem2.p1.7.m7.4.4.2.3.1" xref="S5.Thmtheorem2.p1.7.m7.4.4.2.3.1.cmml">^</mo></mover><mo id="S5.Thmtheorem2.p1.7.m7.4.4.2.2" xref="S5.Thmtheorem2.p1.7.m7.4.4.2.2.cmml">⁢</mo><mrow id="S5.Thmtheorem2.p1.7.m7.4.4.2.1.1" xref="S5.Thmtheorem2.p1.7.m7.4.4.2.1.2.cmml"><mo id="S5.Thmtheorem2.p1.7.m7.4.4.2.1.1.2" stretchy="false" xref="S5.Thmtheorem2.p1.7.m7.4.4.2.1.2.cmml">(</mo><mi id="S5.Thmtheorem2.p1.7.m7.2.2" xref="S5.Thmtheorem2.p1.7.m7.2.2.cmml">X</mi><mo id="S5.Thmtheorem2.p1.7.m7.4.4.2.1.1.3" xref="S5.Thmtheorem2.p1.7.m7.4.4.2.1.2.cmml">,</mo><msup id="S5.Thmtheorem2.p1.7.m7.4.4.2.1.1.1" xref="S5.Thmtheorem2.p1.7.m7.4.4.2.1.1.1.cmml"><mi id="S5.Thmtheorem2.p1.7.m7.4.4.2.1.1.1.2" xref="S5.Thmtheorem2.p1.7.m7.4.4.2.1.1.1.2.cmml">D</mi><mi id="S5.Thmtheorem2.p1.7.m7.4.4.2.1.1.1.3" xref="S5.Thmtheorem2.p1.7.m7.4.4.2.1.1.1.3.cmml">X</mi></msup><mo id="S5.Thmtheorem2.p1.7.m7.4.4.2.1.1.4" stretchy="false" xref="S5.Thmtheorem2.p1.7.m7.4.4.2.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem2.p1.7.m7.4b"><apply id="S5.Thmtheorem2.p1.7.m7.4.4.cmml" xref="S5.Thmtheorem2.p1.7.m7.4.4"><setdiff id="S5.Thmtheorem2.p1.7.m7.4.4.3.cmml" xref="S5.Thmtheorem2.p1.7.m7.4.4.3"></setdiff><apply id="S5.Thmtheorem2.p1.7.m7.3.3.1.cmml" xref="S5.Thmtheorem2.p1.7.m7.3.3.1"><times id="S5.Thmtheorem2.p1.7.m7.3.3.1.2.cmml" xref="S5.Thmtheorem2.p1.7.m7.3.3.1.2"></times><apply id="S5.Thmtheorem2.p1.7.m7.3.3.1.3.cmml" xref="S5.Thmtheorem2.p1.7.m7.3.3.1.3"><ci id="S5.Thmtheorem2.p1.7.m7.3.3.1.3.1.cmml" xref="S5.Thmtheorem2.p1.7.m7.3.3.1.3.1">^</ci><ci id="S5.Thmtheorem2.p1.7.m7.3.3.1.3.2.cmml" xref="S5.Thmtheorem2.p1.7.m7.3.3.1.3.2">ℒ</ci></apply><interval closure="open" id="S5.Thmtheorem2.p1.7.m7.3.3.1.1.2.cmml" xref="S5.Thmtheorem2.p1.7.m7.3.3.1.1.1"><ci id="S5.Thmtheorem2.p1.7.m7.1.1.cmml" xref="S5.Thmtheorem2.p1.7.m7.1.1">𝐴</ci><apply id="S5.Thmtheorem2.p1.7.m7.3.3.1.1.1.1.cmml" xref="S5.Thmtheorem2.p1.7.m7.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem2.p1.7.m7.3.3.1.1.1.1.1.cmml" xref="S5.Thmtheorem2.p1.7.m7.3.3.1.1.1.1">superscript</csymbol><ci id="S5.Thmtheorem2.p1.7.m7.3.3.1.1.1.1.2.cmml" xref="S5.Thmtheorem2.p1.7.m7.3.3.1.1.1.1.2">𝐷</ci><ci id="S5.Thmtheorem2.p1.7.m7.3.3.1.1.1.1.3.cmml" xref="S5.Thmtheorem2.p1.7.m7.3.3.1.1.1.1.3">𝐴</ci></apply></interval></apply><apply id="S5.Thmtheorem2.p1.7.m7.4.4.2.cmml" xref="S5.Thmtheorem2.p1.7.m7.4.4.2"><times id="S5.Thmtheorem2.p1.7.m7.4.4.2.2.cmml" xref="S5.Thmtheorem2.p1.7.m7.4.4.2.2"></times><apply id="S5.Thmtheorem2.p1.7.m7.4.4.2.3.cmml" xref="S5.Thmtheorem2.p1.7.m7.4.4.2.3"><ci id="S5.Thmtheorem2.p1.7.m7.4.4.2.3.1.cmml" xref="S5.Thmtheorem2.p1.7.m7.4.4.2.3.1">^</ci><ci id="S5.Thmtheorem2.p1.7.m7.4.4.2.3.2.cmml" xref="S5.Thmtheorem2.p1.7.m7.4.4.2.3.2">ℒ</ci></apply><interval closure="open" id="S5.Thmtheorem2.p1.7.m7.4.4.2.1.2.cmml" xref="S5.Thmtheorem2.p1.7.m7.4.4.2.1.1"><ci id="S5.Thmtheorem2.p1.7.m7.2.2.cmml" xref="S5.Thmtheorem2.p1.7.m7.2.2">𝑋</ci><apply id="S5.Thmtheorem2.p1.7.m7.4.4.2.1.1.1.cmml" xref="S5.Thmtheorem2.p1.7.m7.4.4.2.1.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem2.p1.7.m7.4.4.2.1.1.1.1.cmml" xref="S5.Thmtheorem2.p1.7.m7.4.4.2.1.1.1">superscript</csymbol><ci id="S5.Thmtheorem2.p1.7.m7.4.4.2.1.1.1.2.cmml" xref="S5.Thmtheorem2.p1.7.m7.4.4.2.1.1.1.2">𝐷</ci><ci id="S5.Thmtheorem2.p1.7.m7.4.4.2.1.1.1.3.cmml" xref="S5.Thmtheorem2.p1.7.m7.4.4.2.1.1.1.3">𝑋</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem2.p1.7.m7.4c">\hat{\mathscr{L}}(A,D^{A})\setminus\hat{\mathscr{L}}(X,D^{X})</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem2.p1.7.m7.4d">over^ start_ARG script_L end_ARG ( italic_A , italic_D start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) ∖ over^ start_ARG script_L end_ARG ( italic_X , italic_D start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT )</annotation></semantics></math> (or <math alttext="\hat{\mathscr{R}}(A,D^{A})\setminus\hat{\mathscr{R}}(X,D^{X})" class="ltx_Math" display="inline" id="S5.Thmtheorem2.p1.8.m8.4"><semantics id="S5.Thmtheorem2.p1.8.m8.4a"><mrow id="S5.Thmtheorem2.p1.8.m8.4.4" xref="S5.Thmtheorem2.p1.8.m8.4.4.cmml"><mrow id="S5.Thmtheorem2.p1.8.m8.3.3.1" xref="S5.Thmtheorem2.p1.8.m8.3.3.1.cmml"><mover accent="true" id="S5.Thmtheorem2.p1.8.m8.3.3.1.3" xref="S5.Thmtheorem2.p1.8.m8.3.3.1.3.cmml"><mi class="ltx_font_mathscript" id="S5.Thmtheorem2.p1.8.m8.3.3.1.3.2" xref="S5.Thmtheorem2.p1.8.m8.3.3.1.3.2.cmml">ℛ</mi><mo id="S5.Thmtheorem2.p1.8.m8.3.3.1.3.1" xref="S5.Thmtheorem2.p1.8.m8.3.3.1.3.1.cmml">^</mo></mover><mo id="S5.Thmtheorem2.p1.8.m8.3.3.1.2" xref="S5.Thmtheorem2.p1.8.m8.3.3.1.2.cmml">⁢</mo><mrow id="S5.Thmtheorem2.p1.8.m8.3.3.1.1.1" xref="S5.Thmtheorem2.p1.8.m8.3.3.1.1.2.cmml"><mo id="S5.Thmtheorem2.p1.8.m8.3.3.1.1.1.2" stretchy="false" xref="S5.Thmtheorem2.p1.8.m8.3.3.1.1.2.cmml">(</mo><mi id="S5.Thmtheorem2.p1.8.m8.1.1" xref="S5.Thmtheorem2.p1.8.m8.1.1.cmml">A</mi><mo id="S5.Thmtheorem2.p1.8.m8.3.3.1.1.1.3" xref="S5.Thmtheorem2.p1.8.m8.3.3.1.1.2.cmml">,</mo><msup id="S5.Thmtheorem2.p1.8.m8.3.3.1.1.1.1" xref="S5.Thmtheorem2.p1.8.m8.3.3.1.1.1.1.cmml"><mi id="S5.Thmtheorem2.p1.8.m8.3.3.1.1.1.1.2" xref="S5.Thmtheorem2.p1.8.m8.3.3.1.1.1.1.2.cmml">D</mi><mi id="S5.Thmtheorem2.p1.8.m8.3.3.1.1.1.1.3" xref="S5.Thmtheorem2.p1.8.m8.3.3.1.1.1.1.3.cmml">A</mi></msup><mo id="S5.Thmtheorem2.p1.8.m8.3.3.1.1.1.4" stretchy="false" xref="S5.Thmtheorem2.p1.8.m8.3.3.1.1.2.cmml">)</mo></mrow></mrow><mo id="S5.Thmtheorem2.p1.8.m8.4.4.3" xref="S5.Thmtheorem2.p1.8.m8.4.4.3.cmml">∖</mo><mrow id="S5.Thmtheorem2.p1.8.m8.4.4.2" xref="S5.Thmtheorem2.p1.8.m8.4.4.2.cmml"><mover accent="true" id="S5.Thmtheorem2.p1.8.m8.4.4.2.3" xref="S5.Thmtheorem2.p1.8.m8.4.4.2.3.cmml"><mi class="ltx_font_mathscript" id="S5.Thmtheorem2.p1.8.m8.4.4.2.3.2" xref="S5.Thmtheorem2.p1.8.m8.4.4.2.3.2.cmml">ℛ</mi><mo id="S5.Thmtheorem2.p1.8.m8.4.4.2.3.1" xref="S5.Thmtheorem2.p1.8.m8.4.4.2.3.1.cmml">^</mo></mover><mo id="S5.Thmtheorem2.p1.8.m8.4.4.2.2" xref="S5.Thmtheorem2.p1.8.m8.4.4.2.2.cmml">⁢</mo><mrow id="S5.Thmtheorem2.p1.8.m8.4.4.2.1.1" xref="S5.Thmtheorem2.p1.8.m8.4.4.2.1.2.cmml"><mo id="S5.Thmtheorem2.p1.8.m8.4.4.2.1.1.2" stretchy="false" xref="S5.Thmtheorem2.p1.8.m8.4.4.2.1.2.cmml">(</mo><mi id="S5.Thmtheorem2.p1.8.m8.2.2" xref="S5.Thmtheorem2.p1.8.m8.2.2.cmml">X</mi><mo id="S5.Thmtheorem2.p1.8.m8.4.4.2.1.1.3" xref="S5.Thmtheorem2.p1.8.m8.4.4.2.1.2.cmml">,</mo><msup id="S5.Thmtheorem2.p1.8.m8.4.4.2.1.1.1" xref="S5.Thmtheorem2.p1.8.m8.4.4.2.1.1.1.cmml"><mi id="S5.Thmtheorem2.p1.8.m8.4.4.2.1.1.1.2" xref="S5.Thmtheorem2.p1.8.m8.4.4.2.1.1.1.2.cmml">D</mi><mi id="S5.Thmtheorem2.p1.8.m8.4.4.2.1.1.1.3" xref="S5.Thmtheorem2.p1.8.m8.4.4.2.1.1.1.3.cmml">X</mi></msup><mo id="S5.Thmtheorem2.p1.8.m8.4.4.2.1.1.4" stretchy="false" xref="S5.Thmtheorem2.p1.8.m8.4.4.2.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem2.p1.8.m8.4b"><apply id="S5.Thmtheorem2.p1.8.m8.4.4.cmml" xref="S5.Thmtheorem2.p1.8.m8.4.4"><setdiff id="S5.Thmtheorem2.p1.8.m8.4.4.3.cmml" xref="S5.Thmtheorem2.p1.8.m8.4.4.3"></setdiff><apply id="S5.Thmtheorem2.p1.8.m8.3.3.1.cmml" xref="S5.Thmtheorem2.p1.8.m8.3.3.1"><times id="S5.Thmtheorem2.p1.8.m8.3.3.1.2.cmml" xref="S5.Thmtheorem2.p1.8.m8.3.3.1.2"></times><apply id="S5.Thmtheorem2.p1.8.m8.3.3.1.3.cmml" xref="S5.Thmtheorem2.p1.8.m8.3.3.1.3"><ci id="S5.Thmtheorem2.p1.8.m8.3.3.1.3.1.cmml" xref="S5.Thmtheorem2.p1.8.m8.3.3.1.3.1">^</ci><ci id="S5.Thmtheorem2.p1.8.m8.3.3.1.3.2.cmml" xref="S5.Thmtheorem2.p1.8.m8.3.3.1.3.2">ℛ</ci></apply><interval closure="open" id="S5.Thmtheorem2.p1.8.m8.3.3.1.1.2.cmml" xref="S5.Thmtheorem2.p1.8.m8.3.3.1.1.1"><ci id="S5.Thmtheorem2.p1.8.m8.1.1.cmml" xref="S5.Thmtheorem2.p1.8.m8.1.1">𝐴</ci><apply id="S5.Thmtheorem2.p1.8.m8.3.3.1.1.1.1.cmml" xref="S5.Thmtheorem2.p1.8.m8.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem2.p1.8.m8.3.3.1.1.1.1.1.cmml" xref="S5.Thmtheorem2.p1.8.m8.3.3.1.1.1.1">superscript</csymbol><ci id="S5.Thmtheorem2.p1.8.m8.3.3.1.1.1.1.2.cmml" xref="S5.Thmtheorem2.p1.8.m8.3.3.1.1.1.1.2">𝐷</ci><ci id="S5.Thmtheorem2.p1.8.m8.3.3.1.1.1.1.3.cmml" xref="S5.Thmtheorem2.p1.8.m8.3.3.1.1.1.1.3">𝐴</ci></apply></interval></apply><apply id="S5.Thmtheorem2.p1.8.m8.4.4.2.cmml" xref="S5.Thmtheorem2.p1.8.m8.4.4.2"><times id="S5.Thmtheorem2.p1.8.m8.4.4.2.2.cmml" xref="S5.Thmtheorem2.p1.8.m8.4.4.2.2"></times><apply id="S5.Thmtheorem2.p1.8.m8.4.4.2.3.cmml" xref="S5.Thmtheorem2.p1.8.m8.4.4.2.3"><ci id="S5.Thmtheorem2.p1.8.m8.4.4.2.3.1.cmml" xref="S5.Thmtheorem2.p1.8.m8.4.4.2.3.1">^</ci><ci id="S5.Thmtheorem2.p1.8.m8.4.4.2.3.2.cmml" xref="S5.Thmtheorem2.p1.8.m8.4.4.2.3.2">ℛ</ci></apply><interval closure="open" id="S5.Thmtheorem2.p1.8.m8.4.4.2.1.2.cmml" xref="S5.Thmtheorem2.p1.8.m8.4.4.2.1.1"><ci id="S5.Thmtheorem2.p1.8.m8.2.2.cmml" xref="S5.Thmtheorem2.p1.8.m8.2.2">𝑋</ci><apply id="S5.Thmtheorem2.p1.8.m8.4.4.2.1.1.1.cmml" xref="S5.Thmtheorem2.p1.8.m8.4.4.2.1.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem2.p1.8.m8.4.4.2.1.1.1.1.cmml" xref="S5.Thmtheorem2.p1.8.m8.4.4.2.1.1.1">superscript</csymbol><ci id="S5.Thmtheorem2.p1.8.m8.4.4.2.1.1.1.2.cmml" xref="S5.Thmtheorem2.p1.8.m8.4.4.2.1.1.1.2">𝐷</ci><ci id="S5.Thmtheorem2.p1.8.m8.4.4.2.1.1.1.3.cmml" xref="S5.Thmtheorem2.p1.8.m8.4.4.2.1.1.1.3">𝑋</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem2.p1.8.m8.4c">\hat{\mathscr{R}}(A,D^{A})\setminus\hat{\mathscr{R}}(X,D^{X})</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem2.p1.8.m8.4d">over^ start_ARG script_R end_ARG ( italic_A , italic_D start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) ∖ over^ start_ARG script_R end_ARG ( italic_X , italic_D start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT )</annotation></semantics></math>) is a stationary subset of <math alttext="\omega_{1}" class="ltx_Math" display="inline" id="S5.Thmtheorem2.p1.9.m9.1"><semantics id="S5.Thmtheorem2.p1.9.m9.1a"><msub id="S5.Thmtheorem2.p1.9.m9.1.1" xref="S5.Thmtheorem2.p1.9.m9.1.1.cmml"><mi id="S5.Thmtheorem2.p1.9.m9.1.1.2" xref="S5.Thmtheorem2.p1.9.m9.1.1.2.cmml">ω</mi><mn id="S5.Thmtheorem2.p1.9.m9.1.1.3" xref="S5.Thmtheorem2.p1.9.m9.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem2.p1.9.m9.1b"><apply id="S5.Thmtheorem2.p1.9.m9.1.1.cmml" xref="S5.Thmtheorem2.p1.9.m9.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem2.p1.9.m9.1.1.1.cmml" xref="S5.Thmtheorem2.p1.9.m9.1.1">subscript</csymbol><ci id="S5.Thmtheorem2.p1.9.m9.1.1.2.cmml" xref="S5.Thmtheorem2.p1.9.m9.1.1.2">𝜔</ci><cn id="S5.Thmtheorem2.p1.9.m9.1.1.3.cmml" type="integer" xref="S5.Thmtheorem2.p1.9.m9.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem2.p1.9.m9.1c">\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem2.p1.9.m9.1d">italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, then <math alttext="A\ntrianglerighteq X" class="ltx_Math" display="inline" id="S5.Thmtheorem2.p1.10.m10.1"><semantics id="S5.Thmtheorem2.p1.10.m10.1a"><mrow id="S5.Thmtheorem2.p1.10.m10.1.1" xref="S5.Thmtheorem2.p1.10.m10.1.1.cmml"><mi id="S5.Thmtheorem2.p1.10.m10.1.1.2" xref="S5.Thmtheorem2.p1.10.m10.1.1.2.cmml">A</mi><mo id="S5.Thmtheorem2.p1.10.m10.1.1.1" xref="S5.Thmtheorem2.p1.10.m10.1.1.1.cmml">⋭</mo><mi id="S5.Thmtheorem2.p1.10.m10.1.1.3" xref="S5.Thmtheorem2.p1.10.m10.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem2.p1.10.m10.1b"><apply id="S5.Thmtheorem2.p1.10.m10.1.1.cmml" xref="S5.Thmtheorem2.p1.10.m10.1.1"><csymbol cd="latexml" id="S5.Thmtheorem2.p1.10.m10.1.1.1.cmml" xref="S5.Thmtheorem2.p1.10.m10.1.1.1">not-contains-nor-equals</csymbol><ci id="S5.Thmtheorem2.p1.10.m10.1.1.2.cmml" xref="S5.Thmtheorem2.p1.10.m10.1.1.2">𝐴</ci><ci id="S5.Thmtheorem2.p1.10.m10.1.1.3.cmml" xref="S5.Thmtheorem2.p1.10.m10.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem2.p1.10.m10.1c">A\ntrianglerighteq X</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem2.p1.10.m10.1d">italic_A ⋭ italic_X</annotation></semantics></math>.</p> </div> </div> <div class="ltx_proof" id="S5.4"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S5.1.p1"> <p class="ltx_p" id="S5.1.p1.19">Towards a contradiction assume that <math alttext="\hat{\mathscr{L}}(A,D^{A})\setminus\hat{\mathscr{L}}(X,D^{X})" class="ltx_Math" display="inline" id="S5.1.p1.1.m1.4"><semantics id="S5.1.p1.1.m1.4a"><mrow id="S5.1.p1.1.m1.4.4" xref="S5.1.p1.1.m1.4.4.cmml"><mrow id="S5.1.p1.1.m1.3.3.1" xref="S5.1.p1.1.m1.3.3.1.cmml"><mover accent="true" id="S5.1.p1.1.m1.3.3.1.3" xref="S5.1.p1.1.m1.3.3.1.3.cmml"><mi class="ltx_font_mathscript" id="S5.1.p1.1.m1.3.3.1.3.2" xref="S5.1.p1.1.m1.3.3.1.3.2.cmml">ℒ</mi><mo id="S5.1.p1.1.m1.3.3.1.3.1" xref="S5.1.p1.1.m1.3.3.1.3.1.cmml">^</mo></mover><mo id="S5.1.p1.1.m1.3.3.1.2" xref="S5.1.p1.1.m1.3.3.1.2.cmml">⁢</mo><mrow id="S5.1.p1.1.m1.3.3.1.1.1" xref="S5.1.p1.1.m1.3.3.1.1.2.cmml"><mo id="S5.1.p1.1.m1.3.3.1.1.1.2" stretchy="false" xref="S5.1.p1.1.m1.3.3.1.1.2.cmml">(</mo><mi id="S5.1.p1.1.m1.1.1" xref="S5.1.p1.1.m1.1.1.cmml">A</mi><mo id="S5.1.p1.1.m1.3.3.1.1.1.3" xref="S5.1.p1.1.m1.3.3.1.1.2.cmml">,</mo><msup id="S5.1.p1.1.m1.3.3.1.1.1.1" xref="S5.1.p1.1.m1.3.3.1.1.1.1.cmml"><mi id="S5.1.p1.1.m1.3.3.1.1.1.1.2" xref="S5.1.p1.1.m1.3.3.1.1.1.1.2.cmml">D</mi><mi id="S5.1.p1.1.m1.3.3.1.1.1.1.3" xref="S5.1.p1.1.m1.3.3.1.1.1.1.3.cmml">A</mi></msup><mo id="S5.1.p1.1.m1.3.3.1.1.1.4" stretchy="false" xref="S5.1.p1.1.m1.3.3.1.1.2.cmml">)</mo></mrow></mrow><mo id="S5.1.p1.1.m1.4.4.3" xref="S5.1.p1.1.m1.4.4.3.cmml">∖</mo><mrow id="S5.1.p1.1.m1.4.4.2" xref="S5.1.p1.1.m1.4.4.2.cmml"><mover accent="true" id="S5.1.p1.1.m1.4.4.2.3" xref="S5.1.p1.1.m1.4.4.2.3.cmml"><mi class="ltx_font_mathscript" id="S5.1.p1.1.m1.4.4.2.3.2" xref="S5.1.p1.1.m1.4.4.2.3.2.cmml">ℒ</mi><mo id="S5.1.p1.1.m1.4.4.2.3.1" xref="S5.1.p1.1.m1.4.4.2.3.1.cmml">^</mo></mover><mo id="S5.1.p1.1.m1.4.4.2.2" xref="S5.1.p1.1.m1.4.4.2.2.cmml">⁢</mo><mrow id="S5.1.p1.1.m1.4.4.2.1.1" xref="S5.1.p1.1.m1.4.4.2.1.2.cmml"><mo id="S5.1.p1.1.m1.4.4.2.1.1.2" stretchy="false" xref="S5.1.p1.1.m1.4.4.2.1.2.cmml">(</mo><mi id="S5.1.p1.1.m1.2.2" xref="S5.1.p1.1.m1.2.2.cmml">X</mi><mo id="S5.1.p1.1.m1.4.4.2.1.1.3" xref="S5.1.p1.1.m1.4.4.2.1.2.cmml">,</mo><msup id="S5.1.p1.1.m1.4.4.2.1.1.1" xref="S5.1.p1.1.m1.4.4.2.1.1.1.cmml"><mi id="S5.1.p1.1.m1.4.4.2.1.1.1.2" xref="S5.1.p1.1.m1.4.4.2.1.1.1.2.cmml">D</mi><mi id="S5.1.p1.1.m1.4.4.2.1.1.1.3" xref="S5.1.p1.1.m1.4.4.2.1.1.1.3.cmml">X</mi></msup><mo id="S5.1.p1.1.m1.4.4.2.1.1.4" stretchy="false" xref="S5.1.p1.1.m1.4.4.2.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.1.p1.1.m1.4b"><apply id="S5.1.p1.1.m1.4.4.cmml" xref="S5.1.p1.1.m1.4.4"><setdiff id="S5.1.p1.1.m1.4.4.3.cmml" xref="S5.1.p1.1.m1.4.4.3"></setdiff><apply id="S5.1.p1.1.m1.3.3.1.cmml" xref="S5.1.p1.1.m1.3.3.1"><times id="S5.1.p1.1.m1.3.3.1.2.cmml" xref="S5.1.p1.1.m1.3.3.1.2"></times><apply id="S5.1.p1.1.m1.3.3.1.3.cmml" xref="S5.1.p1.1.m1.3.3.1.3"><ci id="S5.1.p1.1.m1.3.3.1.3.1.cmml" xref="S5.1.p1.1.m1.3.3.1.3.1">^</ci><ci id="S5.1.p1.1.m1.3.3.1.3.2.cmml" xref="S5.1.p1.1.m1.3.3.1.3.2">ℒ</ci></apply><interval closure="open" id="S5.1.p1.1.m1.3.3.1.1.2.cmml" xref="S5.1.p1.1.m1.3.3.1.1.1"><ci id="S5.1.p1.1.m1.1.1.cmml" xref="S5.1.p1.1.m1.1.1">𝐴</ci><apply id="S5.1.p1.1.m1.3.3.1.1.1.1.cmml" xref="S5.1.p1.1.m1.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S5.1.p1.1.m1.3.3.1.1.1.1.1.cmml" xref="S5.1.p1.1.m1.3.3.1.1.1.1">superscript</csymbol><ci id="S5.1.p1.1.m1.3.3.1.1.1.1.2.cmml" xref="S5.1.p1.1.m1.3.3.1.1.1.1.2">𝐷</ci><ci id="S5.1.p1.1.m1.3.3.1.1.1.1.3.cmml" xref="S5.1.p1.1.m1.3.3.1.1.1.1.3">𝐴</ci></apply></interval></apply><apply id="S5.1.p1.1.m1.4.4.2.cmml" xref="S5.1.p1.1.m1.4.4.2"><times id="S5.1.p1.1.m1.4.4.2.2.cmml" xref="S5.1.p1.1.m1.4.4.2.2"></times><apply id="S5.1.p1.1.m1.4.4.2.3.cmml" xref="S5.1.p1.1.m1.4.4.2.3"><ci id="S5.1.p1.1.m1.4.4.2.3.1.cmml" xref="S5.1.p1.1.m1.4.4.2.3.1">^</ci><ci id="S5.1.p1.1.m1.4.4.2.3.2.cmml" xref="S5.1.p1.1.m1.4.4.2.3.2">ℒ</ci></apply><interval closure="open" id="S5.1.p1.1.m1.4.4.2.1.2.cmml" xref="S5.1.p1.1.m1.4.4.2.1.1"><ci id="S5.1.p1.1.m1.2.2.cmml" xref="S5.1.p1.1.m1.2.2">𝑋</ci><apply id="S5.1.p1.1.m1.4.4.2.1.1.1.cmml" xref="S5.1.p1.1.m1.4.4.2.1.1.1"><csymbol cd="ambiguous" id="S5.1.p1.1.m1.4.4.2.1.1.1.1.cmml" xref="S5.1.p1.1.m1.4.4.2.1.1.1">superscript</csymbol><ci id="S5.1.p1.1.m1.4.4.2.1.1.1.2.cmml" xref="S5.1.p1.1.m1.4.4.2.1.1.1.2">𝐷</ci><ci id="S5.1.p1.1.m1.4.4.2.1.1.1.3.cmml" xref="S5.1.p1.1.m1.4.4.2.1.1.1.3">𝑋</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.1.m1.4c">\hat{\mathscr{L}}(A,D^{A})\setminus\hat{\mathscr{L}}(X,D^{X})</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.1.m1.4d">over^ start_ARG script_L end_ARG ( italic_A , italic_D start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) ∖ over^ start_ARG script_L end_ARG ( italic_X , italic_D start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT )</annotation></semantics></math> is stationary and that <math alttext="A\trianglerighteq X" class="ltx_Math" display="inline" id="S5.1.p1.2.m2.1"><semantics id="S5.1.p1.2.m2.1a"><mrow id="S5.1.p1.2.m2.1.1" xref="S5.1.p1.2.m2.1.1.cmml"><mi id="S5.1.p1.2.m2.1.1.2" xref="S5.1.p1.2.m2.1.1.2.cmml">A</mi><mo id="S5.1.p1.2.m2.1.1.1" xref="S5.1.p1.2.m2.1.1.1.cmml">⁢</mo><mi id="S5.1.p1.2.m2.1.1.3" mathvariant="normal" xref="S5.1.p1.2.m2.1.1.3.cmml">⊵</mi><mo id="S5.1.p1.2.m2.1.1.1a" xref="S5.1.p1.2.m2.1.1.1.cmml">⁢</mo><mi id="S5.1.p1.2.m2.1.1.4" xref="S5.1.p1.2.m2.1.1.4.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.1.p1.2.m2.1b"><apply id="S5.1.p1.2.m2.1.1.cmml" xref="S5.1.p1.2.m2.1.1"><times id="S5.1.p1.2.m2.1.1.1.cmml" xref="S5.1.p1.2.m2.1.1.1"></times><ci id="S5.1.p1.2.m2.1.1.2.cmml" xref="S5.1.p1.2.m2.1.1.2">𝐴</ci><ci id="S5.1.p1.2.m2.1.1.3.cmml" xref="S5.1.p1.2.m2.1.1.3">⊵</ci><ci id="S5.1.p1.2.m2.1.1.4.cmml" xref="S5.1.p1.2.m2.1.1.4">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.2.m2.1c">A\trianglerighteq X</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.2.m2.1d">italic_A ⊵ italic_X</annotation></semantics></math>. Let <math alttext="f:A\twoheadrightarrow X" class="ltx_Math" display="inline" id="S5.1.p1.3.m3.1"><semantics id="S5.1.p1.3.m3.1a"><mrow id="S5.1.p1.3.m3.1.1" xref="S5.1.p1.3.m3.1.1.cmml"><mi id="S5.1.p1.3.m3.1.1.2" xref="S5.1.p1.3.m3.1.1.2.cmml">f</mi><mo id="S5.1.p1.3.m3.1.1.1" lspace="0.278em" rspace="0.278em" xref="S5.1.p1.3.m3.1.1.1.cmml">:</mo><mrow id="S5.1.p1.3.m3.1.1.3" xref="S5.1.p1.3.m3.1.1.3.cmml"><mi id="S5.1.p1.3.m3.1.1.3.2" xref="S5.1.p1.3.m3.1.1.3.2.cmml">A</mi><mo id="S5.1.p1.3.m3.1.1.3.1" stretchy="false" xref="S5.1.p1.3.m3.1.1.3.1.cmml">↠</mo><mi id="S5.1.p1.3.m3.1.1.3.3" xref="S5.1.p1.3.m3.1.1.3.3.cmml">X</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.1.p1.3.m3.1b"><apply id="S5.1.p1.3.m3.1.1.cmml" xref="S5.1.p1.3.m3.1.1"><ci id="S5.1.p1.3.m3.1.1.1.cmml" xref="S5.1.p1.3.m3.1.1.1">:</ci><ci id="S5.1.p1.3.m3.1.1.2.cmml" xref="S5.1.p1.3.m3.1.1.2">𝑓</ci><apply id="S5.1.p1.3.m3.1.1.3.cmml" xref="S5.1.p1.3.m3.1.1.3"><ci id="S5.1.p1.3.m3.1.1.3.1.cmml" xref="S5.1.p1.3.m3.1.1.3.1">↠</ci><ci id="S5.1.p1.3.m3.1.1.3.2.cmml" xref="S5.1.p1.3.m3.1.1.3.2">𝐴</ci><ci id="S5.1.p1.3.m3.1.1.3.3.cmml" xref="S5.1.p1.3.m3.1.1.3.3">𝑋</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.3.m3.1c">f:A\twoheadrightarrow X</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.3.m3.1d">italic_f : italic_A ↠ italic_X</annotation></semantics></math> be an epimorphism and fix <math alttext="g:X\to A" class="ltx_Math" display="inline" id="S5.1.p1.4.m4.1"><semantics id="S5.1.p1.4.m4.1a"><mrow id="S5.1.p1.4.m4.1.1" xref="S5.1.p1.4.m4.1.1.cmml"><mi id="S5.1.p1.4.m4.1.1.2" xref="S5.1.p1.4.m4.1.1.2.cmml">g</mi><mo id="S5.1.p1.4.m4.1.1.1" lspace="0.278em" rspace="0.278em" xref="S5.1.p1.4.m4.1.1.1.cmml">:</mo><mrow id="S5.1.p1.4.m4.1.1.3" xref="S5.1.p1.4.m4.1.1.3.cmml"><mi id="S5.1.p1.4.m4.1.1.3.2" xref="S5.1.p1.4.m4.1.1.3.2.cmml">X</mi><mo id="S5.1.p1.4.m4.1.1.3.1" stretchy="false" xref="S5.1.p1.4.m4.1.1.3.1.cmml">→</mo><mi id="S5.1.p1.4.m4.1.1.3.3" xref="S5.1.p1.4.m4.1.1.3.3.cmml">A</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.1.p1.4.m4.1b"><apply id="S5.1.p1.4.m4.1.1.cmml" xref="S5.1.p1.4.m4.1.1"><ci id="S5.1.p1.4.m4.1.1.1.cmml" xref="S5.1.p1.4.m4.1.1.1">:</ci><ci id="S5.1.p1.4.m4.1.1.2.cmml" xref="S5.1.p1.4.m4.1.1.2">𝑔</ci><apply id="S5.1.p1.4.m4.1.1.3.cmml" xref="S5.1.p1.4.m4.1.1.3"><ci id="S5.1.p1.4.m4.1.1.3.1.cmml" xref="S5.1.p1.4.m4.1.1.3.1">→</ci><ci id="S5.1.p1.4.m4.1.1.3.2.cmml" xref="S5.1.p1.4.m4.1.1.3.2">𝑋</ci><ci id="S5.1.p1.4.m4.1.1.3.3.cmml" xref="S5.1.p1.4.m4.1.1.3.3">𝐴</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.4.m4.1c">g:X\to A</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.4.m4.1d">italic_g : italic_X → italic_A</annotation></semantics></math> some inverse of <math alttext="f" class="ltx_Math" display="inline" id="S5.1.p1.5.m5.1"><semantics id="S5.1.p1.5.m5.1a"><mi id="S5.1.p1.5.m5.1.1" xref="S5.1.p1.5.m5.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S5.1.p1.5.m5.1b"><ci id="S5.1.p1.5.m5.1.1.cmml" xref="S5.1.p1.5.m5.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.5.m5.1c">f</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.5.m5.1d">italic_f</annotation></semantics></math>. We may assume that <math alttext="A" class="ltx_Math" display="inline" id="S5.1.p1.6.m6.1"><semantics id="S5.1.p1.6.m6.1a"><mi id="S5.1.p1.6.m6.1.1" xref="S5.1.p1.6.m6.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S5.1.p1.6.m6.1b"><ci id="S5.1.p1.6.m6.1.1.cmml" xref="S5.1.p1.6.m6.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.6.m6.1c">A</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.6.m6.1d">italic_A</annotation></semantics></math> and <math alttext="X" class="ltx_Math" display="inline" id="S5.1.p1.7.m7.1"><semantics id="S5.1.p1.7.m7.1a"><mi id="S5.1.p1.7.m7.1.1" xref="S5.1.p1.7.m7.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.1.p1.7.m7.1b"><ci id="S5.1.p1.7.m7.1.1.cmml" xref="S5.1.p1.7.m7.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.7.m7.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.7.m7.1d">italic_X</annotation></semantics></math> are <math alttext="\omega_{1}" class="ltx_Math" display="inline" id="S5.1.p1.8.m8.1"><semantics id="S5.1.p1.8.m8.1a"><msub id="S5.1.p1.8.m8.1.1" xref="S5.1.p1.8.m8.1.1.cmml"><mi id="S5.1.p1.8.m8.1.1.2" xref="S5.1.p1.8.m8.1.1.2.cmml">ω</mi><mn id="S5.1.p1.8.m8.1.1.3" xref="S5.1.p1.8.m8.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S5.1.p1.8.m8.1b"><apply id="S5.1.p1.8.m8.1.1.cmml" xref="S5.1.p1.8.m8.1.1"><csymbol cd="ambiguous" id="S5.1.p1.8.m8.1.1.1.cmml" xref="S5.1.p1.8.m8.1.1">subscript</csymbol><ci id="S5.1.p1.8.m8.1.1.2.cmml" xref="S5.1.p1.8.m8.1.1.2">𝜔</ci><cn id="S5.1.p1.8.m8.1.1.3.cmml" type="integer" xref="S5.1.p1.8.m8.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.8.m8.1c">\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.8.m8.1d">italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> as sets. Now let <math alttext="E\subseteq\omega_{1}" class="ltx_Math" display="inline" id="S5.1.p1.9.m9.1"><semantics id="S5.1.p1.9.m9.1a"><mrow id="S5.1.p1.9.m9.1.1" xref="S5.1.p1.9.m9.1.1.cmml"><mi id="S5.1.p1.9.m9.1.1.2" xref="S5.1.p1.9.m9.1.1.2.cmml">E</mi><mo id="S5.1.p1.9.m9.1.1.1" xref="S5.1.p1.9.m9.1.1.1.cmml">⊆</mo><msub id="S5.1.p1.9.m9.1.1.3" xref="S5.1.p1.9.m9.1.1.3.cmml"><mi id="S5.1.p1.9.m9.1.1.3.2" xref="S5.1.p1.9.m9.1.1.3.2.cmml">ω</mi><mn id="S5.1.p1.9.m9.1.1.3.3" xref="S5.1.p1.9.m9.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.1.p1.9.m9.1b"><apply id="S5.1.p1.9.m9.1.1.cmml" xref="S5.1.p1.9.m9.1.1"><subset id="S5.1.p1.9.m9.1.1.1.cmml" xref="S5.1.p1.9.m9.1.1.1"></subset><ci id="S5.1.p1.9.m9.1.1.2.cmml" xref="S5.1.p1.9.m9.1.1.2">𝐸</ci><apply id="S5.1.p1.9.m9.1.1.3.cmml" xref="S5.1.p1.9.m9.1.1.3"><csymbol cd="ambiguous" id="S5.1.p1.9.m9.1.1.3.1.cmml" xref="S5.1.p1.9.m9.1.1.3">subscript</csymbol><ci id="S5.1.p1.9.m9.1.1.3.2.cmml" xref="S5.1.p1.9.m9.1.1.3.2">𝜔</ci><cn id="S5.1.p1.9.m9.1.1.3.3.cmml" type="integer" xref="S5.1.p1.9.m9.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.9.m9.1c">E\subseteq\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.9.m9.1d">italic_E ⊆ italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> be a club of limit ordinals closed under <math alttext="f" class="ltx_Math" display="inline" id="S5.1.p1.10.m10.1"><semantics id="S5.1.p1.10.m10.1a"><mi id="S5.1.p1.10.m10.1.1" xref="S5.1.p1.10.m10.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S5.1.p1.10.m10.1b"><ci id="S5.1.p1.10.m10.1.1.cmml" xref="S5.1.p1.10.m10.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.10.m10.1c">f</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.10.m10.1d">italic_f</annotation></semantics></math> and <math alttext="g" class="ltx_Math" display="inline" id="S5.1.p1.11.m11.1"><semantics id="S5.1.p1.11.m11.1a"><mi id="S5.1.p1.11.m11.1.1" xref="S5.1.p1.11.m11.1.1.cmml">g</mi><annotation-xml encoding="MathML-Content" id="S5.1.p1.11.m11.1b"><ci id="S5.1.p1.11.m11.1.1.cmml" xref="S5.1.p1.11.m11.1.1">𝑔</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.11.m11.1c">g</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.11.m11.1d">italic_g</annotation></semantics></math>, and such that if <math alttext="\nu\in E" class="ltx_Math" display="inline" id="S5.1.p1.12.m12.1"><semantics id="S5.1.p1.12.m12.1a"><mrow id="S5.1.p1.12.m12.1.1" xref="S5.1.p1.12.m12.1.1.cmml"><mi id="S5.1.p1.12.m12.1.1.2" xref="S5.1.p1.12.m12.1.1.2.cmml">ν</mi><mo id="S5.1.p1.12.m12.1.1.1" xref="S5.1.p1.12.m12.1.1.1.cmml">∈</mo><mi id="S5.1.p1.12.m12.1.1.3" xref="S5.1.p1.12.m12.1.1.3.cmml">E</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.1.p1.12.m12.1b"><apply id="S5.1.p1.12.m12.1.1.cmml" xref="S5.1.p1.12.m12.1.1"><in id="S5.1.p1.12.m12.1.1.1.cmml" xref="S5.1.p1.12.m12.1.1.1"></in><ci id="S5.1.p1.12.m12.1.1.2.cmml" xref="S5.1.p1.12.m12.1.1.2">𝜈</ci><ci id="S5.1.p1.12.m12.1.1.3.cmml" xref="S5.1.p1.12.m12.1.1.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.12.m12.1c">\nu\in E</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.12.m12.1d">italic_ν ∈ italic_E</annotation></semantics></math>, then <math alttext="\nu" class="ltx_Math" display="inline" id="S5.1.p1.13.m13.1"><semantics id="S5.1.p1.13.m13.1a"><mi id="S5.1.p1.13.m13.1.1" xref="S5.1.p1.13.m13.1.1.cmml">ν</mi><annotation-xml encoding="MathML-Content" id="S5.1.p1.13.m13.1b"><ci id="S5.1.p1.13.m13.1.1.cmml" xref="S5.1.p1.13.m13.1.1">𝜈</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.13.m13.1c">\nu</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.13.m13.1d">italic_ν</annotation></semantics></math> approximates all its members in the sense of both <math alttext="<_{A}" class="ltx_Math" display="inline" id="S5.1.p1.14.m14.1"><semantics id="S5.1.p1.14.m14.1a"><msub id="S5.1.p1.14.m14.1.1" xref="S5.1.p1.14.m14.1.1.cmml"><mo id="S5.1.p1.14.m14.1.1.2" xref="S5.1.p1.14.m14.1.1.2.cmml"><</mo><mi id="S5.1.p1.14.m14.1.1.3" xref="S5.1.p1.14.m14.1.1.3.cmml">A</mi></msub><annotation-xml encoding="MathML-Content" id="S5.1.p1.14.m14.1b"><apply id="S5.1.p1.14.m14.1.1.cmml" xref="S5.1.p1.14.m14.1.1"><csymbol cd="ambiguous" id="S5.1.p1.14.m14.1.1.1.cmml" xref="S5.1.p1.14.m14.1.1">subscript</csymbol><lt id="S5.1.p1.14.m14.1.1.2.cmml" xref="S5.1.p1.14.m14.1.1.2"></lt><ci id="S5.1.p1.14.m14.1.1.3.cmml" xref="S5.1.p1.14.m14.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.14.m14.1c"><_{A}</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.14.m14.1d">< start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="<_{X}" class="ltx_Math" display="inline" id="S5.1.p1.15.m15.1"><semantics id="S5.1.p1.15.m15.1a"><msub id="S5.1.p1.15.m15.1.1" xref="S5.1.p1.15.m15.1.1.cmml"><mo id="S5.1.p1.15.m15.1.1.2" xref="S5.1.p1.15.m15.1.1.2.cmml"><</mo><mi id="S5.1.p1.15.m15.1.1.3" xref="S5.1.p1.15.m15.1.1.3.cmml">X</mi></msub><annotation-xml encoding="MathML-Content" id="S5.1.p1.15.m15.1b"><apply id="S5.1.p1.15.m15.1.1.cmml" xref="S5.1.p1.15.m15.1.1"><csymbol cd="ambiguous" id="S5.1.p1.15.m15.1.1.1.cmml" xref="S5.1.p1.15.m15.1.1">subscript</csymbol><lt id="S5.1.p1.15.m15.1.1.2.cmml" xref="S5.1.p1.15.m15.1.1.2"></lt><ci id="S5.1.p1.15.m15.1.1.3.cmml" xref="S5.1.p1.15.m15.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.15.m15.1c"><_{X}</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.15.m15.1d">< start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT</annotation></semantics></math>. This is possible since <math alttext="A" class="ltx_Math" display="inline" id="S5.1.p1.16.m16.1"><semantics id="S5.1.p1.16.m16.1a"><mi id="S5.1.p1.16.m16.1.1" xref="S5.1.p1.16.m16.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S5.1.p1.16.m16.1b"><ci id="S5.1.p1.16.m16.1.1.cmml" xref="S5.1.p1.16.m16.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.16.m16.1c">A</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.16.m16.1d">italic_A</annotation></semantics></math> and <math alttext="X" class="ltx_Math" display="inline" id="S5.1.p1.17.m17.1"><semantics id="S5.1.p1.17.m17.1a"><mi id="S5.1.p1.17.m17.1.1" xref="S5.1.p1.17.m17.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.1.p1.17.m17.1b"><ci id="S5.1.p1.17.m17.1.1.cmml" xref="S5.1.p1.17.m17.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.17.m17.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.17.m17.1d">italic_X</annotation></semantics></math> are Aronszajn lines and thus short. Moreover, by <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S4.Thmtheorem1" title="Lemma 4.1. ‣ 4. Aronszajn line decompositions ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">4.1</span></a> we may also assume that for <math alttext="\nu\in E" class="ltx_Math" display="inline" id="S5.1.p1.18.m18.1"><semantics id="S5.1.p1.18.m18.1a"><mrow id="S5.1.p1.18.m18.1.1" xref="S5.1.p1.18.m18.1.1.cmml"><mi id="S5.1.p1.18.m18.1.1.2" xref="S5.1.p1.18.m18.1.1.2.cmml">ν</mi><mo id="S5.1.p1.18.m18.1.1.1" xref="S5.1.p1.18.m18.1.1.1.cmml">∈</mo><mi id="S5.1.p1.18.m18.1.1.3" xref="S5.1.p1.18.m18.1.1.3.cmml">E</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.1.p1.18.m18.1b"><apply id="S5.1.p1.18.m18.1.1.cmml" xref="S5.1.p1.18.m18.1.1"><in id="S5.1.p1.18.m18.1.1.1.cmml" xref="S5.1.p1.18.m18.1.1.1"></in><ci id="S5.1.p1.18.m18.1.1.2.cmml" xref="S5.1.p1.18.m18.1.1.2">𝜈</ci><ci id="S5.1.p1.18.m18.1.1.3.cmml" xref="S5.1.p1.18.m18.1.1.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.18.m18.1c">\nu\in E</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.18.m18.1d">italic_ν ∈ italic_E</annotation></semantics></math>, <math alttext="D_{\nu}=\nu=E_{\nu}" class="ltx_Math" display="inline" id="S5.1.p1.19.m19.1"><semantics id="S5.1.p1.19.m19.1a"><mrow id="S5.1.p1.19.m19.1.1" xref="S5.1.p1.19.m19.1.1.cmml"><msub id="S5.1.p1.19.m19.1.1.2" xref="S5.1.p1.19.m19.1.1.2.cmml"><mi id="S5.1.p1.19.m19.1.1.2.2" xref="S5.1.p1.19.m19.1.1.2.2.cmml">D</mi><mi id="S5.1.p1.19.m19.1.1.2.3" xref="S5.1.p1.19.m19.1.1.2.3.cmml">ν</mi></msub><mo id="S5.1.p1.19.m19.1.1.3" xref="S5.1.p1.19.m19.1.1.3.cmml">=</mo><mi id="S5.1.p1.19.m19.1.1.4" xref="S5.1.p1.19.m19.1.1.4.cmml">ν</mi><mo id="S5.1.p1.19.m19.1.1.5" xref="S5.1.p1.19.m19.1.1.5.cmml">=</mo><msub id="S5.1.p1.19.m19.1.1.6" xref="S5.1.p1.19.m19.1.1.6.cmml"><mi id="S5.1.p1.19.m19.1.1.6.2" xref="S5.1.p1.19.m19.1.1.6.2.cmml">E</mi><mi id="S5.1.p1.19.m19.1.1.6.3" xref="S5.1.p1.19.m19.1.1.6.3.cmml">ν</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.1.p1.19.m19.1b"><apply id="S5.1.p1.19.m19.1.1.cmml" xref="S5.1.p1.19.m19.1.1"><and id="S5.1.p1.19.m19.1.1a.cmml" xref="S5.1.p1.19.m19.1.1"></and><apply id="S5.1.p1.19.m19.1.1b.cmml" xref="S5.1.p1.19.m19.1.1"><eq id="S5.1.p1.19.m19.1.1.3.cmml" xref="S5.1.p1.19.m19.1.1.3"></eq><apply id="S5.1.p1.19.m19.1.1.2.cmml" xref="S5.1.p1.19.m19.1.1.2"><csymbol cd="ambiguous" id="S5.1.p1.19.m19.1.1.2.1.cmml" xref="S5.1.p1.19.m19.1.1.2">subscript</csymbol><ci id="S5.1.p1.19.m19.1.1.2.2.cmml" xref="S5.1.p1.19.m19.1.1.2.2">𝐷</ci><ci id="S5.1.p1.19.m19.1.1.2.3.cmml" xref="S5.1.p1.19.m19.1.1.2.3">𝜈</ci></apply><ci id="S5.1.p1.19.m19.1.1.4.cmml" xref="S5.1.p1.19.m19.1.1.4">𝜈</ci></apply><apply id="S5.1.p1.19.m19.1.1c.cmml" xref="S5.1.p1.19.m19.1.1"><eq id="S5.1.p1.19.m19.1.1.5.cmml" xref="S5.1.p1.19.m19.1.1.5"></eq><share href="https://arxiv.org/html/2503.13728v1#S5.1.p1.19.m19.1.1.4.cmml" id="S5.1.p1.19.m19.1.1d.cmml" xref="S5.1.p1.19.m19.1.1"></share><apply id="S5.1.p1.19.m19.1.1.6.cmml" xref="S5.1.p1.19.m19.1.1.6"><csymbol cd="ambiguous" id="S5.1.p1.19.m19.1.1.6.1.cmml" xref="S5.1.p1.19.m19.1.1.6">subscript</csymbol><ci id="S5.1.p1.19.m19.1.1.6.2.cmml" xref="S5.1.p1.19.m19.1.1.6.2">𝐸</ci><ci id="S5.1.p1.19.m19.1.1.6.3.cmml" xref="S5.1.p1.19.m19.1.1.6.3">𝜈</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.19.m19.1c">D_{\nu}=\nu=E_{\nu}</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.19.m19.1d">italic_D start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT = italic_ν = italic_E start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S5.2.p2"> <p class="ltx_p" id="S5.2.p2.13">By assumption there is <math alttext="\nu\in E\cap(\hat{\mathscr{L}}(A,D^{A})\setminus\hat{\mathscr{L}}(X,D^{X}))" class="ltx_Math" display="inline" id="S5.2.p2.1.m1.3"><semantics id="S5.2.p2.1.m1.3a"><mrow id="S5.2.p2.1.m1.3.3" xref="S5.2.p2.1.m1.3.3.cmml"><mi id="S5.2.p2.1.m1.3.3.3" xref="S5.2.p2.1.m1.3.3.3.cmml">ν</mi><mo id="S5.2.p2.1.m1.3.3.2" xref="S5.2.p2.1.m1.3.3.2.cmml">∈</mo><mrow id="S5.2.p2.1.m1.3.3.1" xref="S5.2.p2.1.m1.3.3.1.cmml"><mi id="S5.2.p2.1.m1.3.3.1.3" xref="S5.2.p2.1.m1.3.3.1.3.cmml">E</mi><mo id="S5.2.p2.1.m1.3.3.1.2" xref="S5.2.p2.1.m1.3.3.1.2.cmml">∩</mo><mrow id="S5.2.p2.1.m1.3.3.1.1.1" xref="S5.2.p2.1.m1.3.3.1.1.1.1.cmml"><mo id="S5.2.p2.1.m1.3.3.1.1.1.2" stretchy="false" xref="S5.2.p2.1.m1.3.3.1.1.1.1.cmml">(</mo><mrow id="S5.2.p2.1.m1.3.3.1.1.1.1" xref="S5.2.p2.1.m1.3.3.1.1.1.1.cmml"><mrow id="S5.2.p2.1.m1.3.3.1.1.1.1.1" xref="S5.2.p2.1.m1.3.3.1.1.1.1.1.cmml"><mover accent="true" id="S5.2.p2.1.m1.3.3.1.1.1.1.1.3" xref="S5.2.p2.1.m1.3.3.1.1.1.1.1.3.cmml"><mi class="ltx_font_mathscript" id="S5.2.p2.1.m1.3.3.1.1.1.1.1.3.2" xref="S5.2.p2.1.m1.3.3.1.1.1.1.1.3.2.cmml">ℒ</mi><mo id="S5.2.p2.1.m1.3.3.1.1.1.1.1.3.1" xref="S5.2.p2.1.m1.3.3.1.1.1.1.1.3.1.cmml">^</mo></mover><mo id="S5.2.p2.1.m1.3.3.1.1.1.1.1.2" xref="S5.2.p2.1.m1.3.3.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S5.2.p2.1.m1.3.3.1.1.1.1.1.1.1" xref="S5.2.p2.1.m1.3.3.1.1.1.1.1.1.2.cmml"><mo id="S5.2.p2.1.m1.3.3.1.1.1.1.1.1.1.2" stretchy="false" xref="S5.2.p2.1.m1.3.3.1.1.1.1.1.1.2.cmml">(</mo><mi id="S5.2.p2.1.m1.1.1" xref="S5.2.p2.1.m1.1.1.cmml">A</mi><mo id="S5.2.p2.1.m1.3.3.1.1.1.1.1.1.1.3" xref="S5.2.p2.1.m1.3.3.1.1.1.1.1.1.2.cmml">,</mo><msup id="S5.2.p2.1.m1.3.3.1.1.1.1.1.1.1.1" xref="S5.2.p2.1.m1.3.3.1.1.1.1.1.1.1.1.cmml"><mi id="S5.2.p2.1.m1.3.3.1.1.1.1.1.1.1.1.2" xref="S5.2.p2.1.m1.3.3.1.1.1.1.1.1.1.1.2.cmml">D</mi><mi id="S5.2.p2.1.m1.3.3.1.1.1.1.1.1.1.1.3" xref="S5.2.p2.1.m1.3.3.1.1.1.1.1.1.1.1.3.cmml">A</mi></msup><mo id="S5.2.p2.1.m1.3.3.1.1.1.1.1.1.1.4" stretchy="false" xref="S5.2.p2.1.m1.3.3.1.1.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S5.2.p2.1.m1.3.3.1.1.1.1.3" xref="S5.2.p2.1.m1.3.3.1.1.1.1.3.cmml">∖</mo><mrow id="S5.2.p2.1.m1.3.3.1.1.1.1.2" xref="S5.2.p2.1.m1.3.3.1.1.1.1.2.cmml"><mover accent="true" id="S5.2.p2.1.m1.3.3.1.1.1.1.2.3" xref="S5.2.p2.1.m1.3.3.1.1.1.1.2.3.cmml"><mi class="ltx_font_mathscript" id="S5.2.p2.1.m1.3.3.1.1.1.1.2.3.2" xref="S5.2.p2.1.m1.3.3.1.1.1.1.2.3.2.cmml">ℒ</mi><mo id="S5.2.p2.1.m1.3.3.1.1.1.1.2.3.1" xref="S5.2.p2.1.m1.3.3.1.1.1.1.2.3.1.cmml">^</mo></mover><mo id="S5.2.p2.1.m1.3.3.1.1.1.1.2.2" xref="S5.2.p2.1.m1.3.3.1.1.1.1.2.2.cmml">⁢</mo><mrow id="S5.2.p2.1.m1.3.3.1.1.1.1.2.1.1" xref="S5.2.p2.1.m1.3.3.1.1.1.1.2.1.2.cmml"><mo id="S5.2.p2.1.m1.3.3.1.1.1.1.2.1.1.2" stretchy="false" xref="S5.2.p2.1.m1.3.3.1.1.1.1.2.1.2.cmml">(</mo><mi id="S5.2.p2.1.m1.2.2" xref="S5.2.p2.1.m1.2.2.cmml">X</mi><mo id="S5.2.p2.1.m1.3.3.1.1.1.1.2.1.1.3" xref="S5.2.p2.1.m1.3.3.1.1.1.1.2.1.2.cmml">,</mo><msup id="S5.2.p2.1.m1.3.3.1.1.1.1.2.1.1.1" xref="S5.2.p2.1.m1.3.3.1.1.1.1.2.1.1.1.cmml"><mi id="S5.2.p2.1.m1.3.3.1.1.1.1.2.1.1.1.2" xref="S5.2.p2.1.m1.3.3.1.1.1.1.2.1.1.1.2.cmml">D</mi><mi id="S5.2.p2.1.m1.3.3.1.1.1.1.2.1.1.1.3" xref="S5.2.p2.1.m1.3.3.1.1.1.1.2.1.1.1.3.cmml">X</mi></msup><mo id="S5.2.p2.1.m1.3.3.1.1.1.1.2.1.1.4" stretchy="false" xref="S5.2.p2.1.m1.3.3.1.1.1.1.2.1.2.cmml">)</mo></mrow></mrow></mrow><mo id="S5.2.p2.1.m1.3.3.1.1.1.3" stretchy="false" xref="S5.2.p2.1.m1.3.3.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.2.p2.1.m1.3b"><apply id="S5.2.p2.1.m1.3.3.cmml" xref="S5.2.p2.1.m1.3.3"><in id="S5.2.p2.1.m1.3.3.2.cmml" xref="S5.2.p2.1.m1.3.3.2"></in><ci id="S5.2.p2.1.m1.3.3.3.cmml" xref="S5.2.p2.1.m1.3.3.3">𝜈</ci><apply id="S5.2.p2.1.m1.3.3.1.cmml" xref="S5.2.p2.1.m1.3.3.1"><intersect id="S5.2.p2.1.m1.3.3.1.2.cmml" xref="S5.2.p2.1.m1.3.3.1.2"></intersect><ci id="S5.2.p2.1.m1.3.3.1.3.cmml" xref="S5.2.p2.1.m1.3.3.1.3">𝐸</ci><apply id="S5.2.p2.1.m1.3.3.1.1.1.1.cmml" xref="S5.2.p2.1.m1.3.3.1.1.1"><setdiff id="S5.2.p2.1.m1.3.3.1.1.1.1.3.cmml" xref="S5.2.p2.1.m1.3.3.1.1.1.1.3"></setdiff><apply id="S5.2.p2.1.m1.3.3.1.1.1.1.1.cmml" xref="S5.2.p2.1.m1.3.3.1.1.1.1.1"><times id="S5.2.p2.1.m1.3.3.1.1.1.1.1.2.cmml" xref="S5.2.p2.1.m1.3.3.1.1.1.1.1.2"></times><apply id="S5.2.p2.1.m1.3.3.1.1.1.1.1.3.cmml" xref="S5.2.p2.1.m1.3.3.1.1.1.1.1.3"><ci id="S5.2.p2.1.m1.3.3.1.1.1.1.1.3.1.cmml" xref="S5.2.p2.1.m1.3.3.1.1.1.1.1.3.1">^</ci><ci id="S5.2.p2.1.m1.3.3.1.1.1.1.1.3.2.cmml" xref="S5.2.p2.1.m1.3.3.1.1.1.1.1.3.2">ℒ</ci></apply><interval closure="open" id="S5.2.p2.1.m1.3.3.1.1.1.1.1.1.2.cmml" xref="S5.2.p2.1.m1.3.3.1.1.1.1.1.1.1"><ci id="S5.2.p2.1.m1.1.1.cmml" xref="S5.2.p2.1.m1.1.1">𝐴</ci><apply id="S5.2.p2.1.m1.3.3.1.1.1.1.1.1.1.1.cmml" xref="S5.2.p2.1.m1.3.3.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.2.p2.1.m1.3.3.1.1.1.1.1.1.1.1.1.cmml" xref="S5.2.p2.1.m1.3.3.1.1.1.1.1.1.1.1">superscript</csymbol><ci id="S5.2.p2.1.m1.3.3.1.1.1.1.1.1.1.1.2.cmml" xref="S5.2.p2.1.m1.3.3.1.1.1.1.1.1.1.1.2">𝐷</ci><ci id="S5.2.p2.1.m1.3.3.1.1.1.1.1.1.1.1.3.cmml" xref="S5.2.p2.1.m1.3.3.1.1.1.1.1.1.1.1.3">𝐴</ci></apply></interval></apply><apply id="S5.2.p2.1.m1.3.3.1.1.1.1.2.cmml" xref="S5.2.p2.1.m1.3.3.1.1.1.1.2"><times id="S5.2.p2.1.m1.3.3.1.1.1.1.2.2.cmml" xref="S5.2.p2.1.m1.3.3.1.1.1.1.2.2"></times><apply id="S5.2.p2.1.m1.3.3.1.1.1.1.2.3.cmml" xref="S5.2.p2.1.m1.3.3.1.1.1.1.2.3"><ci id="S5.2.p2.1.m1.3.3.1.1.1.1.2.3.1.cmml" xref="S5.2.p2.1.m1.3.3.1.1.1.1.2.3.1">^</ci><ci id="S5.2.p2.1.m1.3.3.1.1.1.1.2.3.2.cmml" xref="S5.2.p2.1.m1.3.3.1.1.1.1.2.3.2">ℒ</ci></apply><interval closure="open" id="S5.2.p2.1.m1.3.3.1.1.1.1.2.1.2.cmml" xref="S5.2.p2.1.m1.3.3.1.1.1.1.2.1.1"><ci id="S5.2.p2.1.m1.2.2.cmml" xref="S5.2.p2.1.m1.2.2">𝑋</ci><apply id="S5.2.p2.1.m1.3.3.1.1.1.1.2.1.1.1.cmml" xref="S5.2.p2.1.m1.3.3.1.1.1.1.2.1.1.1"><csymbol cd="ambiguous" id="S5.2.p2.1.m1.3.3.1.1.1.1.2.1.1.1.1.cmml" xref="S5.2.p2.1.m1.3.3.1.1.1.1.2.1.1.1">superscript</csymbol><ci id="S5.2.p2.1.m1.3.3.1.1.1.1.2.1.1.1.2.cmml" xref="S5.2.p2.1.m1.3.3.1.1.1.1.2.1.1.1.2">𝐷</ci><ci id="S5.2.p2.1.m1.3.3.1.1.1.1.2.1.1.1.3.cmml" xref="S5.2.p2.1.m1.3.3.1.1.1.1.2.1.1.1.3">𝑋</ci></apply></interval></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.1.m1.3c">\nu\in E\cap(\hat{\mathscr{L}}(A,D^{A})\setminus\hat{\mathscr{L}}(X,D^{X}))</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.1.m1.3d">italic_ν ∈ italic_E ∩ ( over^ start_ARG script_L end_ARG ( italic_A , italic_D start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) ∖ over^ start_ARG script_L end_ARG ( italic_X , italic_D start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT ) )</annotation></semantics></math>. This implies that some complementary interval of <math alttext="X\setminus\nu" class="ltx_Math" display="inline" id="S5.2.p2.2.m2.1"><semantics id="S5.2.p2.2.m2.1a"><mrow id="S5.2.p2.2.m2.1.1" xref="S5.2.p2.2.m2.1.1.cmml"><mi id="S5.2.p2.2.m2.1.1.2" xref="S5.2.p2.2.m2.1.1.2.cmml">X</mi><mo id="S5.2.p2.2.m2.1.1.1" xref="S5.2.p2.2.m2.1.1.1.cmml">∖</mo><mi id="S5.2.p2.2.m2.1.1.3" xref="S5.2.p2.2.m2.1.1.3.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.2.p2.2.m2.1b"><apply id="S5.2.p2.2.m2.1.1.cmml" xref="S5.2.p2.2.m2.1.1"><setdiff id="S5.2.p2.2.m2.1.1.1.cmml" xref="S5.2.p2.2.m2.1.1.1"></setdiff><ci id="S5.2.p2.2.m2.1.1.2.cmml" xref="S5.2.p2.2.m2.1.1.2">𝑋</ci><ci id="S5.2.p2.2.m2.1.1.3.cmml" xref="S5.2.p2.2.m2.1.1.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.2.m2.1c">X\setminus\nu</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.2.m2.1d">italic_X ∖ italic_ν</annotation></semantics></math> has no left endpoint. Fix such an interval <math alttext="I" class="ltx_Math" display="inline" id="S5.2.p2.3.m3.1"><semantics id="S5.2.p2.3.m3.1a"><mi id="S5.2.p2.3.m3.1.1" xref="S5.2.p2.3.m3.1.1.cmml">I</mi><annotation-xml encoding="MathML-Content" id="S5.2.p2.3.m3.1b"><ci id="S5.2.p2.3.m3.1.1.cmml" xref="S5.2.p2.3.m3.1.1">𝐼</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.3.m3.1c">I</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.3.m3.1d">italic_I</annotation></semantics></math> and let <math alttext="x\in I" class="ltx_Math" display="inline" id="S5.2.p2.4.m4.1"><semantics id="S5.2.p2.4.m4.1a"><mrow id="S5.2.p2.4.m4.1.1" xref="S5.2.p2.4.m4.1.1.cmml"><mi id="S5.2.p2.4.m4.1.1.2" xref="S5.2.p2.4.m4.1.1.2.cmml">x</mi><mo id="S5.2.p2.4.m4.1.1.1" xref="S5.2.p2.4.m4.1.1.1.cmml">∈</mo><mi id="S5.2.p2.4.m4.1.1.3" xref="S5.2.p2.4.m4.1.1.3.cmml">I</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.2.p2.4.m4.1b"><apply id="S5.2.p2.4.m4.1.1.cmml" xref="S5.2.p2.4.m4.1.1"><in id="S5.2.p2.4.m4.1.1.1.cmml" xref="S5.2.p2.4.m4.1.1.1"></in><ci id="S5.2.p2.4.m4.1.1.2.cmml" xref="S5.2.p2.4.m4.1.1.2">𝑥</ci><ci id="S5.2.p2.4.m4.1.1.3.cmml" xref="S5.2.p2.4.m4.1.1.3">𝐼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.4.m4.1c">x\in I</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.4.m4.1d">italic_x ∈ italic_I</annotation></semantics></math>. Then clearly <math alttext="x\geq\nu" class="ltx_Math" display="inline" id="S5.2.p2.5.m5.1"><semantics id="S5.2.p2.5.m5.1a"><mrow id="S5.2.p2.5.m5.1.1" xref="S5.2.p2.5.m5.1.1.cmml"><mi id="S5.2.p2.5.m5.1.1.2" xref="S5.2.p2.5.m5.1.1.2.cmml">x</mi><mo id="S5.2.p2.5.m5.1.1.1" xref="S5.2.p2.5.m5.1.1.1.cmml">≥</mo><mi id="S5.2.p2.5.m5.1.1.3" xref="S5.2.p2.5.m5.1.1.3.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.2.p2.5.m5.1b"><apply id="S5.2.p2.5.m5.1.1.cmml" xref="S5.2.p2.5.m5.1.1"><geq id="S5.2.p2.5.m5.1.1.1.cmml" xref="S5.2.p2.5.m5.1.1.1"></geq><ci id="S5.2.p2.5.m5.1.1.2.cmml" xref="S5.2.p2.5.m5.1.1.2">𝑥</ci><ci id="S5.2.p2.5.m5.1.1.3.cmml" xref="S5.2.p2.5.m5.1.1.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.5.m5.1c">x\geq\nu</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.5.m5.1d">italic_x ≥ italic_ν</annotation></semantics></math>, and thus <math alttext="g(x)\geq\nu" class="ltx_Math" display="inline" id="S5.2.p2.6.m6.1"><semantics id="S5.2.p2.6.m6.1a"><mrow id="S5.2.p2.6.m6.1.2" xref="S5.2.p2.6.m6.1.2.cmml"><mrow id="S5.2.p2.6.m6.1.2.2" xref="S5.2.p2.6.m6.1.2.2.cmml"><mi id="S5.2.p2.6.m6.1.2.2.2" xref="S5.2.p2.6.m6.1.2.2.2.cmml">g</mi><mo id="S5.2.p2.6.m6.1.2.2.1" xref="S5.2.p2.6.m6.1.2.2.1.cmml">⁢</mo><mrow id="S5.2.p2.6.m6.1.2.2.3.2" xref="S5.2.p2.6.m6.1.2.2.cmml"><mo id="S5.2.p2.6.m6.1.2.2.3.2.1" stretchy="false" xref="S5.2.p2.6.m6.1.2.2.cmml">(</mo><mi id="S5.2.p2.6.m6.1.1" xref="S5.2.p2.6.m6.1.1.cmml">x</mi><mo id="S5.2.p2.6.m6.1.2.2.3.2.2" stretchy="false" xref="S5.2.p2.6.m6.1.2.2.cmml">)</mo></mrow></mrow><mo id="S5.2.p2.6.m6.1.2.1" xref="S5.2.p2.6.m6.1.2.1.cmml">≥</mo><mi id="S5.2.p2.6.m6.1.2.3" xref="S5.2.p2.6.m6.1.2.3.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.2.p2.6.m6.1b"><apply id="S5.2.p2.6.m6.1.2.cmml" xref="S5.2.p2.6.m6.1.2"><geq id="S5.2.p2.6.m6.1.2.1.cmml" xref="S5.2.p2.6.m6.1.2.1"></geq><apply id="S5.2.p2.6.m6.1.2.2.cmml" xref="S5.2.p2.6.m6.1.2.2"><times id="S5.2.p2.6.m6.1.2.2.1.cmml" xref="S5.2.p2.6.m6.1.2.2.1"></times><ci id="S5.2.p2.6.m6.1.2.2.2.cmml" xref="S5.2.p2.6.m6.1.2.2.2">𝑔</ci><ci id="S5.2.p2.6.m6.1.1.cmml" xref="S5.2.p2.6.m6.1.1">𝑥</ci></apply><ci id="S5.2.p2.6.m6.1.2.3.cmml" xref="S5.2.p2.6.m6.1.2.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.6.m6.1c">g(x)\geq\nu</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.6.m6.1d">italic_g ( italic_x ) ≥ italic_ν</annotation></semantics></math> (otherwise <math alttext="x=f(g(x))<\nu" class="ltx_Math" display="inline" id="S5.2.p2.7.m7.2"><semantics id="S5.2.p2.7.m7.2a"><mrow id="S5.2.p2.7.m7.2.2" xref="S5.2.p2.7.m7.2.2.cmml"><mi id="S5.2.p2.7.m7.2.2.3" xref="S5.2.p2.7.m7.2.2.3.cmml">x</mi><mo id="S5.2.p2.7.m7.2.2.4" xref="S5.2.p2.7.m7.2.2.4.cmml">=</mo><mrow id="S5.2.p2.7.m7.2.2.1" xref="S5.2.p2.7.m7.2.2.1.cmml"><mi id="S5.2.p2.7.m7.2.2.1.3" xref="S5.2.p2.7.m7.2.2.1.3.cmml">f</mi><mo id="S5.2.p2.7.m7.2.2.1.2" xref="S5.2.p2.7.m7.2.2.1.2.cmml">⁢</mo><mrow id="S5.2.p2.7.m7.2.2.1.1.1" xref="S5.2.p2.7.m7.2.2.1.1.1.1.cmml"><mo id="S5.2.p2.7.m7.2.2.1.1.1.2" stretchy="false" xref="S5.2.p2.7.m7.2.2.1.1.1.1.cmml">(</mo><mrow id="S5.2.p2.7.m7.2.2.1.1.1.1" xref="S5.2.p2.7.m7.2.2.1.1.1.1.cmml"><mi id="S5.2.p2.7.m7.2.2.1.1.1.1.2" xref="S5.2.p2.7.m7.2.2.1.1.1.1.2.cmml">g</mi><mo id="S5.2.p2.7.m7.2.2.1.1.1.1.1" xref="S5.2.p2.7.m7.2.2.1.1.1.1.1.cmml">⁢</mo><mrow id="S5.2.p2.7.m7.2.2.1.1.1.1.3.2" xref="S5.2.p2.7.m7.2.2.1.1.1.1.cmml"><mo id="S5.2.p2.7.m7.2.2.1.1.1.1.3.2.1" stretchy="false" xref="S5.2.p2.7.m7.2.2.1.1.1.1.cmml">(</mo><mi id="S5.2.p2.7.m7.1.1" xref="S5.2.p2.7.m7.1.1.cmml">x</mi><mo id="S5.2.p2.7.m7.2.2.1.1.1.1.3.2.2" stretchy="false" xref="S5.2.p2.7.m7.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.2.p2.7.m7.2.2.1.1.1.3" stretchy="false" xref="S5.2.p2.7.m7.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.2.p2.7.m7.2.2.5" xref="S5.2.p2.7.m7.2.2.5.cmml"><</mo><mi id="S5.2.p2.7.m7.2.2.6" xref="S5.2.p2.7.m7.2.2.6.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.2.p2.7.m7.2b"><apply id="S5.2.p2.7.m7.2.2.cmml" xref="S5.2.p2.7.m7.2.2"><and id="S5.2.p2.7.m7.2.2a.cmml" xref="S5.2.p2.7.m7.2.2"></and><apply id="S5.2.p2.7.m7.2.2b.cmml" xref="S5.2.p2.7.m7.2.2"><eq id="S5.2.p2.7.m7.2.2.4.cmml" xref="S5.2.p2.7.m7.2.2.4"></eq><ci id="S5.2.p2.7.m7.2.2.3.cmml" xref="S5.2.p2.7.m7.2.2.3">𝑥</ci><apply id="S5.2.p2.7.m7.2.2.1.cmml" xref="S5.2.p2.7.m7.2.2.1"><times id="S5.2.p2.7.m7.2.2.1.2.cmml" xref="S5.2.p2.7.m7.2.2.1.2"></times><ci id="S5.2.p2.7.m7.2.2.1.3.cmml" xref="S5.2.p2.7.m7.2.2.1.3">𝑓</ci><apply id="S5.2.p2.7.m7.2.2.1.1.1.1.cmml" xref="S5.2.p2.7.m7.2.2.1.1.1"><times id="S5.2.p2.7.m7.2.2.1.1.1.1.1.cmml" xref="S5.2.p2.7.m7.2.2.1.1.1.1.1"></times><ci id="S5.2.p2.7.m7.2.2.1.1.1.1.2.cmml" xref="S5.2.p2.7.m7.2.2.1.1.1.1.2">𝑔</ci><ci id="S5.2.p2.7.m7.1.1.cmml" xref="S5.2.p2.7.m7.1.1">𝑥</ci></apply></apply></apply><apply id="S5.2.p2.7.m7.2.2c.cmml" xref="S5.2.p2.7.m7.2.2"><lt id="S5.2.p2.7.m7.2.2.5.cmml" xref="S5.2.p2.7.m7.2.2.5"></lt><share href="https://arxiv.org/html/2503.13728v1#S5.2.p2.7.m7.2.2.1.cmml" id="S5.2.p2.7.m7.2.2d.cmml" xref="S5.2.p2.7.m7.2.2"></share><ci id="S5.2.p2.7.m7.2.2.6.cmml" xref="S5.2.p2.7.m7.2.2.6">𝜈</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.7.m7.2c">x=f(g(x))<\nu</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.7.m7.2d">italic_x = italic_f ( italic_g ( italic_x ) ) < italic_ν</annotation></semantics></math>). Let <math alttext="J" class="ltx_Math" display="inline" id="S5.2.p2.8.m8.1"><semantics id="S5.2.p2.8.m8.1a"><mi id="S5.2.p2.8.m8.1.1" xref="S5.2.p2.8.m8.1.1.cmml">J</mi><annotation-xml encoding="MathML-Content" id="S5.2.p2.8.m8.1b"><ci id="S5.2.p2.8.m8.1.1.cmml" xref="S5.2.p2.8.m8.1.1">𝐽</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.8.m8.1c">J</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.8.m8.1d">italic_J</annotation></semantics></math> be the complementary interval of <math alttext="A\setminus\nu" class="ltx_Math" display="inline" id="S5.2.p2.9.m9.1"><semantics id="S5.2.p2.9.m9.1a"><mrow id="S5.2.p2.9.m9.1.1" xref="S5.2.p2.9.m9.1.1.cmml"><mi id="S5.2.p2.9.m9.1.1.2" xref="S5.2.p2.9.m9.1.1.2.cmml">A</mi><mo id="S5.2.p2.9.m9.1.1.1" xref="S5.2.p2.9.m9.1.1.1.cmml">∖</mo><mi id="S5.2.p2.9.m9.1.1.3" xref="S5.2.p2.9.m9.1.1.3.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.2.p2.9.m9.1b"><apply id="S5.2.p2.9.m9.1.1.cmml" xref="S5.2.p2.9.m9.1.1"><setdiff id="S5.2.p2.9.m9.1.1.1.cmml" xref="S5.2.p2.9.m9.1.1.1"></setdiff><ci id="S5.2.p2.9.m9.1.1.2.cmml" xref="S5.2.p2.9.m9.1.1.2">𝐴</ci><ci id="S5.2.p2.9.m9.1.1.3.cmml" xref="S5.2.p2.9.m9.1.1.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.9.m9.1c">A\setminus\nu</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.9.m9.1d">italic_A ∖ italic_ν</annotation></semantics></math> in which <math alttext="g(x)" class="ltx_Math" display="inline" id="S5.2.p2.10.m10.1"><semantics id="S5.2.p2.10.m10.1a"><mrow id="S5.2.p2.10.m10.1.2" xref="S5.2.p2.10.m10.1.2.cmml"><mi id="S5.2.p2.10.m10.1.2.2" xref="S5.2.p2.10.m10.1.2.2.cmml">g</mi><mo id="S5.2.p2.10.m10.1.2.1" xref="S5.2.p2.10.m10.1.2.1.cmml">⁢</mo><mrow id="S5.2.p2.10.m10.1.2.3.2" xref="S5.2.p2.10.m10.1.2.cmml"><mo id="S5.2.p2.10.m10.1.2.3.2.1" stretchy="false" xref="S5.2.p2.10.m10.1.2.cmml">(</mo><mi id="S5.2.p2.10.m10.1.1" xref="S5.2.p2.10.m10.1.1.cmml">x</mi><mo id="S5.2.p2.10.m10.1.2.3.2.2" stretchy="false" xref="S5.2.p2.10.m10.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.2.p2.10.m10.1b"><apply id="S5.2.p2.10.m10.1.2.cmml" xref="S5.2.p2.10.m10.1.2"><times id="S5.2.p2.10.m10.1.2.1.cmml" xref="S5.2.p2.10.m10.1.2.1"></times><ci id="S5.2.p2.10.m10.1.2.2.cmml" xref="S5.2.p2.10.m10.1.2.2">𝑔</ci><ci id="S5.2.p2.10.m10.1.1.cmml" xref="S5.2.p2.10.m10.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.10.m10.1c">g(x)</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.10.m10.1d">italic_g ( italic_x )</annotation></semantics></math> is. Since <math alttext="\nu\in\hat{\mathscr{L}}(A,D^{A})" class="ltx_Math" display="inline" id="S5.2.p2.11.m11.2"><semantics id="S5.2.p2.11.m11.2a"><mrow id="S5.2.p2.11.m11.2.2" xref="S5.2.p2.11.m11.2.2.cmml"><mi id="S5.2.p2.11.m11.2.2.3" xref="S5.2.p2.11.m11.2.2.3.cmml">ν</mi><mo id="S5.2.p2.11.m11.2.2.2" xref="S5.2.p2.11.m11.2.2.2.cmml">∈</mo><mrow id="S5.2.p2.11.m11.2.2.1" xref="S5.2.p2.11.m11.2.2.1.cmml"><mover accent="true" id="S5.2.p2.11.m11.2.2.1.3" xref="S5.2.p2.11.m11.2.2.1.3.cmml"><mi class="ltx_font_mathscript" id="S5.2.p2.11.m11.2.2.1.3.2" xref="S5.2.p2.11.m11.2.2.1.3.2.cmml">ℒ</mi><mo id="S5.2.p2.11.m11.2.2.1.3.1" xref="S5.2.p2.11.m11.2.2.1.3.1.cmml">^</mo></mover><mo id="S5.2.p2.11.m11.2.2.1.2" xref="S5.2.p2.11.m11.2.2.1.2.cmml">⁢</mo><mrow id="S5.2.p2.11.m11.2.2.1.1.1" xref="S5.2.p2.11.m11.2.2.1.1.2.cmml"><mo id="S5.2.p2.11.m11.2.2.1.1.1.2" stretchy="false" xref="S5.2.p2.11.m11.2.2.1.1.2.cmml">(</mo><mi id="S5.2.p2.11.m11.1.1" xref="S5.2.p2.11.m11.1.1.cmml">A</mi><mo id="S5.2.p2.11.m11.2.2.1.1.1.3" xref="S5.2.p2.11.m11.2.2.1.1.2.cmml">,</mo><msup id="S5.2.p2.11.m11.2.2.1.1.1.1" xref="S5.2.p2.11.m11.2.2.1.1.1.1.cmml"><mi id="S5.2.p2.11.m11.2.2.1.1.1.1.2" xref="S5.2.p2.11.m11.2.2.1.1.1.1.2.cmml">D</mi><mi id="S5.2.p2.11.m11.2.2.1.1.1.1.3" xref="S5.2.p2.11.m11.2.2.1.1.1.1.3.cmml">A</mi></msup><mo id="S5.2.p2.11.m11.2.2.1.1.1.4" stretchy="false" xref="S5.2.p2.11.m11.2.2.1.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.2.p2.11.m11.2b"><apply id="S5.2.p2.11.m11.2.2.cmml" xref="S5.2.p2.11.m11.2.2"><in id="S5.2.p2.11.m11.2.2.2.cmml" xref="S5.2.p2.11.m11.2.2.2"></in><ci id="S5.2.p2.11.m11.2.2.3.cmml" xref="S5.2.p2.11.m11.2.2.3">𝜈</ci><apply id="S5.2.p2.11.m11.2.2.1.cmml" xref="S5.2.p2.11.m11.2.2.1"><times id="S5.2.p2.11.m11.2.2.1.2.cmml" xref="S5.2.p2.11.m11.2.2.1.2"></times><apply id="S5.2.p2.11.m11.2.2.1.3.cmml" xref="S5.2.p2.11.m11.2.2.1.3"><ci id="S5.2.p2.11.m11.2.2.1.3.1.cmml" xref="S5.2.p2.11.m11.2.2.1.3.1">^</ci><ci id="S5.2.p2.11.m11.2.2.1.3.2.cmml" xref="S5.2.p2.11.m11.2.2.1.3.2">ℒ</ci></apply><interval closure="open" id="S5.2.p2.11.m11.2.2.1.1.2.cmml" xref="S5.2.p2.11.m11.2.2.1.1.1"><ci id="S5.2.p2.11.m11.1.1.cmml" xref="S5.2.p2.11.m11.1.1">𝐴</ci><apply id="S5.2.p2.11.m11.2.2.1.1.1.1.cmml" xref="S5.2.p2.11.m11.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S5.2.p2.11.m11.2.2.1.1.1.1.1.cmml" xref="S5.2.p2.11.m11.2.2.1.1.1.1">superscript</csymbol><ci id="S5.2.p2.11.m11.2.2.1.1.1.1.2.cmml" xref="S5.2.p2.11.m11.2.2.1.1.1.1.2">𝐷</ci><ci id="S5.2.p2.11.m11.2.2.1.1.1.1.3.cmml" xref="S5.2.p2.11.m11.2.2.1.1.1.1.3">𝐴</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.11.m11.2c">\nu\in\hat{\mathscr{L}}(A,D^{A})</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.11.m11.2d">italic_ν ∈ over^ start_ARG script_L end_ARG ( italic_A , italic_D start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT )</annotation></semantics></math>, <math alttext="J" class="ltx_Math" display="inline" id="S5.2.p2.12.m12.1"><semantics id="S5.2.p2.12.m12.1a"><mi id="S5.2.p2.12.m12.1.1" xref="S5.2.p2.12.m12.1.1.cmml">J</mi><annotation-xml encoding="MathML-Content" id="S5.2.p2.12.m12.1b"><ci id="S5.2.p2.12.m12.1.1.cmml" xref="S5.2.p2.12.m12.1.1">𝐽</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.12.m12.1c">J</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.12.m12.1d">italic_J</annotation></semantics></math> has a left endpoint, call it <math alttext="a" class="ltx_Math" display="inline" id="S5.2.p2.13.m13.1"><semantics id="S5.2.p2.13.m13.1a"><mi id="S5.2.p2.13.m13.1.1" xref="S5.2.p2.13.m13.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S5.2.p2.13.m13.1b"><ci id="S5.2.p2.13.m13.1.1.cmml" xref="S5.2.p2.13.m13.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.13.m13.1c">a</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.13.m13.1d">italic_a</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S5.3.p3"> <p class="ltx_p" id="S5.3.p3.15">We first claim that <math alttext="f(a)\in I" class="ltx_Math" display="inline" id="S5.3.p3.1.m1.1"><semantics id="S5.3.p3.1.m1.1a"><mrow id="S5.3.p3.1.m1.1.2" xref="S5.3.p3.1.m1.1.2.cmml"><mrow id="S5.3.p3.1.m1.1.2.2" xref="S5.3.p3.1.m1.1.2.2.cmml"><mi id="S5.3.p3.1.m1.1.2.2.2" xref="S5.3.p3.1.m1.1.2.2.2.cmml">f</mi><mo id="S5.3.p3.1.m1.1.2.2.1" xref="S5.3.p3.1.m1.1.2.2.1.cmml">⁢</mo><mrow id="S5.3.p3.1.m1.1.2.2.3.2" xref="S5.3.p3.1.m1.1.2.2.cmml"><mo id="S5.3.p3.1.m1.1.2.2.3.2.1" stretchy="false" xref="S5.3.p3.1.m1.1.2.2.cmml">(</mo><mi id="S5.3.p3.1.m1.1.1" xref="S5.3.p3.1.m1.1.1.cmml">a</mi><mo id="S5.3.p3.1.m1.1.2.2.3.2.2" stretchy="false" xref="S5.3.p3.1.m1.1.2.2.cmml">)</mo></mrow></mrow><mo id="S5.3.p3.1.m1.1.2.1" xref="S5.3.p3.1.m1.1.2.1.cmml">∈</mo><mi id="S5.3.p3.1.m1.1.2.3" xref="S5.3.p3.1.m1.1.2.3.cmml">I</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.3.p3.1.m1.1b"><apply id="S5.3.p3.1.m1.1.2.cmml" xref="S5.3.p3.1.m1.1.2"><in id="S5.3.p3.1.m1.1.2.1.cmml" xref="S5.3.p3.1.m1.1.2.1"></in><apply id="S5.3.p3.1.m1.1.2.2.cmml" xref="S5.3.p3.1.m1.1.2.2"><times id="S5.3.p3.1.m1.1.2.2.1.cmml" xref="S5.3.p3.1.m1.1.2.2.1"></times><ci id="S5.3.p3.1.m1.1.2.2.2.cmml" xref="S5.3.p3.1.m1.1.2.2.2">𝑓</ci><ci id="S5.3.p3.1.m1.1.1.cmml" xref="S5.3.p3.1.m1.1.1">𝑎</ci></apply><ci id="S5.3.p3.1.m1.1.2.3.cmml" xref="S5.3.p3.1.m1.1.2.3">𝐼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.3.p3.1.m1.1c">f(a)\in I</annotation><annotation encoding="application/x-llamapun" id="S5.3.p3.1.m1.1d">italic_f ( italic_a ) ∈ italic_I</annotation></semantics></math>. If not (including when <math alttext="f(a)<\nu" class="ltx_Math" display="inline" id="S5.3.p3.2.m2.1"><semantics id="S5.3.p3.2.m2.1a"><mrow id="S5.3.p3.2.m2.1.2" xref="S5.3.p3.2.m2.1.2.cmml"><mrow id="S5.3.p3.2.m2.1.2.2" xref="S5.3.p3.2.m2.1.2.2.cmml"><mi id="S5.3.p3.2.m2.1.2.2.2" xref="S5.3.p3.2.m2.1.2.2.2.cmml">f</mi><mo id="S5.3.p3.2.m2.1.2.2.1" xref="S5.3.p3.2.m2.1.2.2.1.cmml">⁢</mo><mrow id="S5.3.p3.2.m2.1.2.2.3.2" xref="S5.3.p3.2.m2.1.2.2.cmml"><mo id="S5.3.p3.2.m2.1.2.2.3.2.1" stretchy="false" xref="S5.3.p3.2.m2.1.2.2.cmml">(</mo><mi id="S5.3.p3.2.m2.1.1" xref="S5.3.p3.2.m2.1.1.cmml">a</mi><mo id="S5.3.p3.2.m2.1.2.2.3.2.2" stretchy="false" xref="S5.3.p3.2.m2.1.2.2.cmml">)</mo></mrow></mrow><mo id="S5.3.p3.2.m2.1.2.1" xref="S5.3.p3.2.m2.1.2.1.cmml"><</mo><mi id="S5.3.p3.2.m2.1.2.3" xref="S5.3.p3.2.m2.1.2.3.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.3.p3.2.m2.1b"><apply id="S5.3.p3.2.m2.1.2.cmml" xref="S5.3.p3.2.m2.1.2"><lt id="S5.3.p3.2.m2.1.2.1.cmml" xref="S5.3.p3.2.m2.1.2.1"></lt><apply id="S5.3.p3.2.m2.1.2.2.cmml" xref="S5.3.p3.2.m2.1.2.2"><times id="S5.3.p3.2.m2.1.2.2.1.cmml" xref="S5.3.p3.2.m2.1.2.2.1"></times><ci id="S5.3.p3.2.m2.1.2.2.2.cmml" xref="S5.3.p3.2.m2.1.2.2.2">𝑓</ci><ci id="S5.3.p3.2.m2.1.1.cmml" xref="S5.3.p3.2.m2.1.1">𝑎</ci></apply><ci id="S5.3.p3.2.m2.1.2.3.cmml" xref="S5.3.p3.2.m2.1.2.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.3.p3.2.m2.1c">f(a)<\nu</annotation><annotation encoding="application/x-llamapun" id="S5.3.p3.2.m2.1d">italic_f ( italic_a ) < italic_ν</annotation></semantics></math>), by definition there <math alttext="x^{\prime}" class="ltx_Math" display="inline" id="S5.3.p3.3.m3.1"><semantics id="S5.3.p3.3.m3.1a"><msup id="S5.3.p3.3.m3.1.1" xref="S5.3.p3.3.m3.1.1.cmml"><mi id="S5.3.p3.3.m3.1.1.2" xref="S5.3.p3.3.m3.1.1.2.cmml">x</mi><mo id="S5.3.p3.3.m3.1.1.3" xref="S5.3.p3.3.m3.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S5.3.p3.3.m3.1b"><apply id="S5.3.p3.3.m3.1.1.cmml" xref="S5.3.p3.3.m3.1.1"><csymbol cd="ambiguous" id="S5.3.p3.3.m3.1.1.1.cmml" xref="S5.3.p3.3.m3.1.1">superscript</csymbol><ci id="S5.3.p3.3.m3.1.1.2.cmml" xref="S5.3.p3.3.m3.1.1.2">𝑥</ci><ci id="S5.3.p3.3.m3.1.1.3.cmml" xref="S5.3.p3.3.m3.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.3.p3.3.m3.1c">x^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S5.3.p3.3.m3.1d">italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> such that <math alttext="f(a)\leq_{X}x^{\prime}<_{X}x" class="ltx_Math" display="inline" id="S5.3.p3.4.m4.1"><semantics id="S5.3.p3.4.m4.1a"><mrow id="S5.3.p3.4.m4.1.2" xref="S5.3.p3.4.m4.1.2.cmml"><mrow id="S5.3.p3.4.m4.1.2.2" xref="S5.3.p3.4.m4.1.2.2.cmml"><mi id="S5.3.p3.4.m4.1.2.2.2" xref="S5.3.p3.4.m4.1.2.2.2.cmml">f</mi><mo id="S5.3.p3.4.m4.1.2.2.1" xref="S5.3.p3.4.m4.1.2.2.1.cmml">⁢</mo><mrow id="S5.3.p3.4.m4.1.2.2.3.2" xref="S5.3.p3.4.m4.1.2.2.cmml"><mo id="S5.3.p3.4.m4.1.2.2.3.2.1" stretchy="false" xref="S5.3.p3.4.m4.1.2.2.cmml">(</mo><mi id="S5.3.p3.4.m4.1.1" xref="S5.3.p3.4.m4.1.1.cmml">a</mi><mo id="S5.3.p3.4.m4.1.2.2.3.2.2" stretchy="false" xref="S5.3.p3.4.m4.1.2.2.cmml">)</mo></mrow></mrow><msub id="S5.3.p3.4.m4.1.2.3" xref="S5.3.p3.4.m4.1.2.3.cmml"><mo id="S5.3.p3.4.m4.1.2.3.2" xref="S5.3.p3.4.m4.1.2.3.2.cmml">≤</mo><mi id="S5.3.p3.4.m4.1.2.3.3" xref="S5.3.p3.4.m4.1.2.3.3.cmml">X</mi></msub><msup id="S5.3.p3.4.m4.1.2.4" xref="S5.3.p3.4.m4.1.2.4.cmml"><mi id="S5.3.p3.4.m4.1.2.4.2" xref="S5.3.p3.4.m4.1.2.4.2.cmml">x</mi><mo id="S5.3.p3.4.m4.1.2.4.3" xref="S5.3.p3.4.m4.1.2.4.3.cmml">′</mo></msup><msub id="S5.3.p3.4.m4.1.2.5" xref="S5.3.p3.4.m4.1.2.5.cmml"><mo id="S5.3.p3.4.m4.1.2.5.2" xref="S5.3.p3.4.m4.1.2.5.2.cmml"><</mo><mi id="S5.3.p3.4.m4.1.2.5.3" xref="S5.3.p3.4.m4.1.2.5.3.cmml">X</mi></msub><mi id="S5.3.p3.4.m4.1.2.6" xref="S5.3.p3.4.m4.1.2.6.cmml">x</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.3.p3.4.m4.1b"><apply id="S5.3.p3.4.m4.1.2.cmml" xref="S5.3.p3.4.m4.1.2"><and id="S5.3.p3.4.m4.1.2a.cmml" xref="S5.3.p3.4.m4.1.2"></and><apply id="S5.3.p3.4.m4.1.2b.cmml" xref="S5.3.p3.4.m4.1.2"><apply id="S5.3.p3.4.m4.1.2.3.cmml" xref="S5.3.p3.4.m4.1.2.3"><csymbol cd="ambiguous" id="S5.3.p3.4.m4.1.2.3.1.cmml" xref="S5.3.p3.4.m4.1.2.3">subscript</csymbol><leq id="S5.3.p3.4.m4.1.2.3.2.cmml" xref="S5.3.p3.4.m4.1.2.3.2"></leq><ci id="S5.3.p3.4.m4.1.2.3.3.cmml" xref="S5.3.p3.4.m4.1.2.3.3">𝑋</ci></apply><apply id="S5.3.p3.4.m4.1.2.2.cmml" xref="S5.3.p3.4.m4.1.2.2"><times id="S5.3.p3.4.m4.1.2.2.1.cmml" xref="S5.3.p3.4.m4.1.2.2.1"></times><ci id="S5.3.p3.4.m4.1.2.2.2.cmml" xref="S5.3.p3.4.m4.1.2.2.2">𝑓</ci><ci id="S5.3.p3.4.m4.1.1.cmml" xref="S5.3.p3.4.m4.1.1">𝑎</ci></apply><apply id="S5.3.p3.4.m4.1.2.4.cmml" xref="S5.3.p3.4.m4.1.2.4"><csymbol cd="ambiguous" id="S5.3.p3.4.m4.1.2.4.1.cmml" xref="S5.3.p3.4.m4.1.2.4">superscript</csymbol><ci id="S5.3.p3.4.m4.1.2.4.2.cmml" xref="S5.3.p3.4.m4.1.2.4.2">𝑥</ci><ci id="S5.3.p3.4.m4.1.2.4.3.cmml" xref="S5.3.p3.4.m4.1.2.4.3">′</ci></apply></apply><apply id="S5.3.p3.4.m4.1.2c.cmml" xref="S5.3.p3.4.m4.1.2"><apply id="S5.3.p3.4.m4.1.2.5.cmml" xref="S5.3.p3.4.m4.1.2.5"><csymbol cd="ambiguous" id="S5.3.p3.4.m4.1.2.5.1.cmml" xref="S5.3.p3.4.m4.1.2.5">subscript</csymbol><lt id="S5.3.p3.4.m4.1.2.5.2.cmml" xref="S5.3.p3.4.m4.1.2.5.2"></lt><ci id="S5.3.p3.4.m4.1.2.5.3.cmml" xref="S5.3.p3.4.m4.1.2.5.3">𝑋</ci></apply><share href="https://arxiv.org/html/2503.13728v1#S5.3.p3.4.m4.1.2.4.cmml" id="S5.3.p3.4.m4.1.2d.cmml" xref="S5.3.p3.4.m4.1.2"></share><ci id="S5.3.p3.4.m4.1.2.6.cmml" xref="S5.3.p3.4.m4.1.2.6">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.3.p3.4.m4.1c">f(a)\leq_{X}x^{\prime}<_{X}x</annotation><annotation encoding="application/x-llamapun" id="S5.3.p3.4.m4.1d">italic_f ( italic_a ) ≤ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT < start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT italic_x</annotation></semantics></math> with <math alttext="x^{\prime}<\nu" class="ltx_Math" display="inline" id="S5.3.p3.5.m5.1"><semantics id="S5.3.p3.5.m5.1a"><mrow id="S5.3.p3.5.m5.1.1" xref="S5.3.p3.5.m5.1.1.cmml"><msup id="S5.3.p3.5.m5.1.1.2" xref="S5.3.p3.5.m5.1.1.2.cmml"><mi id="S5.3.p3.5.m5.1.1.2.2" xref="S5.3.p3.5.m5.1.1.2.2.cmml">x</mi><mo id="S5.3.p3.5.m5.1.1.2.3" xref="S5.3.p3.5.m5.1.1.2.3.cmml">′</mo></msup><mo id="S5.3.p3.5.m5.1.1.1" xref="S5.3.p3.5.m5.1.1.1.cmml"><</mo><mi id="S5.3.p3.5.m5.1.1.3" xref="S5.3.p3.5.m5.1.1.3.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.3.p3.5.m5.1b"><apply id="S5.3.p3.5.m5.1.1.cmml" xref="S5.3.p3.5.m5.1.1"><lt id="S5.3.p3.5.m5.1.1.1.cmml" xref="S5.3.p3.5.m5.1.1.1"></lt><apply id="S5.3.p3.5.m5.1.1.2.cmml" xref="S5.3.p3.5.m5.1.1.2"><csymbol cd="ambiguous" id="S5.3.p3.5.m5.1.1.2.1.cmml" xref="S5.3.p3.5.m5.1.1.2">superscript</csymbol><ci id="S5.3.p3.5.m5.1.1.2.2.cmml" xref="S5.3.p3.5.m5.1.1.2.2">𝑥</ci><ci id="S5.3.p3.5.m5.1.1.2.3.cmml" xref="S5.3.p3.5.m5.1.1.2.3">′</ci></apply><ci id="S5.3.p3.5.m5.1.1.3.cmml" xref="S5.3.p3.5.m5.1.1.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.3.p3.5.m5.1c">x^{\prime}<\nu</annotation><annotation encoding="application/x-llamapun" id="S5.3.p3.5.m5.1d">italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT < italic_ν</annotation></semantics></math>. Since <math alttext="\nu" class="ltx_Math" display="inline" id="S5.3.p3.6.m6.1"><semantics id="S5.3.p3.6.m6.1a"><mi id="S5.3.p3.6.m6.1.1" xref="S5.3.p3.6.m6.1.1.cmml">ν</mi><annotation-xml encoding="MathML-Content" id="S5.3.p3.6.m6.1b"><ci id="S5.3.p3.6.m6.1.1.cmml" xref="S5.3.p3.6.m6.1.1">𝜈</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.3.p3.6.m6.1c">\nu</annotation><annotation encoding="application/x-llamapun" id="S5.3.p3.6.m6.1d">italic_ν</annotation></semantics></math> approximates <math alttext="x^{\prime}" class="ltx_Math" display="inline" id="S5.3.p3.7.m7.1"><semantics id="S5.3.p3.7.m7.1a"><msup id="S5.3.p3.7.m7.1.1" xref="S5.3.p3.7.m7.1.1.cmml"><mi id="S5.3.p3.7.m7.1.1.2" xref="S5.3.p3.7.m7.1.1.2.cmml">x</mi><mo id="S5.3.p3.7.m7.1.1.3" xref="S5.3.p3.7.m7.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S5.3.p3.7.m7.1b"><apply id="S5.3.p3.7.m7.1.1.cmml" xref="S5.3.p3.7.m7.1.1"><csymbol cd="ambiguous" id="S5.3.p3.7.m7.1.1.1.cmml" xref="S5.3.p3.7.m7.1.1">superscript</csymbol><ci id="S5.3.p3.7.m7.1.1.2.cmml" xref="S5.3.p3.7.m7.1.1.2">𝑥</ci><ci id="S5.3.p3.7.m7.1.1.3.cmml" xref="S5.3.p3.7.m7.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.3.p3.7.m7.1c">x^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S5.3.p3.7.m7.1d">italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="x\notin\nu" class="ltx_Math" display="inline" id="S5.3.p3.8.m8.1"><semantics id="S5.3.p3.8.m8.1a"><mrow id="S5.3.p3.8.m8.1.1" xref="S5.3.p3.8.m8.1.1.cmml"><mi id="S5.3.p3.8.m8.1.1.2" xref="S5.3.p3.8.m8.1.1.2.cmml">x</mi><mo id="S5.3.p3.8.m8.1.1.1" xref="S5.3.p3.8.m8.1.1.1.cmml">∉</mo><mi id="S5.3.p3.8.m8.1.1.3" xref="S5.3.p3.8.m8.1.1.3.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.3.p3.8.m8.1b"><apply id="S5.3.p3.8.m8.1.1.cmml" xref="S5.3.p3.8.m8.1.1"><notin id="S5.3.p3.8.m8.1.1.1.cmml" xref="S5.3.p3.8.m8.1.1.1"></notin><ci id="S5.3.p3.8.m8.1.1.2.cmml" xref="S5.3.p3.8.m8.1.1.2">𝑥</ci><ci id="S5.3.p3.8.m8.1.1.3.cmml" xref="S5.3.p3.8.m8.1.1.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.3.p3.8.m8.1c">x\notin\nu</annotation><annotation encoding="application/x-llamapun" id="S5.3.p3.8.m8.1d">italic_x ∉ italic_ν</annotation></semantics></math>, we find <math alttext="x^{\prime\prime}<\nu" class="ltx_Math" display="inline" id="S5.3.p3.9.m9.1"><semantics id="S5.3.p3.9.m9.1a"><mrow id="S5.3.p3.9.m9.1.1" xref="S5.3.p3.9.m9.1.1.cmml"><msup id="S5.3.p3.9.m9.1.1.2" xref="S5.3.p3.9.m9.1.1.2.cmml"><mi id="S5.3.p3.9.m9.1.1.2.2" xref="S5.3.p3.9.m9.1.1.2.2.cmml">x</mi><mo id="S5.3.p3.9.m9.1.1.2.3" xref="S5.3.p3.9.m9.1.1.2.3.cmml">′′</mo></msup><mo id="S5.3.p3.9.m9.1.1.1" xref="S5.3.p3.9.m9.1.1.1.cmml"><</mo><mi id="S5.3.p3.9.m9.1.1.3" xref="S5.3.p3.9.m9.1.1.3.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.3.p3.9.m9.1b"><apply id="S5.3.p3.9.m9.1.1.cmml" xref="S5.3.p3.9.m9.1.1"><lt id="S5.3.p3.9.m9.1.1.1.cmml" xref="S5.3.p3.9.m9.1.1.1"></lt><apply id="S5.3.p3.9.m9.1.1.2.cmml" xref="S5.3.p3.9.m9.1.1.2"><csymbol cd="ambiguous" id="S5.3.p3.9.m9.1.1.2.1.cmml" xref="S5.3.p3.9.m9.1.1.2">superscript</csymbol><ci id="S5.3.p3.9.m9.1.1.2.2.cmml" xref="S5.3.p3.9.m9.1.1.2.2">𝑥</ci><ci id="S5.3.p3.9.m9.1.1.2.3.cmml" xref="S5.3.p3.9.m9.1.1.2.3">′′</ci></apply><ci id="S5.3.p3.9.m9.1.1.3.cmml" xref="S5.3.p3.9.m9.1.1.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.3.p3.9.m9.1c">x^{\prime\prime}<\nu</annotation><annotation encoding="application/x-llamapun" id="S5.3.p3.9.m9.1d">italic_x start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT < italic_ν</annotation></semantics></math> such that <math alttext="x^{\prime}<_{X}x^{\prime\prime}<_{X}x" class="ltx_Math" display="inline" id="S5.3.p3.10.m10.1"><semantics id="S5.3.p3.10.m10.1a"><mrow id="S5.3.p3.10.m10.1.1" xref="S5.3.p3.10.m10.1.1.cmml"><msup id="S5.3.p3.10.m10.1.1.2" xref="S5.3.p3.10.m10.1.1.2.cmml"><mi id="S5.3.p3.10.m10.1.1.2.2" xref="S5.3.p3.10.m10.1.1.2.2.cmml">x</mi><mo id="S5.3.p3.10.m10.1.1.2.3" xref="S5.3.p3.10.m10.1.1.2.3.cmml">′</mo></msup><msub id="S5.3.p3.10.m10.1.1.3" xref="S5.3.p3.10.m10.1.1.3.cmml"><mo id="S5.3.p3.10.m10.1.1.3.2" xref="S5.3.p3.10.m10.1.1.3.2.cmml"><</mo><mi id="S5.3.p3.10.m10.1.1.3.3" xref="S5.3.p3.10.m10.1.1.3.3.cmml">X</mi></msub><msup id="S5.3.p3.10.m10.1.1.4" xref="S5.3.p3.10.m10.1.1.4.cmml"><mi id="S5.3.p3.10.m10.1.1.4.2" xref="S5.3.p3.10.m10.1.1.4.2.cmml">x</mi><mo id="S5.3.p3.10.m10.1.1.4.3" xref="S5.3.p3.10.m10.1.1.4.3.cmml">′′</mo></msup><msub id="S5.3.p3.10.m10.1.1.5" xref="S5.3.p3.10.m10.1.1.5.cmml"><mo id="S5.3.p3.10.m10.1.1.5.2" xref="S5.3.p3.10.m10.1.1.5.2.cmml"><</mo><mi id="S5.3.p3.10.m10.1.1.5.3" xref="S5.3.p3.10.m10.1.1.5.3.cmml">X</mi></msub><mi id="S5.3.p3.10.m10.1.1.6" xref="S5.3.p3.10.m10.1.1.6.cmml">x</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.3.p3.10.m10.1b"><apply id="S5.3.p3.10.m10.1.1.cmml" xref="S5.3.p3.10.m10.1.1"><and id="S5.3.p3.10.m10.1.1a.cmml" xref="S5.3.p3.10.m10.1.1"></and><apply id="S5.3.p3.10.m10.1.1b.cmml" xref="S5.3.p3.10.m10.1.1"><apply id="S5.3.p3.10.m10.1.1.3.cmml" xref="S5.3.p3.10.m10.1.1.3"><csymbol cd="ambiguous" id="S5.3.p3.10.m10.1.1.3.1.cmml" xref="S5.3.p3.10.m10.1.1.3">subscript</csymbol><lt id="S5.3.p3.10.m10.1.1.3.2.cmml" xref="S5.3.p3.10.m10.1.1.3.2"></lt><ci id="S5.3.p3.10.m10.1.1.3.3.cmml" xref="S5.3.p3.10.m10.1.1.3.3">𝑋</ci></apply><apply id="S5.3.p3.10.m10.1.1.2.cmml" xref="S5.3.p3.10.m10.1.1.2"><csymbol cd="ambiguous" id="S5.3.p3.10.m10.1.1.2.1.cmml" xref="S5.3.p3.10.m10.1.1.2">superscript</csymbol><ci id="S5.3.p3.10.m10.1.1.2.2.cmml" xref="S5.3.p3.10.m10.1.1.2.2">𝑥</ci><ci id="S5.3.p3.10.m10.1.1.2.3.cmml" xref="S5.3.p3.10.m10.1.1.2.3">′</ci></apply><apply id="S5.3.p3.10.m10.1.1.4.cmml" xref="S5.3.p3.10.m10.1.1.4"><csymbol cd="ambiguous" id="S5.3.p3.10.m10.1.1.4.1.cmml" xref="S5.3.p3.10.m10.1.1.4">superscript</csymbol><ci id="S5.3.p3.10.m10.1.1.4.2.cmml" xref="S5.3.p3.10.m10.1.1.4.2">𝑥</ci><ci id="S5.3.p3.10.m10.1.1.4.3.cmml" xref="S5.3.p3.10.m10.1.1.4.3">′′</ci></apply></apply><apply id="S5.3.p3.10.m10.1.1c.cmml" xref="S5.3.p3.10.m10.1.1"><apply id="S5.3.p3.10.m10.1.1.5.cmml" xref="S5.3.p3.10.m10.1.1.5"><csymbol cd="ambiguous" id="S5.3.p3.10.m10.1.1.5.1.cmml" xref="S5.3.p3.10.m10.1.1.5">subscript</csymbol><lt id="S5.3.p3.10.m10.1.1.5.2.cmml" xref="S5.3.p3.10.m10.1.1.5.2"></lt><ci id="S5.3.p3.10.m10.1.1.5.3.cmml" xref="S5.3.p3.10.m10.1.1.5.3">𝑋</ci></apply><share href="https://arxiv.org/html/2503.13728v1#S5.3.p3.10.m10.1.1.4.cmml" id="S5.3.p3.10.m10.1.1d.cmml" xref="S5.3.p3.10.m10.1.1"></share><ci id="S5.3.p3.10.m10.1.1.6.cmml" xref="S5.3.p3.10.m10.1.1.6">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.3.p3.10.m10.1c">x^{\prime}<_{X}x^{\prime\prime}<_{X}x</annotation><annotation encoding="application/x-llamapun" id="S5.3.p3.10.m10.1d">italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT < start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT italic_x start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT < start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT italic_x</annotation></semantics></math>. But then <math alttext="a<_{A}g(x^{\prime\prime})<_{A}g(x)" class="ltx_Math" display="inline" id="S5.3.p3.11.m11.2"><semantics id="S5.3.p3.11.m11.2a"><mrow id="S5.3.p3.11.m11.2.2" xref="S5.3.p3.11.m11.2.2.cmml"><mi id="S5.3.p3.11.m11.2.2.3" xref="S5.3.p3.11.m11.2.2.3.cmml">a</mi><msub id="S5.3.p3.11.m11.2.2.4" xref="S5.3.p3.11.m11.2.2.4.cmml"><mo id="S5.3.p3.11.m11.2.2.4.2" xref="S5.3.p3.11.m11.2.2.4.2.cmml"><</mo><mi id="S5.3.p3.11.m11.2.2.4.3" xref="S5.3.p3.11.m11.2.2.4.3.cmml">A</mi></msub><mrow id="S5.3.p3.11.m11.2.2.1" xref="S5.3.p3.11.m11.2.2.1.cmml"><mi id="S5.3.p3.11.m11.2.2.1.3" xref="S5.3.p3.11.m11.2.2.1.3.cmml">g</mi><mo id="S5.3.p3.11.m11.2.2.1.2" xref="S5.3.p3.11.m11.2.2.1.2.cmml">⁢</mo><mrow id="S5.3.p3.11.m11.2.2.1.1.1" xref="S5.3.p3.11.m11.2.2.1.1.1.1.cmml"><mo id="S5.3.p3.11.m11.2.2.1.1.1.2" stretchy="false" xref="S5.3.p3.11.m11.2.2.1.1.1.1.cmml">(</mo><msup id="S5.3.p3.11.m11.2.2.1.1.1.1" xref="S5.3.p3.11.m11.2.2.1.1.1.1.cmml"><mi id="S5.3.p3.11.m11.2.2.1.1.1.1.2" xref="S5.3.p3.11.m11.2.2.1.1.1.1.2.cmml">x</mi><mo id="S5.3.p3.11.m11.2.2.1.1.1.1.3" xref="S5.3.p3.11.m11.2.2.1.1.1.1.3.cmml">′′</mo></msup><mo id="S5.3.p3.11.m11.2.2.1.1.1.3" stretchy="false" xref="S5.3.p3.11.m11.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><msub id="S5.3.p3.11.m11.2.2.5" xref="S5.3.p3.11.m11.2.2.5.cmml"><mo id="S5.3.p3.11.m11.2.2.5.2" xref="S5.3.p3.11.m11.2.2.5.2.cmml"><</mo><mi id="S5.3.p3.11.m11.2.2.5.3" xref="S5.3.p3.11.m11.2.2.5.3.cmml">A</mi></msub><mrow id="S5.3.p3.11.m11.2.2.6" xref="S5.3.p3.11.m11.2.2.6.cmml"><mi id="S5.3.p3.11.m11.2.2.6.2" xref="S5.3.p3.11.m11.2.2.6.2.cmml">g</mi><mo id="S5.3.p3.11.m11.2.2.6.1" xref="S5.3.p3.11.m11.2.2.6.1.cmml">⁢</mo><mrow id="S5.3.p3.11.m11.2.2.6.3.2" xref="S5.3.p3.11.m11.2.2.6.cmml"><mo id="S5.3.p3.11.m11.2.2.6.3.2.1" stretchy="false" xref="S5.3.p3.11.m11.2.2.6.cmml">(</mo><mi id="S5.3.p3.11.m11.1.1" xref="S5.3.p3.11.m11.1.1.cmml">x</mi><mo id="S5.3.p3.11.m11.2.2.6.3.2.2" stretchy="false" xref="S5.3.p3.11.m11.2.2.6.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.3.p3.11.m11.2b"><apply id="S5.3.p3.11.m11.2.2.cmml" xref="S5.3.p3.11.m11.2.2"><and id="S5.3.p3.11.m11.2.2a.cmml" xref="S5.3.p3.11.m11.2.2"></and><apply id="S5.3.p3.11.m11.2.2b.cmml" xref="S5.3.p3.11.m11.2.2"><apply id="S5.3.p3.11.m11.2.2.4.cmml" xref="S5.3.p3.11.m11.2.2.4"><csymbol cd="ambiguous" id="S5.3.p3.11.m11.2.2.4.1.cmml" xref="S5.3.p3.11.m11.2.2.4">subscript</csymbol><lt id="S5.3.p3.11.m11.2.2.4.2.cmml" xref="S5.3.p3.11.m11.2.2.4.2"></lt><ci id="S5.3.p3.11.m11.2.2.4.3.cmml" xref="S5.3.p3.11.m11.2.2.4.3">𝐴</ci></apply><ci id="S5.3.p3.11.m11.2.2.3.cmml" xref="S5.3.p3.11.m11.2.2.3">𝑎</ci><apply id="S5.3.p3.11.m11.2.2.1.cmml" xref="S5.3.p3.11.m11.2.2.1"><times id="S5.3.p3.11.m11.2.2.1.2.cmml" xref="S5.3.p3.11.m11.2.2.1.2"></times><ci id="S5.3.p3.11.m11.2.2.1.3.cmml" xref="S5.3.p3.11.m11.2.2.1.3">𝑔</ci><apply id="S5.3.p3.11.m11.2.2.1.1.1.1.cmml" xref="S5.3.p3.11.m11.2.2.1.1.1"><csymbol cd="ambiguous" id="S5.3.p3.11.m11.2.2.1.1.1.1.1.cmml" xref="S5.3.p3.11.m11.2.2.1.1.1">superscript</csymbol><ci id="S5.3.p3.11.m11.2.2.1.1.1.1.2.cmml" xref="S5.3.p3.11.m11.2.2.1.1.1.1.2">𝑥</ci><ci id="S5.3.p3.11.m11.2.2.1.1.1.1.3.cmml" xref="S5.3.p3.11.m11.2.2.1.1.1.1.3">′′</ci></apply></apply></apply><apply id="S5.3.p3.11.m11.2.2c.cmml" xref="S5.3.p3.11.m11.2.2"><apply id="S5.3.p3.11.m11.2.2.5.cmml" xref="S5.3.p3.11.m11.2.2.5"><csymbol cd="ambiguous" id="S5.3.p3.11.m11.2.2.5.1.cmml" xref="S5.3.p3.11.m11.2.2.5">subscript</csymbol><lt id="S5.3.p3.11.m11.2.2.5.2.cmml" xref="S5.3.p3.11.m11.2.2.5.2"></lt><ci id="S5.3.p3.11.m11.2.2.5.3.cmml" xref="S5.3.p3.11.m11.2.2.5.3">𝐴</ci></apply><share href="https://arxiv.org/html/2503.13728v1#S5.3.p3.11.m11.2.2.1.cmml" id="S5.3.p3.11.m11.2.2d.cmml" xref="S5.3.p3.11.m11.2.2"></share><apply id="S5.3.p3.11.m11.2.2.6.cmml" xref="S5.3.p3.11.m11.2.2.6"><times id="S5.3.p3.11.m11.2.2.6.1.cmml" xref="S5.3.p3.11.m11.2.2.6.1"></times><ci id="S5.3.p3.11.m11.2.2.6.2.cmml" xref="S5.3.p3.11.m11.2.2.6.2">𝑔</ci><ci id="S5.3.p3.11.m11.1.1.cmml" xref="S5.3.p3.11.m11.1.1">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.3.p3.11.m11.2c">a<_{A}g(x^{\prime\prime})<_{A}g(x)</annotation><annotation encoding="application/x-llamapun" id="S5.3.p3.11.m11.2d">italic_a < start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_g ( italic_x start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) < start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_g ( italic_x )</annotation></semantics></math>, and <math alttext="g(x^{\prime\prime})<\nu" class="ltx_Math" display="inline" id="S5.3.p3.12.m12.1"><semantics id="S5.3.p3.12.m12.1a"><mrow id="S5.3.p3.12.m12.1.1" xref="S5.3.p3.12.m12.1.1.cmml"><mrow id="S5.3.p3.12.m12.1.1.1" xref="S5.3.p3.12.m12.1.1.1.cmml"><mi id="S5.3.p3.12.m12.1.1.1.3" xref="S5.3.p3.12.m12.1.1.1.3.cmml">g</mi><mo id="S5.3.p3.12.m12.1.1.1.2" xref="S5.3.p3.12.m12.1.1.1.2.cmml">⁢</mo><mrow id="S5.3.p3.12.m12.1.1.1.1.1" xref="S5.3.p3.12.m12.1.1.1.1.1.1.cmml"><mo id="S5.3.p3.12.m12.1.1.1.1.1.2" stretchy="false" xref="S5.3.p3.12.m12.1.1.1.1.1.1.cmml">(</mo><msup id="S5.3.p3.12.m12.1.1.1.1.1.1" xref="S5.3.p3.12.m12.1.1.1.1.1.1.cmml"><mi id="S5.3.p3.12.m12.1.1.1.1.1.1.2" xref="S5.3.p3.12.m12.1.1.1.1.1.1.2.cmml">x</mi><mo id="S5.3.p3.12.m12.1.1.1.1.1.1.3" xref="S5.3.p3.12.m12.1.1.1.1.1.1.3.cmml">′′</mo></msup><mo id="S5.3.p3.12.m12.1.1.1.1.1.3" stretchy="false" xref="S5.3.p3.12.m12.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.3.p3.12.m12.1.1.2" xref="S5.3.p3.12.m12.1.1.2.cmml"><</mo><mi id="S5.3.p3.12.m12.1.1.3" xref="S5.3.p3.12.m12.1.1.3.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.3.p3.12.m12.1b"><apply id="S5.3.p3.12.m12.1.1.cmml" xref="S5.3.p3.12.m12.1.1"><lt id="S5.3.p3.12.m12.1.1.2.cmml" xref="S5.3.p3.12.m12.1.1.2"></lt><apply id="S5.3.p3.12.m12.1.1.1.cmml" xref="S5.3.p3.12.m12.1.1.1"><times id="S5.3.p3.12.m12.1.1.1.2.cmml" xref="S5.3.p3.12.m12.1.1.1.2"></times><ci id="S5.3.p3.12.m12.1.1.1.3.cmml" xref="S5.3.p3.12.m12.1.1.1.3">𝑔</ci><apply id="S5.3.p3.12.m12.1.1.1.1.1.1.cmml" xref="S5.3.p3.12.m12.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.3.p3.12.m12.1.1.1.1.1.1.1.cmml" xref="S5.3.p3.12.m12.1.1.1.1.1">superscript</csymbol><ci id="S5.3.p3.12.m12.1.1.1.1.1.1.2.cmml" xref="S5.3.p3.12.m12.1.1.1.1.1.1.2">𝑥</ci><ci id="S5.3.p3.12.m12.1.1.1.1.1.1.3.cmml" xref="S5.3.p3.12.m12.1.1.1.1.1.1.3">′′</ci></apply></apply><ci id="S5.3.p3.12.m12.1.1.3.cmml" xref="S5.3.p3.12.m12.1.1.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.3.p3.12.m12.1c">g(x^{\prime\prime})<\nu</annotation><annotation encoding="application/x-llamapun" id="S5.3.p3.12.m12.1d">italic_g ( italic_x start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) < italic_ν</annotation></semantics></math>, which contradicts that <math alttext="a" class="ltx_Math" display="inline" id="S5.3.p3.13.m13.1"><semantics id="S5.3.p3.13.m13.1a"><mi id="S5.3.p3.13.m13.1.1" xref="S5.3.p3.13.m13.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S5.3.p3.13.m13.1b"><ci id="S5.3.p3.13.m13.1.1.cmml" xref="S5.3.p3.13.m13.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.3.p3.13.m13.1c">a</annotation><annotation encoding="application/x-llamapun" id="S5.3.p3.13.m13.1d">italic_a</annotation></semantics></math> and <math alttext="g(x)" class="ltx_Math" display="inline" id="S5.3.p3.14.m14.1"><semantics id="S5.3.p3.14.m14.1a"><mrow id="S5.3.p3.14.m14.1.2" xref="S5.3.p3.14.m14.1.2.cmml"><mi id="S5.3.p3.14.m14.1.2.2" xref="S5.3.p3.14.m14.1.2.2.cmml">g</mi><mo id="S5.3.p3.14.m14.1.2.1" xref="S5.3.p3.14.m14.1.2.1.cmml">⁢</mo><mrow id="S5.3.p3.14.m14.1.2.3.2" xref="S5.3.p3.14.m14.1.2.cmml"><mo id="S5.3.p3.14.m14.1.2.3.2.1" stretchy="false" xref="S5.3.p3.14.m14.1.2.cmml">(</mo><mi id="S5.3.p3.14.m14.1.1" xref="S5.3.p3.14.m14.1.1.cmml">x</mi><mo id="S5.3.p3.14.m14.1.2.3.2.2" stretchy="false" xref="S5.3.p3.14.m14.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.3.p3.14.m14.1b"><apply id="S5.3.p3.14.m14.1.2.cmml" xref="S5.3.p3.14.m14.1.2"><times id="S5.3.p3.14.m14.1.2.1.cmml" xref="S5.3.p3.14.m14.1.2.1"></times><ci id="S5.3.p3.14.m14.1.2.2.cmml" xref="S5.3.p3.14.m14.1.2.2">𝑔</ci><ci id="S5.3.p3.14.m14.1.1.cmml" xref="S5.3.p3.14.m14.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.3.p3.14.m14.1c">g(x)</annotation><annotation encoding="application/x-llamapun" id="S5.3.p3.14.m14.1d">italic_g ( italic_x )</annotation></semantics></math> are in the same complementary interval of <math alttext="A\setminus\nu" class="ltx_Math" display="inline" id="S5.3.p3.15.m15.1"><semantics id="S5.3.p3.15.m15.1a"><mrow id="S5.3.p3.15.m15.1.1" xref="S5.3.p3.15.m15.1.1.cmml"><mi id="S5.3.p3.15.m15.1.1.2" xref="S5.3.p3.15.m15.1.1.2.cmml">A</mi><mo id="S5.3.p3.15.m15.1.1.1" xref="S5.3.p3.15.m15.1.1.1.cmml">∖</mo><mi id="S5.3.p3.15.m15.1.1.3" xref="S5.3.p3.15.m15.1.1.3.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.3.p3.15.m15.1b"><apply id="S5.3.p3.15.m15.1.1.cmml" xref="S5.3.p3.15.m15.1.1"><setdiff id="S5.3.p3.15.m15.1.1.1.cmml" xref="S5.3.p3.15.m15.1.1.1"></setdiff><ci id="S5.3.p3.15.m15.1.1.2.cmml" xref="S5.3.p3.15.m15.1.1.2">𝐴</ci><ci id="S5.3.p3.15.m15.1.1.3.cmml" xref="S5.3.p3.15.m15.1.1.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.3.p3.15.m15.1c">A\setminus\nu</annotation><annotation encoding="application/x-llamapun" id="S5.3.p3.15.m15.1d">italic_A ∖ italic_ν</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S5.4.p4"> <p class="ltx_p" id="S5.4.p4.13">Since <math alttext="f(a)" class="ltx_Math" display="inline" id="S5.4.p4.1.m1.1"><semantics id="S5.4.p4.1.m1.1a"><mrow id="S5.4.p4.1.m1.1.2" xref="S5.4.p4.1.m1.1.2.cmml"><mi id="S5.4.p4.1.m1.1.2.2" xref="S5.4.p4.1.m1.1.2.2.cmml">f</mi><mo id="S5.4.p4.1.m1.1.2.1" xref="S5.4.p4.1.m1.1.2.1.cmml">⁢</mo><mrow id="S5.4.p4.1.m1.1.2.3.2" xref="S5.4.p4.1.m1.1.2.cmml"><mo id="S5.4.p4.1.m1.1.2.3.2.1" stretchy="false" xref="S5.4.p4.1.m1.1.2.cmml">(</mo><mi id="S5.4.p4.1.m1.1.1" xref="S5.4.p4.1.m1.1.1.cmml">a</mi><mo id="S5.4.p4.1.m1.1.2.3.2.2" stretchy="false" xref="S5.4.p4.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.4.p4.1.m1.1b"><apply id="S5.4.p4.1.m1.1.2.cmml" xref="S5.4.p4.1.m1.1.2"><times id="S5.4.p4.1.m1.1.2.1.cmml" xref="S5.4.p4.1.m1.1.2.1"></times><ci id="S5.4.p4.1.m1.1.2.2.cmml" xref="S5.4.p4.1.m1.1.2.2">𝑓</ci><ci id="S5.4.p4.1.m1.1.1.cmml" xref="S5.4.p4.1.m1.1.1">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p4.1.m1.1c">f(a)</annotation><annotation encoding="application/x-llamapun" id="S5.4.p4.1.m1.1d">italic_f ( italic_a )</annotation></semantics></math> is in <math alttext="I" class="ltx_Math" display="inline" id="S5.4.p4.2.m2.1"><semantics id="S5.4.p4.2.m2.1a"><mi id="S5.4.p4.2.m2.1.1" xref="S5.4.p4.2.m2.1.1.cmml">I</mi><annotation-xml encoding="MathML-Content" id="S5.4.p4.2.m2.1b"><ci id="S5.4.p4.2.m2.1.1.cmml" xref="S5.4.p4.2.m2.1.1">𝐼</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p4.2.m2.1c">I</annotation><annotation encoding="application/x-llamapun" id="S5.4.p4.2.m2.1d">italic_I</annotation></semantics></math>, and <math alttext="I" class="ltx_Math" display="inline" id="S5.4.p4.3.m3.1"><semantics id="S5.4.p4.3.m3.1a"><mi id="S5.4.p4.3.m3.1.1" xref="S5.4.p4.3.m3.1.1.cmml">I</mi><annotation-xml encoding="MathML-Content" id="S5.4.p4.3.m3.1b"><ci id="S5.4.p4.3.m3.1.1.cmml" xref="S5.4.p4.3.m3.1.1">𝐼</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p4.3.m3.1c">I</annotation><annotation encoding="application/x-llamapun" id="S5.4.p4.3.m3.1d">italic_I</annotation></semantics></math> does not have a left endpoint, one easily finds <math alttext="x^{\prime}\in I" class="ltx_Math" display="inline" id="S5.4.p4.4.m4.1"><semantics id="S5.4.p4.4.m4.1a"><mrow id="S5.4.p4.4.m4.1.1" xref="S5.4.p4.4.m4.1.1.cmml"><msup id="S5.4.p4.4.m4.1.1.2" xref="S5.4.p4.4.m4.1.1.2.cmml"><mi id="S5.4.p4.4.m4.1.1.2.2" xref="S5.4.p4.4.m4.1.1.2.2.cmml">x</mi><mo id="S5.4.p4.4.m4.1.1.2.3" xref="S5.4.p4.4.m4.1.1.2.3.cmml">′</mo></msup><mo id="S5.4.p4.4.m4.1.1.1" xref="S5.4.p4.4.m4.1.1.1.cmml">∈</mo><mi id="S5.4.p4.4.m4.1.1.3" xref="S5.4.p4.4.m4.1.1.3.cmml">I</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.4.p4.4.m4.1b"><apply id="S5.4.p4.4.m4.1.1.cmml" xref="S5.4.p4.4.m4.1.1"><in id="S5.4.p4.4.m4.1.1.1.cmml" xref="S5.4.p4.4.m4.1.1.1"></in><apply id="S5.4.p4.4.m4.1.1.2.cmml" xref="S5.4.p4.4.m4.1.1.2"><csymbol cd="ambiguous" id="S5.4.p4.4.m4.1.1.2.1.cmml" xref="S5.4.p4.4.m4.1.1.2">superscript</csymbol><ci id="S5.4.p4.4.m4.1.1.2.2.cmml" xref="S5.4.p4.4.m4.1.1.2.2">𝑥</ci><ci id="S5.4.p4.4.m4.1.1.2.3.cmml" xref="S5.4.p4.4.m4.1.1.2.3">′</ci></apply><ci id="S5.4.p4.4.m4.1.1.3.cmml" xref="S5.4.p4.4.m4.1.1.3">𝐼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p4.4.m4.1c">x^{\prime}\in I</annotation><annotation encoding="application/x-llamapun" id="S5.4.p4.4.m4.1d">italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_I</annotation></semantics></math> such that <math alttext="x^{\prime}<_{X}f(a)" class="ltx_Math" display="inline" id="S5.4.p4.5.m5.1"><semantics id="S5.4.p4.5.m5.1a"><mrow id="S5.4.p4.5.m5.1.2" xref="S5.4.p4.5.m5.1.2.cmml"><msup id="S5.4.p4.5.m5.1.2.2" xref="S5.4.p4.5.m5.1.2.2.cmml"><mi id="S5.4.p4.5.m5.1.2.2.2" xref="S5.4.p4.5.m5.1.2.2.2.cmml">x</mi><mo id="S5.4.p4.5.m5.1.2.2.3" xref="S5.4.p4.5.m5.1.2.2.3.cmml">′</mo></msup><msub id="S5.4.p4.5.m5.1.2.1" xref="S5.4.p4.5.m5.1.2.1.cmml"><mo id="S5.4.p4.5.m5.1.2.1.2" xref="S5.4.p4.5.m5.1.2.1.2.cmml"><</mo><mi id="S5.4.p4.5.m5.1.2.1.3" xref="S5.4.p4.5.m5.1.2.1.3.cmml">X</mi></msub><mrow id="S5.4.p4.5.m5.1.2.3" xref="S5.4.p4.5.m5.1.2.3.cmml"><mi id="S5.4.p4.5.m5.1.2.3.2" xref="S5.4.p4.5.m5.1.2.3.2.cmml">f</mi><mo id="S5.4.p4.5.m5.1.2.3.1" xref="S5.4.p4.5.m5.1.2.3.1.cmml">⁢</mo><mrow id="S5.4.p4.5.m5.1.2.3.3.2" xref="S5.4.p4.5.m5.1.2.3.cmml"><mo id="S5.4.p4.5.m5.1.2.3.3.2.1" stretchy="false" xref="S5.4.p4.5.m5.1.2.3.cmml">(</mo><mi id="S5.4.p4.5.m5.1.1" xref="S5.4.p4.5.m5.1.1.cmml">a</mi><mo id="S5.4.p4.5.m5.1.2.3.3.2.2" stretchy="false" xref="S5.4.p4.5.m5.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.4.p4.5.m5.1b"><apply id="S5.4.p4.5.m5.1.2.cmml" xref="S5.4.p4.5.m5.1.2"><apply id="S5.4.p4.5.m5.1.2.1.cmml" xref="S5.4.p4.5.m5.1.2.1"><csymbol cd="ambiguous" id="S5.4.p4.5.m5.1.2.1.1.cmml" xref="S5.4.p4.5.m5.1.2.1">subscript</csymbol><lt id="S5.4.p4.5.m5.1.2.1.2.cmml" xref="S5.4.p4.5.m5.1.2.1.2"></lt><ci id="S5.4.p4.5.m5.1.2.1.3.cmml" xref="S5.4.p4.5.m5.1.2.1.3">𝑋</ci></apply><apply id="S5.4.p4.5.m5.1.2.2.cmml" xref="S5.4.p4.5.m5.1.2.2"><csymbol cd="ambiguous" id="S5.4.p4.5.m5.1.2.2.1.cmml" xref="S5.4.p4.5.m5.1.2.2">superscript</csymbol><ci id="S5.4.p4.5.m5.1.2.2.2.cmml" xref="S5.4.p4.5.m5.1.2.2.2">𝑥</ci><ci id="S5.4.p4.5.m5.1.2.2.3.cmml" xref="S5.4.p4.5.m5.1.2.2.3">′</ci></apply><apply id="S5.4.p4.5.m5.1.2.3.cmml" xref="S5.4.p4.5.m5.1.2.3"><times id="S5.4.p4.5.m5.1.2.3.1.cmml" xref="S5.4.p4.5.m5.1.2.3.1"></times><ci id="S5.4.p4.5.m5.1.2.3.2.cmml" xref="S5.4.p4.5.m5.1.2.3.2">𝑓</ci><ci id="S5.4.p4.5.m5.1.1.cmml" xref="S5.4.p4.5.m5.1.1">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p4.5.m5.1c">x^{\prime}<_{X}f(a)</annotation><annotation encoding="application/x-llamapun" id="S5.4.p4.5.m5.1d">italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT < start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT italic_f ( italic_a )</annotation></semantics></math>, and thus that <math alttext="g(x^{\prime})<_{A}a" class="ltx_Math" display="inline" id="S5.4.p4.6.m6.1"><semantics id="S5.4.p4.6.m6.1a"><mrow id="S5.4.p4.6.m6.1.1" xref="S5.4.p4.6.m6.1.1.cmml"><mrow id="S5.4.p4.6.m6.1.1.1" xref="S5.4.p4.6.m6.1.1.1.cmml"><mi id="S5.4.p4.6.m6.1.1.1.3" xref="S5.4.p4.6.m6.1.1.1.3.cmml">g</mi><mo id="S5.4.p4.6.m6.1.1.1.2" xref="S5.4.p4.6.m6.1.1.1.2.cmml">⁢</mo><mrow id="S5.4.p4.6.m6.1.1.1.1.1" xref="S5.4.p4.6.m6.1.1.1.1.1.1.cmml"><mo id="S5.4.p4.6.m6.1.1.1.1.1.2" stretchy="false" xref="S5.4.p4.6.m6.1.1.1.1.1.1.cmml">(</mo><msup id="S5.4.p4.6.m6.1.1.1.1.1.1" xref="S5.4.p4.6.m6.1.1.1.1.1.1.cmml"><mi id="S5.4.p4.6.m6.1.1.1.1.1.1.2" xref="S5.4.p4.6.m6.1.1.1.1.1.1.2.cmml">x</mi><mo id="S5.4.p4.6.m6.1.1.1.1.1.1.3" xref="S5.4.p4.6.m6.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S5.4.p4.6.m6.1.1.1.1.1.3" stretchy="false" xref="S5.4.p4.6.m6.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><msub id="S5.4.p4.6.m6.1.1.2" xref="S5.4.p4.6.m6.1.1.2.cmml"><mo id="S5.4.p4.6.m6.1.1.2.2" xref="S5.4.p4.6.m6.1.1.2.2.cmml"><</mo><mi id="S5.4.p4.6.m6.1.1.2.3" xref="S5.4.p4.6.m6.1.1.2.3.cmml">A</mi></msub><mi id="S5.4.p4.6.m6.1.1.3" xref="S5.4.p4.6.m6.1.1.3.cmml">a</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.4.p4.6.m6.1b"><apply id="S5.4.p4.6.m6.1.1.cmml" xref="S5.4.p4.6.m6.1.1"><apply id="S5.4.p4.6.m6.1.1.2.cmml" xref="S5.4.p4.6.m6.1.1.2"><csymbol cd="ambiguous" id="S5.4.p4.6.m6.1.1.2.1.cmml" xref="S5.4.p4.6.m6.1.1.2">subscript</csymbol><lt id="S5.4.p4.6.m6.1.1.2.2.cmml" xref="S5.4.p4.6.m6.1.1.2.2"></lt><ci id="S5.4.p4.6.m6.1.1.2.3.cmml" xref="S5.4.p4.6.m6.1.1.2.3">𝐴</ci></apply><apply id="S5.4.p4.6.m6.1.1.1.cmml" xref="S5.4.p4.6.m6.1.1.1"><times id="S5.4.p4.6.m6.1.1.1.2.cmml" xref="S5.4.p4.6.m6.1.1.1.2"></times><ci id="S5.4.p4.6.m6.1.1.1.3.cmml" xref="S5.4.p4.6.m6.1.1.1.3">𝑔</ci><apply id="S5.4.p4.6.m6.1.1.1.1.1.1.cmml" xref="S5.4.p4.6.m6.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.4.p4.6.m6.1.1.1.1.1.1.1.cmml" xref="S5.4.p4.6.m6.1.1.1.1.1">superscript</csymbol><ci id="S5.4.p4.6.m6.1.1.1.1.1.1.2.cmml" xref="S5.4.p4.6.m6.1.1.1.1.1.1.2">𝑥</ci><ci id="S5.4.p4.6.m6.1.1.1.1.1.1.3.cmml" xref="S5.4.p4.6.m6.1.1.1.1.1.1.3">′</ci></apply></apply><ci id="S5.4.p4.6.m6.1.1.3.cmml" xref="S5.4.p4.6.m6.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p4.6.m6.1c">g(x^{\prime})<_{A}a</annotation><annotation encoding="application/x-llamapun" id="S5.4.p4.6.m6.1d">italic_g ( italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) < start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_a</annotation></semantics></math>. Since <math alttext="a" class="ltx_Math" display="inline" id="S5.4.p4.7.m7.1"><semantics id="S5.4.p4.7.m7.1a"><mi id="S5.4.p4.7.m7.1.1" xref="S5.4.p4.7.m7.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S5.4.p4.7.m7.1b"><ci id="S5.4.p4.7.m7.1.1.cmml" xref="S5.4.p4.7.m7.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p4.7.m7.1c">a</annotation><annotation encoding="application/x-llamapun" id="S5.4.p4.7.m7.1d">italic_a</annotation></semantics></math> is the left endpoint of <math alttext="J" class="ltx_Math" display="inline" id="S5.4.p4.8.m8.1"><semantics id="S5.4.p4.8.m8.1a"><mi id="S5.4.p4.8.m8.1.1" xref="S5.4.p4.8.m8.1.1.cmml">J</mi><annotation-xml encoding="MathML-Content" id="S5.4.p4.8.m8.1b"><ci id="S5.4.p4.8.m8.1.1.cmml" xref="S5.4.p4.8.m8.1.1">𝐽</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p4.8.m8.1c">J</annotation><annotation encoding="application/x-llamapun" id="S5.4.p4.8.m8.1d">italic_J</annotation></semantics></math>, there is <math alttext="a^{\prime}<\nu" class="ltx_Math" display="inline" id="S5.4.p4.9.m9.1"><semantics id="S5.4.p4.9.m9.1a"><mrow id="S5.4.p4.9.m9.1.1" xref="S5.4.p4.9.m9.1.1.cmml"><msup id="S5.4.p4.9.m9.1.1.2" xref="S5.4.p4.9.m9.1.1.2.cmml"><mi id="S5.4.p4.9.m9.1.1.2.2" xref="S5.4.p4.9.m9.1.1.2.2.cmml">a</mi><mo id="S5.4.p4.9.m9.1.1.2.3" xref="S5.4.p4.9.m9.1.1.2.3.cmml">′</mo></msup><mo id="S5.4.p4.9.m9.1.1.1" xref="S5.4.p4.9.m9.1.1.1.cmml"><</mo><mi id="S5.4.p4.9.m9.1.1.3" xref="S5.4.p4.9.m9.1.1.3.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.4.p4.9.m9.1b"><apply id="S5.4.p4.9.m9.1.1.cmml" xref="S5.4.p4.9.m9.1.1"><lt id="S5.4.p4.9.m9.1.1.1.cmml" xref="S5.4.p4.9.m9.1.1.1"></lt><apply id="S5.4.p4.9.m9.1.1.2.cmml" xref="S5.4.p4.9.m9.1.1.2"><csymbol cd="ambiguous" id="S5.4.p4.9.m9.1.1.2.1.cmml" xref="S5.4.p4.9.m9.1.1.2">superscript</csymbol><ci id="S5.4.p4.9.m9.1.1.2.2.cmml" xref="S5.4.p4.9.m9.1.1.2.2">𝑎</ci><ci id="S5.4.p4.9.m9.1.1.2.3.cmml" xref="S5.4.p4.9.m9.1.1.2.3">′</ci></apply><ci id="S5.4.p4.9.m9.1.1.3.cmml" xref="S5.4.p4.9.m9.1.1.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p4.9.m9.1c">a^{\prime}<\nu</annotation><annotation encoding="application/x-llamapun" id="S5.4.p4.9.m9.1d">italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT < italic_ν</annotation></semantics></math> such that <math alttext="g(x^{\prime})\leq_{A}a^{\prime}<_{A}a" class="ltx_Math" display="inline" id="S5.4.p4.10.m10.1"><semantics id="S5.4.p4.10.m10.1a"><mrow id="S5.4.p4.10.m10.1.1" xref="S5.4.p4.10.m10.1.1.cmml"><mrow id="S5.4.p4.10.m10.1.1.1" xref="S5.4.p4.10.m10.1.1.1.cmml"><mi id="S5.4.p4.10.m10.1.1.1.3" xref="S5.4.p4.10.m10.1.1.1.3.cmml">g</mi><mo id="S5.4.p4.10.m10.1.1.1.2" xref="S5.4.p4.10.m10.1.1.1.2.cmml">⁢</mo><mrow id="S5.4.p4.10.m10.1.1.1.1.1" xref="S5.4.p4.10.m10.1.1.1.1.1.1.cmml"><mo id="S5.4.p4.10.m10.1.1.1.1.1.2" stretchy="false" xref="S5.4.p4.10.m10.1.1.1.1.1.1.cmml">(</mo><msup id="S5.4.p4.10.m10.1.1.1.1.1.1" xref="S5.4.p4.10.m10.1.1.1.1.1.1.cmml"><mi id="S5.4.p4.10.m10.1.1.1.1.1.1.2" xref="S5.4.p4.10.m10.1.1.1.1.1.1.2.cmml">x</mi><mo id="S5.4.p4.10.m10.1.1.1.1.1.1.3" xref="S5.4.p4.10.m10.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S5.4.p4.10.m10.1.1.1.1.1.3" stretchy="false" xref="S5.4.p4.10.m10.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><msub id="S5.4.p4.10.m10.1.1.3" xref="S5.4.p4.10.m10.1.1.3.cmml"><mo id="S5.4.p4.10.m10.1.1.3.2" xref="S5.4.p4.10.m10.1.1.3.2.cmml">≤</mo><mi id="S5.4.p4.10.m10.1.1.3.3" xref="S5.4.p4.10.m10.1.1.3.3.cmml">A</mi></msub><msup id="S5.4.p4.10.m10.1.1.4" xref="S5.4.p4.10.m10.1.1.4.cmml"><mi id="S5.4.p4.10.m10.1.1.4.2" xref="S5.4.p4.10.m10.1.1.4.2.cmml">a</mi><mo id="S5.4.p4.10.m10.1.1.4.3" xref="S5.4.p4.10.m10.1.1.4.3.cmml">′</mo></msup><msub id="S5.4.p4.10.m10.1.1.5" xref="S5.4.p4.10.m10.1.1.5.cmml"><mo id="S5.4.p4.10.m10.1.1.5.2" xref="S5.4.p4.10.m10.1.1.5.2.cmml"><</mo><mi id="S5.4.p4.10.m10.1.1.5.3" xref="S5.4.p4.10.m10.1.1.5.3.cmml">A</mi></msub><mi id="S5.4.p4.10.m10.1.1.6" xref="S5.4.p4.10.m10.1.1.6.cmml">a</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.4.p4.10.m10.1b"><apply id="S5.4.p4.10.m10.1.1.cmml" xref="S5.4.p4.10.m10.1.1"><and id="S5.4.p4.10.m10.1.1a.cmml" xref="S5.4.p4.10.m10.1.1"></and><apply id="S5.4.p4.10.m10.1.1b.cmml" xref="S5.4.p4.10.m10.1.1"><apply id="S5.4.p4.10.m10.1.1.3.cmml" xref="S5.4.p4.10.m10.1.1.3"><csymbol cd="ambiguous" id="S5.4.p4.10.m10.1.1.3.1.cmml" xref="S5.4.p4.10.m10.1.1.3">subscript</csymbol><leq id="S5.4.p4.10.m10.1.1.3.2.cmml" xref="S5.4.p4.10.m10.1.1.3.2"></leq><ci id="S5.4.p4.10.m10.1.1.3.3.cmml" xref="S5.4.p4.10.m10.1.1.3.3">𝐴</ci></apply><apply id="S5.4.p4.10.m10.1.1.1.cmml" xref="S5.4.p4.10.m10.1.1.1"><times id="S5.4.p4.10.m10.1.1.1.2.cmml" xref="S5.4.p4.10.m10.1.1.1.2"></times><ci id="S5.4.p4.10.m10.1.1.1.3.cmml" xref="S5.4.p4.10.m10.1.1.1.3">𝑔</ci><apply id="S5.4.p4.10.m10.1.1.1.1.1.1.cmml" xref="S5.4.p4.10.m10.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.4.p4.10.m10.1.1.1.1.1.1.1.cmml" xref="S5.4.p4.10.m10.1.1.1.1.1">superscript</csymbol><ci id="S5.4.p4.10.m10.1.1.1.1.1.1.2.cmml" xref="S5.4.p4.10.m10.1.1.1.1.1.1.2">𝑥</ci><ci id="S5.4.p4.10.m10.1.1.1.1.1.1.3.cmml" xref="S5.4.p4.10.m10.1.1.1.1.1.1.3">′</ci></apply></apply><apply id="S5.4.p4.10.m10.1.1.4.cmml" xref="S5.4.p4.10.m10.1.1.4"><csymbol cd="ambiguous" id="S5.4.p4.10.m10.1.1.4.1.cmml" xref="S5.4.p4.10.m10.1.1.4">superscript</csymbol><ci id="S5.4.p4.10.m10.1.1.4.2.cmml" xref="S5.4.p4.10.m10.1.1.4.2">𝑎</ci><ci id="S5.4.p4.10.m10.1.1.4.3.cmml" xref="S5.4.p4.10.m10.1.1.4.3">′</ci></apply></apply><apply id="S5.4.p4.10.m10.1.1c.cmml" xref="S5.4.p4.10.m10.1.1"><apply id="S5.4.p4.10.m10.1.1.5.cmml" xref="S5.4.p4.10.m10.1.1.5"><csymbol cd="ambiguous" id="S5.4.p4.10.m10.1.1.5.1.cmml" xref="S5.4.p4.10.m10.1.1.5">subscript</csymbol><lt id="S5.4.p4.10.m10.1.1.5.2.cmml" xref="S5.4.p4.10.m10.1.1.5.2"></lt><ci id="S5.4.p4.10.m10.1.1.5.3.cmml" xref="S5.4.p4.10.m10.1.1.5.3">𝐴</ci></apply><share href="https://arxiv.org/html/2503.13728v1#S5.4.p4.10.m10.1.1.4.cmml" id="S5.4.p4.10.m10.1.1d.cmml" xref="S5.4.p4.10.m10.1.1"></share><ci id="S5.4.p4.10.m10.1.1.6.cmml" xref="S5.4.p4.10.m10.1.1.6">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p4.10.m10.1c">g(x^{\prime})\leq_{A}a^{\prime}<_{A}a</annotation><annotation encoding="application/x-llamapun" id="S5.4.p4.10.m10.1d">italic_g ( italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ≤ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT < start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_a</annotation></semantics></math>, and thus <math alttext="x^{\prime}\leq_{X}f(a^{\prime})\leq_{X}f(a)" class="ltx_Math" display="inline" id="S5.4.p4.11.m11.2"><semantics id="S5.4.p4.11.m11.2a"><mrow id="S5.4.p4.11.m11.2.2" xref="S5.4.p4.11.m11.2.2.cmml"><msup id="S5.4.p4.11.m11.2.2.3" xref="S5.4.p4.11.m11.2.2.3.cmml"><mi id="S5.4.p4.11.m11.2.2.3.2" xref="S5.4.p4.11.m11.2.2.3.2.cmml">x</mi><mo id="S5.4.p4.11.m11.2.2.3.3" xref="S5.4.p4.11.m11.2.2.3.3.cmml">′</mo></msup><msub id="S5.4.p4.11.m11.2.2.4" xref="S5.4.p4.11.m11.2.2.4.cmml"><mo id="S5.4.p4.11.m11.2.2.4.2" xref="S5.4.p4.11.m11.2.2.4.2.cmml">≤</mo><mi id="S5.4.p4.11.m11.2.2.4.3" xref="S5.4.p4.11.m11.2.2.4.3.cmml">X</mi></msub><mrow id="S5.4.p4.11.m11.2.2.1" xref="S5.4.p4.11.m11.2.2.1.cmml"><mi id="S5.4.p4.11.m11.2.2.1.3" xref="S5.4.p4.11.m11.2.2.1.3.cmml">f</mi><mo id="S5.4.p4.11.m11.2.2.1.2" xref="S5.4.p4.11.m11.2.2.1.2.cmml">⁢</mo><mrow id="S5.4.p4.11.m11.2.2.1.1.1" xref="S5.4.p4.11.m11.2.2.1.1.1.1.cmml"><mo id="S5.4.p4.11.m11.2.2.1.1.1.2" stretchy="false" xref="S5.4.p4.11.m11.2.2.1.1.1.1.cmml">(</mo><msup id="S5.4.p4.11.m11.2.2.1.1.1.1" xref="S5.4.p4.11.m11.2.2.1.1.1.1.cmml"><mi id="S5.4.p4.11.m11.2.2.1.1.1.1.2" xref="S5.4.p4.11.m11.2.2.1.1.1.1.2.cmml">a</mi><mo id="S5.4.p4.11.m11.2.2.1.1.1.1.3" xref="S5.4.p4.11.m11.2.2.1.1.1.1.3.cmml">′</mo></msup><mo id="S5.4.p4.11.m11.2.2.1.1.1.3" stretchy="false" xref="S5.4.p4.11.m11.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><msub id="S5.4.p4.11.m11.2.2.5" xref="S5.4.p4.11.m11.2.2.5.cmml"><mo id="S5.4.p4.11.m11.2.2.5.2" xref="S5.4.p4.11.m11.2.2.5.2.cmml">≤</mo><mi id="S5.4.p4.11.m11.2.2.5.3" xref="S5.4.p4.11.m11.2.2.5.3.cmml">X</mi></msub><mrow id="S5.4.p4.11.m11.2.2.6" xref="S5.4.p4.11.m11.2.2.6.cmml"><mi id="S5.4.p4.11.m11.2.2.6.2" xref="S5.4.p4.11.m11.2.2.6.2.cmml">f</mi><mo id="S5.4.p4.11.m11.2.2.6.1" xref="S5.4.p4.11.m11.2.2.6.1.cmml">⁢</mo><mrow id="S5.4.p4.11.m11.2.2.6.3.2" xref="S5.4.p4.11.m11.2.2.6.cmml"><mo id="S5.4.p4.11.m11.2.2.6.3.2.1" stretchy="false" xref="S5.4.p4.11.m11.2.2.6.cmml">(</mo><mi id="S5.4.p4.11.m11.1.1" xref="S5.4.p4.11.m11.1.1.cmml">a</mi><mo id="S5.4.p4.11.m11.2.2.6.3.2.2" stretchy="false" xref="S5.4.p4.11.m11.2.2.6.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.4.p4.11.m11.2b"><apply id="S5.4.p4.11.m11.2.2.cmml" xref="S5.4.p4.11.m11.2.2"><and id="S5.4.p4.11.m11.2.2a.cmml" xref="S5.4.p4.11.m11.2.2"></and><apply id="S5.4.p4.11.m11.2.2b.cmml" xref="S5.4.p4.11.m11.2.2"><apply id="S5.4.p4.11.m11.2.2.4.cmml" xref="S5.4.p4.11.m11.2.2.4"><csymbol cd="ambiguous" id="S5.4.p4.11.m11.2.2.4.1.cmml" xref="S5.4.p4.11.m11.2.2.4">subscript</csymbol><leq id="S5.4.p4.11.m11.2.2.4.2.cmml" xref="S5.4.p4.11.m11.2.2.4.2"></leq><ci id="S5.4.p4.11.m11.2.2.4.3.cmml" xref="S5.4.p4.11.m11.2.2.4.3">𝑋</ci></apply><apply id="S5.4.p4.11.m11.2.2.3.cmml" xref="S5.4.p4.11.m11.2.2.3"><csymbol cd="ambiguous" id="S5.4.p4.11.m11.2.2.3.1.cmml" xref="S5.4.p4.11.m11.2.2.3">superscript</csymbol><ci id="S5.4.p4.11.m11.2.2.3.2.cmml" xref="S5.4.p4.11.m11.2.2.3.2">𝑥</ci><ci id="S5.4.p4.11.m11.2.2.3.3.cmml" xref="S5.4.p4.11.m11.2.2.3.3">′</ci></apply><apply id="S5.4.p4.11.m11.2.2.1.cmml" xref="S5.4.p4.11.m11.2.2.1"><times id="S5.4.p4.11.m11.2.2.1.2.cmml" xref="S5.4.p4.11.m11.2.2.1.2"></times><ci id="S5.4.p4.11.m11.2.2.1.3.cmml" xref="S5.4.p4.11.m11.2.2.1.3">𝑓</ci><apply id="S5.4.p4.11.m11.2.2.1.1.1.1.cmml" xref="S5.4.p4.11.m11.2.2.1.1.1"><csymbol cd="ambiguous" id="S5.4.p4.11.m11.2.2.1.1.1.1.1.cmml" xref="S5.4.p4.11.m11.2.2.1.1.1">superscript</csymbol><ci id="S5.4.p4.11.m11.2.2.1.1.1.1.2.cmml" xref="S5.4.p4.11.m11.2.2.1.1.1.1.2">𝑎</ci><ci id="S5.4.p4.11.m11.2.2.1.1.1.1.3.cmml" xref="S5.4.p4.11.m11.2.2.1.1.1.1.3">′</ci></apply></apply></apply><apply id="S5.4.p4.11.m11.2.2c.cmml" xref="S5.4.p4.11.m11.2.2"><apply id="S5.4.p4.11.m11.2.2.5.cmml" xref="S5.4.p4.11.m11.2.2.5"><csymbol cd="ambiguous" id="S5.4.p4.11.m11.2.2.5.1.cmml" xref="S5.4.p4.11.m11.2.2.5">subscript</csymbol><leq id="S5.4.p4.11.m11.2.2.5.2.cmml" xref="S5.4.p4.11.m11.2.2.5.2"></leq><ci id="S5.4.p4.11.m11.2.2.5.3.cmml" xref="S5.4.p4.11.m11.2.2.5.3">𝑋</ci></apply><share href="https://arxiv.org/html/2503.13728v1#S5.4.p4.11.m11.2.2.1.cmml" id="S5.4.p4.11.m11.2.2d.cmml" xref="S5.4.p4.11.m11.2.2"></share><apply id="S5.4.p4.11.m11.2.2.6.cmml" xref="S5.4.p4.11.m11.2.2.6"><times id="S5.4.p4.11.m11.2.2.6.1.cmml" xref="S5.4.p4.11.m11.2.2.6.1"></times><ci id="S5.4.p4.11.m11.2.2.6.2.cmml" xref="S5.4.p4.11.m11.2.2.6.2">𝑓</ci><ci id="S5.4.p4.11.m11.1.1.cmml" xref="S5.4.p4.11.m11.1.1">𝑎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p4.11.m11.2c">x^{\prime}\leq_{X}f(a^{\prime})\leq_{X}f(a)</annotation><annotation encoding="application/x-llamapun" id="S5.4.p4.11.m11.2d">italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT italic_f ( italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ≤ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT italic_f ( italic_a )</annotation></semantics></math>, which implies that <math alttext="f(a^{\prime})\in I" class="ltx_Math" display="inline" id="S5.4.p4.12.m12.1"><semantics id="S5.4.p4.12.m12.1a"><mrow id="S5.4.p4.12.m12.1.1" xref="S5.4.p4.12.m12.1.1.cmml"><mrow id="S5.4.p4.12.m12.1.1.1" xref="S5.4.p4.12.m12.1.1.1.cmml"><mi id="S5.4.p4.12.m12.1.1.1.3" xref="S5.4.p4.12.m12.1.1.1.3.cmml">f</mi><mo id="S5.4.p4.12.m12.1.1.1.2" xref="S5.4.p4.12.m12.1.1.1.2.cmml">⁢</mo><mrow id="S5.4.p4.12.m12.1.1.1.1.1" xref="S5.4.p4.12.m12.1.1.1.1.1.1.cmml"><mo id="S5.4.p4.12.m12.1.1.1.1.1.2" stretchy="false" xref="S5.4.p4.12.m12.1.1.1.1.1.1.cmml">(</mo><msup id="S5.4.p4.12.m12.1.1.1.1.1.1" xref="S5.4.p4.12.m12.1.1.1.1.1.1.cmml"><mi id="S5.4.p4.12.m12.1.1.1.1.1.1.2" xref="S5.4.p4.12.m12.1.1.1.1.1.1.2.cmml">a</mi><mo id="S5.4.p4.12.m12.1.1.1.1.1.1.3" xref="S5.4.p4.12.m12.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S5.4.p4.12.m12.1.1.1.1.1.3" stretchy="false" xref="S5.4.p4.12.m12.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.4.p4.12.m12.1.1.2" xref="S5.4.p4.12.m12.1.1.2.cmml">∈</mo><mi id="S5.4.p4.12.m12.1.1.3" xref="S5.4.p4.12.m12.1.1.3.cmml">I</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.4.p4.12.m12.1b"><apply id="S5.4.p4.12.m12.1.1.cmml" xref="S5.4.p4.12.m12.1.1"><in id="S5.4.p4.12.m12.1.1.2.cmml" xref="S5.4.p4.12.m12.1.1.2"></in><apply id="S5.4.p4.12.m12.1.1.1.cmml" xref="S5.4.p4.12.m12.1.1.1"><times id="S5.4.p4.12.m12.1.1.1.2.cmml" xref="S5.4.p4.12.m12.1.1.1.2"></times><ci id="S5.4.p4.12.m12.1.1.1.3.cmml" xref="S5.4.p4.12.m12.1.1.1.3">𝑓</ci><apply id="S5.4.p4.12.m12.1.1.1.1.1.1.cmml" xref="S5.4.p4.12.m12.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.4.p4.12.m12.1.1.1.1.1.1.1.cmml" xref="S5.4.p4.12.m12.1.1.1.1.1">superscript</csymbol><ci id="S5.4.p4.12.m12.1.1.1.1.1.1.2.cmml" xref="S5.4.p4.12.m12.1.1.1.1.1.1.2">𝑎</ci><ci id="S5.4.p4.12.m12.1.1.1.1.1.1.3.cmml" xref="S5.4.p4.12.m12.1.1.1.1.1.1.3">′</ci></apply></apply><ci id="S5.4.p4.12.m12.1.1.3.cmml" xref="S5.4.p4.12.m12.1.1.3">𝐼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p4.12.m12.1c">f(a^{\prime})\in I</annotation><annotation encoding="application/x-llamapun" id="S5.4.p4.12.m12.1d">italic_f ( italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ∈ italic_I</annotation></semantics></math>. This contradicts the fact that <math alttext="a^{\prime}<\nu" class="ltx_Math" display="inline" id="S5.4.p4.13.m13.1"><semantics id="S5.4.p4.13.m13.1a"><mrow id="S5.4.p4.13.m13.1.1" xref="S5.4.p4.13.m13.1.1.cmml"><msup id="S5.4.p4.13.m13.1.1.2" xref="S5.4.p4.13.m13.1.1.2.cmml"><mi id="S5.4.p4.13.m13.1.1.2.2" xref="S5.4.p4.13.m13.1.1.2.2.cmml">a</mi><mo id="S5.4.p4.13.m13.1.1.2.3" xref="S5.4.p4.13.m13.1.1.2.3.cmml">′</mo></msup><mo id="S5.4.p4.13.m13.1.1.1" xref="S5.4.p4.13.m13.1.1.1.cmml"><</mo><mi id="S5.4.p4.13.m13.1.1.3" xref="S5.4.p4.13.m13.1.1.3.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.4.p4.13.m13.1b"><apply id="S5.4.p4.13.m13.1.1.cmml" xref="S5.4.p4.13.m13.1.1"><lt id="S5.4.p4.13.m13.1.1.1.cmml" xref="S5.4.p4.13.m13.1.1.1"></lt><apply id="S5.4.p4.13.m13.1.1.2.cmml" xref="S5.4.p4.13.m13.1.1.2"><csymbol cd="ambiguous" id="S5.4.p4.13.m13.1.1.2.1.cmml" xref="S5.4.p4.13.m13.1.1.2">superscript</csymbol><ci id="S5.4.p4.13.m13.1.1.2.2.cmml" xref="S5.4.p4.13.m13.1.1.2.2">𝑎</ci><ci id="S5.4.p4.13.m13.1.1.2.3.cmml" xref="S5.4.p4.13.m13.1.1.2.3">′</ci></apply><ci id="S5.4.p4.13.m13.1.1.3.cmml" xref="S5.4.p4.13.m13.1.1.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p4.13.m13.1c">a^{\prime}<\nu</annotation><annotation encoding="application/x-llamapun" id="S5.4.p4.13.m13.1d">italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT < italic_ν</annotation></semantics></math>. ∎</p> </div> </div> <div class="ltx_para" id="S5.p3"> <p class="ltx_p" id="S5.p3.4">We now proceed to construct our antichain. This is an adaptation of the classical way to construct <math alttext="2^{\aleph_{1}}" class="ltx_Math" display="inline" id="S5.p3.1.m1.1"><semantics id="S5.p3.1.m1.1a"><msup id="S5.p3.1.m1.1.1" xref="S5.p3.1.m1.1.1.cmml"><mn id="S5.p3.1.m1.1.1.2" xref="S5.p3.1.m1.1.1.2.cmml">2</mn><msub id="S5.p3.1.m1.1.1.3" xref="S5.p3.1.m1.1.1.3.cmml"><mi id="S5.p3.1.m1.1.1.3.2" mathvariant="normal" xref="S5.p3.1.m1.1.1.3.2.cmml">ℵ</mi><mn id="S5.p3.1.m1.1.1.3.3" xref="S5.p3.1.m1.1.1.3.3.cmml">1</mn></msub></msup><annotation-xml encoding="MathML-Content" id="S5.p3.1.m1.1b"><apply id="S5.p3.1.m1.1.1.cmml" xref="S5.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S5.p3.1.m1.1.1.1.cmml" xref="S5.p3.1.m1.1.1">superscript</csymbol><cn id="S5.p3.1.m1.1.1.2.cmml" type="integer" xref="S5.p3.1.m1.1.1.2">2</cn><apply id="S5.p3.1.m1.1.1.3.cmml" xref="S5.p3.1.m1.1.1.3"><csymbol cd="ambiguous" id="S5.p3.1.m1.1.1.3.1.cmml" xref="S5.p3.1.m1.1.1.3">subscript</csymbol><ci id="S5.p3.1.m1.1.1.3.2.cmml" xref="S5.p3.1.m1.1.1.3.2">ℵ</ci><cn id="S5.p3.1.m1.1.1.3.3.cmml" type="integer" xref="S5.p3.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p3.1.m1.1c">2^{\aleph_{1}}</annotation><annotation encoding="application/x-llamapun" id="S5.p3.1.m1.1d">2 start_POSTSUPERSCRIPT roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> many <math alttext="\aleph_{1}" class="ltx_Math" display="inline" id="S5.p3.2.m2.1"><semantics id="S5.p3.2.m2.1a"><msub id="S5.p3.2.m2.1.1" xref="S5.p3.2.m2.1.1.cmml"><mi id="S5.p3.2.m2.1.1.2" mathvariant="normal" xref="S5.p3.2.m2.1.1.2.cmml">ℵ</mi><mn id="S5.p3.2.m2.1.1.3" xref="S5.p3.2.m2.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S5.p3.2.m2.1b"><apply id="S5.p3.2.m2.1.1.cmml" xref="S5.p3.2.m2.1.1"><csymbol cd="ambiguous" id="S5.p3.2.m2.1.1.1.cmml" xref="S5.p3.2.m2.1.1">subscript</csymbol><ci id="S5.p3.2.m2.1.1.2.cmml" xref="S5.p3.2.m2.1.1.2">ℵ</ci><cn id="S5.p3.2.m2.1.1.3.cmml" type="integer" xref="S5.p3.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p3.2.m2.1c">\aleph_{1}</annotation><annotation encoding="application/x-llamapun" id="S5.p3.2.m2.1d">roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-dense non isomorphic Aronszajn lines (see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib3" title="">3</a>, Theorem 4.8]</cite>), and it shows that one of the main differences between <math alttext="\preceq" class="ltx_Math" display="inline" id="S5.p3.3.m3.1"><semantics id="S5.p3.3.m3.1a"><mo id="S5.p3.3.m3.1.1" xref="S5.p3.3.m3.1.1.cmml">⪯</mo><annotation-xml encoding="MathML-Content" id="S5.p3.3.m3.1b"><csymbol cd="latexml" id="S5.p3.3.m3.1.1.cmml" xref="S5.p3.3.m3.1.1">precedes-or-equals</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S5.p3.3.m3.1c">\preceq</annotation><annotation encoding="application/x-llamapun" id="S5.p3.3.m3.1d">⪯</annotation></semantics></math> and <math alttext="\trianglerighteq" class="ltx_Math" display="inline" id="S5.p3.4.m4.1"><semantics id="S5.p3.4.m4.1a"><mi id="S5.p3.4.m4.1.1" mathvariant="normal" xref="S5.p3.4.m4.1.1.cmml">⊵</mi><annotation-xml encoding="MathML-Content" id="S5.p3.4.m4.1b"><ci id="S5.p3.4.m4.1.1.cmml" xref="S5.p3.4.m4.1.1">⊵</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p3.4.m4.1c">\trianglerighteq</annotation><annotation encoding="application/x-llamapun" id="S5.p3.4.m4.1d">⊵</annotation></semantics></math> is that epimorphisms are continuous, while embeddings need not to.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S5.Thmtheorem3"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem3.1.1.1">Theorem 5.3</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem3.2.2">.</span> </h6> <div class="ltx_para" id="S5.Thmtheorem3.p1"> <p class="ltx_p" id="S5.Thmtheorem3.p1.3"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem3.p1.3.3">There is an <math alttext="\trianglelefteq" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.1.1.m1.1"><semantics id="S5.Thmtheorem3.p1.1.1.m1.1a"><mi id="S5.Thmtheorem3.p1.1.1.m1.1.1" mathvariant="normal" xref="S5.Thmtheorem3.p1.1.1.m1.1.1.cmml">⊴</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.1.1.m1.1b"><ci id="S5.Thmtheorem3.p1.1.1.m1.1.1.cmml" xref="S5.Thmtheorem3.p1.1.1.m1.1.1">⊴</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.1.1.m1.1c">\trianglelefteq</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.1.1.m1.1d">⊴</annotation></semantics></math>-antichain of <math alttext="\aleph_{1}" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.2.2.m2.1"><semantics id="S5.Thmtheorem3.p1.2.2.m2.1a"><msub id="S5.Thmtheorem3.p1.2.2.m2.1.1" xref="S5.Thmtheorem3.p1.2.2.m2.1.1.cmml"><mi id="S5.Thmtheorem3.p1.2.2.m2.1.1.2" mathvariant="normal" xref="S5.Thmtheorem3.p1.2.2.m2.1.1.2.cmml">ℵ</mi><mn id="S5.Thmtheorem3.p1.2.2.m2.1.1.3" xref="S5.Thmtheorem3.p1.2.2.m2.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.2.2.m2.1b"><apply id="S5.Thmtheorem3.p1.2.2.m2.1.1.cmml" xref="S5.Thmtheorem3.p1.2.2.m2.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p1.2.2.m2.1.1.1.cmml" xref="S5.Thmtheorem3.p1.2.2.m2.1.1">subscript</csymbol><ci id="S5.Thmtheorem3.p1.2.2.m2.1.1.2.cmml" xref="S5.Thmtheorem3.p1.2.2.m2.1.1.2">ℵ</ci><cn id="S5.Thmtheorem3.p1.2.2.m2.1.1.3.cmml" type="integer" xref="S5.Thmtheorem3.p1.2.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.2.2.m2.1c">\aleph_{1}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.2.2.m2.1d">roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-dense Aronszajn lines of size <math alttext="2^{\aleph_{1}}" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.3.3.m3.1"><semantics id="S5.Thmtheorem3.p1.3.3.m3.1a"><msup id="S5.Thmtheorem3.p1.3.3.m3.1.1" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.cmml"><mn id="S5.Thmtheorem3.p1.3.3.m3.1.1.2" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.2.cmml">2</mn><msub id="S5.Thmtheorem3.p1.3.3.m3.1.1.3" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.3.cmml"><mi id="S5.Thmtheorem3.p1.3.3.m3.1.1.3.2" mathvariant="normal" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.3.2.cmml">ℵ</mi><mn id="S5.Thmtheorem3.p1.3.3.m3.1.1.3.3" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.3.3.cmml">1</mn></msub></msup><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.3.3.m3.1b"><apply id="S5.Thmtheorem3.p1.3.3.m3.1.1.cmml" xref="S5.Thmtheorem3.p1.3.3.m3.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p1.3.3.m3.1.1.1.cmml" xref="S5.Thmtheorem3.p1.3.3.m3.1.1">superscript</csymbol><cn id="S5.Thmtheorem3.p1.3.3.m3.1.1.2.cmml" type="integer" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.2">2</cn><apply id="S5.Thmtheorem3.p1.3.3.m3.1.1.3.cmml" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.3"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p1.3.3.m3.1.1.3.1.cmml" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.3">subscript</csymbol><ci id="S5.Thmtheorem3.p1.3.3.m3.1.1.3.2.cmml" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.3.2">ℵ</ci><cn id="S5.Thmtheorem3.p1.3.3.m3.1.1.3.3.cmml" type="integer" xref="S5.Thmtheorem3.p1.3.3.m3.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.3.3.m3.1c">2^{\aleph_{1}}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.3.3.m3.1d">2 start_POSTSUPERSCRIPT roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S5.6"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S5.5.p1"> <p class="ltx_p" id="S5.5.p1.11">Let <math alttext="\langle S_{\xi}:\xi<\omega_{1}\rangle" class="ltx_math_unparsed" display="inline" id="S5.5.p1.1.m1.1"><semantics id="S5.5.p1.1.m1.1a"><mrow id="S5.5.p1.1.m1.1b"><mo id="S5.5.p1.1.m1.1.1" stretchy="false">⟨</mo><msub id="S5.5.p1.1.m1.1.2"><mi id="S5.5.p1.1.m1.1.2.2">S</mi><mi id="S5.5.p1.1.m1.1.2.3">ξ</mi></msub><mo id="S5.5.p1.1.m1.1.3" lspace="0.278em" rspace="0.278em">:</mo><mi id="S5.5.p1.1.m1.1.4">ξ</mi><mo id="S5.5.p1.1.m1.1.5"><</mo><msub id="S5.5.p1.1.m1.1.6"><mi id="S5.5.p1.1.m1.1.6.2">ω</mi><mn id="S5.5.p1.1.m1.1.6.3">1</mn></msub><mo id="S5.5.p1.1.m1.1.7" stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex" id="S5.5.p1.1.m1.1c">\langle S_{\xi}:\xi<\omega_{1}\rangle</annotation><annotation encoding="application/x-llamapun" id="S5.5.p1.1.m1.1d">⟨ italic_S start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT : italic_ξ < italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⟩</annotation></semantics></math> be a family of disjoint stationary subsets of <math alttext="\omega_{1}" class="ltx_Math" display="inline" id="S5.5.p1.2.m2.1"><semantics id="S5.5.p1.2.m2.1a"><msub id="S5.5.p1.2.m2.1.1" xref="S5.5.p1.2.m2.1.1.cmml"><mi id="S5.5.p1.2.m2.1.1.2" xref="S5.5.p1.2.m2.1.1.2.cmml">ω</mi><mn id="S5.5.p1.2.m2.1.1.3" xref="S5.5.p1.2.m2.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S5.5.p1.2.m2.1b"><apply id="S5.5.p1.2.m2.1.1.cmml" xref="S5.5.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S5.5.p1.2.m2.1.1.1.cmml" xref="S5.5.p1.2.m2.1.1">subscript</csymbol><ci id="S5.5.p1.2.m2.1.1.2.cmml" xref="S5.5.p1.2.m2.1.1.2">𝜔</ci><cn id="S5.5.p1.2.m2.1.1.3.cmml" type="integer" xref="S5.5.p1.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p1.2.m2.1c">\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S5.5.p1.2.m2.1d">italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, and for <math alttext="Z\subseteq\omega_{1}" class="ltx_Math" display="inline" id="S5.5.p1.3.m3.1"><semantics id="S5.5.p1.3.m3.1a"><mrow id="S5.5.p1.3.m3.1.1" xref="S5.5.p1.3.m3.1.1.cmml"><mi id="S5.5.p1.3.m3.1.1.2" xref="S5.5.p1.3.m3.1.1.2.cmml">Z</mi><mo id="S5.5.p1.3.m3.1.1.1" xref="S5.5.p1.3.m3.1.1.1.cmml">⊆</mo><msub id="S5.5.p1.3.m3.1.1.3" xref="S5.5.p1.3.m3.1.1.3.cmml"><mi id="S5.5.p1.3.m3.1.1.3.2" xref="S5.5.p1.3.m3.1.1.3.2.cmml">ω</mi><mn id="S5.5.p1.3.m3.1.1.3.3" xref="S5.5.p1.3.m3.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.5.p1.3.m3.1b"><apply id="S5.5.p1.3.m3.1.1.cmml" xref="S5.5.p1.3.m3.1.1"><subset id="S5.5.p1.3.m3.1.1.1.cmml" xref="S5.5.p1.3.m3.1.1.1"></subset><ci id="S5.5.p1.3.m3.1.1.2.cmml" xref="S5.5.p1.3.m3.1.1.2">𝑍</ci><apply id="S5.5.p1.3.m3.1.1.3.cmml" xref="S5.5.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S5.5.p1.3.m3.1.1.3.1.cmml" xref="S5.5.p1.3.m3.1.1.3">subscript</csymbol><ci id="S5.5.p1.3.m3.1.1.3.2.cmml" xref="S5.5.p1.3.m3.1.1.3.2">𝜔</ci><cn id="S5.5.p1.3.m3.1.1.3.3.cmml" type="integer" xref="S5.5.p1.3.m3.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p1.3.m3.1c">Z\subseteq\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S5.5.p1.3.m3.1d">italic_Z ⊆ italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> let <math alttext="L(Z):=\bigcup_{\xi\in Z}S_{\xi}" class="ltx_Math" display="inline" id="S5.5.p1.4.m4.1"><semantics id="S5.5.p1.4.m4.1a"><mrow id="S5.5.p1.4.m4.1.2" xref="S5.5.p1.4.m4.1.2.cmml"><mrow id="S5.5.p1.4.m4.1.2.2" xref="S5.5.p1.4.m4.1.2.2.cmml"><mi id="S5.5.p1.4.m4.1.2.2.2" xref="S5.5.p1.4.m4.1.2.2.2.cmml">L</mi><mo id="S5.5.p1.4.m4.1.2.2.1" xref="S5.5.p1.4.m4.1.2.2.1.cmml">⁢</mo><mrow id="S5.5.p1.4.m4.1.2.2.3.2" xref="S5.5.p1.4.m4.1.2.2.cmml"><mo id="S5.5.p1.4.m4.1.2.2.3.2.1" stretchy="false" xref="S5.5.p1.4.m4.1.2.2.cmml">(</mo><mi id="S5.5.p1.4.m4.1.1" xref="S5.5.p1.4.m4.1.1.cmml">Z</mi><mo id="S5.5.p1.4.m4.1.2.2.3.2.2" rspace="0.278em" stretchy="false" xref="S5.5.p1.4.m4.1.2.2.cmml">)</mo></mrow></mrow><mo id="S5.5.p1.4.m4.1.2.1" rspace="0.111em" xref="S5.5.p1.4.m4.1.2.1.cmml">:=</mo><mrow id="S5.5.p1.4.m4.1.2.3" xref="S5.5.p1.4.m4.1.2.3.cmml"><msub id="S5.5.p1.4.m4.1.2.3.1" xref="S5.5.p1.4.m4.1.2.3.1.cmml"><mo id="S5.5.p1.4.m4.1.2.3.1.2" xref="S5.5.p1.4.m4.1.2.3.1.2.cmml">⋃</mo><mrow id="S5.5.p1.4.m4.1.2.3.1.3" xref="S5.5.p1.4.m4.1.2.3.1.3.cmml"><mi id="S5.5.p1.4.m4.1.2.3.1.3.2" xref="S5.5.p1.4.m4.1.2.3.1.3.2.cmml">ξ</mi><mo id="S5.5.p1.4.m4.1.2.3.1.3.1" xref="S5.5.p1.4.m4.1.2.3.1.3.1.cmml">∈</mo><mi id="S5.5.p1.4.m4.1.2.3.1.3.3" xref="S5.5.p1.4.m4.1.2.3.1.3.3.cmml">Z</mi></mrow></msub><msub id="S5.5.p1.4.m4.1.2.3.2" xref="S5.5.p1.4.m4.1.2.3.2.cmml"><mi id="S5.5.p1.4.m4.1.2.3.2.2" xref="S5.5.p1.4.m4.1.2.3.2.2.cmml">S</mi><mi id="S5.5.p1.4.m4.1.2.3.2.3" xref="S5.5.p1.4.m4.1.2.3.2.3.cmml">ξ</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.5.p1.4.m4.1b"><apply id="S5.5.p1.4.m4.1.2.cmml" xref="S5.5.p1.4.m4.1.2"><csymbol cd="latexml" id="S5.5.p1.4.m4.1.2.1.cmml" xref="S5.5.p1.4.m4.1.2.1">assign</csymbol><apply id="S5.5.p1.4.m4.1.2.2.cmml" xref="S5.5.p1.4.m4.1.2.2"><times id="S5.5.p1.4.m4.1.2.2.1.cmml" xref="S5.5.p1.4.m4.1.2.2.1"></times><ci id="S5.5.p1.4.m4.1.2.2.2.cmml" xref="S5.5.p1.4.m4.1.2.2.2">𝐿</ci><ci id="S5.5.p1.4.m4.1.1.cmml" xref="S5.5.p1.4.m4.1.1">𝑍</ci></apply><apply id="S5.5.p1.4.m4.1.2.3.cmml" xref="S5.5.p1.4.m4.1.2.3"><apply id="S5.5.p1.4.m4.1.2.3.1.cmml" xref="S5.5.p1.4.m4.1.2.3.1"><csymbol cd="ambiguous" id="S5.5.p1.4.m4.1.2.3.1.1.cmml" xref="S5.5.p1.4.m4.1.2.3.1">subscript</csymbol><union id="S5.5.p1.4.m4.1.2.3.1.2.cmml" xref="S5.5.p1.4.m4.1.2.3.1.2"></union><apply id="S5.5.p1.4.m4.1.2.3.1.3.cmml" xref="S5.5.p1.4.m4.1.2.3.1.3"><in id="S5.5.p1.4.m4.1.2.3.1.3.1.cmml" xref="S5.5.p1.4.m4.1.2.3.1.3.1"></in><ci id="S5.5.p1.4.m4.1.2.3.1.3.2.cmml" xref="S5.5.p1.4.m4.1.2.3.1.3.2">𝜉</ci><ci id="S5.5.p1.4.m4.1.2.3.1.3.3.cmml" xref="S5.5.p1.4.m4.1.2.3.1.3.3">𝑍</ci></apply></apply><apply id="S5.5.p1.4.m4.1.2.3.2.cmml" xref="S5.5.p1.4.m4.1.2.3.2"><csymbol cd="ambiguous" id="S5.5.p1.4.m4.1.2.3.2.1.cmml" xref="S5.5.p1.4.m4.1.2.3.2">subscript</csymbol><ci id="S5.5.p1.4.m4.1.2.3.2.2.cmml" xref="S5.5.p1.4.m4.1.2.3.2.2">𝑆</ci><ci id="S5.5.p1.4.m4.1.2.3.2.3.cmml" xref="S5.5.p1.4.m4.1.2.3.2.3">𝜉</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p1.4.m4.1c">L(Z):=\bigcup_{\xi\in Z}S_{\xi}</annotation><annotation encoding="application/x-llamapun" id="S5.5.p1.4.m4.1d">italic_L ( italic_Z ) := ⋃ start_POSTSUBSCRIPT italic_ξ ∈ italic_Z end_POSTSUBSCRIPT italic_S start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="R(Z):=\bigcup_{\xi\in(\omega_{1}\setminus Z)}S_{\xi}" class="ltx_Math" display="inline" id="S5.5.p1.5.m5.2"><semantics id="S5.5.p1.5.m5.2a"><mrow id="S5.5.p1.5.m5.2.3" xref="S5.5.p1.5.m5.2.3.cmml"><mrow id="S5.5.p1.5.m5.2.3.2" xref="S5.5.p1.5.m5.2.3.2.cmml"><mi id="S5.5.p1.5.m5.2.3.2.2" xref="S5.5.p1.5.m5.2.3.2.2.cmml">R</mi><mo id="S5.5.p1.5.m5.2.3.2.1" xref="S5.5.p1.5.m5.2.3.2.1.cmml">⁢</mo><mrow id="S5.5.p1.5.m5.2.3.2.3.2" xref="S5.5.p1.5.m5.2.3.2.cmml"><mo id="S5.5.p1.5.m5.2.3.2.3.2.1" stretchy="false" xref="S5.5.p1.5.m5.2.3.2.cmml">(</mo><mi id="S5.5.p1.5.m5.2.2" xref="S5.5.p1.5.m5.2.2.cmml">Z</mi><mo id="S5.5.p1.5.m5.2.3.2.3.2.2" rspace="0.278em" stretchy="false" xref="S5.5.p1.5.m5.2.3.2.cmml">)</mo></mrow></mrow><mo id="S5.5.p1.5.m5.2.3.1" rspace="0.111em" xref="S5.5.p1.5.m5.2.3.1.cmml">:=</mo><mrow id="S5.5.p1.5.m5.2.3.3" xref="S5.5.p1.5.m5.2.3.3.cmml"><msub id="S5.5.p1.5.m5.2.3.3.1" xref="S5.5.p1.5.m5.2.3.3.1.cmml"><mo id="S5.5.p1.5.m5.2.3.3.1.2" xref="S5.5.p1.5.m5.2.3.3.1.2.cmml">⋃</mo><mrow id="S5.5.p1.5.m5.1.1.1" xref="S5.5.p1.5.m5.1.1.1.cmml"><mi id="S5.5.p1.5.m5.1.1.1.3" xref="S5.5.p1.5.m5.1.1.1.3.cmml">ξ</mi><mo id="S5.5.p1.5.m5.1.1.1.2" xref="S5.5.p1.5.m5.1.1.1.2.cmml">∈</mo><mrow id="S5.5.p1.5.m5.1.1.1.1.1" xref="S5.5.p1.5.m5.1.1.1.1.1.1.cmml"><mo id="S5.5.p1.5.m5.1.1.1.1.1.2" stretchy="false" xref="S5.5.p1.5.m5.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.5.p1.5.m5.1.1.1.1.1.1" xref="S5.5.p1.5.m5.1.1.1.1.1.1.cmml"><msub id="S5.5.p1.5.m5.1.1.1.1.1.1.2" xref="S5.5.p1.5.m5.1.1.1.1.1.1.2.cmml"><mi id="S5.5.p1.5.m5.1.1.1.1.1.1.2.2" xref="S5.5.p1.5.m5.1.1.1.1.1.1.2.2.cmml">ω</mi><mn id="S5.5.p1.5.m5.1.1.1.1.1.1.2.3" xref="S5.5.p1.5.m5.1.1.1.1.1.1.2.3.cmml">1</mn></msub><mo id="S5.5.p1.5.m5.1.1.1.1.1.1.1" xref="S5.5.p1.5.m5.1.1.1.1.1.1.1.cmml">∖</mo><mi id="S5.5.p1.5.m5.1.1.1.1.1.1.3" xref="S5.5.p1.5.m5.1.1.1.1.1.1.3.cmml">Z</mi></mrow><mo id="S5.5.p1.5.m5.1.1.1.1.1.3" stretchy="false" xref="S5.5.p1.5.m5.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></msub><msub id="S5.5.p1.5.m5.2.3.3.2" xref="S5.5.p1.5.m5.2.3.3.2.cmml"><mi id="S5.5.p1.5.m5.2.3.3.2.2" xref="S5.5.p1.5.m5.2.3.3.2.2.cmml">S</mi><mi id="S5.5.p1.5.m5.2.3.3.2.3" xref="S5.5.p1.5.m5.2.3.3.2.3.cmml">ξ</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.5.p1.5.m5.2b"><apply id="S5.5.p1.5.m5.2.3.cmml" xref="S5.5.p1.5.m5.2.3"><csymbol cd="latexml" id="S5.5.p1.5.m5.2.3.1.cmml" xref="S5.5.p1.5.m5.2.3.1">assign</csymbol><apply id="S5.5.p1.5.m5.2.3.2.cmml" xref="S5.5.p1.5.m5.2.3.2"><times id="S5.5.p1.5.m5.2.3.2.1.cmml" xref="S5.5.p1.5.m5.2.3.2.1"></times><ci id="S5.5.p1.5.m5.2.3.2.2.cmml" xref="S5.5.p1.5.m5.2.3.2.2">𝑅</ci><ci id="S5.5.p1.5.m5.2.2.cmml" xref="S5.5.p1.5.m5.2.2">𝑍</ci></apply><apply id="S5.5.p1.5.m5.2.3.3.cmml" xref="S5.5.p1.5.m5.2.3.3"><apply id="S5.5.p1.5.m5.2.3.3.1.cmml" xref="S5.5.p1.5.m5.2.3.3.1"><csymbol cd="ambiguous" id="S5.5.p1.5.m5.2.3.3.1.1.cmml" xref="S5.5.p1.5.m5.2.3.3.1">subscript</csymbol><union id="S5.5.p1.5.m5.2.3.3.1.2.cmml" xref="S5.5.p1.5.m5.2.3.3.1.2"></union><apply id="S5.5.p1.5.m5.1.1.1.cmml" xref="S5.5.p1.5.m5.1.1.1"><in id="S5.5.p1.5.m5.1.1.1.2.cmml" xref="S5.5.p1.5.m5.1.1.1.2"></in><ci id="S5.5.p1.5.m5.1.1.1.3.cmml" xref="S5.5.p1.5.m5.1.1.1.3">𝜉</ci><apply id="S5.5.p1.5.m5.1.1.1.1.1.1.cmml" xref="S5.5.p1.5.m5.1.1.1.1.1"><setdiff id="S5.5.p1.5.m5.1.1.1.1.1.1.1.cmml" xref="S5.5.p1.5.m5.1.1.1.1.1.1.1"></setdiff><apply id="S5.5.p1.5.m5.1.1.1.1.1.1.2.cmml" xref="S5.5.p1.5.m5.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S5.5.p1.5.m5.1.1.1.1.1.1.2.1.cmml" xref="S5.5.p1.5.m5.1.1.1.1.1.1.2">subscript</csymbol><ci id="S5.5.p1.5.m5.1.1.1.1.1.1.2.2.cmml" xref="S5.5.p1.5.m5.1.1.1.1.1.1.2.2">𝜔</ci><cn id="S5.5.p1.5.m5.1.1.1.1.1.1.2.3.cmml" type="integer" xref="S5.5.p1.5.m5.1.1.1.1.1.1.2.3">1</cn></apply><ci id="S5.5.p1.5.m5.1.1.1.1.1.1.3.cmml" xref="S5.5.p1.5.m5.1.1.1.1.1.1.3">𝑍</ci></apply></apply></apply><apply id="S5.5.p1.5.m5.2.3.3.2.cmml" xref="S5.5.p1.5.m5.2.3.3.2"><csymbol cd="ambiguous" id="S5.5.p1.5.m5.2.3.3.2.1.cmml" xref="S5.5.p1.5.m5.2.3.3.2">subscript</csymbol><ci id="S5.5.p1.5.m5.2.3.3.2.2.cmml" xref="S5.5.p1.5.m5.2.3.3.2.2">𝑆</ci><ci id="S5.5.p1.5.m5.2.3.3.2.3.cmml" xref="S5.5.p1.5.m5.2.3.3.2.3">𝜉</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p1.5.m5.2c">R(Z):=\bigcup_{\xi\in(\omega_{1}\setminus Z)}S_{\xi}</annotation><annotation encoding="application/x-llamapun" id="S5.5.p1.5.m5.2d">italic_R ( italic_Z ) := ⋃ start_POSTSUBSCRIPT italic_ξ ∈ ( italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∖ italic_Z ) end_POSTSUBSCRIPT italic_S start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT</annotation></semantics></math>. Using <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S4.Thmtheorem5" title="Theorem 4.5. ‣ 4. Aronszajn line decompositions ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">4.5</span></a>, let <math alttext="A^{Z}" class="ltx_Math" display="inline" id="S5.5.p1.6.m6.1"><semantics id="S5.5.p1.6.m6.1a"><msup id="S5.5.p1.6.m6.1.1" xref="S5.5.p1.6.m6.1.1.cmml"><mi id="S5.5.p1.6.m6.1.1.2" xref="S5.5.p1.6.m6.1.1.2.cmml">A</mi><mi id="S5.5.p1.6.m6.1.1.3" xref="S5.5.p1.6.m6.1.1.3.cmml">Z</mi></msup><annotation-xml encoding="MathML-Content" id="S5.5.p1.6.m6.1b"><apply id="S5.5.p1.6.m6.1.1.cmml" xref="S5.5.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S5.5.p1.6.m6.1.1.1.cmml" xref="S5.5.p1.6.m6.1.1">superscript</csymbol><ci id="S5.5.p1.6.m6.1.1.2.cmml" xref="S5.5.p1.6.m6.1.1.2">𝐴</ci><ci id="S5.5.p1.6.m6.1.1.3.cmml" xref="S5.5.p1.6.m6.1.1.3">𝑍</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p1.6.m6.1c">A^{Z}</annotation><annotation encoding="application/x-llamapun" id="S5.5.p1.6.m6.1d">italic_A start_POSTSUPERSCRIPT italic_Z end_POSTSUPERSCRIPT</annotation></semantics></math> be an Aronszajn line and <math alttext="D^{Z}" class="ltx_Math" display="inline" id="S5.5.p1.7.m7.1"><semantics id="S5.5.p1.7.m7.1a"><msup id="S5.5.p1.7.m7.1.1" xref="S5.5.p1.7.m7.1.1.cmml"><mi id="S5.5.p1.7.m7.1.1.2" xref="S5.5.p1.7.m7.1.1.2.cmml">D</mi><mi id="S5.5.p1.7.m7.1.1.3" xref="S5.5.p1.7.m7.1.1.3.cmml">Z</mi></msup><annotation-xml encoding="MathML-Content" id="S5.5.p1.7.m7.1b"><apply id="S5.5.p1.7.m7.1.1.cmml" xref="S5.5.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S5.5.p1.7.m7.1.1.1.cmml" xref="S5.5.p1.7.m7.1.1">superscript</csymbol><ci id="S5.5.p1.7.m7.1.1.2.cmml" xref="S5.5.p1.7.m7.1.1.2">𝐷</ci><ci id="S5.5.p1.7.m7.1.1.3.cmml" xref="S5.5.p1.7.m7.1.1.3">𝑍</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p1.7.m7.1c">D^{Z}</annotation><annotation encoding="application/x-llamapun" id="S5.5.p1.7.m7.1d">italic_D start_POSTSUPERSCRIPT italic_Z end_POSTSUPERSCRIPT</annotation></semantics></math> a decomposition for <math alttext="A^{Z}" class="ltx_Math" display="inline" id="S5.5.p1.8.m8.1"><semantics id="S5.5.p1.8.m8.1a"><msup id="S5.5.p1.8.m8.1.1" xref="S5.5.p1.8.m8.1.1.cmml"><mi id="S5.5.p1.8.m8.1.1.2" xref="S5.5.p1.8.m8.1.1.2.cmml">A</mi><mi id="S5.5.p1.8.m8.1.1.3" xref="S5.5.p1.8.m8.1.1.3.cmml">Z</mi></msup><annotation-xml encoding="MathML-Content" id="S5.5.p1.8.m8.1b"><apply id="S5.5.p1.8.m8.1.1.cmml" xref="S5.5.p1.8.m8.1.1"><csymbol cd="ambiguous" id="S5.5.p1.8.m8.1.1.1.cmml" xref="S5.5.p1.8.m8.1.1">superscript</csymbol><ci id="S5.5.p1.8.m8.1.1.2.cmml" xref="S5.5.p1.8.m8.1.1.2">𝐴</ci><ci id="S5.5.p1.8.m8.1.1.3.cmml" xref="S5.5.p1.8.m8.1.1.3">𝑍</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p1.8.m8.1c">A^{Z}</annotation><annotation encoding="application/x-llamapun" id="S5.5.p1.8.m8.1d">italic_A start_POSTSUPERSCRIPT italic_Z end_POSTSUPERSCRIPT</annotation></semantics></math> such that <math alttext="\hat{\mathscr{L}}(A^{Z},D^{Z})=L(Z)" class="ltx_Math" display="inline" id="S5.5.p1.9.m9.3"><semantics id="S5.5.p1.9.m9.3a"><mrow id="S5.5.p1.9.m9.3.3" xref="S5.5.p1.9.m9.3.3.cmml"><mrow id="S5.5.p1.9.m9.3.3.2" xref="S5.5.p1.9.m9.3.3.2.cmml"><mover accent="true" id="S5.5.p1.9.m9.3.3.2.4" xref="S5.5.p1.9.m9.3.3.2.4.cmml"><mi class="ltx_font_mathscript" id="S5.5.p1.9.m9.3.3.2.4.2" xref="S5.5.p1.9.m9.3.3.2.4.2.cmml">ℒ</mi><mo id="S5.5.p1.9.m9.3.3.2.4.1" xref="S5.5.p1.9.m9.3.3.2.4.1.cmml">^</mo></mover><mo id="S5.5.p1.9.m9.3.3.2.3" xref="S5.5.p1.9.m9.3.3.2.3.cmml">⁢</mo><mrow id="S5.5.p1.9.m9.3.3.2.2.2" xref="S5.5.p1.9.m9.3.3.2.2.3.cmml"><mo id="S5.5.p1.9.m9.3.3.2.2.2.3" stretchy="false" xref="S5.5.p1.9.m9.3.3.2.2.3.cmml">(</mo><msup id="S5.5.p1.9.m9.2.2.1.1.1.1" xref="S5.5.p1.9.m9.2.2.1.1.1.1.cmml"><mi id="S5.5.p1.9.m9.2.2.1.1.1.1.2" xref="S5.5.p1.9.m9.2.2.1.1.1.1.2.cmml">A</mi><mi id="S5.5.p1.9.m9.2.2.1.1.1.1.3" xref="S5.5.p1.9.m9.2.2.1.1.1.1.3.cmml">Z</mi></msup><mo id="S5.5.p1.9.m9.3.3.2.2.2.4" xref="S5.5.p1.9.m9.3.3.2.2.3.cmml">,</mo><msup id="S5.5.p1.9.m9.3.3.2.2.2.2" xref="S5.5.p1.9.m9.3.3.2.2.2.2.cmml"><mi id="S5.5.p1.9.m9.3.3.2.2.2.2.2" xref="S5.5.p1.9.m9.3.3.2.2.2.2.2.cmml">D</mi><mi id="S5.5.p1.9.m9.3.3.2.2.2.2.3" xref="S5.5.p1.9.m9.3.3.2.2.2.2.3.cmml">Z</mi></msup><mo id="S5.5.p1.9.m9.3.3.2.2.2.5" stretchy="false" xref="S5.5.p1.9.m9.3.3.2.2.3.cmml">)</mo></mrow></mrow><mo id="S5.5.p1.9.m9.3.3.3" xref="S5.5.p1.9.m9.3.3.3.cmml">=</mo><mrow id="S5.5.p1.9.m9.3.3.4" xref="S5.5.p1.9.m9.3.3.4.cmml"><mi id="S5.5.p1.9.m9.3.3.4.2" xref="S5.5.p1.9.m9.3.3.4.2.cmml">L</mi><mo id="S5.5.p1.9.m9.3.3.4.1" xref="S5.5.p1.9.m9.3.3.4.1.cmml">⁢</mo><mrow id="S5.5.p1.9.m9.3.3.4.3.2" xref="S5.5.p1.9.m9.3.3.4.cmml"><mo id="S5.5.p1.9.m9.3.3.4.3.2.1" stretchy="false" xref="S5.5.p1.9.m9.3.3.4.cmml">(</mo><mi id="S5.5.p1.9.m9.1.1" xref="S5.5.p1.9.m9.1.1.cmml">Z</mi><mo id="S5.5.p1.9.m9.3.3.4.3.2.2" stretchy="false" xref="S5.5.p1.9.m9.3.3.4.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.5.p1.9.m9.3b"><apply id="S5.5.p1.9.m9.3.3.cmml" xref="S5.5.p1.9.m9.3.3"><eq id="S5.5.p1.9.m9.3.3.3.cmml" xref="S5.5.p1.9.m9.3.3.3"></eq><apply id="S5.5.p1.9.m9.3.3.2.cmml" xref="S5.5.p1.9.m9.3.3.2"><times id="S5.5.p1.9.m9.3.3.2.3.cmml" xref="S5.5.p1.9.m9.3.3.2.3"></times><apply id="S5.5.p1.9.m9.3.3.2.4.cmml" xref="S5.5.p1.9.m9.3.3.2.4"><ci id="S5.5.p1.9.m9.3.3.2.4.1.cmml" xref="S5.5.p1.9.m9.3.3.2.4.1">^</ci><ci id="S5.5.p1.9.m9.3.3.2.4.2.cmml" xref="S5.5.p1.9.m9.3.3.2.4.2">ℒ</ci></apply><interval closure="open" id="S5.5.p1.9.m9.3.3.2.2.3.cmml" xref="S5.5.p1.9.m9.3.3.2.2.2"><apply id="S5.5.p1.9.m9.2.2.1.1.1.1.cmml" xref="S5.5.p1.9.m9.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S5.5.p1.9.m9.2.2.1.1.1.1.1.cmml" xref="S5.5.p1.9.m9.2.2.1.1.1.1">superscript</csymbol><ci id="S5.5.p1.9.m9.2.2.1.1.1.1.2.cmml" xref="S5.5.p1.9.m9.2.2.1.1.1.1.2">𝐴</ci><ci id="S5.5.p1.9.m9.2.2.1.1.1.1.3.cmml" xref="S5.5.p1.9.m9.2.2.1.1.1.1.3">𝑍</ci></apply><apply id="S5.5.p1.9.m9.3.3.2.2.2.2.cmml" xref="S5.5.p1.9.m9.3.3.2.2.2.2"><csymbol cd="ambiguous" id="S5.5.p1.9.m9.3.3.2.2.2.2.1.cmml" xref="S5.5.p1.9.m9.3.3.2.2.2.2">superscript</csymbol><ci id="S5.5.p1.9.m9.3.3.2.2.2.2.2.cmml" xref="S5.5.p1.9.m9.3.3.2.2.2.2.2">𝐷</ci><ci id="S5.5.p1.9.m9.3.3.2.2.2.2.3.cmml" xref="S5.5.p1.9.m9.3.3.2.2.2.2.3">𝑍</ci></apply></interval></apply><apply id="S5.5.p1.9.m9.3.3.4.cmml" xref="S5.5.p1.9.m9.3.3.4"><times id="S5.5.p1.9.m9.3.3.4.1.cmml" xref="S5.5.p1.9.m9.3.3.4.1"></times><ci id="S5.5.p1.9.m9.3.3.4.2.cmml" xref="S5.5.p1.9.m9.3.3.4.2">𝐿</ci><ci id="S5.5.p1.9.m9.1.1.cmml" xref="S5.5.p1.9.m9.1.1">𝑍</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p1.9.m9.3c">\hat{\mathscr{L}}(A^{Z},D^{Z})=L(Z)</annotation><annotation encoding="application/x-llamapun" id="S5.5.p1.9.m9.3d">over^ start_ARG script_L end_ARG ( italic_A start_POSTSUPERSCRIPT italic_Z end_POSTSUPERSCRIPT , italic_D start_POSTSUPERSCRIPT italic_Z end_POSTSUPERSCRIPT ) = italic_L ( italic_Z )</annotation></semantics></math> and <math alttext="\hat{\mathscr{R}}(A^{Z},D^{Z})=R(Z)" class="ltx_Math" display="inline" id="S5.5.p1.10.m10.3"><semantics id="S5.5.p1.10.m10.3a"><mrow id="S5.5.p1.10.m10.3.3" xref="S5.5.p1.10.m10.3.3.cmml"><mrow id="S5.5.p1.10.m10.3.3.2" xref="S5.5.p1.10.m10.3.3.2.cmml"><mover accent="true" id="S5.5.p1.10.m10.3.3.2.4" xref="S5.5.p1.10.m10.3.3.2.4.cmml"><mi class="ltx_font_mathscript" id="S5.5.p1.10.m10.3.3.2.4.2" xref="S5.5.p1.10.m10.3.3.2.4.2.cmml">ℛ</mi><mo id="S5.5.p1.10.m10.3.3.2.4.1" xref="S5.5.p1.10.m10.3.3.2.4.1.cmml">^</mo></mover><mo id="S5.5.p1.10.m10.3.3.2.3" xref="S5.5.p1.10.m10.3.3.2.3.cmml">⁢</mo><mrow id="S5.5.p1.10.m10.3.3.2.2.2" xref="S5.5.p1.10.m10.3.3.2.2.3.cmml"><mo id="S5.5.p1.10.m10.3.3.2.2.2.3" stretchy="false" xref="S5.5.p1.10.m10.3.3.2.2.3.cmml">(</mo><msup id="S5.5.p1.10.m10.2.2.1.1.1.1" xref="S5.5.p1.10.m10.2.2.1.1.1.1.cmml"><mi id="S5.5.p1.10.m10.2.2.1.1.1.1.2" xref="S5.5.p1.10.m10.2.2.1.1.1.1.2.cmml">A</mi><mi id="S5.5.p1.10.m10.2.2.1.1.1.1.3" xref="S5.5.p1.10.m10.2.2.1.1.1.1.3.cmml">Z</mi></msup><mo id="S5.5.p1.10.m10.3.3.2.2.2.4" xref="S5.5.p1.10.m10.3.3.2.2.3.cmml">,</mo><msup id="S5.5.p1.10.m10.3.3.2.2.2.2" xref="S5.5.p1.10.m10.3.3.2.2.2.2.cmml"><mi id="S5.5.p1.10.m10.3.3.2.2.2.2.2" xref="S5.5.p1.10.m10.3.3.2.2.2.2.2.cmml">D</mi><mi id="S5.5.p1.10.m10.3.3.2.2.2.2.3" xref="S5.5.p1.10.m10.3.3.2.2.2.2.3.cmml">Z</mi></msup><mo id="S5.5.p1.10.m10.3.3.2.2.2.5" stretchy="false" xref="S5.5.p1.10.m10.3.3.2.2.3.cmml">)</mo></mrow></mrow><mo id="S5.5.p1.10.m10.3.3.3" xref="S5.5.p1.10.m10.3.3.3.cmml">=</mo><mrow id="S5.5.p1.10.m10.3.3.4" xref="S5.5.p1.10.m10.3.3.4.cmml"><mi id="S5.5.p1.10.m10.3.3.4.2" xref="S5.5.p1.10.m10.3.3.4.2.cmml">R</mi><mo id="S5.5.p1.10.m10.3.3.4.1" xref="S5.5.p1.10.m10.3.3.4.1.cmml">⁢</mo><mrow id="S5.5.p1.10.m10.3.3.4.3.2" xref="S5.5.p1.10.m10.3.3.4.cmml"><mo id="S5.5.p1.10.m10.3.3.4.3.2.1" stretchy="false" xref="S5.5.p1.10.m10.3.3.4.cmml">(</mo><mi id="S5.5.p1.10.m10.1.1" xref="S5.5.p1.10.m10.1.1.cmml">Z</mi><mo id="S5.5.p1.10.m10.3.3.4.3.2.2" stretchy="false" xref="S5.5.p1.10.m10.3.3.4.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.5.p1.10.m10.3b"><apply id="S5.5.p1.10.m10.3.3.cmml" xref="S5.5.p1.10.m10.3.3"><eq id="S5.5.p1.10.m10.3.3.3.cmml" xref="S5.5.p1.10.m10.3.3.3"></eq><apply id="S5.5.p1.10.m10.3.3.2.cmml" xref="S5.5.p1.10.m10.3.3.2"><times id="S5.5.p1.10.m10.3.3.2.3.cmml" xref="S5.5.p1.10.m10.3.3.2.3"></times><apply id="S5.5.p1.10.m10.3.3.2.4.cmml" xref="S5.5.p1.10.m10.3.3.2.4"><ci id="S5.5.p1.10.m10.3.3.2.4.1.cmml" xref="S5.5.p1.10.m10.3.3.2.4.1">^</ci><ci id="S5.5.p1.10.m10.3.3.2.4.2.cmml" xref="S5.5.p1.10.m10.3.3.2.4.2">ℛ</ci></apply><interval closure="open" id="S5.5.p1.10.m10.3.3.2.2.3.cmml" xref="S5.5.p1.10.m10.3.3.2.2.2"><apply id="S5.5.p1.10.m10.2.2.1.1.1.1.cmml" xref="S5.5.p1.10.m10.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S5.5.p1.10.m10.2.2.1.1.1.1.1.cmml" xref="S5.5.p1.10.m10.2.2.1.1.1.1">superscript</csymbol><ci id="S5.5.p1.10.m10.2.2.1.1.1.1.2.cmml" xref="S5.5.p1.10.m10.2.2.1.1.1.1.2">𝐴</ci><ci id="S5.5.p1.10.m10.2.2.1.1.1.1.3.cmml" xref="S5.5.p1.10.m10.2.2.1.1.1.1.3">𝑍</ci></apply><apply id="S5.5.p1.10.m10.3.3.2.2.2.2.cmml" xref="S5.5.p1.10.m10.3.3.2.2.2.2"><csymbol cd="ambiguous" id="S5.5.p1.10.m10.3.3.2.2.2.2.1.cmml" xref="S5.5.p1.10.m10.3.3.2.2.2.2">superscript</csymbol><ci id="S5.5.p1.10.m10.3.3.2.2.2.2.2.cmml" xref="S5.5.p1.10.m10.3.3.2.2.2.2.2">𝐷</ci><ci id="S5.5.p1.10.m10.3.3.2.2.2.2.3.cmml" xref="S5.5.p1.10.m10.3.3.2.2.2.2.3">𝑍</ci></apply></interval></apply><apply id="S5.5.p1.10.m10.3.3.4.cmml" xref="S5.5.p1.10.m10.3.3.4"><times id="S5.5.p1.10.m10.3.3.4.1.cmml" xref="S5.5.p1.10.m10.3.3.4.1"></times><ci id="S5.5.p1.10.m10.3.3.4.2.cmml" xref="S5.5.p1.10.m10.3.3.4.2">𝑅</ci><ci id="S5.5.p1.10.m10.1.1.cmml" xref="S5.5.p1.10.m10.1.1">𝑍</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p1.10.m10.3c">\hat{\mathscr{R}}(A^{Z},D^{Z})=R(Z)</annotation><annotation encoding="application/x-llamapun" id="S5.5.p1.10.m10.3d">over^ start_ARG script_R end_ARG ( italic_A start_POSTSUPERSCRIPT italic_Z end_POSTSUPERSCRIPT , italic_D start_POSTSUPERSCRIPT italic_Z end_POSTSUPERSCRIPT ) = italic_R ( italic_Z )</annotation></semantics></math>. We claim that <math alttext="\{A^{Z}:Z\subseteq\omega_{1}\}" class="ltx_Math" display="inline" id="S5.5.p1.11.m11.2"><semantics id="S5.5.p1.11.m11.2a"><mrow id="S5.5.p1.11.m11.2.2.2" xref="S5.5.p1.11.m11.2.2.3.cmml"><mo id="S5.5.p1.11.m11.2.2.2.3" stretchy="false" xref="S5.5.p1.11.m11.2.2.3.1.cmml">{</mo><msup id="S5.5.p1.11.m11.1.1.1.1" xref="S5.5.p1.11.m11.1.1.1.1.cmml"><mi id="S5.5.p1.11.m11.1.1.1.1.2" xref="S5.5.p1.11.m11.1.1.1.1.2.cmml">A</mi><mi id="S5.5.p1.11.m11.1.1.1.1.3" xref="S5.5.p1.11.m11.1.1.1.1.3.cmml">Z</mi></msup><mo id="S5.5.p1.11.m11.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S5.5.p1.11.m11.2.2.3.1.cmml">:</mo><mrow id="S5.5.p1.11.m11.2.2.2.2" xref="S5.5.p1.11.m11.2.2.2.2.cmml"><mi id="S5.5.p1.11.m11.2.2.2.2.2" xref="S5.5.p1.11.m11.2.2.2.2.2.cmml">Z</mi><mo id="S5.5.p1.11.m11.2.2.2.2.1" xref="S5.5.p1.11.m11.2.2.2.2.1.cmml">⊆</mo><msub id="S5.5.p1.11.m11.2.2.2.2.3" xref="S5.5.p1.11.m11.2.2.2.2.3.cmml"><mi id="S5.5.p1.11.m11.2.2.2.2.3.2" xref="S5.5.p1.11.m11.2.2.2.2.3.2.cmml">ω</mi><mn id="S5.5.p1.11.m11.2.2.2.2.3.3" xref="S5.5.p1.11.m11.2.2.2.2.3.3.cmml">1</mn></msub></mrow><mo id="S5.5.p1.11.m11.2.2.2.5" stretchy="false" xref="S5.5.p1.11.m11.2.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.5.p1.11.m11.2b"><apply id="S5.5.p1.11.m11.2.2.3.cmml" xref="S5.5.p1.11.m11.2.2.2"><csymbol cd="latexml" id="S5.5.p1.11.m11.2.2.3.1.cmml" xref="S5.5.p1.11.m11.2.2.2.3">conditional-set</csymbol><apply id="S5.5.p1.11.m11.1.1.1.1.cmml" xref="S5.5.p1.11.m11.1.1.1.1"><csymbol cd="ambiguous" id="S5.5.p1.11.m11.1.1.1.1.1.cmml" xref="S5.5.p1.11.m11.1.1.1.1">superscript</csymbol><ci id="S5.5.p1.11.m11.1.1.1.1.2.cmml" xref="S5.5.p1.11.m11.1.1.1.1.2">𝐴</ci><ci id="S5.5.p1.11.m11.1.1.1.1.3.cmml" xref="S5.5.p1.11.m11.1.1.1.1.3">𝑍</ci></apply><apply id="S5.5.p1.11.m11.2.2.2.2.cmml" xref="S5.5.p1.11.m11.2.2.2.2"><subset id="S5.5.p1.11.m11.2.2.2.2.1.cmml" xref="S5.5.p1.11.m11.2.2.2.2.1"></subset><ci id="S5.5.p1.11.m11.2.2.2.2.2.cmml" xref="S5.5.p1.11.m11.2.2.2.2.2">𝑍</ci><apply id="S5.5.p1.11.m11.2.2.2.2.3.cmml" xref="S5.5.p1.11.m11.2.2.2.2.3"><csymbol cd="ambiguous" id="S5.5.p1.11.m11.2.2.2.2.3.1.cmml" xref="S5.5.p1.11.m11.2.2.2.2.3">subscript</csymbol><ci id="S5.5.p1.11.m11.2.2.2.2.3.2.cmml" xref="S5.5.p1.11.m11.2.2.2.2.3.2">𝜔</ci><cn id="S5.5.p1.11.m11.2.2.2.2.3.3.cmml" type="integer" xref="S5.5.p1.11.m11.2.2.2.2.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p1.11.m11.2c">\{A^{Z}:Z\subseteq\omega_{1}\}</annotation><annotation encoding="application/x-llamapun" id="S5.5.p1.11.m11.2d">{ italic_A start_POSTSUPERSCRIPT italic_Z end_POSTSUPERSCRIPT : italic_Z ⊆ italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT }</annotation></semantics></math> is our desired antichain.</p> </div> <div class="ltx_para" id="S5.6.p2"> <p class="ltx_p" id="S5.6.p2.10">Let <math alttext="Z_{0}" class="ltx_Math" display="inline" id="S5.6.p2.1.m1.1"><semantics id="S5.6.p2.1.m1.1a"><msub id="S5.6.p2.1.m1.1.1" xref="S5.6.p2.1.m1.1.1.cmml"><mi id="S5.6.p2.1.m1.1.1.2" xref="S5.6.p2.1.m1.1.1.2.cmml">Z</mi><mn id="S5.6.p2.1.m1.1.1.3" xref="S5.6.p2.1.m1.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S5.6.p2.1.m1.1b"><apply id="S5.6.p2.1.m1.1.1.cmml" xref="S5.6.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S5.6.p2.1.m1.1.1.1.cmml" xref="S5.6.p2.1.m1.1.1">subscript</csymbol><ci id="S5.6.p2.1.m1.1.1.2.cmml" xref="S5.6.p2.1.m1.1.1.2">𝑍</ci><cn id="S5.6.p2.1.m1.1.1.3.cmml" type="integer" xref="S5.6.p2.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.6.p2.1.m1.1c">Z_{0}</annotation><annotation encoding="application/x-llamapun" id="S5.6.p2.1.m1.1d">italic_Z start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="Z_{1}" class="ltx_Math" display="inline" id="S5.6.p2.2.m2.1"><semantics id="S5.6.p2.2.m2.1a"><msub id="S5.6.p2.2.m2.1.1" xref="S5.6.p2.2.m2.1.1.cmml"><mi id="S5.6.p2.2.m2.1.1.2" xref="S5.6.p2.2.m2.1.1.2.cmml">Z</mi><mn id="S5.6.p2.2.m2.1.1.3" xref="S5.6.p2.2.m2.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S5.6.p2.2.m2.1b"><apply id="S5.6.p2.2.m2.1.1.cmml" xref="S5.6.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S5.6.p2.2.m2.1.1.1.cmml" xref="S5.6.p2.2.m2.1.1">subscript</csymbol><ci id="S5.6.p2.2.m2.1.1.2.cmml" xref="S5.6.p2.2.m2.1.1.2">𝑍</ci><cn id="S5.6.p2.2.m2.1.1.3.cmml" type="integer" xref="S5.6.p2.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.6.p2.2.m2.1c">Z_{1}</annotation><annotation encoding="application/x-llamapun" id="S5.6.p2.2.m2.1d">italic_Z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> be distinct subsets of <math alttext="\omega_{1}" class="ltx_Math" display="inline" id="S5.6.p2.3.m3.1"><semantics id="S5.6.p2.3.m3.1a"><msub id="S5.6.p2.3.m3.1.1" xref="S5.6.p2.3.m3.1.1.cmml"><mi id="S5.6.p2.3.m3.1.1.2" xref="S5.6.p2.3.m3.1.1.2.cmml">ω</mi><mn id="S5.6.p2.3.m3.1.1.3" xref="S5.6.p2.3.m3.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S5.6.p2.3.m3.1b"><apply id="S5.6.p2.3.m3.1.1.cmml" xref="S5.6.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S5.6.p2.3.m3.1.1.1.cmml" xref="S5.6.p2.3.m3.1.1">subscript</csymbol><ci id="S5.6.p2.3.m3.1.1.2.cmml" xref="S5.6.p2.3.m3.1.1.2">𝜔</ci><cn id="S5.6.p2.3.m3.1.1.3.cmml" type="integer" xref="S5.6.p2.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.6.p2.3.m3.1c">\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S5.6.p2.3.m3.1d">italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>. We may assume that <math alttext="Z_{0}" class="ltx_Math" display="inline" id="S5.6.p2.4.m4.1"><semantics id="S5.6.p2.4.m4.1a"><msub id="S5.6.p2.4.m4.1.1" xref="S5.6.p2.4.m4.1.1.cmml"><mi id="S5.6.p2.4.m4.1.1.2" xref="S5.6.p2.4.m4.1.1.2.cmml">Z</mi><mn id="S5.6.p2.4.m4.1.1.3" xref="S5.6.p2.4.m4.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S5.6.p2.4.m4.1b"><apply id="S5.6.p2.4.m4.1.1.cmml" xref="S5.6.p2.4.m4.1.1"><csymbol cd="ambiguous" id="S5.6.p2.4.m4.1.1.1.cmml" xref="S5.6.p2.4.m4.1.1">subscript</csymbol><ci id="S5.6.p2.4.m4.1.1.2.cmml" xref="S5.6.p2.4.m4.1.1.2">𝑍</ci><cn id="S5.6.p2.4.m4.1.1.3.cmml" type="integer" xref="S5.6.p2.4.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.6.p2.4.m4.1c">Z_{0}</annotation><annotation encoding="application/x-llamapun" id="S5.6.p2.4.m4.1d">italic_Z start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> is not contained in <math alttext="Z_{1}" class="ltx_Math" display="inline" id="S5.6.p2.5.m5.1"><semantics id="S5.6.p2.5.m5.1a"><msub id="S5.6.p2.5.m5.1.1" xref="S5.6.p2.5.m5.1.1.cmml"><mi id="S5.6.p2.5.m5.1.1.2" xref="S5.6.p2.5.m5.1.1.2.cmml">Z</mi><mn id="S5.6.p2.5.m5.1.1.3" xref="S5.6.p2.5.m5.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S5.6.p2.5.m5.1b"><apply id="S5.6.p2.5.m5.1.1.cmml" xref="S5.6.p2.5.m5.1.1"><csymbol cd="ambiguous" id="S5.6.p2.5.m5.1.1.1.cmml" xref="S5.6.p2.5.m5.1.1">subscript</csymbol><ci id="S5.6.p2.5.m5.1.1.2.cmml" xref="S5.6.p2.5.m5.1.1.2">𝑍</ci><cn id="S5.6.p2.5.m5.1.1.3.cmml" type="integer" xref="S5.6.p2.5.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.6.p2.5.m5.1c">Z_{1}</annotation><annotation encoding="application/x-llamapun" id="S5.6.p2.5.m5.1d">italic_Z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, and thus let <math alttext="\alpha\in Z_{0}\setminus Z_{1}" class="ltx_Math" display="inline" id="S5.6.p2.6.m6.1"><semantics id="S5.6.p2.6.m6.1a"><mrow id="S5.6.p2.6.m6.1.1" xref="S5.6.p2.6.m6.1.1.cmml"><mi id="S5.6.p2.6.m6.1.1.2" xref="S5.6.p2.6.m6.1.1.2.cmml">α</mi><mo id="S5.6.p2.6.m6.1.1.1" xref="S5.6.p2.6.m6.1.1.1.cmml">∈</mo><mrow id="S5.6.p2.6.m6.1.1.3" xref="S5.6.p2.6.m6.1.1.3.cmml"><msub id="S5.6.p2.6.m6.1.1.3.2" xref="S5.6.p2.6.m6.1.1.3.2.cmml"><mi id="S5.6.p2.6.m6.1.1.3.2.2" xref="S5.6.p2.6.m6.1.1.3.2.2.cmml">Z</mi><mn id="S5.6.p2.6.m6.1.1.3.2.3" xref="S5.6.p2.6.m6.1.1.3.2.3.cmml">0</mn></msub><mo id="S5.6.p2.6.m6.1.1.3.1" xref="S5.6.p2.6.m6.1.1.3.1.cmml">∖</mo><msub id="S5.6.p2.6.m6.1.1.3.3" xref="S5.6.p2.6.m6.1.1.3.3.cmml"><mi id="S5.6.p2.6.m6.1.1.3.3.2" xref="S5.6.p2.6.m6.1.1.3.3.2.cmml">Z</mi><mn id="S5.6.p2.6.m6.1.1.3.3.3" xref="S5.6.p2.6.m6.1.1.3.3.3.cmml">1</mn></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.6.p2.6.m6.1b"><apply id="S5.6.p2.6.m6.1.1.cmml" xref="S5.6.p2.6.m6.1.1"><in id="S5.6.p2.6.m6.1.1.1.cmml" xref="S5.6.p2.6.m6.1.1.1"></in><ci id="S5.6.p2.6.m6.1.1.2.cmml" xref="S5.6.p2.6.m6.1.1.2">𝛼</ci><apply id="S5.6.p2.6.m6.1.1.3.cmml" xref="S5.6.p2.6.m6.1.1.3"><setdiff id="S5.6.p2.6.m6.1.1.3.1.cmml" xref="S5.6.p2.6.m6.1.1.3.1"></setdiff><apply id="S5.6.p2.6.m6.1.1.3.2.cmml" xref="S5.6.p2.6.m6.1.1.3.2"><csymbol cd="ambiguous" id="S5.6.p2.6.m6.1.1.3.2.1.cmml" xref="S5.6.p2.6.m6.1.1.3.2">subscript</csymbol><ci id="S5.6.p2.6.m6.1.1.3.2.2.cmml" xref="S5.6.p2.6.m6.1.1.3.2.2">𝑍</ci><cn id="S5.6.p2.6.m6.1.1.3.2.3.cmml" type="integer" xref="S5.6.p2.6.m6.1.1.3.2.3">0</cn></apply><apply id="S5.6.p2.6.m6.1.1.3.3.cmml" xref="S5.6.p2.6.m6.1.1.3.3"><csymbol cd="ambiguous" id="S5.6.p2.6.m6.1.1.3.3.1.cmml" xref="S5.6.p2.6.m6.1.1.3.3">subscript</csymbol><ci id="S5.6.p2.6.m6.1.1.3.3.2.cmml" xref="S5.6.p2.6.m6.1.1.3.3.2">𝑍</ci><cn id="S5.6.p2.6.m6.1.1.3.3.3.cmml" type="integer" xref="S5.6.p2.6.m6.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.6.p2.6.m6.1c">\alpha\in Z_{0}\setminus Z_{1}</annotation><annotation encoding="application/x-llamapun" id="S5.6.p2.6.m6.1d">italic_α ∈ italic_Z start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∖ italic_Z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>. Then <math alttext="\hat{\mathscr{L}}(A^{Z_{0}},D^{Z_{0}})\setminus\hat{\mathscr{L}}(A^{Z_{1}},D^{% Z_{1}})=L(Z_{0})\setminus L(Z_{1})\supseteq S_{\alpha}" class="ltx_Math" display="inline" id="S5.6.p2.7.m7.6"><semantics id="S5.6.p2.7.m7.6a"><mrow id="S5.6.p2.7.m7.6.6" xref="S5.6.p2.7.m7.6.6.cmml"><mrow id="S5.6.p2.7.m7.4.4.4" xref="S5.6.p2.7.m7.4.4.4.cmml"><mrow id="S5.6.p2.7.m7.2.2.2.2" xref="S5.6.p2.7.m7.2.2.2.2.cmml"><mover accent="true" id="S5.6.p2.7.m7.2.2.2.2.4" xref="S5.6.p2.7.m7.2.2.2.2.4.cmml"><mi class="ltx_font_mathscript" id="S5.6.p2.7.m7.2.2.2.2.4.2" xref="S5.6.p2.7.m7.2.2.2.2.4.2.cmml">ℒ</mi><mo id="S5.6.p2.7.m7.2.2.2.2.4.1" xref="S5.6.p2.7.m7.2.2.2.2.4.1.cmml">^</mo></mover><mo id="S5.6.p2.7.m7.2.2.2.2.3" xref="S5.6.p2.7.m7.2.2.2.2.3.cmml">⁢</mo><mrow id="S5.6.p2.7.m7.2.2.2.2.2.2" xref="S5.6.p2.7.m7.2.2.2.2.2.3.cmml"><mo id="S5.6.p2.7.m7.2.2.2.2.2.2.3" stretchy="false" xref="S5.6.p2.7.m7.2.2.2.2.2.3.cmml">(</mo><msup id="S5.6.p2.7.m7.1.1.1.1.1.1.1" xref="S5.6.p2.7.m7.1.1.1.1.1.1.1.cmml"><mi id="S5.6.p2.7.m7.1.1.1.1.1.1.1.2" xref="S5.6.p2.7.m7.1.1.1.1.1.1.1.2.cmml">A</mi><msub id="S5.6.p2.7.m7.1.1.1.1.1.1.1.3" xref="S5.6.p2.7.m7.1.1.1.1.1.1.1.3.cmml"><mi id="S5.6.p2.7.m7.1.1.1.1.1.1.1.3.2" xref="S5.6.p2.7.m7.1.1.1.1.1.1.1.3.2.cmml">Z</mi><mn id="S5.6.p2.7.m7.1.1.1.1.1.1.1.3.3" xref="S5.6.p2.7.m7.1.1.1.1.1.1.1.3.3.cmml">0</mn></msub></msup><mo id="S5.6.p2.7.m7.2.2.2.2.2.2.4" xref="S5.6.p2.7.m7.2.2.2.2.2.3.cmml">,</mo><msup id="S5.6.p2.7.m7.2.2.2.2.2.2.2" xref="S5.6.p2.7.m7.2.2.2.2.2.2.2.cmml"><mi id="S5.6.p2.7.m7.2.2.2.2.2.2.2.2" xref="S5.6.p2.7.m7.2.2.2.2.2.2.2.2.cmml">D</mi><msub id="S5.6.p2.7.m7.2.2.2.2.2.2.2.3" xref="S5.6.p2.7.m7.2.2.2.2.2.2.2.3.cmml"><mi id="S5.6.p2.7.m7.2.2.2.2.2.2.2.3.2" xref="S5.6.p2.7.m7.2.2.2.2.2.2.2.3.2.cmml">Z</mi><mn id="S5.6.p2.7.m7.2.2.2.2.2.2.2.3.3" xref="S5.6.p2.7.m7.2.2.2.2.2.2.2.3.3.cmml">0</mn></msub></msup><mo id="S5.6.p2.7.m7.2.2.2.2.2.2.5" stretchy="false" xref="S5.6.p2.7.m7.2.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S5.6.p2.7.m7.4.4.4.5" xref="S5.6.p2.7.m7.4.4.4.5.cmml">∖</mo><mrow id="S5.6.p2.7.m7.4.4.4.4" xref="S5.6.p2.7.m7.4.4.4.4.cmml"><mover accent="true" id="S5.6.p2.7.m7.4.4.4.4.4" xref="S5.6.p2.7.m7.4.4.4.4.4.cmml"><mi class="ltx_font_mathscript" id="S5.6.p2.7.m7.4.4.4.4.4.2" xref="S5.6.p2.7.m7.4.4.4.4.4.2.cmml">ℒ</mi><mo id="S5.6.p2.7.m7.4.4.4.4.4.1" xref="S5.6.p2.7.m7.4.4.4.4.4.1.cmml">^</mo></mover><mo id="S5.6.p2.7.m7.4.4.4.4.3" xref="S5.6.p2.7.m7.4.4.4.4.3.cmml">⁢</mo><mrow id="S5.6.p2.7.m7.4.4.4.4.2.2" xref="S5.6.p2.7.m7.4.4.4.4.2.3.cmml"><mo id="S5.6.p2.7.m7.4.4.4.4.2.2.3" stretchy="false" xref="S5.6.p2.7.m7.4.4.4.4.2.3.cmml">(</mo><msup id="S5.6.p2.7.m7.3.3.3.3.1.1.1" xref="S5.6.p2.7.m7.3.3.3.3.1.1.1.cmml"><mi id="S5.6.p2.7.m7.3.3.3.3.1.1.1.2" xref="S5.6.p2.7.m7.3.3.3.3.1.1.1.2.cmml">A</mi><msub id="S5.6.p2.7.m7.3.3.3.3.1.1.1.3" xref="S5.6.p2.7.m7.3.3.3.3.1.1.1.3.cmml"><mi id="S5.6.p2.7.m7.3.3.3.3.1.1.1.3.2" xref="S5.6.p2.7.m7.3.3.3.3.1.1.1.3.2.cmml">Z</mi><mn id="S5.6.p2.7.m7.3.3.3.3.1.1.1.3.3" xref="S5.6.p2.7.m7.3.3.3.3.1.1.1.3.3.cmml">1</mn></msub></msup><mo id="S5.6.p2.7.m7.4.4.4.4.2.2.4" xref="S5.6.p2.7.m7.4.4.4.4.2.3.cmml">,</mo><msup id="S5.6.p2.7.m7.4.4.4.4.2.2.2" xref="S5.6.p2.7.m7.4.4.4.4.2.2.2.cmml"><mi id="S5.6.p2.7.m7.4.4.4.4.2.2.2.2" xref="S5.6.p2.7.m7.4.4.4.4.2.2.2.2.cmml">D</mi><msub id="S5.6.p2.7.m7.4.4.4.4.2.2.2.3" xref="S5.6.p2.7.m7.4.4.4.4.2.2.2.3.cmml"><mi id="S5.6.p2.7.m7.4.4.4.4.2.2.2.3.2" xref="S5.6.p2.7.m7.4.4.4.4.2.2.2.3.2.cmml">Z</mi><mn id="S5.6.p2.7.m7.4.4.4.4.2.2.2.3.3" xref="S5.6.p2.7.m7.4.4.4.4.2.2.2.3.3.cmml">1</mn></msub></msup><mo id="S5.6.p2.7.m7.4.4.4.4.2.2.5" stretchy="false" xref="S5.6.p2.7.m7.4.4.4.4.2.3.cmml">)</mo></mrow></mrow></mrow><mo id="S5.6.p2.7.m7.6.6.8" xref="S5.6.p2.7.m7.6.6.8.cmml">=</mo><mrow id="S5.6.p2.7.m7.6.6.6" xref="S5.6.p2.7.m7.6.6.6.cmml"><mrow id="S5.6.p2.7.m7.5.5.5.1" xref="S5.6.p2.7.m7.5.5.5.1.cmml"><mi id="S5.6.p2.7.m7.5.5.5.1.3" xref="S5.6.p2.7.m7.5.5.5.1.3.cmml">L</mi><mo id="S5.6.p2.7.m7.5.5.5.1.2" xref="S5.6.p2.7.m7.5.5.5.1.2.cmml">⁢</mo><mrow id="S5.6.p2.7.m7.5.5.5.1.1.1" xref="S5.6.p2.7.m7.5.5.5.1.1.1.1.cmml"><mo id="S5.6.p2.7.m7.5.5.5.1.1.1.2" stretchy="false" xref="S5.6.p2.7.m7.5.5.5.1.1.1.1.cmml">(</mo><msub id="S5.6.p2.7.m7.5.5.5.1.1.1.1" xref="S5.6.p2.7.m7.5.5.5.1.1.1.1.cmml"><mi id="S5.6.p2.7.m7.5.5.5.1.1.1.1.2" xref="S5.6.p2.7.m7.5.5.5.1.1.1.1.2.cmml">Z</mi><mn id="S5.6.p2.7.m7.5.5.5.1.1.1.1.3" xref="S5.6.p2.7.m7.5.5.5.1.1.1.1.3.cmml">0</mn></msub><mo id="S5.6.p2.7.m7.5.5.5.1.1.1.3" stretchy="false" xref="S5.6.p2.7.m7.5.5.5.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.6.p2.7.m7.6.6.6.3" xref="S5.6.p2.7.m7.6.6.6.3.cmml">∖</mo><mrow id="S5.6.p2.7.m7.6.6.6.2" xref="S5.6.p2.7.m7.6.6.6.2.cmml"><mi id="S5.6.p2.7.m7.6.6.6.2.3" xref="S5.6.p2.7.m7.6.6.6.2.3.cmml">L</mi><mo id="S5.6.p2.7.m7.6.6.6.2.2" xref="S5.6.p2.7.m7.6.6.6.2.2.cmml">⁢</mo><mrow id="S5.6.p2.7.m7.6.6.6.2.1.1" xref="S5.6.p2.7.m7.6.6.6.2.1.1.1.cmml"><mo id="S5.6.p2.7.m7.6.6.6.2.1.1.2" stretchy="false" xref="S5.6.p2.7.m7.6.6.6.2.1.1.1.cmml">(</mo><msub id="S5.6.p2.7.m7.6.6.6.2.1.1.1" xref="S5.6.p2.7.m7.6.6.6.2.1.1.1.cmml"><mi id="S5.6.p2.7.m7.6.6.6.2.1.1.1.2" xref="S5.6.p2.7.m7.6.6.6.2.1.1.1.2.cmml">Z</mi><mn id="S5.6.p2.7.m7.6.6.6.2.1.1.1.3" xref="S5.6.p2.7.m7.6.6.6.2.1.1.1.3.cmml">1</mn></msub><mo id="S5.6.p2.7.m7.6.6.6.2.1.1.3" stretchy="false" xref="S5.6.p2.7.m7.6.6.6.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S5.6.p2.7.m7.6.6.9" xref="S5.6.p2.7.m7.6.6.9.cmml">⊇</mo><msub id="S5.6.p2.7.m7.6.6.10" xref="S5.6.p2.7.m7.6.6.10.cmml"><mi id="S5.6.p2.7.m7.6.6.10.2" xref="S5.6.p2.7.m7.6.6.10.2.cmml">S</mi><mi id="S5.6.p2.7.m7.6.6.10.3" xref="S5.6.p2.7.m7.6.6.10.3.cmml">α</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.6.p2.7.m7.6b"><apply id="S5.6.p2.7.m7.6.6.cmml" xref="S5.6.p2.7.m7.6.6"><and id="S5.6.p2.7.m7.6.6a.cmml" xref="S5.6.p2.7.m7.6.6"></and><apply id="S5.6.p2.7.m7.6.6b.cmml" xref="S5.6.p2.7.m7.6.6"><eq id="S5.6.p2.7.m7.6.6.8.cmml" xref="S5.6.p2.7.m7.6.6.8"></eq><apply id="S5.6.p2.7.m7.4.4.4.cmml" xref="S5.6.p2.7.m7.4.4.4"><setdiff id="S5.6.p2.7.m7.4.4.4.5.cmml" xref="S5.6.p2.7.m7.4.4.4.5"></setdiff><apply id="S5.6.p2.7.m7.2.2.2.2.cmml" xref="S5.6.p2.7.m7.2.2.2.2"><times id="S5.6.p2.7.m7.2.2.2.2.3.cmml" xref="S5.6.p2.7.m7.2.2.2.2.3"></times><apply id="S5.6.p2.7.m7.2.2.2.2.4.cmml" xref="S5.6.p2.7.m7.2.2.2.2.4"><ci id="S5.6.p2.7.m7.2.2.2.2.4.1.cmml" xref="S5.6.p2.7.m7.2.2.2.2.4.1">^</ci><ci id="S5.6.p2.7.m7.2.2.2.2.4.2.cmml" xref="S5.6.p2.7.m7.2.2.2.2.4.2">ℒ</ci></apply><interval closure="open" id="S5.6.p2.7.m7.2.2.2.2.2.3.cmml" xref="S5.6.p2.7.m7.2.2.2.2.2.2"><apply id="S5.6.p2.7.m7.1.1.1.1.1.1.1.cmml" xref="S5.6.p2.7.m7.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.6.p2.7.m7.1.1.1.1.1.1.1.1.cmml" xref="S5.6.p2.7.m7.1.1.1.1.1.1.1">superscript</csymbol><ci id="S5.6.p2.7.m7.1.1.1.1.1.1.1.2.cmml" xref="S5.6.p2.7.m7.1.1.1.1.1.1.1.2">𝐴</ci><apply id="S5.6.p2.7.m7.1.1.1.1.1.1.1.3.cmml" xref="S5.6.p2.7.m7.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S5.6.p2.7.m7.1.1.1.1.1.1.1.3.1.cmml" xref="S5.6.p2.7.m7.1.1.1.1.1.1.1.3">subscript</csymbol><ci id="S5.6.p2.7.m7.1.1.1.1.1.1.1.3.2.cmml" xref="S5.6.p2.7.m7.1.1.1.1.1.1.1.3.2">𝑍</ci><cn id="S5.6.p2.7.m7.1.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S5.6.p2.7.m7.1.1.1.1.1.1.1.3.3">0</cn></apply></apply><apply id="S5.6.p2.7.m7.2.2.2.2.2.2.2.cmml" xref="S5.6.p2.7.m7.2.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S5.6.p2.7.m7.2.2.2.2.2.2.2.1.cmml" xref="S5.6.p2.7.m7.2.2.2.2.2.2.2">superscript</csymbol><ci id="S5.6.p2.7.m7.2.2.2.2.2.2.2.2.cmml" xref="S5.6.p2.7.m7.2.2.2.2.2.2.2.2">𝐷</ci><apply id="S5.6.p2.7.m7.2.2.2.2.2.2.2.3.cmml" xref="S5.6.p2.7.m7.2.2.2.2.2.2.2.3"><csymbol cd="ambiguous" id="S5.6.p2.7.m7.2.2.2.2.2.2.2.3.1.cmml" xref="S5.6.p2.7.m7.2.2.2.2.2.2.2.3">subscript</csymbol><ci id="S5.6.p2.7.m7.2.2.2.2.2.2.2.3.2.cmml" xref="S5.6.p2.7.m7.2.2.2.2.2.2.2.3.2">𝑍</ci><cn id="S5.6.p2.7.m7.2.2.2.2.2.2.2.3.3.cmml" type="integer" xref="S5.6.p2.7.m7.2.2.2.2.2.2.2.3.3">0</cn></apply></apply></interval></apply><apply id="S5.6.p2.7.m7.4.4.4.4.cmml" xref="S5.6.p2.7.m7.4.4.4.4"><times id="S5.6.p2.7.m7.4.4.4.4.3.cmml" xref="S5.6.p2.7.m7.4.4.4.4.3"></times><apply id="S5.6.p2.7.m7.4.4.4.4.4.cmml" xref="S5.6.p2.7.m7.4.4.4.4.4"><ci id="S5.6.p2.7.m7.4.4.4.4.4.1.cmml" xref="S5.6.p2.7.m7.4.4.4.4.4.1">^</ci><ci id="S5.6.p2.7.m7.4.4.4.4.4.2.cmml" xref="S5.6.p2.7.m7.4.4.4.4.4.2">ℒ</ci></apply><interval closure="open" id="S5.6.p2.7.m7.4.4.4.4.2.3.cmml" xref="S5.6.p2.7.m7.4.4.4.4.2.2"><apply id="S5.6.p2.7.m7.3.3.3.3.1.1.1.cmml" xref="S5.6.p2.7.m7.3.3.3.3.1.1.1"><csymbol cd="ambiguous" id="S5.6.p2.7.m7.3.3.3.3.1.1.1.1.cmml" xref="S5.6.p2.7.m7.3.3.3.3.1.1.1">superscript</csymbol><ci id="S5.6.p2.7.m7.3.3.3.3.1.1.1.2.cmml" xref="S5.6.p2.7.m7.3.3.3.3.1.1.1.2">𝐴</ci><apply id="S5.6.p2.7.m7.3.3.3.3.1.1.1.3.cmml" xref="S5.6.p2.7.m7.3.3.3.3.1.1.1.3"><csymbol cd="ambiguous" id="S5.6.p2.7.m7.3.3.3.3.1.1.1.3.1.cmml" xref="S5.6.p2.7.m7.3.3.3.3.1.1.1.3">subscript</csymbol><ci id="S5.6.p2.7.m7.3.3.3.3.1.1.1.3.2.cmml" xref="S5.6.p2.7.m7.3.3.3.3.1.1.1.3.2">𝑍</ci><cn id="S5.6.p2.7.m7.3.3.3.3.1.1.1.3.3.cmml" type="integer" xref="S5.6.p2.7.m7.3.3.3.3.1.1.1.3.3">1</cn></apply></apply><apply id="S5.6.p2.7.m7.4.4.4.4.2.2.2.cmml" xref="S5.6.p2.7.m7.4.4.4.4.2.2.2"><csymbol cd="ambiguous" id="S5.6.p2.7.m7.4.4.4.4.2.2.2.1.cmml" xref="S5.6.p2.7.m7.4.4.4.4.2.2.2">superscript</csymbol><ci id="S5.6.p2.7.m7.4.4.4.4.2.2.2.2.cmml" xref="S5.6.p2.7.m7.4.4.4.4.2.2.2.2">𝐷</ci><apply id="S5.6.p2.7.m7.4.4.4.4.2.2.2.3.cmml" xref="S5.6.p2.7.m7.4.4.4.4.2.2.2.3"><csymbol cd="ambiguous" id="S5.6.p2.7.m7.4.4.4.4.2.2.2.3.1.cmml" xref="S5.6.p2.7.m7.4.4.4.4.2.2.2.3">subscript</csymbol><ci id="S5.6.p2.7.m7.4.4.4.4.2.2.2.3.2.cmml" xref="S5.6.p2.7.m7.4.4.4.4.2.2.2.3.2">𝑍</ci><cn id="S5.6.p2.7.m7.4.4.4.4.2.2.2.3.3.cmml" type="integer" xref="S5.6.p2.7.m7.4.4.4.4.2.2.2.3.3">1</cn></apply></apply></interval></apply></apply><apply id="S5.6.p2.7.m7.6.6.6.cmml" xref="S5.6.p2.7.m7.6.6.6"><setdiff id="S5.6.p2.7.m7.6.6.6.3.cmml" xref="S5.6.p2.7.m7.6.6.6.3"></setdiff><apply id="S5.6.p2.7.m7.5.5.5.1.cmml" xref="S5.6.p2.7.m7.5.5.5.1"><times id="S5.6.p2.7.m7.5.5.5.1.2.cmml" xref="S5.6.p2.7.m7.5.5.5.1.2"></times><ci id="S5.6.p2.7.m7.5.5.5.1.3.cmml" xref="S5.6.p2.7.m7.5.5.5.1.3">𝐿</ci><apply id="S5.6.p2.7.m7.5.5.5.1.1.1.1.cmml" xref="S5.6.p2.7.m7.5.5.5.1.1.1"><csymbol cd="ambiguous" id="S5.6.p2.7.m7.5.5.5.1.1.1.1.1.cmml" xref="S5.6.p2.7.m7.5.5.5.1.1.1">subscript</csymbol><ci id="S5.6.p2.7.m7.5.5.5.1.1.1.1.2.cmml" xref="S5.6.p2.7.m7.5.5.5.1.1.1.1.2">𝑍</ci><cn id="S5.6.p2.7.m7.5.5.5.1.1.1.1.3.cmml" type="integer" xref="S5.6.p2.7.m7.5.5.5.1.1.1.1.3">0</cn></apply></apply><apply id="S5.6.p2.7.m7.6.6.6.2.cmml" xref="S5.6.p2.7.m7.6.6.6.2"><times id="S5.6.p2.7.m7.6.6.6.2.2.cmml" xref="S5.6.p2.7.m7.6.6.6.2.2"></times><ci id="S5.6.p2.7.m7.6.6.6.2.3.cmml" xref="S5.6.p2.7.m7.6.6.6.2.3">𝐿</ci><apply id="S5.6.p2.7.m7.6.6.6.2.1.1.1.cmml" xref="S5.6.p2.7.m7.6.6.6.2.1.1"><csymbol cd="ambiguous" id="S5.6.p2.7.m7.6.6.6.2.1.1.1.1.cmml" xref="S5.6.p2.7.m7.6.6.6.2.1.1">subscript</csymbol><ci id="S5.6.p2.7.m7.6.6.6.2.1.1.1.2.cmml" xref="S5.6.p2.7.m7.6.6.6.2.1.1.1.2">𝑍</ci><cn id="S5.6.p2.7.m7.6.6.6.2.1.1.1.3.cmml" type="integer" xref="S5.6.p2.7.m7.6.6.6.2.1.1.1.3">1</cn></apply></apply></apply></apply><apply id="S5.6.p2.7.m7.6.6c.cmml" xref="S5.6.p2.7.m7.6.6"><csymbol cd="latexml" id="S5.6.p2.7.m7.6.6.9.cmml" xref="S5.6.p2.7.m7.6.6.9">superset-of-or-equals</csymbol><share href="https://arxiv.org/html/2503.13728v1#S5.6.p2.7.m7.6.6.6.cmml" id="S5.6.p2.7.m7.6.6d.cmml" xref="S5.6.p2.7.m7.6.6"></share><apply id="S5.6.p2.7.m7.6.6.10.cmml" xref="S5.6.p2.7.m7.6.6.10"><csymbol cd="ambiguous" id="S5.6.p2.7.m7.6.6.10.1.cmml" xref="S5.6.p2.7.m7.6.6.10">subscript</csymbol><ci id="S5.6.p2.7.m7.6.6.10.2.cmml" xref="S5.6.p2.7.m7.6.6.10.2">𝑆</ci><ci id="S5.6.p2.7.m7.6.6.10.3.cmml" xref="S5.6.p2.7.m7.6.6.10.3">𝛼</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.6.p2.7.m7.6c">\hat{\mathscr{L}}(A^{Z_{0}},D^{Z_{0}})\setminus\hat{\mathscr{L}}(A^{Z_{1}},D^{% Z_{1}})=L(Z_{0})\setminus L(Z_{1})\supseteq S_{\alpha}</annotation><annotation encoding="application/x-llamapun" id="S5.6.p2.7.m7.6d">over^ start_ARG script_L end_ARG ( italic_A start_POSTSUPERSCRIPT italic_Z start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT , italic_D start_POSTSUPERSCRIPT italic_Z start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ) ∖ over^ start_ARG script_L end_ARG ( italic_A start_POSTSUPERSCRIPT italic_Z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT , italic_D start_POSTSUPERSCRIPT italic_Z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ) = italic_L ( italic_Z start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ∖ italic_L ( italic_Z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ⊇ italic_S start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT</annotation></semantics></math>, which is stationary. Thus by <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S5.Thmtheorem2" title="Lemma 5.2. ‣ 5. An infinite antichain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">5.2</span></a> <math alttext="A^{Z_{0}}\ntrianglerighteq A^{Z_{1}}" class="ltx_Math" display="inline" id="S5.6.p2.8.m8.1"><semantics id="S5.6.p2.8.m8.1a"><mrow id="S5.6.p2.8.m8.1.1" xref="S5.6.p2.8.m8.1.1.cmml"><msup id="S5.6.p2.8.m8.1.1.2" xref="S5.6.p2.8.m8.1.1.2.cmml"><mi id="S5.6.p2.8.m8.1.1.2.2" xref="S5.6.p2.8.m8.1.1.2.2.cmml">A</mi><msub id="S5.6.p2.8.m8.1.1.2.3" xref="S5.6.p2.8.m8.1.1.2.3.cmml"><mi id="S5.6.p2.8.m8.1.1.2.3.2" xref="S5.6.p2.8.m8.1.1.2.3.2.cmml">Z</mi><mn id="S5.6.p2.8.m8.1.1.2.3.3" xref="S5.6.p2.8.m8.1.1.2.3.3.cmml">0</mn></msub></msup><mo id="S5.6.p2.8.m8.1.1.1" xref="S5.6.p2.8.m8.1.1.1.cmml">⋭</mo><msup id="S5.6.p2.8.m8.1.1.3" xref="S5.6.p2.8.m8.1.1.3.cmml"><mi id="S5.6.p2.8.m8.1.1.3.2" xref="S5.6.p2.8.m8.1.1.3.2.cmml">A</mi><msub id="S5.6.p2.8.m8.1.1.3.3" xref="S5.6.p2.8.m8.1.1.3.3.cmml"><mi id="S5.6.p2.8.m8.1.1.3.3.2" xref="S5.6.p2.8.m8.1.1.3.3.2.cmml">Z</mi><mn id="S5.6.p2.8.m8.1.1.3.3.3" xref="S5.6.p2.8.m8.1.1.3.3.3.cmml">1</mn></msub></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.6.p2.8.m8.1b"><apply id="S5.6.p2.8.m8.1.1.cmml" xref="S5.6.p2.8.m8.1.1"><csymbol cd="latexml" id="S5.6.p2.8.m8.1.1.1.cmml" xref="S5.6.p2.8.m8.1.1.1">not-contains-nor-equals</csymbol><apply id="S5.6.p2.8.m8.1.1.2.cmml" xref="S5.6.p2.8.m8.1.1.2"><csymbol cd="ambiguous" id="S5.6.p2.8.m8.1.1.2.1.cmml" xref="S5.6.p2.8.m8.1.1.2">superscript</csymbol><ci id="S5.6.p2.8.m8.1.1.2.2.cmml" xref="S5.6.p2.8.m8.1.1.2.2">𝐴</ci><apply id="S5.6.p2.8.m8.1.1.2.3.cmml" xref="S5.6.p2.8.m8.1.1.2.3"><csymbol cd="ambiguous" id="S5.6.p2.8.m8.1.1.2.3.1.cmml" xref="S5.6.p2.8.m8.1.1.2.3">subscript</csymbol><ci id="S5.6.p2.8.m8.1.1.2.3.2.cmml" xref="S5.6.p2.8.m8.1.1.2.3.2">𝑍</ci><cn id="S5.6.p2.8.m8.1.1.2.3.3.cmml" type="integer" xref="S5.6.p2.8.m8.1.1.2.3.3">0</cn></apply></apply><apply id="S5.6.p2.8.m8.1.1.3.cmml" xref="S5.6.p2.8.m8.1.1.3"><csymbol cd="ambiguous" id="S5.6.p2.8.m8.1.1.3.1.cmml" xref="S5.6.p2.8.m8.1.1.3">superscript</csymbol><ci id="S5.6.p2.8.m8.1.1.3.2.cmml" xref="S5.6.p2.8.m8.1.1.3.2">𝐴</ci><apply id="S5.6.p2.8.m8.1.1.3.3.cmml" xref="S5.6.p2.8.m8.1.1.3.3"><csymbol cd="ambiguous" id="S5.6.p2.8.m8.1.1.3.3.1.cmml" xref="S5.6.p2.8.m8.1.1.3.3">subscript</csymbol><ci id="S5.6.p2.8.m8.1.1.3.3.2.cmml" xref="S5.6.p2.8.m8.1.1.3.3.2">𝑍</ci><cn id="S5.6.p2.8.m8.1.1.3.3.3.cmml" type="integer" xref="S5.6.p2.8.m8.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.6.p2.8.m8.1c">A^{Z_{0}}\ntrianglerighteq A^{Z_{1}}</annotation><annotation encoding="application/x-llamapun" id="S5.6.p2.8.m8.1d">italic_A start_POSTSUPERSCRIPT italic_Z start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ⋭ italic_A start_POSTSUPERSCRIPT italic_Z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math>. Similarly, <math alttext="\hat{\mathscr{R}}(A^{Z_{1}},D^{Z_{1}})\setminus\hat{\mathscr{R}}(A^{Z_{0}},D^{% Z_{0}})=R(Z_{1})\setminus R(Z_{0})\supseteq S_{\alpha}" class="ltx_Math" display="inline" id="S5.6.p2.9.m9.6"><semantics id="S5.6.p2.9.m9.6a"><mrow id="S5.6.p2.9.m9.6.6" xref="S5.6.p2.9.m9.6.6.cmml"><mrow id="S5.6.p2.9.m9.4.4.4" xref="S5.6.p2.9.m9.4.4.4.cmml"><mrow id="S5.6.p2.9.m9.2.2.2.2" xref="S5.6.p2.9.m9.2.2.2.2.cmml"><mover accent="true" id="S5.6.p2.9.m9.2.2.2.2.4" xref="S5.6.p2.9.m9.2.2.2.2.4.cmml"><mi class="ltx_font_mathscript" id="S5.6.p2.9.m9.2.2.2.2.4.2" xref="S5.6.p2.9.m9.2.2.2.2.4.2.cmml">ℛ</mi><mo id="S5.6.p2.9.m9.2.2.2.2.4.1" xref="S5.6.p2.9.m9.2.2.2.2.4.1.cmml">^</mo></mover><mo id="S5.6.p2.9.m9.2.2.2.2.3" xref="S5.6.p2.9.m9.2.2.2.2.3.cmml">⁢</mo><mrow id="S5.6.p2.9.m9.2.2.2.2.2.2" xref="S5.6.p2.9.m9.2.2.2.2.2.3.cmml"><mo id="S5.6.p2.9.m9.2.2.2.2.2.2.3" stretchy="false" xref="S5.6.p2.9.m9.2.2.2.2.2.3.cmml">(</mo><msup id="S5.6.p2.9.m9.1.1.1.1.1.1.1" xref="S5.6.p2.9.m9.1.1.1.1.1.1.1.cmml"><mi id="S5.6.p2.9.m9.1.1.1.1.1.1.1.2" xref="S5.6.p2.9.m9.1.1.1.1.1.1.1.2.cmml">A</mi><msub id="S5.6.p2.9.m9.1.1.1.1.1.1.1.3" xref="S5.6.p2.9.m9.1.1.1.1.1.1.1.3.cmml"><mi id="S5.6.p2.9.m9.1.1.1.1.1.1.1.3.2" xref="S5.6.p2.9.m9.1.1.1.1.1.1.1.3.2.cmml">Z</mi><mn id="S5.6.p2.9.m9.1.1.1.1.1.1.1.3.3" xref="S5.6.p2.9.m9.1.1.1.1.1.1.1.3.3.cmml">1</mn></msub></msup><mo id="S5.6.p2.9.m9.2.2.2.2.2.2.4" xref="S5.6.p2.9.m9.2.2.2.2.2.3.cmml">,</mo><msup id="S5.6.p2.9.m9.2.2.2.2.2.2.2" xref="S5.6.p2.9.m9.2.2.2.2.2.2.2.cmml"><mi id="S5.6.p2.9.m9.2.2.2.2.2.2.2.2" xref="S5.6.p2.9.m9.2.2.2.2.2.2.2.2.cmml">D</mi><msub id="S5.6.p2.9.m9.2.2.2.2.2.2.2.3" xref="S5.6.p2.9.m9.2.2.2.2.2.2.2.3.cmml"><mi id="S5.6.p2.9.m9.2.2.2.2.2.2.2.3.2" xref="S5.6.p2.9.m9.2.2.2.2.2.2.2.3.2.cmml">Z</mi><mn id="S5.6.p2.9.m9.2.2.2.2.2.2.2.3.3" xref="S5.6.p2.9.m9.2.2.2.2.2.2.2.3.3.cmml">1</mn></msub></msup><mo id="S5.6.p2.9.m9.2.2.2.2.2.2.5" stretchy="false" xref="S5.6.p2.9.m9.2.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S5.6.p2.9.m9.4.4.4.5" xref="S5.6.p2.9.m9.4.4.4.5.cmml">∖</mo><mrow id="S5.6.p2.9.m9.4.4.4.4" xref="S5.6.p2.9.m9.4.4.4.4.cmml"><mover accent="true" id="S5.6.p2.9.m9.4.4.4.4.4" xref="S5.6.p2.9.m9.4.4.4.4.4.cmml"><mi class="ltx_font_mathscript" id="S5.6.p2.9.m9.4.4.4.4.4.2" xref="S5.6.p2.9.m9.4.4.4.4.4.2.cmml">ℛ</mi><mo id="S5.6.p2.9.m9.4.4.4.4.4.1" xref="S5.6.p2.9.m9.4.4.4.4.4.1.cmml">^</mo></mover><mo id="S5.6.p2.9.m9.4.4.4.4.3" xref="S5.6.p2.9.m9.4.4.4.4.3.cmml">⁢</mo><mrow id="S5.6.p2.9.m9.4.4.4.4.2.2" xref="S5.6.p2.9.m9.4.4.4.4.2.3.cmml"><mo id="S5.6.p2.9.m9.4.4.4.4.2.2.3" stretchy="false" xref="S5.6.p2.9.m9.4.4.4.4.2.3.cmml">(</mo><msup id="S5.6.p2.9.m9.3.3.3.3.1.1.1" xref="S5.6.p2.9.m9.3.3.3.3.1.1.1.cmml"><mi id="S5.6.p2.9.m9.3.3.3.3.1.1.1.2" xref="S5.6.p2.9.m9.3.3.3.3.1.1.1.2.cmml">A</mi><msub id="S5.6.p2.9.m9.3.3.3.3.1.1.1.3" xref="S5.6.p2.9.m9.3.3.3.3.1.1.1.3.cmml"><mi id="S5.6.p2.9.m9.3.3.3.3.1.1.1.3.2" xref="S5.6.p2.9.m9.3.3.3.3.1.1.1.3.2.cmml">Z</mi><mn id="S5.6.p2.9.m9.3.3.3.3.1.1.1.3.3" xref="S5.6.p2.9.m9.3.3.3.3.1.1.1.3.3.cmml">0</mn></msub></msup><mo id="S5.6.p2.9.m9.4.4.4.4.2.2.4" xref="S5.6.p2.9.m9.4.4.4.4.2.3.cmml">,</mo><msup id="S5.6.p2.9.m9.4.4.4.4.2.2.2" xref="S5.6.p2.9.m9.4.4.4.4.2.2.2.cmml"><mi id="S5.6.p2.9.m9.4.4.4.4.2.2.2.2" xref="S5.6.p2.9.m9.4.4.4.4.2.2.2.2.cmml">D</mi><msub id="S5.6.p2.9.m9.4.4.4.4.2.2.2.3" xref="S5.6.p2.9.m9.4.4.4.4.2.2.2.3.cmml"><mi id="S5.6.p2.9.m9.4.4.4.4.2.2.2.3.2" xref="S5.6.p2.9.m9.4.4.4.4.2.2.2.3.2.cmml">Z</mi><mn id="S5.6.p2.9.m9.4.4.4.4.2.2.2.3.3" xref="S5.6.p2.9.m9.4.4.4.4.2.2.2.3.3.cmml">0</mn></msub></msup><mo id="S5.6.p2.9.m9.4.4.4.4.2.2.5" stretchy="false" xref="S5.6.p2.9.m9.4.4.4.4.2.3.cmml">)</mo></mrow></mrow></mrow><mo id="S5.6.p2.9.m9.6.6.8" xref="S5.6.p2.9.m9.6.6.8.cmml">=</mo><mrow id="S5.6.p2.9.m9.6.6.6" xref="S5.6.p2.9.m9.6.6.6.cmml"><mrow id="S5.6.p2.9.m9.5.5.5.1" xref="S5.6.p2.9.m9.5.5.5.1.cmml"><mi id="S5.6.p2.9.m9.5.5.5.1.3" xref="S5.6.p2.9.m9.5.5.5.1.3.cmml">R</mi><mo id="S5.6.p2.9.m9.5.5.5.1.2" xref="S5.6.p2.9.m9.5.5.5.1.2.cmml">⁢</mo><mrow id="S5.6.p2.9.m9.5.5.5.1.1.1" xref="S5.6.p2.9.m9.5.5.5.1.1.1.1.cmml"><mo id="S5.6.p2.9.m9.5.5.5.1.1.1.2" stretchy="false" xref="S5.6.p2.9.m9.5.5.5.1.1.1.1.cmml">(</mo><msub id="S5.6.p2.9.m9.5.5.5.1.1.1.1" xref="S5.6.p2.9.m9.5.5.5.1.1.1.1.cmml"><mi id="S5.6.p2.9.m9.5.5.5.1.1.1.1.2" xref="S5.6.p2.9.m9.5.5.5.1.1.1.1.2.cmml">Z</mi><mn id="S5.6.p2.9.m9.5.5.5.1.1.1.1.3" xref="S5.6.p2.9.m9.5.5.5.1.1.1.1.3.cmml">1</mn></msub><mo id="S5.6.p2.9.m9.5.5.5.1.1.1.3" stretchy="false" xref="S5.6.p2.9.m9.5.5.5.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.6.p2.9.m9.6.6.6.3" xref="S5.6.p2.9.m9.6.6.6.3.cmml">∖</mo><mrow id="S5.6.p2.9.m9.6.6.6.2" xref="S5.6.p2.9.m9.6.6.6.2.cmml"><mi id="S5.6.p2.9.m9.6.6.6.2.3" xref="S5.6.p2.9.m9.6.6.6.2.3.cmml">R</mi><mo id="S5.6.p2.9.m9.6.6.6.2.2" xref="S5.6.p2.9.m9.6.6.6.2.2.cmml">⁢</mo><mrow id="S5.6.p2.9.m9.6.6.6.2.1.1" xref="S5.6.p2.9.m9.6.6.6.2.1.1.1.cmml"><mo id="S5.6.p2.9.m9.6.6.6.2.1.1.2" stretchy="false" xref="S5.6.p2.9.m9.6.6.6.2.1.1.1.cmml">(</mo><msub id="S5.6.p2.9.m9.6.6.6.2.1.1.1" xref="S5.6.p2.9.m9.6.6.6.2.1.1.1.cmml"><mi id="S5.6.p2.9.m9.6.6.6.2.1.1.1.2" xref="S5.6.p2.9.m9.6.6.6.2.1.1.1.2.cmml">Z</mi><mn id="S5.6.p2.9.m9.6.6.6.2.1.1.1.3" xref="S5.6.p2.9.m9.6.6.6.2.1.1.1.3.cmml">0</mn></msub><mo id="S5.6.p2.9.m9.6.6.6.2.1.1.3" stretchy="false" xref="S5.6.p2.9.m9.6.6.6.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S5.6.p2.9.m9.6.6.9" xref="S5.6.p2.9.m9.6.6.9.cmml">⊇</mo><msub id="S5.6.p2.9.m9.6.6.10" xref="S5.6.p2.9.m9.6.6.10.cmml"><mi id="S5.6.p2.9.m9.6.6.10.2" xref="S5.6.p2.9.m9.6.6.10.2.cmml">S</mi><mi id="S5.6.p2.9.m9.6.6.10.3" xref="S5.6.p2.9.m9.6.6.10.3.cmml">α</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.6.p2.9.m9.6b"><apply id="S5.6.p2.9.m9.6.6.cmml" xref="S5.6.p2.9.m9.6.6"><and id="S5.6.p2.9.m9.6.6a.cmml" xref="S5.6.p2.9.m9.6.6"></and><apply id="S5.6.p2.9.m9.6.6b.cmml" xref="S5.6.p2.9.m9.6.6"><eq id="S5.6.p2.9.m9.6.6.8.cmml" xref="S5.6.p2.9.m9.6.6.8"></eq><apply id="S5.6.p2.9.m9.4.4.4.cmml" xref="S5.6.p2.9.m9.4.4.4"><setdiff id="S5.6.p2.9.m9.4.4.4.5.cmml" xref="S5.6.p2.9.m9.4.4.4.5"></setdiff><apply id="S5.6.p2.9.m9.2.2.2.2.cmml" xref="S5.6.p2.9.m9.2.2.2.2"><times id="S5.6.p2.9.m9.2.2.2.2.3.cmml" xref="S5.6.p2.9.m9.2.2.2.2.3"></times><apply id="S5.6.p2.9.m9.2.2.2.2.4.cmml" xref="S5.6.p2.9.m9.2.2.2.2.4"><ci id="S5.6.p2.9.m9.2.2.2.2.4.1.cmml" xref="S5.6.p2.9.m9.2.2.2.2.4.1">^</ci><ci id="S5.6.p2.9.m9.2.2.2.2.4.2.cmml" xref="S5.6.p2.9.m9.2.2.2.2.4.2">ℛ</ci></apply><interval closure="open" id="S5.6.p2.9.m9.2.2.2.2.2.3.cmml" xref="S5.6.p2.9.m9.2.2.2.2.2.2"><apply id="S5.6.p2.9.m9.1.1.1.1.1.1.1.cmml" xref="S5.6.p2.9.m9.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.6.p2.9.m9.1.1.1.1.1.1.1.1.cmml" xref="S5.6.p2.9.m9.1.1.1.1.1.1.1">superscript</csymbol><ci id="S5.6.p2.9.m9.1.1.1.1.1.1.1.2.cmml" xref="S5.6.p2.9.m9.1.1.1.1.1.1.1.2">𝐴</ci><apply id="S5.6.p2.9.m9.1.1.1.1.1.1.1.3.cmml" xref="S5.6.p2.9.m9.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S5.6.p2.9.m9.1.1.1.1.1.1.1.3.1.cmml" xref="S5.6.p2.9.m9.1.1.1.1.1.1.1.3">subscript</csymbol><ci id="S5.6.p2.9.m9.1.1.1.1.1.1.1.3.2.cmml" xref="S5.6.p2.9.m9.1.1.1.1.1.1.1.3.2">𝑍</ci><cn id="S5.6.p2.9.m9.1.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S5.6.p2.9.m9.1.1.1.1.1.1.1.3.3">1</cn></apply></apply><apply id="S5.6.p2.9.m9.2.2.2.2.2.2.2.cmml" xref="S5.6.p2.9.m9.2.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S5.6.p2.9.m9.2.2.2.2.2.2.2.1.cmml" xref="S5.6.p2.9.m9.2.2.2.2.2.2.2">superscript</csymbol><ci id="S5.6.p2.9.m9.2.2.2.2.2.2.2.2.cmml" xref="S5.6.p2.9.m9.2.2.2.2.2.2.2.2">𝐷</ci><apply id="S5.6.p2.9.m9.2.2.2.2.2.2.2.3.cmml" xref="S5.6.p2.9.m9.2.2.2.2.2.2.2.3"><csymbol cd="ambiguous" id="S5.6.p2.9.m9.2.2.2.2.2.2.2.3.1.cmml" xref="S5.6.p2.9.m9.2.2.2.2.2.2.2.3">subscript</csymbol><ci id="S5.6.p2.9.m9.2.2.2.2.2.2.2.3.2.cmml" xref="S5.6.p2.9.m9.2.2.2.2.2.2.2.3.2">𝑍</ci><cn id="S5.6.p2.9.m9.2.2.2.2.2.2.2.3.3.cmml" type="integer" xref="S5.6.p2.9.m9.2.2.2.2.2.2.2.3.3">1</cn></apply></apply></interval></apply><apply id="S5.6.p2.9.m9.4.4.4.4.cmml" xref="S5.6.p2.9.m9.4.4.4.4"><times id="S5.6.p2.9.m9.4.4.4.4.3.cmml" xref="S5.6.p2.9.m9.4.4.4.4.3"></times><apply id="S5.6.p2.9.m9.4.4.4.4.4.cmml" xref="S5.6.p2.9.m9.4.4.4.4.4"><ci id="S5.6.p2.9.m9.4.4.4.4.4.1.cmml" xref="S5.6.p2.9.m9.4.4.4.4.4.1">^</ci><ci id="S5.6.p2.9.m9.4.4.4.4.4.2.cmml" xref="S5.6.p2.9.m9.4.4.4.4.4.2">ℛ</ci></apply><interval closure="open" id="S5.6.p2.9.m9.4.4.4.4.2.3.cmml" xref="S5.6.p2.9.m9.4.4.4.4.2.2"><apply id="S5.6.p2.9.m9.3.3.3.3.1.1.1.cmml" xref="S5.6.p2.9.m9.3.3.3.3.1.1.1"><csymbol cd="ambiguous" id="S5.6.p2.9.m9.3.3.3.3.1.1.1.1.cmml" xref="S5.6.p2.9.m9.3.3.3.3.1.1.1">superscript</csymbol><ci id="S5.6.p2.9.m9.3.3.3.3.1.1.1.2.cmml" xref="S5.6.p2.9.m9.3.3.3.3.1.1.1.2">𝐴</ci><apply id="S5.6.p2.9.m9.3.3.3.3.1.1.1.3.cmml" xref="S5.6.p2.9.m9.3.3.3.3.1.1.1.3"><csymbol cd="ambiguous" id="S5.6.p2.9.m9.3.3.3.3.1.1.1.3.1.cmml" xref="S5.6.p2.9.m9.3.3.3.3.1.1.1.3">subscript</csymbol><ci id="S5.6.p2.9.m9.3.3.3.3.1.1.1.3.2.cmml" xref="S5.6.p2.9.m9.3.3.3.3.1.1.1.3.2">𝑍</ci><cn id="S5.6.p2.9.m9.3.3.3.3.1.1.1.3.3.cmml" type="integer" xref="S5.6.p2.9.m9.3.3.3.3.1.1.1.3.3">0</cn></apply></apply><apply id="S5.6.p2.9.m9.4.4.4.4.2.2.2.cmml" xref="S5.6.p2.9.m9.4.4.4.4.2.2.2"><csymbol cd="ambiguous" id="S5.6.p2.9.m9.4.4.4.4.2.2.2.1.cmml" xref="S5.6.p2.9.m9.4.4.4.4.2.2.2">superscript</csymbol><ci id="S5.6.p2.9.m9.4.4.4.4.2.2.2.2.cmml" xref="S5.6.p2.9.m9.4.4.4.4.2.2.2.2">𝐷</ci><apply id="S5.6.p2.9.m9.4.4.4.4.2.2.2.3.cmml" xref="S5.6.p2.9.m9.4.4.4.4.2.2.2.3"><csymbol cd="ambiguous" id="S5.6.p2.9.m9.4.4.4.4.2.2.2.3.1.cmml" xref="S5.6.p2.9.m9.4.4.4.4.2.2.2.3">subscript</csymbol><ci id="S5.6.p2.9.m9.4.4.4.4.2.2.2.3.2.cmml" xref="S5.6.p2.9.m9.4.4.4.4.2.2.2.3.2">𝑍</ci><cn id="S5.6.p2.9.m9.4.4.4.4.2.2.2.3.3.cmml" type="integer" xref="S5.6.p2.9.m9.4.4.4.4.2.2.2.3.3">0</cn></apply></apply></interval></apply></apply><apply id="S5.6.p2.9.m9.6.6.6.cmml" xref="S5.6.p2.9.m9.6.6.6"><setdiff id="S5.6.p2.9.m9.6.6.6.3.cmml" xref="S5.6.p2.9.m9.6.6.6.3"></setdiff><apply id="S5.6.p2.9.m9.5.5.5.1.cmml" xref="S5.6.p2.9.m9.5.5.5.1"><times id="S5.6.p2.9.m9.5.5.5.1.2.cmml" xref="S5.6.p2.9.m9.5.5.5.1.2"></times><ci id="S5.6.p2.9.m9.5.5.5.1.3.cmml" xref="S5.6.p2.9.m9.5.5.5.1.3">𝑅</ci><apply id="S5.6.p2.9.m9.5.5.5.1.1.1.1.cmml" xref="S5.6.p2.9.m9.5.5.5.1.1.1"><csymbol cd="ambiguous" id="S5.6.p2.9.m9.5.5.5.1.1.1.1.1.cmml" xref="S5.6.p2.9.m9.5.5.5.1.1.1">subscript</csymbol><ci id="S5.6.p2.9.m9.5.5.5.1.1.1.1.2.cmml" xref="S5.6.p2.9.m9.5.5.5.1.1.1.1.2">𝑍</ci><cn id="S5.6.p2.9.m9.5.5.5.1.1.1.1.3.cmml" type="integer" xref="S5.6.p2.9.m9.5.5.5.1.1.1.1.3">1</cn></apply></apply><apply id="S5.6.p2.9.m9.6.6.6.2.cmml" xref="S5.6.p2.9.m9.6.6.6.2"><times id="S5.6.p2.9.m9.6.6.6.2.2.cmml" xref="S5.6.p2.9.m9.6.6.6.2.2"></times><ci id="S5.6.p2.9.m9.6.6.6.2.3.cmml" xref="S5.6.p2.9.m9.6.6.6.2.3">𝑅</ci><apply id="S5.6.p2.9.m9.6.6.6.2.1.1.1.cmml" xref="S5.6.p2.9.m9.6.6.6.2.1.1"><csymbol cd="ambiguous" id="S5.6.p2.9.m9.6.6.6.2.1.1.1.1.cmml" xref="S5.6.p2.9.m9.6.6.6.2.1.1">subscript</csymbol><ci id="S5.6.p2.9.m9.6.6.6.2.1.1.1.2.cmml" xref="S5.6.p2.9.m9.6.6.6.2.1.1.1.2">𝑍</ci><cn id="S5.6.p2.9.m9.6.6.6.2.1.1.1.3.cmml" type="integer" xref="S5.6.p2.9.m9.6.6.6.2.1.1.1.3">0</cn></apply></apply></apply></apply><apply id="S5.6.p2.9.m9.6.6c.cmml" xref="S5.6.p2.9.m9.6.6"><csymbol cd="latexml" id="S5.6.p2.9.m9.6.6.9.cmml" xref="S5.6.p2.9.m9.6.6.9">superset-of-or-equals</csymbol><share href="https://arxiv.org/html/2503.13728v1#S5.6.p2.9.m9.6.6.6.cmml" id="S5.6.p2.9.m9.6.6d.cmml" xref="S5.6.p2.9.m9.6.6"></share><apply id="S5.6.p2.9.m9.6.6.10.cmml" xref="S5.6.p2.9.m9.6.6.10"><csymbol cd="ambiguous" id="S5.6.p2.9.m9.6.6.10.1.cmml" xref="S5.6.p2.9.m9.6.6.10">subscript</csymbol><ci id="S5.6.p2.9.m9.6.6.10.2.cmml" xref="S5.6.p2.9.m9.6.6.10.2">𝑆</ci><ci id="S5.6.p2.9.m9.6.6.10.3.cmml" xref="S5.6.p2.9.m9.6.6.10.3">𝛼</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.6.p2.9.m9.6c">\hat{\mathscr{R}}(A^{Z_{1}},D^{Z_{1}})\setminus\hat{\mathscr{R}}(A^{Z_{0}},D^{% Z_{0}})=R(Z_{1})\setminus R(Z_{0})\supseteq S_{\alpha}</annotation><annotation encoding="application/x-llamapun" id="S5.6.p2.9.m9.6d">over^ start_ARG script_R end_ARG ( italic_A start_POSTSUPERSCRIPT italic_Z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT , italic_D start_POSTSUPERSCRIPT italic_Z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ) ∖ over^ start_ARG script_R end_ARG ( italic_A start_POSTSUPERSCRIPT italic_Z start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT , italic_D start_POSTSUPERSCRIPT italic_Z start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ) = italic_R ( italic_Z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ∖ italic_R ( italic_Z start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ⊇ italic_S start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT</annotation></semantics></math>, thus by <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S5.Thmtheorem2" title="Lemma 5.2. ‣ 5. An infinite antichain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">5.2</span></a> <math alttext="A^{Z_{1}}\ntrianglerighteq A^{Z_{0}}" class="ltx_Math" display="inline" id="S5.6.p2.10.m10.1"><semantics id="S5.6.p2.10.m10.1a"><mrow id="S5.6.p2.10.m10.1.1" xref="S5.6.p2.10.m10.1.1.cmml"><msup id="S5.6.p2.10.m10.1.1.2" xref="S5.6.p2.10.m10.1.1.2.cmml"><mi id="S5.6.p2.10.m10.1.1.2.2" xref="S5.6.p2.10.m10.1.1.2.2.cmml">A</mi><msub id="S5.6.p2.10.m10.1.1.2.3" xref="S5.6.p2.10.m10.1.1.2.3.cmml"><mi id="S5.6.p2.10.m10.1.1.2.3.2" xref="S5.6.p2.10.m10.1.1.2.3.2.cmml">Z</mi><mn id="S5.6.p2.10.m10.1.1.2.3.3" xref="S5.6.p2.10.m10.1.1.2.3.3.cmml">1</mn></msub></msup><mo id="S5.6.p2.10.m10.1.1.1" xref="S5.6.p2.10.m10.1.1.1.cmml">⋭</mo><msup id="S5.6.p2.10.m10.1.1.3" xref="S5.6.p2.10.m10.1.1.3.cmml"><mi id="S5.6.p2.10.m10.1.1.3.2" xref="S5.6.p2.10.m10.1.1.3.2.cmml">A</mi><msub id="S5.6.p2.10.m10.1.1.3.3" xref="S5.6.p2.10.m10.1.1.3.3.cmml"><mi id="S5.6.p2.10.m10.1.1.3.3.2" xref="S5.6.p2.10.m10.1.1.3.3.2.cmml">Z</mi><mn id="S5.6.p2.10.m10.1.1.3.3.3" xref="S5.6.p2.10.m10.1.1.3.3.3.cmml">0</mn></msub></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.6.p2.10.m10.1b"><apply id="S5.6.p2.10.m10.1.1.cmml" xref="S5.6.p2.10.m10.1.1"><csymbol cd="latexml" id="S5.6.p2.10.m10.1.1.1.cmml" xref="S5.6.p2.10.m10.1.1.1">not-contains-nor-equals</csymbol><apply id="S5.6.p2.10.m10.1.1.2.cmml" xref="S5.6.p2.10.m10.1.1.2"><csymbol cd="ambiguous" id="S5.6.p2.10.m10.1.1.2.1.cmml" xref="S5.6.p2.10.m10.1.1.2">superscript</csymbol><ci id="S5.6.p2.10.m10.1.1.2.2.cmml" xref="S5.6.p2.10.m10.1.1.2.2">𝐴</ci><apply id="S5.6.p2.10.m10.1.1.2.3.cmml" xref="S5.6.p2.10.m10.1.1.2.3"><csymbol cd="ambiguous" id="S5.6.p2.10.m10.1.1.2.3.1.cmml" xref="S5.6.p2.10.m10.1.1.2.3">subscript</csymbol><ci id="S5.6.p2.10.m10.1.1.2.3.2.cmml" xref="S5.6.p2.10.m10.1.1.2.3.2">𝑍</ci><cn id="S5.6.p2.10.m10.1.1.2.3.3.cmml" type="integer" xref="S5.6.p2.10.m10.1.1.2.3.3">1</cn></apply></apply><apply id="S5.6.p2.10.m10.1.1.3.cmml" xref="S5.6.p2.10.m10.1.1.3"><csymbol cd="ambiguous" id="S5.6.p2.10.m10.1.1.3.1.cmml" xref="S5.6.p2.10.m10.1.1.3">superscript</csymbol><ci id="S5.6.p2.10.m10.1.1.3.2.cmml" xref="S5.6.p2.10.m10.1.1.3.2">𝐴</ci><apply id="S5.6.p2.10.m10.1.1.3.3.cmml" xref="S5.6.p2.10.m10.1.1.3.3"><csymbol cd="ambiguous" id="S5.6.p2.10.m10.1.1.3.3.1.cmml" xref="S5.6.p2.10.m10.1.1.3.3">subscript</csymbol><ci id="S5.6.p2.10.m10.1.1.3.3.2.cmml" xref="S5.6.p2.10.m10.1.1.3.3.2">𝑍</ci><cn id="S5.6.p2.10.m10.1.1.3.3.3.cmml" type="integer" xref="S5.6.p2.10.m10.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.6.p2.10.m10.1c">A^{Z_{1}}\ntrianglerighteq A^{Z_{0}}</annotation><annotation encoding="application/x-llamapun" id="S5.6.p2.10.m10.1d">italic_A start_POSTSUPERSCRIPT italic_Z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ⋭ italic_A start_POSTSUPERSCRIPT italic_Z start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math>. ∎</p> </div> </div> <div class="ltx_para" id="S5.p4"> <p class="ltx_p" id="S5.p4.1">Note that <math alttext="2^{\aleph_{1}}" class="ltx_Math" display="inline" id="S5.p4.1.m1.1"><semantics id="S5.p4.1.m1.1a"><msup id="S5.p4.1.m1.1.1" xref="S5.p4.1.m1.1.1.cmml"><mn id="S5.p4.1.m1.1.1.2" xref="S5.p4.1.m1.1.1.2.cmml">2</mn><msub id="S5.p4.1.m1.1.1.3" xref="S5.p4.1.m1.1.1.3.cmml"><mi id="S5.p4.1.m1.1.1.3.2" mathvariant="normal" xref="S5.p4.1.m1.1.1.3.2.cmml">ℵ</mi><mn id="S5.p4.1.m1.1.1.3.3" xref="S5.p4.1.m1.1.1.3.3.cmml">1</mn></msub></msup><annotation-xml encoding="MathML-Content" id="S5.p4.1.m1.1b"><apply id="S5.p4.1.m1.1.1.cmml" xref="S5.p4.1.m1.1.1"><csymbol cd="ambiguous" id="S5.p4.1.m1.1.1.1.cmml" xref="S5.p4.1.m1.1.1">superscript</csymbol><cn id="S5.p4.1.m1.1.1.2.cmml" type="integer" xref="S5.p4.1.m1.1.1.2">2</cn><apply id="S5.p4.1.m1.1.1.3.cmml" xref="S5.p4.1.m1.1.1.3"><csymbol cd="ambiguous" id="S5.p4.1.m1.1.1.3.1.cmml" xref="S5.p4.1.m1.1.1.3">subscript</csymbol><ci id="S5.p4.1.m1.1.1.3.2.cmml" xref="S5.p4.1.m1.1.1.3.2">ℵ</ci><cn id="S5.p4.1.m1.1.1.3.3.cmml" type="integer" xref="S5.p4.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p4.1.m1.1c">2^{\aleph_{1}}</annotation><annotation encoding="application/x-llamapun" id="S5.p4.1.m1.1d">2 start_POSTSUPERSCRIPT roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> is the largest possible size of an antichain. Also note that <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S4.Thmtheorem8" title="Corollary 4.8. ‣ 4. Aronszajn line decompositions ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Corollary</span> <span class="ltx_text ltx_ref_tag">4.8</span></a> easily implies the following.</p> </div> <div class="ltx_theorem ltx_theorem_corollary" id="S5.Thmtheorem4"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem4.1.1.1">Corollary 5.4</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem4.2.2">.</span> </h6> <div class="ltx_para" id="S5.Thmtheorem4.p1"> <p class="ltx_p" id="S5.Thmtheorem4.p1.4">Assume <math alttext="\mathsf{MA}_{\aleph_{1}}" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p1.1.m1.1"><semantics id="S5.Thmtheorem4.p1.1.m1.1a"><msub id="S5.Thmtheorem4.p1.1.m1.1.1" xref="S5.Thmtheorem4.p1.1.m1.1.1.cmml"><mi id="S5.Thmtheorem4.p1.1.m1.1.1.2" xref="S5.Thmtheorem4.p1.1.m1.1.1.2.cmml">𝖬𝖠</mi><msub id="S5.Thmtheorem4.p1.1.m1.1.1.3" xref="S5.Thmtheorem4.p1.1.m1.1.1.3.cmml"><mi id="S5.Thmtheorem4.p1.1.m1.1.1.3.2" mathvariant="normal" xref="S5.Thmtheorem4.p1.1.m1.1.1.3.2.cmml">ℵ</mi><mn id="S5.Thmtheorem4.p1.1.m1.1.1.3.3" xref="S5.Thmtheorem4.p1.1.m1.1.1.3.3.cmml">1</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p1.1.m1.1b"><apply id="S5.Thmtheorem4.p1.1.m1.1.1.cmml" xref="S5.Thmtheorem4.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem4.p1.1.m1.1.1.1.cmml" xref="S5.Thmtheorem4.p1.1.m1.1.1">subscript</csymbol><ci id="S5.Thmtheorem4.p1.1.m1.1.1.2.cmml" xref="S5.Thmtheorem4.p1.1.m1.1.1.2">𝖬𝖠</ci><apply id="S5.Thmtheorem4.p1.1.m1.1.1.3.cmml" xref="S5.Thmtheorem4.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S5.Thmtheorem4.p1.1.m1.1.1.3.1.cmml" xref="S5.Thmtheorem4.p1.1.m1.1.1.3">subscript</csymbol><ci id="S5.Thmtheorem4.p1.1.m1.1.1.3.2.cmml" xref="S5.Thmtheorem4.p1.1.m1.1.1.3.2">ℵ</ci><cn id="S5.Thmtheorem4.p1.1.m1.1.1.3.3.cmml" type="integer" xref="S5.Thmtheorem4.p1.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p1.1.m1.1c">\mathsf{MA}_{\aleph_{1}}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p1.1.m1.1d">sansserif_MA start_POSTSUBSCRIPT roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. There is an <math alttext="\trianglelefteq" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p1.2.m2.1"><semantics id="S5.Thmtheorem4.p1.2.m2.1a"><mi id="S5.Thmtheorem4.p1.2.m2.1.1" mathvariant="normal" xref="S5.Thmtheorem4.p1.2.m2.1.1.cmml">⊴</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p1.2.m2.1b"><ci id="S5.Thmtheorem4.p1.2.m2.1.1.cmml" xref="S5.Thmtheorem4.p1.2.m2.1.1">⊴</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p1.2.m2.1c">\trianglelefteq</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p1.2.m2.1d">⊴</annotation></semantics></math>-antichain of <math alttext="\aleph_{1}" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p1.3.m3.1"><semantics id="S5.Thmtheorem4.p1.3.m3.1a"><msub id="S5.Thmtheorem4.p1.3.m3.1.1" xref="S5.Thmtheorem4.p1.3.m3.1.1.cmml"><mi id="S5.Thmtheorem4.p1.3.m3.1.1.2" mathvariant="normal" xref="S5.Thmtheorem4.p1.3.m3.1.1.2.cmml">ℵ</mi><mn id="S5.Thmtheorem4.p1.3.m3.1.1.3" xref="S5.Thmtheorem4.p1.3.m3.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p1.3.m3.1b"><apply id="S5.Thmtheorem4.p1.3.m3.1.1.cmml" xref="S5.Thmtheorem4.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem4.p1.3.m3.1.1.1.cmml" xref="S5.Thmtheorem4.p1.3.m3.1.1">subscript</csymbol><ci id="S5.Thmtheorem4.p1.3.m3.1.1.2.cmml" xref="S5.Thmtheorem4.p1.3.m3.1.1.2">ℵ</ci><cn id="S5.Thmtheorem4.p1.3.m3.1.1.3.cmml" type="integer" xref="S5.Thmtheorem4.p1.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p1.3.m3.1c">\aleph_{1}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p1.3.m3.1d">roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-dense Countryman lines of size <math alttext="2^{\aleph_{1}}" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p1.4.m4.1"><semantics id="S5.Thmtheorem4.p1.4.m4.1a"><msup id="S5.Thmtheorem4.p1.4.m4.1.1" xref="S5.Thmtheorem4.p1.4.m4.1.1.cmml"><mn id="S5.Thmtheorem4.p1.4.m4.1.1.2" xref="S5.Thmtheorem4.p1.4.m4.1.1.2.cmml">2</mn><msub id="S5.Thmtheorem4.p1.4.m4.1.1.3" xref="S5.Thmtheorem4.p1.4.m4.1.1.3.cmml"><mi id="S5.Thmtheorem4.p1.4.m4.1.1.3.2" mathvariant="normal" xref="S5.Thmtheorem4.p1.4.m4.1.1.3.2.cmml">ℵ</mi><mn id="S5.Thmtheorem4.p1.4.m4.1.1.3.3" xref="S5.Thmtheorem4.p1.4.m4.1.1.3.3.cmml">1</mn></msub></msup><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p1.4.m4.1b"><apply id="S5.Thmtheorem4.p1.4.m4.1.1.cmml" xref="S5.Thmtheorem4.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem4.p1.4.m4.1.1.1.cmml" xref="S5.Thmtheorem4.p1.4.m4.1.1">superscript</csymbol><cn id="S5.Thmtheorem4.p1.4.m4.1.1.2.cmml" type="integer" xref="S5.Thmtheorem4.p1.4.m4.1.1.2">2</cn><apply id="S5.Thmtheorem4.p1.4.m4.1.1.3.cmml" xref="S5.Thmtheorem4.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="S5.Thmtheorem4.p1.4.m4.1.1.3.1.cmml" xref="S5.Thmtheorem4.p1.4.m4.1.1.3">subscript</csymbol><ci id="S5.Thmtheorem4.p1.4.m4.1.1.3.2.cmml" xref="S5.Thmtheorem4.p1.4.m4.1.1.3.2">ℵ</ci><cn id="S5.Thmtheorem4.p1.4.m4.1.1.3.3.cmml" type="integer" xref="S5.Thmtheorem4.p1.4.m4.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p1.4.m4.1c">2^{\aleph_{1}}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p1.4.m4.1d">2 start_POSTSUPERSCRIPT roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> </div> <div class="ltx_para" id="S5.p5"> <p class="ltx_p" id="S5.p5.1">By <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S4.Thmtheorem9" title="Remark 4.9. ‣ 4. Aronszajn line decompositions ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Remark</span> <span class="ltx_text ltx_ref_tag">4.9</span></a>, with a little more work this can also be achieved in <math alttext="\mathsf{ZFC}" class="ltx_Math" display="inline" id="S5.p5.1.m1.1"><semantics id="S5.p5.1.m1.1a"><mi id="S5.p5.1.m1.1.1" xref="S5.p5.1.m1.1.1.cmml">𝖹𝖥𝖢</mi><annotation-xml encoding="MathML-Content" id="S5.p5.1.m1.1b"><ci id="S5.p5.1.m1.1.1.cmml" xref="S5.p5.1.m1.1.1">𝖹𝖥𝖢</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p5.1.m1.1c">\mathsf{ZFC}</annotation><annotation encoding="application/x-llamapun" id="S5.p5.1.m1.1d">sansserif_ZFC</annotation></semantics></math>, and even without it, one can also get an infinite antichain of Countryman lines at the cost of the size.</p> </div> <div class="ltx_theorem ltx_theorem_corollary" id="S5.Thmtheorem5"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem5.1.1.1">Corollary 5.5</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem5.2.2">.</span> </h6> <div class="ltx_para" id="S5.Thmtheorem5.p1"> <p class="ltx_p" id="S5.Thmtheorem5.p1.3">There is an <math alttext="\trianglelefteq" class="ltx_Math" display="inline" id="S5.Thmtheorem5.p1.1.m1.1"><semantics id="S5.Thmtheorem5.p1.1.m1.1a"><mi id="S5.Thmtheorem5.p1.1.m1.1.1" mathvariant="normal" xref="S5.Thmtheorem5.p1.1.m1.1.1.cmml">⊴</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem5.p1.1.m1.1b"><ci id="S5.Thmtheorem5.p1.1.m1.1.1.cmml" xref="S5.Thmtheorem5.p1.1.m1.1.1">⊴</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem5.p1.1.m1.1c">\trianglelefteq</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem5.p1.1.m1.1d">⊴</annotation></semantics></math>-antichain of <math alttext="\aleph_{1}" class="ltx_Math" display="inline" id="S5.Thmtheorem5.p1.2.m2.1"><semantics id="S5.Thmtheorem5.p1.2.m2.1a"><msub id="S5.Thmtheorem5.p1.2.m2.1.1" xref="S5.Thmtheorem5.p1.2.m2.1.1.cmml"><mi id="S5.Thmtheorem5.p1.2.m2.1.1.2" mathvariant="normal" xref="S5.Thmtheorem5.p1.2.m2.1.1.2.cmml">ℵ</mi><mn id="S5.Thmtheorem5.p1.2.m2.1.1.3" xref="S5.Thmtheorem5.p1.2.m2.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem5.p1.2.m2.1b"><apply id="S5.Thmtheorem5.p1.2.m2.1.1.cmml" xref="S5.Thmtheorem5.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem5.p1.2.m2.1.1.1.cmml" xref="S5.Thmtheorem5.p1.2.m2.1.1">subscript</csymbol><ci id="S5.Thmtheorem5.p1.2.m2.1.1.2.cmml" xref="S5.Thmtheorem5.p1.2.m2.1.1.2">ℵ</ci><cn id="S5.Thmtheorem5.p1.2.m2.1.1.3.cmml" type="integer" xref="S5.Thmtheorem5.p1.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem5.p1.2.m2.1c">\aleph_{1}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem5.p1.2.m2.1d">roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-dense Countryman lines of size <math alttext="\aleph_{1}" class="ltx_Math" display="inline" id="S5.Thmtheorem5.p1.3.m3.1"><semantics id="S5.Thmtheorem5.p1.3.m3.1a"><msub id="S5.Thmtheorem5.p1.3.m3.1.1" xref="S5.Thmtheorem5.p1.3.m3.1.1.cmml"><mi id="S5.Thmtheorem5.p1.3.m3.1.1.2" mathvariant="normal" xref="S5.Thmtheorem5.p1.3.m3.1.1.2.cmml">ℵ</mi><mn id="S5.Thmtheorem5.p1.3.m3.1.1.3" xref="S5.Thmtheorem5.p1.3.m3.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem5.p1.3.m3.1b"><apply id="S5.Thmtheorem5.p1.3.m3.1.1.cmml" xref="S5.Thmtheorem5.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem5.p1.3.m3.1.1.1.cmml" xref="S5.Thmtheorem5.p1.3.m3.1.1">subscript</csymbol><ci id="S5.Thmtheorem5.p1.3.m3.1.1.2.cmml" xref="S5.Thmtheorem5.p1.3.m3.1.1.2">ℵ</ci><cn id="S5.Thmtheorem5.p1.3.m3.1.1.3.cmml" type="integer" xref="S5.Thmtheorem5.p1.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem5.p1.3.m3.1c">\aleph_{1}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem5.p1.3.m3.1d">roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> </div> <div class="ltx_proof" id="S5.7"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S5.7.p1"> <p class="ltx_p" id="S5.7.p1.6">Let <math alttext="\langle S_{\xi}:\xi<\omega_{1}\rangle" class="ltx_math_unparsed" display="inline" id="S5.7.p1.1.m1.1"><semantics id="S5.7.p1.1.m1.1a"><mrow id="S5.7.p1.1.m1.1b"><mo id="S5.7.p1.1.m1.1.1" stretchy="false">⟨</mo><msub id="S5.7.p1.1.m1.1.2"><mi id="S5.7.p1.1.m1.1.2.2">S</mi><mi id="S5.7.p1.1.m1.1.2.3">ξ</mi></msub><mo id="S5.7.p1.1.m1.1.3" lspace="0.278em" rspace="0.278em">:</mo><mi id="S5.7.p1.1.m1.1.4">ξ</mi><mo id="S5.7.p1.1.m1.1.5"><</mo><msub id="S5.7.p1.1.m1.1.6"><mi id="S5.7.p1.1.m1.1.6.2">ω</mi><mn id="S5.7.p1.1.m1.1.6.3">1</mn></msub><mo id="S5.7.p1.1.m1.1.7" stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex" id="S5.7.p1.1.m1.1c">\langle S_{\xi}:\xi<\omega_{1}\rangle</annotation><annotation encoding="application/x-llamapun" id="S5.7.p1.1.m1.1d">⟨ italic_S start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT : italic_ξ < italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⟩</annotation></semantics></math> be a family of disjoint stationary subsets of <math alttext="\omega_{1}" class="ltx_Math" display="inline" id="S5.7.p1.2.m2.1"><semantics id="S5.7.p1.2.m2.1a"><msub id="S5.7.p1.2.m2.1.1" xref="S5.7.p1.2.m2.1.1.cmml"><mi id="S5.7.p1.2.m2.1.1.2" xref="S5.7.p1.2.m2.1.1.2.cmml">ω</mi><mn id="S5.7.p1.2.m2.1.1.3" xref="S5.7.p1.2.m2.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S5.7.p1.2.m2.1b"><apply id="S5.7.p1.2.m2.1.1.cmml" xref="S5.7.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S5.7.p1.2.m2.1.1.1.cmml" xref="S5.7.p1.2.m2.1.1">subscript</csymbol><ci id="S5.7.p1.2.m2.1.1.2.cmml" xref="S5.7.p1.2.m2.1.1.2">𝜔</ci><cn id="S5.7.p1.2.m2.1.1.3.cmml" type="integer" xref="S5.7.p1.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.7.p1.2.m2.1c">\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S5.7.p1.2.m2.1d">italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and using <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S4.Thmtheorem6" title="Corollary 4.6. ‣ 4. Aronszajn line decompositions ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Corollary</span> <span class="ltx_text ltx_ref_tag">4.6</span></a> let <math alttext="C^{S_{\xi}}" class="ltx_Math" display="inline" id="S5.7.p1.3.m3.1"><semantics id="S5.7.p1.3.m3.1a"><msup id="S5.7.p1.3.m3.1.1" xref="S5.7.p1.3.m3.1.1.cmml"><mi id="S5.7.p1.3.m3.1.1.2" xref="S5.7.p1.3.m3.1.1.2.cmml">C</mi><msub id="S5.7.p1.3.m3.1.1.3" xref="S5.7.p1.3.m3.1.1.3.cmml"><mi id="S5.7.p1.3.m3.1.1.3.2" xref="S5.7.p1.3.m3.1.1.3.2.cmml">S</mi><mi id="S5.7.p1.3.m3.1.1.3.3" xref="S5.7.p1.3.m3.1.1.3.3.cmml">ξ</mi></msub></msup><annotation-xml encoding="MathML-Content" id="S5.7.p1.3.m3.1b"><apply id="S5.7.p1.3.m3.1.1.cmml" xref="S5.7.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S5.7.p1.3.m3.1.1.1.cmml" xref="S5.7.p1.3.m3.1.1">superscript</csymbol><ci id="S5.7.p1.3.m3.1.1.2.cmml" xref="S5.7.p1.3.m3.1.1.2">𝐶</ci><apply id="S5.7.p1.3.m3.1.1.3.cmml" xref="S5.7.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S5.7.p1.3.m3.1.1.3.1.cmml" xref="S5.7.p1.3.m3.1.1.3">subscript</csymbol><ci id="S5.7.p1.3.m3.1.1.3.2.cmml" xref="S5.7.p1.3.m3.1.1.3.2">𝑆</ci><ci id="S5.7.p1.3.m3.1.1.3.3.cmml" xref="S5.7.p1.3.m3.1.1.3.3">𝜉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.7.p1.3.m3.1c">C^{S_{\xi}}</annotation><annotation encoding="application/x-llamapun" id="S5.7.p1.3.m3.1d">italic_C start_POSTSUPERSCRIPT italic_S start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="D^{S_{\xi}}" class="ltx_Math" display="inline" id="S5.7.p1.4.m4.1"><semantics id="S5.7.p1.4.m4.1a"><msup id="S5.7.p1.4.m4.1.1" xref="S5.7.p1.4.m4.1.1.cmml"><mi id="S5.7.p1.4.m4.1.1.2" xref="S5.7.p1.4.m4.1.1.2.cmml">D</mi><msub id="S5.7.p1.4.m4.1.1.3" xref="S5.7.p1.4.m4.1.1.3.cmml"><mi id="S5.7.p1.4.m4.1.1.3.2" xref="S5.7.p1.4.m4.1.1.3.2.cmml">S</mi><mi id="S5.7.p1.4.m4.1.1.3.3" xref="S5.7.p1.4.m4.1.1.3.3.cmml">ξ</mi></msub></msup><annotation-xml encoding="MathML-Content" id="S5.7.p1.4.m4.1b"><apply id="S5.7.p1.4.m4.1.1.cmml" xref="S5.7.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S5.7.p1.4.m4.1.1.1.cmml" xref="S5.7.p1.4.m4.1.1">superscript</csymbol><ci id="S5.7.p1.4.m4.1.1.2.cmml" xref="S5.7.p1.4.m4.1.1.2">𝐷</ci><apply id="S5.7.p1.4.m4.1.1.3.cmml" xref="S5.7.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="S5.7.p1.4.m4.1.1.3.1.cmml" xref="S5.7.p1.4.m4.1.1.3">subscript</csymbol><ci id="S5.7.p1.4.m4.1.1.3.2.cmml" xref="S5.7.p1.4.m4.1.1.3.2">𝑆</ci><ci id="S5.7.p1.4.m4.1.1.3.3.cmml" xref="S5.7.p1.4.m4.1.1.3.3">𝜉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.7.p1.4.m4.1c">D^{S_{\xi}}</annotation><annotation encoding="application/x-llamapun" id="S5.7.p1.4.m4.1d">italic_D start_POSTSUPERSCRIPT italic_S start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> be such that <math alttext="\hat{\mathscr{L}}(C^{S_{\xi}},D^{S_{\xi}})=S_{\xi}" class="ltx_Math" display="inline" id="S5.7.p1.5.m5.2"><semantics id="S5.7.p1.5.m5.2a"><mrow id="S5.7.p1.5.m5.2.2" xref="S5.7.p1.5.m5.2.2.cmml"><mrow id="S5.7.p1.5.m5.2.2.2" xref="S5.7.p1.5.m5.2.2.2.cmml"><mover accent="true" id="S5.7.p1.5.m5.2.2.2.4" xref="S5.7.p1.5.m5.2.2.2.4.cmml"><mi class="ltx_font_mathscript" id="S5.7.p1.5.m5.2.2.2.4.2" xref="S5.7.p1.5.m5.2.2.2.4.2.cmml">ℒ</mi><mo id="S5.7.p1.5.m5.2.2.2.4.1" xref="S5.7.p1.5.m5.2.2.2.4.1.cmml">^</mo></mover><mo id="S5.7.p1.5.m5.2.2.2.3" xref="S5.7.p1.5.m5.2.2.2.3.cmml">⁢</mo><mrow id="S5.7.p1.5.m5.2.2.2.2.2" xref="S5.7.p1.5.m5.2.2.2.2.3.cmml"><mo id="S5.7.p1.5.m5.2.2.2.2.2.3" stretchy="false" xref="S5.7.p1.5.m5.2.2.2.2.3.cmml">(</mo><msup id="S5.7.p1.5.m5.1.1.1.1.1.1" xref="S5.7.p1.5.m5.1.1.1.1.1.1.cmml"><mi id="S5.7.p1.5.m5.1.1.1.1.1.1.2" xref="S5.7.p1.5.m5.1.1.1.1.1.1.2.cmml">C</mi><msub id="S5.7.p1.5.m5.1.1.1.1.1.1.3" xref="S5.7.p1.5.m5.1.1.1.1.1.1.3.cmml"><mi id="S5.7.p1.5.m5.1.1.1.1.1.1.3.2" xref="S5.7.p1.5.m5.1.1.1.1.1.1.3.2.cmml">S</mi><mi id="S5.7.p1.5.m5.1.1.1.1.1.1.3.3" xref="S5.7.p1.5.m5.1.1.1.1.1.1.3.3.cmml">ξ</mi></msub></msup><mo id="S5.7.p1.5.m5.2.2.2.2.2.4" xref="S5.7.p1.5.m5.2.2.2.2.3.cmml">,</mo><msup id="S5.7.p1.5.m5.2.2.2.2.2.2" xref="S5.7.p1.5.m5.2.2.2.2.2.2.cmml"><mi id="S5.7.p1.5.m5.2.2.2.2.2.2.2" xref="S5.7.p1.5.m5.2.2.2.2.2.2.2.cmml">D</mi><msub id="S5.7.p1.5.m5.2.2.2.2.2.2.3" xref="S5.7.p1.5.m5.2.2.2.2.2.2.3.cmml"><mi id="S5.7.p1.5.m5.2.2.2.2.2.2.3.2" xref="S5.7.p1.5.m5.2.2.2.2.2.2.3.2.cmml">S</mi><mi id="S5.7.p1.5.m5.2.2.2.2.2.2.3.3" xref="S5.7.p1.5.m5.2.2.2.2.2.2.3.3.cmml">ξ</mi></msub></msup><mo id="S5.7.p1.5.m5.2.2.2.2.2.5" stretchy="false" xref="S5.7.p1.5.m5.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S5.7.p1.5.m5.2.2.3" xref="S5.7.p1.5.m5.2.2.3.cmml">=</mo><msub id="S5.7.p1.5.m5.2.2.4" xref="S5.7.p1.5.m5.2.2.4.cmml"><mi id="S5.7.p1.5.m5.2.2.4.2" xref="S5.7.p1.5.m5.2.2.4.2.cmml">S</mi><mi id="S5.7.p1.5.m5.2.2.4.3" xref="S5.7.p1.5.m5.2.2.4.3.cmml">ξ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.7.p1.5.m5.2b"><apply id="S5.7.p1.5.m5.2.2.cmml" xref="S5.7.p1.5.m5.2.2"><eq id="S5.7.p1.5.m5.2.2.3.cmml" xref="S5.7.p1.5.m5.2.2.3"></eq><apply id="S5.7.p1.5.m5.2.2.2.cmml" xref="S5.7.p1.5.m5.2.2.2"><times id="S5.7.p1.5.m5.2.2.2.3.cmml" xref="S5.7.p1.5.m5.2.2.2.3"></times><apply id="S5.7.p1.5.m5.2.2.2.4.cmml" xref="S5.7.p1.5.m5.2.2.2.4"><ci id="S5.7.p1.5.m5.2.2.2.4.1.cmml" xref="S5.7.p1.5.m5.2.2.2.4.1">^</ci><ci id="S5.7.p1.5.m5.2.2.2.4.2.cmml" xref="S5.7.p1.5.m5.2.2.2.4.2">ℒ</ci></apply><interval closure="open" id="S5.7.p1.5.m5.2.2.2.2.3.cmml" xref="S5.7.p1.5.m5.2.2.2.2.2"><apply id="S5.7.p1.5.m5.1.1.1.1.1.1.cmml" xref="S5.7.p1.5.m5.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.7.p1.5.m5.1.1.1.1.1.1.1.cmml" xref="S5.7.p1.5.m5.1.1.1.1.1.1">superscript</csymbol><ci id="S5.7.p1.5.m5.1.1.1.1.1.1.2.cmml" xref="S5.7.p1.5.m5.1.1.1.1.1.1.2">𝐶</ci><apply id="S5.7.p1.5.m5.1.1.1.1.1.1.3.cmml" xref="S5.7.p1.5.m5.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S5.7.p1.5.m5.1.1.1.1.1.1.3.1.cmml" xref="S5.7.p1.5.m5.1.1.1.1.1.1.3">subscript</csymbol><ci id="S5.7.p1.5.m5.1.1.1.1.1.1.3.2.cmml" xref="S5.7.p1.5.m5.1.1.1.1.1.1.3.2">𝑆</ci><ci id="S5.7.p1.5.m5.1.1.1.1.1.1.3.3.cmml" xref="S5.7.p1.5.m5.1.1.1.1.1.1.3.3">𝜉</ci></apply></apply><apply id="S5.7.p1.5.m5.2.2.2.2.2.2.cmml" xref="S5.7.p1.5.m5.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S5.7.p1.5.m5.2.2.2.2.2.2.1.cmml" xref="S5.7.p1.5.m5.2.2.2.2.2.2">superscript</csymbol><ci id="S5.7.p1.5.m5.2.2.2.2.2.2.2.cmml" xref="S5.7.p1.5.m5.2.2.2.2.2.2.2">𝐷</ci><apply id="S5.7.p1.5.m5.2.2.2.2.2.2.3.cmml" xref="S5.7.p1.5.m5.2.2.2.2.2.2.3"><csymbol cd="ambiguous" id="S5.7.p1.5.m5.2.2.2.2.2.2.3.1.cmml" xref="S5.7.p1.5.m5.2.2.2.2.2.2.3">subscript</csymbol><ci id="S5.7.p1.5.m5.2.2.2.2.2.2.3.2.cmml" xref="S5.7.p1.5.m5.2.2.2.2.2.2.3.2">𝑆</ci><ci id="S5.7.p1.5.m5.2.2.2.2.2.2.3.3.cmml" xref="S5.7.p1.5.m5.2.2.2.2.2.2.3.3">𝜉</ci></apply></apply></interval></apply><apply id="S5.7.p1.5.m5.2.2.4.cmml" xref="S5.7.p1.5.m5.2.2.4"><csymbol cd="ambiguous" id="S5.7.p1.5.m5.2.2.4.1.cmml" xref="S5.7.p1.5.m5.2.2.4">subscript</csymbol><ci id="S5.7.p1.5.m5.2.2.4.2.cmml" xref="S5.7.p1.5.m5.2.2.4.2">𝑆</ci><ci id="S5.7.p1.5.m5.2.2.4.3.cmml" xref="S5.7.p1.5.m5.2.2.4.3">𝜉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.7.p1.5.m5.2c">\hat{\mathscr{L}}(C^{S_{\xi}},D^{S_{\xi}})=S_{\xi}</annotation><annotation encoding="application/x-llamapun" id="S5.7.p1.5.m5.2d">over^ start_ARG script_L end_ARG ( italic_C start_POSTSUPERSCRIPT italic_S start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT end_POSTSUPERSCRIPT , italic_D start_POSTSUPERSCRIPT italic_S start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ) = italic_S start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT</annotation></semantics></math>. Then one proves that <math alttext="\{C^{\xi}:\xi<\omega_{1}\}" class="ltx_Math" display="inline" id="S5.7.p1.6.m6.2"><semantics id="S5.7.p1.6.m6.2a"><mrow id="S5.7.p1.6.m6.2.2.2" xref="S5.7.p1.6.m6.2.2.3.cmml"><mo id="S5.7.p1.6.m6.2.2.2.3" stretchy="false" xref="S5.7.p1.6.m6.2.2.3.1.cmml">{</mo><msup id="S5.7.p1.6.m6.1.1.1.1" xref="S5.7.p1.6.m6.1.1.1.1.cmml"><mi id="S5.7.p1.6.m6.1.1.1.1.2" xref="S5.7.p1.6.m6.1.1.1.1.2.cmml">C</mi><mi id="S5.7.p1.6.m6.1.1.1.1.3" xref="S5.7.p1.6.m6.1.1.1.1.3.cmml">ξ</mi></msup><mo id="S5.7.p1.6.m6.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S5.7.p1.6.m6.2.2.3.1.cmml">:</mo><mrow id="S5.7.p1.6.m6.2.2.2.2" xref="S5.7.p1.6.m6.2.2.2.2.cmml"><mi id="S5.7.p1.6.m6.2.2.2.2.2" xref="S5.7.p1.6.m6.2.2.2.2.2.cmml">ξ</mi><mo id="S5.7.p1.6.m6.2.2.2.2.1" xref="S5.7.p1.6.m6.2.2.2.2.1.cmml"><</mo><msub id="S5.7.p1.6.m6.2.2.2.2.3" xref="S5.7.p1.6.m6.2.2.2.2.3.cmml"><mi id="S5.7.p1.6.m6.2.2.2.2.3.2" xref="S5.7.p1.6.m6.2.2.2.2.3.2.cmml">ω</mi><mn id="S5.7.p1.6.m6.2.2.2.2.3.3" xref="S5.7.p1.6.m6.2.2.2.2.3.3.cmml">1</mn></msub></mrow><mo id="S5.7.p1.6.m6.2.2.2.5" stretchy="false" xref="S5.7.p1.6.m6.2.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.7.p1.6.m6.2b"><apply id="S5.7.p1.6.m6.2.2.3.cmml" xref="S5.7.p1.6.m6.2.2.2"><csymbol cd="latexml" id="S5.7.p1.6.m6.2.2.3.1.cmml" xref="S5.7.p1.6.m6.2.2.2.3">conditional-set</csymbol><apply id="S5.7.p1.6.m6.1.1.1.1.cmml" xref="S5.7.p1.6.m6.1.1.1.1"><csymbol cd="ambiguous" id="S5.7.p1.6.m6.1.1.1.1.1.cmml" xref="S5.7.p1.6.m6.1.1.1.1">superscript</csymbol><ci id="S5.7.p1.6.m6.1.1.1.1.2.cmml" xref="S5.7.p1.6.m6.1.1.1.1.2">𝐶</ci><ci id="S5.7.p1.6.m6.1.1.1.1.3.cmml" xref="S5.7.p1.6.m6.1.1.1.1.3">𝜉</ci></apply><apply id="S5.7.p1.6.m6.2.2.2.2.cmml" xref="S5.7.p1.6.m6.2.2.2.2"><lt id="S5.7.p1.6.m6.2.2.2.2.1.cmml" xref="S5.7.p1.6.m6.2.2.2.2.1"></lt><ci id="S5.7.p1.6.m6.2.2.2.2.2.cmml" xref="S5.7.p1.6.m6.2.2.2.2.2">𝜉</ci><apply id="S5.7.p1.6.m6.2.2.2.2.3.cmml" xref="S5.7.p1.6.m6.2.2.2.2.3"><csymbol cd="ambiguous" id="S5.7.p1.6.m6.2.2.2.2.3.1.cmml" xref="S5.7.p1.6.m6.2.2.2.2.3">subscript</csymbol><ci id="S5.7.p1.6.m6.2.2.2.2.3.2.cmml" xref="S5.7.p1.6.m6.2.2.2.2.3.2">𝜔</ci><cn id="S5.7.p1.6.m6.2.2.2.2.3.3.cmml" type="integer" xref="S5.7.p1.6.m6.2.2.2.2.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.7.p1.6.m6.2c">\{C^{\xi}:\xi<\omega_{1}\}</annotation><annotation encoding="application/x-llamapun" id="S5.7.p1.6.m6.2d">{ italic_C start_POSTSUPERSCRIPT italic_ξ end_POSTSUPERSCRIPT : italic_ξ < italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT }</annotation></semantics></math> is the desired antichain exactly as in the proof of the previous theorem. ∎</p> </div> </div> </section> <section class="ltx_section" id="S6"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">6. </span>An infinite decreasing chain</h2> <div class="ltx_para" id="S6.p1"> <p class="ltx_p" id="S6.p1.3">In this section we show that an adaptation of Moore’s forcing in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib16" title="">16</a>]</cite>, allows to introduce several epimorphisms between Countryman lines under <math alttext="\mathsf{MA}_{\aleph_{1}}" class="ltx_Math" display="inline" id="S6.p1.1.m1.1"><semantics id="S6.p1.1.m1.1a"><msub id="S6.p1.1.m1.1.1" xref="S6.p1.1.m1.1.1.cmml"><mi id="S6.p1.1.m1.1.1.2" xref="S6.p1.1.m1.1.1.2.cmml">𝖬𝖠</mi><msub id="S6.p1.1.m1.1.1.3" xref="S6.p1.1.m1.1.1.3.cmml"><mi id="S6.p1.1.m1.1.1.3.2" mathvariant="normal" xref="S6.p1.1.m1.1.1.3.2.cmml">ℵ</mi><mn id="S6.p1.1.m1.1.1.3.3" xref="S6.p1.1.m1.1.1.3.3.cmml">1</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S6.p1.1.m1.1b"><apply id="S6.p1.1.m1.1.1.cmml" xref="S6.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S6.p1.1.m1.1.1.1.cmml" xref="S6.p1.1.m1.1.1">subscript</csymbol><ci id="S6.p1.1.m1.1.1.2.cmml" xref="S6.p1.1.m1.1.1.2">𝖬𝖠</ci><apply id="S6.p1.1.m1.1.1.3.cmml" xref="S6.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S6.p1.1.m1.1.1.3.1.cmml" xref="S6.p1.1.m1.1.1.3">subscript</csymbol><ci id="S6.p1.1.m1.1.1.3.2.cmml" xref="S6.p1.1.m1.1.1.3.2">ℵ</ci><cn id="S6.p1.1.m1.1.1.3.3.cmml" type="integer" xref="S6.p1.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p1.1.m1.1c">\mathsf{MA}_{\aleph_{1}}</annotation><annotation encoding="application/x-llamapun" id="S6.p1.1.m1.1d">sansserif_MA start_POSTSUBSCRIPT roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. In particular we show that under this axiom, there is an infinite <math alttext="\trianglelefteq" class="ltx_Math" display="inline" id="S6.p1.2.m2.1"><semantics id="S6.p1.2.m2.1a"><mi id="S6.p1.2.m2.1.1" mathvariant="normal" xref="S6.p1.2.m2.1.1.cmml">⊴</mi><annotation-xml encoding="MathML-Content" id="S6.p1.2.m2.1b"><ci id="S6.p1.2.m2.1.1.cmml" xref="S6.p1.2.m2.1.1">⊴</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.p1.2.m2.1c">\trianglelefteq</annotation><annotation encoding="application/x-llamapun" id="S6.p1.2.m2.1d">⊴</annotation></semantics></math>-decreasing chain of order type <math alttext="\omega_{1}" class="ltx_Math" display="inline" id="S6.p1.3.m3.1"><semantics id="S6.p1.3.m3.1a"><msub id="S6.p1.3.m3.1.1" xref="S6.p1.3.m3.1.1.cmml"><mi id="S6.p1.3.m3.1.1.2" xref="S6.p1.3.m3.1.1.2.cmml">ω</mi><mn id="S6.p1.3.m3.1.1.3" xref="S6.p1.3.m3.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S6.p1.3.m3.1b"><apply id="S6.p1.3.m3.1.1.cmml" xref="S6.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S6.p1.3.m3.1.1.1.cmml" xref="S6.p1.3.m3.1.1">subscript</csymbol><ci id="S6.p1.3.m3.1.1.2.cmml" xref="S6.p1.3.m3.1.1.2">𝜔</ci><cn id="S6.p1.3.m3.1.1.3.cmml" type="integer" xref="S6.p1.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p1.3.m3.1c">\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S6.p1.3.m3.1d">italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S6.p2"> <p class="ltx_p" id="S6.p2.12">Let <math alttext="\langle S_{\alpha}:\alpha<\omega_{1}\rangle" class="ltx_math_unparsed" display="inline" id="S6.p2.1.m1.1"><semantics id="S6.p2.1.m1.1a"><mrow id="S6.p2.1.m1.1b"><mo id="S6.p2.1.m1.1.1" stretchy="false">⟨</mo><msub id="S6.p2.1.m1.1.2"><mi id="S6.p2.1.m1.1.2.2">S</mi><mi id="S6.p2.1.m1.1.2.3">α</mi></msub><mo id="S6.p2.1.m1.1.3" lspace="0.278em" rspace="0.278em">:</mo><mi id="S6.p2.1.m1.1.4">α</mi><mo id="S6.p2.1.m1.1.5"><</mo><msub id="S6.p2.1.m1.1.6"><mi id="S6.p2.1.m1.1.6.2">ω</mi><mn id="S6.p2.1.m1.1.6.3">1</mn></msub><mo id="S6.p2.1.m1.1.7" stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex" id="S6.p2.1.m1.1c">\langle S_{\alpha}:\alpha<\omega_{1}\rangle</annotation><annotation encoding="application/x-llamapun" id="S6.p2.1.m1.1d">⟨ italic_S start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT : italic_α < italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⟩</annotation></semantics></math> be a family of disjoint stationary subsets of <math alttext="\omega_{1}" class="ltx_Math" display="inline" id="S6.p2.2.m2.1"><semantics id="S6.p2.2.m2.1a"><msub id="S6.p2.2.m2.1.1" xref="S6.p2.2.m2.1.1.cmml"><mi id="S6.p2.2.m2.1.1.2" xref="S6.p2.2.m2.1.1.2.cmml">ω</mi><mn id="S6.p2.2.m2.1.1.3" xref="S6.p2.2.m2.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S6.p2.2.m2.1b"><apply id="S6.p2.2.m2.1.1.cmml" xref="S6.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S6.p2.2.m2.1.1.1.cmml" xref="S6.p2.2.m2.1.1">subscript</csymbol><ci id="S6.p2.2.m2.1.1.2.cmml" xref="S6.p2.2.m2.1.1.2">𝜔</ci><cn id="S6.p2.2.m2.1.1.3.cmml" type="integer" xref="S6.p2.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.2.m2.1c">\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S6.p2.2.m2.1d">italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, and for each <math alttext="\alpha<\omega_{1}" class="ltx_Math" display="inline" id="S6.p2.3.m3.1"><semantics id="S6.p2.3.m3.1a"><mrow id="S6.p2.3.m3.1.1" xref="S6.p2.3.m3.1.1.cmml"><mi id="S6.p2.3.m3.1.1.2" xref="S6.p2.3.m3.1.1.2.cmml">α</mi><mo id="S6.p2.3.m3.1.1.1" xref="S6.p2.3.m3.1.1.1.cmml"><</mo><msub id="S6.p2.3.m3.1.1.3" xref="S6.p2.3.m3.1.1.3.cmml"><mi id="S6.p2.3.m3.1.1.3.2" xref="S6.p2.3.m3.1.1.3.2.cmml">ω</mi><mn id="S6.p2.3.m3.1.1.3.3" xref="S6.p2.3.m3.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.p2.3.m3.1b"><apply id="S6.p2.3.m3.1.1.cmml" xref="S6.p2.3.m3.1.1"><lt id="S6.p2.3.m3.1.1.1.cmml" xref="S6.p2.3.m3.1.1.1"></lt><ci id="S6.p2.3.m3.1.1.2.cmml" xref="S6.p2.3.m3.1.1.2">𝛼</ci><apply id="S6.p2.3.m3.1.1.3.cmml" xref="S6.p2.3.m3.1.1.3"><csymbol cd="ambiguous" id="S6.p2.3.m3.1.1.3.1.cmml" xref="S6.p2.3.m3.1.1.3">subscript</csymbol><ci id="S6.p2.3.m3.1.1.3.2.cmml" xref="S6.p2.3.m3.1.1.3.2">𝜔</ci><cn id="S6.p2.3.m3.1.1.3.3.cmml" type="integer" xref="S6.p2.3.m3.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.3.m3.1c">\alpha<\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S6.p2.3.m3.1d">italic_α < italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, using <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S4.Thmtheorem6" title="Corollary 4.6. ‣ 4. Aronszajn line decompositions ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Corollary</span> <span class="ltx_text ltx_ref_tag">4.6</span></a>, let <math alttext="C^{\alpha}" class="ltx_Math" display="inline" id="S6.p2.4.m4.1"><semantics id="S6.p2.4.m4.1a"><msup id="S6.p2.4.m4.1.1" xref="S6.p2.4.m4.1.1.cmml"><mi id="S6.p2.4.m4.1.1.2" xref="S6.p2.4.m4.1.1.2.cmml">C</mi><mi id="S6.p2.4.m4.1.1.3" xref="S6.p2.4.m4.1.1.3.cmml">α</mi></msup><annotation-xml encoding="MathML-Content" id="S6.p2.4.m4.1b"><apply id="S6.p2.4.m4.1.1.cmml" xref="S6.p2.4.m4.1.1"><csymbol cd="ambiguous" id="S6.p2.4.m4.1.1.1.cmml" xref="S6.p2.4.m4.1.1">superscript</csymbol><ci id="S6.p2.4.m4.1.1.2.cmml" xref="S6.p2.4.m4.1.1.2">𝐶</ci><ci id="S6.p2.4.m4.1.1.3.cmml" xref="S6.p2.4.m4.1.1.3">𝛼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.4.m4.1c">C^{\alpha}</annotation><annotation encoding="application/x-llamapun" id="S6.p2.4.m4.1d">italic_C start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT</annotation></semantics></math> be a Countryman line and <math alttext="D^{\alpha}" class="ltx_Math" display="inline" id="S6.p2.5.m5.1"><semantics id="S6.p2.5.m5.1a"><msup id="S6.p2.5.m5.1.1" xref="S6.p2.5.m5.1.1.cmml"><mi id="S6.p2.5.m5.1.1.2" xref="S6.p2.5.m5.1.1.2.cmml">D</mi><mi id="S6.p2.5.m5.1.1.3" xref="S6.p2.5.m5.1.1.3.cmml">α</mi></msup><annotation-xml encoding="MathML-Content" id="S6.p2.5.m5.1b"><apply id="S6.p2.5.m5.1.1.cmml" xref="S6.p2.5.m5.1.1"><csymbol cd="ambiguous" id="S6.p2.5.m5.1.1.1.cmml" xref="S6.p2.5.m5.1.1">superscript</csymbol><ci id="S6.p2.5.m5.1.1.2.cmml" xref="S6.p2.5.m5.1.1.2">𝐷</ci><ci id="S6.p2.5.m5.1.1.3.cmml" xref="S6.p2.5.m5.1.1.3">𝛼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.5.m5.1c">D^{\alpha}</annotation><annotation encoding="application/x-llamapun" id="S6.p2.5.m5.1d">italic_D start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT</annotation></semantics></math> a decomposition for it such that <math alttext="\hat{\mathscr{L}}(C^{\alpha},D^{\alpha})=\mathscr{L}(C^{\alpha},D^{\alpha})=% \bigcup_{\xi<\alpha}S_{\xi}" class="ltx_Math" display="inline" id="S6.p2.6.m6.4"><semantics id="S6.p2.6.m6.4a"><mrow id="S6.p2.6.m6.4.4" xref="S6.p2.6.m6.4.4.cmml"><mrow id="S6.p2.6.m6.2.2.2" xref="S6.p2.6.m6.2.2.2.cmml"><mover accent="true" id="S6.p2.6.m6.2.2.2.4" xref="S6.p2.6.m6.2.2.2.4.cmml"><mi class="ltx_font_mathscript" id="S6.p2.6.m6.2.2.2.4.2" xref="S6.p2.6.m6.2.2.2.4.2.cmml">ℒ</mi><mo id="S6.p2.6.m6.2.2.2.4.1" xref="S6.p2.6.m6.2.2.2.4.1.cmml">^</mo></mover><mo id="S6.p2.6.m6.2.2.2.3" xref="S6.p2.6.m6.2.2.2.3.cmml">⁢</mo><mrow id="S6.p2.6.m6.2.2.2.2.2" xref="S6.p2.6.m6.2.2.2.2.3.cmml"><mo id="S6.p2.6.m6.2.2.2.2.2.3" stretchy="false" xref="S6.p2.6.m6.2.2.2.2.3.cmml">(</mo><msup id="S6.p2.6.m6.1.1.1.1.1.1" xref="S6.p2.6.m6.1.1.1.1.1.1.cmml"><mi id="S6.p2.6.m6.1.1.1.1.1.1.2" xref="S6.p2.6.m6.1.1.1.1.1.1.2.cmml">C</mi><mi id="S6.p2.6.m6.1.1.1.1.1.1.3" xref="S6.p2.6.m6.1.1.1.1.1.1.3.cmml">α</mi></msup><mo id="S6.p2.6.m6.2.2.2.2.2.4" xref="S6.p2.6.m6.2.2.2.2.3.cmml">,</mo><msup id="S6.p2.6.m6.2.2.2.2.2.2" xref="S6.p2.6.m6.2.2.2.2.2.2.cmml"><mi id="S6.p2.6.m6.2.2.2.2.2.2.2" xref="S6.p2.6.m6.2.2.2.2.2.2.2.cmml">D</mi><mi id="S6.p2.6.m6.2.2.2.2.2.2.3" xref="S6.p2.6.m6.2.2.2.2.2.2.3.cmml">α</mi></msup><mo id="S6.p2.6.m6.2.2.2.2.2.5" stretchy="false" xref="S6.p2.6.m6.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.p2.6.m6.4.4.6" xref="S6.p2.6.m6.4.4.6.cmml">=</mo><mrow id="S6.p2.6.m6.4.4.4" xref="S6.p2.6.m6.4.4.4.cmml"><mi class="ltx_font_mathscript" id="S6.p2.6.m6.4.4.4.4" xref="S6.p2.6.m6.4.4.4.4.cmml">ℒ</mi><mo id="S6.p2.6.m6.4.4.4.3" xref="S6.p2.6.m6.4.4.4.3.cmml">⁢</mo><mrow id="S6.p2.6.m6.4.4.4.2.2" xref="S6.p2.6.m6.4.4.4.2.3.cmml"><mo id="S6.p2.6.m6.4.4.4.2.2.3" stretchy="false" xref="S6.p2.6.m6.4.4.4.2.3.cmml">(</mo><msup id="S6.p2.6.m6.3.3.3.1.1.1" xref="S6.p2.6.m6.3.3.3.1.1.1.cmml"><mi id="S6.p2.6.m6.3.3.3.1.1.1.2" xref="S6.p2.6.m6.3.3.3.1.1.1.2.cmml">C</mi><mi id="S6.p2.6.m6.3.3.3.1.1.1.3" xref="S6.p2.6.m6.3.3.3.1.1.1.3.cmml">α</mi></msup><mo id="S6.p2.6.m6.4.4.4.2.2.4" xref="S6.p2.6.m6.4.4.4.2.3.cmml">,</mo><msup id="S6.p2.6.m6.4.4.4.2.2.2" xref="S6.p2.6.m6.4.4.4.2.2.2.cmml"><mi id="S6.p2.6.m6.4.4.4.2.2.2.2" xref="S6.p2.6.m6.4.4.4.2.2.2.2.cmml">D</mi><mi id="S6.p2.6.m6.4.4.4.2.2.2.3" xref="S6.p2.6.m6.4.4.4.2.2.2.3.cmml">α</mi></msup><mo id="S6.p2.6.m6.4.4.4.2.2.5" stretchy="false" xref="S6.p2.6.m6.4.4.4.2.3.cmml">)</mo></mrow></mrow><mo id="S6.p2.6.m6.4.4.7" rspace="0.111em" xref="S6.p2.6.m6.4.4.7.cmml">=</mo><mrow id="S6.p2.6.m6.4.4.8" xref="S6.p2.6.m6.4.4.8.cmml"><msub id="S6.p2.6.m6.4.4.8.1" xref="S6.p2.6.m6.4.4.8.1.cmml"><mo id="S6.p2.6.m6.4.4.8.1.2" xref="S6.p2.6.m6.4.4.8.1.2.cmml">⋃</mo><mrow id="S6.p2.6.m6.4.4.8.1.3" xref="S6.p2.6.m6.4.4.8.1.3.cmml"><mi id="S6.p2.6.m6.4.4.8.1.3.2" xref="S6.p2.6.m6.4.4.8.1.3.2.cmml">ξ</mi><mo id="S6.p2.6.m6.4.4.8.1.3.1" xref="S6.p2.6.m6.4.4.8.1.3.1.cmml"><</mo><mi id="S6.p2.6.m6.4.4.8.1.3.3" xref="S6.p2.6.m6.4.4.8.1.3.3.cmml">α</mi></mrow></msub><msub id="S6.p2.6.m6.4.4.8.2" xref="S6.p2.6.m6.4.4.8.2.cmml"><mi id="S6.p2.6.m6.4.4.8.2.2" xref="S6.p2.6.m6.4.4.8.2.2.cmml">S</mi><mi id="S6.p2.6.m6.4.4.8.2.3" xref="S6.p2.6.m6.4.4.8.2.3.cmml">ξ</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.p2.6.m6.4b"><apply id="S6.p2.6.m6.4.4.cmml" xref="S6.p2.6.m6.4.4"><and id="S6.p2.6.m6.4.4a.cmml" xref="S6.p2.6.m6.4.4"></and><apply id="S6.p2.6.m6.4.4b.cmml" xref="S6.p2.6.m6.4.4"><eq id="S6.p2.6.m6.4.4.6.cmml" xref="S6.p2.6.m6.4.4.6"></eq><apply id="S6.p2.6.m6.2.2.2.cmml" xref="S6.p2.6.m6.2.2.2"><times id="S6.p2.6.m6.2.2.2.3.cmml" xref="S6.p2.6.m6.2.2.2.3"></times><apply id="S6.p2.6.m6.2.2.2.4.cmml" xref="S6.p2.6.m6.2.2.2.4"><ci id="S6.p2.6.m6.2.2.2.4.1.cmml" xref="S6.p2.6.m6.2.2.2.4.1">^</ci><ci id="S6.p2.6.m6.2.2.2.4.2.cmml" xref="S6.p2.6.m6.2.2.2.4.2">ℒ</ci></apply><interval closure="open" id="S6.p2.6.m6.2.2.2.2.3.cmml" xref="S6.p2.6.m6.2.2.2.2.2"><apply id="S6.p2.6.m6.1.1.1.1.1.1.cmml" xref="S6.p2.6.m6.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.p2.6.m6.1.1.1.1.1.1.1.cmml" xref="S6.p2.6.m6.1.1.1.1.1.1">superscript</csymbol><ci id="S6.p2.6.m6.1.1.1.1.1.1.2.cmml" xref="S6.p2.6.m6.1.1.1.1.1.1.2">𝐶</ci><ci id="S6.p2.6.m6.1.1.1.1.1.1.3.cmml" xref="S6.p2.6.m6.1.1.1.1.1.1.3">𝛼</ci></apply><apply id="S6.p2.6.m6.2.2.2.2.2.2.cmml" xref="S6.p2.6.m6.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.p2.6.m6.2.2.2.2.2.2.1.cmml" xref="S6.p2.6.m6.2.2.2.2.2.2">superscript</csymbol><ci id="S6.p2.6.m6.2.2.2.2.2.2.2.cmml" xref="S6.p2.6.m6.2.2.2.2.2.2.2">𝐷</ci><ci id="S6.p2.6.m6.2.2.2.2.2.2.3.cmml" xref="S6.p2.6.m6.2.2.2.2.2.2.3">𝛼</ci></apply></interval></apply><apply id="S6.p2.6.m6.4.4.4.cmml" xref="S6.p2.6.m6.4.4.4"><times id="S6.p2.6.m6.4.4.4.3.cmml" xref="S6.p2.6.m6.4.4.4.3"></times><ci id="S6.p2.6.m6.4.4.4.4.cmml" xref="S6.p2.6.m6.4.4.4.4">ℒ</ci><interval closure="open" id="S6.p2.6.m6.4.4.4.2.3.cmml" xref="S6.p2.6.m6.4.4.4.2.2"><apply id="S6.p2.6.m6.3.3.3.1.1.1.cmml" xref="S6.p2.6.m6.3.3.3.1.1.1"><csymbol cd="ambiguous" id="S6.p2.6.m6.3.3.3.1.1.1.1.cmml" xref="S6.p2.6.m6.3.3.3.1.1.1">superscript</csymbol><ci id="S6.p2.6.m6.3.3.3.1.1.1.2.cmml" xref="S6.p2.6.m6.3.3.3.1.1.1.2">𝐶</ci><ci id="S6.p2.6.m6.3.3.3.1.1.1.3.cmml" xref="S6.p2.6.m6.3.3.3.1.1.1.3">𝛼</ci></apply><apply id="S6.p2.6.m6.4.4.4.2.2.2.cmml" xref="S6.p2.6.m6.4.4.4.2.2.2"><csymbol cd="ambiguous" id="S6.p2.6.m6.4.4.4.2.2.2.1.cmml" xref="S6.p2.6.m6.4.4.4.2.2.2">superscript</csymbol><ci id="S6.p2.6.m6.4.4.4.2.2.2.2.cmml" xref="S6.p2.6.m6.4.4.4.2.2.2.2">𝐷</ci><ci id="S6.p2.6.m6.4.4.4.2.2.2.3.cmml" xref="S6.p2.6.m6.4.4.4.2.2.2.3">𝛼</ci></apply></interval></apply></apply><apply id="S6.p2.6.m6.4.4c.cmml" xref="S6.p2.6.m6.4.4"><eq id="S6.p2.6.m6.4.4.7.cmml" xref="S6.p2.6.m6.4.4.7"></eq><share href="https://arxiv.org/html/2503.13728v1#S6.p2.6.m6.4.4.4.cmml" id="S6.p2.6.m6.4.4d.cmml" xref="S6.p2.6.m6.4.4"></share><apply id="S6.p2.6.m6.4.4.8.cmml" xref="S6.p2.6.m6.4.4.8"><apply id="S6.p2.6.m6.4.4.8.1.cmml" xref="S6.p2.6.m6.4.4.8.1"><csymbol cd="ambiguous" id="S6.p2.6.m6.4.4.8.1.1.cmml" xref="S6.p2.6.m6.4.4.8.1">subscript</csymbol><union id="S6.p2.6.m6.4.4.8.1.2.cmml" xref="S6.p2.6.m6.4.4.8.1.2"></union><apply id="S6.p2.6.m6.4.4.8.1.3.cmml" xref="S6.p2.6.m6.4.4.8.1.3"><lt id="S6.p2.6.m6.4.4.8.1.3.1.cmml" xref="S6.p2.6.m6.4.4.8.1.3.1"></lt><ci id="S6.p2.6.m6.4.4.8.1.3.2.cmml" xref="S6.p2.6.m6.4.4.8.1.3.2">𝜉</ci><ci id="S6.p2.6.m6.4.4.8.1.3.3.cmml" xref="S6.p2.6.m6.4.4.8.1.3.3">𝛼</ci></apply></apply><apply id="S6.p2.6.m6.4.4.8.2.cmml" xref="S6.p2.6.m6.4.4.8.2"><csymbol cd="ambiguous" id="S6.p2.6.m6.4.4.8.2.1.cmml" xref="S6.p2.6.m6.4.4.8.2">subscript</csymbol><ci id="S6.p2.6.m6.4.4.8.2.2.cmml" xref="S6.p2.6.m6.4.4.8.2.2">𝑆</ci><ci id="S6.p2.6.m6.4.4.8.2.3.cmml" xref="S6.p2.6.m6.4.4.8.2.3">𝜉</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.6.m6.4c">\hat{\mathscr{L}}(C^{\alpha},D^{\alpha})=\mathscr{L}(C^{\alpha},D^{\alpha})=% \bigcup_{\xi<\alpha}S_{\xi}</annotation><annotation encoding="application/x-llamapun" id="S6.p2.6.m6.4d">over^ start_ARG script_L end_ARG ( italic_C start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT , italic_D start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT ) = script_L ( italic_C start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT , italic_D start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT ) = ⋃ start_POSTSUBSCRIPT italic_ξ < italic_α end_POSTSUBSCRIPT italic_S start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\mathscr{R}(C^{\alpha},D^{\alpha})=\varnothing" class="ltx_Math" display="inline" id="S6.p2.7.m7.2"><semantics id="S6.p2.7.m7.2a"><mrow id="S6.p2.7.m7.2.2" xref="S6.p2.7.m7.2.2.cmml"><mrow id="S6.p2.7.m7.2.2.2" xref="S6.p2.7.m7.2.2.2.cmml"><mi class="ltx_font_mathscript" id="S6.p2.7.m7.2.2.2.4" xref="S6.p2.7.m7.2.2.2.4.cmml">ℛ</mi><mo id="S6.p2.7.m7.2.2.2.3" xref="S6.p2.7.m7.2.2.2.3.cmml">⁢</mo><mrow id="S6.p2.7.m7.2.2.2.2.2" xref="S6.p2.7.m7.2.2.2.2.3.cmml"><mo id="S6.p2.7.m7.2.2.2.2.2.3" stretchy="false" xref="S6.p2.7.m7.2.2.2.2.3.cmml">(</mo><msup id="S6.p2.7.m7.1.1.1.1.1.1" xref="S6.p2.7.m7.1.1.1.1.1.1.cmml"><mi id="S6.p2.7.m7.1.1.1.1.1.1.2" xref="S6.p2.7.m7.1.1.1.1.1.1.2.cmml">C</mi><mi id="S6.p2.7.m7.1.1.1.1.1.1.3" xref="S6.p2.7.m7.1.1.1.1.1.1.3.cmml">α</mi></msup><mo id="S6.p2.7.m7.2.2.2.2.2.4" xref="S6.p2.7.m7.2.2.2.2.3.cmml">,</mo><msup id="S6.p2.7.m7.2.2.2.2.2.2" xref="S6.p2.7.m7.2.2.2.2.2.2.cmml"><mi id="S6.p2.7.m7.2.2.2.2.2.2.2" xref="S6.p2.7.m7.2.2.2.2.2.2.2.cmml">D</mi><mi id="S6.p2.7.m7.2.2.2.2.2.2.3" xref="S6.p2.7.m7.2.2.2.2.2.2.3.cmml">α</mi></msup><mo id="S6.p2.7.m7.2.2.2.2.2.5" stretchy="false" xref="S6.p2.7.m7.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.p2.7.m7.2.2.3" xref="S6.p2.7.m7.2.2.3.cmml">=</mo><mi id="S6.p2.7.m7.2.2.4" mathvariant="normal" xref="S6.p2.7.m7.2.2.4.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.p2.7.m7.2b"><apply id="S6.p2.7.m7.2.2.cmml" xref="S6.p2.7.m7.2.2"><eq id="S6.p2.7.m7.2.2.3.cmml" xref="S6.p2.7.m7.2.2.3"></eq><apply id="S6.p2.7.m7.2.2.2.cmml" xref="S6.p2.7.m7.2.2.2"><times id="S6.p2.7.m7.2.2.2.3.cmml" xref="S6.p2.7.m7.2.2.2.3"></times><ci id="S6.p2.7.m7.2.2.2.4.cmml" xref="S6.p2.7.m7.2.2.2.4">ℛ</ci><interval closure="open" id="S6.p2.7.m7.2.2.2.2.3.cmml" xref="S6.p2.7.m7.2.2.2.2.2"><apply id="S6.p2.7.m7.1.1.1.1.1.1.cmml" xref="S6.p2.7.m7.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.p2.7.m7.1.1.1.1.1.1.1.cmml" xref="S6.p2.7.m7.1.1.1.1.1.1">superscript</csymbol><ci id="S6.p2.7.m7.1.1.1.1.1.1.2.cmml" xref="S6.p2.7.m7.1.1.1.1.1.1.2">𝐶</ci><ci id="S6.p2.7.m7.1.1.1.1.1.1.3.cmml" xref="S6.p2.7.m7.1.1.1.1.1.1.3">𝛼</ci></apply><apply id="S6.p2.7.m7.2.2.2.2.2.2.cmml" xref="S6.p2.7.m7.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.p2.7.m7.2.2.2.2.2.2.1.cmml" xref="S6.p2.7.m7.2.2.2.2.2.2">superscript</csymbol><ci id="S6.p2.7.m7.2.2.2.2.2.2.2.cmml" xref="S6.p2.7.m7.2.2.2.2.2.2.2">𝐷</ci><ci id="S6.p2.7.m7.2.2.2.2.2.2.3.cmml" xref="S6.p2.7.m7.2.2.2.2.2.2.3">𝛼</ci></apply></interval></apply><emptyset id="S6.p2.7.m7.2.2.4.cmml" xref="S6.p2.7.m7.2.2.4"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.7.m7.2c">\mathscr{R}(C^{\alpha},D^{\alpha})=\varnothing</annotation><annotation encoding="application/x-llamapun" id="S6.p2.7.m7.2d">script_R ( italic_C start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT , italic_D start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT ) = ∅</annotation></semantics></math>. <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S5.Thmtheorem2" title="Lemma 5.2. ‣ 5. An infinite antichain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">5.2</span></a> implies that if <math alttext="\alpha<\beta" class="ltx_Math" display="inline" id="S6.p2.8.m8.1"><semantics id="S6.p2.8.m8.1a"><mrow id="S6.p2.8.m8.1.1" xref="S6.p2.8.m8.1.1.cmml"><mi id="S6.p2.8.m8.1.1.2" xref="S6.p2.8.m8.1.1.2.cmml">α</mi><mo id="S6.p2.8.m8.1.1.1" xref="S6.p2.8.m8.1.1.1.cmml"><</mo><mi id="S6.p2.8.m8.1.1.3" xref="S6.p2.8.m8.1.1.3.cmml">β</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.p2.8.m8.1b"><apply id="S6.p2.8.m8.1.1.cmml" xref="S6.p2.8.m8.1.1"><lt id="S6.p2.8.m8.1.1.1.cmml" xref="S6.p2.8.m8.1.1.1"></lt><ci id="S6.p2.8.m8.1.1.2.cmml" xref="S6.p2.8.m8.1.1.2">𝛼</ci><ci id="S6.p2.8.m8.1.1.3.cmml" xref="S6.p2.8.m8.1.1.3">𝛽</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.8.m8.1c">\alpha<\beta</annotation><annotation encoding="application/x-llamapun" id="S6.p2.8.m8.1d">italic_α < italic_β</annotation></semantics></math>, then <math alttext="C^{\beta}\ntrianglerighteq C^{\alpha}" class="ltx_Math" display="inline" id="S6.p2.9.m9.1"><semantics id="S6.p2.9.m9.1a"><mrow id="S6.p2.9.m9.1.1" xref="S6.p2.9.m9.1.1.cmml"><msup id="S6.p2.9.m9.1.1.2" xref="S6.p2.9.m9.1.1.2.cmml"><mi id="S6.p2.9.m9.1.1.2.2" xref="S6.p2.9.m9.1.1.2.2.cmml">C</mi><mi id="S6.p2.9.m9.1.1.2.3" xref="S6.p2.9.m9.1.1.2.3.cmml">β</mi></msup><mo id="S6.p2.9.m9.1.1.1" xref="S6.p2.9.m9.1.1.1.cmml">⋭</mo><msup id="S6.p2.9.m9.1.1.3" xref="S6.p2.9.m9.1.1.3.cmml"><mi id="S6.p2.9.m9.1.1.3.2" xref="S6.p2.9.m9.1.1.3.2.cmml">C</mi><mi id="S6.p2.9.m9.1.1.3.3" xref="S6.p2.9.m9.1.1.3.3.cmml">α</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S6.p2.9.m9.1b"><apply id="S6.p2.9.m9.1.1.cmml" xref="S6.p2.9.m9.1.1"><csymbol cd="latexml" id="S6.p2.9.m9.1.1.1.cmml" xref="S6.p2.9.m9.1.1.1">not-contains-nor-equals</csymbol><apply id="S6.p2.9.m9.1.1.2.cmml" xref="S6.p2.9.m9.1.1.2"><csymbol cd="ambiguous" id="S6.p2.9.m9.1.1.2.1.cmml" xref="S6.p2.9.m9.1.1.2">superscript</csymbol><ci id="S6.p2.9.m9.1.1.2.2.cmml" xref="S6.p2.9.m9.1.1.2.2">𝐶</ci><ci id="S6.p2.9.m9.1.1.2.3.cmml" xref="S6.p2.9.m9.1.1.2.3">𝛽</ci></apply><apply id="S6.p2.9.m9.1.1.3.cmml" xref="S6.p2.9.m9.1.1.3"><csymbol cd="ambiguous" id="S6.p2.9.m9.1.1.3.1.cmml" xref="S6.p2.9.m9.1.1.3">superscript</csymbol><ci id="S6.p2.9.m9.1.1.3.2.cmml" xref="S6.p2.9.m9.1.1.3.2">𝐶</ci><ci id="S6.p2.9.m9.1.1.3.3.cmml" xref="S6.p2.9.m9.1.1.3.3">𝛼</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.9.m9.1c">C^{\beta}\ntrianglerighteq C^{\alpha}</annotation><annotation encoding="application/x-llamapun" id="S6.p2.9.m9.1d">italic_C start_POSTSUPERSCRIPT italic_β end_POSTSUPERSCRIPT ⋭ italic_C start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT</annotation></semantics></math>, and <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S3.Thmtheorem2" title="Corollary 3.2. ‣ 3. Strongly surjective Aronszajn lines ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Corollary</span> <span class="ltx_text ltx_ref_tag">3.2</span></a> that <math alttext="C^{0}\trianglerighteq C^{\alpha}" class="ltx_Math" display="inline" id="S6.p2.10.m10.1"><semantics id="S6.p2.10.m10.1a"><mrow id="S6.p2.10.m10.1.1" xref="S6.p2.10.m10.1.1.cmml"><msup id="S6.p2.10.m10.1.1.2" xref="S6.p2.10.m10.1.1.2.cmml"><mi id="S6.p2.10.m10.1.1.2.2" xref="S6.p2.10.m10.1.1.2.2.cmml">C</mi><mn id="S6.p2.10.m10.1.1.2.3" xref="S6.p2.10.m10.1.1.2.3.cmml">0</mn></msup><mo id="S6.p2.10.m10.1.1.1" xref="S6.p2.10.m10.1.1.1.cmml">⁢</mo><mi id="S6.p2.10.m10.1.1.3" mathvariant="normal" xref="S6.p2.10.m10.1.1.3.cmml">⊵</mi><mo id="S6.p2.10.m10.1.1.1a" xref="S6.p2.10.m10.1.1.1.cmml">⁢</mo><msup id="S6.p2.10.m10.1.1.4" xref="S6.p2.10.m10.1.1.4.cmml"><mi id="S6.p2.10.m10.1.1.4.2" xref="S6.p2.10.m10.1.1.4.2.cmml">C</mi><mi id="S6.p2.10.m10.1.1.4.3" xref="S6.p2.10.m10.1.1.4.3.cmml">α</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S6.p2.10.m10.1b"><apply id="S6.p2.10.m10.1.1.cmml" xref="S6.p2.10.m10.1.1"><times id="S6.p2.10.m10.1.1.1.cmml" xref="S6.p2.10.m10.1.1.1"></times><apply id="S6.p2.10.m10.1.1.2.cmml" xref="S6.p2.10.m10.1.1.2"><csymbol cd="ambiguous" id="S6.p2.10.m10.1.1.2.1.cmml" xref="S6.p2.10.m10.1.1.2">superscript</csymbol><ci id="S6.p2.10.m10.1.1.2.2.cmml" xref="S6.p2.10.m10.1.1.2.2">𝐶</ci><cn id="S6.p2.10.m10.1.1.2.3.cmml" type="integer" xref="S6.p2.10.m10.1.1.2.3">0</cn></apply><ci id="S6.p2.10.m10.1.1.3.cmml" xref="S6.p2.10.m10.1.1.3">⊵</ci><apply id="S6.p2.10.m10.1.1.4.cmml" xref="S6.p2.10.m10.1.1.4"><csymbol cd="ambiguous" id="S6.p2.10.m10.1.1.4.1.cmml" xref="S6.p2.10.m10.1.1.4">superscript</csymbol><ci id="S6.p2.10.m10.1.1.4.2.cmml" xref="S6.p2.10.m10.1.1.4.2">𝐶</ci><ci id="S6.p2.10.m10.1.1.4.3.cmml" xref="S6.p2.10.m10.1.1.4.3">𝛼</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.10.m10.1c">C^{0}\trianglerighteq C^{\alpha}</annotation><annotation encoding="application/x-llamapun" id="S6.p2.10.m10.1d">italic_C start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT ⊵ italic_C start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT</annotation></semantics></math> for every <math alttext="\alpha<\omega_{1}" class="ltx_Math" display="inline" id="S6.p2.11.m11.1"><semantics id="S6.p2.11.m11.1a"><mrow id="S6.p2.11.m11.1.1" xref="S6.p2.11.m11.1.1.cmml"><mi id="S6.p2.11.m11.1.1.2" xref="S6.p2.11.m11.1.1.2.cmml">α</mi><mo id="S6.p2.11.m11.1.1.1" xref="S6.p2.11.m11.1.1.1.cmml"><</mo><msub id="S6.p2.11.m11.1.1.3" xref="S6.p2.11.m11.1.1.3.cmml"><mi id="S6.p2.11.m11.1.1.3.2" xref="S6.p2.11.m11.1.1.3.2.cmml">ω</mi><mn id="S6.p2.11.m11.1.1.3.3" xref="S6.p2.11.m11.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.p2.11.m11.1b"><apply id="S6.p2.11.m11.1.1.cmml" xref="S6.p2.11.m11.1.1"><lt id="S6.p2.11.m11.1.1.1.cmml" xref="S6.p2.11.m11.1.1.1"></lt><ci id="S6.p2.11.m11.1.1.2.cmml" xref="S6.p2.11.m11.1.1.2">𝛼</ci><apply id="S6.p2.11.m11.1.1.3.cmml" xref="S6.p2.11.m11.1.1.3"><csymbol cd="ambiguous" id="S6.p2.11.m11.1.1.3.1.cmml" xref="S6.p2.11.m11.1.1.3">subscript</csymbol><ci id="S6.p2.11.m11.1.1.3.2.cmml" xref="S6.p2.11.m11.1.1.3.2">𝜔</ci><cn id="S6.p2.11.m11.1.1.3.3.cmml" type="integer" xref="S6.p2.11.m11.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.11.m11.1c">\alpha<\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S6.p2.11.m11.1d">italic_α < italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> under <math alttext="\mathsf{MA}_{\aleph_{1}}" class="ltx_Math" display="inline" id="S6.p2.12.m12.1"><semantics id="S6.p2.12.m12.1a"><msub id="S6.p2.12.m12.1.1" xref="S6.p2.12.m12.1.1.cmml"><mi id="S6.p2.12.m12.1.1.2" xref="S6.p2.12.m12.1.1.2.cmml">𝖬𝖠</mi><msub id="S6.p2.12.m12.1.1.3" xref="S6.p2.12.m12.1.1.3.cmml"><mi id="S6.p2.12.m12.1.1.3.2" mathvariant="normal" xref="S6.p2.12.m12.1.1.3.2.cmml">ℵ</mi><mn id="S6.p2.12.m12.1.1.3.3" xref="S6.p2.12.m12.1.1.3.3.cmml">1</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S6.p2.12.m12.1b"><apply id="S6.p2.12.m12.1.1.cmml" xref="S6.p2.12.m12.1.1"><csymbol cd="ambiguous" id="S6.p2.12.m12.1.1.1.cmml" xref="S6.p2.12.m12.1.1">subscript</csymbol><ci id="S6.p2.12.m12.1.1.2.cmml" xref="S6.p2.12.m12.1.1.2">𝖬𝖠</ci><apply id="S6.p2.12.m12.1.1.3.cmml" xref="S6.p2.12.m12.1.1.3"><csymbol cd="ambiguous" id="S6.p2.12.m12.1.1.3.1.cmml" xref="S6.p2.12.m12.1.1.3">subscript</csymbol><ci id="S6.p2.12.m12.1.1.3.2.cmml" xref="S6.p2.12.m12.1.1.3.2">ℵ</ci><cn id="S6.p2.12.m12.1.1.3.3.cmml" type="integer" xref="S6.p2.12.m12.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.12.m12.1c">\mathsf{MA}_{\aleph_{1}}</annotation><annotation encoding="application/x-llamapun" id="S6.p2.12.m12.1d">sansserif_MA start_POSTSUBSCRIPT roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. We claim that this generalizes.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="S6.Thmtheorem1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem1.1.1.1">Proposition 6.1</span></span><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem1.2.2">.</span> </h6> <div class="ltx_para" id="S6.Thmtheorem1.p1"> <p class="ltx_p" id="S6.Thmtheorem1.p1.3">Assume <math alttext="\mathsf{MA}_{\aleph_{1}}" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p1.1.m1.1"><semantics id="S6.Thmtheorem1.p1.1.m1.1a"><msub id="S6.Thmtheorem1.p1.1.m1.1.1" xref="S6.Thmtheorem1.p1.1.m1.1.1.cmml"><mi id="S6.Thmtheorem1.p1.1.m1.1.1.2" xref="S6.Thmtheorem1.p1.1.m1.1.1.2.cmml">𝖬𝖠</mi><msub id="S6.Thmtheorem1.p1.1.m1.1.1.3" xref="S6.Thmtheorem1.p1.1.m1.1.1.3.cmml"><mi id="S6.Thmtheorem1.p1.1.m1.1.1.3.2" mathvariant="normal" xref="S6.Thmtheorem1.p1.1.m1.1.1.3.2.cmml">ℵ</mi><mn id="S6.Thmtheorem1.p1.1.m1.1.1.3.3" xref="S6.Thmtheorem1.p1.1.m1.1.1.3.3.cmml">1</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p1.1.m1.1b"><apply id="S6.Thmtheorem1.p1.1.m1.1.1.cmml" xref="S6.Thmtheorem1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p1.1.m1.1.1.1.cmml" xref="S6.Thmtheorem1.p1.1.m1.1.1">subscript</csymbol><ci id="S6.Thmtheorem1.p1.1.m1.1.1.2.cmml" xref="S6.Thmtheorem1.p1.1.m1.1.1.2">𝖬𝖠</ci><apply id="S6.Thmtheorem1.p1.1.m1.1.1.3.cmml" xref="S6.Thmtheorem1.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p1.1.m1.1.1.3.1.cmml" xref="S6.Thmtheorem1.p1.1.m1.1.1.3">subscript</csymbol><ci id="S6.Thmtheorem1.p1.1.m1.1.1.3.2.cmml" xref="S6.Thmtheorem1.p1.1.m1.1.1.3.2">ℵ</ci><cn id="S6.Thmtheorem1.p1.1.m1.1.1.3.3.cmml" type="integer" xref="S6.Thmtheorem1.p1.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p1.1.m1.1c">\mathsf{MA}_{\aleph_{1}}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p1.1.m1.1d">sansserif_MA start_POSTSUBSCRIPT roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. For all <math alttext="\alpha<\beta" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p1.2.m2.1"><semantics id="S6.Thmtheorem1.p1.2.m2.1a"><mrow id="S6.Thmtheorem1.p1.2.m2.1.1" xref="S6.Thmtheorem1.p1.2.m2.1.1.cmml"><mi id="S6.Thmtheorem1.p1.2.m2.1.1.2" xref="S6.Thmtheorem1.p1.2.m2.1.1.2.cmml">α</mi><mo id="S6.Thmtheorem1.p1.2.m2.1.1.1" xref="S6.Thmtheorem1.p1.2.m2.1.1.1.cmml"><</mo><mi id="S6.Thmtheorem1.p1.2.m2.1.1.3" xref="S6.Thmtheorem1.p1.2.m2.1.1.3.cmml">β</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p1.2.m2.1b"><apply id="S6.Thmtheorem1.p1.2.m2.1.1.cmml" xref="S6.Thmtheorem1.p1.2.m2.1.1"><lt id="S6.Thmtheorem1.p1.2.m2.1.1.1.cmml" xref="S6.Thmtheorem1.p1.2.m2.1.1.1"></lt><ci id="S6.Thmtheorem1.p1.2.m2.1.1.2.cmml" xref="S6.Thmtheorem1.p1.2.m2.1.1.2">𝛼</ci><ci id="S6.Thmtheorem1.p1.2.m2.1.1.3.cmml" xref="S6.Thmtheorem1.p1.2.m2.1.1.3">𝛽</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p1.2.m2.1c">\alpha<\beta</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p1.2.m2.1d">italic_α < italic_β</annotation></semantics></math>, <math alttext="C^{\alpha}\vartriangleright C^{\beta}" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p1.3.m3.1"><semantics id="S6.Thmtheorem1.p1.3.m3.1a"><mrow id="S6.Thmtheorem1.p1.3.m3.1.1" xref="S6.Thmtheorem1.p1.3.m3.1.1.cmml"><msup id="S6.Thmtheorem1.p1.3.m3.1.1.2" xref="S6.Thmtheorem1.p1.3.m3.1.1.2.cmml"><mi id="S6.Thmtheorem1.p1.3.m3.1.1.2.2" xref="S6.Thmtheorem1.p1.3.m3.1.1.2.2.cmml">C</mi><mi id="S6.Thmtheorem1.p1.3.m3.1.1.2.3" xref="S6.Thmtheorem1.p1.3.m3.1.1.2.3.cmml">α</mi></msup><mo id="S6.Thmtheorem1.p1.3.m3.1.1.1" xref="S6.Thmtheorem1.p1.3.m3.1.1.1.cmml">⁢</mo><mi id="S6.Thmtheorem1.p1.3.m3.1.1.3" mathvariant="normal" xref="S6.Thmtheorem1.p1.3.m3.1.1.3.cmml">⊳</mi><mo id="S6.Thmtheorem1.p1.3.m3.1.1.1a" xref="S6.Thmtheorem1.p1.3.m3.1.1.1.cmml">⁢</mo><msup id="S6.Thmtheorem1.p1.3.m3.1.1.4" xref="S6.Thmtheorem1.p1.3.m3.1.1.4.cmml"><mi id="S6.Thmtheorem1.p1.3.m3.1.1.4.2" xref="S6.Thmtheorem1.p1.3.m3.1.1.4.2.cmml">C</mi><mi id="S6.Thmtheorem1.p1.3.m3.1.1.4.3" xref="S6.Thmtheorem1.p1.3.m3.1.1.4.3.cmml">β</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p1.3.m3.1b"><apply id="S6.Thmtheorem1.p1.3.m3.1.1.cmml" xref="S6.Thmtheorem1.p1.3.m3.1.1"><times id="S6.Thmtheorem1.p1.3.m3.1.1.1.cmml" xref="S6.Thmtheorem1.p1.3.m3.1.1.1"></times><apply id="S6.Thmtheorem1.p1.3.m3.1.1.2.cmml" xref="S6.Thmtheorem1.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p1.3.m3.1.1.2.1.cmml" xref="S6.Thmtheorem1.p1.3.m3.1.1.2">superscript</csymbol><ci id="S6.Thmtheorem1.p1.3.m3.1.1.2.2.cmml" xref="S6.Thmtheorem1.p1.3.m3.1.1.2.2">𝐶</ci><ci id="S6.Thmtheorem1.p1.3.m3.1.1.2.3.cmml" xref="S6.Thmtheorem1.p1.3.m3.1.1.2.3">𝛼</ci></apply><ci id="S6.Thmtheorem1.p1.3.m3.1.1.3.cmml" xref="S6.Thmtheorem1.p1.3.m3.1.1.3">⊳</ci><apply id="S6.Thmtheorem1.p1.3.m3.1.1.4.cmml" xref="S6.Thmtheorem1.p1.3.m3.1.1.4"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p1.3.m3.1.1.4.1.cmml" xref="S6.Thmtheorem1.p1.3.m3.1.1.4">superscript</csymbol><ci id="S6.Thmtheorem1.p1.3.m3.1.1.4.2.cmml" xref="S6.Thmtheorem1.p1.3.m3.1.1.4.2">𝐶</ci><ci id="S6.Thmtheorem1.p1.3.m3.1.1.4.3.cmml" xref="S6.Thmtheorem1.p1.3.m3.1.1.4.3">𝛽</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p1.3.m3.1c">C^{\alpha}\vartriangleright C^{\beta}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p1.3.m3.1d">italic_C start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT ⊳ italic_C start_POSTSUPERSCRIPT italic_β end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> </div> <div class="ltx_para" id="S6.p3"> <p class="ltx_p" id="S6.p3.1">This would indeed give us an infinite <math alttext="\trianglelefteq" class="ltx_Math" display="inline" id="S6.p3.1.m1.1"><semantics id="S6.p3.1.m1.1a"><mi id="S6.p3.1.m1.1.1" mathvariant="normal" xref="S6.p3.1.m1.1.1.cmml">⊴</mi><annotation-xml encoding="MathML-Content" id="S6.p3.1.m1.1b"><ci id="S6.p3.1.m1.1.1.cmml" xref="S6.p3.1.m1.1.1">⊴</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.p3.1.m1.1c">\trianglelefteq</annotation><annotation encoding="application/x-llamapun" id="S6.p3.1.m1.1d">⊴</annotation></semantics></math>-decreasing chain of Countryman lines. We will prove a slightly more general result, given by the following.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S6.Thmtheorem2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem2.1.1.1">Theorem 6.2</span></span><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem2.2.2">.</span> </h6> <div class="ltx_para" id="S6.Thmtheorem2.p1"> <p class="ltx_p" id="S6.Thmtheorem2.p1.10"><span class="ltx_text ltx_font_italic" id="S6.Thmtheorem2.p1.10.10">Assume <math alttext="\mathsf{MA}_{\aleph_{1}}" class="ltx_Math" display="inline" id="S6.Thmtheorem2.p1.1.1.m1.1"><semantics id="S6.Thmtheorem2.p1.1.1.m1.1a"><msub id="S6.Thmtheorem2.p1.1.1.m1.1.1" xref="S6.Thmtheorem2.p1.1.1.m1.1.1.cmml"><mi id="S6.Thmtheorem2.p1.1.1.m1.1.1.2" xref="S6.Thmtheorem2.p1.1.1.m1.1.1.2.cmml">𝖬𝖠</mi><msub id="S6.Thmtheorem2.p1.1.1.m1.1.1.3" xref="S6.Thmtheorem2.p1.1.1.m1.1.1.3.cmml"><mi id="S6.Thmtheorem2.p1.1.1.m1.1.1.3.2" mathvariant="normal" xref="S6.Thmtheorem2.p1.1.1.m1.1.1.3.2.cmml">ℵ</mi><mn id="S6.Thmtheorem2.p1.1.1.m1.1.1.3.3" xref="S6.Thmtheorem2.p1.1.1.m1.1.1.3.3.cmml">1</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem2.p1.1.1.m1.1b"><apply id="S6.Thmtheorem2.p1.1.1.m1.1.1.cmml" xref="S6.Thmtheorem2.p1.1.1.m1.1.1"><csymbol cd="ambiguous" id="S6.Thmtheorem2.p1.1.1.m1.1.1.1.cmml" xref="S6.Thmtheorem2.p1.1.1.m1.1.1">subscript</csymbol><ci id="S6.Thmtheorem2.p1.1.1.m1.1.1.2.cmml" xref="S6.Thmtheorem2.p1.1.1.m1.1.1.2">𝖬𝖠</ci><apply id="S6.Thmtheorem2.p1.1.1.m1.1.1.3.cmml" xref="S6.Thmtheorem2.p1.1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S6.Thmtheorem2.p1.1.1.m1.1.1.3.1.cmml" xref="S6.Thmtheorem2.p1.1.1.m1.1.1.3">subscript</csymbol><ci id="S6.Thmtheorem2.p1.1.1.m1.1.1.3.2.cmml" xref="S6.Thmtheorem2.p1.1.1.m1.1.1.3.2">ℵ</ci><cn id="S6.Thmtheorem2.p1.1.1.m1.1.1.3.3.cmml" type="integer" xref="S6.Thmtheorem2.p1.1.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem2.p1.1.1.m1.1c">\mathsf{MA}_{\aleph_{1}}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem2.p1.1.1.m1.1d">sansserif_MA start_POSTSUBSCRIPT roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. Let <math alttext="A" class="ltx_Math" display="inline" id="S6.Thmtheorem2.p1.2.2.m2.1"><semantics id="S6.Thmtheorem2.p1.2.2.m2.1a"><mi id="S6.Thmtheorem2.p1.2.2.m2.1.1" xref="S6.Thmtheorem2.p1.2.2.m2.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem2.p1.2.2.m2.1b"><ci id="S6.Thmtheorem2.p1.2.2.m2.1.1.cmml" xref="S6.Thmtheorem2.p1.2.2.m2.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem2.p1.2.2.m2.1c">A</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem2.p1.2.2.m2.1d">italic_A</annotation></semantics></math> and <math alttext="X" class="ltx_Math" display="inline" id="S6.Thmtheorem2.p1.3.3.m3.1"><semantics id="S6.Thmtheorem2.p1.3.3.m3.1a"><mi id="S6.Thmtheorem2.p1.3.3.m3.1.1" xref="S6.Thmtheorem2.p1.3.3.m3.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem2.p1.3.3.m3.1b"><ci id="S6.Thmtheorem2.p1.3.3.m3.1.1.cmml" xref="S6.Thmtheorem2.p1.3.3.m3.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem2.p1.3.3.m3.1c">X</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem2.p1.3.3.m3.1d">italic_X</annotation></semantics></math> be <math alttext="\preceq" class="ltx_Math" display="inline" id="S6.Thmtheorem2.p1.4.4.m4.1"><semantics id="S6.Thmtheorem2.p1.4.4.m4.1a"><mo id="S6.Thmtheorem2.p1.4.4.m4.1.1" xref="S6.Thmtheorem2.p1.4.4.m4.1.1.cmml">⪯</mo><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem2.p1.4.4.m4.1b"><csymbol cd="latexml" id="S6.Thmtheorem2.p1.4.4.m4.1.1.cmml" xref="S6.Thmtheorem2.p1.4.4.m4.1.1">precedes-or-equals</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem2.p1.4.4.m4.1c">\preceq</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem2.p1.4.4.m4.1d">⪯</annotation></semantics></math>-equivalent <math alttext="\aleph_{1}" class="ltx_Math" display="inline" id="S6.Thmtheorem2.p1.5.5.m5.1"><semantics id="S6.Thmtheorem2.p1.5.5.m5.1a"><msub id="S6.Thmtheorem2.p1.5.5.m5.1.1" xref="S6.Thmtheorem2.p1.5.5.m5.1.1.cmml"><mi id="S6.Thmtheorem2.p1.5.5.m5.1.1.2" mathvariant="normal" xref="S6.Thmtheorem2.p1.5.5.m5.1.1.2.cmml">ℵ</mi><mn id="S6.Thmtheorem2.p1.5.5.m5.1.1.3" xref="S6.Thmtheorem2.p1.5.5.m5.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem2.p1.5.5.m5.1b"><apply id="S6.Thmtheorem2.p1.5.5.m5.1.1.cmml" xref="S6.Thmtheorem2.p1.5.5.m5.1.1"><csymbol cd="ambiguous" id="S6.Thmtheorem2.p1.5.5.m5.1.1.1.cmml" xref="S6.Thmtheorem2.p1.5.5.m5.1.1">subscript</csymbol><ci id="S6.Thmtheorem2.p1.5.5.m5.1.1.2.cmml" xref="S6.Thmtheorem2.p1.5.5.m5.1.1.2">ℵ</ci><cn id="S6.Thmtheorem2.p1.5.5.m5.1.1.3.cmml" type="integer" xref="S6.Thmtheorem2.p1.5.5.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem2.p1.5.5.m5.1c">\aleph_{1}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem2.p1.5.5.m5.1d">roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-dense Countryman lines. If for some decomposition <math alttext="D^{A}" class="ltx_Math" display="inline" id="S6.Thmtheorem2.p1.6.6.m6.1"><semantics id="S6.Thmtheorem2.p1.6.6.m6.1a"><msup id="S6.Thmtheorem2.p1.6.6.m6.1.1" xref="S6.Thmtheorem2.p1.6.6.m6.1.1.cmml"><mi id="S6.Thmtheorem2.p1.6.6.m6.1.1.2" xref="S6.Thmtheorem2.p1.6.6.m6.1.1.2.cmml">D</mi><mi id="S6.Thmtheorem2.p1.6.6.m6.1.1.3" xref="S6.Thmtheorem2.p1.6.6.m6.1.1.3.cmml">A</mi></msup><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem2.p1.6.6.m6.1b"><apply id="S6.Thmtheorem2.p1.6.6.m6.1.1.cmml" xref="S6.Thmtheorem2.p1.6.6.m6.1.1"><csymbol cd="ambiguous" id="S6.Thmtheorem2.p1.6.6.m6.1.1.1.cmml" xref="S6.Thmtheorem2.p1.6.6.m6.1.1">superscript</csymbol><ci id="S6.Thmtheorem2.p1.6.6.m6.1.1.2.cmml" xref="S6.Thmtheorem2.p1.6.6.m6.1.1.2">𝐷</ci><ci id="S6.Thmtheorem2.p1.6.6.m6.1.1.3.cmml" xref="S6.Thmtheorem2.p1.6.6.m6.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem2.p1.6.6.m6.1c">D^{A}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem2.p1.6.6.m6.1d">italic_D start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="D^{X}" class="ltx_Math" display="inline" id="S6.Thmtheorem2.p1.7.7.m7.1"><semantics id="S6.Thmtheorem2.p1.7.7.m7.1a"><msup id="S6.Thmtheorem2.p1.7.7.m7.1.1" xref="S6.Thmtheorem2.p1.7.7.m7.1.1.cmml"><mi id="S6.Thmtheorem2.p1.7.7.m7.1.1.2" xref="S6.Thmtheorem2.p1.7.7.m7.1.1.2.cmml">D</mi><mi id="S6.Thmtheorem2.p1.7.7.m7.1.1.3" xref="S6.Thmtheorem2.p1.7.7.m7.1.1.3.cmml">X</mi></msup><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem2.p1.7.7.m7.1b"><apply id="S6.Thmtheorem2.p1.7.7.m7.1.1.cmml" xref="S6.Thmtheorem2.p1.7.7.m7.1.1"><csymbol cd="ambiguous" id="S6.Thmtheorem2.p1.7.7.m7.1.1.1.cmml" xref="S6.Thmtheorem2.p1.7.7.m7.1.1">superscript</csymbol><ci id="S6.Thmtheorem2.p1.7.7.m7.1.1.2.cmml" xref="S6.Thmtheorem2.p1.7.7.m7.1.1.2">𝐷</ci><ci id="S6.Thmtheorem2.p1.7.7.m7.1.1.3.cmml" xref="S6.Thmtheorem2.p1.7.7.m7.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem2.p1.7.7.m7.1c">D^{X}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem2.p1.7.7.m7.1d">italic_D start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="\mathscr{L}(A,D^{A})\subseteq\hat{\mathscr{L}}(X,D^{X})" class="ltx_Math" display="inline" id="S6.Thmtheorem2.p1.8.8.m8.4"><semantics id="S6.Thmtheorem2.p1.8.8.m8.4a"><mrow id="S6.Thmtheorem2.p1.8.8.m8.4.4" xref="S6.Thmtheorem2.p1.8.8.m8.4.4.cmml"><mrow id="S6.Thmtheorem2.p1.8.8.m8.3.3.1" xref="S6.Thmtheorem2.p1.8.8.m8.3.3.1.cmml"><mi class="ltx_font_mathscript" id="S6.Thmtheorem2.p1.8.8.m8.3.3.1.3" xref="S6.Thmtheorem2.p1.8.8.m8.3.3.1.3.cmml">ℒ</mi><mo id="S6.Thmtheorem2.p1.8.8.m8.3.3.1.2" xref="S6.Thmtheorem2.p1.8.8.m8.3.3.1.2.cmml">⁢</mo><mrow id="S6.Thmtheorem2.p1.8.8.m8.3.3.1.1.1" xref="S6.Thmtheorem2.p1.8.8.m8.3.3.1.1.2.cmml"><mo id="S6.Thmtheorem2.p1.8.8.m8.3.3.1.1.1.2" stretchy="false" xref="S6.Thmtheorem2.p1.8.8.m8.3.3.1.1.2.cmml">(</mo><mi id="S6.Thmtheorem2.p1.8.8.m8.1.1" xref="S6.Thmtheorem2.p1.8.8.m8.1.1.cmml">A</mi><mo id="S6.Thmtheorem2.p1.8.8.m8.3.3.1.1.1.3" xref="S6.Thmtheorem2.p1.8.8.m8.3.3.1.1.2.cmml">,</mo><msup id="S6.Thmtheorem2.p1.8.8.m8.3.3.1.1.1.1" xref="S6.Thmtheorem2.p1.8.8.m8.3.3.1.1.1.1.cmml"><mi id="S6.Thmtheorem2.p1.8.8.m8.3.3.1.1.1.1.2" xref="S6.Thmtheorem2.p1.8.8.m8.3.3.1.1.1.1.2.cmml">D</mi><mi id="S6.Thmtheorem2.p1.8.8.m8.3.3.1.1.1.1.3" xref="S6.Thmtheorem2.p1.8.8.m8.3.3.1.1.1.1.3.cmml">A</mi></msup><mo id="S6.Thmtheorem2.p1.8.8.m8.3.3.1.1.1.4" stretchy="false" xref="S6.Thmtheorem2.p1.8.8.m8.3.3.1.1.2.cmml">)</mo></mrow></mrow><mo id="S6.Thmtheorem2.p1.8.8.m8.4.4.3" xref="S6.Thmtheorem2.p1.8.8.m8.4.4.3.cmml">⊆</mo><mrow id="S6.Thmtheorem2.p1.8.8.m8.4.4.2" xref="S6.Thmtheorem2.p1.8.8.m8.4.4.2.cmml"><mover accent="true" id="S6.Thmtheorem2.p1.8.8.m8.4.4.2.3" xref="S6.Thmtheorem2.p1.8.8.m8.4.4.2.3.cmml"><mi class="ltx_font_mathscript" id="S6.Thmtheorem2.p1.8.8.m8.4.4.2.3.2" xref="S6.Thmtheorem2.p1.8.8.m8.4.4.2.3.2.cmml">ℒ</mi><mo id="S6.Thmtheorem2.p1.8.8.m8.4.4.2.3.1" xref="S6.Thmtheorem2.p1.8.8.m8.4.4.2.3.1.cmml">^</mo></mover><mo id="S6.Thmtheorem2.p1.8.8.m8.4.4.2.2" xref="S6.Thmtheorem2.p1.8.8.m8.4.4.2.2.cmml">⁢</mo><mrow id="S6.Thmtheorem2.p1.8.8.m8.4.4.2.1.1" xref="S6.Thmtheorem2.p1.8.8.m8.4.4.2.1.2.cmml"><mo id="S6.Thmtheorem2.p1.8.8.m8.4.4.2.1.1.2" stretchy="false" xref="S6.Thmtheorem2.p1.8.8.m8.4.4.2.1.2.cmml">(</mo><mi id="S6.Thmtheorem2.p1.8.8.m8.2.2" xref="S6.Thmtheorem2.p1.8.8.m8.2.2.cmml">X</mi><mo id="S6.Thmtheorem2.p1.8.8.m8.4.4.2.1.1.3" xref="S6.Thmtheorem2.p1.8.8.m8.4.4.2.1.2.cmml">,</mo><msup id="S6.Thmtheorem2.p1.8.8.m8.4.4.2.1.1.1" xref="S6.Thmtheorem2.p1.8.8.m8.4.4.2.1.1.1.cmml"><mi id="S6.Thmtheorem2.p1.8.8.m8.4.4.2.1.1.1.2" xref="S6.Thmtheorem2.p1.8.8.m8.4.4.2.1.1.1.2.cmml">D</mi><mi id="S6.Thmtheorem2.p1.8.8.m8.4.4.2.1.1.1.3" xref="S6.Thmtheorem2.p1.8.8.m8.4.4.2.1.1.1.3.cmml">X</mi></msup><mo id="S6.Thmtheorem2.p1.8.8.m8.4.4.2.1.1.4" stretchy="false" xref="S6.Thmtheorem2.p1.8.8.m8.4.4.2.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem2.p1.8.8.m8.4b"><apply id="S6.Thmtheorem2.p1.8.8.m8.4.4.cmml" xref="S6.Thmtheorem2.p1.8.8.m8.4.4"><subset id="S6.Thmtheorem2.p1.8.8.m8.4.4.3.cmml" xref="S6.Thmtheorem2.p1.8.8.m8.4.4.3"></subset><apply id="S6.Thmtheorem2.p1.8.8.m8.3.3.1.cmml" xref="S6.Thmtheorem2.p1.8.8.m8.3.3.1"><times id="S6.Thmtheorem2.p1.8.8.m8.3.3.1.2.cmml" xref="S6.Thmtheorem2.p1.8.8.m8.3.3.1.2"></times><ci id="S6.Thmtheorem2.p1.8.8.m8.3.3.1.3.cmml" xref="S6.Thmtheorem2.p1.8.8.m8.3.3.1.3">ℒ</ci><interval closure="open" id="S6.Thmtheorem2.p1.8.8.m8.3.3.1.1.2.cmml" xref="S6.Thmtheorem2.p1.8.8.m8.3.3.1.1.1"><ci id="S6.Thmtheorem2.p1.8.8.m8.1.1.cmml" xref="S6.Thmtheorem2.p1.8.8.m8.1.1">𝐴</ci><apply id="S6.Thmtheorem2.p1.8.8.m8.3.3.1.1.1.1.cmml" xref="S6.Thmtheorem2.p1.8.8.m8.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S6.Thmtheorem2.p1.8.8.m8.3.3.1.1.1.1.1.cmml" xref="S6.Thmtheorem2.p1.8.8.m8.3.3.1.1.1.1">superscript</csymbol><ci id="S6.Thmtheorem2.p1.8.8.m8.3.3.1.1.1.1.2.cmml" xref="S6.Thmtheorem2.p1.8.8.m8.3.3.1.1.1.1.2">𝐷</ci><ci id="S6.Thmtheorem2.p1.8.8.m8.3.3.1.1.1.1.3.cmml" xref="S6.Thmtheorem2.p1.8.8.m8.3.3.1.1.1.1.3">𝐴</ci></apply></interval></apply><apply id="S6.Thmtheorem2.p1.8.8.m8.4.4.2.cmml" xref="S6.Thmtheorem2.p1.8.8.m8.4.4.2"><times id="S6.Thmtheorem2.p1.8.8.m8.4.4.2.2.cmml" xref="S6.Thmtheorem2.p1.8.8.m8.4.4.2.2"></times><apply id="S6.Thmtheorem2.p1.8.8.m8.4.4.2.3.cmml" xref="S6.Thmtheorem2.p1.8.8.m8.4.4.2.3"><ci id="S6.Thmtheorem2.p1.8.8.m8.4.4.2.3.1.cmml" xref="S6.Thmtheorem2.p1.8.8.m8.4.4.2.3.1">^</ci><ci id="S6.Thmtheorem2.p1.8.8.m8.4.4.2.3.2.cmml" xref="S6.Thmtheorem2.p1.8.8.m8.4.4.2.3.2">ℒ</ci></apply><interval closure="open" id="S6.Thmtheorem2.p1.8.8.m8.4.4.2.1.2.cmml" xref="S6.Thmtheorem2.p1.8.8.m8.4.4.2.1.1"><ci id="S6.Thmtheorem2.p1.8.8.m8.2.2.cmml" xref="S6.Thmtheorem2.p1.8.8.m8.2.2">𝑋</ci><apply id="S6.Thmtheorem2.p1.8.8.m8.4.4.2.1.1.1.cmml" xref="S6.Thmtheorem2.p1.8.8.m8.4.4.2.1.1.1"><csymbol cd="ambiguous" id="S6.Thmtheorem2.p1.8.8.m8.4.4.2.1.1.1.1.cmml" xref="S6.Thmtheorem2.p1.8.8.m8.4.4.2.1.1.1">superscript</csymbol><ci id="S6.Thmtheorem2.p1.8.8.m8.4.4.2.1.1.1.2.cmml" xref="S6.Thmtheorem2.p1.8.8.m8.4.4.2.1.1.1.2">𝐷</ci><ci id="S6.Thmtheorem2.p1.8.8.m8.4.4.2.1.1.1.3.cmml" xref="S6.Thmtheorem2.p1.8.8.m8.4.4.2.1.1.1.3">𝑋</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem2.p1.8.8.m8.4c">\mathscr{L}(A,D^{A})\subseteq\hat{\mathscr{L}}(X,D^{X})</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem2.p1.8.8.m8.4d">script_L ( italic_A , italic_D start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) ⊆ over^ start_ARG script_L end_ARG ( italic_X , italic_D start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT )</annotation></semantics></math> and <math alttext="\mathscr{R}(A,D^{A})\subseteq\hat{\mathscr{R}}(X,D^{X})" class="ltx_Math" display="inline" id="S6.Thmtheorem2.p1.9.9.m9.4"><semantics id="S6.Thmtheorem2.p1.9.9.m9.4a"><mrow id="S6.Thmtheorem2.p1.9.9.m9.4.4" xref="S6.Thmtheorem2.p1.9.9.m9.4.4.cmml"><mrow id="S6.Thmtheorem2.p1.9.9.m9.3.3.1" xref="S6.Thmtheorem2.p1.9.9.m9.3.3.1.cmml"><mi class="ltx_font_mathscript" id="S6.Thmtheorem2.p1.9.9.m9.3.3.1.3" xref="S6.Thmtheorem2.p1.9.9.m9.3.3.1.3.cmml">ℛ</mi><mo id="S6.Thmtheorem2.p1.9.9.m9.3.3.1.2" xref="S6.Thmtheorem2.p1.9.9.m9.3.3.1.2.cmml">⁢</mo><mrow id="S6.Thmtheorem2.p1.9.9.m9.3.3.1.1.1" xref="S6.Thmtheorem2.p1.9.9.m9.3.3.1.1.2.cmml"><mo id="S6.Thmtheorem2.p1.9.9.m9.3.3.1.1.1.2" stretchy="false" xref="S6.Thmtheorem2.p1.9.9.m9.3.3.1.1.2.cmml">(</mo><mi id="S6.Thmtheorem2.p1.9.9.m9.1.1" xref="S6.Thmtheorem2.p1.9.9.m9.1.1.cmml">A</mi><mo id="S6.Thmtheorem2.p1.9.9.m9.3.3.1.1.1.3" xref="S6.Thmtheorem2.p1.9.9.m9.3.3.1.1.2.cmml">,</mo><msup id="S6.Thmtheorem2.p1.9.9.m9.3.3.1.1.1.1" xref="S6.Thmtheorem2.p1.9.9.m9.3.3.1.1.1.1.cmml"><mi id="S6.Thmtheorem2.p1.9.9.m9.3.3.1.1.1.1.2" xref="S6.Thmtheorem2.p1.9.9.m9.3.3.1.1.1.1.2.cmml">D</mi><mi id="S6.Thmtheorem2.p1.9.9.m9.3.3.1.1.1.1.3" xref="S6.Thmtheorem2.p1.9.9.m9.3.3.1.1.1.1.3.cmml">A</mi></msup><mo id="S6.Thmtheorem2.p1.9.9.m9.3.3.1.1.1.4" stretchy="false" xref="S6.Thmtheorem2.p1.9.9.m9.3.3.1.1.2.cmml">)</mo></mrow></mrow><mo id="S6.Thmtheorem2.p1.9.9.m9.4.4.3" xref="S6.Thmtheorem2.p1.9.9.m9.4.4.3.cmml">⊆</mo><mrow id="S6.Thmtheorem2.p1.9.9.m9.4.4.2" xref="S6.Thmtheorem2.p1.9.9.m9.4.4.2.cmml"><mover accent="true" id="S6.Thmtheorem2.p1.9.9.m9.4.4.2.3" xref="S6.Thmtheorem2.p1.9.9.m9.4.4.2.3.cmml"><mi class="ltx_font_mathscript" id="S6.Thmtheorem2.p1.9.9.m9.4.4.2.3.2" xref="S6.Thmtheorem2.p1.9.9.m9.4.4.2.3.2.cmml">ℛ</mi><mo id="S6.Thmtheorem2.p1.9.9.m9.4.4.2.3.1" xref="S6.Thmtheorem2.p1.9.9.m9.4.4.2.3.1.cmml">^</mo></mover><mo id="S6.Thmtheorem2.p1.9.9.m9.4.4.2.2" xref="S6.Thmtheorem2.p1.9.9.m9.4.4.2.2.cmml">⁢</mo><mrow id="S6.Thmtheorem2.p1.9.9.m9.4.4.2.1.1" xref="S6.Thmtheorem2.p1.9.9.m9.4.4.2.1.2.cmml"><mo id="S6.Thmtheorem2.p1.9.9.m9.4.4.2.1.1.2" stretchy="false" xref="S6.Thmtheorem2.p1.9.9.m9.4.4.2.1.2.cmml">(</mo><mi id="S6.Thmtheorem2.p1.9.9.m9.2.2" xref="S6.Thmtheorem2.p1.9.9.m9.2.2.cmml">X</mi><mo id="S6.Thmtheorem2.p1.9.9.m9.4.4.2.1.1.3" xref="S6.Thmtheorem2.p1.9.9.m9.4.4.2.1.2.cmml">,</mo><msup id="S6.Thmtheorem2.p1.9.9.m9.4.4.2.1.1.1" xref="S6.Thmtheorem2.p1.9.9.m9.4.4.2.1.1.1.cmml"><mi id="S6.Thmtheorem2.p1.9.9.m9.4.4.2.1.1.1.2" xref="S6.Thmtheorem2.p1.9.9.m9.4.4.2.1.1.1.2.cmml">D</mi><mi id="S6.Thmtheorem2.p1.9.9.m9.4.4.2.1.1.1.3" xref="S6.Thmtheorem2.p1.9.9.m9.4.4.2.1.1.1.3.cmml">X</mi></msup><mo id="S6.Thmtheorem2.p1.9.9.m9.4.4.2.1.1.4" stretchy="false" xref="S6.Thmtheorem2.p1.9.9.m9.4.4.2.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem2.p1.9.9.m9.4b"><apply id="S6.Thmtheorem2.p1.9.9.m9.4.4.cmml" xref="S6.Thmtheorem2.p1.9.9.m9.4.4"><subset id="S6.Thmtheorem2.p1.9.9.m9.4.4.3.cmml" xref="S6.Thmtheorem2.p1.9.9.m9.4.4.3"></subset><apply id="S6.Thmtheorem2.p1.9.9.m9.3.3.1.cmml" xref="S6.Thmtheorem2.p1.9.9.m9.3.3.1"><times id="S6.Thmtheorem2.p1.9.9.m9.3.3.1.2.cmml" xref="S6.Thmtheorem2.p1.9.9.m9.3.3.1.2"></times><ci id="S6.Thmtheorem2.p1.9.9.m9.3.3.1.3.cmml" xref="S6.Thmtheorem2.p1.9.9.m9.3.3.1.3">ℛ</ci><interval closure="open" id="S6.Thmtheorem2.p1.9.9.m9.3.3.1.1.2.cmml" xref="S6.Thmtheorem2.p1.9.9.m9.3.3.1.1.1"><ci id="S6.Thmtheorem2.p1.9.9.m9.1.1.cmml" xref="S6.Thmtheorem2.p1.9.9.m9.1.1">𝐴</ci><apply id="S6.Thmtheorem2.p1.9.9.m9.3.3.1.1.1.1.cmml" xref="S6.Thmtheorem2.p1.9.9.m9.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S6.Thmtheorem2.p1.9.9.m9.3.3.1.1.1.1.1.cmml" xref="S6.Thmtheorem2.p1.9.9.m9.3.3.1.1.1.1">superscript</csymbol><ci id="S6.Thmtheorem2.p1.9.9.m9.3.3.1.1.1.1.2.cmml" xref="S6.Thmtheorem2.p1.9.9.m9.3.3.1.1.1.1.2">𝐷</ci><ci id="S6.Thmtheorem2.p1.9.9.m9.3.3.1.1.1.1.3.cmml" xref="S6.Thmtheorem2.p1.9.9.m9.3.3.1.1.1.1.3">𝐴</ci></apply></interval></apply><apply id="S6.Thmtheorem2.p1.9.9.m9.4.4.2.cmml" xref="S6.Thmtheorem2.p1.9.9.m9.4.4.2"><times id="S6.Thmtheorem2.p1.9.9.m9.4.4.2.2.cmml" xref="S6.Thmtheorem2.p1.9.9.m9.4.4.2.2"></times><apply id="S6.Thmtheorem2.p1.9.9.m9.4.4.2.3.cmml" xref="S6.Thmtheorem2.p1.9.9.m9.4.4.2.3"><ci id="S6.Thmtheorem2.p1.9.9.m9.4.4.2.3.1.cmml" xref="S6.Thmtheorem2.p1.9.9.m9.4.4.2.3.1">^</ci><ci id="S6.Thmtheorem2.p1.9.9.m9.4.4.2.3.2.cmml" xref="S6.Thmtheorem2.p1.9.9.m9.4.4.2.3.2">ℛ</ci></apply><interval closure="open" id="S6.Thmtheorem2.p1.9.9.m9.4.4.2.1.2.cmml" xref="S6.Thmtheorem2.p1.9.9.m9.4.4.2.1.1"><ci id="S6.Thmtheorem2.p1.9.9.m9.2.2.cmml" xref="S6.Thmtheorem2.p1.9.9.m9.2.2">𝑋</ci><apply id="S6.Thmtheorem2.p1.9.9.m9.4.4.2.1.1.1.cmml" xref="S6.Thmtheorem2.p1.9.9.m9.4.4.2.1.1.1"><csymbol cd="ambiguous" id="S6.Thmtheorem2.p1.9.9.m9.4.4.2.1.1.1.1.cmml" xref="S6.Thmtheorem2.p1.9.9.m9.4.4.2.1.1.1">superscript</csymbol><ci id="S6.Thmtheorem2.p1.9.9.m9.4.4.2.1.1.1.2.cmml" xref="S6.Thmtheorem2.p1.9.9.m9.4.4.2.1.1.1.2">𝐷</ci><ci id="S6.Thmtheorem2.p1.9.9.m9.4.4.2.1.1.1.3.cmml" xref="S6.Thmtheorem2.p1.9.9.m9.4.4.2.1.1.1.3">𝑋</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem2.p1.9.9.m9.4c">\mathscr{R}(A,D^{A})\subseteq\hat{\mathscr{R}}(X,D^{X})</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem2.p1.9.9.m9.4d">script_R ( italic_A , italic_D start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) ⊆ over^ start_ARG script_R end_ARG ( italic_X , italic_D start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT )</annotation></semantics></math>, then <math alttext="A\trianglerighteq X" class="ltx_Math" display="inline" id="S6.Thmtheorem2.p1.10.10.m10.1"><semantics id="S6.Thmtheorem2.p1.10.10.m10.1a"><mrow id="S6.Thmtheorem2.p1.10.10.m10.1.1" xref="S6.Thmtheorem2.p1.10.10.m10.1.1.cmml"><mi id="S6.Thmtheorem2.p1.10.10.m10.1.1.2" xref="S6.Thmtheorem2.p1.10.10.m10.1.1.2.cmml">A</mi><mo id="S6.Thmtheorem2.p1.10.10.m10.1.1.1" xref="S6.Thmtheorem2.p1.10.10.m10.1.1.1.cmml">⁢</mo><mi id="S6.Thmtheorem2.p1.10.10.m10.1.1.3" mathvariant="normal" xref="S6.Thmtheorem2.p1.10.10.m10.1.1.3.cmml">⊵</mi><mo id="S6.Thmtheorem2.p1.10.10.m10.1.1.1a" xref="S6.Thmtheorem2.p1.10.10.m10.1.1.1.cmml">⁢</mo><mi id="S6.Thmtheorem2.p1.10.10.m10.1.1.4" xref="S6.Thmtheorem2.p1.10.10.m10.1.1.4.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem2.p1.10.10.m10.1b"><apply id="S6.Thmtheorem2.p1.10.10.m10.1.1.cmml" xref="S6.Thmtheorem2.p1.10.10.m10.1.1"><times id="S6.Thmtheorem2.p1.10.10.m10.1.1.1.cmml" xref="S6.Thmtheorem2.p1.10.10.m10.1.1.1"></times><ci id="S6.Thmtheorem2.p1.10.10.m10.1.1.2.cmml" xref="S6.Thmtheorem2.p1.10.10.m10.1.1.2">𝐴</ci><ci id="S6.Thmtheorem2.p1.10.10.m10.1.1.3.cmml" xref="S6.Thmtheorem2.p1.10.10.m10.1.1.3">⊵</ci><ci id="S6.Thmtheorem2.p1.10.10.m10.1.1.4.cmml" xref="S6.Thmtheorem2.p1.10.10.m10.1.1.4">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem2.p1.10.10.m10.1c">A\trianglerighteq X</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem2.p1.10.10.m10.1d">italic_A ⊵ italic_X</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S6.p4"> <p class="ltx_p" id="S6.p4.3">Observe that if <math alttext="\alpha<\beta" class="ltx_Math" display="inline" id="S6.p4.1.m1.1"><semantics id="S6.p4.1.m1.1a"><mrow id="S6.p4.1.m1.1.1" xref="S6.p4.1.m1.1.1.cmml"><mi id="S6.p4.1.m1.1.1.2" xref="S6.p4.1.m1.1.1.2.cmml">α</mi><mo id="S6.p4.1.m1.1.1.1" xref="S6.p4.1.m1.1.1.1.cmml"><</mo><mi id="S6.p4.1.m1.1.1.3" xref="S6.p4.1.m1.1.1.3.cmml">β</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.p4.1.m1.1b"><apply id="S6.p4.1.m1.1.1.cmml" xref="S6.p4.1.m1.1.1"><lt id="S6.p4.1.m1.1.1.1.cmml" xref="S6.p4.1.m1.1.1.1"></lt><ci id="S6.p4.1.m1.1.1.2.cmml" xref="S6.p4.1.m1.1.1.2">𝛼</ci><ci id="S6.p4.1.m1.1.1.3.cmml" xref="S6.p4.1.m1.1.1.3">𝛽</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p4.1.m1.1c">\alpha<\beta</annotation><annotation encoding="application/x-llamapun" id="S6.p4.1.m1.1d">italic_α < italic_β</annotation></semantics></math>, then the hypothesis of the theorem are satisfied letting <math alttext="A=C^{\alpha}" class="ltx_Math" display="inline" id="S6.p4.2.m2.1"><semantics id="S6.p4.2.m2.1a"><mrow id="S6.p4.2.m2.1.1" xref="S6.p4.2.m2.1.1.cmml"><mi id="S6.p4.2.m2.1.1.2" xref="S6.p4.2.m2.1.1.2.cmml">A</mi><mo id="S6.p4.2.m2.1.1.1" xref="S6.p4.2.m2.1.1.1.cmml">=</mo><msup id="S6.p4.2.m2.1.1.3" xref="S6.p4.2.m2.1.1.3.cmml"><mi id="S6.p4.2.m2.1.1.3.2" xref="S6.p4.2.m2.1.1.3.2.cmml">C</mi><mi id="S6.p4.2.m2.1.1.3.3" xref="S6.p4.2.m2.1.1.3.3.cmml">α</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S6.p4.2.m2.1b"><apply id="S6.p4.2.m2.1.1.cmml" xref="S6.p4.2.m2.1.1"><eq id="S6.p4.2.m2.1.1.1.cmml" xref="S6.p4.2.m2.1.1.1"></eq><ci id="S6.p4.2.m2.1.1.2.cmml" xref="S6.p4.2.m2.1.1.2">𝐴</ci><apply id="S6.p4.2.m2.1.1.3.cmml" xref="S6.p4.2.m2.1.1.3"><csymbol cd="ambiguous" id="S6.p4.2.m2.1.1.3.1.cmml" xref="S6.p4.2.m2.1.1.3">superscript</csymbol><ci id="S6.p4.2.m2.1.1.3.2.cmml" xref="S6.p4.2.m2.1.1.3.2">𝐶</ci><ci id="S6.p4.2.m2.1.1.3.3.cmml" xref="S6.p4.2.m2.1.1.3.3">𝛼</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p4.2.m2.1c">A=C^{\alpha}</annotation><annotation encoding="application/x-llamapun" id="S6.p4.2.m2.1d">italic_A = italic_C start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="X=C^{\beta}" class="ltx_Math" display="inline" id="S6.p4.3.m3.1"><semantics id="S6.p4.3.m3.1a"><mrow id="S6.p4.3.m3.1.1" xref="S6.p4.3.m3.1.1.cmml"><mi id="S6.p4.3.m3.1.1.2" xref="S6.p4.3.m3.1.1.2.cmml">X</mi><mo id="S6.p4.3.m3.1.1.1" xref="S6.p4.3.m3.1.1.1.cmml">=</mo><msup id="S6.p4.3.m3.1.1.3" xref="S6.p4.3.m3.1.1.3.cmml"><mi id="S6.p4.3.m3.1.1.3.2" xref="S6.p4.3.m3.1.1.3.2.cmml">C</mi><mi id="S6.p4.3.m3.1.1.3.3" xref="S6.p4.3.m3.1.1.3.3.cmml">β</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S6.p4.3.m3.1b"><apply id="S6.p4.3.m3.1.1.cmml" xref="S6.p4.3.m3.1.1"><eq id="S6.p4.3.m3.1.1.1.cmml" xref="S6.p4.3.m3.1.1.1"></eq><ci id="S6.p4.3.m3.1.1.2.cmml" xref="S6.p4.3.m3.1.1.2">𝑋</ci><apply id="S6.p4.3.m3.1.1.3.cmml" xref="S6.p4.3.m3.1.1.3"><csymbol cd="ambiguous" id="S6.p4.3.m3.1.1.3.1.cmml" xref="S6.p4.3.m3.1.1.3">superscript</csymbol><ci id="S6.p4.3.m3.1.1.3.2.cmml" xref="S6.p4.3.m3.1.1.3.2">𝐶</ci><ci id="S6.p4.3.m3.1.1.3.3.cmml" xref="S6.p4.3.m3.1.1.3.3">𝛽</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p4.3.m3.1c">X=C^{\beta}</annotation><annotation encoding="application/x-llamapun" id="S6.p4.3.m3.1d">italic_X = italic_C start_POSTSUPERSCRIPT italic_β end_POSTSUPERSCRIPT</annotation></semantics></math>. As a consequence we have another proof of <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S3.Thmtheorem2" title="Corollary 3.2. ‣ 3. Strongly surjective Aronszajn lines ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Corollary</span> <span class="ltx_text ltx_ref_tag">3.2</span></a>.</p> </div> <div class="ltx_theorem ltx_theorem_corollary" id="S6.Thmtheorem3"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem3.1.1.1">Corollary 6.3</span></span><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem3.2.2">.</span> </h6> <div class="ltx_para" id="S6.Thmtheorem3.p1"> <p class="ltx_p" id="S6.Thmtheorem3.p1.1">Assume <math alttext="\mathsf{MA}_{\aleph_{1}}" class="ltx_Math" display="inline" id="S6.Thmtheorem3.p1.1.m1.1"><semantics id="S6.Thmtheorem3.p1.1.m1.1a"><msub id="S6.Thmtheorem3.p1.1.m1.1.1" xref="S6.Thmtheorem3.p1.1.m1.1.1.cmml"><mi id="S6.Thmtheorem3.p1.1.m1.1.1.2" xref="S6.Thmtheorem3.p1.1.m1.1.1.2.cmml">𝖬𝖠</mi><msub id="S6.Thmtheorem3.p1.1.m1.1.1.3" xref="S6.Thmtheorem3.p1.1.m1.1.1.3.cmml"><mi id="S6.Thmtheorem3.p1.1.m1.1.1.3.2" mathvariant="normal" xref="S6.Thmtheorem3.p1.1.m1.1.1.3.2.cmml">ℵ</mi><mn id="S6.Thmtheorem3.p1.1.m1.1.1.3.3" xref="S6.Thmtheorem3.p1.1.m1.1.1.3.3.cmml">1</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem3.p1.1.m1.1b"><apply id="S6.Thmtheorem3.p1.1.m1.1.1.cmml" xref="S6.Thmtheorem3.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S6.Thmtheorem3.p1.1.m1.1.1.1.cmml" xref="S6.Thmtheorem3.p1.1.m1.1.1">subscript</csymbol><ci id="S6.Thmtheorem3.p1.1.m1.1.1.2.cmml" xref="S6.Thmtheorem3.p1.1.m1.1.1.2">𝖬𝖠</ci><apply id="S6.Thmtheorem3.p1.1.m1.1.1.3.cmml" xref="S6.Thmtheorem3.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S6.Thmtheorem3.p1.1.m1.1.1.3.1.cmml" xref="S6.Thmtheorem3.p1.1.m1.1.1.3">subscript</csymbol><ci id="S6.Thmtheorem3.p1.1.m1.1.1.3.2.cmml" xref="S6.Thmtheorem3.p1.1.m1.1.1.3.2">ℵ</ci><cn id="S6.Thmtheorem3.p1.1.m1.1.1.3.3.cmml" type="integer" xref="S6.Thmtheorem3.p1.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem3.p1.1.m1.1c">\mathsf{MA}_{\aleph_{1}}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem3.p1.1.m1.1d">sansserif_MA start_POSTSUBSCRIPT roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. Every normal Countryman line is strongly surjective.</p> </div> </div> <div class="ltx_proof" id="S6.1"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S6.1.p1"> <p class="ltx_p" id="S6.1.p1.9">If <math alttext="C" class="ltx_Math" display="inline" id="S6.1.p1.1.m1.1"><semantics id="S6.1.p1.1.m1.1a"><mi id="S6.1.p1.1.m1.1.1" xref="S6.1.p1.1.m1.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S6.1.p1.1.m1.1b"><ci id="S6.1.p1.1.m1.1.1.cmml" xref="S6.1.p1.1.m1.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.1.p1.1.m1.1c">C</annotation><annotation encoding="application/x-llamapun" id="S6.1.p1.1.m1.1d">italic_C</annotation></semantics></math> is normal, then in there is a decomposition <math alttext="D" class="ltx_Math" display="inline" id="S6.1.p1.2.m2.1"><semantics id="S6.1.p1.2.m2.1a"><mi id="S6.1.p1.2.m2.1.1" xref="S6.1.p1.2.m2.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S6.1.p1.2.m2.1b"><ci id="S6.1.p1.2.m2.1.1.cmml" xref="S6.1.p1.2.m2.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.1.p1.2.m2.1c">D</annotation><annotation encoding="application/x-llamapun" id="S6.1.p1.2.m2.1d">italic_D</annotation></semantics></math> for which which <math alttext="\mathscr{L}(C,D)=\mathscr{R}(C,D)=\varnothing" class="ltx_Math" display="inline" id="S6.1.p1.3.m3.4"><semantics id="S6.1.p1.3.m3.4a"><mrow id="S6.1.p1.3.m3.4.5" xref="S6.1.p1.3.m3.4.5.cmml"><mrow id="S6.1.p1.3.m3.4.5.2" xref="S6.1.p1.3.m3.4.5.2.cmml"><mi class="ltx_font_mathscript" id="S6.1.p1.3.m3.4.5.2.2" xref="S6.1.p1.3.m3.4.5.2.2.cmml">ℒ</mi><mo id="S6.1.p1.3.m3.4.5.2.1" xref="S6.1.p1.3.m3.4.5.2.1.cmml">⁢</mo><mrow id="S6.1.p1.3.m3.4.5.2.3.2" xref="S6.1.p1.3.m3.4.5.2.3.1.cmml"><mo id="S6.1.p1.3.m3.4.5.2.3.2.1" stretchy="false" xref="S6.1.p1.3.m3.4.5.2.3.1.cmml">(</mo><mi id="S6.1.p1.3.m3.1.1" xref="S6.1.p1.3.m3.1.1.cmml">C</mi><mo id="S6.1.p1.3.m3.4.5.2.3.2.2" xref="S6.1.p1.3.m3.4.5.2.3.1.cmml">,</mo><mi id="S6.1.p1.3.m3.2.2" xref="S6.1.p1.3.m3.2.2.cmml">D</mi><mo id="S6.1.p1.3.m3.4.5.2.3.2.3" stretchy="false" xref="S6.1.p1.3.m3.4.5.2.3.1.cmml">)</mo></mrow></mrow><mo id="S6.1.p1.3.m3.4.5.3" xref="S6.1.p1.3.m3.4.5.3.cmml">=</mo><mrow id="S6.1.p1.3.m3.4.5.4" xref="S6.1.p1.3.m3.4.5.4.cmml"><mi class="ltx_font_mathscript" id="S6.1.p1.3.m3.4.5.4.2" xref="S6.1.p1.3.m3.4.5.4.2.cmml">ℛ</mi><mo id="S6.1.p1.3.m3.4.5.4.1" xref="S6.1.p1.3.m3.4.5.4.1.cmml">⁢</mo><mrow id="S6.1.p1.3.m3.4.5.4.3.2" xref="S6.1.p1.3.m3.4.5.4.3.1.cmml"><mo id="S6.1.p1.3.m3.4.5.4.3.2.1" stretchy="false" xref="S6.1.p1.3.m3.4.5.4.3.1.cmml">(</mo><mi id="S6.1.p1.3.m3.3.3" xref="S6.1.p1.3.m3.3.3.cmml">C</mi><mo id="S6.1.p1.3.m3.4.5.4.3.2.2" xref="S6.1.p1.3.m3.4.5.4.3.1.cmml">,</mo><mi id="S6.1.p1.3.m3.4.4" xref="S6.1.p1.3.m3.4.4.cmml">D</mi><mo id="S6.1.p1.3.m3.4.5.4.3.2.3" stretchy="false" xref="S6.1.p1.3.m3.4.5.4.3.1.cmml">)</mo></mrow></mrow><mo id="S6.1.p1.3.m3.4.5.5" xref="S6.1.p1.3.m3.4.5.5.cmml">=</mo><mi id="S6.1.p1.3.m3.4.5.6" mathvariant="normal" xref="S6.1.p1.3.m3.4.5.6.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.1.p1.3.m3.4b"><apply id="S6.1.p1.3.m3.4.5.cmml" xref="S6.1.p1.3.m3.4.5"><and id="S6.1.p1.3.m3.4.5a.cmml" xref="S6.1.p1.3.m3.4.5"></and><apply id="S6.1.p1.3.m3.4.5b.cmml" xref="S6.1.p1.3.m3.4.5"><eq id="S6.1.p1.3.m3.4.5.3.cmml" xref="S6.1.p1.3.m3.4.5.3"></eq><apply id="S6.1.p1.3.m3.4.5.2.cmml" xref="S6.1.p1.3.m3.4.5.2"><times id="S6.1.p1.3.m3.4.5.2.1.cmml" xref="S6.1.p1.3.m3.4.5.2.1"></times><ci id="S6.1.p1.3.m3.4.5.2.2.cmml" xref="S6.1.p1.3.m3.4.5.2.2">ℒ</ci><interval closure="open" id="S6.1.p1.3.m3.4.5.2.3.1.cmml" xref="S6.1.p1.3.m3.4.5.2.3.2"><ci id="S6.1.p1.3.m3.1.1.cmml" xref="S6.1.p1.3.m3.1.1">𝐶</ci><ci id="S6.1.p1.3.m3.2.2.cmml" xref="S6.1.p1.3.m3.2.2">𝐷</ci></interval></apply><apply id="S6.1.p1.3.m3.4.5.4.cmml" xref="S6.1.p1.3.m3.4.5.4"><times id="S6.1.p1.3.m3.4.5.4.1.cmml" xref="S6.1.p1.3.m3.4.5.4.1"></times><ci id="S6.1.p1.3.m3.4.5.4.2.cmml" xref="S6.1.p1.3.m3.4.5.4.2">ℛ</ci><interval closure="open" id="S6.1.p1.3.m3.4.5.4.3.1.cmml" xref="S6.1.p1.3.m3.4.5.4.3.2"><ci id="S6.1.p1.3.m3.3.3.cmml" xref="S6.1.p1.3.m3.3.3">𝐶</ci><ci id="S6.1.p1.3.m3.4.4.cmml" xref="S6.1.p1.3.m3.4.4">𝐷</ci></interval></apply></apply><apply id="S6.1.p1.3.m3.4.5c.cmml" xref="S6.1.p1.3.m3.4.5"><eq id="S6.1.p1.3.m3.4.5.5.cmml" xref="S6.1.p1.3.m3.4.5.5"></eq><share href="https://arxiv.org/html/2503.13728v1#S6.1.p1.3.m3.4.5.4.cmml" id="S6.1.p1.3.m3.4.5d.cmml" xref="S6.1.p1.3.m3.4.5"></share><emptyset id="S6.1.p1.3.m3.4.5.6.cmml" xref="S6.1.p1.3.m3.4.5.6"></emptyset></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.1.p1.3.m3.4c">\mathscr{L}(C,D)=\mathscr{R}(C,D)=\varnothing</annotation><annotation encoding="application/x-llamapun" id="S6.1.p1.3.m3.4d">script_L ( italic_C , italic_D ) = script_R ( italic_C , italic_D ) = ∅</annotation></semantics></math>. Then, if <math alttext="A\preceq C" class="ltx_Math" display="inline" id="S6.1.p1.4.m4.1"><semantics id="S6.1.p1.4.m4.1a"><mrow id="S6.1.p1.4.m4.1.1" xref="S6.1.p1.4.m4.1.1.cmml"><mi id="S6.1.p1.4.m4.1.1.2" xref="S6.1.p1.4.m4.1.1.2.cmml">A</mi><mo id="S6.1.p1.4.m4.1.1.1" xref="S6.1.p1.4.m4.1.1.1.cmml">⪯</mo><mi id="S6.1.p1.4.m4.1.1.3" xref="S6.1.p1.4.m4.1.1.3.cmml">C</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.1.p1.4.m4.1b"><apply id="S6.1.p1.4.m4.1.1.cmml" xref="S6.1.p1.4.m4.1.1"><csymbol cd="latexml" id="S6.1.p1.4.m4.1.1.1.cmml" xref="S6.1.p1.4.m4.1.1.1">precedes-or-equals</csymbol><ci id="S6.1.p1.4.m4.1.1.2.cmml" xref="S6.1.p1.4.m4.1.1.2">𝐴</ci><ci id="S6.1.p1.4.m4.1.1.3.cmml" xref="S6.1.p1.4.m4.1.1.3">𝐶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.1.p1.4.m4.1c">A\preceq C</annotation><annotation encoding="application/x-llamapun" id="S6.1.p1.4.m4.1d">italic_A ⪯ italic_C</annotation></semantics></math> is nonempty, <math alttext="A\times C" class="ltx_Math" display="inline" id="S6.1.p1.5.m5.1"><semantics id="S6.1.p1.5.m5.1a"><mrow id="S6.1.p1.5.m5.1.1" xref="S6.1.p1.5.m5.1.1.cmml"><mi id="S6.1.p1.5.m5.1.1.2" xref="S6.1.p1.5.m5.1.1.2.cmml">A</mi><mo id="S6.1.p1.5.m5.1.1.1" lspace="0.222em" rspace="0.222em" xref="S6.1.p1.5.m5.1.1.1.cmml">×</mo><mi id="S6.1.p1.5.m5.1.1.3" xref="S6.1.p1.5.m5.1.1.3.cmml">C</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.1.p1.5.m5.1b"><apply id="S6.1.p1.5.m5.1.1.cmml" xref="S6.1.p1.5.m5.1.1"><times id="S6.1.p1.5.m5.1.1.1.cmml" xref="S6.1.p1.5.m5.1.1.1"></times><ci id="S6.1.p1.5.m5.1.1.2.cmml" xref="S6.1.p1.5.m5.1.1.2">𝐴</ci><ci id="S6.1.p1.5.m5.1.1.3.cmml" xref="S6.1.p1.5.m5.1.1.3">𝐶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.1.p1.5.m5.1c">A\times C</annotation><annotation encoding="application/x-llamapun" id="S6.1.p1.5.m5.1d">italic_A × italic_C</annotation></semantics></math> is an <math alttext="\aleph_{1}" class="ltx_Math" display="inline" id="S6.1.p1.6.m6.1"><semantics id="S6.1.p1.6.m6.1a"><msub id="S6.1.p1.6.m6.1.1" xref="S6.1.p1.6.m6.1.1.cmml"><mi id="S6.1.p1.6.m6.1.1.2" mathvariant="normal" xref="S6.1.p1.6.m6.1.1.2.cmml">ℵ</mi><mn id="S6.1.p1.6.m6.1.1.3" xref="S6.1.p1.6.m6.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S6.1.p1.6.m6.1b"><apply id="S6.1.p1.6.m6.1.1.cmml" xref="S6.1.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S6.1.p1.6.m6.1.1.1.cmml" xref="S6.1.p1.6.m6.1.1">subscript</csymbol><ci id="S6.1.p1.6.m6.1.1.2.cmml" xref="S6.1.p1.6.m6.1.1.2">ℵ</ci><cn id="S6.1.p1.6.m6.1.1.3.cmml" type="integer" xref="S6.1.p1.6.m6.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.1.p1.6.m6.1c">\aleph_{1}</annotation><annotation encoding="application/x-llamapun" id="S6.1.p1.6.m6.1d">roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-dense Countryman line, which is <math alttext="\preceq" class="ltx_Math" display="inline" id="S6.1.p1.7.m7.1"><semantics id="S6.1.p1.7.m7.1a"><mo id="S6.1.p1.7.m7.1.1" xref="S6.1.p1.7.m7.1.1.cmml">⪯</mo><annotation-xml encoding="MathML-Content" id="S6.1.p1.7.m7.1b"><csymbol cd="latexml" id="S6.1.p1.7.m7.1.1.cmml" xref="S6.1.p1.7.m7.1.1">precedes-or-equals</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S6.1.p1.7.m7.1c">\preceq</annotation><annotation encoding="application/x-llamapun" id="S6.1.p1.7.m7.1d">⪯</annotation></semantics></math>-equivalent to <math alttext="C" class="ltx_Math" display="inline" id="S6.1.p1.8.m8.1"><semantics id="S6.1.p1.8.m8.1a"><mi id="S6.1.p1.8.m8.1.1" xref="S6.1.p1.8.m8.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S6.1.p1.8.m8.1b"><ci id="S6.1.p1.8.m8.1.1.cmml" xref="S6.1.p1.8.m8.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.1.p1.8.m8.1c">C</annotation><annotation encoding="application/x-llamapun" id="S6.1.p1.8.m8.1d">italic_C</annotation></semantics></math>. Thus <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.Thmtheorem2" title="Theorem 6.2. ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">6.2</span></a> implies that <math alttext="C\trianglerighteq A\times C" class="ltx_Math" display="inline" id="S6.1.p1.9.m9.1"><semantics id="S6.1.p1.9.m9.1a"><mrow id="S6.1.p1.9.m9.1.1" xref="S6.1.p1.9.m9.1.1.cmml"><mrow id="S6.1.p1.9.m9.1.1.2" xref="S6.1.p1.9.m9.1.1.2.cmml"><mi id="S6.1.p1.9.m9.1.1.2.2" xref="S6.1.p1.9.m9.1.1.2.2.cmml">C</mi><mo id="S6.1.p1.9.m9.1.1.2.1" xref="S6.1.p1.9.m9.1.1.2.1.cmml">⁢</mo><mi id="S6.1.p1.9.m9.1.1.2.3" mathvariant="normal" xref="S6.1.p1.9.m9.1.1.2.3.cmml">⊵</mi><mo id="S6.1.p1.9.m9.1.1.2.1a" xref="S6.1.p1.9.m9.1.1.2.1.cmml">⁢</mo><mi id="S6.1.p1.9.m9.1.1.2.4" xref="S6.1.p1.9.m9.1.1.2.4.cmml">A</mi></mrow><mo id="S6.1.p1.9.m9.1.1.1" lspace="0.222em" rspace="0.222em" xref="S6.1.p1.9.m9.1.1.1.cmml">×</mo><mi id="S6.1.p1.9.m9.1.1.3" xref="S6.1.p1.9.m9.1.1.3.cmml">C</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.1.p1.9.m9.1b"><apply id="S6.1.p1.9.m9.1.1.cmml" xref="S6.1.p1.9.m9.1.1"><times id="S6.1.p1.9.m9.1.1.1.cmml" xref="S6.1.p1.9.m9.1.1.1"></times><apply id="S6.1.p1.9.m9.1.1.2.cmml" xref="S6.1.p1.9.m9.1.1.2"><times id="S6.1.p1.9.m9.1.1.2.1.cmml" xref="S6.1.p1.9.m9.1.1.2.1"></times><ci id="S6.1.p1.9.m9.1.1.2.2.cmml" xref="S6.1.p1.9.m9.1.1.2.2">𝐶</ci><ci id="S6.1.p1.9.m9.1.1.2.3.cmml" xref="S6.1.p1.9.m9.1.1.2.3">⊵</ci><ci id="S6.1.p1.9.m9.1.1.2.4.cmml" xref="S6.1.p1.9.m9.1.1.2.4">𝐴</ci></apply><ci id="S6.1.p1.9.m9.1.1.3.cmml" xref="S6.1.p1.9.m9.1.1.3">𝐶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.1.p1.9.m9.1c">C\trianglerighteq A\times C</annotation><annotation encoding="application/x-llamapun" id="S6.1.p1.9.m9.1d">italic_C ⊵ italic_A × italic_C</annotation></semantics></math>. ∎</p> </div> </div> <div class="ltx_para" id="S6.p5"> <p class="ltx_p" id="S6.p5.1">The proof of <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.Thmtheorem2" title="Theorem 6.2. ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">6.2</span></a> will be strongly based on Moore’s proof of his Theorem 1.1 (<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S2.Thmtheorem8" title="Theorem 2.8. ‣ 2. Aronszajn and Countryman lines ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">2.8</span></a>).</p> </div> <section class="ltx_subsection" id="S6.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">6.1. </span>Moore’s forcing</h3> <div class="ltx_para" id="S6.SS1.p1"> <p class="ltx_p" id="S6.SS1.p1.1">In this section we give the necessary details that we need from Moore’s forcing. All definitions and results in this section are from <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib16" title="">16</a>]</cite>.</p> </div> <div class="ltx_para" id="S6.SS1.p2"> <p class="ltx_p" id="S6.SS1.p2.17">Let <math alttext="A" class="ltx_Math" display="inline" id="S6.SS1.p2.1.m1.1"><semantics id="S6.SS1.p2.1.m1.1a"><mi id="S6.SS1.p2.1.m1.1.1" xref="S6.SS1.p2.1.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S6.SS1.p2.1.m1.1b"><ci id="S6.SS1.p2.1.m1.1.1.cmml" xref="S6.SS1.p2.1.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p2.1.m1.1c">A</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p2.1.m1.1d">italic_A</annotation></semantics></math> be an Aronszajn line and assume that <math alttext="A" class="ltx_Math" display="inline" id="S6.SS1.p2.2.m2.1"><semantics id="S6.SS1.p2.2.m2.1a"><mi id="S6.SS1.p2.2.m2.1.1" xref="S6.SS1.p2.2.m2.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S6.SS1.p2.2.m2.1b"><ci id="S6.SS1.p2.2.m2.1.1.cmml" xref="S6.SS1.p2.2.m2.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p2.2.m2.1c">A</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p2.2.m2.1d">italic_A</annotation></semantics></math> is <math alttext="\omega_{1}" class="ltx_Math" display="inline" id="S6.SS1.p2.3.m3.1"><semantics id="S6.SS1.p2.3.m3.1a"><msub id="S6.SS1.p2.3.m3.1.1" xref="S6.SS1.p2.3.m3.1.1.cmml"><mi id="S6.SS1.p2.3.m3.1.1.2" xref="S6.SS1.p2.3.m3.1.1.2.cmml">ω</mi><mn id="S6.SS1.p2.3.m3.1.1.3" xref="S6.SS1.p2.3.m3.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S6.SS1.p2.3.m3.1b"><apply id="S6.SS1.p2.3.m3.1.1.cmml" xref="S6.SS1.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S6.SS1.p2.3.m3.1.1.1.cmml" xref="S6.SS1.p2.3.m3.1.1">subscript</csymbol><ci id="S6.SS1.p2.3.m3.1.1.2.cmml" xref="S6.SS1.p2.3.m3.1.1.2">𝜔</ci><cn id="S6.SS1.p2.3.m3.1.1.3.cmml" type="integer" xref="S6.SS1.p2.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p2.3.m3.1c">\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p2.3.m3.1d">italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> as a set. For <math alttext="a\in A" class="ltx_Math" display="inline" id="S6.SS1.p2.4.m4.1"><semantics id="S6.SS1.p2.4.m4.1a"><mrow id="S6.SS1.p2.4.m4.1.1" xref="S6.SS1.p2.4.m4.1.1.cmml"><mi id="S6.SS1.p2.4.m4.1.1.2" xref="S6.SS1.p2.4.m4.1.1.2.cmml">a</mi><mo id="S6.SS1.p2.4.m4.1.1.1" xref="S6.SS1.p2.4.m4.1.1.1.cmml">∈</mo><mi id="S6.SS1.p2.4.m4.1.1.3" xref="S6.SS1.p2.4.m4.1.1.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p2.4.m4.1b"><apply id="S6.SS1.p2.4.m4.1.1.cmml" xref="S6.SS1.p2.4.m4.1.1"><in id="S6.SS1.p2.4.m4.1.1.1.cmml" xref="S6.SS1.p2.4.m4.1.1.1"></in><ci id="S6.SS1.p2.4.m4.1.1.2.cmml" xref="S6.SS1.p2.4.m4.1.1.2">𝑎</ci><ci id="S6.SS1.p2.4.m4.1.1.3.cmml" xref="S6.SS1.p2.4.m4.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p2.4.m4.1c">a\in A</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p2.4.m4.1d">italic_a ∈ italic_A</annotation></semantics></math> let <math alttext="\tilde{a}\in 2^{\omega_{1}}" class="ltx_Math" display="inline" id="S6.SS1.p2.5.m5.1"><semantics id="S6.SS1.p2.5.m5.1a"><mrow id="S6.SS1.p2.5.m5.1.1" xref="S6.SS1.p2.5.m5.1.1.cmml"><mover accent="true" id="S6.SS1.p2.5.m5.1.1.2" xref="S6.SS1.p2.5.m5.1.1.2.cmml"><mi id="S6.SS1.p2.5.m5.1.1.2.2" xref="S6.SS1.p2.5.m5.1.1.2.2.cmml">a</mi><mo id="S6.SS1.p2.5.m5.1.1.2.1" xref="S6.SS1.p2.5.m5.1.1.2.1.cmml">~</mo></mover><mo id="S6.SS1.p2.5.m5.1.1.1" xref="S6.SS1.p2.5.m5.1.1.1.cmml">∈</mo><msup id="S6.SS1.p2.5.m5.1.1.3" xref="S6.SS1.p2.5.m5.1.1.3.cmml"><mn id="S6.SS1.p2.5.m5.1.1.3.2" xref="S6.SS1.p2.5.m5.1.1.3.2.cmml">2</mn><msub id="S6.SS1.p2.5.m5.1.1.3.3" xref="S6.SS1.p2.5.m5.1.1.3.3.cmml"><mi id="S6.SS1.p2.5.m5.1.1.3.3.2" xref="S6.SS1.p2.5.m5.1.1.3.3.2.cmml">ω</mi><mn id="S6.SS1.p2.5.m5.1.1.3.3.3" xref="S6.SS1.p2.5.m5.1.1.3.3.3.cmml">1</mn></msub></msup></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p2.5.m5.1b"><apply id="S6.SS1.p2.5.m5.1.1.cmml" xref="S6.SS1.p2.5.m5.1.1"><in id="S6.SS1.p2.5.m5.1.1.1.cmml" xref="S6.SS1.p2.5.m5.1.1.1"></in><apply id="S6.SS1.p2.5.m5.1.1.2.cmml" xref="S6.SS1.p2.5.m5.1.1.2"><ci id="S6.SS1.p2.5.m5.1.1.2.1.cmml" xref="S6.SS1.p2.5.m5.1.1.2.1">~</ci><ci id="S6.SS1.p2.5.m5.1.1.2.2.cmml" xref="S6.SS1.p2.5.m5.1.1.2.2">𝑎</ci></apply><apply id="S6.SS1.p2.5.m5.1.1.3.cmml" xref="S6.SS1.p2.5.m5.1.1.3"><csymbol cd="ambiguous" id="S6.SS1.p2.5.m5.1.1.3.1.cmml" xref="S6.SS1.p2.5.m5.1.1.3">superscript</csymbol><cn id="S6.SS1.p2.5.m5.1.1.3.2.cmml" type="integer" xref="S6.SS1.p2.5.m5.1.1.3.2">2</cn><apply id="S6.SS1.p2.5.m5.1.1.3.3.cmml" xref="S6.SS1.p2.5.m5.1.1.3.3"><csymbol cd="ambiguous" id="S6.SS1.p2.5.m5.1.1.3.3.1.cmml" xref="S6.SS1.p2.5.m5.1.1.3.3">subscript</csymbol><ci id="S6.SS1.p2.5.m5.1.1.3.3.2.cmml" xref="S6.SS1.p2.5.m5.1.1.3.3.2">𝜔</ci><cn id="S6.SS1.p2.5.m5.1.1.3.3.3.cmml" type="integer" xref="S6.SS1.p2.5.m5.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p2.5.m5.1c">\tilde{a}\in 2^{\omega_{1}}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p2.5.m5.1d">over~ start_ARG italic_a end_ARG ∈ 2 start_POSTSUPERSCRIPT italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> be the characteristic function of <math alttext="\{b\in A:b<_{A}a\}" class="ltx_Math" display="inline" id="S6.SS1.p2.6.m6.2"><semantics id="S6.SS1.p2.6.m6.2a"><mrow id="S6.SS1.p2.6.m6.2.2.2" xref="S6.SS1.p2.6.m6.2.2.3.cmml"><mo id="S6.SS1.p2.6.m6.2.2.2.3" stretchy="false" xref="S6.SS1.p2.6.m6.2.2.3.1.cmml">{</mo><mrow id="S6.SS1.p2.6.m6.1.1.1.1" xref="S6.SS1.p2.6.m6.1.1.1.1.cmml"><mi id="S6.SS1.p2.6.m6.1.1.1.1.2" xref="S6.SS1.p2.6.m6.1.1.1.1.2.cmml">b</mi><mo id="S6.SS1.p2.6.m6.1.1.1.1.1" xref="S6.SS1.p2.6.m6.1.1.1.1.1.cmml">∈</mo><mi id="S6.SS1.p2.6.m6.1.1.1.1.3" xref="S6.SS1.p2.6.m6.1.1.1.1.3.cmml">A</mi></mrow><mo id="S6.SS1.p2.6.m6.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S6.SS1.p2.6.m6.2.2.3.1.cmml">:</mo><mrow id="S6.SS1.p2.6.m6.2.2.2.2" xref="S6.SS1.p2.6.m6.2.2.2.2.cmml"><mi id="S6.SS1.p2.6.m6.2.2.2.2.2" xref="S6.SS1.p2.6.m6.2.2.2.2.2.cmml">b</mi><msub id="S6.SS1.p2.6.m6.2.2.2.2.1" xref="S6.SS1.p2.6.m6.2.2.2.2.1.cmml"><mo id="S6.SS1.p2.6.m6.2.2.2.2.1.2" xref="S6.SS1.p2.6.m6.2.2.2.2.1.2.cmml"><</mo><mi id="S6.SS1.p2.6.m6.2.2.2.2.1.3" xref="S6.SS1.p2.6.m6.2.2.2.2.1.3.cmml">A</mi></msub><mi id="S6.SS1.p2.6.m6.2.2.2.2.3" xref="S6.SS1.p2.6.m6.2.2.2.2.3.cmml">a</mi></mrow><mo id="S6.SS1.p2.6.m6.2.2.2.5" stretchy="false" xref="S6.SS1.p2.6.m6.2.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p2.6.m6.2b"><apply id="S6.SS1.p2.6.m6.2.2.3.cmml" xref="S6.SS1.p2.6.m6.2.2.2"><csymbol cd="latexml" id="S6.SS1.p2.6.m6.2.2.3.1.cmml" xref="S6.SS1.p2.6.m6.2.2.2.3">conditional-set</csymbol><apply id="S6.SS1.p2.6.m6.1.1.1.1.cmml" xref="S6.SS1.p2.6.m6.1.1.1.1"><in id="S6.SS1.p2.6.m6.1.1.1.1.1.cmml" xref="S6.SS1.p2.6.m6.1.1.1.1.1"></in><ci id="S6.SS1.p2.6.m6.1.1.1.1.2.cmml" xref="S6.SS1.p2.6.m6.1.1.1.1.2">𝑏</ci><ci id="S6.SS1.p2.6.m6.1.1.1.1.3.cmml" xref="S6.SS1.p2.6.m6.1.1.1.1.3">𝐴</ci></apply><apply id="S6.SS1.p2.6.m6.2.2.2.2.cmml" xref="S6.SS1.p2.6.m6.2.2.2.2"><apply id="S6.SS1.p2.6.m6.2.2.2.2.1.cmml" xref="S6.SS1.p2.6.m6.2.2.2.2.1"><csymbol cd="ambiguous" id="S6.SS1.p2.6.m6.2.2.2.2.1.1.cmml" xref="S6.SS1.p2.6.m6.2.2.2.2.1">subscript</csymbol><lt id="S6.SS1.p2.6.m6.2.2.2.2.1.2.cmml" xref="S6.SS1.p2.6.m6.2.2.2.2.1.2"></lt><ci id="S6.SS1.p2.6.m6.2.2.2.2.1.3.cmml" xref="S6.SS1.p2.6.m6.2.2.2.2.1.3">𝐴</ci></apply><ci id="S6.SS1.p2.6.m6.2.2.2.2.2.cmml" xref="S6.SS1.p2.6.m6.2.2.2.2.2">𝑏</ci><ci id="S6.SS1.p2.6.m6.2.2.2.2.3.cmml" xref="S6.SS1.p2.6.m6.2.2.2.2.3">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p2.6.m6.2c">\{b\in A:b<_{A}a\}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p2.6.m6.2d">{ italic_b ∈ italic_A : italic_b < start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_a }</annotation></semantics></math>. One easily sees that <math alttext="a\mapsto\tilde{a}" class="ltx_Math" display="inline" id="S6.SS1.p2.7.m7.1"><semantics id="S6.SS1.p2.7.m7.1a"><mrow id="S6.SS1.p2.7.m7.1.1" xref="S6.SS1.p2.7.m7.1.1.cmml"><mi id="S6.SS1.p2.7.m7.1.1.2" xref="S6.SS1.p2.7.m7.1.1.2.cmml">a</mi><mo id="S6.SS1.p2.7.m7.1.1.1" stretchy="false" xref="S6.SS1.p2.7.m7.1.1.1.cmml">↦</mo><mover accent="true" id="S6.SS1.p2.7.m7.1.1.3" xref="S6.SS1.p2.7.m7.1.1.3.cmml"><mi id="S6.SS1.p2.7.m7.1.1.3.2" xref="S6.SS1.p2.7.m7.1.1.3.2.cmml">a</mi><mo id="S6.SS1.p2.7.m7.1.1.3.1" xref="S6.SS1.p2.7.m7.1.1.3.1.cmml">~</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p2.7.m7.1b"><apply id="S6.SS1.p2.7.m7.1.1.cmml" xref="S6.SS1.p2.7.m7.1.1"><csymbol cd="latexml" id="S6.SS1.p2.7.m7.1.1.1.cmml" xref="S6.SS1.p2.7.m7.1.1.1">maps-to</csymbol><ci id="S6.SS1.p2.7.m7.1.1.2.cmml" xref="S6.SS1.p2.7.m7.1.1.2">𝑎</ci><apply id="S6.SS1.p2.7.m7.1.1.3.cmml" xref="S6.SS1.p2.7.m7.1.1.3"><ci id="S6.SS1.p2.7.m7.1.1.3.1.cmml" xref="S6.SS1.p2.7.m7.1.1.3.1">~</ci><ci id="S6.SS1.p2.7.m7.1.1.3.2.cmml" xref="S6.SS1.p2.7.m7.1.1.3.2">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p2.7.m7.1c">a\mapsto\tilde{a}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p2.7.m7.1d">italic_a ↦ over~ start_ARG italic_a end_ARG</annotation></semantics></math> is an isomorphism between <math alttext="A" class="ltx_Math" display="inline" id="S6.SS1.p2.8.m8.1"><semantics id="S6.SS1.p2.8.m8.1a"><mi id="S6.SS1.p2.8.m8.1.1" xref="S6.SS1.p2.8.m8.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S6.SS1.p2.8.m8.1b"><ci id="S6.SS1.p2.8.m8.1.1.cmml" xref="S6.SS1.p2.8.m8.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p2.8.m8.1c">A</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p2.8.m8.1d">italic_A</annotation></semantics></math> and a subset of <math alttext="2^{\omega_{1}}" class="ltx_Math" display="inline" id="S6.SS1.p2.9.m9.1"><semantics id="S6.SS1.p2.9.m9.1a"><msup id="S6.SS1.p2.9.m9.1.1" xref="S6.SS1.p2.9.m9.1.1.cmml"><mn id="S6.SS1.p2.9.m9.1.1.2" xref="S6.SS1.p2.9.m9.1.1.2.cmml">2</mn><msub id="S6.SS1.p2.9.m9.1.1.3" xref="S6.SS1.p2.9.m9.1.1.3.cmml"><mi id="S6.SS1.p2.9.m9.1.1.3.2" xref="S6.SS1.p2.9.m9.1.1.3.2.cmml">ω</mi><mn id="S6.SS1.p2.9.m9.1.1.3.3" xref="S6.SS1.p2.9.m9.1.1.3.3.cmml">1</mn></msub></msup><annotation-xml encoding="MathML-Content" id="S6.SS1.p2.9.m9.1b"><apply id="S6.SS1.p2.9.m9.1.1.cmml" xref="S6.SS1.p2.9.m9.1.1"><csymbol cd="ambiguous" id="S6.SS1.p2.9.m9.1.1.1.cmml" xref="S6.SS1.p2.9.m9.1.1">superscript</csymbol><cn id="S6.SS1.p2.9.m9.1.1.2.cmml" type="integer" xref="S6.SS1.p2.9.m9.1.1.2">2</cn><apply id="S6.SS1.p2.9.m9.1.1.3.cmml" xref="S6.SS1.p2.9.m9.1.1.3"><csymbol cd="ambiguous" id="S6.SS1.p2.9.m9.1.1.3.1.cmml" xref="S6.SS1.p2.9.m9.1.1.3">subscript</csymbol><ci id="S6.SS1.p2.9.m9.1.1.3.2.cmml" xref="S6.SS1.p2.9.m9.1.1.3.2">𝜔</ci><cn id="S6.SS1.p2.9.m9.1.1.3.3.cmml" type="integer" xref="S6.SS1.p2.9.m9.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p2.9.m9.1c">2^{\omega_{1}}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p2.9.m9.1d">2 start_POSTSUPERSCRIPT italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> ordered lexicographical. For <math alttext="a\neq b" class="ltx_Math" display="inline" id="S6.SS1.p2.10.m10.1"><semantics id="S6.SS1.p2.10.m10.1a"><mrow id="S6.SS1.p2.10.m10.1.1" xref="S6.SS1.p2.10.m10.1.1.cmml"><mi id="S6.SS1.p2.10.m10.1.1.2" xref="S6.SS1.p2.10.m10.1.1.2.cmml">a</mi><mo id="S6.SS1.p2.10.m10.1.1.1" xref="S6.SS1.p2.10.m10.1.1.1.cmml">≠</mo><mi id="S6.SS1.p2.10.m10.1.1.3" xref="S6.SS1.p2.10.m10.1.1.3.cmml">b</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p2.10.m10.1b"><apply id="S6.SS1.p2.10.m10.1.1.cmml" xref="S6.SS1.p2.10.m10.1.1"><neq id="S6.SS1.p2.10.m10.1.1.1.cmml" xref="S6.SS1.p2.10.m10.1.1.1"></neq><ci id="S6.SS1.p2.10.m10.1.1.2.cmml" xref="S6.SS1.p2.10.m10.1.1.2">𝑎</ci><ci id="S6.SS1.p2.10.m10.1.1.3.cmml" xref="S6.SS1.p2.10.m10.1.1.3">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p2.10.m10.1c">a\neq b</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p2.10.m10.1d">italic_a ≠ italic_b</annotation></semantics></math> in <math alttext="A" class="ltx_Math" display="inline" id="S6.SS1.p2.11.m11.1"><semantics id="S6.SS1.p2.11.m11.1a"><mi id="S6.SS1.p2.11.m11.1.1" xref="S6.SS1.p2.11.m11.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S6.SS1.p2.11.m11.1b"><ci id="S6.SS1.p2.11.m11.1.1.cmml" xref="S6.SS1.p2.11.m11.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p2.11.m11.1c">A</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p2.11.m11.1d">italic_A</annotation></semantics></math>, let <math alttext="\Delta_{A}(a,b)" class="ltx_Math" display="inline" id="S6.SS1.p2.12.m12.2"><semantics id="S6.SS1.p2.12.m12.2a"><mrow id="S6.SS1.p2.12.m12.2.3" xref="S6.SS1.p2.12.m12.2.3.cmml"><msub id="S6.SS1.p2.12.m12.2.3.2" xref="S6.SS1.p2.12.m12.2.3.2.cmml"><mi id="S6.SS1.p2.12.m12.2.3.2.2" mathvariant="normal" xref="S6.SS1.p2.12.m12.2.3.2.2.cmml">Δ</mi><mi id="S6.SS1.p2.12.m12.2.3.2.3" xref="S6.SS1.p2.12.m12.2.3.2.3.cmml">A</mi></msub><mo id="S6.SS1.p2.12.m12.2.3.1" xref="S6.SS1.p2.12.m12.2.3.1.cmml">⁢</mo><mrow id="S6.SS1.p2.12.m12.2.3.3.2" xref="S6.SS1.p2.12.m12.2.3.3.1.cmml"><mo id="S6.SS1.p2.12.m12.2.3.3.2.1" stretchy="false" xref="S6.SS1.p2.12.m12.2.3.3.1.cmml">(</mo><mi id="S6.SS1.p2.12.m12.1.1" xref="S6.SS1.p2.12.m12.1.1.cmml">a</mi><mo id="S6.SS1.p2.12.m12.2.3.3.2.2" xref="S6.SS1.p2.12.m12.2.3.3.1.cmml">,</mo><mi id="S6.SS1.p2.12.m12.2.2" xref="S6.SS1.p2.12.m12.2.2.cmml">b</mi><mo id="S6.SS1.p2.12.m12.2.3.3.2.3" stretchy="false" xref="S6.SS1.p2.12.m12.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p2.12.m12.2b"><apply id="S6.SS1.p2.12.m12.2.3.cmml" xref="S6.SS1.p2.12.m12.2.3"><times id="S6.SS1.p2.12.m12.2.3.1.cmml" xref="S6.SS1.p2.12.m12.2.3.1"></times><apply id="S6.SS1.p2.12.m12.2.3.2.cmml" xref="S6.SS1.p2.12.m12.2.3.2"><csymbol cd="ambiguous" id="S6.SS1.p2.12.m12.2.3.2.1.cmml" xref="S6.SS1.p2.12.m12.2.3.2">subscript</csymbol><ci id="S6.SS1.p2.12.m12.2.3.2.2.cmml" xref="S6.SS1.p2.12.m12.2.3.2.2">Δ</ci><ci id="S6.SS1.p2.12.m12.2.3.2.3.cmml" xref="S6.SS1.p2.12.m12.2.3.2.3">𝐴</ci></apply><interval closure="open" id="S6.SS1.p2.12.m12.2.3.3.1.cmml" xref="S6.SS1.p2.12.m12.2.3.3.2"><ci id="S6.SS1.p2.12.m12.1.1.cmml" xref="S6.SS1.p2.12.m12.1.1">𝑎</ci><ci id="S6.SS1.p2.12.m12.2.2.cmml" xref="S6.SS1.p2.12.m12.2.2">𝑏</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p2.12.m12.2c">\Delta_{A}(a,b)</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p2.12.m12.2d">roman_Δ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_a , italic_b )</annotation></semantics></math> be the least ordinal <math alttext="\xi" class="ltx_Math" display="inline" id="S6.SS1.p2.13.m13.1"><semantics id="S6.SS1.p2.13.m13.1a"><mi id="S6.SS1.p2.13.m13.1.1" xref="S6.SS1.p2.13.m13.1.1.cmml">ξ</mi><annotation-xml encoding="MathML-Content" id="S6.SS1.p2.13.m13.1b"><ci id="S6.SS1.p2.13.m13.1.1.cmml" xref="S6.SS1.p2.13.m13.1.1">𝜉</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p2.13.m13.1c">\xi</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p2.13.m13.1d">italic_ξ</annotation></semantics></math> such that <math alttext="\tilde{a}(\xi)\neq\tilde{b}(\xi)" class="ltx_Math" display="inline" id="S6.SS1.p2.14.m14.2"><semantics id="S6.SS1.p2.14.m14.2a"><mrow id="S6.SS1.p2.14.m14.2.3" xref="S6.SS1.p2.14.m14.2.3.cmml"><mrow id="S6.SS1.p2.14.m14.2.3.2" xref="S6.SS1.p2.14.m14.2.3.2.cmml"><mover accent="true" id="S6.SS1.p2.14.m14.2.3.2.2" xref="S6.SS1.p2.14.m14.2.3.2.2.cmml"><mi id="S6.SS1.p2.14.m14.2.3.2.2.2" xref="S6.SS1.p2.14.m14.2.3.2.2.2.cmml">a</mi><mo id="S6.SS1.p2.14.m14.2.3.2.2.1" xref="S6.SS1.p2.14.m14.2.3.2.2.1.cmml">~</mo></mover><mo id="S6.SS1.p2.14.m14.2.3.2.1" xref="S6.SS1.p2.14.m14.2.3.2.1.cmml">⁢</mo><mrow id="S6.SS1.p2.14.m14.2.3.2.3.2" xref="S6.SS1.p2.14.m14.2.3.2.cmml"><mo id="S6.SS1.p2.14.m14.2.3.2.3.2.1" stretchy="false" xref="S6.SS1.p2.14.m14.2.3.2.cmml">(</mo><mi id="S6.SS1.p2.14.m14.1.1" xref="S6.SS1.p2.14.m14.1.1.cmml">ξ</mi><mo id="S6.SS1.p2.14.m14.2.3.2.3.2.2" stretchy="false" xref="S6.SS1.p2.14.m14.2.3.2.cmml">)</mo></mrow></mrow><mo id="S6.SS1.p2.14.m14.2.3.1" xref="S6.SS1.p2.14.m14.2.3.1.cmml">≠</mo><mrow id="S6.SS1.p2.14.m14.2.3.3" xref="S6.SS1.p2.14.m14.2.3.3.cmml"><mover accent="true" id="S6.SS1.p2.14.m14.2.3.3.2" xref="S6.SS1.p2.14.m14.2.3.3.2.cmml"><mi id="S6.SS1.p2.14.m14.2.3.3.2.2" xref="S6.SS1.p2.14.m14.2.3.3.2.2.cmml">b</mi><mo id="S6.SS1.p2.14.m14.2.3.3.2.1" xref="S6.SS1.p2.14.m14.2.3.3.2.1.cmml">~</mo></mover><mo id="S6.SS1.p2.14.m14.2.3.3.1" xref="S6.SS1.p2.14.m14.2.3.3.1.cmml">⁢</mo><mrow id="S6.SS1.p2.14.m14.2.3.3.3.2" xref="S6.SS1.p2.14.m14.2.3.3.cmml"><mo id="S6.SS1.p2.14.m14.2.3.3.3.2.1" stretchy="false" xref="S6.SS1.p2.14.m14.2.3.3.cmml">(</mo><mi id="S6.SS1.p2.14.m14.2.2" xref="S6.SS1.p2.14.m14.2.2.cmml">ξ</mi><mo id="S6.SS1.p2.14.m14.2.3.3.3.2.2" stretchy="false" xref="S6.SS1.p2.14.m14.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p2.14.m14.2b"><apply id="S6.SS1.p2.14.m14.2.3.cmml" xref="S6.SS1.p2.14.m14.2.3"><neq id="S6.SS1.p2.14.m14.2.3.1.cmml" xref="S6.SS1.p2.14.m14.2.3.1"></neq><apply id="S6.SS1.p2.14.m14.2.3.2.cmml" xref="S6.SS1.p2.14.m14.2.3.2"><times id="S6.SS1.p2.14.m14.2.3.2.1.cmml" xref="S6.SS1.p2.14.m14.2.3.2.1"></times><apply id="S6.SS1.p2.14.m14.2.3.2.2.cmml" xref="S6.SS1.p2.14.m14.2.3.2.2"><ci id="S6.SS1.p2.14.m14.2.3.2.2.1.cmml" xref="S6.SS1.p2.14.m14.2.3.2.2.1">~</ci><ci id="S6.SS1.p2.14.m14.2.3.2.2.2.cmml" xref="S6.SS1.p2.14.m14.2.3.2.2.2">𝑎</ci></apply><ci id="S6.SS1.p2.14.m14.1.1.cmml" xref="S6.SS1.p2.14.m14.1.1">𝜉</ci></apply><apply id="S6.SS1.p2.14.m14.2.3.3.cmml" xref="S6.SS1.p2.14.m14.2.3.3"><times id="S6.SS1.p2.14.m14.2.3.3.1.cmml" xref="S6.SS1.p2.14.m14.2.3.3.1"></times><apply id="S6.SS1.p2.14.m14.2.3.3.2.cmml" xref="S6.SS1.p2.14.m14.2.3.3.2"><ci id="S6.SS1.p2.14.m14.2.3.3.2.1.cmml" xref="S6.SS1.p2.14.m14.2.3.3.2.1">~</ci><ci id="S6.SS1.p2.14.m14.2.3.3.2.2.cmml" xref="S6.SS1.p2.14.m14.2.3.3.2.2">𝑏</ci></apply><ci id="S6.SS1.p2.14.m14.2.2.cmml" xref="S6.SS1.p2.14.m14.2.2">𝜉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p2.14.m14.2c">\tilde{a}(\xi)\neq\tilde{b}(\xi)</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p2.14.m14.2d">over~ start_ARG italic_a end_ARG ( italic_ξ ) ≠ over~ start_ARG italic_b end_ARG ( italic_ξ )</annotation></semantics></math>, and note then that <math alttext="a<_{A}b" class="ltx_Math" display="inline" id="S6.SS1.p2.15.m15.1"><semantics id="S6.SS1.p2.15.m15.1a"><mrow id="S6.SS1.p2.15.m15.1.1" xref="S6.SS1.p2.15.m15.1.1.cmml"><mi id="S6.SS1.p2.15.m15.1.1.2" xref="S6.SS1.p2.15.m15.1.1.2.cmml">a</mi><msub id="S6.SS1.p2.15.m15.1.1.1" xref="S6.SS1.p2.15.m15.1.1.1.cmml"><mo id="S6.SS1.p2.15.m15.1.1.1.2" xref="S6.SS1.p2.15.m15.1.1.1.2.cmml"><</mo><mi id="S6.SS1.p2.15.m15.1.1.1.3" xref="S6.SS1.p2.15.m15.1.1.1.3.cmml">A</mi></msub><mi id="S6.SS1.p2.15.m15.1.1.3" xref="S6.SS1.p2.15.m15.1.1.3.cmml">b</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p2.15.m15.1b"><apply id="S6.SS1.p2.15.m15.1.1.cmml" xref="S6.SS1.p2.15.m15.1.1"><apply id="S6.SS1.p2.15.m15.1.1.1.cmml" xref="S6.SS1.p2.15.m15.1.1.1"><csymbol cd="ambiguous" id="S6.SS1.p2.15.m15.1.1.1.1.cmml" xref="S6.SS1.p2.15.m15.1.1.1">subscript</csymbol><lt id="S6.SS1.p2.15.m15.1.1.1.2.cmml" xref="S6.SS1.p2.15.m15.1.1.1.2"></lt><ci id="S6.SS1.p2.15.m15.1.1.1.3.cmml" xref="S6.SS1.p2.15.m15.1.1.1.3">𝐴</ci></apply><ci id="S6.SS1.p2.15.m15.1.1.2.cmml" xref="S6.SS1.p2.15.m15.1.1.2">𝑎</ci><ci id="S6.SS1.p2.15.m15.1.1.3.cmml" xref="S6.SS1.p2.15.m15.1.1.3">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p2.15.m15.1c">a<_{A}b</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p2.15.m15.1d">italic_a < start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_b</annotation></semantics></math> iff <math alttext="\tilde{a}(\Delta_{A}(a,b))=0" class="ltx_Math" display="inline" id="S6.SS1.p2.16.m16.3"><semantics id="S6.SS1.p2.16.m16.3a"><mrow id="S6.SS1.p2.16.m16.3.3" xref="S6.SS1.p2.16.m16.3.3.cmml"><mrow id="S6.SS1.p2.16.m16.3.3.1" xref="S6.SS1.p2.16.m16.3.3.1.cmml"><mover accent="true" id="S6.SS1.p2.16.m16.3.3.1.3" xref="S6.SS1.p2.16.m16.3.3.1.3.cmml"><mi id="S6.SS1.p2.16.m16.3.3.1.3.2" xref="S6.SS1.p2.16.m16.3.3.1.3.2.cmml">a</mi><mo id="S6.SS1.p2.16.m16.3.3.1.3.1" xref="S6.SS1.p2.16.m16.3.3.1.3.1.cmml">~</mo></mover><mo id="S6.SS1.p2.16.m16.3.3.1.2" xref="S6.SS1.p2.16.m16.3.3.1.2.cmml">⁢</mo><mrow id="S6.SS1.p2.16.m16.3.3.1.1.1" xref="S6.SS1.p2.16.m16.3.3.1.1.1.1.cmml"><mo id="S6.SS1.p2.16.m16.3.3.1.1.1.2" stretchy="false" xref="S6.SS1.p2.16.m16.3.3.1.1.1.1.cmml">(</mo><mrow id="S6.SS1.p2.16.m16.3.3.1.1.1.1" xref="S6.SS1.p2.16.m16.3.3.1.1.1.1.cmml"><msub id="S6.SS1.p2.16.m16.3.3.1.1.1.1.2" xref="S6.SS1.p2.16.m16.3.3.1.1.1.1.2.cmml"><mi id="S6.SS1.p2.16.m16.3.3.1.1.1.1.2.2" mathvariant="normal" xref="S6.SS1.p2.16.m16.3.3.1.1.1.1.2.2.cmml">Δ</mi><mi id="S6.SS1.p2.16.m16.3.3.1.1.1.1.2.3" xref="S6.SS1.p2.16.m16.3.3.1.1.1.1.2.3.cmml">A</mi></msub><mo id="S6.SS1.p2.16.m16.3.3.1.1.1.1.1" xref="S6.SS1.p2.16.m16.3.3.1.1.1.1.1.cmml">⁢</mo><mrow id="S6.SS1.p2.16.m16.3.3.1.1.1.1.3.2" xref="S6.SS1.p2.16.m16.3.3.1.1.1.1.3.1.cmml"><mo id="S6.SS1.p2.16.m16.3.3.1.1.1.1.3.2.1" stretchy="false" xref="S6.SS1.p2.16.m16.3.3.1.1.1.1.3.1.cmml">(</mo><mi id="S6.SS1.p2.16.m16.1.1" xref="S6.SS1.p2.16.m16.1.1.cmml">a</mi><mo id="S6.SS1.p2.16.m16.3.3.1.1.1.1.3.2.2" xref="S6.SS1.p2.16.m16.3.3.1.1.1.1.3.1.cmml">,</mo><mi id="S6.SS1.p2.16.m16.2.2" xref="S6.SS1.p2.16.m16.2.2.cmml">b</mi><mo id="S6.SS1.p2.16.m16.3.3.1.1.1.1.3.2.3" stretchy="false" xref="S6.SS1.p2.16.m16.3.3.1.1.1.1.3.1.cmml">)</mo></mrow></mrow><mo id="S6.SS1.p2.16.m16.3.3.1.1.1.3" stretchy="false" xref="S6.SS1.p2.16.m16.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.SS1.p2.16.m16.3.3.2" xref="S6.SS1.p2.16.m16.3.3.2.cmml">=</mo><mn id="S6.SS1.p2.16.m16.3.3.3" xref="S6.SS1.p2.16.m16.3.3.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p2.16.m16.3b"><apply id="S6.SS1.p2.16.m16.3.3.cmml" xref="S6.SS1.p2.16.m16.3.3"><eq id="S6.SS1.p2.16.m16.3.3.2.cmml" xref="S6.SS1.p2.16.m16.3.3.2"></eq><apply id="S6.SS1.p2.16.m16.3.3.1.cmml" xref="S6.SS1.p2.16.m16.3.3.1"><times id="S6.SS1.p2.16.m16.3.3.1.2.cmml" xref="S6.SS1.p2.16.m16.3.3.1.2"></times><apply id="S6.SS1.p2.16.m16.3.3.1.3.cmml" xref="S6.SS1.p2.16.m16.3.3.1.3"><ci id="S6.SS1.p2.16.m16.3.3.1.3.1.cmml" xref="S6.SS1.p2.16.m16.3.3.1.3.1">~</ci><ci id="S6.SS1.p2.16.m16.3.3.1.3.2.cmml" xref="S6.SS1.p2.16.m16.3.3.1.3.2">𝑎</ci></apply><apply id="S6.SS1.p2.16.m16.3.3.1.1.1.1.cmml" xref="S6.SS1.p2.16.m16.3.3.1.1.1"><times id="S6.SS1.p2.16.m16.3.3.1.1.1.1.1.cmml" xref="S6.SS1.p2.16.m16.3.3.1.1.1.1.1"></times><apply id="S6.SS1.p2.16.m16.3.3.1.1.1.1.2.cmml" xref="S6.SS1.p2.16.m16.3.3.1.1.1.1.2"><csymbol cd="ambiguous" id="S6.SS1.p2.16.m16.3.3.1.1.1.1.2.1.cmml" xref="S6.SS1.p2.16.m16.3.3.1.1.1.1.2">subscript</csymbol><ci id="S6.SS1.p2.16.m16.3.3.1.1.1.1.2.2.cmml" xref="S6.SS1.p2.16.m16.3.3.1.1.1.1.2.2">Δ</ci><ci id="S6.SS1.p2.16.m16.3.3.1.1.1.1.2.3.cmml" xref="S6.SS1.p2.16.m16.3.3.1.1.1.1.2.3">𝐴</ci></apply><interval closure="open" id="S6.SS1.p2.16.m16.3.3.1.1.1.1.3.1.cmml" xref="S6.SS1.p2.16.m16.3.3.1.1.1.1.3.2"><ci id="S6.SS1.p2.16.m16.1.1.cmml" xref="S6.SS1.p2.16.m16.1.1">𝑎</ci><ci id="S6.SS1.p2.16.m16.2.2.cmml" xref="S6.SS1.p2.16.m16.2.2">𝑏</ci></interval></apply></apply><cn id="S6.SS1.p2.16.m16.3.3.3.cmml" type="integer" xref="S6.SS1.p2.16.m16.3.3.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p2.16.m16.3c">\tilde{a}(\Delta_{A}(a,b))=0</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p2.16.m16.3d">over~ start_ARG italic_a end_ARG ( roman_Δ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_a , italic_b ) ) = 0</annotation></semantics></math> and <math alttext="\tilde{b}(\Delta_{A}(a,b))=1" class="ltx_Math" display="inline" id="S6.SS1.p2.17.m17.3"><semantics id="S6.SS1.p2.17.m17.3a"><mrow id="S6.SS1.p2.17.m17.3.3" xref="S6.SS1.p2.17.m17.3.3.cmml"><mrow id="S6.SS1.p2.17.m17.3.3.1" xref="S6.SS1.p2.17.m17.3.3.1.cmml"><mover accent="true" id="S6.SS1.p2.17.m17.3.3.1.3" xref="S6.SS1.p2.17.m17.3.3.1.3.cmml"><mi id="S6.SS1.p2.17.m17.3.3.1.3.2" xref="S6.SS1.p2.17.m17.3.3.1.3.2.cmml">b</mi><mo id="S6.SS1.p2.17.m17.3.3.1.3.1" xref="S6.SS1.p2.17.m17.3.3.1.3.1.cmml">~</mo></mover><mo id="S6.SS1.p2.17.m17.3.3.1.2" xref="S6.SS1.p2.17.m17.3.3.1.2.cmml">⁢</mo><mrow id="S6.SS1.p2.17.m17.3.3.1.1.1" xref="S6.SS1.p2.17.m17.3.3.1.1.1.1.cmml"><mo id="S6.SS1.p2.17.m17.3.3.1.1.1.2" stretchy="false" xref="S6.SS1.p2.17.m17.3.3.1.1.1.1.cmml">(</mo><mrow id="S6.SS1.p2.17.m17.3.3.1.1.1.1" xref="S6.SS1.p2.17.m17.3.3.1.1.1.1.cmml"><msub id="S6.SS1.p2.17.m17.3.3.1.1.1.1.2" xref="S6.SS1.p2.17.m17.3.3.1.1.1.1.2.cmml"><mi id="S6.SS1.p2.17.m17.3.3.1.1.1.1.2.2" mathvariant="normal" xref="S6.SS1.p2.17.m17.3.3.1.1.1.1.2.2.cmml">Δ</mi><mi id="S6.SS1.p2.17.m17.3.3.1.1.1.1.2.3" xref="S6.SS1.p2.17.m17.3.3.1.1.1.1.2.3.cmml">A</mi></msub><mo id="S6.SS1.p2.17.m17.3.3.1.1.1.1.1" xref="S6.SS1.p2.17.m17.3.3.1.1.1.1.1.cmml">⁢</mo><mrow id="S6.SS1.p2.17.m17.3.3.1.1.1.1.3.2" xref="S6.SS1.p2.17.m17.3.3.1.1.1.1.3.1.cmml"><mo id="S6.SS1.p2.17.m17.3.3.1.1.1.1.3.2.1" stretchy="false" xref="S6.SS1.p2.17.m17.3.3.1.1.1.1.3.1.cmml">(</mo><mi id="S6.SS1.p2.17.m17.1.1" xref="S6.SS1.p2.17.m17.1.1.cmml">a</mi><mo id="S6.SS1.p2.17.m17.3.3.1.1.1.1.3.2.2" xref="S6.SS1.p2.17.m17.3.3.1.1.1.1.3.1.cmml">,</mo><mi id="S6.SS1.p2.17.m17.2.2" xref="S6.SS1.p2.17.m17.2.2.cmml">b</mi><mo id="S6.SS1.p2.17.m17.3.3.1.1.1.1.3.2.3" stretchy="false" xref="S6.SS1.p2.17.m17.3.3.1.1.1.1.3.1.cmml">)</mo></mrow></mrow><mo id="S6.SS1.p2.17.m17.3.3.1.1.1.3" stretchy="false" xref="S6.SS1.p2.17.m17.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.SS1.p2.17.m17.3.3.2" xref="S6.SS1.p2.17.m17.3.3.2.cmml">=</mo><mn id="S6.SS1.p2.17.m17.3.3.3" xref="S6.SS1.p2.17.m17.3.3.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p2.17.m17.3b"><apply id="S6.SS1.p2.17.m17.3.3.cmml" xref="S6.SS1.p2.17.m17.3.3"><eq id="S6.SS1.p2.17.m17.3.3.2.cmml" xref="S6.SS1.p2.17.m17.3.3.2"></eq><apply id="S6.SS1.p2.17.m17.3.3.1.cmml" xref="S6.SS1.p2.17.m17.3.3.1"><times id="S6.SS1.p2.17.m17.3.3.1.2.cmml" xref="S6.SS1.p2.17.m17.3.3.1.2"></times><apply id="S6.SS1.p2.17.m17.3.3.1.3.cmml" xref="S6.SS1.p2.17.m17.3.3.1.3"><ci id="S6.SS1.p2.17.m17.3.3.1.3.1.cmml" xref="S6.SS1.p2.17.m17.3.3.1.3.1">~</ci><ci id="S6.SS1.p2.17.m17.3.3.1.3.2.cmml" xref="S6.SS1.p2.17.m17.3.3.1.3.2">𝑏</ci></apply><apply id="S6.SS1.p2.17.m17.3.3.1.1.1.1.cmml" xref="S6.SS1.p2.17.m17.3.3.1.1.1"><times id="S6.SS1.p2.17.m17.3.3.1.1.1.1.1.cmml" xref="S6.SS1.p2.17.m17.3.3.1.1.1.1.1"></times><apply id="S6.SS1.p2.17.m17.3.3.1.1.1.1.2.cmml" xref="S6.SS1.p2.17.m17.3.3.1.1.1.1.2"><csymbol cd="ambiguous" id="S6.SS1.p2.17.m17.3.3.1.1.1.1.2.1.cmml" xref="S6.SS1.p2.17.m17.3.3.1.1.1.1.2">subscript</csymbol><ci id="S6.SS1.p2.17.m17.3.3.1.1.1.1.2.2.cmml" xref="S6.SS1.p2.17.m17.3.3.1.1.1.1.2.2">Δ</ci><ci id="S6.SS1.p2.17.m17.3.3.1.1.1.1.2.3.cmml" xref="S6.SS1.p2.17.m17.3.3.1.1.1.1.2.3">𝐴</ci></apply><interval closure="open" id="S6.SS1.p2.17.m17.3.3.1.1.1.1.3.1.cmml" xref="S6.SS1.p2.17.m17.3.3.1.1.1.1.3.2"><ci id="S6.SS1.p2.17.m17.1.1.cmml" xref="S6.SS1.p2.17.m17.1.1">𝑎</ci><ci id="S6.SS1.p2.17.m17.2.2.cmml" xref="S6.SS1.p2.17.m17.2.2">𝑏</ci></interval></apply></apply><cn id="S6.SS1.p2.17.m17.3.3.3.cmml" type="integer" xref="S6.SS1.p2.17.m17.3.3.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p2.17.m17.3c">\tilde{b}(\Delta_{A}(a,b))=1</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p2.17.m17.3d">over~ start_ARG italic_b end_ARG ( roman_Δ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_a , italic_b ) ) = 1</annotation></semantics></math>. More generally, the following are easily seen to be true.</p> </div> <div class="ltx_para" id="S6.SS1.p3"> <ul class="ltx_itemize" id="S6.I1"> <li class="ltx_item" id="S6.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S6.I1.i1.p1"> <p class="ltx_p" id="S6.I1.i1.p1.4">If <math alttext="a<_{A}b" class="ltx_Math" display="inline" id="S6.I1.i1.p1.1.m1.1"><semantics id="S6.I1.i1.p1.1.m1.1a"><mrow id="S6.I1.i1.p1.1.m1.1.1" xref="S6.I1.i1.p1.1.m1.1.1.cmml"><mi id="S6.I1.i1.p1.1.m1.1.1.2" xref="S6.I1.i1.p1.1.m1.1.1.2.cmml">a</mi><msub id="S6.I1.i1.p1.1.m1.1.1.1" xref="S6.I1.i1.p1.1.m1.1.1.1.cmml"><mo id="S6.I1.i1.p1.1.m1.1.1.1.2" xref="S6.I1.i1.p1.1.m1.1.1.1.2.cmml"><</mo><mi id="S6.I1.i1.p1.1.m1.1.1.1.3" xref="S6.I1.i1.p1.1.m1.1.1.1.3.cmml">A</mi></msub><mi id="S6.I1.i1.p1.1.m1.1.1.3" xref="S6.I1.i1.p1.1.m1.1.1.3.cmml">b</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.I1.i1.p1.1.m1.1b"><apply id="S6.I1.i1.p1.1.m1.1.1.cmml" xref="S6.I1.i1.p1.1.m1.1.1"><apply id="S6.I1.i1.p1.1.m1.1.1.1.cmml" xref="S6.I1.i1.p1.1.m1.1.1.1"><csymbol cd="ambiguous" id="S6.I1.i1.p1.1.m1.1.1.1.1.cmml" xref="S6.I1.i1.p1.1.m1.1.1.1">subscript</csymbol><lt id="S6.I1.i1.p1.1.m1.1.1.1.2.cmml" xref="S6.I1.i1.p1.1.m1.1.1.1.2"></lt><ci id="S6.I1.i1.p1.1.m1.1.1.1.3.cmml" xref="S6.I1.i1.p1.1.m1.1.1.1.3">𝐴</ci></apply><ci id="S6.I1.i1.p1.1.m1.1.1.2.cmml" xref="S6.I1.i1.p1.1.m1.1.1.2">𝑎</ci><ci id="S6.I1.i1.p1.1.m1.1.1.3.cmml" xref="S6.I1.i1.p1.1.m1.1.1.3">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I1.i1.p1.1.m1.1c">a<_{A}b</annotation><annotation encoding="application/x-llamapun" id="S6.I1.i1.p1.1.m1.1d">italic_a < start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_b</annotation></semantics></math>, then <math alttext="\Delta_{A}(a,b)" class="ltx_Math" display="inline" id="S6.I1.i1.p1.2.m2.2"><semantics id="S6.I1.i1.p1.2.m2.2a"><mrow id="S6.I1.i1.p1.2.m2.2.3" xref="S6.I1.i1.p1.2.m2.2.3.cmml"><msub id="S6.I1.i1.p1.2.m2.2.3.2" xref="S6.I1.i1.p1.2.m2.2.3.2.cmml"><mi id="S6.I1.i1.p1.2.m2.2.3.2.2" mathvariant="normal" xref="S6.I1.i1.p1.2.m2.2.3.2.2.cmml">Δ</mi><mi id="S6.I1.i1.p1.2.m2.2.3.2.3" xref="S6.I1.i1.p1.2.m2.2.3.2.3.cmml">A</mi></msub><mo id="S6.I1.i1.p1.2.m2.2.3.1" xref="S6.I1.i1.p1.2.m2.2.3.1.cmml">⁢</mo><mrow id="S6.I1.i1.p1.2.m2.2.3.3.2" xref="S6.I1.i1.p1.2.m2.2.3.3.1.cmml"><mo id="S6.I1.i1.p1.2.m2.2.3.3.2.1" stretchy="false" xref="S6.I1.i1.p1.2.m2.2.3.3.1.cmml">(</mo><mi id="S6.I1.i1.p1.2.m2.1.1" xref="S6.I1.i1.p1.2.m2.1.1.cmml">a</mi><mo id="S6.I1.i1.p1.2.m2.2.3.3.2.2" xref="S6.I1.i1.p1.2.m2.2.3.3.1.cmml">,</mo><mi id="S6.I1.i1.p1.2.m2.2.2" xref="S6.I1.i1.p1.2.m2.2.2.cmml">b</mi><mo id="S6.I1.i1.p1.2.m2.2.3.3.2.3" stretchy="false" xref="S6.I1.i1.p1.2.m2.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.I1.i1.p1.2.m2.2b"><apply id="S6.I1.i1.p1.2.m2.2.3.cmml" xref="S6.I1.i1.p1.2.m2.2.3"><times id="S6.I1.i1.p1.2.m2.2.3.1.cmml" xref="S6.I1.i1.p1.2.m2.2.3.1"></times><apply id="S6.I1.i1.p1.2.m2.2.3.2.cmml" xref="S6.I1.i1.p1.2.m2.2.3.2"><csymbol cd="ambiguous" id="S6.I1.i1.p1.2.m2.2.3.2.1.cmml" xref="S6.I1.i1.p1.2.m2.2.3.2">subscript</csymbol><ci id="S6.I1.i1.p1.2.m2.2.3.2.2.cmml" xref="S6.I1.i1.p1.2.m2.2.3.2.2">Δ</ci><ci id="S6.I1.i1.p1.2.m2.2.3.2.3.cmml" xref="S6.I1.i1.p1.2.m2.2.3.2.3">𝐴</ci></apply><interval closure="open" id="S6.I1.i1.p1.2.m2.2.3.3.1.cmml" xref="S6.I1.i1.p1.2.m2.2.3.3.2"><ci id="S6.I1.i1.p1.2.m2.1.1.cmml" xref="S6.I1.i1.p1.2.m2.1.1">𝑎</ci><ci id="S6.I1.i1.p1.2.m2.2.2.cmml" xref="S6.I1.i1.p1.2.m2.2.2">𝑏</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I1.i1.p1.2.m2.2c">\Delta_{A}(a,b)</annotation><annotation encoding="application/x-llamapun" id="S6.I1.i1.p1.2.m2.2d">roman_Δ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_a , italic_b )</annotation></semantics></math> is the least ordinal <math alttext="\xi" class="ltx_Math" display="inline" id="S6.I1.i1.p1.3.m3.1"><semantics id="S6.I1.i1.p1.3.m3.1a"><mi id="S6.I1.i1.p1.3.m3.1.1" xref="S6.I1.i1.p1.3.m3.1.1.cmml">ξ</mi><annotation-xml encoding="MathML-Content" id="S6.I1.i1.p1.3.m3.1b"><ci id="S6.I1.i1.p1.3.m3.1.1.cmml" xref="S6.I1.i1.p1.3.m3.1.1">𝜉</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.I1.i1.p1.3.m3.1c">\xi</annotation><annotation encoding="application/x-llamapun" id="S6.I1.i1.p1.3.m3.1d">italic_ξ</annotation></semantics></math> such that <math alttext="a\leq_{A}\xi<_{A}b" class="ltx_Math" display="inline" id="S6.I1.i1.p1.4.m4.1"><semantics id="S6.I1.i1.p1.4.m4.1a"><mrow id="S6.I1.i1.p1.4.m4.1.1" xref="S6.I1.i1.p1.4.m4.1.1.cmml"><mi id="S6.I1.i1.p1.4.m4.1.1.2" xref="S6.I1.i1.p1.4.m4.1.1.2.cmml">a</mi><msub id="S6.I1.i1.p1.4.m4.1.1.3" xref="S6.I1.i1.p1.4.m4.1.1.3.cmml"><mo id="S6.I1.i1.p1.4.m4.1.1.3.2" xref="S6.I1.i1.p1.4.m4.1.1.3.2.cmml">≤</mo><mi id="S6.I1.i1.p1.4.m4.1.1.3.3" xref="S6.I1.i1.p1.4.m4.1.1.3.3.cmml">A</mi></msub><mi id="S6.I1.i1.p1.4.m4.1.1.4" xref="S6.I1.i1.p1.4.m4.1.1.4.cmml">ξ</mi><msub id="S6.I1.i1.p1.4.m4.1.1.5" xref="S6.I1.i1.p1.4.m4.1.1.5.cmml"><mo id="S6.I1.i1.p1.4.m4.1.1.5.2" xref="S6.I1.i1.p1.4.m4.1.1.5.2.cmml"><</mo><mi id="S6.I1.i1.p1.4.m4.1.1.5.3" xref="S6.I1.i1.p1.4.m4.1.1.5.3.cmml">A</mi></msub><mi id="S6.I1.i1.p1.4.m4.1.1.6" xref="S6.I1.i1.p1.4.m4.1.1.6.cmml">b</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.I1.i1.p1.4.m4.1b"><apply id="S6.I1.i1.p1.4.m4.1.1.cmml" xref="S6.I1.i1.p1.4.m4.1.1"><and id="S6.I1.i1.p1.4.m4.1.1a.cmml" xref="S6.I1.i1.p1.4.m4.1.1"></and><apply id="S6.I1.i1.p1.4.m4.1.1b.cmml" xref="S6.I1.i1.p1.4.m4.1.1"><apply id="S6.I1.i1.p1.4.m4.1.1.3.cmml" xref="S6.I1.i1.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="S6.I1.i1.p1.4.m4.1.1.3.1.cmml" xref="S6.I1.i1.p1.4.m4.1.1.3">subscript</csymbol><leq id="S6.I1.i1.p1.4.m4.1.1.3.2.cmml" xref="S6.I1.i1.p1.4.m4.1.1.3.2"></leq><ci id="S6.I1.i1.p1.4.m4.1.1.3.3.cmml" xref="S6.I1.i1.p1.4.m4.1.1.3.3">𝐴</ci></apply><ci id="S6.I1.i1.p1.4.m4.1.1.2.cmml" xref="S6.I1.i1.p1.4.m4.1.1.2">𝑎</ci><ci id="S6.I1.i1.p1.4.m4.1.1.4.cmml" xref="S6.I1.i1.p1.4.m4.1.1.4">𝜉</ci></apply><apply id="S6.I1.i1.p1.4.m4.1.1c.cmml" xref="S6.I1.i1.p1.4.m4.1.1"><apply id="S6.I1.i1.p1.4.m4.1.1.5.cmml" xref="S6.I1.i1.p1.4.m4.1.1.5"><csymbol cd="ambiguous" id="S6.I1.i1.p1.4.m4.1.1.5.1.cmml" xref="S6.I1.i1.p1.4.m4.1.1.5">subscript</csymbol><lt id="S6.I1.i1.p1.4.m4.1.1.5.2.cmml" xref="S6.I1.i1.p1.4.m4.1.1.5.2"></lt><ci id="S6.I1.i1.p1.4.m4.1.1.5.3.cmml" xref="S6.I1.i1.p1.4.m4.1.1.5.3">𝐴</ci></apply><share href="https://arxiv.org/html/2503.13728v1#S6.I1.i1.p1.4.m4.1.1.4.cmml" id="S6.I1.i1.p1.4.m4.1.1d.cmml" xref="S6.I1.i1.p1.4.m4.1.1"></share><ci id="S6.I1.i1.p1.4.m4.1.1.6.cmml" xref="S6.I1.i1.p1.4.m4.1.1.6">𝑏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I1.i1.p1.4.m4.1c">a\leq_{A}\xi<_{A}b</annotation><annotation encoding="application/x-llamapun" id="S6.I1.i1.p1.4.m4.1d">italic_a ≤ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_ξ < start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_b</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S6.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S6.I1.i2.p1"> <p class="ltx_p" id="S6.I1.i2.p1.2">If <math alttext="a\leq_{A}a^{\prime}<_{A}b^{\prime}\leq_{A}b" class="ltx_Math" display="inline" id="S6.I1.i2.p1.1.m1.1"><semantics id="S6.I1.i2.p1.1.m1.1a"><mrow id="S6.I1.i2.p1.1.m1.1.1" xref="S6.I1.i2.p1.1.m1.1.1.cmml"><mi id="S6.I1.i2.p1.1.m1.1.1.2" xref="S6.I1.i2.p1.1.m1.1.1.2.cmml">a</mi><msub id="S6.I1.i2.p1.1.m1.1.1.3" xref="S6.I1.i2.p1.1.m1.1.1.3.cmml"><mo id="S6.I1.i2.p1.1.m1.1.1.3.2" xref="S6.I1.i2.p1.1.m1.1.1.3.2.cmml">≤</mo><mi id="S6.I1.i2.p1.1.m1.1.1.3.3" xref="S6.I1.i2.p1.1.m1.1.1.3.3.cmml">A</mi></msub><msup id="S6.I1.i2.p1.1.m1.1.1.4" xref="S6.I1.i2.p1.1.m1.1.1.4.cmml"><mi id="S6.I1.i2.p1.1.m1.1.1.4.2" xref="S6.I1.i2.p1.1.m1.1.1.4.2.cmml">a</mi><mo id="S6.I1.i2.p1.1.m1.1.1.4.3" xref="S6.I1.i2.p1.1.m1.1.1.4.3.cmml">′</mo></msup><msub id="S6.I1.i2.p1.1.m1.1.1.5" xref="S6.I1.i2.p1.1.m1.1.1.5.cmml"><mo id="S6.I1.i2.p1.1.m1.1.1.5.2" xref="S6.I1.i2.p1.1.m1.1.1.5.2.cmml"><</mo><mi id="S6.I1.i2.p1.1.m1.1.1.5.3" xref="S6.I1.i2.p1.1.m1.1.1.5.3.cmml">A</mi></msub><msup id="S6.I1.i2.p1.1.m1.1.1.6" xref="S6.I1.i2.p1.1.m1.1.1.6.cmml"><mi id="S6.I1.i2.p1.1.m1.1.1.6.2" xref="S6.I1.i2.p1.1.m1.1.1.6.2.cmml">b</mi><mo id="S6.I1.i2.p1.1.m1.1.1.6.3" xref="S6.I1.i2.p1.1.m1.1.1.6.3.cmml">′</mo></msup><msub id="S6.I1.i2.p1.1.m1.1.1.7" xref="S6.I1.i2.p1.1.m1.1.1.7.cmml"><mo id="S6.I1.i2.p1.1.m1.1.1.7.2" xref="S6.I1.i2.p1.1.m1.1.1.7.2.cmml">≤</mo><mi id="S6.I1.i2.p1.1.m1.1.1.7.3" xref="S6.I1.i2.p1.1.m1.1.1.7.3.cmml">A</mi></msub><mi id="S6.I1.i2.p1.1.m1.1.1.8" xref="S6.I1.i2.p1.1.m1.1.1.8.cmml">b</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.I1.i2.p1.1.m1.1b"><apply id="S6.I1.i2.p1.1.m1.1.1.cmml" xref="S6.I1.i2.p1.1.m1.1.1"><and id="S6.I1.i2.p1.1.m1.1.1a.cmml" xref="S6.I1.i2.p1.1.m1.1.1"></and><apply id="S6.I1.i2.p1.1.m1.1.1b.cmml" xref="S6.I1.i2.p1.1.m1.1.1"><apply id="S6.I1.i2.p1.1.m1.1.1.3.cmml" xref="S6.I1.i2.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S6.I1.i2.p1.1.m1.1.1.3.1.cmml" xref="S6.I1.i2.p1.1.m1.1.1.3">subscript</csymbol><leq id="S6.I1.i2.p1.1.m1.1.1.3.2.cmml" xref="S6.I1.i2.p1.1.m1.1.1.3.2"></leq><ci id="S6.I1.i2.p1.1.m1.1.1.3.3.cmml" xref="S6.I1.i2.p1.1.m1.1.1.3.3">𝐴</ci></apply><ci id="S6.I1.i2.p1.1.m1.1.1.2.cmml" xref="S6.I1.i2.p1.1.m1.1.1.2">𝑎</ci><apply id="S6.I1.i2.p1.1.m1.1.1.4.cmml" xref="S6.I1.i2.p1.1.m1.1.1.4"><csymbol cd="ambiguous" id="S6.I1.i2.p1.1.m1.1.1.4.1.cmml" xref="S6.I1.i2.p1.1.m1.1.1.4">superscript</csymbol><ci id="S6.I1.i2.p1.1.m1.1.1.4.2.cmml" xref="S6.I1.i2.p1.1.m1.1.1.4.2">𝑎</ci><ci id="S6.I1.i2.p1.1.m1.1.1.4.3.cmml" xref="S6.I1.i2.p1.1.m1.1.1.4.3">′</ci></apply></apply><apply id="S6.I1.i2.p1.1.m1.1.1c.cmml" xref="S6.I1.i2.p1.1.m1.1.1"><apply id="S6.I1.i2.p1.1.m1.1.1.5.cmml" xref="S6.I1.i2.p1.1.m1.1.1.5"><csymbol cd="ambiguous" id="S6.I1.i2.p1.1.m1.1.1.5.1.cmml" xref="S6.I1.i2.p1.1.m1.1.1.5">subscript</csymbol><lt id="S6.I1.i2.p1.1.m1.1.1.5.2.cmml" xref="S6.I1.i2.p1.1.m1.1.1.5.2"></lt><ci id="S6.I1.i2.p1.1.m1.1.1.5.3.cmml" xref="S6.I1.i2.p1.1.m1.1.1.5.3">𝐴</ci></apply><share href="https://arxiv.org/html/2503.13728v1#S6.I1.i2.p1.1.m1.1.1.4.cmml" id="S6.I1.i2.p1.1.m1.1.1d.cmml" xref="S6.I1.i2.p1.1.m1.1.1"></share><apply id="S6.I1.i2.p1.1.m1.1.1.6.cmml" xref="S6.I1.i2.p1.1.m1.1.1.6"><csymbol cd="ambiguous" id="S6.I1.i2.p1.1.m1.1.1.6.1.cmml" xref="S6.I1.i2.p1.1.m1.1.1.6">superscript</csymbol><ci id="S6.I1.i2.p1.1.m1.1.1.6.2.cmml" xref="S6.I1.i2.p1.1.m1.1.1.6.2">𝑏</ci><ci id="S6.I1.i2.p1.1.m1.1.1.6.3.cmml" xref="S6.I1.i2.p1.1.m1.1.1.6.3">′</ci></apply></apply><apply id="S6.I1.i2.p1.1.m1.1.1e.cmml" xref="S6.I1.i2.p1.1.m1.1.1"><apply id="S6.I1.i2.p1.1.m1.1.1.7.cmml" xref="S6.I1.i2.p1.1.m1.1.1.7"><csymbol cd="ambiguous" id="S6.I1.i2.p1.1.m1.1.1.7.1.cmml" xref="S6.I1.i2.p1.1.m1.1.1.7">subscript</csymbol><leq id="S6.I1.i2.p1.1.m1.1.1.7.2.cmml" xref="S6.I1.i2.p1.1.m1.1.1.7.2"></leq><ci id="S6.I1.i2.p1.1.m1.1.1.7.3.cmml" xref="S6.I1.i2.p1.1.m1.1.1.7.3">𝐴</ci></apply><share href="https://arxiv.org/html/2503.13728v1#S6.I1.i2.p1.1.m1.1.1.6.cmml" id="S6.I1.i2.p1.1.m1.1.1f.cmml" xref="S6.I1.i2.p1.1.m1.1.1"></share><ci id="S6.I1.i2.p1.1.m1.1.1.8.cmml" xref="S6.I1.i2.p1.1.m1.1.1.8">𝑏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I1.i2.p1.1.m1.1c">a\leq_{A}a^{\prime}<_{A}b^{\prime}\leq_{A}b</annotation><annotation encoding="application/x-llamapun" id="S6.I1.i2.p1.1.m1.1d">italic_a ≤ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT < start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_b</annotation></semantics></math>, then <math alttext="\Delta_{A}(a,b)\leq\Delta_{A}(a^{\prime},b^{\prime})" class="ltx_Math" display="inline" id="S6.I1.i2.p1.2.m2.4"><semantics id="S6.I1.i2.p1.2.m2.4a"><mrow id="S6.I1.i2.p1.2.m2.4.4" xref="S6.I1.i2.p1.2.m2.4.4.cmml"><mrow id="S6.I1.i2.p1.2.m2.4.4.4" xref="S6.I1.i2.p1.2.m2.4.4.4.cmml"><msub id="S6.I1.i2.p1.2.m2.4.4.4.2" xref="S6.I1.i2.p1.2.m2.4.4.4.2.cmml"><mi id="S6.I1.i2.p1.2.m2.4.4.4.2.2" mathvariant="normal" xref="S6.I1.i2.p1.2.m2.4.4.4.2.2.cmml">Δ</mi><mi id="S6.I1.i2.p1.2.m2.4.4.4.2.3" xref="S6.I1.i2.p1.2.m2.4.4.4.2.3.cmml">A</mi></msub><mo id="S6.I1.i2.p1.2.m2.4.4.4.1" xref="S6.I1.i2.p1.2.m2.4.4.4.1.cmml">⁢</mo><mrow id="S6.I1.i2.p1.2.m2.4.4.4.3.2" xref="S6.I1.i2.p1.2.m2.4.4.4.3.1.cmml"><mo id="S6.I1.i2.p1.2.m2.4.4.4.3.2.1" stretchy="false" xref="S6.I1.i2.p1.2.m2.4.4.4.3.1.cmml">(</mo><mi id="S6.I1.i2.p1.2.m2.1.1" xref="S6.I1.i2.p1.2.m2.1.1.cmml">a</mi><mo id="S6.I1.i2.p1.2.m2.4.4.4.3.2.2" xref="S6.I1.i2.p1.2.m2.4.4.4.3.1.cmml">,</mo><mi id="S6.I1.i2.p1.2.m2.2.2" xref="S6.I1.i2.p1.2.m2.2.2.cmml">b</mi><mo id="S6.I1.i2.p1.2.m2.4.4.4.3.2.3" stretchy="false" xref="S6.I1.i2.p1.2.m2.4.4.4.3.1.cmml">)</mo></mrow></mrow><mo id="S6.I1.i2.p1.2.m2.4.4.3" xref="S6.I1.i2.p1.2.m2.4.4.3.cmml">≤</mo><mrow id="S6.I1.i2.p1.2.m2.4.4.2" xref="S6.I1.i2.p1.2.m2.4.4.2.cmml"><msub id="S6.I1.i2.p1.2.m2.4.4.2.4" xref="S6.I1.i2.p1.2.m2.4.4.2.4.cmml"><mi id="S6.I1.i2.p1.2.m2.4.4.2.4.2" mathvariant="normal" xref="S6.I1.i2.p1.2.m2.4.4.2.4.2.cmml">Δ</mi><mi id="S6.I1.i2.p1.2.m2.4.4.2.4.3" xref="S6.I1.i2.p1.2.m2.4.4.2.4.3.cmml">A</mi></msub><mo id="S6.I1.i2.p1.2.m2.4.4.2.3" xref="S6.I1.i2.p1.2.m2.4.4.2.3.cmml">⁢</mo><mrow id="S6.I1.i2.p1.2.m2.4.4.2.2.2" xref="S6.I1.i2.p1.2.m2.4.4.2.2.3.cmml"><mo id="S6.I1.i2.p1.2.m2.4.4.2.2.2.3" stretchy="false" xref="S6.I1.i2.p1.2.m2.4.4.2.2.3.cmml">(</mo><msup id="S6.I1.i2.p1.2.m2.3.3.1.1.1.1" xref="S6.I1.i2.p1.2.m2.3.3.1.1.1.1.cmml"><mi id="S6.I1.i2.p1.2.m2.3.3.1.1.1.1.2" xref="S6.I1.i2.p1.2.m2.3.3.1.1.1.1.2.cmml">a</mi><mo id="S6.I1.i2.p1.2.m2.3.3.1.1.1.1.3" xref="S6.I1.i2.p1.2.m2.3.3.1.1.1.1.3.cmml">′</mo></msup><mo id="S6.I1.i2.p1.2.m2.4.4.2.2.2.4" xref="S6.I1.i2.p1.2.m2.4.4.2.2.3.cmml">,</mo><msup id="S6.I1.i2.p1.2.m2.4.4.2.2.2.2" xref="S6.I1.i2.p1.2.m2.4.4.2.2.2.2.cmml"><mi id="S6.I1.i2.p1.2.m2.4.4.2.2.2.2.2" xref="S6.I1.i2.p1.2.m2.4.4.2.2.2.2.2.cmml">b</mi><mo id="S6.I1.i2.p1.2.m2.4.4.2.2.2.2.3" xref="S6.I1.i2.p1.2.m2.4.4.2.2.2.2.3.cmml">′</mo></msup><mo id="S6.I1.i2.p1.2.m2.4.4.2.2.2.5" stretchy="false" xref="S6.I1.i2.p1.2.m2.4.4.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.I1.i2.p1.2.m2.4b"><apply id="S6.I1.i2.p1.2.m2.4.4.cmml" xref="S6.I1.i2.p1.2.m2.4.4"><leq id="S6.I1.i2.p1.2.m2.4.4.3.cmml" xref="S6.I1.i2.p1.2.m2.4.4.3"></leq><apply id="S6.I1.i2.p1.2.m2.4.4.4.cmml" xref="S6.I1.i2.p1.2.m2.4.4.4"><times id="S6.I1.i2.p1.2.m2.4.4.4.1.cmml" xref="S6.I1.i2.p1.2.m2.4.4.4.1"></times><apply id="S6.I1.i2.p1.2.m2.4.4.4.2.cmml" xref="S6.I1.i2.p1.2.m2.4.4.4.2"><csymbol cd="ambiguous" id="S6.I1.i2.p1.2.m2.4.4.4.2.1.cmml" xref="S6.I1.i2.p1.2.m2.4.4.4.2">subscript</csymbol><ci id="S6.I1.i2.p1.2.m2.4.4.4.2.2.cmml" xref="S6.I1.i2.p1.2.m2.4.4.4.2.2">Δ</ci><ci id="S6.I1.i2.p1.2.m2.4.4.4.2.3.cmml" xref="S6.I1.i2.p1.2.m2.4.4.4.2.3">𝐴</ci></apply><interval closure="open" id="S6.I1.i2.p1.2.m2.4.4.4.3.1.cmml" xref="S6.I1.i2.p1.2.m2.4.4.4.3.2"><ci id="S6.I1.i2.p1.2.m2.1.1.cmml" xref="S6.I1.i2.p1.2.m2.1.1">𝑎</ci><ci id="S6.I1.i2.p1.2.m2.2.2.cmml" xref="S6.I1.i2.p1.2.m2.2.2">𝑏</ci></interval></apply><apply id="S6.I1.i2.p1.2.m2.4.4.2.cmml" xref="S6.I1.i2.p1.2.m2.4.4.2"><times id="S6.I1.i2.p1.2.m2.4.4.2.3.cmml" xref="S6.I1.i2.p1.2.m2.4.4.2.3"></times><apply id="S6.I1.i2.p1.2.m2.4.4.2.4.cmml" xref="S6.I1.i2.p1.2.m2.4.4.2.4"><csymbol cd="ambiguous" id="S6.I1.i2.p1.2.m2.4.4.2.4.1.cmml" xref="S6.I1.i2.p1.2.m2.4.4.2.4">subscript</csymbol><ci id="S6.I1.i2.p1.2.m2.4.4.2.4.2.cmml" xref="S6.I1.i2.p1.2.m2.4.4.2.4.2">Δ</ci><ci id="S6.I1.i2.p1.2.m2.4.4.2.4.3.cmml" xref="S6.I1.i2.p1.2.m2.4.4.2.4.3">𝐴</ci></apply><interval closure="open" id="S6.I1.i2.p1.2.m2.4.4.2.2.3.cmml" xref="S6.I1.i2.p1.2.m2.4.4.2.2.2"><apply id="S6.I1.i2.p1.2.m2.3.3.1.1.1.1.cmml" xref="S6.I1.i2.p1.2.m2.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S6.I1.i2.p1.2.m2.3.3.1.1.1.1.1.cmml" xref="S6.I1.i2.p1.2.m2.3.3.1.1.1.1">superscript</csymbol><ci id="S6.I1.i2.p1.2.m2.3.3.1.1.1.1.2.cmml" xref="S6.I1.i2.p1.2.m2.3.3.1.1.1.1.2">𝑎</ci><ci id="S6.I1.i2.p1.2.m2.3.3.1.1.1.1.3.cmml" xref="S6.I1.i2.p1.2.m2.3.3.1.1.1.1.3">′</ci></apply><apply id="S6.I1.i2.p1.2.m2.4.4.2.2.2.2.cmml" xref="S6.I1.i2.p1.2.m2.4.4.2.2.2.2"><csymbol cd="ambiguous" id="S6.I1.i2.p1.2.m2.4.4.2.2.2.2.1.cmml" xref="S6.I1.i2.p1.2.m2.4.4.2.2.2.2">superscript</csymbol><ci id="S6.I1.i2.p1.2.m2.4.4.2.2.2.2.2.cmml" xref="S6.I1.i2.p1.2.m2.4.4.2.2.2.2.2">𝑏</ci><ci id="S6.I1.i2.p1.2.m2.4.4.2.2.2.2.3.cmml" xref="S6.I1.i2.p1.2.m2.4.4.2.2.2.2.3">′</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I1.i2.p1.2.m2.4c">\Delta_{A}(a,b)\leq\Delta_{A}(a^{\prime},b^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S6.I1.i2.p1.2.m2.4d">roman_Δ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_a , italic_b ) ≤ roman_Δ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S6.I1.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S6.I1.i3.p1"> <p class="ltx_p" id="S6.I1.i3.p1.2">For every <math alttext="a\in A" class="ltx_Math" display="inline" id="S6.I1.i3.p1.1.m1.1"><semantics id="S6.I1.i3.p1.1.m1.1a"><mrow id="S6.I1.i3.p1.1.m1.1.1" xref="S6.I1.i3.p1.1.m1.1.1.cmml"><mi id="S6.I1.i3.p1.1.m1.1.1.2" xref="S6.I1.i3.p1.1.m1.1.1.2.cmml">a</mi><mo id="S6.I1.i3.p1.1.m1.1.1.1" xref="S6.I1.i3.p1.1.m1.1.1.1.cmml">∈</mo><mi id="S6.I1.i3.p1.1.m1.1.1.3" xref="S6.I1.i3.p1.1.m1.1.1.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.I1.i3.p1.1.m1.1b"><apply id="S6.I1.i3.p1.1.m1.1.1.cmml" xref="S6.I1.i3.p1.1.m1.1.1"><in id="S6.I1.i3.p1.1.m1.1.1.1.cmml" xref="S6.I1.i3.p1.1.m1.1.1.1"></in><ci id="S6.I1.i3.p1.1.m1.1.1.2.cmml" xref="S6.I1.i3.p1.1.m1.1.1.2">𝑎</ci><ci id="S6.I1.i3.p1.1.m1.1.1.3.cmml" xref="S6.I1.i3.p1.1.m1.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I1.i3.p1.1.m1.1c">a\in A</annotation><annotation encoding="application/x-llamapun" id="S6.I1.i3.p1.1.m1.1d">italic_a ∈ italic_A</annotation></semantics></math>, <math alttext="\delta_{A}(a):=\sup\{\Delta_{A}(a,b)+1:b\in A\setminus\{a\}\}" class="ltx_Math" display="inline" id="S6.I1.i3.p1.2.m2.6"><semantics id="S6.I1.i3.p1.2.m2.6a"><mrow id="S6.I1.i3.p1.2.m2.6.6" xref="S6.I1.i3.p1.2.m2.6.6.cmml"><mrow id="S6.I1.i3.p1.2.m2.6.6.4" xref="S6.I1.i3.p1.2.m2.6.6.4.cmml"><msub id="S6.I1.i3.p1.2.m2.6.6.4.2" xref="S6.I1.i3.p1.2.m2.6.6.4.2.cmml"><mi id="S6.I1.i3.p1.2.m2.6.6.4.2.2" xref="S6.I1.i3.p1.2.m2.6.6.4.2.2.cmml">δ</mi><mi id="S6.I1.i3.p1.2.m2.6.6.4.2.3" xref="S6.I1.i3.p1.2.m2.6.6.4.2.3.cmml">A</mi></msub><mo id="S6.I1.i3.p1.2.m2.6.6.4.1" xref="S6.I1.i3.p1.2.m2.6.6.4.1.cmml">⁢</mo><mrow id="S6.I1.i3.p1.2.m2.6.6.4.3.2" xref="S6.I1.i3.p1.2.m2.6.6.4.cmml"><mo id="S6.I1.i3.p1.2.m2.6.6.4.3.2.1" stretchy="false" xref="S6.I1.i3.p1.2.m2.6.6.4.cmml">(</mo><mi id="S6.I1.i3.p1.2.m2.1.1" xref="S6.I1.i3.p1.2.m2.1.1.cmml">a</mi><mo id="S6.I1.i3.p1.2.m2.6.6.4.3.2.2" rspace="0.278em" stretchy="false" xref="S6.I1.i3.p1.2.m2.6.6.4.cmml">)</mo></mrow></mrow><mo id="S6.I1.i3.p1.2.m2.6.6.3" xref="S6.I1.i3.p1.2.m2.6.6.3.cmml">:=</mo><mrow id="S6.I1.i3.p1.2.m2.6.6.2" xref="S6.I1.i3.p1.2.m2.6.6.2.cmml"><mo id="S6.I1.i3.p1.2.m2.6.6.2.3" lspace="0.111em" rspace="0em" xref="S6.I1.i3.p1.2.m2.6.6.2.3.cmml">sup</mo><mrow id="S6.I1.i3.p1.2.m2.6.6.2.2.2" xref="S6.I1.i3.p1.2.m2.6.6.2.2.3.cmml"><mo id="S6.I1.i3.p1.2.m2.6.6.2.2.2.3" stretchy="false" xref="S6.I1.i3.p1.2.m2.6.6.2.2.3.1.cmml">{</mo><mrow id="S6.I1.i3.p1.2.m2.5.5.1.1.1.1" xref="S6.I1.i3.p1.2.m2.5.5.1.1.1.1.cmml"><mrow id="S6.I1.i3.p1.2.m2.5.5.1.1.1.1.2" xref="S6.I1.i3.p1.2.m2.5.5.1.1.1.1.2.cmml"><msub id="S6.I1.i3.p1.2.m2.5.5.1.1.1.1.2.2" xref="S6.I1.i3.p1.2.m2.5.5.1.1.1.1.2.2.cmml"><mi id="S6.I1.i3.p1.2.m2.5.5.1.1.1.1.2.2.2" mathvariant="normal" xref="S6.I1.i3.p1.2.m2.5.5.1.1.1.1.2.2.2.cmml">Δ</mi><mi id="S6.I1.i3.p1.2.m2.5.5.1.1.1.1.2.2.3" xref="S6.I1.i3.p1.2.m2.5.5.1.1.1.1.2.2.3.cmml">A</mi></msub><mo id="S6.I1.i3.p1.2.m2.5.5.1.1.1.1.2.1" xref="S6.I1.i3.p1.2.m2.5.5.1.1.1.1.2.1.cmml">⁢</mo><mrow id="S6.I1.i3.p1.2.m2.5.5.1.1.1.1.2.3.2" xref="S6.I1.i3.p1.2.m2.5.5.1.1.1.1.2.3.1.cmml"><mo id="S6.I1.i3.p1.2.m2.5.5.1.1.1.1.2.3.2.1" stretchy="false" xref="S6.I1.i3.p1.2.m2.5.5.1.1.1.1.2.3.1.cmml">(</mo><mi id="S6.I1.i3.p1.2.m2.2.2" xref="S6.I1.i3.p1.2.m2.2.2.cmml">a</mi><mo id="S6.I1.i3.p1.2.m2.5.5.1.1.1.1.2.3.2.2" xref="S6.I1.i3.p1.2.m2.5.5.1.1.1.1.2.3.1.cmml">,</mo><mi id="S6.I1.i3.p1.2.m2.3.3" xref="S6.I1.i3.p1.2.m2.3.3.cmml">b</mi><mo id="S6.I1.i3.p1.2.m2.5.5.1.1.1.1.2.3.2.3" stretchy="false" xref="S6.I1.i3.p1.2.m2.5.5.1.1.1.1.2.3.1.cmml">)</mo></mrow></mrow><mo id="S6.I1.i3.p1.2.m2.5.5.1.1.1.1.1" xref="S6.I1.i3.p1.2.m2.5.5.1.1.1.1.1.cmml">+</mo><mn id="S6.I1.i3.p1.2.m2.5.5.1.1.1.1.3" xref="S6.I1.i3.p1.2.m2.5.5.1.1.1.1.3.cmml">1</mn></mrow><mo id="S6.I1.i3.p1.2.m2.6.6.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S6.I1.i3.p1.2.m2.6.6.2.2.3.1.cmml">:</mo><mrow id="S6.I1.i3.p1.2.m2.6.6.2.2.2.2" xref="S6.I1.i3.p1.2.m2.6.6.2.2.2.2.cmml"><mi id="S6.I1.i3.p1.2.m2.6.6.2.2.2.2.2" xref="S6.I1.i3.p1.2.m2.6.6.2.2.2.2.2.cmml">b</mi><mo id="S6.I1.i3.p1.2.m2.6.6.2.2.2.2.1" xref="S6.I1.i3.p1.2.m2.6.6.2.2.2.2.1.cmml">∈</mo><mrow id="S6.I1.i3.p1.2.m2.6.6.2.2.2.2.3" xref="S6.I1.i3.p1.2.m2.6.6.2.2.2.2.3.cmml"><mi id="S6.I1.i3.p1.2.m2.6.6.2.2.2.2.3.2" xref="S6.I1.i3.p1.2.m2.6.6.2.2.2.2.3.2.cmml">A</mi><mo id="S6.I1.i3.p1.2.m2.6.6.2.2.2.2.3.1" xref="S6.I1.i3.p1.2.m2.6.6.2.2.2.2.3.1.cmml">∖</mo><mrow id="S6.I1.i3.p1.2.m2.6.6.2.2.2.2.3.3.2" xref="S6.I1.i3.p1.2.m2.6.6.2.2.2.2.3.3.1.cmml"><mo id="S6.I1.i3.p1.2.m2.6.6.2.2.2.2.3.3.2.1" stretchy="false" xref="S6.I1.i3.p1.2.m2.6.6.2.2.2.2.3.3.1.cmml">{</mo><mi id="S6.I1.i3.p1.2.m2.4.4" xref="S6.I1.i3.p1.2.m2.4.4.cmml">a</mi><mo id="S6.I1.i3.p1.2.m2.6.6.2.2.2.2.3.3.2.2" stretchy="false" xref="S6.I1.i3.p1.2.m2.6.6.2.2.2.2.3.3.1.cmml">}</mo></mrow></mrow></mrow><mo id="S6.I1.i3.p1.2.m2.6.6.2.2.2.5" stretchy="false" xref="S6.I1.i3.p1.2.m2.6.6.2.2.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.I1.i3.p1.2.m2.6b"><apply id="S6.I1.i3.p1.2.m2.6.6.cmml" xref="S6.I1.i3.p1.2.m2.6.6"><csymbol cd="latexml" id="S6.I1.i3.p1.2.m2.6.6.3.cmml" xref="S6.I1.i3.p1.2.m2.6.6.3">assign</csymbol><apply id="S6.I1.i3.p1.2.m2.6.6.4.cmml" xref="S6.I1.i3.p1.2.m2.6.6.4"><times id="S6.I1.i3.p1.2.m2.6.6.4.1.cmml" xref="S6.I1.i3.p1.2.m2.6.6.4.1"></times><apply id="S6.I1.i3.p1.2.m2.6.6.4.2.cmml" xref="S6.I1.i3.p1.2.m2.6.6.4.2"><csymbol cd="ambiguous" id="S6.I1.i3.p1.2.m2.6.6.4.2.1.cmml" xref="S6.I1.i3.p1.2.m2.6.6.4.2">subscript</csymbol><ci id="S6.I1.i3.p1.2.m2.6.6.4.2.2.cmml" xref="S6.I1.i3.p1.2.m2.6.6.4.2.2">𝛿</ci><ci id="S6.I1.i3.p1.2.m2.6.6.4.2.3.cmml" xref="S6.I1.i3.p1.2.m2.6.6.4.2.3">𝐴</ci></apply><ci id="S6.I1.i3.p1.2.m2.1.1.cmml" xref="S6.I1.i3.p1.2.m2.1.1">𝑎</ci></apply><apply id="S6.I1.i3.p1.2.m2.6.6.2.cmml" xref="S6.I1.i3.p1.2.m2.6.6.2"><csymbol cd="latexml" id="S6.I1.i3.p1.2.m2.6.6.2.3.cmml" xref="S6.I1.i3.p1.2.m2.6.6.2.3">supremum</csymbol><apply id="S6.I1.i3.p1.2.m2.6.6.2.2.3.cmml" xref="S6.I1.i3.p1.2.m2.6.6.2.2.2"><csymbol cd="latexml" id="S6.I1.i3.p1.2.m2.6.6.2.2.3.1.cmml" xref="S6.I1.i3.p1.2.m2.6.6.2.2.2.3">conditional-set</csymbol><apply id="S6.I1.i3.p1.2.m2.5.5.1.1.1.1.cmml" xref="S6.I1.i3.p1.2.m2.5.5.1.1.1.1"><plus id="S6.I1.i3.p1.2.m2.5.5.1.1.1.1.1.cmml" xref="S6.I1.i3.p1.2.m2.5.5.1.1.1.1.1"></plus><apply id="S6.I1.i3.p1.2.m2.5.5.1.1.1.1.2.cmml" xref="S6.I1.i3.p1.2.m2.5.5.1.1.1.1.2"><times id="S6.I1.i3.p1.2.m2.5.5.1.1.1.1.2.1.cmml" xref="S6.I1.i3.p1.2.m2.5.5.1.1.1.1.2.1"></times><apply id="S6.I1.i3.p1.2.m2.5.5.1.1.1.1.2.2.cmml" xref="S6.I1.i3.p1.2.m2.5.5.1.1.1.1.2.2"><csymbol cd="ambiguous" id="S6.I1.i3.p1.2.m2.5.5.1.1.1.1.2.2.1.cmml" xref="S6.I1.i3.p1.2.m2.5.5.1.1.1.1.2.2">subscript</csymbol><ci id="S6.I1.i3.p1.2.m2.5.5.1.1.1.1.2.2.2.cmml" xref="S6.I1.i3.p1.2.m2.5.5.1.1.1.1.2.2.2">Δ</ci><ci id="S6.I1.i3.p1.2.m2.5.5.1.1.1.1.2.2.3.cmml" xref="S6.I1.i3.p1.2.m2.5.5.1.1.1.1.2.2.3">𝐴</ci></apply><interval closure="open" id="S6.I1.i3.p1.2.m2.5.5.1.1.1.1.2.3.1.cmml" xref="S6.I1.i3.p1.2.m2.5.5.1.1.1.1.2.3.2"><ci id="S6.I1.i3.p1.2.m2.2.2.cmml" xref="S6.I1.i3.p1.2.m2.2.2">𝑎</ci><ci id="S6.I1.i3.p1.2.m2.3.3.cmml" xref="S6.I1.i3.p1.2.m2.3.3">𝑏</ci></interval></apply><cn id="S6.I1.i3.p1.2.m2.5.5.1.1.1.1.3.cmml" type="integer" xref="S6.I1.i3.p1.2.m2.5.5.1.1.1.1.3">1</cn></apply><apply id="S6.I1.i3.p1.2.m2.6.6.2.2.2.2.cmml" xref="S6.I1.i3.p1.2.m2.6.6.2.2.2.2"><in id="S6.I1.i3.p1.2.m2.6.6.2.2.2.2.1.cmml" xref="S6.I1.i3.p1.2.m2.6.6.2.2.2.2.1"></in><ci id="S6.I1.i3.p1.2.m2.6.6.2.2.2.2.2.cmml" xref="S6.I1.i3.p1.2.m2.6.6.2.2.2.2.2">𝑏</ci><apply id="S6.I1.i3.p1.2.m2.6.6.2.2.2.2.3.cmml" xref="S6.I1.i3.p1.2.m2.6.6.2.2.2.2.3"><setdiff id="S6.I1.i3.p1.2.m2.6.6.2.2.2.2.3.1.cmml" xref="S6.I1.i3.p1.2.m2.6.6.2.2.2.2.3.1"></setdiff><ci id="S6.I1.i3.p1.2.m2.6.6.2.2.2.2.3.2.cmml" xref="S6.I1.i3.p1.2.m2.6.6.2.2.2.2.3.2">𝐴</ci><set id="S6.I1.i3.p1.2.m2.6.6.2.2.2.2.3.3.1.cmml" xref="S6.I1.i3.p1.2.m2.6.6.2.2.2.2.3.3.2"><ci id="S6.I1.i3.p1.2.m2.4.4.cmml" xref="S6.I1.i3.p1.2.m2.4.4">𝑎</ci></set></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I1.i3.p1.2.m2.6c">\delta_{A}(a):=\sup\{\Delta_{A}(a,b)+1:b\in A\setminus\{a\}\}</annotation><annotation encoding="application/x-llamapun" id="S6.I1.i3.p1.2.m2.6d">italic_δ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_a ) := roman_sup { roman_Δ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_a , italic_b ) + 1 : italic_b ∈ italic_A ∖ { italic_a } }</annotation></semantics></math> is a countable ordinal.</p> </div> </li> </ul> </div> <div class="ltx_para" id="S6.SS1.p4"> <p class="ltx_p" id="S6.SS1.p4.9">Now let <math alttext="E\subseteq\omega_{1}" class="ltx_Math" display="inline" id="S6.SS1.p4.1.m1.1"><semantics id="S6.SS1.p4.1.m1.1a"><mrow id="S6.SS1.p4.1.m1.1.1" xref="S6.SS1.p4.1.m1.1.1.cmml"><mi id="S6.SS1.p4.1.m1.1.1.2" xref="S6.SS1.p4.1.m1.1.1.2.cmml">E</mi><mo id="S6.SS1.p4.1.m1.1.1.1" xref="S6.SS1.p4.1.m1.1.1.1.cmml">⊆</mo><msub id="S6.SS1.p4.1.m1.1.1.3" xref="S6.SS1.p4.1.m1.1.1.3.cmml"><mi id="S6.SS1.p4.1.m1.1.1.3.2" xref="S6.SS1.p4.1.m1.1.1.3.2.cmml">ω</mi><mn id="S6.SS1.p4.1.m1.1.1.3.3" xref="S6.SS1.p4.1.m1.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p4.1.m1.1b"><apply id="S6.SS1.p4.1.m1.1.1.cmml" xref="S6.SS1.p4.1.m1.1.1"><subset id="S6.SS1.p4.1.m1.1.1.1.cmml" xref="S6.SS1.p4.1.m1.1.1.1"></subset><ci id="S6.SS1.p4.1.m1.1.1.2.cmml" xref="S6.SS1.p4.1.m1.1.1.2">𝐸</ci><apply id="S6.SS1.p4.1.m1.1.1.3.cmml" xref="S6.SS1.p4.1.m1.1.1.3"><csymbol cd="ambiguous" id="S6.SS1.p4.1.m1.1.1.3.1.cmml" xref="S6.SS1.p4.1.m1.1.1.3">subscript</csymbol><ci id="S6.SS1.p4.1.m1.1.1.3.2.cmml" xref="S6.SS1.p4.1.m1.1.1.3.2">𝜔</ci><cn id="S6.SS1.p4.1.m1.1.1.3.3.cmml" type="integer" xref="S6.SS1.p4.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p4.1.m1.1c">E\subseteq\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p4.1.m1.1d">italic_E ⊆ italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> be a club of limit ordinals closed under <math alttext="\delta_{A}" class="ltx_Math" display="inline" id="S6.SS1.p4.2.m2.1"><semantics id="S6.SS1.p4.2.m2.1a"><msub id="S6.SS1.p4.2.m2.1.1" xref="S6.SS1.p4.2.m2.1.1.cmml"><mi id="S6.SS1.p4.2.m2.1.1.2" xref="S6.SS1.p4.2.m2.1.1.2.cmml">δ</mi><mi id="S6.SS1.p4.2.m2.1.1.3" xref="S6.SS1.p4.2.m2.1.1.3.cmml">A</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS1.p4.2.m2.1b"><apply id="S6.SS1.p4.2.m2.1.1.cmml" xref="S6.SS1.p4.2.m2.1.1"><csymbol cd="ambiguous" id="S6.SS1.p4.2.m2.1.1.1.cmml" xref="S6.SS1.p4.2.m2.1.1">subscript</csymbol><ci id="S6.SS1.p4.2.m2.1.1.2.cmml" xref="S6.SS1.p4.2.m2.1.1.2">𝛿</ci><ci id="S6.SS1.p4.2.m2.1.1.3.cmml" xref="S6.SS1.p4.2.m2.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p4.2.m2.1c">\delta_{A}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p4.2.m2.1d">italic_δ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT</annotation></semantics></math> and let <math alttext="T_{A}:=\{\tilde{a}|_{\xi}:\xi\leq\min(E\setminus(\delta_{A}(a)+1))\}" class="ltx_math_unparsed" display="inline" id="S6.SS1.p4.3.m3.2"><semantics id="S6.SS1.p4.3.m3.2a"><mrow id="S6.SS1.p4.3.m3.2b"><msub id="S6.SS1.p4.3.m3.2.3"><mi id="S6.SS1.p4.3.m3.2.3.2">T</mi><mi id="S6.SS1.p4.3.m3.2.3.3">A</mi></msub><mo id="S6.SS1.p4.3.m3.2.4" lspace="0.278em" rspace="0.278em">:=</mo><mrow id="S6.SS1.p4.3.m3.2.5"><mo id="S6.SS1.p4.3.m3.2.5.1" stretchy="false">{</mo><mover accent="true" id="S6.SS1.p4.3.m3.1.1"><mi id="S6.SS1.p4.3.m3.1.1.2">a</mi><mo id="S6.SS1.p4.3.m3.1.1.1">~</mo></mover><msub id="S6.SS1.p4.3.m3.2.5.2"><mo fence="false" id="S6.SS1.p4.3.m3.2.5.2.2" rspace="0.167em" stretchy="false">|</mo><mi id="S6.SS1.p4.3.m3.2.2.1">ξ</mi></msub><mo id="S6.SS1.p4.3.m3.2.5.3" rspace="0.278em">:</mo><mi id="S6.SS1.p4.3.m3.2.5.4">ξ</mi><mo id="S6.SS1.p4.3.m3.2.5.5">≤</mo><mi id="S6.SS1.p4.3.m3.2.5.6">min</mi><mrow id="S6.SS1.p4.3.m3.2.5.7"><mo id="S6.SS1.p4.3.m3.2.5.7.1" stretchy="false">(</mo><mi id="S6.SS1.p4.3.m3.2.5.7.2">E</mi><mo id="S6.SS1.p4.3.m3.2.5.7.3">∖</mo><mrow id="S6.SS1.p4.3.m3.2.5.7.4"><mo id="S6.SS1.p4.3.m3.2.5.7.4.1" stretchy="false">(</mo><msub id="S6.SS1.p4.3.m3.2.5.7.4.2"><mi id="S6.SS1.p4.3.m3.2.5.7.4.2.2">δ</mi><mi id="S6.SS1.p4.3.m3.2.5.7.4.2.3">A</mi></msub><mrow id="S6.SS1.p4.3.m3.2.5.7.4.3"><mo id="S6.SS1.p4.3.m3.2.5.7.4.3.1" stretchy="false">(</mo><mi id="S6.SS1.p4.3.m3.2.5.7.4.3.2">a</mi><mo id="S6.SS1.p4.3.m3.2.5.7.4.3.3" stretchy="false">)</mo></mrow><mo id="S6.SS1.p4.3.m3.2.5.7.4.4">+</mo><mn id="S6.SS1.p4.3.m3.2.5.7.4.5">1</mn><mo id="S6.SS1.p4.3.m3.2.5.7.4.6" stretchy="false">)</mo></mrow><mo id="S6.SS1.p4.3.m3.2.5.7.5" stretchy="false">)</mo></mrow><mo id="S6.SS1.p4.3.m3.2.5.8" stretchy="false">}</mo></mrow></mrow><annotation encoding="application/x-tex" id="S6.SS1.p4.3.m3.2c">T_{A}:=\{\tilde{a}|_{\xi}:\xi\leq\min(E\setminus(\delta_{A}(a)+1))\}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p4.3.m3.2d">italic_T start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT := { over~ start_ARG italic_a end_ARG | start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT : italic_ξ ≤ roman_min ( italic_E ∖ ( italic_δ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_a ) + 1 ) ) }</annotation></semantics></math>. Note that <math alttext="a\mapsto\tilde{a}|_{\delta_{A}(a)}" class="ltx_Math" display="inline" id="S6.SS1.p4.4.m4.2"><semantics id="S6.SS1.p4.4.m4.2a"><mrow id="S6.SS1.p4.4.m4.2.3" xref="S6.SS1.p4.4.m4.2.3.cmml"><mi id="S6.SS1.p4.4.m4.2.3.2" xref="S6.SS1.p4.4.m4.2.3.2.cmml">a</mi><mo id="S6.SS1.p4.4.m4.2.3.1" stretchy="false" xref="S6.SS1.p4.4.m4.2.3.1.cmml">↦</mo><msub id="S6.SS1.p4.4.m4.2.3.3.2" xref="S6.SS1.p4.4.m4.2.3.3.1.cmml"><mrow id="S6.SS1.p4.4.m4.2.3.3.2.2" xref="S6.SS1.p4.4.m4.2.3.3.1.cmml"><mover accent="true" id="S6.SS1.p4.4.m4.2.2" xref="S6.SS1.p4.4.m4.2.2.cmml"><mi id="S6.SS1.p4.4.m4.2.2.2" xref="S6.SS1.p4.4.m4.2.2.2.cmml">a</mi><mo id="S6.SS1.p4.4.m4.2.2.1" xref="S6.SS1.p4.4.m4.2.2.1.cmml">~</mo></mover><mo id="S6.SS1.p4.4.m4.2.3.3.2.2.1" stretchy="false" xref="S6.SS1.p4.4.m4.2.3.3.1.1.cmml">|</mo></mrow><mrow id="S6.SS1.p4.4.m4.1.1.1" xref="S6.SS1.p4.4.m4.1.1.1.cmml"><msub id="S6.SS1.p4.4.m4.1.1.1.3" xref="S6.SS1.p4.4.m4.1.1.1.3.cmml"><mi id="S6.SS1.p4.4.m4.1.1.1.3.2" xref="S6.SS1.p4.4.m4.1.1.1.3.2.cmml">δ</mi><mi id="S6.SS1.p4.4.m4.1.1.1.3.3" xref="S6.SS1.p4.4.m4.1.1.1.3.3.cmml">A</mi></msub><mo id="S6.SS1.p4.4.m4.1.1.1.2" xref="S6.SS1.p4.4.m4.1.1.1.2.cmml">⁢</mo><mrow id="S6.SS1.p4.4.m4.1.1.1.4.2" xref="S6.SS1.p4.4.m4.1.1.1.cmml"><mo id="S6.SS1.p4.4.m4.1.1.1.4.2.1" stretchy="false" xref="S6.SS1.p4.4.m4.1.1.1.cmml">(</mo><mi id="S6.SS1.p4.4.m4.1.1.1.1" xref="S6.SS1.p4.4.m4.1.1.1.1.cmml">a</mi><mo id="S6.SS1.p4.4.m4.1.1.1.4.2.2" stretchy="false" xref="S6.SS1.p4.4.m4.1.1.1.cmml">)</mo></mrow></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p4.4.m4.2b"><apply id="S6.SS1.p4.4.m4.2.3.cmml" xref="S6.SS1.p4.4.m4.2.3"><csymbol cd="latexml" id="S6.SS1.p4.4.m4.2.3.1.cmml" xref="S6.SS1.p4.4.m4.2.3.1">maps-to</csymbol><ci id="S6.SS1.p4.4.m4.2.3.2.cmml" xref="S6.SS1.p4.4.m4.2.3.2">𝑎</ci><apply id="S6.SS1.p4.4.m4.2.3.3.1.cmml" xref="S6.SS1.p4.4.m4.2.3.3.2"><csymbol cd="latexml" id="S6.SS1.p4.4.m4.2.3.3.1.1.cmml" xref="S6.SS1.p4.4.m4.2.3.3.2.2.1">evaluated-at</csymbol><apply id="S6.SS1.p4.4.m4.2.2.cmml" xref="S6.SS1.p4.4.m4.2.2"><ci id="S6.SS1.p4.4.m4.2.2.1.cmml" xref="S6.SS1.p4.4.m4.2.2.1">~</ci><ci id="S6.SS1.p4.4.m4.2.2.2.cmml" xref="S6.SS1.p4.4.m4.2.2.2">𝑎</ci></apply><apply id="S6.SS1.p4.4.m4.1.1.1.cmml" xref="S6.SS1.p4.4.m4.1.1.1"><times id="S6.SS1.p4.4.m4.1.1.1.2.cmml" xref="S6.SS1.p4.4.m4.1.1.1.2"></times><apply id="S6.SS1.p4.4.m4.1.1.1.3.cmml" xref="S6.SS1.p4.4.m4.1.1.1.3"><csymbol cd="ambiguous" id="S6.SS1.p4.4.m4.1.1.1.3.1.cmml" xref="S6.SS1.p4.4.m4.1.1.1.3">subscript</csymbol><ci id="S6.SS1.p4.4.m4.1.1.1.3.2.cmml" xref="S6.SS1.p4.4.m4.1.1.1.3.2">𝛿</ci><ci id="S6.SS1.p4.4.m4.1.1.1.3.3.cmml" xref="S6.SS1.p4.4.m4.1.1.1.3.3">𝐴</ci></apply><ci id="S6.SS1.p4.4.m4.1.1.1.1.cmml" xref="S6.SS1.p4.4.m4.1.1.1.1">𝑎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p4.4.m4.2c">a\mapsto\tilde{a}|_{\delta_{A}(a)}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p4.4.m4.2d">italic_a ↦ over~ start_ARG italic_a end_ARG | start_POSTSUBSCRIPT italic_δ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_a ) end_POSTSUBSCRIPT</annotation></semantics></math> maps <math alttext="A" class="ltx_Math" display="inline" id="S6.SS1.p4.5.m5.1"><semantics id="S6.SS1.p4.5.m5.1a"><mi id="S6.SS1.p4.5.m5.1.1" xref="S6.SS1.p4.5.m5.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S6.SS1.p4.5.m5.1b"><ci id="S6.SS1.p4.5.m5.1.1.cmml" xref="S6.SS1.p4.5.m5.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p4.5.m5.1c">A</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p4.5.m5.1d">italic_A</annotation></semantics></math> to an antichain of <math alttext="T_{A}" class="ltx_Math" display="inline" id="S6.SS1.p4.6.m6.1"><semantics id="S6.SS1.p4.6.m6.1a"><msub id="S6.SS1.p4.6.m6.1.1" xref="S6.SS1.p4.6.m6.1.1.cmml"><mi id="S6.SS1.p4.6.m6.1.1.2" xref="S6.SS1.p4.6.m6.1.1.2.cmml">T</mi><mi id="S6.SS1.p4.6.m6.1.1.3" xref="S6.SS1.p4.6.m6.1.1.3.cmml">A</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS1.p4.6.m6.1b"><apply id="S6.SS1.p4.6.m6.1.1.cmml" xref="S6.SS1.p4.6.m6.1.1"><csymbol cd="ambiguous" id="S6.SS1.p4.6.m6.1.1.1.cmml" xref="S6.SS1.p4.6.m6.1.1">subscript</csymbol><ci id="S6.SS1.p4.6.m6.1.1.2.cmml" xref="S6.SS1.p4.6.m6.1.1.2">𝑇</ci><ci id="S6.SS1.p4.6.m6.1.1.3.cmml" xref="S6.SS1.p4.6.m6.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p4.6.m6.1c">T_{A}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p4.6.m6.1d">italic_T start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT</annotation></semantics></math>, so <math alttext="A" class="ltx_Math" display="inline" id="S6.SS1.p4.7.m7.1"><semantics id="S6.SS1.p4.7.m7.1a"><mi id="S6.SS1.p4.7.m7.1.1" xref="S6.SS1.p4.7.m7.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S6.SS1.p4.7.m7.1b"><ci id="S6.SS1.p4.7.m7.1.1.cmml" xref="S6.SS1.p4.7.m7.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p4.7.m7.1c">A</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p4.7.m7.1d">italic_A</annotation></semantics></math> can be thought of as the lexicographic ordering of the leafs of <math alttext="T_{A}" class="ltx_Math" display="inline" id="S6.SS1.p4.8.m8.1"><semantics id="S6.SS1.p4.8.m8.1a"><msub id="S6.SS1.p4.8.m8.1.1" xref="S6.SS1.p4.8.m8.1.1.cmml"><mi id="S6.SS1.p4.8.m8.1.1.2" xref="S6.SS1.p4.8.m8.1.1.2.cmml">T</mi><mi id="S6.SS1.p4.8.m8.1.1.3" xref="S6.SS1.p4.8.m8.1.1.3.cmml">A</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS1.p4.8.m8.1b"><apply id="S6.SS1.p4.8.m8.1.1.cmml" xref="S6.SS1.p4.8.m8.1.1"><csymbol cd="ambiguous" id="S6.SS1.p4.8.m8.1.1.1.cmml" xref="S6.SS1.p4.8.m8.1.1">subscript</csymbol><ci id="S6.SS1.p4.8.m8.1.1.2.cmml" xref="S6.SS1.p4.8.m8.1.1.2">𝑇</ci><ci id="S6.SS1.p4.8.m8.1.1.3.cmml" xref="S6.SS1.p4.8.m8.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p4.8.m8.1c">T_{A}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p4.8.m8.1d">italic_T start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT</annotation></semantics></math>. Needless to say, <math alttext="T_{A}" class="ltx_Math" display="inline" id="S6.SS1.p4.9.m9.1"><semantics id="S6.SS1.p4.9.m9.1a"><msub id="S6.SS1.p4.9.m9.1.1" xref="S6.SS1.p4.9.m9.1.1.cmml"><mi id="S6.SS1.p4.9.m9.1.1.2" xref="S6.SS1.p4.9.m9.1.1.2.cmml">T</mi><mi id="S6.SS1.p4.9.m9.1.1.3" xref="S6.SS1.p4.9.m9.1.1.3.cmml">A</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS1.p4.9.m9.1b"><apply id="S6.SS1.p4.9.m9.1.1.cmml" xref="S6.SS1.p4.9.m9.1.1"><csymbol cd="ambiguous" id="S6.SS1.p4.9.m9.1.1.1.cmml" xref="S6.SS1.p4.9.m9.1.1">subscript</csymbol><ci id="S6.SS1.p4.9.m9.1.1.2.cmml" xref="S6.SS1.p4.9.m9.1.1.2">𝑇</ci><ci id="S6.SS1.p4.9.m9.1.1.3.cmml" xref="S6.SS1.p4.9.m9.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p4.9.m9.1c">T_{A}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p4.9.m9.1d">italic_T start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT</annotation></semantics></math> is an Aronszajn tree.</p> </div> <div class="ltx_para" id="S6.SS1.p5"> <p class="ltx_p" id="S6.SS1.p5.8">For the rest of this section fix <math alttext="A" class="ltx_Math" display="inline" id="S6.SS1.p5.1.m1.1"><semantics id="S6.SS1.p5.1.m1.1a"><mi id="S6.SS1.p5.1.m1.1.1" xref="S6.SS1.p5.1.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S6.SS1.p5.1.m1.1b"><ci id="S6.SS1.p5.1.m1.1.1.cmml" xref="S6.SS1.p5.1.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p5.1.m1.1c">A</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p5.1.m1.1d">italic_A</annotation></semantics></math> and <math alttext="X" class="ltx_Math" display="inline" id="S6.SS1.p5.2.m2.1"><semantics id="S6.SS1.p5.2.m2.1a"><mi id="S6.SS1.p5.2.m2.1.1" xref="S6.SS1.p5.2.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S6.SS1.p5.2.m2.1b"><ci id="S6.SS1.p5.2.m2.1.1.cmml" xref="S6.SS1.p5.2.m2.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p5.2.m2.1c">X</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p5.2.m2.1d">italic_X</annotation></semantics></math> Countryman lines with underlying set <math alttext="\omega_{1}" class="ltx_Math" display="inline" id="S6.SS1.p5.3.m3.1"><semantics id="S6.SS1.p5.3.m3.1a"><msub id="S6.SS1.p5.3.m3.1.1" xref="S6.SS1.p5.3.m3.1.1.cmml"><mi id="S6.SS1.p5.3.m3.1.1.2" xref="S6.SS1.p5.3.m3.1.1.2.cmml">ω</mi><mn id="S6.SS1.p5.3.m3.1.1.3" xref="S6.SS1.p5.3.m3.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S6.SS1.p5.3.m3.1b"><apply id="S6.SS1.p5.3.m3.1.1.cmml" xref="S6.SS1.p5.3.m3.1.1"><csymbol cd="ambiguous" id="S6.SS1.p5.3.m3.1.1.1.cmml" xref="S6.SS1.p5.3.m3.1.1">subscript</csymbol><ci id="S6.SS1.p5.3.m3.1.1.2.cmml" xref="S6.SS1.p5.3.m3.1.1.2">𝜔</ci><cn id="S6.SS1.p5.3.m3.1.1.3.cmml" type="integer" xref="S6.SS1.p5.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p5.3.m3.1c">\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p5.3.m3.1d">italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>. The natural forcing for adding an isomorphism between <math alttext="A" class="ltx_Math" display="inline" id="S6.SS1.p5.4.m4.1"><semantics id="S6.SS1.p5.4.m4.1a"><mi id="S6.SS1.p5.4.m4.1.1" xref="S6.SS1.p5.4.m4.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S6.SS1.p5.4.m4.1b"><ci id="S6.SS1.p5.4.m4.1.1.cmml" xref="S6.SS1.p5.4.m4.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p5.4.m4.1c">A</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p5.4.m4.1d">italic_A</annotation></semantics></math> and <math alttext="X" class="ltx_Math" display="inline" id="S6.SS1.p5.5.m5.1"><semantics id="S6.SS1.p5.5.m5.1a"><mi id="S6.SS1.p5.5.m5.1.1" xref="S6.SS1.p5.5.m5.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S6.SS1.p5.5.m5.1b"><ci id="S6.SS1.p5.5.m5.1.1.cmml" xref="S6.SS1.p5.5.m5.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p5.5.m5.1c">X</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p5.5.m5.1d">italic_X</annotation></semantics></math>, would be to consider the set of finite partial functions from <math alttext="A" class="ltx_Math" display="inline" id="S6.SS1.p5.6.m6.1"><semantics id="S6.SS1.p5.6.m6.1a"><mi id="S6.SS1.p5.6.m6.1.1" xref="S6.SS1.p5.6.m6.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S6.SS1.p5.6.m6.1b"><ci id="S6.SS1.p5.6.m6.1.1.cmml" xref="S6.SS1.p5.6.m6.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p5.6.m6.1c">A</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p5.6.m6.1d">italic_A</annotation></semantics></math> to <math alttext="X" class="ltx_Math" display="inline" id="S6.SS1.p5.7.m7.1"><semantics id="S6.SS1.p5.7.m7.1a"><mi id="S6.SS1.p5.7.m7.1.1" xref="S6.SS1.p5.7.m7.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S6.SS1.p5.7.m7.1b"><ci id="S6.SS1.p5.7.m7.1.1.cmml" xref="S6.SS1.p5.7.m7.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p5.7.m7.1c">X</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p5.7.m7.1d">italic_X</annotation></semantics></math> that are increasing. However this is easily seen to collapse <math alttext="\omega_{1}" class="ltx_Math" display="inline" id="S6.SS1.p5.8.m8.1"><semantics id="S6.SS1.p5.8.m8.1a"><msub id="S6.SS1.p5.8.m8.1.1" xref="S6.SS1.p5.8.m8.1.1.cmml"><mi id="S6.SS1.p5.8.m8.1.1.2" xref="S6.SS1.p5.8.m8.1.1.2.cmml">ω</mi><mn id="S6.SS1.p5.8.m8.1.1.3" xref="S6.SS1.p5.8.m8.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S6.SS1.p5.8.m8.1b"><apply id="S6.SS1.p5.8.m8.1.1.cmml" xref="S6.SS1.p5.8.m8.1.1"><csymbol cd="ambiguous" id="S6.SS1.p5.8.m8.1.1.1.cmml" xref="S6.SS1.p5.8.m8.1.1">subscript</csymbol><ci id="S6.SS1.p5.8.m8.1.1.2.cmml" xref="S6.SS1.p5.8.m8.1.1.2">𝜔</ci><cn id="S6.SS1.p5.8.m8.1.1.3.cmml" type="integer" xref="S6.SS1.p5.8.m8.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p5.8.m8.1c">\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p5.8.m8.1d">italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="S6.Thmtheorem4"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem4.1.1.1">Definition 6.4</span></span><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem4.2.2">.</span> </h6> <div class="ltx_para" id="S6.Thmtheorem4.p1"> <p class="ltx_p" id="S6.Thmtheorem4.p1.5">For <math alttext="E\subseteq\omega_{1}" class="ltx_Math" display="inline" id="S6.Thmtheorem4.p1.1.m1.1"><semantics id="S6.Thmtheorem4.p1.1.m1.1a"><mrow id="S6.Thmtheorem4.p1.1.m1.1.1" xref="S6.Thmtheorem4.p1.1.m1.1.1.cmml"><mi id="S6.Thmtheorem4.p1.1.m1.1.1.2" xref="S6.Thmtheorem4.p1.1.m1.1.1.2.cmml">E</mi><mo id="S6.Thmtheorem4.p1.1.m1.1.1.1" xref="S6.Thmtheorem4.p1.1.m1.1.1.1.cmml">⊆</mo><msub id="S6.Thmtheorem4.p1.1.m1.1.1.3" xref="S6.Thmtheorem4.p1.1.m1.1.1.3.cmml"><mi id="S6.Thmtheorem4.p1.1.m1.1.1.3.2" xref="S6.Thmtheorem4.p1.1.m1.1.1.3.2.cmml">ω</mi><mn id="S6.Thmtheorem4.p1.1.m1.1.1.3.3" xref="S6.Thmtheorem4.p1.1.m1.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem4.p1.1.m1.1b"><apply id="S6.Thmtheorem4.p1.1.m1.1.1.cmml" xref="S6.Thmtheorem4.p1.1.m1.1.1"><subset id="S6.Thmtheorem4.p1.1.m1.1.1.1.cmml" xref="S6.Thmtheorem4.p1.1.m1.1.1.1"></subset><ci id="S6.Thmtheorem4.p1.1.m1.1.1.2.cmml" xref="S6.Thmtheorem4.p1.1.m1.1.1.2">𝐸</ci><apply id="S6.Thmtheorem4.p1.1.m1.1.1.3.cmml" xref="S6.Thmtheorem4.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S6.Thmtheorem4.p1.1.m1.1.1.3.1.cmml" xref="S6.Thmtheorem4.p1.1.m1.1.1.3">subscript</csymbol><ci id="S6.Thmtheorem4.p1.1.m1.1.1.3.2.cmml" xref="S6.Thmtheorem4.p1.1.m1.1.1.3.2">𝜔</ci><cn id="S6.Thmtheorem4.p1.1.m1.1.1.3.3.cmml" type="integer" xref="S6.Thmtheorem4.p1.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem4.p1.1.m1.1c">E\subseteq\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem4.p1.1.m1.1d">italic_E ⊆ italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, the forcing <math alttext="Q_{E}:=Q_{E}(A,X)" class="ltx_Math" display="inline" id="S6.Thmtheorem4.p1.2.m2.2"><semantics id="S6.Thmtheorem4.p1.2.m2.2a"><mrow id="S6.Thmtheorem4.p1.2.m2.2.3" xref="S6.Thmtheorem4.p1.2.m2.2.3.cmml"><msub id="S6.Thmtheorem4.p1.2.m2.2.3.2" xref="S6.Thmtheorem4.p1.2.m2.2.3.2.cmml"><mi id="S6.Thmtheorem4.p1.2.m2.2.3.2.2" xref="S6.Thmtheorem4.p1.2.m2.2.3.2.2.cmml">Q</mi><mi id="S6.Thmtheorem4.p1.2.m2.2.3.2.3" xref="S6.Thmtheorem4.p1.2.m2.2.3.2.3.cmml">E</mi></msub><mo id="S6.Thmtheorem4.p1.2.m2.2.3.1" lspace="0.278em" rspace="0.278em" xref="S6.Thmtheorem4.p1.2.m2.2.3.1.cmml">:=</mo><mrow id="S6.Thmtheorem4.p1.2.m2.2.3.3" xref="S6.Thmtheorem4.p1.2.m2.2.3.3.cmml"><msub id="S6.Thmtheorem4.p1.2.m2.2.3.3.2" xref="S6.Thmtheorem4.p1.2.m2.2.3.3.2.cmml"><mi id="S6.Thmtheorem4.p1.2.m2.2.3.3.2.2" xref="S6.Thmtheorem4.p1.2.m2.2.3.3.2.2.cmml">Q</mi><mi id="S6.Thmtheorem4.p1.2.m2.2.3.3.2.3" xref="S6.Thmtheorem4.p1.2.m2.2.3.3.2.3.cmml">E</mi></msub><mo id="S6.Thmtheorem4.p1.2.m2.2.3.3.1" xref="S6.Thmtheorem4.p1.2.m2.2.3.3.1.cmml">⁢</mo><mrow id="S6.Thmtheorem4.p1.2.m2.2.3.3.3.2" xref="S6.Thmtheorem4.p1.2.m2.2.3.3.3.1.cmml"><mo id="S6.Thmtheorem4.p1.2.m2.2.3.3.3.2.1" stretchy="false" xref="S6.Thmtheorem4.p1.2.m2.2.3.3.3.1.cmml">(</mo><mi id="S6.Thmtheorem4.p1.2.m2.1.1" xref="S6.Thmtheorem4.p1.2.m2.1.1.cmml">A</mi><mo id="S6.Thmtheorem4.p1.2.m2.2.3.3.3.2.2" xref="S6.Thmtheorem4.p1.2.m2.2.3.3.3.1.cmml">,</mo><mi id="S6.Thmtheorem4.p1.2.m2.2.2" xref="S6.Thmtheorem4.p1.2.m2.2.2.cmml">X</mi><mo id="S6.Thmtheorem4.p1.2.m2.2.3.3.3.2.3" stretchy="false" xref="S6.Thmtheorem4.p1.2.m2.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem4.p1.2.m2.2b"><apply id="S6.Thmtheorem4.p1.2.m2.2.3.cmml" xref="S6.Thmtheorem4.p1.2.m2.2.3"><csymbol cd="latexml" id="S6.Thmtheorem4.p1.2.m2.2.3.1.cmml" xref="S6.Thmtheorem4.p1.2.m2.2.3.1">assign</csymbol><apply id="S6.Thmtheorem4.p1.2.m2.2.3.2.cmml" xref="S6.Thmtheorem4.p1.2.m2.2.3.2"><csymbol cd="ambiguous" id="S6.Thmtheorem4.p1.2.m2.2.3.2.1.cmml" xref="S6.Thmtheorem4.p1.2.m2.2.3.2">subscript</csymbol><ci id="S6.Thmtheorem4.p1.2.m2.2.3.2.2.cmml" xref="S6.Thmtheorem4.p1.2.m2.2.3.2.2">𝑄</ci><ci id="S6.Thmtheorem4.p1.2.m2.2.3.2.3.cmml" xref="S6.Thmtheorem4.p1.2.m2.2.3.2.3">𝐸</ci></apply><apply id="S6.Thmtheorem4.p1.2.m2.2.3.3.cmml" xref="S6.Thmtheorem4.p1.2.m2.2.3.3"><times id="S6.Thmtheorem4.p1.2.m2.2.3.3.1.cmml" xref="S6.Thmtheorem4.p1.2.m2.2.3.3.1"></times><apply id="S6.Thmtheorem4.p1.2.m2.2.3.3.2.cmml" xref="S6.Thmtheorem4.p1.2.m2.2.3.3.2"><csymbol cd="ambiguous" id="S6.Thmtheorem4.p1.2.m2.2.3.3.2.1.cmml" xref="S6.Thmtheorem4.p1.2.m2.2.3.3.2">subscript</csymbol><ci id="S6.Thmtheorem4.p1.2.m2.2.3.3.2.2.cmml" xref="S6.Thmtheorem4.p1.2.m2.2.3.3.2.2">𝑄</ci><ci id="S6.Thmtheorem4.p1.2.m2.2.3.3.2.3.cmml" xref="S6.Thmtheorem4.p1.2.m2.2.3.3.2.3">𝐸</ci></apply><interval closure="open" id="S6.Thmtheorem4.p1.2.m2.2.3.3.3.1.cmml" xref="S6.Thmtheorem4.p1.2.m2.2.3.3.3.2"><ci id="S6.Thmtheorem4.p1.2.m2.1.1.cmml" xref="S6.Thmtheorem4.p1.2.m2.1.1">𝐴</ci><ci id="S6.Thmtheorem4.p1.2.m2.2.2.cmml" xref="S6.Thmtheorem4.p1.2.m2.2.2">𝑋</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem4.p1.2.m2.2c">Q_{E}:=Q_{E}(A,X)</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem4.p1.2.m2.2d">italic_Q start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT := italic_Q start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT ( italic_A , italic_X )</annotation></semantics></math> consists of the <math alttext="q" class="ltx_Math" display="inline" id="S6.Thmtheorem4.p1.3.m3.1"><semantics id="S6.Thmtheorem4.p1.3.m3.1a"><mi id="S6.Thmtheorem4.p1.3.m3.1.1" xref="S6.Thmtheorem4.p1.3.m3.1.1.cmml">q</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem4.p1.3.m3.1b"><ci id="S6.Thmtheorem4.p1.3.m3.1.1.cmml" xref="S6.Thmtheorem4.p1.3.m3.1.1">𝑞</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem4.p1.3.m3.1c">q</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem4.p1.3.m3.1d">italic_q</annotation></semantics></math> that are finite partial functions from <math alttext="A" class="ltx_Math" display="inline" id="S6.Thmtheorem4.p1.4.m4.1"><semantics id="S6.Thmtheorem4.p1.4.m4.1a"><mi id="S6.Thmtheorem4.p1.4.m4.1.1" xref="S6.Thmtheorem4.p1.4.m4.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem4.p1.4.m4.1b"><ci id="S6.Thmtheorem4.p1.4.m4.1.1.cmml" xref="S6.Thmtheorem4.p1.4.m4.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem4.p1.4.m4.1c">A</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem4.p1.4.m4.1d">italic_A</annotation></semantics></math> to <math alttext="X" class="ltx_Math" display="inline" id="S6.Thmtheorem4.p1.5.m5.1"><semantics id="S6.Thmtheorem4.p1.5.m5.1a"><mi id="S6.Thmtheorem4.p1.5.m5.1.1" xref="S6.Thmtheorem4.p1.5.m5.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem4.p1.5.m5.1b"><ci id="S6.Thmtheorem4.p1.5.m5.1.1.cmml" xref="S6.Thmtheorem4.p1.5.m5.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem4.p1.5.m5.1c">X</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem4.p1.5.m5.1d">italic_X</annotation></semantics></math> such that,</p> <ul class="ltx_itemize" id="S6.I2"> <li class="ltx_item" id="S6.I2.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S6.I2.i1.p1"> <p class="ltx_p" id="S6.I2.i1.p1.4"><math alttext="q" class="ltx_Math" display="inline" id="S6.I2.i1.p1.1.m1.1"><semantics id="S6.I2.i1.p1.1.m1.1a"><mi id="S6.I2.i1.p1.1.m1.1.1" xref="S6.I2.i1.p1.1.m1.1.1.cmml">q</mi><annotation-xml encoding="MathML-Content" id="S6.I2.i1.p1.1.m1.1b"><ci id="S6.I2.i1.p1.1.m1.1.1.cmml" xref="S6.I2.i1.p1.1.m1.1.1">𝑞</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.I2.i1.p1.1.m1.1c">q</annotation><annotation encoding="application/x-llamapun" id="S6.I2.i1.p1.1.m1.1d">italic_q</annotation></semantics></math> is increasing in the sense that if <math alttext="a<_{A}b" class="ltx_Math" display="inline" id="S6.I2.i1.p1.2.m2.1"><semantics id="S6.I2.i1.p1.2.m2.1a"><mrow id="S6.I2.i1.p1.2.m2.1.1" xref="S6.I2.i1.p1.2.m2.1.1.cmml"><mi id="S6.I2.i1.p1.2.m2.1.1.2" xref="S6.I2.i1.p1.2.m2.1.1.2.cmml">a</mi><msub id="S6.I2.i1.p1.2.m2.1.1.1" xref="S6.I2.i1.p1.2.m2.1.1.1.cmml"><mo id="S6.I2.i1.p1.2.m2.1.1.1.2" xref="S6.I2.i1.p1.2.m2.1.1.1.2.cmml"><</mo><mi id="S6.I2.i1.p1.2.m2.1.1.1.3" xref="S6.I2.i1.p1.2.m2.1.1.1.3.cmml">A</mi></msub><mi id="S6.I2.i1.p1.2.m2.1.1.3" xref="S6.I2.i1.p1.2.m2.1.1.3.cmml">b</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.I2.i1.p1.2.m2.1b"><apply id="S6.I2.i1.p1.2.m2.1.1.cmml" xref="S6.I2.i1.p1.2.m2.1.1"><apply id="S6.I2.i1.p1.2.m2.1.1.1.cmml" xref="S6.I2.i1.p1.2.m2.1.1.1"><csymbol cd="ambiguous" id="S6.I2.i1.p1.2.m2.1.1.1.1.cmml" xref="S6.I2.i1.p1.2.m2.1.1.1">subscript</csymbol><lt id="S6.I2.i1.p1.2.m2.1.1.1.2.cmml" xref="S6.I2.i1.p1.2.m2.1.1.1.2"></lt><ci id="S6.I2.i1.p1.2.m2.1.1.1.3.cmml" xref="S6.I2.i1.p1.2.m2.1.1.1.3">𝐴</ci></apply><ci id="S6.I2.i1.p1.2.m2.1.1.2.cmml" xref="S6.I2.i1.p1.2.m2.1.1.2">𝑎</ci><ci id="S6.I2.i1.p1.2.m2.1.1.3.cmml" xref="S6.I2.i1.p1.2.m2.1.1.3">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I2.i1.p1.2.m2.1c">a<_{A}b</annotation><annotation encoding="application/x-llamapun" id="S6.I2.i1.p1.2.m2.1d">italic_a < start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_b</annotation></semantics></math> are in <math alttext="\operatorname{dom}(q)" class="ltx_Math" display="inline" id="S6.I2.i1.p1.3.m3.2"><semantics id="S6.I2.i1.p1.3.m3.2a"><mrow id="S6.I2.i1.p1.3.m3.2.3.2" xref="S6.I2.i1.p1.3.m3.2.3.1.cmml"><mi id="S6.I2.i1.p1.3.m3.1.1" xref="S6.I2.i1.p1.3.m3.1.1.cmml">dom</mi><mo id="S6.I2.i1.p1.3.m3.2.3.2a" xref="S6.I2.i1.p1.3.m3.2.3.1.cmml">⁡</mo><mrow id="S6.I2.i1.p1.3.m3.2.3.2.1" xref="S6.I2.i1.p1.3.m3.2.3.1.cmml"><mo id="S6.I2.i1.p1.3.m3.2.3.2.1.1" stretchy="false" xref="S6.I2.i1.p1.3.m3.2.3.1.cmml">(</mo><mi id="S6.I2.i1.p1.3.m3.2.2" xref="S6.I2.i1.p1.3.m3.2.2.cmml">q</mi><mo id="S6.I2.i1.p1.3.m3.2.3.2.1.2" stretchy="false" xref="S6.I2.i1.p1.3.m3.2.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.I2.i1.p1.3.m3.2b"><apply id="S6.I2.i1.p1.3.m3.2.3.1.cmml" xref="S6.I2.i1.p1.3.m3.2.3.2"><ci id="S6.I2.i1.p1.3.m3.1.1.cmml" xref="S6.I2.i1.p1.3.m3.1.1">dom</ci><ci id="S6.I2.i1.p1.3.m3.2.2.cmml" xref="S6.I2.i1.p1.3.m3.2.2">𝑞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I2.i1.p1.3.m3.2c">\operatorname{dom}(q)</annotation><annotation encoding="application/x-llamapun" id="S6.I2.i1.p1.3.m3.2d">roman_dom ( italic_q )</annotation></semantics></math>, then <math alttext="q(a)<_{A}q(b)" class="ltx_Math" display="inline" id="S6.I2.i1.p1.4.m4.2"><semantics id="S6.I2.i1.p1.4.m4.2a"><mrow id="S6.I2.i1.p1.4.m4.2.3" xref="S6.I2.i1.p1.4.m4.2.3.cmml"><mrow id="S6.I2.i1.p1.4.m4.2.3.2" xref="S6.I2.i1.p1.4.m4.2.3.2.cmml"><mi id="S6.I2.i1.p1.4.m4.2.3.2.2" xref="S6.I2.i1.p1.4.m4.2.3.2.2.cmml">q</mi><mo id="S6.I2.i1.p1.4.m4.2.3.2.1" xref="S6.I2.i1.p1.4.m4.2.3.2.1.cmml">⁢</mo><mrow id="S6.I2.i1.p1.4.m4.2.3.2.3.2" xref="S6.I2.i1.p1.4.m4.2.3.2.cmml"><mo id="S6.I2.i1.p1.4.m4.2.3.2.3.2.1" stretchy="false" xref="S6.I2.i1.p1.4.m4.2.3.2.cmml">(</mo><mi id="S6.I2.i1.p1.4.m4.1.1" xref="S6.I2.i1.p1.4.m4.1.1.cmml">a</mi><mo id="S6.I2.i1.p1.4.m4.2.3.2.3.2.2" stretchy="false" xref="S6.I2.i1.p1.4.m4.2.3.2.cmml">)</mo></mrow></mrow><msub id="S6.I2.i1.p1.4.m4.2.3.1" xref="S6.I2.i1.p1.4.m4.2.3.1.cmml"><mo id="S6.I2.i1.p1.4.m4.2.3.1.2" xref="S6.I2.i1.p1.4.m4.2.3.1.2.cmml"><</mo><mi id="S6.I2.i1.p1.4.m4.2.3.1.3" xref="S6.I2.i1.p1.4.m4.2.3.1.3.cmml">A</mi></msub><mrow id="S6.I2.i1.p1.4.m4.2.3.3" xref="S6.I2.i1.p1.4.m4.2.3.3.cmml"><mi id="S6.I2.i1.p1.4.m4.2.3.3.2" xref="S6.I2.i1.p1.4.m4.2.3.3.2.cmml">q</mi><mo id="S6.I2.i1.p1.4.m4.2.3.3.1" xref="S6.I2.i1.p1.4.m4.2.3.3.1.cmml">⁢</mo><mrow id="S6.I2.i1.p1.4.m4.2.3.3.3.2" xref="S6.I2.i1.p1.4.m4.2.3.3.cmml"><mo id="S6.I2.i1.p1.4.m4.2.3.3.3.2.1" stretchy="false" xref="S6.I2.i1.p1.4.m4.2.3.3.cmml">(</mo><mi id="S6.I2.i1.p1.4.m4.2.2" xref="S6.I2.i1.p1.4.m4.2.2.cmml">b</mi><mo id="S6.I2.i1.p1.4.m4.2.3.3.3.2.2" stretchy="false" xref="S6.I2.i1.p1.4.m4.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.I2.i1.p1.4.m4.2b"><apply id="S6.I2.i1.p1.4.m4.2.3.cmml" xref="S6.I2.i1.p1.4.m4.2.3"><apply id="S6.I2.i1.p1.4.m4.2.3.1.cmml" xref="S6.I2.i1.p1.4.m4.2.3.1"><csymbol cd="ambiguous" id="S6.I2.i1.p1.4.m4.2.3.1.1.cmml" xref="S6.I2.i1.p1.4.m4.2.3.1">subscript</csymbol><lt id="S6.I2.i1.p1.4.m4.2.3.1.2.cmml" xref="S6.I2.i1.p1.4.m4.2.3.1.2"></lt><ci id="S6.I2.i1.p1.4.m4.2.3.1.3.cmml" xref="S6.I2.i1.p1.4.m4.2.3.1.3">𝐴</ci></apply><apply id="S6.I2.i1.p1.4.m4.2.3.2.cmml" xref="S6.I2.i1.p1.4.m4.2.3.2"><times id="S6.I2.i1.p1.4.m4.2.3.2.1.cmml" xref="S6.I2.i1.p1.4.m4.2.3.2.1"></times><ci id="S6.I2.i1.p1.4.m4.2.3.2.2.cmml" xref="S6.I2.i1.p1.4.m4.2.3.2.2">𝑞</ci><ci id="S6.I2.i1.p1.4.m4.1.1.cmml" xref="S6.I2.i1.p1.4.m4.1.1">𝑎</ci></apply><apply id="S6.I2.i1.p1.4.m4.2.3.3.cmml" xref="S6.I2.i1.p1.4.m4.2.3.3"><times id="S6.I2.i1.p1.4.m4.2.3.3.1.cmml" xref="S6.I2.i1.p1.4.m4.2.3.3.1"></times><ci id="S6.I2.i1.p1.4.m4.2.3.3.2.cmml" xref="S6.I2.i1.p1.4.m4.2.3.3.2">𝑞</ci><ci id="S6.I2.i1.p1.4.m4.2.2.cmml" xref="S6.I2.i1.p1.4.m4.2.2">𝑏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I2.i1.p1.4.m4.2c">q(a)<_{A}q(b)</annotation><annotation encoding="application/x-llamapun" id="S6.I2.i1.p1.4.m4.2d">italic_q ( italic_a ) < start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_q ( italic_b )</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S6.I2.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S6.I2.i2.p1"> <p class="ltx_p" id="S6.I2.i2.p1.3">For all <math alttext="\nu\in E" class="ltx_Math" display="inline" id="S6.I2.i2.p1.1.m1.1"><semantics id="S6.I2.i2.p1.1.m1.1a"><mrow id="S6.I2.i2.p1.1.m1.1.1" xref="S6.I2.i2.p1.1.m1.1.1.cmml"><mi id="S6.I2.i2.p1.1.m1.1.1.2" xref="S6.I2.i2.p1.1.m1.1.1.2.cmml">ν</mi><mo id="S6.I2.i2.p1.1.m1.1.1.1" xref="S6.I2.i2.p1.1.m1.1.1.1.cmml">∈</mo><mi id="S6.I2.i2.p1.1.m1.1.1.3" xref="S6.I2.i2.p1.1.m1.1.1.3.cmml">E</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.I2.i2.p1.1.m1.1b"><apply id="S6.I2.i2.p1.1.m1.1.1.cmml" xref="S6.I2.i2.p1.1.m1.1.1"><in id="S6.I2.i2.p1.1.m1.1.1.1.cmml" xref="S6.I2.i2.p1.1.m1.1.1.1"></in><ci id="S6.I2.i2.p1.1.m1.1.1.2.cmml" xref="S6.I2.i2.p1.1.m1.1.1.2">𝜈</ci><ci id="S6.I2.i2.p1.1.m1.1.1.3.cmml" xref="S6.I2.i2.p1.1.m1.1.1.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I2.i2.p1.1.m1.1c">\nu\in E</annotation><annotation encoding="application/x-llamapun" id="S6.I2.i2.p1.1.m1.1d">italic_ν ∈ italic_E</annotation></semantics></math> and <math alttext="a\in\operatorname{dom}(q)" class="ltx_Math" display="inline" id="S6.I2.i2.p1.2.m2.2"><semantics id="S6.I2.i2.p1.2.m2.2a"><mrow id="S6.I2.i2.p1.2.m2.2.3" xref="S6.I2.i2.p1.2.m2.2.3.cmml"><mi id="S6.I2.i2.p1.2.m2.2.3.2" xref="S6.I2.i2.p1.2.m2.2.3.2.cmml">a</mi><mo id="S6.I2.i2.p1.2.m2.2.3.1" xref="S6.I2.i2.p1.2.m2.2.3.1.cmml">∈</mo><mrow id="S6.I2.i2.p1.2.m2.2.3.3.2" xref="S6.I2.i2.p1.2.m2.2.3.3.1.cmml"><mi id="S6.I2.i2.p1.2.m2.1.1" xref="S6.I2.i2.p1.2.m2.1.1.cmml">dom</mi><mo id="S6.I2.i2.p1.2.m2.2.3.3.2a" xref="S6.I2.i2.p1.2.m2.2.3.3.1.cmml">⁡</mo><mrow id="S6.I2.i2.p1.2.m2.2.3.3.2.1" xref="S6.I2.i2.p1.2.m2.2.3.3.1.cmml"><mo id="S6.I2.i2.p1.2.m2.2.3.3.2.1.1" stretchy="false" xref="S6.I2.i2.p1.2.m2.2.3.3.1.cmml">(</mo><mi id="S6.I2.i2.p1.2.m2.2.2" xref="S6.I2.i2.p1.2.m2.2.2.cmml">q</mi><mo id="S6.I2.i2.p1.2.m2.2.3.3.2.1.2" stretchy="false" xref="S6.I2.i2.p1.2.m2.2.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.I2.i2.p1.2.m2.2b"><apply id="S6.I2.i2.p1.2.m2.2.3.cmml" xref="S6.I2.i2.p1.2.m2.2.3"><in id="S6.I2.i2.p1.2.m2.2.3.1.cmml" xref="S6.I2.i2.p1.2.m2.2.3.1"></in><ci id="S6.I2.i2.p1.2.m2.2.3.2.cmml" xref="S6.I2.i2.p1.2.m2.2.3.2">𝑎</ci><apply id="S6.I2.i2.p1.2.m2.2.3.3.1.cmml" xref="S6.I2.i2.p1.2.m2.2.3.3.2"><ci id="S6.I2.i2.p1.2.m2.1.1.cmml" xref="S6.I2.i2.p1.2.m2.1.1">dom</ci><ci id="S6.I2.i2.p1.2.m2.2.2.cmml" xref="S6.I2.i2.p1.2.m2.2.2">𝑞</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I2.i2.p1.2.m2.2c">a\in\operatorname{dom}(q)</annotation><annotation encoding="application/x-llamapun" id="S6.I2.i2.p1.2.m2.2d">italic_a ∈ roman_dom ( italic_q )</annotation></semantics></math>, <math alttext="a<\nu\leftrightarrow q(a)<\nu" class="ltx_Math" display="inline" id="S6.I2.i2.p1.3.m3.1"><semantics id="S6.I2.i2.p1.3.m3.1a"><mrow id="S6.I2.i2.p1.3.m3.1.2" xref="S6.I2.i2.p1.3.m3.1.2.cmml"><mrow id="S6.I2.i2.p1.3.m3.1.2.2" xref="S6.I2.i2.p1.3.m3.1.2.2.cmml"><mi id="S6.I2.i2.p1.3.m3.1.2.2.2" xref="S6.I2.i2.p1.3.m3.1.2.2.2.cmml">a</mi><mo id="S6.I2.i2.p1.3.m3.1.2.2.1" xref="S6.I2.i2.p1.3.m3.1.2.2.1.cmml"><</mo><mi id="S6.I2.i2.p1.3.m3.1.2.2.3" xref="S6.I2.i2.p1.3.m3.1.2.2.3.cmml">ν</mi></mrow><mo id="S6.I2.i2.p1.3.m3.1.2.1" stretchy="false" xref="S6.I2.i2.p1.3.m3.1.2.1.cmml">↔</mo><mrow id="S6.I2.i2.p1.3.m3.1.2.3" xref="S6.I2.i2.p1.3.m3.1.2.3.cmml"><mrow id="S6.I2.i2.p1.3.m3.1.2.3.2" xref="S6.I2.i2.p1.3.m3.1.2.3.2.cmml"><mi id="S6.I2.i2.p1.3.m3.1.2.3.2.2" xref="S6.I2.i2.p1.3.m3.1.2.3.2.2.cmml">q</mi><mo id="S6.I2.i2.p1.3.m3.1.2.3.2.1" xref="S6.I2.i2.p1.3.m3.1.2.3.2.1.cmml">⁢</mo><mrow id="S6.I2.i2.p1.3.m3.1.2.3.2.3.2" xref="S6.I2.i2.p1.3.m3.1.2.3.2.cmml"><mo id="S6.I2.i2.p1.3.m3.1.2.3.2.3.2.1" stretchy="false" xref="S6.I2.i2.p1.3.m3.1.2.3.2.cmml">(</mo><mi id="S6.I2.i2.p1.3.m3.1.1" xref="S6.I2.i2.p1.3.m3.1.1.cmml">a</mi><mo id="S6.I2.i2.p1.3.m3.1.2.3.2.3.2.2" stretchy="false" xref="S6.I2.i2.p1.3.m3.1.2.3.2.cmml">)</mo></mrow></mrow><mo id="S6.I2.i2.p1.3.m3.1.2.3.1" xref="S6.I2.i2.p1.3.m3.1.2.3.1.cmml"><</mo><mi id="S6.I2.i2.p1.3.m3.1.2.3.3" xref="S6.I2.i2.p1.3.m3.1.2.3.3.cmml">ν</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.I2.i2.p1.3.m3.1b"><apply id="S6.I2.i2.p1.3.m3.1.2.cmml" xref="S6.I2.i2.p1.3.m3.1.2"><ci id="S6.I2.i2.p1.3.m3.1.2.1.cmml" xref="S6.I2.i2.p1.3.m3.1.2.1">↔</ci><apply id="S6.I2.i2.p1.3.m3.1.2.2.cmml" xref="S6.I2.i2.p1.3.m3.1.2.2"><lt id="S6.I2.i2.p1.3.m3.1.2.2.1.cmml" xref="S6.I2.i2.p1.3.m3.1.2.2.1"></lt><ci id="S6.I2.i2.p1.3.m3.1.2.2.2.cmml" xref="S6.I2.i2.p1.3.m3.1.2.2.2">𝑎</ci><ci id="S6.I2.i2.p1.3.m3.1.2.2.3.cmml" xref="S6.I2.i2.p1.3.m3.1.2.2.3">𝜈</ci></apply><apply id="S6.I2.i2.p1.3.m3.1.2.3.cmml" xref="S6.I2.i2.p1.3.m3.1.2.3"><lt id="S6.I2.i2.p1.3.m3.1.2.3.1.cmml" xref="S6.I2.i2.p1.3.m3.1.2.3.1"></lt><apply id="S6.I2.i2.p1.3.m3.1.2.3.2.cmml" xref="S6.I2.i2.p1.3.m3.1.2.3.2"><times id="S6.I2.i2.p1.3.m3.1.2.3.2.1.cmml" xref="S6.I2.i2.p1.3.m3.1.2.3.2.1"></times><ci id="S6.I2.i2.p1.3.m3.1.2.3.2.2.cmml" xref="S6.I2.i2.p1.3.m3.1.2.3.2.2">𝑞</ci><ci id="S6.I2.i2.p1.3.m3.1.1.cmml" xref="S6.I2.i2.p1.3.m3.1.1">𝑎</ci></apply><ci id="S6.I2.i2.p1.3.m3.1.2.3.3.cmml" xref="S6.I2.i2.p1.3.m3.1.2.3.3">𝜈</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I2.i2.p1.3.m3.1c">a<\nu\leftrightarrow q(a)<\nu</annotation><annotation encoding="application/x-llamapun" id="S6.I2.i2.p1.3.m3.1d">italic_a < italic_ν ↔ italic_q ( italic_a ) < italic_ν</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S6.I2.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S6.I2.i3.p1"> <p class="ltx_p" id="S6.I2.i3.p1.3">For all <math alttext="\nu\in E" class="ltx_Math" display="inline" id="S6.I2.i3.p1.1.m1.1"><semantics id="S6.I2.i3.p1.1.m1.1a"><mrow id="S6.I2.i3.p1.1.m1.1.1" xref="S6.I2.i3.p1.1.m1.1.1.cmml"><mi id="S6.I2.i3.p1.1.m1.1.1.2" xref="S6.I2.i3.p1.1.m1.1.1.2.cmml">ν</mi><mo id="S6.I2.i3.p1.1.m1.1.1.1" xref="S6.I2.i3.p1.1.m1.1.1.1.cmml">∈</mo><mi id="S6.I2.i3.p1.1.m1.1.1.3" xref="S6.I2.i3.p1.1.m1.1.1.3.cmml">E</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.I2.i3.p1.1.m1.1b"><apply id="S6.I2.i3.p1.1.m1.1.1.cmml" xref="S6.I2.i3.p1.1.m1.1.1"><in id="S6.I2.i3.p1.1.m1.1.1.1.cmml" xref="S6.I2.i3.p1.1.m1.1.1.1"></in><ci id="S6.I2.i3.p1.1.m1.1.1.2.cmml" xref="S6.I2.i3.p1.1.m1.1.1.2">𝜈</ci><ci id="S6.I2.i3.p1.1.m1.1.1.3.cmml" xref="S6.I2.i3.p1.1.m1.1.1.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I2.i3.p1.1.m1.1c">\nu\in E</annotation><annotation encoding="application/x-llamapun" id="S6.I2.i3.p1.1.m1.1d">italic_ν ∈ italic_E</annotation></semantics></math> and <math alttext="a\neq b\in\operatorname{dom}(q)" class="ltx_Math" display="inline" id="S6.I2.i3.p1.2.m2.2"><semantics id="S6.I2.i3.p1.2.m2.2a"><mrow id="S6.I2.i3.p1.2.m2.2.3" xref="S6.I2.i3.p1.2.m2.2.3.cmml"><mi id="S6.I2.i3.p1.2.m2.2.3.2" xref="S6.I2.i3.p1.2.m2.2.3.2.cmml">a</mi><mo id="S6.I2.i3.p1.2.m2.2.3.3" xref="S6.I2.i3.p1.2.m2.2.3.3.cmml">≠</mo><mi id="S6.I2.i3.p1.2.m2.2.3.4" xref="S6.I2.i3.p1.2.m2.2.3.4.cmml">b</mi><mo id="S6.I2.i3.p1.2.m2.2.3.5" xref="S6.I2.i3.p1.2.m2.2.3.5.cmml">∈</mo><mrow id="S6.I2.i3.p1.2.m2.2.3.6.2" xref="S6.I2.i3.p1.2.m2.2.3.6.1.cmml"><mi id="S6.I2.i3.p1.2.m2.1.1" xref="S6.I2.i3.p1.2.m2.1.1.cmml">dom</mi><mo id="S6.I2.i3.p1.2.m2.2.3.6.2a" xref="S6.I2.i3.p1.2.m2.2.3.6.1.cmml">⁡</mo><mrow id="S6.I2.i3.p1.2.m2.2.3.6.2.1" xref="S6.I2.i3.p1.2.m2.2.3.6.1.cmml"><mo id="S6.I2.i3.p1.2.m2.2.3.6.2.1.1" stretchy="false" xref="S6.I2.i3.p1.2.m2.2.3.6.1.cmml">(</mo><mi id="S6.I2.i3.p1.2.m2.2.2" xref="S6.I2.i3.p1.2.m2.2.2.cmml">q</mi><mo id="S6.I2.i3.p1.2.m2.2.3.6.2.1.2" stretchy="false" xref="S6.I2.i3.p1.2.m2.2.3.6.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.I2.i3.p1.2.m2.2b"><apply id="S6.I2.i3.p1.2.m2.2.3.cmml" xref="S6.I2.i3.p1.2.m2.2.3"><and id="S6.I2.i3.p1.2.m2.2.3a.cmml" xref="S6.I2.i3.p1.2.m2.2.3"></and><apply id="S6.I2.i3.p1.2.m2.2.3b.cmml" xref="S6.I2.i3.p1.2.m2.2.3"><neq id="S6.I2.i3.p1.2.m2.2.3.3.cmml" xref="S6.I2.i3.p1.2.m2.2.3.3"></neq><ci id="S6.I2.i3.p1.2.m2.2.3.2.cmml" xref="S6.I2.i3.p1.2.m2.2.3.2">𝑎</ci><ci id="S6.I2.i3.p1.2.m2.2.3.4.cmml" xref="S6.I2.i3.p1.2.m2.2.3.4">𝑏</ci></apply><apply id="S6.I2.i3.p1.2.m2.2.3c.cmml" xref="S6.I2.i3.p1.2.m2.2.3"><in id="S6.I2.i3.p1.2.m2.2.3.5.cmml" xref="S6.I2.i3.p1.2.m2.2.3.5"></in><share href="https://arxiv.org/html/2503.13728v1#S6.I2.i3.p1.2.m2.2.3.4.cmml" id="S6.I2.i3.p1.2.m2.2.3d.cmml" xref="S6.I2.i3.p1.2.m2.2.3"></share><apply id="S6.I2.i3.p1.2.m2.2.3.6.1.cmml" xref="S6.I2.i3.p1.2.m2.2.3.6.2"><ci id="S6.I2.i3.p1.2.m2.1.1.cmml" xref="S6.I2.i3.p1.2.m2.1.1">dom</ci><ci id="S6.I2.i3.p1.2.m2.2.2.cmml" xref="S6.I2.i3.p1.2.m2.2.2">𝑞</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I2.i3.p1.2.m2.2c">a\neq b\in\operatorname{dom}(q)</annotation><annotation encoding="application/x-llamapun" id="S6.I2.i3.p1.2.m2.2d">italic_a ≠ italic_b ∈ roman_dom ( italic_q )</annotation></semantics></math>, <math alttext="\Delta_{A}(a,b)<\nu\leftrightarrow\Delta_{X}(q(a),q(b))<\nu" class="ltx_Math" display="inline" id="S6.I2.i3.p1.3.m3.6"><semantics id="S6.I2.i3.p1.3.m3.6a"><mrow id="S6.I2.i3.p1.3.m3.6.6" xref="S6.I2.i3.p1.3.m3.6.6.cmml"><mrow id="S6.I2.i3.p1.3.m3.6.6.4" xref="S6.I2.i3.p1.3.m3.6.6.4.cmml"><mrow id="S6.I2.i3.p1.3.m3.6.6.4.2" xref="S6.I2.i3.p1.3.m3.6.6.4.2.cmml"><msub id="S6.I2.i3.p1.3.m3.6.6.4.2.2" xref="S6.I2.i3.p1.3.m3.6.6.4.2.2.cmml"><mi id="S6.I2.i3.p1.3.m3.6.6.4.2.2.2" mathvariant="normal" xref="S6.I2.i3.p1.3.m3.6.6.4.2.2.2.cmml">Δ</mi><mi id="S6.I2.i3.p1.3.m3.6.6.4.2.2.3" xref="S6.I2.i3.p1.3.m3.6.6.4.2.2.3.cmml">A</mi></msub><mo id="S6.I2.i3.p1.3.m3.6.6.4.2.1" xref="S6.I2.i3.p1.3.m3.6.6.4.2.1.cmml">⁢</mo><mrow id="S6.I2.i3.p1.3.m3.6.6.4.2.3.2" xref="S6.I2.i3.p1.3.m3.6.6.4.2.3.1.cmml"><mo id="S6.I2.i3.p1.3.m3.6.6.4.2.3.2.1" stretchy="false" xref="S6.I2.i3.p1.3.m3.6.6.4.2.3.1.cmml">(</mo><mi id="S6.I2.i3.p1.3.m3.1.1" xref="S6.I2.i3.p1.3.m3.1.1.cmml">a</mi><mo id="S6.I2.i3.p1.3.m3.6.6.4.2.3.2.2" xref="S6.I2.i3.p1.3.m3.6.6.4.2.3.1.cmml">,</mo><mi id="S6.I2.i3.p1.3.m3.2.2" xref="S6.I2.i3.p1.3.m3.2.2.cmml">b</mi><mo id="S6.I2.i3.p1.3.m3.6.6.4.2.3.2.3" stretchy="false" xref="S6.I2.i3.p1.3.m3.6.6.4.2.3.1.cmml">)</mo></mrow></mrow><mo id="S6.I2.i3.p1.3.m3.6.6.4.1" xref="S6.I2.i3.p1.3.m3.6.6.4.1.cmml"><</mo><mi id="S6.I2.i3.p1.3.m3.6.6.4.3" xref="S6.I2.i3.p1.3.m3.6.6.4.3.cmml">ν</mi></mrow><mo id="S6.I2.i3.p1.3.m3.6.6.3" stretchy="false" xref="S6.I2.i3.p1.3.m3.6.6.3.cmml">↔</mo><mrow id="S6.I2.i3.p1.3.m3.6.6.2" xref="S6.I2.i3.p1.3.m3.6.6.2.cmml"><mrow id="S6.I2.i3.p1.3.m3.6.6.2.2" xref="S6.I2.i3.p1.3.m3.6.6.2.2.cmml"><msub id="S6.I2.i3.p1.3.m3.6.6.2.2.4" xref="S6.I2.i3.p1.3.m3.6.6.2.2.4.cmml"><mi id="S6.I2.i3.p1.3.m3.6.6.2.2.4.2" mathvariant="normal" xref="S6.I2.i3.p1.3.m3.6.6.2.2.4.2.cmml">Δ</mi><mi id="S6.I2.i3.p1.3.m3.6.6.2.2.4.3" xref="S6.I2.i3.p1.3.m3.6.6.2.2.4.3.cmml">X</mi></msub><mo id="S6.I2.i3.p1.3.m3.6.6.2.2.3" xref="S6.I2.i3.p1.3.m3.6.6.2.2.3.cmml">⁢</mo><mrow id="S6.I2.i3.p1.3.m3.6.6.2.2.2.2" xref="S6.I2.i3.p1.3.m3.6.6.2.2.2.3.cmml"><mo id="S6.I2.i3.p1.3.m3.6.6.2.2.2.2.3" stretchy="false" xref="S6.I2.i3.p1.3.m3.6.6.2.2.2.3.cmml">(</mo><mrow id="S6.I2.i3.p1.3.m3.5.5.1.1.1.1.1" xref="S6.I2.i3.p1.3.m3.5.5.1.1.1.1.1.cmml"><mi id="S6.I2.i3.p1.3.m3.5.5.1.1.1.1.1.2" xref="S6.I2.i3.p1.3.m3.5.5.1.1.1.1.1.2.cmml">q</mi><mo id="S6.I2.i3.p1.3.m3.5.5.1.1.1.1.1.1" xref="S6.I2.i3.p1.3.m3.5.5.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S6.I2.i3.p1.3.m3.5.5.1.1.1.1.1.3.2" xref="S6.I2.i3.p1.3.m3.5.5.1.1.1.1.1.cmml"><mo id="S6.I2.i3.p1.3.m3.5.5.1.1.1.1.1.3.2.1" stretchy="false" xref="S6.I2.i3.p1.3.m3.5.5.1.1.1.1.1.cmml">(</mo><mi id="S6.I2.i3.p1.3.m3.3.3" xref="S6.I2.i3.p1.3.m3.3.3.cmml">a</mi><mo id="S6.I2.i3.p1.3.m3.5.5.1.1.1.1.1.3.2.2" stretchy="false" xref="S6.I2.i3.p1.3.m3.5.5.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.I2.i3.p1.3.m3.6.6.2.2.2.2.4" xref="S6.I2.i3.p1.3.m3.6.6.2.2.2.3.cmml">,</mo><mrow id="S6.I2.i3.p1.3.m3.6.6.2.2.2.2.2" xref="S6.I2.i3.p1.3.m3.6.6.2.2.2.2.2.cmml"><mi id="S6.I2.i3.p1.3.m3.6.6.2.2.2.2.2.2" xref="S6.I2.i3.p1.3.m3.6.6.2.2.2.2.2.2.cmml">q</mi><mo id="S6.I2.i3.p1.3.m3.6.6.2.2.2.2.2.1" xref="S6.I2.i3.p1.3.m3.6.6.2.2.2.2.2.1.cmml">⁢</mo><mrow id="S6.I2.i3.p1.3.m3.6.6.2.2.2.2.2.3.2" xref="S6.I2.i3.p1.3.m3.6.6.2.2.2.2.2.cmml"><mo id="S6.I2.i3.p1.3.m3.6.6.2.2.2.2.2.3.2.1" stretchy="false" xref="S6.I2.i3.p1.3.m3.6.6.2.2.2.2.2.cmml">(</mo><mi id="S6.I2.i3.p1.3.m3.4.4" xref="S6.I2.i3.p1.3.m3.4.4.cmml">b</mi><mo id="S6.I2.i3.p1.3.m3.6.6.2.2.2.2.2.3.2.2" stretchy="false" xref="S6.I2.i3.p1.3.m3.6.6.2.2.2.2.2.cmml">)</mo></mrow></mrow><mo id="S6.I2.i3.p1.3.m3.6.6.2.2.2.2.5" stretchy="false" xref="S6.I2.i3.p1.3.m3.6.6.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.I2.i3.p1.3.m3.6.6.2.3" xref="S6.I2.i3.p1.3.m3.6.6.2.3.cmml"><</mo><mi id="S6.I2.i3.p1.3.m3.6.6.2.4" xref="S6.I2.i3.p1.3.m3.6.6.2.4.cmml">ν</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.I2.i3.p1.3.m3.6b"><apply id="S6.I2.i3.p1.3.m3.6.6.cmml" xref="S6.I2.i3.p1.3.m3.6.6"><ci id="S6.I2.i3.p1.3.m3.6.6.3.cmml" xref="S6.I2.i3.p1.3.m3.6.6.3">↔</ci><apply id="S6.I2.i3.p1.3.m3.6.6.4.cmml" xref="S6.I2.i3.p1.3.m3.6.6.4"><lt id="S6.I2.i3.p1.3.m3.6.6.4.1.cmml" xref="S6.I2.i3.p1.3.m3.6.6.4.1"></lt><apply id="S6.I2.i3.p1.3.m3.6.6.4.2.cmml" xref="S6.I2.i3.p1.3.m3.6.6.4.2"><times id="S6.I2.i3.p1.3.m3.6.6.4.2.1.cmml" xref="S6.I2.i3.p1.3.m3.6.6.4.2.1"></times><apply id="S6.I2.i3.p1.3.m3.6.6.4.2.2.cmml" xref="S6.I2.i3.p1.3.m3.6.6.4.2.2"><csymbol cd="ambiguous" id="S6.I2.i3.p1.3.m3.6.6.4.2.2.1.cmml" xref="S6.I2.i3.p1.3.m3.6.6.4.2.2">subscript</csymbol><ci id="S6.I2.i3.p1.3.m3.6.6.4.2.2.2.cmml" xref="S6.I2.i3.p1.3.m3.6.6.4.2.2.2">Δ</ci><ci id="S6.I2.i3.p1.3.m3.6.6.4.2.2.3.cmml" xref="S6.I2.i3.p1.3.m3.6.6.4.2.2.3">𝐴</ci></apply><interval closure="open" id="S6.I2.i3.p1.3.m3.6.6.4.2.3.1.cmml" xref="S6.I2.i3.p1.3.m3.6.6.4.2.3.2"><ci id="S6.I2.i3.p1.3.m3.1.1.cmml" xref="S6.I2.i3.p1.3.m3.1.1">𝑎</ci><ci id="S6.I2.i3.p1.3.m3.2.2.cmml" xref="S6.I2.i3.p1.3.m3.2.2">𝑏</ci></interval></apply><ci id="S6.I2.i3.p1.3.m3.6.6.4.3.cmml" xref="S6.I2.i3.p1.3.m3.6.6.4.3">𝜈</ci></apply><apply id="S6.I2.i3.p1.3.m3.6.6.2.cmml" xref="S6.I2.i3.p1.3.m3.6.6.2"><lt id="S6.I2.i3.p1.3.m3.6.6.2.3.cmml" xref="S6.I2.i3.p1.3.m3.6.6.2.3"></lt><apply id="S6.I2.i3.p1.3.m3.6.6.2.2.cmml" xref="S6.I2.i3.p1.3.m3.6.6.2.2"><times id="S6.I2.i3.p1.3.m3.6.6.2.2.3.cmml" xref="S6.I2.i3.p1.3.m3.6.6.2.2.3"></times><apply id="S6.I2.i3.p1.3.m3.6.6.2.2.4.cmml" xref="S6.I2.i3.p1.3.m3.6.6.2.2.4"><csymbol cd="ambiguous" id="S6.I2.i3.p1.3.m3.6.6.2.2.4.1.cmml" xref="S6.I2.i3.p1.3.m3.6.6.2.2.4">subscript</csymbol><ci id="S6.I2.i3.p1.3.m3.6.6.2.2.4.2.cmml" xref="S6.I2.i3.p1.3.m3.6.6.2.2.4.2">Δ</ci><ci id="S6.I2.i3.p1.3.m3.6.6.2.2.4.3.cmml" xref="S6.I2.i3.p1.3.m3.6.6.2.2.4.3">𝑋</ci></apply><interval closure="open" id="S6.I2.i3.p1.3.m3.6.6.2.2.2.3.cmml" xref="S6.I2.i3.p1.3.m3.6.6.2.2.2.2"><apply id="S6.I2.i3.p1.3.m3.5.5.1.1.1.1.1.cmml" xref="S6.I2.i3.p1.3.m3.5.5.1.1.1.1.1"><times id="S6.I2.i3.p1.3.m3.5.5.1.1.1.1.1.1.cmml" xref="S6.I2.i3.p1.3.m3.5.5.1.1.1.1.1.1"></times><ci id="S6.I2.i3.p1.3.m3.5.5.1.1.1.1.1.2.cmml" xref="S6.I2.i3.p1.3.m3.5.5.1.1.1.1.1.2">𝑞</ci><ci id="S6.I2.i3.p1.3.m3.3.3.cmml" xref="S6.I2.i3.p1.3.m3.3.3">𝑎</ci></apply><apply id="S6.I2.i3.p1.3.m3.6.6.2.2.2.2.2.cmml" xref="S6.I2.i3.p1.3.m3.6.6.2.2.2.2.2"><times id="S6.I2.i3.p1.3.m3.6.6.2.2.2.2.2.1.cmml" xref="S6.I2.i3.p1.3.m3.6.6.2.2.2.2.2.1"></times><ci id="S6.I2.i3.p1.3.m3.6.6.2.2.2.2.2.2.cmml" xref="S6.I2.i3.p1.3.m3.6.6.2.2.2.2.2.2">𝑞</ci><ci id="S6.I2.i3.p1.3.m3.4.4.cmml" xref="S6.I2.i3.p1.3.m3.4.4">𝑏</ci></apply></interval></apply><ci id="S6.I2.i3.p1.3.m3.6.6.2.4.cmml" xref="S6.I2.i3.p1.3.m3.6.6.2.4">𝜈</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I2.i3.p1.3.m3.6c">\Delta_{A}(a,b)<\nu\leftrightarrow\Delta_{X}(q(a),q(b))<\nu</annotation><annotation encoding="application/x-llamapun" id="S6.I2.i3.p1.3.m3.6d">roman_Δ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_a , italic_b ) < italic_ν ↔ roman_Δ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ( italic_q ( italic_a ) , italic_q ( italic_b ) ) < italic_ν</annotation></semantics></math>.</p> </div> </li> </ul> </div> </div> <div class="ltx_para" id="S6.SS1.p6"> <p class="ltx_p" id="S6.SS1.p6.1">Moore proves the following (see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib16" title="">16</a>, Section 3]</cite>).</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S6.Thmtheorem5"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem5.1.1.1">Lemma 6.5</span></span><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem5.2.2">.</span> </h6> <div class="ltx_para" id="S6.Thmtheorem5.p1"> <p class="ltx_p" id="S6.Thmtheorem5.p1.6">If <math alttext="A" class="ltx_Math" display="inline" id="S6.Thmtheorem5.p1.1.m1.1"><semantics id="S6.Thmtheorem5.p1.1.m1.1a"><mi id="S6.Thmtheorem5.p1.1.m1.1.1" xref="S6.Thmtheorem5.p1.1.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem5.p1.1.m1.1b"><ci id="S6.Thmtheorem5.p1.1.m1.1.1.cmml" xref="S6.Thmtheorem5.p1.1.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem5.p1.1.m1.1c">A</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem5.p1.1.m1.1d">italic_A</annotation></semantics></math> and <math alttext="X" class="ltx_Math" display="inline" id="S6.Thmtheorem5.p1.2.m2.1"><semantics id="S6.Thmtheorem5.p1.2.m2.1a"><mi id="S6.Thmtheorem5.p1.2.m2.1.1" xref="S6.Thmtheorem5.p1.2.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem5.p1.2.m2.1b"><ci id="S6.Thmtheorem5.p1.2.m2.1.1.cmml" xref="S6.Thmtheorem5.p1.2.m2.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem5.p1.2.m2.1c">X</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem5.p1.2.m2.1d">italic_X</annotation></semantics></math> are <math alttext="\preceq" class="ltx_Math" display="inline" id="S6.Thmtheorem5.p1.3.m3.1"><semantics id="S6.Thmtheorem5.p1.3.m3.1a"><mo id="S6.Thmtheorem5.p1.3.m3.1.1" xref="S6.Thmtheorem5.p1.3.m3.1.1.cmml">⪯</mo><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem5.p1.3.m3.1b"><csymbol cd="latexml" id="S6.Thmtheorem5.p1.3.m3.1.1.cmml" xref="S6.Thmtheorem5.p1.3.m3.1.1">precedes-or-equals</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem5.p1.3.m3.1c">\preceq</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem5.p1.3.m3.1d">⪯</annotation></semantics></math>-comparable Countryman lines. Then there is a club <math alttext="E" class="ltx_Math" display="inline" id="S6.Thmtheorem5.p1.4.m4.1"><semantics id="S6.Thmtheorem5.p1.4.m4.1a"><mi id="S6.Thmtheorem5.p1.4.m4.1.1" xref="S6.Thmtheorem5.p1.4.m4.1.1.cmml">E</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem5.p1.4.m4.1b"><ci id="S6.Thmtheorem5.p1.4.m4.1.1.cmml" xref="S6.Thmtheorem5.p1.4.m4.1.1">𝐸</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem5.p1.4.m4.1c">E</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem5.p1.4.m4.1d">italic_E</annotation></semantics></math> such that <math alttext="Q_{E^{\prime}}" class="ltx_Math" display="inline" id="S6.Thmtheorem5.p1.5.m5.1"><semantics id="S6.Thmtheorem5.p1.5.m5.1a"><msub id="S6.Thmtheorem5.p1.5.m5.1.1" xref="S6.Thmtheorem5.p1.5.m5.1.1.cmml"><mi id="S6.Thmtheorem5.p1.5.m5.1.1.2" xref="S6.Thmtheorem5.p1.5.m5.1.1.2.cmml">Q</mi><msup id="S6.Thmtheorem5.p1.5.m5.1.1.3" xref="S6.Thmtheorem5.p1.5.m5.1.1.3.cmml"><mi id="S6.Thmtheorem5.p1.5.m5.1.1.3.2" xref="S6.Thmtheorem5.p1.5.m5.1.1.3.2.cmml">E</mi><mo id="S6.Thmtheorem5.p1.5.m5.1.1.3.3" xref="S6.Thmtheorem5.p1.5.m5.1.1.3.3.cmml">′</mo></msup></msub><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem5.p1.5.m5.1b"><apply id="S6.Thmtheorem5.p1.5.m5.1.1.cmml" xref="S6.Thmtheorem5.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S6.Thmtheorem5.p1.5.m5.1.1.1.cmml" xref="S6.Thmtheorem5.p1.5.m5.1.1">subscript</csymbol><ci id="S6.Thmtheorem5.p1.5.m5.1.1.2.cmml" xref="S6.Thmtheorem5.p1.5.m5.1.1.2">𝑄</ci><apply id="S6.Thmtheorem5.p1.5.m5.1.1.3.cmml" xref="S6.Thmtheorem5.p1.5.m5.1.1.3"><csymbol cd="ambiguous" id="S6.Thmtheorem5.p1.5.m5.1.1.3.1.cmml" xref="S6.Thmtheorem5.p1.5.m5.1.1.3">superscript</csymbol><ci id="S6.Thmtheorem5.p1.5.m5.1.1.3.2.cmml" xref="S6.Thmtheorem5.p1.5.m5.1.1.3.2">𝐸</ci><ci id="S6.Thmtheorem5.p1.5.m5.1.1.3.3.cmml" xref="S6.Thmtheorem5.p1.5.m5.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem5.p1.5.m5.1c">Q_{E^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem5.p1.5.m5.1d">italic_Q start_POSTSUBSCRIPT italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> is ccc whenever <math alttext="E^{\prime}\subseteq E" class="ltx_Math" display="inline" id="S6.Thmtheorem5.p1.6.m6.1"><semantics id="S6.Thmtheorem5.p1.6.m6.1a"><mrow id="S6.Thmtheorem5.p1.6.m6.1.1" xref="S6.Thmtheorem5.p1.6.m6.1.1.cmml"><msup id="S6.Thmtheorem5.p1.6.m6.1.1.2" xref="S6.Thmtheorem5.p1.6.m6.1.1.2.cmml"><mi id="S6.Thmtheorem5.p1.6.m6.1.1.2.2" xref="S6.Thmtheorem5.p1.6.m6.1.1.2.2.cmml">E</mi><mo id="S6.Thmtheorem5.p1.6.m6.1.1.2.3" xref="S6.Thmtheorem5.p1.6.m6.1.1.2.3.cmml">′</mo></msup><mo id="S6.Thmtheorem5.p1.6.m6.1.1.1" xref="S6.Thmtheorem5.p1.6.m6.1.1.1.cmml">⊆</mo><mi id="S6.Thmtheorem5.p1.6.m6.1.1.3" xref="S6.Thmtheorem5.p1.6.m6.1.1.3.cmml">E</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem5.p1.6.m6.1b"><apply id="S6.Thmtheorem5.p1.6.m6.1.1.cmml" xref="S6.Thmtheorem5.p1.6.m6.1.1"><subset id="S6.Thmtheorem5.p1.6.m6.1.1.1.cmml" xref="S6.Thmtheorem5.p1.6.m6.1.1.1"></subset><apply id="S6.Thmtheorem5.p1.6.m6.1.1.2.cmml" xref="S6.Thmtheorem5.p1.6.m6.1.1.2"><csymbol cd="ambiguous" id="S6.Thmtheorem5.p1.6.m6.1.1.2.1.cmml" xref="S6.Thmtheorem5.p1.6.m6.1.1.2">superscript</csymbol><ci id="S6.Thmtheorem5.p1.6.m6.1.1.2.2.cmml" xref="S6.Thmtheorem5.p1.6.m6.1.1.2.2">𝐸</ci><ci id="S6.Thmtheorem5.p1.6.m6.1.1.2.3.cmml" xref="S6.Thmtheorem5.p1.6.m6.1.1.2.3">′</ci></apply><ci id="S6.Thmtheorem5.p1.6.m6.1.1.3.cmml" xref="S6.Thmtheorem5.p1.6.m6.1.1.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem5.p1.6.m6.1c">E^{\prime}\subseteq E</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem5.p1.6.m6.1d">italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⊆ italic_E</annotation></semantics></math> is club.</p> </div> </div> <div class="ltx_theorem ltx_theorem_definition" id="S6.Thmtheorem6"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem6.1.1.1">Definition 6.6</span></span><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem6.2.2">.</span> </h6> <div class="ltx_para" id="S6.Thmtheorem6.p1"> <p class="ltx_p" id="S6.Thmtheorem6.p1.5">If <math alttext="S\in H(\omega_{2})" class="ltx_Math" display="inline" id="S6.Thmtheorem6.p1.1.m1.1"><semantics id="S6.Thmtheorem6.p1.1.m1.1a"><mrow id="S6.Thmtheorem6.p1.1.m1.1.1" xref="S6.Thmtheorem6.p1.1.m1.1.1.cmml"><mi id="S6.Thmtheorem6.p1.1.m1.1.1.3" xref="S6.Thmtheorem6.p1.1.m1.1.1.3.cmml">S</mi><mo id="S6.Thmtheorem6.p1.1.m1.1.1.2" xref="S6.Thmtheorem6.p1.1.m1.1.1.2.cmml">∈</mo><mrow id="S6.Thmtheorem6.p1.1.m1.1.1.1" xref="S6.Thmtheorem6.p1.1.m1.1.1.1.cmml"><mi id="S6.Thmtheorem6.p1.1.m1.1.1.1.3" xref="S6.Thmtheorem6.p1.1.m1.1.1.1.3.cmml">H</mi><mo id="S6.Thmtheorem6.p1.1.m1.1.1.1.2" xref="S6.Thmtheorem6.p1.1.m1.1.1.1.2.cmml">⁢</mo><mrow id="S6.Thmtheorem6.p1.1.m1.1.1.1.1.1" xref="S6.Thmtheorem6.p1.1.m1.1.1.1.1.1.1.cmml"><mo id="S6.Thmtheorem6.p1.1.m1.1.1.1.1.1.2" stretchy="false" xref="S6.Thmtheorem6.p1.1.m1.1.1.1.1.1.1.cmml">(</mo><msub id="S6.Thmtheorem6.p1.1.m1.1.1.1.1.1.1" xref="S6.Thmtheorem6.p1.1.m1.1.1.1.1.1.1.cmml"><mi id="S6.Thmtheorem6.p1.1.m1.1.1.1.1.1.1.2" xref="S6.Thmtheorem6.p1.1.m1.1.1.1.1.1.1.2.cmml">ω</mi><mn id="S6.Thmtheorem6.p1.1.m1.1.1.1.1.1.1.3" xref="S6.Thmtheorem6.p1.1.m1.1.1.1.1.1.1.3.cmml">2</mn></msub><mo id="S6.Thmtheorem6.p1.1.m1.1.1.1.1.1.3" stretchy="false" xref="S6.Thmtheorem6.p1.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem6.p1.1.m1.1b"><apply id="S6.Thmtheorem6.p1.1.m1.1.1.cmml" xref="S6.Thmtheorem6.p1.1.m1.1.1"><in id="S6.Thmtheorem6.p1.1.m1.1.1.2.cmml" xref="S6.Thmtheorem6.p1.1.m1.1.1.2"></in><ci id="S6.Thmtheorem6.p1.1.m1.1.1.3.cmml" xref="S6.Thmtheorem6.p1.1.m1.1.1.3">𝑆</ci><apply id="S6.Thmtheorem6.p1.1.m1.1.1.1.cmml" xref="S6.Thmtheorem6.p1.1.m1.1.1.1"><times id="S6.Thmtheorem6.p1.1.m1.1.1.1.2.cmml" xref="S6.Thmtheorem6.p1.1.m1.1.1.1.2"></times><ci id="S6.Thmtheorem6.p1.1.m1.1.1.1.3.cmml" xref="S6.Thmtheorem6.p1.1.m1.1.1.1.3">𝐻</ci><apply id="S6.Thmtheorem6.p1.1.m1.1.1.1.1.1.1.cmml" xref="S6.Thmtheorem6.p1.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.Thmtheorem6.p1.1.m1.1.1.1.1.1.1.1.cmml" xref="S6.Thmtheorem6.p1.1.m1.1.1.1.1.1">subscript</csymbol><ci id="S6.Thmtheorem6.p1.1.m1.1.1.1.1.1.1.2.cmml" xref="S6.Thmtheorem6.p1.1.m1.1.1.1.1.1.1.2">𝜔</ci><cn id="S6.Thmtheorem6.p1.1.m1.1.1.1.1.1.1.3.cmml" type="integer" xref="S6.Thmtheorem6.p1.1.m1.1.1.1.1.1.1.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem6.p1.1.m1.1c">S\in H(\omega_{2})</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem6.p1.1.m1.1d">italic_S ∈ italic_H ( italic_ω start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT )</annotation></semantics></math> we say that <math alttext="E" class="ltx_Math" display="inline" id="S6.Thmtheorem6.p1.2.m2.1"><semantics id="S6.Thmtheorem6.p1.2.m2.1a"><mi id="S6.Thmtheorem6.p1.2.m2.1.1" xref="S6.Thmtheorem6.p1.2.m2.1.1.cmml">E</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem6.p1.2.m2.1b"><ci id="S6.Thmtheorem6.p1.2.m2.1.1.cmml" xref="S6.Thmtheorem6.p1.2.m2.1.1">𝐸</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem6.p1.2.m2.1c">E</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem6.p1.2.m2.1d">italic_E</annotation></semantics></math> is an <em class="ltx_emph ltx_font_italic" id="S6.Thmtheorem6.p1.5.1">elementary club</em> for <math alttext="S" class="ltx_Math" display="inline" id="S6.Thmtheorem6.p1.3.m3.1"><semantics id="S6.Thmtheorem6.p1.3.m3.1a"><mi id="S6.Thmtheorem6.p1.3.m3.1.1" xref="S6.Thmtheorem6.p1.3.m3.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem6.p1.3.m3.1b"><ci id="S6.Thmtheorem6.p1.3.m3.1.1.cmml" xref="S6.Thmtheorem6.p1.3.m3.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem6.p1.3.m3.1c">S</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem6.p1.3.m3.1d">italic_S</annotation></semantics></math> if <math alttext="E" class="ltx_Math" display="inline" id="S6.Thmtheorem6.p1.4.m4.1"><semantics id="S6.Thmtheorem6.p1.4.m4.1a"><mi id="S6.Thmtheorem6.p1.4.m4.1.1" xref="S6.Thmtheorem6.p1.4.m4.1.1.cmml">E</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem6.p1.4.m4.1b"><ci id="S6.Thmtheorem6.p1.4.m4.1.1.cmml" xref="S6.Thmtheorem6.p1.4.m4.1.1">𝐸</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem6.p1.4.m4.1c">E</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem6.p1.4.m4.1d">italic_E</annotation></semantics></math> is contained in <math alttext="\{N\cap\omega_{1}:N\prec H(\omega_{2}),|N|=\aleph_{0},S\in N\}" class="ltx_Math" display="inline" id="S6.Thmtheorem6.p1.5.m5.3"><semantics id="S6.Thmtheorem6.p1.5.m5.3a"><mrow id="S6.Thmtheorem6.p1.5.m5.3.3.2" xref="S6.Thmtheorem6.p1.5.m5.3.3.3.cmml"><mo id="S6.Thmtheorem6.p1.5.m5.3.3.2.3" stretchy="false" xref="S6.Thmtheorem6.p1.5.m5.3.3.3.1.cmml">{</mo><mrow id="S6.Thmtheorem6.p1.5.m5.2.2.1.1" xref="S6.Thmtheorem6.p1.5.m5.2.2.1.1.cmml"><mi id="S6.Thmtheorem6.p1.5.m5.2.2.1.1.2" xref="S6.Thmtheorem6.p1.5.m5.2.2.1.1.2.cmml">N</mi><mo id="S6.Thmtheorem6.p1.5.m5.2.2.1.1.1" xref="S6.Thmtheorem6.p1.5.m5.2.2.1.1.1.cmml">∩</mo><msub id="S6.Thmtheorem6.p1.5.m5.2.2.1.1.3" xref="S6.Thmtheorem6.p1.5.m5.2.2.1.1.3.cmml"><mi id="S6.Thmtheorem6.p1.5.m5.2.2.1.1.3.2" xref="S6.Thmtheorem6.p1.5.m5.2.2.1.1.3.2.cmml">ω</mi><mn id="S6.Thmtheorem6.p1.5.m5.2.2.1.1.3.3" xref="S6.Thmtheorem6.p1.5.m5.2.2.1.1.3.3.cmml">1</mn></msub></mrow><mo id="S6.Thmtheorem6.p1.5.m5.3.3.2.4" lspace="0.278em" rspace="0.278em" xref="S6.Thmtheorem6.p1.5.m5.3.3.3.1.cmml">:</mo><mrow id="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2" xref="S6.Thmtheorem6.p1.5.m5.3.3.2.2.3.cmml"><mrow id="S6.Thmtheorem6.p1.5.m5.3.3.2.2.1.1" xref="S6.Thmtheorem6.p1.5.m5.3.3.2.2.1.1.cmml"><mi id="S6.Thmtheorem6.p1.5.m5.3.3.2.2.1.1.3" xref="S6.Thmtheorem6.p1.5.m5.3.3.2.2.1.1.3.cmml">N</mi><mo id="S6.Thmtheorem6.p1.5.m5.3.3.2.2.1.1.2" xref="S6.Thmtheorem6.p1.5.m5.3.3.2.2.1.1.2.cmml">≺</mo><mrow id="S6.Thmtheorem6.p1.5.m5.3.3.2.2.1.1.1" xref="S6.Thmtheorem6.p1.5.m5.3.3.2.2.1.1.1.cmml"><mi id="S6.Thmtheorem6.p1.5.m5.3.3.2.2.1.1.1.3" xref="S6.Thmtheorem6.p1.5.m5.3.3.2.2.1.1.1.3.cmml">H</mi><mo id="S6.Thmtheorem6.p1.5.m5.3.3.2.2.1.1.1.2" xref="S6.Thmtheorem6.p1.5.m5.3.3.2.2.1.1.1.2.cmml">⁢</mo><mrow id="S6.Thmtheorem6.p1.5.m5.3.3.2.2.1.1.1.1.1" xref="S6.Thmtheorem6.p1.5.m5.3.3.2.2.1.1.1.1.1.1.cmml"><mo id="S6.Thmtheorem6.p1.5.m5.3.3.2.2.1.1.1.1.1.2" stretchy="false" xref="S6.Thmtheorem6.p1.5.m5.3.3.2.2.1.1.1.1.1.1.cmml">(</mo><msub id="S6.Thmtheorem6.p1.5.m5.3.3.2.2.1.1.1.1.1.1" xref="S6.Thmtheorem6.p1.5.m5.3.3.2.2.1.1.1.1.1.1.cmml"><mi id="S6.Thmtheorem6.p1.5.m5.3.3.2.2.1.1.1.1.1.1.2" xref="S6.Thmtheorem6.p1.5.m5.3.3.2.2.1.1.1.1.1.1.2.cmml">ω</mi><mn id="S6.Thmtheorem6.p1.5.m5.3.3.2.2.1.1.1.1.1.1.3" xref="S6.Thmtheorem6.p1.5.m5.3.3.2.2.1.1.1.1.1.1.3.cmml">2</mn></msub><mo id="S6.Thmtheorem6.p1.5.m5.3.3.2.2.1.1.1.1.1.3" stretchy="false" xref="S6.Thmtheorem6.p1.5.m5.3.3.2.2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.3" xref="S6.Thmtheorem6.p1.5.m5.3.3.2.2.3a.cmml">,</mo><mrow id="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.2.2" xref="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.2.3.cmml"><mrow id="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.2.1.1" xref="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.2.1.1.cmml"><mrow id="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.2.1.1.2.2" xref="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.2.1.1.2.1.cmml"><mo id="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.2.1.1.2.2.1" stretchy="false" xref="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.2.1.1.2.1.1.cmml">|</mo><mi id="S6.Thmtheorem6.p1.5.m5.1.1" xref="S6.Thmtheorem6.p1.5.m5.1.1.cmml">N</mi><mo id="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.2.1.1.2.2.2" stretchy="false" xref="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.2.1.1.2.1.1.cmml">|</mo></mrow><mo id="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.2.1.1.1" xref="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.2.1.1.1.cmml">=</mo><msub id="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.2.1.1.3" xref="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.2.1.1.3.cmml"><mi id="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.2.1.1.3.2" mathvariant="normal" xref="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.2.1.1.3.2.cmml">ℵ</mi><mn id="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.2.1.1.3.3" xref="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.2.1.1.3.3.cmml">0</mn></msub></mrow><mo id="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.2.2.3" xref="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.2.3a.cmml">,</mo><mrow id="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.2.2.2" xref="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.2.2.2.cmml"><mi id="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.2.2.2.2" xref="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.2.2.2.2.cmml">S</mi><mo id="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.2.2.2.1" xref="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.2.2.2.1.cmml">∈</mo><mi id="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.2.2.2.3" xref="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.2.2.2.3.cmml">N</mi></mrow></mrow></mrow><mo id="S6.Thmtheorem6.p1.5.m5.3.3.2.5" stretchy="false" xref="S6.Thmtheorem6.p1.5.m5.3.3.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem6.p1.5.m5.3b"><apply id="S6.Thmtheorem6.p1.5.m5.3.3.3.cmml" xref="S6.Thmtheorem6.p1.5.m5.3.3.2"><csymbol cd="latexml" id="S6.Thmtheorem6.p1.5.m5.3.3.3.1.cmml" xref="S6.Thmtheorem6.p1.5.m5.3.3.2.3">conditional-set</csymbol><apply id="S6.Thmtheorem6.p1.5.m5.2.2.1.1.cmml" xref="S6.Thmtheorem6.p1.5.m5.2.2.1.1"><intersect id="S6.Thmtheorem6.p1.5.m5.2.2.1.1.1.cmml" xref="S6.Thmtheorem6.p1.5.m5.2.2.1.1.1"></intersect><ci id="S6.Thmtheorem6.p1.5.m5.2.2.1.1.2.cmml" xref="S6.Thmtheorem6.p1.5.m5.2.2.1.1.2">𝑁</ci><apply id="S6.Thmtheorem6.p1.5.m5.2.2.1.1.3.cmml" xref="S6.Thmtheorem6.p1.5.m5.2.2.1.1.3"><csymbol cd="ambiguous" id="S6.Thmtheorem6.p1.5.m5.2.2.1.1.3.1.cmml" xref="S6.Thmtheorem6.p1.5.m5.2.2.1.1.3">subscript</csymbol><ci id="S6.Thmtheorem6.p1.5.m5.2.2.1.1.3.2.cmml" xref="S6.Thmtheorem6.p1.5.m5.2.2.1.1.3.2">𝜔</ci><cn id="S6.Thmtheorem6.p1.5.m5.2.2.1.1.3.3.cmml" type="integer" xref="S6.Thmtheorem6.p1.5.m5.2.2.1.1.3.3">1</cn></apply></apply><apply id="S6.Thmtheorem6.p1.5.m5.3.3.2.2.3.cmml" xref="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2"><csymbol cd="ambiguous" id="S6.Thmtheorem6.p1.5.m5.3.3.2.2.3a.cmml" xref="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.3">formulae-sequence</csymbol><apply id="S6.Thmtheorem6.p1.5.m5.3.3.2.2.1.1.cmml" xref="S6.Thmtheorem6.p1.5.m5.3.3.2.2.1.1"><csymbol cd="latexml" id="S6.Thmtheorem6.p1.5.m5.3.3.2.2.1.1.2.cmml" xref="S6.Thmtheorem6.p1.5.m5.3.3.2.2.1.1.2">precedes</csymbol><ci id="S6.Thmtheorem6.p1.5.m5.3.3.2.2.1.1.3.cmml" xref="S6.Thmtheorem6.p1.5.m5.3.3.2.2.1.1.3">𝑁</ci><apply id="S6.Thmtheorem6.p1.5.m5.3.3.2.2.1.1.1.cmml" xref="S6.Thmtheorem6.p1.5.m5.3.3.2.2.1.1.1"><times id="S6.Thmtheorem6.p1.5.m5.3.3.2.2.1.1.1.2.cmml" xref="S6.Thmtheorem6.p1.5.m5.3.3.2.2.1.1.1.2"></times><ci id="S6.Thmtheorem6.p1.5.m5.3.3.2.2.1.1.1.3.cmml" xref="S6.Thmtheorem6.p1.5.m5.3.3.2.2.1.1.1.3">𝐻</ci><apply id="S6.Thmtheorem6.p1.5.m5.3.3.2.2.1.1.1.1.1.1.cmml" xref="S6.Thmtheorem6.p1.5.m5.3.3.2.2.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.Thmtheorem6.p1.5.m5.3.3.2.2.1.1.1.1.1.1.1.cmml" xref="S6.Thmtheorem6.p1.5.m5.3.3.2.2.1.1.1.1.1">subscript</csymbol><ci id="S6.Thmtheorem6.p1.5.m5.3.3.2.2.1.1.1.1.1.1.2.cmml" xref="S6.Thmtheorem6.p1.5.m5.3.3.2.2.1.1.1.1.1.1.2">𝜔</ci><cn id="S6.Thmtheorem6.p1.5.m5.3.3.2.2.1.1.1.1.1.1.3.cmml" type="integer" xref="S6.Thmtheorem6.p1.5.m5.3.3.2.2.1.1.1.1.1.1.3">2</cn></apply></apply></apply><apply id="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.2.3.cmml" xref="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.2.3a.cmml" xref="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.2.2.3">formulae-sequence</csymbol><apply id="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.2.1.1.cmml" xref="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.2.1.1"><eq id="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.2.1.1.1.cmml" xref="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.2.1.1.1"></eq><apply id="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.2.1.1.2.1.cmml" xref="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.2.1.1.2.2"><abs id="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.2.1.1.2.1.1.cmml" xref="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.2.1.1.2.2.1"></abs><ci id="S6.Thmtheorem6.p1.5.m5.1.1.cmml" xref="S6.Thmtheorem6.p1.5.m5.1.1">𝑁</ci></apply><apply id="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.2.1.1.3.cmml" xref="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.2.1.1.3"><csymbol cd="ambiguous" id="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.2.1.1.3.1.cmml" xref="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.2.1.1.3">subscript</csymbol><ci id="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.2.1.1.3.2.cmml" xref="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.2.1.1.3.2">ℵ</ci><cn id="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.2.1.1.3.3.cmml" type="integer" xref="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.2.1.1.3.3">0</cn></apply></apply><apply id="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.2.2.2.cmml" xref="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.2.2.2"><in id="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.2.2.2.1.cmml" xref="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.2.2.2.1"></in><ci id="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.2.2.2.2.cmml" xref="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.2.2.2.2">𝑆</ci><ci id="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.2.2.2.3.cmml" xref="S6.Thmtheorem6.p1.5.m5.3.3.2.2.2.2.2.2.3">𝑁</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem6.p1.5.m5.3c">\{N\cap\omega_{1}:N\prec H(\omega_{2}),|N|=\aleph_{0},S\in N\}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem6.p1.5.m5.3d">{ italic_N ∩ italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT : italic_N ≺ italic_H ( italic_ω start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) , | italic_N | = roman_ℵ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_S ∈ italic_N }</annotation></semantics></math>.</p> </div> </div> <div class="ltx_para" id="S6.SS1.p7"> <p class="ltx_p" id="S6.SS1.p7.2">It is well known that for any <math alttext="S\in H(\omega_{2})" class="ltx_Math" display="inline" id="S6.SS1.p7.1.m1.1"><semantics id="S6.SS1.p7.1.m1.1a"><mrow id="S6.SS1.p7.1.m1.1.1" xref="S6.SS1.p7.1.m1.1.1.cmml"><mi id="S6.SS1.p7.1.m1.1.1.3" xref="S6.SS1.p7.1.m1.1.1.3.cmml">S</mi><mo id="S6.SS1.p7.1.m1.1.1.2" xref="S6.SS1.p7.1.m1.1.1.2.cmml">∈</mo><mrow id="S6.SS1.p7.1.m1.1.1.1" xref="S6.SS1.p7.1.m1.1.1.1.cmml"><mi id="S6.SS1.p7.1.m1.1.1.1.3" xref="S6.SS1.p7.1.m1.1.1.1.3.cmml">H</mi><mo id="S6.SS1.p7.1.m1.1.1.1.2" xref="S6.SS1.p7.1.m1.1.1.1.2.cmml">⁢</mo><mrow id="S6.SS1.p7.1.m1.1.1.1.1.1" xref="S6.SS1.p7.1.m1.1.1.1.1.1.1.cmml"><mo id="S6.SS1.p7.1.m1.1.1.1.1.1.2" stretchy="false" xref="S6.SS1.p7.1.m1.1.1.1.1.1.1.cmml">(</mo><msub id="S6.SS1.p7.1.m1.1.1.1.1.1.1" xref="S6.SS1.p7.1.m1.1.1.1.1.1.1.cmml"><mi id="S6.SS1.p7.1.m1.1.1.1.1.1.1.2" xref="S6.SS1.p7.1.m1.1.1.1.1.1.1.2.cmml">ω</mi><mn id="S6.SS1.p7.1.m1.1.1.1.1.1.1.3" xref="S6.SS1.p7.1.m1.1.1.1.1.1.1.3.cmml">2</mn></msub><mo id="S6.SS1.p7.1.m1.1.1.1.1.1.3" stretchy="false" xref="S6.SS1.p7.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p7.1.m1.1b"><apply id="S6.SS1.p7.1.m1.1.1.cmml" xref="S6.SS1.p7.1.m1.1.1"><in id="S6.SS1.p7.1.m1.1.1.2.cmml" xref="S6.SS1.p7.1.m1.1.1.2"></in><ci id="S6.SS1.p7.1.m1.1.1.3.cmml" xref="S6.SS1.p7.1.m1.1.1.3">𝑆</ci><apply id="S6.SS1.p7.1.m1.1.1.1.cmml" xref="S6.SS1.p7.1.m1.1.1.1"><times id="S6.SS1.p7.1.m1.1.1.1.2.cmml" xref="S6.SS1.p7.1.m1.1.1.1.2"></times><ci id="S6.SS1.p7.1.m1.1.1.1.3.cmml" xref="S6.SS1.p7.1.m1.1.1.1.3">𝐻</ci><apply id="S6.SS1.p7.1.m1.1.1.1.1.1.1.cmml" xref="S6.SS1.p7.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS1.p7.1.m1.1.1.1.1.1.1.1.cmml" xref="S6.SS1.p7.1.m1.1.1.1.1.1">subscript</csymbol><ci id="S6.SS1.p7.1.m1.1.1.1.1.1.1.2.cmml" xref="S6.SS1.p7.1.m1.1.1.1.1.1.1.2">𝜔</ci><cn id="S6.SS1.p7.1.m1.1.1.1.1.1.1.3.cmml" type="integer" xref="S6.SS1.p7.1.m1.1.1.1.1.1.1.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p7.1.m1.1c">S\in H(\omega_{2})</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p7.1.m1.1d">italic_S ∈ italic_H ( italic_ω start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT )</annotation></semantics></math> there is an elementary club for <math alttext="X" class="ltx_Math" display="inline" id="S6.SS1.p7.2.m2.1"><semantics id="S6.SS1.p7.2.m2.1a"><mi id="S6.SS1.p7.2.m2.1.1" xref="S6.SS1.p7.2.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S6.SS1.p7.2.m2.1b"><ci id="S6.SS1.p7.2.m2.1.1.cmml" xref="S6.SS1.p7.2.m2.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p7.2.m2.1c">X</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p7.2.m2.1d">italic_X</annotation></semantics></math>. Moore also proves the following.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S6.Thmtheorem7"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem7.1.1.1">Lemma 6.7</span></span><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem7.2.2">.</span> </h6> <div class="ltx_para" id="S6.Thmtheorem7.p1"> <p class="ltx_p" id="S6.Thmtheorem7.p1.8">If <math alttext="A" class="ltx_Math" display="inline" id="S6.Thmtheorem7.p1.1.m1.1"><semantics id="S6.Thmtheorem7.p1.1.m1.1a"><mi id="S6.Thmtheorem7.p1.1.m1.1.1" xref="S6.Thmtheorem7.p1.1.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem7.p1.1.m1.1b"><ci id="S6.Thmtheorem7.p1.1.m1.1.1.cmml" xref="S6.Thmtheorem7.p1.1.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem7.p1.1.m1.1c">A</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem7.p1.1.m1.1d">italic_A</annotation></semantics></math> and <math alttext="X" class="ltx_Math" display="inline" id="S6.Thmtheorem7.p1.2.m2.1"><semantics id="S6.Thmtheorem7.p1.2.m2.1a"><mi id="S6.Thmtheorem7.p1.2.m2.1.1" xref="S6.Thmtheorem7.p1.2.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem7.p1.2.m2.1b"><ci id="S6.Thmtheorem7.p1.2.m2.1.1.cmml" xref="S6.Thmtheorem7.p1.2.m2.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem7.p1.2.m2.1c">X</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem7.p1.2.m2.1d">italic_X</annotation></semantics></math> are normal Aronszajn lines, and <math alttext="E" class="ltx_Math" display="inline" id="S6.Thmtheorem7.p1.3.m3.1"><semantics id="S6.Thmtheorem7.p1.3.m3.1a"><mi id="S6.Thmtheorem7.p1.3.m3.1.1" xref="S6.Thmtheorem7.p1.3.m3.1.1.cmml">E</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem7.p1.3.m3.1b"><ci id="S6.Thmtheorem7.p1.3.m3.1.1.cmml" xref="S6.Thmtheorem7.p1.3.m3.1.1">𝐸</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem7.p1.3.m3.1c">E</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem7.p1.3.m3.1d">italic_E</annotation></semantics></math> is an elementary cub for <math alttext="A" class="ltx_Math" display="inline" id="S6.Thmtheorem7.p1.4.m4.1"><semantics id="S6.Thmtheorem7.p1.4.m4.1a"><mi id="S6.Thmtheorem7.p1.4.m4.1.1" xref="S6.Thmtheorem7.p1.4.m4.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem7.p1.4.m4.1b"><ci id="S6.Thmtheorem7.p1.4.m4.1.1.cmml" xref="S6.Thmtheorem7.p1.4.m4.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem7.p1.4.m4.1c">A</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem7.p1.4.m4.1d">italic_A</annotation></semantics></math> and <math alttext="X" class="ltx_Math" display="inline" id="S6.Thmtheorem7.p1.5.m5.1"><semantics id="S6.Thmtheorem7.p1.5.m5.1a"><mi id="S6.Thmtheorem7.p1.5.m5.1.1" xref="S6.Thmtheorem7.p1.5.m5.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem7.p1.5.m5.1b"><ci id="S6.Thmtheorem7.p1.5.m5.1.1.cmml" xref="S6.Thmtheorem7.p1.5.m5.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem7.p1.5.m5.1c">X</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem7.p1.5.m5.1d">italic_X</annotation></semantics></math>, then for every <math alttext="a\in A" class="ltx_Math" display="inline" id="S6.Thmtheorem7.p1.6.m6.1"><semantics id="S6.Thmtheorem7.p1.6.m6.1a"><mrow id="S6.Thmtheorem7.p1.6.m6.1.1" xref="S6.Thmtheorem7.p1.6.m6.1.1.cmml"><mi id="S6.Thmtheorem7.p1.6.m6.1.1.2" xref="S6.Thmtheorem7.p1.6.m6.1.1.2.cmml">a</mi><mo id="S6.Thmtheorem7.p1.6.m6.1.1.1" xref="S6.Thmtheorem7.p1.6.m6.1.1.1.cmml">∈</mo><mi id="S6.Thmtheorem7.p1.6.m6.1.1.3" xref="S6.Thmtheorem7.p1.6.m6.1.1.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem7.p1.6.m6.1b"><apply id="S6.Thmtheorem7.p1.6.m6.1.1.cmml" xref="S6.Thmtheorem7.p1.6.m6.1.1"><in id="S6.Thmtheorem7.p1.6.m6.1.1.1.cmml" xref="S6.Thmtheorem7.p1.6.m6.1.1.1"></in><ci id="S6.Thmtheorem7.p1.6.m6.1.1.2.cmml" xref="S6.Thmtheorem7.p1.6.m6.1.1.2">𝑎</ci><ci id="S6.Thmtheorem7.p1.6.m6.1.1.3.cmml" xref="S6.Thmtheorem7.p1.6.m6.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem7.p1.6.m6.1c">a\in A</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem7.p1.6.m6.1d">italic_a ∈ italic_A</annotation></semantics></math> and <math alttext="x\in X" class="ltx_Math" display="inline" id="S6.Thmtheorem7.p1.7.m7.1"><semantics id="S6.Thmtheorem7.p1.7.m7.1a"><mrow id="S6.Thmtheorem7.p1.7.m7.1.1" xref="S6.Thmtheorem7.p1.7.m7.1.1.cmml"><mi id="S6.Thmtheorem7.p1.7.m7.1.1.2" xref="S6.Thmtheorem7.p1.7.m7.1.1.2.cmml">x</mi><mo id="S6.Thmtheorem7.p1.7.m7.1.1.1" xref="S6.Thmtheorem7.p1.7.m7.1.1.1.cmml">∈</mo><mi id="S6.Thmtheorem7.p1.7.m7.1.1.3" xref="S6.Thmtheorem7.p1.7.m7.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem7.p1.7.m7.1b"><apply id="S6.Thmtheorem7.p1.7.m7.1.1.cmml" xref="S6.Thmtheorem7.p1.7.m7.1.1"><in id="S6.Thmtheorem7.p1.7.m7.1.1.1.cmml" xref="S6.Thmtheorem7.p1.7.m7.1.1.1"></in><ci id="S6.Thmtheorem7.p1.7.m7.1.1.2.cmml" xref="S6.Thmtheorem7.p1.7.m7.1.1.2">𝑥</ci><ci id="S6.Thmtheorem7.p1.7.m7.1.1.3.cmml" xref="S6.Thmtheorem7.p1.7.m7.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem7.p1.7.m7.1c">x\in X</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem7.p1.7.m7.1d">italic_x ∈ italic_X</annotation></semantics></math>, <math alttext="\{q\in Q_{E}:a\in\operatorname{dom}(q),x\in\operatorname{ran}(q)\}" class="ltx_Math" display="inline" id="S6.Thmtheorem7.p1.8.m8.6"><semantics id="S6.Thmtheorem7.p1.8.m8.6a"><mrow id="S6.Thmtheorem7.p1.8.m8.6.6.2" xref="S6.Thmtheorem7.p1.8.m8.6.6.3.cmml"><mo id="S6.Thmtheorem7.p1.8.m8.6.6.2.3" stretchy="false" xref="S6.Thmtheorem7.p1.8.m8.6.6.3.1.cmml">{</mo><mrow id="S6.Thmtheorem7.p1.8.m8.5.5.1.1" xref="S6.Thmtheorem7.p1.8.m8.5.5.1.1.cmml"><mi id="S6.Thmtheorem7.p1.8.m8.5.5.1.1.2" xref="S6.Thmtheorem7.p1.8.m8.5.5.1.1.2.cmml">q</mi><mo id="S6.Thmtheorem7.p1.8.m8.5.5.1.1.1" xref="S6.Thmtheorem7.p1.8.m8.5.5.1.1.1.cmml">∈</mo><msub id="S6.Thmtheorem7.p1.8.m8.5.5.1.1.3" xref="S6.Thmtheorem7.p1.8.m8.5.5.1.1.3.cmml"><mi id="S6.Thmtheorem7.p1.8.m8.5.5.1.1.3.2" xref="S6.Thmtheorem7.p1.8.m8.5.5.1.1.3.2.cmml">Q</mi><mi id="S6.Thmtheorem7.p1.8.m8.5.5.1.1.3.3" xref="S6.Thmtheorem7.p1.8.m8.5.5.1.1.3.3.cmml">E</mi></msub></mrow><mo id="S6.Thmtheorem7.p1.8.m8.6.6.2.4" lspace="0.278em" rspace="0.278em" xref="S6.Thmtheorem7.p1.8.m8.6.6.3.1.cmml">:</mo><mrow id="S6.Thmtheorem7.p1.8.m8.6.6.2.2.2" xref="S6.Thmtheorem7.p1.8.m8.6.6.2.2.3.cmml"><mrow id="S6.Thmtheorem7.p1.8.m8.6.6.2.2.1.1" xref="S6.Thmtheorem7.p1.8.m8.6.6.2.2.1.1.cmml"><mi id="S6.Thmtheorem7.p1.8.m8.6.6.2.2.1.1.2" xref="S6.Thmtheorem7.p1.8.m8.6.6.2.2.1.1.2.cmml">a</mi><mo id="S6.Thmtheorem7.p1.8.m8.6.6.2.2.1.1.1" xref="S6.Thmtheorem7.p1.8.m8.6.6.2.2.1.1.1.cmml">∈</mo><mrow id="S6.Thmtheorem7.p1.8.m8.6.6.2.2.1.1.3.2" xref="S6.Thmtheorem7.p1.8.m8.6.6.2.2.1.1.3.1.cmml"><mi id="S6.Thmtheorem7.p1.8.m8.1.1" xref="S6.Thmtheorem7.p1.8.m8.1.1.cmml">dom</mi><mo id="S6.Thmtheorem7.p1.8.m8.6.6.2.2.1.1.3.2a" xref="S6.Thmtheorem7.p1.8.m8.6.6.2.2.1.1.3.1.cmml">⁡</mo><mrow id="S6.Thmtheorem7.p1.8.m8.6.6.2.2.1.1.3.2.1" xref="S6.Thmtheorem7.p1.8.m8.6.6.2.2.1.1.3.1.cmml"><mo id="S6.Thmtheorem7.p1.8.m8.6.6.2.2.1.1.3.2.1.1" stretchy="false" xref="S6.Thmtheorem7.p1.8.m8.6.6.2.2.1.1.3.1.cmml">(</mo><mi id="S6.Thmtheorem7.p1.8.m8.2.2" xref="S6.Thmtheorem7.p1.8.m8.2.2.cmml">q</mi><mo id="S6.Thmtheorem7.p1.8.m8.6.6.2.2.1.1.3.2.1.2" stretchy="false" xref="S6.Thmtheorem7.p1.8.m8.6.6.2.2.1.1.3.1.cmml">)</mo></mrow></mrow></mrow><mo id="S6.Thmtheorem7.p1.8.m8.6.6.2.2.2.3" xref="S6.Thmtheorem7.p1.8.m8.6.6.2.2.3a.cmml">,</mo><mrow id="S6.Thmtheorem7.p1.8.m8.6.6.2.2.2.2" xref="S6.Thmtheorem7.p1.8.m8.6.6.2.2.2.2.cmml"><mi id="S6.Thmtheorem7.p1.8.m8.6.6.2.2.2.2.2" xref="S6.Thmtheorem7.p1.8.m8.6.6.2.2.2.2.2.cmml">x</mi><mo id="S6.Thmtheorem7.p1.8.m8.6.6.2.2.2.2.1" xref="S6.Thmtheorem7.p1.8.m8.6.6.2.2.2.2.1.cmml">∈</mo><mrow id="S6.Thmtheorem7.p1.8.m8.6.6.2.2.2.2.3.2" xref="S6.Thmtheorem7.p1.8.m8.6.6.2.2.2.2.3.1.cmml"><mi id="S6.Thmtheorem7.p1.8.m8.3.3" xref="S6.Thmtheorem7.p1.8.m8.3.3.cmml">ran</mi><mo id="S6.Thmtheorem7.p1.8.m8.6.6.2.2.2.2.3.2a" xref="S6.Thmtheorem7.p1.8.m8.6.6.2.2.2.2.3.1.cmml">⁡</mo><mrow id="S6.Thmtheorem7.p1.8.m8.6.6.2.2.2.2.3.2.1" xref="S6.Thmtheorem7.p1.8.m8.6.6.2.2.2.2.3.1.cmml"><mo id="S6.Thmtheorem7.p1.8.m8.6.6.2.2.2.2.3.2.1.1" stretchy="false" xref="S6.Thmtheorem7.p1.8.m8.6.6.2.2.2.2.3.1.cmml">(</mo><mi id="S6.Thmtheorem7.p1.8.m8.4.4" xref="S6.Thmtheorem7.p1.8.m8.4.4.cmml">q</mi><mo id="S6.Thmtheorem7.p1.8.m8.6.6.2.2.2.2.3.2.1.2" stretchy="false" xref="S6.Thmtheorem7.p1.8.m8.6.6.2.2.2.2.3.1.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S6.Thmtheorem7.p1.8.m8.6.6.2.5" stretchy="false" xref="S6.Thmtheorem7.p1.8.m8.6.6.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem7.p1.8.m8.6b"><apply id="S6.Thmtheorem7.p1.8.m8.6.6.3.cmml" xref="S6.Thmtheorem7.p1.8.m8.6.6.2"><csymbol cd="latexml" id="S6.Thmtheorem7.p1.8.m8.6.6.3.1.cmml" xref="S6.Thmtheorem7.p1.8.m8.6.6.2.3">conditional-set</csymbol><apply id="S6.Thmtheorem7.p1.8.m8.5.5.1.1.cmml" xref="S6.Thmtheorem7.p1.8.m8.5.5.1.1"><in id="S6.Thmtheorem7.p1.8.m8.5.5.1.1.1.cmml" xref="S6.Thmtheorem7.p1.8.m8.5.5.1.1.1"></in><ci id="S6.Thmtheorem7.p1.8.m8.5.5.1.1.2.cmml" xref="S6.Thmtheorem7.p1.8.m8.5.5.1.1.2">𝑞</ci><apply id="S6.Thmtheorem7.p1.8.m8.5.5.1.1.3.cmml" xref="S6.Thmtheorem7.p1.8.m8.5.5.1.1.3"><csymbol cd="ambiguous" id="S6.Thmtheorem7.p1.8.m8.5.5.1.1.3.1.cmml" xref="S6.Thmtheorem7.p1.8.m8.5.5.1.1.3">subscript</csymbol><ci id="S6.Thmtheorem7.p1.8.m8.5.5.1.1.3.2.cmml" xref="S6.Thmtheorem7.p1.8.m8.5.5.1.1.3.2">𝑄</ci><ci id="S6.Thmtheorem7.p1.8.m8.5.5.1.1.3.3.cmml" xref="S6.Thmtheorem7.p1.8.m8.5.5.1.1.3.3">𝐸</ci></apply></apply><apply id="S6.Thmtheorem7.p1.8.m8.6.6.2.2.3.cmml" xref="S6.Thmtheorem7.p1.8.m8.6.6.2.2.2"><csymbol cd="ambiguous" id="S6.Thmtheorem7.p1.8.m8.6.6.2.2.3a.cmml" xref="S6.Thmtheorem7.p1.8.m8.6.6.2.2.2.3">formulae-sequence</csymbol><apply id="S6.Thmtheorem7.p1.8.m8.6.6.2.2.1.1.cmml" xref="S6.Thmtheorem7.p1.8.m8.6.6.2.2.1.1"><in id="S6.Thmtheorem7.p1.8.m8.6.6.2.2.1.1.1.cmml" xref="S6.Thmtheorem7.p1.8.m8.6.6.2.2.1.1.1"></in><ci id="S6.Thmtheorem7.p1.8.m8.6.6.2.2.1.1.2.cmml" xref="S6.Thmtheorem7.p1.8.m8.6.6.2.2.1.1.2">𝑎</ci><apply id="S6.Thmtheorem7.p1.8.m8.6.6.2.2.1.1.3.1.cmml" xref="S6.Thmtheorem7.p1.8.m8.6.6.2.2.1.1.3.2"><ci id="S6.Thmtheorem7.p1.8.m8.1.1.cmml" xref="S6.Thmtheorem7.p1.8.m8.1.1">dom</ci><ci id="S6.Thmtheorem7.p1.8.m8.2.2.cmml" xref="S6.Thmtheorem7.p1.8.m8.2.2">𝑞</ci></apply></apply><apply id="S6.Thmtheorem7.p1.8.m8.6.6.2.2.2.2.cmml" xref="S6.Thmtheorem7.p1.8.m8.6.6.2.2.2.2"><in id="S6.Thmtheorem7.p1.8.m8.6.6.2.2.2.2.1.cmml" xref="S6.Thmtheorem7.p1.8.m8.6.6.2.2.2.2.1"></in><ci id="S6.Thmtheorem7.p1.8.m8.6.6.2.2.2.2.2.cmml" xref="S6.Thmtheorem7.p1.8.m8.6.6.2.2.2.2.2">𝑥</ci><apply id="S6.Thmtheorem7.p1.8.m8.6.6.2.2.2.2.3.1.cmml" xref="S6.Thmtheorem7.p1.8.m8.6.6.2.2.2.2.3.2"><ci id="S6.Thmtheorem7.p1.8.m8.3.3.cmml" xref="S6.Thmtheorem7.p1.8.m8.3.3">ran</ci><ci id="S6.Thmtheorem7.p1.8.m8.4.4.cmml" xref="S6.Thmtheorem7.p1.8.m8.4.4">𝑞</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem7.p1.8.m8.6c">\{q\in Q_{E}:a\in\operatorname{dom}(q),x\in\operatorname{ran}(q)\}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem7.p1.8.m8.6d">{ italic_q ∈ italic_Q start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT : italic_a ∈ roman_dom ( italic_q ) , italic_x ∈ roman_ran ( italic_q ) }</annotation></semantics></math> is dense.</p> </div> </div> <div class="ltx_para" id="S6.SS1.p8"> <p class="ltx_p" id="S6.SS1.p8.1">Combining <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S2.Thmtheorem7" title="Theorem 2.7. ‣ 2. Aronszajn and Countryman lines ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">2.7</span></a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.Thmtheorem5" title="Lemma 6.5. ‣ 6.1. Moore’s forcing ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">6.5</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.Thmtheorem7" title="Lemma 6.7. ‣ 6.1. Moore’s forcing ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">6.7</span></a> one easily obtains a proof of <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S2.Thmtheorem8" title="Theorem 2.8. ‣ 2. Aronszajn and Countryman lines ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">2.8</span></a>.</p> </div> <div class="ltx_para" id="S6.SS1.p9"> <p class="ltx_p" id="S6.SS1.p9.1">We will also need the following, which is stated without proof in Moore’s paper (see the proof of <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib16" title="">16</a>, Lemma 3.4]</cite>).</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S6.Thmtheorem8"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem8.1.1.1">Lemma 6.8</span></span><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem8.2.2">.</span> </h6> <div class="ltx_para" id="S6.Thmtheorem8.p1"> <p class="ltx_p" id="S6.Thmtheorem8.p1.6">Assume <math alttext="A" class="ltx_Math" display="inline" id="S6.Thmtheorem8.p1.1.m1.1"><semantics id="S6.Thmtheorem8.p1.1.m1.1a"><mi id="S6.Thmtheorem8.p1.1.m1.1.1" xref="S6.Thmtheorem8.p1.1.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem8.p1.1.m1.1b"><ci id="S6.Thmtheorem8.p1.1.m1.1.1.cmml" xref="S6.Thmtheorem8.p1.1.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem8.p1.1.m1.1c">A</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem8.p1.1.m1.1d">italic_A</annotation></semantics></math> is Countryman. If <math alttext="\langle q_{\xi}:\xi<\omega_{1}\rangle" class="ltx_math_unparsed" display="inline" id="S6.Thmtheorem8.p1.2.m2.1"><semantics id="S6.Thmtheorem8.p1.2.m2.1a"><mrow id="S6.Thmtheorem8.p1.2.m2.1b"><mo id="S6.Thmtheorem8.p1.2.m2.1.1" stretchy="false">⟨</mo><msub id="S6.Thmtheorem8.p1.2.m2.1.2"><mi id="S6.Thmtheorem8.p1.2.m2.1.2.2">q</mi><mi id="S6.Thmtheorem8.p1.2.m2.1.2.3">ξ</mi></msub><mo id="S6.Thmtheorem8.p1.2.m2.1.3" lspace="0.278em" rspace="0.278em">:</mo><mi id="S6.Thmtheorem8.p1.2.m2.1.4">ξ</mi><mo id="S6.Thmtheorem8.p1.2.m2.1.5"><</mo><msub id="S6.Thmtheorem8.p1.2.m2.1.6"><mi id="S6.Thmtheorem8.p1.2.m2.1.6.2">ω</mi><mn id="S6.Thmtheorem8.p1.2.m2.1.6.3">1</mn></msub><mo id="S6.Thmtheorem8.p1.2.m2.1.7" stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex" id="S6.Thmtheorem8.p1.2.m2.1c">\langle q_{\xi}:\xi<\omega_{1}\rangle</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem8.p1.2.m2.1d">⟨ italic_q start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT : italic_ξ < italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⟩</annotation></semantics></math> is a sequence of elements of <math alttext="Q_{\varnothing}" class="ltx_Math" display="inline" id="S6.Thmtheorem8.p1.3.m3.1"><semantics id="S6.Thmtheorem8.p1.3.m3.1a"><msub id="S6.Thmtheorem8.p1.3.m3.1.1" xref="S6.Thmtheorem8.p1.3.m3.1.1.cmml"><mi id="S6.Thmtheorem8.p1.3.m3.1.1.2" xref="S6.Thmtheorem8.p1.3.m3.1.1.2.cmml">Q</mi><mi id="S6.Thmtheorem8.p1.3.m3.1.1.3" mathvariant="normal" xref="S6.Thmtheorem8.p1.3.m3.1.1.3.cmml">∅</mi></msub><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem8.p1.3.m3.1b"><apply id="S6.Thmtheorem8.p1.3.m3.1.1.cmml" xref="S6.Thmtheorem8.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S6.Thmtheorem8.p1.3.m3.1.1.1.cmml" xref="S6.Thmtheorem8.p1.3.m3.1.1">subscript</csymbol><ci id="S6.Thmtheorem8.p1.3.m3.1.1.2.cmml" xref="S6.Thmtheorem8.p1.3.m3.1.1.2">𝑄</ci><emptyset id="S6.Thmtheorem8.p1.3.m3.1.1.3.cmml" xref="S6.Thmtheorem8.p1.3.m3.1.1.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem8.p1.3.m3.1c">Q_{\varnothing}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem8.p1.3.m3.1d">italic_Q start_POSTSUBSCRIPT ∅ end_POSTSUBSCRIPT</annotation></semantics></math> with pairwise disjoint domains, then there is <math alttext="n<\omega" class="ltx_Math" display="inline" id="S6.Thmtheorem8.p1.4.m4.1"><semantics id="S6.Thmtheorem8.p1.4.m4.1a"><mrow id="S6.Thmtheorem8.p1.4.m4.1.1" xref="S6.Thmtheorem8.p1.4.m4.1.1.cmml"><mi id="S6.Thmtheorem8.p1.4.m4.1.1.2" xref="S6.Thmtheorem8.p1.4.m4.1.1.2.cmml">n</mi><mo id="S6.Thmtheorem8.p1.4.m4.1.1.1" xref="S6.Thmtheorem8.p1.4.m4.1.1.1.cmml"><</mo><mi id="S6.Thmtheorem8.p1.4.m4.1.1.3" xref="S6.Thmtheorem8.p1.4.m4.1.1.3.cmml">ω</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem8.p1.4.m4.1b"><apply id="S6.Thmtheorem8.p1.4.m4.1.1.cmml" xref="S6.Thmtheorem8.p1.4.m4.1.1"><lt id="S6.Thmtheorem8.p1.4.m4.1.1.1.cmml" xref="S6.Thmtheorem8.p1.4.m4.1.1.1"></lt><ci id="S6.Thmtheorem8.p1.4.m4.1.1.2.cmml" xref="S6.Thmtheorem8.p1.4.m4.1.1.2">𝑛</ci><ci id="S6.Thmtheorem8.p1.4.m4.1.1.3.cmml" xref="S6.Thmtheorem8.p1.4.m4.1.1.3">𝜔</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem8.p1.4.m4.1c">n<\omega</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem8.p1.4.m4.1d">italic_n < italic_ω</annotation></semantics></math>, an uncountable <math alttext="\Gamma\subseteq\omega_{1}" class="ltx_Math" display="inline" id="S6.Thmtheorem8.p1.5.m5.1"><semantics id="S6.Thmtheorem8.p1.5.m5.1a"><mrow id="S6.Thmtheorem8.p1.5.m5.1.1" xref="S6.Thmtheorem8.p1.5.m5.1.1.cmml"><mi id="S6.Thmtheorem8.p1.5.m5.1.1.2" mathvariant="normal" xref="S6.Thmtheorem8.p1.5.m5.1.1.2.cmml">Γ</mi><mo id="S6.Thmtheorem8.p1.5.m5.1.1.1" xref="S6.Thmtheorem8.p1.5.m5.1.1.1.cmml">⊆</mo><msub id="S6.Thmtheorem8.p1.5.m5.1.1.3" xref="S6.Thmtheorem8.p1.5.m5.1.1.3.cmml"><mi id="S6.Thmtheorem8.p1.5.m5.1.1.3.2" xref="S6.Thmtheorem8.p1.5.m5.1.1.3.2.cmml">ω</mi><mn id="S6.Thmtheorem8.p1.5.m5.1.1.3.3" xref="S6.Thmtheorem8.p1.5.m5.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem8.p1.5.m5.1b"><apply id="S6.Thmtheorem8.p1.5.m5.1.1.cmml" xref="S6.Thmtheorem8.p1.5.m5.1.1"><subset id="S6.Thmtheorem8.p1.5.m5.1.1.1.cmml" xref="S6.Thmtheorem8.p1.5.m5.1.1.1"></subset><ci id="S6.Thmtheorem8.p1.5.m5.1.1.2.cmml" xref="S6.Thmtheorem8.p1.5.m5.1.1.2">Γ</ci><apply id="S6.Thmtheorem8.p1.5.m5.1.1.3.cmml" xref="S6.Thmtheorem8.p1.5.m5.1.1.3"><csymbol cd="ambiguous" id="S6.Thmtheorem8.p1.5.m5.1.1.3.1.cmml" xref="S6.Thmtheorem8.p1.5.m5.1.1.3">subscript</csymbol><ci id="S6.Thmtheorem8.p1.5.m5.1.1.3.2.cmml" xref="S6.Thmtheorem8.p1.5.m5.1.1.3.2">𝜔</ci><cn id="S6.Thmtheorem8.p1.5.m5.1.1.3.3.cmml" type="integer" xref="S6.Thmtheorem8.p1.5.m5.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem8.p1.5.m5.1c">\Gamma\subseteq\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem8.p1.5.m5.1d">roman_Γ ⊆ italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\gamma<\min(\Gamma)" class="ltx_Math" display="inline" id="S6.Thmtheorem8.p1.6.m6.2"><semantics id="S6.Thmtheorem8.p1.6.m6.2a"><mrow id="S6.Thmtheorem8.p1.6.m6.2.3" xref="S6.Thmtheorem8.p1.6.m6.2.3.cmml"><mi id="S6.Thmtheorem8.p1.6.m6.2.3.2" xref="S6.Thmtheorem8.p1.6.m6.2.3.2.cmml">γ</mi><mo id="S6.Thmtheorem8.p1.6.m6.2.3.1" xref="S6.Thmtheorem8.p1.6.m6.2.3.1.cmml"><</mo><mrow id="S6.Thmtheorem8.p1.6.m6.2.3.3.2" xref="S6.Thmtheorem8.p1.6.m6.2.3.3.1.cmml"><mi id="S6.Thmtheorem8.p1.6.m6.1.1" xref="S6.Thmtheorem8.p1.6.m6.1.1.cmml">min</mi><mo id="S6.Thmtheorem8.p1.6.m6.2.3.3.2a" xref="S6.Thmtheorem8.p1.6.m6.2.3.3.1.cmml">⁡</mo><mrow id="S6.Thmtheorem8.p1.6.m6.2.3.3.2.1" xref="S6.Thmtheorem8.p1.6.m6.2.3.3.1.cmml"><mo id="S6.Thmtheorem8.p1.6.m6.2.3.3.2.1.1" stretchy="false" xref="S6.Thmtheorem8.p1.6.m6.2.3.3.1.cmml">(</mo><mi id="S6.Thmtheorem8.p1.6.m6.2.2" mathvariant="normal" xref="S6.Thmtheorem8.p1.6.m6.2.2.cmml">Γ</mi><mo id="S6.Thmtheorem8.p1.6.m6.2.3.3.2.1.2" stretchy="false" xref="S6.Thmtheorem8.p1.6.m6.2.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem8.p1.6.m6.2b"><apply id="S6.Thmtheorem8.p1.6.m6.2.3.cmml" xref="S6.Thmtheorem8.p1.6.m6.2.3"><lt id="S6.Thmtheorem8.p1.6.m6.2.3.1.cmml" xref="S6.Thmtheorem8.p1.6.m6.2.3.1"></lt><ci id="S6.Thmtheorem8.p1.6.m6.2.3.2.cmml" xref="S6.Thmtheorem8.p1.6.m6.2.3.2">𝛾</ci><apply id="S6.Thmtheorem8.p1.6.m6.2.3.3.1.cmml" xref="S6.Thmtheorem8.p1.6.m6.2.3.3.2"><min id="S6.Thmtheorem8.p1.6.m6.1.1.cmml" xref="S6.Thmtheorem8.p1.6.m6.1.1"></min><ci id="S6.Thmtheorem8.p1.6.m6.2.2.cmml" xref="S6.Thmtheorem8.p1.6.m6.2.2">Γ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem8.p1.6.m6.2c">\gamma<\min(\Gamma)</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem8.p1.6.m6.2d">italic_γ < roman_min ( roman_Γ )</annotation></semantics></math> such that,</p> <ol class="ltx_enumerate" id="S6.I3"> <li class="ltx_item" id="S6.I3.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(1)</span> <div class="ltx_para" id="S6.I3.i1.p1"> <p class="ltx_p" id="S6.I3.i1.p1.2">For all <math alttext="\xi\in\Gamma" class="ltx_Math" display="inline" id="S6.I3.i1.p1.1.m1.1"><semantics id="S6.I3.i1.p1.1.m1.1a"><mrow id="S6.I3.i1.p1.1.m1.1.1" xref="S6.I3.i1.p1.1.m1.1.1.cmml"><mi id="S6.I3.i1.p1.1.m1.1.1.2" xref="S6.I3.i1.p1.1.m1.1.1.2.cmml">ξ</mi><mo id="S6.I3.i1.p1.1.m1.1.1.1" xref="S6.I3.i1.p1.1.m1.1.1.1.cmml">∈</mo><mi id="S6.I3.i1.p1.1.m1.1.1.3" mathvariant="normal" xref="S6.I3.i1.p1.1.m1.1.1.3.cmml">Γ</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.I3.i1.p1.1.m1.1b"><apply id="S6.I3.i1.p1.1.m1.1.1.cmml" xref="S6.I3.i1.p1.1.m1.1.1"><in id="S6.I3.i1.p1.1.m1.1.1.1.cmml" xref="S6.I3.i1.p1.1.m1.1.1.1"></in><ci id="S6.I3.i1.p1.1.m1.1.1.2.cmml" xref="S6.I3.i1.p1.1.m1.1.1.2">𝜉</ci><ci id="S6.I3.i1.p1.1.m1.1.1.3.cmml" xref="S6.I3.i1.p1.1.m1.1.1.3">Γ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I3.i1.p1.1.m1.1c">\xi\in\Gamma</annotation><annotation encoding="application/x-llamapun" id="S6.I3.i1.p1.1.m1.1d">italic_ξ ∈ roman_Γ</annotation></semantics></math>, <math alttext="|\operatorname{dom}(q_{\xi})|=n" class="ltx_Math" display="inline" id="S6.I3.i1.p1.2.m2.2"><semantics id="S6.I3.i1.p1.2.m2.2a"><mrow id="S6.I3.i1.p1.2.m2.2.2" xref="S6.I3.i1.p1.2.m2.2.2.cmml"><mrow id="S6.I3.i1.p1.2.m2.2.2.1.1" xref="S6.I3.i1.p1.2.m2.2.2.1.2.cmml"><mo id="S6.I3.i1.p1.2.m2.2.2.1.1.2" stretchy="false" xref="S6.I3.i1.p1.2.m2.2.2.1.2.1.cmml">|</mo><mrow id="S6.I3.i1.p1.2.m2.2.2.1.1.1.1" xref="S6.I3.i1.p1.2.m2.2.2.1.1.1.2.cmml"><mi id="S6.I3.i1.p1.2.m2.1.1" xref="S6.I3.i1.p1.2.m2.1.1.cmml">dom</mi><mo id="S6.I3.i1.p1.2.m2.2.2.1.1.1.1a" xref="S6.I3.i1.p1.2.m2.2.2.1.1.1.2.cmml">⁡</mo><mrow id="S6.I3.i1.p1.2.m2.2.2.1.1.1.1.1" xref="S6.I3.i1.p1.2.m2.2.2.1.1.1.2.cmml"><mo id="S6.I3.i1.p1.2.m2.2.2.1.1.1.1.1.2" stretchy="false" xref="S6.I3.i1.p1.2.m2.2.2.1.1.1.2.cmml">(</mo><msub id="S6.I3.i1.p1.2.m2.2.2.1.1.1.1.1.1" xref="S6.I3.i1.p1.2.m2.2.2.1.1.1.1.1.1.cmml"><mi id="S6.I3.i1.p1.2.m2.2.2.1.1.1.1.1.1.2" xref="S6.I3.i1.p1.2.m2.2.2.1.1.1.1.1.1.2.cmml">q</mi><mi id="S6.I3.i1.p1.2.m2.2.2.1.1.1.1.1.1.3" xref="S6.I3.i1.p1.2.m2.2.2.1.1.1.1.1.1.3.cmml">ξ</mi></msub><mo id="S6.I3.i1.p1.2.m2.2.2.1.1.1.1.1.3" stretchy="false" xref="S6.I3.i1.p1.2.m2.2.2.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S6.I3.i1.p1.2.m2.2.2.1.1.3" stretchy="false" xref="S6.I3.i1.p1.2.m2.2.2.1.2.1.cmml">|</mo></mrow><mo id="S6.I3.i1.p1.2.m2.2.2.2" xref="S6.I3.i1.p1.2.m2.2.2.2.cmml">=</mo><mi id="S6.I3.i1.p1.2.m2.2.2.3" xref="S6.I3.i1.p1.2.m2.2.2.3.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.I3.i1.p1.2.m2.2b"><apply id="S6.I3.i1.p1.2.m2.2.2.cmml" xref="S6.I3.i1.p1.2.m2.2.2"><eq id="S6.I3.i1.p1.2.m2.2.2.2.cmml" xref="S6.I3.i1.p1.2.m2.2.2.2"></eq><apply id="S6.I3.i1.p1.2.m2.2.2.1.2.cmml" xref="S6.I3.i1.p1.2.m2.2.2.1.1"><abs id="S6.I3.i1.p1.2.m2.2.2.1.2.1.cmml" xref="S6.I3.i1.p1.2.m2.2.2.1.1.2"></abs><apply id="S6.I3.i1.p1.2.m2.2.2.1.1.1.2.cmml" xref="S6.I3.i1.p1.2.m2.2.2.1.1.1.1"><ci id="S6.I3.i1.p1.2.m2.1.1.cmml" xref="S6.I3.i1.p1.2.m2.1.1">dom</ci><apply id="S6.I3.i1.p1.2.m2.2.2.1.1.1.1.1.1.cmml" xref="S6.I3.i1.p1.2.m2.2.2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.I3.i1.p1.2.m2.2.2.1.1.1.1.1.1.1.cmml" xref="S6.I3.i1.p1.2.m2.2.2.1.1.1.1.1.1">subscript</csymbol><ci id="S6.I3.i1.p1.2.m2.2.2.1.1.1.1.1.1.2.cmml" xref="S6.I3.i1.p1.2.m2.2.2.1.1.1.1.1.1.2">𝑞</ci><ci id="S6.I3.i1.p1.2.m2.2.2.1.1.1.1.1.1.3.cmml" xref="S6.I3.i1.p1.2.m2.2.2.1.1.1.1.1.1.3">𝜉</ci></apply></apply></apply><ci id="S6.I3.i1.p1.2.m2.2.2.3.cmml" xref="S6.I3.i1.p1.2.m2.2.2.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I3.i1.p1.2.m2.2c">|\operatorname{dom}(q_{\xi})|=n</annotation><annotation encoding="application/x-llamapun" id="S6.I3.i1.p1.2.m2.2d">| roman_dom ( italic_q start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT ) | = italic_n</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S6.I3.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(2)</span> <div class="ltx_para" id="S6.I3.i2.p1"> <p class="ltx_p" id="S6.I3.i2.p1.4">For all <math alttext="\xi\in\Gamma" class="ltx_Math" display="inline" id="S6.I3.i2.p1.1.m1.1"><semantics id="S6.I3.i2.p1.1.m1.1a"><mrow id="S6.I3.i2.p1.1.m1.1.1" xref="S6.I3.i2.p1.1.m1.1.1.cmml"><mi id="S6.I3.i2.p1.1.m1.1.1.2" xref="S6.I3.i2.p1.1.m1.1.1.2.cmml">ξ</mi><mo id="S6.I3.i2.p1.1.m1.1.1.1" xref="S6.I3.i2.p1.1.m1.1.1.1.cmml">∈</mo><mi id="S6.I3.i2.p1.1.m1.1.1.3" mathvariant="normal" xref="S6.I3.i2.p1.1.m1.1.1.3.cmml">Γ</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.I3.i2.p1.1.m1.1b"><apply id="S6.I3.i2.p1.1.m1.1.1.cmml" xref="S6.I3.i2.p1.1.m1.1.1"><in id="S6.I3.i2.p1.1.m1.1.1.1.cmml" xref="S6.I3.i2.p1.1.m1.1.1.1"></in><ci id="S6.I3.i2.p1.1.m1.1.1.2.cmml" xref="S6.I3.i2.p1.1.m1.1.1.2">𝜉</ci><ci id="S6.I3.i2.p1.1.m1.1.1.3.cmml" xref="S6.I3.i2.p1.1.m1.1.1.3">Γ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I3.i2.p1.1.m1.1c">\xi\in\Gamma</annotation><annotation encoding="application/x-llamapun" id="S6.I3.i2.p1.1.m1.1d">italic_ξ ∈ roman_Γ</annotation></semantics></math> and <math alttext="a\neq b\in\operatorname{dom}(q_{\xi})" class="ltx_Math" display="inline" id="S6.I3.i2.p1.2.m2.2"><semantics id="S6.I3.i2.p1.2.m2.2a"><mrow id="S6.I3.i2.p1.2.m2.2.2" xref="S6.I3.i2.p1.2.m2.2.2.cmml"><mi id="S6.I3.i2.p1.2.m2.2.2.3" xref="S6.I3.i2.p1.2.m2.2.2.3.cmml">a</mi><mo id="S6.I3.i2.p1.2.m2.2.2.4" xref="S6.I3.i2.p1.2.m2.2.2.4.cmml">≠</mo><mi id="S6.I3.i2.p1.2.m2.2.2.5" xref="S6.I3.i2.p1.2.m2.2.2.5.cmml">b</mi><mo id="S6.I3.i2.p1.2.m2.2.2.6" xref="S6.I3.i2.p1.2.m2.2.2.6.cmml">∈</mo><mrow id="S6.I3.i2.p1.2.m2.2.2.1.1" xref="S6.I3.i2.p1.2.m2.2.2.1.2.cmml"><mi id="S6.I3.i2.p1.2.m2.1.1" xref="S6.I3.i2.p1.2.m2.1.1.cmml">dom</mi><mo id="S6.I3.i2.p1.2.m2.2.2.1.1a" xref="S6.I3.i2.p1.2.m2.2.2.1.2.cmml">⁡</mo><mrow id="S6.I3.i2.p1.2.m2.2.2.1.1.1" xref="S6.I3.i2.p1.2.m2.2.2.1.2.cmml"><mo id="S6.I3.i2.p1.2.m2.2.2.1.1.1.2" stretchy="false" xref="S6.I3.i2.p1.2.m2.2.2.1.2.cmml">(</mo><msub id="S6.I3.i2.p1.2.m2.2.2.1.1.1.1" xref="S6.I3.i2.p1.2.m2.2.2.1.1.1.1.cmml"><mi id="S6.I3.i2.p1.2.m2.2.2.1.1.1.1.2" xref="S6.I3.i2.p1.2.m2.2.2.1.1.1.1.2.cmml">q</mi><mi id="S6.I3.i2.p1.2.m2.2.2.1.1.1.1.3" xref="S6.I3.i2.p1.2.m2.2.2.1.1.1.1.3.cmml">ξ</mi></msub><mo id="S6.I3.i2.p1.2.m2.2.2.1.1.1.3" stretchy="false" xref="S6.I3.i2.p1.2.m2.2.2.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.I3.i2.p1.2.m2.2b"><apply id="S6.I3.i2.p1.2.m2.2.2.cmml" xref="S6.I3.i2.p1.2.m2.2.2"><and id="S6.I3.i2.p1.2.m2.2.2a.cmml" xref="S6.I3.i2.p1.2.m2.2.2"></and><apply id="S6.I3.i2.p1.2.m2.2.2b.cmml" xref="S6.I3.i2.p1.2.m2.2.2"><neq id="S6.I3.i2.p1.2.m2.2.2.4.cmml" xref="S6.I3.i2.p1.2.m2.2.2.4"></neq><ci id="S6.I3.i2.p1.2.m2.2.2.3.cmml" xref="S6.I3.i2.p1.2.m2.2.2.3">𝑎</ci><ci id="S6.I3.i2.p1.2.m2.2.2.5.cmml" xref="S6.I3.i2.p1.2.m2.2.2.5">𝑏</ci></apply><apply id="S6.I3.i2.p1.2.m2.2.2c.cmml" xref="S6.I3.i2.p1.2.m2.2.2"><in id="S6.I3.i2.p1.2.m2.2.2.6.cmml" xref="S6.I3.i2.p1.2.m2.2.2.6"></in><share href="https://arxiv.org/html/2503.13728v1#S6.I3.i2.p1.2.m2.2.2.5.cmml" id="S6.I3.i2.p1.2.m2.2.2d.cmml" xref="S6.I3.i2.p1.2.m2.2.2"></share><apply id="S6.I3.i2.p1.2.m2.2.2.1.2.cmml" xref="S6.I3.i2.p1.2.m2.2.2.1.1"><ci id="S6.I3.i2.p1.2.m2.1.1.cmml" xref="S6.I3.i2.p1.2.m2.1.1">dom</ci><apply id="S6.I3.i2.p1.2.m2.2.2.1.1.1.1.cmml" xref="S6.I3.i2.p1.2.m2.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S6.I3.i2.p1.2.m2.2.2.1.1.1.1.1.cmml" xref="S6.I3.i2.p1.2.m2.2.2.1.1.1.1">subscript</csymbol><ci id="S6.I3.i2.p1.2.m2.2.2.1.1.1.1.2.cmml" xref="S6.I3.i2.p1.2.m2.2.2.1.1.1.1.2">𝑞</ci><ci id="S6.I3.i2.p1.2.m2.2.2.1.1.1.1.3.cmml" xref="S6.I3.i2.p1.2.m2.2.2.1.1.1.1.3">𝜉</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I3.i2.p1.2.m2.2c">a\neq b\in\operatorname{dom}(q_{\xi})</annotation><annotation encoding="application/x-llamapun" id="S6.I3.i2.p1.2.m2.2d">italic_a ≠ italic_b ∈ roman_dom ( italic_q start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT )</annotation></semantics></math>, if <math alttext="\Delta_{A}(a,b)<\xi" class="ltx_Math" display="inline" id="S6.I3.i2.p1.3.m3.2"><semantics id="S6.I3.i2.p1.3.m3.2a"><mrow id="S6.I3.i2.p1.3.m3.2.3" xref="S6.I3.i2.p1.3.m3.2.3.cmml"><mrow id="S6.I3.i2.p1.3.m3.2.3.2" xref="S6.I3.i2.p1.3.m3.2.3.2.cmml"><msub id="S6.I3.i2.p1.3.m3.2.3.2.2" xref="S6.I3.i2.p1.3.m3.2.3.2.2.cmml"><mi id="S6.I3.i2.p1.3.m3.2.3.2.2.2" mathvariant="normal" xref="S6.I3.i2.p1.3.m3.2.3.2.2.2.cmml">Δ</mi><mi id="S6.I3.i2.p1.3.m3.2.3.2.2.3" xref="S6.I3.i2.p1.3.m3.2.3.2.2.3.cmml">A</mi></msub><mo id="S6.I3.i2.p1.3.m3.2.3.2.1" xref="S6.I3.i2.p1.3.m3.2.3.2.1.cmml">⁢</mo><mrow id="S6.I3.i2.p1.3.m3.2.3.2.3.2" xref="S6.I3.i2.p1.3.m3.2.3.2.3.1.cmml"><mo id="S6.I3.i2.p1.3.m3.2.3.2.3.2.1" stretchy="false" xref="S6.I3.i2.p1.3.m3.2.3.2.3.1.cmml">(</mo><mi id="S6.I3.i2.p1.3.m3.1.1" xref="S6.I3.i2.p1.3.m3.1.1.cmml">a</mi><mo id="S6.I3.i2.p1.3.m3.2.3.2.3.2.2" xref="S6.I3.i2.p1.3.m3.2.3.2.3.1.cmml">,</mo><mi id="S6.I3.i2.p1.3.m3.2.2" xref="S6.I3.i2.p1.3.m3.2.2.cmml">b</mi><mo id="S6.I3.i2.p1.3.m3.2.3.2.3.2.3" stretchy="false" xref="S6.I3.i2.p1.3.m3.2.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="S6.I3.i2.p1.3.m3.2.3.1" xref="S6.I3.i2.p1.3.m3.2.3.1.cmml"><</mo><mi id="S6.I3.i2.p1.3.m3.2.3.3" xref="S6.I3.i2.p1.3.m3.2.3.3.cmml">ξ</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.I3.i2.p1.3.m3.2b"><apply id="S6.I3.i2.p1.3.m3.2.3.cmml" xref="S6.I3.i2.p1.3.m3.2.3"><lt id="S6.I3.i2.p1.3.m3.2.3.1.cmml" xref="S6.I3.i2.p1.3.m3.2.3.1"></lt><apply id="S6.I3.i2.p1.3.m3.2.3.2.cmml" xref="S6.I3.i2.p1.3.m3.2.3.2"><times id="S6.I3.i2.p1.3.m3.2.3.2.1.cmml" xref="S6.I3.i2.p1.3.m3.2.3.2.1"></times><apply id="S6.I3.i2.p1.3.m3.2.3.2.2.cmml" xref="S6.I3.i2.p1.3.m3.2.3.2.2"><csymbol cd="ambiguous" id="S6.I3.i2.p1.3.m3.2.3.2.2.1.cmml" xref="S6.I3.i2.p1.3.m3.2.3.2.2">subscript</csymbol><ci id="S6.I3.i2.p1.3.m3.2.3.2.2.2.cmml" xref="S6.I3.i2.p1.3.m3.2.3.2.2.2">Δ</ci><ci id="S6.I3.i2.p1.3.m3.2.3.2.2.3.cmml" xref="S6.I3.i2.p1.3.m3.2.3.2.2.3">𝐴</ci></apply><interval closure="open" id="S6.I3.i2.p1.3.m3.2.3.2.3.1.cmml" xref="S6.I3.i2.p1.3.m3.2.3.2.3.2"><ci id="S6.I3.i2.p1.3.m3.1.1.cmml" xref="S6.I3.i2.p1.3.m3.1.1">𝑎</ci><ci id="S6.I3.i2.p1.3.m3.2.2.cmml" xref="S6.I3.i2.p1.3.m3.2.2">𝑏</ci></interval></apply><ci id="S6.I3.i2.p1.3.m3.2.3.3.cmml" xref="S6.I3.i2.p1.3.m3.2.3.3">𝜉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I3.i2.p1.3.m3.2c">\Delta_{A}(a,b)<\xi</annotation><annotation encoding="application/x-llamapun" id="S6.I3.i2.p1.3.m3.2d">roman_Δ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_a , italic_b ) < italic_ξ</annotation></semantics></math> then <math alttext="\Delta_{A}(a,b)<\gamma" class="ltx_Math" display="inline" id="S6.I3.i2.p1.4.m4.2"><semantics id="S6.I3.i2.p1.4.m4.2a"><mrow id="S6.I3.i2.p1.4.m4.2.3" xref="S6.I3.i2.p1.4.m4.2.3.cmml"><mrow id="S6.I3.i2.p1.4.m4.2.3.2" xref="S6.I3.i2.p1.4.m4.2.3.2.cmml"><msub id="S6.I3.i2.p1.4.m4.2.3.2.2" xref="S6.I3.i2.p1.4.m4.2.3.2.2.cmml"><mi id="S6.I3.i2.p1.4.m4.2.3.2.2.2" mathvariant="normal" xref="S6.I3.i2.p1.4.m4.2.3.2.2.2.cmml">Δ</mi><mi id="S6.I3.i2.p1.4.m4.2.3.2.2.3" xref="S6.I3.i2.p1.4.m4.2.3.2.2.3.cmml">A</mi></msub><mo id="S6.I3.i2.p1.4.m4.2.3.2.1" xref="S6.I3.i2.p1.4.m4.2.3.2.1.cmml">⁢</mo><mrow id="S6.I3.i2.p1.4.m4.2.3.2.3.2" xref="S6.I3.i2.p1.4.m4.2.3.2.3.1.cmml"><mo id="S6.I3.i2.p1.4.m4.2.3.2.3.2.1" stretchy="false" xref="S6.I3.i2.p1.4.m4.2.3.2.3.1.cmml">(</mo><mi id="S6.I3.i2.p1.4.m4.1.1" xref="S6.I3.i2.p1.4.m4.1.1.cmml">a</mi><mo id="S6.I3.i2.p1.4.m4.2.3.2.3.2.2" xref="S6.I3.i2.p1.4.m4.2.3.2.3.1.cmml">,</mo><mi id="S6.I3.i2.p1.4.m4.2.2" xref="S6.I3.i2.p1.4.m4.2.2.cmml">b</mi><mo id="S6.I3.i2.p1.4.m4.2.3.2.3.2.3" stretchy="false" xref="S6.I3.i2.p1.4.m4.2.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="S6.I3.i2.p1.4.m4.2.3.1" xref="S6.I3.i2.p1.4.m4.2.3.1.cmml"><</mo><mi id="S6.I3.i2.p1.4.m4.2.3.3" xref="S6.I3.i2.p1.4.m4.2.3.3.cmml">γ</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.I3.i2.p1.4.m4.2b"><apply id="S6.I3.i2.p1.4.m4.2.3.cmml" xref="S6.I3.i2.p1.4.m4.2.3"><lt id="S6.I3.i2.p1.4.m4.2.3.1.cmml" xref="S6.I3.i2.p1.4.m4.2.3.1"></lt><apply id="S6.I3.i2.p1.4.m4.2.3.2.cmml" xref="S6.I3.i2.p1.4.m4.2.3.2"><times id="S6.I3.i2.p1.4.m4.2.3.2.1.cmml" xref="S6.I3.i2.p1.4.m4.2.3.2.1"></times><apply id="S6.I3.i2.p1.4.m4.2.3.2.2.cmml" xref="S6.I3.i2.p1.4.m4.2.3.2.2"><csymbol cd="ambiguous" id="S6.I3.i2.p1.4.m4.2.3.2.2.1.cmml" xref="S6.I3.i2.p1.4.m4.2.3.2.2">subscript</csymbol><ci id="S6.I3.i2.p1.4.m4.2.3.2.2.2.cmml" xref="S6.I3.i2.p1.4.m4.2.3.2.2.2">Δ</ci><ci id="S6.I3.i2.p1.4.m4.2.3.2.2.3.cmml" xref="S6.I3.i2.p1.4.m4.2.3.2.2.3">𝐴</ci></apply><interval closure="open" id="S6.I3.i2.p1.4.m4.2.3.2.3.1.cmml" xref="S6.I3.i2.p1.4.m4.2.3.2.3.2"><ci id="S6.I3.i2.p1.4.m4.1.1.cmml" xref="S6.I3.i2.p1.4.m4.1.1">𝑎</ci><ci id="S6.I3.i2.p1.4.m4.2.2.cmml" xref="S6.I3.i2.p1.4.m4.2.2">𝑏</ci></interval></apply><ci id="S6.I3.i2.p1.4.m4.2.3.3.cmml" xref="S6.I3.i2.p1.4.m4.2.3.3">𝛾</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I3.i2.p1.4.m4.2c">\Delta_{A}(a,b)<\gamma</annotation><annotation encoding="application/x-llamapun" id="S6.I3.i2.p1.4.m4.2d">roman_Δ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_a , italic_b ) < italic_γ</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S6.I3.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(3)</span> <div class="ltx_para" id="S6.I3.i3.p1"> <p class="ltx_p" id="S6.I3.i3.p1.10">For all <math alttext="\xi<\eta\in\Gamma" class="ltx_Math" display="inline" id="S6.I3.i3.p1.1.m1.1"><semantics id="S6.I3.i3.p1.1.m1.1a"><mrow id="S6.I3.i3.p1.1.m1.1.1" xref="S6.I3.i3.p1.1.m1.1.1.cmml"><mi id="S6.I3.i3.p1.1.m1.1.1.2" xref="S6.I3.i3.p1.1.m1.1.1.2.cmml">ξ</mi><mo id="S6.I3.i3.p1.1.m1.1.1.3" xref="S6.I3.i3.p1.1.m1.1.1.3.cmml"><</mo><mi id="S6.I3.i3.p1.1.m1.1.1.4" xref="S6.I3.i3.p1.1.m1.1.1.4.cmml">η</mi><mo id="S6.I3.i3.p1.1.m1.1.1.5" xref="S6.I3.i3.p1.1.m1.1.1.5.cmml">∈</mo><mi id="S6.I3.i3.p1.1.m1.1.1.6" mathvariant="normal" xref="S6.I3.i3.p1.1.m1.1.1.6.cmml">Γ</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.I3.i3.p1.1.m1.1b"><apply id="S6.I3.i3.p1.1.m1.1.1.cmml" xref="S6.I3.i3.p1.1.m1.1.1"><and id="S6.I3.i3.p1.1.m1.1.1a.cmml" xref="S6.I3.i3.p1.1.m1.1.1"></and><apply id="S6.I3.i3.p1.1.m1.1.1b.cmml" xref="S6.I3.i3.p1.1.m1.1.1"><lt id="S6.I3.i3.p1.1.m1.1.1.3.cmml" xref="S6.I3.i3.p1.1.m1.1.1.3"></lt><ci id="S6.I3.i3.p1.1.m1.1.1.2.cmml" xref="S6.I3.i3.p1.1.m1.1.1.2">𝜉</ci><ci id="S6.I3.i3.p1.1.m1.1.1.4.cmml" xref="S6.I3.i3.p1.1.m1.1.1.4">𝜂</ci></apply><apply id="S6.I3.i3.p1.1.m1.1.1c.cmml" xref="S6.I3.i3.p1.1.m1.1.1"><in id="S6.I3.i3.p1.1.m1.1.1.5.cmml" xref="S6.I3.i3.p1.1.m1.1.1.5"></in><share href="https://arxiv.org/html/2503.13728v1#S6.I3.i3.p1.1.m1.1.1.4.cmml" id="S6.I3.i3.p1.1.m1.1.1d.cmml" xref="S6.I3.i3.p1.1.m1.1.1"></share><ci id="S6.I3.i3.p1.1.m1.1.1.6.cmml" xref="S6.I3.i3.p1.1.m1.1.1.6">Γ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I3.i3.p1.1.m1.1c">\xi<\eta\in\Gamma</annotation><annotation encoding="application/x-llamapun" id="S6.I3.i3.p1.1.m1.1d">italic_ξ < italic_η ∈ roman_Γ</annotation></semantics></math> and <math alttext="i<n" class="ltx_Math" display="inline" id="S6.I3.i3.p1.2.m2.1"><semantics id="S6.I3.i3.p1.2.m2.1a"><mrow id="S6.I3.i3.p1.2.m2.1.1" xref="S6.I3.i3.p1.2.m2.1.1.cmml"><mi id="S6.I3.i3.p1.2.m2.1.1.2" xref="S6.I3.i3.p1.2.m2.1.1.2.cmml">i</mi><mo id="S6.I3.i3.p1.2.m2.1.1.1" xref="S6.I3.i3.p1.2.m2.1.1.1.cmml"><</mo><mi id="S6.I3.i3.p1.2.m2.1.1.3" xref="S6.I3.i3.p1.2.m2.1.1.3.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.I3.i3.p1.2.m2.1b"><apply id="S6.I3.i3.p1.2.m2.1.1.cmml" xref="S6.I3.i3.p1.2.m2.1.1"><lt id="S6.I3.i3.p1.2.m2.1.1.1.cmml" xref="S6.I3.i3.p1.2.m2.1.1.1"></lt><ci id="S6.I3.i3.p1.2.m2.1.1.2.cmml" xref="S6.I3.i3.p1.2.m2.1.1.2">𝑖</ci><ci id="S6.I3.i3.p1.2.m2.1.1.3.cmml" xref="S6.I3.i3.p1.2.m2.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I3.i3.p1.2.m2.1c">i<n</annotation><annotation encoding="application/x-llamapun" id="S6.I3.i3.p1.2.m2.1d">italic_i < italic_n</annotation></semantics></math>, if <math alttext="a" class="ltx_Math" display="inline" id="S6.I3.i3.p1.3.m3.1"><semantics id="S6.I3.i3.p1.3.m3.1a"><mi id="S6.I3.i3.p1.3.m3.1.1" xref="S6.I3.i3.p1.3.m3.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S6.I3.i3.p1.3.m3.1b"><ci id="S6.I3.i3.p1.3.m3.1.1.cmml" xref="S6.I3.i3.p1.3.m3.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.I3.i3.p1.3.m3.1c">a</annotation><annotation encoding="application/x-llamapun" id="S6.I3.i3.p1.3.m3.1d">italic_a</annotation></semantics></math> is the <math alttext="i" class="ltx_Math" display="inline" id="S6.I3.i3.p1.4.m4.1"><semantics id="S6.I3.i3.p1.4.m4.1a"><mi id="S6.I3.i3.p1.4.m4.1.1" xref="S6.I3.i3.p1.4.m4.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S6.I3.i3.p1.4.m4.1b"><ci id="S6.I3.i3.p1.4.m4.1.1.cmml" xref="S6.I3.i3.p1.4.m4.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.I3.i3.p1.4.m4.1c">i</annotation><annotation encoding="application/x-llamapun" id="S6.I3.i3.p1.4.m4.1d">italic_i</annotation></semantics></math>-th element in the <math alttext="<_{A}" class="ltx_Math" display="inline" id="S6.I3.i3.p1.5.m5.1"><semantics id="S6.I3.i3.p1.5.m5.1a"><msub id="S6.I3.i3.p1.5.m5.1.1" xref="S6.I3.i3.p1.5.m5.1.1.cmml"><mo id="S6.I3.i3.p1.5.m5.1.1.2" xref="S6.I3.i3.p1.5.m5.1.1.2.cmml"><</mo><mi id="S6.I3.i3.p1.5.m5.1.1.3" xref="S6.I3.i3.p1.5.m5.1.1.3.cmml">A</mi></msub><annotation-xml encoding="MathML-Content" id="S6.I3.i3.p1.5.m5.1b"><apply id="S6.I3.i3.p1.5.m5.1.1.cmml" xref="S6.I3.i3.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S6.I3.i3.p1.5.m5.1.1.1.cmml" xref="S6.I3.i3.p1.5.m5.1.1">subscript</csymbol><lt id="S6.I3.i3.p1.5.m5.1.1.2.cmml" xref="S6.I3.i3.p1.5.m5.1.1.2"></lt><ci id="S6.I3.i3.p1.5.m5.1.1.3.cmml" xref="S6.I3.i3.p1.5.m5.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I3.i3.p1.5.m5.1c"><_{A}</annotation><annotation encoding="application/x-llamapun" id="S6.I3.i3.p1.5.m5.1d">< start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT</annotation></semantics></math>-increasing enumeration of <math alttext="\operatorname{dom}(q_{\xi})" class="ltx_Math" display="inline" id="S6.I3.i3.p1.6.m6.2"><semantics id="S6.I3.i3.p1.6.m6.2a"><mrow id="S6.I3.i3.p1.6.m6.2.2.1" xref="S6.I3.i3.p1.6.m6.2.2.2.cmml"><mi id="S6.I3.i3.p1.6.m6.1.1" xref="S6.I3.i3.p1.6.m6.1.1.cmml">dom</mi><mo id="S6.I3.i3.p1.6.m6.2.2.1a" xref="S6.I3.i3.p1.6.m6.2.2.2.cmml">⁡</mo><mrow id="S6.I3.i3.p1.6.m6.2.2.1.1" xref="S6.I3.i3.p1.6.m6.2.2.2.cmml"><mo id="S6.I3.i3.p1.6.m6.2.2.1.1.2" stretchy="false" xref="S6.I3.i3.p1.6.m6.2.2.2.cmml">(</mo><msub id="S6.I3.i3.p1.6.m6.2.2.1.1.1" xref="S6.I3.i3.p1.6.m6.2.2.1.1.1.cmml"><mi id="S6.I3.i3.p1.6.m6.2.2.1.1.1.2" xref="S6.I3.i3.p1.6.m6.2.2.1.1.1.2.cmml">q</mi><mi id="S6.I3.i3.p1.6.m6.2.2.1.1.1.3" xref="S6.I3.i3.p1.6.m6.2.2.1.1.1.3.cmml">ξ</mi></msub><mo id="S6.I3.i3.p1.6.m6.2.2.1.1.3" stretchy="false" xref="S6.I3.i3.p1.6.m6.2.2.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.I3.i3.p1.6.m6.2b"><apply id="S6.I3.i3.p1.6.m6.2.2.2.cmml" xref="S6.I3.i3.p1.6.m6.2.2.1"><ci id="S6.I3.i3.p1.6.m6.1.1.cmml" xref="S6.I3.i3.p1.6.m6.1.1">dom</ci><apply id="S6.I3.i3.p1.6.m6.2.2.1.1.1.cmml" xref="S6.I3.i3.p1.6.m6.2.2.1.1.1"><csymbol cd="ambiguous" id="S6.I3.i3.p1.6.m6.2.2.1.1.1.1.cmml" xref="S6.I3.i3.p1.6.m6.2.2.1.1.1">subscript</csymbol><ci id="S6.I3.i3.p1.6.m6.2.2.1.1.1.2.cmml" xref="S6.I3.i3.p1.6.m6.2.2.1.1.1.2">𝑞</ci><ci id="S6.I3.i3.p1.6.m6.2.2.1.1.1.3.cmml" xref="S6.I3.i3.p1.6.m6.2.2.1.1.1.3">𝜉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I3.i3.p1.6.m6.2c">\operatorname{dom}(q_{\xi})</annotation><annotation encoding="application/x-llamapun" id="S6.I3.i3.p1.6.m6.2d">roman_dom ( italic_q start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT )</annotation></semantics></math>, and <math alttext="b" class="ltx_Math" display="inline" id="S6.I3.i3.p1.7.m7.1"><semantics id="S6.I3.i3.p1.7.m7.1a"><mi id="S6.I3.i3.p1.7.m7.1.1" xref="S6.I3.i3.p1.7.m7.1.1.cmml">b</mi><annotation-xml encoding="MathML-Content" id="S6.I3.i3.p1.7.m7.1b"><ci id="S6.I3.i3.p1.7.m7.1.1.cmml" xref="S6.I3.i3.p1.7.m7.1.1">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.I3.i3.p1.7.m7.1c">b</annotation><annotation encoding="application/x-llamapun" id="S6.I3.i3.p1.7.m7.1d">italic_b</annotation></semantics></math> the <math alttext="i" class="ltx_Math" display="inline" id="S6.I3.i3.p1.8.m8.1"><semantics id="S6.I3.i3.p1.8.m8.1a"><mi id="S6.I3.i3.p1.8.m8.1.1" xref="S6.I3.i3.p1.8.m8.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S6.I3.i3.p1.8.m8.1b"><ci id="S6.I3.i3.p1.8.m8.1.1.cmml" xref="S6.I3.i3.p1.8.m8.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.I3.i3.p1.8.m8.1c">i</annotation><annotation encoding="application/x-llamapun" id="S6.I3.i3.p1.8.m8.1d">italic_i</annotation></semantics></math>-the element in the increasing enumeration of <math alttext="\operatorname{dom}(q_{\eta})" class="ltx_Math" display="inline" id="S6.I3.i3.p1.9.m9.2"><semantics id="S6.I3.i3.p1.9.m9.2a"><mrow id="S6.I3.i3.p1.9.m9.2.2.1" xref="S6.I3.i3.p1.9.m9.2.2.2.cmml"><mi id="S6.I3.i3.p1.9.m9.1.1" xref="S6.I3.i3.p1.9.m9.1.1.cmml">dom</mi><mo id="S6.I3.i3.p1.9.m9.2.2.1a" xref="S6.I3.i3.p1.9.m9.2.2.2.cmml">⁡</mo><mrow id="S6.I3.i3.p1.9.m9.2.2.1.1" xref="S6.I3.i3.p1.9.m9.2.2.2.cmml"><mo id="S6.I3.i3.p1.9.m9.2.2.1.1.2" stretchy="false" xref="S6.I3.i3.p1.9.m9.2.2.2.cmml">(</mo><msub id="S6.I3.i3.p1.9.m9.2.2.1.1.1" xref="S6.I3.i3.p1.9.m9.2.2.1.1.1.cmml"><mi id="S6.I3.i3.p1.9.m9.2.2.1.1.1.2" xref="S6.I3.i3.p1.9.m9.2.2.1.1.1.2.cmml">q</mi><mi id="S6.I3.i3.p1.9.m9.2.2.1.1.1.3" xref="S6.I3.i3.p1.9.m9.2.2.1.1.1.3.cmml">η</mi></msub><mo id="S6.I3.i3.p1.9.m9.2.2.1.1.3" stretchy="false" xref="S6.I3.i3.p1.9.m9.2.2.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.I3.i3.p1.9.m9.2b"><apply id="S6.I3.i3.p1.9.m9.2.2.2.cmml" xref="S6.I3.i3.p1.9.m9.2.2.1"><ci id="S6.I3.i3.p1.9.m9.1.1.cmml" xref="S6.I3.i3.p1.9.m9.1.1">dom</ci><apply id="S6.I3.i3.p1.9.m9.2.2.1.1.1.cmml" xref="S6.I3.i3.p1.9.m9.2.2.1.1.1"><csymbol cd="ambiguous" id="S6.I3.i3.p1.9.m9.2.2.1.1.1.1.cmml" xref="S6.I3.i3.p1.9.m9.2.2.1.1.1">subscript</csymbol><ci id="S6.I3.i3.p1.9.m9.2.2.1.1.1.2.cmml" xref="S6.I3.i3.p1.9.m9.2.2.1.1.1.2">𝑞</ci><ci id="S6.I3.i3.p1.9.m9.2.2.1.1.1.3.cmml" xref="S6.I3.i3.p1.9.m9.2.2.1.1.1.3">𝜂</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I3.i3.p1.9.m9.2c">\operatorname{dom}(q_{\eta})</annotation><annotation encoding="application/x-llamapun" id="S6.I3.i3.p1.9.m9.2d">roman_dom ( italic_q start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT )</annotation></semantics></math>, then <math alttext="\gamma<\Delta_{A}(a,b)<\xi" class="ltx_Math" display="inline" id="S6.I3.i3.p1.10.m10.2"><semantics id="S6.I3.i3.p1.10.m10.2a"><mrow id="S6.I3.i3.p1.10.m10.2.3" xref="S6.I3.i3.p1.10.m10.2.3.cmml"><mi id="S6.I3.i3.p1.10.m10.2.3.2" xref="S6.I3.i3.p1.10.m10.2.3.2.cmml">γ</mi><mo id="S6.I3.i3.p1.10.m10.2.3.3" xref="S6.I3.i3.p1.10.m10.2.3.3.cmml"><</mo><mrow id="S6.I3.i3.p1.10.m10.2.3.4" xref="S6.I3.i3.p1.10.m10.2.3.4.cmml"><msub id="S6.I3.i3.p1.10.m10.2.3.4.2" xref="S6.I3.i3.p1.10.m10.2.3.4.2.cmml"><mi id="S6.I3.i3.p1.10.m10.2.3.4.2.2" mathvariant="normal" xref="S6.I3.i3.p1.10.m10.2.3.4.2.2.cmml">Δ</mi><mi id="S6.I3.i3.p1.10.m10.2.3.4.2.3" xref="S6.I3.i3.p1.10.m10.2.3.4.2.3.cmml">A</mi></msub><mo id="S6.I3.i3.p1.10.m10.2.3.4.1" xref="S6.I3.i3.p1.10.m10.2.3.4.1.cmml">⁢</mo><mrow id="S6.I3.i3.p1.10.m10.2.3.4.3.2" xref="S6.I3.i3.p1.10.m10.2.3.4.3.1.cmml"><mo id="S6.I3.i3.p1.10.m10.2.3.4.3.2.1" stretchy="false" xref="S6.I3.i3.p1.10.m10.2.3.4.3.1.cmml">(</mo><mi id="S6.I3.i3.p1.10.m10.1.1" xref="S6.I3.i3.p1.10.m10.1.1.cmml">a</mi><mo id="S6.I3.i3.p1.10.m10.2.3.4.3.2.2" xref="S6.I3.i3.p1.10.m10.2.3.4.3.1.cmml">,</mo><mi id="S6.I3.i3.p1.10.m10.2.2" xref="S6.I3.i3.p1.10.m10.2.2.cmml">b</mi><mo id="S6.I3.i3.p1.10.m10.2.3.4.3.2.3" stretchy="false" xref="S6.I3.i3.p1.10.m10.2.3.4.3.1.cmml">)</mo></mrow></mrow><mo id="S6.I3.i3.p1.10.m10.2.3.5" xref="S6.I3.i3.p1.10.m10.2.3.5.cmml"><</mo><mi id="S6.I3.i3.p1.10.m10.2.3.6" xref="S6.I3.i3.p1.10.m10.2.3.6.cmml">ξ</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.I3.i3.p1.10.m10.2b"><apply id="S6.I3.i3.p1.10.m10.2.3.cmml" xref="S6.I3.i3.p1.10.m10.2.3"><and id="S6.I3.i3.p1.10.m10.2.3a.cmml" xref="S6.I3.i3.p1.10.m10.2.3"></and><apply id="S6.I3.i3.p1.10.m10.2.3b.cmml" xref="S6.I3.i3.p1.10.m10.2.3"><lt id="S6.I3.i3.p1.10.m10.2.3.3.cmml" xref="S6.I3.i3.p1.10.m10.2.3.3"></lt><ci id="S6.I3.i3.p1.10.m10.2.3.2.cmml" xref="S6.I3.i3.p1.10.m10.2.3.2">𝛾</ci><apply id="S6.I3.i3.p1.10.m10.2.3.4.cmml" xref="S6.I3.i3.p1.10.m10.2.3.4"><times id="S6.I3.i3.p1.10.m10.2.3.4.1.cmml" xref="S6.I3.i3.p1.10.m10.2.3.4.1"></times><apply id="S6.I3.i3.p1.10.m10.2.3.4.2.cmml" xref="S6.I3.i3.p1.10.m10.2.3.4.2"><csymbol cd="ambiguous" id="S6.I3.i3.p1.10.m10.2.3.4.2.1.cmml" xref="S6.I3.i3.p1.10.m10.2.3.4.2">subscript</csymbol><ci id="S6.I3.i3.p1.10.m10.2.3.4.2.2.cmml" xref="S6.I3.i3.p1.10.m10.2.3.4.2.2">Δ</ci><ci id="S6.I3.i3.p1.10.m10.2.3.4.2.3.cmml" xref="S6.I3.i3.p1.10.m10.2.3.4.2.3">𝐴</ci></apply><interval closure="open" id="S6.I3.i3.p1.10.m10.2.3.4.3.1.cmml" xref="S6.I3.i3.p1.10.m10.2.3.4.3.2"><ci id="S6.I3.i3.p1.10.m10.1.1.cmml" xref="S6.I3.i3.p1.10.m10.1.1">𝑎</ci><ci id="S6.I3.i3.p1.10.m10.2.2.cmml" xref="S6.I3.i3.p1.10.m10.2.2">𝑏</ci></interval></apply></apply><apply id="S6.I3.i3.p1.10.m10.2.3c.cmml" xref="S6.I3.i3.p1.10.m10.2.3"><lt id="S6.I3.i3.p1.10.m10.2.3.5.cmml" xref="S6.I3.i3.p1.10.m10.2.3.5"></lt><share href="https://arxiv.org/html/2503.13728v1#S6.I3.i3.p1.10.m10.2.3.4.cmml" id="S6.I3.i3.p1.10.m10.2.3d.cmml" xref="S6.I3.i3.p1.10.m10.2.3"></share><ci id="S6.I3.i3.p1.10.m10.2.3.6.cmml" xref="S6.I3.i3.p1.10.m10.2.3.6">𝜉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I3.i3.p1.10.m10.2c">\gamma<\Delta_{A}(a,b)<\xi</annotation><annotation encoding="application/x-llamapun" id="S6.I3.i3.p1.10.m10.2d">italic_γ < roman_Δ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_a , italic_b ) < italic_ξ</annotation></semantics></math>.</p> </div> </li> </ol> </div> </div> <div class="ltx_proof" id="S6.SS1.2"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S6.SS1.1.p1"> <p class="ltx_p" id="S6.SS1.1.p1.3">One easily finds stationary <math alttext="\Gamma^{\prime}\subseteq\omega_{1}" class="ltx_Math" display="inline" id="S6.SS1.1.p1.1.m1.1"><semantics id="S6.SS1.1.p1.1.m1.1a"><mrow id="S6.SS1.1.p1.1.m1.1.1" xref="S6.SS1.1.p1.1.m1.1.1.cmml"><msup id="S6.SS1.1.p1.1.m1.1.1.2" xref="S6.SS1.1.p1.1.m1.1.1.2.cmml"><mi id="S6.SS1.1.p1.1.m1.1.1.2.2" mathvariant="normal" xref="S6.SS1.1.p1.1.m1.1.1.2.2.cmml">Γ</mi><mo id="S6.SS1.1.p1.1.m1.1.1.2.3" xref="S6.SS1.1.p1.1.m1.1.1.2.3.cmml">′</mo></msup><mo id="S6.SS1.1.p1.1.m1.1.1.1" xref="S6.SS1.1.p1.1.m1.1.1.1.cmml">⊆</mo><msub id="S6.SS1.1.p1.1.m1.1.1.3" xref="S6.SS1.1.p1.1.m1.1.1.3.cmml"><mi id="S6.SS1.1.p1.1.m1.1.1.3.2" xref="S6.SS1.1.p1.1.m1.1.1.3.2.cmml">ω</mi><mn id="S6.SS1.1.p1.1.m1.1.1.3.3" xref="S6.SS1.1.p1.1.m1.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.1.p1.1.m1.1b"><apply id="S6.SS1.1.p1.1.m1.1.1.cmml" xref="S6.SS1.1.p1.1.m1.1.1"><subset id="S6.SS1.1.p1.1.m1.1.1.1.cmml" xref="S6.SS1.1.p1.1.m1.1.1.1"></subset><apply id="S6.SS1.1.p1.1.m1.1.1.2.cmml" xref="S6.SS1.1.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S6.SS1.1.p1.1.m1.1.1.2.1.cmml" xref="S6.SS1.1.p1.1.m1.1.1.2">superscript</csymbol><ci id="S6.SS1.1.p1.1.m1.1.1.2.2.cmml" xref="S6.SS1.1.p1.1.m1.1.1.2.2">Γ</ci><ci id="S6.SS1.1.p1.1.m1.1.1.2.3.cmml" xref="S6.SS1.1.p1.1.m1.1.1.2.3">′</ci></apply><apply id="S6.SS1.1.p1.1.m1.1.1.3.cmml" xref="S6.SS1.1.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S6.SS1.1.p1.1.m1.1.1.3.1.cmml" xref="S6.SS1.1.p1.1.m1.1.1.3">subscript</csymbol><ci id="S6.SS1.1.p1.1.m1.1.1.3.2.cmml" xref="S6.SS1.1.p1.1.m1.1.1.3.2">𝜔</ci><cn id="S6.SS1.1.p1.1.m1.1.1.3.3.cmml" type="integer" xref="S6.SS1.1.p1.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.1.p1.1.m1.1c">\Gamma^{\prime}\subseteq\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.1.p1.1.m1.1d">roman_Γ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⊆ italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> such that (1) holds. We may assume that <math alttext="\Gamma^{\prime}" class="ltx_Math" display="inline" id="S6.SS1.1.p1.2.m2.1"><semantics id="S6.SS1.1.p1.2.m2.1a"><msup id="S6.SS1.1.p1.2.m2.1.1" xref="S6.SS1.1.p1.2.m2.1.1.cmml"><mi id="S6.SS1.1.p1.2.m2.1.1.2" mathvariant="normal" xref="S6.SS1.1.p1.2.m2.1.1.2.cmml">Γ</mi><mo id="S6.SS1.1.p1.2.m2.1.1.3" xref="S6.SS1.1.p1.2.m2.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S6.SS1.1.p1.2.m2.1b"><apply id="S6.SS1.1.p1.2.m2.1.1.cmml" xref="S6.SS1.1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S6.SS1.1.p1.2.m2.1.1.1.cmml" xref="S6.SS1.1.p1.2.m2.1.1">superscript</csymbol><ci id="S6.SS1.1.p1.2.m2.1.1.2.cmml" xref="S6.SS1.1.p1.2.m2.1.1.2">Γ</ci><ci id="S6.SS1.1.p1.2.m2.1.1.3.cmml" xref="S6.SS1.1.p1.2.m2.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.1.p1.2.m2.1c">\Gamma^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.1.p1.2.m2.1d">roman_Γ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> consists only of limit ordinals. Now define <math alttext="f:\Gamma^{\prime}\to\omega_{1}" class="ltx_Math" display="inline" id="S6.SS1.1.p1.3.m3.1"><semantics id="S6.SS1.1.p1.3.m3.1a"><mrow id="S6.SS1.1.p1.3.m3.1.1" xref="S6.SS1.1.p1.3.m3.1.1.cmml"><mi id="S6.SS1.1.p1.3.m3.1.1.2" xref="S6.SS1.1.p1.3.m3.1.1.2.cmml">f</mi><mo id="S6.SS1.1.p1.3.m3.1.1.1" lspace="0.278em" rspace="0.278em" xref="S6.SS1.1.p1.3.m3.1.1.1.cmml">:</mo><mrow id="S6.SS1.1.p1.3.m3.1.1.3" xref="S6.SS1.1.p1.3.m3.1.1.3.cmml"><msup id="S6.SS1.1.p1.3.m3.1.1.3.2" xref="S6.SS1.1.p1.3.m3.1.1.3.2.cmml"><mi id="S6.SS1.1.p1.3.m3.1.1.3.2.2" mathvariant="normal" xref="S6.SS1.1.p1.3.m3.1.1.3.2.2.cmml">Γ</mi><mo id="S6.SS1.1.p1.3.m3.1.1.3.2.3" xref="S6.SS1.1.p1.3.m3.1.1.3.2.3.cmml">′</mo></msup><mo id="S6.SS1.1.p1.3.m3.1.1.3.1" stretchy="false" xref="S6.SS1.1.p1.3.m3.1.1.3.1.cmml">→</mo><msub id="S6.SS1.1.p1.3.m3.1.1.3.3" xref="S6.SS1.1.p1.3.m3.1.1.3.3.cmml"><mi id="S6.SS1.1.p1.3.m3.1.1.3.3.2" xref="S6.SS1.1.p1.3.m3.1.1.3.3.2.cmml">ω</mi><mn id="S6.SS1.1.p1.3.m3.1.1.3.3.3" xref="S6.SS1.1.p1.3.m3.1.1.3.3.3.cmml">1</mn></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.1.p1.3.m3.1b"><apply id="S6.SS1.1.p1.3.m3.1.1.cmml" xref="S6.SS1.1.p1.3.m3.1.1"><ci id="S6.SS1.1.p1.3.m3.1.1.1.cmml" xref="S6.SS1.1.p1.3.m3.1.1.1">:</ci><ci id="S6.SS1.1.p1.3.m3.1.1.2.cmml" xref="S6.SS1.1.p1.3.m3.1.1.2">𝑓</ci><apply id="S6.SS1.1.p1.3.m3.1.1.3.cmml" xref="S6.SS1.1.p1.3.m3.1.1.3"><ci id="S6.SS1.1.p1.3.m3.1.1.3.1.cmml" xref="S6.SS1.1.p1.3.m3.1.1.3.1">→</ci><apply id="S6.SS1.1.p1.3.m3.1.1.3.2.cmml" xref="S6.SS1.1.p1.3.m3.1.1.3.2"><csymbol cd="ambiguous" id="S6.SS1.1.p1.3.m3.1.1.3.2.1.cmml" xref="S6.SS1.1.p1.3.m3.1.1.3.2">superscript</csymbol><ci id="S6.SS1.1.p1.3.m3.1.1.3.2.2.cmml" xref="S6.SS1.1.p1.3.m3.1.1.3.2.2">Γ</ci><ci id="S6.SS1.1.p1.3.m3.1.1.3.2.3.cmml" xref="S6.SS1.1.p1.3.m3.1.1.3.2.3">′</ci></apply><apply id="S6.SS1.1.p1.3.m3.1.1.3.3.cmml" xref="S6.SS1.1.p1.3.m3.1.1.3.3"><csymbol cd="ambiguous" id="S6.SS1.1.p1.3.m3.1.1.3.3.1.cmml" xref="S6.SS1.1.p1.3.m3.1.1.3.3">subscript</csymbol><ci id="S6.SS1.1.p1.3.m3.1.1.3.3.2.cmml" xref="S6.SS1.1.p1.3.m3.1.1.3.3.2">𝜔</ci><cn id="S6.SS1.1.p1.3.m3.1.1.3.3.3.cmml" type="integer" xref="S6.SS1.1.p1.3.m3.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.1.p1.3.m3.1c">f:\Gamma^{\prime}\to\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.1.p1.3.m3.1d">italic_f : roman_Γ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT → italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> by</p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex12"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="f(\xi):=\sup(\{0\}\cup\{\Delta_{A}(a,b)+1:a\neq b\in\operatorname{dom}(q_{\xi}% ),\Delta_{A}(a,b)<\xi\})." class="ltx_Math" display="block" id="S6.Ex12.m1.8"><semantics id="S6.Ex12.m1.8a"><mrow id="S6.Ex12.m1.8.8.1" xref="S6.Ex12.m1.8.8.1.1.cmml"><mrow id="S6.Ex12.m1.8.8.1.1" xref="S6.Ex12.m1.8.8.1.1.cmml"><mrow id="S6.Ex12.m1.8.8.1.1.3" xref="S6.Ex12.m1.8.8.1.1.3.cmml"><mi id="S6.Ex12.m1.8.8.1.1.3.2" xref="S6.Ex12.m1.8.8.1.1.3.2.cmml">f</mi><mo id="S6.Ex12.m1.8.8.1.1.3.1" xref="S6.Ex12.m1.8.8.1.1.3.1.cmml">⁢</mo><mrow id="S6.Ex12.m1.8.8.1.1.3.3.2" xref="S6.Ex12.m1.8.8.1.1.3.cmml"><mo id="S6.Ex12.m1.8.8.1.1.3.3.2.1" stretchy="false" xref="S6.Ex12.m1.8.8.1.1.3.cmml">(</mo><mi id="S6.Ex12.m1.1.1" xref="S6.Ex12.m1.1.1.cmml">ξ</mi><mo id="S6.Ex12.m1.8.8.1.1.3.3.2.2" rspace="0.278em" stretchy="false" xref="S6.Ex12.m1.8.8.1.1.3.cmml">)</mo></mrow></mrow><mo id="S6.Ex12.m1.8.8.1.1.2" xref="S6.Ex12.m1.8.8.1.1.2.cmml">:=</mo><mrow id="S6.Ex12.m1.8.8.1.1.1" xref="S6.Ex12.m1.8.8.1.1.1.cmml"><mo id="S6.Ex12.m1.8.8.1.1.1.2" lspace="0.111em" movablelimits="false" rspace="0em" xref="S6.Ex12.m1.8.8.1.1.1.2.cmml">sup</mo><mrow id="S6.Ex12.m1.8.8.1.1.1.1.1" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.cmml"><mo id="S6.Ex12.m1.8.8.1.1.1.1.1.2" stretchy="false" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.cmml">(</mo><mrow id="S6.Ex12.m1.8.8.1.1.1.1.1.1" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.cmml"><mrow id="S6.Ex12.m1.8.8.1.1.1.1.1.1.4.2" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.4.1.cmml"><mo id="S6.Ex12.m1.8.8.1.1.1.1.1.1.4.2.1" stretchy="false" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.4.1.cmml">{</mo><mn id="S6.Ex12.m1.2.2" xref="S6.Ex12.m1.2.2.cmml">0</mn><mo id="S6.Ex12.m1.8.8.1.1.1.1.1.1.4.2.2" stretchy="false" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.4.1.cmml">}</mo></mrow><mo id="S6.Ex12.m1.8.8.1.1.1.1.1.1.3" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.3.cmml">∪</mo><mrow id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.3.cmml"><mo id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.3" stretchy="false" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.3.1.cmml">{</mo><mrow id="S6.Ex12.m1.8.8.1.1.1.1.1.1.1.1.1" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.1.1.1.cmml"><mrow id="S6.Ex12.m1.8.8.1.1.1.1.1.1.1.1.1.2" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.1.1.1.2.cmml"><msub id="S6.Ex12.m1.8.8.1.1.1.1.1.1.1.1.1.2.2" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.1.1.1.2.2.cmml"><mi id="S6.Ex12.m1.8.8.1.1.1.1.1.1.1.1.1.2.2.2" mathvariant="normal" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.1.1.1.2.2.2.cmml">Δ</mi><mi id="S6.Ex12.m1.8.8.1.1.1.1.1.1.1.1.1.2.2.3" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.1.1.1.2.2.3.cmml">A</mi></msub><mo id="S6.Ex12.m1.8.8.1.1.1.1.1.1.1.1.1.2.1" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.1.1.1.2.1.cmml">⁢</mo><mrow id="S6.Ex12.m1.8.8.1.1.1.1.1.1.1.1.1.2.3.2" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.1.1.1.2.3.1.cmml"><mo id="S6.Ex12.m1.8.8.1.1.1.1.1.1.1.1.1.2.3.2.1" stretchy="false" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.1.1.1.2.3.1.cmml">(</mo><mi id="S6.Ex12.m1.3.3" xref="S6.Ex12.m1.3.3.cmml">a</mi><mo id="S6.Ex12.m1.8.8.1.1.1.1.1.1.1.1.1.2.3.2.2" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.1.1.1.2.3.1.cmml">,</mo><mi id="S6.Ex12.m1.4.4" xref="S6.Ex12.m1.4.4.cmml">b</mi><mo id="S6.Ex12.m1.8.8.1.1.1.1.1.1.1.1.1.2.3.2.3" stretchy="false" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.1.1.1.2.3.1.cmml">)</mo></mrow></mrow><mo id="S6.Ex12.m1.8.8.1.1.1.1.1.1.1.1.1.1" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.1.1.1.1.cmml">+</mo><mn id="S6.Ex12.m1.8.8.1.1.1.1.1.1.1.1.1.3" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.4" lspace="0.278em" rspace="0.278em" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.3.1.cmml">:</mo><mrow id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.2" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.3.cmml"><mrow id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.1.1" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.1.1.cmml"><mi id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.1.1.3" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.1.1.3.cmml">a</mi><mo id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.1.1.4" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.1.1.4.cmml">≠</mo><mi id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.1.1.5" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.1.1.5.cmml">b</mi><mo id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.1.1.6" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.1.1.6.cmml">∈</mo><mrow id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.1.1.1.1" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.1.1.1.2.cmml"><mi id="S6.Ex12.m1.5.5" xref="S6.Ex12.m1.5.5.cmml">dom</mi><mo id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.1.1.1.1a" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.1.1.1.2.cmml">⁡</mo><mrow id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.1.1.1.1.1" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.1.1.1.2.cmml"><mo id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.1.1.1.1.1.2" stretchy="false" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.1.1.1.2.cmml">(</mo><msub id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.1.1.1.1.1.1" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.1.1.1.1.1.1.cmml"><mi id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.1.1.1.1.1.1.2" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.1.1.1.1.1.1.2.cmml">q</mi><mi id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.1.1.1.1.1.1.3" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.1.1.1.1.1.1.3.cmml">ξ</mi></msub><mo id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.1.1.1.1.1.3" stretchy="false" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.1.1.1.2.cmml">)</mo></mrow></mrow></mrow><mo id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.2.3" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.3a.cmml">,</mo><mrow id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.2.2" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.2.2.cmml"><mrow id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.2.2.2" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.2.2.2.cmml"><msub id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.2.2.2.2" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.2.2.2.2.cmml"><mi id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.2.2.2.2.2" mathvariant="normal" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.2.2.2.2.2.cmml">Δ</mi><mi id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.2.2.2.2.3" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.2.2.2.2.3.cmml">A</mi></msub><mo id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.2.2.2.1" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.2.2.2.1.cmml">⁢</mo><mrow id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.2.2.2.3.2" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.2.2.2.3.1.cmml"><mo id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.2.2.2.3.2.1" stretchy="false" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.2.2.2.3.1.cmml">(</mo><mi id="S6.Ex12.m1.6.6" xref="S6.Ex12.m1.6.6.cmml">a</mi><mo id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.2.2.2.3.2.2" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.2.2.2.3.1.cmml">,</mo><mi id="S6.Ex12.m1.7.7" xref="S6.Ex12.m1.7.7.cmml">b</mi><mo id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.2.2.2.3.2.3" stretchy="false" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.2.2.2.3.1.cmml">)</mo></mrow></mrow><mo id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.2.2.1" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.2.2.1.cmml"><</mo><mi id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.2.2.3" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.2.2.3.cmml">ξ</mi></mrow></mrow><mo id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.5" stretchy="false" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.3.1.cmml">}</mo></mrow></mrow><mo id="S6.Ex12.m1.8.8.1.1.1.1.1.3" stretchy="false" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S6.Ex12.m1.8.8.1.2" lspace="0em" xref="S6.Ex12.m1.8.8.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.Ex12.m1.8b"><apply id="S6.Ex12.m1.8.8.1.1.cmml" xref="S6.Ex12.m1.8.8.1"><csymbol cd="latexml" id="S6.Ex12.m1.8.8.1.1.2.cmml" xref="S6.Ex12.m1.8.8.1.1.2">assign</csymbol><apply id="S6.Ex12.m1.8.8.1.1.3.cmml" xref="S6.Ex12.m1.8.8.1.1.3"><times id="S6.Ex12.m1.8.8.1.1.3.1.cmml" xref="S6.Ex12.m1.8.8.1.1.3.1"></times><ci id="S6.Ex12.m1.8.8.1.1.3.2.cmml" xref="S6.Ex12.m1.8.8.1.1.3.2">𝑓</ci><ci id="S6.Ex12.m1.1.1.cmml" xref="S6.Ex12.m1.1.1">𝜉</ci></apply><apply id="S6.Ex12.m1.8.8.1.1.1.cmml" xref="S6.Ex12.m1.8.8.1.1.1"><csymbol cd="latexml" id="S6.Ex12.m1.8.8.1.1.1.2.cmml" xref="S6.Ex12.m1.8.8.1.1.1.2">supremum</csymbol><apply id="S6.Ex12.m1.8.8.1.1.1.1.1.1.cmml" xref="S6.Ex12.m1.8.8.1.1.1.1.1"><union id="S6.Ex12.m1.8.8.1.1.1.1.1.1.3.cmml" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.3"></union><set id="S6.Ex12.m1.8.8.1.1.1.1.1.1.4.1.cmml" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.4.2"><cn id="S6.Ex12.m1.2.2.cmml" type="integer" xref="S6.Ex12.m1.2.2">0</cn></set><apply id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.3.cmml" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2"><csymbol cd="latexml" id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.3.1.cmml" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.3">conditional-set</csymbol><apply id="S6.Ex12.m1.8.8.1.1.1.1.1.1.1.1.1.cmml" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.1.1.1"><plus id="S6.Ex12.m1.8.8.1.1.1.1.1.1.1.1.1.1.cmml" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.1.1.1.1"></plus><apply id="S6.Ex12.m1.8.8.1.1.1.1.1.1.1.1.1.2.cmml" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.1.1.1.2"><times id="S6.Ex12.m1.8.8.1.1.1.1.1.1.1.1.1.2.1.cmml" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.1.1.1.2.1"></times><apply id="S6.Ex12.m1.8.8.1.1.1.1.1.1.1.1.1.2.2.cmml" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.1.1.1.2.2"><csymbol cd="ambiguous" id="S6.Ex12.m1.8.8.1.1.1.1.1.1.1.1.1.2.2.1.cmml" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.1.1.1.2.2">subscript</csymbol><ci id="S6.Ex12.m1.8.8.1.1.1.1.1.1.1.1.1.2.2.2.cmml" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.1.1.1.2.2.2">Δ</ci><ci id="S6.Ex12.m1.8.8.1.1.1.1.1.1.1.1.1.2.2.3.cmml" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.1.1.1.2.2.3">𝐴</ci></apply><interval closure="open" id="S6.Ex12.m1.8.8.1.1.1.1.1.1.1.1.1.2.3.1.cmml" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.1.1.1.2.3.2"><ci id="S6.Ex12.m1.3.3.cmml" xref="S6.Ex12.m1.3.3">𝑎</ci><ci id="S6.Ex12.m1.4.4.cmml" xref="S6.Ex12.m1.4.4">𝑏</ci></interval></apply><cn id="S6.Ex12.m1.8.8.1.1.1.1.1.1.1.1.1.3.cmml" type="integer" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.1.1.1.3">1</cn></apply><apply id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.3.cmml" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.2"><csymbol cd="ambiguous" id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.3a.cmml" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.2.3">formulae-sequence</csymbol><apply id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.1.1.cmml" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.1.1"><and id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.1.1a.cmml" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.1.1"></and><apply id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.1.1b.cmml" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.1.1"><neq id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.1.1.4.cmml" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.1.1.4"></neq><ci id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.1.1.3.cmml" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.1.1.3">𝑎</ci><ci id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.1.1.5.cmml" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.1.1.5">𝑏</ci></apply><apply id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.1.1c.cmml" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.1.1"><in id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.1.1.6.cmml" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.1.1.6"></in><share href="https://arxiv.org/html/2503.13728v1#S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.1.1.5.cmml" id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.1.1d.cmml" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.1.1"></share><apply id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.1.1.1.2.cmml" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.1.1.1.1"><ci id="S6.Ex12.m1.5.5.cmml" xref="S6.Ex12.m1.5.5">dom</ci><apply id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.1.1.1.1.1.1.cmml" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.1.1.1.1.1.1.1.cmml" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.1.1.1.1.1.1">subscript</csymbol><ci id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.1.1.1.1.1.1.2.cmml" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.1.1.1.1.1.1.2">𝑞</ci><ci id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.1.1.1.1.1.1.3.cmml" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.1.1.1.1.1.1.3">𝜉</ci></apply></apply></apply></apply><apply id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.2.2.cmml" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.2.2"><lt id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.2.2.1.cmml" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.2.2.1"></lt><apply id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.2.2.2.cmml" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.2.2.2"><times id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.2.2.2.1.cmml" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.2.2.2.1"></times><apply id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.2.2.2.2.cmml" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.2.2.2.2.1.cmml" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.2.2.2.2">subscript</csymbol><ci id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.2.2.2.2.2.cmml" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.2.2.2.2.2">Δ</ci><ci id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.2.2.2.2.3.cmml" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.2.2.2.2.3">𝐴</ci></apply><interval closure="open" id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.2.2.2.3.1.cmml" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.2.2.2.3.2"><ci id="S6.Ex12.m1.6.6.cmml" xref="S6.Ex12.m1.6.6">𝑎</ci><ci id="S6.Ex12.m1.7.7.cmml" xref="S6.Ex12.m1.7.7">𝑏</ci></interval></apply><ci id="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.2.2.3.cmml" xref="S6.Ex12.m1.8.8.1.1.1.1.1.1.2.2.2.2.2.3">𝜉</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex12.m1.8c">f(\xi):=\sup(\{0\}\cup\{\Delta_{A}(a,b)+1:a\neq b\in\operatorname{dom}(q_{\xi}% ),\Delta_{A}(a,b)<\xi\}).</annotation><annotation encoding="application/x-llamapun" id="S6.Ex12.m1.8d">italic_f ( italic_ξ ) := roman_sup ( { 0 } ∪ { roman_Δ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_a , italic_b ) + 1 : italic_a ≠ italic_b ∈ roman_dom ( italic_q start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT ) , roman_Δ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_a , italic_b ) < italic_ξ } ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.SS1.1.p1.8">By Fodor’s lemma, there is stationary <math alttext="\Gamma^{\prime\prime}\subseteq\Gamma^{\prime}" class="ltx_Math" display="inline" id="S6.SS1.1.p1.4.m1.1"><semantics id="S6.SS1.1.p1.4.m1.1a"><mrow id="S6.SS1.1.p1.4.m1.1.1" xref="S6.SS1.1.p1.4.m1.1.1.cmml"><msup id="S6.SS1.1.p1.4.m1.1.1.2" xref="S6.SS1.1.p1.4.m1.1.1.2.cmml"><mi id="S6.SS1.1.p1.4.m1.1.1.2.2" mathvariant="normal" xref="S6.SS1.1.p1.4.m1.1.1.2.2.cmml">Γ</mi><mo id="S6.SS1.1.p1.4.m1.1.1.2.3" xref="S6.SS1.1.p1.4.m1.1.1.2.3.cmml">′′</mo></msup><mo id="S6.SS1.1.p1.4.m1.1.1.1" xref="S6.SS1.1.p1.4.m1.1.1.1.cmml">⊆</mo><msup id="S6.SS1.1.p1.4.m1.1.1.3" xref="S6.SS1.1.p1.4.m1.1.1.3.cmml"><mi id="S6.SS1.1.p1.4.m1.1.1.3.2" mathvariant="normal" xref="S6.SS1.1.p1.4.m1.1.1.3.2.cmml">Γ</mi><mo id="S6.SS1.1.p1.4.m1.1.1.3.3" xref="S6.SS1.1.p1.4.m1.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.1.p1.4.m1.1b"><apply id="S6.SS1.1.p1.4.m1.1.1.cmml" xref="S6.SS1.1.p1.4.m1.1.1"><subset id="S6.SS1.1.p1.4.m1.1.1.1.cmml" xref="S6.SS1.1.p1.4.m1.1.1.1"></subset><apply id="S6.SS1.1.p1.4.m1.1.1.2.cmml" xref="S6.SS1.1.p1.4.m1.1.1.2"><csymbol cd="ambiguous" id="S6.SS1.1.p1.4.m1.1.1.2.1.cmml" xref="S6.SS1.1.p1.4.m1.1.1.2">superscript</csymbol><ci id="S6.SS1.1.p1.4.m1.1.1.2.2.cmml" xref="S6.SS1.1.p1.4.m1.1.1.2.2">Γ</ci><ci id="S6.SS1.1.p1.4.m1.1.1.2.3.cmml" xref="S6.SS1.1.p1.4.m1.1.1.2.3">′′</ci></apply><apply id="S6.SS1.1.p1.4.m1.1.1.3.cmml" xref="S6.SS1.1.p1.4.m1.1.1.3"><csymbol cd="ambiguous" id="S6.SS1.1.p1.4.m1.1.1.3.1.cmml" xref="S6.SS1.1.p1.4.m1.1.1.3">superscript</csymbol><ci id="S6.SS1.1.p1.4.m1.1.1.3.2.cmml" xref="S6.SS1.1.p1.4.m1.1.1.3.2">Γ</ci><ci id="S6.SS1.1.p1.4.m1.1.1.3.3.cmml" xref="S6.SS1.1.p1.4.m1.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.1.p1.4.m1.1c">\Gamma^{\prime\prime}\subseteq\Gamma^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.1.p1.4.m1.1d">roman_Γ start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ⊆ roman_Γ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> such that <math alttext="f" class="ltx_Math" display="inline" id="S6.SS1.1.p1.5.m2.1"><semantics id="S6.SS1.1.p1.5.m2.1a"><mi id="S6.SS1.1.p1.5.m2.1.1" xref="S6.SS1.1.p1.5.m2.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S6.SS1.1.p1.5.m2.1b"><ci id="S6.SS1.1.p1.5.m2.1.1.cmml" xref="S6.SS1.1.p1.5.m2.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.1.p1.5.m2.1c">f</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.1.p1.5.m2.1d">italic_f</annotation></semantics></math> is constant equal to <math alttext="\gamma" class="ltx_Math" display="inline" id="S6.SS1.1.p1.6.m3.1"><semantics id="S6.SS1.1.p1.6.m3.1a"><mi id="S6.SS1.1.p1.6.m3.1.1" xref="S6.SS1.1.p1.6.m3.1.1.cmml">γ</mi><annotation-xml encoding="MathML-Content" id="S6.SS1.1.p1.6.m3.1b"><ci id="S6.SS1.1.p1.6.m3.1.1.cmml" xref="S6.SS1.1.p1.6.m3.1.1">𝛾</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.1.p1.6.m3.1c">\gamma</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.1.p1.6.m3.1d">italic_γ</annotation></semantics></math> on <math alttext="\Gamma^{\prime\prime}" class="ltx_Math" display="inline" id="S6.SS1.1.p1.7.m4.1"><semantics id="S6.SS1.1.p1.7.m4.1a"><msup id="S6.SS1.1.p1.7.m4.1.1" xref="S6.SS1.1.p1.7.m4.1.1.cmml"><mi id="S6.SS1.1.p1.7.m4.1.1.2" mathvariant="normal" xref="S6.SS1.1.p1.7.m4.1.1.2.cmml">Γ</mi><mo id="S6.SS1.1.p1.7.m4.1.1.3" xref="S6.SS1.1.p1.7.m4.1.1.3.cmml">′′</mo></msup><annotation-xml encoding="MathML-Content" id="S6.SS1.1.p1.7.m4.1b"><apply id="S6.SS1.1.p1.7.m4.1.1.cmml" xref="S6.SS1.1.p1.7.m4.1.1"><csymbol cd="ambiguous" id="S6.SS1.1.p1.7.m4.1.1.1.cmml" xref="S6.SS1.1.p1.7.m4.1.1">superscript</csymbol><ci id="S6.SS1.1.p1.7.m4.1.1.2.cmml" xref="S6.SS1.1.p1.7.m4.1.1.2">Γ</ci><ci id="S6.SS1.1.p1.7.m4.1.1.3.cmml" xref="S6.SS1.1.p1.7.m4.1.1.3">′′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.1.p1.7.m4.1c">\Gamma^{\prime\prime}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.1.p1.7.m4.1d">roman_Γ start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT</annotation></semantics></math>. Then <math alttext="\Gamma^{\prime\prime\prime}:=\Gamma^{\prime\prime}\setminus(\gamma+1)" class="ltx_Math" display="inline" id="S6.SS1.1.p1.8.m5.1"><semantics id="S6.SS1.1.p1.8.m5.1a"><mrow id="S6.SS1.1.p1.8.m5.1.1" xref="S6.SS1.1.p1.8.m5.1.1.cmml"><msup id="S6.SS1.1.p1.8.m5.1.1.3" xref="S6.SS1.1.p1.8.m5.1.1.3.cmml"><mi id="S6.SS1.1.p1.8.m5.1.1.3.2" mathvariant="normal" xref="S6.SS1.1.p1.8.m5.1.1.3.2.cmml">Γ</mi><mo id="S6.SS1.1.p1.8.m5.1.1.3.3" xref="S6.SS1.1.p1.8.m5.1.1.3.3.cmml">′′′</mo></msup><mo id="S6.SS1.1.p1.8.m5.1.1.2" lspace="0.278em" rspace="0.278em" xref="S6.SS1.1.p1.8.m5.1.1.2.cmml">:=</mo><mrow id="S6.SS1.1.p1.8.m5.1.1.1" xref="S6.SS1.1.p1.8.m5.1.1.1.cmml"><msup id="S6.SS1.1.p1.8.m5.1.1.1.3" xref="S6.SS1.1.p1.8.m5.1.1.1.3.cmml"><mi id="S6.SS1.1.p1.8.m5.1.1.1.3.2" mathvariant="normal" xref="S6.SS1.1.p1.8.m5.1.1.1.3.2.cmml">Γ</mi><mo id="S6.SS1.1.p1.8.m5.1.1.1.3.3" xref="S6.SS1.1.p1.8.m5.1.1.1.3.3.cmml">′′</mo></msup><mo id="S6.SS1.1.p1.8.m5.1.1.1.2" xref="S6.SS1.1.p1.8.m5.1.1.1.2.cmml">∖</mo><mrow id="S6.SS1.1.p1.8.m5.1.1.1.1.1" xref="S6.SS1.1.p1.8.m5.1.1.1.1.1.1.cmml"><mo id="S6.SS1.1.p1.8.m5.1.1.1.1.1.2" stretchy="false" xref="S6.SS1.1.p1.8.m5.1.1.1.1.1.1.cmml">(</mo><mrow id="S6.SS1.1.p1.8.m5.1.1.1.1.1.1" xref="S6.SS1.1.p1.8.m5.1.1.1.1.1.1.cmml"><mi id="S6.SS1.1.p1.8.m5.1.1.1.1.1.1.2" xref="S6.SS1.1.p1.8.m5.1.1.1.1.1.1.2.cmml">γ</mi><mo id="S6.SS1.1.p1.8.m5.1.1.1.1.1.1.1" xref="S6.SS1.1.p1.8.m5.1.1.1.1.1.1.1.cmml">+</mo><mn id="S6.SS1.1.p1.8.m5.1.1.1.1.1.1.3" xref="S6.SS1.1.p1.8.m5.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S6.SS1.1.p1.8.m5.1.1.1.1.1.3" stretchy="false" xref="S6.SS1.1.p1.8.m5.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.1.p1.8.m5.1b"><apply id="S6.SS1.1.p1.8.m5.1.1.cmml" xref="S6.SS1.1.p1.8.m5.1.1"><csymbol cd="latexml" id="S6.SS1.1.p1.8.m5.1.1.2.cmml" xref="S6.SS1.1.p1.8.m5.1.1.2">assign</csymbol><apply id="S6.SS1.1.p1.8.m5.1.1.3.cmml" xref="S6.SS1.1.p1.8.m5.1.1.3"><csymbol cd="ambiguous" id="S6.SS1.1.p1.8.m5.1.1.3.1.cmml" xref="S6.SS1.1.p1.8.m5.1.1.3">superscript</csymbol><ci id="S6.SS1.1.p1.8.m5.1.1.3.2.cmml" xref="S6.SS1.1.p1.8.m5.1.1.3.2">Γ</ci><ci id="S6.SS1.1.p1.8.m5.1.1.3.3.cmml" xref="S6.SS1.1.p1.8.m5.1.1.3.3">′′′</ci></apply><apply id="S6.SS1.1.p1.8.m5.1.1.1.cmml" xref="S6.SS1.1.p1.8.m5.1.1.1"><setdiff id="S6.SS1.1.p1.8.m5.1.1.1.2.cmml" xref="S6.SS1.1.p1.8.m5.1.1.1.2"></setdiff><apply id="S6.SS1.1.p1.8.m5.1.1.1.3.cmml" xref="S6.SS1.1.p1.8.m5.1.1.1.3"><csymbol cd="ambiguous" id="S6.SS1.1.p1.8.m5.1.1.1.3.1.cmml" xref="S6.SS1.1.p1.8.m5.1.1.1.3">superscript</csymbol><ci id="S6.SS1.1.p1.8.m5.1.1.1.3.2.cmml" xref="S6.SS1.1.p1.8.m5.1.1.1.3.2">Γ</ci><ci id="S6.SS1.1.p1.8.m5.1.1.1.3.3.cmml" xref="S6.SS1.1.p1.8.m5.1.1.1.3.3">′′</ci></apply><apply id="S6.SS1.1.p1.8.m5.1.1.1.1.1.1.cmml" xref="S6.SS1.1.p1.8.m5.1.1.1.1.1"><plus id="S6.SS1.1.p1.8.m5.1.1.1.1.1.1.1.cmml" xref="S6.SS1.1.p1.8.m5.1.1.1.1.1.1.1"></plus><ci id="S6.SS1.1.p1.8.m5.1.1.1.1.1.1.2.cmml" xref="S6.SS1.1.p1.8.m5.1.1.1.1.1.1.2">𝛾</ci><cn id="S6.SS1.1.p1.8.m5.1.1.1.1.1.1.3.cmml" type="integer" xref="S6.SS1.1.p1.8.m5.1.1.1.1.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.1.p1.8.m5.1c">\Gamma^{\prime\prime\prime}:=\Gamma^{\prime\prime}\setminus(\gamma+1)</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.1.p1.8.m5.1d">roman_Γ start_POSTSUPERSCRIPT ′ ′ ′ end_POSTSUPERSCRIPT := roman_Γ start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ∖ ( italic_γ + 1 )</annotation></semantics></math> satisfies (2).</p> </div> <div class="ltx_para" id="S6.SS1.2.p2"> <p class="ltx_p" id="S6.SS1.2.p2.24">We now turn to satisfy (3). For <math alttext="\xi\in\Gamma^{\prime\prime\prime}" class="ltx_Math" display="inline" id="S6.SS1.2.p2.1.m1.1"><semantics id="S6.SS1.2.p2.1.m1.1a"><mrow id="S6.SS1.2.p2.1.m1.1.1" xref="S6.SS1.2.p2.1.m1.1.1.cmml"><mi id="S6.SS1.2.p2.1.m1.1.1.2" xref="S6.SS1.2.p2.1.m1.1.1.2.cmml">ξ</mi><mo id="S6.SS1.2.p2.1.m1.1.1.1" xref="S6.SS1.2.p2.1.m1.1.1.1.cmml">∈</mo><msup id="S6.SS1.2.p2.1.m1.1.1.3" xref="S6.SS1.2.p2.1.m1.1.1.3.cmml"><mi id="S6.SS1.2.p2.1.m1.1.1.3.2" mathvariant="normal" xref="S6.SS1.2.p2.1.m1.1.1.3.2.cmml">Γ</mi><mo id="S6.SS1.2.p2.1.m1.1.1.3.3" xref="S6.SS1.2.p2.1.m1.1.1.3.3.cmml">′′′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.2.p2.1.m1.1b"><apply id="S6.SS1.2.p2.1.m1.1.1.cmml" xref="S6.SS1.2.p2.1.m1.1.1"><in id="S6.SS1.2.p2.1.m1.1.1.1.cmml" xref="S6.SS1.2.p2.1.m1.1.1.1"></in><ci id="S6.SS1.2.p2.1.m1.1.1.2.cmml" xref="S6.SS1.2.p2.1.m1.1.1.2">𝜉</ci><apply id="S6.SS1.2.p2.1.m1.1.1.3.cmml" xref="S6.SS1.2.p2.1.m1.1.1.3"><csymbol cd="ambiguous" id="S6.SS1.2.p2.1.m1.1.1.3.1.cmml" xref="S6.SS1.2.p2.1.m1.1.1.3">superscript</csymbol><ci id="S6.SS1.2.p2.1.m1.1.1.3.2.cmml" xref="S6.SS1.2.p2.1.m1.1.1.3.2">Γ</ci><ci id="S6.SS1.2.p2.1.m1.1.1.3.3.cmml" xref="S6.SS1.2.p2.1.m1.1.1.3.3">′′′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.2.p2.1.m1.1c">\xi\in\Gamma^{\prime\prime\prime}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.2.p2.1.m1.1d">italic_ξ ∈ roman_Γ start_POSTSUPERSCRIPT ′ ′ ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, let <math alttext="\xi_{0},\dots,\xi_{n-1}" class="ltx_Math" display="inline" id="S6.SS1.2.p2.2.m2.3"><semantics id="S6.SS1.2.p2.2.m2.3a"><mrow id="S6.SS1.2.p2.2.m2.3.3.2" xref="S6.SS1.2.p2.2.m2.3.3.3.cmml"><msub id="S6.SS1.2.p2.2.m2.2.2.1.1" xref="S6.SS1.2.p2.2.m2.2.2.1.1.cmml"><mi id="S6.SS1.2.p2.2.m2.2.2.1.1.2" xref="S6.SS1.2.p2.2.m2.2.2.1.1.2.cmml">ξ</mi><mn id="S6.SS1.2.p2.2.m2.2.2.1.1.3" xref="S6.SS1.2.p2.2.m2.2.2.1.1.3.cmml">0</mn></msub><mo id="S6.SS1.2.p2.2.m2.3.3.2.3" xref="S6.SS1.2.p2.2.m2.3.3.3.cmml">,</mo><mi id="S6.SS1.2.p2.2.m2.1.1" mathvariant="normal" xref="S6.SS1.2.p2.2.m2.1.1.cmml">…</mi><mo id="S6.SS1.2.p2.2.m2.3.3.2.4" xref="S6.SS1.2.p2.2.m2.3.3.3.cmml">,</mo><msub id="S6.SS1.2.p2.2.m2.3.3.2.2" xref="S6.SS1.2.p2.2.m2.3.3.2.2.cmml"><mi id="S6.SS1.2.p2.2.m2.3.3.2.2.2" xref="S6.SS1.2.p2.2.m2.3.3.2.2.2.cmml">ξ</mi><mrow id="S6.SS1.2.p2.2.m2.3.3.2.2.3" xref="S6.SS1.2.p2.2.m2.3.3.2.2.3.cmml"><mi id="S6.SS1.2.p2.2.m2.3.3.2.2.3.2" xref="S6.SS1.2.p2.2.m2.3.3.2.2.3.2.cmml">n</mi><mo id="S6.SS1.2.p2.2.m2.3.3.2.2.3.1" xref="S6.SS1.2.p2.2.m2.3.3.2.2.3.1.cmml">−</mo><mn id="S6.SS1.2.p2.2.m2.3.3.2.2.3.3" xref="S6.SS1.2.p2.2.m2.3.3.2.2.3.3.cmml">1</mn></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.2.p2.2.m2.3b"><list id="S6.SS1.2.p2.2.m2.3.3.3.cmml" xref="S6.SS1.2.p2.2.m2.3.3.2"><apply id="S6.SS1.2.p2.2.m2.2.2.1.1.cmml" xref="S6.SS1.2.p2.2.m2.2.2.1.1"><csymbol cd="ambiguous" id="S6.SS1.2.p2.2.m2.2.2.1.1.1.cmml" xref="S6.SS1.2.p2.2.m2.2.2.1.1">subscript</csymbol><ci id="S6.SS1.2.p2.2.m2.2.2.1.1.2.cmml" xref="S6.SS1.2.p2.2.m2.2.2.1.1.2">𝜉</ci><cn id="S6.SS1.2.p2.2.m2.2.2.1.1.3.cmml" type="integer" xref="S6.SS1.2.p2.2.m2.2.2.1.1.3">0</cn></apply><ci id="S6.SS1.2.p2.2.m2.1.1.cmml" xref="S6.SS1.2.p2.2.m2.1.1">…</ci><apply id="S6.SS1.2.p2.2.m2.3.3.2.2.cmml" xref="S6.SS1.2.p2.2.m2.3.3.2.2"><csymbol cd="ambiguous" id="S6.SS1.2.p2.2.m2.3.3.2.2.1.cmml" xref="S6.SS1.2.p2.2.m2.3.3.2.2">subscript</csymbol><ci id="S6.SS1.2.p2.2.m2.3.3.2.2.2.cmml" xref="S6.SS1.2.p2.2.m2.3.3.2.2.2">𝜉</ci><apply id="S6.SS1.2.p2.2.m2.3.3.2.2.3.cmml" xref="S6.SS1.2.p2.2.m2.3.3.2.2.3"><minus id="S6.SS1.2.p2.2.m2.3.3.2.2.3.1.cmml" xref="S6.SS1.2.p2.2.m2.3.3.2.2.3.1"></minus><ci id="S6.SS1.2.p2.2.m2.3.3.2.2.3.2.cmml" xref="S6.SS1.2.p2.2.m2.3.3.2.2.3.2">𝑛</ci><cn id="S6.SS1.2.p2.2.m2.3.3.2.2.3.3.cmml" type="integer" xref="S6.SS1.2.p2.2.m2.3.3.2.2.3.3">1</cn></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.2.p2.2.m2.3c">\xi_{0},\dots,\xi_{n-1}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.2.p2.2.m2.3d">italic_ξ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_ξ start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT</annotation></semantics></math> be the <math alttext="<_{A}" class="ltx_Math" display="inline" id="S6.SS1.2.p2.3.m3.1"><semantics id="S6.SS1.2.p2.3.m3.1a"><msub id="S6.SS1.2.p2.3.m3.1.1" xref="S6.SS1.2.p2.3.m3.1.1.cmml"><mo id="S6.SS1.2.p2.3.m3.1.1.2" xref="S6.SS1.2.p2.3.m3.1.1.2.cmml"><</mo><mi id="S6.SS1.2.p2.3.m3.1.1.3" xref="S6.SS1.2.p2.3.m3.1.1.3.cmml">A</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS1.2.p2.3.m3.1b"><apply id="S6.SS1.2.p2.3.m3.1.1.cmml" xref="S6.SS1.2.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S6.SS1.2.p2.3.m3.1.1.1.cmml" xref="S6.SS1.2.p2.3.m3.1.1">subscript</csymbol><lt id="S6.SS1.2.p2.3.m3.1.1.2.cmml" xref="S6.SS1.2.p2.3.m3.1.1.2"></lt><ci id="S6.SS1.2.p2.3.m3.1.1.3.cmml" xref="S6.SS1.2.p2.3.m3.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.2.p2.3.m3.1c"><_{A}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.2.p2.3.m3.1d">< start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT</annotation></semantics></math> increasing enumeration of <math alttext="\operatorname{dom}(q_{\xi})" class="ltx_Math" display="inline" id="S6.SS1.2.p2.4.m4.2"><semantics id="S6.SS1.2.p2.4.m4.2a"><mrow id="S6.SS1.2.p2.4.m4.2.2.1" xref="S6.SS1.2.p2.4.m4.2.2.2.cmml"><mi id="S6.SS1.2.p2.4.m4.1.1" xref="S6.SS1.2.p2.4.m4.1.1.cmml">dom</mi><mo id="S6.SS1.2.p2.4.m4.2.2.1a" xref="S6.SS1.2.p2.4.m4.2.2.2.cmml">⁡</mo><mrow id="S6.SS1.2.p2.4.m4.2.2.1.1" xref="S6.SS1.2.p2.4.m4.2.2.2.cmml"><mo id="S6.SS1.2.p2.4.m4.2.2.1.1.2" stretchy="false" xref="S6.SS1.2.p2.4.m4.2.2.2.cmml">(</mo><msub id="S6.SS1.2.p2.4.m4.2.2.1.1.1" xref="S6.SS1.2.p2.4.m4.2.2.1.1.1.cmml"><mi id="S6.SS1.2.p2.4.m4.2.2.1.1.1.2" xref="S6.SS1.2.p2.4.m4.2.2.1.1.1.2.cmml">q</mi><mi id="S6.SS1.2.p2.4.m4.2.2.1.1.1.3" xref="S6.SS1.2.p2.4.m4.2.2.1.1.1.3.cmml">ξ</mi></msub><mo id="S6.SS1.2.p2.4.m4.2.2.1.1.3" stretchy="false" xref="S6.SS1.2.p2.4.m4.2.2.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.2.p2.4.m4.2b"><apply id="S6.SS1.2.p2.4.m4.2.2.2.cmml" xref="S6.SS1.2.p2.4.m4.2.2.1"><ci id="S6.SS1.2.p2.4.m4.1.1.cmml" xref="S6.SS1.2.p2.4.m4.1.1">dom</ci><apply id="S6.SS1.2.p2.4.m4.2.2.1.1.1.cmml" xref="S6.SS1.2.p2.4.m4.2.2.1.1.1"><csymbol cd="ambiguous" id="S6.SS1.2.p2.4.m4.2.2.1.1.1.1.cmml" xref="S6.SS1.2.p2.4.m4.2.2.1.1.1">subscript</csymbol><ci id="S6.SS1.2.p2.4.m4.2.2.1.1.1.2.cmml" xref="S6.SS1.2.p2.4.m4.2.2.1.1.1.2">𝑞</ci><ci id="S6.SS1.2.p2.4.m4.2.2.1.1.1.3.cmml" xref="S6.SS1.2.p2.4.m4.2.2.1.1.1.3">𝜉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.2.p2.4.m4.2c">\operatorname{dom}(q_{\xi})</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.2.p2.4.m4.2d">roman_dom ( italic_q start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT )</annotation></semantics></math>. Fix <math alttext="i<n" class="ltx_Math" display="inline" id="S6.SS1.2.p2.5.m5.1"><semantics id="S6.SS1.2.p2.5.m5.1a"><mrow id="S6.SS1.2.p2.5.m5.1.1" xref="S6.SS1.2.p2.5.m5.1.1.cmml"><mi id="S6.SS1.2.p2.5.m5.1.1.2" xref="S6.SS1.2.p2.5.m5.1.1.2.cmml">i</mi><mo id="S6.SS1.2.p2.5.m5.1.1.1" xref="S6.SS1.2.p2.5.m5.1.1.1.cmml"><</mo><mi id="S6.SS1.2.p2.5.m5.1.1.3" xref="S6.SS1.2.p2.5.m5.1.1.3.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.2.p2.5.m5.1b"><apply id="S6.SS1.2.p2.5.m5.1.1.cmml" xref="S6.SS1.2.p2.5.m5.1.1"><lt id="S6.SS1.2.p2.5.m5.1.1.1.cmml" xref="S6.SS1.2.p2.5.m5.1.1.1"></lt><ci id="S6.SS1.2.p2.5.m5.1.1.2.cmml" xref="S6.SS1.2.p2.5.m5.1.1.2">𝑖</ci><ci id="S6.SS1.2.p2.5.m5.1.1.3.cmml" xref="S6.SS1.2.p2.5.m5.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.2.p2.5.m5.1c">i<n</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.2.p2.5.m5.1d">italic_i < italic_n</annotation></semantics></math>. Let <math alttext="T=\{t\in T:t\sqsubseteq\tilde{\xi}_{i}\text{ for some $\xi\in\Gamma$}\}" class="ltx_Math" display="inline" id="S6.SS1.2.p2.6.m6.3"><semantics id="S6.SS1.2.p2.6.m6.3a"><mrow id="S6.SS1.2.p2.6.m6.3.3" xref="S6.SS1.2.p2.6.m6.3.3.cmml"><mi id="S6.SS1.2.p2.6.m6.3.3.4" xref="S6.SS1.2.p2.6.m6.3.3.4.cmml">T</mi><mo id="S6.SS1.2.p2.6.m6.3.3.3" xref="S6.SS1.2.p2.6.m6.3.3.3.cmml">=</mo><mrow id="S6.SS1.2.p2.6.m6.3.3.2.2" xref="S6.SS1.2.p2.6.m6.3.3.2.3.cmml"><mo id="S6.SS1.2.p2.6.m6.3.3.2.2.3" stretchy="false" xref="S6.SS1.2.p2.6.m6.3.3.2.3.1.cmml">{</mo><mrow id="S6.SS1.2.p2.6.m6.2.2.1.1.1" xref="S6.SS1.2.p2.6.m6.2.2.1.1.1.cmml"><mi id="S6.SS1.2.p2.6.m6.2.2.1.1.1.2" xref="S6.SS1.2.p2.6.m6.2.2.1.1.1.2.cmml">t</mi><mo id="S6.SS1.2.p2.6.m6.2.2.1.1.1.1" xref="S6.SS1.2.p2.6.m6.2.2.1.1.1.1.cmml">∈</mo><mi id="S6.SS1.2.p2.6.m6.2.2.1.1.1.3" xref="S6.SS1.2.p2.6.m6.2.2.1.1.1.3.cmml">T</mi></mrow><mo id="S6.SS1.2.p2.6.m6.3.3.2.2.4" lspace="0.278em" rspace="0.278em" xref="S6.SS1.2.p2.6.m6.3.3.2.3.1.cmml">:</mo><mrow id="S6.SS1.2.p2.6.m6.3.3.2.2.2" xref="S6.SS1.2.p2.6.m6.3.3.2.2.2.cmml"><mi id="S6.SS1.2.p2.6.m6.3.3.2.2.2.2" xref="S6.SS1.2.p2.6.m6.3.3.2.2.2.2.cmml">t</mi><mo id="S6.SS1.2.p2.6.m6.3.3.2.2.2.1" xref="S6.SS1.2.p2.6.m6.3.3.2.2.2.1.cmml">⊑</mo><mrow id="S6.SS1.2.p2.6.m6.3.3.2.2.2.3" xref="S6.SS1.2.p2.6.m6.3.3.2.2.2.3.cmml"><msub id="S6.SS1.2.p2.6.m6.3.3.2.2.2.3.2" xref="S6.SS1.2.p2.6.m6.3.3.2.2.2.3.2.cmml"><mover accent="true" id="S6.SS1.2.p2.6.m6.3.3.2.2.2.3.2.2" xref="S6.SS1.2.p2.6.m6.3.3.2.2.2.3.2.2.cmml"><mi id="S6.SS1.2.p2.6.m6.3.3.2.2.2.3.2.2.2" xref="S6.SS1.2.p2.6.m6.3.3.2.2.2.3.2.2.2.cmml">ξ</mi><mo id="S6.SS1.2.p2.6.m6.3.3.2.2.2.3.2.2.1" xref="S6.SS1.2.p2.6.m6.3.3.2.2.2.3.2.2.1.cmml">~</mo></mover><mi id="S6.SS1.2.p2.6.m6.3.3.2.2.2.3.2.3" xref="S6.SS1.2.p2.6.m6.3.3.2.2.2.3.2.3.cmml">i</mi></msub><mo id="S6.SS1.2.p2.6.m6.3.3.2.2.2.3.1" xref="S6.SS1.2.p2.6.m6.3.3.2.2.2.3.1.cmml">⁢</mo><mrow id="S6.SS1.2.p2.6.m6.1.1.1" xref="S6.SS1.2.p2.6.m6.1.1.1b.cmml"><mtext id="S6.SS1.2.p2.6.m6.1.1.1a" xref="S6.SS1.2.p2.6.m6.1.1.1b.cmml"> for some </mtext><mrow id="S6.SS1.2.p2.6.m6.1.1.1.m1.1.1" xref="S6.SS1.2.p2.6.m6.1.1.1.m1.1.1.cmml"><mi id="S6.SS1.2.p2.6.m6.1.1.1.m1.1.1.2" xref="S6.SS1.2.p2.6.m6.1.1.1.m1.1.1.2.cmml">ξ</mi><mo id="S6.SS1.2.p2.6.m6.1.1.1.m1.1.1.1" xref="S6.SS1.2.p2.6.m6.1.1.1.m1.1.1.1.cmml">∈</mo><mi id="S6.SS1.2.p2.6.m6.1.1.1.m1.1.1.3" mathvariant="normal" xref="S6.SS1.2.p2.6.m6.1.1.1.m1.1.1.3.cmml">Γ</mi></mrow></mrow></mrow></mrow><mo id="S6.SS1.2.p2.6.m6.3.3.2.2.5" stretchy="false" xref="S6.SS1.2.p2.6.m6.3.3.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.2.p2.6.m6.3b"><apply id="S6.SS1.2.p2.6.m6.3.3.cmml" xref="S6.SS1.2.p2.6.m6.3.3"><eq id="S6.SS1.2.p2.6.m6.3.3.3.cmml" xref="S6.SS1.2.p2.6.m6.3.3.3"></eq><ci id="S6.SS1.2.p2.6.m6.3.3.4.cmml" xref="S6.SS1.2.p2.6.m6.3.3.4">𝑇</ci><apply id="S6.SS1.2.p2.6.m6.3.3.2.3.cmml" xref="S6.SS1.2.p2.6.m6.3.3.2.2"><csymbol cd="latexml" id="S6.SS1.2.p2.6.m6.3.3.2.3.1.cmml" xref="S6.SS1.2.p2.6.m6.3.3.2.2.3">conditional-set</csymbol><apply id="S6.SS1.2.p2.6.m6.2.2.1.1.1.cmml" xref="S6.SS1.2.p2.6.m6.2.2.1.1.1"><in id="S6.SS1.2.p2.6.m6.2.2.1.1.1.1.cmml" xref="S6.SS1.2.p2.6.m6.2.2.1.1.1.1"></in><ci id="S6.SS1.2.p2.6.m6.2.2.1.1.1.2.cmml" xref="S6.SS1.2.p2.6.m6.2.2.1.1.1.2">𝑡</ci><ci id="S6.SS1.2.p2.6.m6.2.2.1.1.1.3.cmml" xref="S6.SS1.2.p2.6.m6.2.2.1.1.1.3">𝑇</ci></apply><apply id="S6.SS1.2.p2.6.m6.3.3.2.2.2.cmml" xref="S6.SS1.2.p2.6.m6.3.3.2.2.2"><csymbol cd="latexml" id="S6.SS1.2.p2.6.m6.3.3.2.2.2.1.cmml" xref="S6.SS1.2.p2.6.m6.3.3.2.2.2.1">square-image-of-or-equals</csymbol><ci id="S6.SS1.2.p2.6.m6.3.3.2.2.2.2.cmml" xref="S6.SS1.2.p2.6.m6.3.3.2.2.2.2">𝑡</ci><apply id="S6.SS1.2.p2.6.m6.3.3.2.2.2.3.cmml" xref="S6.SS1.2.p2.6.m6.3.3.2.2.2.3"><times id="S6.SS1.2.p2.6.m6.3.3.2.2.2.3.1.cmml" xref="S6.SS1.2.p2.6.m6.3.3.2.2.2.3.1"></times><apply id="S6.SS1.2.p2.6.m6.3.3.2.2.2.3.2.cmml" xref="S6.SS1.2.p2.6.m6.3.3.2.2.2.3.2"><csymbol cd="ambiguous" id="S6.SS1.2.p2.6.m6.3.3.2.2.2.3.2.1.cmml" xref="S6.SS1.2.p2.6.m6.3.3.2.2.2.3.2">subscript</csymbol><apply id="S6.SS1.2.p2.6.m6.3.3.2.2.2.3.2.2.cmml" xref="S6.SS1.2.p2.6.m6.3.3.2.2.2.3.2.2"><ci id="S6.SS1.2.p2.6.m6.3.3.2.2.2.3.2.2.1.cmml" xref="S6.SS1.2.p2.6.m6.3.3.2.2.2.3.2.2.1">~</ci><ci id="S6.SS1.2.p2.6.m6.3.3.2.2.2.3.2.2.2.cmml" xref="S6.SS1.2.p2.6.m6.3.3.2.2.2.3.2.2.2">𝜉</ci></apply><ci id="S6.SS1.2.p2.6.m6.3.3.2.2.2.3.2.3.cmml" xref="S6.SS1.2.p2.6.m6.3.3.2.2.2.3.2.3">𝑖</ci></apply><ci id="S6.SS1.2.p2.6.m6.1.1.1b.cmml" xref="S6.SS1.2.p2.6.m6.1.1.1"><mrow id="S6.SS1.2.p2.6.m6.1.1.1.cmml" xref="S6.SS1.2.p2.6.m6.1.1.1"><mtext id="S6.SS1.2.p2.6.m6.1.1.1a.cmml" xref="S6.SS1.2.p2.6.m6.1.1.1"> for some </mtext><mrow id="S6.SS1.2.p2.6.m6.1.1.1.m1.1.1.cmml" xref="S6.SS1.2.p2.6.m6.1.1.1.m1.1.1"><mi id="S6.SS1.2.p2.6.m6.1.1.1.m1.1.1.2.cmml" xref="S6.SS1.2.p2.6.m6.1.1.1.m1.1.1.2">ξ</mi><mo id="S6.SS1.2.p2.6.m6.1.1.1.m1.1.1.1.cmml" xref="S6.SS1.2.p2.6.m6.1.1.1.m1.1.1.1">∈</mo><mi id="S6.SS1.2.p2.6.m6.1.1.1.m1.1.1.3.cmml" mathvariant="normal" xref="S6.SS1.2.p2.6.m6.1.1.1.m1.1.1.3">Γ</mi></mrow></mrow></ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.2.p2.6.m6.3c">T=\{t\in T:t\sqsubseteq\tilde{\xi}_{i}\text{ for some $\xi\in\Gamma$}\}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.2.p2.6.m6.3d">italic_T = { italic_t ∈ italic_T : italic_t ⊑ over~ start_ARG italic_ξ end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT for some italic_ξ ∈ roman_Γ }</annotation></semantics></math>, where <math alttext="\sqsubset" class="ltx_Math" display="inline" id="S6.SS1.2.p2.7.m7.1"><semantics id="S6.SS1.2.p2.7.m7.1a"><mo id="S6.SS1.2.p2.7.m7.1.1" xref="S6.SS1.2.p2.7.m7.1.1.cmml">⊏</mo><annotation-xml encoding="MathML-Content" id="S6.SS1.2.p2.7.m7.1b"><csymbol cd="latexml" id="S6.SS1.2.p2.7.m7.1.1.cmml" xref="S6.SS1.2.p2.7.m7.1.1">square-image-of</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.2.p2.7.m7.1c">\sqsubset</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.2.p2.7.m7.1d">⊏</annotation></semantics></math> denotes sequence extension. Since <math alttext="T" class="ltx_Math" display="inline" id="S6.SS1.2.p2.8.m8.1"><semantics id="S6.SS1.2.p2.8.m8.1a"><mi id="S6.SS1.2.p2.8.m8.1.1" xref="S6.SS1.2.p2.8.m8.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S6.SS1.2.p2.8.m8.1b"><ci id="S6.SS1.2.p2.8.m8.1.1.cmml" xref="S6.SS1.2.p2.8.m8.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.2.p2.8.m8.1c">T</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.2.p2.8.m8.1d">italic_T</annotation></semantics></math> is Aronszajn there is <math alttext="t_{0}" class="ltx_Math" display="inline" id="S6.SS1.2.p2.9.m9.1"><semantics id="S6.SS1.2.p2.9.m9.1a"><msub id="S6.SS1.2.p2.9.m9.1.1" xref="S6.SS1.2.p2.9.m9.1.1.cmml"><mi id="S6.SS1.2.p2.9.m9.1.1.2" xref="S6.SS1.2.p2.9.m9.1.1.2.cmml">t</mi><mn id="S6.SS1.2.p2.9.m9.1.1.3" xref="S6.SS1.2.p2.9.m9.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S6.SS1.2.p2.9.m9.1b"><apply id="S6.SS1.2.p2.9.m9.1.1.cmml" xref="S6.SS1.2.p2.9.m9.1.1"><csymbol cd="ambiguous" id="S6.SS1.2.p2.9.m9.1.1.1.cmml" xref="S6.SS1.2.p2.9.m9.1.1">subscript</csymbol><ci id="S6.SS1.2.p2.9.m9.1.1.2.cmml" xref="S6.SS1.2.p2.9.m9.1.1.2">𝑡</ci><cn id="S6.SS1.2.p2.9.m9.1.1.3.cmml" type="integer" xref="S6.SS1.2.p2.9.m9.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.2.p2.9.m9.1c">t_{0}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.2.p2.9.m9.1d">italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> at a level <math alttext="\gamma+1" class="ltx_Math" display="inline" id="S6.SS1.2.p2.10.m10.1"><semantics id="S6.SS1.2.p2.10.m10.1a"><mrow id="S6.SS1.2.p2.10.m10.1.1" xref="S6.SS1.2.p2.10.m10.1.1.cmml"><mi id="S6.SS1.2.p2.10.m10.1.1.2" xref="S6.SS1.2.p2.10.m10.1.1.2.cmml">γ</mi><mo id="S6.SS1.2.p2.10.m10.1.1.1" xref="S6.SS1.2.p2.10.m10.1.1.1.cmml">+</mo><mn id="S6.SS1.2.p2.10.m10.1.1.3" xref="S6.SS1.2.p2.10.m10.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.2.p2.10.m10.1b"><apply id="S6.SS1.2.p2.10.m10.1.1.cmml" xref="S6.SS1.2.p2.10.m10.1.1"><plus id="S6.SS1.2.p2.10.m10.1.1.1.cmml" xref="S6.SS1.2.p2.10.m10.1.1.1"></plus><ci id="S6.SS1.2.p2.10.m10.1.1.2.cmml" xref="S6.SS1.2.p2.10.m10.1.1.2">𝛾</ci><cn id="S6.SS1.2.p2.10.m10.1.1.3.cmml" type="integer" xref="S6.SS1.2.p2.10.m10.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.2.p2.10.m10.1c">\gamma+1</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.2.p2.10.m10.1d">italic_γ + 1</annotation></semantics></math> with extensions at every level. Let <math alttext="T^{\prime}" class="ltx_Math" display="inline" id="S6.SS1.2.p2.11.m11.1"><semantics id="S6.SS1.2.p2.11.m11.1a"><msup id="S6.SS1.2.p2.11.m11.1.1" xref="S6.SS1.2.p2.11.m11.1.1.cmml"><mi id="S6.SS1.2.p2.11.m11.1.1.2" xref="S6.SS1.2.p2.11.m11.1.1.2.cmml">T</mi><mo id="S6.SS1.2.p2.11.m11.1.1.3" xref="S6.SS1.2.p2.11.m11.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S6.SS1.2.p2.11.m11.1b"><apply id="S6.SS1.2.p2.11.m11.1.1.cmml" xref="S6.SS1.2.p2.11.m11.1.1"><csymbol cd="ambiguous" id="S6.SS1.2.p2.11.m11.1.1.1.cmml" xref="S6.SS1.2.p2.11.m11.1.1">superscript</csymbol><ci id="S6.SS1.2.p2.11.m11.1.1.2.cmml" xref="S6.SS1.2.p2.11.m11.1.1.2">𝑇</ci><ci id="S6.SS1.2.p2.11.m11.1.1.3.cmml" xref="S6.SS1.2.p2.11.m11.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.2.p2.11.m11.1c">T^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.2.p2.11.m11.1d">italic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> be the subtree of <math alttext="T" class="ltx_Math" display="inline" id="S6.SS1.2.p2.12.m12.1"><semantics id="S6.SS1.2.p2.12.m12.1a"><mi id="S6.SS1.2.p2.12.m12.1.1" xref="S6.SS1.2.p2.12.m12.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S6.SS1.2.p2.12.m12.1b"><ci id="S6.SS1.2.p2.12.m12.1.1.cmml" xref="S6.SS1.2.p2.12.m12.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.2.p2.12.m12.1c">T</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.2.p2.12.m12.1d">italic_T</annotation></semantics></math> of those who are comparable with <math alttext="t_{0}" class="ltx_Math" display="inline" id="S6.SS1.2.p2.13.m13.1"><semantics id="S6.SS1.2.p2.13.m13.1a"><msub id="S6.SS1.2.p2.13.m13.1.1" xref="S6.SS1.2.p2.13.m13.1.1.cmml"><mi id="S6.SS1.2.p2.13.m13.1.1.2" xref="S6.SS1.2.p2.13.m13.1.1.2.cmml">t</mi><mn id="S6.SS1.2.p2.13.m13.1.1.3" xref="S6.SS1.2.p2.13.m13.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S6.SS1.2.p2.13.m13.1b"><apply id="S6.SS1.2.p2.13.m13.1.1.cmml" xref="S6.SS1.2.p2.13.m13.1.1"><csymbol cd="ambiguous" id="S6.SS1.2.p2.13.m13.1.1.1.cmml" xref="S6.SS1.2.p2.13.m13.1.1">subscript</csymbol><ci id="S6.SS1.2.p2.13.m13.1.1.2.cmml" xref="S6.SS1.2.p2.13.m13.1.1.2">𝑡</ci><cn id="S6.SS1.2.p2.13.m13.1.1.3.cmml" type="integer" xref="S6.SS1.2.p2.13.m13.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.2.p2.13.m13.1c">t_{0}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.2.p2.13.m13.1d">italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> and have uncountable extensions, and let <math alttext="H\subseteq T^{\prime}" class="ltx_Math" display="inline" id="S6.SS1.2.p2.14.m14.1"><semantics id="S6.SS1.2.p2.14.m14.1a"><mrow id="S6.SS1.2.p2.14.m14.1.1" xref="S6.SS1.2.p2.14.m14.1.1.cmml"><mi id="S6.SS1.2.p2.14.m14.1.1.2" xref="S6.SS1.2.p2.14.m14.1.1.2.cmml">H</mi><mo id="S6.SS1.2.p2.14.m14.1.1.1" xref="S6.SS1.2.p2.14.m14.1.1.1.cmml">⊆</mo><msup id="S6.SS1.2.p2.14.m14.1.1.3" xref="S6.SS1.2.p2.14.m14.1.1.3.cmml"><mi id="S6.SS1.2.p2.14.m14.1.1.3.2" xref="S6.SS1.2.p2.14.m14.1.1.3.2.cmml">T</mi><mo id="S6.SS1.2.p2.14.m14.1.1.3.3" xref="S6.SS1.2.p2.14.m14.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.2.p2.14.m14.1b"><apply id="S6.SS1.2.p2.14.m14.1.1.cmml" xref="S6.SS1.2.p2.14.m14.1.1"><subset id="S6.SS1.2.p2.14.m14.1.1.1.cmml" xref="S6.SS1.2.p2.14.m14.1.1.1"></subset><ci id="S6.SS1.2.p2.14.m14.1.1.2.cmml" xref="S6.SS1.2.p2.14.m14.1.1.2">𝐻</ci><apply id="S6.SS1.2.p2.14.m14.1.1.3.cmml" xref="S6.SS1.2.p2.14.m14.1.1.3"><csymbol cd="ambiguous" id="S6.SS1.2.p2.14.m14.1.1.3.1.cmml" xref="S6.SS1.2.p2.14.m14.1.1.3">superscript</csymbol><ci id="S6.SS1.2.p2.14.m14.1.1.3.2.cmml" xref="S6.SS1.2.p2.14.m14.1.1.3.2">𝑇</ci><ci id="S6.SS1.2.p2.14.m14.1.1.3.3.cmml" xref="S6.SS1.2.p2.14.m14.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.2.p2.14.m14.1c">H\subseteq T^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.2.p2.14.m14.1d">italic_H ⊆ italic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> be an uncountable antichain. Then for any <math alttext="t\in H" class="ltx_Math" display="inline" id="S6.SS1.2.p2.15.m15.1"><semantics id="S6.SS1.2.p2.15.m15.1a"><mrow id="S6.SS1.2.p2.15.m15.1.1" xref="S6.SS1.2.p2.15.m15.1.1.cmml"><mi id="S6.SS1.2.p2.15.m15.1.1.2" xref="S6.SS1.2.p2.15.m15.1.1.2.cmml">t</mi><mo id="S6.SS1.2.p2.15.m15.1.1.1" xref="S6.SS1.2.p2.15.m15.1.1.1.cmml">∈</mo><mi id="S6.SS1.2.p2.15.m15.1.1.3" xref="S6.SS1.2.p2.15.m15.1.1.3.cmml">H</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.2.p2.15.m15.1b"><apply id="S6.SS1.2.p2.15.m15.1.1.cmml" xref="S6.SS1.2.p2.15.m15.1.1"><in id="S6.SS1.2.p2.15.m15.1.1.1.cmml" xref="S6.SS1.2.p2.15.m15.1.1.1"></in><ci id="S6.SS1.2.p2.15.m15.1.1.2.cmml" xref="S6.SS1.2.p2.15.m15.1.1.2">𝑡</ci><ci id="S6.SS1.2.p2.15.m15.1.1.3.cmml" xref="S6.SS1.2.p2.15.m15.1.1.3">𝐻</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.2.p2.15.m15.1c">t\in H</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.2.p2.15.m15.1d">italic_t ∈ italic_H</annotation></semantics></math> there is <math alttext="\xi^{(t)}\in\Gamma^{\prime\prime\prime}" class="ltx_Math" display="inline" id="S6.SS1.2.p2.16.m16.1"><semantics id="S6.SS1.2.p2.16.m16.1a"><mrow id="S6.SS1.2.p2.16.m16.1.2" xref="S6.SS1.2.p2.16.m16.1.2.cmml"><msup id="S6.SS1.2.p2.16.m16.1.2.2" xref="S6.SS1.2.p2.16.m16.1.2.2.cmml"><mi id="S6.SS1.2.p2.16.m16.1.2.2.2" xref="S6.SS1.2.p2.16.m16.1.2.2.2.cmml">ξ</mi><mrow id="S6.SS1.2.p2.16.m16.1.1.1.3" xref="S6.SS1.2.p2.16.m16.1.2.2.cmml"><mo id="S6.SS1.2.p2.16.m16.1.1.1.3.1" stretchy="false" xref="S6.SS1.2.p2.16.m16.1.2.2.cmml">(</mo><mi id="S6.SS1.2.p2.16.m16.1.1.1.1" xref="S6.SS1.2.p2.16.m16.1.1.1.1.cmml">t</mi><mo id="S6.SS1.2.p2.16.m16.1.1.1.3.2" stretchy="false" xref="S6.SS1.2.p2.16.m16.1.2.2.cmml">)</mo></mrow></msup><mo id="S6.SS1.2.p2.16.m16.1.2.1" xref="S6.SS1.2.p2.16.m16.1.2.1.cmml">∈</mo><msup id="S6.SS1.2.p2.16.m16.1.2.3" xref="S6.SS1.2.p2.16.m16.1.2.3.cmml"><mi id="S6.SS1.2.p2.16.m16.1.2.3.2" mathvariant="normal" xref="S6.SS1.2.p2.16.m16.1.2.3.2.cmml">Γ</mi><mo id="S6.SS1.2.p2.16.m16.1.2.3.3" xref="S6.SS1.2.p2.16.m16.1.2.3.3.cmml">′′′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.2.p2.16.m16.1b"><apply id="S6.SS1.2.p2.16.m16.1.2.cmml" xref="S6.SS1.2.p2.16.m16.1.2"><in id="S6.SS1.2.p2.16.m16.1.2.1.cmml" xref="S6.SS1.2.p2.16.m16.1.2.1"></in><apply id="S6.SS1.2.p2.16.m16.1.2.2.cmml" xref="S6.SS1.2.p2.16.m16.1.2.2"><csymbol cd="ambiguous" id="S6.SS1.2.p2.16.m16.1.2.2.1.cmml" xref="S6.SS1.2.p2.16.m16.1.2.2">superscript</csymbol><ci id="S6.SS1.2.p2.16.m16.1.2.2.2.cmml" xref="S6.SS1.2.p2.16.m16.1.2.2.2">𝜉</ci><ci id="S6.SS1.2.p2.16.m16.1.1.1.1.cmml" xref="S6.SS1.2.p2.16.m16.1.1.1.1">𝑡</ci></apply><apply id="S6.SS1.2.p2.16.m16.1.2.3.cmml" xref="S6.SS1.2.p2.16.m16.1.2.3"><csymbol cd="ambiguous" id="S6.SS1.2.p2.16.m16.1.2.3.1.cmml" xref="S6.SS1.2.p2.16.m16.1.2.3">superscript</csymbol><ci id="S6.SS1.2.p2.16.m16.1.2.3.2.cmml" xref="S6.SS1.2.p2.16.m16.1.2.3.2">Γ</ci><ci id="S6.SS1.2.p2.16.m16.1.2.3.3.cmml" xref="S6.SS1.2.p2.16.m16.1.2.3.3">′′′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.2.p2.16.m16.1c">\xi^{(t)}\in\Gamma^{\prime\prime\prime}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.2.p2.16.m16.1d">italic_ξ start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT ∈ roman_Γ start_POSTSUPERSCRIPT ′ ′ ′ end_POSTSUPERSCRIPT</annotation></semantics></math> such that <math alttext="\tilde{\xi}^{(t)}_{i}\sqsupset t" class="ltx_Math" display="inline" id="S6.SS1.2.p2.17.m17.1"><semantics id="S6.SS1.2.p2.17.m17.1a"><mrow id="S6.SS1.2.p2.17.m17.1.2" xref="S6.SS1.2.p2.17.m17.1.2.cmml"><msubsup id="S6.SS1.2.p2.17.m17.1.2.2" xref="S6.SS1.2.p2.17.m17.1.2.2.cmml"><mover accent="true" id="S6.SS1.2.p2.17.m17.1.2.2.2.2" xref="S6.SS1.2.p2.17.m17.1.2.2.2.2.cmml"><mi id="S6.SS1.2.p2.17.m17.1.2.2.2.2.2" xref="S6.SS1.2.p2.17.m17.1.2.2.2.2.2.cmml">ξ</mi><mo id="S6.SS1.2.p2.17.m17.1.2.2.2.2.1" xref="S6.SS1.2.p2.17.m17.1.2.2.2.2.1.cmml">~</mo></mover><mi id="S6.SS1.2.p2.17.m17.1.2.2.3" xref="S6.SS1.2.p2.17.m17.1.2.2.3.cmml">i</mi><mrow id="S6.SS1.2.p2.17.m17.1.1.1.3" xref="S6.SS1.2.p2.17.m17.1.2.2.cmml"><mo id="S6.SS1.2.p2.17.m17.1.1.1.3.1" stretchy="false" xref="S6.SS1.2.p2.17.m17.1.2.2.cmml">(</mo><mi id="S6.SS1.2.p2.17.m17.1.1.1.1" xref="S6.SS1.2.p2.17.m17.1.1.1.1.cmml">t</mi><mo id="S6.SS1.2.p2.17.m17.1.1.1.3.2" stretchy="false" xref="S6.SS1.2.p2.17.m17.1.2.2.cmml">)</mo></mrow></msubsup><mo id="S6.SS1.2.p2.17.m17.1.2.1" xref="S6.SS1.2.p2.17.m17.1.2.1.cmml">⊐</mo><mi id="S6.SS1.2.p2.17.m17.1.2.3" xref="S6.SS1.2.p2.17.m17.1.2.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.2.p2.17.m17.1b"><apply id="S6.SS1.2.p2.17.m17.1.2.cmml" xref="S6.SS1.2.p2.17.m17.1.2"><csymbol cd="latexml" id="S6.SS1.2.p2.17.m17.1.2.1.cmml" xref="S6.SS1.2.p2.17.m17.1.2.1">square-original-of</csymbol><apply id="S6.SS1.2.p2.17.m17.1.2.2.cmml" xref="S6.SS1.2.p2.17.m17.1.2.2"><csymbol cd="ambiguous" id="S6.SS1.2.p2.17.m17.1.2.2.1.cmml" xref="S6.SS1.2.p2.17.m17.1.2.2">subscript</csymbol><apply id="S6.SS1.2.p2.17.m17.1.2.2.2.cmml" xref="S6.SS1.2.p2.17.m17.1.2.2"><csymbol cd="ambiguous" id="S6.SS1.2.p2.17.m17.1.2.2.2.1.cmml" xref="S6.SS1.2.p2.17.m17.1.2.2">superscript</csymbol><apply id="S6.SS1.2.p2.17.m17.1.2.2.2.2.cmml" xref="S6.SS1.2.p2.17.m17.1.2.2.2.2"><ci id="S6.SS1.2.p2.17.m17.1.2.2.2.2.1.cmml" xref="S6.SS1.2.p2.17.m17.1.2.2.2.2.1">~</ci><ci id="S6.SS1.2.p2.17.m17.1.2.2.2.2.2.cmml" xref="S6.SS1.2.p2.17.m17.1.2.2.2.2.2">𝜉</ci></apply><ci id="S6.SS1.2.p2.17.m17.1.1.1.1.cmml" xref="S6.SS1.2.p2.17.m17.1.1.1.1">𝑡</ci></apply><ci id="S6.SS1.2.p2.17.m17.1.2.2.3.cmml" xref="S6.SS1.2.p2.17.m17.1.2.2.3">𝑖</ci></apply><ci id="S6.SS1.2.p2.17.m17.1.2.3.cmml" xref="S6.SS1.2.p2.17.m17.1.2.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.2.p2.17.m17.1c">\tilde{\xi}^{(t)}_{i}\sqsupset t</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.2.p2.17.m17.1d">over~ start_ARG italic_ξ end_ARG start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊐ italic_t</annotation></semantics></math>, and <math alttext="\xi^{(t)}>\operatorname{ht}(t)" class="ltx_Math" display="inline" id="S6.SS1.2.p2.18.m18.3"><semantics id="S6.SS1.2.p2.18.m18.3a"><mrow id="S6.SS1.2.p2.18.m18.3.4" xref="S6.SS1.2.p2.18.m18.3.4.cmml"><msup id="S6.SS1.2.p2.18.m18.3.4.2" xref="S6.SS1.2.p2.18.m18.3.4.2.cmml"><mi id="S6.SS1.2.p2.18.m18.3.4.2.2" xref="S6.SS1.2.p2.18.m18.3.4.2.2.cmml">ξ</mi><mrow id="S6.SS1.2.p2.18.m18.1.1.1.3" xref="S6.SS1.2.p2.18.m18.3.4.2.cmml"><mo id="S6.SS1.2.p2.18.m18.1.1.1.3.1" stretchy="false" xref="S6.SS1.2.p2.18.m18.3.4.2.cmml">(</mo><mi id="S6.SS1.2.p2.18.m18.1.1.1.1" xref="S6.SS1.2.p2.18.m18.1.1.1.1.cmml">t</mi><mo id="S6.SS1.2.p2.18.m18.1.1.1.3.2" stretchy="false" xref="S6.SS1.2.p2.18.m18.3.4.2.cmml">)</mo></mrow></msup><mo id="S6.SS1.2.p2.18.m18.3.4.1" xref="S6.SS1.2.p2.18.m18.3.4.1.cmml">></mo><mrow id="S6.SS1.2.p2.18.m18.3.4.3.2" xref="S6.SS1.2.p2.18.m18.3.4.3.1.cmml"><mi id="S6.SS1.2.p2.18.m18.2.2" xref="S6.SS1.2.p2.18.m18.2.2.cmml">ht</mi><mo id="S6.SS1.2.p2.18.m18.3.4.3.2a" xref="S6.SS1.2.p2.18.m18.3.4.3.1.cmml">⁡</mo><mrow id="S6.SS1.2.p2.18.m18.3.4.3.2.1" xref="S6.SS1.2.p2.18.m18.3.4.3.1.cmml"><mo id="S6.SS1.2.p2.18.m18.3.4.3.2.1.1" stretchy="false" xref="S6.SS1.2.p2.18.m18.3.4.3.1.cmml">(</mo><mi id="S6.SS1.2.p2.18.m18.3.3" xref="S6.SS1.2.p2.18.m18.3.3.cmml">t</mi><mo id="S6.SS1.2.p2.18.m18.3.4.3.2.1.2" stretchy="false" xref="S6.SS1.2.p2.18.m18.3.4.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.2.p2.18.m18.3b"><apply id="S6.SS1.2.p2.18.m18.3.4.cmml" xref="S6.SS1.2.p2.18.m18.3.4"><gt id="S6.SS1.2.p2.18.m18.3.4.1.cmml" xref="S6.SS1.2.p2.18.m18.3.4.1"></gt><apply id="S6.SS1.2.p2.18.m18.3.4.2.cmml" xref="S6.SS1.2.p2.18.m18.3.4.2"><csymbol cd="ambiguous" id="S6.SS1.2.p2.18.m18.3.4.2.1.cmml" xref="S6.SS1.2.p2.18.m18.3.4.2">superscript</csymbol><ci id="S6.SS1.2.p2.18.m18.3.4.2.2.cmml" xref="S6.SS1.2.p2.18.m18.3.4.2.2">𝜉</ci><ci id="S6.SS1.2.p2.18.m18.1.1.1.1.cmml" xref="S6.SS1.2.p2.18.m18.1.1.1.1">𝑡</ci></apply><apply id="S6.SS1.2.p2.18.m18.3.4.3.1.cmml" xref="S6.SS1.2.p2.18.m18.3.4.3.2"><ci id="S6.SS1.2.p2.18.m18.2.2.cmml" xref="S6.SS1.2.p2.18.m18.2.2">ht</ci><ci id="S6.SS1.2.p2.18.m18.3.3.cmml" xref="S6.SS1.2.p2.18.m18.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.2.p2.18.m18.3c">\xi^{(t)}>\operatorname{ht}(t)</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.2.p2.18.m18.3d">italic_ξ start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT > roman_ht ( italic_t )</annotation></semantics></math>. Finally let <math alttext="\Gamma=\{\xi^{t}:t\in H\}" class="ltx_Math" display="inline" id="S6.SS1.2.p2.19.m19.2"><semantics id="S6.SS1.2.p2.19.m19.2a"><mrow id="S6.SS1.2.p2.19.m19.2.2" xref="S6.SS1.2.p2.19.m19.2.2.cmml"><mi id="S6.SS1.2.p2.19.m19.2.2.4" mathvariant="normal" xref="S6.SS1.2.p2.19.m19.2.2.4.cmml">Γ</mi><mo id="S6.SS1.2.p2.19.m19.2.2.3" xref="S6.SS1.2.p2.19.m19.2.2.3.cmml">=</mo><mrow id="S6.SS1.2.p2.19.m19.2.2.2.2" xref="S6.SS1.2.p2.19.m19.2.2.2.3.cmml"><mo id="S6.SS1.2.p2.19.m19.2.2.2.2.3" stretchy="false" xref="S6.SS1.2.p2.19.m19.2.2.2.3.1.cmml">{</mo><msup id="S6.SS1.2.p2.19.m19.1.1.1.1.1" xref="S6.SS1.2.p2.19.m19.1.1.1.1.1.cmml"><mi id="S6.SS1.2.p2.19.m19.1.1.1.1.1.2" xref="S6.SS1.2.p2.19.m19.1.1.1.1.1.2.cmml">ξ</mi><mi id="S6.SS1.2.p2.19.m19.1.1.1.1.1.3" xref="S6.SS1.2.p2.19.m19.1.1.1.1.1.3.cmml">t</mi></msup><mo id="S6.SS1.2.p2.19.m19.2.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S6.SS1.2.p2.19.m19.2.2.2.3.1.cmml">:</mo><mrow id="S6.SS1.2.p2.19.m19.2.2.2.2.2" xref="S6.SS1.2.p2.19.m19.2.2.2.2.2.cmml"><mi id="S6.SS1.2.p2.19.m19.2.2.2.2.2.2" xref="S6.SS1.2.p2.19.m19.2.2.2.2.2.2.cmml">t</mi><mo id="S6.SS1.2.p2.19.m19.2.2.2.2.2.1" xref="S6.SS1.2.p2.19.m19.2.2.2.2.2.1.cmml">∈</mo><mi id="S6.SS1.2.p2.19.m19.2.2.2.2.2.3" xref="S6.SS1.2.p2.19.m19.2.2.2.2.2.3.cmml">H</mi></mrow><mo id="S6.SS1.2.p2.19.m19.2.2.2.2.5" stretchy="false" xref="S6.SS1.2.p2.19.m19.2.2.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.2.p2.19.m19.2b"><apply id="S6.SS1.2.p2.19.m19.2.2.cmml" xref="S6.SS1.2.p2.19.m19.2.2"><eq id="S6.SS1.2.p2.19.m19.2.2.3.cmml" xref="S6.SS1.2.p2.19.m19.2.2.3"></eq><ci id="S6.SS1.2.p2.19.m19.2.2.4.cmml" xref="S6.SS1.2.p2.19.m19.2.2.4">Γ</ci><apply id="S6.SS1.2.p2.19.m19.2.2.2.3.cmml" xref="S6.SS1.2.p2.19.m19.2.2.2.2"><csymbol cd="latexml" id="S6.SS1.2.p2.19.m19.2.2.2.3.1.cmml" xref="S6.SS1.2.p2.19.m19.2.2.2.2.3">conditional-set</csymbol><apply id="S6.SS1.2.p2.19.m19.1.1.1.1.1.cmml" xref="S6.SS1.2.p2.19.m19.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS1.2.p2.19.m19.1.1.1.1.1.1.cmml" xref="S6.SS1.2.p2.19.m19.1.1.1.1.1">superscript</csymbol><ci id="S6.SS1.2.p2.19.m19.1.1.1.1.1.2.cmml" xref="S6.SS1.2.p2.19.m19.1.1.1.1.1.2">𝜉</ci><ci id="S6.SS1.2.p2.19.m19.1.1.1.1.1.3.cmml" xref="S6.SS1.2.p2.19.m19.1.1.1.1.1.3">𝑡</ci></apply><apply id="S6.SS1.2.p2.19.m19.2.2.2.2.2.cmml" xref="S6.SS1.2.p2.19.m19.2.2.2.2.2"><in id="S6.SS1.2.p2.19.m19.2.2.2.2.2.1.cmml" xref="S6.SS1.2.p2.19.m19.2.2.2.2.2.1"></in><ci id="S6.SS1.2.p2.19.m19.2.2.2.2.2.2.cmml" xref="S6.SS1.2.p2.19.m19.2.2.2.2.2.2">𝑡</ci><ci id="S6.SS1.2.p2.19.m19.2.2.2.2.2.3.cmml" xref="S6.SS1.2.p2.19.m19.2.2.2.2.2.3">𝐻</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.2.p2.19.m19.2c">\Gamma=\{\xi^{t}:t\in H\}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.2.p2.19.m19.2d">roman_Γ = { italic_ξ start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT : italic_t ∈ italic_H }</annotation></semantics></math> and note that for <math alttext="\xi<\eta" class="ltx_Math" display="inline" id="S6.SS1.2.p2.20.m20.1"><semantics id="S6.SS1.2.p2.20.m20.1a"><mrow id="S6.SS1.2.p2.20.m20.1.1" xref="S6.SS1.2.p2.20.m20.1.1.cmml"><mi id="S6.SS1.2.p2.20.m20.1.1.2" xref="S6.SS1.2.p2.20.m20.1.1.2.cmml">ξ</mi><mo id="S6.SS1.2.p2.20.m20.1.1.1" xref="S6.SS1.2.p2.20.m20.1.1.1.cmml"><</mo><mi id="S6.SS1.2.p2.20.m20.1.1.3" xref="S6.SS1.2.p2.20.m20.1.1.3.cmml">η</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.2.p2.20.m20.1b"><apply id="S6.SS1.2.p2.20.m20.1.1.cmml" xref="S6.SS1.2.p2.20.m20.1.1"><lt id="S6.SS1.2.p2.20.m20.1.1.1.cmml" xref="S6.SS1.2.p2.20.m20.1.1.1"></lt><ci id="S6.SS1.2.p2.20.m20.1.1.2.cmml" xref="S6.SS1.2.p2.20.m20.1.1.2">𝜉</ci><ci id="S6.SS1.2.p2.20.m20.1.1.3.cmml" xref="S6.SS1.2.p2.20.m20.1.1.3">𝜂</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.2.p2.20.m20.1c">\xi<\eta</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.2.p2.20.m20.1d">italic_ξ < italic_η</annotation></semantics></math> in <math alttext="\Gamma" class="ltx_Math" display="inline" id="S6.SS1.2.p2.21.m21.1"><semantics id="S6.SS1.2.p2.21.m21.1a"><mi id="S6.SS1.2.p2.21.m21.1.1" mathvariant="normal" xref="S6.SS1.2.p2.21.m21.1.1.cmml">Γ</mi><annotation-xml encoding="MathML-Content" id="S6.SS1.2.p2.21.m21.1b"><ci id="S6.SS1.2.p2.21.m21.1.1.cmml" xref="S6.SS1.2.p2.21.m21.1.1">Γ</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.2.p2.21.m21.1c">\Gamma</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.2.p2.21.m21.1d">roman_Γ</annotation></semantics></math>, <math alttext="\gamma<\Delta_{A}(\xi_{i},\eta_{i})<\xi" class="ltx_Math" display="inline" id="S6.SS1.2.p2.22.m22.2"><semantics id="S6.SS1.2.p2.22.m22.2a"><mrow id="S6.SS1.2.p2.22.m22.2.2" xref="S6.SS1.2.p2.22.m22.2.2.cmml"><mi id="S6.SS1.2.p2.22.m22.2.2.4" xref="S6.SS1.2.p2.22.m22.2.2.4.cmml">γ</mi><mo id="S6.SS1.2.p2.22.m22.2.2.5" xref="S6.SS1.2.p2.22.m22.2.2.5.cmml"><</mo><mrow id="S6.SS1.2.p2.22.m22.2.2.2" xref="S6.SS1.2.p2.22.m22.2.2.2.cmml"><msub id="S6.SS1.2.p2.22.m22.2.2.2.4" xref="S6.SS1.2.p2.22.m22.2.2.2.4.cmml"><mi id="S6.SS1.2.p2.22.m22.2.2.2.4.2" mathvariant="normal" xref="S6.SS1.2.p2.22.m22.2.2.2.4.2.cmml">Δ</mi><mi id="S6.SS1.2.p2.22.m22.2.2.2.4.3" xref="S6.SS1.2.p2.22.m22.2.2.2.4.3.cmml">A</mi></msub><mo id="S6.SS1.2.p2.22.m22.2.2.2.3" xref="S6.SS1.2.p2.22.m22.2.2.2.3.cmml">⁢</mo><mrow id="S6.SS1.2.p2.22.m22.2.2.2.2.2" xref="S6.SS1.2.p2.22.m22.2.2.2.2.3.cmml"><mo id="S6.SS1.2.p2.22.m22.2.2.2.2.2.3" stretchy="false" xref="S6.SS1.2.p2.22.m22.2.2.2.2.3.cmml">(</mo><msub id="S6.SS1.2.p2.22.m22.1.1.1.1.1.1" xref="S6.SS1.2.p2.22.m22.1.1.1.1.1.1.cmml"><mi id="S6.SS1.2.p2.22.m22.1.1.1.1.1.1.2" xref="S6.SS1.2.p2.22.m22.1.1.1.1.1.1.2.cmml">ξ</mi><mi id="S6.SS1.2.p2.22.m22.1.1.1.1.1.1.3" xref="S6.SS1.2.p2.22.m22.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S6.SS1.2.p2.22.m22.2.2.2.2.2.4" xref="S6.SS1.2.p2.22.m22.2.2.2.2.3.cmml">,</mo><msub id="S6.SS1.2.p2.22.m22.2.2.2.2.2.2" xref="S6.SS1.2.p2.22.m22.2.2.2.2.2.2.cmml"><mi id="S6.SS1.2.p2.22.m22.2.2.2.2.2.2.2" xref="S6.SS1.2.p2.22.m22.2.2.2.2.2.2.2.cmml">η</mi><mi id="S6.SS1.2.p2.22.m22.2.2.2.2.2.2.3" xref="S6.SS1.2.p2.22.m22.2.2.2.2.2.2.3.cmml">i</mi></msub><mo id="S6.SS1.2.p2.22.m22.2.2.2.2.2.5" stretchy="false" xref="S6.SS1.2.p2.22.m22.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.SS1.2.p2.22.m22.2.2.6" xref="S6.SS1.2.p2.22.m22.2.2.6.cmml"><</mo><mi id="S6.SS1.2.p2.22.m22.2.2.7" xref="S6.SS1.2.p2.22.m22.2.2.7.cmml">ξ</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.2.p2.22.m22.2b"><apply id="S6.SS1.2.p2.22.m22.2.2.cmml" xref="S6.SS1.2.p2.22.m22.2.2"><and id="S6.SS1.2.p2.22.m22.2.2a.cmml" xref="S6.SS1.2.p2.22.m22.2.2"></and><apply id="S6.SS1.2.p2.22.m22.2.2b.cmml" xref="S6.SS1.2.p2.22.m22.2.2"><lt id="S6.SS1.2.p2.22.m22.2.2.5.cmml" xref="S6.SS1.2.p2.22.m22.2.2.5"></lt><ci id="S6.SS1.2.p2.22.m22.2.2.4.cmml" xref="S6.SS1.2.p2.22.m22.2.2.4">𝛾</ci><apply id="S6.SS1.2.p2.22.m22.2.2.2.cmml" xref="S6.SS1.2.p2.22.m22.2.2.2"><times id="S6.SS1.2.p2.22.m22.2.2.2.3.cmml" xref="S6.SS1.2.p2.22.m22.2.2.2.3"></times><apply id="S6.SS1.2.p2.22.m22.2.2.2.4.cmml" xref="S6.SS1.2.p2.22.m22.2.2.2.4"><csymbol cd="ambiguous" id="S6.SS1.2.p2.22.m22.2.2.2.4.1.cmml" xref="S6.SS1.2.p2.22.m22.2.2.2.4">subscript</csymbol><ci id="S6.SS1.2.p2.22.m22.2.2.2.4.2.cmml" xref="S6.SS1.2.p2.22.m22.2.2.2.4.2">Δ</ci><ci id="S6.SS1.2.p2.22.m22.2.2.2.4.3.cmml" xref="S6.SS1.2.p2.22.m22.2.2.2.4.3">𝐴</ci></apply><interval closure="open" id="S6.SS1.2.p2.22.m22.2.2.2.2.3.cmml" xref="S6.SS1.2.p2.22.m22.2.2.2.2.2"><apply id="S6.SS1.2.p2.22.m22.1.1.1.1.1.1.cmml" xref="S6.SS1.2.p2.22.m22.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS1.2.p2.22.m22.1.1.1.1.1.1.1.cmml" xref="S6.SS1.2.p2.22.m22.1.1.1.1.1.1">subscript</csymbol><ci id="S6.SS1.2.p2.22.m22.1.1.1.1.1.1.2.cmml" xref="S6.SS1.2.p2.22.m22.1.1.1.1.1.1.2">𝜉</ci><ci id="S6.SS1.2.p2.22.m22.1.1.1.1.1.1.3.cmml" xref="S6.SS1.2.p2.22.m22.1.1.1.1.1.1.3">𝑖</ci></apply><apply id="S6.SS1.2.p2.22.m22.2.2.2.2.2.2.cmml" xref="S6.SS1.2.p2.22.m22.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS1.2.p2.22.m22.2.2.2.2.2.2.1.cmml" xref="S6.SS1.2.p2.22.m22.2.2.2.2.2.2">subscript</csymbol><ci id="S6.SS1.2.p2.22.m22.2.2.2.2.2.2.2.cmml" xref="S6.SS1.2.p2.22.m22.2.2.2.2.2.2.2">𝜂</ci><ci id="S6.SS1.2.p2.22.m22.2.2.2.2.2.2.3.cmml" xref="S6.SS1.2.p2.22.m22.2.2.2.2.2.2.3">𝑖</ci></apply></interval></apply></apply><apply id="S6.SS1.2.p2.22.m22.2.2c.cmml" xref="S6.SS1.2.p2.22.m22.2.2"><lt id="S6.SS1.2.p2.22.m22.2.2.6.cmml" xref="S6.SS1.2.p2.22.m22.2.2.6"></lt><share href="https://arxiv.org/html/2503.13728v1#S6.SS1.2.p2.22.m22.2.2.2.cmml" id="S6.SS1.2.p2.22.m22.2.2d.cmml" xref="S6.SS1.2.p2.22.m22.2.2"></share><ci id="S6.SS1.2.p2.22.m22.2.2.7.cmml" xref="S6.SS1.2.p2.22.m22.2.2.7">𝜉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.2.p2.22.m22.2c">\gamma<\Delta_{A}(\xi_{i},\eta_{i})<\xi</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.2.p2.22.m22.2d">italic_γ < roman_Δ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ξ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_η start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) < italic_ξ</annotation></semantics></math>. Repeating this for all <math alttext="i" class="ltx_Math" display="inline" id="S6.SS1.2.p2.23.m23.1"><semantics id="S6.SS1.2.p2.23.m23.1a"><mi id="S6.SS1.2.p2.23.m23.1.1" xref="S6.SS1.2.p2.23.m23.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S6.SS1.2.p2.23.m23.1b"><ci id="S6.SS1.2.p2.23.m23.1.1.cmml" xref="S6.SS1.2.p2.23.m23.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.2.p2.23.m23.1c">i</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.2.p2.23.m23.1d">italic_i</annotation></semantics></math> we may assume that <math alttext="\Gamma" class="ltx_Math" display="inline" id="S6.SS1.2.p2.24.m24.1"><semantics id="S6.SS1.2.p2.24.m24.1a"><mi id="S6.SS1.2.p2.24.m24.1.1" mathvariant="normal" xref="S6.SS1.2.p2.24.m24.1.1.cmml">Γ</mi><annotation-xml encoding="MathML-Content" id="S6.SS1.2.p2.24.m24.1b"><ci id="S6.SS1.2.p2.24.m24.1.1.cmml" xref="S6.SS1.2.p2.24.m24.1.1">Γ</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.2.p2.24.m24.1c">\Gamma</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.2.p2.24.m24.1d">roman_Γ</annotation></semantics></math> satisfies (3). ∎</p> </div> </div> </section> <section class="ltx_subsection" id="S6.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">6.2. </span>Forcing epimorphisms</h3> <div class="ltx_para" id="S6.SS2.p1"> <p class="ltx_p" id="S6.SS2.p1.16">Fix <math alttext="A" class="ltx_Math" display="inline" id="S6.SS2.p1.1.m1.1"><semantics id="S6.SS2.p1.1.m1.1a"><mi id="S6.SS2.p1.1.m1.1.1" xref="S6.SS2.p1.1.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.1.m1.1b"><ci id="S6.SS2.p1.1.m1.1.1.cmml" xref="S6.SS2.p1.1.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.1.m1.1c">A</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.1.m1.1d">italic_A</annotation></semantics></math>, <math alttext="X" class="ltx_Math" display="inline" id="S6.SS2.p1.2.m2.1"><semantics id="S6.SS2.p1.2.m2.1a"><mi id="S6.SS2.p1.2.m2.1.1" xref="S6.SS2.p1.2.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.2.m2.1b"><ci id="S6.SS2.p1.2.m2.1.1.cmml" xref="S6.SS2.p1.2.m2.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.2.m2.1c">X</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.2.m2.1d">italic_X</annotation></semantics></math>, <math alttext="D^{A}" class="ltx_Math" display="inline" id="S6.SS2.p1.3.m3.1"><semantics id="S6.SS2.p1.3.m3.1a"><msup id="S6.SS2.p1.3.m3.1.1" xref="S6.SS2.p1.3.m3.1.1.cmml"><mi id="S6.SS2.p1.3.m3.1.1.2" xref="S6.SS2.p1.3.m3.1.1.2.cmml">D</mi><mi id="S6.SS2.p1.3.m3.1.1.3" xref="S6.SS2.p1.3.m3.1.1.3.cmml">A</mi></msup><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.3.m3.1b"><apply id="S6.SS2.p1.3.m3.1.1.cmml" xref="S6.SS2.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S6.SS2.p1.3.m3.1.1.1.cmml" xref="S6.SS2.p1.3.m3.1.1">superscript</csymbol><ci id="S6.SS2.p1.3.m3.1.1.2.cmml" xref="S6.SS2.p1.3.m3.1.1.2">𝐷</ci><ci id="S6.SS2.p1.3.m3.1.1.3.cmml" xref="S6.SS2.p1.3.m3.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.3.m3.1c">D^{A}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.3.m3.1d">italic_D start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="D^{X}" class="ltx_Math" display="inline" id="S6.SS2.p1.4.m4.1"><semantics id="S6.SS2.p1.4.m4.1a"><msup id="S6.SS2.p1.4.m4.1.1" xref="S6.SS2.p1.4.m4.1.1.cmml"><mi id="S6.SS2.p1.4.m4.1.1.2" xref="S6.SS2.p1.4.m4.1.1.2.cmml">D</mi><mi id="S6.SS2.p1.4.m4.1.1.3" xref="S6.SS2.p1.4.m4.1.1.3.cmml">X</mi></msup><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.4.m4.1b"><apply id="S6.SS2.p1.4.m4.1.1.cmml" xref="S6.SS2.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S6.SS2.p1.4.m4.1.1.1.cmml" xref="S6.SS2.p1.4.m4.1.1">superscript</csymbol><ci id="S6.SS2.p1.4.m4.1.1.2.cmml" xref="S6.SS2.p1.4.m4.1.1.2">𝐷</ci><ci id="S6.SS2.p1.4.m4.1.1.3.cmml" xref="S6.SS2.p1.4.m4.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.4.m4.1c">D^{X}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.4.m4.1d">italic_D start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT</annotation></semantics></math> as in the statement of <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.Thmtheorem2" title="Theorem 6.2. ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">6.2</span></a>, we may assume that <math alttext="A" class="ltx_Math" display="inline" id="S6.SS2.p1.5.m5.1"><semantics id="S6.SS2.p1.5.m5.1a"><mi id="S6.SS2.p1.5.m5.1.1" xref="S6.SS2.p1.5.m5.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.5.m5.1b"><ci id="S6.SS2.p1.5.m5.1.1.cmml" xref="S6.SS2.p1.5.m5.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.5.m5.1c">A</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.5.m5.1d">italic_A</annotation></semantics></math> and <math alttext="X" class="ltx_Math" display="inline" id="S6.SS2.p1.6.m6.1"><semantics id="S6.SS2.p1.6.m6.1a"><mi id="S6.SS2.p1.6.m6.1.1" xref="S6.SS2.p1.6.m6.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.6.m6.1b"><ci id="S6.SS2.p1.6.m6.1.1.cmml" xref="S6.SS2.p1.6.m6.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.6.m6.1c">X</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.6.m6.1d">italic_X</annotation></semantics></math> are <math alttext="\omega_{1}" class="ltx_Math" display="inline" id="S6.SS2.p1.7.m7.1"><semantics id="S6.SS2.p1.7.m7.1a"><msub id="S6.SS2.p1.7.m7.1.1" xref="S6.SS2.p1.7.m7.1.1.cmml"><mi id="S6.SS2.p1.7.m7.1.1.2" xref="S6.SS2.p1.7.m7.1.1.2.cmml">ω</mi><mn id="S6.SS2.p1.7.m7.1.1.3" xref="S6.SS2.p1.7.m7.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.7.m7.1b"><apply id="S6.SS2.p1.7.m7.1.1.cmml" xref="S6.SS2.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S6.SS2.p1.7.m7.1.1.1.cmml" xref="S6.SS2.p1.7.m7.1.1">subscript</csymbol><ci id="S6.SS2.p1.7.m7.1.1.2.cmml" xref="S6.SS2.p1.7.m7.1.1.2">𝜔</ci><cn id="S6.SS2.p1.7.m7.1.1.3.cmml" type="integer" xref="S6.SS2.p1.7.m7.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.7.m7.1c">\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.7.m7.1d">italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> as sets. We will define a variant on Moore’s forcing <math alttext="Q_{E}" class="ltx_Math" display="inline" id="S6.SS2.p1.8.m8.1"><semantics id="S6.SS2.p1.8.m8.1a"><msub id="S6.SS2.p1.8.m8.1.1" xref="S6.SS2.p1.8.m8.1.1.cmml"><mi id="S6.SS2.p1.8.m8.1.1.2" xref="S6.SS2.p1.8.m8.1.1.2.cmml">Q</mi><mi id="S6.SS2.p1.8.m8.1.1.3" xref="S6.SS2.p1.8.m8.1.1.3.cmml">E</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.8.m8.1b"><apply id="S6.SS2.p1.8.m8.1.1.cmml" xref="S6.SS2.p1.8.m8.1.1"><csymbol cd="ambiguous" id="S6.SS2.p1.8.m8.1.1.1.cmml" xref="S6.SS2.p1.8.m8.1.1">subscript</csymbol><ci id="S6.SS2.p1.8.m8.1.1.2.cmml" xref="S6.SS2.p1.8.m8.1.1.2">𝑄</ci><ci id="S6.SS2.p1.8.m8.1.1.3.cmml" xref="S6.SS2.p1.8.m8.1.1.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.8.m8.1c">Q_{E}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.8.m8.1d">italic_Q start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT</annotation></semantics></math> to introduce an epimorphism from <math alttext="A" class="ltx_Math" display="inline" id="S6.SS2.p1.9.m9.1"><semantics id="S6.SS2.p1.9.m9.1a"><mi id="S6.SS2.p1.9.m9.1.1" xref="S6.SS2.p1.9.m9.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.9.m9.1b"><ci id="S6.SS2.p1.9.m9.1.1.cmml" xref="S6.SS2.p1.9.m9.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.9.m9.1c">A</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.9.m9.1d">italic_A</annotation></semantics></math> onto <math alttext="X" class="ltx_Math" display="inline" id="S6.SS2.p1.10.m10.1"><semantics id="S6.SS2.p1.10.m10.1a"><mi id="S6.SS2.p1.10.m10.1.1" xref="S6.SS2.p1.10.m10.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.10.m10.1b"><ci id="S6.SS2.p1.10.m10.1.1.cmml" xref="S6.SS2.p1.10.m10.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.10.m10.1c">X</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.10.m10.1d">italic_X</annotation></semantics></math>. As in Moore’s forcing, the fact that <math alttext="A" class="ltx_Math" display="inline" id="S6.SS2.p1.11.m11.1"><semantics id="S6.SS2.p1.11.m11.1a"><mi id="S6.SS2.p1.11.m11.1.1" xref="S6.SS2.p1.11.m11.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.11.m11.1b"><ci id="S6.SS2.p1.11.m11.1.1.cmml" xref="S6.SS2.p1.11.m11.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.11.m11.1c">A</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.11.m11.1d">italic_A</annotation></semantics></math> and <math alttext="X" class="ltx_Math" display="inline" id="S6.SS2.p1.12.m12.1"><semantics id="S6.SS2.p1.12.m12.1a"><mi id="S6.SS2.p1.12.m12.1.1" xref="S6.SS2.p1.12.m12.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.12.m12.1b"><ci id="S6.SS2.p1.12.m12.1.1.cmml" xref="S6.SS2.p1.12.m12.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.12.m12.1c">X</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.12.m12.1d">italic_X</annotation></semantics></math> are <math alttext="\preceq" class="ltx_Math" display="inline" id="S6.SS2.p1.13.m13.1"><semantics id="S6.SS2.p1.13.m13.1a"><mo id="S6.SS2.p1.13.m13.1.1" xref="S6.SS2.p1.13.m13.1.1.cmml">⪯</mo><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.13.m13.1b"><csymbol cd="latexml" id="S6.SS2.p1.13.m13.1.1.cmml" xref="S6.SS2.p1.13.m13.1.1">precedes-or-equals</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.13.m13.1c">\preceq</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.13.m13.1d">⪯</annotation></semantics></math>-comparable Countryman lines will be used in obtaining the ccc, and the <math alttext="\aleph_{1}" class="ltx_Math" display="inline" id="S6.SS2.p1.14.m14.1"><semantics id="S6.SS2.p1.14.m14.1a"><msub id="S6.SS2.p1.14.m14.1.1" xref="S6.SS2.p1.14.m14.1.1.cmml"><mi id="S6.SS2.p1.14.m14.1.1.2" mathvariant="normal" xref="S6.SS2.p1.14.m14.1.1.2.cmml">ℵ</mi><mn id="S6.SS2.p1.14.m14.1.1.3" xref="S6.SS2.p1.14.m14.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.14.m14.1b"><apply id="S6.SS2.p1.14.m14.1.1.cmml" xref="S6.SS2.p1.14.m14.1.1"><csymbol cd="ambiguous" id="S6.SS2.p1.14.m14.1.1.1.cmml" xref="S6.SS2.p1.14.m14.1.1">subscript</csymbol><ci id="S6.SS2.p1.14.m14.1.1.2.cmml" xref="S6.SS2.p1.14.m14.1.1.2">ℵ</ci><cn id="S6.SS2.p1.14.m14.1.1.3.cmml" type="integer" xref="S6.SS2.p1.14.m14.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.14.m14.1c">\aleph_{1}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.14.m14.1d">roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-density plus the hypothesis on <math alttext="\mathscr{L}" class="ltx_Math" display="inline" id="S6.SS2.p1.15.m15.1"><semantics id="S6.SS2.p1.15.m15.1a"><mi class="ltx_font_mathscript" id="S6.SS2.p1.15.m15.1.1" xref="S6.SS2.p1.15.m15.1.1.cmml">ℒ</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.15.m15.1b"><ci id="S6.SS2.p1.15.m15.1.1.cmml" xref="S6.SS2.p1.15.m15.1.1">ℒ</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.15.m15.1c">\mathscr{L}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.15.m15.1d">script_L</annotation></semantics></math> and <math alttext="\mathscr{R}" class="ltx_Math" display="inline" id="S6.SS2.p1.16.m16.1"><semantics id="S6.SS2.p1.16.m16.1a"><mi class="ltx_font_mathscript" id="S6.SS2.p1.16.m16.1.1" xref="S6.SS2.p1.16.m16.1.1.cmml">ℛ</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.16.m16.1b"><ci id="S6.SS2.p1.16.m16.1.1.cmml" xref="S6.SS2.p1.16.m16.1.1">ℛ</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.16.m16.1c">\mathscr{R}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.16.m16.1d">script_R</annotation></semantics></math> in obtaining enough dense sets.</p> </div> <div class="ltx_para" id="S6.SS2.p2"> <p class="ltx_p" id="S6.SS2.p2.26">Let <math alttext="\bar{A}=\{(a,b)\in A^{2}:a<_{A}b\}" class="ltx_Math" display="inline" id="S6.SS2.p2.1.m1.4"><semantics id="S6.SS2.p2.1.m1.4a"><mrow id="S6.SS2.p2.1.m1.4.4" xref="S6.SS2.p2.1.m1.4.4.cmml"><mover accent="true" id="S6.SS2.p2.1.m1.4.4.4" xref="S6.SS2.p2.1.m1.4.4.4.cmml"><mi id="S6.SS2.p2.1.m1.4.4.4.2" xref="S6.SS2.p2.1.m1.4.4.4.2.cmml">A</mi><mo id="S6.SS2.p2.1.m1.4.4.4.1" xref="S6.SS2.p2.1.m1.4.4.4.1.cmml">¯</mo></mover><mo id="S6.SS2.p2.1.m1.4.4.3" xref="S6.SS2.p2.1.m1.4.4.3.cmml">=</mo><mrow id="S6.SS2.p2.1.m1.4.4.2.2" xref="S6.SS2.p2.1.m1.4.4.2.3.cmml"><mo id="S6.SS2.p2.1.m1.4.4.2.2.3" stretchy="false" xref="S6.SS2.p2.1.m1.4.4.2.3.1.cmml">{</mo><mrow id="S6.SS2.p2.1.m1.3.3.1.1.1" xref="S6.SS2.p2.1.m1.3.3.1.1.1.cmml"><mrow id="S6.SS2.p2.1.m1.3.3.1.1.1.2.2" xref="S6.SS2.p2.1.m1.3.3.1.1.1.2.1.cmml"><mo id="S6.SS2.p2.1.m1.3.3.1.1.1.2.2.1" stretchy="false" xref="S6.SS2.p2.1.m1.3.3.1.1.1.2.1.cmml">(</mo><mi id="S6.SS2.p2.1.m1.1.1" xref="S6.SS2.p2.1.m1.1.1.cmml">a</mi><mo id="S6.SS2.p2.1.m1.3.3.1.1.1.2.2.2" xref="S6.SS2.p2.1.m1.3.3.1.1.1.2.1.cmml">,</mo><mi id="S6.SS2.p2.1.m1.2.2" xref="S6.SS2.p2.1.m1.2.2.cmml">b</mi><mo id="S6.SS2.p2.1.m1.3.3.1.1.1.2.2.3" stretchy="false" xref="S6.SS2.p2.1.m1.3.3.1.1.1.2.1.cmml">)</mo></mrow><mo id="S6.SS2.p2.1.m1.3.3.1.1.1.1" xref="S6.SS2.p2.1.m1.3.3.1.1.1.1.cmml">∈</mo><msup id="S6.SS2.p2.1.m1.3.3.1.1.1.3" xref="S6.SS2.p2.1.m1.3.3.1.1.1.3.cmml"><mi id="S6.SS2.p2.1.m1.3.3.1.1.1.3.2" xref="S6.SS2.p2.1.m1.3.3.1.1.1.3.2.cmml">A</mi><mn id="S6.SS2.p2.1.m1.3.3.1.1.1.3.3" xref="S6.SS2.p2.1.m1.3.3.1.1.1.3.3.cmml">2</mn></msup></mrow><mo id="S6.SS2.p2.1.m1.4.4.2.2.4" lspace="0.278em" rspace="0.278em" xref="S6.SS2.p2.1.m1.4.4.2.3.1.cmml">:</mo><mrow id="S6.SS2.p2.1.m1.4.4.2.2.2" xref="S6.SS2.p2.1.m1.4.4.2.2.2.cmml"><mi id="S6.SS2.p2.1.m1.4.4.2.2.2.2" xref="S6.SS2.p2.1.m1.4.4.2.2.2.2.cmml">a</mi><msub id="S6.SS2.p2.1.m1.4.4.2.2.2.1" xref="S6.SS2.p2.1.m1.4.4.2.2.2.1.cmml"><mo id="S6.SS2.p2.1.m1.4.4.2.2.2.1.2" xref="S6.SS2.p2.1.m1.4.4.2.2.2.1.2.cmml"><</mo><mi id="S6.SS2.p2.1.m1.4.4.2.2.2.1.3" xref="S6.SS2.p2.1.m1.4.4.2.2.2.1.3.cmml">A</mi></msub><mi id="S6.SS2.p2.1.m1.4.4.2.2.2.3" xref="S6.SS2.p2.1.m1.4.4.2.2.2.3.cmml">b</mi></mrow><mo id="S6.SS2.p2.1.m1.4.4.2.2.5" stretchy="false" xref="S6.SS2.p2.1.m1.4.4.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.1.m1.4b"><apply id="S6.SS2.p2.1.m1.4.4.cmml" xref="S6.SS2.p2.1.m1.4.4"><eq id="S6.SS2.p2.1.m1.4.4.3.cmml" xref="S6.SS2.p2.1.m1.4.4.3"></eq><apply id="S6.SS2.p2.1.m1.4.4.4.cmml" xref="S6.SS2.p2.1.m1.4.4.4"><ci id="S6.SS2.p2.1.m1.4.4.4.1.cmml" xref="S6.SS2.p2.1.m1.4.4.4.1">¯</ci><ci id="S6.SS2.p2.1.m1.4.4.4.2.cmml" xref="S6.SS2.p2.1.m1.4.4.4.2">𝐴</ci></apply><apply id="S6.SS2.p2.1.m1.4.4.2.3.cmml" xref="S6.SS2.p2.1.m1.4.4.2.2"><csymbol cd="latexml" id="S6.SS2.p2.1.m1.4.4.2.3.1.cmml" xref="S6.SS2.p2.1.m1.4.4.2.2.3">conditional-set</csymbol><apply id="S6.SS2.p2.1.m1.3.3.1.1.1.cmml" xref="S6.SS2.p2.1.m1.3.3.1.1.1"><in id="S6.SS2.p2.1.m1.3.3.1.1.1.1.cmml" xref="S6.SS2.p2.1.m1.3.3.1.1.1.1"></in><interval closure="open" id="S6.SS2.p2.1.m1.3.3.1.1.1.2.1.cmml" xref="S6.SS2.p2.1.m1.3.3.1.1.1.2.2"><ci id="S6.SS2.p2.1.m1.1.1.cmml" xref="S6.SS2.p2.1.m1.1.1">𝑎</ci><ci id="S6.SS2.p2.1.m1.2.2.cmml" xref="S6.SS2.p2.1.m1.2.2">𝑏</ci></interval><apply id="S6.SS2.p2.1.m1.3.3.1.1.1.3.cmml" xref="S6.SS2.p2.1.m1.3.3.1.1.1.3"><csymbol cd="ambiguous" id="S6.SS2.p2.1.m1.3.3.1.1.1.3.1.cmml" xref="S6.SS2.p2.1.m1.3.3.1.1.1.3">superscript</csymbol><ci id="S6.SS2.p2.1.m1.3.3.1.1.1.3.2.cmml" xref="S6.SS2.p2.1.m1.3.3.1.1.1.3.2">𝐴</ci><cn id="S6.SS2.p2.1.m1.3.3.1.1.1.3.3.cmml" type="integer" xref="S6.SS2.p2.1.m1.3.3.1.1.1.3.3">2</cn></apply></apply><apply id="S6.SS2.p2.1.m1.4.4.2.2.2.cmml" xref="S6.SS2.p2.1.m1.4.4.2.2.2"><apply id="S6.SS2.p2.1.m1.4.4.2.2.2.1.cmml" xref="S6.SS2.p2.1.m1.4.4.2.2.2.1"><csymbol cd="ambiguous" id="S6.SS2.p2.1.m1.4.4.2.2.2.1.1.cmml" xref="S6.SS2.p2.1.m1.4.4.2.2.2.1">subscript</csymbol><lt id="S6.SS2.p2.1.m1.4.4.2.2.2.1.2.cmml" xref="S6.SS2.p2.1.m1.4.4.2.2.2.1.2"></lt><ci id="S6.SS2.p2.1.m1.4.4.2.2.2.1.3.cmml" xref="S6.SS2.p2.1.m1.4.4.2.2.2.1.3">𝐴</ci></apply><ci id="S6.SS2.p2.1.m1.4.4.2.2.2.2.cmml" xref="S6.SS2.p2.1.m1.4.4.2.2.2.2">𝑎</ci><ci id="S6.SS2.p2.1.m1.4.4.2.2.2.3.cmml" xref="S6.SS2.p2.1.m1.4.4.2.2.2.3">𝑏</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.1.m1.4c">\bar{A}=\{(a,b)\in A^{2}:a<_{A}b\}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.1.m1.4d">over¯ start_ARG italic_A end_ARG = { ( italic_a , italic_b ) ∈ italic_A start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT : italic_a < start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_b }</annotation></semantics></math> and for <math alttext="\bar{a}\in\bar{A}" class="ltx_Math" display="inline" id="S6.SS2.p2.2.m2.1"><semantics id="S6.SS2.p2.2.m2.1a"><mrow id="S6.SS2.p2.2.m2.1.1" xref="S6.SS2.p2.2.m2.1.1.cmml"><mover accent="true" id="S6.SS2.p2.2.m2.1.1.2" xref="S6.SS2.p2.2.m2.1.1.2.cmml"><mi id="S6.SS2.p2.2.m2.1.1.2.2" xref="S6.SS2.p2.2.m2.1.1.2.2.cmml">a</mi><mo id="S6.SS2.p2.2.m2.1.1.2.1" xref="S6.SS2.p2.2.m2.1.1.2.1.cmml">¯</mo></mover><mo id="S6.SS2.p2.2.m2.1.1.1" xref="S6.SS2.p2.2.m2.1.1.1.cmml">∈</mo><mover accent="true" id="S6.SS2.p2.2.m2.1.1.3" xref="S6.SS2.p2.2.m2.1.1.3.cmml"><mi id="S6.SS2.p2.2.m2.1.1.3.2" xref="S6.SS2.p2.2.m2.1.1.3.2.cmml">A</mi><mo id="S6.SS2.p2.2.m2.1.1.3.1" xref="S6.SS2.p2.2.m2.1.1.3.1.cmml">¯</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.2.m2.1b"><apply id="S6.SS2.p2.2.m2.1.1.cmml" xref="S6.SS2.p2.2.m2.1.1"><in id="S6.SS2.p2.2.m2.1.1.1.cmml" xref="S6.SS2.p2.2.m2.1.1.1"></in><apply id="S6.SS2.p2.2.m2.1.1.2.cmml" xref="S6.SS2.p2.2.m2.1.1.2"><ci id="S6.SS2.p2.2.m2.1.1.2.1.cmml" xref="S6.SS2.p2.2.m2.1.1.2.1">¯</ci><ci id="S6.SS2.p2.2.m2.1.1.2.2.cmml" xref="S6.SS2.p2.2.m2.1.1.2.2">𝑎</ci></apply><apply id="S6.SS2.p2.2.m2.1.1.3.cmml" xref="S6.SS2.p2.2.m2.1.1.3"><ci id="S6.SS2.p2.2.m2.1.1.3.1.cmml" xref="S6.SS2.p2.2.m2.1.1.3.1">¯</ci><ci id="S6.SS2.p2.2.m2.1.1.3.2.cmml" xref="S6.SS2.p2.2.m2.1.1.3.2">𝐴</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.2.m2.1c">\bar{a}\in\bar{A}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.2.m2.1d">over¯ start_ARG italic_a end_ARG ∈ over¯ start_ARG italic_A end_ARG</annotation></semantics></math> we write <math alttext="\bar{a}=(a_{l},a_{r})" class="ltx_Math" display="inline" id="S6.SS2.p2.3.m3.2"><semantics id="S6.SS2.p2.3.m3.2a"><mrow id="S6.SS2.p2.3.m3.2.2" xref="S6.SS2.p2.3.m3.2.2.cmml"><mover accent="true" id="S6.SS2.p2.3.m3.2.2.4" xref="S6.SS2.p2.3.m3.2.2.4.cmml"><mi id="S6.SS2.p2.3.m3.2.2.4.2" xref="S6.SS2.p2.3.m3.2.2.4.2.cmml">a</mi><mo id="S6.SS2.p2.3.m3.2.2.4.1" xref="S6.SS2.p2.3.m3.2.2.4.1.cmml">¯</mo></mover><mo id="S6.SS2.p2.3.m3.2.2.3" xref="S6.SS2.p2.3.m3.2.2.3.cmml">=</mo><mrow id="S6.SS2.p2.3.m3.2.2.2.2" xref="S6.SS2.p2.3.m3.2.2.2.3.cmml"><mo id="S6.SS2.p2.3.m3.2.2.2.2.3" stretchy="false" xref="S6.SS2.p2.3.m3.2.2.2.3.cmml">(</mo><msub id="S6.SS2.p2.3.m3.1.1.1.1.1" xref="S6.SS2.p2.3.m3.1.1.1.1.1.cmml"><mi id="S6.SS2.p2.3.m3.1.1.1.1.1.2" xref="S6.SS2.p2.3.m3.1.1.1.1.1.2.cmml">a</mi><mi id="S6.SS2.p2.3.m3.1.1.1.1.1.3" xref="S6.SS2.p2.3.m3.1.1.1.1.1.3.cmml">l</mi></msub><mo id="S6.SS2.p2.3.m3.2.2.2.2.4" xref="S6.SS2.p2.3.m3.2.2.2.3.cmml">,</mo><msub id="S6.SS2.p2.3.m3.2.2.2.2.2" xref="S6.SS2.p2.3.m3.2.2.2.2.2.cmml"><mi id="S6.SS2.p2.3.m3.2.2.2.2.2.2" xref="S6.SS2.p2.3.m3.2.2.2.2.2.2.cmml">a</mi><mi id="S6.SS2.p2.3.m3.2.2.2.2.2.3" xref="S6.SS2.p2.3.m3.2.2.2.2.2.3.cmml">r</mi></msub><mo id="S6.SS2.p2.3.m3.2.2.2.2.5" stretchy="false" xref="S6.SS2.p2.3.m3.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.3.m3.2b"><apply id="S6.SS2.p2.3.m3.2.2.cmml" xref="S6.SS2.p2.3.m3.2.2"><eq id="S6.SS2.p2.3.m3.2.2.3.cmml" xref="S6.SS2.p2.3.m3.2.2.3"></eq><apply id="S6.SS2.p2.3.m3.2.2.4.cmml" xref="S6.SS2.p2.3.m3.2.2.4"><ci id="S6.SS2.p2.3.m3.2.2.4.1.cmml" xref="S6.SS2.p2.3.m3.2.2.4.1">¯</ci><ci id="S6.SS2.p2.3.m3.2.2.4.2.cmml" xref="S6.SS2.p2.3.m3.2.2.4.2">𝑎</ci></apply><interval closure="open" id="S6.SS2.p2.3.m3.2.2.2.3.cmml" xref="S6.SS2.p2.3.m3.2.2.2.2"><apply id="S6.SS2.p2.3.m3.1.1.1.1.1.cmml" xref="S6.SS2.p2.3.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.p2.3.m3.1.1.1.1.1.1.cmml" xref="S6.SS2.p2.3.m3.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.p2.3.m3.1.1.1.1.1.2.cmml" xref="S6.SS2.p2.3.m3.1.1.1.1.1.2">𝑎</ci><ci id="S6.SS2.p2.3.m3.1.1.1.1.1.3.cmml" xref="S6.SS2.p2.3.m3.1.1.1.1.1.3">𝑙</ci></apply><apply id="S6.SS2.p2.3.m3.2.2.2.2.2.cmml" xref="S6.SS2.p2.3.m3.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.p2.3.m3.2.2.2.2.2.1.cmml" xref="S6.SS2.p2.3.m3.2.2.2.2.2">subscript</csymbol><ci id="S6.SS2.p2.3.m3.2.2.2.2.2.2.cmml" xref="S6.SS2.p2.3.m3.2.2.2.2.2.2">𝑎</ci><ci id="S6.SS2.p2.3.m3.2.2.2.2.2.3.cmml" xref="S6.SS2.p2.3.m3.2.2.2.2.2.3">𝑟</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.3.m3.2c">\bar{a}=(a_{l},a_{r})</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.3.m3.2d">over¯ start_ARG italic_a end_ARG = ( italic_a start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT )</annotation></semantics></math> and define <math alttext="a_{m}:=\Delta_{A}(a_{l},a_{r})" class="ltx_Math" display="inline" id="S6.SS2.p2.4.m4.2"><semantics id="S6.SS2.p2.4.m4.2a"><mrow id="S6.SS2.p2.4.m4.2.2" xref="S6.SS2.p2.4.m4.2.2.cmml"><msub id="S6.SS2.p2.4.m4.2.2.4" xref="S6.SS2.p2.4.m4.2.2.4.cmml"><mi id="S6.SS2.p2.4.m4.2.2.4.2" xref="S6.SS2.p2.4.m4.2.2.4.2.cmml">a</mi><mi id="S6.SS2.p2.4.m4.2.2.4.3" xref="S6.SS2.p2.4.m4.2.2.4.3.cmml">m</mi></msub><mo id="S6.SS2.p2.4.m4.2.2.3" lspace="0.278em" rspace="0.278em" xref="S6.SS2.p2.4.m4.2.2.3.cmml">:=</mo><mrow id="S6.SS2.p2.4.m4.2.2.2" xref="S6.SS2.p2.4.m4.2.2.2.cmml"><msub id="S6.SS2.p2.4.m4.2.2.2.4" xref="S6.SS2.p2.4.m4.2.2.2.4.cmml"><mi id="S6.SS2.p2.4.m4.2.2.2.4.2" mathvariant="normal" xref="S6.SS2.p2.4.m4.2.2.2.4.2.cmml">Δ</mi><mi id="S6.SS2.p2.4.m4.2.2.2.4.3" xref="S6.SS2.p2.4.m4.2.2.2.4.3.cmml">A</mi></msub><mo id="S6.SS2.p2.4.m4.2.2.2.3" xref="S6.SS2.p2.4.m4.2.2.2.3.cmml">⁢</mo><mrow id="S6.SS2.p2.4.m4.2.2.2.2.2" xref="S6.SS2.p2.4.m4.2.2.2.2.3.cmml"><mo id="S6.SS2.p2.4.m4.2.2.2.2.2.3" stretchy="false" xref="S6.SS2.p2.4.m4.2.2.2.2.3.cmml">(</mo><msub id="S6.SS2.p2.4.m4.1.1.1.1.1.1" xref="S6.SS2.p2.4.m4.1.1.1.1.1.1.cmml"><mi id="S6.SS2.p2.4.m4.1.1.1.1.1.1.2" xref="S6.SS2.p2.4.m4.1.1.1.1.1.1.2.cmml">a</mi><mi id="S6.SS2.p2.4.m4.1.1.1.1.1.1.3" xref="S6.SS2.p2.4.m4.1.1.1.1.1.1.3.cmml">l</mi></msub><mo id="S6.SS2.p2.4.m4.2.2.2.2.2.4" xref="S6.SS2.p2.4.m4.2.2.2.2.3.cmml">,</mo><msub id="S6.SS2.p2.4.m4.2.2.2.2.2.2" xref="S6.SS2.p2.4.m4.2.2.2.2.2.2.cmml"><mi id="S6.SS2.p2.4.m4.2.2.2.2.2.2.2" xref="S6.SS2.p2.4.m4.2.2.2.2.2.2.2.cmml">a</mi><mi id="S6.SS2.p2.4.m4.2.2.2.2.2.2.3" xref="S6.SS2.p2.4.m4.2.2.2.2.2.2.3.cmml">r</mi></msub><mo id="S6.SS2.p2.4.m4.2.2.2.2.2.5" stretchy="false" xref="S6.SS2.p2.4.m4.2.2.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.4.m4.2b"><apply id="S6.SS2.p2.4.m4.2.2.cmml" xref="S6.SS2.p2.4.m4.2.2"><csymbol cd="latexml" id="S6.SS2.p2.4.m4.2.2.3.cmml" xref="S6.SS2.p2.4.m4.2.2.3">assign</csymbol><apply id="S6.SS2.p2.4.m4.2.2.4.cmml" xref="S6.SS2.p2.4.m4.2.2.4"><csymbol cd="ambiguous" id="S6.SS2.p2.4.m4.2.2.4.1.cmml" xref="S6.SS2.p2.4.m4.2.2.4">subscript</csymbol><ci id="S6.SS2.p2.4.m4.2.2.4.2.cmml" xref="S6.SS2.p2.4.m4.2.2.4.2">𝑎</ci><ci id="S6.SS2.p2.4.m4.2.2.4.3.cmml" xref="S6.SS2.p2.4.m4.2.2.4.3">𝑚</ci></apply><apply id="S6.SS2.p2.4.m4.2.2.2.cmml" xref="S6.SS2.p2.4.m4.2.2.2"><times id="S6.SS2.p2.4.m4.2.2.2.3.cmml" xref="S6.SS2.p2.4.m4.2.2.2.3"></times><apply id="S6.SS2.p2.4.m4.2.2.2.4.cmml" xref="S6.SS2.p2.4.m4.2.2.2.4"><csymbol cd="ambiguous" id="S6.SS2.p2.4.m4.2.2.2.4.1.cmml" xref="S6.SS2.p2.4.m4.2.2.2.4">subscript</csymbol><ci id="S6.SS2.p2.4.m4.2.2.2.4.2.cmml" xref="S6.SS2.p2.4.m4.2.2.2.4.2">Δ</ci><ci id="S6.SS2.p2.4.m4.2.2.2.4.3.cmml" xref="S6.SS2.p2.4.m4.2.2.2.4.3">𝐴</ci></apply><interval closure="open" id="S6.SS2.p2.4.m4.2.2.2.2.3.cmml" xref="S6.SS2.p2.4.m4.2.2.2.2.2"><apply id="S6.SS2.p2.4.m4.1.1.1.1.1.1.cmml" xref="S6.SS2.p2.4.m4.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.p2.4.m4.1.1.1.1.1.1.1.cmml" xref="S6.SS2.p2.4.m4.1.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.p2.4.m4.1.1.1.1.1.1.2.cmml" xref="S6.SS2.p2.4.m4.1.1.1.1.1.1.2">𝑎</ci><ci id="S6.SS2.p2.4.m4.1.1.1.1.1.1.3.cmml" xref="S6.SS2.p2.4.m4.1.1.1.1.1.1.3">𝑙</ci></apply><apply id="S6.SS2.p2.4.m4.2.2.2.2.2.2.cmml" xref="S6.SS2.p2.4.m4.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.p2.4.m4.2.2.2.2.2.2.1.cmml" xref="S6.SS2.p2.4.m4.2.2.2.2.2.2">subscript</csymbol><ci id="S6.SS2.p2.4.m4.2.2.2.2.2.2.2.cmml" xref="S6.SS2.p2.4.m4.2.2.2.2.2.2.2">𝑎</ci><ci id="S6.SS2.p2.4.m4.2.2.2.2.2.2.3.cmml" xref="S6.SS2.p2.4.m4.2.2.2.2.2.2.3">𝑟</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.4.m4.2c">a_{m}:=\Delta_{A}(a_{l},a_{r})</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.4.m4.2d">italic_a start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT := roman_Δ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT )</annotation></semantics></math> so that <math alttext="a_{l}\leq_{A}a_{m}\leq_{A}a_{r}" class="ltx_Math" display="inline" id="S6.SS2.p2.5.m5.1"><semantics id="S6.SS2.p2.5.m5.1a"><mrow id="S6.SS2.p2.5.m5.1.1" xref="S6.SS2.p2.5.m5.1.1.cmml"><msub id="S6.SS2.p2.5.m5.1.1.2" xref="S6.SS2.p2.5.m5.1.1.2.cmml"><mi id="S6.SS2.p2.5.m5.1.1.2.2" xref="S6.SS2.p2.5.m5.1.1.2.2.cmml">a</mi><mi id="S6.SS2.p2.5.m5.1.1.2.3" xref="S6.SS2.p2.5.m5.1.1.2.3.cmml">l</mi></msub><msub id="S6.SS2.p2.5.m5.1.1.3" xref="S6.SS2.p2.5.m5.1.1.3.cmml"><mo id="S6.SS2.p2.5.m5.1.1.3.2" xref="S6.SS2.p2.5.m5.1.1.3.2.cmml">≤</mo><mi id="S6.SS2.p2.5.m5.1.1.3.3" xref="S6.SS2.p2.5.m5.1.1.3.3.cmml">A</mi></msub><msub id="S6.SS2.p2.5.m5.1.1.4" xref="S6.SS2.p2.5.m5.1.1.4.cmml"><mi id="S6.SS2.p2.5.m5.1.1.4.2" xref="S6.SS2.p2.5.m5.1.1.4.2.cmml">a</mi><mi id="S6.SS2.p2.5.m5.1.1.4.3" xref="S6.SS2.p2.5.m5.1.1.4.3.cmml">m</mi></msub><msub id="S6.SS2.p2.5.m5.1.1.5" xref="S6.SS2.p2.5.m5.1.1.5.cmml"><mo id="S6.SS2.p2.5.m5.1.1.5.2" xref="S6.SS2.p2.5.m5.1.1.5.2.cmml">≤</mo><mi id="S6.SS2.p2.5.m5.1.1.5.3" xref="S6.SS2.p2.5.m5.1.1.5.3.cmml">A</mi></msub><msub id="S6.SS2.p2.5.m5.1.1.6" xref="S6.SS2.p2.5.m5.1.1.6.cmml"><mi id="S6.SS2.p2.5.m5.1.1.6.2" xref="S6.SS2.p2.5.m5.1.1.6.2.cmml">a</mi><mi id="S6.SS2.p2.5.m5.1.1.6.3" xref="S6.SS2.p2.5.m5.1.1.6.3.cmml">r</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.5.m5.1b"><apply id="S6.SS2.p2.5.m5.1.1.cmml" xref="S6.SS2.p2.5.m5.1.1"><and id="S6.SS2.p2.5.m5.1.1a.cmml" xref="S6.SS2.p2.5.m5.1.1"></and><apply id="S6.SS2.p2.5.m5.1.1b.cmml" xref="S6.SS2.p2.5.m5.1.1"><apply id="S6.SS2.p2.5.m5.1.1.3.cmml" xref="S6.SS2.p2.5.m5.1.1.3"><csymbol cd="ambiguous" id="S6.SS2.p2.5.m5.1.1.3.1.cmml" xref="S6.SS2.p2.5.m5.1.1.3">subscript</csymbol><leq id="S6.SS2.p2.5.m5.1.1.3.2.cmml" xref="S6.SS2.p2.5.m5.1.1.3.2"></leq><ci id="S6.SS2.p2.5.m5.1.1.3.3.cmml" xref="S6.SS2.p2.5.m5.1.1.3.3">𝐴</ci></apply><apply id="S6.SS2.p2.5.m5.1.1.2.cmml" xref="S6.SS2.p2.5.m5.1.1.2"><csymbol cd="ambiguous" id="S6.SS2.p2.5.m5.1.1.2.1.cmml" xref="S6.SS2.p2.5.m5.1.1.2">subscript</csymbol><ci id="S6.SS2.p2.5.m5.1.1.2.2.cmml" xref="S6.SS2.p2.5.m5.1.1.2.2">𝑎</ci><ci id="S6.SS2.p2.5.m5.1.1.2.3.cmml" xref="S6.SS2.p2.5.m5.1.1.2.3">𝑙</ci></apply><apply id="S6.SS2.p2.5.m5.1.1.4.cmml" xref="S6.SS2.p2.5.m5.1.1.4"><csymbol cd="ambiguous" id="S6.SS2.p2.5.m5.1.1.4.1.cmml" xref="S6.SS2.p2.5.m5.1.1.4">subscript</csymbol><ci id="S6.SS2.p2.5.m5.1.1.4.2.cmml" xref="S6.SS2.p2.5.m5.1.1.4.2">𝑎</ci><ci id="S6.SS2.p2.5.m5.1.1.4.3.cmml" xref="S6.SS2.p2.5.m5.1.1.4.3">𝑚</ci></apply></apply><apply id="S6.SS2.p2.5.m5.1.1c.cmml" xref="S6.SS2.p2.5.m5.1.1"><apply id="S6.SS2.p2.5.m5.1.1.5.cmml" xref="S6.SS2.p2.5.m5.1.1.5"><csymbol cd="ambiguous" id="S6.SS2.p2.5.m5.1.1.5.1.cmml" xref="S6.SS2.p2.5.m5.1.1.5">subscript</csymbol><leq id="S6.SS2.p2.5.m5.1.1.5.2.cmml" xref="S6.SS2.p2.5.m5.1.1.5.2"></leq><ci id="S6.SS2.p2.5.m5.1.1.5.3.cmml" xref="S6.SS2.p2.5.m5.1.1.5.3">𝐴</ci></apply><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.p2.5.m5.1.1.4.cmml" id="S6.SS2.p2.5.m5.1.1d.cmml" xref="S6.SS2.p2.5.m5.1.1"></share><apply id="S6.SS2.p2.5.m5.1.1.6.cmml" xref="S6.SS2.p2.5.m5.1.1.6"><csymbol cd="ambiguous" id="S6.SS2.p2.5.m5.1.1.6.1.cmml" xref="S6.SS2.p2.5.m5.1.1.6">subscript</csymbol><ci id="S6.SS2.p2.5.m5.1.1.6.2.cmml" xref="S6.SS2.p2.5.m5.1.1.6.2">𝑎</ci><ci id="S6.SS2.p2.5.m5.1.1.6.3.cmml" xref="S6.SS2.p2.5.m5.1.1.6.3">𝑟</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.5.m5.1c">a_{l}\leq_{A}a_{m}\leq_{A}a_{r}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.5.m5.1d">italic_a start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ≤ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ≤ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT</annotation></semantics></math>. We let <math alttext="<_{b}" class="ltx_Math" display="inline" id="S6.SS2.p2.6.m6.1"><semantics id="S6.SS2.p2.6.m6.1a"><msub id="S6.SS2.p2.6.m6.1.1" xref="S6.SS2.p2.6.m6.1.1.cmml"><mo id="S6.SS2.p2.6.m6.1.1.2" xref="S6.SS2.p2.6.m6.1.1.2.cmml"><</mo><mi id="S6.SS2.p2.6.m6.1.1.3" xref="S6.SS2.p2.6.m6.1.1.3.cmml">b</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.6.m6.1b"><apply id="S6.SS2.p2.6.m6.1.1.cmml" xref="S6.SS2.p2.6.m6.1.1"><csymbol cd="ambiguous" id="S6.SS2.p2.6.m6.1.1.1.cmml" xref="S6.SS2.p2.6.m6.1.1">subscript</csymbol><lt id="S6.SS2.p2.6.m6.1.1.2.cmml" xref="S6.SS2.p2.6.m6.1.1.2"></lt><ci id="S6.SS2.p2.6.m6.1.1.3.cmml" xref="S6.SS2.p2.6.m6.1.1.3">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.6.m6.1c"><_{b}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.6.m6.1d">< start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT</annotation></semantics></math> be the partial order on <math alttext="\bar{A}" class="ltx_Math" display="inline" id="S6.SS2.p2.7.m7.1"><semantics id="S6.SS2.p2.7.m7.1a"><mover accent="true" id="S6.SS2.p2.7.m7.1.1" xref="S6.SS2.p2.7.m7.1.1.cmml"><mi id="S6.SS2.p2.7.m7.1.1.2" xref="S6.SS2.p2.7.m7.1.1.2.cmml">A</mi><mo id="S6.SS2.p2.7.m7.1.1.1" xref="S6.SS2.p2.7.m7.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.7.m7.1b"><apply id="S6.SS2.p2.7.m7.1.1.cmml" xref="S6.SS2.p2.7.m7.1.1"><ci id="S6.SS2.p2.7.m7.1.1.1.cmml" xref="S6.SS2.p2.7.m7.1.1.1">¯</ci><ci id="S6.SS2.p2.7.m7.1.1.2.cmml" xref="S6.SS2.p2.7.m7.1.1.2">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.7.m7.1c">\bar{A}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.7.m7.1d">over¯ start_ARG italic_A end_ARG</annotation></semantics></math> defined by letting <math alttext="\bar{a}<_{b}\bar{b}" class="ltx_Math" display="inline" id="S6.SS2.p2.8.m8.1"><semantics id="S6.SS2.p2.8.m8.1a"><mrow id="S6.SS2.p2.8.m8.1.1" xref="S6.SS2.p2.8.m8.1.1.cmml"><mover accent="true" id="S6.SS2.p2.8.m8.1.1.2" xref="S6.SS2.p2.8.m8.1.1.2.cmml"><mi id="S6.SS2.p2.8.m8.1.1.2.2" xref="S6.SS2.p2.8.m8.1.1.2.2.cmml">a</mi><mo id="S6.SS2.p2.8.m8.1.1.2.1" xref="S6.SS2.p2.8.m8.1.1.2.1.cmml">¯</mo></mover><msub id="S6.SS2.p2.8.m8.1.1.1" xref="S6.SS2.p2.8.m8.1.1.1.cmml"><mo id="S6.SS2.p2.8.m8.1.1.1.2" xref="S6.SS2.p2.8.m8.1.1.1.2.cmml"><</mo><mi id="S6.SS2.p2.8.m8.1.1.1.3" xref="S6.SS2.p2.8.m8.1.1.1.3.cmml">b</mi></msub><mover accent="true" id="S6.SS2.p2.8.m8.1.1.3" xref="S6.SS2.p2.8.m8.1.1.3.cmml"><mi id="S6.SS2.p2.8.m8.1.1.3.2" xref="S6.SS2.p2.8.m8.1.1.3.2.cmml">b</mi><mo id="S6.SS2.p2.8.m8.1.1.3.1" xref="S6.SS2.p2.8.m8.1.1.3.1.cmml">¯</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.8.m8.1b"><apply id="S6.SS2.p2.8.m8.1.1.cmml" xref="S6.SS2.p2.8.m8.1.1"><apply id="S6.SS2.p2.8.m8.1.1.1.cmml" xref="S6.SS2.p2.8.m8.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.p2.8.m8.1.1.1.1.cmml" xref="S6.SS2.p2.8.m8.1.1.1">subscript</csymbol><lt id="S6.SS2.p2.8.m8.1.1.1.2.cmml" xref="S6.SS2.p2.8.m8.1.1.1.2"></lt><ci id="S6.SS2.p2.8.m8.1.1.1.3.cmml" xref="S6.SS2.p2.8.m8.1.1.1.3">𝑏</ci></apply><apply id="S6.SS2.p2.8.m8.1.1.2.cmml" xref="S6.SS2.p2.8.m8.1.1.2"><ci id="S6.SS2.p2.8.m8.1.1.2.1.cmml" xref="S6.SS2.p2.8.m8.1.1.2.1">¯</ci><ci id="S6.SS2.p2.8.m8.1.1.2.2.cmml" xref="S6.SS2.p2.8.m8.1.1.2.2">𝑎</ci></apply><apply id="S6.SS2.p2.8.m8.1.1.3.cmml" xref="S6.SS2.p2.8.m8.1.1.3"><ci id="S6.SS2.p2.8.m8.1.1.3.1.cmml" xref="S6.SS2.p2.8.m8.1.1.3.1">¯</ci><ci id="S6.SS2.p2.8.m8.1.1.3.2.cmml" xref="S6.SS2.p2.8.m8.1.1.3.2">𝑏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.8.m8.1c">\bar{a}<_{b}\bar{b}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.8.m8.1d">over¯ start_ARG italic_a end_ARG < start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT over¯ start_ARG italic_b end_ARG</annotation></semantics></math> iff <math alttext="a_{r}<_{A}b_{l}" class="ltx_Math" display="inline" id="S6.SS2.p2.9.m9.1"><semantics id="S6.SS2.p2.9.m9.1a"><mrow id="S6.SS2.p2.9.m9.1.1" xref="S6.SS2.p2.9.m9.1.1.cmml"><msub id="S6.SS2.p2.9.m9.1.1.2" xref="S6.SS2.p2.9.m9.1.1.2.cmml"><mi id="S6.SS2.p2.9.m9.1.1.2.2" xref="S6.SS2.p2.9.m9.1.1.2.2.cmml">a</mi><mi id="S6.SS2.p2.9.m9.1.1.2.3" xref="S6.SS2.p2.9.m9.1.1.2.3.cmml">r</mi></msub><msub id="S6.SS2.p2.9.m9.1.1.1" xref="S6.SS2.p2.9.m9.1.1.1.cmml"><mo id="S6.SS2.p2.9.m9.1.1.1.2" xref="S6.SS2.p2.9.m9.1.1.1.2.cmml"><</mo><mi id="S6.SS2.p2.9.m9.1.1.1.3" xref="S6.SS2.p2.9.m9.1.1.1.3.cmml">A</mi></msub><msub id="S6.SS2.p2.9.m9.1.1.3" xref="S6.SS2.p2.9.m9.1.1.3.cmml"><mi id="S6.SS2.p2.9.m9.1.1.3.2" xref="S6.SS2.p2.9.m9.1.1.3.2.cmml">b</mi><mi id="S6.SS2.p2.9.m9.1.1.3.3" xref="S6.SS2.p2.9.m9.1.1.3.3.cmml">l</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.9.m9.1b"><apply id="S6.SS2.p2.9.m9.1.1.cmml" xref="S6.SS2.p2.9.m9.1.1"><apply id="S6.SS2.p2.9.m9.1.1.1.cmml" xref="S6.SS2.p2.9.m9.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.p2.9.m9.1.1.1.1.cmml" xref="S6.SS2.p2.9.m9.1.1.1">subscript</csymbol><lt id="S6.SS2.p2.9.m9.1.1.1.2.cmml" xref="S6.SS2.p2.9.m9.1.1.1.2"></lt><ci id="S6.SS2.p2.9.m9.1.1.1.3.cmml" xref="S6.SS2.p2.9.m9.1.1.1.3">𝐴</ci></apply><apply id="S6.SS2.p2.9.m9.1.1.2.cmml" xref="S6.SS2.p2.9.m9.1.1.2"><csymbol cd="ambiguous" id="S6.SS2.p2.9.m9.1.1.2.1.cmml" xref="S6.SS2.p2.9.m9.1.1.2">subscript</csymbol><ci id="S6.SS2.p2.9.m9.1.1.2.2.cmml" xref="S6.SS2.p2.9.m9.1.1.2.2">𝑎</ci><ci id="S6.SS2.p2.9.m9.1.1.2.3.cmml" xref="S6.SS2.p2.9.m9.1.1.2.3">𝑟</ci></apply><apply id="S6.SS2.p2.9.m9.1.1.3.cmml" xref="S6.SS2.p2.9.m9.1.1.3"><csymbol cd="ambiguous" id="S6.SS2.p2.9.m9.1.1.3.1.cmml" xref="S6.SS2.p2.9.m9.1.1.3">subscript</csymbol><ci id="S6.SS2.p2.9.m9.1.1.3.2.cmml" xref="S6.SS2.p2.9.m9.1.1.3.2">𝑏</ci><ci id="S6.SS2.p2.9.m9.1.1.3.3.cmml" xref="S6.SS2.p2.9.m9.1.1.3.3">𝑙</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.9.m9.1c">a_{r}<_{A}b_{l}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.9.m9.1d">italic_a start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT < start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT</annotation></semantics></math>, that is, if the interval <math alttext="{[a_{l},a_{r}]}_{A}" class="ltx_Math" display="inline" id="S6.SS2.p2.10.m10.2"><semantics id="S6.SS2.p2.10.m10.2a"><msub id="S6.SS2.p2.10.m10.2.2" xref="S6.SS2.p2.10.m10.2.2.cmml"><mrow id="S6.SS2.p2.10.m10.2.2.2.2" xref="S6.SS2.p2.10.m10.2.2.2.3.cmml"><mo id="S6.SS2.p2.10.m10.2.2.2.2.3" stretchy="false" xref="S6.SS2.p2.10.m10.2.2.2.3.cmml">[</mo><msub id="S6.SS2.p2.10.m10.1.1.1.1.1" xref="S6.SS2.p2.10.m10.1.1.1.1.1.cmml"><mi id="S6.SS2.p2.10.m10.1.1.1.1.1.2" xref="S6.SS2.p2.10.m10.1.1.1.1.1.2.cmml">a</mi><mi id="S6.SS2.p2.10.m10.1.1.1.1.1.3" xref="S6.SS2.p2.10.m10.1.1.1.1.1.3.cmml">l</mi></msub><mo id="S6.SS2.p2.10.m10.2.2.2.2.4" xref="S6.SS2.p2.10.m10.2.2.2.3.cmml">,</mo><msub id="S6.SS2.p2.10.m10.2.2.2.2.2" xref="S6.SS2.p2.10.m10.2.2.2.2.2.cmml"><mi id="S6.SS2.p2.10.m10.2.2.2.2.2.2" xref="S6.SS2.p2.10.m10.2.2.2.2.2.2.cmml">a</mi><mi id="S6.SS2.p2.10.m10.2.2.2.2.2.3" xref="S6.SS2.p2.10.m10.2.2.2.2.2.3.cmml">r</mi></msub><mo id="S6.SS2.p2.10.m10.2.2.2.2.5" stretchy="false" xref="S6.SS2.p2.10.m10.2.2.2.3.cmml">]</mo></mrow><mi id="S6.SS2.p2.10.m10.2.2.4" xref="S6.SS2.p2.10.m10.2.2.4.cmml">A</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.10.m10.2b"><apply id="S6.SS2.p2.10.m10.2.2.cmml" xref="S6.SS2.p2.10.m10.2.2"><csymbol cd="ambiguous" id="S6.SS2.p2.10.m10.2.2.3.cmml" xref="S6.SS2.p2.10.m10.2.2">subscript</csymbol><interval closure="closed" id="S6.SS2.p2.10.m10.2.2.2.3.cmml" xref="S6.SS2.p2.10.m10.2.2.2.2"><apply id="S6.SS2.p2.10.m10.1.1.1.1.1.cmml" xref="S6.SS2.p2.10.m10.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.p2.10.m10.1.1.1.1.1.1.cmml" xref="S6.SS2.p2.10.m10.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.p2.10.m10.1.1.1.1.1.2.cmml" xref="S6.SS2.p2.10.m10.1.1.1.1.1.2">𝑎</ci><ci id="S6.SS2.p2.10.m10.1.1.1.1.1.3.cmml" xref="S6.SS2.p2.10.m10.1.1.1.1.1.3">𝑙</ci></apply><apply id="S6.SS2.p2.10.m10.2.2.2.2.2.cmml" xref="S6.SS2.p2.10.m10.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.p2.10.m10.2.2.2.2.2.1.cmml" xref="S6.SS2.p2.10.m10.2.2.2.2.2">subscript</csymbol><ci id="S6.SS2.p2.10.m10.2.2.2.2.2.2.cmml" xref="S6.SS2.p2.10.m10.2.2.2.2.2.2">𝑎</ci><ci id="S6.SS2.p2.10.m10.2.2.2.2.2.3.cmml" xref="S6.SS2.p2.10.m10.2.2.2.2.2.3">𝑟</ci></apply></interval><ci id="S6.SS2.p2.10.m10.2.2.4.cmml" xref="S6.SS2.p2.10.m10.2.2.4">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.10.m10.2c">{[a_{l},a_{r}]}_{A}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.10.m10.2d">[ italic_a start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ] start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT</annotation></semantics></math> comes strictly before <math alttext="{[b_{l},b_{r}]}_{A}" class="ltx_Math" display="inline" id="S6.SS2.p2.11.m11.2"><semantics id="S6.SS2.p2.11.m11.2a"><msub id="S6.SS2.p2.11.m11.2.2" xref="S6.SS2.p2.11.m11.2.2.cmml"><mrow id="S6.SS2.p2.11.m11.2.2.2.2" xref="S6.SS2.p2.11.m11.2.2.2.3.cmml"><mo id="S6.SS2.p2.11.m11.2.2.2.2.3" stretchy="false" xref="S6.SS2.p2.11.m11.2.2.2.3.cmml">[</mo><msub id="S6.SS2.p2.11.m11.1.1.1.1.1" xref="S6.SS2.p2.11.m11.1.1.1.1.1.cmml"><mi id="S6.SS2.p2.11.m11.1.1.1.1.1.2" xref="S6.SS2.p2.11.m11.1.1.1.1.1.2.cmml">b</mi><mi id="S6.SS2.p2.11.m11.1.1.1.1.1.3" xref="S6.SS2.p2.11.m11.1.1.1.1.1.3.cmml">l</mi></msub><mo id="S6.SS2.p2.11.m11.2.2.2.2.4" xref="S6.SS2.p2.11.m11.2.2.2.3.cmml">,</mo><msub id="S6.SS2.p2.11.m11.2.2.2.2.2" xref="S6.SS2.p2.11.m11.2.2.2.2.2.cmml"><mi id="S6.SS2.p2.11.m11.2.2.2.2.2.2" xref="S6.SS2.p2.11.m11.2.2.2.2.2.2.cmml">b</mi><mi id="S6.SS2.p2.11.m11.2.2.2.2.2.3" xref="S6.SS2.p2.11.m11.2.2.2.2.2.3.cmml">r</mi></msub><mo id="S6.SS2.p2.11.m11.2.2.2.2.5" stretchy="false" xref="S6.SS2.p2.11.m11.2.2.2.3.cmml">]</mo></mrow><mi id="S6.SS2.p2.11.m11.2.2.4" xref="S6.SS2.p2.11.m11.2.2.4.cmml">A</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.11.m11.2b"><apply id="S6.SS2.p2.11.m11.2.2.cmml" xref="S6.SS2.p2.11.m11.2.2"><csymbol cd="ambiguous" id="S6.SS2.p2.11.m11.2.2.3.cmml" xref="S6.SS2.p2.11.m11.2.2">subscript</csymbol><interval closure="closed" id="S6.SS2.p2.11.m11.2.2.2.3.cmml" xref="S6.SS2.p2.11.m11.2.2.2.2"><apply id="S6.SS2.p2.11.m11.1.1.1.1.1.cmml" xref="S6.SS2.p2.11.m11.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.p2.11.m11.1.1.1.1.1.1.cmml" xref="S6.SS2.p2.11.m11.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.p2.11.m11.1.1.1.1.1.2.cmml" xref="S6.SS2.p2.11.m11.1.1.1.1.1.2">𝑏</ci><ci id="S6.SS2.p2.11.m11.1.1.1.1.1.3.cmml" xref="S6.SS2.p2.11.m11.1.1.1.1.1.3">𝑙</ci></apply><apply id="S6.SS2.p2.11.m11.2.2.2.2.2.cmml" xref="S6.SS2.p2.11.m11.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.p2.11.m11.2.2.2.2.2.1.cmml" xref="S6.SS2.p2.11.m11.2.2.2.2.2">subscript</csymbol><ci id="S6.SS2.p2.11.m11.2.2.2.2.2.2.cmml" xref="S6.SS2.p2.11.m11.2.2.2.2.2.2">𝑏</ci><ci id="S6.SS2.p2.11.m11.2.2.2.2.2.3.cmml" xref="S6.SS2.p2.11.m11.2.2.2.2.2.3">𝑟</ci></apply></interval><ci id="S6.SS2.p2.11.m11.2.2.4.cmml" xref="S6.SS2.p2.11.m11.2.2.4">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.11.m11.2c">{[b_{l},b_{r}]}_{A}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.11.m11.2d">[ italic_b start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ] start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT</annotation></semantics></math> in the block order. We always think of elements of <math alttext="\bar{A}" class="ltx_Math" display="inline" id="S6.SS2.p2.12.m12.1"><semantics id="S6.SS2.p2.12.m12.1a"><mover accent="true" id="S6.SS2.p2.12.m12.1.1" xref="S6.SS2.p2.12.m12.1.1.cmml"><mi id="S6.SS2.p2.12.m12.1.1.2" xref="S6.SS2.p2.12.m12.1.1.2.cmml">A</mi><mo id="S6.SS2.p2.12.m12.1.1.1" xref="S6.SS2.p2.12.m12.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.12.m12.1b"><apply id="S6.SS2.p2.12.m12.1.1.cmml" xref="S6.SS2.p2.12.m12.1.1"><ci id="S6.SS2.p2.12.m12.1.1.1.cmml" xref="S6.SS2.p2.12.m12.1.1.1">¯</ci><ci id="S6.SS2.p2.12.m12.1.1.2.cmml" xref="S6.SS2.p2.12.m12.1.1.2">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.12.m12.1c">\bar{A}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.12.m12.1d">over¯ start_ARG italic_A end_ARG</annotation></semantics></math> as coding closed intervals. As a set we let <math alttext="P:=P(A,X)" class="ltx_Math" display="inline" id="S6.SS2.p2.13.m13.2"><semantics id="S6.SS2.p2.13.m13.2a"><mrow id="S6.SS2.p2.13.m13.2.3" xref="S6.SS2.p2.13.m13.2.3.cmml"><mi id="S6.SS2.p2.13.m13.2.3.2" xref="S6.SS2.p2.13.m13.2.3.2.cmml">P</mi><mo id="S6.SS2.p2.13.m13.2.3.1" lspace="0.278em" rspace="0.278em" xref="S6.SS2.p2.13.m13.2.3.1.cmml">:=</mo><mrow id="S6.SS2.p2.13.m13.2.3.3" xref="S6.SS2.p2.13.m13.2.3.3.cmml"><mi id="S6.SS2.p2.13.m13.2.3.3.2" xref="S6.SS2.p2.13.m13.2.3.3.2.cmml">P</mi><mo id="S6.SS2.p2.13.m13.2.3.3.1" xref="S6.SS2.p2.13.m13.2.3.3.1.cmml">⁢</mo><mrow id="S6.SS2.p2.13.m13.2.3.3.3.2" xref="S6.SS2.p2.13.m13.2.3.3.3.1.cmml"><mo id="S6.SS2.p2.13.m13.2.3.3.3.2.1" stretchy="false" xref="S6.SS2.p2.13.m13.2.3.3.3.1.cmml">(</mo><mi id="S6.SS2.p2.13.m13.1.1" xref="S6.SS2.p2.13.m13.1.1.cmml">A</mi><mo id="S6.SS2.p2.13.m13.2.3.3.3.2.2" xref="S6.SS2.p2.13.m13.2.3.3.3.1.cmml">,</mo><mi id="S6.SS2.p2.13.m13.2.2" xref="S6.SS2.p2.13.m13.2.2.cmml">X</mi><mo id="S6.SS2.p2.13.m13.2.3.3.3.2.3" stretchy="false" xref="S6.SS2.p2.13.m13.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.13.m13.2b"><apply id="S6.SS2.p2.13.m13.2.3.cmml" xref="S6.SS2.p2.13.m13.2.3"><csymbol cd="latexml" id="S6.SS2.p2.13.m13.2.3.1.cmml" xref="S6.SS2.p2.13.m13.2.3.1">assign</csymbol><ci id="S6.SS2.p2.13.m13.2.3.2.cmml" xref="S6.SS2.p2.13.m13.2.3.2">𝑃</ci><apply id="S6.SS2.p2.13.m13.2.3.3.cmml" xref="S6.SS2.p2.13.m13.2.3.3"><times id="S6.SS2.p2.13.m13.2.3.3.1.cmml" xref="S6.SS2.p2.13.m13.2.3.3.1"></times><ci id="S6.SS2.p2.13.m13.2.3.3.2.cmml" xref="S6.SS2.p2.13.m13.2.3.3.2">𝑃</ci><interval closure="open" id="S6.SS2.p2.13.m13.2.3.3.3.1.cmml" xref="S6.SS2.p2.13.m13.2.3.3.3.2"><ci id="S6.SS2.p2.13.m13.1.1.cmml" xref="S6.SS2.p2.13.m13.1.1">𝐴</ci><ci id="S6.SS2.p2.13.m13.2.2.cmml" xref="S6.SS2.p2.13.m13.2.2">𝑋</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.13.m13.2c">P:=P(A,X)</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.13.m13.2d">italic_P := italic_P ( italic_A , italic_X )</annotation></semantics></math> be the set of finite partial functions from <math alttext="\bar{A}" class="ltx_Math" display="inline" id="S6.SS2.p2.14.m14.1"><semantics id="S6.SS2.p2.14.m14.1a"><mover accent="true" id="S6.SS2.p2.14.m14.1.1" xref="S6.SS2.p2.14.m14.1.1.cmml"><mi id="S6.SS2.p2.14.m14.1.1.2" xref="S6.SS2.p2.14.m14.1.1.2.cmml">A</mi><mo id="S6.SS2.p2.14.m14.1.1.1" xref="S6.SS2.p2.14.m14.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.14.m14.1b"><apply id="S6.SS2.p2.14.m14.1.1.cmml" xref="S6.SS2.p2.14.m14.1.1"><ci id="S6.SS2.p2.14.m14.1.1.1.cmml" xref="S6.SS2.p2.14.m14.1.1.1">¯</ci><ci id="S6.SS2.p2.14.m14.1.1.2.cmml" xref="S6.SS2.p2.14.m14.1.1.2">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.14.m14.1c">\bar{A}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.14.m14.1d">over¯ start_ARG italic_A end_ARG</annotation></semantics></math> to <math alttext="X" class="ltx_Math" display="inline" id="S6.SS2.p2.15.m15.1"><semantics id="S6.SS2.p2.15.m15.1a"><mi id="S6.SS2.p2.15.m15.1.1" xref="S6.SS2.p2.15.m15.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.15.m15.1b"><ci id="S6.SS2.p2.15.m15.1.1.cmml" xref="S6.SS2.p2.15.m15.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.15.m15.1c">X</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.15.m15.1d">italic_X</annotation></semantics></math> that have domain linearly ordered and are increasing. Every <math alttext="p\in P" class="ltx_Math" display="inline" id="S6.SS2.p2.16.m16.1"><semantics id="S6.SS2.p2.16.m16.1a"><mrow id="S6.SS2.p2.16.m16.1.1" xref="S6.SS2.p2.16.m16.1.1.cmml"><mi id="S6.SS2.p2.16.m16.1.1.2" xref="S6.SS2.p2.16.m16.1.1.2.cmml">p</mi><mo id="S6.SS2.p2.16.m16.1.1.1" xref="S6.SS2.p2.16.m16.1.1.1.cmml">∈</mo><mi id="S6.SS2.p2.16.m16.1.1.3" xref="S6.SS2.p2.16.m16.1.1.3.cmml">P</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.16.m16.1b"><apply id="S6.SS2.p2.16.m16.1.1.cmml" xref="S6.SS2.p2.16.m16.1.1"><in id="S6.SS2.p2.16.m16.1.1.1.cmml" xref="S6.SS2.p2.16.m16.1.1.1"></in><ci id="S6.SS2.p2.16.m16.1.1.2.cmml" xref="S6.SS2.p2.16.m16.1.1.2">𝑝</ci><ci id="S6.SS2.p2.16.m16.1.1.3.cmml" xref="S6.SS2.p2.16.m16.1.1.3">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.16.m16.1c">p\in P</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.16.m16.1d">italic_p ∈ italic_P</annotation></semantics></math> naturally codes a partial epimorphism from <math alttext="A" class="ltx_Math" display="inline" id="S6.SS2.p2.17.m17.1"><semantics id="S6.SS2.p2.17.m17.1a"><mi id="S6.SS2.p2.17.m17.1.1" xref="S6.SS2.p2.17.m17.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.17.m17.1b"><ci id="S6.SS2.p2.17.m17.1.1.cmml" xref="S6.SS2.p2.17.m17.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.17.m17.1c">A</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.17.m17.1d">italic_A</annotation></semantics></math> to <math alttext="X" class="ltx_Math" display="inline" id="S6.SS2.p2.18.m18.1"><semantics id="S6.SS2.p2.18.m18.1a"><mi id="S6.SS2.p2.18.m18.1.1" xref="S6.SS2.p2.18.m18.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.18.m18.1b"><ci id="S6.SS2.p2.18.m18.1.1.cmml" xref="S6.SS2.p2.18.m18.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.18.m18.1c">X</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.18.m18.1d">italic_X</annotation></semantics></math> defined by <math alttext="f_{p}(a)=x" class="ltx_Math" display="inline" id="S6.SS2.p2.19.m19.1"><semantics id="S6.SS2.p2.19.m19.1a"><mrow id="S6.SS2.p2.19.m19.1.2" xref="S6.SS2.p2.19.m19.1.2.cmml"><mrow id="S6.SS2.p2.19.m19.1.2.2" xref="S6.SS2.p2.19.m19.1.2.2.cmml"><msub id="S6.SS2.p2.19.m19.1.2.2.2" xref="S6.SS2.p2.19.m19.1.2.2.2.cmml"><mi id="S6.SS2.p2.19.m19.1.2.2.2.2" xref="S6.SS2.p2.19.m19.1.2.2.2.2.cmml">f</mi><mi id="S6.SS2.p2.19.m19.1.2.2.2.3" xref="S6.SS2.p2.19.m19.1.2.2.2.3.cmml">p</mi></msub><mo id="S6.SS2.p2.19.m19.1.2.2.1" xref="S6.SS2.p2.19.m19.1.2.2.1.cmml">⁢</mo><mrow id="S6.SS2.p2.19.m19.1.2.2.3.2" xref="S6.SS2.p2.19.m19.1.2.2.cmml"><mo id="S6.SS2.p2.19.m19.1.2.2.3.2.1" stretchy="false" xref="S6.SS2.p2.19.m19.1.2.2.cmml">(</mo><mi id="S6.SS2.p2.19.m19.1.1" xref="S6.SS2.p2.19.m19.1.1.cmml">a</mi><mo id="S6.SS2.p2.19.m19.1.2.2.3.2.2" stretchy="false" xref="S6.SS2.p2.19.m19.1.2.2.cmml">)</mo></mrow></mrow><mo id="S6.SS2.p2.19.m19.1.2.1" xref="S6.SS2.p2.19.m19.1.2.1.cmml">=</mo><mi id="S6.SS2.p2.19.m19.1.2.3" xref="S6.SS2.p2.19.m19.1.2.3.cmml">x</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.19.m19.1b"><apply id="S6.SS2.p2.19.m19.1.2.cmml" xref="S6.SS2.p2.19.m19.1.2"><eq id="S6.SS2.p2.19.m19.1.2.1.cmml" xref="S6.SS2.p2.19.m19.1.2.1"></eq><apply id="S6.SS2.p2.19.m19.1.2.2.cmml" xref="S6.SS2.p2.19.m19.1.2.2"><times id="S6.SS2.p2.19.m19.1.2.2.1.cmml" xref="S6.SS2.p2.19.m19.1.2.2.1"></times><apply id="S6.SS2.p2.19.m19.1.2.2.2.cmml" xref="S6.SS2.p2.19.m19.1.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.p2.19.m19.1.2.2.2.1.cmml" xref="S6.SS2.p2.19.m19.1.2.2.2">subscript</csymbol><ci id="S6.SS2.p2.19.m19.1.2.2.2.2.cmml" xref="S6.SS2.p2.19.m19.1.2.2.2.2">𝑓</ci><ci id="S6.SS2.p2.19.m19.1.2.2.2.3.cmml" xref="S6.SS2.p2.19.m19.1.2.2.2.3">𝑝</ci></apply><ci id="S6.SS2.p2.19.m19.1.1.cmml" xref="S6.SS2.p2.19.m19.1.1">𝑎</ci></apply><ci id="S6.SS2.p2.19.m19.1.2.3.cmml" xref="S6.SS2.p2.19.m19.1.2.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.19.m19.1c">f_{p}(a)=x</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.19.m19.1d">italic_f start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( italic_a ) = italic_x</annotation></semantics></math> iff for some <math alttext="\bar{a}\in\operatorname{dom}(p)" class="ltx_Math" display="inline" id="S6.SS2.p2.20.m20.2"><semantics id="S6.SS2.p2.20.m20.2a"><mrow id="S6.SS2.p2.20.m20.2.3" xref="S6.SS2.p2.20.m20.2.3.cmml"><mover accent="true" id="S6.SS2.p2.20.m20.2.3.2" xref="S6.SS2.p2.20.m20.2.3.2.cmml"><mi id="S6.SS2.p2.20.m20.2.3.2.2" xref="S6.SS2.p2.20.m20.2.3.2.2.cmml">a</mi><mo id="S6.SS2.p2.20.m20.2.3.2.1" xref="S6.SS2.p2.20.m20.2.3.2.1.cmml">¯</mo></mover><mo id="S6.SS2.p2.20.m20.2.3.1" xref="S6.SS2.p2.20.m20.2.3.1.cmml">∈</mo><mrow id="S6.SS2.p2.20.m20.2.3.3.2" xref="S6.SS2.p2.20.m20.2.3.3.1.cmml"><mi id="S6.SS2.p2.20.m20.1.1" xref="S6.SS2.p2.20.m20.1.1.cmml">dom</mi><mo id="S6.SS2.p2.20.m20.2.3.3.2a" xref="S6.SS2.p2.20.m20.2.3.3.1.cmml">⁡</mo><mrow id="S6.SS2.p2.20.m20.2.3.3.2.1" xref="S6.SS2.p2.20.m20.2.3.3.1.cmml"><mo id="S6.SS2.p2.20.m20.2.3.3.2.1.1" stretchy="false" xref="S6.SS2.p2.20.m20.2.3.3.1.cmml">(</mo><mi id="S6.SS2.p2.20.m20.2.2" xref="S6.SS2.p2.20.m20.2.2.cmml">p</mi><mo id="S6.SS2.p2.20.m20.2.3.3.2.1.2" stretchy="false" xref="S6.SS2.p2.20.m20.2.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.20.m20.2b"><apply id="S6.SS2.p2.20.m20.2.3.cmml" xref="S6.SS2.p2.20.m20.2.3"><in id="S6.SS2.p2.20.m20.2.3.1.cmml" xref="S6.SS2.p2.20.m20.2.3.1"></in><apply id="S6.SS2.p2.20.m20.2.3.2.cmml" xref="S6.SS2.p2.20.m20.2.3.2"><ci id="S6.SS2.p2.20.m20.2.3.2.1.cmml" xref="S6.SS2.p2.20.m20.2.3.2.1">¯</ci><ci id="S6.SS2.p2.20.m20.2.3.2.2.cmml" xref="S6.SS2.p2.20.m20.2.3.2.2">𝑎</ci></apply><apply id="S6.SS2.p2.20.m20.2.3.3.1.cmml" xref="S6.SS2.p2.20.m20.2.3.3.2"><ci id="S6.SS2.p2.20.m20.1.1.cmml" xref="S6.SS2.p2.20.m20.1.1">dom</ci><ci id="S6.SS2.p2.20.m20.2.2.cmml" xref="S6.SS2.p2.20.m20.2.2">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.20.m20.2c">\bar{a}\in\operatorname{dom}(p)</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.20.m20.2d">over¯ start_ARG italic_a end_ARG ∈ roman_dom ( italic_p )</annotation></semantics></math>, <math alttext="a\in[a_{l},a_{r}]" class="ltx_Math" display="inline" id="S6.SS2.p2.21.m21.2"><semantics id="S6.SS2.p2.21.m21.2a"><mrow id="S6.SS2.p2.21.m21.2.2" xref="S6.SS2.p2.21.m21.2.2.cmml"><mi id="S6.SS2.p2.21.m21.2.2.4" xref="S6.SS2.p2.21.m21.2.2.4.cmml">a</mi><mo id="S6.SS2.p2.21.m21.2.2.3" xref="S6.SS2.p2.21.m21.2.2.3.cmml">∈</mo><mrow id="S6.SS2.p2.21.m21.2.2.2.2" xref="S6.SS2.p2.21.m21.2.2.2.3.cmml"><mo id="S6.SS2.p2.21.m21.2.2.2.2.3" stretchy="false" xref="S6.SS2.p2.21.m21.2.2.2.3.cmml">[</mo><msub id="S6.SS2.p2.21.m21.1.1.1.1.1" xref="S6.SS2.p2.21.m21.1.1.1.1.1.cmml"><mi id="S6.SS2.p2.21.m21.1.1.1.1.1.2" xref="S6.SS2.p2.21.m21.1.1.1.1.1.2.cmml">a</mi><mi id="S6.SS2.p2.21.m21.1.1.1.1.1.3" xref="S6.SS2.p2.21.m21.1.1.1.1.1.3.cmml">l</mi></msub><mo id="S6.SS2.p2.21.m21.2.2.2.2.4" xref="S6.SS2.p2.21.m21.2.2.2.3.cmml">,</mo><msub id="S6.SS2.p2.21.m21.2.2.2.2.2" xref="S6.SS2.p2.21.m21.2.2.2.2.2.cmml"><mi id="S6.SS2.p2.21.m21.2.2.2.2.2.2" xref="S6.SS2.p2.21.m21.2.2.2.2.2.2.cmml">a</mi><mi id="S6.SS2.p2.21.m21.2.2.2.2.2.3" xref="S6.SS2.p2.21.m21.2.2.2.2.2.3.cmml">r</mi></msub><mo id="S6.SS2.p2.21.m21.2.2.2.2.5" stretchy="false" xref="S6.SS2.p2.21.m21.2.2.2.3.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.21.m21.2b"><apply id="S6.SS2.p2.21.m21.2.2.cmml" xref="S6.SS2.p2.21.m21.2.2"><in id="S6.SS2.p2.21.m21.2.2.3.cmml" xref="S6.SS2.p2.21.m21.2.2.3"></in><ci id="S6.SS2.p2.21.m21.2.2.4.cmml" xref="S6.SS2.p2.21.m21.2.2.4">𝑎</ci><interval closure="closed" id="S6.SS2.p2.21.m21.2.2.2.3.cmml" xref="S6.SS2.p2.21.m21.2.2.2.2"><apply id="S6.SS2.p2.21.m21.1.1.1.1.1.cmml" xref="S6.SS2.p2.21.m21.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.p2.21.m21.1.1.1.1.1.1.cmml" xref="S6.SS2.p2.21.m21.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.p2.21.m21.1.1.1.1.1.2.cmml" xref="S6.SS2.p2.21.m21.1.1.1.1.1.2">𝑎</ci><ci id="S6.SS2.p2.21.m21.1.1.1.1.1.3.cmml" xref="S6.SS2.p2.21.m21.1.1.1.1.1.3">𝑙</ci></apply><apply id="S6.SS2.p2.21.m21.2.2.2.2.2.cmml" xref="S6.SS2.p2.21.m21.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.p2.21.m21.2.2.2.2.2.1.cmml" xref="S6.SS2.p2.21.m21.2.2.2.2.2">subscript</csymbol><ci id="S6.SS2.p2.21.m21.2.2.2.2.2.2.cmml" xref="S6.SS2.p2.21.m21.2.2.2.2.2.2">𝑎</ci><ci id="S6.SS2.p2.21.m21.2.2.2.2.2.3.cmml" xref="S6.SS2.p2.21.m21.2.2.2.2.2.3">𝑟</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.21.m21.2c">a\in[a_{l},a_{r}]</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.21.m21.2d">italic_a ∈ [ italic_a start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ]</annotation></semantics></math> and <math alttext="p(\bar{a})=x" class="ltx_Math" display="inline" id="S6.SS2.p2.22.m22.1"><semantics id="S6.SS2.p2.22.m22.1a"><mrow id="S6.SS2.p2.22.m22.1.2" xref="S6.SS2.p2.22.m22.1.2.cmml"><mrow id="S6.SS2.p2.22.m22.1.2.2" xref="S6.SS2.p2.22.m22.1.2.2.cmml"><mi id="S6.SS2.p2.22.m22.1.2.2.2" xref="S6.SS2.p2.22.m22.1.2.2.2.cmml">p</mi><mo id="S6.SS2.p2.22.m22.1.2.2.1" xref="S6.SS2.p2.22.m22.1.2.2.1.cmml">⁢</mo><mrow id="S6.SS2.p2.22.m22.1.2.2.3.2" xref="S6.SS2.p2.22.m22.1.1.cmml"><mo id="S6.SS2.p2.22.m22.1.2.2.3.2.1" stretchy="false" xref="S6.SS2.p2.22.m22.1.1.cmml">(</mo><mover accent="true" id="S6.SS2.p2.22.m22.1.1" xref="S6.SS2.p2.22.m22.1.1.cmml"><mi id="S6.SS2.p2.22.m22.1.1.2" xref="S6.SS2.p2.22.m22.1.1.2.cmml">a</mi><mo id="S6.SS2.p2.22.m22.1.1.1" xref="S6.SS2.p2.22.m22.1.1.1.cmml">¯</mo></mover><mo id="S6.SS2.p2.22.m22.1.2.2.3.2.2" stretchy="false" xref="S6.SS2.p2.22.m22.1.1.cmml">)</mo></mrow></mrow><mo id="S6.SS2.p2.22.m22.1.2.1" xref="S6.SS2.p2.22.m22.1.2.1.cmml">=</mo><mi id="S6.SS2.p2.22.m22.1.2.3" xref="S6.SS2.p2.22.m22.1.2.3.cmml">x</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.22.m22.1b"><apply id="S6.SS2.p2.22.m22.1.2.cmml" xref="S6.SS2.p2.22.m22.1.2"><eq id="S6.SS2.p2.22.m22.1.2.1.cmml" xref="S6.SS2.p2.22.m22.1.2.1"></eq><apply id="S6.SS2.p2.22.m22.1.2.2.cmml" xref="S6.SS2.p2.22.m22.1.2.2"><times id="S6.SS2.p2.22.m22.1.2.2.1.cmml" xref="S6.SS2.p2.22.m22.1.2.2.1"></times><ci id="S6.SS2.p2.22.m22.1.2.2.2.cmml" xref="S6.SS2.p2.22.m22.1.2.2.2">𝑝</ci><apply id="S6.SS2.p2.22.m22.1.1.cmml" xref="S6.SS2.p2.22.m22.1.2.2.3.2"><ci id="S6.SS2.p2.22.m22.1.1.1.cmml" xref="S6.SS2.p2.22.m22.1.1.1">¯</ci><ci id="S6.SS2.p2.22.m22.1.1.2.cmml" xref="S6.SS2.p2.22.m22.1.1.2">𝑎</ci></apply></apply><ci id="S6.SS2.p2.22.m22.1.2.3.cmml" xref="S6.SS2.p2.22.m22.1.2.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.22.m22.1c">p(\bar{a})=x</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.22.m22.1d">italic_p ( over¯ start_ARG italic_a end_ARG ) = italic_x</annotation></semantics></math>. We let <math alttext="q\leq p" class="ltx_Math" display="inline" id="S6.SS2.p2.23.m23.1"><semantics id="S6.SS2.p2.23.m23.1a"><mrow id="S6.SS2.p2.23.m23.1.1" xref="S6.SS2.p2.23.m23.1.1.cmml"><mi id="S6.SS2.p2.23.m23.1.1.2" xref="S6.SS2.p2.23.m23.1.1.2.cmml">q</mi><mo id="S6.SS2.p2.23.m23.1.1.1" xref="S6.SS2.p2.23.m23.1.1.1.cmml">≤</mo><mi id="S6.SS2.p2.23.m23.1.1.3" xref="S6.SS2.p2.23.m23.1.1.3.cmml">p</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.23.m23.1b"><apply id="S6.SS2.p2.23.m23.1.1.cmml" xref="S6.SS2.p2.23.m23.1.1"><leq id="S6.SS2.p2.23.m23.1.1.1.cmml" xref="S6.SS2.p2.23.m23.1.1.1"></leq><ci id="S6.SS2.p2.23.m23.1.1.2.cmml" xref="S6.SS2.p2.23.m23.1.1.2">𝑞</ci><ci id="S6.SS2.p2.23.m23.1.1.3.cmml" xref="S6.SS2.p2.23.m23.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.23.m23.1c">q\leq p</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.23.m23.1d">italic_q ≤ italic_p</annotation></semantics></math> iff <math alttext="f_{q}" class="ltx_Math" display="inline" id="S6.SS2.p2.24.m24.1"><semantics id="S6.SS2.p2.24.m24.1a"><msub id="S6.SS2.p2.24.m24.1.1" xref="S6.SS2.p2.24.m24.1.1.cmml"><mi id="S6.SS2.p2.24.m24.1.1.2" xref="S6.SS2.p2.24.m24.1.1.2.cmml">f</mi><mi id="S6.SS2.p2.24.m24.1.1.3" xref="S6.SS2.p2.24.m24.1.1.3.cmml">q</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.24.m24.1b"><apply id="S6.SS2.p2.24.m24.1.1.cmml" xref="S6.SS2.p2.24.m24.1.1"><csymbol cd="ambiguous" id="S6.SS2.p2.24.m24.1.1.1.cmml" xref="S6.SS2.p2.24.m24.1.1">subscript</csymbol><ci id="S6.SS2.p2.24.m24.1.1.2.cmml" xref="S6.SS2.p2.24.m24.1.1.2">𝑓</ci><ci id="S6.SS2.p2.24.m24.1.1.3.cmml" xref="S6.SS2.p2.24.m24.1.1.3">𝑞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.24.m24.1c">f_{q}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.24.m24.1d">italic_f start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT</annotation></semantics></math> extends <math alttext="f_{p}" class="ltx_Math" display="inline" id="S6.SS2.p2.25.m25.1"><semantics id="S6.SS2.p2.25.m25.1a"><msub id="S6.SS2.p2.25.m25.1.1" xref="S6.SS2.p2.25.m25.1.1.cmml"><mi id="S6.SS2.p2.25.m25.1.1.2" xref="S6.SS2.p2.25.m25.1.1.2.cmml">f</mi><mi id="S6.SS2.p2.25.m25.1.1.3" xref="S6.SS2.p2.25.m25.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.25.m25.1b"><apply id="S6.SS2.p2.25.m25.1.1.cmml" xref="S6.SS2.p2.25.m25.1.1"><csymbol cd="ambiguous" id="S6.SS2.p2.25.m25.1.1.1.cmml" xref="S6.SS2.p2.25.m25.1.1">subscript</csymbol><ci id="S6.SS2.p2.25.m25.1.1.2.cmml" xref="S6.SS2.p2.25.m25.1.1.2">𝑓</ci><ci id="S6.SS2.p2.25.m25.1.1.3.cmml" xref="S6.SS2.p2.25.m25.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.25.m25.1c">f_{p}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.25.m25.1d">italic_f start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>. This way, a condition <math alttext="p" class="ltx_Math" display="inline" id="S6.SS2.p2.26.m26.1"><semantics id="S6.SS2.p2.26.m26.1a"><mi id="S6.SS2.p2.26.m26.1.1" xref="S6.SS2.p2.26.m26.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.26.m26.1b"><ci id="S6.SS2.p2.26.m26.1.1.cmml" xref="S6.SS2.p2.26.m26.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.26.m26.1c">p</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p2.26.m26.1d">italic_p</annotation></semantics></math> can be extended either by adding new intervals, or by enlarging an existing one.</p> </div> <div class="ltx_para" id="S6.SS2.p3"> <p class="ltx_p" id="S6.SS2.p3.10">The <math alttext="\aleph_{1}" class="ltx_Math" display="inline" id="S6.SS2.p3.1.m1.1"><semantics id="S6.SS2.p3.1.m1.1a"><msub id="S6.SS2.p3.1.m1.1.1" xref="S6.SS2.p3.1.m1.1.1.cmml"><mi id="S6.SS2.p3.1.m1.1.1.2" mathvariant="normal" xref="S6.SS2.p3.1.m1.1.1.2.cmml">ℵ</mi><mn id="S6.SS2.p3.1.m1.1.1.3" xref="S6.SS2.p3.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.p3.1.m1.1b"><apply id="S6.SS2.p3.1.m1.1.1.cmml" xref="S6.SS2.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S6.SS2.p3.1.m1.1.1.1.cmml" xref="S6.SS2.p3.1.m1.1.1">subscript</csymbol><ci id="S6.SS2.p3.1.m1.1.1.2.cmml" xref="S6.SS2.p3.1.m1.1.1.2">ℵ</ci><cn id="S6.SS2.p3.1.m1.1.1.3.cmml" type="integer" xref="S6.SS2.p3.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p3.1.m1.1c">\aleph_{1}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p3.1.m1.1d">roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-density of <math alttext="A" class="ltx_Math" display="inline" id="S6.SS2.p3.2.m2.1"><semantics id="S6.SS2.p3.2.m2.1a"><mi id="S6.SS2.p3.2.m2.1.1" xref="S6.SS2.p3.2.m2.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.p3.2.m2.1b"><ci id="S6.SS2.p3.2.m2.1.1.cmml" xref="S6.SS2.p3.2.m2.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p3.2.m2.1c">A</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p3.2.m2.1d">italic_A</annotation></semantics></math> and <math alttext="X" class="ltx_Math" display="inline" id="S6.SS2.p3.3.m3.1"><semantics id="S6.SS2.p3.3.m3.1a"><mi id="S6.SS2.p3.3.m3.1.1" xref="S6.SS2.p3.3.m3.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.p3.3.m3.1b"><ci id="S6.SS2.p3.3.m3.1.1.cmml" xref="S6.SS2.p3.3.m3.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p3.3.m3.1c">X</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p3.3.m3.1d">italic_X</annotation></semantics></math> implies that for every <math alttext="p\in P" class="ltx_Math" display="inline" id="S6.SS2.p3.4.m4.1"><semantics id="S6.SS2.p3.4.m4.1a"><mrow id="S6.SS2.p3.4.m4.1.1" xref="S6.SS2.p3.4.m4.1.1.cmml"><mi id="S6.SS2.p3.4.m4.1.1.2" xref="S6.SS2.p3.4.m4.1.1.2.cmml">p</mi><mo id="S6.SS2.p3.4.m4.1.1.1" xref="S6.SS2.p3.4.m4.1.1.1.cmml">∈</mo><mi id="S6.SS2.p3.4.m4.1.1.3" xref="S6.SS2.p3.4.m4.1.1.3.cmml">P</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p3.4.m4.1b"><apply id="S6.SS2.p3.4.m4.1.1.cmml" xref="S6.SS2.p3.4.m4.1.1"><in id="S6.SS2.p3.4.m4.1.1.1.cmml" xref="S6.SS2.p3.4.m4.1.1.1"></in><ci id="S6.SS2.p3.4.m4.1.1.2.cmml" xref="S6.SS2.p3.4.m4.1.1.2">𝑝</ci><ci id="S6.SS2.p3.4.m4.1.1.3.cmml" xref="S6.SS2.p3.4.m4.1.1.3">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p3.4.m4.1c">p\in P</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p3.4.m4.1d">italic_p ∈ italic_P</annotation></semantics></math>, <math alttext="\{p\in P:a\in\operatorname{dom}(f_{p}),x\in\operatorname{ran}(f_{p})\}" class="ltx_Math" display="inline" id="S6.SS2.p3.5.m5.4"><semantics id="S6.SS2.p3.5.m5.4a"><mrow id="S6.SS2.p3.5.m5.4.4.2" xref="S6.SS2.p3.5.m5.4.4.3.cmml"><mo id="S6.SS2.p3.5.m5.4.4.2.3" stretchy="false" xref="S6.SS2.p3.5.m5.4.4.3.1.cmml">{</mo><mrow id="S6.SS2.p3.5.m5.3.3.1.1" xref="S6.SS2.p3.5.m5.3.3.1.1.cmml"><mi id="S6.SS2.p3.5.m5.3.3.1.1.2" xref="S6.SS2.p3.5.m5.3.3.1.1.2.cmml">p</mi><mo id="S6.SS2.p3.5.m5.3.3.1.1.1" xref="S6.SS2.p3.5.m5.3.3.1.1.1.cmml">∈</mo><mi id="S6.SS2.p3.5.m5.3.3.1.1.3" xref="S6.SS2.p3.5.m5.3.3.1.1.3.cmml">P</mi></mrow><mo id="S6.SS2.p3.5.m5.4.4.2.4" lspace="0.278em" rspace="0.278em" xref="S6.SS2.p3.5.m5.4.4.3.1.cmml">:</mo><mrow id="S6.SS2.p3.5.m5.4.4.2.2.2" xref="S6.SS2.p3.5.m5.4.4.2.2.3.cmml"><mrow id="S6.SS2.p3.5.m5.4.4.2.2.1.1" xref="S6.SS2.p3.5.m5.4.4.2.2.1.1.cmml"><mi id="S6.SS2.p3.5.m5.4.4.2.2.1.1.3" xref="S6.SS2.p3.5.m5.4.4.2.2.1.1.3.cmml">a</mi><mo id="S6.SS2.p3.5.m5.4.4.2.2.1.1.2" xref="S6.SS2.p3.5.m5.4.4.2.2.1.1.2.cmml">∈</mo><mrow id="S6.SS2.p3.5.m5.4.4.2.2.1.1.1.1" xref="S6.SS2.p3.5.m5.4.4.2.2.1.1.1.2.cmml"><mi id="S6.SS2.p3.5.m5.1.1" xref="S6.SS2.p3.5.m5.1.1.cmml">dom</mi><mo id="S6.SS2.p3.5.m5.4.4.2.2.1.1.1.1a" xref="S6.SS2.p3.5.m5.4.4.2.2.1.1.1.2.cmml">⁡</mo><mrow id="S6.SS2.p3.5.m5.4.4.2.2.1.1.1.1.1" xref="S6.SS2.p3.5.m5.4.4.2.2.1.1.1.2.cmml"><mo id="S6.SS2.p3.5.m5.4.4.2.2.1.1.1.1.1.2" stretchy="false" xref="S6.SS2.p3.5.m5.4.4.2.2.1.1.1.2.cmml">(</mo><msub id="S6.SS2.p3.5.m5.4.4.2.2.1.1.1.1.1.1" xref="S6.SS2.p3.5.m5.4.4.2.2.1.1.1.1.1.1.cmml"><mi id="S6.SS2.p3.5.m5.4.4.2.2.1.1.1.1.1.1.2" xref="S6.SS2.p3.5.m5.4.4.2.2.1.1.1.1.1.1.2.cmml">f</mi><mi id="S6.SS2.p3.5.m5.4.4.2.2.1.1.1.1.1.1.3" xref="S6.SS2.p3.5.m5.4.4.2.2.1.1.1.1.1.1.3.cmml">p</mi></msub><mo id="S6.SS2.p3.5.m5.4.4.2.2.1.1.1.1.1.3" stretchy="false" xref="S6.SS2.p3.5.m5.4.4.2.2.1.1.1.2.cmml">)</mo></mrow></mrow></mrow><mo id="S6.SS2.p3.5.m5.4.4.2.2.2.3" xref="S6.SS2.p3.5.m5.4.4.2.2.3a.cmml">,</mo><mrow id="S6.SS2.p3.5.m5.4.4.2.2.2.2" xref="S6.SS2.p3.5.m5.4.4.2.2.2.2.cmml"><mi id="S6.SS2.p3.5.m5.4.4.2.2.2.2.3" xref="S6.SS2.p3.5.m5.4.4.2.2.2.2.3.cmml">x</mi><mo id="S6.SS2.p3.5.m5.4.4.2.2.2.2.2" xref="S6.SS2.p3.5.m5.4.4.2.2.2.2.2.cmml">∈</mo><mrow id="S6.SS2.p3.5.m5.4.4.2.2.2.2.1.1" xref="S6.SS2.p3.5.m5.4.4.2.2.2.2.1.2.cmml"><mi id="S6.SS2.p3.5.m5.2.2" xref="S6.SS2.p3.5.m5.2.2.cmml">ran</mi><mo id="S6.SS2.p3.5.m5.4.4.2.2.2.2.1.1a" xref="S6.SS2.p3.5.m5.4.4.2.2.2.2.1.2.cmml">⁡</mo><mrow id="S6.SS2.p3.5.m5.4.4.2.2.2.2.1.1.1" xref="S6.SS2.p3.5.m5.4.4.2.2.2.2.1.2.cmml"><mo id="S6.SS2.p3.5.m5.4.4.2.2.2.2.1.1.1.2" stretchy="false" xref="S6.SS2.p3.5.m5.4.4.2.2.2.2.1.2.cmml">(</mo><msub id="S6.SS2.p3.5.m5.4.4.2.2.2.2.1.1.1.1" xref="S6.SS2.p3.5.m5.4.4.2.2.2.2.1.1.1.1.cmml"><mi id="S6.SS2.p3.5.m5.4.4.2.2.2.2.1.1.1.1.2" xref="S6.SS2.p3.5.m5.4.4.2.2.2.2.1.1.1.1.2.cmml">f</mi><mi id="S6.SS2.p3.5.m5.4.4.2.2.2.2.1.1.1.1.3" xref="S6.SS2.p3.5.m5.4.4.2.2.2.2.1.1.1.1.3.cmml">p</mi></msub><mo id="S6.SS2.p3.5.m5.4.4.2.2.2.2.1.1.1.3" stretchy="false" xref="S6.SS2.p3.5.m5.4.4.2.2.2.2.1.2.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S6.SS2.p3.5.m5.4.4.2.5" stretchy="false" xref="S6.SS2.p3.5.m5.4.4.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p3.5.m5.4b"><apply id="S6.SS2.p3.5.m5.4.4.3.cmml" xref="S6.SS2.p3.5.m5.4.4.2"><csymbol cd="latexml" id="S6.SS2.p3.5.m5.4.4.3.1.cmml" xref="S6.SS2.p3.5.m5.4.4.2.3">conditional-set</csymbol><apply id="S6.SS2.p3.5.m5.3.3.1.1.cmml" xref="S6.SS2.p3.5.m5.3.3.1.1"><in id="S6.SS2.p3.5.m5.3.3.1.1.1.cmml" xref="S6.SS2.p3.5.m5.3.3.1.1.1"></in><ci id="S6.SS2.p3.5.m5.3.3.1.1.2.cmml" xref="S6.SS2.p3.5.m5.3.3.1.1.2">𝑝</ci><ci id="S6.SS2.p3.5.m5.3.3.1.1.3.cmml" xref="S6.SS2.p3.5.m5.3.3.1.1.3">𝑃</ci></apply><apply id="S6.SS2.p3.5.m5.4.4.2.2.3.cmml" xref="S6.SS2.p3.5.m5.4.4.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.p3.5.m5.4.4.2.2.3a.cmml" xref="S6.SS2.p3.5.m5.4.4.2.2.2.3">formulae-sequence</csymbol><apply id="S6.SS2.p3.5.m5.4.4.2.2.1.1.cmml" xref="S6.SS2.p3.5.m5.4.4.2.2.1.1"><in id="S6.SS2.p3.5.m5.4.4.2.2.1.1.2.cmml" xref="S6.SS2.p3.5.m5.4.4.2.2.1.1.2"></in><ci id="S6.SS2.p3.5.m5.4.4.2.2.1.1.3.cmml" xref="S6.SS2.p3.5.m5.4.4.2.2.1.1.3">𝑎</ci><apply id="S6.SS2.p3.5.m5.4.4.2.2.1.1.1.2.cmml" xref="S6.SS2.p3.5.m5.4.4.2.2.1.1.1.1"><ci id="S6.SS2.p3.5.m5.1.1.cmml" xref="S6.SS2.p3.5.m5.1.1">dom</ci><apply id="S6.SS2.p3.5.m5.4.4.2.2.1.1.1.1.1.1.cmml" xref="S6.SS2.p3.5.m5.4.4.2.2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.p3.5.m5.4.4.2.2.1.1.1.1.1.1.1.cmml" xref="S6.SS2.p3.5.m5.4.4.2.2.1.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.p3.5.m5.4.4.2.2.1.1.1.1.1.1.2.cmml" xref="S6.SS2.p3.5.m5.4.4.2.2.1.1.1.1.1.1.2">𝑓</ci><ci id="S6.SS2.p3.5.m5.4.4.2.2.1.1.1.1.1.1.3.cmml" xref="S6.SS2.p3.5.m5.4.4.2.2.1.1.1.1.1.1.3">𝑝</ci></apply></apply></apply><apply id="S6.SS2.p3.5.m5.4.4.2.2.2.2.cmml" xref="S6.SS2.p3.5.m5.4.4.2.2.2.2"><in id="S6.SS2.p3.5.m5.4.4.2.2.2.2.2.cmml" xref="S6.SS2.p3.5.m5.4.4.2.2.2.2.2"></in><ci id="S6.SS2.p3.5.m5.4.4.2.2.2.2.3.cmml" xref="S6.SS2.p3.5.m5.4.4.2.2.2.2.3">𝑥</ci><apply id="S6.SS2.p3.5.m5.4.4.2.2.2.2.1.2.cmml" xref="S6.SS2.p3.5.m5.4.4.2.2.2.2.1.1"><ci id="S6.SS2.p3.5.m5.2.2.cmml" xref="S6.SS2.p3.5.m5.2.2">ran</ci><apply id="S6.SS2.p3.5.m5.4.4.2.2.2.2.1.1.1.1.cmml" xref="S6.SS2.p3.5.m5.4.4.2.2.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.p3.5.m5.4.4.2.2.2.2.1.1.1.1.1.cmml" xref="S6.SS2.p3.5.m5.4.4.2.2.2.2.1.1.1.1">subscript</csymbol><ci id="S6.SS2.p3.5.m5.4.4.2.2.2.2.1.1.1.1.2.cmml" xref="S6.SS2.p3.5.m5.4.4.2.2.2.2.1.1.1.1.2">𝑓</ci><ci id="S6.SS2.p3.5.m5.4.4.2.2.2.2.1.1.1.1.3.cmml" xref="S6.SS2.p3.5.m5.4.4.2.2.2.2.1.1.1.1.3">𝑝</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p3.5.m5.4c">\{p\in P:a\in\operatorname{dom}(f_{p}),x\in\operatorname{ran}(f_{p})\}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p3.5.m5.4d">{ italic_p ∈ italic_P : italic_a ∈ roman_dom ( italic_f start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ) , italic_x ∈ roman_ran ( italic_f start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ) }</annotation></semantics></math> is dense. Thus any generic filter for <math alttext="P" class="ltx_Math" display="inline" id="S6.SS2.p3.6.m6.1"><semantics id="S6.SS2.p3.6.m6.1a"><mi id="S6.SS2.p3.6.m6.1.1" xref="S6.SS2.p3.6.m6.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.p3.6.m6.1b"><ci id="S6.SS2.p3.6.m6.1.1.cmml" xref="S6.SS2.p3.6.m6.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p3.6.m6.1c">P</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p3.6.m6.1d">italic_P</annotation></semantics></math> introduces an epimorphism from <math alttext="A" class="ltx_Math" display="inline" id="S6.SS2.p3.7.m7.1"><semantics id="S6.SS2.p3.7.m7.1a"><mi id="S6.SS2.p3.7.m7.1.1" xref="S6.SS2.p3.7.m7.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.p3.7.m7.1b"><ci id="S6.SS2.p3.7.m7.1.1.cmml" xref="S6.SS2.p3.7.m7.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p3.7.m7.1c">A</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p3.7.m7.1d">italic_A</annotation></semantics></math> onto <math alttext="X" class="ltx_Math" display="inline" id="S6.SS2.p3.8.m8.1"><semantics id="S6.SS2.p3.8.m8.1a"><mi id="S6.SS2.p3.8.m8.1.1" xref="S6.SS2.p3.8.m8.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.p3.8.m8.1b"><ci id="S6.SS2.p3.8.m8.1.1.cmml" xref="S6.SS2.p3.8.m8.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p3.8.m8.1c">X</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p3.8.m8.1d">italic_X</annotation></semantics></math>. However it is easily seen that <math alttext="P" class="ltx_Math" display="inline" id="S6.SS2.p3.9.m9.1"><semantics id="S6.SS2.p3.9.m9.1a"><mi id="S6.SS2.p3.9.m9.1.1" xref="S6.SS2.p3.9.m9.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.p3.9.m9.1b"><ci id="S6.SS2.p3.9.m9.1.1.cmml" xref="S6.SS2.p3.9.m9.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p3.9.m9.1c">P</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p3.9.m9.1d">italic_P</annotation></semantics></math> is never ccc, and in fact collapses <math alttext="\omega_{1}" class="ltx_Math" display="inline" id="S6.SS2.p3.10.m10.1"><semantics id="S6.SS2.p3.10.m10.1a"><msub id="S6.SS2.p3.10.m10.1.1" xref="S6.SS2.p3.10.m10.1.1.cmml"><mi id="S6.SS2.p3.10.m10.1.1.2" xref="S6.SS2.p3.10.m10.1.1.2.cmml">ω</mi><mn id="S6.SS2.p3.10.m10.1.1.3" xref="S6.SS2.p3.10.m10.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.p3.10.m10.1b"><apply id="S6.SS2.p3.10.m10.1.1.cmml" xref="S6.SS2.p3.10.m10.1.1"><csymbol cd="ambiguous" id="S6.SS2.p3.10.m10.1.1.1.cmml" xref="S6.SS2.p3.10.m10.1.1">subscript</csymbol><ci id="S6.SS2.p3.10.m10.1.1.2.cmml" xref="S6.SS2.p3.10.m10.1.1.2">𝜔</ci><cn id="S6.SS2.p3.10.m10.1.1.3.cmml" type="integer" xref="S6.SS2.p3.10.m10.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p3.10.m10.1c">\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p3.10.m10.1d">italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>. As in Moore’s forcing, it is necessary to refine it to a ccc forcing.</p> </div> <div class="ltx_para" id="S6.SS2.p4"> <p class="ltx_p" id="S6.SS2.p4.11">For a club <math alttext="E" class="ltx_Math" display="inline" id="S6.SS2.p4.1.m1.1"><semantics id="S6.SS2.p4.1.m1.1a"><mi id="S6.SS2.p4.1.m1.1.1" xref="S6.SS2.p4.1.m1.1.1.cmml">E</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.p4.1.m1.1b"><ci id="S6.SS2.p4.1.m1.1.1.cmml" xref="S6.SS2.p4.1.m1.1.1">𝐸</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p4.1.m1.1c">E</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p4.1.m1.1d">italic_E</annotation></semantics></math> and <math alttext="\alpha<\omega_{1}" class="ltx_Math" display="inline" id="S6.SS2.p4.2.m2.1"><semantics id="S6.SS2.p4.2.m2.1a"><mrow id="S6.SS2.p4.2.m2.1.1" xref="S6.SS2.p4.2.m2.1.1.cmml"><mi id="S6.SS2.p4.2.m2.1.1.2" xref="S6.SS2.p4.2.m2.1.1.2.cmml">α</mi><mo id="S6.SS2.p4.2.m2.1.1.1" xref="S6.SS2.p4.2.m2.1.1.1.cmml"><</mo><msub id="S6.SS2.p4.2.m2.1.1.3" xref="S6.SS2.p4.2.m2.1.1.3.cmml"><mi id="S6.SS2.p4.2.m2.1.1.3.2" xref="S6.SS2.p4.2.m2.1.1.3.2.cmml">ω</mi><mn id="S6.SS2.p4.2.m2.1.1.3.3" xref="S6.SS2.p4.2.m2.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p4.2.m2.1b"><apply id="S6.SS2.p4.2.m2.1.1.cmml" xref="S6.SS2.p4.2.m2.1.1"><lt id="S6.SS2.p4.2.m2.1.1.1.cmml" xref="S6.SS2.p4.2.m2.1.1.1"></lt><ci id="S6.SS2.p4.2.m2.1.1.2.cmml" xref="S6.SS2.p4.2.m2.1.1.2">𝛼</ci><apply id="S6.SS2.p4.2.m2.1.1.3.cmml" xref="S6.SS2.p4.2.m2.1.1.3"><csymbol cd="ambiguous" id="S6.SS2.p4.2.m2.1.1.3.1.cmml" xref="S6.SS2.p4.2.m2.1.1.3">subscript</csymbol><ci id="S6.SS2.p4.2.m2.1.1.3.2.cmml" xref="S6.SS2.p4.2.m2.1.1.3.2">𝜔</ci><cn id="S6.SS2.p4.2.m2.1.1.3.3.cmml" type="integer" xref="S6.SS2.p4.2.m2.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p4.2.m2.1c">\alpha<\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p4.2.m2.1d">italic_α < italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, we let <math alttext="\nu_{E}(\alpha)" class="ltx_Math" display="inline" id="S6.SS2.p4.3.m3.1"><semantics id="S6.SS2.p4.3.m3.1a"><mrow id="S6.SS2.p4.3.m3.1.2" xref="S6.SS2.p4.3.m3.1.2.cmml"><msub id="S6.SS2.p4.3.m3.1.2.2" xref="S6.SS2.p4.3.m3.1.2.2.cmml"><mi id="S6.SS2.p4.3.m3.1.2.2.2" xref="S6.SS2.p4.3.m3.1.2.2.2.cmml">ν</mi><mi id="S6.SS2.p4.3.m3.1.2.2.3" xref="S6.SS2.p4.3.m3.1.2.2.3.cmml">E</mi></msub><mo id="S6.SS2.p4.3.m3.1.2.1" xref="S6.SS2.p4.3.m3.1.2.1.cmml">⁢</mo><mrow id="S6.SS2.p4.3.m3.1.2.3.2" xref="S6.SS2.p4.3.m3.1.2.cmml"><mo id="S6.SS2.p4.3.m3.1.2.3.2.1" stretchy="false" xref="S6.SS2.p4.3.m3.1.2.cmml">(</mo><mi id="S6.SS2.p4.3.m3.1.1" xref="S6.SS2.p4.3.m3.1.1.cmml">α</mi><mo id="S6.SS2.p4.3.m3.1.2.3.2.2" stretchy="false" xref="S6.SS2.p4.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p4.3.m3.1b"><apply id="S6.SS2.p4.3.m3.1.2.cmml" xref="S6.SS2.p4.3.m3.1.2"><times id="S6.SS2.p4.3.m3.1.2.1.cmml" xref="S6.SS2.p4.3.m3.1.2.1"></times><apply id="S6.SS2.p4.3.m3.1.2.2.cmml" xref="S6.SS2.p4.3.m3.1.2.2"><csymbol cd="ambiguous" id="S6.SS2.p4.3.m3.1.2.2.1.cmml" xref="S6.SS2.p4.3.m3.1.2.2">subscript</csymbol><ci id="S6.SS2.p4.3.m3.1.2.2.2.cmml" xref="S6.SS2.p4.3.m3.1.2.2.2">𝜈</ci><ci id="S6.SS2.p4.3.m3.1.2.2.3.cmml" xref="S6.SS2.p4.3.m3.1.2.2.3">𝐸</ci></apply><ci id="S6.SS2.p4.3.m3.1.1.cmml" xref="S6.SS2.p4.3.m3.1.1">𝛼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p4.3.m3.1c">\nu_{E}(\alpha)</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p4.3.m3.1d">italic_ν start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT ( italic_α )</annotation></semantics></math> be the greatest ordinal in <math alttext="\xi\in E\cup\{0\}" class="ltx_Math" display="inline" id="S6.SS2.p4.4.m4.1"><semantics id="S6.SS2.p4.4.m4.1a"><mrow id="S6.SS2.p4.4.m4.1.2" xref="S6.SS2.p4.4.m4.1.2.cmml"><mi id="S6.SS2.p4.4.m4.1.2.2" xref="S6.SS2.p4.4.m4.1.2.2.cmml">ξ</mi><mo id="S6.SS2.p4.4.m4.1.2.1" xref="S6.SS2.p4.4.m4.1.2.1.cmml">∈</mo><mrow id="S6.SS2.p4.4.m4.1.2.3" xref="S6.SS2.p4.4.m4.1.2.3.cmml"><mi id="S6.SS2.p4.4.m4.1.2.3.2" xref="S6.SS2.p4.4.m4.1.2.3.2.cmml">E</mi><mo id="S6.SS2.p4.4.m4.1.2.3.1" xref="S6.SS2.p4.4.m4.1.2.3.1.cmml">∪</mo><mrow id="S6.SS2.p4.4.m4.1.2.3.3.2" xref="S6.SS2.p4.4.m4.1.2.3.3.1.cmml"><mo id="S6.SS2.p4.4.m4.1.2.3.3.2.1" stretchy="false" xref="S6.SS2.p4.4.m4.1.2.3.3.1.cmml">{</mo><mn id="S6.SS2.p4.4.m4.1.1" xref="S6.SS2.p4.4.m4.1.1.cmml">0</mn><mo id="S6.SS2.p4.4.m4.1.2.3.3.2.2" stretchy="false" xref="S6.SS2.p4.4.m4.1.2.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p4.4.m4.1b"><apply id="S6.SS2.p4.4.m4.1.2.cmml" xref="S6.SS2.p4.4.m4.1.2"><in id="S6.SS2.p4.4.m4.1.2.1.cmml" xref="S6.SS2.p4.4.m4.1.2.1"></in><ci id="S6.SS2.p4.4.m4.1.2.2.cmml" xref="S6.SS2.p4.4.m4.1.2.2">𝜉</ci><apply id="S6.SS2.p4.4.m4.1.2.3.cmml" xref="S6.SS2.p4.4.m4.1.2.3"><union id="S6.SS2.p4.4.m4.1.2.3.1.cmml" xref="S6.SS2.p4.4.m4.1.2.3.1"></union><ci id="S6.SS2.p4.4.m4.1.2.3.2.cmml" xref="S6.SS2.p4.4.m4.1.2.3.2">𝐸</ci><set id="S6.SS2.p4.4.m4.1.2.3.3.1.cmml" xref="S6.SS2.p4.4.m4.1.2.3.3.2"><cn id="S6.SS2.p4.4.m4.1.1.cmml" type="integer" xref="S6.SS2.p4.4.m4.1.1">0</cn></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p4.4.m4.1c">\xi\in E\cup\{0\}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p4.4.m4.1d">italic_ξ ∈ italic_E ∪ { 0 }</annotation></semantics></math> such that <math alttext="\xi\leq\alpha" class="ltx_Math" display="inline" id="S6.SS2.p4.5.m5.1"><semantics id="S6.SS2.p4.5.m5.1a"><mrow id="S6.SS2.p4.5.m5.1.1" xref="S6.SS2.p4.5.m5.1.1.cmml"><mi id="S6.SS2.p4.5.m5.1.1.2" xref="S6.SS2.p4.5.m5.1.1.2.cmml">ξ</mi><mo id="S6.SS2.p4.5.m5.1.1.1" xref="S6.SS2.p4.5.m5.1.1.1.cmml">≤</mo><mi id="S6.SS2.p4.5.m5.1.1.3" xref="S6.SS2.p4.5.m5.1.1.3.cmml">α</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p4.5.m5.1b"><apply id="S6.SS2.p4.5.m5.1.1.cmml" xref="S6.SS2.p4.5.m5.1.1"><leq id="S6.SS2.p4.5.m5.1.1.1.cmml" xref="S6.SS2.p4.5.m5.1.1.1"></leq><ci id="S6.SS2.p4.5.m5.1.1.2.cmml" xref="S6.SS2.p4.5.m5.1.1.2">𝜉</ci><ci id="S6.SS2.p4.5.m5.1.1.3.cmml" xref="S6.SS2.p4.5.m5.1.1.3">𝛼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p4.5.m5.1c">\xi\leq\alpha</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p4.5.m5.1d">italic_ξ ≤ italic_α</annotation></semantics></math>. When the club is clear from the context we drop the subscript and we let <math alttext="\alpha^{+}" class="ltx_Math" display="inline" id="S6.SS2.p4.6.m6.1"><semantics id="S6.SS2.p4.6.m6.1a"><msup id="S6.SS2.p4.6.m6.1.1" xref="S6.SS2.p4.6.m6.1.1.cmml"><mi id="S6.SS2.p4.6.m6.1.1.2" xref="S6.SS2.p4.6.m6.1.1.2.cmml">α</mi><mo id="S6.SS2.p4.6.m6.1.1.3" xref="S6.SS2.p4.6.m6.1.1.3.cmml">+</mo></msup><annotation-xml encoding="MathML-Content" id="S6.SS2.p4.6.m6.1b"><apply id="S6.SS2.p4.6.m6.1.1.cmml" xref="S6.SS2.p4.6.m6.1.1"><csymbol cd="ambiguous" id="S6.SS2.p4.6.m6.1.1.1.cmml" xref="S6.SS2.p4.6.m6.1.1">superscript</csymbol><ci id="S6.SS2.p4.6.m6.1.1.2.cmml" xref="S6.SS2.p4.6.m6.1.1.2">𝛼</ci><plus id="S6.SS2.p4.6.m6.1.1.3.cmml" xref="S6.SS2.p4.6.m6.1.1.3"></plus></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p4.6.m6.1c">\alpha^{+}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p4.6.m6.1d">italic_α start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> stand for the least ordinal <math alttext="\xi\in E" class="ltx_Math" display="inline" id="S6.SS2.p4.7.m7.1"><semantics id="S6.SS2.p4.7.m7.1a"><mrow id="S6.SS2.p4.7.m7.1.1" xref="S6.SS2.p4.7.m7.1.1.cmml"><mi id="S6.SS2.p4.7.m7.1.1.2" xref="S6.SS2.p4.7.m7.1.1.2.cmml">ξ</mi><mo id="S6.SS2.p4.7.m7.1.1.1" xref="S6.SS2.p4.7.m7.1.1.1.cmml">∈</mo><mi id="S6.SS2.p4.7.m7.1.1.3" xref="S6.SS2.p4.7.m7.1.1.3.cmml">E</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p4.7.m7.1b"><apply id="S6.SS2.p4.7.m7.1.1.cmml" xref="S6.SS2.p4.7.m7.1.1"><in id="S6.SS2.p4.7.m7.1.1.1.cmml" xref="S6.SS2.p4.7.m7.1.1.1"></in><ci id="S6.SS2.p4.7.m7.1.1.2.cmml" xref="S6.SS2.p4.7.m7.1.1.2">𝜉</ci><ci id="S6.SS2.p4.7.m7.1.1.3.cmml" xref="S6.SS2.p4.7.m7.1.1.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p4.7.m7.1c">\xi\in E</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p4.7.m7.1d">italic_ξ ∈ italic_E</annotation></semantics></math> such that <math alttext="\alpha<\xi" class="ltx_Math" display="inline" id="S6.SS2.p4.8.m8.1"><semantics id="S6.SS2.p4.8.m8.1a"><mrow id="S6.SS2.p4.8.m8.1.1" xref="S6.SS2.p4.8.m8.1.1.cmml"><mi id="S6.SS2.p4.8.m8.1.1.2" xref="S6.SS2.p4.8.m8.1.1.2.cmml">α</mi><mo id="S6.SS2.p4.8.m8.1.1.1" xref="S6.SS2.p4.8.m8.1.1.1.cmml"><</mo><mi id="S6.SS2.p4.8.m8.1.1.3" xref="S6.SS2.p4.8.m8.1.1.3.cmml">ξ</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p4.8.m8.1b"><apply id="S6.SS2.p4.8.m8.1.1.cmml" xref="S6.SS2.p4.8.m8.1.1"><lt id="S6.SS2.p4.8.m8.1.1.1.cmml" xref="S6.SS2.p4.8.m8.1.1.1"></lt><ci id="S6.SS2.p4.8.m8.1.1.2.cmml" xref="S6.SS2.p4.8.m8.1.1.2">𝛼</ci><ci id="S6.SS2.p4.8.m8.1.1.3.cmml" xref="S6.SS2.p4.8.m8.1.1.3">𝜉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p4.8.m8.1c">\alpha<\xi</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p4.8.m8.1d">italic_α < italic_ξ</annotation></semantics></math>. Also when it is clear from context <math alttext="\nu(a,b)" class="ltx_Math" display="inline" id="S6.SS2.p4.9.m9.2"><semantics id="S6.SS2.p4.9.m9.2a"><mrow id="S6.SS2.p4.9.m9.2.3" xref="S6.SS2.p4.9.m9.2.3.cmml"><mi id="S6.SS2.p4.9.m9.2.3.2" xref="S6.SS2.p4.9.m9.2.3.2.cmml">ν</mi><mo id="S6.SS2.p4.9.m9.2.3.1" xref="S6.SS2.p4.9.m9.2.3.1.cmml">⁢</mo><mrow id="S6.SS2.p4.9.m9.2.3.3.2" xref="S6.SS2.p4.9.m9.2.3.3.1.cmml"><mo id="S6.SS2.p4.9.m9.2.3.3.2.1" stretchy="false" xref="S6.SS2.p4.9.m9.2.3.3.1.cmml">(</mo><mi id="S6.SS2.p4.9.m9.1.1" xref="S6.SS2.p4.9.m9.1.1.cmml">a</mi><mo id="S6.SS2.p4.9.m9.2.3.3.2.2" xref="S6.SS2.p4.9.m9.2.3.3.1.cmml">,</mo><mi id="S6.SS2.p4.9.m9.2.2" xref="S6.SS2.p4.9.m9.2.2.cmml">b</mi><mo id="S6.SS2.p4.9.m9.2.3.3.2.3" stretchy="false" xref="S6.SS2.p4.9.m9.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p4.9.m9.2b"><apply id="S6.SS2.p4.9.m9.2.3.cmml" xref="S6.SS2.p4.9.m9.2.3"><times id="S6.SS2.p4.9.m9.2.3.1.cmml" xref="S6.SS2.p4.9.m9.2.3.1"></times><ci id="S6.SS2.p4.9.m9.2.3.2.cmml" xref="S6.SS2.p4.9.m9.2.3.2">𝜈</ci><interval closure="open" id="S6.SS2.p4.9.m9.2.3.3.1.cmml" xref="S6.SS2.p4.9.m9.2.3.3.2"><ci id="S6.SS2.p4.9.m9.1.1.cmml" xref="S6.SS2.p4.9.m9.1.1">𝑎</ci><ci id="S6.SS2.p4.9.m9.2.2.cmml" xref="S6.SS2.p4.9.m9.2.2">𝑏</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p4.9.m9.2c">\nu(a,b)</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p4.9.m9.2d">italic_ν ( italic_a , italic_b )</annotation></semantics></math> will abbreviate <math alttext="\nu(\Delta_{A}(a,b))" class="ltx_Math" display="inline" id="S6.SS2.p4.10.m10.3"><semantics id="S6.SS2.p4.10.m10.3a"><mrow id="S6.SS2.p4.10.m10.3.3" xref="S6.SS2.p4.10.m10.3.3.cmml"><mi id="S6.SS2.p4.10.m10.3.3.3" xref="S6.SS2.p4.10.m10.3.3.3.cmml">ν</mi><mo id="S6.SS2.p4.10.m10.3.3.2" xref="S6.SS2.p4.10.m10.3.3.2.cmml">⁢</mo><mrow id="S6.SS2.p4.10.m10.3.3.1.1" xref="S6.SS2.p4.10.m10.3.3.1.1.1.cmml"><mo id="S6.SS2.p4.10.m10.3.3.1.1.2" stretchy="false" xref="S6.SS2.p4.10.m10.3.3.1.1.1.cmml">(</mo><mrow id="S6.SS2.p4.10.m10.3.3.1.1.1" xref="S6.SS2.p4.10.m10.3.3.1.1.1.cmml"><msub id="S6.SS2.p4.10.m10.3.3.1.1.1.2" xref="S6.SS2.p4.10.m10.3.3.1.1.1.2.cmml"><mi id="S6.SS2.p4.10.m10.3.3.1.1.1.2.2" mathvariant="normal" xref="S6.SS2.p4.10.m10.3.3.1.1.1.2.2.cmml">Δ</mi><mi id="S6.SS2.p4.10.m10.3.3.1.1.1.2.3" xref="S6.SS2.p4.10.m10.3.3.1.1.1.2.3.cmml">A</mi></msub><mo id="S6.SS2.p4.10.m10.3.3.1.1.1.1" xref="S6.SS2.p4.10.m10.3.3.1.1.1.1.cmml">⁢</mo><mrow id="S6.SS2.p4.10.m10.3.3.1.1.1.3.2" xref="S6.SS2.p4.10.m10.3.3.1.1.1.3.1.cmml"><mo id="S6.SS2.p4.10.m10.3.3.1.1.1.3.2.1" stretchy="false" xref="S6.SS2.p4.10.m10.3.3.1.1.1.3.1.cmml">(</mo><mi id="S6.SS2.p4.10.m10.1.1" xref="S6.SS2.p4.10.m10.1.1.cmml">a</mi><mo id="S6.SS2.p4.10.m10.3.3.1.1.1.3.2.2" xref="S6.SS2.p4.10.m10.3.3.1.1.1.3.1.cmml">,</mo><mi id="S6.SS2.p4.10.m10.2.2" xref="S6.SS2.p4.10.m10.2.2.cmml">b</mi><mo id="S6.SS2.p4.10.m10.3.3.1.1.1.3.2.3" stretchy="false" xref="S6.SS2.p4.10.m10.3.3.1.1.1.3.1.cmml">)</mo></mrow></mrow><mo id="S6.SS2.p4.10.m10.3.3.1.1.3" stretchy="false" xref="S6.SS2.p4.10.m10.3.3.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p4.10.m10.3b"><apply id="S6.SS2.p4.10.m10.3.3.cmml" xref="S6.SS2.p4.10.m10.3.3"><times id="S6.SS2.p4.10.m10.3.3.2.cmml" xref="S6.SS2.p4.10.m10.3.3.2"></times><ci id="S6.SS2.p4.10.m10.3.3.3.cmml" xref="S6.SS2.p4.10.m10.3.3.3">𝜈</ci><apply id="S6.SS2.p4.10.m10.3.3.1.1.1.cmml" xref="S6.SS2.p4.10.m10.3.3.1.1"><times id="S6.SS2.p4.10.m10.3.3.1.1.1.1.cmml" xref="S6.SS2.p4.10.m10.3.3.1.1.1.1"></times><apply id="S6.SS2.p4.10.m10.3.3.1.1.1.2.cmml" xref="S6.SS2.p4.10.m10.3.3.1.1.1.2"><csymbol cd="ambiguous" id="S6.SS2.p4.10.m10.3.3.1.1.1.2.1.cmml" xref="S6.SS2.p4.10.m10.3.3.1.1.1.2">subscript</csymbol><ci id="S6.SS2.p4.10.m10.3.3.1.1.1.2.2.cmml" xref="S6.SS2.p4.10.m10.3.3.1.1.1.2.2">Δ</ci><ci id="S6.SS2.p4.10.m10.3.3.1.1.1.2.3.cmml" xref="S6.SS2.p4.10.m10.3.3.1.1.1.2.3">𝐴</ci></apply><interval closure="open" id="S6.SS2.p4.10.m10.3.3.1.1.1.3.1.cmml" xref="S6.SS2.p4.10.m10.3.3.1.1.1.3.2"><ci id="S6.SS2.p4.10.m10.1.1.cmml" xref="S6.SS2.p4.10.m10.1.1">𝑎</ci><ci id="S6.SS2.p4.10.m10.2.2.cmml" xref="S6.SS2.p4.10.m10.2.2">𝑏</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p4.10.m10.3c">\nu(\Delta_{A}(a,b))</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p4.10.m10.3d">italic_ν ( roman_Δ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_a , italic_b ) )</annotation></semantics></math>, and <math alttext="\nu(x,y)=\nu(\Delta_{X}(x,y))" class="ltx_Math" display="inline" id="S6.SS2.p4.11.m11.5"><semantics id="S6.SS2.p4.11.m11.5a"><mrow id="S6.SS2.p4.11.m11.5.5" xref="S6.SS2.p4.11.m11.5.5.cmml"><mrow id="S6.SS2.p4.11.m11.5.5.3" xref="S6.SS2.p4.11.m11.5.5.3.cmml"><mi id="S6.SS2.p4.11.m11.5.5.3.2" xref="S6.SS2.p4.11.m11.5.5.3.2.cmml">ν</mi><mo id="S6.SS2.p4.11.m11.5.5.3.1" xref="S6.SS2.p4.11.m11.5.5.3.1.cmml">⁢</mo><mrow id="S6.SS2.p4.11.m11.5.5.3.3.2" xref="S6.SS2.p4.11.m11.5.5.3.3.1.cmml"><mo id="S6.SS2.p4.11.m11.5.5.3.3.2.1" stretchy="false" xref="S6.SS2.p4.11.m11.5.5.3.3.1.cmml">(</mo><mi id="S6.SS2.p4.11.m11.1.1" xref="S6.SS2.p4.11.m11.1.1.cmml">x</mi><mo id="S6.SS2.p4.11.m11.5.5.3.3.2.2" xref="S6.SS2.p4.11.m11.5.5.3.3.1.cmml">,</mo><mi id="S6.SS2.p4.11.m11.2.2" xref="S6.SS2.p4.11.m11.2.2.cmml">y</mi><mo id="S6.SS2.p4.11.m11.5.5.3.3.2.3" stretchy="false" xref="S6.SS2.p4.11.m11.5.5.3.3.1.cmml">)</mo></mrow></mrow><mo id="S6.SS2.p4.11.m11.5.5.2" xref="S6.SS2.p4.11.m11.5.5.2.cmml">=</mo><mrow id="S6.SS2.p4.11.m11.5.5.1" xref="S6.SS2.p4.11.m11.5.5.1.cmml"><mi id="S6.SS2.p4.11.m11.5.5.1.3" xref="S6.SS2.p4.11.m11.5.5.1.3.cmml">ν</mi><mo id="S6.SS2.p4.11.m11.5.5.1.2" xref="S6.SS2.p4.11.m11.5.5.1.2.cmml">⁢</mo><mrow id="S6.SS2.p4.11.m11.5.5.1.1.1" xref="S6.SS2.p4.11.m11.5.5.1.1.1.1.cmml"><mo id="S6.SS2.p4.11.m11.5.5.1.1.1.2" stretchy="false" xref="S6.SS2.p4.11.m11.5.5.1.1.1.1.cmml">(</mo><mrow id="S6.SS2.p4.11.m11.5.5.1.1.1.1" xref="S6.SS2.p4.11.m11.5.5.1.1.1.1.cmml"><msub id="S6.SS2.p4.11.m11.5.5.1.1.1.1.2" xref="S6.SS2.p4.11.m11.5.5.1.1.1.1.2.cmml"><mi id="S6.SS2.p4.11.m11.5.5.1.1.1.1.2.2" mathvariant="normal" xref="S6.SS2.p4.11.m11.5.5.1.1.1.1.2.2.cmml">Δ</mi><mi id="S6.SS2.p4.11.m11.5.5.1.1.1.1.2.3" xref="S6.SS2.p4.11.m11.5.5.1.1.1.1.2.3.cmml">X</mi></msub><mo id="S6.SS2.p4.11.m11.5.5.1.1.1.1.1" xref="S6.SS2.p4.11.m11.5.5.1.1.1.1.1.cmml">⁢</mo><mrow id="S6.SS2.p4.11.m11.5.5.1.1.1.1.3.2" xref="S6.SS2.p4.11.m11.5.5.1.1.1.1.3.1.cmml"><mo id="S6.SS2.p4.11.m11.5.5.1.1.1.1.3.2.1" stretchy="false" xref="S6.SS2.p4.11.m11.5.5.1.1.1.1.3.1.cmml">(</mo><mi id="S6.SS2.p4.11.m11.3.3" xref="S6.SS2.p4.11.m11.3.3.cmml">x</mi><mo id="S6.SS2.p4.11.m11.5.5.1.1.1.1.3.2.2" xref="S6.SS2.p4.11.m11.5.5.1.1.1.1.3.1.cmml">,</mo><mi id="S6.SS2.p4.11.m11.4.4" xref="S6.SS2.p4.11.m11.4.4.cmml">y</mi><mo id="S6.SS2.p4.11.m11.5.5.1.1.1.1.3.2.3" stretchy="false" xref="S6.SS2.p4.11.m11.5.5.1.1.1.1.3.1.cmml">)</mo></mrow></mrow><mo id="S6.SS2.p4.11.m11.5.5.1.1.1.3" stretchy="false" xref="S6.SS2.p4.11.m11.5.5.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p4.11.m11.5b"><apply id="S6.SS2.p4.11.m11.5.5.cmml" xref="S6.SS2.p4.11.m11.5.5"><eq id="S6.SS2.p4.11.m11.5.5.2.cmml" xref="S6.SS2.p4.11.m11.5.5.2"></eq><apply id="S6.SS2.p4.11.m11.5.5.3.cmml" xref="S6.SS2.p4.11.m11.5.5.3"><times id="S6.SS2.p4.11.m11.5.5.3.1.cmml" xref="S6.SS2.p4.11.m11.5.5.3.1"></times><ci id="S6.SS2.p4.11.m11.5.5.3.2.cmml" xref="S6.SS2.p4.11.m11.5.5.3.2">𝜈</ci><interval closure="open" id="S6.SS2.p4.11.m11.5.5.3.3.1.cmml" xref="S6.SS2.p4.11.m11.5.5.3.3.2"><ci id="S6.SS2.p4.11.m11.1.1.cmml" xref="S6.SS2.p4.11.m11.1.1">𝑥</ci><ci id="S6.SS2.p4.11.m11.2.2.cmml" xref="S6.SS2.p4.11.m11.2.2">𝑦</ci></interval></apply><apply id="S6.SS2.p4.11.m11.5.5.1.cmml" xref="S6.SS2.p4.11.m11.5.5.1"><times id="S6.SS2.p4.11.m11.5.5.1.2.cmml" xref="S6.SS2.p4.11.m11.5.5.1.2"></times><ci id="S6.SS2.p4.11.m11.5.5.1.3.cmml" xref="S6.SS2.p4.11.m11.5.5.1.3">𝜈</ci><apply id="S6.SS2.p4.11.m11.5.5.1.1.1.1.cmml" xref="S6.SS2.p4.11.m11.5.5.1.1.1"><times id="S6.SS2.p4.11.m11.5.5.1.1.1.1.1.cmml" xref="S6.SS2.p4.11.m11.5.5.1.1.1.1.1"></times><apply id="S6.SS2.p4.11.m11.5.5.1.1.1.1.2.cmml" xref="S6.SS2.p4.11.m11.5.5.1.1.1.1.2"><csymbol cd="ambiguous" id="S6.SS2.p4.11.m11.5.5.1.1.1.1.2.1.cmml" xref="S6.SS2.p4.11.m11.5.5.1.1.1.1.2">subscript</csymbol><ci id="S6.SS2.p4.11.m11.5.5.1.1.1.1.2.2.cmml" xref="S6.SS2.p4.11.m11.5.5.1.1.1.1.2.2">Δ</ci><ci id="S6.SS2.p4.11.m11.5.5.1.1.1.1.2.3.cmml" xref="S6.SS2.p4.11.m11.5.5.1.1.1.1.2.3">𝑋</ci></apply><interval closure="open" id="S6.SS2.p4.11.m11.5.5.1.1.1.1.3.1.cmml" xref="S6.SS2.p4.11.m11.5.5.1.1.1.1.3.2"><ci id="S6.SS2.p4.11.m11.3.3.cmml" xref="S6.SS2.p4.11.m11.3.3">𝑥</ci><ci id="S6.SS2.p4.11.m11.4.4.cmml" xref="S6.SS2.p4.11.m11.4.4">𝑦</ci></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p4.11.m11.5c">\nu(x,y)=\nu(\Delta_{X}(x,y))</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p4.11.m11.5d">italic_ν ( italic_x , italic_y ) = italic_ν ( roman_Δ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ( italic_x , italic_y ) )</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="S6.Thmtheorem9"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem9.1.1.1">Definition 6.9</span></span><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem9.2.2">.</span> </h6> <div class="ltx_para" id="S6.Thmtheorem9.p1"> <p class="ltx_p" id="S6.Thmtheorem9.p1.3">For <math alttext="E\subseteq\omega_{1}" class="ltx_Math" display="inline" id="S6.Thmtheorem9.p1.1.m1.1"><semantics id="S6.Thmtheorem9.p1.1.m1.1a"><mrow id="S6.Thmtheorem9.p1.1.m1.1.1" xref="S6.Thmtheorem9.p1.1.m1.1.1.cmml"><mi id="S6.Thmtheorem9.p1.1.m1.1.1.2" xref="S6.Thmtheorem9.p1.1.m1.1.1.2.cmml">E</mi><mo id="S6.Thmtheorem9.p1.1.m1.1.1.1" xref="S6.Thmtheorem9.p1.1.m1.1.1.1.cmml">⊆</mo><msub id="S6.Thmtheorem9.p1.1.m1.1.1.3" xref="S6.Thmtheorem9.p1.1.m1.1.1.3.cmml"><mi id="S6.Thmtheorem9.p1.1.m1.1.1.3.2" xref="S6.Thmtheorem9.p1.1.m1.1.1.3.2.cmml">ω</mi><mn id="S6.Thmtheorem9.p1.1.m1.1.1.3.3" xref="S6.Thmtheorem9.p1.1.m1.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem9.p1.1.m1.1b"><apply id="S6.Thmtheorem9.p1.1.m1.1.1.cmml" xref="S6.Thmtheorem9.p1.1.m1.1.1"><subset id="S6.Thmtheorem9.p1.1.m1.1.1.1.cmml" xref="S6.Thmtheorem9.p1.1.m1.1.1.1"></subset><ci id="S6.Thmtheorem9.p1.1.m1.1.1.2.cmml" xref="S6.Thmtheorem9.p1.1.m1.1.1.2">𝐸</ci><apply id="S6.Thmtheorem9.p1.1.m1.1.1.3.cmml" xref="S6.Thmtheorem9.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S6.Thmtheorem9.p1.1.m1.1.1.3.1.cmml" xref="S6.Thmtheorem9.p1.1.m1.1.1.3">subscript</csymbol><ci id="S6.Thmtheorem9.p1.1.m1.1.1.3.2.cmml" xref="S6.Thmtheorem9.p1.1.m1.1.1.3.2">𝜔</ci><cn id="S6.Thmtheorem9.p1.1.m1.1.1.3.3.cmml" type="integer" xref="S6.Thmtheorem9.p1.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem9.p1.1.m1.1c">E\subseteq\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem9.p1.1.m1.1d">italic_E ⊆ italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> let <math alttext="P_{E}:=P_{E}(A,X)" class="ltx_Math" display="inline" id="S6.Thmtheorem9.p1.2.m2.2"><semantics id="S6.Thmtheorem9.p1.2.m2.2a"><mrow id="S6.Thmtheorem9.p1.2.m2.2.3" xref="S6.Thmtheorem9.p1.2.m2.2.3.cmml"><msub id="S6.Thmtheorem9.p1.2.m2.2.3.2" xref="S6.Thmtheorem9.p1.2.m2.2.3.2.cmml"><mi id="S6.Thmtheorem9.p1.2.m2.2.3.2.2" xref="S6.Thmtheorem9.p1.2.m2.2.3.2.2.cmml">P</mi><mi id="S6.Thmtheorem9.p1.2.m2.2.3.2.3" xref="S6.Thmtheorem9.p1.2.m2.2.3.2.3.cmml">E</mi></msub><mo id="S6.Thmtheorem9.p1.2.m2.2.3.1" lspace="0.278em" rspace="0.278em" xref="S6.Thmtheorem9.p1.2.m2.2.3.1.cmml">:=</mo><mrow id="S6.Thmtheorem9.p1.2.m2.2.3.3" xref="S6.Thmtheorem9.p1.2.m2.2.3.3.cmml"><msub id="S6.Thmtheorem9.p1.2.m2.2.3.3.2" xref="S6.Thmtheorem9.p1.2.m2.2.3.3.2.cmml"><mi id="S6.Thmtheorem9.p1.2.m2.2.3.3.2.2" xref="S6.Thmtheorem9.p1.2.m2.2.3.3.2.2.cmml">P</mi><mi id="S6.Thmtheorem9.p1.2.m2.2.3.3.2.3" xref="S6.Thmtheorem9.p1.2.m2.2.3.3.2.3.cmml">E</mi></msub><mo id="S6.Thmtheorem9.p1.2.m2.2.3.3.1" xref="S6.Thmtheorem9.p1.2.m2.2.3.3.1.cmml">⁢</mo><mrow id="S6.Thmtheorem9.p1.2.m2.2.3.3.3.2" xref="S6.Thmtheorem9.p1.2.m2.2.3.3.3.1.cmml"><mo id="S6.Thmtheorem9.p1.2.m2.2.3.3.3.2.1" stretchy="false" xref="S6.Thmtheorem9.p1.2.m2.2.3.3.3.1.cmml">(</mo><mi id="S6.Thmtheorem9.p1.2.m2.1.1" xref="S6.Thmtheorem9.p1.2.m2.1.1.cmml">A</mi><mo id="S6.Thmtheorem9.p1.2.m2.2.3.3.3.2.2" xref="S6.Thmtheorem9.p1.2.m2.2.3.3.3.1.cmml">,</mo><mi id="S6.Thmtheorem9.p1.2.m2.2.2" xref="S6.Thmtheorem9.p1.2.m2.2.2.cmml">X</mi><mo id="S6.Thmtheorem9.p1.2.m2.2.3.3.3.2.3" stretchy="false" xref="S6.Thmtheorem9.p1.2.m2.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem9.p1.2.m2.2b"><apply id="S6.Thmtheorem9.p1.2.m2.2.3.cmml" xref="S6.Thmtheorem9.p1.2.m2.2.3"><csymbol cd="latexml" id="S6.Thmtheorem9.p1.2.m2.2.3.1.cmml" xref="S6.Thmtheorem9.p1.2.m2.2.3.1">assign</csymbol><apply id="S6.Thmtheorem9.p1.2.m2.2.3.2.cmml" xref="S6.Thmtheorem9.p1.2.m2.2.3.2"><csymbol cd="ambiguous" id="S6.Thmtheorem9.p1.2.m2.2.3.2.1.cmml" xref="S6.Thmtheorem9.p1.2.m2.2.3.2">subscript</csymbol><ci id="S6.Thmtheorem9.p1.2.m2.2.3.2.2.cmml" xref="S6.Thmtheorem9.p1.2.m2.2.3.2.2">𝑃</ci><ci id="S6.Thmtheorem9.p1.2.m2.2.3.2.3.cmml" xref="S6.Thmtheorem9.p1.2.m2.2.3.2.3">𝐸</ci></apply><apply id="S6.Thmtheorem9.p1.2.m2.2.3.3.cmml" xref="S6.Thmtheorem9.p1.2.m2.2.3.3"><times id="S6.Thmtheorem9.p1.2.m2.2.3.3.1.cmml" xref="S6.Thmtheorem9.p1.2.m2.2.3.3.1"></times><apply id="S6.Thmtheorem9.p1.2.m2.2.3.3.2.cmml" xref="S6.Thmtheorem9.p1.2.m2.2.3.3.2"><csymbol cd="ambiguous" id="S6.Thmtheorem9.p1.2.m2.2.3.3.2.1.cmml" xref="S6.Thmtheorem9.p1.2.m2.2.3.3.2">subscript</csymbol><ci id="S6.Thmtheorem9.p1.2.m2.2.3.3.2.2.cmml" xref="S6.Thmtheorem9.p1.2.m2.2.3.3.2.2">𝑃</ci><ci id="S6.Thmtheorem9.p1.2.m2.2.3.3.2.3.cmml" xref="S6.Thmtheorem9.p1.2.m2.2.3.3.2.3">𝐸</ci></apply><interval closure="open" id="S6.Thmtheorem9.p1.2.m2.2.3.3.3.1.cmml" xref="S6.Thmtheorem9.p1.2.m2.2.3.3.3.2"><ci id="S6.Thmtheorem9.p1.2.m2.1.1.cmml" xref="S6.Thmtheorem9.p1.2.m2.1.1">𝐴</ci><ci id="S6.Thmtheorem9.p1.2.m2.2.2.cmml" xref="S6.Thmtheorem9.p1.2.m2.2.2">𝑋</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem9.p1.2.m2.2c">P_{E}:=P_{E}(A,X)</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem9.p1.2.m2.2d">italic_P start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT := italic_P start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT ( italic_A , italic_X )</annotation></semantics></math> be defined as the set of <math alttext="p\in P" class="ltx_Math" display="inline" id="S6.Thmtheorem9.p1.3.m3.1"><semantics id="S6.Thmtheorem9.p1.3.m3.1a"><mrow id="S6.Thmtheorem9.p1.3.m3.1.1" xref="S6.Thmtheorem9.p1.3.m3.1.1.cmml"><mi id="S6.Thmtheorem9.p1.3.m3.1.1.2" xref="S6.Thmtheorem9.p1.3.m3.1.1.2.cmml">p</mi><mo id="S6.Thmtheorem9.p1.3.m3.1.1.1" xref="S6.Thmtheorem9.p1.3.m3.1.1.1.cmml">∈</mo><mi id="S6.Thmtheorem9.p1.3.m3.1.1.3" xref="S6.Thmtheorem9.p1.3.m3.1.1.3.cmml">P</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem9.p1.3.m3.1b"><apply id="S6.Thmtheorem9.p1.3.m3.1.1.cmml" xref="S6.Thmtheorem9.p1.3.m3.1.1"><in id="S6.Thmtheorem9.p1.3.m3.1.1.1.cmml" xref="S6.Thmtheorem9.p1.3.m3.1.1.1"></in><ci id="S6.Thmtheorem9.p1.3.m3.1.1.2.cmml" xref="S6.Thmtheorem9.p1.3.m3.1.1.2">𝑝</ci><ci id="S6.Thmtheorem9.p1.3.m3.1.1.3.cmml" xref="S6.Thmtheorem9.p1.3.m3.1.1.3">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem9.p1.3.m3.1c">p\in P</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem9.p1.3.m3.1d">italic_p ∈ italic_P</annotation></semantics></math> such that,</p> <ol class="ltx_enumerate" id="S6.I4"> <li class="ltx_item" id="S6.I4.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(i)</span> <div class="ltx_para" id="S6.I4.i1.p1"> <p class="ltx_p" id="S6.I4.i1.p1.3"><math alttext="\nu(p(\bar{a}))=\nu(a_{l})=\nu(a_{r})=\nu(a_{m})" class="ltx_Math" display="inline" id="S6.I4.i1.p1.1.m1.5"><semantics id="S6.I4.i1.p1.1.m1.5a"><mrow id="S6.I4.i1.p1.1.m1.5.5" xref="S6.I4.i1.p1.1.m1.5.5.cmml"><mrow id="S6.I4.i1.p1.1.m1.2.2.1" xref="S6.I4.i1.p1.1.m1.2.2.1.cmml"><mi id="S6.I4.i1.p1.1.m1.2.2.1.3" xref="S6.I4.i1.p1.1.m1.2.2.1.3.cmml">ν</mi><mo id="S6.I4.i1.p1.1.m1.2.2.1.2" xref="S6.I4.i1.p1.1.m1.2.2.1.2.cmml">⁢</mo><mrow id="S6.I4.i1.p1.1.m1.2.2.1.1.1" xref="S6.I4.i1.p1.1.m1.2.2.1.1.1.1.cmml"><mo id="S6.I4.i1.p1.1.m1.2.2.1.1.1.2" stretchy="false" xref="S6.I4.i1.p1.1.m1.2.2.1.1.1.1.cmml">(</mo><mrow id="S6.I4.i1.p1.1.m1.2.2.1.1.1.1" xref="S6.I4.i1.p1.1.m1.2.2.1.1.1.1.cmml"><mi id="S6.I4.i1.p1.1.m1.2.2.1.1.1.1.2" xref="S6.I4.i1.p1.1.m1.2.2.1.1.1.1.2.cmml">p</mi><mo id="S6.I4.i1.p1.1.m1.2.2.1.1.1.1.1" xref="S6.I4.i1.p1.1.m1.2.2.1.1.1.1.1.cmml">⁢</mo><mrow id="S6.I4.i1.p1.1.m1.2.2.1.1.1.1.3.2" xref="S6.I4.i1.p1.1.m1.1.1.cmml"><mo id="S6.I4.i1.p1.1.m1.2.2.1.1.1.1.3.2.1" stretchy="false" xref="S6.I4.i1.p1.1.m1.1.1.cmml">(</mo><mover accent="true" id="S6.I4.i1.p1.1.m1.1.1" xref="S6.I4.i1.p1.1.m1.1.1.cmml"><mi id="S6.I4.i1.p1.1.m1.1.1.2" xref="S6.I4.i1.p1.1.m1.1.1.2.cmml">a</mi><mo id="S6.I4.i1.p1.1.m1.1.1.1" xref="S6.I4.i1.p1.1.m1.1.1.1.cmml">¯</mo></mover><mo id="S6.I4.i1.p1.1.m1.2.2.1.1.1.1.3.2.2" stretchy="false" xref="S6.I4.i1.p1.1.m1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.I4.i1.p1.1.m1.2.2.1.1.1.3" stretchy="false" xref="S6.I4.i1.p1.1.m1.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.I4.i1.p1.1.m1.5.5.6" xref="S6.I4.i1.p1.1.m1.5.5.6.cmml">=</mo><mrow id="S6.I4.i1.p1.1.m1.3.3.2" xref="S6.I4.i1.p1.1.m1.3.3.2.cmml"><mi id="S6.I4.i1.p1.1.m1.3.3.2.3" xref="S6.I4.i1.p1.1.m1.3.3.2.3.cmml">ν</mi><mo id="S6.I4.i1.p1.1.m1.3.3.2.2" xref="S6.I4.i1.p1.1.m1.3.3.2.2.cmml">⁢</mo><mrow id="S6.I4.i1.p1.1.m1.3.3.2.1.1" xref="S6.I4.i1.p1.1.m1.3.3.2.1.1.1.cmml"><mo id="S6.I4.i1.p1.1.m1.3.3.2.1.1.2" stretchy="false" xref="S6.I4.i1.p1.1.m1.3.3.2.1.1.1.cmml">(</mo><msub id="S6.I4.i1.p1.1.m1.3.3.2.1.1.1" xref="S6.I4.i1.p1.1.m1.3.3.2.1.1.1.cmml"><mi id="S6.I4.i1.p1.1.m1.3.3.2.1.1.1.2" xref="S6.I4.i1.p1.1.m1.3.3.2.1.1.1.2.cmml">a</mi><mi id="S6.I4.i1.p1.1.m1.3.3.2.1.1.1.3" xref="S6.I4.i1.p1.1.m1.3.3.2.1.1.1.3.cmml">l</mi></msub><mo id="S6.I4.i1.p1.1.m1.3.3.2.1.1.3" stretchy="false" xref="S6.I4.i1.p1.1.m1.3.3.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.I4.i1.p1.1.m1.5.5.7" xref="S6.I4.i1.p1.1.m1.5.5.7.cmml">=</mo><mrow id="S6.I4.i1.p1.1.m1.4.4.3" xref="S6.I4.i1.p1.1.m1.4.4.3.cmml"><mi id="S6.I4.i1.p1.1.m1.4.4.3.3" xref="S6.I4.i1.p1.1.m1.4.4.3.3.cmml">ν</mi><mo id="S6.I4.i1.p1.1.m1.4.4.3.2" xref="S6.I4.i1.p1.1.m1.4.4.3.2.cmml">⁢</mo><mrow id="S6.I4.i1.p1.1.m1.4.4.3.1.1" xref="S6.I4.i1.p1.1.m1.4.4.3.1.1.1.cmml"><mo id="S6.I4.i1.p1.1.m1.4.4.3.1.1.2" stretchy="false" xref="S6.I4.i1.p1.1.m1.4.4.3.1.1.1.cmml">(</mo><msub id="S6.I4.i1.p1.1.m1.4.4.3.1.1.1" xref="S6.I4.i1.p1.1.m1.4.4.3.1.1.1.cmml"><mi id="S6.I4.i1.p1.1.m1.4.4.3.1.1.1.2" xref="S6.I4.i1.p1.1.m1.4.4.3.1.1.1.2.cmml">a</mi><mi id="S6.I4.i1.p1.1.m1.4.4.3.1.1.1.3" xref="S6.I4.i1.p1.1.m1.4.4.3.1.1.1.3.cmml">r</mi></msub><mo id="S6.I4.i1.p1.1.m1.4.4.3.1.1.3" stretchy="false" xref="S6.I4.i1.p1.1.m1.4.4.3.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.I4.i1.p1.1.m1.5.5.8" xref="S6.I4.i1.p1.1.m1.5.5.8.cmml">=</mo><mrow id="S6.I4.i1.p1.1.m1.5.5.4" xref="S6.I4.i1.p1.1.m1.5.5.4.cmml"><mi id="S6.I4.i1.p1.1.m1.5.5.4.3" xref="S6.I4.i1.p1.1.m1.5.5.4.3.cmml">ν</mi><mo id="S6.I4.i1.p1.1.m1.5.5.4.2" xref="S6.I4.i1.p1.1.m1.5.5.4.2.cmml">⁢</mo><mrow id="S6.I4.i1.p1.1.m1.5.5.4.1.1" xref="S6.I4.i1.p1.1.m1.5.5.4.1.1.1.cmml"><mo id="S6.I4.i1.p1.1.m1.5.5.4.1.1.2" stretchy="false" xref="S6.I4.i1.p1.1.m1.5.5.4.1.1.1.cmml">(</mo><msub id="S6.I4.i1.p1.1.m1.5.5.4.1.1.1" xref="S6.I4.i1.p1.1.m1.5.5.4.1.1.1.cmml"><mi id="S6.I4.i1.p1.1.m1.5.5.4.1.1.1.2" xref="S6.I4.i1.p1.1.m1.5.5.4.1.1.1.2.cmml">a</mi><mi id="S6.I4.i1.p1.1.m1.5.5.4.1.1.1.3" xref="S6.I4.i1.p1.1.m1.5.5.4.1.1.1.3.cmml">m</mi></msub><mo id="S6.I4.i1.p1.1.m1.5.5.4.1.1.3" stretchy="false" xref="S6.I4.i1.p1.1.m1.5.5.4.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.I4.i1.p1.1.m1.5b"><apply id="S6.I4.i1.p1.1.m1.5.5.cmml" xref="S6.I4.i1.p1.1.m1.5.5"><and id="S6.I4.i1.p1.1.m1.5.5a.cmml" xref="S6.I4.i1.p1.1.m1.5.5"></and><apply id="S6.I4.i1.p1.1.m1.5.5b.cmml" xref="S6.I4.i1.p1.1.m1.5.5"><eq id="S6.I4.i1.p1.1.m1.5.5.6.cmml" xref="S6.I4.i1.p1.1.m1.5.5.6"></eq><apply id="S6.I4.i1.p1.1.m1.2.2.1.cmml" xref="S6.I4.i1.p1.1.m1.2.2.1"><times id="S6.I4.i1.p1.1.m1.2.2.1.2.cmml" xref="S6.I4.i1.p1.1.m1.2.2.1.2"></times><ci id="S6.I4.i1.p1.1.m1.2.2.1.3.cmml" xref="S6.I4.i1.p1.1.m1.2.2.1.3">𝜈</ci><apply id="S6.I4.i1.p1.1.m1.2.2.1.1.1.1.cmml" xref="S6.I4.i1.p1.1.m1.2.2.1.1.1"><times id="S6.I4.i1.p1.1.m1.2.2.1.1.1.1.1.cmml" xref="S6.I4.i1.p1.1.m1.2.2.1.1.1.1.1"></times><ci id="S6.I4.i1.p1.1.m1.2.2.1.1.1.1.2.cmml" xref="S6.I4.i1.p1.1.m1.2.2.1.1.1.1.2">𝑝</ci><apply id="S6.I4.i1.p1.1.m1.1.1.cmml" xref="S6.I4.i1.p1.1.m1.2.2.1.1.1.1.3.2"><ci id="S6.I4.i1.p1.1.m1.1.1.1.cmml" xref="S6.I4.i1.p1.1.m1.1.1.1">¯</ci><ci id="S6.I4.i1.p1.1.m1.1.1.2.cmml" xref="S6.I4.i1.p1.1.m1.1.1.2">𝑎</ci></apply></apply></apply><apply id="S6.I4.i1.p1.1.m1.3.3.2.cmml" xref="S6.I4.i1.p1.1.m1.3.3.2"><times id="S6.I4.i1.p1.1.m1.3.3.2.2.cmml" xref="S6.I4.i1.p1.1.m1.3.3.2.2"></times><ci id="S6.I4.i1.p1.1.m1.3.3.2.3.cmml" xref="S6.I4.i1.p1.1.m1.3.3.2.3">𝜈</ci><apply id="S6.I4.i1.p1.1.m1.3.3.2.1.1.1.cmml" xref="S6.I4.i1.p1.1.m1.3.3.2.1.1"><csymbol cd="ambiguous" id="S6.I4.i1.p1.1.m1.3.3.2.1.1.1.1.cmml" xref="S6.I4.i1.p1.1.m1.3.3.2.1.1">subscript</csymbol><ci id="S6.I4.i1.p1.1.m1.3.3.2.1.1.1.2.cmml" xref="S6.I4.i1.p1.1.m1.3.3.2.1.1.1.2">𝑎</ci><ci id="S6.I4.i1.p1.1.m1.3.3.2.1.1.1.3.cmml" xref="S6.I4.i1.p1.1.m1.3.3.2.1.1.1.3">𝑙</ci></apply></apply></apply><apply id="S6.I4.i1.p1.1.m1.5.5c.cmml" xref="S6.I4.i1.p1.1.m1.5.5"><eq id="S6.I4.i1.p1.1.m1.5.5.7.cmml" xref="S6.I4.i1.p1.1.m1.5.5.7"></eq><share href="https://arxiv.org/html/2503.13728v1#S6.I4.i1.p1.1.m1.3.3.2.cmml" id="S6.I4.i1.p1.1.m1.5.5d.cmml" xref="S6.I4.i1.p1.1.m1.5.5"></share><apply id="S6.I4.i1.p1.1.m1.4.4.3.cmml" xref="S6.I4.i1.p1.1.m1.4.4.3"><times id="S6.I4.i1.p1.1.m1.4.4.3.2.cmml" xref="S6.I4.i1.p1.1.m1.4.4.3.2"></times><ci id="S6.I4.i1.p1.1.m1.4.4.3.3.cmml" xref="S6.I4.i1.p1.1.m1.4.4.3.3">𝜈</ci><apply id="S6.I4.i1.p1.1.m1.4.4.3.1.1.1.cmml" xref="S6.I4.i1.p1.1.m1.4.4.3.1.1"><csymbol cd="ambiguous" id="S6.I4.i1.p1.1.m1.4.4.3.1.1.1.1.cmml" xref="S6.I4.i1.p1.1.m1.4.4.3.1.1">subscript</csymbol><ci id="S6.I4.i1.p1.1.m1.4.4.3.1.1.1.2.cmml" xref="S6.I4.i1.p1.1.m1.4.4.3.1.1.1.2">𝑎</ci><ci id="S6.I4.i1.p1.1.m1.4.4.3.1.1.1.3.cmml" xref="S6.I4.i1.p1.1.m1.4.4.3.1.1.1.3">𝑟</ci></apply></apply></apply><apply id="S6.I4.i1.p1.1.m1.5.5e.cmml" xref="S6.I4.i1.p1.1.m1.5.5"><eq id="S6.I4.i1.p1.1.m1.5.5.8.cmml" xref="S6.I4.i1.p1.1.m1.5.5.8"></eq><share href="https://arxiv.org/html/2503.13728v1#S6.I4.i1.p1.1.m1.4.4.3.cmml" id="S6.I4.i1.p1.1.m1.5.5f.cmml" xref="S6.I4.i1.p1.1.m1.5.5"></share><apply id="S6.I4.i1.p1.1.m1.5.5.4.cmml" xref="S6.I4.i1.p1.1.m1.5.5.4"><times id="S6.I4.i1.p1.1.m1.5.5.4.2.cmml" xref="S6.I4.i1.p1.1.m1.5.5.4.2"></times><ci id="S6.I4.i1.p1.1.m1.5.5.4.3.cmml" xref="S6.I4.i1.p1.1.m1.5.5.4.3">𝜈</ci><apply id="S6.I4.i1.p1.1.m1.5.5.4.1.1.1.cmml" xref="S6.I4.i1.p1.1.m1.5.5.4.1.1"><csymbol cd="ambiguous" id="S6.I4.i1.p1.1.m1.5.5.4.1.1.1.1.cmml" xref="S6.I4.i1.p1.1.m1.5.5.4.1.1">subscript</csymbol><ci id="S6.I4.i1.p1.1.m1.5.5.4.1.1.1.2.cmml" xref="S6.I4.i1.p1.1.m1.5.5.4.1.1.1.2">𝑎</ci><ci id="S6.I4.i1.p1.1.m1.5.5.4.1.1.1.3.cmml" xref="S6.I4.i1.p1.1.m1.5.5.4.1.1.1.3">𝑚</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I4.i1.p1.1.m1.5c">\nu(p(\bar{a}))=\nu(a_{l})=\nu(a_{r})=\nu(a_{m})</annotation><annotation encoding="application/x-llamapun" id="S6.I4.i1.p1.1.m1.5d">italic_ν ( italic_p ( over¯ start_ARG italic_a end_ARG ) ) = italic_ν ( italic_a start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ) = italic_ν ( italic_a start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) = italic_ν ( italic_a start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT )</annotation></semantics></math>, for all <math alttext="\bar{a}\in\operatorname{dom}(p)" class="ltx_Math" display="inline" id="S6.I4.i1.p1.2.m2.2"><semantics id="S6.I4.i1.p1.2.m2.2a"><mrow id="S6.I4.i1.p1.2.m2.2.3" xref="S6.I4.i1.p1.2.m2.2.3.cmml"><mover accent="true" id="S6.I4.i1.p1.2.m2.2.3.2" xref="S6.I4.i1.p1.2.m2.2.3.2.cmml"><mi id="S6.I4.i1.p1.2.m2.2.3.2.2" xref="S6.I4.i1.p1.2.m2.2.3.2.2.cmml">a</mi><mo id="S6.I4.i1.p1.2.m2.2.3.2.1" xref="S6.I4.i1.p1.2.m2.2.3.2.1.cmml">¯</mo></mover><mo id="S6.I4.i1.p1.2.m2.2.3.1" xref="S6.I4.i1.p1.2.m2.2.3.1.cmml">∈</mo><mrow id="S6.I4.i1.p1.2.m2.2.3.3.2" xref="S6.I4.i1.p1.2.m2.2.3.3.1.cmml"><mi id="S6.I4.i1.p1.2.m2.1.1" xref="S6.I4.i1.p1.2.m2.1.1.cmml">dom</mi><mo id="S6.I4.i1.p1.2.m2.2.3.3.2a" xref="S6.I4.i1.p1.2.m2.2.3.3.1.cmml">⁡</mo><mrow id="S6.I4.i1.p1.2.m2.2.3.3.2.1" xref="S6.I4.i1.p1.2.m2.2.3.3.1.cmml"><mo id="S6.I4.i1.p1.2.m2.2.3.3.2.1.1" stretchy="false" xref="S6.I4.i1.p1.2.m2.2.3.3.1.cmml">(</mo><mi id="S6.I4.i1.p1.2.m2.2.2" xref="S6.I4.i1.p1.2.m2.2.2.cmml">p</mi><mo id="S6.I4.i1.p1.2.m2.2.3.3.2.1.2" stretchy="false" xref="S6.I4.i1.p1.2.m2.2.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.I4.i1.p1.2.m2.2b"><apply id="S6.I4.i1.p1.2.m2.2.3.cmml" xref="S6.I4.i1.p1.2.m2.2.3"><in id="S6.I4.i1.p1.2.m2.2.3.1.cmml" xref="S6.I4.i1.p1.2.m2.2.3.1"></in><apply id="S6.I4.i1.p1.2.m2.2.3.2.cmml" xref="S6.I4.i1.p1.2.m2.2.3.2"><ci id="S6.I4.i1.p1.2.m2.2.3.2.1.cmml" xref="S6.I4.i1.p1.2.m2.2.3.2.1">¯</ci><ci id="S6.I4.i1.p1.2.m2.2.3.2.2.cmml" xref="S6.I4.i1.p1.2.m2.2.3.2.2">𝑎</ci></apply><apply id="S6.I4.i1.p1.2.m2.2.3.3.1.cmml" xref="S6.I4.i1.p1.2.m2.2.3.3.2"><ci id="S6.I4.i1.p1.2.m2.1.1.cmml" xref="S6.I4.i1.p1.2.m2.1.1">dom</ci><ci id="S6.I4.i1.p1.2.m2.2.2.cmml" xref="S6.I4.i1.p1.2.m2.2.2">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I4.i1.p1.2.m2.2c">\bar{a}\in\operatorname{dom}(p)</annotation><annotation encoding="application/x-llamapun" id="S6.I4.i1.p1.2.m2.2d">over¯ start_ARG italic_a end_ARG ∈ roman_dom ( italic_p )</annotation></semantics></math>. And thus <math alttext="\nu(\bar{a})" class="ltx_Math" display="inline" id="S6.I4.i1.p1.3.m3.1"><semantics id="S6.I4.i1.p1.3.m3.1a"><mrow id="S6.I4.i1.p1.3.m3.1.2" xref="S6.I4.i1.p1.3.m3.1.2.cmml"><mi id="S6.I4.i1.p1.3.m3.1.2.2" xref="S6.I4.i1.p1.3.m3.1.2.2.cmml">ν</mi><mo id="S6.I4.i1.p1.3.m3.1.2.1" xref="S6.I4.i1.p1.3.m3.1.2.1.cmml">⁢</mo><mrow id="S6.I4.i1.p1.3.m3.1.2.3.2" xref="S6.I4.i1.p1.3.m3.1.1.cmml"><mo id="S6.I4.i1.p1.3.m3.1.2.3.2.1" stretchy="false" xref="S6.I4.i1.p1.3.m3.1.1.cmml">(</mo><mover accent="true" id="S6.I4.i1.p1.3.m3.1.1" xref="S6.I4.i1.p1.3.m3.1.1.cmml"><mi id="S6.I4.i1.p1.3.m3.1.1.2" xref="S6.I4.i1.p1.3.m3.1.1.2.cmml">a</mi><mo id="S6.I4.i1.p1.3.m3.1.1.1" xref="S6.I4.i1.p1.3.m3.1.1.1.cmml">¯</mo></mover><mo id="S6.I4.i1.p1.3.m3.1.2.3.2.2" stretchy="false" xref="S6.I4.i1.p1.3.m3.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.I4.i1.p1.3.m3.1b"><apply id="S6.I4.i1.p1.3.m3.1.2.cmml" xref="S6.I4.i1.p1.3.m3.1.2"><times id="S6.I4.i1.p1.3.m3.1.2.1.cmml" xref="S6.I4.i1.p1.3.m3.1.2.1"></times><ci id="S6.I4.i1.p1.3.m3.1.2.2.cmml" xref="S6.I4.i1.p1.3.m3.1.2.2">𝜈</ci><apply id="S6.I4.i1.p1.3.m3.1.1.cmml" xref="S6.I4.i1.p1.3.m3.1.2.3.2"><ci id="S6.I4.i1.p1.3.m3.1.1.1.cmml" xref="S6.I4.i1.p1.3.m3.1.1.1">¯</ci><ci id="S6.I4.i1.p1.3.m3.1.1.2.cmml" xref="S6.I4.i1.p1.3.m3.1.1.2">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I4.i1.p1.3.m3.1c">\nu(\bar{a})</annotation><annotation encoding="application/x-llamapun" id="S6.I4.i1.p1.3.m3.1d">italic_ν ( over¯ start_ARG italic_a end_ARG )</annotation></semantics></math> is defined as this common value.</p> </div> </li> <li class="ltx_item" id="S6.I4.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(ii)</span> <div class="ltx_para" id="S6.I4.i2.p1"> <p class="ltx_p" id="S6.I4.i2.p1.3"><math alttext="\nu(a_{r},b_{l})=\nu(p(\bar{a}),p(\bar{b}))" class="ltx_Math" display="inline" id="S6.I4.i2.p1.1.m1.6"><semantics id="S6.I4.i2.p1.1.m1.6a"><mrow id="S6.I4.i2.p1.1.m1.6.6" xref="S6.I4.i2.p1.1.m1.6.6.cmml"><mrow id="S6.I4.i2.p1.1.m1.4.4.2" xref="S6.I4.i2.p1.1.m1.4.4.2.cmml"><mi id="S6.I4.i2.p1.1.m1.4.4.2.4" xref="S6.I4.i2.p1.1.m1.4.4.2.4.cmml">ν</mi><mo id="S6.I4.i2.p1.1.m1.4.4.2.3" xref="S6.I4.i2.p1.1.m1.4.4.2.3.cmml">⁢</mo><mrow id="S6.I4.i2.p1.1.m1.4.4.2.2.2" xref="S6.I4.i2.p1.1.m1.4.4.2.2.3.cmml"><mo id="S6.I4.i2.p1.1.m1.4.4.2.2.2.3" stretchy="false" xref="S6.I4.i2.p1.1.m1.4.4.2.2.3.cmml">(</mo><msub id="S6.I4.i2.p1.1.m1.3.3.1.1.1.1" xref="S6.I4.i2.p1.1.m1.3.3.1.1.1.1.cmml"><mi id="S6.I4.i2.p1.1.m1.3.3.1.1.1.1.2" xref="S6.I4.i2.p1.1.m1.3.3.1.1.1.1.2.cmml">a</mi><mi id="S6.I4.i2.p1.1.m1.3.3.1.1.1.1.3" xref="S6.I4.i2.p1.1.m1.3.3.1.1.1.1.3.cmml">r</mi></msub><mo id="S6.I4.i2.p1.1.m1.4.4.2.2.2.4" xref="S6.I4.i2.p1.1.m1.4.4.2.2.3.cmml">,</mo><msub id="S6.I4.i2.p1.1.m1.4.4.2.2.2.2" xref="S6.I4.i2.p1.1.m1.4.4.2.2.2.2.cmml"><mi id="S6.I4.i2.p1.1.m1.4.4.2.2.2.2.2" xref="S6.I4.i2.p1.1.m1.4.4.2.2.2.2.2.cmml">b</mi><mi id="S6.I4.i2.p1.1.m1.4.4.2.2.2.2.3" xref="S6.I4.i2.p1.1.m1.4.4.2.2.2.2.3.cmml">l</mi></msub><mo id="S6.I4.i2.p1.1.m1.4.4.2.2.2.5" stretchy="false" xref="S6.I4.i2.p1.1.m1.4.4.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.I4.i2.p1.1.m1.6.6.5" xref="S6.I4.i2.p1.1.m1.6.6.5.cmml">=</mo><mrow id="S6.I4.i2.p1.1.m1.6.6.4" xref="S6.I4.i2.p1.1.m1.6.6.4.cmml"><mi id="S6.I4.i2.p1.1.m1.6.6.4.4" xref="S6.I4.i2.p1.1.m1.6.6.4.4.cmml">ν</mi><mo id="S6.I4.i2.p1.1.m1.6.6.4.3" xref="S6.I4.i2.p1.1.m1.6.6.4.3.cmml">⁢</mo><mrow id="S6.I4.i2.p1.1.m1.6.6.4.2.2" xref="S6.I4.i2.p1.1.m1.6.6.4.2.3.cmml"><mo id="S6.I4.i2.p1.1.m1.6.6.4.2.2.3" stretchy="false" xref="S6.I4.i2.p1.1.m1.6.6.4.2.3.cmml">(</mo><mrow id="S6.I4.i2.p1.1.m1.5.5.3.1.1.1" xref="S6.I4.i2.p1.1.m1.5.5.3.1.1.1.cmml"><mi id="S6.I4.i2.p1.1.m1.5.5.3.1.1.1.2" xref="S6.I4.i2.p1.1.m1.5.5.3.1.1.1.2.cmml">p</mi><mo id="S6.I4.i2.p1.1.m1.5.5.3.1.1.1.1" xref="S6.I4.i2.p1.1.m1.5.5.3.1.1.1.1.cmml">⁢</mo><mrow id="S6.I4.i2.p1.1.m1.5.5.3.1.1.1.3.2" xref="S6.I4.i2.p1.1.m1.1.1.cmml"><mo id="S6.I4.i2.p1.1.m1.5.5.3.1.1.1.3.2.1" stretchy="false" xref="S6.I4.i2.p1.1.m1.1.1.cmml">(</mo><mover accent="true" id="S6.I4.i2.p1.1.m1.1.1" xref="S6.I4.i2.p1.1.m1.1.1.cmml"><mi id="S6.I4.i2.p1.1.m1.1.1.2" xref="S6.I4.i2.p1.1.m1.1.1.2.cmml">a</mi><mo id="S6.I4.i2.p1.1.m1.1.1.1" xref="S6.I4.i2.p1.1.m1.1.1.1.cmml">¯</mo></mover><mo id="S6.I4.i2.p1.1.m1.5.5.3.1.1.1.3.2.2" stretchy="false" xref="S6.I4.i2.p1.1.m1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.I4.i2.p1.1.m1.6.6.4.2.2.4" xref="S6.I4.i2.p1.1.m1.6.6.4.2.3.cmml">,</mo><mrow id="S6.I4.i2.p1.1.m1.6.6.4.2.2.2" xref="S6.I4.i2.p1.1.m1.6.6.4.2.2.2.cmml"><mi id="S6.I4.i2.p1.1.m1.6.6.4.2.2.2.2" xref="S6.I4.i2.p1.1.m1.6.6.4.2.2.2.2.cmml">p</mi><mo id="S6.I4.i2.p1.1.m1.6.6.4.2.2.2.1" xref="S6.I4.i2.p1.1.m1.6.6.4.2.2.2.1.cmml">⁢</mo><mrow id="S6.I4.i2.p1.1.m1.6.6.4.2.2.2.3.2" xref="S6.I4.i2.p1.1.m1.2.2.cmml"><mo id="S6.I4.i2.p1.1.m1.6.6.4.2.2.2.3.2.1" stretchy="false" xref="S6.I4.i2.p1.1.m1.2.2.cmml">(</mo><mover accent="true" id="S6.I4.i2.p1.1.m1.2.2" xref="S6.I4.i2.p1.1.m1.2.2.cmml"><mi id="S6.I4.i2.p1.1.m1.2.2.2" xref="S6.I4.i2.p1.1.m1.2.2.2.cmml">b</mi><mo id="S6.I4.i2.p1.1.m1.2.2.1" xref="S6.I4.i2.p1.1.m1.2.2.1.cmml">¯</mo></mover><mo id="S6.I4.i2.p1.1.m1.6.6.4.2.2.2.3.2.2" stretchy="false" xref="S6.I4.i2.p1.1.m1.2.2.cmml">)</mo></mrow></mrow><mo id="S6.I4.i2.p1.1.m1.6.6.4.2.2.5" stretchy="false" xref="S6.I4.i2.p1.1.m1.6.6.4.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.I4.i2.p1.1.m1.6b"><apply id="S6.I4.i2.p1.1.m1.6.6.cmml" xref="S6.I4.i2.p1.1.m1.6.6"><eq id="S6.I4.i2.p1.1.m1.6.6.5.cmml" xref="S6.I4.i2.p1.1.m1.6.6.5"></eq><apply id="S6.I4.i2.p1.1.m1.4.4.2.cmml" xref="S6.I4.i2.p1.1.m1.4.4.2"><times id="S6.I4.i2.p1.1.m1.4.4.2.3.cmml" xref="S6.I4.i2.p1.1.m1.4.4.2.3"></times><ci id="S6.I4.i2.p1.1.m1.4.4.2.4.cmml" xref="S6.I4.i2.p1.1.m1.4.4.2.4">𝜈</ci><interval closure="open" id="S6.I4.i2.p1.1.m1.4.4.2.2.3.cmml" xref="S6.I4.i2.p1.1.m1.4.4.2.2.2"><apply id="S6.I4.i2.p1.1.m1.3.3.1.1.1.1.cmml" xref="S6.I4.i2.p1.1.m1.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S6.I4.i2.p1.1.m1.3.3.1.1.1.1.1.cmml" xref="S6.I4.i2.p1.1.m1.3.3.1.1.1.1">subscript</csymbol><ci id="S6.I4.i2.p1.1.m1.3.3.1.1.1.1.2.cmml" xref="S6.I4.i2.p1.1.m1.3.3.1.1.1.1.2">𝑎</ci><ci id="S6.I4.i2.p1.1.m1.3.3.1.1.1.1.3.cmml" xref="S6.I4.i2.p1.1.m1.3.3.1.1.1.1.3">𝑟</ci></apply><apply id="S6.I4.i2.p1.1.m1.4.4.2.2.2.2.cmml" xref="S6.I4.i2.p1.1.m1.4.4.2.2.2.2"><csymbol cd="ambiguous" id="S6.I4.i2.p1.1.m1.4.4.2.2.2.2.1.cmml" xref="S6.I4.i2.p1.1.m1.4.4.2.2.2.2">subscript</csymbol><ci id="S6.I4.i2.p1.1.m1.4.4.2.2.2.2.2.cmml" xref="S6.I4.i2.p1.1.m1.4.4.2.2.2.2.2">𝑏</ci><ci id="S6.I4.i2.p1.1.m1.4.4.2.2.2.2.3.cmml" xref="S6.I4.i2.p1.1.m1.4.4.2.2.2.2.3">𝑙</ci></apply></interval></apply><apply id="S6.I4.i2.p1.1.m1.6.6.4.cmml" xref="S6.I4.i2.p1.1.m1.6.6.4"><times id="S6.I4.i2.p1.1.m1.6.6.4.3.cmml" xref="S6.I4.i2.p1.1.m1.6.6.4.3"></times><ci id="S6.I4.i2.p1.1.m1.6.6.4.4.cmml" xref="S6.I4.i2.p1.1.m1.6.6.4.4">𝜈</ci><interval closure="open" id="S6.I4.i2.p1.1.m1.6.6.4.2.3.cmml" xref="S6.I4.i2.p1.1.m1.6.6.4.2.2"><apply id="S6.I4.i2.p1.1.m1.5.5.3.1.1.1.cmml" xref="S6.I4.i2.p1.1.m1.5.5.3.1.1.1"><times id="S6.I4.i2.p1.1.m1.5.5.3.1.1.1.1.cmml" xref="S6.I4.i2.p1.1.m1.5.5.3.1.1.1.1"></times><ci id="S6.I4.i2.p1.1.m1.5.5.3.1.1.1.2.cmml" xref="S6.I4.i2.p1.1.m1.5.5.3.1.1.1.2">𝑝</ci><apply id="S6.I4.i2.p1.1.m1.1.1.cmml" xref="S6.I4.i2.p1.1.m1.5.5.3.1.1.1.3.2"><ci id="S6.I4.i2.p1.1.m1.1.1.1.cmml" xref="S6.I4.i2.p1.1.m1.1.1.1">¯</ci><ci id="S6.I4.i2.p1.1.m1.1.1.2.cmml" xref="S6.I4.i2.p1.1.m1.1.1.2">𝑎</ci></apply></apply><apply id="S6.I4.i2.p1.1.m1.6.6.4.2.2.2.cmml" xref="S6.I4.i2.p1.1.m1.6.6.4.2.2.2"><times id="S6.I4.i2.p1.1.m1.6.6.4.2.2.2.1.cmml" xref="S6.I4.i2.p1.1.m1.6.6.4.2.2.2.1"></times><ci id="S6.I4.i2.p1.1.m1.6.6.4.2.2.2.2.cmml" xref="S6.I4.i2.p1.1.m1.6.6.4.2.2.2.2">𝑝</ci><apply id="S6.I4.i2.p1.1.m1.2.2.cmml" xref="S6.I4.i2.p1.1.m1.6.6.4.2.2.2.3.2"><ci id="S6.I4.i2.p1.1.m1.2.2.1.cmml" xref="S6.I4.i2.p1.1.m1.2.2.1">¯</ci><ci id="S6.I4.i2.p1.1.m1.2.2.2.cmml" xref="S6.I4.i2.p1.1.m1.2.2.2">𝑏</ci></apply></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I4.i2.p1.1.m1.6c">\nu(a_{r},b_{l})=\nu(p(\bar{a}),p(\bar{b}))</annotation><annotation encoding="application/x-llamapun" id="S6.I4.i2.p1.1.m1.6d">italic_ν ( italic_a start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ) = italic_ν ( italic_p ( over¯ start_ARG italic_a end_ARG ) , italic_p ( over¯ start_ARG italic_b end_ARG ) )</annotation></semantics></math> for all <math alttext="\bar{a}<_{b}\bar{b}" class="ltx_Math" display="inline" id="S6.I4.i2.p1.2.m2.1"><semantics id="S6.I4.i2.p1.2.m2.1a"><mrow id="S6.I4.i2.p1.2.m2.1.1" xref="S6.I4.i2.p1.2.m2.1.1.cmml"><mover accent="true" id="S6.I4.i2.p1.2.m2.1.1.2" xref="S6.I4.i2.p1.2.m2.1.1.2.cmml"><mi id="S6.I4.i2.p1.2.m2.1.1.2.2" xref="S6.I4.i2.p1.2.m2.1.1.2.2.cmml">a</mi><mo id="S6.I4.i2.p1.2.m2.1.1.2.1" xref="S6.I4.i2.p1.2.m2.1.1.2.1.cmml">¯</mo></mover><msub id="S6.I4.i2.p1.2.m2.1.1.1" xref="S6.I4.i2.p1.2.m2.1.1.1.cmml"><mo id="S6.I4.i2.p1.2.m2.1.1.1.2" xref="S6.I4.i2.p1.2.m2.1.1.1.2.cmml"><</mo><mi id="S6.I4.i2.p1.2.m2.1.1.1.3" xref="S6.I4.i2.p1.2.m2.1.1.1.3.cmml">b</mi></msub><mover accent="true" id="S6.I4.i2.p1.2.m2.1.1.3" xref="S6.I4.i2.p1.2.m2.1.1.3.cmml"><mi id="S6.I4.i2.p1.2.m2.1.1.3.2" xref="S6.I4.i2.p1.2.m2.1.1.3.2.cmml">b</mi><mo id="S6.I4.i2.p1.2.m2.1.1.3.1" xref="S6.I4.i2.p1.2.m2.1.1.3.1.cmml">¯</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S6.I4.i2.p1.2.m2.1b"><apply id="S6.I4.i2.p1.2.m2.1.1.cmml" xref="S6.I4.i2.p1.2.m2.1.1"><apply id="S6.I4.i2.p1.2.m2.1.1.1.cmml" xref="S6.I4.i2.p1.2.m2.1.1.1"><csymbol cd="ambiguous" id="S6.I4.i2.p1.2.m2.1.1.1.1.cmml" xref="S6.I4.i2.p1.2.m2.1.1.1">subscript</csymbol><lt id="S6.I4.i2.p1.2.m2.1.1.1.2.cmml" xref="S6.I4.i2.p1.2.m2.1.1.1.2"></lt><ci id="S6.I4.i2.p1.2.m2.1.1.1.3.cmml" xref="S6.I4.i2.p1.2.m2.1.1.1.3">𝑏</ci></apply><apply id="S6.I4.i2.p1.2.m2.1.1.2.cmml" xref="S6.I4.i2.p1.2.m2.1.1.2"><ci id="S6.I4.i2.p1.2.m2.1.1.2.1.cmml" xref="S6.I4.i2.p1.2.m2.1.1.2.1">¯</ci><ci id="S6.I4.i2.p1.2.m2.1.1.2.2.cmml" xref="S6.I4.i2.p1.2.m2.1.1.2.2">𝑎</ci></apply><apply id="S6.I4.i2.p1.2.m2.1.1.3.cmml" xref="S6.I4.i2.p1.2.m2.1.1.3"><ci id="S6.I4.i2.p1.2.m2.1.1.3.1.cmml" xref="S6.I4.i2.p1.2.m2.1.1.3.1">¯</ci><ci id="S6.I4.i2.p1.2.m2.1.1.3.2.cmml" xref="S6.I4.i2.p1.2.m2.1.1.3.2">𝑏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I4.i2.p1.2.m2.1c">\bar{a}<_{b}\bar{b}</annotation><annotation encoding="application/x-llamapun" id="S6.I4.i2.p1.2.m2.1d">over¯ start_ARG italic_a end_ARG < start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT over¯ start_ARG italic_b end_ARG</annotation></semantics></math> in <math alttext="\operatorname{dom}(p)" class="ltx_Math" display="inline" id="S6.I4.i2.p1.3.m3.2"><semantics id="S6.I4.i2.p1.3.m3.2a"><mrow id="S6.I4.i2.p1.3.m3.2.3.2" xref="S6.I4.i2.p1.3.m3.2.3.1.cmml"><mi id="S6.I4.i2.p1.3.m3.1.1" xref="S6.I4.i2.p1.3.m3.1.1.cmml">dom</mi><mo id="S6.I4.i2.p1.3.m3.2.3.2a" xref="S6.I4.i2.p1.3.m3.2.3.1.cmml">⁡</mo><mrow id="S6.I4.i2.p1.3.m3.2.3.2.1" xref="S6.I4.i2.p1.3.m3.2.3.1.cmml"><mo id="S6.I4.i2.p1.3.m3.2.3.2.1.1" stretchy="false" xref="S6.I4.i2.p1.3.m3.2.3.1.cmml">(</mo><mi id="S6.I4.i2.p1.3.m3.2.2" xref="S6.I4.i2.p1.3.m3.2.2.cmml">p</mi><mo id="S6.I4.i2.p1.3.m3.2.3.2.1.2" stretchy="false" xref="S6.I4.i2.p1.3.m3.2.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.I4.i2.p1.3.m3.2b"><apply id="S6.I4.i2.p1.3.m3.2.3.1.cmml" xref="S6.I4.i2.p1.3.m3.2.3.2"><ci id="S6.I4.i2.p1.3.m3.1.1.cmml" xref="S6.I4.i2.p1.3.m3.1.1">dom</ci><ci id="S6.I4.i2.p1.3.m3.2.2.cmml" xref="S6.I4.i2.p1.3.m3.2.2">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I4.i2.p1.3.m3.2c">\operatorname{dom}(p)</annotation><annotation encoding="application/x-llamapun" id="S6.I4.i2.p1.3.m3.2d">roman_dom ( italic_p )</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S6.I4.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(iii)</span> <div class="ltx_para" id="S6.I4.i3.p1"> <p class="ltx_p" id="S6.I4.i3.p1.8">Let <math alttext="\bar{a}\in\operatorname{dom}(p)" class="ltx_Math" display="inline" id="S6.I4.i3.p1.1.m1.2"><semantics id="S6.I4.i3.p1.1.m1.2a"><mrow id="S6.I4.i3.p1.1.m1.2.3" xref="S6.I4.i3.p1.1.m1.2.3.cmml"><mover accent="true" id="S6.I4.i3.p1.1.m1.2.3.2" xref="S6.I4.i3.p1.1.m1.2.3.2.cmml"><mi id="S6.I4.i3.p1.1.m1.2.3.2.2" xref="S6.I4.i3.p1.1.m1.2.3.2.2.cmml">a</mi><mo id="S6.I4.i3.p1.1.m1.2.3.2.1" xref="S6.I4.i3.p1.1.m1.2.3.2.1.cmml">¯</mo></mover><mo id="S6.I4.i3.p1.1.m1.2.3.1" xref="S6.I4.i3.p1.1.m1.2.3.1.cmml">∈</mo><mrow id="S6.I4.i3.p1.1.m1.2.3.3.2" xref="S6.I4.i3.p1.1.m1.2.3.3.1.cmml"><mi id="S6.I4.i3.p1.1.m1.1.1" xref="S6.I4.i3.p1.1.m1.1.1.cmml">dom</mi><mo id="S6.I4.i3.p1.1.m1.2.3.3.2a" xref="S6.I4.i3.p1.1.m1.2.3.3.1.cmml">⁡</mo><mrow id="S6.I4.i3.p1.1.m1.2.3.3.2.1" xref="S6.I4.i3.p1.1.m1.2.3.3.1.cmml"><mo id="S6.I4.i3.p1.1.m1.2.3.3.2.1.1" stretchy="false" xref="S6.I4.i3.p1.1.m1.2.3.3.1.cmml">(</mo><mi id="S6.I4.i3.p1.1.m1.2.2" xref="S6.I4.i3.p1.1.m1.2.2.cmml">p</mi><mo id="S6.I4.i3.p1.1.m1.2.3.3.2.1.2" stretchy="false" xref="S6.I4.i3.p1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.I4.i3.p1.1.m1.2b"><apply id="S6.I4.i3.p1.1.m1.2.3.cmml" xref="S6.I4.i3.p1.1.m1.2.3"><in id="S6.I4.i3.p1.1.m1.2.3.1.cmml" xref="S6.I4.i3.p1.1.m1.2.3.1"></in><apply id="S6.I4.i3.p1.1.m1.2.3.2.cmml" xref="S6.I4.i3.p1.1.m1.2.3.2"><ci id="S6.I4.i3.p1.1.m1.2.3.2.1.cmml" xref="S6.I4.i3.p1.1.m1.2.3.2.1">¯</ci><ci id="S6.I4.i3.p1.1.m1.2.3.2.2.cmml" xref="S6.I4.i3.p1.1.m1.2.3.2.2">𝑎</ci></apply><apply id="S6.I4.i3.p1.1.m1.2.3.3.1.cmml" xref="S6.I4.i3.p1.1.m1.2.3.3.2"><ci id="S6.I4.i3.p1.1.m1.1.1.cmml" xref="S6.I4.i3.p1.1.m1.1.1">dom</ci><ci id="S6.I4.i3.p1.1.m1.2.2.cmml" xref="S6.I4.i3.p1.1.m1.2.2">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I4.i3.p1.1.m1.2c">\bar{a}\in\operatorname{dom}(p)</annotation><annotation encoding="application/x-llamapun" id="S6.I4.i3.p1.1.m1.2d">over¯ start_ARG italic_a end_ARG ∈ roman_dom ( italic_p )</annotation></semantics></math>, <math alttext="I" class="ltx_Math" display="inline" id="S6.I4.i3.p1.2.m2.1"><semantics id="S6.I4.i3.p1.2.m2.1a"><mi id="S6.I4.i3.p1.2.m2.1.1" xref="S6.I4.i3.p1.2.m2.1.1.cmml">I</mi><annotation-xml encoding="MathML-Content" id="S6.I4.i3.p1.2.m2.1b"><ci id="S6.I4.i3.p1.2.m2.1.1.cmml" xref="S6.I4.i3.p1.2.m2.1.1">𝐼</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.I4.i3.p1.2.m2.1c">I</annotation><annotation encoding="application/x-llamapun" id="S6.I4.i3.p1.2.m2.1d">italic_I</annotation></semantics></math> be the complementary interval <math alttext="A\setminus\nu(\bar{a})" class="ltx_Math" display="inline" id="S6.I4.i3.p1.3.m3.1"><semantics id="S6.I4.i3.p1.3.m3.1a"><mrow id="S6.I4.i3.p1.3.m3.1.2" xref="S6.I4.i3.p1.3.m3.1.2.cmml"><mi id="S6.I4.i3.p1.3.m3.1.2.2" xref="S6.I4.i3.p1.3.m3.1.2.2.cmml">A</mi><mo id="S6.I4.i3.p1.3.m3.1.2.1" xref="S6.I4.i3.p1.3.m3.1.2.1.cmml">∖</mo><mrow id="S6.I4.i3.p1.3.m3.1.2.3" xref="S6.I4.i3.p1.3.m3.1.2.3.cmml"><mi id="S6.I4.i3.p1.3.m3.1.2.3.2" xref="S6.I4.i3.p1.3.m3.1.2.3.2.cmml">ν</mi><mo id="S6.I4.i3.p1.3.m3.1.2.3.1" xref="S6.I4.i3.p1.3.m3.1.2.3.1.cmml">⁢</mo><mrow id="S6.I4.i3.p1.3.m3.1.2.3.3.2" xref="S6.I4.i3.p1.3.m3.1.1.cmml"><mo id="S6.I4.i3.p1.3.m3.1.2.3.3.2.1" stretchy="false" xref="S6.I4.i3.p1.3.m3.1.1.cmml">(</mo><mover accent="true" id="S6.I4.i3.p1.3.m3.1.1" xref="S6.I4.i3.p1.3.m3.1.1.cmml"><mi id="S6.I4.i3.p1.3.m3.1.1.2" xref="S6.I4.i3.p1.3.m3.1.1.2.cmml">a</mi><mo id="S6.I4.i3.p1.3.m3.1.1.1" xref="S6.I4.i3.p1.3.m3.1.1.1.cmml">¯</mo></mover><mo id="S6.I4.i3.p1.3.m3.1.2.3.3.2.2" stretchy="false" xref="S6.I4.i3.p1.3.m3.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.I4.i3.p1.3.m3.1b"><apply id="S6.I4.i3.p1.3.m3.1.2.cmml" xref="S6.I4.i3.p1.3.m3.1.2"><setdiff id="S6.I4.i3.p1.3.m3.1.2.1.cmml" xref="S6.I4.i3.p1.3.m3.1.2.1"></setdiff><ci id="S6.I4.i3.p1.3.m3.1.2.2.cmml" xref="S6.I4.i3.p1.3.m3.1.2.2">𝐴</ci><apply id="S6.I4.i3.p1.3.m3.1.2.3.cmml" xref="S6.I4.i3.p1.3.m3.1.2.3"><times id="S6.I4.i3.p1.3.m3.1.2.3.1.cmml" xref="S6.I4.i3.p1.3.m3.1.2.3.1"></times><ci id="S6.I4.i3.p1.3.m3.1.2.3.2.cmml" xref="S6.I4.i3.p1.3.m3.1.2.3.2">𝜈</ci><apply id="S6.I4.i3.p1.3.m3.1.1.cmml" xref="S6.I4.i3.p1.3.m3.1.2.3.3.2"><ci id="S6.I4.i3.p1.3.m3.1.1.1.cmml" xref="S6.I4.i3.p1.3.m3.1.1.1">¯</ci><ci id="S6.I4.i3.p1.3.m3.1.1.2.cmml" xref="S6.I4.i3.p1.3.m3.1.1.2">𝑎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I4.i3.p1.3.m3.1c">A\setminus\nu(\bar{a})</annotation><annotation encoding="application/x-llamapun" id="S6.I4.i3.p1.3.m3.1d">italic_A ∖ italic_ν ( over¯ start_ARG italic_a end_ARG )</annotation></semantics></math> in which <math alttext="a_{l}" class="ltx_Math" display="inline" id="S6.I4.i3.p1.4.m4.1"><semantics id="S6.I4.i3.p1.4.m4.1a"><msub id="S6.I4.i3.p1.4.m4.1.1" xref="S6.I4.i3.p1.4.m4.1.1.cmml"><mi id="S6.I4.i3.p1.4.m4.1.1.2" xref="S6.I4.i3.p1.4.m4.1.1.2.cmml">a</mi><mi id="S6.I4.i3.p1.4.m4.1.1.3" xref="S6.I4.i3.p1.4.m4.1.1.3.cmml">l</mi></msub><annotation-xml encoding="MathML-Content" id="S6.I4.i3.p1.4.m4.1b"><apply id="S6.I4.i3.p1.4.m4.1.1.cmml" xref="S6.I4.i3.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S6.I4.i3.p1.4.m4.1.1.1.cmml" xref="S6.I4.i3.p1.4.m4.1.1">subscript</csymbol><ci id="S6.I4.i3.p1.4.m4.1.1.2.cmml" xref="S6.I4.i3.p1.4.m4.1.1.2">𝑎</ci><ci id="S6.I4.i3.p1.4.m4.1.1.3.cmml" xref="S6.I4.i3.p1.4.m4.1.1.3">𝑙</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I4.i3.p1.4.m4.1c">a_{l}</annotation><annotation encoding="application/x-llamapun" id="S6.I4.i3.p1.4.m4.1d">italic_a start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT</annotation></semantics></math> is (and then by <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.I4.i1" title="Item (i) ‣ Definition 6.9. ‣ 6.2. Forcing epimorphisms ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">(i)</span></a> also <math alttext="a_{r}" class="ltx_Math" display="inline" id="S6.I4.i3.p1.5.m5.1"><semantics id="S6.I4.i3.p1.5.m5.1a"><msub id="S6.I4.i3.p1.5.m5.1.1" xref="S6.I4.i3.p1.5.m5.1.1.cmml"><mi id="S6.I4.i3.p1.5.m5.1.1.2" xref="S6.I4.i3.p1.5.m5.1.1.2.cmml">a</mi><mi id="S6.I4.i3.p1.5.m5.1.1.3" xref="S6.I4.i3.p1.5.m5.1.1.3.cmml">r</mi></msub><annotation-xml encoding="MathML-Content" id="S6.I4.i3.p1.5.m5.1b"><apply id="S6.I4.i3.p1.5.m5.1.1.cmml" xref="S6.I4.i3.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S6.I4.i3.p1.5.m5.1.1.1.cmml" xref="S6.I4.i3.p1.5.m5.1.1">subscript</csymbol><ci id="S6.I4.i3.p1.5.m5.1.1.2.cmml" xref="S6.I4.i3.p1.5.m5.1.1.2">𝑎</ci><ci id="S6.I4.i3.p1.5.m5.1.1.3.cmml" xref="S6.I4.i3.p1.5.m5.1.1.3">𝑟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I4.i3.p1.5.m5.1c">a_{r}</annotation><annotation encoding="application/x-llamapun" id="S6.I4.i3.p1.5.m5.1d">italic_a start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT</annotation></semantics></math>) and <math alttext="J" class="ltx_Math" display="inline" id="S6.I4.i3.p1.6.m6.1"><semantics id="S6.I4.i3.p1.6.m6.1a"><mi id="S6.I4.i3.p1.6.m6.1.1" xref="S6.I4.i3.p1.6.m6.1.1.cmml">J</mi><annotation-xml encoding="MathML-Content" id="S6.I4.i3.p1.6.m6.1b"><ci id="S6.I4.i3.p1.6.m6.1.1.cmml" xref="S6.I4.i3.p1.6.m6.1.1">𝐽</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.I4.i3.p1.6.m6.1c">J</annotation><annotation encoding="application/x-llamapun" id="S6.I4.i3.p1.6.m6.1d">italic_J</annotation></semantics></math> be the complementary interval of <math alttext="X\setminus\nu(\bar{a})" class="ltx_Math" display="inline" id="S6.I4.i3.p1.7.m7.1"><semantics id="S6.I4.i3.p1.7.m7.1a"><mrow id="S6.I4.i3.p1.7.m7.1.2" xref="S6.I4.i3.p1.7.m7.1.2.cmml"><mi id="S6.I4.i3.p1.7.m7.1.2.2" xref="S6.I4.i3.p1.7.m7.1.2.2.cmml">X</mi><mo id="S6.I4.i3.p1.7.m7.1.2.1" xref="S6.I4.i3.p1.7.m7.1.2.1.cmml">∖</mo><mrow id="S6.I4.i3.p1.7.m7.1.2.3" xref="S6.I4.i3.p1.7.m7.1.2.3.cmml"><mi id="S6.I4.i3.p1.7.m7.1.2.3.2" xref="S6.I4.i3.p1.7.m7.1.2.3.2.cmml">ν</mi><mo id="S6.I4.i3.p1.7.m7.1.2.3.1" xref="S6.I4.i3.p1.7.m7.1.2.3.1.cmml">⁢</mo><mrow id="S6.I4.i3.p1.7.m7.1.2.3.3.2" xref="S6.I4.i3.p1.7.m7.1.1.cmml"><mo id="S6.I4.i3.p1.7.m7.1.2.3.3.2.1" stretchy="false" xref="S6.I4.i3.p1.7.m7.1.1.cmml">(</mo><mover accent="true" id="S6.I4.i3.p1.7.m7.1.1" xref="S6.I4.i3.p1.7.m7.1.1.cmml"><mi id="S6.I4.i3.p1.7.m7.1.1.2" xref="S6.I4.i3.p1.7.m7.1.1.2.cmml">a</mi><mo id="S6.I4.i3.p1.7.m7.1.1.1" xref="S6.I4.i3.p1.7.m7.1.1.1.cmml">¯</mo></mover><mo id="S6.I4.i3.p1.7.m7.1.2.3.3.2.2" stretchy="false" xref="S6.I4.i3.p1.7.m7.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.I4.i3.p1.7.m7.1b"><apply id="S6.I4.i3.p1.7.m7.1.2.cmml" xref="S6.I4.i3.p1.7.m7.1.2"><setdiff id="S6.I4.i3.p1.7.m7.1.2.1.cmml" xref="S6.I4.i3.p1.7.m7.1.2.1"></setdiff><ci id="S6.I4.i3.p1.7.m7.1.2.2.cmml" xref="S6.I4.i3.p1.7.m7.1.2.2">𝑋</ci><apply id="S6.I4.i3.p1.7.m7.1.2.3.cmml" xref="S6.I4.i3.p1.7.m7.1.2.3"><times id="S6.I4.i3.p1.7.m7.1.2.3.1.cmml" xref="S6.I4.i3.p1.7.m7.1.2.3.1"></times><ci id="S6.I4.i3.p1.7.m7.1.2.3.2.cmml" xref="S6.I4.i3.p1.7.m7.1.2.3.2">𝜈</ci><apply id="S6.I4.i3.p1.7.m7.1.1.cmml" xref="S6.I4.i3.p1.7.m7.1.2.3.3.2"><ci id="S6.I4.i3.p1.7.m7.1.1.1.cmml" xref="S6.I4.i3.p1.7.m7.1.1.1">¯</ci><ci id="S6.I4.i3.p1.7.m7.1.1.2.cmml" xref="S6.I4.i3.p1.7.m7.1.1.2">𝑎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I4.i3.p1.7.m7.1c">X\setminus\nu(\bar{a})</annotation><annotation encoding="application/x-llamapun" id="S6.I4.i3.p1.7.m7.1d">italic_X ∖ italic_ν ( over¯ start_ARG italic_a end_ARG )</annotation></semantics></math> in which <math alttext="p(\bar{a})" class="ltx_Math" display="inline" id="S6.I4.i3.p1.8.m8.1"><semantics id="S6.I4.i3.p1.8.m8.1a"><mrow id="S6.I4.i3.p1.8.m8.1.2" xref="S6.I4.i3.p1.8.m8.1.2.cmml"><mi id="S6.I4.i3.p1.8.m8.1.2.2" xref="S6.I4.i3.p1.8.m8.1.2.2.cmml">p</mi><mo id="S6.I4.i3.p1.8.m8.1.2.1" xref="S6.I4.i3.p1.8.m8.1.2.1.cmml">⁢</mo><mrow id="S6.I4.i3.p1.8.m8.1.2.3.2" xref="S6.I4.i3.p1.8.m8.1.1.cmml"><mo id="S6.I4.i3.p1.8.m8.1.2.3.2.1" stretchy="false" xref="S6.I4.i3.p1.8.m8.1.1.cmml">(</mo><mover accent="true" id="S6.I4.i3.p1.8.m8.1.1" xref="S6.I4.i3.p1.8.m8.1.1.cmml"><mi id="S6.I4.i3.p1.8.m8.1.1.2" xref="S6.I4.i3.p1.8.m8.1.1.2.cmml">a</mi><mo id="S6.I4.i3.p1.8.m8.1.1.1" xref="S6.I4.i3.p1.8.m8.1.1.1.cmml">¯</mo></mover><mo id="S6.I4.i3.p1.8.m8.1.2.3.2.2" stretchy="false" xref="S6.I4.i3.p1.8.m8.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.I4.i3.p1.8.m8.1b"><apply id="S6.I4.i3.p1.8.m8.1.2.cmml" xref="S6.I4.i3.p1.8.m8.1.2"><times id="S6.I4.i3.p1.8.m8.1.2.1.cmml" xref="S6.I4.i3.p1.8.m8.1.2.1"></times><ci id="S6.I4.i3.p1.8.m8.1.2.2.cmml" xref="S6.I4.i3.p1.8.m8.1.2.2">𝑝</ci><apply id="S6.I4.i3.p1.8.m8.1.1.cmml" xref="S6.I4.i3.p1.8.m8.1.2.3.2"><ci id="S6.I4.i3.p1.8.m8.1.1.1.cmml" xref="S6.I4.i3.p1.8.m8.1.1.1">¯</ci><ci id="S6.I4.i3.p1.8.m8.1.1.2.cmml" xref="S6.I4.i3.p1.8.m8.1.1.2">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I4.i3.p1.8.m8.1c">p(\bar{a})</annotation><annotation encoding="application/x-llamapun" id="S6.I4.i3.p1.8.m8.1d">italic_p ( over¯ start_ARG italic_a end_ARG )</annotation></semantics></math> is. Then,</p> <ul class="ltx_itemize" id="S6.I4.i3.I1"> <li class="ltx_item" id="S6.I4.i3.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S6.I4.i3.I1.i1.p1"> <p class="ltx_p" id="S6.I4.i3.I1.i1.p1.4">If <math alttext="a_{l}" class="ltx_Math" display="inline" id="S6.I4.i3.I1.i1.p1.1.m1.1"><semantics id="S6.I4.i3.I1.i1.p1.1.m1.1a"><msub id="S6.I4.i3.I1.i1.p1.1.m1.1.1" xref="S6.I4.i3.I1.i1.p1.1.m1.1.1.cmml"><mi id="S6.I4.i3.I1.i1.p1.1.m1.1.1.2" xref="S6.I4.i3.I1.i1.p1.1.m1.1.1.2.cmml">a</mi><mi id="S6.I4.i3.I1.i1.p1.1.m1.1.1.3" xref="S6.I4.i3.I1.i1.p1.1.m1.1.1.3.cmml">l</mi></msub><annotation-xml encoding="MathML-Content" id="S6.I4.i3.I1.i1.p1.1.m1.1b"><apply id="S6.I4.i3.I1.i1.p1.1.m1.1.1.cmml" xref="S6.I4.i3.I1.i1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S6.I4.i3.I1.i1.p1.1.m1.1.1.1.cmml" xref="S6.I4.i3.I1.i1.p1.1.m1.1.1">subscript</csymbol><ci id="S6.I4.i3.I1.i1.p1.1.m1.1.1.2.cmml" xref="S6.I4.i3.I1.i1.p1.1.m1.1.1.2">𝑎</ci><ci id="S6.I4.i3.I1.i1.p1.1.m1.1.1.3.cmml" xref="S6.I4.i3.I1.i1.p1.1.m1.1.1.3">𝑙</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I4.i3.I1.i1.p1.1.m1.1c">a_{l}</annotation><annotation encoding="application/x-llamapun" id="S6.I4.i3.I1.i1.p1.1.m1.1d">italic_a start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT</annotation></semantics></math> is the left endpoint of <math alttext="I" class="ltx_Math" display="inline" id="S6.I4.i3.I1.i1.p1.2.m2.1"><semantics id="S6.I4.i3.I1.i1.p1.2.m2.1a"><mi id="S6.I4.i3.I1.i1.p1.2.m2.1.1" xref="S6.I4.i3.I1.i1.p1.2.m2.1.1.cmml">I</mi><annotation-xml encoding="MathML-Content" id="S6.I4.i3.I1.i1.p1.2.m2.1b"><ci id="S6.I4.i3.I1.i1.p1.2.m2.1.1.cmml" xref="S6.I4.i3.I1.i1.p1.2.m2.1.1">𝐼</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.I4.i3.I1.i1.p1.2.m2.1c">I</annotation><annotation encoding="application/x-llamapun" id="S6.I4.i3.I1.i1.p1.2.m2.1d">italic_I</annotation></semantics></math> then so is <math alttext="p(\bar{a})" class="ltx_Math" display="inline" id="S6.I4.i3.I1.i1.p1.3.m3.1"><semantics id="S6.I4.i3.I1.i1.p1.3.m3.1a"><mrow id="S6.I4.i3.I1.i1.p1.3.m3.1.2" xref="S6.I4.i3.I1.i1.p1.3.m3.1.2.cmml"><mi id="S6.I4.i3.I1.i1.p1.3.m3.1.2.2" xref="S6.I4.i3.I1.i1.p1.3.m3.1.2.2.cmml">p</mi><mo id="S6.I4.i3.I1.i1.p1.3.m3.1.2.1" xref="S6.I4.i3.I1.i1.p1.3.m3.1.2.1.cmml">⁢</mo><mrow id="S6.I4.i3.I1.i1.p1.3.m3.1.2.3.2" xref="S6.I4.i3.I1.i1.p1.3.m3.1.1.cmml"><mo id="S6.I4.i3.I1.i1.p1.3.m3.1.2.3.2.1" stretchy="false" xref="S6.I4.i3.I1.i1.p1.3.m3.1.1.cmml">(</mo><mover accent="true" id="S6.I4.i3.I1.i1.p1.3.m3.1.1" xref="S6.I4.i3.I1.i1.p1.3.m3.1.1.cmml"><mi id="S6.I4.i3.I1.i1.p1.3.m3.1.1.2" xref="S6.I4.i3.I1.i1.p1.3.m3.1.1.2.cmml">a</mi><mo id="S6.I4.i3.I1.i1.p1.3.m3.1.1.1" xref="S6.I4.i3.I1.i1.p1.3.m3.1.1.1.cmml">¯</mo></mover><mo id="S6.I4.i3.I1.i1.p1.3.m3.1.2.3.2.2" stretchy="false" xref="S6.I4.i3.I1.i1.p1.3.m3.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.I4.i3.I1.i1.p1.3.m3.1b"><apply id="S6.I4.i3.I1.i1.p1.3.m3.1.2.cmml" xref="S6.I4.i3.I1.i1.p1.3.m3.1.2"><times id="S6.I4.i3.I1.i1.p1.3.m3.1.2.1.cmml" xref="S6.I4.i3.I1.i1.p1.3.m3.1.2.1"></times><ci id="S6.I4.i3.I1.i1.p1.3.m3.1.2.2.cmml" xref="S6.I4.i3.I1.i1.p1.3.m3.1.2.2">𝑝</ci><apply id="S6.I4.i3.I1.i1.p1.3.m3.1.1.cmml" xref="S6.I4.i3.I1.i1.p1.3.m3.1.2.3.2"><ci id="S6.I4.i3.I1.i1.p1.3.m3.1.1.1.cmml" xref="S6.I4.i3.I1.i1.p1.3.m3.1.1.1">¯</ci><ci id="S6.I4.i3.I1.i1.p1.3.m3.1.1.2.cmml" xref="S6.I4.i3.I1.i1.p1.3.m3.1.1.2">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I4.i3.I1.i1.p1.3.m3.1c">p(\bar{a})</annotation><annotation encoding="application/x-llamapun" id="S6.I4.i3.I1.i1.p1.3.m3.1d">italic_p ( over¯ start_ARG italic_a end_ARG )</annotation></semantics></math> of <math alttext="J" class="ltx_Math" display="inline" id="S6.I4.i3.I1.i1.p1.4.m4.1"><semantics id="S6.I4.i3.I1.i1.p1.4.m4.1a"><mi id="S6.I4.i3.I1.i1.p1.4.m4.1.1" xref="S6.I4.i3.I1.i1.p1.4.m4.1.1.cmml">J</mi><annotation-xml encoding="MathML-Content" id="S6.I4.i3.I1.i1.p1.4.m4.1b"><ci id="S6.I4.i3.I1.i1.p1.4.m4.1.1.cmml" xref="S6.I4.i3.I1.i1.p1.4.m4.1.1">𝐽</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.I4.i3.I1.i1.p1.4.m4.1c">J</annotation><annotation encoding="application/x-llamapun" id="S6.I4.i3.I1.i1.p1.4.m4.1d">italic_J</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S6.I4.i3.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S6.I4.i3.I1.i2.p1"> <p class="ltx_p" id="S6.I4.i3.I1.i2.p1.4">If <math alttext="a_{r}" class="ltx_Math" display="inline" id="S6.I4.i3.I1.i2.p1.1.m1.1"><semantics id="S6.I4.i3.I1.i2.p1.1.m1.1a"><msub id="S6.I4.i3.I1.i2.p1.1.m1.1.1" xref="S6.I4.i3.I1.i2.p1.1.m1.1.1.cmml"><mi id="S6.I4.i3.I1.i2.p1.1.m1.1.1.2" xref="S6.I4.i3.I1.i2.p1.1.m1.1.1.2.cmml">a</mi><mi id="S6.I4.i3.I1.i2.p1.1.m1.1.1.3" xref="S6.I4.i3.I1.i2.p1.1.m1.1.1.3.cmml">r</mi></msub><annotation-xml encoding="MathML-Content" id="S6.I4.i3.I1.i2.p1.1.m1.1b"><apply id="S6.I4.i3.I1.i2.p1.1.m1.1.1.cmml" xref="S6.I4.i3.I1.i2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S6.I4.i3.I1.i2.p1.1.m1.1.1.1.cmml" xref="S6.I4.i3.I1.i2.p1.1.m1.1.1">subscript</csymbol><ci id="S6.I4.i3.I1.i2.p1.1.m1.1.1.2.cmml" xref="S6.I4.i3.I1.i2.p1.1.m1.1.1.2">𝑎</ci><ci id="S6.I4.i3.I1.i2.p1.1.m1.1.1.3.cmml" xref="S6.I4.i3.I1.i2.p1.1.m1.1.1.3">𝑟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I4.i3.I1.i2.p1.1.m1.1c">a_{r}</annotation><annotation encoding="application/x-llamapun" id="S6.I4.i3.I1.i2.p1.1.m1.1d">italic_a start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT</annotation></semantics></math> is the right endpoint of <math alttext="I" class="ltx_Math" display="inline" id="S6.I4.i3.I1.i2.p1.2.m2.1"><semantics id="S6.I4.i3.I1.i2.p1.2.m2.1a"><mi id="S6.I4.i3.I1.i2.p1.2.m2.1.1" xref="S6.I4.i3.I1.i2.p1.2.m2.1.1.cmml">I</mi><annotation-xml encoding="MathML-Content" id="S6.I4.i3.I1.i2.p1.2.m2.1b"><ci id="S6.I4.i3.I1.i2.p1.2.m2.1.1.cmml" xref="S6.I4.i3.I1.i2.p1.2.m2.1.1">𝐼</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.I4.i3.I1.i2.p1.2.m2.1c">I</annotation><annotation encoding="application/x-llamapun" id="S6.I4.i3.I1.i2.p1.2.m2.1d">italic_I</annotation></semantics></math> then so is <math alttext="p(\bar{a})" class="ltx_Math" display="inline" id="S6.I4.i3.I1.i2.p1.3.m3.1"><semantics id="S6.I4.i3.I1.i2.p1.3.m3.1a"><mrow id="S6.I4.i3.I1.i2.p1.3.m3.1.2" xref="S6.I4.i3.I1.i2.p1.3.m3.1.2.cmml"><mi id="S6.I4.i3.I1.i2.p1.3.m3.1.2.2" xref="S6.I4.i3.I1.i2.p1.3.m3.1.2.2.cmml">p</mi><mo id="S6.I4.i3.I1.i2.p1.3.m3.1.2.1" xref="S6.I4.i3.I1.i2.p1.3.m3.1.2.1.cmml">⁢</mo><mrow id="S6.I4.i3.I1.i2.p1.3.m3.1.2.3.2" xref="S6.I4.i3.I1.i2.p1.3.m3.1.1.cmml"><mo id="S6.I4.i3.I1.i2.p1.3.m3.1.2.3.2.1" stretchy="false" xref="S6.I4.i3.I1.i2.p1.3.m3.1.1.cmml">(</mo><mover accent="true" id="S6.I4.i3.I1.i2.p1.3.m3.1.1" xref="S6.I4.i3.I1.i2.p1.3.m3.1.1.cmml"><mi id="S6.I4.i3.I1.i2.p1.3.m3.1.1.2" xref="S6.I4.i3.I1.i2.p1.3.m3.1.1.2.cmml">a</mi><mo id="S6.I4.i3.I1.i2.p1.3.m3.1.1.1" xref="S6.I4.i3.I1.i2.p1.3.m3.1.1.1.cmml">¯</mo></mover><mo id="S6.I4.i3.I1.i2.p1.3.m3.1.2.3.2.2" stretchy="false" xref="S6.I4.i3.I1.i2.p1.3.m3.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.I4.i3.I1.i2.p1.3.m3.1b"><apply id="S6.I4.i3.I1.i2.p1.3.m3.1.2.cmml" xref="S6.I4.i3.I1.i2.p1.3.m3.1.2"><times id="S6.I4.i3.I1.i2.p1.3.m3.1.2.1.cmml" xref="S6.I4.i3.I1.i2.p1.3.m3.1.2.1"></times><ci id="S6.I4.i3.I1.i2.p1.3.m3.1.2.2.cmml" xref="S6.I4.i3.I1.i2.p1.3.m3.1.2.2">𝑝</ci><apply id="S6.I4.i3.I1.i2.p1.3.m3.1.1.cmml" xref="S6.I4.i3.I1.i2.p1.3.m3.1.2.3.2"><ci id="S6.I4.i3.I1.i2.p1.3.m3.1.1.1.cmml" xref="S6.I4.i3.I1.i2.p1.3.m3.1.1.1">¯</ci><ci id="S6.I4.i3.I1.i2.p1.3.m3.1.1.2.cmml" xref="S6.I4.i3.I1.i2.p1.3.m3.1.1.2">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I4.i3.I1.i2.p1.3.m3.1c">p(\bar{a})</annotation><annotation encoding="application/x-llamapun" id="S6.I4.i3.I1.i2.p1.3.m3.1d">italic_p ( over¯ start_ARG italic_a end_ARG )</annotation></semantics></math> of <math alttext="J" class="ltx_Math" display="inline" id="S6.I4.i3.I1.i2.p1.4.m4.1"><semantics id="S6.I4.i3.I1.i2.p1.4.m4.1a"><mi id="S6.I4.i3.I1.i2.p1.4.m4.1.1" xref="S6.I4.i3.I1.i2.p1.4.m4.1.1.cmml">J</mi><annotation-xml encoding="MathML-Content" id="S6.I4.i3.I1.i2.p1.4.m4.1b"><ci id="S6.I4.i3.I1.i2.p1.4.m4.1.1.cmml" xref="S6.I4.i3.I1.i2.p1.4.m4.1.1">𝐽</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.I4.i3.I1.i2.p1.4.m4.1c">J</annotation><annotation encoding="application/x-llamapun" id="S6.I4.i3.I1.i2.p1.4.m4.1d">italic_J</annotation></semantics></math>.</p> </div> </li> </ul> <p class="ltx_p" id="S6.I4.i3.p1.10">Note than that if <math alttext="I=[a_{l},a_{r}]" class="ltx_Math" display="inline" id="S6.I4.i3.p1.9.m1.2"><semantics id="S6.I4.i3.p1.9.m1.2a"><mrow id="S6.I4.i3.p1.9.m1.2.2" xref="S6.I4.i3.p1.9.m1.2.2.cmml"><mi id="S6.I4.i3.p1.9.m1.2.2.4" xref="S6.I4.i3.p1.9.m1.2.2.4.cmml">I</mi><mo id="S6.I4.i3.p1.9.m1.2.2.3" xref="S6.I4.i3.p1.9.m1.2.2.3.cmml">=</mo><mrow id="S6.I4.i3.p1.9.m1.2.2.2.2" xref="S6.I4.i3.p1.9.m1.2.2.2.3.cmml"><mo id="S6.I4.i3.p1.9.m1.2.2.2.2.3" stretchy="false" xref="S6.I4.i3.p1.9.m1.2.2.2.3.cmml">[</mo><msub id="S6.I4.i3.p1.9.m1.1.1.1.1.1" xref="S6.I4.i3.p1.9.m1.1.1.1.1.1.cmml"><mi id="S6.I4.i3.p1.9.m1.1.1.1.1.1.2" xref="S6.I4.i3.p1.9.m1.1.1.1.1.1.2.cmml">a</mi><mi id="S6.I4.i3.p1.9.m1.1.1.1.1.1.3" xref="S6.I4.i3.p1.9.m1.1.1.1.1.1.3.cmml">l</mi></msub><mo id="S6.I4.i3.p1.9.m1.2.2.2.2.4" xref="S6.I4.i3.p1.9.m1.2.2.2.3.cmml">,</mo><msub id="S6.I4.i3.p1.9.m1.2.2.2.2.2" xref="S6.I4.i3.p1.9.m1.2.2.2.2.2.cmml"><mi id="S6.I4.i3.p1.9.m1.2.2.2.2.2.2" xref="S6.I4.i3.p1.9.m1.2.2.2.2.2.2.cmml">a</mi><mi id="S6.I4.i3.p1.9.m1.2.2.2.2.2.3" xref="S6.I4.i3.p1.9.m1.2.2.2.2.2.3.cmml">r</mi></msub><mo id="S6.I4.i3.p1.9.m1.2.2.2.2.5" stretchy="false" xref="S6.I4.i3.p1.9.m1.2.2.2.3.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.I4.i3.p1.9.m1.2b"><apply id="S6.I4.i3.p1.9.m1.2.2.cmml" xref="S6.I4.i3.p1.9.m1.2.2"><eq id="S6.I4.i3.p1.9.m1.2.2.3.cmml" xref="S6.I4.i3.p1.9.m1.2.2.3"></eq><ci id="S6.I4.i3.p1.9.m1.2.2.4.cmml" xref="S6.I4.i3.p1.9.m1.2.2.4">𝐼</ci><interval closure="closed" id="S6.I4.i3.p1.9.m1.2.2.2.3.cmml" xref="S6.I4.i3.p1.9.m1.2.2.2.2"><apply id="S6.I4.i3.p1.9.m1.1.1.1.1.1.cmml" xref="S6.I4.i3.p1.9.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.I4.i3.p1.9.m1.1.1.1.1.1.1.cmml" xref="S6.I4.i3.p1.9.m1.1.1.1.1.1">subscript</csymbol><ci id="S6.I4.i3.p1.9.m1.1.1.1.1.1.2.cmml" xref="S6.I4.i3.p1.9.m1.1.1.1.1.1.2">𝑎</ci><ci id="S6.I4.i3.p1.9.m1.1.1.1.1.1.3.cmml" xref="S6.I4.i3.p1.9.m1.1.1.1.1.1.3">𝑙</ci></apply><apply id="S6.I4.i3.p1.9.m1.2.2.2.2.2.cmml" xref="S6.I4.i3.p1.9.m1.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.I4.i3.p1.9.m1.2.2.2.2.2.1.cmml" xref="S6.I4.i3.p1.9.m1.2.2.2.2.2">subscript</csymbol><ci id="S6.I4.i3.p1.9.m1.2.2.2.2.2.2.cmml" xref="S6.I4.i3.p1.9.m1.2.2.2.2.2.2">𝑎</ci><ci id="S6.I4.i3.p1.9.m1.2.2.2.2.2.3.cmml" xref="S6.I4.i3.p1.9.m1.2.2.2.2.2.3">𝑟</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I4.i3.p1.9.m1.2c">I=[a_{l},a_{r}]</annotation><annotation encoding="application/x-llamapun" id="S6.I4.i3.p1.9.m1.2d">italic_I = [ italic_a start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ]</annotation></semantics></math>, then <math alttext="J" class="ltx_Math" display="inline" id="S6.I4.i3.p1.10.m2.1"><semantics id="S6.I4.i3.p1.10.m2.1a"><mi id="S6.I4.i3.p1.10.m2.1.1" xref="S6.I4.i3.p1.10.m2.1.1.cmml">J</mi><annotation-xml encoding="MathML-Content" id="S6.I4.i3.p1.10.m2.1b"><ci id="S6.I4.i3.p1.10.m2.1.1.cmml" xref="S6.I4.i3.p1.10.m2.1.1">𝐽</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.I4.i3.p1.10.m2.1c">J</annotation><annotation encoding="application/x-llamapun" id="S6.I4.i3.p1.10.m2.1d">italic_J</annotation></semantics></math> must be singleton.</p> </div> </li> </ol> </div> </div> <div class="ltx_para" id="S6.SS2.p5"> <p class="ltx_p" id="S6.SS2.p5.11">To prove <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.Thmtheorem2" title="Theorem 6.2. ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">6.2</span></a> we will find a club <math alttext="E" class="ltx_Math" display="inline" id="S6.SS2.p5.1.m1.1"><semantics id="S6.SS2.p5.1.m1.1a"><mi id="S6.SS2.p5.1.m1.1.1" xref="S6.SS2.p5.1.m1.1.1.cmml">E</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.p5.1.m1.1b"><ci id="S6.SS2.p5.1.m1.1.1.cmml" xref="S6.SS2.p5.1.m1.1.1">𝐸</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p5.1.m1.1c">E</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p5.1.m1.1d">italic_E</annotation></semantics></math> such that <math alttext="P_{E}" class="ltx_Math" display="inline" id="S6.SS2.p5.2.m2.1"><semantics id="S6.SS2.p5.2.m2.1a"><msub id="S6.SS2.p5.2.m2.1.1" xref="S6.SS2.p5.2.m2.1.1.cmml"><mi id="S6.SS2.p5.2.m2.1.1.2" xref="S6.SS2.p5.2.m2.1.1.2.cmml">P</mi><mi id="S6.SS2.p5.2.m2.1.1.3" xref="S6.SS2.p5.2.m2.1.1.3.cmml">E</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.p5.2.m2.1b"><apply id="S6.SS2.p5.2.m2.1.1.cmml" xref="S6.SS2.p5.2.m2.1.1"><csymbol cd="ambiguous" id="S6.SS2.p5.2.m2.1.1.1.cmml" xref="S6.SS2.p5.2.m2.1.1">subscript</csymbol><ci id="S6.SS2.p5.2.m2.1.1.2.cmml" xref="S6.SS2.p5.2.m2.1.1.2">𝑃</ci><ci id="S6.SS2.p5.2.m2.1.1.3.cmml" xref="S6.SS2.p5.2.m2.1.1.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p5.2.m2.1c">P_{E}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p5.2.m2.1d">italic_P start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT</annotation></semantics></math> is ccc, and such that for all <math alttext="a\in A" class="ltx_Math" display="inline" id="S6.SS2.p5.3.m3.1"><semantics id="S6.SS2.p5.3.m3.1a"><mrow id="S6.SS2.p5.3.m3.1.1" xref="S6.SS2.p5.3.m3.1.1.cmml"><mi id="S6.SS2.p5.3.m3.1.1.2" xref="S6.SS2.p5.3.m3.1.1.2.cmml">a</mi><mo id="S6.SS2.p5.3.m3.1.1.1" xref="S6.SS2.p5.3.m3.1.1.1.cmml">∈</mo><mi id="S6.SS2.p5.3.m3.1.1.3" xref="S6.SS2.p5.3.m3.1.1.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p5.3.m3.1b"><apply id="S6.SS2.p5.3.m3.1.1.cmml" xref="S6.SS2.p5.3.m3.1.1"><in id="S6.SS2.p5.3.m3.1.1.1.cmml" xref="S6.SS2.p5.3.m3.1.1.1"></in><ci id="S6.SS2.p5.3.m3.1.1.2.cmml" xref="S6.SS2.p5.3.m3.1.1.2">𝑎</ci><ci id="S6.SS2.p5.3.m3.1.1.3.cmml" xref="S6.SS2.p5.3.m3.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p5.3.m3.1c">a\in A</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p5.3.m3.1d">italic_a ∈ italic_A</annotation></semantics></math> and <math alttext="x\in X" class="ltx_Math" display="inline" id="S6.SS2.p5.4.m4.1"><semantics id="S6.SS2.p5.4.m4.1a"><mrow id="S6.SS2.p5.4.m4.1.1" xref="S6.SS2.p5.4.m4.1.1.cmml"><mi id="S6.SS2.p5.4.m4.1.1.2" xref="S6.SS2.p5.4.m4.1.1.2.cmml">x</mi><mo id="S6.SS2.p5.4.m4.1.1.1" xref="S6.SS2.p5.4.m4.1.1.1.cmml">∈</mo><mi id="S6.SS2.p5.4.m4.1.1.3" xref="S6.SS2.p5.4.m4.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p5.4.m4.1b"><apply id="S6.SS2.p5.4.m4.1.1.cmml" xref="S6.SS2.p5.4.m4.1.1"><in id="S6.SS2.p5.4.m4.1.1.1.cmml" xref="S6.SS2.p5.4.m4.1.1.1"></in><ci id="S6.SS2.p5.4.m4.1.1.2.cmml" xref="S6.SS2.p5.4.m4.1.1.2">𝑥</ci><ci id="S6.SS2.p5.4.m4.1.1.3.cmml" xref="S6.SS2.p5.4.m4.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p5.4.m4.1c">x\in X</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p5.4.m4.1d">italic_x ∈ italic_X</annotation></semantics></math>, <math alttext="\{p\in P_{E}:a\in\operatorname{dom}(f_{p}),x\in\operatorname{ran}(f_{p})\}" class="ltx_Math" display="inline" id="S6.SS2.p5.5.m5.4"><semantics id="S6.SS2.p5.5.m5.4a"><mrow id="S6.SS2.p5.5.m5.4.4.2" xref="S6.SS2.p5.5.m5.4.4.3.cmml"><mo id="S6.SS2.p5.5.m5.4.4.2.3" stretchy="false" xref="S6.SS2.p5.5.m5.4.4.3.1.cmml">{</mo><mrow id="S6.SS2.p5.5.m5.3.3.1.1" xref="S6.SS2.p5.5.m5.3.3.1.1.cmml"><mi id="S6.SS2.p5.5.m5.3.3.1.1.2" xref="S6.SS2.p5.5.m5.3.3.1.1.2.cmml">p</mi><mo id="S6.SS2.p5.5.m5.3.3.1.1.1" xref="S6.SS2.p5.5.m5.3.3.1.1.1.cmml">∈</mo><msub id="S6.SS2.p5.5.m5.3.3.1.1.3" xref="S6.SS2.p5.5.m5.3.3.1.1.3.cmml"><mi id="S6.SS2.p5.5.m5.3.3.1.1.3.2" xref="S6.SS2.p5.5.m5.3.3.1.1.3.2.cmml">P</mi><mi id="S6.SS2.p5.5.m5.3.3.1.1.3.3" xref="S6.SS2.p5.5.m5.3.3.1.1.3.3.cmml">E</mi></msub></mrow><mo id="S6.SS2.p5.5.m5.4.4.2.4" lspace="0.278em" rspace="0.278em" xref="S6.SS2.p5.5.m5.4.4.3.1.cmml">:</mo><mrow id="S6.SS2.p5.5.m5.4.4.2.2.2" xref="S6.SS2.p5.5.m5.4.4.2.2.3.cmml"><mrow id="S6.SS2.p5.5.m5.4.4.2.2.1.1" xref="S6.SS2.p5.5.m5.4.4.2.2.1.1.cmml"><mi id="S6.SS2.p5.5.m5.4.4.2.2.1.1.3" xref="S6.SS2.p5.5.m5.4.4.2.2.1.1.3.cmml">a</mi><mo id="S6.SS2.p5.5.m5.4.4.2.2.1.1.2" xref="S6.SS2.p5.5.m5.4.4.2.2.1.1.2.cmml">∈</mo><mrow id="S6.SS2.p5.5.m5.4.4.2.2.1.1.1.1" xref="S6.SS2.p5.5.m5.4.4.2.2.1.1.1.2.cmml"><mi id="S6.SS2.p5.5.m5.1.1" xref="S6.SS2.p5.5.m5.1.1.cmml">dom</mi><mo id="S6.SS2.p5.5.m5.4.4.2.2.1.1.1.1a" xref="S6.SS2.p5.5.m5.4.4.2.2.1.1.1.2.cmml">⁡</mo><mrow id="S6.SS2.p5.5.m5.4.4.2.2.1.1.1.1.1" xref="S6.SS2.p5.5.m5.4.4.2.2.1.1.1.2.cmml"><mo id="S6.SS2.p5.5.m5.4.4.2.2.1.1.1.1.1.2" stretchy="false" xref="S6.SS2.p5.5.m5.4.4.2.2.1.1.1.2.cmml">(</mo><msub id="S6.SS2.p5.5.m5.4.4.2.2.1.1.1.1.1.1" xref="S6.SS2.p5.5.m5.4.4.2.2.1.1.1.1.1.1.cmml"><mi id="S6.SS2.p5.5.m5.4.4.2.2.1.1.1.1.1.1.2" xref="S6.SS2.p5.5.m5.4.4.2.2.1.1.1.1.1.1.2.cmml">f</mi><mi id="S6.SS2.p5.5.m5.4.4.2.2.1.1.1.1.1.1.3" xref="S6.SS2.p5.5.m5.4.4.2.2.1.1.1.1.1.1.3.cmml">p</mi></msub><mo id="S6.SS2.p5.5.m5.4.4.2.2.1.1.1.1.1.3" stretchy="false" xref="S6.SS2.p5.5.m5.4.4.2.2.1.1.1.2.cmml">)</mo></mrow></mrow></mrow><mo id="S6.SS2.p5.5.m5.4.4.2.2.2.3" xref="S6.SS2.p5.5.m5.4.4.2.2.3a.cmml">,</mo><mrow id="S6.SS2.p5.5.m5.4.4.2.2.2.2" xref="S6.SS2.p5.5.m5.4.4.2.2.2.2.cmml"><mi id="S6.SS2.p5.5.m5.4.4.2.2.2.2.3" xref="S6.SS2.p5.5.m5.4.4.2.2.2.2.3.cmml">x</mi><mo id="S6.SS2.p5.5.m5.4.4.2.2.2.2.2" xref="S6.SS2.p5.5.m5.4.4.2.2.2.2.2.cmml">∈</mo><mrow id="S6.SS2.p5.5.m5.4.4.2.2.2.2.1.1" xref="S6.SS2.p5.5.m5.4.4.2.2.2.2.1.2.cmml"><mi id="S6.SS2.p5.5.m5.2.2" xref="S6.SS2.p5.5.m5.2.2.cmml">ran</mi><mo id="S6.SS2.p5.5.m5.4.4.2.2.2.2.1.1a" xref="S6.SS2.p5.5.m5.4.4.2.2.2.2.1.2.cmml">⁡</mo><mrow id="S6.SS2.p5.5.m5.4.4.2.2.2.2.1.1.1" xref="S6.SS2.p5.5.m5.4.4.2.2.2.2.1.2.cmml"><mo id="S6.SS2.p5.5.m5.4.4.2.2.2.2.1.1.1.2" stretchy="false" xref="S6.SS2.p5.5.m5.4.4.2.2.2.2.1.2.cmml">(</mo><msub id="S6.SS2.p5.5.m5.4.4.2.2.2.2.1.1.1.1" xref="S6.SS2.p5.5.m5.4.4.2.2.2.2.1.1.1.1.cmml"><mi id="S6.SS2.p5.5.m5.4.4.2.2.2.2.1.1.1.1.2" xref="S6.SS2.p5.5.m5.4.4.2.2.2.2.1.1.1.1.2.cmml">f</mi><mi id="S6.SS2.p5.5.m5.4.4.2.2.2.2.1.1.1.1.3" xref="S6.SS2.p5.5.m5.4.4.2.2.2.2.1.1.1.1.3.cmml">p</mi></msub><mo id="S6.SS2.p5.5.m5.4.4.2.2.2.2.1.1.1.3" stretchy="false" xref="S6.SS2.p5.5.m5.4.4.2.2.2.2.1.2.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S6.SS2.p5.5.m5.4.4.2.5" stretchy="false" xref="S6.SS2.p5.5.m5.4.4.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p5.5.m5.4b"><apply id="S6.SS2.p5.5.m5.4.4.3.cmml" xref="S6.SS2.p5.5.m5.4.4.2"><csymbol cd="latexml" id="S6.SS2.p5.5.m5.4.4.3.1.cmml" xref="S6.SS2.p5.5.m5.4.4.2.3">conditional-set</csymbol><apply id="S6.SS2.p5.5.m5.3.3.1.1.cmml" xref="S6.SS2.p5.5.m5.3.3.1.1"><in id="S6.SS2.p5.5.m5.3.3.1.1.1.cmml" xref="S6.SS2.p5.5.m5.3.3.1.1.1"></in><ci id="S6.SS2.p5.5.m5.3.3.1.1.2.cmml" xref="S6.SS2.p5.5.m5.3.3.1.1.2">𝑝</ci><apply id="S6.SS2.p5.5.m5.3.3.1.1.3.cmml" xref="S6.SS2.p5.5.m5.3.3.1.1.3"><csymbol cd="ambiguous" id="S6.SS2.p5.5.m5.3.3.1.1.3.1.cmml" xref="S6.SS2.p5.5.m5.3.3.1.1.3">subscript</csymbol><ci id="S6.SS2.p5.5.m5.3.3.1.1.3.2.cmml" xref="S6.SS2.p5.5.m5.3.3.1.1.3.2">𝑃</ci><ci id="S6.SS2.p5.5.m5.3.3.1.1.3.3.cmml" xref="S6.SS2.p5.5.m5.3.3.1.1.3.3">𝐸</ci></apply></apply><apply id="S6.SS2.p5.5.m5.4.4.2.2.3.cmml" xref="S6.SS2.p5.5.m5.4.4.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.p5.5.m5.4.4.2.2.3a.cmml" xref="S6.SS2.p5.5.m5.4.4.2.2.2.3">formulae-sequence</csymbol><apply id="S6.SS2.p5.5.m5.4.4.2.2.1.1.cmml" xref="S6.SS2.p5.5.m5.4.4.2.2.1.1"><in id="S6.SS2.p5.5.m5.4.4.2.2.1.1.2.cmml" xref="S6.SS2.p5.5.m5.4.4.2.2.1.1.2"></in><ci id="S6.SS2.p5.5.m5.4.4.2.2.1.1.3.cmml" xref="S6.SS2.p5.5.m5.4.4.2.2.1.1.3">𝑎</ci><apply id="S6.SS2.p5.5.m5.4.4.2.2.1.1.1.2.cmml" xref="S6.SS2.p5.5.m5.4.4.2.2.1.1.1.1"><ci id="S6.SS2.p5.5.m5.1.1.cmml" xref="S6.SS2.p5.5.m5.1.1">dom</ci><apply id="S6.SS2.p5.5.m5.4.4.2.2.1.1.1.1.1.1.cmml" xref="S6.SS2.p5.5.m5.4.4.2.2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.p5.5.m5.4.4.2.2.1.1.1.1.1.1.1.cmml" xref="S6.SS2.p5.5.m5.4.4.2.2.1.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.p5.5.m5.4.4.2.2.1.1.1.1.1.1.2.cmml" xref="S6.SS2.p5.5.m5.4.4.2.2.1.1.1.1.1.1.2">𝑓</ci><ci id="S6.SS2.p5.5.m5.4.4.2.2.1.1.1.1.1.1.3.cmml" xref="S6.SS2.p5.5.m5.4.4.2.2.1.1.1.1.1.1.3">𝑝</ci></apply></apply></apply><apply id="S6.SS2.p5.5.m5.4.4.2.2.2.2.cmml" xref="S6.SS2.p5.5.m5.4.4.2.2.2.2"><in id="S6.SS2.p5.5.m5.4.4.2.2.2.2.2.cmml" xref="S6.SS2.p5.5.m5.4.4.2.2.2.2.2"></in><ci id="S6.SS2.p5.5.m5.4.4.2.2.2.2.3.cmml" xref="S6.SS2.p5.5.m5.4.4.2.2.2.2.3">𝑥</ci><apply id="S6.SS2.p5.5.m5.4.4.2.2.2.2.1.2.cmml" xref="S6.SS2.p5.5.m5.4.4.2.2.2.2.1.1"><ci id="S6.SS2.p5.5.m5.2.2.cmml" xref="S6.SS2.p5.5.m5.2.2">ran</ci><apply id="S6.SS2.p5.5.m5.4.4.2.2.2.2.1.1.1.1.cmml" xref="S6.SS2.p5.5.m5.4.4.2.2.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.p5.5.m5.4.4.2.2.2.2.1.1.1.1.1.cmml" xref="S6.SS2.p5.5.m5.4.4.2.2.2.2.1.1.1.1">subscript</csymbol><ci id="S6.SS2.p5.5.m5.4.4.2.2.2.2.1.1.1.1.2.cmml" xref="S6.SS2.p5.5.m5.4.4.2.2.2.2.1.1.1.1.2">𝑓</ci><ci id="S6.SS2.p5.5.m5.4.4.2.2.2.2.1.1.1.1.3.cmml" xref="S6.SS2.p5.5.m5.4.4.2.2.2.2.1.1.1.1.3">𝑝</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p5.5.m5.4c">\{p\in P_{E}:a\in\operatorname{dom}(f_{p}),x\in\operatorname{ran}(f_{p})\}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p5.5.m5.4d">{ italic_p ∈ italic_P start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT : italic_a ∈ roman_dom ( italic_f start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ) , italic_x ∈ roman_ran ( italic_f start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ) }</annotation></semantics></math> is dense. Then an application of <math alttext="\mathsf{MA}_{\aleph_{1}}" class="ltx_Math" display="inline" id="S6.SS2.p5.6.m6.1"><semantics id="S6.SS2.p5.6.m6.1a"><msub id="S6.SS2.p5.6.m6.1.1" xref="S6.SS2.p5.6.m6.1.1.cmml"><mi id="S6.SS2.p5.6.m6.1.1.2" xref="S6.SS2.p5.6.m6.1.1.2.cmml">𝖬𝖠</mi><msub id="S6.SS2.p5.6.m6.1.1.3" xref="S6.SS2.p5.6.m6.1.1.3.cmml"><mi id="S6.SS2.p5.6.m6.1.1.3.2" mathvariant="normal" xref="S6.SS2.p5.6.m6.1.1.3.2.cmml">ℵ</mi><mn id="S6.SS2.p5.6.m6.1.1.3.3" xref="S6.SS2.p5.6.m6.1.1.3.3.cmml">1</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.p5.6.m6.1b"><apply id="S6.SS2.p5.6.m6.1.1.cmml" xref="S6.SS2.p5.6.m6.1.1"><csymbol cd="ambiguous" id="S6.SS2.p5.6.m6.1.1.1.cmml" xref="S6.SS2.p5.6.m6.1.1">subscript</csymbol><ci id="S6.SS2.p5.6.m6.1.1.2.cmml" xref="S6.SS2.p5.6.m6.1.1.2">𝖬𝖠</ci><apply id="S6.SS2.p5.6.m6.1.1.3.cmml" xref="S6.SS2.p5.6.m6.1.1.3"><csymbol cd="ambiguous" id="S6.SS2.p5.6.m6.1.1.3.1.cmml" xref="S6.SS2.p5.6.m6.1.1.3">subscript</csymbol><ci id="S6.SS2.p5.6.m6.1.1.3.2.cmml" xref="S6.SS2.p5.6.m6.1.1.3.2">ℵ</ci><cn id="S6.SS2.p5.6.m6.1.1.3.3.cmml" type="integer" xref="S6.SS2.p5.6.m6.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p5.6.m6.1c">\mathsf{MA}_{\aleph_{1}}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p5.6.m6.1d">sansserif_MA start_POSTSUBSCRIPT roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> gives <math alttext="G" class="ltx_Math" display="inline" id="S6.SS2.p5.7.m7.1"><semantics id="S6.SS2.p5.7.m7.1a"><mi id="S6.SS2.p5.7.m7.1.1" xref="S6.SS2.p5.7.m7.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.p5.7.m7.1b"><ci id="S6.SS2.p5.7.m7.1.1.cmml" xref="S6.SS2.p5.7.m7.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p5.7.m7.1c">G</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p5.7.m7.1d">italic_G</annotation></semantics></math> a generic filter of <math alttext="P_{E}" class="ltx_Math" display="inline" id="S6.SS2.p5.8.m8.1"><semantics id="S6.SS2.p5.8.m8.1a"><msub id="S6.SS2.p5.8.m8.1.1" xref="S6.SS2.p5.8.m8.1.1.cmml"><mi id="S6.SS2.p5.8.m8.1.1.2" xref="S6.SS2.p5.8.m8.1.1.2.cmml">P</mi><mi id="S6.SS2.p5.8.m8.1.1.3" xref="S6.SS2.p5.8.m8.1.1.3.cmml">E</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.p5.8.m8.1b"><apply id="S6.SS2.p5.8.m8.1.1.cmml" xref="S6.SS2.p5.8.m8.1.1"><csymbol cd="ambiguous" id="S6.SS2.p5.8.m8.1.1.1.cmml" xref="S6.SS2.p5.8.m8.1.1">subscript</csymbol><ci id="S6.SS2.p5.8.m8.1.1.2.cmml" xref="S6.SS2.p5.8.m8.1.1.2">𝑃</ci><ci id="S6.SS2.p5.8.m8.1.1.3.cmml" xref="S6.SS2.p5.8.m8.1.1.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p5.8.m8.1c">P_{E}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p5.8.m8.1d">italic_P start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT</annotation></semantics></math>, and one sees that <math alttext="\bigcup_{p\in G}f_{p}" class="ltx_Math" display="inline" id="S6.SS2.p5.9.m9.1"><semantics id="S6.SS2.p5.9.m9.1a"><mrow id="S6.SS2.p5.9.m9.1.1" xref="S6.SS2.p5.9.m9.1.1.cmml"><msub id="S6.SS2.p5.9.m9.1.1.1" xref="S6.SS2.p5.9.m9.1.1.1.cmml"><mo id="S6.SS2.p5.9.m9.1.1.1.2" xref="S6.SS2.p5.9.m9.1.1.1.2.cmml">⋃</mo><mrow id="S6.SS2.p5.9.m9.1.1.1.3" xref="S6.SS2.p5.9.m9.1.1.1.3.cmml"><mi id="S6.SS2.p5.9.m9.1.1.1.3.2" xref="S6.SS2.p5.9.m9.1.1.1.3.2.cmml">p</mi><mo id="S6.SS2.p5.9.m9.1.1.1.3.1" xref="S6.SS2.p5.9.m9.1.1.1.3.1.cmml">∈</mo><mi id="S6.SS2.p5.9.m9.1.1.1.3.3" xref="S6.SS2.p5.9.m9.1.1.1.3.3.cmml">G</mi></mrow></msub><msub id="S6.SS2.p5.9.m9.1.1.2" xref="S6.SS2.p5.9.m9.1.1.2.cmml"><mi id="S6.SS2.p5.9.m9.1.1.2.2" xref="S6.SS2.p5.9.m9.1.1.2.2.cmml">f</mi><mi id="S6.SS2.p5.9.m9.1.1.2.3" xref="S6.SS2.p5.9.m9.1.1.2.3.cmml">p</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p5.9.m9.1b"><apply id="S6.SS2.p5.9.m9.1.1.cmml" xref="S6.SS2.p5.9.m9.1.1"><apply id="S6.SS2.p5.9.m9.1.1.1.cmml" xref="S6.SS2.p5.9.m9.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.p5.9.m9.1.1.1.1.cmml" xref="S6.SS2.p5.9.m9.1.1.1">subscript</csymbol><union id="S6.SS2.p5.9.m9.1.1.1.2.cmml" xref="S6.SS2.p5.9.m9.1.1.1.2"></union><apply id="S6.SS2.p5.9.m9.1.1.1.3.cmml" xref="S6.SS2.p5.9.m9.1.1.1.3"><in id="S6.SS2.p5.9.m9.1.1.1.3.1.cmml" xref="S6.SS2.p5.9.m9.1.1.1.3.1"></in><ci id="S6.SS2.p5.9.m9.1.1.1.3.2.cmml" xref="S6.SS2.p5.9.m9.1.1.1.3.2">𝑝</ci><ci id="S6.SS2.p5.9.m9.1.1.1.3.3.cmml" xref="S6.SS2.p5.9.m9.1.1.1.3.3">𝐺</ci></apply></apply><apply id="S6.SS2.p5.9.m9.1.1.2.cmml" xref="S6.SS2.p5.9.m9.1.1.2"><csymbol cd="ambiguous" id="S6.SS2.p5.9.m9.1.1.2.1.cmml" xref="S6.SS2.p5.9.m9.1.1.2">subscript</csymbol><ci id="S6.SS2.p5.9.m9.1.1.2.2.cmml" xref="S6.SS2.p5.9.m9.1.1.2.2">𝑓</ci><ci id="S6.SS2.p5.9.m9.1.1.2.3.cmml" xref="S6.SS2.p5.9.m9.1.1.2.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p5.9.m9.1c">\bigcup_{p\in G}f_{p}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p5.9.m9.1d">⋃ start_POSTSUBSCRIPT italic_p ∈ italic_G end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> is an epimorphism from <math alttext="A" class="ltx_Math" display="inline" id="S6.SS2.p5.10.m10.1"><semantics id="S6.SS2.p5.10.m10.1a"><mi id="S6.SS2.p5.10.m10.1.1" xref="S6.SS2.p5.10.m10.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.p5.10.m10.1b"><ci id="S6.SS2.p5.10.m10.1.1.cmml" xref="S6.SS2.p5.10.m10.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p5.10.m10.1c">A</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p5.10.m10.1d">italic_A</annotation></semantics></math> onto <math alttext="X" class="ltx_Math" display="inline" id="S6.SS2.p5.11.m11.1"><semantics id="S6.SS2.p5.11.m11.1a"><mi id="S6.SS2.p5.11.m11.1.1" xref="S6.SS2.p5.11.m11.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.p5.11.m11.1b"><ci id="S6.SS2.p5.11.m11.1.1.cmml" xref="S6.SS2.p5.11.m11.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p5.11.m11.1c">X</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p5.11.m11.1d">italic_X</annotation></semantics></math>. We now turn to the task of finding such a club.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S6.Thmtheorem10"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem10.1.1.1">Lemma 6.10</span></span><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem10.2.2">.</span> </h6> <div class="ltx_para" id="S6.Thmtheorem10.p1"> <p class="ltx_p" id="S6.Thmtheorem10.p1.1">There is a club <math alttext="E" class="ltx_Math" display="inline" id="S6.Thmtheorem10.p1.1.m1.1"><semantics id="S6.Thmtheorem10.p1.1.m1.1a"><mi id="S6.Thmtheorem10.p1.1.m1.1.1" xref="S6.Thmtheorem10.p1.1.m1.1.1.cmml">E</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem10.p1.1.m1.1b"><ci id="S6.Thmtheorem10.p1.1.m1.1.1.cmml" xref="S6.Thmtheorem10.p1.1.m1.1.1">𝐸</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem10.p1.1.m1.1c">E</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem10.p1.1.m1.1d">italic_E</annotation></semantics></math> such that</p> <ol class="ltx_enumerate" id="S6.I5"> <li class="ltx_item" id="S6.I5.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(1)</span> <div class="ltx_para" id="S6.I5.i1.p1"> <p class="ltx_p" id="S6.I5.i1.p1.2"><math alttext="E" class="ltx_Math" display="inline" id="S6.I5.i1.p1.1.m1.1"><semantics id="S6.I5.i1.p1.1.m1.1a"><mi id="S6.I5.i1.p1.1.m1.1.1" xref="S6.I5.i1.p1.1.m1.1.1.cmml">E</mi><annotation-xml encoding="MathML-Content" id="S6.I5.i1.p1.1.m1.1b"><ci id="S6.I5.i1.p1.1.m1.1.1.cmml" xref="S6.I5.i1.p1.1.m1.1.1">𝐸</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.I5.i1.p1.1.m1.1c">E</annotation><annotation encoding="application/x-llamapun" id="S6.I5.i1.p1.1.m1.1d">italic_E</annotation></semantics></math> makes Moore’s forcing <math alttext="Q_{E}" class="ltx_Math" display="inline" id="S6.I5.i1.p1.2.m2.1"><semantics id="S6.I5.i1.p1.2.m2.1a"><msub id="S6.I5.i1.p1.2.m2.1.1" xref="S6.I5.i1.p1.2.m2.1.1.cmml"><mi id="S6.I5.i1.p1.2.m2.1.1.2" xref="S6.I5.i1.p1.2.m2.1.1.2.cmml">Q</mi><mi id="S6.I5.i1.p1.2.m2.1.1.3" xref="S6.I5.i1.p1.2.m2.1.1.3.cmml">E</mi></msub><annotation-xml encoding="MathML-Content" id="S6.I5.i1.p1.2.m2.1b"><apply id="S6.I5.i1.p1.2.m2.1.1.cmml" xref="S6.I5.i1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S6.I5.i1.p1.2.m2.1.1.1.cmml" xref="S6.I5.i1.p1.2.m2.1.1">subscript</csymbol><ci id="S6.I5.i1.p1.2.m2.1.1.2.cmml" xref="S6.I5.i1.p1.2.m2.1.1.2">𝑄</ci><ci id="S6.I5.i1.p1.2.m2.1.1.3.cmml" xref="S6.I5.i1.p1.2.m2.1.1.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I5.i1.p1.2.m2.1c">Q_{E}</annotation><annotation encoding="application/x-llamapun" id="S6.I5.i1.p1.2.m2.1d">italic_Q start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT</annotation></semantics></math> ccc.</p> </div> </li> <li class="ltx_item" id="S6.I5.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(2)</span> <div class="ltx_para" id="S6.I5.i2.p1"> <p class="ltx_p" id="S6.I5.i2.p1.5"><math alttext="E" class="ltx_Math" display="inline" id="S6.I5.i2.p1.1.m1.1"><semantics id="S6.I5.i2.p1.1.m1.1a"><mi id="S6.I5.i2.p1.1.m1.1.1" xref="S6.I5.i2.p1.1.m1.1.1.cmml">E</mi><annotation-xml encoding="MathML-Content" id="S6.I5.i2.p1.1.m1.1b"><ci id="S6.I5.i2.p1.1.m1.1.1.cmml" xref="S6.I5.i2.p1.1.m1.1.1">𝐸</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.I5.i2.p1.1.m1.1c">E</annotation><annotation encoding="application/x-llamapun" id="S6.I5.i2.p1.1.m1.1d">italic_E</annotation></semantics></math> is elementary for <math alttext="A" class="ltx_Math" display="inline" id="S6.I5.i2.p1.2.m2.1"><semantics id="S6.I5.i2.p1.2.m2.1a"><mi id="S6.I5.i2.p1.2.m2.1.1" xref="S6.I5.i2.p1.2.m2.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S6.I5.i2.p1.2.m2.1b"><ci id="S6.I5.i2.p1.2.m2.1.1.cmml" xref="S6.I5.i2.p1.2.m2.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.I5.i2.p1.2.m2.1c">A</annotation><annotation encoding="application/x-llamapun" id="S6.I5.i2.p1.2.m2.1d">italic_A</annotation></semantics></math>, <math alttext="X" class="ltx_Math" display="inline" id="S6.I5.i2.p1.3.m3.1"><semantics id="S6.I5.i2.p1.3.m3.1a"><mi id="S6.I5.i2.p1.3.m3.1.1" xref="S6.I5.i2.p1.3.m3.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S6.I5.i2.p1.3.m3.1b"><ci id="S6.I5.i2.p1.3.m3.1.1.cmml" xref="S6.I5.i2.p1.3.m3.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.I5.i2.p1.3.m3.1c">X</annotation><annotation encoding="application/x-llamapun" id="S6.I5.i2.p1.3.m3.1d">italic_X</annotation></semantics></math>, <math alttext="D^{A}" class="ltx_Math" display="inline" id="S6.I5.i2.p1.4.m4.1"><semantics id="S6.I5.i2.p1.4.m4.1a"><msup id="S6.I5.i2.p1.4.m4.1.1" xref="S6.I5.i2.p1.4.m4.1.1.cmml"><mi id="S6.I5.i2.p1.4.m4.1.1.2" xref="S6.I5.i2.p1.4.m4.1.1.2.cmml">D</mi><mi id="S6.I5.i2.p1.4.m4.1.1.3" xref="S6.I5.i2.p1.4.m4.1.1.3.cmml">A</mi></msup><annotation-xml encoding="MathML-Content" id="S6.I5.i2.p1.4.m4.1b"><apply id="S6.I5.i2.p1.4.m4.1.1.cmml" xref="S6.I5.i2.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S6.I5.i2.p1.4.m4.1.1.1.cmml" xref="S6.I5.i2.p1.4.m4.1.1">superscript</csymbol><ci id="S6.I5.i2.p1.4.m4.1.1.2.cmml" xref="S6.I5.i2.p1.4.m4.1.1.2">𝐷</ci><ci id="S6.I5.i2.p1.4.m4.1.1.3.cmml" xref="S6.I5.i2.p1.4.m4.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I5.i2.p1.4.m4.1c">D^{A}</annotation><annotation encoding="application/x-llamapun" id="S6.I5.i2.p1.4.m4.1d">italic_D start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="D^{X}" class="ltx_Math" display="inline" id="S6.I5.i2.p1.5.m5.1"><semantics id="S6.I5.i2.p1.5.m5.1a"><msup id="S6.I5.i2.p1.5.m5.1.1" xref="S6.I5.i2.p1.5.m5.1.1.cmml"><mi id="S6.I5.i2.p1.5.m5.1.1.2" xref="S6.I5.i2.p1.5.m5.1.1.2.cmml">D</mi><mi id="S6.I5.i2.p1.5.m5.1.1.3" xref="S6.I5.i2.p1.5.m5.1.1.3.cmml">X</mi></msup><annotation-xml encoding="MathML-Content" id="S6.I5.i2.p1.5.m5.1b"><apply id="S6.I5.i2.p1.5.m5.1.1.cmml" xref="S6.I5.i2.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S6.I5.i2.p1.5.m5.1.1.1.cmml" xref="S6.I5.i2.p1.5.m5.1.1">superscript</csymbol><ci id="S6.I5.i2.p1.5.m5.1.1.2.cmml" xref="S6.I5.i2.p1.5.m5.1.1.2">𝐷</ci><ci id="S6.I5.i2.p1.5.m5.1.1.3.cmml" xref="S6.I5.i2.p1.5.m5.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I5.i2.p1.5.m5.1c">D^{X}</annotation><annotation encoding="application/x-llamapun" id="S6.I5.i2.p1.5.m5.1d">italic_D start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S6.I5.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(3)</span> <div class="ltx_para" id="S6.I5.i3.p1"> <p class="ltx_p" id="S6.I5.i3.p1.4"><math alttext="E" class="ltx_Math" display="inline" id="S6.I5.i3.p1.1.m1.1"><semantics id="S6.I5.i3.p1.1.m1.1a"><mi id="S6.I5.i3.p1.1.m1.1.1" xref="S6.I5.i3.p1.1.m1.1.1.cmml">E</mi><annotation-xml encoding="MathML-Content" id="S6.I5.i3.p1.1.m1.1b"><ci id="S6.I5.i3.p1.1.m1.1.1.cmml" xref="S6.I5.i3.p1.1.m1.1.1">𝐸</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.I5.i3.p1.1.m1.1c">E</annotation><annotation encoding="application/x-llamapun" id="S6.I5.i3.p1.1.m1.1d">italic_E</annotation></semantics></math> consists of ordinals closed under <math alttext="\delta_{A}" class="ltx_Math" display="inline" id="S6.I5.i3.p1.2.m2.1"><semantics id="S6.I5.i3.p1.2.m2.1a"><msub id="S6.I5.i3.p1.2.m2.1.1" xref="S6.I5.i3.p1.2.m2.1.1.cmml"><mi id="S6.I5.i3.p1.2.m2.1.1.2" xref="S6.I5.i3.p1.2.m2.1.1.2.cmml">δ</mi><mi id="S6.I5.i3.p1.2.m2.1.1.3" xref="S6.I5.i3.p1.2.m2.1.1.3.cmml">A</mi></msub><annotation-xml encoding="MathML-Content" id="S6.I5.i3.p1.2.m2.1b"><apply id="S6.I5.i3.p1.2.m2.1.1.cmml" xref="S6.I5.i3.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S6.I5.i3.p1.2.m2.1.1.1.cmml" xref="S6.I5.i3.p1.2.m2.1.1">subscript</csymbol><ci id="S6.I5.i3.p1.2.m2.1.1.2.cmml" xref="S6.I5.i3.p1.2.m2.1.1.2">𝛿</ci><ci id="S6.I5.i3.p1.2.m2.1.1.3.cmml" xref="S6.I5.i3.p1.2.m2.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I5.i3.p1.2.m2.1c">\delta_{A}</annotation><annotation encoding="application/x-llamapun" id="S6.I5.i3.p1.2.m2.1d">italic_δ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\delta_{X}" class="ltx_Math" display="inline" id="S6.I5.i3.p1.3.m3.1"><semantics id="S6.I5.i3.p1.3.m3.1a"><msub id="S6.I5.i3.p1.3.m3.1.1" xref="S6.I5.i3.p1.3.m3.1.1.cmml"><mi id="S6.I5.i3.p1.3.m3.1.1.2" xref="S6.I5.i3.p1.3.m3.1.1.2.cmml">δ</mi><mi id="S6.I5.i3.p1.3.m3.1.1.3" xref="S6.I5.i3.p1.3.m3.1.1.3.cmml">X</mi></msub><annotation-xml encoding="MathML-Content" id="S6.I5.i3.p1.3.m3.1b"><apply id="S6.I5.i3.p1.3.m3.1.1.cmml" xref="S6.I5.i3.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S6.I5.i3.p1.3.m3.1.1.1.cmml" xref="S6.I5.i3.p1.3.m3.1.1">subscript</csymbol><ci id="S6.I5.i3.p1.3.m3.1.1.2.cmml" xref="S6.I5.i3.p1.3.m3.1.1.2">𝛿</ci><ci id="S6.I5.i3.p1.3.m3.1.1.3.cmml" xref="S6.I5.i3.p1.3.m3.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I5.i3.p1.3.m3.1c">\delta_{X}</annotation><annotation encoding="application/x-llamapun" id="S6.I5.i3.p1.3.m3.1d">italic_δ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT</annotation></semantics></math> (recall the definition of <math alttext="\delta_{A}" class="ltx_Math" display="inline" id="S6.I5.i3.p1.4.m4.1"><semantics id="S6.I5.i3.p1.4.m4.1a"><msub id="S6.I5.i3.p1.4.m4.1.1" xref="S6.I5.i3.p1.4.m4.1.1.cmml"><mi id="S6.I5.i3.p1.4.m4.1.1.2" xref="S6.I5.i3.p1.4.m4.1.1.2.cmml">δ</mi><mi id="S6.I5.i3.p1.4.m4.1.1.3" xref="S6.I5.i3.p1.4.m4.1.1.3.cmml">A</mi></msub><annotation-xml encoding="MathML-Content" id="S6.I5.i3.p1.4.m4.1b"><apply id="S6.I5.i3.p1.4.m4.1.1.cmml" xref="S6.I5.i3.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S6.I5.i3.p1.4.m4.1.1.1.cmml" xref="S6.I5.i3.p1.4.m4.1.1">subscript</csymbol><ci id="S6.I5.i3.p1.4.m4.1.1.2.cmml" xref="S6.I5.i3.p1.4.m4.1.1.2">𝛿</ci><ci id="S6.I5.i3.p1.4.m4.1.1.3.cmml" xref="S6.I5.i3.p1.4.m4.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I5.i3.p1.4.m4.1c">\delta_{A}</annotation><annotation encoding="application/x-llamapun" id="S6.I5.i3.p1.4.m4.1d">italic_δ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT</annotation></semantics></math> from <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.SS1" title="6.1. Moore’s forcing ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">6.1</span></a>).</p> </div> </li> </ol> </div> </div> <div class="ltx_proof" id="S6.SS2.2"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S6.SS2.1.p1"> <p class="ltx_p" id="S6.SS2.1.p1.5">Using <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.Thmtheorem5" title="Lemma 6.5. ‣ 6.1. Moore’s forcing ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">6.5</span></a> let <math alttext="E_{0}" class="ltx_Math" display="inline" id="S6.SS2.1.p1.1.m1.1"><semantics id="S6.SS2.1.p1.1.m1.1a"><msub id="S6.SS2.1.p1.1.m1.1.1" xref="S6.SS2.1.p1.1.m1.1.1.cmml"><mi id="S6.SS2.1.p1.1.m1.1.1.2" xref="S6.SS2.1.p1.1.m1.1.1.2.cmml">E</mi><mn id="S6.SS2.1.p1.1.m1.1.1.3" xref="S6.SS2.1.p1.1.m1.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.1.p1.1.m1.1b"><apply id="S6.SS2.1.p1.1.m1.1.1.cmml" xref="S6.SS2.1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S6.SS2.1.p1.1.m1.1.1.1.cmml" xref="S6.SS2.1.p1.1.m1.1.1">subscript</csymbol><ci id="S6.SS2.1.p1.1.m1.1.1.2.cmml" xref="S6.SS2.1.p1.1.m1.1.1.2">𝐸</ci><cn id="S6.SS2.1.p1.1.m1.1.1.3.cmml" type="integer" xref="S6.SS2.1.p1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.1.p1.1.m1.1c">E_{0}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.1.p1.1.m1.1d">italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> be a club such that <math alttext="Q_{E_{0}}" class="ltx_Math" display="inline" id="S6.SS2.1.p1.2.m2.1"><semantics id="S6.SS2.1.p1.2.m2.1a"><msub id="S6.SS2.1.p1.2.m2.1.1" xref="S6.SS2.1.p1.2.m2.1.1.cmml"><mi id="S6.SS2.1.p1.2.m2.1.1.2" xref="S6.SS2.1.p1.2.m2.1.1.2.cmml">Q</mi><msub id="S6.SS2.1.p1.2.m2.1.1.3" xref="S6.SS2.1.p1.2.m2.1.1.3.cmml"><mi id="S6.SS2.1.p1.2.m2.1.1.3.2" xref="S6.SS2.1.p1.2.m2.1.1.3.2.cmml">E</mi><mn id="S6.SS2.1.p1.2.m2.1.1.3.3" xref="S6.SS2.1.p1.2.m2.1.1.3.3.cmml">0</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.1.p1.2.m2.1b"><apply id="S6.SS2.1.p1.2.m2.1.1.cmml" xref="S6.SS2.1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S6.SS2.1.p1.2.m2.1.1.1.cmml" xref="S6.SS2.1.p1.2.m2.1.1">subscript</csymbol><ci id="S6.SS2.1.p1.2.m2.1.1.2.cmml" xref="S6.SS2.1.p1.2.m2.1.1.2">𝑄</ci><apply id="S6.SS2.1.p1.2.m2.1.1.3.cmml" xref="S6.SS2.1.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S6.SS2.1.p1.2.m2.1.1.3.1.cmml" xref="S6.SS2.1.p1.2.m2.1.1.3">subscript</csymbol><ci id="S6.SS2.1.p1.2.m2.1.1.3.2.cmml" xref="S6.SS2.1.p1.2.m2.1.1.3.2">𝐸</ci><cn id="S6.SS2.1.p1.2.m2.1.1.3.3.cmml" type="integer" xref="S6.SS2.1.p1.2.m2.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.1.p1.2.m2.1c">Q_{E_{0}}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.1.p1.2.m2.1d">italic_Q start_POSTSUBSCRIPT italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> is ccc, let <math alttext="E_{1}" class="ltx_Math" display="inline" id="S6.SS2.1.p1.3.m3.1"><semantics id="S6.SS2.1.p1.3.m3.1a"><msub id="S6.SS2.1.p1.3.m3.1.1" xref="S6.SS2.1.p1.3.m3.1.1.cmml"><mi id="S6.SS2.1.p1.3.m3.1.1.2" xref="S6.SS2.1.p1.3.m3.1.1.2.cmml">E</mi><mn id="S6.SS2.1.p1.3.m3.1.1.3" xref="S6.SS2.1.p1.3.m3.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.1.p1.3.m3.1b"><apply id="S6.SS2.1.p1.3.m3.1.1.cmml" xref="S6.SS2.1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S6.SS2.1.p1.3.m3.1.1.1.cmml" xref="S6.SS2.1.p1.3.m3.1.1">subscript</csymbol><ci id="S6.SS2.1.p1.3.m3.1.1.2.cmml" xref="S6.SS2.1.p1.3.m3.1.1.2">𝐸</ci><cn id="S6.SS2.1.p1.3.m3.1.1.3.cmml" type="integer" xref="S6.SS2.1.p1.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.1.p1.3.m3.1c">E_{1}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.1.p1.3.m3.1d">italic_E start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> be a club elementary for all the relevant objects, and let <math alttext="E_{2}" class="ltx_Math" display="inline" id="S6.SS2.1.p1.4.m4.1"><semantics id="S6.SS2.1.p1.4.m4.1a"><msub id="S6.SS2.1.p1.4.m4.1.1" xref="S6.SS2.1.p1.4.m4.1.1.cmml"><mi id="S6.SS2.1.p1.4.m4.1.1.2" xref="S6.SS2.1.p1.4.m4.1.1.2.cmml">E</mi><mn id="S6.SS2.1.p1.4.m4.1.1.3" xref="S6.SS2.1.p1.4.m4.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.1.p1.4.m4.1b"><apply id="S6.SS2.1.p1.4.m4.1.1.cmml" xref="S6.SS2.1.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S6.SS2.1.p1.4.m4.1.1.1.cmml" xref="S6.SS2.1.p1.4.m4.1.1">subscript</csymbol><ci id="S6.SS2.1.p1.4.m4.1.1.2.cmml" xref="S6.SS2.1.p1.4.m4.1.1.2">𝐸</ci><cn id="S6.SS2.1.p1.4.m4.1.1.3.cmml" type="integer" xref="S6.SS2.1.p1.4.m4.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.1.p1.4.m4.1c">E_{2}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.1.p1.4.m4.1d">italic_E start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> be any satisfying (3). We claim that <math alttext="E:=E_{0}\cap E_{1}\cap E_{2}" class="ltx_Math" display="inline" id="S6.SS2.1.p1.5.m5.1"><semantics id="S6.SS2.1.p1.5.m5.1a"><mrow id="S6.SS2.1.p1.5.m5.1.1" xref="S6.SS2.1.p1.5.m5.1.1.cmml"><mi id="S6.SS2.1.p1.5.m5.1.1.2" xref="S6.SS2.1.p1.5.m5.1.1.2.cmml">E</mi><mo id="S6.SS2.1.p1.5.m5.1.1.1" lspace="0.278em" rspace="0.278em" xref="S6.SS2.1.p1.5.m5.1.1.1.cmml">:=</mo><mrow id="S6.SS2.1.p1.5.m5.1.1.3" xref="S6.SS2.1.p1.5.m5.1.1.3.cmml"><msub id="S6.SS2.1.p1.5.m5.1.1.3.2" xref="S6.SS2.1.p1.5.m5.1.1.3.2.cmml"><mi id="S6.SS2.1.p1.5.m5.1.1.3.2.2" xref="S6.SS2.1.p1.5.m5.1.1.3.2.2.cmml">E</mi><mn id="S6.SS2.1.p1.5.m5.1.1.3.2.3" xref="S6.SS2.1.p1.5.m5.1.1.3.2.3.cmml">0</mn></msub><mo id="S6.SS2.1.p1.5.m5.1.1.3.1" xref="S6.SS2.1.p1.5.m5.1.1.3.1.cmml">∩</mo><msub id="S6.SS2.1.p1.5.m5.1.1.3.3" xref="S6.SS2.1.p1.5.m5.1.1.3.3.cmml"><mi id="S6.SS2.1.p1.5.m5.1.1.3.3.2" xref="S6.SS2.1.p1.5.m5.1.1.3.3.2.cmml">E</mi><mn id="S6.SS2.1.p1.5.m5.1.1.3.3.3" xref="S6.SS2.1.p1.5.m5.1.1.3.3.3.cmml">1</mn></msub><mo id="S6.SS2.1.p1.5.m5.1.1.3.1a" xref="S6.SS2.1.p1.5.m5.1.1.3.1.cmml">∩</mo><msub id="S6.SS2.1.p1.5.m5.1.1.3.4" xref="S6.SS2.1.p1.5.m5.1.1.3.4.cmml"><mi id="S6.SS2.1.p1.5.m5.1.1.3.4.2" xref="S6.SS2.1.p1.5.m5.1.1.3.4.2.cmml">E</mi><mn id="S6.SS2.1.p1.5.m5.1.1.3.4.3" xref="S6.SS2.1.p1.5.m5.1.1.3.4.3.cmml">2</mn></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.1.p1.5.m5.1b"><apply id="S6.SS2.1.p1.5.m5.1.1.cmml" xref="S6.SS2.1.p1.5.m5.1.1"><csymbol cd="latexml" id="S6.SS2.1.p1.5.m5.1.1.1.cmml" xref="S6.SS2.1.p1.5.m5.1.1.1">assign</csymbol><ci id="S6.SS2.1.p1.5.m5.1.1.2.cmml" xref="S6.SS2.1.p1.5.m5.1.1.2">𝐸</ci><apply id="S6.SS2.1.p1.5.m5.1.1.3.cmml" xref="S6.SS2.1.p1.5.m5.1.1.3"><intersect id="S6.SS2.1.p1.5.m5.1.1.3.1.cmml" xref="S6.SS2.1.p1.5.m5.1.1.3.1"></intersect><apply id="S6.SS2.1.p1.5.m5.1.1.3.2.cmml" xref="S6.SS2.1.p1.5.m5.1.1.3.2"><csymbol cd="ambiguous" id="S6.SS2.1.p1.5.m5.1.1.3.2.1.cmml" xref="S6.SS2.1.p1.5.m5.1.1.3.2">subscript</csymbol><ci id="S6.SS2.1.p1.5.m5.1.1.3.2.2.cmml" xref="S6.SS2.1.p1.5.m5.1.1.3.2.2">𝐸</ci><cn id="S6.SS2.1.p1.5.m5.1.1.3.2.3.cmml" type="integer" xref="S6.SS2.1.p1.5.m5.1.1.3.2.3">0</cn></apply><apply id="S6.SS2.1.p1.5.m5.1.1.3.3.cmml" xref="S6.SS2.1.p1.5.m5.1.1.3.3"><csymbol cd="ambiguous" id="S6.SS2.1.p1.5.m5.1.1.3.3.1.cmml" xref="S6.SS2.1.p1.5.m5.1.1.3.3">subscript</csymbol><ci id="S6.SS2.1.p1.5.m5.1.1.3.3.2.cmml" xref="S6.SS2.1.p1.5.m5.1.1.3.3.2">𝐸</ci><cn id="S6.SS2.1.p1.5.m5.1.1.3.3.3.cmml" type="integer" xref="S6.SS2.1.p1.5.m5.1.1.3.3.3">1</cn></apply><apply id="S6.SS2.1.p1.5.m5.1.1.3.4.cmml" xref="S6.SS2.1.p1.5.m5.1.1.3.4"><csymbol cd="ambiguous" id="S6.SS2.1.p1.5.m5.1.1.3.4.1.cmml" xref="S6.SS2.1.p1.5.m5.1.1.3.4">subscript</csymbol><ci id="S6.SS2.1.p1.5.m5.1.1.3.4.2.cmml" xref="S6.SS2.1.p1.5.m5.1.1.3.4.2">𝐸</ci><cn id="S6.SS2.1.p1.5.m5.1.1.3.4.3.cmml" type="integer" xref="S6.SS2.1.p1.5.m5.1.1.3.4.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.1.p1.5.m5.1c">E:=E_{0}\cap E_{1}\cap E_{2}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.1.p1.5.m5.1d">italic_E := italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∩ italic_E start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∩ italic_E start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> work.</p> </div> <div class="ltx_para" id="S6.SS2.2.p2"> <p class="ltx_p" id="S6.SS2.2.p2.1">Conditions (2) and (3) are trivially preserved by taking subclubs, and condition (1) is preserved by the choice of <math alttext="E_{0}" class="ltx_Math" display="inline" id="S6.SS2.2.p2.1.m1.1"><semantics id="S6.SS2.2.p2.1.m1.1a"><msub id="S6.SS2.2.p2.1.m1.1.1" xref="S6.SS2.2.p2.1.m1.1.1.cmml"><mi id="S6.SS2.2.p2.1.m1.1.1.2" xref="S6.SS2.2.p2.1.m1.1.1.2.cmml">E</mi><mn id="S6.SS2.2.p2.1.m1.1.1.3" xref="S6.SS2.2.p2.1.m1.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.2.p2.1.m1.1b"><apply id="S6.SS2.2.p2.1.m1.1.1.cmml" xref="S6.SS2.2.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S6.SS2.2.p2.1.m1.1.1.1.cmml" xref="S6.SS2.2.p2.1.m1.1.1">subscript</csymbol><ci id="S6.SS2.2.p2.1.m1.1.1.2.cmml" xref="S6.SS2.2.p2.1.m1.1.1.2">𝐸</ci><cn id="S6.SS2.2.p2.1.m1.1.1.3.cmml" type="integer" xref="S6.SS2.2.p2.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.2.p2.1.m1.1c">E_{0}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.2.p2.1.m1.1d">italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> (see <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.Thmtheorem5" title="Lemma 6.5. ‣ 6.1. Moore’s forcing ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">6.5</span></a>). ∎</p> </div> </div> <div class="ltx_para" id="S6.SS2.p6"> <p class="ltx_p" id="S6.SS2.p6.4">We will show that such a club <math alttext="E" class="ltx_Math" display="inline" id="S6.SS2.p6.1.m1.1"><semantics id="S6.SS2.p6.1.m1.1a"><mi id="S6.SS2.p6.1.m1.1.1" xref="S6.SS2.p6.1.m1.1.1.cmml">E</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.p6.1.m1.1b"><ci id="S6.SS2.p6.1.m1.1.1.cmml" xref="S6.SS2.p6.1.m1.1.1">𝐸</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p6.1.m1.1c">E</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p6.1.m1.1d">italic_E</annotation></semantics></math> makes <math alttext="P_{E}" class="ltx_Math" display="inline" id="S6.SS2.p6.2.m2.1"><semantics id="S6.SS2.p6.2.m2.1a"><msub id="S6.SS2.p6.2.m2.1.1" xref="S6.SS2.p6.2.m2.1.1.cmml"><mi id="S6.SS2.p6.2.m2.1.1.2" xref="S6.SS2.p6.2.m2.1.1.2.cmml">P</mi><mi id="S6.SS2.p6.2.m2.1.1.3" xref="S6.SS2.p6.2.m2.1.1.3.cmml">E</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.p6.2.m2.1b"><apply id="S6.SS2.p6.2.m2.1.1.cmml" xref="S6.SS2.p6.2.m2.1.1"><csymbol cd="ambiguous" id="S6.SS2.p6.2.m2.1.1.1.cmml" xref="S6.SS2.p6.2.m2.1.1">subscript</csymbol><ci id="S6.SS2.p6.2.m2.1.1.2.cmml" xref="S6.SS2.p6.2.m2.1.1.2">𝑃</ci><ci id="S6.SS2.p6.2.m2.1.1.3.cmml" xref="S6.SS2.p6.2.m2.1.1.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p6.2.m2.1c">P_{E}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p6.2.m2.1d">italic_P start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT</annotation></semantics></math> ccc and all the needed sets dense. First we prove a lemma that gives explicit connections between <math alttext="P_{E}" class="ltx_Math" display="inline" id="S6.SS2.p6.3.m3.1"><semantics id="S6.SS2.p6.3.m3.1a"><msub id="S6.SS2.p6.3.m3.1.1" xref="S6.SS2.p6.3.m3.1.1.cmml"><mi id="S6.SS2.p6.3.m3.1.1.2" xref="S6.SS2.p6.3.m3.1.1.2.cmml">P</mi><mi id="S6.SS2.p6.3.m3.1.1.3" xref="S6.SS2.p6.3.m3.1.1.3.cmml">E</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.p6.3.m3.1b"><apply id="S6.SS2.p6.3.m3.1.1.cmml" xref="S6.SS2.p6.3.m3.1.1"><csymbol cd="ambiguous" id="S6.SS2.p6.3.m3.1.1.1.cmml" xref="S6.SS2.p6.3.m3.1.1">subscript</csymbol><ci id="S6.SS2.p6.3.m3.1.1.2.cmml" xref="S6.SS2.p6.3.m3.1.1.2">𝑃</ci><ci id="S6.SS2.p6.3.m3.1.1.3.cmml" xref="S6.SS2.p6.3.m3.1.1.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p6.3.m3.1c">P_{E}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p6.3.m3.1d">italic_P start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="Q_{E}" class="ltx_Math" display="inline" id="S6.SS2.p6.4.m4.1"><semantics id="S6.SS2.p6.4.m4.1a"><msub id="S6.SS2.p6.4.m4.1.1" xref="S6.SS2.p6.4.m4.1.1.cmml"><mi id="S6.SS2.p6.4.m4.1.1.2" xref="S6.SS2.p6.4.m4.1.1.2.cmml">Q</mi><mi id="S6.SS2.p6.4.m4.1.1.3" xref="S6.SS2.p6.4.m4.1.1.3.cmml">E</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.p6.4.m4.1b"><apply id="S6.SS2.p6.4.m4.1.1.cmml" xref="S6.SS2.p6.4.m4.1.1"><csymbol cd="ambiguous" id="S6.SS2.p6.4.m4.1.1.1.cmml" xref="S6.SS2.p6.4.m4.1.1">subscript</csymbol><ci id="S6.SS2.p6.4.m4.1.1.2.cmml" xref="S6.SS2.p6.4.m4.1.1.2">𝑄</ci><ci id="S6.SS2.p6.4.m4.1.1.3.cmml" xref="S6.SS2.p6.4.m4.1.1.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p6.4.m4.1c">Q_{E}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p6.4.m4.1d">italic_Q start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT</annotation></semantics></math>. We will often use (a) without mention.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S6.Thmtheorem11"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem11.1.1.1">Lemma 6.11</span></span><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem11.2.2">.</span> </h6> <div class="ltx_para" id="S6.Thmtheorem11.p1"> <p class="ltx_p" id="S6.Thmtheorem11.p1.2">If <math alttext="E" class="ltx_Math" display="inline" id="S6.Thmtheorem11.p1.1.m1.1"><semantics id="S6.Thmtheorem11.p1.1.m1.1a"><mi id="S6.Thmtheorem11.p1.1.m1.1.1" xref="S6.Thmtheorem11.p1.1.m1.1.1.cmml">E</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem11.p1.1.m1.1b"><ci id="S6.Thmtheorem11.p1.1.m1.1.1.cmml" xref="S6.Thmtheorem11.p1.1.m1.1.1">𝐸</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem11.p1.1.m1.1c">E</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem11.p1.1.m1.1d">italic_E</annotation></semantics></math> is a club of ordinals closed under <math alttext="\delta_{A}" class="ltx_Math" display="inline" id="S6.Thmtheorem11.p1.2.m2.1"><semantics id="S6.Thmtheorem11.p1.2.m2.1a"><msub id="S6.Thmtheorem11.p1.2.m2.1.1" xref="S6.Thmtheorem11.p1.2.m2.1.1.cmml"><mi id="S6.Thmtheorem11.p1.2.m2.1.1.2" xref="S6.Thmtheorem11.p1.2.m2.1.1.2.cmml">δ</mi><mi id="S6.Thmtheorem11.p1.2.m2.1.1.3" xref="S6.Thmtheorem11.p1.2.m2.1.1.3.cmml">A</mi></msub><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem11.p1.2.m2.1b"><apply id="S6.Thmtheorem11.p1.2.m2.1.1.cmml" xref="S6.Thmtheorem11.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S6.Thmtheorem11.p1.2.m2.1.1.1.cmml" xref="S6.Thmtheorem11.p1.2.m2.1.1">subscript</csymbol><ci id="S6.Thmtheorem11.p1.2.m2.1.1.2.cmml" xref="S6.Thmtheorem11.p1.2.m2.1.1.2">𝛿</ci><ci id="S6.Thmtheorem11.p1.2.m2.1.1.3.cmml" xref="S6.Thmtheorem11.p1.2.m2.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem11.p1.2.m2.1c">\delta_{A}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem11.p1.2.m2.1d">italic_δ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT</annotation></semantics></math>, then,</p> <ol class="ltx_enumerate" id="S6.I6"> <li class="ltx_item" id="S6.I6.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(a)</span> <div class="ltx_para" id="S6.I6.i1.p1"> <p class="ltx_p" id="S6.I6.i1.p1.2">For all <math alttext="a,b\in A" class="ltx_Math" display="inline" id="S6.I6.i1.p1.1.m1.2"><semantics id="S6.I6.i1.p1.1.m1.2a"><mrow id="S6.I6.i1.p1.1.m1.2.3" xref="S6.I6.i1.p1.1.m1.2.3.cmml"><mrow id="S6.I6.i1.p1.1.m1.2.3.2.2" xref="S6.I6.i1.p1.1.m1.2.3.2.1.cmml"><mi id="S6.I6.i1.p1.1.m1.1.1" xref="S6.I6.i1.p1.1.m1.1.1.cmml">a</mi><mo id="S6.I6.i1.p1.1.m1.2.3.2.2.1" xref="S6.I6.i1.p1.1.m1.2.3.2.1.cmml">,</mo><mi id="S6.I6.i1.p1.1.m1.2.2" xref="S6.I6.i1.p1.1.m1.2.2.cmml">b</mi></mrow><mo id="S6.I6.i1.p1.1.m1.2.3.1" xref="S6.I6.i1.p1.1.m1.2.3.1.cmml">∈</mo><mi id="S6.I6.i1.p1.1.m1.2.3.3" xref="S6.I6.i1.p1.1.m1.2.3.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.I6.i1.p1.1.m1.2b"><apply id="S6.I6.i1.p1.1.m1.2.3.cmml" xref="S6.I6.i1.p1.1.m1.2.3"><in id="S6.I6.i1.p1.1.m1.2.3.1.cmml" xref="S6.I6.i1.p1.1.m1.2.3.1"></in><list id="S6.I6.i1.p1.1.m1.2.3.2.1.cmml" xref="S6.I6.i1.p1.1.m1.2.3.2.2"><ci id="S6.I6.i1.p1.1.m1.1.1.cmml" xref="S6.I6.i1.p1.1.m1.1.1">𝑎</ci><ci id="S6.I6.i1.p1.1.m1.2.2.cmml" xref="S6.I6.i1.p1.1.m1.2.2">𝑏</ci></list><ci id="S6.I6.i1.p1.1.m1.2.3.3.cmml" xref="S6.I6.i1.p1.1.m1.2.3.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I6.i1.p1.1.m1.2c">a,b\in A</annotation><annotation encoding="application/x-llamapun" id="S6.I6.i1.p1.1.m1.2d">italic_a , italic_b ∈ italic_A</annotation></semantics></math>, <math alttext="\nu(a,b)\leq\nu(a),\nu(b)" class="ltx_Math" display="inline" id="S6.I6.i1.p1.2.m2.6"><semantics id="S6.I6.i1.p1.2.m2.6a"><mrow id="S6.I6.i1.p1.2.m2.6.6" xref="S6.I6.i1.p1.2.m2.6.6.cmml"><mrow id="S6.I6.i1.p1.2.m2.6.6.4" xref="S6.I6.i1.p1.2.m2.6.6.4.cmml"><mi id="S6.I6.i1.p1.2.m2.6.6.4.2" xref="S6.I6.i1.p1.2.m2.6.6.4.2.cmml">ν</mi><mo id="S6.I6.i1.p1.2.m2.6.6.4.1" xref="S6.I6.i1.p1.2.m2.6.6.4.1.cmml">⁢</mo><mrow id="S6.I6.i1.p1.2.m2.6.6.4.3.2" xref="S6.I6.i1.p1.2.m2.6.6.4.3.1.cmml"><mo id="S6.I6.i1.p1.2.m2.6.6.4.3.2.1" stretchy="false" xref="S6.I6.i1.p1.2.m2.6.6.4.3.1.cmml">(</mo><mi id="S6.I6.i1.p1.2.m2.1.1" xref="S6.I6.i1.p1.2.m2.1.1.cmml">a</mi><mo id="S6.I6.i1.p1.2.m2.6.6.4.3.2.2" xref="S6.I6.i1.p1.2.m2.6.6.4.3.1.cmml">,</mo><mi id="S6.I6.i1.p1.2.m2.2.2" xref="S6.I6.i1.p1.2.m2.2.2.cmml">b</mi><mo id="S6.I6.i1.p1.2.m2.6.6.4.3.2.3" stretchy="false" xref="S6.I6.i1.p1.2.m2.6.6.4.3.1.cmml">)</mo></mrow></mrow><mo id="S6.I6.i1.p1.2.m2.6.6.3" xref="S6.I6.i1.p1.2.m2.6.6.3.cmml">≤</mo><mrow id="S6.I6.i1.p1.2.m2.6.6.2.2" xref="S6.I6.i1.p1.2.m2.6.6.2.3.cmml"><mrow id="S6.I6.i1.p1.2.m2.5.5.1.1.1" xref="S6.I6.i1.p1.2.m2.5.5.1.1.1.cmml"><mi id="S6.I6.i1.p1.2.m2.5.5.1.1.1.2" xref="S6.I6.i1.p1.2.m2.5.5.1.1.1.2.cmml">ν</mi><mo id="S6.I6.i1.p1.2.m2.5.5.1.1.1.1" xref="S6.I6.i1.p1.2.m2.5.5.1.1.1.1.cmml">⁢</mo><mrow id="S6.I6.i1.p1.2.m2.5.5.1.1.1.3.2" xref="S6.I6.i1.p1.2.m2.5.5.1.1.1.cmml"><mo id="S6.I6.i1.p1.2.m2.5.5.1.1.1.3.2.1" stretchy="false" xref="S6.I6.i1.p1.2.m2.5.5.1.1.1.cmml">(</mo><mi id="S6.I6.i1.p1.2.m2.3.3" xref="S6.I6.i1.p1.2.m2.3.3.cmml">a</mi><mo id="S6.I6.i1.p1.2.m2.5.5.1.1.1.3.2.2" stretchy="false" xref="S6.I6.i1.p1.2.m2.5.5.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.I6.i1.p1.2.m2.6.6.2.2.3" xref="S6.I6.i1.p1.2.m2.6.6.2.3.cmml">,</mo><mrow id="S6.I6.i1.p1.2.m2.6.6.2.2.2" xref="S6.I6.i1.p1.2.m2.6.6.2.2.2.cmml"><mi id="S6.I6.i1.p1.2.m2.6.6.2.2.2.2" xref="S6.I6.i1.p1.2.m2.6.6.2.2.2.2.cmml">ν</mi><mo id="S6.I6.i1.p1.2.m2.6.6.2.2.2.1" xref="S6.I6.i1.p1.2.m2.6.6.2.2.2.1.cmml">⁢</mo><mrow id="S6.I6.i1.p1.2.m2.6.6.2.2.2.3.2" xref="S6.I6.i1.p1.2.m2.6.6.2.2.2.cmml"><mo id="S6.I6.i1.p1.2.m2.6.6.2.2.2.3.2.1" stretchy="false" xref="S6.I6.i1.p1.2.m2.6.6.2.2.2.cmml">(</mo><mi id="S6.I6.i1.p1.2.m2.4.4" xref="S6.I6.i1.p1.2.m2.4.4.cmml">b</mi><mo id="S6.I6.i1.p1.2.m2.6.6.2.2.2.3.2.2" stretchy="false" xref="S6.I6.i1.p1.2.m2.6.6.2.2.2.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.I6.i1.p1.2.m2.6b"><apply id="S6.I6.i1.p1.2.m2.6.6.cmml" xref="S6.I6.i1.p1.2.m2.6.6"><leq id="S6.I6.i1.p1.2.m2.6.6.3.cmml" xref="S6.I6.i1.p1.2.m2.6.6.3"></leq><apply id="S6.I6.i1.p1.2.m2.6.6.4.cmml" xref="S6.I6.i1.p1.2.m2.6.6.4"><times id="S6.I6.i1.p1.2.m2.6.6.4.1.cmml" xref="S6.I6.i1.p1.2.m2.6.6.4.1"></times><ci id="S6.I6.i1.p1.2.m2.6.6.4.2.cmml" xref="S6.I6.i1.p1.2.m2.6.6.4.2">𝜈</ci><interval closure="open" id="S6.I6.i1.p1.2.m2.6.6.4.3.1.cmml" xref="S6.I6.i1.p1.2.m2.6.6.4.3.2"><ci id="S6.I6.i1.p1.2.m2.1.1.cmml" xref="S6.I6.i1.p1.2.m2.1.1">𝑎</ci><ci id="S6.I6.i1.p1.2.m2.2.2.cmml" xref="S6.I6.i1.p1.2.m2.2.2">𝑏</ci></interval></apply><list id="S6.I6.i1.p1.2.m2.6.6.2.3.cmml" xref="S6.I6.i1.p1.2.m2.6.6.2.2"><apply id="S6.I6.i1.p1.2.m2.5.5.1.1.1.cmml" xref="S6.I6.i1.p1.2.m2.5.5.1.1.1"><times id="S6.I6.i1.p1.2.m2.5.5.1.1.1.1.cmml" xref="S6.I6.i1.p1.2.m2.5.5.1.1.1.1"></times><ci id="S6.I6.i1.p1.2.m2.5.5.1.1.1.2.cmml" xref="S6.I6.i1.p1.2.m2.5.5.1.1.1.2">𝜈</ci><ci id="S6.I6.i1.p1.2.m2.3.3.cmml" xref="S6.I6.i1.p1.2.m2.3.3">𝑎</ci></apply><apply id="S6.I6.i1.p1.2.m2.6.6.2.2.2.cmml" xref="S6.I6.i1.p1.2.m2.6.6.2.2.2"><times id="S6.I6.i1.p1.2.m2.6.6.2.2.2.1.cmml" xref="S6.I6.i1.p1.2.m2.6.6.2.2.2.1"></times><ci id="S6.I6.i1.p1.2.m2.6.6.2.2.2.2.cmml" xref="S6.I6.i1.p1.2.m2.6.6.2.2.2.2">𝜈</ci><ci id="S6.I6.i1.p1.2.m2.4.4.cmml" xref="S6.I6.i1.p1.2.m2.4.4">𝑏</ci></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I6.i1.p1.2.m2.6c">\nu(a,b)\leq\nu(a),\nu(b)</annotation><annotation encoding="application/x-llamapun" id="S6.I6.i1.p1.2.m2.6d">italic_ν ( italic_a , italic_b ) ≤ italic_ν ( italic_a ) , italic_ν ( italic_b )</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S6.I6.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(b)</span> <div class="ltx_para" id="S6.I6.i2.p1"> <p class="ltx_p" id="S6.I6.i2.p1.4">For all <math alttext="p\in P" class="ltx_Math" display="inline" id="S6.I6.i2.p1.1.m1.1"><semantics id="S6.I6.i2.p1.1.m1.1a"><mrow id="S6.I6.i2.p1.1.m1.1.1" xref="S6.I6.i2.p1.1.m1.1.1.cmml"><mi id="S6.I6.i2.p1.1.m1.1.1.2" xref="S6.I6.i2.p1.1.m1.1.1.2.cmml">p</mi><mo id="S6.I6.i2.p1.1.m1.1.1.1" xref="S6.I6.i2.p1.1.m1.1.1.1.cmml">∈</mo><mi id="S6.I6.i2.p1.1.m1.1.1.3" xref="S6.I6.i2.p1.1.m1.1.1.3.cmml">P</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.I6.i2.p1.1.m1.1b"><apply id="S6.I6.i2.p1.1.m1.1.1.cmml" xref="S6.I6.i2.p1.1.m1.1.1"><in id="S6.I6.i2.p1.1.m1.1.1.1.cmml" xref="S6.I6.i2.p1.1.m1.1.1.1"></in><ci id="S6.I6.i2.p1.1.m1.1.1.2.cmml" xref="S6.I6.i2.p1.1.m1.1.1.2">𝑝</ci><ci id="S6.I6.i2.p1.1.m1.1.1.3.cmml" xref="S6.I6.i2.p1.1.m1.1.1.3">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I6.i2.p1.1.m1.1c">p\in P</annotation><annotation encoding="application/x-llamapun" id="S6.I6.i2.p1.1.m1.1d">italic_p ∈ italic_P</annotation></semantics></math> and <math alttext="\bar{a}\neq\bar{b}\in\operatorname{dom}(p)" class="ltx_Math" display="inline" id="S6.I6.i2.p1.2.m2.2"><semantics id="S6.I6.i2.p1.2.m2.2a"><mrow id="S6.I6.i2.p1.2.m2.2.3" xref="S6.I6.i2.p1.2.m2.2.3.cmml"><mover accent="true" id="S6.I6.i2.p1.2.m2.2.3.2" xref="S6.I6.i2.p1.2.m2.2.3.2.cmml"><mi id="S6.I6.i2.p1.2.m2.2.3.2.2" xref="S6.I6.i2.p1.2.m2.2.3.2.2.cmml">a</mi><mo id="S6.I6.i2.p1.2.m2.2.3.2.1" xref="S6.I6.i2.p1.2.m2.2.3.2.1.cmml">¯</mo></mover><mo id="S6.I6.i2.p1.2.m2.2.3.3" xref="S6.I6.i2.p1.2.m2.2.3.3.cmml">≠</mo><mover accent="true" id="S6.I6.i2.p1.2.m2.2.3.4" xref="S6.I6.i2.p1.2.m2.2.3.4.cmml"><mi id="S6.I6.i2.p1.2.m2.2.3.4.2" xref="S6.I6.i2.p1.2.m2.2.3.4.2.cmml">b</mi><mo id="S6.I6.i2.p1.2.m2.2.3.4.1" xref="S6.I6.i2.p1.2.m2.2.3.4.1.cmml">¯</mo></mover><mo id="S6.I6.i2.p1.2.m2.2.3.5" xref="S6.I6.i2.p1.2.m2.2.3.5.cmml">∈</mo><mrow id="S6.I6.i2.p1.2.m2.2.3.6.2" xref="S6.I6.i2.p1.2.m2.2.3.6.1.cmml"><mi id="S6.I6.i2.p1.2.m2.1.1" xref="S6.I6.i2.p1.2.m2.1.1.cmml">dom</mi><mo id="S6.I6.i2.p1.2.m2.2.3.6.2a" xref="S6.I6.i2.p1.2.m2.2.3.6.1.cmml">⁡</mo><mrow id="S6.I6.i2.p1.2.m2.2.3.6.2.1" xref="S6.I6.i2.p1.2.m2.2.3.6.1.cmml"><mo id="S6.I6.i2.p1.2.m2.2.3.6.2.1.1" stretchy="false" xref="S6.I6.i2.p1.2.m2.2.3.6.1.cmml">(</mo><mi id="S6.I6.i2.p1.2.m2.2.2" xref="S6.I6.i2.p1.2.m2.2.2.cmml">p</mi><mo id="S6.I6.i2.p1.2.m2.2.3.6.2.1.2" stretchy="false" xref="S6.I6.i2.p1.2.m2.2.3.6.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.I6.i2.p1.2.m2.2b"><apply id="S6.I6.i2.p1.2.m2.2.3.cmml" xref="S6.I6.i2.p1.2.m2.2.3"><and id="S6.I6.i2.p1.2.m2.2.3a.cmml" xref="S6.I6.i2.p1.2.m2.2.3"></and><apply id="S6.I6.i2.p1.2.m2.2.3b.cmml" xref="S6.I6.i2.p1.2.m2.2.3"><neq id="S6.I6.i2.p1.2.m2.2.3.3.cmml" xref="S6.I6.i2.p1.2.m2.2.3.3"></neq><apply id="S6.I6.i2.p1.2.m2.2.3.2.cmml" xref="S6.I6.i2.p1.2.m2.2.3.2"><ci id="S6.I6.i2.p1.2.m2.2.3.2.1.cmml" xref="S6.I6.i2.p1.2.m2.2.3.2.1">¯</ci><ci id="S6.I6.i2.p1.2.m2.2.3.2.2.cmml" xref="S6.I6.i2.p1.2.m2.2.3.2.2">𝑎</ci></apply><apply id="S6.I6.i2.p1.2.m2.2.3.4.cmml" xref="S6.I6.i2.p1.2.m2.2.3.4"><ci id="S6.I6.i2.p1.2.m2.2.3.4.1.cmml" xref="S6.I6.i2.p1.2.m2.2.3.4.1">¯</ci><ci id="S6.I6.i2.p1.2.m2.2.3.4.2.cmml" xref="S6.I6.i2.p1.2.m2.2.3.4.2">𝑏</ci></apply></apply><apply id="S6.I6.i2.p1.2.m2.2.3c.cmml" xref="S6.I6.i2.p1.2.m2.2.3"><in id="S6.I6.i2.p1.2.m2.2.3.5.cmml" xref="S6.I6.i2.p1.2.m2.2.3.5"></in><share href="https://arxiv.org/html/2503.13728v1#S6.I6.i2.p1.2.m2.2.3.4.cmml" id="S6.I6.i2.p1.2.m2.2.3d.cmml" xref="S6.I6.i2.p1.2.m2.2.3"></share><apply id="S6.I6.i2.p1.2.m2.2.3.6.1.cmml" xref="S6.I6.i2.p1.2.m2.2.3.6.2"><ci id="S6.I6.i2.p1.2.m2.1.1.cmml" xref="S6.I6.i2.p1.2.m2.1.1">dom</ci><ci id="S6.I6.i2.p1.2.m2.2.2.cmml" xref="S6.I6.i2.p1.2.m2.2.2">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I6.i2.p1.2.m2.2c">\bar{a}\neq\bar{b}\in\operatorname{dom}(p)</annotation><annotation encoding="application/x-llamapun" id="S6.I6.i2.p1.2.m2.2d">over¯ start_ARG italic_a end_ARG ≠ over¯ start_ARG italic_b end_ARG ∈ roman_dom ( italic_p )</annotation></semantics></math>, if <math alttext="p" class="ltx_Math" display="inline" id="S6.I6.i2.p1.3.m3.1"><semantics id="S6.I6.i2.p1.3.m3.1a"><mi id="S6.I6.i2.p1.3.m3.1.1" xref="S6.I6.i2.p1.3.m3.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S6.I6.i2.p1.3.m3.1b"><ci id="S6.I6.i2.p1.3.m3.1.1.cmml" xref="S6.I6.i2.p1.3.m3.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.I6.i2.p1.3.m3.1c">p</annotation><annotation encoding="application/x-llamapun" id="S6.I6.i2.p1.3.m3.1d">italic_p</annotation></semantics></math> satisfies <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.Thmtheorem9" title="Definition 6.9. ‣ 6.2. Forcing epimorphisms ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">6.9</span></a> <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.I4.i1" title="Item (i) ‣ Definition 6.9. ‣ 6.2. Forcing epimorphisms ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">(i)</span></a>, then <math alttext="\nu(a_{l},b_{r})=\nu(a_{m},b_{m})=\nu(a_{r},b_{l})" class="ltx_Math" display="inline" id="S6.I6.i2.p1.4.m4.6"><semantics id="S6.I6.i2.p1.4.m4.6a"><mrow id="S6.I6.i2.p1.4.m4.6.6" xref="S6.I6.i2.p1.4.m4.6.6.cmml"><mrow id="S6.I6.i2.p1.4.m4.2.2.2" xref="S6.I6.i2.p1.4.m4.2.2.2.cmml"><mi id="S6.I6.i2.p1.4.m4.2.2.2.4" xref="S6.I6.i2.p1.4.m4.2.2.2.4.cmml">ν</mi><mo id="S6.I6.i2.p1.4.m4.2.2.2.3" xref="S6.I6.i2.p1.4.m4.2.2.2.3.cmml">⁢</mo><mrow id="S6.I6.i2.p1.4.m4.2.2.2.2.2" xref="S6.I6.i2.p1.4.m4.2.2.2.2.3.cmml"><mo id="S6.I6.i2.p1.4.m4.2.2.2.2.2.3" stretchy="false" xref="S6.I6.i2.p1.4.m4.2.2.2.2.3.cmml">(</mo><msub id="S6.I6.i2.p1.4.m4.1.1.1.1.1.1" xref="S6.I6.i2.p1.4.m4.1.1.1.1.1.1.cmml"><mi id="S6.I6.i2.p1.4.m4.1.1.1.1.1.1.2" xref="S6.I6.i2.p1.4.m4.1.1.1.1.1.1.2.cmml">a</mi><mi id="S6.I6.i2.p1.4.m4.1.1.1.1.1.1.3" xref="S6.I6.i2.p1.4.m4.1.1.1.1.1.1.3.cmml">l</mi></msub><mo id="S6.I6.i2.p1.4.m4.2.2.2.2.2.4" xref="S6.I6.i2.p1.4.m4.2.2.2.2.3.cmml">,</mo><msub id="S6.I6.i2.p1.4.m4.2.2.2.2.2.2" xref="S6.I6.i2.p1.4.m4.2.2.2.2.2.2.cmml"><mi id="S6.I6.i2.p1.4.m4.2.2.2.2.2.2.2" xref="S6.I6.i2.p1.4.m4.2.2.2.2.2.2.2.cmml">b</mi><mi id="S6.I6.i2.p1.4.m4.2.2.2.2.2.2.3" xref="S6.I6.i2.p1.4.m4.2.2.2.2.2.2.3.cmml">r</mi></msub><mo id="S6.I6.i2.p1.4.m4.2.2.2.2.2.5" stretchy="false" xref="S6.I6.i2.p1.4.m4.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.I6.i2.p1.4.m4.6.6.8" xref="S6.I6.i2.p1.4.m4.6.6.8.cmml">=</mo><mrow id="S6.I6.i2.p1.4.m4.4.4.4" xref="S6.I6.i2.p1.4.m4.4.4.4.cmml"><mi id="S6.I6.i2.p1.4.m4.4.4.4.4" xref="S6.I6.i2.p1.4.m4.4.4.4.4.cmml">ν</mi><mo id="S6.I6.i2.p1.4.m4.4.4.4.3" xref="S6.I6.i2.p1.4.m4.4.4.4.3.cmml">⁢</mo><mrow id="S6.I6.i2.p1.4.m4.4.4.4.2.2" xref="S6.I6.i2.p1.4.m4.4.4.4.2.3.cmml"><mo id="S6.I6.i2.p1.4.m4.4.4.4.2.2.3" stretchy="false" xref="S6.I6.i2.p1.4.m4.4.4.4.2.3.cmml">(</mo><msub id="S6.I6.i2.p1.4.m4.3.3.3.1.1.1" xref="S6.I6.i2.p1.4.m4.3.3.3.1.1.1.cmml"><mi id="S6.I6.i2.p1.4.m4.3.3.3.1.1.1.2" xref="S6.I6.i2.p1.4.m4.3.3.3.1.1.1.2.cmml">a</mi><mi id="S6.I6.i2.p1.4.m4.3.3.3.1.1.1.3" xref="S6.I6.i2.p1.4.m4.3.3.3.1.1.1.3.cmml">m</mi></msub><mo id="S6.I6.i2.p1.4.m4.4.4.4.2.2.4" xref="S6.I6.i2.p1.4.m4.4.4.4.2.3.cmml">,</mo><msub id="S6.I6.i2.p1.4.m4.4.4.4.2.2.2" xref="S6.I6.i2.p1.4.m4.4.4.4.2.2.2.cmml"><mi id="S6.I6.i2.p1.4.m4.4.4.4.2.2.2.2" xref="S6.I6.i2.p1.4.m4.4.4.4.2.2.2.2.cmml">b</mi><mi id="S6.I6.i2.p1.4.m4.4.4.4.2.2.2.3" xref="S6.I6.i2.p1.4.m4.4.4.4.2.2.2.3.cmml">m</mi></msub><mo id="S6.I6.i2.p1.4.m4.4.4.4.2.2.5" stretchy="false" xref="S6.I6.i2.p1.4.m4.4.4.4.2.3.cmml">)</mo></mrow></mrow><mo id="S6.I6.i2.p1.4.m4.6.6.9" xref="S6.I6.i2.p1.4.m4.6.6.9.cmml">=</mo><mrow id="S6.I6.i2.p1.4.m4.6.6.6" xref="S6.I6.i2.p1.4.m4.6.6.6.cmml"><mi id="S6.I6.i2.p1.4.m4.6.6.6.4" xref="S6.I6.i2.p1.4.m4.6.6.6.4.cmml">ν</mi><mo id="S6.I6.i2.p1.4.m4.6.6.6.3" xref="S6.I6.i2.p1.4.m4.6.6.6.3.cmml">⁢</mo><mrow id="S6.I6.i2.p1.4.m4.6.6.6.2.2" xref="S6.I6.i2.p1.4.m4.6.6.6.2.3.cmml"><mo id="S6.I6.i2.p1.4.m4.6.6.6.2.2.3" stretchy="false" xref="S6.I6.i2.p1.4.m4.6.6.6.2.3.cmml">(</mo><msub id="S6.I6.i2.p1.4.m4.5.5.5.1.1.1" xref="S6.I6.i2.p1.4.m4.5.5.5.1.1.1.cmml"><mi id="S6.I6.i2.p1.4.m4.5.5.5.1.1.1.2" xref="S6.I6.i2.p1.4.m4.5.5.5.1.1.1.2.cmml">a</mi><mi id="S6.I6.i2.p1.4.m4.5.5.5.1.1.1.3" xref="S6.I6.i2.p1.4.m4.5.5.5.1.1.1.3.cmml">r</mi></msub><mo id="S6.I6.i2.p1.4.m4.6.6.6.2.2.4" xref="S6.I6.i2.p1.4.m4.6.6.6.2.3.cmml">,</mo><msub id="S6.I6.i2.p1.4.m4.6.6.6.2.2.2" xref="S6.I6.i2.p1.4.m4.6.6.6.2.2.2.cmml"><mi id="S6.I6.i2.p1.4.m4.6.6.6.2.2.2.2" xref="S6.I6.i2.p1.4.m4.6.6.6.2.2.2.2.cmml">b</mi><mi id="S6.I6.i2.p1.4.m4.6.6.6.2.2.2.3" xref="S6.I6.i2.p1.4.m4.6.6.6.2.2.2.3.cmml">l</mi></msub><mo id="S6.I6.i2.p1.4.m4.6.6.6.2.2.5" stretchy="false" xref="S6.I6.i2.p1.4.m4.6.6.6.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.I6.i2.p1.4.m4.6b"><apply id="S6.I6.i2.p1.4.m4.6.6.cmml" xref="S6.I6.i2.p1.4.m4.6.6"><and id="S6.I6.i2.p1.4.m4.6.6a.cmml" xref="S6.I6.i2.p1.4.m4.6.6"></and><apply id="S6.I6.i2.p1.4.m4.6.6b.cmml" xref="S6.I6.i2.p1.4.m4.6.6"><eq id="S6.I6.i2.p1.4.m4.6.6.8.cmml" xref="S6.I6.i2.p1.4.m4.6.6.8"></eq><apply id="S6.I6.i2.p1.4.m4.2.2.2.cmml" xref="S6.I6.i2.p1.4.m4.2.2.2"><times id="S6.I6.i2.p1.4.m4.2.2.2.3.cmml" xref="S6.I6.i2.p1.4.m4.2.2.2.3"></times><ci id="S6.I6.i2.p1.4.m4.2.2.2.4.cmml" xref="S6.I6.i2.p1.4.m4.2.2.2.4">𝜈</ci><interval closure="open" id="S6.I6.i2.p1.4.m4.2.2.2.2.3.cmml" xref="S6.I6.i2.p1.4.m4.2.2.2.2.2"><apply id="S6.I6.i2.p1.4.m4.1.1.1.1.1.1.cmml" xref="S6.I6.i2.p1.4.m4.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.I6.i2.p1.4.m4.1.1.1.1.1.1.1.cmml" xref="S6.I6.i2.p1.4.m4.1.1.1.1.1.1">subscript</csymbol><ci id="S6.I6.i2.p1.4.m4.1.1.1.1.1.1.2.cmml" xref="S6.I6.i2.p1.4.m4.1.1.1.1.1.1.2">𝑎</ci><ci id="S6.I6.i2.p1.4.m4.1.1.1.1.1.1.3.cmml" xref="S6.I6.i2.p1.4.m4.1.1.1.1.1.1.3">𝑙</ci></apply><apply id="S6.I6.i2.p1.4.m4.2.2.2.2.2.2.cmml" xref="S6.I6.i2.p1.4.m4.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.I6.i2.p1.4.m4.2.2.2.2.2.2.1.cmml" xref="S6.I6.i2.p1.4.m4.2.2.2.2.2.2">subscript</csymbol><ci id="S6.I6.i2.p1.4.m4.2.2.2.2.2.2.2.cmml" xref="S6.I6.i2.p1.4.m4.2.2.2.2.2.2.2">𝑏</ci><ci id="S6.I6.i2.p1.4.m4.2.2.2.2.2.2.3.cmml" xref="S6.I6.i2.p1.4.m4.2.2.2.2.2.2.3">𝑟</ci></apply></interval></apply><apply id="S6.I6.i2.p1.4.m4.4.4.4.cmml" xref="S6.I6.i2.p1.4.m4.4.4.4"><times id="S6.I6.i2.p1.4.m4.4.4.4.3.cmml" xref="S6.I6.i2.p1.4.m4.4.4.4.3"></times><ci id="S6.I6.i2.p1.4.m4.4.4.4.4.cmml" xref="S6.I6.i2.p1.4.m4.4.4.4.4">𝜈</ci><interval closure="open" id="S6.I6.i2.p1.4.m4.4.4.4.2.3.cmml" xref="S6.I6.i2.p1.4.m4.4.4.4.2.2"><apply id="S6.I6.i2.p1.4.m4.3.3.3.1.1.1.cmml" xref="S6.I6.i2.p1.4.m4.3.3.3.1.1.1"><csymbol cd="ambiguous" id="S6.I6.i2.p1.4.m4.3.3.3.1.1.1.1.cmml" xref="S6.I6.i2.p1.4.m4.3.3.3.1.1.1">subscript</csymbol><ci id="S6.I6.i2.p1.4.m4.3.3.3.1.1.1.2.cmml" xref="S6.I6.i2.p1.4.m4.3.3.3.1.1.1.2">𝑎</ci><ci id="S6.I6.i2.p1.4.m4.3.3.3.1.1.1.3.cmml" xref="S6.I6.i2.p1.4.m4.3.3.3.1.1.1.3">𝑚</ci></apply><apply id="S6.I6.i2.p1.4.m4.4.4.4.2.2.2.cmml" xref="S6.I6.i2.p1.4.m4.4.4.4.2.2.2"><csymbol cd="ambiguous" id="S6.I6.i2.p1.4.m4.4.4.4.2.2.2.1.cmml" xref="S6.I6.i2.p1.4.m4.4.4.4.2.2.2">subscript</csymbol><ci id="S6.I6.i2.p1.4.m4.4.4.4.2.2.2.2.cmml" xref="S6.I6.i2.p1.4.m4.4.4.4.2.2.2.2">𝑏</ci><ci id="S6.I6.i2.p1.4.m4.4.4.4.2.2.2.3.cmml" xref="S6.I6.i2.p1.4.m4.4.4.4.2.2.2.3">𝑚</ci></apply></interval></apply></apply><apply id="S6.I6.i2.p1.4.m4.6.6c.cmml" xref="S6.I6.i2.p1.4.m4.6.6"><eq id="S6.I6.i2.p1.4.m4.6.6.9.cmml" xref="S6.I6.i2.p1.4.m4.6.6.9"></eq><share href="https://arxiv.org/html/2503.13728v1#S6.I6.i2.p1.4.m4.4.4.4.cmml" id="S6.I6.i2.p1.4.m4.6.6d.cmml" xref="S6.I6.i2.p1.4.m4.6.6"></share><apply id="S6.I6.i2.p1.4.m4.6.6.6.cmml" xref="S6.I6.i2.p1.4.m4.6.6.6"><times id="S6.I6.i2.p1.4.m4.6.6.6.3.cmml" xref="S6.I6.i2.p1.4.m4.6.6.6.3"></times><ci id="S6.I6.i2.p1.4.m4.6.6.6.4.cmml" xref="S6.I6.i2.p1.4.m4.6.6.6.4">𝜈</ci><interval closure="open" id="S6.I6.i2.p1.4.m4.6.6.6.2.3.cmml" xref="S6.I6.i2.p1.4.m4.6.6.6.2.2"><apply id="S6.I6.i2.p1.4.m4.5.5.5.1.1.1.cmml" xref="S6.I6.i2.p1.4.m4.5.5.5.1.1.1"><csymbol cd="ambiguous" id="S6.I6.i2.p1.4.m4.5.5.5.1.1.1.1.cmml" xref="S6.I6.i2.p1.4.m4.5.5.5.1.1.1">subscript</csymbol><ci id="S6.I6.i2.p1.4.m4.5.5.5.1.1.1.2.cmml" xref="S6.I6.i2.p1.4.m4.5.5.5.1.1.1.2">𝑎</ci><ci id="S6.I6.i2.p1.4.m4.5.5.5.1.1.1.3.cmml" xref="S6.I6.i2.p1.4.m4.5.5.5.1.1.1.3">𝑟</ci></apply><apply id="S6.I6.i2.p1.4.m4.6.6.6.2.2.2.cmml" xref="S6.I6.i2.p1.4.m4.6.6.6.2.2.2"><csymbol cd="ambiguous" id="S6.I6.i2.p1.4.m4.6.6.6.2.2.2.1.cmml" xref="S6.I6.i2.p1.4.m4.6.6.6.2.2.2">subscript</csymbol><ci id="S6.I6.i2.p1.4.m4.6.6.6.2.2.2.2.cmml" xref="S6.I6.i2.p1.4.m4.6.6.6.2.2.2.2">𝑏</ci><ci id="S6.I6.i2.p1.4.m4.6.6.6.2.2.2.3.cmml" xref="S6.I6.i2.p1.4.m4.6.6.6.2.2.2.3">𝑙</ci></apply></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I6.i2.p1.4.m4.6c">\nu(a_{l},b_{r})=\nu(a_{m},b_{m})=\nu(a_{r},b_{l})</annotation><annotation encoding="application/x-llamapun" id="S6.I6.i2.p1.4.m4.6d">italic_ν ( italic_a start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) = italic_ν ( italic_a start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) = italic_ν ( italic_a start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT )</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S6.I6.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(c)</span> <div class="ltx_para" id="S6.I6.i3.p1"> <p class="ltx_p" id="S6.I6.i3.p1.5">For all <math alttext="p\in P" class="ltx_Math" display="inline" id="S6.I6.i3.p1.1.m1.1"><semantics id="S6.I6.i3.p1.1.m1.1a"><mrow id="S6.I6.i3.p1.1.m1.1.1" xref="S6.I6.i3.p1.1.m1.1.1.cmml"><mi id="S6.I6.i3.p1.1.m1.1.1.2" xref="S6.I6.i3.p1.1.m1.1.1.2.cmml">p</mi><mo id="S6.I6.i3.p1.1.m1.1.1.1" xref="S6.I6.i3.p1.1.m1.1.1.1.cmml">∈</mo><mi id="S6.I6.i3.p1.1.m1.1.1.3" xref="S6.I6.i3.p1.1.m1.1.1.3.cmml">P</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.I6.i3.p1.1.m1.1b"><apply id="S6.I6.i3.p1.1.m1.1.1.cmml" xref="S6.I6.i3.p1.1.m1.1.1"><in id="S6.I6.i3.p1.1.m1.1.1.1.cmml" xref="S6.I6.i3.p1.1.m1.1.1.1"></in><ci id="S6.I6.i3.p1.1.m1.1.1.2.cmml" xref="S6.I6.i3.p1.1.m1.1.1.2">𝑝</ci><ci id="S6.I6.i3.p1.1.m1.1.1.3.cmml" xref="S6.I6.i3.p1.1.m1.1.1.3">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I6.i3.p1.1.m1.1c">p\in P</annotation><annotation encoding="application/x-llamapun" id="S6.I6.i3.p1.1.m1.1d">italic_p ∈ italic_P</annotation></semantics></math>, <math alttext="p\in P_{E}" class="ltx_Math" display="inline" id="S6.I6.i3.p1.2.m2.1"><semantics id="S6.I6.i3.p1.2.m2.1a"><mrow id="S6.I6.i3.p1.2.m2.1.1" xref="S6.I6.i3.p1.2.m2.1.1.cmml"><mi id="S6.I6.i3.p1.2.m2.1.1.2" xref="S6.I6.i3.p1.2.m2.1.1.2.cmml">p</mi><mo id="S6.I6.i3.p1.2.m2.1.1.1" xref="S6.I6.i3.p1.2.m2.1.1.1.cmml">∈</mo><msub id="S6.I6.i3.p1.2.m2.1.1.3" xref="S6.I6.i3.p1.2.m2.1.1.3.cmml"><mi id="S6.I6.i3.p1.2.m2.1.1.3.2" xref="S6.I6.i3.p1.2.m2.1.1.3.2.cmml">P</mi><mi id="S6.I6.i3.p1.2.m2.1.1.3.3" xref="S6.I6.i3.p1.2.m2.1.1.3.3.cmml">E</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.I6.i3.p1.2.m2.1b"><apply id="S6.I6.i3.p1.2.m2.1.1.cmml" xref="S6.I6.i3.p1.2.m2.1.1"><in id="S6.I6.i3.p1.2.m2.1.1.1.cmml" xref="S6.I6.i3.p1.2.m2.1.1.1"></in><ci id="S6.I6.i3.p1.2.m2.1.1.2.cmml" xref="S6.I6.i3.p1.2.m2.1.1.2">𝑝</ci><apply id="S6.I6.i3.p1.2.m2.1.1.3.cmml" xref="S6.I6.i3.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S6.I6.i3.p1.2.m2.1.1.3.1.cmml" xref="S6.I6.i3.p1.2.m2.1.1.3">subscript</csymbol><ci id="S6.I6.i3.p1.2.m2.1.1.3.2.cmml" xref="S6.I6.i3.p1.2.m2.1.1.3.2">𝑃</ci><ci id="S6.I6.i3.p1.2.m2.1.1.3.3.cmml" xref="S6.I6.i3.p1.2.m2.1.1.3.3">𝐸</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I6.i3.p1.2.m2.1c">p\in P_{E}</annotation><annotation encoding="application/x-llamapun" id="S6.I6.i3.p1.2.m2.1d">italic_p ∈ italic_P start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT</annotation></semantics></math> iff <math alttext="q(p):=\{(a_{m},p(\bar{a})):\bar{a}\in\operatorname{dom}(p)\}" class="ltx_Math" display="inline" id="S6.I6.i3.p1.3.m3.6"><semantics id="S6.I6.i3.p1.3.m3.6a"><mrow id="S6.I6.i3.p1.3.m3.6.6" xref="S6.I6.i3.p1.3.m3.6.6.cmml"><mrow id="S6.I6.i3.p1.3.m3.6.6.4" xref="S6.I6.i3.p1.3.m3.6.6.4.cmml"><mi id="S6.I6.i3.p1.3.m3.6.6.4.2" xref="S6.I6.i3.p1.3.m3.6.6.4.2.cmml">q</mi><mo id="S6.I6.i3.p1.3.m3.6.6.4.1" xref="S6.I6.i3.p1.3.m3.6.6.4.1.cmml">⁢</mo><mrow id="S6.I6.i3.p1.3.m3.6.6.4.3.2" xref="S6.I6.i3.p1.3.m3.6.6.4.cmml"><mo id="S6.I6.i3.p1.3.m3.6.6.4.3.2.1" stretchy="false" xref="S6.I6.i3.p1.3.m3.6.6.4.cmml">(</mo><mi id="S6.I6.i3.p1.3.m3.1.1" xref="S6.I6.i3.p1.3.m3.1.1.cmml">p</mi><mo id="S6.I6.i3.p1.3.m3.6.6.4.3.2.2" rspace="0.278em" stretchy="false" xref="S6.I6.i3.p1.3.m3.6.6.4.cmml">)</mo></mrow></mrow><mo id="S6.I6.i3.p1.3.m3.6.6.3" rspace="0.278em" xref="S6.I6.i3.p1.3.m3.6.6.3.cmml">:=</mo><mrow id="S6.I6.i3.p1.3.m3.6.6.2.2" xref="S6.I6.i3.p1.3.m3.6.6.2.3.cmml"><mo id="S6.I6.i3.p1.3.m3.6.6.2.2.3" stretchy="false" xref="S6.I6.i3.p1.3.m3.6.6.2.3.1.cmml">{</mo><mrow id="S6.I6.i3.p1.3.m3.5.5.1.1.1.2" xref="S6.I6.i3.p1.3.m3.5.5.1.1.1.3.cmml"><mo id="S6.I6.i3.p1.3.m3.5.5.1.1.1.2.3" stretchy="false" xref="S6.I6.i3.p1.3.m3.5.5.1.1.1.3.cmml">(</mo><msub id="S6.I6.i3.p1.3.m3.5.5.1.1.1.1.1" xref="S6.I6.i3.p1.3.m3.5.5.1.1.1.1.1.cmml"><mi id="S6.I6.i3.p1.3.m3.5.5.1.1.1.1.1.2" xref="S6.I6.i3.p1.3.m3.5.5.1.1.1.1.1.2.cmml">a</mi><mi id="S6.I6.i3.p1.3.m3.5.5.1.1.1.1.1.3" xref="S6.I6.i3.p1.3.m3.5.5.1.1.1.1.1.3.cmml">m</mi></msub><mo id="S6.I6.i3.p1.3.m3.5.5.1.1.1.2.4" xref="S6.I6.i3.p1.3.m3.5.5.1.1.1.3.cmml">,</mo><mrow id="S6.I6.i3.p1.3.m3.5.5.1.1.1.2.2" xref="S6.I6.i3.p1.3.m3.5.5.1.1.1.2.2.cmml"><mi id="S6.I6.i3.p1.3.m3.5.5.1.1.1.2.2.2" xref="S6.I6.i3.p1.3.m3.5.5.1.1.1.2.2.2.cmml">p</mi><mo id="S6.I6.i3.p1.3.m3.5.5.1.1.1.2.2.1" xref="S6.I6.i3.p1.3.m3.5.5.1.1.1.2.2.1.cmml">⁢</mo><mrow id="S6.I6.i3.p1.3.m3.5.5.1.1.1.2.2.3.2" xref="S6.I6.i3.p1.3.m3.2.2.cmml"><mo id="S6.I6.i3.p1.3.m3.5.5.1.1.1.2.2.3.2.1" stretchy="false" xref="S6.I6.i3.p1.3.m3.2.2.cmml">(</mo><mover accent="true" id="S6.I6.i3.p1.3.m3.2.2" xref="S6.I6.i3.p1.3.m3.2.2.cmml"><mi id="S6.I6.i3.p1.3.m3.2.2.2" xref="S6.I6.i3.p1.3.m3.2.2.2.cmml">a</mi><mo id="S6.I6.i3.p1.3.m3.2.2.1" xref="S6.I6.i3.p1.3.m3.2.2.1.cmml">¯</mo></mover><mo id="S6.I6.i3.p1.3.m3.5.5.1.1.1.2.2.3.2.2" stretchy="false" xref="S6.I6.i3.p1.3.m3.2.2.cmml">)</mo></mrow></mrow><mo id="S6.I6.i3.p1.3.m3.5.5.1.1.1.2.5" rspace="0.278em" stretchy="false" xref="S6.I6.i3.p1.3.m3.5.5.1.1.1.3.cmml">)</mo></mrow><mo id="S6.I6.i3.p1.3.m3.6.6.2.2.4" rspace="0.278em" xref="S6.I6.i3.p1.3.m3.6.6.2.3.1.cmml">:</mo><mrow id="S6.I6.i3.p1.3.m3.6.6.2.2.2" xref="S6.I6.i3.p1.3.m3.6.6.2.2.2.cmml"><mover accent="true" id="S6.I6.i3.p1.3.m3.6.6.2.2.2.2" xref="S6.I6.i3.p1.3.m3.6.6.2.2.2.2.cmml"><mi id="S6.I6.i3.p1.3.m3.6.6.2.2.2.2.2" xref="S6.I6.i3.p1.3.m3.6.6.2.2.2.2.2.cmml">a</mi><mo id="S6.I6.i3.p1.3.m3.6.6.2.2.2.2.1" xref="S6.I6.i3.p1.3.m3.6.6.2.2.2.2.1.cmml">¯</mo></mover><mo id="S6.I6.i3.p1.3.m3.6.6.2.2.2.1" xref="S6.I6.i3.p1.3.m3.6.6.2.2.2.1.cmml">∈</mo><mrow id="S6.I6.i3.p1.3.m3.6.6.2.2.2.3.2" xref="S6.I6.i3.p1.3.m3.6.6.2.2.2.3.1.cmml"><mi id="S6.I6.i3.p1.3.m3.3.3" xref="S6.I6.i3.p1.3.m3.3.3.cmml">dom</mi><mo id="S6.I6.i3.p1.3.m3.6.6.2.2.2.3.2a" xref="S6.I6.i3.p1.3.m3.6.6.2.2.2.3.1.cmml">⁡</mo><mrow id="S6.I6.i3.p1.3.m3.6.6.2.2.2.3.2.1" xref="S6.I6.i3.p1.3.m3.6.6.2.2.2.3.1.cmml"><mo id="S6.I6.i3.p1.3.m3.6.6.2.2.2.3.2.1.1" stretchy="false" xref="S6.I6.i3.p1.3.m3.6.6.2.2.2.3.1.cmml">(</mo><mi id="S6.I6.i3.p1.3.m3.4.4" xref="S6.I6.i3.p1.3.m3.4.4.cmml">p</mi><mo id="S6.I6.i3.p1.3.m3.6.6.2.2.2.3.2.1.2" stretchy="false" xref="S6.I6.i3.p1.3.m3.6.6.2.2.2.3.1.cmml">)</mo></mrow></mrow></mrow><mo id="S6.I6.i3.p1.3.m3.6.6.2.2.5" stretchy="false" xref="S6.I6.i3.p1.3.m3.6.6.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.I6.i3.p1.3.m3.6b"><apply id="S6.I6.i3.p1.3.m3.6.6.cmml" xref="S6.I6.i3.p1.3.m3.6.6"><csymbol cd="latexml" id="S6.I6.i3.p1.3.m3.6.6.3.cmml" xref="S6.I6.i3.p1.3.m3.6.6.3">assign</csymbol><apply id="S6.I6.i3.p1.3.m3.6.6.4.cmml" xref="S6.I6.i3.p1.3.m3.6.6.4"><times id="S6.I6.i3.p1.3.m3.6.6.4.1.cmml" xref="S6.I6.i3.p1.3.m3.6.6.4.1"></times><ci id="S6.I6.i3.p1.3.m3.6.6.4.2.cmml" xref="S6.I6.i3.p1.3.m3.6.6.4.2">𝑞</ci><ci id="S6.I6.i3.p1.3.m3.1.1.cmml" xref="S6.I6.i3.p1.3.m3.1.1">𝑝</ci></apply><apply id="S6.I6.i3.p1.3.m3.6.6.2.3.cmml" xref="S6.I6.i3.p1.3.m3.6.6.2.2"><csymbol cd="latexml" id="S6.I6.i3.p1.3.m3.6.6.2.3.1.cmml" xref="S6.I6.i3.p1.3.m3.6.6.2.2.3">conditional-set</csymbol><interval closure="open" id="S6.I6.i3.p1.3.m3.5.5.1.1.1.3.cmml" xref="S6.I6.i3.p1.3.m3.5.5.1.1.1.2"><apply id="S6.I6.i3.p1.3.m3.5.5.1.1.1.1.1.cmml" xref="S6.I6.i3.p1.3.m3.5.5.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.I6.i3.p1.3.m3.5.5.1.1.1.1.1.1.cmml" xref="S6.I6.i3.p1.3.m3.5.5.1.1.1.1.1">subscript</csymbol><ci id="S6.I6.i3.p1.3.m3.5.5.1.1.1.1.1.2.cmml" xref="S6.I6.i3.p1.3.m3.5.5.1.1.1.1.1.2">𝑎</ci><ci id="S6.I6.i3.p1.3.m3.5.5.1.1.1.1.1.3.cmml" xref="S6.I6.i3.p1.3.m3.5.5.1.1.1.1.1.3">𝑚</ci></apply><apply id="S6.I6.i3.p1.3.m3.5.5.1.1.1.2.2.cmml" xref="S6.I6.i3.p1.3.m3.5.5.1.1.1.2.2"><times id="S6.I6.i3.p1.3.m3.5.5.1.1.1.2.2.1.cmml" xref="S6.I6.i3.p1.3.m3.5.5.1.1.1.2.2.1"></times><ci id="S6.I6.i3.p1.3.m3.5.5.1.1.1.2.2.2.cmml" xref="S6.I6.i3.p1.3.m3.5.5.1.1.1.2.2.2">𝑝</ci><apply id="S6.I6.i3.p1.3.m3.2.2.cmml" xref="S6.I6.i3.p1.3.m3.5.5.1.1.1.2.2.3.2"><ci id="S6.I6.i3.p1.3.m3.2.2.1.cmml" xref="S6.I6.i3.p1.3.m3.2.2.1">¯</ci><ci id="S6.I6.i3.p1.3.m3.2.2.2.cmml" xref="S6.I6.i3.p1.3.m3.2.2.2">𝑎</ci></apply></apply></interval><apply id="S6.I6.i3.p1.3.m3.6.6.2.2.2.cmml" xref="S6.I6.i3.p1.3.m3.6.6.2.2.2"><in id="S6.I6.i3.p1.3.m3.6.6.2.2.2.1.cmml" xref="S6.I6.i3.p1.3.m3.6.6.2.2.2.1"></in><apply id="S6.I6.i3.p1.3.m3.6.6.2.2.2.2.cmml" xref="S6.I6.i3.p1.3.m3.6.6.2.2.2.2"><ci id="S6.I6.i3.p1.3.m3.6.6.2.2.2.2.1.cmml" xref="S6.I6.i3.p1.3.m3.6.6.2.2.2.2.1">¯</ci><ci id="S6.I6.i3.p1.3.m3.6.6.2.2.2.2.2.cmml" xref="S6.I6.i3.p1.3.m3.6.6.2.2.2.2.2">𝑎</ci></apply><apply id="S6.I6.i3.p1.3.m3.6.6.2.2.2.3.1.cmml" xref="S6.I6.i3.p1.3.m3.6.6.2.2.2.3.2"><ci id="S6.I6.i3.p1.3.m3.3.3.cmml" xref="S6.I6.i3.p1.3.m3.3.3">dom</ci><ci id="S6.I6.i3.p1.3.m3.4.4.cmml" xref="S6.I6.i3.p1.3.m3.4.4">𝑝</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I6.i3.p1.3.m3.6c">q(p):=\{(a_{m},p(\bar{a})):\bar{a}\in\operatorname{dom}(p)\}</annotation><annotation encoding="application/x-llamapun" id="S6.I6.i3.p1.3.m3.6d">italic_q ( italic_p ) := { ( italic_a start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT , italic_p ( over¯ start_ARG italic_a end_ARG ) ) : over¯ start_ARG italic_a end_ARG ∈ roman_dom ( italic_p ) }</annotation></semantics></math> is in <math alttext="Q_{E}" class="ltx_Math" display="inline" id="S6.I6.i3.p1.4.m4.1"><semantics id="S6.I6.i3.p1.4.m4.1a"><msub id="S6.I6.i3.p1.4.m4.1.1" xref="S6.I6.i3.p1.4.m4.1.1.cmml"><mi id="S6.I6.i3.p1.4.m4.1.1.2" xref="S6.I6.i3.p1.4.m4.1.1.2.cmml">Q</mi><mi id="S6.I6.i3.p1.4.m4.1.1.3" xref="S6.I6.i3.p1.4.m4.1.1.3.cmml">E</mi></msub><annotation-xml encoding="MathML-Content" id="S6.I6.i3.p1.4.m4.1b"><apply id="S6.I6.i3.p1.4.m4.1.1.cmml" xref="S6.I6.i3.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S6.I6.i3.p1.4.m4.1.1.1.cmml" xref="S6.I6.i3.p1.4.m4.1.1">subscript</csymbol><ci id="S6.I6.i3.p1.4.m4.1.1.2.cmml" xref="S6.I6.i3.p1.4.m4.1.1.2">𝑄</ci><ci id="S6.I6.i3.p1.4.m4.1.1.3.cmml" xref="S6.I6.i3.p1.4.m4.1.1.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I6.i3.p1.4.m4.1c">Q_{E}</annotation><annotation encoding="application/x-llamapun" id="S6.I6.i3.p1.4.m4.1d">italic_Q start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="p" class="ltx_Math" display="inline" id="S6.I6.i3.p1.5.m5.1"><semantics id="S6.I6.i3.p1.5.m5.1a"><mi id="S6.I6.i3.p1.5.m5.1.1" xref="S6.I6.i3.p1.5.m5.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S6.I6.i3.p1.5.m5.1b"><ci id="S6.I6.i3.p1.5.m5.1.1.cmml" xref="S6.I6.i3.p1.5.m5.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.I6.i3.p1.5.m5.1c">p</annotation><annotation encoding="application/x-llamapun" id="S6.I6.i3.p1.5.m5.1d">italic_p</annotation></semantics></math> satisfies <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.Thmtheorem9" title="Definition 6.9. ‣ 6.2. Forcing epimorphisms ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">6.9</span></a> <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.I4.i3" title="Item (iii) ‣ Definition 6.9. ‣ 6.2. Forcing epimorphisms ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">(iii)</span></a>.</p> </div> </li> </ol> </div> </div> <div class="ltx_proof" id="S6.SS2.5"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S6.SS2.3.p1"> <p class="ltx_p" id="S6.SS2.3.p1.6">(a). Assume <math alttext="a<_{A}b" class="ltx_Math" display="inline" id="S6.SS2.3.p1.1.m1.1"><semantics id="S6.SS2.3.p1.1.m1.1a"><mrow id="S6.SS2.3.p1.1.m1.1.1" xref="S6.SS2.3.p1.1.m1.1.1.cmml"><mi id="S6.SS2.3.p1.1.m1.1.1.2" xref="S6.SS2.3.p1.1.m1.1.1.2.cmml">a</mi><msub id="S6.SS2.3.p1.1.m1.1.1.1" xref="S6.SS2.3.p1.1.m1.1.1.1.cmml"><mo id="S6.SS2.3.p1.1.m1.1.1.1.2" xref="S6.SS2.3.p1.1.m1.1.1.1.2.cmml"><</mo><mi id="S6.SS2.3.p1.1.m1.1.1.1.3" xref="S6.SS2.3.p1.1.m1.1.1.1.3.cmml">A</mi></msub><mi id="S6.SS2.3.p1.1.m1.1.1.3" xref="S6.SS2.3.p1.1.m1.1.1.3.cmml">b</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.3.p1.1.m1.1b"><apply id="S6.SS2.3.p1.1.m1.1.1.cmml" xref="S6.SS2.3.p1.1.m1.1.1"><apply id="S6.SS2.3.p1.1.m1.1.1.1.cmml" xref="S6.SS2.3.p1.1.m1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.3.p1.1.m1.1.1.1.1.cmml" xref="S6.SS2.3.p1.1.m1.1.1.1">subscript</csymbol><lt id="S6.SS2.3.p1.1.m1.1.1.1.2.cmml" xref="S6.SS2.3.p1.1.m1.1.1.1.2"></lt><ci id="S6.SS2.3.p1.1.m1.1.1.1.3.cmml" xref="S6.SS2.3.p1.1.m1.1.1.1.3">𝐴</ci></apply><ci id="S6.SS2.3.p1.1.m1.1.1.2.cmml" xref="S6.SS2.3.p1.1.m1.1.1.2">𝑎</ci><ci id="S6.SS2.3.p1.1.m1.1.1.3.cmml" xref="S6.SS2.3.p1.1.m1.1.1.3">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.3.p1.1.m1.1c">a<_{A}b</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.3.p1.1.m1.1d">italic_a < start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_b</annotation></semantics></math>. By definition <math alttext="\Delta_{A}(a,b)\leq a" class="ltx_Math" display="inline" id="S6.SS2.3.p1.2.m2.2"><semantics id="S6.SS2.3.p1.2.m2.2a"><mrow id="S6.SS2.3.p1.2.m2.2.3" xref="S6.SS2.3.p1.2.m2.2.3.cmml"><mrow id="S6.SS2.3.p1.2.m2.2.3.2" xref="S6.SS2.3.p1.2.m2.2.3.2.cmml"><msub id="S6.SS2.3.p1.2.m2.2.3.2.2" xref="S6.SS2.3.p1.2.m2.2.3.2.2.cmml"><mi id="S6.SS2.3.p1.2.m2.2.3.2.2.2" mathvariant="normal" xref="S6.SS2.3.p1.2.m2.2.3.2.2.2.cmml">Δ</mi><mi id="S6.SS2.3.p1.2.m2.2.3.2.2.3" xref="S6.SS2.3.p1.2.m2.2.3.2.2.3.cmml">A</mi></msub><mo id="S6.SS2.3.p1.2.m2.2.3.2.1" xref="S6.SS2.3.p1.2.m2.2.3.2.1.cmml">⁢</mo><mrow id="S6.SS2.3.p1.2.m2.2.3.2.3.2" xref="S6.SS2.3.p1.2.m2.2.3.2.3.1.cmml"><mo id="S6.SS2.3.p1.2.m2.2.3.2.3.2.1" stretchy="false" xref="S6.SS2.3.p1.2.m2.2.3.2.3.1.cmml">(</mo><mi id="S6.SS2.3.p1.2.m2.1.1" xref="S6.SS2.3.p1.2.m2.1.1.cmml">a</mi><mo id="S6.SS2.3.p1.2.m2.2.3.2.3.2.2" xref="S6.SS2.3.p1.2.m2.2.3.2.3.1.cmml">,</mo><mi id="S6.SS2.3.p1.2.m2.2.2" xref="S6.SS2.3.p1.2.m2.2.2.cmml">b</mi><mo id="S6.SS2.3.p1.2.m2.2.3.2.3.2.3" stretchy="false" xref="S6.SS2.3.p1.2.m2.2.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="S6.SS2.3.p1.2.m2.2.3.1" xref="S6.SS2.3.p1.2.m2.2.3.1.cmml">≤</mo><mi id="S6.SS2.3.p1.2.m2.2.3.3" xref="S6.SS2.3.p1.2.m2.2.3.3.cmml">a</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.3.p1.2.m2.2b"><apply id="S6.SS2.3.p1.2.m2.2.3.cmml" xref="S6.SS2.3.p1.2.m2.2.3"><leq id="S6.SS2.3.p1.2.m2.2.3.1.cmml" xref="S6.SS2.3.p1.2.m2.2.3.1"></leq><apply id="S6.SS2.3.p1.2.m2.2.3.2.cmml" xref="S6.SS2.3.p1.2.m2.2.3.2"><times id="S6.SS2.3.p1.2.m2.2.3.2.1.cmml" xref="S6.SS2.3.p1.2.m2.2.3.2.1"></times><apply id="S6.SS2.3.p1.2.m2.2.3.2.2.cmml" xref="S6.SS2.3.p1.2.m2.2.3.2.2"><csymbol cd="ambiguous" id="S6.SS2.3.p1.2.m2.2.3.2.2.1.cmml" xref="S6.SS2.3.p1.2.m2.2.3.2.2">subscript</csymbol><ci id="S6.SS2.3.p1.2.m2.2.3.2.2.2.cmml" xref="S6.SS2.3.p1.2.m2.2.3.2.2.2">Δ</ci><ci id="S6.SS2.3.p1.2.m2.2.3.2.2.3.cmml" xref="S6.SS2.3.p1.2.m2.2.3.2.2.3">𝐴</ci></apply><interval closure="open" id="S6.SS2.3.p1.2.m2.2.3.2.3.1.cmml" xref="S6.SS2.3.p1.2.m2.2.3.2.3.2"><ci id="S6.SS2.3.p1.2.m2.1.1.cmml" xref="S6.SS2.3.p1.2.m2.1.1">𝑎</ci><ci id="S6.SS2.3.p1.2.m2.2.2.cmml" xref="S6.SS2.3.p1.2.m2.2.2">𝑏</ci></interval></apply><ci id="S6.SS2.3.p1.2.m2.2.3.3.cmml" xref="S6.SS2.3.p1.2.m2.2.3.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.3.p1.2.m2.2c">\Delta_{A}(a,b)\leq a</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.3.p1.2.m2.2d">roman_Δ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_a , italic_b ) ≤ italic_a</annotation></semantics></math>, thus we need to prove that <math alttext="\nu(a,b)\leq\nu(b)" class="ltx_Math" display="inline" id="S6.SS2.3.p1.3.m3.3"><semantics id="S6.SS2.3.p1.3.m3.3a"><mrow id="S6.SS2.3.p1.3.m3.3.4" xref="S6.SS2.3.p1.3.m3.3.4.cmml"><mrow id="S6.SS2.3.p1.3.m3.3.4.2" xref="S6.SS2.3.p1.3.m3.3.4.2.cmml"><mi id="S6.SS2.3.p1.3.m3.3.4.2.2" xref="S6.SS2.3.p1.3.m3.3.4.2.2.cmml">ν</mi><mo id="S6.SS2.3.p1.3.m3.3.4.2.1" xref="S6.SS2.3.p1.3.m3.3.4.2.1.cmml">⁢</mo><mrow id="S6.SS2.3.p1.3.m3.3.4.2.3.2" xref="S6.SS2.3.p1.3.m3.3.4.2.3.1.cmml"><mo id="S6.SS2.3.p1.3.m3.3.4.2.3.2.1" stretchy="false" xref="S6.SS2.3.p1.3.m3.3.4.2.3.1.cmml">(</mo><mi id="S6.SS2.3.p1.3.m3.1.1" xref="S6.SS2.3.p1.3.m3.1.1.cmml">a</mi><mo id="S6.SS2.3.p1.3.m3.3.4.2.3.2.2" xref="S6.SS2.3.p1.3.m3.3.4.2.3.1.cmml">,</mo><mi id="S6.SS2.3.p1.3.m3.2.2" xref="S6.SS2.3.p1.3.m3.2.2.cmml">b</mi><mo id="S6.SS2.3.p1.3.m3.3.4.2.3.2.3" stretchy="false" xref="S6.SS2.3.p1.3.m3.3.4.2.3.1.cmml">)</mo></mrow></mrow><mo id="S6.SS2.3.p1.3.m3.3.4.1" xref="S6.SS2.3.p1.3.m3.3.4.1.cmml">≤</mo><mrow id="S6.SS2.3.p1.3.m3.3.4.3" xref="S6.SS2.3.p1.3.m3.3.4.3.cmml"><mi id="S6.SS2.3.p1.3.m3.3.4.3.2" xref="S6.SS2.3.p1.3.m3.3.4.3.2.cmml">ν</mi><mo id="S6.SS2.3.p1.3.m3.3.4.3.1" xref="S6.SS2.3.p1.3.m3.3.4.3.1.cmml">⁢</mo><mrow id="S6.SS2.3.p1.3.m3.3.4.3.3.2" xref="S6.SS2.3.p1.3.m3.3.4.3.cmml"><mo id="S6.SS2.3.p1.3.m3.3.4.3.3.2.1" stretchy="false" xref="S6.SS2.3.p1.3.m3.3.4.3.cmml">(</mo><mi id="S6.SS2.3.p1.3.m3.3.3" xref="S6.SS2.3.p1.3.m3.3.3.cmml">b</mi><mo id="S6.SS2.3.p1.3.m3.3.4.3.3.2.2" stretchy="false" xref="S6.SS2.3.p1.3.m3.3.4.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.3.p1.3.m3.3b"><apply id="S6.SS2.3.p1.3.m3.3.4.cmml" xref="S6.SS2.3.p1.3.m3.3.4"><leq id="S6.SS2.3.p1.3.m3.3.4.1.cmml" xref="S6.SS2.3.p1.3.m3.3.4.1"></leq><apply id="S6.SS2.3.p1.3.m3.3.4.2.cmml" xref="S6.SS2.3.p1.3.m3.3.4.2"><times id="S6.SS2.3.p1.3.m3.3.4.2.1.cmml" xref="S6.SS2.3.p1.3.m3.3.4.2.1"></times><ci id="S6.SS2.3.p1.3.m3.3.4.2.2.cmml" xref="S6.SS2.3.p1.3.m3.3.4.2.2">𝜈</ci><interval closure="open" id="S6.SS2.3.p1.3.m3.3.4.2.3.1.cmml" xref="S6.SS2.3.p1.3.m3.3.4.2.3.2"><ci id="S6.SS2.3.p1.3.m3.1.1.cmml" xref="S6.SS2.3.p1.3.m3.1.1">𝑎</ci><ci id="S6.SS2.3.p1.3.m3.2.2.cmml" xref="S6.SS2.3.p1.3.m3.2.2">𝑏</ci></interval></apply><apply id="S6.SS2.3.p1.3.m3.3.4.3.cmml" xref="S6.SS2.3.p1.3.m3.3.4.3"><times id="S6.SS2.3.p1.3.m3.3.4.3.1.cmml" xref="S6.SS2.3.p1.3.m3.3.4.3.1"></times><ci id="S6.SS2.3.p1.3.m3.3.4.3.2.cmml" xref="S6.SS2.3.p1.3.m3.3.4.3.2">𝜈</ci><ci id="S6.SS2.3.p1.3.m3.3.3.cmml" xref="S6.SS2.3.p1.3.m3.3.3">𝑏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.3.p1.3.m3.3c">\nu(a,b)\leq\nu(b)</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.3.p1.3.m3.3d">italic_ν ( italic_a , italic_b ) ≤ italic_ν ( italic_b )</annotation></semantics></math>. Observe that <math alttext="b\in{\nu(b)}^{+}" class="ltx_Math" display="inline" id="S6.SS2.3.p1.4.m4.1"><semantics id="S6.SS2.3.p1.4.m4.1a"><mrow id="S6.SS2.3.p1.4.m4.1.2" xref="S6.SS2.3.p1.4.m4.1.2.cmml"><mi id="S6.SS2.3.p1.4.m4.1.2.2" xref="S6.SS2.3.p1.4.m4.1.2.2.cmml">b</mi><mo id="S6.SS2.3.p1.4.m4.1.2.1" xref="S6.SS2.3.p1.4.m4.1.2.1.cmml">∈</mo><mrow id="S6.SS2.3.p1.4.m4.1.2.3" xref="S6.SS2.3.p1.4.m4.1.2.3.cmml"><mi id="S6.SS2.3.p1.4.m4.1.2.3.2" xref="S6.SS2.3.p1.4.m4.1.2.3.2.cmml">ν</mi><mo id="S6.SS2.3.p1.4.m4.1.2.3.1" xref="S6.SS2.3.p1.4.m4.1.2.3.1.cmml">⁢</mo><msup id="S6.SS2.3.p1.4.m4.1.2.3.3" xref="S6.SS2.3.p1.4.m4.1.2.3.3.cmml"><mrow id="S6.SS2.3.p1.4.m4.1.2.3.3.2.2" xref="S6.SS2.3.p1.4.m4.1.2.3.3.cmml"><mo id="S6.SS2.3.p1.4.m4.1.2.3.3.2.2.1" stretchy="false" xref="S6.SS2.3.p1.4.m4.1.2.3.3.cmml">(</mo><mi id="S6.SS2.3.p1.4.m4.1.1" xref="S6.SS2.3.p1.4.m4.1.1.cmml">b</mi><mo id="S6.SS2.3.p1.4.m4.1.2.3.3.2.2.2" stretchy="false" xref="S6.SS2.3.p1.4.m4.1.2.3.3.cmml">)</mo></mrow><mo id="S6.SS2.3.p1.4.m4.1.2.3.3.3" xref="S6.SS2.3.p1.4.m4.1.2.3.3.3.cmml">+</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.3.p1.4.m4.1b"><apply id="S6.SS2.3.p1.4.m4.1.2.cmml" xref="S6.SS2.3.p1.4.m4.1.2"><in id="S6.SS2.3.p1.4.m4.1.2.1.cmml" xref="S6.SS2.3.p1.4.m4.1.2.1"></in><ci id="S6.SS2.3.p1.4.m4.1.2.2.cmml" xref="S6.SS2.3.p1.4.m4.1.2.2">𝑏</ci><apply id="S6.SS2.3.p1.4.m4.1.2.3.cmml" xref="S6.SS2.3.p1.4.m4.1.2.3"><times id="S6.SS2.3.p1.4.m4.1.2.3.1.cmml" xref="S6.SS2.3.p1.4.m4.1.2.3.1"></times><ci id="S6.SS2.3.p1.4.m4.1.2.3.2.cmml" xref="S6.SS2.3.p1.4.m4.1.2.3.2">𝜈</ci><apply id="S6.SS2.3.p1.4.m4.1.2.3.3.cmml" xref="S6.SS2.3.p1.4.m4.1.2.3.3"><csymbol cd="ambiguous" id="S6.SS2.3.p1.4.m4.1.2.3.3.1.cmml" xref="S6.SS2.3.p1.4.m4.1.2.3.3">superscript</csymbol><ci id="S6.SS2.3.p1.4.m4.1.1.cmml" xref="S6.SS2.3.p1.4.m4.1.1">𝑏</ci><plus id="S6.SS2.3.p1.4.m4.1.2.3.3.3.cmml" xref="S6.SS2.3.p1.4.m4.1.2.3.3.3"></plus></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.3.p1.4.m4.1c">b\in{\nu(b)}^{+}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.3.p1.4.m4.1d">italic_b ∈ italic_ν ( italic_b ) start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math>, and thus <math alttext="\delta_{A}(b)<\nu(b)^{+}" class="ltx_Math" display="inline" id="S6.SS2.3.p1.5.m5.2"><semantics id="S6.SS2.3.p1.5.m5.2a"><mrow id="S6.SS2.3.p1.5.m5.2.3" xref="S6.SS2.3.p1.5.m5.2.3.cmml"><mrow id="S6.SS2.3.p1.5.m5.2.3.2" xref="S6.SS2.3.p1.5.m5.2.3.2.cmml"><msub id="S6.SS2.3.p1.5.m5.2.3.2.2" xref="S6.SS2.3.p1.5.m5.2.3.2.2.cmml"><mi id="S6.SS2.3.p1.5.m5.2.3.2.2.2" xref="S6.SS2.3.p1.5.m5.2.3.2.2.2.cmml">δ</mi><mi id="S6.SS2.3.p1.5.m5.2.3.2.2.3" xref="S6.SS2.3.p1.5.m5.2.3.2.2.3.cmml">A</mi></msub><mo id="S6.SS2.3.p1.5.m5.2.3.2.1" xref="S6.SS2.3.p1.5.m5.2.3.2.1.cmml">⁢</mo><mrow id="S6.SS2.3.p1.5.m5.2.3.2.3.2" xref="S6.SS2.3.p1.5.m5.2.3.2.cmml"><mo id="S6.SS2.3.p1.5.m5.2.3.2.3.2.1" stretchy="false" xref="S6.SS2.3.p1.5.m5.2.3.2.cmml">(</mo><mi id="S6.SS2.3.p1.5.m5.1.1" xref="S6.SS2.3.p1.5.m5.1.1.cmml">b</mi><mo id="S6.SS2.3.p1.5.m5.2.3.2.3.2.2" stretchy="false" xref="S6.SS2.3.p1.5.m5.2.3.2.cmml">)</mo></mrow></mrow><mo id="S6.SS2.3.p1.5.m5.2.3.1" xref="S6.SS2.3.p1.5.m5.2.3.1.cmml"><</mo><mrow id="S6.SS2.3.p1.5.m5.2.3.3" xref="S6.SS2.3.p1.5.m5.2.3.3.cmml"><mi id="S6.SS2.3.p1.5.m5.2.3.3.2" xref="S6.SS2.3.p1.5.m5.2.3.3.2.cmml">ν</mi><mo id="S6.SS2.3.p1.5.m5.2.3.3.1" xref="S6.SS2.3.p1.5.m5.2.3.3.1.cmml">⁢</mo><msup id="S6.SS2.3.p1.5.m5.2.3.3.3" xref="S6.SS2.3.p1.5.m5.2.3.3.3.cmml"><mrow id="S6.SS2.3.p1.5.m5.2.3.3.3.2.2" xref="S6.SS2.3.p1.5.m5.2.3.3.3.cmml"><mo id="S6.SS2.3.p1.5.m5.2.3.3.3.2.2.1" stretchy="false" xref="S6.SS2.3.p1.5.m5.2.3.3.3.cmml">(</mo><mi id="S6.SS2.3.p1.5.m5.2.2" xref="S6.SS2.3.p1.5.m5.2.2.cmml">b</mi><mo id="S6.SS2.3.p1.5.m5.2.3.3.3.2.2.2" stretchy="false" xref="S6.SS2.3.p1.5.m5.2.3.3.3.cmml">)</mo></mrow><mo id="S6.SS2.3.p1.5.m5.2.3.3.3.3" xref="S6.SS2.3.p1.5.m5.2.3.3.3.3.cmml">+</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.3.p1.5.m5.2b"><apply id="S6.SS2.3.p1.5.m5.2.3.cmml" xref="S6.SS2.3.p1.5.m5.2.3"><lt id="S6.SS2.3.p1.5.m5.2.3.1.cmml" xref="S6.SS2.3.p1.5.m5.2.3.1"></lt><apply id="S6.SS2.3.p1.5.m5.2.3.2.cmml" xref="S6.SS2.3.p1.5.m5.2.3.2"><times id="S6.SS2.3.p1.5.m5.2.3.2.1.cmml" xref="S6.SS2.3.p1.5.m5.2.3.2.1"></times><apply id="S6.SS2.3.p1.5.m5.2.3.2.2.cmml" xref="S6.SS2.3.p1.5.m5.2.3.2.2"><csymbol cd="ambiguous" id="S6.SS2.3.p1.5.m5.2.3.2.2.1.cmml" xref="S6.SS2.3.p1.5.m5.2.3.2.2">subscript</csymbol><ci id="S6.SS2.3.p1.5.m5.2.3.2.2.2.cmml" xref="S6.SS2.3.p1.5.m5.2.3.2.2.2">𝛿</ci><ci id="S6.SS2.3.p1.5.m5.2.3.2.2.3.cmml" xref="S6.SS2.3.p1.5.m5.2.3.2.2.3">𝐴</ci></apply><ci id="S6.SS2.3.p1.5.m5.1.1.cmml" xref="S6.SS2.3.p1.5.m5.1.1">𝑏</ci></apply><apply id="S6.SS2.3.p1.5.m5.2.3.3.cmml" xref="S6.SS2.3.p1.5.m5.2.3.3"><times id="S6.SS2.3.p1.5.m5.2.3.3.1.cmml" xref="S6.SS2.3.p1.5.m5.2.3.3.1"></times><ci id="S6.SS2.3.p1.5.m5.2.3.3.2.cmml" xref="S6.SS2.3.p1.5.m5.2.3.3.2">𝜈</ci><apply id="S6.SS2.3.p1.5.m5.2.3.3.3.cmml" xref="S6.SS2.3.p1.5.m5.2.3.3.3"><csymbol cd="ambiguous" id="S6.SS2.3.p1.5.m5.2.3.3.3.1.cmml" xref="S6.SS2.3.p1.5.m5.2.3.3.3">superscript</csymbol><ci id="S6.SS2.3.p1.5.m5.2.2.cmml" xref="S6.SS2.3.p1.5.m5.2.2">𝑏</ci><plus id="S6.SS2.3.p1.5.m5.2.3.3.3.3.cmml" xref="S6.SS2.3.p1.5.m5.2.3.3.3.3"></plus></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.3.p1.5.m5.2c">\delta_{A}(b)<\nu(b)^{+}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.3.p1.5.m5.2d">italic_δ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_b ) < italic_ν ( italic_b ) start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math>. Then clearly <math alttext="\nu(a,b)\leq\nu(b)" class="ltx_Math" display="inline" id="S6.SS2.3.p1.6.m6.3"><semantics id="S6.SS2.3.p1.6.m6.3a"><mrow id="S6.SS2.3.p1.6.m6.3.4" xref="S6.SS2.3.p1.6.m6.3.4.cmml"><mrow id="S6.SS2.3.p1.6.m6.3.4.2" xref="S6.SS2.3.p1.6.m6.3.4.2.cmml"><mi id="S6.SS2.3.p1.6.m6.3.4.2.2" xref="S6.SS2.3.p1.6.m6.3.4.2.2.cmml">ν</mi><mo id="S6.SS2.3.p1.6.m6.3.4.2.1" xref="S6.SS2.3.p1.6.m6.3.4.2.1.cmml">⁢</mo><mrow id="S6.SS2.3.p1.6.m6.3.4.2.3.2" xref="S6.SS2.3.p1.6.m6.3.4.2.3.1.cmml"><mo id="S6.SS2.3.p1.6.m6.3.4.2.3.2.1" stretchy="false" xref="S6.SS2.3.p1.6.m6.3.4.2.3.1.cmml">(</mo><mi id="S6.SS2.3.p1.6.m6.1.1" xref="S6.SS2.3.p1.6.m6.1.1.cmml">a</mi><mo id="S6.SS2.3.p1.6.m6.3.4.2.3.2.2" xref="S6.SS2.3.p1.6.m6.3.4.2.3.1.cmml">,</mo><mi id="S6.SS2.3.p1.6.m6.2.2" xref="S6.SS2.3.p1.6.m6.2.2.cmml">b</mi><mo id="S6.SS2.3.p1.6.m6.3.4.2.3.2.3" stretchy="false" xref="S6.SS2.3.p1.6.m6.3.4.2.3.1.cmml">)</mo></mrow></mrow><mo id="S6.SS2.3.p1.6.m6.3.4.1" xref="S6.SS2.3.p1.6.m6.3.4.1.cmml">≤</mo><mrow id="S6.SS2.3.p1.6.m6.3.4.3" xref="S6.SS2.3.p1.6.m6.3.4.3.cmml"><mi id="S6.SS2.3.p1.6.m6.3.4.3.2" xref="S6.SS2.3.p1.6.m6.3.4.3.2.cmml">ν</mi><mo id="S6.SS2.3.p1.6.m6.3.4.3.1" xref="S6.SS2.3.p1.6.m6.3.4.3.1.cmml">⁢</mo><mrow id="S6.SS2.3.p1.6.m6.3.4.3.3.2" xref="S6.SS2.3.p1.6.m6.3.4.3.cmml"><mo id="S6.SS2.3.p1.6.m6.3.4.3.3.2.1" stretchy="false" xref="S6.SS2.3.p1.6.m6.3.4.3.cmml">(</mo><mi id="S6.SS2.3.p1.6.m6.3.3" xref="S6.SS2.3.p1.6.m6.3.3.cmml">b</mi><mo id="S6.SS2.3.p1.6.m6.3.4.3.3.2.2" stretchy="false" xref="S6.SS2.3.p1.6.m6.3.4.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.3.p1.6.m6.3b"><apply id="S6.SS2.3.p1.6.m6.3.4.cmml" xref="S6.SS2.3.p1.6.m6.3.4"><leq id="S6.SS2.3.p1.6.m6.3.4.1.cmml" xref="S6.SS2.3.p1.6.m6.3.4.1"></leq><apply id="S6.SS2.3.p1.6.m6.3.4.2.cmml" xref="S6.SS2.3.p1.6.m6.3.4.2"><times id="S6.SS2.3.p1.6.m6.3.4.2.1.cmml" xref="S6.SS2.3.p1.6.m6.3.4.2.1"></times><ci id="S6.SS2.3.p1.6.m6.3.4.2.2.cmml" xref="S6.SS2.3.p1.6.m6.3.4.2.2">𝜈</ci><interval closure="open" id="S6.SS2.3.p1.6.m6.3.4.2.3.1.cmml" xref="S6.SS2.3.p1.6.m6.3.4.2.3.2"><ci id="S6.SS2.3.p1.6.m6.1.1.cmml" xref="S6.SS2.3.p1.6.m6.1.1">𝑎</ci><ci id="S6.SS2.3.p1.6.m6.2.2.cmml" xref="S6.SS2.3.p1.6.m6.2.2">𝑏</ci></interval></apply><apply id="S6.SS2.3.p1.6.m6.3.4.3.cmml" xref="S6.SS2.3.p1.6.m6.3.4.3"><times id="S6.SS2.3.p1.6.m6.3.4.3.1.cmml" xref="S6.SS2.3.p1.6.m6.3.4.3.1"></times><ci id="S6.SS2.3.p1.6.m6.3.4.3.2.cmml" xref="S6.SS2.3.p1.6.m6.3.4.3.2">𝜈</ci><ci id="S6.SS2.3.p1.6.m6.3.3.cmml" xref="S6.SS2.3.p1.6.m6.3.3">𝑏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.3.p1.6.m6.3c">\nu(a,b)\leq\nu(b)</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.3.p1.6.m6.3d">italic_ν ( italic_a , italic_b ) ≤ italic_ν ( italic_b )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S6.SS2.4.p2"> <p class="ltx_p" id="S6.SS2.4.p2.11">(b). Assume <math alttext="\bar{a}<_{b}\bar{b}" class="ltx_Math" display="inline" id="S6.SS2.4.p2.1.m1.1"><semantics id="S6.SS2.4.p2.1.m1.1a"><mrow id="S6.SS2.4.p2.1.m1.1.1" xref="S6.SS2.4.p2.1.m1.1.1.cmml"><mover accent="true" id="S6.SS2.4.p2.1.m1.1.1.2" xref="S6.SS2.4.p2.1.m1.1.1.2.cmml"><mi id="S6.SS2.4.p2.1.m1.1.1.2.2" xref="S6.SS2.4.p2.1.m1.1.1.2.2.cmml">a</mi><mo id="S6.SS2.4.p2.1.m1.1.1.2.1" xref="S6.SS2.4.p2.1.m1.1.1.2.1.cmml">¯</mo></mover><msub id="S6.SS2.4.p2.1.m1.1.1.1" xref="S6.SS2.4.p2.1.m1.1.1.1.cmml"><mo id="S6.SS2.4.p2.1.m1.1.1.1.2" xref="S6.SS2.4.p2.1.m1.1.1.1.2.cmml"><</mo><mi id="S6.SS2.4.p2.1.m1.1.1.1.3" xref="S6.SS2.4.p2.1.m1.1.1.1.3.cmml">b</mi></msub><mover accent="true" id="S6.SS2.4.p2.1.m1.1.1.3" xref="S6.SS2.4.p2.1.m1.1.1.3.cmml"><mi id="S6.SS2.4.p2.1.m1.1.1.3.2" xref="S6.SS2.4.p2.1.m1.1.1.3.2.cmml">b</mi><mo id="S6.SS2.4.p2.1.m1.1.1.3.1" xref="S6.SS2.4.p2.1.m1.1.1.3.1.cmml">¯</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.4.p2.1.m1.1b"><apply id="S6.SS2.4.p2.1.m1.1.1.cmml" xref="S6.SS2.4.p2.1.m1.1.1"><apply id="S6.SS2.4.p2.1.m1.1.1.1.cmml" xref="S6.SS2.4.p2.1.m1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.4.p2.1.m1.1.1.1.1.cmml" xref="S6.SS2.4.p2.1.m1.1.1.1">subscript</csymbol><lt id="S6.SS2.4.p2.1.m1.1.1.1.2.cmml" xref="S6.SS2.4.p2.1.m1.1.1.1.2"></lt><ci id="S6.SS2.4.p2.1.m1.1.1.1.3.cmml" xref="S6.SS2.4.p2.1.m1.1.1.1.3">𝑏</ci></apply><apply id="S6.SS2.4.p2.1.m1.1.1.2.cmml" xref="S6.SS2.4.p2.1.m1.1.1.2"><ci id="S6.SS2.4.p2.1.m1.1.1.2.1.cmml" xref="S6.SS2.4.p2.1.m1.1.1.2.1">¯</ci><ci id="S6.SS2.4.p2.1.m1.1.1.2.2.cmml" xref="S6.SS2.4.p2.1.m1.1.1.2.2">𝑎</ci></apply><apply id="S6.SS2.4.p2.1.m1.1.1.3.cmml" xref="S6.SS2.4.p2.1.m1.1.1.3"><ci id="S6.SS2.4.p2.1.m1.1.1.3.1.cmml" xref="S6.SS2.4.p2.1.m1.1.1.3.1">¯</ci><ci id="S6.SS2.4.p2.1.m1.1.1.3.2.cmml" xref="S6.SS2.4.p2.1.m1.1.1.3.2">𝑏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.4.p2.1.m1.1c">\bar{a}<_{b}\bar{b}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.4.p2.1.m1.1d">over¯ start_ARG italic_a end_ARG < start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT over¯ start_ARG italic_b end_ARG</annotation></semantics></math>. Then clearly <math alttext="a_{l}\leq_{A}a_{m}\leq_{A}a_{r}<_{A}b_{l}\leq_{A}b_{m}\leq_{A}b_{r}" class="ltx_Math" display="inline" id="S6.SS2.4.p2.2.m2.1"><semantics id="S6.SS2.4.p2.2.m2.1a"><mrow id="S6.SS2.4.p2.2.m2.1.1" xref="S6.SS2.4.p2.2.m2.1.1.cmml"><msub id="S6.SS2.4.p2.2.m2.1.1.2" xref="S6.SS2.4.p2.2.m2.1.1.2.cmml"><mi id="S6.SS2.4.p2.2.m2.1.1.2.2" xref="S6.SS2.4.p2.2.m2.1.1.2.2.cmml">a</mi><mi id="S6.SS2.4.p2.2.m2.1.1.2.3" xref="S6.SS2.4.p2.2.m2.1.1.2.3.cmml">l</mi></msub><msub id="S6.SS2.4.p2.2.m2.1.1.3" xref="S6.SS2.4.p2.2.m2.1.1.3.cmml"><mo id="S6.SS2.4.p2.2.m2.1.1.3.2" xref="S6.SS2.4.p2.2.m2.1.1.3.2.cmml">≤</mo><mi id="S6.SS2.4.p2.2.m2.1.1.3.3" xref="S6.SS2.4.p2.2.m2.1.1.3.3.cmml">A</mi></msub><msub id="S6.SS2.4.p2.2.m2.1.1.4" xref="S6.SS2.4.p2.2.m2.1.1.4.cmml"><mi id="S6.SS2.4.p2.2.m2.1.1.4.2" xref="S6.SS2.4.p2.2.m2.1.1.4.2.cmml">a</mi><mi id="S6.SS2.4.p2.2.m2.1.1.4.3" xref="S6.SS2.4.p2.2.m2.1.1.4.3.cmml">m</mi></msub><msub id="S6.SS2.4.p2.2.m2.1.1.5" xref="S6.SS2.4.p2.2.m2.1.1.5.cmml"><mo id="S6.SS2.4.p2.2.m2.1.1.5.2" xref="S6.SS2.4.p2.2.m2.1.1.5.2.cmml">≤</mo><mi id="S6.SS2.4.p2.2.m2.1.1.5.3" xref="S6.SS2.4.p2.2.m2.1.1.5.3.cmml">A</mi></msub><msub id="S6.SS2.4.p2.2.m2.1.1.6" xref="S6.SS2.4.p2.2.m2.1.1.6.cmml"><mi id="S6.SS2.4.p2.2.m2.1.1.6.2" xref="S6.SS2.4.p2.2.m2.1.1.6.2.cmml">a</mi><mi id="S6.SS2.4.p2.2.m2.1.1.6.3" xref="S6.SS2.4.p2.2.m2.1.1.6.3.cmml">r</mi></msub><msub id="S6.SS2.4.p2.2.m2.1.1.7" xref="S6.SS2.4.p2.2.m2.1.1.7.cmml"><mo id="S6.SS2.4.p2.2.m2.1.1.7.2" xref="S6.SS2.4.p2.2.m2.1.1.7.2.cmml"><</mo><mi id="S6.SS2.4.p2.2.m2.1.1.7.3" xref="S6.SS2.4.p2.2.m2.1.1.7.3.cmml">A</mi></msub><msub id="S6.SS2.4.p2.2.m2.1.1.8" xref="S6.SS2.4.p2.2.m2.1.1.8.cmml"><mi id="S6.SS2.4.p2.2.m2.1.1.8.2" xref="S6.SS2.4.p2.2.m2.1.1.8.2.cmml">b</mi><mi id="S6.SS2.4.p2.2.m2.1.1.8.3" xref="S6.SS2.4.p2.2.m2.1.1.8.3.cmml">l</mi></msub><msub id="S6.SS2.4.p2.2.m2.1.1.9" xref="S6.SS2.4.p2.2.m2.1.1.9.cmml"><mo id="S6.SS2.4.p2.2.m2.1.1.9.2" xref="S6.SS2.4.p2.2.m2.1.1.9.2.cmml">≤</mo><mi id="S6.SS2.4.p2.2.m2.1.1.9.3" xref="S6.SS2.4.p2.2.m2.1.1.9.3.cmml">A</mi></msub><msub id="S6.SS2.4.p2.2.m2.1.1.10" xref="S6.SS2.4.p2.2.m2.1.1.10.cmml"><mi id="S6.SS2.4.p2.2.m2.1.1.10.2" xref="S6.SS2.4.p2.2.m2.1.1.10.2.cmml">b</mi><mi id="S6.SS2.4.p2.2.m2.1.1.10.3" xref="S6.SS2.4.p2.2.m2.1.1.10.3.cmml">m</mi></msub><msub id="S6.SS2.4.p2.2.m2.1.1.11" xref="S6.SS2.4.p2.2.m2.1.1.11.cmml"><mo id="S6.SS2.4.p2.2.m2.1.1.11.2" xref="S6.SS2.4.p2.2.m2.1.1.11.2.cmml">≤</mo><mi id="S6.SS2.4.p2.2.m2.1.1.11.3" xref="S6.SS2.4.p2.2.m2.1.1.11.3.cmml">A</mi></msub><msub id="S6.SS2.4.p2.2.m2.1.1.12" xref="S6.SS2.4.p2.2.m2.1.1.12.cmml"><mi id="S6.SS2.4.p2.2.m2.1.1.12.2" xref="S6.SS2.4.p2.2.m2.1.1.12.2.cmml">b</mi><mi id="S6.SS2.4.p2.2.m2.1.1.12.3" xref="S6.SS2.4.p2.2.m2.1.1.12.3.cmml">r</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.4.p2.2.m2.1b"><apply id="S6.SS2.4.p2.2.m2.1.1.cmml" xref="S6.SS2.4.p2.2.m2.1.1"><and id="S6.SS2.4.p2.2.m2.1.1a.cmml" xref="S6.SS2.4.p2.2.m2.1.1"></and><apply id="S6.SS2.4.p2.2.m2.1.1b.cmml" xref="S6.SS2.4.p2.2.m2.1.1"><apply id="S6.SS2.4.p2.2.m2.1.1.3.cmml" xref="S6.SS2.4.p2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S6.SS2.4.p2.2.m2.1.1.3.1.cmml" xref="S6.SS2.4.p2.2.m2.1.1.3">subscript</csymbol><leq id="S6.SS2.4.p2.2.m2.1.1.3.2.cmml" xref="S6.SS2.4.p2.2.m2.1.1.3.2"></leq><ci id="S6.SS2.4.p2.2.m2.1.1.3.3.cmml" xref="S6.SS2.4.p2.2.m2.1.1.3.3">𝐴</ci></apply><apply id="S6.SS2.4.p2.2.m2.1.1.2.cmml" xref="S6.SS2.4.p2.2.m2.1.1.2"><csymbol cd="ambiguous" id="S6.SS2.4.p2.2.m2.1.1.2.1.cmml" xref="S6.SS2.4.p2.2.m2.1.1.2">subscript</csymbol><ci id="S6.SS2.4.p2.2.m2.1.1.2.2.cmml" xref="S6.SS2.4.p2.2.m2.1.1.2.2">𝑎</ci><ci id="S6.SS2.4.p2.2.m2.1.1.2.3.cmml" xref="S6.SS2.4.p2.2.m2.1.1.2.3">𝑙</ci></apply><apply id="S6.SS2.4.p2.2.m2.1.1.4.cmml" xref="S6.SS2.4.p2.2.m2.1.1.4"><csymbol cd="ambiguous" id="S6.SS2.4.p2.2.m2.1.1.4.1.cmml" xref="S6.SS2.4.p2.2.m2.1.1.4">subscript</csymbol><ci id="S6.SS2.4.p2.2.m2.1.1.4.2.cmml" xref="S6.SS2.4.p2.2.m2.1.1.4.2">𝑎</ci><ci id="S6.SS2.4.p2.2.m2.1.1.4.3.cmml" xref="S6.SS2.4.p2.2.m2.1.1.4.3">𝑚</ci></apply></apply><apply id="S6.SS2.4.p2.2.m2.1.1c.cmml" xref="S6.SS2.4.p2.2.m2.1.1"><apply id="S6.SS2.4.p2.2.m2.1.1.5.cmml" xref="S6.SS2.4.p2.2.m2.1.1.5"><csymbol cd="ambiguous" id="S6.SS2.4.p2.2.m2.1.1.5.1.cmml" xref="S6.SS2.4.p2.2.m2.1.1.5">subscript</csymbol><leq id="S6.SS2.4.p2.2.m2.1.1.5.2.cmml" xref="S6.SS2.4.p2.2.m2.1.1.5.2"></leq><ci id="S6.SS2.4.p2.2.m2.1.1.5.3.cmml" xref="S6.SS2.4.p2.2.m2.1.1.5.3">𝐴</ci></apply><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.4.p2.2.m2.1.1.4.cmml" id="S6.SS2.4.p2.2.m2.1.1d.cmml" xref="S6.SS2.4.p2.2.m2.1.1"></share><apply id="S6.SS2.4.p2.2.m2.1.1.6.cmml" xref="S6.SS2.4.p2.2.m2.1.1.6"><csymbol cd="ambiguous" id="S6.SS2.4.p2.2.m2.1.1.6.1.cmml" xref="S6.SS2.4.p2.2.m2.1.1.6">subscript</csymbol><ci id="S6.SS2.4.p2.2.m2.1.1.6.2.cmml" xref="S6.SS2.4.p2.2.m2.1.1.6.2">𝑎</ci><ci id="S6.SS2.4.p2.2.m2.1.1.6.3.cmml" xref="S6.SS2.4.p2.2.m2.1.1.6.3">𝑟</ci></apply></apply><apply id="S6.SS2.4.p2.2.m2.1.1e.cmml" xref="S6.SS2.4.p2.2.m2.1.1"><apply id="S6.SS2.4.p2.2.m2.1.1.7.cmml" xref="S6.SS2.4.p2.2.m2.1.1.7"><csymbol cd="ambiguous" id="S6.SS2.4.p2.2.m2.1.1.7.1.cmml" xref="S6.SS2.4.p2.2.m2.1.1.7">subscript</csymbol><lt id="S6.SS2.4.p2.2.m2.1.1.7.2.cmml" xref="S6.SS2.4.p2.2.m2.1.1.7.2"></lt><ci id="S6.SS2.4.p2.2.m2.1.1.7.3.cmml" xref="S6.SS2.4.p2.2.m2.1.1.7.3">𝐴</ci></apply><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.4.p2.2.m2.1.1.6.cmml" id="S6.SS2.4.p2.2.m2.1.1f.cmml" xref="S6.SS2.4.p2.2.m2.1.1"></share><apply id="S6.SS2.4.p2.2.m2.1.1.8.cmml" xref="S6.SS2.4.p2.2.m2.1.1.8"><csymbol cd="ambiguous" id="S6.SS2.4.p2.2.m2.1.1.8.1.cmml" xref="S6.SS2.4.p2.2.m2.1.1.8">subscript</csymbol><ci id="S6.SS2.4.p2.2.m2.1.1.8.2.cmml" xref="S6.SS2.4.p2.2.m2.1.1.8.2">𝑏</ci><ci id="S6.SS2.4.p2.2.m2.1.1.8.3.cmml" xref="S6.SS2.4.p2.2.m2.1.1.8.3">𝑙</ci></apply></apply><apply id="S6.SS2.4.p2.2.m2.1.1g.cmml" xref="S6.SS2.4.p2.2.m2.1.1"><apply id="S6.SS2.4.p2.2.m2.1.1.9.cmml" xref="S6.SS2.4.p2.2.m2.1.1.9"><csymbol cd="ambiguous" id="S6.SS2.4.p2.2.m2.1.1.9.1.cmml" xref="S6.SS2.4.p2.2.m2.1.1.9">subscript</csymbol><leq id="S6.SS2.4.p2.2.m2.1.1.9.2.cmml" xref="S6.SS2.4.p2.2.m2.1.1.9.2"></leq><ci id="S6.SS2.4.p2.2.m2.1.1.9.3.cmml" xref="S6.SS2.4.p2.2.m2.1.1.9.3">𝐴</ci></apply><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.4.p2.2.m2.1.1.8.cmml" id="S6.SS2.4.p2.2.m2.1.1h.cmml" xref="S6.SS2.4.p2.2.m2.1.1"></share><apply id="S6.SS2.4.p2.2.m2.1.1.10.cmml" xref="S6.SS2.4.p2.2.m2.1.1.10"><csymbol cd="ambiguous" id="S6.SS2.4.p2.2.m2.1.1.10.1.cmml" xref="S6.SS2.4.p2.2.m2.1.1.10">subscript</csymbol><ci id="S6.SS2.4.p2.2.m2.1.1.10.2.cmml" xref="S6.SS2.4.p2.2.m2.1.1.10.2">𝑏</ci><ci id="S6.SS2.4.p2.2.m2.1.1.10.3.cmml" xref="S6.SS2.4.p2.2.m2.1.1.10.3">𝑚</ci></apply></apply><apply id="S6.SS2.4.p2.2.m2.1.1i.cmml" xref="S6.SS2.4.p2.2.m2.1.1"><apply id="S6.SS2.4.p2.2.m2.1.1.11.cmml" xref="S6.SS2.4.p2.2.m2.1.1.11"><csymbol cd="ambiguous" id="S6.SS2.4.p2.2.m2.1.1.11.1.cmml" xref="S6.SS2.4.p2.2.m2.1.1.11">subscript</csymbol><leq id="S6.SS2.4.p2.2.m2.1.1.11.2.cmml" xref="S6.SS2.4.p2.2.m2.1.1.11.2"></leq><ci id="S6.SS2.4.p2.2.m2.1.1.11.3.cmml" xref="S6.SS2.4.p2.2.m2.1.1.11.3">𝐴</ci></apply><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.4.p2.2.m2.1.1.10.cmml" id="S6.SS2.4.p2.2.m2.1.1j.cmml" xref="S6.SS2.4.p2.2.m2.1.1"></share><apply id="S6.SS2.4.p2.2.m2.1.1.12.cmml" xref="S6.SS2.4.p2.2.m2.1.1.12"><csymbol cd="ambiguous" id="S6.SS2.4.p2.2.m2.1.1.12.1.cmml" xref="S6.SS2.4.p2.2.m2.1.1.12">subscript</csymbol><ci id="S6.SS2.4.p2.2.m2.1.1.12.2.cmml" xref="S6.SS2.4.p2.2.m2.1.1.12.2">𝑏</ci><ci id="S6.SS2.4.p2.2.m2.1.1.12.3.cmml" xref="S6.SS2.4.p2.2.m2.1.1.12.3">𝑟</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.4.p2.2.m2.1c">a_{l}\leq_{A}a_{m}\leq_{A}a_{r}<_{A}b_{l}\leq_{A}b_{m}\leq_{A}b_{r}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.4.p2.2.m2.1d">italic_a start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ≤ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ≤ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT < start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ≤ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ≤ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT</annotation></semantics></math>, and thus <math alttext="\Delta_{A}(a_{l},b_{r})\leq\Delta_{A}(a_{m},b_{m})\leq\Delta_{A}(a_{r},b_{l})" class="ltx_Math" display="inline" id="S6.SS2.4.p2.3.m3.6"><semantics id="S6.SS2.4.p2.3.m3.6a"><mrow id="S6.SS2.4.p2.3.m3.6.6" xref="S6.SS2.4.p2.3.m3.6.6.cmml"><mrow id="S6.SS2.4.p2.3.m3.2.2.2" xref="S6.SS2.4.p2.3.m3.2.2.2.cmml"><msub id="S6.SS2.4.p2.3.m3.2.2.2.4" xref="S6.SS2.4.p2.3.m3.2.2.2.4.cmml"><mi id="S6.SS2.4.p2.3.m3.2.2.2.4.2" mathvariant="normal" xref="S6.SS2.4.p2.3.m3.2.2.2.4.2.cmml">Δ</mi><mi id="S6.SS2.4.p2.3.m3.2.2.2.4.3" xref="S6.SS2.4.p2.3.m3.2.2.2.4.3.cmml">A</mi></msub><mo id="S6.SS2.4.p2.3.m3.2.2.2.3" xref="S6.SS2.4.p2.3.m3.2.2.2.3.cmml">⁢</mo><mrow id="S6.SS2.4.p2.3.m3.2.2.2.2.2" xref="S6.SS2.4.p2.3.m3.2.2.2.2.3.cmml"><mo id="S6.SS2.4.p2.3.m3.2.2.2.2.2.3" stretchy="false" xref="S6.SS2.4.p2.3.m3.2.2.2.2.3.cmml">(</mo><msub id="S6.SS2.4.p2.3.m3.1.1.1.1.1.1" xref="S6.SS2.4.p2.3.m3.1.1.1.1.1.1.cmml"><mi id="S6.SS2.4.p2.3.m3.1.1.1.1.1.1.2" xref="S6.SS2.4.p2.3.m3.1.1.1.1.1.1.2.cmml">a</mi><mi id="S6.SS2.4.p2.3.m3.1.1.1.1.1.1.3" xref="S6.SS2.4.p2.3.m3.1.1.1.1.1.1.3.cmml">l</mi></msub><mo id="S6.SS2.4.p2.3.m3.2.2.2.2.2.4" xref="S6.SS2.4.p2.3.m3.2.2.2.2.3.cmml">,</mo><msub id="S6.SS2.4.p2.3.m3.2.2.2.2.2.2" xref="S6.SS2.4.p2.3.m3.2.2.2.2.2.2.cmml"><mi id="S6.SS2.4.p2.3.m3.2.2.2.2.2.2.2" xref="S6.SS2.4.p2.3.m3.2.2.2.2.2.2.2.cmml">b</mi><mi id="S6.SS2.4.p2.3.m3.2.2.2.2.2.2.3" xref="S6.SS2.4.p2.3.m3.2.2.2.2.2.2.3.cmml">r</mi></msub><mo id="S6.SS2.4.p2.3.m3.2.2.2.2.2.5" stretchy="false" xref="S6.SS2.4.p2.3.m3.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.SS2.4.p2.3.m3.6.6.8" xref="S6.SS2.4.p2.3.m3.6.6.8.cmml">≤</mo><mrow id="S6.SS2.4.p2.3.m3.4.4.4" xref="S6.SS2.4.p2.3.m3.4.4.4.cmml"><msub id="S6.SS2.4.p2.3.m3.4.4.4.4" xref="S6.SS2.4.p2.3.m3.4.4.4.4.cmml"><mi id="S6.SS2.4.p2.3.m3.4.4.4.4.2" mathvariant="normal" xref="S6.SS2.4.p2.3.m3.4.4.4.4.2.cmml">Δ</mi><mi id="S6.SS2.4.p2.3.m3.4.4.4.4.3" xref="S6.SS2.4.p2.3.m3.4.4.4.4.3.cmml">A</mi></msub><mo id="S6.SS2.4.p2.3.m3.4.4.4.3" xref="S6.SS2.4.p2.3.m3.4.4.4.3.cmml">⁢</mo><mrow id="S6.SS2.4.p2.3.m3.4.4.4.2.2" xref="S6.SS2.4.p2.3.m3.4.4.4.2.3.cmml"><mo id="S6.SS2.4.p2.3.m3.4.4.4.2.2.3" stretchy="false" xref="S6.SS2.4.p2.3.m3.4.4.4.2.3.cmml">(</mo><msub id="S6.SS2.4.p2.3.m3.3.3.3.1.1.1" xref="S6.SS2.4.p2.3.m3.3.3.3.1.1.1.cmml"><mi id="S6.SS2.4.p2.3.m3.3.3.3.1.1.1.2" xref="S6.SS2.4.p2.3.m3.3.3.3.1.1.1.2.cmml">a</mi><mi id="S6.SS2.4.p2.3.m3.3.3.3.1.1.1.3" xref="S6.SS2.4.p2.3.m3.3.3.3.1.1.1.3.cmml">m</mi></msub><mo id="S6.SS2.4.p2.3.m3.4.4.4.2.2.4" xref="S6.SS2.4.p2.3.m3.4.4.4.2.3.cmml">,</mo><msub id="S6.SS2.4.p2.3.m3.4.4.4.2.2.2" xref="S6.SS2.4.p2.3.m3.4.4.4.2.2.2.cmml"><mi id="S6.SS2.4.p2.3.m3.4.4.4.2.2.2.2" xref="S6.SS2.4.p2.3.m3.4.4.4.2.2.2.2.cmml">b</mi><mi id="S6.SS2.4.p2.3.m3.4.4.4.2.2.2.3" xref="S6.SS2.4.p2.3.m3.4.4.4.2.2.2.3.cmml">m</mi></msub><mo id="S6.SS2.4.p2.3.m3.4.4.4.2.2.5" stretchy="false" xref="S6.SS2.4.p2.3.m3.4.4.4.2.3.cmml">)</mo></mrow></mrow><mo id="S6.SS2.4.p2.3.m3.6.6.9" xref="S6.SS2.4.p2.3.m3.6.6.9.cmml">≤</mo><mrow id="S6.SS2.4.p2.3.m3.6.6.6" xref="S6.SS2.4.p2.3.m3.6.6.6.cmml"><msub id="S6.SS2.4.p2.3.m3.6.6.6.4" xref="S6.SS2.4.p2.3.m3.6.6.6.4.cmml"><mi id="S6.SS2.4.p2.3.m3.6.6.6.4.2" mathvariant="normal" xref="S6.SS2.4.p2.3.m3.6.6.6.4.2.cmml">Δ</mi><mi id="S6.SS2.4.p2.3.m3.6.6.6.4.3" xref="S6.SS2.4.p2.3.m3.6.6.6.4.3.cmml">A</mi></msub><mo id="S6.SS2.4.p2.3.m3.6.6.6.3" xref="S6.SS2.4.p2.3.m3.6.6.6.3.cmml">⁢</mo><mrow id="S6.SS2.4.p2.3.m3.6.6.6.2.2" xref="S6.SS2.4.p2.3.m3.6.6.6.2.3.cmml"><mo id="S6.SS2.4.p2.3.m3.6.6.6.2.2.3" stretchy="false" xref="S6.SS2.4.p2.3.m3.6.6.6.2.3.cmml">(</mo><msub id="S6.SS2.4.p2.3.m3.5.5.5.1.1.1" xref="S6.SS2.4.p2.3.m3.5.5.5.1.1.1.cmml"><mi id="S6.SS2.4.p2.3.m3.5.5.5.1.1.1.2" xref="S6.SS2.4.p2.3.m3.5.5.5.1.1.1.2.cmml">a</mi><mi id="S6.SS2.4.p2.3.m3.5.5.5.1.1.1.3" xref="S6.SS2.4.p2.3.m3.5.5.5.1.1.1.3.cmml">r</mi></msub><mo id="S6.SS2.4.p2.3.m3.6.6.6.2.2.4" xref="S6.SS2.4.p2.3.m3.6.6.6.2.3.cmml">,</mo><msub id="S6.SS2.4.p2.3.m3.6.6.6.2.2.2" xref="S6.SS2.4.p2.3.m3.6.6.6.2.2.2.cmml"><mi id="S6.SS2.4.p2.3.m3.6.6.6.2.2.2.2" xref="S6.SS2.4.p2.3.m3.6.6.6.2.2.2.2.cmml">b</mi><mi id="S6.SS2.4.p2.3.m3.6.6.6.2.2.2.3" xref="S6.SS2.4.p2.3.m3.6.6.6.2.2.2.3.cmml">l</mi></msub><mo id="S6.SS2.4.p2.3.m3.6.6.6.2.2.5" stretchy="false" xref="S6.SS2.4.p2.3.m3.6.6.6.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.4.p2.3.m3.6b"><apply id="S6.SS2.4.p2.3.m3.6.6.cmml" xref="S6.SS2.4.p2.3.m3.6.6"><and id="S6.SS2.4.p2.3.m3.6.6a.cmml" xref="S6.SS2.4.p2.3.m3.6.6"></and><apply id="S6.SS2.4.p2.3.m3.6.6b.cmml" xref="S6.SS2.4.p2.3.m3.6.6"><leq id="S6.SS2.4.p2.3.m3.6.6.8.cmml" xref="S6.SS2.4.p2.3.m3.6.6.8"></leq><apply id="S6.SS2.4.p2.3.m3.2.2.2.cmml" xref="S6.SS2.4.p2.3.m3.2.2.2"><times id="S6.SS2.4.p2.3.m3.2.2.2.3.cmml" xref="S6.SS2.4.p2.3.m3.2.2.2.3"></times><apply id="S6.SS2.4.p2.3.m3.2.2.2.4.cmml" xref="S6.SS2.4.p2.3.m3.2.2.2.4"><csymbol cd="ambiguous" id="S6.SS2.4.p2.3.m3.2.2.2.4.1.cmml" xref="S6.SS2.4.p2.3.m3.2.2.2.4">subscript</csymbol><ci id="S6.SS2.4.p2.3.m3.2.2.2.4.2.cmml" xref="S6.SS2.4.p2.3.m3.2.2.2.4.2">Δ</ci><ci id="S6.SS2.4.p2.3.m3.2.2.2.4.3.cmml" xref="S6.SS2.4.p2.3.m3.2.2.2.4.3">𝐴</ci></apply><interval closure="open" id="S6.SS2.4.p2.3.m3.2.2.2.2.3.cmml" xref="S6.SS2.4.p2.3.m3.2.2.2.2.2"><apply id="S6.SS2.4.p2.3.m3.1.1.1.1.1.1.cmml" xref="S6.SS2.4.p2.3.m3.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.4.p2.3.m3.1.1.1.1.1.1.1.cmml" xref="S6.SS2.4.p2.3.m3.1.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.4.p2.3.m3.1.1.1.1.1.1.2.cmml" xref="S6.SS2.4.p2.3.m3.1.1.1.1.1.1.2">𝑎</ci><ci id="S6.SS2.4.p2.3.m3.1.1.1.1.1.1.3.cmml" xref="S6.SS2.4.p2.3.m3.1.1.1.1.1.1.3">𝑙</ci></apply><apply id="S6.SS2.4.p2.3.m3.2.2.2.2.2.2.cmml" xref="S6.SS2.4.p2.3.m3.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.4.p2.3.m3.2.2.2.2.2.2.1.cmml" xref="S6.SS2.4.p2.3.m3.2.2.2.2.2.2">subscript</csymbol><ci id="S6.SS2.4.p2.3.m3.2.2.2.2.2.2.2.cmml" xref="S6.SS2.4.p2.3.m3.2.2.2.2.2.2.2">𝑏</ci><ci id="S6.SS2.4.p2.3.m3.2.2.2.2.2.2.3.cmml" xref="S6.SS2.4.p2.3.m3.2.2.2.2.2.2.3">𝑟</ci></apply></interval></apply><apply id="S6.SS2.4.p2.3.m3.4.4.4.cmml" xref="S6.SS2.4.p2.3.m3.4.4.4"><times id="S6.SS2.4.p2.3.m3.4.4.4.3.cmml" xref="S6.SS2.4.p2.3.m3.4.4.4.3"></times><apply id="S6.SS2.4.p2.3.m3.4.4.4.4.cmml" xref="S6.SS2.4.p2.3.m3.4.4.4.4"><csymbol cd="ambiguous" id="S6.SS2.4.p2.3.m3.4.4.4.4.1.cmml" xref="S6.SS2.4.p2.3.m3.4.4.4.4">subscript</csymbol><ci id="S6.SS2.4.p2.3.m3.4.4.4.4.2.cmml" xref="S6.SS2.4.p2.3.m3.4.4.4.4.2">Δ</ci><ci id="S6.SS2.4.p2.3.m3.4.4.4.4.3.cmml" xref="S6.SS2.4.p2.3.m3.4.4.4.4.3">𝐴</ci></apply><interval closure="open" id="S6.SS2.4.p2.3.m3.4.4.4.2.3.cmml" xref="S6.SS2.4.p2.3.m3.4.4.4.2.2"><apply id="S6.SS2.4.p2.3.m3.3.3.3.1.1.1.cmml" xref="S6.SS2.4.p2.3.m3.3.3.3.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.4.p2.3.m3.3.3.3.1.1.1.1.cmml" xref="S6.SS2.4.p2.3.m3.3.3.3.1.1.1">subscript</csymbol><ci id="S6.SS2.4.p2.3.m3.3.3.3.1.1.1.2.cmml" xref="S6.SS2.4.p2.3.m3.3.3.3.1.1.1.2">𝑎</ci><ci id="S6.SS2.4.p2.3.m3.3.3.3.1.1.1.3.cmml" xref="S6.SS2.4.p2.3.m3.3.3.3.1.1.1.3">𝑚</ci></apply><apply id="S6.SS2.4.p2.3.m3.4.4.4.2.2.2.cmml" xref="S6.SS2.4.p2.3.m3.4.4.4.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.4.p2.3.m3.4.4.4.2.2.2.1.cmml" xref="S6.SS2.4.p2.3.m3.4.4.4.2.2.2">subscript</csymbol><ci id="S6.SS2.4.p2.3.m3.4.4.4.2.2.2.2.cmml" xref="S6.SS2.4.p2.3.m3.4.4.4.2.2.2.2">𝑏</ci><ci id="S6.SS2.4.p2.3.m3.4.4.4.2.2.2.3.cmml" xref="S6.SS2.4.p2.3.m3.4.4.4.2.2.2.3">𝑚</ci></apply></interval></apply></apply><apply id="S6.SS2.4.p2.3.m3.6.6c.cmml" xref="S6.SS2.4.p2.3.m3.6.6"><leq id="S6.SS2.4.p2.3.m3.6.6.9.cmml" xref="S6.SS2.4.p2.3.m3.6.6.9"></leq><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.4.p2.3.m3.4.4.4.cmml" id="S6.SS2.4.p2.3.m3.6.6d.cmml" xref="S6.SS2.4.p2.3.m3.6.6"></share><apply id="S6.SS2.4.p2.3.m3.6.6.6.cmml" xref="S6.SS2.4.p2.3.m3.6.6.6"><times id="S6.SS2.4.p2.3.m3.6.6.6.3.cmml" xref="S6.SS2.4.p2.3.m3.6.6.6.3"></times><apply id="S6.SS2.4.p2.3.m3.6.6.6.4.cmml" xref="S6.SS2.4.p2.3.m3.6.6.6.4"><csymbol cd="ambiguous" id="S6.SS2.4.p2.3.m3.6.6.6.4.1.cmml" xref="S6.SS2.4.p2.3.m3.6.6.6.4">subscript</csymbol><ci id="S6.SS2.4.p2.3.m3.6.6.6.4.2.cmml" xref="S6.SS2.4.p2.3.m3.6.6.6.4.2">Δ</ci><ci id="S6.SS2.4.p2.3.m3.6.6.6.4.3.cmml" xref="S6.SS2.4.p2.3.m3.6.6.6.4.3">𝐴</ci></apply><interval closure="open" id="S6.SS2.4.p2.3.m3.6.6.6.2.3.cmml" xref="S6.SS2.4.p2.3.m3.6.6.6.2.2"><apply id="S6.SS2.4.p2.3.m3.5.5.5.1.1.1.cmml" xref="S6.SS2.4.p2.3.m3.5.5.5.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.4.p2.3.m3.5.5.5.1.1.1.1.cmml" xref="S6.SS2.4.p2.3.m3.5.5.5.1.1.1">subscript</csymbol><ci id="S6.SS2.4.p2.3.m3.5.5.5.1.1.1.2.cmml" xref="S6.SS2.4.p2.3.m3.5.5.5.1.1.1.2">𝑎</ci><ci id="S6.SS2.4.p2.3.m3.5.5.5.1.1.1.3.cmml" xref="S6.SS2.4.p2.3.m3.5.5.5.1.1.1.3">𝑟</ci></apply><apply id="S6.SS2.4.p2.3.m3.6.6.6.2.2.2.cmml" xref="S6.SS2.4.p2.3.m3.6.6.6.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.4.p2.3.m3.6.6.6.2.2.2.1.cmml" xref="S6.SS2.4.p2.3.m3.6.6.6.2.2.2">subscript</csymbol><ci id="S6.SS2.4.p2.3.m3.6.6.6.2.2.2.2.cmml" xref="S6.SS2.4.p2.3.m3.6.6.6.2.2.2.2">𝑏</ci><ci id="S6.SS2.4.p2.3.m3.6.6.6.2.2.2.3.cmml" xref="S6.SS2.4.p2.3.m3.6.6.6.2.2.2.3">𝑙</ci></apply></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.4.p2.3.m3.6c">\Delta_{A}(a_{l},b_{r})\leq\Delta_{A}(a_{m},b_{m})\leq\Delta_{A}(a_{r},b_{l})</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.4.p2.3.m3.6d">roman_Δ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) ≤ roman_Δ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) ≤ roman_Δ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT )</annotation></semantics></math>. Therefore we need only to prove that <math alttext="\nu:=\nu(a_{l},b_{r})\geq\nu(a_{r},b_{l}):=\mu" class="ltx_Math" display="inline" id="S6.SS2.4.p2.4.m4.4"><semantics id="S6.SS2.4.p2.4.m4.4a"><mrow id="S6.SS2.4.p2.4.m4.4.4" xref="S6.SS2.4.p2.4.m4.4.4.cmml"><mi id="S6.SS2.4.p2.4.m4.4.4.6" xref="S6.SS2.4.p2.4.m4.4.4.6.cmml">ν</mi><mo id="S6.SS2.4.p2.4.m4.4.4.7" lspace="0.278em" rspace="0.278em" xref="S6.SS2.4.p2.4.m4.4.4.7.cmml">:=</mo><mrow id="S6.SS2.4.p2.4.m4.2.2.2" xref="S6.SS2.4.p2.4.m4.2.2.2.cmml"><mi id="S6.SS2.4.p2.4.m4.2.2.2.4" xref="S6.SS2.4.p2.4.m4.2.2.2.4.cmml">ν</mi><mo id="S6.SS2.4.p2.4.m4.2.2.2.3" xref="S6.SS2.4.p2.4.m4.2.2.2.3.cmml">⁢</mo><mrow id="S6.SS2.4.p2.4.m4.2.2.2.2.2" xref="S6.SS2.4.p2.4.m4.2.2.2.2.3.cmml"><mo id="S6.SS2.4.p2.4.m4.2.2.2.2.2.3" stretchy="false" xref="S6.SS2.4.p2.4.m4.2.2.2.2.3.cmml">(</mo><msub id="S6.SS2.4.p2.4.m4.1.1.1.1.1.1" xref="S6.SS2.4.p2.4.m4.1.1.1.1.1.1.cmml"><mi id="S6.SS2.4.p2.4.m4.1.1.1.1.1.1.2" xref="S6.SS2.4.p2.4.m4.1.1.1.1.1.1.2.cmml">a</mi><mi id="S6.SS2.4.p2.4.m4.1.1.1.1.1.1.3" xref="S6.SS2.4.p2.4.m4.1.1.1.1.1.1.3.cmml">l</mi></msub><mo id="S6.SS2.4.p2.4.m4.2.2.2.2.2.4" xref="S6.SS2.4.p2.4.m4.2.2.2.2.3.cmml">,</mo><msub id="S6.SS2.4.p2.4.m4.2.2.2.2.2.2" xref="S6.SS2.4.p2.4.m4.2.2.2.2.2.2.cmml"><mi id="S6.SS2.4.p2.4.m4.2.2.2.2.2.2.2" xref="S6.SS2.4.p2.4.m4.2.2.2.2.2.2.2.cmml">b</mi><mi id="S6.SS2.4.p2.4.m4.2.2.2.2.2.2.3" xref="S6.SS2.4.p2.4.m4.2.2.2.2.2.2.3.cmml">r</mi></msub><mo id="S6.SS2.4.p2.4.m4.2.2.2.2.2.5" stretchy="false" xref="S6.SS2.4.p2.4.m4.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.SS2.4.p2.4.m4.4.4.8" xref="S6.SS2.4.p2.4.m4.4.4.8.cmml">≥</mo><mrow id="S6.SS2.4.p2.4.m4.4.4.4" xref="S6.SS2.4.p2.4.m4.4.4.4.cmml"><mi id="S6.SS2.4.p2.4.m4.4.4.4.4" xref="S6.SS2.4.p2.4.m4.4.4.4.4.cmml">ν</mi><mo id="S6.SS2.4.p2.4.m4.4.4.4.3" xref="S6.SS2.4.p2.4.m4.4.4.4.3.cmml">⁢</mo><mrow id="S6.SS2.4.p2.4.m4.4.4.4.2.2" xref="S6.SS2.4.p2.4.m4.4.4.4.2.3.cmml"><mo id="S6.SS2.4.p2.4.m4.4.4.4.2.2.3" stretchy="false" xref="S6.SS2.4.p2.4.m4.4.4.4.2.3.cmml">(</mo><msub id="S6.SS2.4.p2.4.m4.3.3.3.1.1.1" xref="S6.SS2.4.p2.4.m4.3.3.3.1.1.1.cmml"><mi id="S6.SS2.4.p2.4.m4.3.3.3.1.1.1.2" xref="S6.SS2.4.p2.4.m4.3.3.3.1.1.1.2.cmml">a</mi><mi id="S6.SS2.4.p2.4.m4.3.3.3.1.1.1.3" xref="S6.SS2.4.p2.4.m4.3.3.3.1.1.1.3.cmml">r</mi></msub><mo id="S6.SS2.4.p2.4.m4.4.4.4.2.2.4" xref="S6.SS2.4.p2.4.m4.4.4.4.2.3.cmml">,</mo><msub id="S6.SS2.4.p2.4.m4.4.4.4.2.2.2" xref="S6.SS2.4.p2.4.m4.4.4.4.2.2.2.cmml"><mi id="S6.SS2.4.p2.4.m4.4.4.4.2.2.2.2" xref="S6.SS2.4.p2.4.m4.4.4.4.2.2.2.2.cmml">b</mi><mi id="S6.SS2.4.p2.4.m4.4.4.4.2.2.2.3" xref="S6.SS2.4.p2.4.m4.4.4.4.2.2.2.3.cmml">l</mi></msub><mo id="S6.SS2.4.p2.4.m4.4.4.4.2.2.5" rspace="0.278em" stretchy="false" xref="S6.SS2.4.p2.4.m4.4.4.4.2.3.cmml">)</mo></mrow></mrow><mo id="S6.SS2.4.p2.4.m4.4.4.9" rspace="0.278em" xref="S6.SS2.4.p2.4.m4.4.4.9.cmml">:=</mo><mi id="S6.SS2.4.p2.4.m4.4.4.10" xref="S6.SS2.4.p2.4.m4.4.4.10.cmml">μ</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.4.p2.4.m4.4b"><apply id="S6.SS2.4.p2.4.m4.4.4.cmml" xref="S6.SS2.4.p2.4.m4.4.4"><and id="S6.SS2.4.p2.4.m4.4.4a.cmml" xref="S6.SS2.4.p2.4.m4.4.4"></and><apply id="S6.SS2.4.p2.4.m4.4.4b.cmml" xref="S6.SS2.4.p2.4.m4.4.4"><csymbol cd="latexml" id="S6.SS2.4.p2.4.m4.4.4.7.cmml" xref="S6.SS2.4.p2.4.m4.4.4.7">assign</csymbol><ci id="S6.SS2.4.p2.4.m4.4.4.6.cmml" xref="S6.SS2.4.p2.4.m4.4.4.6">𝜈</ci><apply id="S6.SS2.4.p2.4.m4.2.2.2.cmml" xref="S6.SS2.4.p2.4.m4.2.2.2"><times id="S6.SS2.4.p2.4.m4.2.2.2.3.cmml" xref="S6.SS2.4.p2.4.m4.2.2.2.3"></times><ci id="S6.SS2.4.p2.4.m4.2.2.2.4.cmml" xref="S6.SS2.4.p2.4.m4.2.2.2.4">𝜈</ci><interval closure="open" id="S6.SS2.4.p2.4.m4.2.2.2.2.3.cmml" xref="S6.SS2.4.p2.4.m4.2.2.2.2.2"><apply id="S6.SS2.4.p2.4.m4.1.1.1.1.1.1.cmml" xref="S6.SS2.4.p2.4.m4.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.4.p2.4.m4.1.1.1.1.1.1.1.cmml" xref="S6.SS2.4.p2.4.m4.1.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.4.p2.4.m4.1.1.1.1.1.1.2.cmml" xref="S6.SS2.4.p2.4.m4.1.1.1.1.1.1.2">𝑎</ci><ci id="S6.SS2.4.p2.4.m4.1.1.1.1.1.1.3.cmml" xref="S6.SS2.4.p2.4.m4.1.1.1.1.1.1.3">𝑙</ci></apply><apply id="S6.SS2.4.p2.4.m4.2.2.2.2.2.2.cmml" xref="S6.SS2.4.p2.4.m4.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.4.p2.4.m4.2.2.2.2.2.2.1.cmml" xref="S6.SS2.4.p2.4.m4.2.2.2.2.2.2">subscript</csymbol><ci id="S6.SS2.4.p2.4.m4.2.2.2.2.2.2.2.cmml" xref="S6.SS2.4.p2.4.m4.2.2.2.2.2.2.2">𝑏</ci><ci id="S6.SS2.4.p2.4.m4.2.2.2.2.2.2.3.cmml" xref="S6.SS2.4.p2.4.m4.2.2.2.2.2.2.3">𝑟</ci></apply></interval></apply></apply><apply id="S6.SS2.4.p2.4.m4.4.4c.cmml" xref="S6.SS2.4.p2.4.m4.4.4"><geq id="S6.SS2.4.p2.4.m4.4.4.8.cmml" xref="S6.SS2.4.p2.4.m4.4.4.8"></geq><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.4.p2.4.m4.2.2.2.cmml" id="S6.SS2.4.p2.4.m4.4.4d.cmml" xref="S6.SS2.4.p2.4.m4.4.4"></share><apply id="S6.SS2.4.p2.4.m4.4.4.4.cmml" xref="S6.SS2.4.p2.4.m4.4.4.4"><times id="S6.SS2.4.p2.4.m4.4.4.4.3.cmml" xref="S6.SS2.4.p2.4.m4.4.4.4.3"></times><ci id="S6.SS2.4.p2.4.m4.4.4.4.4.cmml" xref="S6.SS2.4.p2.4.m4.4.4.4.4">𝜈</ci><interval closure="open" id="S6.SS2.4.p2.4.m4.4.4.4.2.3.cmml" xref="S6.SS2.4.p2.4.m4.4.4.4.2.2"><apply id="S6.SS2.4.p2.4.m4.3.3.3.1.1.1.cmml" xref="S6.SS2.4.p2.4.m4.3.3.3.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.4.p2.4.m4.3.3.3.1.1.1.1.cmml" xref="S6.SS2.4.p2.4.m4.3.3.3.1.1.1">subscript</csymbol><ci id="S6.SS2.4.p2.4.m4.3.3.3.1.1.1.2.cmml" xref="S6.SS2.4.p2.4.m4.3.3.3.1.1.1.2">𝑎</ci><ci id="S6.SS2.4.p2.4.m4.3.3.3.1.1.1.3.cmml" xref="S6.SS2.4.p2.4.m4.3.3.3.1.1.1.3">𝑟</ci></apply><apply id="S6.SS2.4.p2.4.m4.4.4.4.2.2.2.cmml" xref="S6.SS2.4.p2.4.m4.4.4.4.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.4.p2.4.m4.4.4.4.2.2.2.1.cmml" xref="S6.SS2.4.p2.4.m4.4.4.4.2.2.2">subscript</csymbol><ci id="S6.SS2.4.p2.4.m4.4.4.4.2.2.2.2.cmml" xref="S6.SS2.4.p2.4.m4.4.4.4.2.2.2.2">𝑏</ci><ci id="S6.SS2.4.p2.4.m4.4.4.4.2.2.2.3.cmml" xref="S6.SS2.4.p2.4.m4.4.4.4.2.2.2.3">𝑙</ci></apply></interval></apply></apply><apply id="S6.SS2.4.p2.4.m4.4.4e.cmml" xref="S6.SS2.4.p2.4.m4.4.4"><csymbol cd="latexml" id="S6.SS2.4.p2.4.m4.4.4.9.cmml" xref="S6.SS2.4.p2.4.m4.4.4.9">assign</csymbol><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.4.p2.4.m4.4.4.4.cmml" id="S6.SS2.4.p2.4.m4.4.4f.cmml" xref="S6.SS2.4.p2.4.m4.4.4"></share><ci id="S6.SS2.4.p2.4.m4.4.4.10.cmml" xref="S6.SS2.4.p2.4.m4.4.4.10">𝜇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.4.p2.4.m4.4c">\nu:=\nu(a_{l},b_{r})\geq\nu(a_{r},b_{l}):=\mu</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.4.p2.4.m4.4d">italic_ν := italic_ν ( italic_a start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) ≥ italic_ν ( italic_a start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ) := italic_μ</annotation></semantics></math>. Suppose not, and thus that <math alttext="\nu<\mu" class="ltx_Math" display="inline" id="S6.SS2.4.p2.5.m5.1"><semantics id="S6.SS2.4.p2.5.m5.1a"><mrow id="S6.SS2.4.p2.5.m5.1.1" xref="S6.SS2.4.p2.5.m5.1.1.cmml"><mi id="S6.SS2.4.p2.5.m5.1.1.2" xref="S6.SS2.4.p2.5.m5.1.1.2.cmml">ν</mi><mo id="S6.SS2.4.p2.5.m5.1.1.1" xref="S6.SS2.4.p2.5.m5.1.1.1.cmml"><</mo><mi id="S6.SS2.4.p2.5.m5.1.1.3" xref="S6.SS2.4.p2.5.m5.1.1.3.cmml">μ</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.4.p2.5.m5.1b"><apply id="S6.SS2.4.p2.5.m5.1.1.cmml" xref="S6.SS2.4.p2.5.m5.1.1"><lt id="S6.SS2.4.p2.5.m5.1.1.1.cmml" xref="S6.SS2.4.p2.5.m5.1.1.1"></lt><ci id="S6.SS2.4.p2.5.m5.1.1.2.cmml" xref="S6.SS2.4.p2.5.m5.1.1.2">𝜈</ci><ci id="S6.SS2.4.p2.5.m5.1.1.3.cmml" xref="S6.SS2.4.p2.5.m5.1.1.3">𝜇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.4.p2.5.m5.1c">\nu<\mu</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.4.p2.5.m5.1d">italic_ν < italic_μ</annotation></semantics></math>. Then either <math alttext="\Delta_{A}(a_{l},a_{r})=\Delta_{A}(a_{l},b_{r})" class="ltx_Math" display="inline" id="S6.SS2.4.p2.6.m6.4"><semantics id="S6.SS2.4.p2.6.m6.4a"><mrow id="S6.SS2.4.p2.6.m6.4.4" xref="S6.SS2.4.p2.6.m6.4.4.cmml"><mrow id="S6.SS2.4.p2.6.m6.2.2.2" xref="S6.SS2.4.p2.6.m6.2.2.2.cmml"><msub id="S6.SS2.4.p2.6.m6.2.2.2.4" xref="S6.SS2.4.p2.6.m6.2.2.2.4.cmml"><mi id="S6.SS2.4.p2.6.m6.2.2.2.4.2" mathvariant="normal" xref="S6.SS2.4.p2.6.m6.2.2.2.4.2.cmml">Δ</mi><mi id="S6.SS2.4.p2.6.m6.2.2.2.4.3" xref="S6.SS2.4.p2.6.m6.2.2.2.4.3.cmml">A</mi></msub><mo id="S6.SS2.4.p2.6.m6.2.2.2.3" xref="S6.SS2.4.p2.6.m6.2.2.2.3.cmml">⁢</mo><mrow id="S6.SS2.4.p2.6.m6.2.2.2.2.2" xref="S6.SS2.4.p2.6.m6.2.2.2.2.3.cmml"><mo id="S6.SS2.4.p2.6.m6.2.2.2.2.2.3" stretchy="false" xref="S6.SS2.4.p2.6.m6.2.2.2.2.3.cmml">(</mo><msub id="S6.SS2.4.p2.6.m6.1.1.1.1.1.1" xref="S6.SS2.4.p2.6.m6.1.1.1.1.1.1.cmml"><mi id="S6.SS2.4.p2.6.m6.1.1.1.1.1.1.2" xref="S6.SS2.4.p2.6.m6.1.1.1.1.1.1.2.cmml">a</mi><mi id="S6.SS2.4.p2.6.m6.1.1.1.1.1.1.3" xref="S6.SS2.4.p2.6.m6.1.1.1.1.1.1.3.cmml">l</mi></msub><mo id="S6.SS2.4.p2.6.m6.2.2.2.2.2.4" xref="S6.SS2.4.p2.6.m6.2.2.2.2.3.cmml">,</mo><msub id="S6.SS2.4.p2.6.m6.2.2.2.2.2.2" xref="S6.SS2.4.p2.6.m6.2.2.2.2.2.2.cmml"><mi id="S6.SS2.4.p2.6.m6.2.2.2.2.2.2.2" xref="S6.SS2.4.p2.6.m6.2.2.2.2.2.2.2.cmml">a</mi><mi id="S6.SS2.4.p2.6.m6.2.2.2.2.2.2.3" xref="S6.SS2.4.p2.6.m6.2.2.2.2.2.2.3.cmml">r</mi></msub><mo id="S6.SS2.4.p2.6.m6.2.2.2.2.2.5" stretchy="false" xref="S6.SS2.4.p2.6.m6.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.SS2.4.p2.6.m6.4.4.5" xref="S6.SS2.4.p2.6.m6.4.4.5.cmml">=</mo><mrow id="S6.SS2.4.p2.6.m6.4.4.4" xref="S6.SS2.4.p2.6.m6.4.4.4.cmml"><msub id="S6.SS2.4.p2.6.m6.4.4.4.4" xref="S6.SS2.4.p2.6.m6.4.4.4.4.cmml"><mi id="S6.SS2.4.p2.6.m6.4.4.4.4.2" mathvariant="normal" xref="S6.SS2.4.p2.6.m6.4.4.4.4.2.cmml">Δ</mi><mi id="S6.SS2.4.p2.6.m6.4.4.4.4.3" xref="S6.SS2.4.p2.6.m6.4.4.4.4.3.cmml">A</mi></msub><mo id="S6.SS2.4.p2.6.m6.4.4.4.3" xref="S6.SS2.4.p2.6.m6.4.4.4.3.cmml">⁢</mo><mrow id="S6.SS2.4.p2.6.m6.4.4.4.2.2" xref="S6.SS2.4.p2.6.m6.4.4.4.2.3.cmml"><mo id="S6.SS2.4.p2.6.m6.4.4.4.2.2.3" stretchy="false" xref="S6.SS2.4.p2.6.m6.4.4.4.2.3.cmml">(</mo><msub id="S6.SS2.4.p2.6.m6.3.3.3.1.1.1" xref="S6.SS2.4.p2.6.m6.3.3.3.1.1.1.cmml"><mi id="S6.SS2.4.p2.6.m6.3.3.3.1.1.1.2" xref="S6.SS2.4.p2.6.m6.3.3.3.1.1.1.2.cmml">a</mi><mi id="S6.SS2.4.p2.6.m6.3.3.3.1.1.1.3" xref="S6.SS2.4.p2.6.m6.3.3.3.1.1.1.3.cmml">l</mi></msub><mo id="S6.SS2.4.p2.6.m6.4.4.4.2.2.4" xref="S6.SS2.4.p2.6.m6.4.4.4.2.3.cmml">,</mo><msub id="S6.SS2.4.p2.6.m6.4.4.4.2.2.2" xref="S6.SS2.4.p2.6.m6.4.4.4.2.2.2.cmml"><mi id="S6.SS2.4.p2.6.m6.4.4.4.2.2.2.2" xref="S6.SS2.4.p2.6.m6.4.4.4.2.2.2.2.cmml">b</mi><mi id="S6.SS2.4.p2.6.m6.4.4.4.2.2.2.3" xref="S6.SS2.4.p2.6.m6.4.4.4.2.2.2.3.cmml">r</mi></msub><mo id="S6.SS2.4.p2.6.m6.4.4.4.2.2.5" stretchy="false" xref="S6.SS2.4.p2.6.m6.4.4.4.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.4.p2.6.m6.4b"><apply id="S6.SS2.4.p2.6.m6.4.4.cmml" xref="S6.SS2.4.p2.6.m6.4.4"><eq id="S6.SS2.4.p2.6.m6.4.4.5.cmml" xref="S6.SS2.4.p2.6.m6.4.4.5"></eq><apply id="S6.SS2.4.p2.6.m6.2.2.2.cmml" xref="S6.SS2.4.p2.6.m6.2.2.2"><times id="S6.SS2.4.p2.6.m6.2.2.2.3.cmml" xref="S6.SS2.4.p2.6.m6.2.2.2.3"></times><apply id="S6.SS2.4.p2.6.m6.2.2.2.4.cmml" xref="S6.SS2.4.p2.6.m6.2.2.2.4"><csymbol cd="ambiguous" id="S6.SS2.4.p2.6.m6.2.2.2.4.1.cmml" xref="S6.SS2.4.p2.6.m6.2.2.2.4">subscript</csymbol><ci id="S6.SS2.4.p2.6.m6.2.2.2.4.2.cmml" xref="S6.SS2.4.p2.6.m6.2.2.2.4.2">Δ</ci><ci id="S6.SS2.4.p2.6.m6.2.2.2.4.3.cmml" xref="S6.SS2.4.p2.6.m6.2.2.2.4.3">𝐴</ci></apply><interval closure="open" id="S6.SS2.4.p2.6.m6.2.2.2.2.3.cmml" xref="S6.SS2.4.p2.6.m6.2.2.2.2.2"><apply id="S6.SS2.4.p2.6.m6.1.1.1.1.1.1.cmml" xref="S6.SS2.4.p2.6.m6.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.4.p2.6.m6.1.1.1.1.1.1.1.cmml" xref="S6.SS2.4.p2.6.m6.1.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.4.p2.6.m6.1.1.1.1.1.1.2.cmml" xref="S6.SS2.4.p2.6.m6.1.1.1.1.1.1.2">𝑎</ci><ci id="S6.SS2.4.p2.6.m6.1.1.1.1.1.1.3.cmml" xref="S6.SS2.4.p2.6.m6.1.1.1.1.1.1.3">𝑙</ci></apply><apply id="S6.SS2.4.p2.6.m6.2.2.2.2.2.2.cmml" xref="S6.SS2.4.p2.6.m6.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.4.p2.6.m6.2.2.2.2.2.2.1.cmml" xref="S6.SS2.4.p2.6.m6.2.2.2.2.2.2">subscript</csymbol><ci id="S6.SS2.4.p2.6.m6.2.2.2.2.2.2.2.cmml" xref="S6.SS2.4.p2.6.m6.2.2.2.2.2.2.2">𝑎</ci><ci id="S6.SS2.4.p2.6.m6.2.2.2.2.2.2.3.cmml" xref="S6.SS2.4.p2.6.m6.2.2.2.2.2.2.3">𝑟</ci></apply></interval></apply><apply id="S6.SS2.4.p2.6.m6.4.4.4.cmml" xref="S6.SS2.4.p2.6.m6.4.4.4"><times id="S6.SS2.4.p2.6.m6.4.4.4.3.cmml" xref="S6.SS2.4.p2.6.m6.4.4.4.3"></times><apply id="S6.SS2.4.p2.6.m6.4.4.4.4.cmml" xref="S6.SS2.4.p2.6.m6.4.4.4.4"><csymbol cd="ambiguous" id="S6.SS2.4.p2.6.m6.4.4.4.4.1.cmml" xref="S6.SS2.4.p2.6.m6.4.4.4.4">subscript</csymbol><ci id="S6.SS2.4.p2.6.m6.4.4.4.4.2.cmml" xref="S6.SS2.4.p2.6.m6.4.4.4.4.2">Δ</ci><ci id="S6.SS2.4.p2.6.m6.4.4.4.4.3.cmml" xref="S6.SS2.4.p2.6.m6.4.4.4.4.3">𝐴</ci></apply><interval closure="open" id="S6.SS2.4.p2.6.m6.4.4.4.2.3.cmml" xref="S6.SS2.4.p2.6.m6.4.4.4.2.2"><apply id="S6.SS2.4.p2.6.m6.3.3.3.1.1.1.cmml" xref="S6.SS2.4.p2.6.m6.3.3.3.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.4.p2.6.m6.3.3.3.1.1.1.1.cmml" xref="S6.SS2.4.p2.6.m6.3.3.3.1.1.1">subscript</csymbol><ci id="S6.SS2.4.p2.6.m6.3.3.3.1.1.1.2.cmml" xref="S6.SS2.4.p2.6.m6.3.3.3.1.1.1.2">𝑎</ci><ci id="S6.SS2.4.p2.6.m6.3.3.3.1.1.1.3.cmml" xref="S6.SS2.4.p2.6.m6.3.3.3.1.1.1.3">𝑙</ci></apply><apply id="S6.SS2.4.p2.6.m6.4.4.4.2.2.2.cmml" xref="S6.SS2.4.p2.6.m6.4.4.4.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.4.p2.6.m6.4.4.4.2.2.2.1.cmml" xref="S6.SS2.4.p2.6.m6.4.4.4.2.2.2">subscript</csymbol><ci id="S6.SS2.4.p2.6.m6.4.4.4.2.2.2.2.cmml" xref="S6.SS2.4.p2.6.m6.4.4.4.2.2.2.2">𝑏</ci><ci id="S6.SS2.4.p2.6.m6.4.4.4.2.2.2.3.cmml" xref="S6.SS2.4.p2.6.m6.4.4.4.2.2.2.3">𝑟</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.4.p2.6.m6.4c">\Delta_{A}(a_{l},a_{r})=\Delta_{A}(a_{l},b_{r})</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.4.p2.6.m6.4d">roman_Δ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) = roman_Δ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT )</annotation></semantics></math> or <math alttext="\Delta_{A}(b_{l},b_{r})=\Delta_{A}(a_{l},b_{r})" class="ltx_Math" display="inline" id="S6.SS2.4.p2.7.m7.4"><semantics id="S6.SS2.4.p2.7.m7.4a"><mrow id="S6.SS2.4.p2.7.m7.4.4" xref="S6.SS2.4.p2.7.m7.4.4.cmml"><mrow id="S6.SS2.4.p2.7.m7.2.2.2" xref="S6.SS2.4.p2.7.m7.2.2.2.cmml"><msub id="S6.SS2.4.p2.7.m7.2.2.2.4" xref="S6.SS2.4.p2.7.m7.2.2.2.4.cmml"><mi id="S6.SS2.4.p2.7.m7.2.2.2.4.2" mathvariant="normal" xref="S6.SS2.4.p2.7.m7.2.2.2.4.2.cmml">Δ</mi><mi id="S6.SS2.4.p2.7.m7.2.2.2.4.3" xref="S6.SS2.4.p2.7.m7.2.2.2.4.3.cmml">A</mi></msub><mo id="S6.SS2.4.p2.7.m7.2.2.2.3" xref="S6.SS2.4.p2.7.m7.2.2.2.3.cmml">⁢</mo><mrow id="S6.SS2.4.p2.7.m7.2.2.2.2.2" xref="S6.SS2.4.p2.7.m7.2.2.2.2.3.cmml"><mo id="S6.SS2.4.p2.7.m7.2.2.2.2.2.3" stretchy="false" xref="S6.SS2.4.p2.7.m7.2.2.2.2.3.cmml">(</mo><msub id="S6.SS2.4.p2.7.m7.1.1.1.1.1.1" xref="S6.SS2.4.p2.7.m7.1.1.1.1.1.1.cmml"><mi id="S6.SS2.4.p2.7.m7.1.1.1.1.1.1.2" xref="S6.SS2.4.p2.7.m7.1.1.1.1.1.1.2.cmml">b</mi><mi id="S6.SS2.4.p2.7.m7.1.1.1.1.1.1.3" xref="S6.SS2.4.p2.7.m7.1.1.1.1.1.1.3.cmml">l</mi></msub><mo id="S6.SS2.4.p2.7.m7.2.2.2.2.2.4" xref="S6.SS2.4.p2.7.m7.2.2.2.2.3.cmml">,</mo><msub id="S6.SS2.4.p2.7.m7.2.2.2.2.2.2" xref="S6.SS2.4.p2.7.m7.2.2.2.2.2.2.cmml"><mi id="S6.SS2.4.p2.7.m7.2.2.2.2.2.2.2" xref="S6.SS2.4.p2.7.m7.2.2.2.2.2.2.2.cmml">b</mi><mi id="S6.SS2.4.p2.7.m7.2.2.2.2.2.2.3" xref="S6.SS2.4.p2.7.m7.2.2.2.2.2.2.3.cmml">r</mi></msub><mo id="S6.SS2.4.p2.7.m7.2.2.2.2.2.5" stretchy="false" xref="S6.SS2.4.p2.7.m7.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.SS2.4.p2.7.m7.4.4.5" xref="S6.SS2.4.p2.7.m7.4.4.5.cmml">=</mo><mrow id="S6.SS2.4.p2.7.m7.4.4.4" xref="S6.SS2.4.p2.7.m7.4.4.4.cmml"><msub id="S6.SS2.4.p2.7.m7.4.4.4.4" xref="S6.SS2.4.p2.7.m7.4.4.4.4.cmml"><mi id="S6.SS2.4.p2.7.m7.4.4.4.4.2" mathvariant="normal" xref="S6.SS2.4.p2.7.m7.4.4.4.4.2.cmml">Δ</mi><mi id="S6.SS2.4.p2.7.m7.4.4.4.4.3" xref="S6.SS2.4.p2.7.m7.4.4.4.4.3.cmml">A</mi></msub><mo id="S6.SS2.4.p2.7.m7.4.4.4.3" xref="S6.SS2.4.p2.7.m7.4.4.4.3.cmml">⁢</mo><mrow id="S6.SS2.4.p2.7.m7.4.4.4.2.2" xref="S6.SS2.4.p2.7.m7.4.4.4.2.3.cmml"><mo id="S6.SS2.4.p2.7.m7.4.4.4.2.2.3" stretchy="false" xref="S6.SS2.4.p2.7.m7.4.4.4.2.3.cmml">(</mo><msub id="S6.SS2.4.p2.7.m7.3.3.3.1.1.1" xref="S6.SS2.4.p2.7.m7.3.3.3.1.1.1.cmml"><mi id="S6.SS2.4.p2.7.m7.3.3.3.1.1.1.2" xref="S6.SS2.4.p2.7.m7.3.3.3.1.1.1.2.cmml">a</mi><mi id="S6.SS2.4.p2.7.m7.3.3.3.1.1.1.3" xref="S6.SS2.4.p2.7.m7.3.3.3.1.1.1.3.cmml">l</mi></msub><mo id="S6.SS2.4.p2.7.m7.4.4.4.2.2.4" xref="S6.SS2.4.p2.7.m7.4.4.4.2.3.cmml">,</mo><msub id="S6.SS2.4.p2.7.m7.4.4.4.2.2.2" xref="S6.SS2.4.p2.7.m7.4.4.4.2.2.2.cmml"><mi id="S6.SS2.4.p2.7.m7.4.4.4.2.2.2.2" xref="S6.SS2.4.p2.7.m7.4.4.4.2.2.2.2.cmml">b</mi><mi id="S6.SS2.4.p2.7.m7.4.4.4.2.2.2.3" xref="S6.SS2.4.p2.7.m7.4.4.4.2.2.2.3.cmml">r</mi></msub><mo id="S6.SS2.4.p2.7.m7.4.4.4.2.2.5" stretchy="false" xref="S6.SS2.4.p2.7.m7.4.4.4.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.4.p2.7.m7.4b"><apply id="S6.SS2.4.p2.7.m7.4.4.cmml" xref="S6.SS2.4.p2.7.m7.4.4"><eq id="S6.SS2.4.p2.7.m7.4.4.5.cmml" xref="S6.SS2.4.p2.7.m7.4.4.5"></eq><apply id="S6.SS2.4.p2.7.m7.2.2.2.cmml" xref="S6.SS2.4.p2.7.m7.2.2.2"><times id="S6.SS2.4.p2.7.m7.2.2.2.3.cmml" xref="S6.SS2.4.p2.7.m7.2.2.2.3"></times><apply id="S6.SS2.4.p2.7.m7.2.2.2.4.cmml" xref="S6.SS2.4.p2.7.m7.2.2.2.4"><csymbol cd="ambiguous" id="S6.SS2.4.p2.7.m7.2.2.2.4.1.cmml" xref="S6.SS2.4.p2.7.m7.2.2.2.4">subscript</csymbol><ci id="S6.SS2.4.p2.7.m7.2.2.2.4.2.cmml" xref="S6.SS2.4.p2.7.m7.2.2.2.4.2">Δ</ci><ci id="S6.SS2.4.p2.7.m7.2.2.2.4.3.cmml" xref="S6.SS2.4.p2.7.m7.2.2.2.4.3">𝐴</ci></apply><interval closure="open" id="S6.SS2.4.p2.7.m7.2.2.2.2.3.cmml" xref="S6.SS2.4.p2.7.m7.2.2.2.2.2"><apply id="S6.SS2.4.p2.7.m7.1.1.1.1.1.1.cmml" xref="S6.SS2.4.p2.7.m7.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.4.p2.7.m7.1.1.1.1.1.1.1.cmml" xref="S6.SS2.4.p2.7.m7.1.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.4.p2.7.m7.1.1.1.1.1.1.2.cmml" xref="S6.SS2.4.p2.7.m7.1.1.1.1.1.1.2">𝑏</ci><ci id="S6.SS2.4.p2.7.m7.1.1.1.1.1.1.3.cmml" xref="S6.SS2.4.p2.7.m7.1.1.1.1.1.1.3">𝑙</ci></apply><apply id="S6.SS2.4.p2.7.m7.2.2.2.2.2.2.cmml" xref="S6.SS2.4.p2.7.m7.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.4.p2.7.m7.2.2.2.2.2.2.1.cmml" xref="S6.SS2.4.p2.7.m7.2.2.2.2.2.2">subscript</csymbol><ci id="S6.SS2.4.p2.7.m7.2.2.2.2.2.2.2.cmml" xref="S6.SS2.4.p2.7.m7.2.2.2.2.2.2.2">𝑏</ci><ci id="S6.SS2.4.p2.7.m7.2.2.2.2.2.2.3.cmml" xref="S6.SS2.4.p2.7.m7.2.2.2.2.2.2.3">𝑟</ci></apply></interval></apply><apply id="S6.SS2.4.p2.7.m7.4.4.4.cmml" xref="S6.SS2.4.p2.7.m7.4.4.4"><times id="S6.SS2.4.p2.7.m7.4.4.4.3.cmml" xref="S6.SS2.4.p2.7.m7.4.4.4.3"></times><apply id="S6.SS2.4.p2.7.m7.4.4.4.4.cmml" xref="S6.SS2.4.p2.7.m7.4.4.4.4"><csymbol cd="ambiguous" id="S6.SS2.4.p2.7.m7.4.4.4.4.1.cmml" xref="S6.SS2.4.p2.7.m7.4.4.4.4">subscript</csymbol><ci id="S6.SS2.4.p2.7.m7.4.4.4.4.2.cmml" xref="S6.SS2.4.p2.7.m7.4.4.4.4.2">Δ</ci><ci id="S6.SS2.4.p2.7.m7.4.4.4.4.3.cmml" xref="S6.SS2.4.p2.7.m7.4.4.4.4.3">𝐴</ci></apply><interval closure="open" id="S6.SS2.4.p2.7.m7.4.4.4.2.3.cmml" xref="S6.SS2.4.p2.7.m7.4.4.4.2.2"><apply id="S6.SS2.4.p2.7.m7.3.3.3.1.1.1.cmml" xref="S6.SS2.4.p2.7.m7.3.3.3.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.4.p2.7.m7.3.3.3.1.1.1.1.cmml" xref="S6.SS2.4.p2.7.m7.3.3.3.1.1.1">subscript</csymbol><ci id="S6.SS2.4.p2.7.m7.3.3.3.1.1.1.2.cmml" xref="S6.SS2.4.p2.7.m7.3.3.3.1.1.1.2">𝑎</ci><ci id="S6.SS2.4.p2.7.m7.3.3.3.1.1.1.3.cmml" xref="S6.SS2.4.p2.7.m7.3.3.3.1.1.1.3">𝑙</ci></apply><apply id="S6.SS2.4.p2.7.m7.4.4.4.2.2.2.cmml" xref="S6.SS2.4.p2.7.m7.4.4.4.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.4.p2.7.m7.4.4.4.2.2.2.1.cmml" xref="S6.SS2.4.p2.7.m7.4.4.4.2.2.2">subscript</csymbol><ci id="S6.SS2.4.p2.7.m7.4.4.4.2.2.2.2.cmml" xref="S6.SS2.4.p2.7.m7.4.4.4.2.2.2.2">𝑏</ci><ci id="S6.SS2.4.p2.7.m7.4.4.4.2.2.2.3.cmml" xref="S6.SS2.4.p2.7.m7.4.4.4.2.2.2.3">𝑟</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.4.p2.7.m7.4c">\Delta_{A}(b_{l},b_{r})=\Delta_{A}(a_{l},b_{r})</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.4.p2.7.m7.4d">roman_Δ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_b start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) = roman_Δ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT )</annotation></semantics></math>. Suppose the second holds, the other case is symmetric. Then in particular <math alttext="\nu(b_{l},b_{r})=\nu" class="ltx_Math" display="inline" id="S6.SS2.4.p2.8.m8.2"><semantics id="S6.SS2.4.p2.8.m8.2a"><mrow id="S6.SS2.4.p2.8.m8.2.2" xref="S6.SS2.4.p2.8.m8.2.2.cmml"><mrow id="S6.SS2.4.p2.8.m8.2.2.2" xref="S6.SS2.4.p2.8.m8.2.2.2.cmml"><mi id="S6.SS2.4.p2.8.m8.2.2.2.4" xref="S6.SS2.4.p2.8.m8.2.2.2.4.cmml">ν</mi><mo id="S6.SS2.4.p2.8.m8.2.2.2.3" xref="S6.SS2.4.p2.8.m8.2.2.2.3.cmml">⁢</mo><mrow id="S6.SS2.4.p2.8.m8.2.2.2.2.2" xref="S6.SS2.4.p2.8.m8.2.2.2.2.3.cmml"><mo id="S6.SS2.4.p2.8.m8.2.2.2.2.2.3" stretchy="false" xref="S6.SS2.4.p2.8.m8.2.2.2.2.3.cmml">(</mo><msub id="S6.SS2.4.p2.8.m8.1.1.1.1.1.1" xref="S6.SS2.4.p2.8.m8.1.1.1.1.1.1.cmml"><mi id="S6.SS2.4.p2.8.m8.1.1.1.1.1.1.2" xref="S6.SS2.4.p2.8.m8.1.1.1.1.1.1.2.cmml">b</mi><mi id="S6.SS2.4.p2.8.m8.1.1.1.1.1.1.3" xref="S6.SS2.4.p2.8.m8.1.1.1.1.1.1.3.cmml">l</mi></msub><mo id="S6.SS2.4.p2.8.m8.2.2.2.2.2.4" xref="S6.SS2.4.p2.8.m8.2.2.2.2.3.cmml">,</mo><msub id="S6.SS2.4.p2.8.m8.2.2.2.2.2.2" xref="S6.SS2.4.p2.8.m8.2.2.2.2.2.2.cmml"><mi id="S6.SS2.4.p2.8.m8.2.2.2.2.2.2.2" xref="S6.SS2.4.p2.8.m8.2.2.2.2.2.2.2.cmml">b</mi><mi id="S6.SS2.4.p2.8.m8.2.2.2.2.2.2.3" xref="S6.SS2.4.p2.8.m8.2.2.2.2.2.2.3.cmml">r</mi></msub><mo id="S6.SS2.4.p2.8.m8.2.2.2.2.2.5" stretchy="false" xref="S6.SS2.4.p2.8.m8.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.SS2.4.p2.8.m8.2.2.3" xref="S6.SS2.4.p2.8.m8.2.2.3.cmml">=</mo><mi id="S6.SS2.4.p2.8.m8.2.2.4" xref="S6.SS2.4.p2.8.m8.2.2.4.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.4.p2.8.m8.2b"><apply id="S6.SS2.4.p2.8.m8.2.2.cmml" xref="S6.SS2.4.p2.8.m8.2.2"><eq id="S6.SS2.4.p2.8.m8.2.2.3.cmml" xref="S6.SS2.4.p2.8.m8.2.2.3"></eq><apply id="S6.SS2.4.p2.8.m8.2.2.2.cmml" xref="S6.SS2.4.p2.8.m8.2.2.2"><times id="S6.SS2.4.p2.8.m8.2.2.2.3.cmml" xref="S6.SS2.4.p2.8.m8.2.2.2.3"></times><ci id="S6.SS2.4.p2.8.m8.2.2.2.4.cmml" xref="S6.SS2.4.p2.8.m8.2.2.2.4">𝜈</ci><interval closure="open" id="S6.SS2.4.p2.8.m8.2.2.2.2.3.cmml" xref="S6.SS2.4.p2.8.m8.2.2.2.2.2"><apply id="S6.SS2.4.p2.8.m8.1.1.1.1.1.1.cmml" xref="S6.SS2.4.p2.8.m8.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.4.p2.8.m8.1.1.1.1.1.1.1.cmml" xref="S6.SS2.4.p2.8.m8.1.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.4.p2.8.m8.1.1.1.1.1.1.2.cmml" xref="S6.SS2.4.p2.8.m8.1.1.1.1.1.1.2">𝑏</ci><ci id="S6.SS2.4.p2.8.m8.1.1.1.1.1.1.3.cmml" xref="S6.SS2.4.p2.8.m8.1.1.1.1.1.1.3">𝑙</ci></apply><apply id="S6.SS2.4.p2.8.m8.2.2.2.2.2.2.cmml" xref="S6.SS2.4.p2.8.m8.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.4.p2.8.m8.2.2.2.2.2.2.1.cmml" xref="S6.SS2.4.p2.8.m8.2.2.2.2.2.2">subscript</csymbol><ci id="S6.SS2.4.p2.8.m8.2.2.2.2.2.2.2.cmml" xref="S6.SS2.4.p2.8.m8.2.2.2.2.2.2.2">𝑏</ci><ci id="S6.SS2.4.p2.8.m8.2.2.2.2.2.2.3.cmml" xref="S6.SS2.4.p2.8.m8.2.2.2.2.2.2.3">𝑟</ci></apply></interval></apply><ci id="S6.SS2.4.p2.8.m8.2.2.4.cmml" xref="S6.SS2.4.p2.8.m8.2.2.4">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.4.p2.8.m8.2c">\nu(b_{l},b_{r})=\nu</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.4.p2.8.m8.2d">italic_ν ( italic_b start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) = italic_ν</annotation></semantics></math>, and since <math alttext="p\in P_{E}" class="ltx_Math" display="inline" id="S6.SS2.4.p2.9.m9.1"><semantics id="S6.SS2.4.p2.9.m9.1a"><mrow id="S6.SS2.4.p2.9.m9.1.1" xref="S6.SS2.4.p2.9.m9.1.1.cmml"><mi id="S6.SS2.4.p2.9.m9.1.1.2" xref="S6.SS2.4.p2.9.m9.1.1.2.cmml">p</mi><mo id="S6.SS2.4.p2.9.m9.1.1.1" xref="S6.SS2.4.p2.9.m9.1.1.1.cmml">∈</mo><msub id="S6.SS2.4.p2.9.m9.1.1.3" xref="S6.SS2.4.p2.9.m9.1.1.3.cmml"><mi id="S6.SS2.4.p2.9.m9.1.1.3.2" xref="S6.SS2.4.p2.9.m9.1.1.3.2.cmml">P</mi><mi id="S6.SS2.4.p2.9.m9.1.1.3.3" xref="S6.SS2.4.p2.9.m9.1.1.3.3.cmml">E</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.4.p2.9.m9.1b"><apply id="S6.SS2.4.p2.9.m9.1.1.cmml" xref="S6.SS2.4.p2.9.m9.1.1"><in id="S6.SS2.4.p2.9.m9.1.1.1.cmml" xref="S6.SS2.4.p2.9.m9.1.1.1"></in><ci id="S6.SS2.4.p2.9.m9.1.1.2.cmml" xref="S6.SS2.4.p2.9.m9.1.1.2">𝑝</ci><apply id="S6.SS2.4.p2.9.m9.1.1.3.cmml" xref="S6.SS2.4.p2.9.m9.1.1.3"><csymbol cd="ambiguous" id="S6.SS2.4.p2.9.m9.1.1.3.1.cmml" xref="S6.SS2.4.p2.9.m9.1.1.3">subscript</csymbol><ci id="S6.SS2.4.p2.9.m9.1.1.3.2.cmml" xref="S6.SS2.4.p2.9.m9.1.1.3.2">𝑃</ci><ci id="S6.SS2.4.p2.9.m9.1.1.3.3.cmml" xref="S6.SS2.4.p2.9.m9.1.1.3.3">𝐸</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.4.p2.9.m9.1c">p\in P_{E}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.4.p2.9.m9.1d">italic_p ∈ italic_P start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\nu(b_{l})=\nu(b_{m})=\nu(b_{l},b_{r})=\nu" class="ltx_Math" display="inline" id="S6.SS2.4.p2.10.m10.4"><semantics id="S6.SS2.4.p2.10.m10.4a"><mrow id="S6.SS2.4.p2.10.m10.4.4" xref="S6.SS2.4.p2.10.m10.4.4.cmml"><mrow id="S6.SS2.4.p2.10.m10.1.1.1" xref="S6.SS2.4.p2.10.m10.1.1.1.cmml"><mi id="S6.SS2.4.p2.10.m10.1.1.1.3" xref="S6.SS2.4.p2.10.m10.1.1.1.3.cmml">ν</mi><mo id="S6.SS2.4.p2.10.m10.1.1.1.2" xref="S6.SS2.4.p2.10.m10.1.1.1.2.cmml">⁢</mo><mrow id="S6.SS2.4.p2.10.m10.1.1.1.1.1" xref="S6.SS2.4.p2.10.m10.1.1.1.1.1.1.cmml"><mo id="S6.SS2.4.p2.10.m10.1.1.1.1.1.2" stretchy="false" xref="S6.SS2.4.p2.10.m10.1.1.1.1.1.1.cmml">(</mo><msub id="S6.SS2.4.p2.10.m10.1.1.1.1.1.1" xref="S6.SS2.4.p2.10.m10.1.1.1.1.1.1.cmml"><mi id="S6.SS2.4.p2.10.m10.1.1.1.1.1.1.2" xref="S6.SS2.4.p2.10.m10.1.1.1.1.1.1.2.cmml">b</mi><mi id="S6.SS2.4.p2.10.m10.1.1.1.1.1.1.3" xref="S6.SS2.4.p2.10.m10.1.1.1.1.1.1.3.cmml">l</mi></msub><mo id="S6.SS2.4.p2.10.m10.1.1.1.1.1.3" stretchy="false" xref="S6.SS2.4.p2.10.m10.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.SS2.4.p2.10.m10.4.4.6" xref="S6.SS2.4.p2.10.m10.4.4.6.cmml">=</mo><mrow id="S6.SS2.4.p2.10.m10.2.2.2" xref="S6.SS2.4.p2.10.m10.2.2.2.cmml"><mi id="S6.SS2.4.p2.10.m10.2.2.2.3" xref="S6.SS2.4.p2.10.m10.2.2.2.3.cmml">ν</mi><mo id="S6.SS2.4.p2.10.m10.2.2.2.2" xref="S6.SS2.4.p2.10.m10.2.2.2.2.cmml">⁢</mo><mrow id="S6.SS2.4.p2.10.m10.2.2.2.1.1" xref="S6.SS2.4.p2.10.m10.2.2.2.1.1.1.cmml"><mo id="S6.SS2.4.p2.10.m10.2.2.2.1.1.2" stretchy="false" xref="S6.SS2.4.p2.10.m10.2.2.2.1.1.1.cmml">(</mo><msub id="S6.SS2.4.p2.10.m10.2.2.2.1.1.1" xref="S6.SS2.4.p2.10.m10.2.2.2.1.1.1.cmml"><mi id="S6.SS2.4.p2.10.m10.2.2.2.1.1.1.2" xref="S6.SS2.4.p2.10.m10.2.2.2.1.1.1.2.cmml">b</mi><mi id="S6.SS2.4.p2.10.m10.2.2.2.1.1.1.3" xref="S6.SS2.4.p2.10.m10.2.2.2.1.1.1.3.cmml">m</mi></msub><mo id="S6.SS2.4.p2.10.m10.2.2.2.1.1.3" stretchy="false" xref="S6.SS2.4.p2.10.m10.2.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.SS2.4.p2.10.m10.4.4.7" xref="S6.SS2.4.p2.10.m10.4.4.7.cmml">=</mo><mrow id="S6.SS2.4.p2.10.m10.4.4.4" xref="S6.SS2.4.p2.10.m10.4.4.4.cmml"><mi id="S6.SS2.4.p2.10.m10.4.4.4.4" xref="S6.SS2.4.p2.10.m10.4.4.4.4.cmml">ν</mi><mo id="S6.SS2.4.p2.10.m10.4.4.4.3" xref="S6.SS2.4.p2.10.m10.4.4.4.3.cmml">⁢</mo><mrow id="S6.SS2.4.p2.10.m10.4.4.4.2.2" xref="S6.SS2.4.p2.10.m10.4.4.4.2.3.cmml"><mo id="S6.SS2.4.p2.10.m10.4.4.4.2.2.3" stretchy="false" xref="S6.SS2.4.p2.10.m10.4.4.4.2.3.cmml">(</mo><msub id="S6.SS2.4.p2.10.m10.3.3.3.1.1.1" xref="S6.SS2.4.p2.10.m10.3.3.3.1.1.1.cmml"><mi id="S6.SS2.4.p2.10.m10.3.3.3.1.1.1.2" xref="S6.SS2.4.p2.10.m10.3.3.3.1.1.1.2.cmml">b</mi><mi id="S6.SS2.4.p2.10.m10.3.3.3.1.1.1.3" xref="S6.SS2.4.p2.10.m10.3.3.3.1.1.1.3.cmml">l</mi></msub><mo id="S6.SS2.4.p2.10.m10.4.4.4.2.2.4" xref="S6.SS2.4.p2.10.m10.4.4.4.2.3.cmml">,</mo><msub id="S6.SS2.4.p2.10.m10.4.4.4.2.2.2" xref="S6.SS2.4.p2.10.m10.4.4.4.2.2.2.cmml"><mi id="S6.SS2.4.p2.10.m10.4.4.4.2.2.2.2" xref="S6.SS2.4.p2.10.m10.4.4.4.2.2.2.2.cmml">b</mi><mi id="S6.SS2.4.p2.10.m10.4.4.4.2.2.2.3" xref="S6.SS2.4.p2.10.m10.4.4.4.2.2.2.3.cmml">r</mi></msub><mo id="S6.SS2.4.p2.10.m10.4.4.4.2.2.5" stretchy="false" xref="S6.SS2.4.p2.10.m10.4.4.4.2.3.cmml">)</mo></mrow></mrow><mo id="S6.SS2.4.p2.10.m10.4.4.8" xref="S6.SS2.4.p2.10.m10.4.4.8.cmml">=</mo><mi id="S6.SS2.4.p2.10.m10.4.4.9" xref="S6.SS2.4.p2.10.m10.4.4.9.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.4.p2.10.m10.4b"><apply id="S6.SS2.4.p2.10.m10.4.4.cmml" xref="S6.SS2.4.p2.10.m10.4.4"><and id="S6.SS2.4.p2.10.m10.4.4a.cmml" xref="S6.SS2.4.p2.10.m10.4.4"></and><apply id="S6.SS2.4.p2.10.m10.4.4b.cmml" xref="S6.SS2.4.p2.10.m10.4.4"><eq id="S6.SS2.4.p2.10.m10.4.4.6.cmml" xref="S6.SS2.4.p2.10.m10.4.4.6"></eq><apply id="S6.SS2.4.p2.10.m10.1.1.1.cmml" xref="S6.SS2.4.p2.10.m10.1.1.1"><times id="S6.SS2.4.p2.10.m10.1.1.1.2.cmml" xref="S6.SS2.4.p2.10.m10.1.1.1.2"></times><ci id="S6.SS2.4.p2.10.m10.1.1.1.3.cmml" xref="S6.SS2.4.p2.10.m10.1.1.1.3">𝜈</ci><apply id="S6.SS2.4.p2.10.m10.1.1.1.1.1.1.cmml" xref="S6.SS2.4.p2.10.m10.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.4.p2.10.m10.1.1.1.1.1.1.1.cmml" xref="S6.SS2.4.p2.10.m10.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.4.p2.10.m10.1.1.1.1.1.1.2.cmml" xref="S6.SS2.4.p2.10.m10.1.1.1.1.1.1.2">𝑏</ci><ci id="S6.SS2.4.p2.10.m10.1.1.1.1.1.1.3.cmml" xref="S6.SS2.4.p2.10.m10.1.1.1.1.1.1.3">𝑙</ci></apply></apply><apply id="S6.SS2.4.p2.10.m10.2.2.2.cmml" xref="S6.SS2.4.p2.10.m10.2.2.2"><times id="S6.SS2.4.p2.10.m10.2.2.2.2.cmml" xref="S6.SS2.4.p2.10.m10.2.2.2.2"></times><ci id="S6.SS2.4.p2.10.m10.2.2.2.3.cmml" xref="S6.SS2.4.p2.10.m10.2.2.2.3">𝜈</ci><apply id="S6.SS2.4.p2.10.m10.2.2.2.1.1.1.cmml" xref="S6.SS2.4.p2.10.m10.2.2.2.1.1"><csymbol cd="ambiguous" id="S6.SS2.4.p2.10.m10.2.2.2.1.1.1.1.cmml" xref="S6.SS2.4.p2.10.m10.2.2.2.1.1">subscript</csymbol><ci id="S6.SS2.4.p2.10.m10.2.2.2.1.1.1.2.cmml" xref="S6.SS2.4.p2.10.m10.2.2.2.1.1.1.2">𝑏</ci><ci id="S6.SS2.4.p2.10.m10.2.2.2.1.1.1.3.cmml" xref="S6.SS2.4.p2.10.m10.2.2.2.1.1.1.3">𝑚</ci></apply></apply></apply><apply id="S6.SS2.4.p2.10.m10.4.4c.cmml" xref="S6.SS2.4.p2.10.m10.4.4"><eq id="S6.SS2.4.p2.10.m10.4.4.7.cmml" xref="S6.SS2.4.p2.10.m10.4.4.7"></eq><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.4.p2.10.m10.2.2.2.cmml" id="S6.SS2.4.p2.10.m10.4.4d.cmml" xref="S6.SS2.4.p2.10.m10.4.4"></share><apply id="S6.SS2.4.p2.10.m10.4.4.4.cmml" xref="S6.SS2.4.p2.10.m10.4.4.4"><times id="S6.SS2.4.p2.10.m10.4.4.4.3.cmml" xref="S6.SS2.4.p2.10.m10.4.4.4.3"></times><ci id="S6.SS2.4.p2.10.m10.4.4.4.4.cmml" xref="S6.SS2.4.p2.10.m10.4.4.4.4">𝜈</ci><interval closure="open" id="S6.SS2.4.p2.10.m10.4.4.4.2.3.cmml" xref="S6.SS2.4.p2.10.m10.4.4.4.2.2"><apply id="S6.SS2.4.p2.10.m10.3.3.3.1.1.1.cmml" xref="S6.SS2.4.p2.10.m10.3.3.3.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.4.p2.10.m10.3.3.3.1.1.1.1.cmml" xref="S6.SS2.4.p2.10.m10.3.3.3.1.1.1">subscript</csymbol><ci id="S6.SS2.4.p2.10.m10.3.3.3.1.1.1.2.cmml" xref="S6.SS2.4.p2.10.m10.3.3.3.1.1.1.2">𝑏</ci><ci id="S6.SS2.4.p2.10.m10.3.3.3.1.1.1.3.cmml" xref="S6.SS2.4.p2.10.m10.3.3.3.1.1.1.3">𝑙</ci></apply><apply id="S6.SS2.4.p2.10.m10.4.4.4.2.2.2.cmml" xref="S6.SS2.4.p2.10.m10.4.4.4.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.4.p2.10.m10.4.4.4.2.2.2.1.cmml" xref="S6.SS2.4.p2.10.m10.4.4.4.2.2.2">subscript</csymbol><ci id="S6.SS2.4.p2.10.m10.4.4.4.2.2.2.2.cmml" xref="S6.SS2.4.p2.10.m10.4.4.4.2.2.2.2">𝑏</ci><ci id="S6.SS2.4.p2.10.m10.4.4.4.2.2.2.3.cmml" xref="S6.SS2.4.p2.10.m10.4.4.4.2.2.2.3">𝑟</ci></apply></interval></apply></apply><apply id="S6.SS2.4.p2.10.m10.4.4e.cmml" xref="S6.SS2.4.p2.10.m10.4.4"><eq id="S6.SS2.4.p2.10.m10.4.4.8.cmml" xref="S6.SS2.4.p2.10.m10.4.4.8"></eq><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.4.p2.10.m10.4.4.4.cmml" id="S6.SS2.4.p2.10.m10.4.4f.cmml" xref="S6.SS2.4.p2.10.m10.4.4"></share><ci id="S6.SS2.4.p2.10.m10.4.4.9.cmml" xref="S6.SS2.4.p2.10.m10.4.4.9">𝜈</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.4.p2.10.m10.4c">\nu(b_{l})=\nu(b_{m})=\nu(b_{l},b_{r})=\nu</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.4.p2.10.m10.4d">italic_ν ( italic_b start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ) = italic_ν ( italic_b start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) = italic_ν ( italic_b start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) = italic_ν</annotation></semantics></math>. Now using (a) we see that <math alttext="\nu<\mu=\nu(a_{r},b_{l})\leq\nu(b_{l})=\nu" class="ltx_Math" display="inline" id="S6.SS2.4.p2.11.m11.3"><semantics id="S6.SS2.4.p2.11.m11.3a"><mrow id="S6.SS2.4.p2.11.m11.3.3" xref="S6.SS2.4.p2.11.m11.3.3.cmml"><mi id="S6.SS2.4.p2.11.m11.3.3.5" xref="S6.SS2.4.p2.11.m11.3.3.5.cmml">ν</mi><mo id="S6.SS2.4.p2.11.m11.3.3.6" xref="S6.SS2.4.p2.11.m11.3.3.6.cmml"><</mo><mi id="S6.SS2.4.p2.11.m11.3.3.7" xref="S6.SS2.4.p2.11.m11.3.3.7.cmml">μ</mi><mo id="S6.SS2.4.p2.11.m11.3.3.8" xref="S6.SS2.4.p2.11.m11.3.3.8.cmml">=</mo><mrow id="S6.SS2.4.p2.11.m11.2.2.2" xref="S6.SS2.4.p2.11.m11.2.2.2.cmml"><mi id="S6.SS2.4.p2.11.m11.2.2.2.4" xref="S6.SS2.4.p2.11.m11.2.2.2.4.cmml">ν</mi><mo id="S6.SS2.4.p2.11.m11.2.2.2.3" xref="S6.SS2.4.p2.11.m11.2.2.2.3.cmml">⁢</mo><mrow id="S6.SS2.4.p2.11.m11.2.2.2.2.2" xref="S6.SS2.4.p2.11.m11.2.2.2.2.3.cmml"><mo id="S6.SS2.4.p2.11.m11.2.2.2.2.2.3" stretchy="false" xref="S6.SS2.4.p2.11.m11.2.2.2.2.3.cmml">(</mo><msub id="S6.SS2.4.p2.11.m11.1.1.1.1.1.1" xref="S6.SS2.4.p2.11.m11.1.1.1.1.1.1.cmml"><mi id="S6.SS2.4.p2.11.m11.1.1.1.1.1.1.2" xref="S6.SS2.4.p2.11.m11.1.1.1.1.1.1.2.cmml">a</mi><mi id="S6.SS2.4.p2.11.m11.1.1.1.1.1.1.3" xref="S6.SS2.4.p2.11.m11.1.1.1.1.1.1.3.cmml">r</mi></msub><mo id="S6.SS2.4.p2.11.m11.2.2.2.2.2.4" xref="S6.SS2.4.p2.11.m11.2.2.2.2.3.cmml">,</mo><msub id="S6.SS2.4.p2.11.m11.2.2.2.2.2.2" xref="S6.SS2.4.p2.11.m11.2.2.2.2.2.2.cmml"><mi id="S6.SS2.4.p2.11.m11.2.2.2.2.2.2.2" xref="S6.SS2.4.p2.11.m11.2.2.2.2.2.2.2.cmml">b</mi><mi id="S6.SS2.4.p2.11.m11.2.2.2.2.2.2.3" xref="S6.SS2.4.p2.11.m11.2.2.2.2.2.2.3.cmml">l</mi></msub><mo id="S6.SS2.4.p2.11.m11.2.2.2.2.2.5" stretchy="false" xref="S6.SS2.4.p2.11.m11.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.SS2.4.p2.11.m11.3.3.9" xref="S6.SS2.4.p2.11.m11.3.3.9.cmml">≤</mo><mrow id="S6.SS2.4.p2.11.m11.3.3.3" xref="S6.SS2.4.p2.11.m11.3.3.3.cmml"><mi id="S6.SS2.4.p2.11.m11.3.3.3.3" xref="S6.SS2.4.p2.11.m11.3.3.3.3.cmml">ν</mi><mo id="S6.SS2.4.p2.11.m11.3.3.3.2" xref="S6.SS2.4.p2.11.m11.3.3.3.2.cmml">⁢</mo><mrow id="S6.SS2.4.p2.11.m11.3.3.3.1.1" xref="S6.SS2.4.p2.11.m11.3.3.3.1.1.1.cmml"><mo id="S6.SS2.4.p2.11.m11.3.3.3.1.1.2" stretchy="false" xref="S6.SS2.4.p2.11.m11.3.3.3.1.1.1.cmml">(</mo><msub id="S6.SS2.4.p2.11.m11.3.3.3.1.1.1" xref="S6.SS2.4.p2.11.m11.3.3.3.1.1.1.cmml"><mi id="S6.SS2.4.p2.11.m11.3.3.3.1.1.1.2" xref="S6.SS2.4.p2.11.m11.3.3.3.1.1.1.2.cmml">b</mi><mi id="S6.SS2.4.p2.11.m11.3.3.3.1.1.1.3" xref="S6.SS2.4.p2.11.m11.3.3.3.1.1.1.3.cmml">l</mi></msub><mo id="S6.SS2.4.p2.11.m11.3.3.3.1.1.3" stretchy="false" xref="S6.SS2.4.p2.11.m11.3.3.3.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.SS2.4.p2.11.m11.3.3.10" xref="S6.SS2.4.p2.11.m11.3.3.10.cmml">=</mo><mi id="S6.SS2.4.p2.11.m11.3.3.11" xref="S6.SS2.4.p2.11.m11.3.3.11.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.4.p2.11.m11.3b"><apply id="S6.SS2.4.p2.11.m11.3.3.cmml" xref="S6.SS2.4.p2.11.m11.3.3"><and id="S6.SS2.4.p2.11.m11.3.3a.cmml" xref="S6.SS2.4.p2.11.m11.3.3"></and><apply id="S6.SS2.4.p2.11.m11.3.3b.cmml" xref="S6.SS2.4.p2.11.m11.3.3"><lt id="S6.SS2.4.p2.11.m11.3.3.6.cmml" xref="S6.SS2.4.p2.11.m11.3.3.6"></lt><ci id="S6.SS2.4.p2.11.m11.3.3.5.cmml" xref="S6.SS2.4.p2.11.m11.3.3.5">𝜈</ci><ci id="S6.SS2.4.p2.11.m11.3.3.7.cmml" xref="S6.SS2.4.p2.11.m11.3.3.7">𝜇</ci></apply><apply id="S6.SS2.4.p2.11.m11.3.3c.cmml" xref="S6.SS2.4.p2.11.m11.3.3"><eq id="S6.SS2.4.p2.11.m11.3.3.8.cmml" xref="S6.SS2.4.p2.11.m11.3.3.8"></eq><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.4.p2.11.m11.3.3.7.cmml" id="S6.SS2.4.p2.11.m11.3.3d.cmml" xref="S6.SS2.4.p2.11.m11.3.3"></share><apply id="S6.SS2.4.p2.11.m11.2.2.2.cmml" xref="S6.SS2.4.p2.11.m11.2.2.2"><times id="S6.SS2.4.p2.11.m11.2.2.2.3.cmml" xref="S6.SS2.4.p2.11.m11.2.2.2.3"></times><ci id="S6.SS2.4.p2.11.m11.2.2.2.4.cmml" xref="S6.SS2.4.p2.11.m11.2.2.2.4">𝜈</ci><interval closure="open" id="S6.SS2.4.p2.11.m11.2.2.2.2.3.cmml" xref="S6.SS2.4.p2.11.m11.2.2.2.2.2"><apply id="S6.SS2.4.p2.11.m11.1.1.1.1.1.1.cmml" xref="S6.SS2.4.p2.11.m11.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.4.p2.11.m11.1.1.1.1.1.1.1.cmml" xref="S6.SS2.4.p2.11.m11.1.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.4.p2.11.m11.1.1.1.1.1.1.2.cmml" xref="S6.SS2.4.p2.11.m11.1.1.1.1.1.1.2">𝑎</ci><ci id="S6.SS2.4.p2.11.m11.1.1.1.1.1.1.3.cmml" xref="S6.SS2.4.p2.11.m11.1.1.1.1.1.1.3">𝑟</ci></apply><apply id="S6.SS2.4.p2.11.m11.2.2.2.2.2.2.cmml" xref="S6.SS2.4.p2.11.m11.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.4.p2.11.m11.2.2.2.2.2.2.1.cmml" xref="S6.SS2.4.p2.11.m11.2.2.2.2.2.2">subscript</csymbol><ci id="S6.SS2.4.p2.11.m11.2.2.2.2.2.2.2.cmml" xref="S6.SS2.4.p2.11.m11.2.2.2.2.2.2.2">𝑏</ci><ci id="S6.SS2.4.p2.11.m11.2.2.2.2.2.2.3.cmml" xref="S6.SS2.4.p2.11.m11.2.2.2.2.2.2.3">𝑙</ci></apply></interval></apply></apply><apply id="S6.SS2.4.p2.11.m11.3.3e.cmml" xref="S6.SS2.4.p2.11.m11.3.3"><leq id="S6.SS2.4.p2.11.m11.3.3.9.cmml" xref="S6.SS2.4.p2.11.m11.3.3.9"></leq><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.4.p2.11.m11.2.2.2.cmml" id="S6.SS2.4.p2.11.m11.3.3f.cmml" xref="S6.SS2.4.p2.11.m11.3.3"></share><apply id="S6.SS2.4.p2.11.m11.3.3.3.cmml" xref="S6.SS2.4.p2.11.m11.3.3.3"><times id="S6.SS2.4.p2.11.m11.3.3.3.2.cmml" xref="S6.SS2.4.p2.11.m11.3.3.3.2"></times><ci id="S6.SS2.4.p2.11.m11.3.3.3.3.cmml" xref="S6.SS2.4.p2.11.m11.3.3.3.3">𝜈</ci><apply id="S6.SS2.4.p2.11.m11.3.3.3.1.1.1.cmml" xref="S6.SS2.4.p2.11.m11.3.3.3.1.1"><csymbol cd="ambiguous" id="S6.SS2.4.p2.11.m11.3.3.3.1.1.1.1.cmml" xref="S6.SS2.4.p2.11.m11.3.3.3.1.1">subscript</csymbol><ci id="S6.SS2.4.p2.11.m11.3.3.3.1.1.1.2.cmml" xref="S6.SS2.4.p2.11.m11.3.3.3.1.1.1.2">𝑏</ci><ci id="S6.SS2.4.p2.11.m11.3.3.3.1.1.1.3.cmml" xref="S6.SS2.4.p2.11.m11.3.3.3.1.1.1.3">𝑙</ci></apply></apply></apply><apply id="S6.SS2.4.p2.11.m11.3.3g.cmml" xref="S6.SS2.4.p2.11.m11.3.3"><eq id="S6.SS2.4.p2.11.m11.3.3.10.cmml" xref="S6.SS2.4.p2.11.m11.3.3.10"></eq><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.4.p2.11.m11.3.3.3.cmml" id="S6.SS2.4.p2.11.m11.3.3h.cmml" xref="S6.SS2.4.p2.11.m11.3.3"></share><ci id="S6.SS2.4.p2.11.m11.3.3.11.cmml" xref="S6.SS2.4.p2.11.m11.3.3.11">𝜈</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.4.p2.11.m11.3c">\nu<\mu=\nu(a_{r},b_{l})\leq\nu(b_{l})=\nu</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.4.p2.11.m11.3d">italic_ν < italic_μ = italic_ν ( italic_a start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ) ≤ italic_ν ( italic_b start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ) = italic_ν</annotation></semantics></math>, which is a contradiction.</p> </div> <div class="ltx_para" id="S6.SS2.5.p3"> <p class="ltx_p" id="S6.SS2.5.p3.1">(c). Follows directly from (b) and the definition of <math alttext="P_{E}" class="ltx_Math" display="inline" id="S6.SS2.5.p3.1.m1.1"><semantics id="S6.SS2.5.p3.1.m1.1a"><msub id="S6.SS2.5.p3.1.m1.1.1" xref="S6.SS2.5.p3.1.m1.1.1.cmml"><mi id="S6.SS2.5.p3.1.m1.1.1.2" xref="S6.SS2.5.p3.1.m1.1.1.2.cmml">P</mi><mi id="S6.SS2.5.p3.1.m1.1.1.3" xref="S6.SS2.5.p3.1.m1.1.1.3.cmml">E</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.5.p3.1.m1.1b"><apply id="S6.SS2.5.p3.1.m1.1.1.cmml" xref="S6.SS2.5.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S6.SS2.5.p3.1.m1.1.1.1.cmml" xref="S6.SS2.5.p3.1.m1.1.1">subscript</csymbol><ci id="S6.SS2.5.p3.1.m1.1.1.2.cmml" xref="S6.SS2.5.p3.1.m1.1.1.2">𝑃</ci><ci id="S6.SS2.5.p3.1.m1.1.1.3.cmml" xref="S6.SS2.5.p3.1.m1.1.1.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.5.p3.1.m1.1c">P_{E}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.5.p3.1.m1.1d">italic_P start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT</annotation></semantics></math>. ∎</p> </div> </div> <div class="ltx_para" id="S6.SS2.p7"> <p class="ltx_p" id="S6.SS2.p7.1">We now prove that the club from <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.Thmtheorem10" title="Lemma 6.10. ‣ 6.2. Forcing epimorphisms ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">6.10</span></a> makes <math alttext="P_{E}" class="ltx_Math" display="inline" id="S6.SS2.p7.1.m1.1"><semantics id="S6.SS2.p7.1.m1.1a"><msub id="S6.SS2.p7.1.m1.1.1" xref="S6.SS2.p7.1.m1.1.1.cmml"><mi id="S6.SS2.p7.1.m1.1.1.2" xref="S6.SS2.p7.1.m1.1.1.2.cmml">P</mi><mi id="S6.SS2.p7.1.m1.1.1.3" xref="S6.SS2.p7.1.m1.1.1.3.cmml">E</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.p7.1.m1.1b"><apply id="S6.SS2.p7.1.m1.1.1.cmml" xref="S6.SS2.p7.1.m1.1.1"><csymbol cd="ambiguous" id="S6.SS2.p7.1.m1.1.1.1.cmml" xref="S6.SS2.p7.1.m1.1.1">subscript</csymbol><ci id="S6.SS2.p7.1.m1.1.1.2.cmml" xref="S6.SS2.p7.1.m1.1.1.2">𝑃</ci><ci id="S6.SS2.p7.1.m1.1.1.3.cmml" xref="S6.SS2.p7.1.m1.1.1.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p7.1.m1.1c">P_{E}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p7.1.m1.1d">italic_P start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT</annotation></semantics></math> ccc. For this we will only use (1) and (3) of <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.Thmtheorem10" title="Lemma 6.10. ‣ 6.2. Forcing epimorphisms ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">6.10</span></a>.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S6.Thmtheorem12"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem12.1.1.1">Theorem 6.12</span></span><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem12.2.2">.</span> </h6> <div class="ltx_para" id="S6.Thmtheorem12.p1"> <p class="ltx_p" id="S6.Thmtheorem12.p1.2"><span class="ltx_text ltx_font_italic" id="S6.Thmtheorem12.p1.2.2">If <math alttext="E" class="ltx_Math" display="inline" id="S6.Thmtheorem12.p1.1.1.m1.1"><semantics id="S6.Thmtheorem12.p1.1.1.m1.1a"><mi id="S6.Thmtheorem12.p1.1.1.m1.1.1" xref="S6.Thmtheorem12.p1.1.1.m1.1.1.cmml">E</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem12.p1.1.1.m1.1b"><ci id="S6.Thmtheorem12.p1.1.1.m1.1.1.cmml" xref="S6.Thmtheorem12.p1.1.1.m1.1.1">𝐸</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem12.p1.1.1.m1.1c">E</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem12.p1.1.1.m1.1d">italic_E</annotation></semantics></math> is a club satisfying (1) and (3) from <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.Thmtheorem10" title="Lemma 6.10. ‣ 6.2. Forcing epimorphisms ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">6.10</span></a>, then <math alttext="P_{E}" class="ltx_Math" display="inline" id="S6.Thmtheorem12.p1.2.2.m2.1"><semantics id="S6.Thmtheorem12.p1.2.2.m2.1a"><msub id="S6.Thmtheorem12.p1.2.2.m2.1.1" xref="S6.Thmtheorem12.p1.2.2.m2.1.1.cmml"><mi id="S6.Thmtheorem12.p1.2.2.m2.1.1.2" xref="S6.Thmtheorem12.p1.2.2.m2.1.1.2.cmml">P</mi><mi id="S6.Thmtheorem12.p1.2.2.m2.1.1.3" xref="S6.Thmtheorem12.p1.2.2.m2.1.1.3.cmml">E</mi></msub><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem12.p1.2.2.m2.1b"><apply id="S6.Thmtheorem12.p1.2.2.m2.1.1.cmml" xref="S6.Thmtheorem12.p1.2.2.m2.1.1"><csymbol cd="ambiguous" id="S6.Thmtheorem12.p1.2.2.m2.1.1.1.cmml" xref="S6.Thmtheorem12.p1.2.2.m2.1.1">subscript</csymbol><ci id="S6.Thmtheorem12.p1.2.2.m2.1.1.2.cmml" xref="S6.Thmtheorem12.p1.2.2.m2.1.1.2">𝑃</ci><ci id="S6.Thmtheorem12.p1.2.2.m2.1.1.3.cmml" xref="S6.Thmtheorem12.p1.2.2.m2.1.1.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem12.p1.2.2.m2.1c">P_{E}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem12.p1.2.2.m2.1d">italic_P start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT</annotation></semantics></math> is ccc.</span></p> </div> </div> <div class="ltx_proof" id="S6.SS2.14"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S6.SS2.6.p1"> <p class="ltx_p" id="S6.SS2.6.p1.18">Suppose towards a contradiction that there is an uncountable antichain <math alttext="H\subseteq P_{E}" class="ltx_Math" display="inline" id="S6.SS2.6.p1.1.m1.1"><semantics id="S6.SS2.6.p1.1.m1.1a"><mrow id="S6.SS2.6.p1.1.m1.1.1" xref="S6.SS2.6.p1.1.m1.1.1.cmml"><mi id="S6.SS2.6.p1.1.m1.1.1.2" xref="S6.SS2.6.p1.1.m1.1.1.2.cmml">H</mi><mo id="S6.SS2.6.p1.1.m1.1.1.1" xref="S6.SS2.6.p1.1.m1.1.1.1.cmml">⊆</mo><msub id="S6.SS2.6.p1.1.m1.1.1.3" xref="S6.SS2.6.p1.1.m1.1.1.3.cmml"><mi id="S6.SS2.6.p1.1.m1.1.1.3.2" xref="S6.SS2.6.p1.1.m1.1.1.3.2.cmml">P</mi><mi id="S6.SS2.6.p1.1.m1.1.1.3.3" xref="S6.SS2.6.p1.1.m1.1.1.3.3.cmml">E</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.6.p1.1.m1.1b"><apply id="S6.SS2.6.p1.1.m1.1.1.cmml" xref="S6.SS2.6.p1.1.m1.1.1"><subset id="S6.SS2.6.p1.1.m1.1.1.1.cmml" xref="S6.SS2.6.p1.1.m1.1.1.1"></subset><ci id="S6.SS2.6.p1.1.m1.1.1.2.cmml" xref="S6.SS2.6.p1.1.m1.1.1.2">𝐻</ci><apply id="S6.SS2.6.p1.1.m1.1.1.3.cmml" xref="S6.SS2.6.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S6.SS2.6.p1.1.m1.1.1.3.1.cmml" xref="S6.SS2.6.p1.1.m1.1.1.3">subscript</csymbol><ci id="S6.SS2.6.p1.1.m1.1.1.3.2.cmml" xref="S6.SS2.6.p1.1.m1.1.1.3.2">𝑃</ci><ci id="S6.SS2.6.p1.1.m1.1.1.3.3.cmml" xref="S6.SS2.6.p1.1.m1.1.1.3.3">𝐸</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.6.p1.1.m1.1c">H\subseteq P_{E}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.6.p1.1.m1.1d">italic_H ⊆ italic_P start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT</annotation></semantics></math>. By going to an uncountable subset of <math alttext="H" class="ltx_Math" display="inline" id="S6.SS2.6.p1.2.m2.1"><semantics id="S6.SS2.6.p1.2.m2.1a"><mi id="S6.SS2.6.p1.2.m2.1.1" xref="S6.SS2.6.p1.2.m2.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.6.p1.2.m2.1b"><ci id="S6.SS2.6.p1.2.m2.1.1.cmml" xref="S6.SS2.6.p1.2.m2.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.6.p1.2.m2.1c">H</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.6.p1.2.m2.1d">italic_H</annotation></semantics></math> we may assume that for all <math alttext="p\in H" class="ltx_Math" display="inline" id="S6.SS2.6.p1.3.m3.1"><semantics id="S6.SS2.6.p1.3.m3.1a"><mrow id="S6.SS2.6.p1.3.m3.1.1" xref="S6.SS2.6.p1.3.m3.1.1.cmml"><mi id="S6.SS2.6.p1.3.m3.1.1.2" xref="S6.SS2.6.p1.3.m3.1.1.2.cmml">p</mi><mo id="S6.SS2.6.p1.3.m3.1.1.1" xref="S6.SS2.6.p1.3.m3.1.1.1.cmml">∈</mo><mi id="S6.SS2.6.p1.3.m3.1.1.3" xref="S6.SS2.6.p1.3.m3.1.1.3.cmml">H</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.6.p1.3.m3.1b"><apply id="S6.SS2.6.p1.3.m3.1.1.cmml" xref="S6.SS2.6.p1.3.m3.1.1"><in id="S6.SS2.6.p1.3.m3.1.1.1.cmml" xref="S6.SS2.6.p1.3.m3.1.1.1"></in><ci id="S6.SS2.6.p1.3.m3.1.1.2.cmml" xref="S6.SS2.6.p1.3.m3.1.1.2">𝑝</ci><ci id="S6.SS2.6.p1.3.m3.1.1.3.cmml" xref="S6.SS2.6.p1.3.m3.1.1.3">𝐻</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.6.p1.3.m3.1c">p\in H</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.6.p1.3.m3.1d">italic_p ∈ italic_H</annotation></semantics></math>, <math alttext="|\operatorname{dom}(p)|=n" class="ltx_Math" display="inline" id="S6.SS2.6.p1.4.m4.3"><semantics id="S6.SS2.6.p1.4.m4.3a"><mrow id="S6.SS2.6.p1.4.m4.3.3" xref="S6.SS2.6.p1.4.m4.3.3.cmml"><mrow id="S6.SS2.6.p1.4.m4.3.3.1.1" xref="S6.SS2.6.p1.4.m4.3.3.1.2.cmml"><mo id="S6.SS2.6.p1.4.m4.3.3.1.1.2" stretchy="false" xref="S6.SS2.6.p1.4.m4.3.3.1.2.1.cmml">|</mo><mrow id="S6.SS2.6.p1.4.m4.3.3.1.1.1.2" xref="S6.SS2.6.p1.4.m4.3.3.1.1.1.1.cmml"><mi id="S6.SS2.6.p1.4.m4.1.1" xref="S6.SS2.6.p1.4.m4.1.1.cmml">dom</mi><mo id="S6.SS2.6.p1.4.m4.3.3.1.1.1.2a" xref="S6.SS2.6.p1.4.m4.3.3.1.1.1.1.cmml">⁡</mo><mrow id="S6.SS2.6.p1.4.m4.3.3.1.1.1.2.1" xref="S6.SS2.6.p1.4.m4.3.3.1.1.1.1.cmml"><mo id="S6.SS2.6.p1.4.m4.3.3.1.1.1.2.1.1" stretchy="false" xref="S6.SS2.6.p1.4.m4.3.3.1.1.1.1.cmml">(</mo><mi id="S6.SS2.6.p1.4.m4.2.2" xref="S6.SS2.6.p1.4.m4.2.2.cmml">p</mi><mo id="S6.SS2.6.p1.4.m4.3.3.1.1.1.2.1.2" stretchy="false" xref="S6.SS2.6.p1.4.m4.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.SS2.6.p1.4.m4.3.3.1.1.3" stretchy="false" xref="S6.SS2.6.p1.4.m4.3.3.1.2.1.cmml">|</mo></mrow><mo id="S6.SS2.6.p1.4.m4.3.3.2" xref="S6.SS2.6.p1.4.m4.3.3.2.cmml">=</mo><mi id="S6.SS2.6.p1.4.m4.3.3.3" xref="S6.SS2.6.p1.4.m4.3.3.3.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.6.p1.4.m4.3b"><apply id="S6.SS2.6.p1.4.m4.3.3.cmml" xref="S6.SS2.6.p1.4.m4.3.3"><eq id="S6.SS2.6.p1.4.m4.3.3.2.cmml" xref="S6.SS2.6.p1.4.m4.3.3.2"></eq><apply id="S6.SS2.6.p1.4.m4.3.3.1.2.cmml" xref="S6.SS2.6.p1.4.m4.3.3.1.1"><abs id="S6.SS2.6.p1.4.m4.3.3.1.2.1.cmml" xref="S6.SS2.6.p1.4.m4.3.3.1.1.2"></abs><apply id="S6.SS2.6.p1.4.m4.3.3.1.1.1.1.cmml" xref="S6.SS2.6.p1.4.m4.3.3.1.1.1.2"><ci id="S6.SS2.6.p1.4.m4.1.1.cmml" xref="S6.SS2.6.p1.4.m4.1.1">dom</ci><ci id="S6.SS2.6.p1.4.m4.2.2.cmml" xref="S6.SS2.6.p1.4.m4.2.2">𝑝</ci></apply></apply><ci id="S6.SS2.6.p1.4.m4.3.3.3.cmml" xref="S6.SS2.6.p1.4.m4.3.3.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.6.p1.4.m4.3c">|\operatorname{dom}(p)|=n</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.6.p1.4.m4.3d">| roman_dom ( italic_p ) | = italic_n</annotation></semantics></math>. Moreover we may assume that <math alttext="H" class="ltx_Math" display="inline" id="S6.SS2.6.p1.5.m5.1"><semantics id="S6.SS2.6.p1.5.m5.1a"><mi id="S6.SS2.6.p1.5.m5.1.1" xref="S6.SS2.6.p1.5.m5.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.6.p1.5.m5.1b"><ci id="S6.SS2.6.p1.5.m5.1.1.cmml" xref="S6.SS2.6.p1.5.m5.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.6.p1.5.m5.1c">H</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.6.p1.5.m5.1d">italic_H</annotation></semantics></math> is such that <math alttext="n" class="ltx_Math" display="inline" id="S6.SS2.6.p1.6.m6.1"><semantics id="S6.SS2.6.p1.6.m6.1a"><mi id="S6.SS2.6.p1.6.m6.1.1" xref="S6.SS2.6.p1.6.m6.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.6.p1.6.m6.1b"><ci id="S6.SS2.6.p1.6.m6.1.1.cmml" xref="S6.SS2.6.p1.6.m6.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.6.p1.6.m6.1c">n</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.6.p1.6.m6.1d">italic_n</annotation></semantics></math> is minimal, in the sense that for every other antichain <math alttext="H^{\prime}" class="ltx_Math" display="inline" id="S6.SS2.6.p1.7.m7.1"><semantics id="S6.SS2.6.p1.7.m7.1a"><msup id="S6.SS2.6.p1.7.m7.1.1" xref="S6.SS2.6.p1.7.m7.1.1.cmml"><mi id="S6.SS2.6.p1.7.m7.1.1.2" xref="S6.SS2.6.p1.7.m7.1.1.2.cmml">H</mi><mo id="S6.SS2.6.p1.7.m7.1.1.3" xref="S6.SS2.6.p1.7.m7.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S6.SS2.6.p1.7.m7.1b"><apply id="S6.SS2.6.p1.7.m7.1.1.cmml" xref="S6.SS2.6.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S6.SS2.6.p1.7.m7.1.1.1.cmml" xref="S6.SS2.6.p1.7.m7.1.1">superscript</csymbol><ci id="S6.SS2.6.p1.7.m7.1.1.2.cmml" xref="S6.SS2.6.p1.7.m7.1.1.2">𝐻</ci><ci id="S6.SS2.6.p1.7.m7.1.1.3.cmml" xref="S6.SS2.6.p1.7.m7.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.6.p1.7.m7.1c">H^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.6.p1.7.m7.1d">italic_H start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, the set of <math alttext="p\in H^{\prime}" class="ltx_Math" display="inline" id="S6.SS2.6.p1.8.m8.1"><semantics id="S6.SS2.6.p1.8.m8.1a"><mrow id="S6.SS2.6.p1.8.m8.1.1" xref="S6.SS2.6.p1.8.m8.1.1.cmml"><mi id="S6.SS2.6.p1.8.m8.1.1.2" xref="S6.SS2.6.p1.8.m8.1.1.2.cmml">p</mi><mo id="S6.SS2.6.p1.8.m8.1.1.1" xref="S6.SS2.6.p1.8.m8.1.1.1.cmml">∈</mo><msup id="S6.SS2.6.p1.8.m8.1.1.3" xref="S6.SS2.6.p1.8.m8.1.1.3.cmml"><mi id="S6.SS2.6.p1.8.m8.1.1.3.2" xref="S6.SS2.6.p1.8.m8.1.1.3.2.cmml">H</mi><mo id="S6.SS2.6.p1.8.m8.1.1.3.3" xref="S6.SS2.6.p1.8.m8.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.6.p1.8.m8.1b"><apply id="S6.SS2.6.p1.8.m8.1.1.cmml" xref="S6.SS2.6.p1.8.m8.1.1"><in id="S6.SS2.6.p1.8.m8.1.1.1.cmml" xref="S6.SS2.6.p1.8.m8.1.1.1"></in><ci id="S6.SS2.6.p1.8.m8.1.1.2.cmml" xref="S6.SS2.6.p1.8.m8.1.1.2">𝑝</ci><apply id="S6.SS2.6.p1.8.m8.1.1.3.cmml" xref="S6.SS2.6.p1.8.m8.1.1.3"><csymbol cd="ambiguous" id="S6.SS2.6.p1.8.m8.1.1.3.1.cmml" xref="S6.SS2.6.p1.8.m8.1.1.3">superscript</csymbol><ci id="S6.SS2.6.p1.8.m8.1.1.3.2.cmml" xref="S6.SS2.6.p1.8.m8.1.1.3.2">𝐻</ci><ci id="S6.SS2.6.p1.8.m8.1.1.3.3.cmml" xref="S6.SS2.6.p1.8.m8.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.6.p1.8.m8.1c">p\in H^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.6.p1.8.m8.1d">italic_p ∈ italic_H start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> with <math alttext="|\operatorname{dom}(p)|<n" class="ltx_Math" display="inline" id="S6.SS2.6.p1.9.m9.3"><semantics id="S6.SS2.6.p1.9.m9.3a"><mrow id="S6.SS2.6.p1.9.m9.3.3" xref="S6.SS2.6.p1.9.m9.3.3.cmml"><mrow id="S6.SS2.6.p1.9.m9.3.3.1.1" xref="S6.SS2.6.p1.9.m9.3.3.1.2.cmml"><mo id="S6.SS2.6.p1.9.m9.3.3.1.1.2" stretchy="false" xref="S6.SS2.6.p1.9.m9.3.3.1.2.1.cmml">|</mo><mrow id="S6.SS2.6.p1.9.m9.3.3.1.1.1.2" xref="S6.SS2.6.p1.9.m9.3.3.1.1.1.1.cmml"><mi id="S6.SS2.6.p1.9.m9.1.1" xref="S6.SS2.6.p1.9.m9.1.1.cmml">dom</mi><mo id="S6.SS2.6.p1.9.m9.3.3.1.1.1.2a" xref="S6.SS2.6.p1.9.m9.3.3.1.1.1.1.cmml">⁡</mo><mrow id="S6.SS2.6.p1.9.m9.3.3.1.1.1.2.1" xref="S6.SS2.6.p1.9.m9.3.3.1.1.1.1.cmml"><mo id="S6.SS2.6.p1.9.m9.3.3.1.1.1.2.1.1" stretchy="false" xref="S6.SS2.6.p1.9.m9.3.3.1.1.1.1.cmml">(</mo><mi id="S6.SS2.6.p1.9.m9.2.2" xref="S6.SS2.6.p1.9.m9.2.2.cmml">p</mi><mo id="S6.SS2.6.p1.9.m9.3.3.1.1.1.2.1.2" stretchy="false" xref="S6.SS2.6.p1.9.m9.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.SS2.6.p1.9.m9.3.3.1.1.3" stretchy="false" xref="S6.SS2.6.p1.9.m9.3.3.1.2.1.cmml">|</mo></mrow><mo id="S6.SS2.6.p1.9.m9.3.3.2" xref="S6.SS2.6.p1.9.m9.3.3.2.cmml"><</mo><mi id="S6.SS2.6.p1.9.m9.3.3.3" xref="S6.SS2.6.p1.9.m9.3.3.3.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.6.p1.9.m9.3b"><apply id="S6.SS2.6.p1.9.m9.3.3.cmml" xref="S6.SS2.6.p1.9.m9.3.3"><lt id="S6.SS2.6.p1.9.m9.3.3.2.cmml" xref="S6.SS2.6.p1.9.m9.3.3.2"></lt><apply id="S6.SS2.6.p1.9.m9.3.3.1.2.cmml" xref="S6.SS2.6.p1.9.m9.3.3.1.1"><abs id="S6.SS2.6.p1.9.m9.3.3.1.2.1.cmml" xref="S6.SS2.6.p1.9.m9.3.3.1.1.2"></abs><apply id="S6.SS2.6.p1.9.m9.3.3.1.1.1.1.cmml" xref="S6.SS2.6.p1.9.m9.3.3.1.1.1.2"><ci id="S6.SS2.6.p1.9.m9.1.1.cmml" xref="S6.SS2.6.p1.9.m9.1.1">dom</ci><ci id="S6.SS2.6.p1.9.m9.2.2.cmml" xref="S6.SS2.6.p1.9.m9.2.2">𝑝</ci></apply></apply><ci id="S6.SS2.6.p1.9.m9.3.3.3.cmml" xref="S6.SS2.6.p1.9.m9.3.3.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.6.p1.9.m9.3c">|\operatorname{dom}(p)|<n</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.6.p1.9.m9.3d">| roman_dom ( italic_p ) | < italic_n</annotation></semantics></math> is countable. From this, a typical <math alttext="\Delta" class="ltx_Math" display="inline" id="S6.SS2.6.p1.10.m10.1"><semantics id="S6.SS2.6.p1.10.m10.1a"><mi id="S6.SS2.6.p1.10.m10.1.1" mathvariant="normal" xref="S6.SS2.6.p1.10.m10.1.1.cmml">Δ</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.6.p1.10.m10.1b"><ci id="S6.SS2.6.p1.10.m10.1.1.cmml" xref="S6.SS2.6.p1.10.m10.1.1">Δ</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.6.p1.10.m10.1c">\Delta</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.6.p1.10.m10.1d">roman_Δ</annotation></semantics></math>-system argument shows that the conditions of <math alttext="H" class="ltx_Math" display="inline" id="S6.SS2.6.p1.11.m11.1"><semantics id="S6.SS2.6.p1.11.m11.1a"><mi id="S6.SS2.6.p1.11.m11.1.1" xref="S6.SS2.6.p1.11.m11.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.6.p1.11.m11.1b"><ci id="S6.SS2.6.p1.11.m11.1.1.cmml" xref="S6.SS2.6.p1.11.m11.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.6.p1.11.m11.1c">H</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.6.p1.11.m11.1d">italic_H</annotation></semantics></math> must have pairwise disjoint domains. We will refine <math alttext="H" class="ltx_Math" display="inline" id="S6.SS2.6.p1.12.m12.1"><semantics id="S6.SS2.6.p1.12.m12.1a"><mi id="S6.SS2.6.p1.12.m12.1.1" xref="S6.SS2.6.p1.12.m12.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.6.p1.12.m12.1b"><ci id="S6.SS2.6.p1.12.m12.1.1.cmml" xref="S6.SS2.6.p1.12.m12.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.6.p1.12.m12.1c">H</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.6.p1.12.m12.1d">italic_H</annotation></semantics></math> as to satisfy that for all <math alttext="p\neq p^{\prime}\in H" class="ltx_Math" display="inline" id="S6.SS2.6.p1.13.m13.1"><semantics id="S6.SS2.6.p1.13.m13.1a"><mrow id="S6.SS2.6.p1.13.m13.1.1" xref="S6.SS2.6.p1.13.m13.1.1.cmml"><mi id="S6.SS2.6.p1.13.m13.1.1.2" xref="S6.SS2.6.p1.13.m13.1.1.2.cmml">p</mi><mo id="S6.SS2.6.p1.13.m13.1.1.3" xref="S6.SS2.6.p1.13.m13.1.1.3.cmml">≠</mo><msup id="S6.SS2.6.p1.13.m13.1.1.4" xref="S6.SS2.6.p1.13.m13.1.1.4.cmml"><mi id="S6.SS2.6.p1.13.m13.1.1.4.2" xref="S6.SS2.6.p1.13.m13.1.1.4.2.cmml">p</mi><mo id="S6.SS2.6.p1.13.m13.1.1.4.3" xref="S6.SS2.6.p1.13.m13.1.1.4.3.cmml">′</mo></msup><mo id="S6.SS2.6.p1.13.m13.1.1.5" xref="S6.SS2.6.p1.13.m13.1.1.5.cmml">∈</mo><mi id="S6.SS2.6.p1.13.m13.1.1.6" xref="S6.SS2.6.p1.13.m13.1.1.6.cmml">H</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.6.p1.13.m13.1b"><apply id="S6.SS2.6.p1.13.m13.1.1.cmml" xref="S6.SS2.6.p1.13.m13.1.1"><and id="S6.SS2.6.p1.13.m13.1.1a.cmml" xref="S6.SS2.6.p1.13.m13.1.1"></and><apply id="S6.SS2.6.p1.13.m13.1.1b.cmml" xref="S6.SS2.6.p1.13.m13.1.1"><neq id="S6.SS2.6.p1.13.m13.1.1.3.cmml" xref="S6.SS2.6.p1.13.m13.1.1.3"></neq><ci id="S6.SS2.6.p1.13.m13.1.1.2.cmml" xref="S6.SS2.6.p1.13.m13.1.1.2">𝑝</ci><apply id="S6.SS2.6.p1.13.m13.1.1.4.cmml" xref="S6.SS2.6.p1.13.m13.1.1.4"><csymbol cd="ambiguous" id="S6.SS2.6.p1.13.m13.1.1.4.1.cmml" xref="S6.SS2.6.p1.13.m13.1.1.4">superscript</csymbol><ci id="S6.SS2.6.p1.13.m13.1.1.4.2.cmml" xref="S6.SS2.6.p1.13.m13.1.1.4.2">𝑝</ci><ci id="S6.SS2.6.p1.13.m13.1.1.4.3.cmml" xref="S6.SS2.6.p1.13.m13.1.1.4.3">′</ci></apply></apply><apply id="S6.SS2.6.p1.13.m13.1.1c.cmml" xref="S6.SS2.6.p1.13.m13.1.1"><in id="S6.SS2.6.p1.13.m13.1.1.5.cmml" xref="S6.SS2.6.p1.13.m13.1.1.5"></in><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.6.p1.13.m13.1.1.4.cmml" id="S6.SS2.6.p1.13.m13.1.1d.cmml" xref="S6.SS2.6.p1.13.m13.1.1"></share><ci id="S6.SS2.6.p1.13.m13.1.1.6.cmml" xref="S6.SS2.6.p1.13.m13.1.1.6">𝐻</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.6.p1.13.m13.1c">p\neq p^{\prime}\in H</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.6.p1.13.m13.1d">italic_p ≠ italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_H</annotation></semantics></math>, if <math alttext="\bar{a}\in\operatorname{dom}(p)" class="ltx_Math" display="inline" id="S6.SS2.6.p1.14.m14.2"><semantics id="S6.SS2.6.p1.14.m14.2a"><mrow id="S6.SS2.6.p1.14.m14.2.3" xref="S6.SS2.6.p1.14.m14.2.3.cmml"><mover accent="true" id="S6.SS2.6.p1.14.m14.2.3.2" xref="S6.SS2.6.p1.14.m14.2.3.2.cmml"><mi id="S6.SS2.6.p1.14.m14.2.3.2.2" xref="S6.SS2.6.p1.14.m14.2.3.2.2.cmml">a</mi><mo id="S6.SS2.6.p1.14.m14.2.3.2.1" xref="S6.SS2.6.p1.14.m14.2.3.2.1.cmml">¯</mo></mover><mo id="S6.SS2.6.p1.14.m14.2.3.1" xref="S6.SS2.6.p1.14.m14.2.3.1.cmml">∈</mo><mrow id="S6.SS2.6.p1.14.m14.2.3.3.2" xref="S6.SS2.6.p1.14.m14.2.3.3.1.cmml"><mi id="S6.SS2.6.p1.14.m14.1.1" xref="S6.SS2.6.p1.14.m14.1.1.cmml">dom</mi><mo id="S6.SS2.6.p1.14.m14.2.3.3.2a" xref="S6.SS2.6.p1.14.m14.2.3.3.1.cmml">⁡</mo><mrow id="S6.SS2.6.p1.14.m14.2.3.3.2.1" xref="S6.SS2.6.p1.14.m14.2.3.3.1.cmml"><mo id="S6.SS2.6.p1.14.m14.2.3.3.2.1.1" stretchy="false" xref="S6.SS2.6.p1.14.m14.2.3.3.1.cmml">(</mo><mi id="S6.SS2.6.p1.14.m14.2.2" xref="S6.SS2.6.p1.14.m14.2.2.cmml">p</mi><mo id="S6.SS2.6.p1.14.m14.2.3.3.2.1.2" stretchy="false" xref="S6.SS2.6.p1.14.m14.2.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.6.p1.14.m14.2b"><apply id="S6.SS2.6.p1.14.m14.2.3.cmml" xref="S6.SS2.6.p1.14.m14.2.3"><in id="S6.SS2.6.p1.14.m14.2.3.1.cmml" xref="S6.SS2.6.p1.14.m14.2.3.1"></in><apply id="S6.SS2.6.p1.14.m14.2.3.2.cmml" xref="S6.SS2.6.p1.14.m14.2.3.2"><ci id="S6.SS2.6.p1.14.m14.2.3.2.1.cmml" xref="S6.SS2.6.p1.14.m14.2.3.2.1">¯</ci><ci id="S6.SS2.6.p1.14.m14.2.3.2.2.cmml" xref="S6.SS2.6.p1.14.m14.2.3.2.2">𝑎</ci></apply><apply id="S6.SS2.6.p1.14.m14.2.3.3.1.cmml" xref="S6.SS2.6.p1.14.m14.2.3.3.2"><ci id="S6.SS2.6.p1.14.m14.1.1.cmml" xref="S6.SS2.6.p1.14.m14.1.1">dom</ci><ci id="S6.SS2.6.p1.14.m14.2.2.cmml" xref="S6.SS2.6.p1.14.m14.2.2">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.6.p1.14.m14.2c">\bar{a}\in\operatorname{dom}(p)</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.6.p1.14.m14.2d">over¯ start_ARG italic_a end_ARG ∈ roman_dom ( italic_p )</annotation></semantics></math> and <math alttext="\bar{b}\in\operatorname{dom}(p^{\prime})" class="ltx_Math" display="inline" id="S6.SS2.6.p1.15.m15.2"><semantics id="S6.SS2.6.p1.15.m15.2a"><mrow id="S6.SS2.6.p1.15.m15.2.2" xref="S6.SS2.6.p1.15.m15.2.2.cmml"><mover accent="true" id="S6.SS2.6.p1.15.m15.2.2.3" xref="S6.SS2.6.p1.15.m15.2.2.3.cmml"><mi id="S6.SS2.6.p1.15.m15.2.2.3.2" xref="S6.SS2.6.p1.15.m15.2.2.3.2.cmml">b</mi><mo id="S6.SS2.6.p1.15.m15.2.2.3.1" xref="S6.SS2.6.p1.15.m15.2.2.3.1.cmml">¯</mo></mover><mo id="S6.SS2.6.p1.15.m15.2.2.2" xref="S6.SS2.6.p1.15.m15.2.2.2.cmml">∈</mo><mrow id="S6.SS2.6.p1.15.m15.2.2.1.1" xref="S6.SS2.6.p1.15.m15.2.2.1.2.cmml"><mi id="S6.SS2.6.p1.15.m15.1.1" xref="S6.SS2.6.p1.15.m15.1.1.cmml">dom</mi><mo id="S6.SS2.6.p1.15.m15.2.2.1.1a" xref="S6.SS2.6.p1.15.m15.2.2.1.2.cmml">⁡</mo><mrow id="S6.SS2.6.p1.15.m15.2.2.1.1.1" xref="S6.SS2.6.p1.15.m15.2.2.1.2.cmml"><mo id="S6.SS2.6.p1.15.m15.2.2.1.1.1.2" stretchy="false" xref="S6.SS2.6.p1.15.m15.2.2.1.2.cmml">(</mo><msup id="S6.SS2.6.p1.15.m15.2.2.1.1.1.1" xref="S6.SS2.6.p1.15.m15.2.2.1.1.1.1.cmml"><mi id="S6.SS2.6.p1.15.m15.2.2.1.1.1.1.2" xref="S6.SS2.6.p1.15.m15.2.2.1.1.1.1.2.cmml">p</mi><mo id="S6.SS2.6.p1.15.m15.2.2.1.1.1.1.3" xref="S6.SS2.6.p1.15.m15.2.2.1.1.1.1.3.cmml">′</mo></msup><mo id="S6.SS2.6.p1.15.m15.2.2.1.1.1.3" stretchy="false" xref="S6.SS2.6.p1.15.m15.2.2.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.6.p1.15.m15.2b"><apply id="S6.SS2.6.p1.15.m15.2.2.cmml" xref="S6.SS2.6.p1.15.m15.2.2"><in id="S6.SS2.6.p1.15.m15.2.2.2.cmml" xref="S6.SS2.6.p1.15.m15.2.2.2"></in><apply id="S6.SS2.6.p1.15.m15.2.2.3.cmml" xref="S6.SS2.6.p1.15.m15.2.2.3"><ci id="S6.SS2.6.p1.15.m15.2.2.3.1.cmml" xref="S6.SS2.6.p1.15.m15.2.2.3.1">¯</ci><ci id="S6.SS2.6.p1.15.m15.2.2.3.2.cmml" xref="S6.SS2.6.p1.15.m15.2.2.3.2">𝑏</ci></apply><apply id="S6.SS2.6.p1.15.m15.2.2.1.2.cmml" xref="S6.SS2.6.p1.15.m15.2.2.1.1"><ci id="S6.SS2.6.p1.15.m15.1.1.cmml" xref="S6.SS2.6.p1.15.m15.1.1">dom</ci><apply id="S6.SS2.6.p1.15.m15.2.2.1.1.1.1.cmml" xref="S6.SS2.6.p1.15.m15.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.6.p1.15.m15.2.2.1.1.1.1.1.cmml" xref="S6.SS2.6.p1.15.m15.2.2.1.1.1.1">superscript</csymbol><ci id="S6.SS2.6.p1.15.m15.2.2.1.1.1.1.2.cmml" xref="S6.SS2.6.p1.15.m15.2.2.1.1.1.1.2">𝑝</ci><ci id="S6.SS2.6.p1.15.m15.2.2.1.1.1.1.3.cmml" xref="S6.SS2.6.p1.15.m15.2.2.1.1.1.1.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.6.p1.15.m15.2c">\bar{b}\in\operatorname{dom}(p^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.6.p1.15.m15.2d">over¯ start_ARG italic_b end_ARG ∈ roman_dom ( italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math>, then <math alttext="[a_{l},a_{r}]\cap[b_{l},b_{r}]=\varnothing" class="ltx_Math" display="inline" id="S6.SS2.6.p1.16.m16.4"><semantics id="S6.SS2.6.p1.16.m16.4a"><mrow id="S6.SS2.6.p1.16.m16.4.4" xref="S6.SS2.6.p1.16.m16.4.4.cmml"><mrow id="S6.SS2.6.p1.16.m16.4.4.4" xref="S6.SS2.6.p1.16.m16.4.4.4.cmml"><mrow id="S6.SS2.6.p1.16.m16.2.2.2.2.2" xref="S6.SS2.6.p1.16.m16.2.2.2.2.3.cmml"><mo id="S6.SS2.6.p1.16.m16.2.2.2.2.2.3" stretchy="false" xref="S6.SS2.6.p1.16.m16.2.2.2.2.3.cmml">[</mo><msub id="S6.SS2.6.p1.16.m16.1.1.1.1.1.1" xref="S6.SS2.6.p1.16.m16.1.1.1.1.1.1.cmml"><mi id="S6.SS2.6.p1.16.m16.1.1.1.1.1.1.2" xref="S6.SS2.6.p1.16.m16.1.1.1.1.1.1.2.cmml">a</mi><mi id="S6.SS2.6.p1.16.m16.1.1.1.1.1.1.3" xref="S6.SS2.6.p1.16.m16.1.1.1.1.1.1.3.cmml">l</mi></msub><mo id="S6.SS2.6.p1.16.m16.2.2.2.2.2.4" xref="S6.SS2.6.p1.16.m16.2.2.2.2.3.cmml">,</mo><msub id="S6.SS2.6.p1.16.m16.2.2.2.2.2.2" xref="S6.SS2.6.p1.16.m16.2.2.2.2.2.2.cmml"><mi id="S6.SS2.6.p1.16.m16.2.2.2.2.2.2.2" xref="S6.SS2.6.p1.16.m16.2.2.2.2.2.2.2.cmml">a</mi><mi id="S6.SS2.6.p1.16.m16.2.2.2.2.2.2.3" xref="S6.SS2.6.p1.16.m16.2.2.2.2.2.2.3.cmml">r</mi></msub><mo id="S6.SS2.6.p1.16.m16.2.2.2.2.2.5" stretchy="false" xref="S6.SS2.6.p1.16.m16.2.2.2.2.3.cmml">]</mo></mrow><mo id="S6.SS2.6.p1.16.m16.4.4.4.5" xref="S6.SS2.6.p1.16.m16.4.4.4.5.cmml">∩</mo><mrow id="S6.SS2.6.p1.16.m16.4.4.4.4.2" xref="S6.SS2.6.p1.16.m16.4.4.4.4.3.cmml"><mo id="S6.SS2.6.p1.16.m16.4.4.4.4.2.3" stretchy="false" xref="S6.SS2.6.p1.16.m16.4.4.4.4.3.cmml">[</mo><msub id="S6.SS2.6.p1.16.m16.3.3.3.3.1.1" xref="S6.SS2.6.p1.16.m16.3.3.3.3.1.1.cmml"><mi id="S6.SS2.6.p1.16.m16.3.3.3.3.1.1.2" xref="S6.SS2.6.p1.16.m16.3.3.3.3.1.1.2.cmml">b</mi><mi id="S6.SS2.6.p1.16.m16.3.3.3.3.1.1.3" xref="S6.SS2.6.p1.16.m16.3.3.3.3.1.1.3.cmml">l</mi></msub><mo id="S6.SS2.6.p1.16.m16.4.4.4.4.2.4" xref="S6.SS2.6.p1.16.m16.4.4.4.4.3.cmml">,</mo><msub id="S6.SS2.6.p1.16.m16.4.4.4.4.2.2" xref="S6.SS2.6.p1.16.m16.4.4.4.4.2.2.cmml"><mi id="S6.SS2.6.p1.16.m16.4.4.4.4.2.2.2" xref="S6.SS2.6.p1.16.m16.4.4.4.4.2.2.2.cmml">b</mi><mi id="S6.SS2.6.p1.16.m16.4.4.4.4.2.2.3" xref="S6.SS2.6.p1.16.m16.4.4.4.4.2.2.3.cmml">r</mi></msub><mo id="S6.SS2.6.p1.16.m16.4.4.4.4.2.5" stretchy="false" xref="S6.SS2.6.p1.16.m16.4.4.4.4.3.cmml">]</mo></mrow></mrow><mo id="S6.SS2.6.p1.16.m16.4.4.5" xref="S6.SS2.6.p1.16.m16.4.4.5.cmml">=</mo><mi id="S6.SS2.6.p1.16.m16.4.4.6" mathvariant="normal" xref="S6.SS2.6.p1.16.m16.4.4.6.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.6.p1.16.m16.4b"><apply id="S6.SS2.6.p1.16.m16.4.4.cmml" xref="S6.SS2.6.p1.16.m16.4.4"><eq id="S6.SS2.6.p1.16.m16.4.4.5.cmml" xref="S6.SS2.6.p1.16.m16.4.4.5"></eq><apply id="S6.SS2.6.p1.16.m16.4.4.4.cmml" xref="S6.SS2.6.p1.16.m16.4.4.4"><intersect id="S6.SS2.6.p1.16.m16.4.4.4.5.cmml" xref="S6.SS2.6.p1.16.m16.4.4.4.5"></intersect><interval closure="closed" id="S6.SS2.6.p1.16.m16.2.2.2.2.3.cmml" xref="S6.SS2.6.p1.16.m16.2.2.2.2.2"><apply id="S6.SS2.6.p1.16.m16.1.1.1.1.1.1.cmml" xref="S6.SS2.6.p1.16.m16.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.6.p1.16.m16.1.1.1.1.1.1.1.cmml" xref="S6.SS2.6.p1.16.m16.1.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.6.p1.16.m16.1.1.1.1.1.1.2.cmml" xref="S6.SS2.6.p1.16.m16.1.1.1.1.1.1.2">𝑎</ci><ci id="S6.SS2.6.p1.16.m16.1.1.1.1.1.1.3.cmml" xref="S6.SS2.6.p1.16.m16.1.1.1.1.1.1.3">𝑙</ci></apply><apply id="S6.SS2.6.p1.16.m16.2.2.2.2.2.2.cmml" xref="S6.SS2.6.p1.16.m16.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.6.p1.16.m16.2.2.2.2.2.2.1.cmml" xref="S6.SS2.6.p1.16.m16.2.2.2.2.2.2">subscript</csymbol><ci id="S6.SS2.6.p1.16.m16.2.2.2.2.2.2.2.cmml" xref="S6.SS2.6.p1.16.m16.2.2.2.2.2.2.2">𝑎</ci><ci id="S6.SS2.6.p1.16.m16.2.2.2.2.2.2.3.cmml" xref="S6.SS2.6.p1.16.m16.2.2.2.2.2.2.3">𝑟</ci></apply></interval><interval closure="closed" id="S6.SS2.6.p1.16.m16.4.4.4.4.3.cmml" xref="S6.SS2.6.p1.16.m16.4.4.4.4.2"><apply id="S6.SS2.6.p1.16.m16.3.3.3.3.1.1.cmml" xref="S6.SS2.6.p1.16.m16.3.3.3.3.1.1"><csymbol cd="ambiguous" id="S6.SS2.6.p1.16.m16.3.3.3.3.1.1.1.cmml" xref="S6.SS2.6.p1.16.m16.3.3.3.3.1.1">subscript</csymbol><ci id="S6.SS2.6.p1.16.m16.3.3.3.3.1.1.2.cmml" xref="S6.SS2.6.p1.16.m16.3.3.3.3.1.1.2">𝑏</ci><ci id="S6.SS2.6.p1.16.m16.3.3.3.3.1.1.3.cmml" xref="S6.SS2.6.p1.16.m16.3.3.3.3.1.1.3">𝑙</ci></apply><apply id="S6.SS2.6.p1.16.m16.4.4.4.4.2.2.cmml" xref="S6.SS2.6.p1.16.m16.4.4.4.4.2.2"><csymbol cd="ambiguous" id="S6.SS2.6.p1.16.m16.4.4.4.4.2.2.1.cmml" xref="S6.SS2.6.p1.16.m16.4.4.4.4.2.2">subscript</csymbol><ci id="S6.SS2.6.p1.16.m16.4.4.4.4.2.2.2.cmml" xref="S6.SS2.6.p1.16.m16.4.4.4.4.2.2.2">𝑏</ci><ci id="S6.SS2.6.p1.16.m16.4.4.4.4.2.2.3.cmml" xref="S6.SS2.6.p1.16.m16.4.4.4.4.2.2.3">𝑟</ci></apply></interval></apply><emptyset id="S6.SS2.6.p1.16.m16.4.4.6.cmml" xref="S6.SS2.6.p1.16.m16.4.4.6"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.6.p1.16.m16.4c">[a_{l},a_{r}]\cap[b_{l},b_{r}]=\varnothing</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.6.p1.16.m16.4d">[ italic_a start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ] ∩ [ italic_b start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ] = ∅</annotation></semantics></math>, in other words, <math alttext="f_{p}" class="ltx_Math" display="inline" id="S6.SS2.6.p1.17.m17.1"><semantics id="S6.SS2.6.p1.17.m17.1a"><msub id="S6.SS2.6.p1.17.m17.1.1" xref="S6.SS2.6.p1.17.m17.1.1.cmml"><mi id="S6.SS2.6.p1.17.m17.1.1.2" xref="S6.SS2.6.p1.17.m17.1.1.2.cmml">f</mi><mi id="S6.SS2.6.p1.17.m17.1.1.3" xref="S6.SS2.6.p1.17.m17.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.6.p1.17.m17.1b"><apply id="S6.SS2.6.p1.17.m17.1.1.cmml" xref="S6.SS2.6.p1.17.m17.1.1"><csymbol cd="ambiguous" id="S6.SS2.6.p1.17.m17.1.1.1.cmml" xref="S6.SS2.6.p1.17.m17.1.1">subscript</csymbol><ci id="S6.SS2.6.p1.17.m17.1.1.2.cmml" xref="S6.SS2.6.p1.17.m17.1.1.2">𝑓</ci><ci id="S6.SS2.6.p1.17.m17.1.1.3.cmml" xref="S6.SS2.6.p1.17.m17.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.6.p1.17.m17.1c">f_{p}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.6.p1.17.m17.1d">italic_f start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="f_{p^{\prime}}" class="ltx_Math" display="inline" id="S6.SS2.6.p1.18.m18.1"><semantics id="S6.SS2.6.p1.18.m18.1a"><msub id="S6.SS2.6.p1.18.m18.1.1" xref="S6.SS2.6.p1.18.m18.1.1.cmml"><mi id="S6.SS2.6.p1.18.m18.1.1.2" xref="S6.SS2.6.p1.18.m18.1.1.2.cmml">f</mi><msup id="S6.SS2.6.p1.18.m18.1.1.3" xref="S6.SS2.6.p1.18.m18.1.1.3.cmml"><mi id="S6.SS2.6.p1.18.m18.1.1.3.2" xref="S6.SS2.6.p1.18.m18.1.1.3.2.cmml">p</mi><mo id="S6.SS2.6.p1.18.m18.1.1.3.3" xref="S6.SS2.6.p1.18.m18.1.1.3.3.cmml">′</mo></msup></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.6.p1.18.m18.1b"><apply id="S6.SS2.6.p1.18.m18.1.1.cmml" xref="S6.SS2.6.p1.18.m18.1.1"><csymbol cd="ambiguous" id="S6.SS2.6.p1.18.m18.1.1.1.cmml" xref="S6.SS2.6.p1.18.m18.1.1">subscript</csymbol><ci id="S6.SS2.6.p1.18.m18.1.1.2.cmml" xref="S6.SS2.6.p1.18.m18.1.1.2">𝑓</ci><apply id="S6.SS2.6.p1.18.m18.1.1.3.cmml" xref="S6.SS2.6.p1.18.m18.1.1.3"><csymbol cd="ambiguous" id="S6.SS2.6.p1.18.m18.1.1.3.1.cmml" xref="S6.SS2.6.p1.18.m18.1.1.3">superscript</csymbol><ci id="S6.SS2.6.p1.18.m18.1.1.3.2.cmml" xref="S6.SS2.6.p1.18.m18.1.1.3.2">𝑝</ci><ci id="S6.SS2.6.p1.18.m18.1.1.3.3.cmml" xref="S6.SS2.6.p1.18.m18.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.6.p1.18.m18.1c">f_{p^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.6.p1.18.m18.1d">italic_f start_POSTSUBSCRIPT italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> have disjoint domains. First we show that this suffices to derive our contradiction.</p> </div> <div class="ltx_theorem ltx_theorem_claim" id="S6.Thmtheorem13"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_italic" id="S6.Thmtheorem13.1.1.1">Claim 6.13</span></span><span class="ltx_text ltx_font_italic" id="S6.Thmtheorem13.2.2">.</span> </h6> <div class="ltx_para" id="S6.Thmtheorem13.p1"> <p class="ltx_p" id="S6.Thmtheorem13.p1.2">If <math alttext="H" class="ltx_Math" display="inline" id="S6.Thmtheorem13.p1.1.m1.1"><semantics id="S6.Thmtheorem13.p1.1.m1.1a"><mi id="S6.Thmtheorem13.p1.1.m1.1.1" xref="S6.Thmtheorem13.p1.1.m1.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem13.p1.1.m1.1b"><ci id="S6.Thmtheorem13.p1.1.m1.1.1.cmml" xref="S6.Thmtheorem13.p1.1.m1.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem13.p1.1.m1.1c">H</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem13.p1.1.m1.1d">italic_H</annotation></semantics></math> satisfies the above, then <math alttext="H" class="ltx_Math" display="inline" id="S6.Thmtheorem13.p1.2.m2.1"><semantics id="S6.Thmtheorem13.p1.2.m2.1a"><mi id="S6.Thmtheorem13.p1.2.m2.1.1" xref="S6.Thmtheorem13.p1.2.m2.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem13.p1.2.m2.1b"><ci id="S6.Thmtheorem13.p1.2.m2.1.1.cmml" xref="S6.Thmtheorem13.p1.2.m2.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem13.p1.2.m2.1c">H</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem13.p1.2.m2.1d">italic_H</annotation></semantics></math> contains two compatible elements.</p> </div> </div> <div class="ltx_proof" id="S6.SS2.8.2"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S6.SS2.7.1.p1"> <p class="ltx_p" id="S6.SS2.7.1.p1.13">Since for each <math alttext="\bar{a}" class="ltx_Math" display="inline" id="S6.SS2.7.1.p1.1.m1.1"><semantics id="S6.SS2.7.1.p1.1.m1.1a"><mover accent="true" id="S6.SS2.7.1.p1.1.m1.1.1" xref="S6.SS2.7.1.p1.1.m1.1.1.cmml"><mi id="S6.SS2.7.1.p1.1.m1.1.1.2" xref="S6.SS2.7.1.p1.1.m1.1.1.2.cmml">a</mi><mo id="S6.SS2.7.1.p1.1.m1.1.1.1" xref="S6.SS2.7.1.p1.1.m1.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S6.SS2.7.1.p1.1.m1.1b"><apply id="S6.SS2.7.1.p1.1.m1.1.1.cmml" xref="S6.SS2.7.1.p1.1.m1.1.1"><ci id="S6.SS2.7.1.p1.1.m1.1.1.1.cmml" xref="S6.SS2.7.1.p1.1.m1.1.1.1">¯</ci><ci id="S6.SS2.7.1.p1.1.m1.1.1.2.cmml" xref="S6.SS2.7.1.p1.1.m1.1.1.2">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.7.1.p1.1.m1.1c">\bar{a}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.7.1.p1.1.m1.1d">over¯ start_ARG italic_a end_ARG</annotation></semantics></math>, <math alttext="a_{l}\leq_{A}a_{m}\leq_{A}a_{r}" class="ltx_Math" display="inline" id="S6.SS2.7.1.p1.2.m2.1"><semantics id="S6.SS2.7.1.p1.2.m2.1a"><mrow id="S6.SS2.7.1.p1.2.m2.1.1" xref="S6.SS2.7.1.p1.2.m2.1.1.cmml"><msub id="S6.SS2.7.1.p1.2.m2.1.1.2" xref="S6.SS2.7.1.p1.2.m2.1.1.2.cmml"><mi id="S6.SS2.7.1.p1.2.m2.1.1.2.2" xref="S6.SS2.7.1.p1.2.m2.1.1.2.2.cmml">a</mi><mi id="S6.SS2.7.1.p1.2.m2.1.1.2.3" xref="S6.SS2.7.1.p1.2.m2.1.1.2.3.cmml">l</mi></msub><msub id="S6.SS2.7.1.p1.2.m2.1.1.3" xref="S6.SS2.7.1.p1.2.m2.1.1.3.cmml"><mo id="S6.SS2.7.1.p1.2.m2.1.1.3.2" xref="S6.SS2.7.1.p1.2.m2.1.1.3.2.cmml">≤</mo><mi id="S6.SS2.7.1.p1.2.m2.1.1.3.3" xref="S6.SS2.7.1.p1.2.m2.1.1.3.3.cmml">A</mi></msub><msub id="S6.SS2.7.1.p1.2.m2.1.1.4" xref="S6.SS2.7.1.p1.2.m2.1.1.4.cmml"><mi id="S6.SS2.7.1.p1.2.m2.1.1.4.2" xref="S6.SS2.7.1.p1.2.m2.1.1.4.2.cmml">a</mi><mi id="S6.SS2.7.1.p1.2.m2.1.1.4.3" xref="S6.SS2.7.1.p1.2.m2.1.1.4.3.cmml">m</mi></msub><msub id="S6.SS2.7.1.p1.2.m2.1.1.5" xref="S6.SS2.7.1.p1.2.m2.1.1.5.cmml"><mo id="S6.SS2.7.1.p1.2.m2.1.1.5.2" xref="S6.SS2.7.1.p1.2.m2.1.1.5.2.cmml">≤</mo><mi id="S6.SS2.7.1.p1.2.m2.1.1.5.3" xref="S6.SS2.7.1.p1.2.m2.1.1.5.3.cmml">A</mi></msub><msub id="S6.SS2.7.1.p1.2.m2.1.1.6" xref="S6.SS2.7.1.p1.2.m2.1.1.6.cmml"><mi id="S6.SS2.7.1.p1.2.m2.1.1.6.2" xref="S6.SS2.7.1.p1.2.m2.1.1.6.2.cmml">a</mi><mi id="S6.SS2.7.1.p1.2.m2.1.1.6.3" xref="S6.SS2.7.1.p1.2.m2.1.1.6.3.cmml">r</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.7.1.p1.2.m2.1b"><apply id="S6.SS2.7.1.p1.2.m2.1.1.cmml" xref="S6.SS2.7.1.p1.2.m2.1.1"><and id="S6.SS2.7.1.p1.2.m2.1.1a.cmml" xref="S6.SS2.7.1.p1.2.m2.1.1"></and><apply id="S6.SS2.7.1.p1.2.m2.1.1b.cmml" xref="S6.SS2.7.1.p1.2.m2.1.1"><apply id="S6.SS2.7.1.p1.2.m2.1.1.3.cmml" xref="S6.SS2.7.1.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S6.SS2.7.1.p1.2.m2.1.1.3.1.cmml" xref="S6.SS2.7.1.p1.2.m2.1.1.3">subscript</csymbol><leq id="S6.SS2.7.1.p1.2.m2.1.1.3.2.cmml" xref="S6.SS2.7.1.p1.2.m2.1.1.3.2"></leq><ci id="S6.SS2.7.1.p1.2.m2.1.1.3.3.cmml" xref="S6.SS2.7.1.p1.2.m2.1.1.3.3">𝐴</ci></apply><apply id="S6.SS2.7.1.p1.2.m2.1.1.2.cmml" xref="S6.SS2.7.1.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="S6.SS2.7.1.p1.2.m2.1.1.2.1.cmml" xref="S6.SS2.7.1.p1.2.m2.1.1.2">subscript</csymbol><ci id="S6.SS2.7.1.p1.2.m2.1.1.2.2.cmml" xref="S6.SS2.7.1.p1.2.m2.1.1.2.2">𝑎</ci><ci id="S6.SS2.7.1.p1.2.m2.1.1.2.3.cmml" xref="S6.SS2.7.1.p1.2.m2.1.1.2.3">𝑙</ci></apply><apply id="S6.SS2.7.1.p1.2.m2.1.1.4.cmml" xref="S6.SS2.7.1.p1.2.m2.1.1.4"><csymbol cd="ambiguous" id="S6.SS2.7.1.p1.2.m2.1.1.4.1.cmml" xref="S6.SS2.7.1.p1.2.m2.1.1.4">subscript</csymbol><ci id="S6.SS2.7.1.p1.2.m2.1.1.4.2.cmml" xref="S6.SS2.7.1.p1.2.m2.1.1.4.2">𝑎</ci><ci id="S6.SS2.7.1.p1.2.m2.1.1.4.3.cmml" xref="S6.SS2.7.1.p1.2.m2.1.1.4.3">𝑚</ci></apply></apply><apply id="S6.SS2.7.1.p1.2.m2.1.1c.cmml" xref="S6.SS2.7.1.p1.2.m2.1.1"><apply id="S6.SS2.7.1.p1.2.m2.1.1.5.cmml" xref="S6.SS2.7.1.p1.2.m2.1.1.5"><csymbol cd="ambiguous" id="S6.SS2.7.1.p1.2.m2.1.1.5.1.cmml" xref="S6.SS2.7.1.p1.2.m2.1.1.5">subscript</csymbol><leq id="S6.SS2.7.1.p1.2.m2.1.1.5.2.cmml" xref="S6.SS2.7.1.p1.2.m2.1.1.5.2"></leq><ci id="S6.SS2.7.1.p1.2.m2.1.1.5.3.cmml" xref="S6.SS2.7.1.p1.2.m2.1.1.5.3">𝐴</ci></apply><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.7.1.p1.2.m2.1.1.4.cmml" id="S6.SS2.7.1.p1.2.m2.1.1d.cmml" xref="S6.SS2.7.1.p1.2.m2.1.1"></share><apply id="S6.SS2.7.1.p1.2.m2.1.1.6.cmml" xref="S6.SS2.7.1.p1.2.m2.1.1.6"><csymbol cd="ambiguous" id="S6.SS2.7.1.p1.2.m2.1.1.6.1.cmml" xref="S6.SS2.7.1.p1.2.m2.1.1.6">subscript</csymbol><ci id="S6.SS2.7.1.p1.2.m2.1.1.6.2.cmml" xref="S6.SS2.7.1.p1.2.m2.1.1.6.2">𝑎</ci><ci id="S6.SS2.7.1.p1.2.m2.1.1.6.3.cmml" xref="S6.SS2.7.1.p1.2.m2.1.1.6.3">𝑟</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.7.1.p1.2.m2.1c">a_{l}\leq_{A}a_{m}\leq_{A}a_{r}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.7.1.p1.2.m2.1d">italic_a start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ≤ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ≤ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT</annotation></semantics></math>, the hypothesis on <math alttext="H" class="ltx_Math" display="inline" id="S6.SS2.7.1.p1.3.m3.1"><semantics id="S6.SS2.7.1.p1.3.m3.1a"><mi id="S6.SS2.7.1.p1.3.m3.1.1" xref="S6.SS2.7.1.p1.3.m3.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.7.1.p1.3.m3.1b"><ci id="S6.SS2.7.1.p1.3.m3.1.1.cmml" xref="S6.SS2.7.1.p1.3.m3.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.7.1.p1.3.m3.1c">H</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.7.1.p1.3.m3.1d">italic_H</annotation></semantics></math> implies that the mapping <math alttext="p\mapsto q(p)" class="ltx_Math" display="inline" id="S6.SS2.7.1.p1.4.m4.1"><semantics id="S6.SS2.7.1.p1.4.m4.1a"><mrow id="S6.SS2.7.1.p1.4.m4.1.2" xref="S6.SS2.7.1.p1.4.m4.1.2.cmml"><mi id="S6.SS2.7.1.p1.4.m4.1.2.2" xref="S6.SS2.7.1.p1.4.m4.1.2.2.cmml">p</mi><mo id="S6.SS2.7.1.p1.4.m4.1.2.1" stretchy="false" xref="S6.SS2.7.1.p1.4.m4.1.2.1.cmml">↦</mo><mrow id="S6.SS2.7.1.p1.4.m4.1.2.3" xref="S6.SS2.7.1.p1.4.m4.1.2.3.cmml"><mi id="S6.SS2.7.1.p1.4.m4.1.2.3.2" xref="S6.SS2.7.1.p1.4.m4.1.2.3.2.cmml">q</mi><mo id="S6.SS2.7.1.p1.4.m4.1.2.3.1" xref="S6.SS2.7.1.p1.4.m4.1.2.3.1.cmml">⁢</mo><mrow id="S6.SS2.7.1.p1.4.m4.1.2.3.3.2" xref="S6.SS2.7.1.p1.4.m4.1.2.3.cmml"><mo id="S6.SS2.7.1.p1.4.m4.1.2.3.3.2.1" stretchy="false" xref="S6.SS2.7.1.p1.4.m4.1.2.3.cmml">(</mo><mi id="S6.SS2.7.1.p1.4.m4.1.1" xref="S6.SS2.7.1.p1.4.m4.1.1.cmml">p</mi><mo id="S6.SS2.7.1.p1.4.m4.1.2.3.3.2.2" stretchy="false" xref="S6.SS2.7.1.p1.4.m4.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.7.1.p1.4.m4.1b"><apply id="S6.SS2.7.1.p1.4.m4.1.2.cmml" xref="S6.SS2.7.1.p1.4.m4.1.2"><csymbol cd="latexml" id="S6.SS2.7.1.p1.4.m4.1.2.1.cmml" xref="S6.SS2.7.1.p1.4.m4.1.2.1">maps-to</csymbol><ci id="S6.SS2.7.1.p1.4.m4.1.2.2.cmml" xref="S6.SS2.7.1.p1.4.m4.1.2.2">𝑝</ci><apply id="S6.SS2.7.1.p1.4.m4.1.2.3.cmml" xref="S6.SS2.7.1.p1.4.m4.1.2.3"><times id="S6.SS2.7.1.p1.4.m4.1.2.3.1.cmml" xref="S6.SS2.7.1.p1.4.m4.1.2.3.1"></times><ci id="S6.SS2.7.1.p1.4.m4.1.2.3.2.cmml" xref="S6.SS2.7.1.p1.4.m4.1.2.3.2">𝑞</ci><ci id="S6.SS2.7.1.p1.4.m4.1.1.cmml" xref="S6.SS2.7.1.p1.4.m4.1.1">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.7.1.p1.4.m4.1c">p\mapsto q(p)</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.7.1.p1.4.m4.1d">italic_p ↦ italic_q ( italic_p )</annotation></semantics></math> is injective when restricted to <math alttext="H" class="ltx_Math" display="inline" id="S6.SS2.7.1.p1.5.m5.1"><semantics id="S6.SS2.7.1.p1.5.m5.1a"><mi id="S6.SS2.7.1.p1.5.m5.1.1" xref="S6.SS2.7.1.p1.5.m5.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.7.1.p1.5.m5.1b"><ci id="S6.SS2.7.1.p1.5.m5.1.1.cmml" xref="S6.SS2.7.1.p1.5.m5.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.7.1.p1.5.m5.1c">H</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.7.1.p1.5.m5.1d">italic_H</annotation></semantics></math>. Thus, using <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.Thmtheorem11" title="Lemma 6.11. ‣ 6.2. Forcing epimorphisms ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">6.11</span></a> (c) and the ccc of <math alttext="Q_{E}" class="ltx_Math" display="inline" id="S6.SS2.7.1.p1.6.m6.1"><semantics id="S6.SS2.7.1.p1.6.m6.1a"><msub id="S6.SS2.7.1.p1.6.m6.1.1" xref="S6.SS2.7.1.p1.6.m6.1.1.cmml"><mi id="S6.SS2.7.1.p1.6.m6.1.1.2" xref="S6.SS2.7.1.p1.6.m6.1.1.2.cmml">Q</mi><mi id="S6.SS2.7.1.p1.6.m6.1.1.3" xref="S6.SS2.7.1.p1.6.m6.1.1.3.cmml">E</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.7.1.p1.6.m6.1b"><apply id="S6.SS2.7.1.p1.6.m6.1.1.cmml" xref="S6.SS2.7.1.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S6.SS2.7.1.p1.6.m6.1.1.1.cmml" xref="S6.SS2.7.1.p1.6.m6.1.1">subscript</csymbol><ci id="S6.SS2.7.1.p1.6.m6.1.1.2.cmml" xref="S6.SS2.7.1.p1.6.m6.1.1.2">𝑄</ci><ci id="S6.SS2.7.1.p1.6.m6.1.1.3.cmml" xref="S6.SS2.7.1.p1.6.m6.1.1.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.7.1.p1.6.m6.1c">Q_{E}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.7.1.p1.6.m6.1d">italic_Q start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT</annotation></semantics></math>, we deduce that for some <math alttext="p\neq p^{\prime}" class="ltx_Math" display="inline" id="S6.SS2.7.1.p1.7.m7.1"><semantics id="S6.SS2.7.1.p1.7.m7.1a"><mrow id="S6.SS2.7.1.p1.7.m7.1.1" xref="S6.SS2.7.1.p1.7.m7.1.1.cmml"><mi id="S6.SS2.7.1.p1.7.m7.1.1.2" xref="S6.SS2.7.1.p1.7.m7.1.1.2.cmml">p</mi><mo id="S6.SS2.7.1.p1.7.m7.1.1.1" xref="S6.SS2.7.1.p1.7.m7.1.1.1.cmml">≠</mo><msup id="S6.SS2.7.1.p1.7.m7.1.1.3" xref="S6.SS2.7.1.p1.7.m7.1.1.3.cmml"><mi id="S6.SS2.7.1.p1.7.m7.1.1.3.2" xref="S6.SS2.7.1.p1.7.m7.1.1.3.2.cmml">p</mi><mo id="S6.SS2.7.1.p1.7.m7.1.1.3.3" xref="S6.SS2.7.1.p1.7.m7.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.7.1.p1.7.m7.1b"><apply id="S6.SS2.7.1.p1.7.m7.1.1.cmml" xref="S6.SS2.7.1.p1.7.m7.1.1"><neq id="S6.SS2.7.1.p1.7.m7.1.1.1.cmml" xref="S6.SS2.7.1.p1.7.m7.1.1.1"></neq><ci id="S6.SS2.7.1.p1.7.m7.1.1.2.cmml" xref="S6.SS2.7.1.p1.7.m7.1.1.2">𝑝</ci><apply id="S6.SS2.7.1.p1.7.m7.1.1.3.cmml" xref="S6.SS2.7.1.p1.7.m7.1.1.3"><csymbol cd="ambiguous" id="S6.SS2.7.1.p1.7.m7.1.1.3.1.cmml" xref="S6.SS2.7.1.p1.7.m7.1.1.3">superscript</csymbol><ci id="S6.SS2.7.1.p1.7.m7.1.1.3.2.cmml" xref="S6.SS2.7.1.p1.7.m7.1.1.3.2">𝑝</ci><ci id="S6.SS2.7.1.p1.7.m7.1.1.3.3.cmml" xref="S6.SS2.7.1.p1.7.m7.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.7.1.p1.7.m7.1c">p\neq p^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.7.1.p1.7.m7.1d">italic_p ≠ italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="q(p)" class="ltx_Math" display="inline" id="S6.SS2.7.1.p1.8.m8.1"><semantics id="S6.SS2.7.1.p1.8.m8.1a"><mrow id="S6.SS2.7.1.p1.8.m8.1.2" xref="S6.SS2.7.1.p1.8.m8.1.2.cmml"><mi id="S6.SS2.7.1.p1.8.m8.1.2.2" xref="S6.SS2.7.1.p1.8.m8.1.2.2.cmml">q</mi><mo id="S6.SS2.7.1.p1.8.m8.1.2.1" xref="S6.SS2.7.1.p1.8.m8.1.2.1.cmml">⁢</mo><mrow id="S6.SS2.7.1.p1.8.m8.1.2.3.2" xref="S6.SS2.7.1.p1.8.m8.1.2.cmml"><mo id="S6.SS2.7.1.p1.8.m8.1.2.3.2.1" stretchy="false" xref="S6.SS2.7.1.p1.8.m8.1.2.cmml">(</mo><mi id="S6.SS2.7.1.p1.8.m8.1.1" xref="S6.SS2.7.1.p1.8.m8.1.1.cmml">p</mi><mo id="S6.SS2.7.1.p1.8.m8.1.2.3.2.2" stretchy="false" xref="S6.SS2.7.1.p1.8.m8.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.7.1.p1.8.m8.1b"><apply id="S6.SS2.7.1.p1.8.m8.1.2.cmml" xref="S6.SS2.7.1.p1.8.m8.1.2"><times id="S6.SS2.7.1.p1.8.m8.1.2.1.cmml" xref="S6.SS2.7.1.p1.8.m8.1.2.1"></times><ci id="S6.SS2.7.1.p1.8.m8.1.2.2.cmml" xref="S6.SS2.7.1.p1.8.m8.1.2.2">𝑞</ci><ci id="S6.SS2.7.1.p1.8.m8.1.1.cmml" xref="S6.SS2.7.1.p1.8.m8.1.1">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.7.1.p1.8.m8.1c">q(p)</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.7.1.p1.8.m8.1d">italic_q ( italic_p )</annotation></semantics></math> and <math alttext="q(p^{\prime})" class="ltx_Math" display="inline" id="S6.SS2.7.1.p1.9.m9.1"><semantics id="S6.SS2.7.1.p1.9.m9.1a"><mrow id="S6.SS2.7.1.p1.9.m9.1.1" xref="S6.SS2.7.1.p1.9.m9.1.1.cmml"><mi id="S6.SS2.7.1.p1.9.m9.1.1.3" xref="S6.SS2.7.1.p1.9.m9.1.1.3.cmml">q</mi><mo id="S6.SS2.7.1.p1.9.m9.1.1.2" xref="S6.SS2.7.1.p1.9.m9.1.1.2.cmml">⁢</mo><mrow id="S6.SS2.7.1.p1.9.m9.1.1.1.1" xref="S6.SS2.7.1.p1.9.m9.1.1.1.1.1.cmml"><mo id="S6.SS2.7.1.p1.9.m9.1.1.1.1.2" stretchy="false" xref="S6.SS2.7.1.p1.9.m9.1.1.1.1.1.cmml">(</mo><msup id="S6.SS2.7.1.p1.9.m9.1.1.1.1.1" xref="S6.SS2.7.1.p1.9.m9.1.1.1.1.1.cmml"><mi id="S6.SS2.7.1.p1.9.m9.1.1.1.1.1.2" xref="S6.SS2.7.1.p1.9.m9.1.1.1.1.1.2.cmml">p</mi><mo id="S6.SS2.7.1.p1.9.m9.1.1.1.1.1.3" xref="S6.SS2.7.1.p1.9.m9.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S6.SS2.7.1.p1.9.m9.1.1.1.1.3" stretchy="false" xref="S6.SS2.7.1.p1.9.m9.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.7.1.p1.9.m9.1b"><apply id="S6.SS2.7.1.p1.9.m9.1.1.cmml" xref="S6.SS2.7.1.p1.9.m9.1.1"><times id="S6.SS2.7.1.p1.9.m9.1.1.2.cmml" xref="S6.SS2.7.1.p1.9.m9.1.1.2"></times><ci id="S6.SS2.7.1.p1.9.m9.1.1.3.cmml" xref="S6.SS2.7.1.p1.9.m9.1.1.3">𝑞</ci><apply id="S6.SS2.7.1.p1.9.m9.1.1.1.1.1.cmml" xref="S6.SS2.7.1.p1.9.m9.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.7.1.p1.9.m9.1.1.1.1.1.1.cmml" xref="S6.SS2.7.1.p1.9.m9.1.1.1.1">superscript</csymbol><ci id="S6.SS2.7.1.p1.9.m9.1.1.1.1.1.2.cmml" xref="S6.SS2.7.1.p1.9.m9.1.1.1.1.1.2">𝑝</ci><ci id="S6.SS2.7.1.p1.9.m9.1.1.1.1.1.3.cmml" xref="S6.SS2.7.1.p1.9.m9.1.1.1.1.1.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.7.1.p1.9.m9.1c">q(p^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.7.1.p1.9.m9.1d">italic_q ( italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> are compatible, therefore <math alttext="q:=q(p)\cup q(p^{\prime})\in Q_{E}" class="ltx_Math" display="inline" id="S6.SS2.7.1.p1.10.m10.2"><semantics id="S6.SS2.7.1.p1.10.m10.2a"><mrow id="S6.SS2.7.1.p1.10.m10.2.2" xref="S6.SS2.7.1.p1.10.m10.2.2.cmml"><mi id="S6.SS2.7.1.p1.10.m10.2.2.3" xref="S6.SS2.7.1.p1.10.m10.2.2.3.cmml">q</mi><mo id="S6.SS2.7.1.p1.10.m10.2.2.4" lspace="0.278em" rspace="0.278em" xref="S6.SS2.7.1.p1.10.m10.2.2.4.cmml">:=</mo><mrow id="S6.SS2.7.1.p1.10.m10.2.2.1" xref="S6.SS2.7.1.p1.10.m10.2.2.1.cmml"><mrow id="S6.SS2.7.1.p1.10.m10.2.2.1.3" xref="S6.SS2.7.1.p1.10.m10.2.2.1.3.cmml"><mi id="S6.SS2.7.1.p1.10.m10.2.2.1.3.2" xref="S6.SS2.7.1.p1.10.m10.2.2.1.3.2.cmml">q</mi><mo id="S6.SS2.7.1.p1.10.m10.2.2.1.3.1" xref="S6.SS2.7.1.p1.10.m10.2.2.1.3.1.cmml">⁢</mo><mrow id="S6.SS2.7.1.p1.10.m10.2.2.1.3.3.2" xref="S6.SS2.7.1.p1.10.m10.2.2.1.3.cmml"><mo id="S6.SS2.7.1.p1.10.m10.2.2.1.3.3.2.1" stretchy="false" xref="S6.SS2.7.1.p1.10.m10.2.2.1.3.cmml">(</mo><mi id="S6.SS2.7.1.p1.10.m10.1.1" xref="S6.SS2.7.1.p1.10.m10.1.1.cmml">p</mi><mo id="S6.SS2.7.1.p1.10.m10.2.2.1.3.3.2.2" stretchy="false" xref="S6.SS2.7.1.p1.10.m10.2.2.1.3.cmml">)</mo></mrow></mrow><mo id="S6.SS2.7.1.p1.10.m10.2.2.1.2" xref="S6.SS2.7.1.p1.10.m10.2.2.1.2.cmml">∪</mo><mrow id="S6.SS2.7.1.p1.10.m10.2.2.1.1" xref="S6.SS2.7.1.p1.10.m10.2.2.1.1.cmml"><mi id="S6.SS2.7.1.p1.10.m10.2.2.1.1.3" xref="S6.SS2.7.1.p1.10.m10.2.2.1.1.3.cmml">q</mi><mo id="S6.SS2.7.1.p1.10.m10.2.2.1.1.2" xref="S6.SS2.7.1.p1.10.m10.2.2.1.1.2.cmml">⁢</mo><mrow id="S6.SS2.7.1.p1.10.m10.2.2.1.1.1.1" xref="S6.SS2.7.1.p1.10.m10.2.2.1.1.1.1.1.cmml"><mo id="S6.SS2.7.1.p1.10.m10.2.2.1.1.1.1.2" stretchy="false" xref="S6.SS2.7.1.p1.10.m10.2.2.1.1.1.1.1.cmml">(</mo><msup id="S6.SS2.7.1.p1.10.m10.2.2.1.1.1.1.1" xref="S6.SS2.7.1.p1.10.m10.2.2.1.1.1.1.1.cmml"><mi id="S6.SS2.7.1.p1.10.m10.2.2.1.1.1.1.1.2" xref="S6.SS2.7.1.p1.10.m10.2.2.1.1.1.1.1.2.cmml">p</mi><mo id="S6.SS2.7.1.p1.10.m10.2.2.1.1.1.1.1.3" xref="S6.SS2.7.1.p1.10.m10.2.2.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S6.SS2.7.1.p1.10.m10.2.2.1.1.1.1.3" stretchy="false" xref="S6.SS2.7.1.p1.10.m10.2.2.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S6.SS2.7.1.p1.10.m10.2.2.5" xref="S6.SS2.7.1.p1.10.m10.2.2.5.cmml">∈</mo><msub id="S6.SS2.7.1.p1.10.m10.2.2.6" xref="S6.SS2.7.1.p1.10.m10.2.2.6.cmml"><mi id="S6.SS2.7.1.p1.10.m10.2.2.6.2" xref="S6.SS2.7.1.p1.10.m10.2.2.6.2.cmml">Q</mi><mi id="S6.SS2.7.1.p1.10.m10.2.2.6.3" xref="S6.SS2.7.1.p1.10.m10.2.2.6.3.cmml">E</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.7.1.p1.10.m10.2b"><apply id="S6.SS2.7.1.p1.10.m10.2.2.cmml" xref="S6.SS2.7.1.p1.10.m10.2.2"><and id="S6.SS2.7.1.p1.10.m10.2.2a.cmml" xref="S6.SS2.7.1.p1.10.m10.2.2"></and><apply id="S6.SS2.7.1.p1.10.m10.2.2b.cmml" xref="S6.SS2.7.1.p1.10.m10.2.2"><csymbol cd="latexml" id="S6.SS2.7.1.p1.10.m10.2.2.4.cmml" xref="S6.SS2.7.1.p1.10.m10.2.2.4">assign</csymbol><ci id="S6.SS2.7.1.p1.10.m10.2.2.3.cmml" xref="S6.SS2.7.1.p1.10.m10.2.2.3">𝑞</ci><apply id="S6.SS2.7.1.p1.10.m10.2.2.1.cmml" xref="S6.SS2.7.1.p1.10.m10.2.2.1"><union id="S6.SS2.7.1.p1.10.m10.2.2.1.2.cmml" xref="S6.SS2.7.1.p1.10.m10.2.2.1.2"></union><apply id="S6.SS2.7.1.p1.10.m10.2.2.1.3.cmml" xref="S6.SS2.7.1.p1.10.m10.2.2.1.3"><times id="S6.SS2.7.1.p1.10.m10.2.2.1.3.1.cmml" xref="S6.SS2.7.1.p1.10.m10.2.2.1.3.1"></times><ci id="S6.SS2.7.1.p1.10.m10.2.2.1.3.2.cmml" xref="S6.SS2.7.1.p1.10.m10.2.2.1.3.2">𝑞</ci><ci id="S6.SS2.7.1.p1.10.m10.1.1.cmml" xref="S6.SS2.7.1.p1.10.m10.1.1">𝑝</ci></apply><apply id="S6.SS2.7.1.p1.10.m10.2.2.1.1.cmml" xref="S6.SS2.7.1.p1.10.m10.2.2.1.1"><times id="S6.SS2.7.1.p1.10.m10.2.2.1.1.2.cmml" xref="S6.SS2.7.1.p1.10.m10.2.2.1.1.2"></times><ci id="S6.SS2.7.1.p1.10.m10.2.2.1.1.3.cmml" xref="S6.SS2.7.1.p1.10.m10.2.2.1.1.3">𝑞</ci><apply id="S6.SS2.7.1.p1.10.m10.2.2.1.1.1.1.1.cmml" xref="S6.SS2.7.1.p1.10.m10.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.7.1.p1.10.m10.2.2.1.1.1.1.1.1.cmml" xref="S6.SS2.7.1.p1.10.m10.2.2.1.1.1.1">superscript</csymbol><ci id="S6.SS2.7.1.p1.10.m10.2.2.1.1.1.1.1.2.cmml" xref="S6.SS2.7.1.p1.10.m10.2.2.1.1.1.1.1.2">𝑝</ci><ci id="S6.SS2.7.1.p1.10.m10.2.2.1.1.1.1.1.3.cmml" xref="S6.SS2.7.1.p1.10.m10.2.2.1.1.1.1.1.3">′</ci></apply></apply></apply></apply><apply id="S6.SS2.7.1.p1.10.m10.2.2c.cmml" xref="S6.SS2.7.1.p1.10.m10.2.2"><in id="S6.SS2.7.1.p1.10.m10.2.2.5.cmml" xref="S6.SS2.7.1.p1.10.m10.2.2.5"></in><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.7.1.p1.10.m10.2.2.1.cmml" id="S6.SS2.7.1.p1.10.m10.2.2d.cmml" xref="S6.SS2.7.1.p1.10.m10.2.2"></share><apply id="S6.SS2.7.1.p1.10.m10.2.2.6.cmml" xref="S6.SS2.7.1.p1.10.m10.2.2.6"><csymbol cd="ambiguous" id="S6.SS2.7.1.p1.10.m10.2.2.6.1.cmml" xref="S6.SS2.7.1.p1.10.m10.2.2.6">subscript</csymbol><ci id="S6.SS2.7.1.p1.10.m10.2.2.6.2.cmml" xref="S6.SS2.7.1.p1.10.m10.2.2.6.2">𝑄</ci><ci id="S6.SS2.7.1.p1.10.m10.2.2.6.3.cmml" xref="S6.SS2.7.1.p1.10.m10.2.2.6.3">𝐸</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.7.1.p1.10.m10.2c">q:=q(p)\cup q(p^{\prime})\in Q_{E}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.7.1.p1.10.m10.2d">italic_q := italic_q ( italic_p ) ∪ italic_q ( italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ∈ italic_Q start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT</annotation></semantics></math>, since any condition witnessing the compatibility of <math alttext="q(p)" class="ltx_Math" display="inline" id="S6.SS2.7.1.p1.11.m11.1"><semantics id="S6.SS2.7.1.p1.11.m11.1a"><mrow id="S6.SS2.7.1.p1.11.m11.1.2" xref="S6.SS2.7.1.p1.11.m11.1.2.cmml"><mi id="S6.SS2.7.1.p1.11.m11.1.2.2" xref="S6.SS2.7.1.p1.11.m11.1.2.2.cmml">q</mi><mo id="S6.SS2.7.1.p1.11.m11.1.2.1" xref="S6.SS2.7.1.p1.11.m11.1.2.1.cmml">⁢</mo><mrow id="S6.SS2.7.1.p1.11.m11.1.2.3.2" xref="S6.SS2.7.1.p1.11.m11.1.2.cmml"><mo id="S6.SS2.7.1.p1.11.m11.1.2.3.2.1" stretchy="false" xref="S6.SS2.7.1.p1.11.m11.1.2.cmml">(</mo><mi id="S6.SS2.7.1.p1.11.m11.1.1" xref="S6.SS2.7.1.p1.11.m11.1.1.cmml">p</mi><mo id="S6.SS2.7.1.p1.11.m11.1.2.3.2.2" stretchy="false" xref="S6.SS2.7.1.p1.11.m11.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.7.1.p1.11.m11.1b"><apply id="S6.SS2.7.1.p1.11.m11.1.2.cmml" xref="S6.SS2.7.1.p1.11.m11.1.2"><times id="S6.SS2.7.1.p1.11.m11.1.2.1.cmml" xref="S6.SS2.7.1.p1.11.m11.1.2.1"></times><ci id="S6.SS2.7.1.p1.11.m11.1.2.2.cmml" xref="S6.SS2.7.1.p1.11.m11.1.2.2">𝑞</ci><ci id="S6.SS2.7.1.p1.11.m11.1.1.cmml" xref="S6.SS2.7.1.p1.11.m11.1.1">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.7.1.p1.11.m11.1c">q(p)</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.7.1.p1.11.m11.1d">italic_q ( italic_p )</annotation></semantics></math> and <math alttext="q(p^{\prime})" class="ltx_Math" display="inline" id="S6.SS2.7.1.p1.12.m12.1"><semantics id="S6.SS2.7.1.p1.12.m12.1a"><mrow id="S6.SS2.7.1.p1.12.m12.1.1" xref="S6.SS2.7.1.p1.12.m12.1.1.cmml"><mi id="S6.SS2.7.1.p1.12.m12.1.1.3" xref="S6.SS2.7.1.p1.12.m12.1.1.3.cmml">q</mi><mo id="S6.SS2.7.1.p1.12.m12.1.1.2" xref="S6.SS2.7.1.p1.12.m12.1.1.2.cmml">⁢</mo><mrow id="S6.SS2.7.1.p1.12.m12.1.1.1.1" xref="S6.SS2.7.1.p1.12.m12.1.1.1.1.1.cmml"><mo id="S6.SS2.7.1.p1.12.m12.1.1.1.1.2" stretchy="false" xref="S6.SS2.7.1.p1.12.m12.1.1.1.1.1.cmml">(</mo><msup id="S6.SS2.7.1.p1.12.m12.1.1.1.1.1" xref="S6.SS2.7.1.p1.12.m12.1.1.1.1.1.cmml"><mi id="S6.SS2.7.1.p1.12.m12.1.1.1.1.1.2" xref="S6.SS2.7.1.p1.12.m12.1.1.1.1.1.2.cmml">p</mi><mo id="S6.SS2.7.1.p1.12.m12.1.1.1.1.1.3" xref="S6.SS2.7.1.p1.12.m12.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S6.SS2.7.1.p1.12.m12.1.1.1.1.3" stretchy="false" xref="S6.SS2.7.1.p1.12.m12.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.7.1.p1.12.m12.1b"><apply id="S6.SS2.7.1.p1.12.m12.1.1.cmml" xref="S6.SS2.7.1.p1.12.m12.1.1"><times id="S6.SS2.7.1.p1.12.m12.1.1.2.cmml" xref="S6.SS2.7.1.p1.12.m12.1.1.2"></times><ci id="S6.SS2.7.1.p1.12.m12.1.1.3.cmml" xref="S6.SS2.7.1.p1.12.m12.1.1.3">𝑞</ci><apply id="S6.SS2.7.1.p1.12.m12.1.1.1.1.1.cmml" xref="S6.SS2.7.1.p1.12.m12.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.7.1.p1.12.m12.1.1.1.1.1.1.cmml" xref="S6.SS2.7.1.p1.12.m12.1.1.1.1">superscript</csymbol><ci id="S6.SS2.7.1.p1.12.m12.1.1.1.1.1.2.cmml" xref="S6.SS2.7.1.p1.12.m12.1.1.1.1.1.2">𝑝</ci><ci id="S6.SS2.7.1.p1.12.m12.1.1.1.1.1.3.cmml" xref="S6.SS2.7.1.p1.12.m12.1.1.1.1.1.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.7.1.p1.12.m12.1c">q(p^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.7.1.p1.12.m12.1d">italic_q ( italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> extends <math alttext="q" class="ltx_Math" display="inline" id="S6.SS2.7.1.p1.13.m13.1"><semantics id="S6.SS2.7.1.p1.13.m13.1a"><mi id="S6.SS2.7.1.p1.13.m13.1.1" xref="S6.SS2.7.1.p1.13.m13.1.1.cmml">q</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.7.1.p1.13.m13.1b"><ci id="S6.SS2.7.1.p1.13.m13.1.1.cmml" xref="S6.SS2.7.1.p1.13.m13.1.1">𝑞</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.7.1.p1.13.m13.1c">q</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.7.1.p1.13.m13.1d">italic_q</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S6.SS2.8.2.p2"> <p class="ltx_p" id="S6.SS2.8.2.p2.13">Now let <math alttext="\hat{p}:=p\cup p^{\prime}" class="ltx_Math" display="inline" id="S6.SS2.8.2.p2.1.m1.1"><semantics id="S6.SS2.8.2.p2.1.m1.1a"><mrow id="S6.SS2.8.2.p2.1.m1.1.1" xref="S6.SS2.8.2.p2.1.m1.1.1.cmml"><mover accent="true" id="S6.SS2.8.2.p2.1.m1.1.1.2" xref="S6.SS2.8.2.p2.1.m1.1.1.2.cmml"><mi id="S6.SS2.8.2.p2.1.m1.1.1.2.2" xref="S6.SS2.8.2.p2.1.m1.1.1.2.2.cmml">p</mi><mo id="S6.SS2.8.2.p2.1.m1.1.1.2.1" xref="S6.SS2.8.2.p2.1.m1.1.1.2.1.cmml">^</mo></mover><mo id="S6.SS2.8.2.p2.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S6.SS2.8.2.p2.1.m1.1.1.1.cmml">:=</mo><mrow id="S6.SS2.8.2.p2.1.m1.1.1.3" xref="S6.SS2.8.2.p2.1.m1.1.1.3.cmml"><mi id="S6.SS2.8.2.p2.1.m1.1.1.3.2" xref="S6.SS2.8.2.p2.1.m1.1.1.3.2.cmml">p</mi><mo id="S6.SS2.8.2.p2.1.m1.1.1.3.1" xref="S6.SS2.8.2.p2.1.m1.1.1.3.1.cmml">∪</mo><msup id="S6.SS2.8.2.p2.1.m1.1.1.3.3" xref="S6.SS2.8.2.p2.1.m1.1.1.3.3.cmml"><mi id="S6.SS2.8.2.p2.1.m1.1.1.3.3.2" xref="S6.SS2.8.2.p2.1.m1.1.1.3.3.2.cmml">p</mi><mo id="S6.SS2.8.2.p2.1.m1.1.1.3.3.3" xref="S6.SS2.8.2.p2.1.m1.1.1.3.3.3.cmml">′</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.8.2.p2.1.m1.1b"><apply id="S6.SS2.8.2.p2.1.m1.1.1.cmml" xref="S6.SS2.8.2.p2.1.m1.1.1"><csymbol cd="latexml" id="S6.SS2.8.2.p2.1.m1.1.1.1.cmml" xref="S6.SS2.8.2.p2.1.m1.1.1.1">assign</csymbol><apply id="S6.SS2.8.2.p2.1.m1.1.1.2.cmml" xref="S6.SS2.8.2.p2.1.m1.1.1.2"><ci id="S6.SS2.8.2.p2.1.m1.1.1.2.1.cmml" xref="S6.SS2.8.2.p2.1.m1.1.1.2.1">^</ci><ci id="S6.SS2.8.2.p2.1.m1.1.1.2.2.cmml" xref="S6.SS2.8.2.p2.1.m1.1.1.2.2">𝑝</ci></apply><apply id="S6.SS2.8.2.p2.1.m1.1.1.3.cmml" xref="S6.SS2.8.2.p2.1.m1.1.1.3"><union id="S6.SS2.8.2.p2.1.m1.1.1.3.1.cmml" xref="S6.SS2.8.2.p2.1.m1.1.1.3.1"></union><ci id="S6.SS2.8.2.p2.1.m1.1.1.3.2.cmml" xref="S6.SS2.8.2.p2.1.m1.1.1.3.2">𝑝</ci><apply id="S6.SS2.8.2.p2.1.m1.1.1.3.3.cmml" xref="S6.SS2.8.2.p2.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S6.SS2.8.2.p2.1.m1.1.1.3.3.1.cmml" xref="S6.SS2.8.2.p2.1.m1.1.1.3.3">superscript</csymbol><ci id="S6.SS2.8.2.p2.1.m1.1.1.3.3.2.cmml" xref="S6.SS2.8.2.p2.1.m1.1.1.3.3.2">𝑝</ci><ci id="S6.SS2.8.2.p2.1.m1.1.1.3.3.3.cmml" xref="S6.SS2.8.2.p2.1.m1.1.1.3.3.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.8.2.p2.1.m1.1c">\hat{p}:=p\cup p^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.8.2.p2.1.m1.1d">over^ start_ARG italic_p end_ARG := italic_p ∪ italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>. Since <math alttext="f_{p}" class="ltx_Math" display="inline" id="S6.SS2.8.2.p2.2.m2.1"><semantics id="S6.SS2.8.2.p2.2.m2.1a"><msub id="S6.SS2.8.2.p2.2.m2.1.1" xref="S6.SS2.8.2.p2.2.m2.1.1.cmml"><mi id="S6.SS2.8.2.p2.2.m2.1.1.2" xref="S6.SS2.8.2.p2.2.m2.1.1.2.cmml">f</mi><mi id="S6.SS2.8.2.p2.2.m2.1.1.3" xref="S6.SS2.8.2.p2.2.m2.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.8.2.p2.2.m2.1b"><apply id="S6.SS2.8.2.p2.2.m2.1.1.cmml" xref="S6.SS2.8.2.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S6.SS2.8.2.p2.2.m2.1.1.1.cmml" xref="S6.SS2.8.2.p2.2.m2.1.1">subscript</csymbol><ci id="S6.SS2.8.2.p2.2.m2.1.1.2.cmml" xref="S6.SS2.8.2.p2.2.m2.1.1.2">𝑓</ci><ci id="S6.SS2.8.2.p2.2.m2.1.1.3.cmml" xref="S6.SS2.8.2.p2.2.m2.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.8.2.p2.2.m2.1c">f_{p}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.8.2.p2.2.m2.1d">italic_f start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="f_{p^{\prime}}" class="ltx_Math" display="inline" id="S6.SS2.8.2.p2.3.m3.1"><semantics id="S6.SS2.8.2.p2.3.m3.1a"><msub id="S6.SS2.8.2.p2.3.m3.1.1" xref="S6.SS2.8.2.p2.3.m3.1.1.cmml"><mi id="S6.SS2.8.2.p2.3.m3.1.1.2" xref="S6.SS2.8.2.p2.3.m3.1.1.2.cmml">f</mi><msup id="S6.SS2.8.2.p2.3.m3.1.1.3" xref="S6.SS2.8.2.p2.3.m3.1.1.3.cmml"><mi id="S6.SS2.8.2.p2.3.m3.1.1.3.2" xref="S6.SS2.8.2.p2.3.m3.1.1.3.2.cmml">p</mi><mo id="S6.SS2.8.2.p2.3.m3.1.1.3.3" xref="S6.SS2.8.2.p2.3.m3.1.1.3.3.cmml">′</mo></msup></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.8.2.p2.3.m3.1b"><apply id="S6.SS2.8.2.p2.3.m3.1.1.cmml" xref="S6.SS2.8.2.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S6.SS2.8.2.p2.3.m3.1.1.1.cmml" xref="S6.SS2.8.2.p2.3.m3.1.1">subscript</csymbol><ci id="S6.SS2.8.2.p2.3.m3.1.1.2.cmml" xref="S6.SS2.8.2.p2.3.m3.1.1.2">𝑓</ci><apply id="S6.SS2.8.2.p2.3.m3.1.1.3.cmml" xref="S6.SS2.8.2.p2.3.m3.1.1.3"><csymbol cd="ambiguous" id="S6.SS2.8.2.p2.3.m3.1.1.3.1.cmml" xref="S6.SS2.8.2.p2.3.m3.1.1.3">superscript</csymbol><ci id="S6.SS2.8.2.p2.3.m3.1.1.3.2.cmml" xref="S6.SS2.8.2.p2.3.m3.1.1.3.2">𝑝</ci><ci id="S6.SS2.8.2.p2.3.m3.1.1.3.3.cmml" xref="S6.SS2.8.2.p2.3.m3.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.8.2.p2.3.m3.1c">f_{p^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.8.2.p2.3.m3.1d">italic_f start_POSTSUBSCRIPT italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> have disjoint domains, it is clear that <math alttext="\operatorname{dom}(\hat{p})" class="ltx_Math" display="inline" id="S6.SS2.8.2.p2.4.m4.2"><semantics id="S6.SS2.8.2.p2.4.m4.2a"><mrow id="S6.SS2.8.2.p2.4.m4.2.3.2" xref="S6.SS2.8.2.p2.4.m4.2.3.1.cmml"><mi id="S6.SS2.8.2.p2.4.m4.1.1" xref="S6.SS2.8.2.p2.4.m4.1.1.cmml">dom</mi><mo id="S6.SS2.8.2.p2.4.m4.2.3.2a" xref="S6.SS2.8.2.p2.4.m4.2.3.1.cmml">⁡</mo><mrow id="S6.SS2.8.2.p2.4.m4.2.3.2.1" xref="S6.SS2.8.2.p2.4.m4.2.3.1.cmml"><mo id="S6.SS2.8.2.p2.4.m4.2.3.2.1.1" stretchy="false" xref="S6.SS2.8.2.p2.4.m4.2.3.1.cmml">(</mo><mover accent="true" id="S6.SS2.8.2.p2.4.m4.2.2" xref="S6.SS2.8.2.p2.4.m4.2.2.cmml"><mi id="S6.SS2.8.2.p2.4.m4.2.2.2" xref="S6.SS2.8.2.p2.4.m4.2.2.2.cmml">p</mi><mo id="S6.SS2.8.2.p2.4.m4.2.2.1" xref="S6.SS2.8.2.p2.4.m4.2.2.1.cmml">^</mo></mover><mo id="S6.SS2.8.2.p2.4.m4.2.3.2.1.2" stretchy="false" xref="S6.SS2.8.2.p2.4.m4.2.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.8.2.p2.4.m4.2b"><apply id="S6.SS2.8.2.p2.4.m4.2.3.1.cmml" xref="S6.SS2.8.2.p2.4.m4.2.3.2"><ci id="S6.SS2.8.2.p2.4.m4.1.1.cmml" xref="S6.SS2.8.2.p2.4.m4.1.1">dom</ci><apply id="S6.SS2.8.2.p2.4.m4.2.2.cmml" xref="S6.SS2.8.2.p2.4.m4.2.2"><ci id="S6.SS2.8.2.p2.4.m4.2.2.1.cmml" xref="S6.SS2.8.2.p2.4.m4.2.2.1">^</ci><ci id="S6.SS2.8.2.p2.4.m4.2.2.2.cmml" xref="S6.SS2.8.2.p2.4.m4.2.2.2">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.8.2.p2.4.m4.2c">\operatorname{dom}(\hat{p})</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.8.2.p2.4.m4.2d">roman_dom ( over^ start_ARG italic_p end_ARG )</annotation></semantics></math> is linearly ordered by <math alttext="<_{b}" class="ltx_Math" display="inline" id="S6.SS2.8.2.p2.5.m5.1"><semantics id="S6.SS2.8.2.p2.5.m5.1a"><msub id="S6.SS2.8.2.p2.5.m5.1.1" xref="S6.SS2.8.2.p2.5.m5.1.1.cmml"><mo id="S6.SS2.8.2.p2.5.m5.1.1.2" xref="S6.SS2.8.2.p2.5.m5.1.1.2.cmml"><</mo><mi id="S6.SS2.8.2.p2.5.m5.1.1.3" xref="S6.SS2.8.2.p2.5.m5.1.1.3.cmml">b</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.8.2.p2.5.m5.1b"><apply id="S6.SS2.8.2.p2.5.m5.1.1.cmml" xref="S6.SS2.8.2.p2.5.m5.1.1"><csymbol cd="ambiguous" id="S6.SS2.8.2.p2.5.m5.1.1.1.cmml" xref="S6.SS2.8.2.p2.5.m5.1.1">subscript</csymbol><lt id="S6.SS2.8.2.p2.5.m5.1.1.2.cmml" xref="S6.SS2.8.2.p2.5.m5.1.1.2"></lt><ci id="S6.SS2.8.2.p2.5.m5.1.1.3.cmml" xref="S6.SS2.8.2.p2.5.m5.1.1.3">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.8.2.p2.5.m5.1c"><_{b}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.8.2.p2.5.m5.1d">< start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT</annotation></semantics></math>. That <math alttext="\hat{p}" class="ltx_Math" display="inline" id="S6.SS2.8.2.p2.6.m6.1"><semantics id="S6.SS2.8.2.p2.6.m6.1a"><mover accent="true" id="S6.SS2.8.2.p2.6.m6.1.1" xref="S6.SS2.8.2.p2.6.m6.1.1.cmml"><mi id="S6.SS2.8.2.p2.6.m6.1.1.2" xref="S6.SS2.8.2.p2.6.m6.1.1.2.cmml">p</mi><mo id="S6.SS2.8.2.p2.6.m6.1.1.1" xref="S6.SS2.8.2.p2.6.m6.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="S6.SS2.8.2.p2.6.m6.1b"><apply id="S6.SS2.8.2.p2.6.m6.1.1.cmml" xref="S6.SS2.8.2.p2.6.m6.1.1"><ci id="S6.SS2.8.2.p2.6.m6.1.1.1.cmml" xref="S6.SS2.8.2.p2.6.m6.1.1.1">^</ci><ci id="S6.SS2.8.2.p2.6.m6.1.1.2.cmml" xref="S6.SS2.8.2.p2.6.m6.1.1.2">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.8.2.p2.6.m6.1c">\hat{p}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.8.2.p2.6.m6.1d">over^ start_ARG italic_p end_ARG</annotation></semantics></math> is increasing follows from the fact that <math alttext="q" class="ltx_Math" display="inline" id="S6.SS2.8.2.p2.7.m7.1"><semantics id="S6.SS2.8.2.p2.7.m7.1a"><mi id="S6.SS2.8.2.p2.7.m7.1.1" xref="S6.SS2.8.2.p2.7.m7.1.1.cmml">q</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.8.2.p2.7.m7.1b"><ci id="S6.SS2.8.2.p2.7.m7.1.1.cmml" xref="S6.SS2.8.2.p2.7.m7.1.1">𝑞</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.8.2.p2.7.m7.1c">q</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.8.2.p2.7.m7.1d">italic_q</annotation></semantics></math> is in <math alttext="Q_{E}" class="ltx_Math" display="inline" id="S6.SS2.8.2.p2.8.m8.1"><semantics id="S6.SS2.8.2.p2.8.m8.1a"><msub id="S6.SS2.8.2.p2.8.m8.1.1" xref="S6.SS2.8.2.p2.8.m8.1.1.cmml"><mi id="S6.SS2.8.2.p2.8.m8.1.1.2" xref="S6.SS2.8.2.p2.8.m8.1.1.2.cmml">Q</mi><mi id="S6.SS2.8.2.p2.8.m8.1.1.3" xref="S6.SS2.8.2.p2.8.m8.1.1.3.cmml">E</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.8.2.p2.8.m8.1b"><apply id="S6.SS2.8.2.p2.8.m8.1.1.cmml" xref="S6.SS2.8.2.p2.8.m8.1.1"><csymbol cd="ambiguous" id="S6.SS2.8.2.p2.8.m8.1.1.1.cmml" xref="S6.SS2.8.2.p2.8.m8.1.1">subscript</csymbol><ci id="S6.SS2.8.2.p2.8.m8.1.1.2.cmml" xref="S6.SS2.8.2.p2.8.m8.1.1.2">𝑄</ci><ci id="S6.SS2.8.2.p2.8.m8.1.1.3.cmml" xref="S6.SS2.8.2.p2.8.m8.1.1.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.8.2.p2.8.m8.1c">Q_{E}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.8.2.p2.8.m8.1d">italic_Q start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT</annotation></semantics></math>. That <math alttext="q\in Q_{E}" class="ltx_Math" display="inline" id="S6.SS2.8.2.p2.9.m9.1"><semantics id="S6.SS2.8.2.p2.9.m9.1a"><mrow id="S6.SS2.8.2.p2.9.m9.1.1" xref="S6.SS2.8.2.p2.9.m9.1.1.cmml"><mi id="S6.SS2.8.2.p2.9.m9.1.1.2" xref="S6.SS2.8.2.p2.9.m9.1.1.2.cmml">q</mi><mo id="S6.SS2.8.2.p2.9.m9.1.1.1" xref="S6.SS2.8.2.p2.9.m9.1.1.1.cmml">∈</mo><msub id="S6.SS2.8.2.p2.9.m9.1.1.3" xref="S6.SS2.8.2.p2.9.m9.1.1.3.cmml"><mi id="S6.SS2.8.2.p2.9.m9.1.1.3.2" xref="S6.SS2.8.2.p2.9.m9.1.1.3.2.cmml">Q</mi><mi id="S6.SS2.8.2.p2.9.m9.1.1.3.3" xref="S6.SS2.8.2.p2.9.m9.1.1.3.3.cmml">E</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.8.2.p2.9.m9.1b"><apply id="S6.SS2.8.2.p2.9.m9.1.1.cmml" xref="S6.SS2.8.2.p2.9.m9.1.1"><in id="S6.SS2.8.2.p2.9.m9.1.1.1.cmml" xref="S6.SS2.8.2.p2.9.m9.1.1.1"></in><ci id="S6.SS2.8.2.p2.9.m9.1.1.2.cmml" xref="S6.SS2.8.2.p2.9.m9.1.1.2">𝑞</ci><apply id="S6.SS2.8.2.p2.9.m9.1.1.3.cmml" xref="S6.SS2.8.2.p2.9.m9.1.1.3"><csymbol cd="ambiguous" id="S6.SS2.8.2.p2.9.m9.1.1.3.1.cmml" xref="S6.SS2.8.2.p2.9.m9.1.1.3">subscript</csymbol><ci id="S6.SS2.8.2.p2.9.m9.1.1.3.2.cmml" xref="S6.SS2.8.2.p2.9.m9.1.1.3.2">𝑄</ci><ci id="S6.SS2.8.2.p2.9.m9.1.1.3.3.cmml" xref="S6.SS2.8.2.p2.9.m9.1.1.3.3">𝐸</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.8.2.p2.9.m9.1c">q\in Q_{E}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.8.2.p2.9.m9.1d">italic_q ∈ italic_Q start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT</annotation></semantics></math> also directly implies that <math alttext="p" class="ltx_Math" display="inline" id="S6.SS2.8.2.p2.10.m10.1"><semantics id="S6.SS2.8.2.p2.10.m10.1a"><mi id="S6.SS2.8.2.p2.10.m10.1.1" xref="S6.SS2.8.2.p2.10.m10.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.8.2.p2.10.m10.1b"><ci id="S6.SS2.8.2.p2.10.m10.1.1.cmml" xref="S6.SS2.8.2.p2.10.m10.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.8.2.p2.10.m10.1c">p</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.8.2.p2.10.m10.1d">italic_p</annotation></semantics></math> satisfies <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.Thmtheorem9" title="Definition 6.9. ‣ 6.2. Forcing epimorphisms ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">6.9</span></a> <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.I4.i1" title="Item (i) ‣ Definition 6.9. ‣ 6.2. Forcing epimorphisms ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">(i)</span></a>, and <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.Thmtheorem9" title="Definition 6.9. ‣ 6.2. Forcing epimorphisms ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">6.9</span></a> <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.I4.i2" title="Item (ii) ‣ Definition 6.9. ‣ 6.2. Forcing epimorphisms ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">(ii)</span></a> follows from the fact that <math alttext="q\in Q_{E}" class="ltx_Math" display="inline" id="S6.SS2.8.2.p2.11.m11.1"><semantics id="S6.SS2.8.2.p2.11.m11.1a"><mrow id="S6.SS2.8.2.p2.11.m11.1.1" xref="S6.SS2.8.2.p2.11.m11.1.1.cmml"><mi id="S6.SS2.8.2.p2.11.m11.1.1.2" xref="S6.SS2.8.2.p2.11.m11.1.1.2.cmml">q</mi><mo id="S6.SS2.8.2.p2.11.m11.1.1.1" xref="S6.SS2.8.2.p2.11.m11.1.1.1.cmml">∈</mo><msub id="S6.SS2.8.2.p2.11.m11.1.1.3" xref="S6.SS2.8.2.p2.11.m11.1.1.3.cmml"><mi id="S6.SS2.8.2.p2.11.m11.1.1.3.2" xref="S6.SS2.8.2.p2.11.m11.1.1.3.2.cmml">Q</mi><mi id="S6.SS2.8.2.p2.11.m11.1.1.3.3" xref="S6.SS2.8.2.p2.11.m11.1.1.3.3.cmml">E</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.8.2.p2.11.m11.1b"><apply id="S6.SS2.8.2.p2.11.m11.1.1.cmml" xref="S6.SS2.8.2.p2.11.m11.1.1"><in id="S6.SS2.8.2.p2.11.m11.1.1.1.cmml" xref="S6.SS2.8.2.p2.11.m11.1.1.1"></in><ci id="S6.SS2.8.2.p2.11.m11.1.1.2.cmml" xref="S6.SS2.8.2.p2.11.m11.1.1.2">𝑞</ci><apply id="S6.SS2.8.2.p2.11.m11.1.1.3.cmml" xref="S6.SS2.8.2.p2.11.m11.1.1.3"><csymbol cd="ambiguous" id="S6.SS2.8.2.p2.11.m11.1.1.3.1.cmml" xref="S6.SS2.8.2.p2.11.m11.1.1.3">subscript</csymbol><ci id="S6.SS2.8.2.p2.11.m11.1.1.3.2.cmml" xref="S6.SS2.8.2.p2.11.m11.1.1.3.2">𝑄</ci><ci id="S6.SS2.8.2.p2.11.m11.1.1.3.3.cmml" xref="S6.SS2.8.2.p2.11.m11.1.1.3.3">𝐸</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.8.2.p2.11.m11.1c">q\in Q_{E}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.8.2.p2.11.m11.1d">italic_q ∈ italic_Q start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT</annotation></semantics></math> and <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.Thmtheorem11" title="Lemma 6.11. ‣ 6.2. Forcing epimorphisms ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">6.11</span></a> (b). Finally <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.Thmtheorem9" title="Definition 6.9. ‣ 6.2. Forcing epimorphisms ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">6.9</span></a> <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.I4.i3" title="Item (iii) ‣ Definition 6.9. ‣ 6.2. Forcing epimorphisms ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">(iii)</span></a> follows trivially from the fact that <math alttext="p" class="ltx_Math" display="inline" id="S6.SS2.8.2.p2.12.m12.1"><semantics id="S6.SS2.8.2.p2.12.m12.1a"><mi id="S6.SS2.8.2.p2.12.m12.1.1" xref="S6.SS2.8.2.p2.12.m12.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.8.2.p2.12.m12.1b"><ci id="S6.SS2.8.2.p2.12.m12.1.1.cmml" xref="S6.SS2.8.2.p2.12.m12.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.8.2.p2.12.m12.1c">p</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.8.2.p2.12.m12.1d">italic_p</annotation></semantics></math> and <math alttext="p^{\prime}" class="ltx_Math" display="inline" id="S6.SS2.8.2.p2.13.m13.1"><semantics id="S6.SS2.8.2.p2.13.m13.1a"><msup id="S6.SS2.8.2.p2.13.m13.1.1" xref="S6.SS2.8.2.p2.13.m13.1.1.cmml"><mi id="S6.SS2.8.2.p2.13.m13.1.1.2" xref="S6.SS2.8.2.p2.13.m13.1.1.2.cmml">p</mi><mo id="S6.SS2.8.2.p2.13.m13.1.1.3" xref="S6.SS2.8.2.p2.13.m13.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S6.SS2.8.2.p2.13.m13.1b"><apply id="S6.SS2.8.2.p2.13.m13.1.1.cmml" xref="S6.SS2.8.2.p2.13.m13.1.1"><csymbol cd="ambiguous" id="S6.SS2.8.2.p2.13.m13.1.1.1.cmml" xref="S6.SS2.8.2.p2.13.m13.1.1">superscript</csymbol><ci id="S6.SS2.8.2.p2.13.m13.1.1.2.cmml" xref="S6.SS2.8.2.p2.13.m13.1.1.2">𝑝</ci><ci id="S6.SS2.8.2.p2.13.m13.1.1.3.cmml" xref="S6.SS2.8.2.p2.13.m13.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.8.2.p2.13.m13.1c">p^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.8.2.p2.13.m13.1d">italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> satisfy it. ∎</p> </div> </div> <div class="ltx_para" id="S6.SS2.9.p2"> <p class="ltx_p" id="S6.SS2.9.p2.1">Now we prove that such a refinement is always possible.</p> </div> <div class="ltx_theorem ltx_theorem_claim" id="S6.Thmtheorem14"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_italic" id="S6.Thmtheorem14.1.1.1">Claim 6.14</span></span><span class="ltx_text ltx_font_italic" id="S6.Thmtheorem14.2.2">.</span> </h6> <div class="ltx_para" id="S6.Thmtheorem14.p1"> <p class="ltx_p" id="S6.Thmtheorem14.p1.3">There is an uncountable <math alttext="H^{\prime}\subseteq H" class="ltx_Math" display="inline" id="S6.Thmtheorem14.p1.1.m1.1"><semantics id="S6.Thmtheorem14.p1.1.m1.1a"><mrow id="S6.Thmtheorem14.p1.1.m1.1.1" xref="S6.Thmtheorem14.p1.1.m1.1.1.cmml"><msup id="S6.Thmtheorem14.p1.1.m1.1.1.2" xref="S6.Thmtheorem14.p1.1.m1.1.1.2.cmml"><mi id="S6.Thmtheorem14.p1.1.m1.1.1.2.2" xref="S6.Thmtheorem14.p1.1.m1.1.1.2.2.cmml">H</mi><mo id="S6.Thmtheorem14.p1.1.m1.1.1.2.3" xref="S6.Thmtheorem14.p1.1.m1.1.1.2.3.cmml">′</mo></msup><mo id="S6.Thmtheorem14.p1.1.m1.1.1.1" xref="S6.Thmtheorem14.p1.1.m1.1.1.1.cmml">⊆</mo><mi id="S6.Thmtheorem14.p1.1.m1.1.1.3" xref="S6.Thmtheorem14.p1.1.m1.1.1.3.cmml">H</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem14.p1.1.m1.1b"><apply id="S6.Thmtheorem14.p1.1.m1.1.1.cmml" xref="S6.Thmtheorem14.p1.1.m1.1.1"><subset id="S6.Thmtheorem14.p1.1.m1.1.1.1.cmml" xref="S6.Thmtheorem14.p1.1.m1.1.1.1"></subset><apply id="S6.Thmtheorem14.p1.1.m1.1.1.2.cmml" xref="S6.Thmtheorem14.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S6.Thmtheorem14.p1.1.m1.1.1.2.1.cmml" xref="S6.Thmtheorem14.p1.1.m1.1.1.2">superscript</csymbol><ci id="S6.Thmtheorem14.p1.1.m1.1.1.2.2.cmml" xref="S6.Thmtheorem14.p1.1.m1.1.1.2.2">𝐻</ci><ci id="S6.Thmtheorem14.p1.1.m1.1.1.2.3.cmml" xref="S6.Thmtheorem14.p1.1.m1.1.1.2.3">′</ci></apply><ci id="S6.Thmtheorem14.p1.1.m1.1.1.3.cmml" xref="S6.Thmtheorem14.p1.1.m1.1.1.3">𝐻</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem14.p1.1.m1.1c">H^{\prime}\subseteq H</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem14.p1.1.m1.1d">italic_H start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⊆ italic_H</annotation></semantics></math> satisfying that for all <math alttext="p\neq p^{\prime}\in H^{\prime}" class="ltx_Math" display="inline" id="S6.Thmtheorem14.p1.2.m2.1"><semantics id="S6.Thmtheorem14.p1.2.m2.1a"><mrow id="S6.Thmtheorem14.p1.2.m2.1.1" xref="S6.Thmtheorem14.p1.2.m2.1.1.cmml"><mi id="S6.Thmtheorem14.p1.2.m2.1.1.2" xref="S6.Thmtheorem14.p1.2.m2.1.1.2.cmml">p</mi><mo id="S6.Thmtheorem14.p1.2.m2.1.1.3" xref="S6.Thmtheorem14.p1.2.m2.1.1.3.cmml">≠</mo><msup id="S6.Thmtheorem14.p1.2.m2.1.1.4" xref="S6.Thmtheorem14.p1.2.m2.1.1.4.cmml"><mi id="S6.Thmtheorem14.p1.2.m2.1.1.4.2" xref="S6.Thmtheorem14.p1.2.m2.1.1.4.2.cmml">p</mi><mo id="S6.Thmtheorem14.p1.2.m2.1.1.4.3" xref="S6.Thmtheorem14.p1.2.m2.1.1.4.3.cmml">′</mo></msup><mo id="S6.Thmtheorem14.p1.2.m2.1.1.5" xref="S6.Thmtheorem14.p1.2.m2.1.1.5.cmml">∈</mo><msup id="S6.Thmtheorem14.p1.2.m2.1.1.6" xref="S6.Thmtheorem14.p1.2.m2.1.1.6.cmml"><mi id="S6.Thmtheorem14.p1.2.m2.1.1.6.2" xref="S6.Thmtheorem14.p1.2.m2.1.1.6.2.cmml">H</mi><mo id="S6.Thmtheorem14.p1.2.m2.1.1.6.3" xref="S6.Thmtheorem14.p1.2.m2.1.1.6.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem14.p1.2.m2.1b"><apply id="S6.Thmtheorem14.p1.2.m2.1.1.cmml" xref="S6.Thmtheorem14.p1.2.m2.1.1"><and id="S6.Thmtheorem14.p1.2.m2.1.1a.cmml" xref="S6.Thmtheorem14.p1.2.m2.1.1"></and><apply id="S6.Thmtheorem14.p1.2.m2.1.1b.cmml" xref="S6.Thmtheorem14.p1.2.m2.1.1"><neq id="S6.Thmtheorem14.p1.2.m2.1.1.3.cmml" xref="S6.Thmtheorem14.p1.2.m2.1.1.3"></neq><ci id="S6.Thmtheorem14.p1.2.m2.1.1.2.cmml" xref="S6.Thmtheorem14.p1.2.m2.1.1.2">𝑝</ci><apply id="S6.Thmtheorem14.p1.2.m2.1.1.4.cmml" xref="S6.Thmtheorem14.p1.2.m2.1.1.4"><csymbol cd="ambiguous" id="S6.Thmtheorem14.p1.2.m2.1.1.4.1.cmml" xref="S6.Thmtheorem14.p1.2.m2.1.1.4">superscript</csymbol><ci id="S6.Thmtheorem14.p1.2.m2.1.1.4.2.cmml" xref="S6.Thmtheorem14.p1.2.m2.1.1.4.2">𝑝</ci><ci id="S6.Thmtheorem14.p1.2.m2.1.1.4.3.cmml" xref="S6.Thmtheorem14.p1.2.m2.1.1.4.3">′</ci></apply></apply><apply id="S6.Thmtheorem14.p1.2.m2.1.1c.cmml" xref="S6.Thmtheorem14.p1.2.m2.1.1"><in id="S6.Thmtheorem14.p1.2.m2.1.1.5.cmml" xref="S6.Thmtheorem14.p1.2.m2.1.1.5"></in><share href="https://arxiv.org/html/2503.13728v1#S6.Thmtheorem14.p1.2.m2.1.1.4.cmml" id="S6.Thmtheorem14.p1.2.m2.1.1d.cmml" xref="S6.Thmtheorem14.p1.2.m2.1.1"></share><apply id="S6.Thmtheorem14.p1.2.m2.1.1.6.cmml" xref="S6.Thmtheorem14.p1.2.m2.1.1.6"><csymbol cd="ambiguous" id="S6.Thmtheorem14.p1.2.m2.1.1.6.1.cmml" xref="S6.Thmtheorem14.p1.2.m2.1.1.6">superscript</csymbol><ci id="S6.Thmtheorem14.p1.2.m2.1.1.6.2.cmml" xref="S6.Thmtheorem14.p1.2.m2.1.1.6.2">𝐻</ci><ci id="S6.Thmtheorem14.p1.2.m2.1.1.6.3.cmml" xref="S6.Thmtheorem14.p1.2.m2.1.1.6.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem14.p1.2.m2.1c">p\neq p^{\prime}\in H^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem14.p1.2.m2.1d">italic_p ≠ italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_H start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="\operatorname{dom}(f_{p})\cap\operatorname{dom}(f_{p^{\prime}})=\varnothing" class="ltx_Math" display="inline" id="S6.Thmtheorem14.p1.3.m3.4"><semantics id="S6.Thmtheorem14.p1.3.m3.4a"><mrow id="S6.Thmtheorem14.p1.3.m3.4.4" xref="S6.Thmtheorem14.p1.3.m3.4.4.cmml"><mrow id="S6.Thmtheorem14.p1.3.m3.4.4.2" xref="S6.Thmtheorem14.p1.3.m3.4.4.2.cmml"><mrow id="S6.Thmtheorem14.p1.3.m3.3.3.1.1.1" xref="S6.Thmtheorem14.p1.3.m3.3.3.1.1.2.cmml"><mi id="S6.Thmtheorem14.p1.3.m3.1.1" xref="S6.Thmtheorem14.p1.3.m3.1.1.cmml">dom</mi><mo id="S6.Thmtheorem14.p1.3.m3.3.3.1.1.1a" xref="S6.Thmtheorem14.p1.3.m3.3.3.1.1.2.cmml">⁡</mo><mrow id="S6.Thmtheorem14.p1.3.m3.3.3.1.1.1.1" xref="S6.Thmtheorem14.p1.3.m3.3.3.1.1.2.cmml"><mo id="S6.Thmtheorem14.p1.3.m3.3.3.1.1.1.1.2" stretchy="false" xref="S6.Thmtheorem14.p1.3.m3.3.3.1.1.2.cmml">(</mo><msub id="S6.Thmtheorem14.p1.3.m3.3.3.1.1.1.1.1" xref="S6.Thmtheorem14.p1.3.m3.3.3.1.1.1.1.1.cmml"><mi id="S6.Thmtheorem14.p1.3.m3.3.3.1.1.1.1.1.2" xref="S6.Thmtheorem14.p1.3.m3.3.3.1.1.1.1.1.2.cmml">f</mi><mi id="S6.Thmtheorem14.p1.3.m3.3.3.1.1.1.1.1.3" xref="S6.Thmtheorem14.p1.3.m3.3.3.1.1.1.1.1.3.cmml">p</mi></msub><mo id="S6.Thmtheorem14.p1.3.m3.3.3.1.1.1.1.3" stretchy="false" xref="S6.Thmtheorem14.p1.3.m3.3.3.1.1.2.cmml">)</mo></mrow></mrow><mo id="S6.Thmtheorem14.p1.3.m3.4.4.2.3" xref="S6.Thmtheorem14.p1.3.m3.4.4.2.3.cmml">∩</mo><mrow id="S6.Thmtheorem14.p1.3.m3.4.4.2.2.1" xref="S6.Thmtheorem14.p1.3.m3.4.4.2.2.2.cmml"><mi id="S6.Thmtheorem14.p1.3.m3.2.2" xref="S6.Thmtheorem14.p1.3.m3.2.2.cmml">dom</mi><mo id="S6.Thmtheorem14.p1.3.m3.4.4.2.2.1a" xref="S6.Thmtheorem14.p1.3.m3.4.4.2.2.2.cmml">⁡</mo><mrow id="S6.Thmtheorem14.p1.3.m3.4.4.2.2.1.1" xref="S6.Thmtheorem14.p1.3.m3.4.4.2.2.2.cmml"><mo id="S6.Thmtheorem14.p1.3.m3.4.4.2.2.1.1.2" stretchy="false" xref="S6.Thmtheorem14.p1.3.m3.4.4.2.2.2.cmml">(</mo><msub id="S6.Thmtheorem14.p1.3.m3.4.4.2.2.1.1.1" xref="S6.Thmtheorem14.p1.3.m3.4.4.2.2.1.1.1.cmml"><mi id="S6.Thmtheorem14.p1.3.m3.4.4.2.2.1.1.1.2" xref="S6.Thmtheorem14.p1.3.m3.4.4.2.2.1.1.1.2.cmml">f</mi><msup id="S6.Thmtheorem14.p1.3.m3.4.4.2.2.1.1.1.3" xref="S6.Thmtheorem14.p1.3.m3.4.4.2.2.1.1.1.3.cmml"><mi id="S6.Thmtheorem14.p1.3.m3.4.4.2.2.1.1.1.3.2" xref="S6.Thmtheorem14.p1.3.m3.4.4.2.2.1.1.1.3.2.cmml">p</mi><mo id="S6.Thmtheorem14.p1.3.m3.4.4.2.2.1.1.1.3.3" xref="S6.Thmtheorem14.p1.3.m3.4.4.2.2.1.1.1.3.3.cmml">′</mo></msup></msub><mo id="S6.Thmtheorem14.p1.3.m3.4.4.2.2.1.1.3" stretchy="false" xref="S6.Thmtheorem14.p1.3.m3.4.4.2.2.2.cmml">)</mo></mrow></mrow></mrow><mo id="S6.Thmtheorem14.p1.3.m3.4.4.3" xref="S6.Thmtheorem14.p1.3.m3.4.4.3.cmml">=</mo><mi id="S6.Thmtheorem14.p1.3.m3.4.4.4" mathvariant="normal" xref="S6.Thmtheorem14.p1.3.m3.4.4.4.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem14.p1.3.m3.4b"><apply id="S6.Thmtheorem14.p1.3.m3.4.4.cmml" xref="S6.Thmtheorem14.p1.3.m3.4.4"><eq id="S6.Thmtheorem14.p1.3.m3.4.4.3.cmml" xref="S6.Thmtheorem14.p1.3.m3.4.4.3"></eq><apply id="S6.Thmtheorem14.p1.3.m3.4.4.2.cmml" xref="S6.Thmtheorem14.p1.3.m3.4.4.2"><intersect id="S6.Thmtheorem14.p1.3.m3.4.4.2.3.cmml" xref="S6.Thmtheorem14.p1.3.m3.4.4.2.3"></intersect><apply id="S6.Thmtheorem14.p1.3.m3.3.3.1.1.2.cmml" xref="S6.Thmtheorem14.p1.3.m3.3.3.1.1.1"><ci id="S6.Thmtheorem14.p1.3.m3.1.1.cmml" xref="S6.Thmtheorem14.p1.3.m3.1.1">dom</ci><apply id="S6.Thmtheorem14.p1.3.m3.3.3.1.1.1.1.1.cmml" xref="S6.Thmtheorem14.p1.3.m3.3.3.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.Thmtheorem14.p1.3.m3.3.3.1.1.1.1.1.1.cmml" xref="S6.Thmtheorem14.p1.3.m3.3.3.1.1.1.1.1">subscript</csymbol><ci id="S6.Thmtheorem14.p1.3.m3.3.3.1.1.1.1.1.2.cmml" xref="S6.Thmtheorem14.p1.3.m3.3.3.1.1.1.1.1.2">𝑓</ci><ci id="S6.Thmtheorem14.p1.3.m3.3.3.1.1.1.1.1.3.cmml" xref="S6.Thmtheorem14.p1.3.m3.3.3.1.1.1.1.1.3">𝑝</ci></apply></apply><apply id="S6.Thmtheorem14.p1.3.m3.4.4.2.2.2.cmml" xref="S6.Thmtheorem14.p1.3.m3.4.4.2.2.1"><ci id="S6.Thmtheorem14.p1.3.m3.2.2.cmml" xref="S6.Thmtheorem14.p1.3.m3.2.2">dom</ci><apply id="S6.Thmtheorem14.p1.3.m3.4.4.2.2.1.1.1.cmml" xref="S6.Thmtheorem14.p1.3.m3.4.4.2.2.1.1.1"><csymbol cd="ambiguous" id="S6.Thmtheorem14.p1.3.m3.4.4.2.2.1.1.1.1.cmml" xref="S6.Thmtheorem14.p1.3.m3.4.4.2.2.1.1.1">subscript</csymbol><ci id="S6.Thmtheorem14.p1.3.m3.4.4.2.2.1.1.1.2.cmml" xref="S6.Thmtheorem14.p1.3.m3.4.4.2.2.1.1.1.2">𝑓</ci><apply id="S6.Thmtheorem14.p1.3.m3.4.4.2.2.1.1.1.3.cmml" xref="S6.Thmtheorem14.p1.3.m3.4.4.2.2.1.1.1.3"><csymbol cd="ambiguous" id="S6.Thmtheorem14.p1.3.m3.4.4.2.2.1.1.1.3.1.cmml" xref="S6.Thmtheorem14.p1.3.m3.4.4.2.2.1.1.1.3">superscript</csymbol><ci id="S6.Thmtheorem14.p1.3.m3.4.4.2.2.1.1.1.3.2.cmml" xref="S6.Thmtheorem14.p1.3.m3.4.4.2.2.1.1.1.3.2">𝑝</ci><ci id="S6.Thmtheorem14.p1.3.m3.4.4.2.2.1.1.1.3.3.cmml" xref="S6.Thmtheorem14.p1.3.m3.4.4.2.2.1.1.1.3.3">′</ci></apply></apply></apply></apply><emptyset id="S6.Thmtheorem14.p1.3.m3.4.4.4.cmml" xref="S6.Thmtheorem14.p1.3.m3.4.4.4"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem14.p1.3.m3.4c">\operatorname{dom}(f_{p})\cap\operatorname{dom}(f_{p^{\prime}})=\varnothing</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem14.p1.3.m3.4d">roman_dom ( italic_f start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ) ∩ roman_dom ( italic_f start_POSTSUBSCRIPT italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) = ∅</annotation></semantics></math>.</p> </div> </div> <div class="ltx_proof" id="S6.SS2.13.6"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S6.SS2.10.3.p1"> <p class="ltx_p" id="S6.SS2.10.3.p1.13">By <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.Thmtheorem9" title="Definition 6.9. ‣ 6.2. Forcing epimorphisms ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">6.9</span></a> <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.I4.i1" title="Item (i) ‣ Definition 6.9. ‣ 6.2. Forcing epimorphisms ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">(i)</span></a> we can find <math alttext="\langle p_{\xi}:\xi<\omega_{1}\rangle" class="ltx_math_unparsed" display="inline" id="S6.SS2.10.3.p1.1.m1.1"><semantics id="S6.SS2.10.3.p1.1.m1.1a"><mrow id="S6.SS2.10.3.p1.1.m1.1b"><mo id="S6.SS2.10.3.p1.1.m1.1.1" stretchy="false">⟨</mo><msub id="S6.SS2.10.3.p1.1.m1.1.2"><mi id="S6.SS2.10.3.p1.1.m1.1.2.2">p</mi><mi id="S6.SS2.10.3.p1.1.m1.1.2.3">ξ</mi></msub><mo id="S6.SS2.10.3.p1.1.m1.1.3" lspace="0.278em" rspace="0.278em">:</mo><mi id="S6.SS2.10.3.p1.1.m1.1.4">ξ</mi><mo id="S6.SS2.10.3.p1.1.m1.1.5"><</mo><msub id="S6.SS2.10.3.p1.1.m1.1.6"><mi id="S6.SS2.10.3.p1.1.m1.1.6.2">ω</mi><mn id="S6.SS2.10.3.p1.1.m1.1.6.3">1</mn></msub><mo id="S6.SS2.10.3.p1.1.m1.1.7" stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex" id="S6.SS2.10.3.p1.1.m1.1c">\langle p_{\xi}:\xi<\omega_{1}\rangle</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.10.3.p1.1.m1.1d">⟨ italic_p start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT : italic_ξ < italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⟩</annotation></semantics></math> contained in <math alttext="H" class="ltx_Math" display="inline" id="S6.SS2.10.3.p1.2.m2.1"><semantics id="S6.SS2.10.3.p1.2.m2.1a"><mi id="S6.SS2.10.3.p1.2.m2.1.1" xref="S6.SS2.10.3.p1.2.m2.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.10.3.p1.2.m2.1b"><ci id="S6.SS2.10.3.p1.2.m2.1.1.cmml" xref="S6.SS2.10.3.p1.2.m2.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.10.3.p1.2.m2.1c">H</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.10.3.p1.2.m2.1d">italic_H</annotation></semantics></math> such that for all <math alttext="\xi<\eta" class="ltx_Math" display="inline" id="S6.SS2.10.3.p1.3.m3.1"><semantics id="S6.SS2.10.3.p1.3.m3.1a"><mrow id="S6.SS2.10.3.p1.3.m3.1.1" xref="S6.SS2.10.3.p1.3.m3.1.1.cmml"><mi id="S6.SS2.10.3.p1.3.m3.1.1.2" xref="S6.SS2.10.3.p1.3.m3.1.1.2.cmml">ξ</mi><mo id="S6.SS2.10.3.p1.3.m3.1.1.1" xref="S6.SS2.10.3.p1.3.m3.1.1.1.cmml"><</mo><mi id="S6.SS2.10.3.p1.3.m3.1.1.3" xref="S6.SS2.10.3.p1.3.m3.1.1.3.cmml">η</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.10.3.p1.3.m3.1b"><apply id="S6.SS2.10.3.p1.3.m3.1.1.cmml" xref="S6.SS2.10.3.p1.3.m3.1.1"><lt id="S6.SS2.10.3.p1.3.m3.1.1.1.cmml" xref="S6.SS2.10.3.p1.3.m3.1.1.1"></lt><ci id="S6.SS2.10.3.p1.3.m3.1.1.2.cmml" xref="S6.SS2.10.3.p1.3.m3.1.1.2">𝜉</ci><ci id="S6.SS2.10.3.p1.3.m3.1.1.3.cmml" xref="S6.SS2.10.3.p1.3.m3.1.1.3">𝜂</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.10.3.p1.3.m3.1c">\xi<\eta</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.10.3.p1.3.m3.1d">italic_ξ < italic_η</annotation></semantics></math>, <math alttext="\max\{\nu(\bar{a}):\bar{a}\in\operatorname{dom}(p_{\xi})\}<\min\{\nu(\bar{a}):% \bar{a}\in\operatorname{dom}(p_{\eta})\}" class="ltx_Math" display="inline" id="S6.SS2.10.3.p1.4.m4.8"><semantics id="S6.SS2.10.3.p1.4.m4.8a"><mrow id="S6.SS2.10.3.p1.4.m4.8.8" xref="S6.SS2.10.3.p1.4.m4.8.8.cmml"><mrow id="S6.SS2.10.3.p1.4.m4.7.7.1.1" xref="S6.SS2.10.3.p1.4.m4.7.7.1.2.cmml"><mi id="S6.SS2.10.3.p1.4.m4.3.3" xref="S6.SS2.10.3.p1.4.m4.3.3.cmml">max</mi><mo id="S6.SS2.10.3.p1.4.m4.7.7.1.1a" xref="S6.SS2.10.3.p1.4.m4.7.7.1.2.cmml">⁡</mo><mrow id="S6.SS2.10.3.p1.4.m4.7.7.1.1.1" xref="S6.SS2.10.3.p1.4.m4.7.7.1.2.cmml"><mo id="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.2" stretchy="false" xref="S6.SS2.10.3.p1.4.m4.7.7.1.2.cmml">{</mo><mrow id="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1" xref="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.cmml"><mrow id="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.3" xref="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.3.cmml"><mi id="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.3.2" xref="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.3.2.cmml">ν</mi><mo id="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.3.1" xref="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.3.1.cmml">⁢</mo><mrow id="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.3.3.2" xref="S6.SS2.10.3.p1.4.m4.1.1.cmml"><mo id="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.3.3.2.1" stretchy="false" xref="S6.SS2.10.3.p1.4.m4.1.1.cmml">(</mo><mover accent="true" id="S6.SS2.10.3.p1.4.m4.1.1" xref="S6.SS2.10.3.p1.4.m4.1.1.cmml"><mi id="S6.SS2.10.3.p1.4.m4.1.1.2" xref="S6.SS2.10.3.p1.4.m4.1.1.2.cmml">a</mi><mo id="S6.SS2.10.3.p1.4.m4.1.1.1" xref="S6.SS2.10.3.p1.4.m4.1.1.1.cmml">¯</mo></mover><mo id="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.3.3.2.2" rspace="0.278em" stretchy="false" xref="S6.SS2.10.3.p1.4.m4.1.1.cmml">)</mo></mrow></mrow><mo id="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.2" rspace="0.278em" xref="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.2.cmml">:</mo><mrow id="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.1" xref="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.1.cmml"><mover accent="true" id="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.1.3" xref="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.1.3.cmml"><mi id="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.1.3.2" xref="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.1.3.2.cmml">a</mi><mo id="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.1.3.1" xref="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.1.3.1.cmml">¯</mo></mover><mo id="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.1.2" xref="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.1.2.cmml">∈</mo><mrow id="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.1.1.1" xref="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.1.1.2.cmml"><mi id="S6.SS2.10.3.p1.4.m4.2.2" xref="S6.SS2.10.3.p1.4.m4.2.2.cmml">dom</mi><mo id="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.1.1.1a" xref="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.1.1.2.cmml">⁡</mo><mrow id="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.1.1.1.1" xref="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.1.1.2.cmml"><mo id="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.1.1.2.cmml">(</mo><msub id="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.1.1.1.1.1" xref="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.1.1.1.1.1.cmml"><mi id="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.1.1.1.1.1.2" xref="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.1.1.1.1.1.2.cmml">p</mi><mi id="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.1.1.1.1.1.3" xref="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.1.1.1.1.1.3.cmml">ξ</mi></msub><mo id="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.1.1.2.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.3" stretchy="false" xref="S6.SS2.10.3.p1.4.m4.7.7.1.2.cmml">}</mo></mrow></mrow><mo id="S6.SS2.10.3.p1.4.m4.8.8.3" xref="S6.SS2.10.3.p1.4.m4.8.8.3.cmml"><</mo><mrow id="S6.SS2.10.3.p1.4.m4.8.8.2.1" xref="S6.SS2.10.3.p1.4.m4.8.8.2.2.cmml"><mi id="S6.SS2.10.3.p1.4.m4.6.6" xref="S6.SS2.10.3.p1.4.m4.6.6.cmml">min</mi><mo id="S6.SS2.10.3.p1.4.m4.8.8.2.1a" xref="S6.SS2.10.3.p1.4.m4.8.8.2.2.cmml">⁡</mo><mrow id="S6.SS2.10.3.p1.4.m4.8.8.2.1.1" xref="S6.SS2.10.3.p1.4.m4.8.8.2.2.cmml"><mo id="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.2" stretchy="false" xref="S6.SS2.10.3.p1.4.m4.8.8.2.2.cmml">{</mo><mrow id="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1" xref="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.cmml"><mrow id="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.3" xref="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.3.cmml"><mi id="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.3.2" xref="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.3.2.cmml">ν</mi><mo id="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.3.1" xref="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.3.1.cmml">⁢</mo><mrow id="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.3.3.2" xref="S6.SS2.10.3.p1.4.m4.4.4.cmml"><mo id="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.3.3.2.1" stretchy="false" xref="S6.SS2.10.3.p1.4.m4.4.4.cmml">(</mo><mover accent="true" id="S6.SS2.10.3.p1.4.m4.4.4" xref="S6.SS2.10.3.p1.4.m4.4.4.cmml"><mi id="S6.SS2.10.3.p1.4.m4.4.4.2" xref="S6.SS2.10.3.p1.4.m4.4.4.2.cmml">a</mi><mo id="S6.SS2.10.3.p1.4.m4.4.4.1" xref="S6.SS2.10.3.p1.4.m4.4.4.1.cmml">¯</mo></mover><mo id="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.3.3.2.2" rspace="0.278em" stretchy="false" xref="S6.SS2.10.3.p1.4.m4.4.4.cmml">)</mo></mrow></mrow><mo id="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.2" rspace="0.278em" xref="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.2.cmml">:</mo><mrow id="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.1" xref="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.1.cmml"><mover accent="true" id="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.1.3" xref="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.1.3.cmml"><mi id="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.1.3.2" xref="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.1.3.2.cmml">a</mi><mo id="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.1.3.1" xref="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.1.3.1.cmml">¯</mo></mover><mo id="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.1.2" xref="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.1.2.cmml">∈</mo><mrow id="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.1.1.1" xref="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.1.1.2.cmml"><mi id="S6.SS2.10.3.p1.4.m4.5.5" xref="S6.SS2.10.3.p1.4.m4.5.5.cmml">dom</mi><mo id="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.1.1.1a" xref="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.1.1.2.cmml">⁡</mo><mrow id="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.1.1.1.1" xref="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.1.1.2.cmml"><mo id="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.1.1.1.1.2" stretchy="false" xref="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.1.1.2.cmml">(</mo><msub id="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.1.1.1.1.1" xref="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.1.1.1.1.1.cmml"><mi id="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.1.1.1.1.1.2" xref="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.1.1.1.1.1.2.cmml">p</mi><mi id="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.1.1.1.1.1.3" xref="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.1.1.1.1.1.3.cmml">η</mi></msub><mo id="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.1.1.1.1.3" stretchy="false" xref="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.1.1.2.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.3" stretchy="false" xref="S6.SS2.10.3.p1.4.m4.8.8.2.2.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.10.3.p1.4.m4.8b"><apply id="S6.SS2.10.3.p1.4.m4.8.8.cmml" xref="S6.SS2.10.3.p1.4.m4.8.8"><lt id="S6.SS2.10.3.p1.4.m4.8.8.3.cmml" xref="S6.SS2.10.3.p1.4.m4.8.8.3"></lt><apply id="S6.SS2.10.3.p1.4.m4.7.7.1.2.cmml" xref="S6.SS2.10.3.p1.4.m4.7.7.1.1"><max id="S6.SS2.10.3.p1.4.m4.3.3.cmml" xref="S6.SS2.10.3.p1.4.m4.3.3"></max><apply id="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.cmml" xref="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1"><ci id="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.2.cmml" xref="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.2">:</ci><apply id="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.3.cmml" xref="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.3"><times id="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.3.1.cmml" xref="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.3.1"></times><ci id="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.3.2.cmml" xref="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.3.2">𝜈</ci><apply id="S6.SS2.10.3.p1.4.m4.1.1.cmml" xref="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.3.3.2"><ci id="S6.SS2.10.3.p1.4.m4.1.1.1.cmml" xref="S6.SS2.10.3.p1.4.m4.1.1.1">¯</ci><ci id="S6.SS2.10.3.p1.4.m4.1.1.2.cmml" xref="S6.SS2.10.3.p1.4.m4.1.1.2">𝑎</ci></apply></apply><apply id="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.1.cmml" xref="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.1"><in id="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.1.2.cmml" xref="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.1.2"></in><apply id="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.1.3.cmml" xref="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.1.3"><ci id="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.1.3.1.cmml" xref="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.1.3.1">¯</ci><ci id="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.1.3.2.cmml" xref="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.1.3.2">𝑎</ci></apply><apply id="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.1.1.2.cmml" xref="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.1.1.1"><ci id="S6.SS2.10.3.p1.4.m4.2.2.cmml" xref="S6.SS2.10.3.p1.4.m4.2.2">dom</ci><apply id="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.1.1.1.1.1.cmml" xref="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.1.1.1.1.1.1.cmml" xref="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.1.1.1.1.1.2.cmml" xref="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.1.1.1.1.1.2">𝑝</ci><ci id="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.1.1.1.1.1.3.cmml" xref="S6.SS2.10.3.p1.4.m4.7.7.1.1.1.1.1.1.1.1.1.3">𝜉</ci></apply></apply></apply></apply></apply><apply id="S6.SS2.10.3.p1.4.m4.8.8.2.2.cmml" xref="S6.SS2.10.3.p1.4.m4.8.8.2.1"><min id="S6.SS2.10.3.p1.4.m4.6.6.cmml" xref="S6.SS2.10.3.p1.4.m4.6.6"></min><apply id="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.cmml" xref="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1"><ci id="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.2.cmml" xref="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.2">:</ci><apply id="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.3.cmml" xref="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.3"><times id="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.3.1.cmml" xref="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.3.1"></times><ci id="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.3.2.cmml" xref="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.3.2">𝜈</ci><apply id="S6.SS2.10.3.p1.4.m4.4.4.cmml" xref="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.3.3.2"><ci id="S6.SS2.10.3.p1.4.m4.4.4.1.cmml" xref="S6.SS2.10.3.p1.4.m4.4.4.1">¯</ci><ci id="S6.SS2.10.3.p1.4.m4.4.4.2.cmml" xref="S6.SS2.10.3.p1.4.m4.4.4.2">𝑎</ci></apply></apply><apply id="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.1.cmml" xref="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.1"><in id="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.1.2.cmml" xref="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.1.2"></in><apply id="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.1.3.cmml" xref="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.1.3"><ci id="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.1.3.1.cmml" xref="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.1.3.1">¯</ci><ci id="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.1.3.2.cmml" xref="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.1.3.2">𝑎</ci></apply><apply id="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.1.1.2.cmml" xref="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.1.1.1"><ci id="S6.SS2.10.3.p1.4.m4.5.5.cmml" xref="S6.SS2.10.3.p1.4.m4.5.5">dom</ci><apply id="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.1.1.1.1.1.cmml" xref="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.1.1.1.1.1.1.cmml" xref="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.1.1.1.1.1.2.cmml" xref="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.1.1.1.1.1.2">𝑝</ci><ci id="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.1.1.1.1.1.3.cmml" xref="S6.SS2.10.3.p1.4.m4.8.8.2.1.1.1.1.1.1.1.1.3">𝜂</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.10.3.p1.4.m4.8c">\max\{\nu(\bar{a}):\bar{a}\in\operatorname{dom}(p_{\xi})\}<\min\{\nu(\bar{a}):% \bar{a}\in\operatorname{dom}(p_{\eta})\}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.10.3.p1.4.m4.8d">roman_max { italic_ν ( over¯ start_ARG italic_a end_ARG ) : over¯ start_ARG italic_a end_ARG ∈ roman_dom ( italic_p start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT ) } < roman_min { italic_ν ( over¯ start_ARG italic_a end_ARG ) : over¯ start_ARG italic_a end_ARG ∈ roman_dom ( italic_p start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT ) }</annotation></semantics></math>. Now for all <math alttext="\xi<\omega_{1}" class="ltx_Math" display="inline" id="S6.SS2.10.3.p1.5.m5.1"><semantics id="S6.SS2.10.3.p1.5.m5.1a"><mrow id="S6.SS2.10.3.p1.5.m5.1.1" xref="S6.SS2.10.3.p1.5.m5.1.1.cmml"><mi id="S6.SS2.10.3.p1.5.m5.1.1.2" xref="S6.SS2.10.3.p1.5.m5.1.1.2.cmml">ξ</mi><mo id="S6.SS2.10.3.p1.5.m5.1.1.1" xref="S6.SS2.10.3.p1.5.m5.1.1.1.cmml"><</mo><msub id="S6.SS2.10.3.p1.5.m5.1.1.3" xref="S6.SS2.10.3.p1.5.m5.1.1.3.cmml"><mi id="S6.SS2.10.3.p1.5.m5.1.1.3.2" xref="S6.SS2.10.3.p1.5.m5.1.1.3.2.cmml">ω</mi><mn id="S6.SS2.10.3.p1.5.m5.1.1.3.3" xref="S6.SS2.10.3.p1.5.m5.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.10.3.p1.5.m5.1b"><apply id="S6.SS2.10.3.p1.5.m5.1.1.cmml" xref="S6.SS2.10.3.p1.5.m5.1.1"><lt id="S6.SS2.10.3.p1.5.m5.1.1.1.cmml" xref="S6.SS2.10.3.p1.5.m5.1.1.1"></lt><ci id="S6.SS2.10.3.p1.5.m5.1.1.2.cmml" xref="S6.SS2.10.3.p1.5.m5.1.1.2">𝜉</ci><apply id="S6.SS2.10.3.p1.5.m5.1.1.3.cmml" xref="S6.SS2.10.3.p1.5.m5.1.1.3"><csymbol cd="ambiguous" id="S6.SS2.10.3.p1.5.m5.1.1.3.1.cmml" xref="S6.SS2.10.3.p1.5.m5.1.1.3">subscript</csymbol><ci id="S6.SS2.10.3.p1.5.m5.1.1.3.2.cmml" xref="S6.SS2.10.3.p1.5.m5.1.1.3.2">𝜔</ci><cn id="S6.SS2.10.3.p1.5.m5.1.1.3.3.cmml" type="integer" xref="S6.SS2.10.3.p1.5.m5.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.10.3.p1.5.m5.1c">\xi<\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.10.3.p1.5.m5.1d">italic_ξ < italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, let <math alttext="q_{\xi}:=q(p_{\xi})" class="ltx_Math" display="inline" id="S6.SS2.10.3.p1.6.m6.1"><semantics id="S6.SS2.10.3.p1.6.m6.1a"><mrow id="S6.SS2.10.3.p1.6.m6.1.1" xref="S6.SS2.10.3.p1.6.m6.1.1.cmml"><msub id="S6.SS2.10.3.p1.6.m6.1.1.3" xref="S6.SS2.10.3.p1.6.m6.1.1.3.cmml"><mi id="S6.SS2.10.3.p1.6.m6.1.1.3.2" xref="S6.SS2.10.3.p1.6.m6.1.1.3.2.cmml">q</mi><mi id="S6.SS2.10.3.p1.6.m6.1.1.3.3" xref="S6.SS2.10.3.p1.6.m6.1.1.3.3.cmml">ξ</mi></msub><mo id="S6.SS2.10.3.p1.6.m6.1.1.2" lspace="0.278em" rspace="0.278em" xref="S6.SS2.10.3.p1.6.m6.1.1.2.cmml">:=</mo><mrow id="S6.SS2.10.3.p1.6.m6.1.1.1" xref="S6.SS2.10.3.p1.6.m6.1.1.1.cmml"><mi id="S6.SS2.10.3.p1.6.m6.1.1.1.3" xref="S6.SS2.10.3.p1.6.m6.1.1.1.3.cmml">q</mi><mo id="S6.SS2.10.3.p1.6.m6.1.1.1.2" xref="S6.SS2.10.3.p1.6.m6.1.1.1.2.cmml">⁢</mo><mrow id="S6.SS2.10.3.p1.6.m6.1.1.1.1.1" xref="S6.SS2.10.3.p1.6.m6.1.1.1.1.1.1.cmml"><mo id="S6.SS2.10.3.p1.6.m6.1.1.1.1.1.2" stretchy="false" xref="S6.SS2.10.3.p1.6.m6.1.1.1.1.1.1.cmml">(</mo><msub id="S6.SS2.10.3.p1.6.m6.1.1.1.1.1.1" xref="S6.SS2.10.3.p1.6.m6.1.1.1.1.1.1.cmml"><mi id="S6.SS2.10.3.p1.6.m6.1.1.1.1.1.1.2" xref="S6.SS2.10.3.p1.6.m6.1.1.1.1.1.1.2.cmml">p</mi><mi id="S6.SS2.10.3.p1.6.m6.1.1.1.1.1.1.3" xref="S6.SS2.10.3.p1.6.m6.1.1.1.1.1.1.3.cmml">ξ</mi></msub><mo id="S6.SS2.10.3.p1.6.m6.1.1.1.1.1.3" stretchy="false" xref="S6.SS2.10.3.p1.6.m6.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.10.3.p1.6.m6.1b"><apply id="S6.SS2.10.3.p1.6.m6.1.1.cmml" xref="S6.SS2.10.3.p1.6.m6.1.1"><csymbol cd="latexml" id="S6.SS2.10.3.p1.6.m6.1.1.2.cmml" xref="S6.SS2.10.3.p1.6.m6.1.1.2">assign</csymbol><apply id="S6.SS2.10.3.p1.6.m6.1.1.3.cmml" xref="S6.SS2.10.3.p1.6.m6.1.1.3"><csymbol cd="ambiguous" id="S6.SS2.10.3.p1.6.m6.1.1.3.1.cmml" xref="S6.SS2.10.3.p1.6.m6.1.1.3">subscript</csymbol><ci id="S6.SS2.10.3.p1.6.m6.1.1.3.2.cmml" xref="S6.SS2.10.3.p1.6.m6.1.1.3.2">𝑞</ci><ci id="S6.SS2.10.3.p1.6.m6.1.1.3.3.cmml" xref="S6.SS2.10.3.p1.6.m6.1.1.3.3">𝜉</ci></apply><apply id="S6.SS2.10.3.p1.6.m6.1.1.1.cmml" xref="S6.SS2.10.3.p1.6.m6.1.1.1"><times id="S6.SS2.10.3.p1.6.m6.1.1.1.2.cmml" xref="S6.SS2.10.3.p1.6.m6.1.1.1.2"></times><ci id="S6.SS2.10.3.p1.6.m6.1.1.1.3.cmml" xref="S6.SS2.10.3.p1.6.m6.1.1.1.3">𝑞</ci><apply id="S6.SS2.10.3.p1.6.m6.1.1.1.1.1.1.cmml" xref="S6.SS2.10.3.p1.6.m6.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.10.3.p1.6.m6.1.1.1.1.1.1.1.cmml" xref="S6.SS2.10.3.p1.6.m6.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.10.3.p1.6.m6.1.1.1.1.1.1.2.cmml" xref="S6.SS2.10.3.p1.6.m6.1.1.1.1.1.1.2">𝑝</ci><ci id="S6.SS2.10.3.p1.6.m6.1.1.1.1.1.1.3.cmml" xref="S6.SS2.10.3.p1.6.m6.1.1.1.1.1.1.3">𝜉</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.10.3.p1.6.m6.1c">q_{\xi}:=q(p_{\xi})</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.10.3.p1.6.m6.1d">italic_q start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT := italic_q ( italic_p start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT )</annotation></semantics></math>, and let <math alttext="\xi_{0}<_{A}\dots<_{A}\xi_{n-1}" class="ltx_Math" display="inline" id="S6.SS2.10.3.p1.7.m7.1"><semantics id="S6.SS2.10.3.p1.7.m7.1a"><mrow id="S6.SS2.10.3.p1.7.m7.1.1" xref="S6.SS2.10.3.p1.7.m7.1.1.cmml"><msub id="S6.SS2.10.3.p1.7.m7.1.1.2" xref="S6.SS2.10.3.p1.7.m7.1.1.2.cmml"><mi id="S6.SS2.10.3.p1.7.m7.1.1.2.2" xref="S6.SS2.10.3.p1.7.m7.1.1.2.2.cmml">ξ</mi><mn id="S6.SS2.10.3.p1.7.m7.1.1.2.3" xref="S6.SS2.10.3.p1.7.m7.1.1.2.3.cmml">0</mn></msub><msub id="S6.SS2.10.3.p1.7.m7.1.1.3" xref="S6.SS2.10.3.p1.7.m7.1.1.3.cmml"><mo id="S6.SS2.10.3.p1.7.m7.1.1.3.2" xref="S6.SS2.10.3.p1.7.m7.1.1.3.2.cmml"><</mo><mi id="S6.SS2.10.3.p1.7.m7.1.1.3.3" xref="S6.SS2.10.3.p1.7.m7.1.1.3.3.cmml">A</mi></msub><mi id="S6.SS2.10.3.p1.7.m7.1.1.4" mathvariant="normal" xref="S6.SS2.10.3.p1.7.m7.1.1.4.cmml">⋯</mi><msub id="S6.SS2.10.3.p1.7.m7.1.1.5" xref="S6.SS2.10.3.p1.7.m7.1.1.5.cmml"><mo id="S6.SS2.10.3.p1.7.m7.1.1.5.2" xref="S6.SS2.10.3.p1.7.m7.1.1.5.2.cmml"><</mo><mi id="S6.SS2.10.3.p1.7.m7.1.1.5.3" xref="S6.SS2.10.3.p1.7.m7.1.1.5.3.cmml">A</mi></msub><msub id="S6.SS2.10.3.p1.7.m7.1.1.6" xref="S6.SS2.10.3.p1.7.m7.1.1.6.cmml"><mi id="S6.SS2.10.3.p1.7.m7.1.1.6.2" xref="S6.SS2.10.3.p1.7.m7.1.1.6.2.cmml">ξ</mi><mrow id="S6.SS2.10.3.p1.7.m7.1.1.6.3" xref="S6.SS2.10.3.p1.7.m7.1.1.6.3.cmml"><mi id="S6.SS2.10.3.p1.7.m7.1.1.6.3.2" xref="S6.SS2.10.3.p1.7.m7.1.1.6.3.2.cmml">n</mi><mo id="S6.SS2.10.3.p1.7.m7.1.1.6.3.1" xref="S6.SS2.10.3.p1.7.m7.1.1.6.3.1.cmml">−</mo><mn id="S6.SS2.10.3.p1.7.m7.1.1.6.3.3" xref="S6.SS2.10.3.p1.7.m7.1.1.6.3.3.cmml">1</mn></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.10.3.p1.7.m7.1b"><apply id="S6.SS2.10.3.p1.7.m7.1.1.cmml" xref="S6.SS2.10.3.p1.7.m7.1.1"><and id="S6.SS2.10.3.p1.7.m7.1.1a.cmml" xref="S6.SS2.10.3.p1.7.m7.1.1"></and><apply id="S6.SS2.10.3.p1.7.m7.1.1b.cmml" xref="S6.SS2.10.3.p1.7.m7.1.1"><apply id="S6.SS2.10.3.p1.7.m7.1.1.3.cmml" xref="S6.SS2.10.3.p1.7.m7.1.1.3"><csymbol cd="ambiguous" id="S6.SS2.10.3.p1.7.m7.1.1.3.1.cmml" xref="S6.SS2.10.3.p1.7.m7.1.1.3">subscript</csymbol><lt id="S6.SS2.10.3.p1.7.m7.1.1.3.2.cmml" xref="S6.SS2.10.3.p1.7.m7.1.1.3.2"></lt><ci id="S6.SS2.10.3.p1.7.m7.1.1.3.3.cmml" xref="S6.SS2.10.3.p1.7.m7.1.1.3.3">𝐴</ci></apply><apply id="S6.SS2.10.3.p1.7.m7.1.1.2.cmml" xref="S6.SS2.10.3.p1.7.m7.1.1.2"><csymbol cd="ambiguous" id="S6.SS2.10.3.p1.7.m7.1.1.2.1.cmml" xref="S6.SS2.10.3.p1.7.m7.1.1.2">subscript</csymbol><ci id="S6.SS2.10.3.p1.7.m7.1.1.2.2.cmml" xref="S6.SS2.10.3.p1.7.m7.1.1.2.2">𝜉</ci><cn id="S6.SS2.10.3.p1.7.m7.1.1.2.3.cmml" type="integer" xref="S6.SS2.10.3.p1.7.m7.1.1.2.3">0</cn></apply><ci id="S6.SS2.10.3.p1.7.m7.1.1.4.cmml" xref="S6.SS2.10.3.p1.7.m7.1.1.4">⋯</ci></apply><apply id="S6.SS2.10.3.p1.7.m7.1.1c.cmml" xref="S6.SS2.10.3.p1.7.m7.1.1"><apply id="S6.SS2.10.3.p1.7.m7.1.1.5.cmml" xref="S6.SS2.10.3.p1.7.m7.1.1.5"><csymbol cd="ambiguous" id="S6.SS2.10.3.p1.7.m7.1.1.5.1.cmml" xref="S6.SS2.10.3.p1.7.m7.1.1.5">subscript</csymbol><lt id="S6.SS2.10.3.p1.7.m7.1.1.5.2.cmml" xref="S6.SS2.10.3.p1.7.m7.1.1.5.2"></lt><ci id="S6.SS2.10.3.p1.7.m7.1.1.5.3.cmml" xref="S6.SS2.10.3.p1.7.m7.1.1.5.3">𝐴</ci></apply><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.10.3.p1.7.m7.1.1.4.cmml" id="S6.SS2.10.3.p1.7.m7.1.1d.cmml" xref="S6.SS2.10.3.p1.7.m7.1.1"></share><apply id="S6.SS2.10.3.p1.7.m7.1.1.6.cmml" xref="S6.SS2.10.3.p1.7.m7.1.1.6"><csymbol cd="ambiguous" id="S6.SS2.10.3.p1.7.m7.1.1.6.1.cmml" xref="S6.SS2.10.3.p1.7.m7.1.1.6">subscript</csymbol><ci id="S6.SS2.10.3.p1.7.m7.1.1.6.2.cmml" xref="S6.SS2.10.3.p1.7.m7.1.1.6.2">𝜉</ci><apply id="S6.SS2.10.3.p1.7.m7.1.1.6.3.cmml" xref="S6.SS2.10.3.p1.7.m7.1.1.6.3"><minus id="S6.SS2.10.3.p1.7.m7.1.1.6.3.1.cmml" xref="S6.SS2.10.3.p1.7.m7.1.1.6.3.1"></minus><ci id="S6.SS2.10.3.p1.7.m7.1.1.6.3.2.cmml" xref="S6.SS2.10.3.p1.7.m7.1.1.6.3.2">𝑛</ci><cn id="S6.SS2.10.3.p1.7.m7.1.1.6.3.3.cmml" type="integer" xref="S6.SS2.10.3.p1.7.m7.1.1.6.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.10.3.p1.7.m7.1c">\xi_{0}<_{A}\dots<_{A}\xi_{n-1}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.10.3.p1.7.m7.1d">italic_ξ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT < start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⋯ < start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_ξ start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT</annotation></semantics></math> enumerate <math alttext="\operatorname{dom}(q_{\xi})" class="ltx_Math" display="inline" id="S6.SS2.10.3.p1.8.m8.2"><semantics id="S6.SS2.10.3.p1.8.m8.2a"><mrow id="S6.SS2.10.3.p1.8.m8.2.2.1" xref="S6.SS2.10.3.p1.8.m8.2.2.2.cmml"><mi id="S6.SS2.10.3.p1.8.m8.1.1" xref="S6.SS2.10.3.p1.8.m8.1.1.cmml">dom</mi><mo id="S6.SS2.10.3.p1.8.m8.2.2.1a" xref="S6.SS2.10.3.p1.8.m8.2.2.2.cmml">⁡</mo><mrow id="S6.SS2.10.3.p1.8.m8.2.2.1.1" xref="S6.SS2.10.3.p1.8.m8.2.2.2.cmml"><mo id="S6.SS2.10.3.p1.8.m8.2.2.1.1.2" stretchy="false" xref="S6.SS2.10.3.p1.8.m8.2.2.2.cmml">(</mo><msub id="S6.SS2.10.3.p1.8.m8.2.2.1.1.1" xref="S6.SS2.10.3.p1.8.m8.2.2.1.1.1.cmml"><mi id="S6.SS2.10.3.p1.8.m8.2.2.1.1.1.2" xref="S6.SS2.10.3.p1.8.m8.2.2.1.1.1.2.cmml">q</mi><mi id="S6.SS2.10.3.p1.8.m8.2.2.1.1.1.3" xref="S6.SS2.10.3.p1.8.m8.2.2.1.1.1.3.cmml">ξ</mi></msub><mo id="S6.SS2.10.3.p1.8.m8.2.2.1.1.3" stretchy="false" xref="S6.SS2.10.3.p1.8.m8.2.2.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.10.3.p1.8.m8.2b"><apply id="S6.SS2.10.3.p1.8.m8.2.2.2.cmml" xref="S6.SS2.10.3.p1.8.m8.2.2.1"><ci id="S6.SS2.10.3.p1.8.m8.1.1.cmml" xref="S6.SS2.10.3.p1.8.m8.1.1">dom</ci><apply id="S6.SS2.10.3.p1.8.m8.2.2.1.1.1.cmml" xref="S6.SS2.10.3.p1.8.m8.2.2.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.10.3.p1.8.m8.2.2.1.1.1.1.cmml" xref="S6.SS2.10.3.p1.8.m8.2.2.1.1.1">subscript</csymbol><ci id="S6.SS2.10.3.p1.8.m8.2.2.1.1.1.2.cmml" xref="S6.SS2.10.3.p1.8.m8.2.2.1.1.1.2">𝑞</ci><ci id="S6.SS2.10.3.p1.8.m8.2.2.1.1.1.3.cmml" xref="S6.SS2.10.3.p1.8.m8.2.2.1.1.1.3">𝜉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.10.3.p1.8.m8.2c">\operatorname{dom}(q_{\xi})</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.10.3.p1.8.m8.2d">roman_dom ( italic_q start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT )</annotation></semantics></math>. Observe that for all <math alttext="i<n" class="ltx_Math" display="inline" id="S6.SS2.10.3.p1.9.m9.1"><semantics id="S6.SS2.10.3.p1.9.m9.1a"><mrow id="S6.SS2.10.3.p1.9.m9.1.1" xref="S6.SS2.10.3.p1.9.m9.1.1.cmml"><mi id="S6.SS2.10.3.p1.9.m9.1.1.2" xref="S6.SS2.10.3.p1.9.m9.1.1.2.cmml">i</mi><mo id="S6.SS2.10.3.p1.9.m9.1.1.1" xref="S6.SS2.10.3.p1.9.m9.1.1.1.cmml"><</mo><mi id="S6.SS2.10.3.p1.9.m9.1.1.3" xref="S6.SS2.10.3.p1.9.m9.1.1.3.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.10.3.p1.9.m9.1b"><apply id="S6.SS2.10.3.p1.9.m9.1.1.cmml" xref="S6.SS2.10.3.p1.9.m9.1.1"><lt id="S6.SS2.10.3.p1.9.m9.1.1.1.cmml" xref="S6.SS2.10.3.p1.9.m9.1.1.1"></lt><ci id="S6.SS2.10.3.p1.9.m9.1.1.2.cmml" xref="S6.SS2.10.3.p1.9.m9.1.1.2">𝑖</ci><ci id="S6.SS2.10.3.p1.9.m9.1.1.3.cmml" xref="S6.SS2.10.3.p1.9.m9.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.10.3.p1.9.m9.1c">i<n</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.10.3.p1.9.m9.1d">italic_i < italic_n</annotation></semantics></math>, <math alttext="\xi_{i}=a_{m}" class="ltx_Math" display="inline" id="S6.SS2.10.3.p1.10.m10.1"><semantics id="S6.SS2.10.3.p1.10.m10.1a"><mrow id="S6.SS2.10.3.p1.10.m10.1.1" xref="S6.SS2.10.3.p1.10.m10.1.1.cmml"><msub id="S6.SS2.10.3.p1.10.m10.1.1.2" xref="S6.SS2.10.3.p1.10.m10.1.1.2.cmml"><mi id="S6.SS2.10.3.p1.10.m10.1.1.2.2" xref="S6.SS2.10.3.p1.10.m10.1.1.2.2.cmml">ξ</mi><mi id="S6.SS2.10.3.p1.10.m10.1.1.2.3" xref="S6.SS2.10.3.p1.10.m10.1.1.2.3.cmml">i</mi></msub><mo id="S6.SS2.10.3.p1.10.m10.1.1.1" xref="S6.SS2.10.3.p1.10.m10.1.1.1.cmml">=</mo><msub id="S6.SS2.10.3.p1.10.m10.1.1.3" xref="S6.SS2.10.3.p1.10.m10.1.1.3.cmml"><mi id="S6.SS2.10.3.p1.10.m10.1.1.3.2" xref="S6.SS2.10.3.p1.10.m10.1.1.3.2.cmml">a</mi><mi id="S6.SS2.10.3.p1.10.m10.1.1.3.3" xref="S6.SS2.10.3.p1.10.m10.1.1.3.3.cmml">m</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.10.3.p1.10.m10.1b"><apply id="S6.SS2.10.3.p1.10.m10.1.1.cmml" xref="S6.SS2.10.3.p1.10.m10.1.1"><eq id="S6.SS2.10.3.p1.10.m10.1.1.1.cmml" xref="S6.SS2.10.3.p1.10.m10.1.1.1"></eq><apply id="S6.SS2.10.3.p1.10.m10.1.1.2.cmml" xref="S6.SS2.10.3.p1.10.m10.1.1.2"><csymbol cd="ambiguous" id="S6.SS2.10.3.p1.10.m10.1.1.2.1.cmml" xref="S6.SS2.10.3.p1.10.m10.1.1.2">subscript</csymbol><ci id="S6.SS2.10.3.p1.10.m10.1.1.2.2.cmml" xref="S6.SS2.10.3.p1.10.m10.1.1.2.2">𝜉</ci><ci id="S6.SS2.10.3.p1.10.m10.1.1.2.3.cmml" xref="S6.SS2.10.3.p1.10.m10.1.1.2.3">𝑖</ci></apply><apply id="S6.SS2.10.3.p1.10.m10.1.1.3.cmml" xref="S6.SS2.10.3.p1.10.m10.1.1.3"><csymbol cd="ambiguous" id="S6.SS2.10.3.p1.10.m10.1.1.3.1.cmml" xref="S6.SS2.10.3.p1.10.m10.1.1.3">subscript</csymbol><ci id="S6.SS2.10.3.p1.10.m10.1.1.3.2.cmml" xref="S6.SS2.10.3.p1.10.m10.1.1.3.2">𝑎</ci><ci id="S6.SS2.10.3.p1.10.m10.1.1.3.3.cmml" xref="S6.SS2.10.3.p1.10.m10.1.1.3.3">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.10.3.p1.10.m10.1c">\xi_{i}=a_{m}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.10.3.p1.10.m10.1d">italic_ξ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_a start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math> for some <math alttext="\bar{a}\in\operatorname{dom}(p_{\xi})" class="ltx_Math" display="inline" id="S6.SS2.10.3.p1.11.m11.2"><semantics id="S6.SS2.10.3.p1.11.m11.2a"><mrow id="S6.SS2.10.3.p1.11.m11.2.2" xref="S6.SS2.10.3.p1.11.m11.2.2.cmml"><mover accent="true" id="S6.SS2.10.3.p1.11.m11.2.2.3" xref="S6.SS2.10.3.p1.11.m11.2.2.3.cmml"><mi id="S6.SS2.10.3.p1.11.m11.2.2.3.2" xref="S6.SS2.10.3.p1.11.m11.2.2.3.2.cmml">a</mi><mo id="S6.SS2.10.3.p1.11.m11.2.2.3.1" xref="S6.SS2.10.3.p1.11.m11.2.2.3.1.cmml">¯</mo></mover><mo id="S6.SS2.10.3.p1.11.m11.2.2.2" xref="S6.SS2.10.3.p1.11.m11.2.2.2.cmml">∈</mo><mrow id="S6.SS2.10.3.p1.11.m11.2.2.1.1" xref="S6.SS2.10.3.p1.11.m11.2.2.1.2.cmml"><mi id="S6.SS2.10.3.p1.11.m11.1.1" xref="S6.SS2.10.3.p1.11.m11.1.1.cmml">dom</mi><mo id="S6.SS2.10.3.p1.11.m11.2.2.1.1a" xref="S6.SS2.10.3.p1.11.m11.2.2.1.2.cmml">⁡</mo><mrow id="S6.SS2.10.3.p1.11.m11.2.2.1.1.1" xref="S6.SS2.10.3.p1.11.m11.2.2.1.2.cmml"><mo id="S6.SS2.10.3.p1.11.m11.2.2.1.1.1.2" stretchy="false" xref="S6.SS2.10.3.p1.11.m11.2.2.1.2.cmml">(</mo><msub id="S6.SS2.10.3.p1.11.m11.2.2.1.1.1.1" xref="S6.SS2.10.3.p1.11.m11.2.2.1.1.1.1.cmml"><mi id="S6.SS2.10.3.p1.11.m11.2.2.1.1.1.1.2" xref="S6.SS2.10.3.p1.11.m11.2.2.1.1.1.1.2.cmml">p</mi><mi id="S6.SS2.10.3.p1.11.m11.2.2.1.1.1.1.3" xref="S6.SS2.10.3.p1.11.m11.2.2.1.1.1.1.3.cmml">ξ</mi></msub><mo id="S6.SS2.10.3.p1.11.m11.2.2.1.1.1.3" stretchy="false" xref="S6.SS2.10.3.p1.11.m11.2.2.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.10.3.p1.11.m11.2b"><apply id="S6.SS2.10.3.p1.11.m11.2.2.cmml" xref="S6.SS2.10.3.p1.11.m11.2.2"><in id="S6.SS2.10.3.p1.11.m11.2.2.2.cmml" xref="S6.SS2.10.3.p1.11.m11.2.2.2"></in><apply id="S6.SS2.10.3.p1.11.m11.2.2.3.cmml" xref="S6.SS2.10.3.p1.11.m11.2.2.3"><ci id="S6.SS2.10.3.p1.11.m11.2.2.3.1.cmml" xref="S6.SS2.10.3.p1.11.m11.2.2.3.1">¯</ci><ci id="S6.SS2.10.3.p1.11.m11.2.2.3.2.cmml" xref="S6.SS2.10.3.p1.11.m11.2.2.3.2">𝑎</ci></apply><apply id="S6.SS2.10.3.p1.11.m11.2.2.1.2.cmml" xref="S6.SS2.10.3.p1.11.m11.2.2.1.1"><ci id="S6.SS2.10.3.p1.11.m11.1.1.cmml" xref="S6.SS2.10.3.p1.11.m11.1.1">dom</ci><apply id="S6.SS2.10.3.p1.11.m11.2.2.1.1.1.1.cmml" xref="S6.SS2.10.3.p1.11.m11.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.10.3.p1.11.m11.2.2.1.1.1.1.1.cmml" xref="S6.SS2.10.3.p1.11.m11.2.2.1.1.1.1">subscript</csymbol><ci id="S6.SS2.10.3.p1.11.m11.2.2.1.1.1.1.2.cmml" xref="S6.SS2.10.3.p1.11.m11.2.2.1.1.1.1.2">𝑝</ci><ci id="S6.SS2.10.3.p1.11.m11.2.2.1.1.1.1.3.cmml" xref="S6.SS2.10.3.p1.11.m11.2.2.1.1.1.1.3">𝜉</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.10.3.p1.11.m11.2c">\bar{a}\in\operatorname{dom}(p_{\xi})</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.10.3.p1.11.m11.2d">over¯ start_ARG italic_a end_ARG ∈ roman_dom ( italic_p start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT )</annotation></semantics></math>. Now we claim that there is an uncountable <math alttext="\Gamma\subseteq\omega_{1}" class="ltx_Math" display="inline" id="S6.SS2.10.3.p1.12.m12.1"><semantics id="S6.SS2.10.3.p1.12.m12.1a"><mrow id="S6.SS2.10.3.p1.12.m12.1.1" xref="S6.SS2.10.3.p1.12.m12.1.1.cmml"><mi id="S6.SS2.10.3.p1.12.m12.1.1.2" mathvariant="normal" xref="S6.SS2.10.3.p1.12.m12.1.1.2.cmml">Γ</mi><mo id="S6.SS2.10.3.p1.12.m12.1.1.1" xref="S6.SS2.10.3.p1.12.m12.1.1.1.cmml">⊆</mo><msub id="S6.SS2.10.3.p1.12.m12.1.1.3" xref="S6.SS2.10.3.p1.12.m12.1.1.3.cmml"><mi id="S6.SS2.10.3.p1.12.m12.1.1.3.2" xref="S6.SS2.10.3.p1.12.m12.1.1.3.2.cmml">ω</mi><mn id="S6.SS2.10.3.p1.12.m12.1.1.3.3" xref="S6.SS2.10.3.p1.12.m12.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.10.3.p1.12.m12.1b"><apply id="S6.SS2.10.3.p1.12.m12.1.1.cmml" xref="S6.SS2.10.3.p1.12.m12.1.1"><subset id="S6.SS2.10.3.p1.12.m12.1.1.1.cmml" xref="S6.SS2.10.3.p1.12.m12.1.1.1"></subset><ci id="S6.SS2.10.3.p1.12.m12.1.1.2.cmml" xref="S6.SS2.10.3.p1.12.m12.1.1.2">Γ</ci><apply id="S6.SS2.10.3.p1.12.m12.1.1.3.cmml" xref="S6.SS2.10.3.p1.12.m12.1.1.3"><csymbol cd="ambiguous" id="S6.SS2.10.3.p1.12.m12.1.1.3.1.cmml" xref="S6.SS2.10.3.p1.12.m12.1.1.3">subscript</csymbol><ci id="S6.SS2.10.3.p1.12.m12.1.1.3.2.cmml" xref="S6.SS2.10.3.p1.12.m12.1.1.3.2">𝜔</ci><cn id="S6.SS2.10.3.p1.12.m12.1.1.3.3.cmml" type="integer" xref="S6.SS2.10.3.p1.12.m12.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.10.3.p1.12.m12.1c">\Gamma\subseteq\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.10.3.p1.12.m12.1d">roman_Γ ⊆ italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\gamma<\min(\Gamma)" class="ltx_Math" display="inline" id="S6.SS2.10.3.p1.13.m13.2"><semantics id="S6.SS2.10.3.p1.13.m13.2a"><mrow id="S6.SS2.10.3.p1.13.m13.2.3" xref="S6.SS2.10.3.p1.13.m13.2.3.cmml"><mi id="S6.SS2.10.3.p1.13.m13.2.3.2" xref="S6.SS2.10.3.p1.13.m13.2.3.2.cmml">γ</mi><mo id="S6.SS2.10.3.p1.13.m13.2.3.1" xref="S6.SS2.10.3.p1.13.m13.2.3.1.cmml"><</mo><mrow id="S6.SS2.10.3.p1.13.m13.2.3.3.2" xref="S6.SS2.10.3.p1.13.m13.2.3.3.1.cmml"><mi id="S6.SS2.10.3.p1.13.m13.1.1" xref="S6.SS2.10.3.p1.13.m13.1.1.cmml">min</mi><mo id="S6.SS2.10.3.p1.13.m13.2.3.3.2a" xref="S6.SS2.10.3.p1.13.m13.2.3.3.1.cmml">⁡</mo><mrow id="S6.SS2.10.3.p1.13.m13.2.3.3.2.1" xref="S6.SS2.10.3.p1.13.m13.2.3.3.1.cmml"><mo id="S6.SS2.10.3.p1.13.m13.2.3.3.2.1.1" stretchy="false" xref="S6.SS2.10.3.p1.13.m13.2.3.3.1.cmml">(</mo><mi id="S6.SS2.10.3.p1.13.m13.2.2" mathvariant="normal" xref="S6.SS2.10.3.p1.13.m13.2.2.cmml">Γ</mi><mo id="S6.SS2.10.3.p1.13.m13.2.3.3.2.1.2" stretchy="false" xref="S6.SS2.10.3.p1.13.m13.2.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.10.3.p1.13.m13.2b"><apply id="S6.SS2.10.3.p1.13.m13.2.3.cmml" xref="S6.SS2.10.3.p1.13.m13.2.3"><lt id="S6.SS2.10.3.p1.13.m13.2.3.1.cmml" xref="S6.SS2.10.3.p1.13.m13.2.3.1"></lt><ci id="S6.SS2.10.3.p1.13.m13.2.3.2.cmml" xref="S6.SS2.10.3.p1.13.m13.2.3.2">𝛾</ci><apply id="S6.SS2.10.3.p1.13.m13.2.3.3.1.cmml" xref="S6.SS2.10.3.p1.13.m13.2.3.3.2"><min id="S6.SS2.10.3.p1.13.m13.1.1.cmml" xref="S6.SS2.10.3.p1.13.m13.1.1"></min><ci id="S6.SS2.10.3.p1.13.m13.2.2.cmml" xref="S6.SS2.10.3.p1.13.m13.2.2">Γ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.10.3.p1.13.m13.2c">\gamma<\min(\Gamma)</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.10.3.p1.13.m13.2d">italic_γ < roman_min ( roman_Γ )</annotation></semantics></math> such that,</p> <ol class="ltx_enumerate" id="S6.I7"> <li class="ltx_item" id="S6.I7.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(a)</span> <div class="ltx_para" id="S6.I7.i1.p1"> <p class="ltx_p" id="S6.I7.i1.p1.1"><math alttext="\xi\leq\min(\operatorname{dom}(q_{\xi}))" class="ltx_Math" display="inline" id="S6.I7.i1.p1.1.m1.3"><semantics id="S6.I7.i1.p1.1.m1.3a"><mrow id="S6.I7.i1.p1.1.m1.3.3" xref="S6.I7.i1.p1.1.m1.3.3.cmml"><mi id="S6.I7.i1.p1.1.m1.3.3.3" xref="S6.I7.i1.p1.1.m1.3.3.3.cmml">ξ</mi><mo id="S6.I7.i1.p1.1.m1.3.3.2" xref="S6.I7.i1.p1.1.m1.3.3.2.cmml">≤</mo><mrow id="S6.I7.i1.p1.1.m1.3.3.1.1" xref="S6.I7.i1.p1.1.m1.3.3.1.2.cmml"><mi id="S6.I7.i1.p1.1.m1.2.2" xref="S6.I7.i1.p1.1.m1.2.2.cmml">min</mi><mo id="S6.I7.i1.p1.1.m1.3.3.1.1a" xref="S6.I7.i1.p1.1.m1.3.3.1.2.cmml">⁡</mo><mrow id="S6.I7.i1.p1.1.m1.3.3.1.1.1" xref="S6.I7.i1.p1.1.m1.3.3.1.2.cmml"><mo id="S6.I7.i1.p1.1.m1.3.3.1.1.1.2" stretchy="false" xref="S6.I7.i1.p1.1.m1.3.3.1.2.cmml">(</mo><mrow id="S6.I7.i1.p1.1.m1.3.3.1.1.1.1.1" xref="S6.I7.i1.p1.1.m1.3.3.1.1.1.1.2.cmml"><mi id="S6.I7.i1.p1.1.m1.1.1" xref="S6.I7.i1.p1.1.m1.1.1.cmml">dom</mi><mo id="S6.I7.i1.p1.1.m1.3.3.1.1.1.1.1a" xref="S6.I7.i1.p1.1.m1.3.3.1.1.1.1.2.cmml">⁡</mo><mrow id="S6.I7.i1.p1.1.m1.3.3.1.1.1.1.1.1" xref="S6.I7.i1.p1.1.m1.3.3.1.1.1.1.2.cmml"><mo id="S6.I7.i1.p1.1.m1.3.3.1.1.1.1.1.1.2" stretchy="false" xref="S6.I7.i1.p1.1.m1.3.3.1.1.1.1.2.cmml">(</mo><msub id="S6.I7.i1.p1.1.m1.3.3.1.1.1.1.1.1.1" xref="S6.I7.i1.p1.1.m1.3.3.1.1.1.1.1.1.1.cmml"><mi id="S6.I7.i1.p1.1.m1.3.3.1.1.1.1.1.1.1.2" xref="S6.I7.i1.p1.1.m1.3.3.1.1.1.1.1.1.1.2.cmml">q</mi><mi id="S6.I7.i1.p1.1.m1.3.3.1.1.1.1.1.1.1.3" xref="S6.I7.i1.p1.1.m1.3.3.1.1.1.1.1.1.1.3.cmml">ξ</mi></msub><mo id="S6.I7.i1.p1.1.m1.3.3.1.1.1.1.1.1.3" stretchy="false" xref="S6.I7.i1.p1.1.m1.3.3.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S6.I7.i1.p1.1.m1.3.3.1.1.1.3" stretchy="false" xref="S6.I7.i1.p1.1.m1.3.3.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.I7.i1.p1.1.m1.3b"><apply id="S6.I7.i1.p1.1.m1.3.3.cmml" xref="S6.I7.i1.p1.1.m1.3.3"><leq id="S6.I7.i1.p1.1.m1.3.3.2.cmml" xref="S6.I7.i1.p1.1.m1.3.3.2"></leq><ci id="S6.I7.i1.p1.1.m1.3.3.3.cmml" xref="S6.I7.i1.p1.1.m1.3.3.3">𝜉</ci><apply id="S6.I7.i1.p1.1.m1.3.3.1.2.cmml" xref="S6.I7.i1.p1.1.m1.3.3.1.1"><min id="S6.I7.i1.p1.1.m1.2.2.cmml" xref="S6.I7.i1.p1.1.m1.2.2"></min><apply id="S6.I7.i1.p1.1.m1.3.3.1.1.1.1.2.cmml" xref="S6.I7.i1.p1.1.m1.3.3.1.1.1.1.1"><ci id="S6.I7.i1.p1.1.m1.1.1.cmml" xref="S6.I7.i1.p1.1.m1.1.1">dom</ci><apply id="S6.I7.i1.p1.1.m1.3.3.1.1.1.1.1.1.1.cmml" xref="S6.I7.i1.p1.1.m1.3.3.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.I7.i1.p1.1.m1.3.3.1.1.1.1.1.1.1.1.cmml" xref="S6.I7.i1.p1.1.m1.3.3.1.1.1.1.1.1.1">subscript</csymbol><ci id="S6.I7.i1.p1.1.m1.3.3.1.1.1.1.1.1.1.2.cmml" xref="S6.I7.i1.p1.1.m1.3.3.1.1.1.1.1.1.1.2">𝑞</ci><ci id="S6.I7.i1.p1.1.m1.3.3.1.1.1.1.1.1.1.3.cmml" xref="S6.I7.i1.p1.1.m1.3.3.1.1.1.1.1.1.1.3">𝜉</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I7.i1.p1.1.m1.3c">\xi\leq\min(\operatorname{dom}(q_{\xi}))</annotation><annotation encoding="application/x-llamapun" id="S6.I7.i1.p1.1.m1.3d">italic_ξ ≤ roman_min ( roman_dom ( italic_q start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT ) )</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S6.I7.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(b)</span> <div class="ltx_para" id="S6.I7.i2.p1"> <p class="ltx_p" id="S6.I7.i2.p1.2">For all <math alttext="i\neq j<n" class="ltx_Math" display="inline" id="S6.I7.i2.p1.1.m1.1"><semantics id="S6.I7.i2.p1.1.m1.1a"><mrow id="S6.I7.i2.p1.1.m1.1.1" xref="S6.I7.i2.p1.1.m1.1.1.cmml"><mi id="S6.I7.i2.p1.1.m1.1.1.2" xref="S6.I7.i2.p1.1.m1.1.1.2.cmml">i</mi><mo id="S6.I7.i2.p1.1.m1.1.1.3" xref="S6.I7.i2.p1.1.m1.1.1.3.cmml">≠</mo><mi id="S6.I7.i2.p1.1.m1.1.1.4" xref="S6.I7.i2.p1.1.m1.1.1.4.cmml">j</mi><mo id="S6.I7.i2.p1.1.m1.1.1.5" xref="S6.I7.i2.p1.1.m1.1.1.5.cmml"><</mo><mi id="S6.I7.i2.p1.1.m1.1.1.6" xref="S6.I7.i2.p1.1.m1.1.1.6.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.I7.i2.p1.1.m1.1b"><apply id="S6.I7.i2.p1.1.m1.1.1.cmml" xref="S6.I7.i2.p1.1.m1.1.1"><and id="S6.I7.i2.p1.1.m1.1.1a.cmml" xref="S6.I7.i2.p1.1.m1.1.1"></and><apply id="S6.I7.i2.p1.1.m1.1.1b.cmml" xref="S6.I7.i2.p1.1.m1.1.1"><neq id="S6.I7.i2.p1.1.m1.1.1.3.cmml" xref="S6.I7.i2.p1.1.m1.1.1.3"></neq><ci id="S6.I7.i2.p1.1.m1.1.1.2.cmml" xref="S6.I7.i2.p1.1.m1.1.1.2">𝑖</ci><ci id="S6.I7.i2.p1.1.m1.1.1.4.cmml" xref="S6.I7.i2.p1.1.m1.1.1.4">𝑗</ci></apply><apply id="S6.I7.i2.p1.1.m1.1.1c.cmml" xref="S6.I7.i2.p1.1.m1.1.1"><lt id="S6.I7.i2.p1.1.m1.1.1.5.cmml" xref="S6.I7.i2.p1.1.m1.1.1.5"></lt><share href="https://arxiv.org/html/2503.13728v1#S6.I7.i2.p1.1.m1.1.1.4.cmml" id="S6.I7.i2.p1.1.m1.1.1d.cmml" xref="S6.I7.i2.p1.1.m1.1.1"></share><ci id="S6.I7.i2.p1.1.m1.1.1.6.cmml" xref="S6.I7.i2.p1.1.m1.1.1.6">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I7.i2.p1.1.m1.1c">i\neq j<n</annotation><annotation encoding="application/x-llamapun" id="S6.I7.i2.p1.1.m1.1d">italic_i ≠ italic_j < italic_n</annotation></semantics></math>, <math alttext="\Delta_{A}(\xi_{i},\xi_{j})<\xi\rightarrow\Delta_{A}(\xi_{i},\xi_{j})<\gamma" class="ltx_Math" display="inline" id="S6.I7.i2.p1.2.m2.4"><semantics id="S6.I7.i2.p1.2.m2.4a"><mrow id="S6.I7.i2.p1.2.m2.4.4" xref="S6.I7.i2.p1.2.m2.4.4.cmml"><mrow id="S6.I7.i2.p1.2.m2.2.2.2" xref="S6.I7.i2.p1.2.m2.2.2.2.cmml"><msub id="S6.I7.i2.p1.2.m2.2.2.2.4" xref="S6.I7.i2.p1.2.m2.2.2.2.4.cmml"><mi id="S6.I7.i2.p1.2.m2.2.2.2.4.2" mathvariant="normal" xref="S6.I7.i2.p1.2.m2.2.2.2.4.2.cmml">Δ</mi><mi id="S6.I7.i2.p1.2.m2.2.2.2.4.3" xref="S6.I7.i2.p1.2.m2.2.2.2.4.3.cmml">A</mi></msub><mo id="S6.I7.i2.p1.2.m2.2.2.2.3" xref="S6.I7.i2.p1.2.m2.2.2.2.3.cmml">⁢</mo><mrow id="S6.I7.i2.p1.2.m2.2.2.2.2.2" xref="S6.I7.i2.p1.2.m2.2.2.2.2.3.cmml"><mo id="S6.I7.i2.p1.2.m2.2.2.2.2.2.3" stretchy="false" xref="S6.I7.i2.p1.2.m2.2.2.2.2.3.cmml">(</mo><msub id="S6.I7.i2.p1.2.m2.1.1.1.1.1.1" xref="S6.I7.i2.p1.2.m2.1.1.1.1.1.1.cmml"><mi id="S6.I7.i2.p1.2.m2.1.1.1.1.1.1.2" xref="S6.I7.i2.p1.2.m2.1.1.1.1.1.1.2.cmml">ξ</mi><mi id="S6.I7.i2.p1.2.m2.1.1.1.1.1.1.3" xref="S6.I7.i2.p1.2.m2.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S6.I7.i2.p1.2.m2.2.2.2.2.2.4" xref="S6.I7.i2.p1.2.m2.2.2.2.2.3.cmml">,</mo><msub id="S6.I7.i2.p1.2.m2.2.2.2.2.2.2" xref="S6.I7.i2.p1.2.m2.2.2.2.2.2.2.cmml"><mi id="S6.I7.i2.p1.2.m2.2.2.2.2.2.2.2" xref="S6.I7.i2.p1.2.m2.2.2.2.2.2.2.2.cmml">ξ</mi><mi id="S6.I7.i2.p1.2.m2.2.2.2.2.2.2.3" xref="S6.I7.i2.p1.2.m2.2.2.2.2.2.2.3.cmml">j</mi></msub><mo id="S6.I7.i2.p1.2.m2.2.2.2.2.2.5" stretchy="false" xref="S6.I7.i2.p1.2.m2.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.I7.i2.p1.2.m2.4.4.6" xref="S6.I7.i2.p1.2.m2.4.4.6.cmml"><</mo><mi id="S6.I7.i2.p1.2.m2.4.4.7" xref="S6.I7.i2.p1.2.m2.4.4.7.cmml">ξ</mi><mo id="S6.I7.i2.p1.2.m2.4.4.8" stretchy="false" xref="S6.I7.i2.p1.2.m2.4.4.8.cmml">→</mo><mrow id="S6.I7.i2.p1.2.m2.4.4.4" xref="S6.I7.i2.p1.2.m2.4.4.4.cmml"><msub id="S6.I7.i2.p1.2.m2.4.4.4.4" xref="S6.I7.i2.p1.2.m2.4.4.4.4.cmml"><mi id="S6.I7.i2.p1.2.m2.4.4.4.4.2" mathvariant="normal" xref="S6.I7.i2.p1.2.m2.4.4.4.4.2.cmml">Δ</mi><mi id="S6.I7.i2.p1.2.m2.4.4.4.4.3" xref="S6.I7.i2.p1.2.m2.4.4.4.4.3.cmml">A</mi></msub><mo id="S6.I7.i2.p1.2.m2.4.4.4.3" xref="S6.I7.i2.p1.2.m2.4.4.4.3.cmml">⁢</mo><mrow id="S6.I7.i2.p1.2.m2.4.4.4.2.2" xref="S6.I7.i2.p1.2.m2.4.4.4.2.3.cmml"><mo id="S6.I7.i2.p1.2.m2.4.4.4.2.2.3" stretchy="false" xref="S6.I7.i2.p1.2.m2.4.4.4.2.3.cmml">(</mo><msub id="S6.I7.i2.p1.2.m2.3.3.3.1.1.1" xref="S6.I7.i2.p1.2.m2.3.3.3.1.1.1.cmml"><mi id="S6.I7.i2.p1.2.m2.3.3.3.1.1.1.2" xref="S6.I7.i2.p1.2.m2.3.3.3.1.1.1.2.cmml">ξ</mi><mi id="S6.I7.i2.p1.2.m2.3.3.3.1.1.1.3" xref="S6.I7.i2.p1.2.m2.3.3.3.1.1.1.3.cmml">i</mi></msub><mo id="S6.I7.i2.p1.2.m2.4.4.4.2.2.4" xref="S6.I7.i2.p1.2.m2.4.4.4.2.3.cmml">,</mo><msub id="S6.I7.i2.p1.2.m2.4.4.4.2.2.2" xref="S6.I7.i2.p1.2.m2.4.4.4.2.2.2.cmml"><mi id="S6.I7.i2.p1.2.m2.4.4.4.2.2.2.2" xref="S6.I7.i2.p1.2.m2.4.4.4.2.2.2.2.cmml">ξ</mi><mi id="S6.I7.i2.p1.2.m2.4.4.4.2.2.2.3" xref="S6.I7.i2.p1.2.m2.4.4.4.2.2.2.3.cmml">j</mi></msub><mo id="S6.I7.i2.p1.2.m2.4.4.4.2.2.5" stretchy="false" xref="S6.I7.i2.p1.2.m2.4.4.4.2.3.cmml">)</mo></mrow></mrow><mo id="S6.I7.i2.p1.2.m2.4.4.9" xref="S6.I7.i2.p1.2.m2.4.4.9.cmml"><</mo><mi id="S6.I7.i2.p1.2.m2.4.4.10" xref="S6.I7.i2.p1.2.m2.4.4.10.cmml">γ</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.I7.i2.p1.2.m2.4b"><apply id="S6.I7.i2.p1.2.m2.4.4.cmml" xref="S6.I7.i2.p1.2.m2.4.4"><and id="S6.I7.i2.p1.2.m2.4.4a.cmml" xref="S6.I7.i2.p1.2.m2.4.4"></and><apply id="S6.I7.i2.p1.2.m2.4.4b.cmml" xref="S6.I7.i2.p1.2.m2.4.4"><lt id="S6.I7.i2.p1.2.m2.4.4.6.cmml" xref="S6.I7.i2.p1.2.m2.4.4.6"></lt><apply id="S6.I7.i2.p1.2.m2.2.2.2.cmml" xref="S6.I7.i2.p1.2.m2.2.2.2"><times id="S6.I7.i2.p1.2.m2.2.2.2.3.cmml" xref="S6.I7.i2.p1.2.m2.2.2.2.3"></times><apply id="S6.I7.i2.p1.2.m2.2.2.2.4.cmml" xref="S6.I7.i2.p1.2.m2.2.2.2.4"><csymbol cd="ambiguous" id="S6.I7.i2.p1.2.m2.2.2.2.4.1.cmml" xref="S6.I7.i2.p1.2.m2.2.2.2.4">subscript</csymbol><ci id="S6.I7.i2.p1.2.m2.2.2.2.4.2.cmml" xref="S6.I7.i2.p1.2.m2.2.2.2.4.2">Δ</ci><ci id="S6.I7.i2.p1.2.m2.2.2.2.4.3.cmml" xref="S6.I7.i2.p1.2.m2.2.2.2.4.3">𝐴</ci></apply><interval closure="open" id="S6.I7.i2.p1.2.m2.2.2.2.2.3.cmml" xref="S6.I7.i2.p1.2.m2.2.2.2.2.2"><apply id="S6.I7.i2.p1.2.m2.1.1.1.1.1.1.cmml" xref="S6.I7.i2.p1.2.m2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.I7.i2.p1.2.m2.1.1.1.1.1.1.1.cmml" xref="S6.I7.i2.p1.2.m2.1.1.1.1.1.1">subscript</csymbol><ci id="S6.I7.i2.p1.2.m2.1.1.1.1.1.1.2.cmml" xref="S6.I7.i2.p1.2.m2.1.1.1.1.1.1.2">𝜉</ci><ci id="S6.I7.i2.p1.2.m2.1.1.1.1.1.1.3.cmml" xref="S6.I7.i2.p1.2.m2.1.1.1.1.1.1.3">𝑖</ci></apply><apply id="S6.I7.i2.p1.2.m2.2.2.2.2.2.2.cmml" xref="S6.I7.i2.p1.2.m2.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.I7.i2.p1.2.m2.2.2.2.2.2.2.1.cmml" xref="S6.I7.i2.p1.2.m2.2.2.2.2.2.2">subscript</csymbol><ci id="S6.I7.i2.p1.2.m2.2.2.2.2.2.2.2.cmml" xref="S6.I7.i2.p1.2.m2.2.2.2.2.2.2.2">𝜉</ci><ci id="S6.I7.i2.p1.2.m2.2.2.2.2.2.2.3.cmml" xref="S6.I7.i2.p1.2.m2.2.2.2.2.2.2.3">𝑗</ci></apply></interval></apply><ci id="S6.I7.i2.p1.2.m2.4.4.7.cmml" xref="S6.I7.i2.p1.2.m2.4.4.7">𝜉</ci></apply><apply id="S6.I7.i2.p1.2.m2.4.4c.cmml" xref="S6.I7.i2.p1.2.m2.4.4"><ci id="S6.I7.i2.p1.2.m2.4.4.8.cmml" xref="S6.I7.i2.p1.2.m2.4.4.8">→</ci><share href="https://arxiv.org/html/2503.13728v1#S6.I7.i2.p1.2.m2.4.4.7.cmml" id="S6.I7.i2.p1.2.m2.4.4d.cmml" xref="S6.I7.i2.p1.2.m2.4.4"></share><apply id="S6.I7.i2.p1.2.m2.4.4.4.cmml" xref="S6.I7.i2.p1.2.m2.4.4.4"><times id="S6.I7.i2.p1.2.m2.4.4.4.3.cmml" xref="S6.I7.i2.p1.2.m2.4.4.4.3"></times><apply id="S6.I7.i2.p1.2.m2.4.4.4.4.cmml" xref="S6.I7.i2.p1.2.m2.4.4.4.4"><csymbol cd="ambiguous" id="S6.I7.i2.p1.2.m2.4.4.4.4.1.cmml" xref="S6.I7.i2.p1.2.m2.4.4.4.4">subscript</csymbol><ci id="S6.I7.i2.p1.2.m2.4.4.4.4.2.cmml" xref="S6.I7.i2.p1.2.m2.4.4.4.4.2">Δ</ci><ci id="S6.I7.i2.p1.2.m2.4.4.4.4.3.cmml" xref="S6.I7.i2.p1.2.m2.4.4.4.4.3">𝐴</ci></apply><interval closure="open" id="S6.I7.i2.p1.2.m2.4.4.4.2.3.cmml" xref="S6.I7.i2.p1.2.m2.4.4.4.2.2"><apply id="S6.I7.i2.p1.2.m2.3.3.3.1.1.1.cmml" xref="S6.I7.i2.p1.2.m2.3.3.3.1.1.1"><csymbol cd="ambiguous" id="S6.I7.i2.p1.2.m2.3.3.3.1.1.1.1.cmml" xref="S6.I7.i2.p1.2.m2.3.3.3.1.1.1">subscript</csymbol><ci id="S6.I7.i2.p1.2.m2.3.3.3.1.1.1.2.cmml" xref="S6.I7.i2.p1.2.m2.3.3.3.1.1.1.2">𝜉</ci><ci id="S6.I7.i2.p1.2.m2.3.3.3.1.1.1.3.cmml" xref="S6.I7.i2.p1.2.m2.3.3.3.1.1.1.3">𝑖</ci></apply><apply id="S6.I7.i2.p1.2.m2.4.4.4.2.2.2.cmml" xref="S6.I7.i2.p1.2.m2.4.4.4.2.2.2"><csymbol cd="ambiguous" id="S6.I7.i2.p1.2.m2.4.4.4.2.2.2.1.cmml" xref="S6.I7.i2.p1.2.m2.4.4.4.2.2.2">subscript</csymbol><ci id="S6.I7.i2.p1.2.m2.4.4.4.2.2.2.2.cmml" xref="S6.I7.i2.p1.2.m2.4.4.4.2.2.2.2">𝜉</ci><ci id="S6.I7.i2.p1.2.m2.4.4.4.2.2.2.3.cmml" xref="S6.I7.i2.p1.2.m2.4.4.4.2.2.2.3">𝑗</ci></apply></interval></apply></apply><apply id="S6.I7.i2.p1.2.m2.4.4e.cmml" xref="S6.I7.i2.p1.2.m2.4.4"><lt id="S6.I7.i2.p1.2.m2.4.4.9.cmml" xref="S6.I7.i2.p1.2.m2.4.4.9"></lt><share href="https://arxiv.org/html/2503.13728v1#S6.I7.i2.p1.2.m2.4.4.4.cmml" id="S6.I7.i2.p1.2.m2.4.4f.cmml" xref="S6.I7.i2.p1.2.m2.4.4"></share><ci id="S6.I7.i2.p1.2.m2.4.4.10.cmml" xref="S6.I7.i2.p1.2.m2.4.4.10">𝛾</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I7.i2.p1.2.m2.4c">\Delta_{A}(\xi_{i},\xi_{j})<\xi\rightarrow\Delta_{A}(\xi_{i},\xi_{j})<\gamma</annotation><annotation encoding="application/x-llamapun" id="S6.I7.i2.p1.2.m2.4d">roman_Δ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ξ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_ξ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) < italic_ξ → roman_Δ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ξ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_ξ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) < italic_γ</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S6.I7.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(c)</span> <div class="ltx_para" id="S6.I7.i3.p1"> <p class="ltx_p" id="S6.I7.i3.p1.2">For all <math alttext="i<n" class="ltx_Math" display="inline" id="S6.I7.i3.p1.1.m1.1"><semantics id="S6.I7.i3.p1.1.m1.1a"><mrow id="S6.I7.i3.p1.1.m1.1.1" xref="S6.I7.i3.p1.1.m1.1.1.cmml"><mi id="S6.I7.i3.p1.1.m1.1.1.2" xref="S6.I7.i3.p1.1.m1.1.1.2.cmml">i</mi><mo id="S6.I7.i3.p1.1.m1.1.1.1" xref="S6.I7.i3.p1.1.m1.1.1.1.cmml"><</mo><mi id="S6.I7.i3.p1.1.m1.1.1.3" xref="S6.I7.i3.p1.1.m1.1.1.3.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.I7.i3.p1.1.m1.1b"><apply id="S6.I7.i3.p1.1.m1.1.1.cmml" xref="S6.I7.i3.p1.1.m1.1.1"><lt id="S6.I7.i3.p1.1.m1.1.1.1.cmml" xref="S6.I7.i3.p1.1.m1.1.1.1"></lt><ci id="S6.I7.i3.p1.1.m1.1.1.2.cmml" xref="S6.I7.i3.p1.1.m1.1.1.2">𝑖</ci><ci id="S6.I7.i3.p1.1.m1.1.1.3.cmml" xref="S6.I7.i3.p1.1.m1.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I7.i3.p1.1.m1.1c">i<n</annotation><annotation encoding="application/x-llamapun" id="S6.I7.i3.p1.1.m1.1d">italic_i < italic_n</annotation></semantics></math>, <math alttext="\gamma<\Delta_{A}(\xi_{i},\eta_{i})<\xi" class="ltx_Math" display="inline" id="S6.I7.i3.p1.2.m2.2"><semantics id="S6.I7.i3.p1.2.m2.2a"><mrow id="S6.I7.i3.p1.2.m2.2.2" xref="S6.I7.i3.p1.2.m2.2.2.cmml"><mi id="S6.I7.i3.p1.2.m2.2.2.4" xref="S6.I7.i3.p1.2.m2.2.2.4.cmml">γ</mi><mo id="S6.I7.i3.p1.2.m2.2.2.5" xref="S6.I7.i3.p1.2.m2.2.2.5.cmml"><</mo><mrow id="S6.I7.i3.p1.2.m2.2.2.2" xref="S6.I7.i3.p1.2.m2.2.2.2.cmml"><msub id="S6.I7.i3.p1.2.m2.2.2.2.4" xref="S6.I7.i3.p1.2.m2.2.2.2.4.cmml"><mi id="S6.I7.i3.p1.2.m2.2.2.2.4.2" mathvariant="normal" xref="S6.I7.i3.p1.2.m2.2.2.2.4.2.cmml">Δ</mi><mi id="S6.I7.i3.p1.2.m2.2.2.2.4.3" xref="S6.I7.i3.p1.2.m2.2.2.2.4.3.cmml">A</mi></msub><mo id="S6.I7.i3.p1.2.m2.2.2.2.3" xref="S6.I7.i3.p1.2.m2.2.2.2.3.cmml">⁢</mo><mrow id="S6.I7.i3.p1.2.m2.2.2.2.2.2" xref="S6.I7.i3.p1.2.m2.2.2.2.2.3.cmml"><mo id="S6.I7.i3.p1.2.m2.2.2.2.2.2.3" stretchy="false" xref="S6.I7.i3.p1.2.m2.2.2.2.2.3.cmml">(</mo><msub id="S6.I7.i3.p1.2.m2.1.1.1.1.1.1" xref="S6.I7.i3.p1.2.m2.1.1.1.1.1.1.cmml"><mi id="S6.I7.i3.p1.2.m2.1.1.1.1.1.1.2" xref="S6.I7.i3.p1.2.m2.1.1.1.1.1.1.2.cmml">ξ</mi><mi id="S6.I7.i3.p1.2.m2.1.1.1.1.1.1.3" xref="S6.I7.i3.p1.2.m2.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S6.I7.i3.p1.2.m2.2.2.2.2.2.4" xref="S6.I7.i3.p1.2.m2.2.2.2.2.3.cmml">,</mo><msub id="S6.I7.i3.p1.2.m2.2.2.2.2.2.2" xref="S6.I7.i3.p1.2.m2.2.2.2.2.2.2.cmml"><mi id="S6.I7.i3.p1.2.m2.2.2.2.2.2.2.2" xref="S6.I7.i3.p1.2.m2.2.2.2.2.2.2.2.cmml">η</mi><mi id="S6.I7.i3.p1.2.m2.2.2.2.2.2.2.3" xref="S6.I7.i3.p1.2.m2.2.2.2.2.2.2.3.cmml">i</mi></msub><mo id="S6.I7.i3.p1.2.m2.2.2.2.2.2.5" stretchy="false" xref="S6.I7.i3.p1.2.m2.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.I7.i3.p1.2.m2.2.2.6" xref="S6.I7.i3.p1.2.m2.2.2.6.cmml"><</mo><mi id="S6.I7.i3.p1.2.m2.2.2.7" xref="S6.I7.i3.p1.2.m2.2.2.7.cmml">ξ</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.I7.i3.p1.2.m2.2b"><apply id="S6.I7.i3.p1.2.m2.2.2.cmml" xref="S6.I7.i3.p1.2.m2.2.2"><and id="S6.I7.i3.p1.2.m2.2.2a.cmml" xref="S6.I7.i3.p1.2.m2.2.2"></and><apply id="S6.I7.i3.p1.2.m2.2.2b.cmml" xref="S6.I7.i3.p1.2.m2.2.2"><lt id="S6.I7.i3.p1.2.m2.2.2.5.cmml" xref="S6.I7.i3.p1.2.m2.2.2.5"></lt><ci id="S6.I7.i3.p1.2.m2.2.2.4.cmml" xref="S6.I7.i3.p1.2.m2.2.2.4">𝛾</ci><apply id="S6.I7.i3.p1.2.m2.2.2.2.cmml" xref="S6.I7.i3.p1.2.m2.2.2.2"><times id="S6.I7.i3.p1.2.m2.2.2.2.3.cmml" xref="S6.I7.i3.p1.2.m2.2.2.2.3"></times><apply id="S6.I7.i3.p1.2.m2.2.2.2.4.cmml" xref="S6.I7.i3.p1.2.m2.2.2.2.4"><csymbol cd="ambiguous" id="S6.I7.i3.p1.2.m2.2.2.2.4.1.cmml" xref="S6.I7.i3.p1.2.m2.2.2.2.4">subscript</csymbol><ci id="S6.I7.i3.p1.2.m2.2.2.2.4.2.cmml" xref="S6.I7.i3.p1.2.m2.2.2.2.4.2">Δ</ci><ci id="S6.I7.i3.p1.2.m2.2.2.2.4.3.cmml" xref="S6.I7.i3.p1.2.m2.2.2.2.4.3">𝐴</ci></apply><interval closure="open" id="S6.I7.i3.p1.2.m2.2.2.2.2.3.cmml" xref="S6.I7.i3.p1.2.m2.2.2.2.2.2"><apply id="S6.I7.i3.p1.2.m2.1.1.1.1.1.1.cmml" xref="S6.I7.i3.p1.2.m2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.I7.i3.p1.2.m2.1.1.1.1.1.1.1.cmml" xref="S6.I7.i3.p1.2.m2.1.1.1.1.1.1">subscript</csymbol><ci id="S6.I7.i3.p1.2.m2.1.1.1.1.1.1.2.cmml" xref="S6.I7.i3.p1.2.m2.1.1.1.1.1.1.2">𝜉</ci><ci id="S6.I7.i3.p1.2.m2.1.1.1.1.1.1.3.cmml" xref="S6.I7.i3.p1.2.m2.1.1.1.1.1.1.3">𝑖</ci></apply><apply id="S6.I7.i3.p1.2.m2.2.2.2.2.2.2.cmml" xref="S6.I7.i3.p1.2.m2.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.I7.i3.p1.2.m2.2.2.2.2.2.2.1.cmml" xref="S6.I7.i3.p1.2.m2.2.2.2.2.2.2">subscript</csymbol><ci id="S6.I7.i3.p1.2.m2.2.2.2.2.2.2.2.cmml" xref="S6.I7.i3.p1.2.m2.2.2.2.2.2.2.2">𝜂</ci><ci id="S6.I7.i3.p1.2.m2.2.2.2.2.2.2.3.cmml" xref="S6.I7.i3.p1.2.m2.2.2.2.2.2.2.3">𝑖</ci></apply></interval></apply></apply><apply id="S6.I7.i3.p1.2.m2.2.2c.cmml" xref="S6.I7.i3.p1.2.m2.2.2"><lt id="S6.I7.i3.p1.2.m2.2.2.6.cmml" xref="S6.I7.i3.p1.2.m2.2.2.6"></lt><share href="https://arxiv.org/html/2503.13728v1#S6.I7.i3.p1.2.m2.2.2.2.cmml" id="S6.I7.i3.p1.2.m2.2.2d.cmml" xref="S6.I7.i3.p1.2.m2.2.2"></share><ci id="S6.I7.i3.p1.2.m2.2.2.7.cmml" xref="S6.I7.i3.p1.2.m2.2.2.7">𝜉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I7.i3.p1.2.m2.2c">\gamma<\Delta_{A}(\xi_{i},\eta_{i})<\xi</annotation><annotation encoding="application/x-llamapun" id="S6.I7.i3.p1.2.m2.2d">italic_γ < roman_Δ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ξ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_η start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) < italic_ξ</annotation></semantics></math>.</p> </div> </li> </ol> <p class="ltx_p" id="S6.SS2.10.3.p1.22">First note that for all <math alttext="\xi<\eta" class="ltx_Math" display="inline" id="S6.SS2.10.3.p1.14.m1.1"><semantics id="S6.SS2.10.3.p1.14.m1.1a"><mrow id="S6.SS2.10.3.p1.14.m1.1.1" xref="S6.SS2.10.3.p1.14.m1.1.1.cmml"><mi id="S6.SS2.10.3.p1.14.m1.1.1.2" xref="S6.SS2.10.3.p1.14.m1.1.1.2.cmml">ξ</mi><mo id="S6.SS2.10.3.p1.14.m1.1.1.1" xref="S6.SS2.10.3.p1.14.m1.1.1.1.cmml"><</mo><mi id="S6.SS2.10.3.p1.14.m1.1.1.3" xref="S6.SS2.10.3.p1.14.m1.1.1.3.cmml">η</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.10.3.p1.14.m1.1b"><apply id="S6.SS2.10.3.p1.14.m1.1.1.cmml" xref="S6.SS2.10.3.p1.14.m1.1.1"><lt id="S6.SS2.10.3.p1.14.m1.1.1.1.cmml" xref="S6.SS2.10.3.p1.14.m1.1.1.1"></lt><ci id="S6.SS2.10.3.p1.14.m1.1.1.2.cmml" xref="S6.SS2.10.3.p1.14.m1.1.1.2">𝜉</ci><ci id="S6.SS2.10.3.p1.14.m1.1.1.3.cmml" xref="S6.SS2.10.3.p1.14.m1.1.1.3">𝜂</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.10.3.p1.14.m1.1c">\xi<\eta</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.10.3.p1.14.m1.1d">italic_ξ < italic_η</annotation></semantics></math>, <math alttext="\bar{a}\in\operatorname{dom}(p_{\xi})" class="ltx_Math" display="inline" id="S6.SS2.10.3.p1.15.m2.2"><semantics id="S6.SS2.10.3.p1.15.m2.2a"><mrow id="S6.SS2.10.3.p1.15.m2.2.2" xref="S6.SS2.10.3.p1.15.m2.2.2.cmml"><mover accent="true" id="S6.SS2.10.3.p1.15.m2.2.2.3" xref="S6.SS2.10.3.p1.15.m2.2.2.3.cmml"><mi id="S6.SS2.10.3.p1.15.m2.2.2.3.2" xref="S6.SS2.10.3.p1.15.m2.2.2.3.2.cmml">a</mi><mo id="S6.SS2.10.3.p1.15.m2.2.2.3.1" xref="S6.SS2.10.3.p1.15.m2.2.2.3.1.cmml">¯</mo></mover><mo id="S6.SS2.10.3.p1.15.m2.2.2.2" xref="S6.SS2.10.3.p1.15.m2.2.2.2.cmml">∈</mo><mrow id="S6.SS2.10.3.p1.15.m2.2.2.1.1" xref="S6.SS2.10.3.p1.15.m2.2.2.1.2.cmml"><mi id="S6.SS2.10.3.p1.15.m2.1.1" xref="S6.SS2.10.3.p1.15.m2.1.1.cmml">dom</mi><mo id="S6.SS2.10.3.p1.15.m2.2.2.1.1a" xref="S6.SS2.10.3.p1.15.m2.2.2.1.2.cmml">⁡</mo><mrow id="S6.SS2.10.3.p1.15.m2.2.2.1.1.1" xref="S6.SS2.10.3.p1.15.m2.2.2.1.2.cmml"><mo id="S6.SS2.10.3.p1.15.m2.2.2.1.1.1.2" stretchy="false" xref="S6.SS2.10.3.p1.15.m2.2.2.1.2.cmml">(</mo><msub id="S6.SS2.10.3.p1.15.m2.2.2.1.1.1.1" xref="S6.SS2.10.3.p1.15.m2.2.2.1.1.1.1.cmml"><mi id="S6.SS2.10.3.p1.15.m2.2.2.1.1.1.1.2" xref="S6.SS2.10.3.p1.15.m2.2.2.1.1.1.1.2.cmml">p</mi><mi id="S6.SS2.10.3.p1.15.m2.2.2.1.1.1.1.3" xref="S6.SS2.10.3.p1.15.m2.2.2.1.1.1.1.3.cmml">ξ</mi></msub><mo id="S6.SS2.10.3.p1.15.m2.2.2.1.1.1.3" stretchy="false" xref="S6.SS2.10.3.p1.15.m2.2.2.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.10.3.p1.15.m2.2b"><apply id="S6.SS2.10.3.p1.15.m2.2.2.cmml" xref="S6.SS2.10.3.p1.15.m2.2.2"><in id="S6.SS2.10.3.p1.15.m2.2.2.2.cmml" xref="S6.SS2.10.3.p1.15.m2.2.2.2"></in><apply id="S6.SS2.10.3.p1.15.m2.2.2.3.cmml" xref="S6.SS2.10.3.p1.15.m2.2.2.3"><ci id="S6.SS2.10.3.p1.15.m2.2.2.3.1.cmml" xref="S6.SS2.10.3.p1.15.m2.2.2.3.1">¯</ci><ci id="S6.SS2.10.3.p1.15.m2.2.2.3.2.cmml" xref="S6.SS2.10.3.p1.15.m2.2.2.3.2">𝑎</ci></apply><apply id="S6.SS2.10.3.p1.15.m2.2.2.1.2.cmml" xref="S6.SS2.10.3.p1.15.m2.2.2.1.1"><ci id="S6.SS2.10.3.p1.15.m2.1.1.cmml" xref="S6.SS2.10.3.p1.15.m2.1.1">dom</ci><apply id="S6.SS2.10.3.p1.15.m2.2.2.1.1.1.1.cmml" xref="S6.SS2.10.3.p1.15.m2.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.10.3.p1.15.m2.2.2.1.1.1.1.1.cmml" xref="S6.SS2.10.3.p1.15.m2.2.2.1.1.1.1">subscript</csymbol><ci id="S6.SS2.10.3.p1.15.m2.2.2.1.1.1.1.2.cmml" xref="S6.SS2.10.3.p1.15.m2.2.2.1.1.1.1.2">𝑝</ci><ci id="S6.SS2.10.3.p1.15.m2.2.2.1.1.1.1.3.cmml" xref="S6.SS2.10.3.p1.15.m2.2.2.1.1.1.1.3">𝜉</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.10.3.p1.15.m2.2c">\bar{a}\in\operatorname{dom}(p_{\xi})</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.10.3.p1.15.m2.2d">over¯ start_ARG italic_a end_ARG ∈ roman_dom ( italic_p start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT )</annotation></semantics></math> and <math alttext="\bar{b}\in\operatorname{dom}(p_{\eta})" class="ltx_Math" display="inline" id="S6.SS2.10.3.p1.16.m3.2"><semantics id="S6.SS2.10.3.p1.16.m3.2a"><mrow id="S6.SS2.10.3.p1.16.m3.2.2" xref="S6.SS2.10.3.p1.16.m3.2.2.cmml"><mover accent="true" id="S6.SS2.10.3.p1.16.m3.2.2.3" xref="S6.SS2.10.3.p1.16.m3.2.2.3.cmml"><mi id="S6.SS2.10.3.p1.16.m3.2.2.3.2" xref="S6.SS2.10.3.p1.16.m3.2.2.3.2.cmml">b</mi><mo id="S6.SS2.10.3.p1.16.m3.2.2.3.1" xref="S6.SS2.10.3.p1.16.m3.2.2.3.1.cmml">¯</mo></mover><mo id="S6.SS2.10.3.p1.16.m3.2.2.2" xref="S6.SS2.10.3.p1.16.m3.2.2.2.cmml">∈</mo><mrow id="S6.SS2.10.3.p1.16.m3.2.2.1.1" xref="S6.SS2.10.3.p1.16.m3.2.2.1.2.cmml"><mi id="S6.SS2.10.3.p1.16.m3.1.1" xref="S6.SS2.10.3.p1.16.m3.1.1.cmml">dom</mi><mo id="S6.SS2.10.3.p1.16.m3.2.2.1.1a" xref="S6.SS2.10.3.p1.16.m3.2.2.1.2.cmml">⁡</mo><mrow id="S6.SS2.10.3.p1.16.m3.2.2.1.1.1" xref="S6.SS2.10.3.p1.16.m3.2.2.1.2.cmml"><mo id="S6.SS2.10.3.p1.16.m3.2.2.1.1.1.2" stretchy="false" xref="S6.SS2.10.3.p1.16.m3.2.2.1.2.cmml">(</mo><msub id="S6.SS2.10.3.p1.16.m3.2.2.1.1.1.1" xref="S6.SS2.10.3.p1.16.m3.2.2.1.1.1.1.cmml"><mi id="S6.SS2.10.3.p1.16.m3.2.2.1.1.1.1.2" xref="S6.SS2.10.3.p1.16.m3.2.2.1.1.1.1.2.cmml">p</mi><mi id="S6.SS2.10.3.p1.16.m3.2.2.1.1.1.1.3" xref="S6.SS2.10.3.p1.16.m3.2.2.1.1.1.1.3.cmml">η</mi></msub><mo id="S6.SS2.10.3.p1.16.m3.2.2.1.1.1.3" stretchy="false" xref="S6.SS2.10.3.p1.16.m3.2.2.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.10.3.p1.16.m3.2b"><apply id="S6.SS2.10.3.p1.16.m3.2.2.cmml" xref="S6.SS2.10.3.p1.16.m3.2.2"><in id="S6.SS2.10.3.p1.16.m3.2.2.2.cmml" xref="S6.SS2.10.3.p1.16.m3.2.2.2"></in><apply id="S6.SS2.10.3.p1.16.m3.2.2.3.cmml" xref="S6.SS2.10.3.p1.16.m3.2.2.3"><ci id="S6.SS2.10.3.p1.16.m3.2.2.3.1.cmml" xref="S6.SS2.10.3.p1.16.m3.2.2.3.1">¯</ci><ci id="S6.SS2.10.3.p1.16.m3.2.2.3.2.cmml" xref="S6.SS2.10.3.p1.16.m3.2.2.3.2">𝑏</ci></apply><apply id="S6.SS2.10.3.p1.16.m3.2.2.1.2.cmml" xref="S6.SS2.10.3.p1.16.m3.2.2.1.1"><ci id="S6.SS2.10.3.p1.16.m3.1.1.cmml" xref="S6.SS2.10.3.p1.16.m3.1.1">dom</ci><apply id="S6.SS2.10.3.p1.16.m3.2.2.1.1.1.1.cmml" xref="S6.SS2.10.3.p1.16.m3.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.10.3.p1.16.m3.2.2.1.1.1.1.1.cmml" xref="S6.SS2.10.3.p1.16.m3.2.2.1.1.1.1">subscript</csymbol><ci id="S6.SS2.10.3.p1.16.m3.2.2.1.1.1.1.2.cmml" xref="S6.SS2.10.3.p1.16.m3.2.2.1.1.1.1.2">𝑝</ci><ci id="S6.SS2.10.3.p1.16.m3.2.2.1.1.1.1.3.cmml" xref="S6.SS2.10.3.p1.16.m3.2.2.1.1.1.1.3">𝜂</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.10.3.p1.16.m3.2c">\bar{b}\in\operatorname{dom}(p_{\eta})</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.10.3.p1.16.m3.2d">over¯ start_ARG italic_b end_ARG ∈ roman_dom ( italic_p start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT )</annotation></semantics></math>, <math alttext="\nu(a_{m})=\nu(\bar{a})<\nu(\bar{b})=\nu(b_{m})" class="ltx_Math" display="inline" id="S6.SS2.10.3.p1.17.m4.4"><semantics id="S6.SS2.10.3.p1.17.m4.4a"><mrow id="S6.SS2.10.3.p1.17.m4.4.4" xref="S6.SS2.10.3.p1.17.m4.4.4.cmml"><mrow id="S6.SS2.10.3.p1.17.m4.3.3.1" xref="S6.SS2.10.3.p1.17.m4.3.3.1.cmml"><mi id="S6.SS2.10.3.p1.17.m4.3.3.1.3" xref="S6.SS2.10.3.p1.17.m4.3.3.1.3.cmml">ν</mi><mo id="S6.SS2.10.3.p1.17.m4.3.3.1.2" xref="S6.SS2.10.3.p1.17.m4.3.3.1.2.cmml">⁢</mo><mrow id="S6.SS2.10.3.p1.17.m4.3.3.1.1.1" xref="S6.SS2.10.3.p1.17.m4.3.3.1.1.1.1.cmml"><mo id="S6.SS2.10.3.p1.17.m4.3.3.1.1.1.2" stretchy="false" xref="S6.SS2.10.3.p1.17.m4.3.3.1.1.1.1.cmml">(</mo><msub id="S6.SS2.10.3.p1.17.m4.3.3.1.1.1.1" xref="S6.SS2.10.3.p1.17.m4.3.3.1.1.1.1.cmml"><mi id="S6.SS2.10.3.p1.17.m4.3.3.1.1.1.1.2" xref="S6.SS2.10.3.p1.17.m4.3.3.1.1.1.1.2.cmml">a</mi><mi id="S6.SS2.10.3.p1.17.m4.3.3.1.1.1.1.3" xref="S6.SS2.10.3.p1.17.m4.3.3.1.1.1.1.3.cmml">m</mi></msub><mo id="S6.SS2.10.3.p1.17.m4.3.3.1.1.1.3" stretchy="false" xref="S6.SS2.10.3.p1.17.m4.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.SS2.10.3.p1.17.m4.4.4.4" xref="S6.SS2.10.3.p1.17.m4.4.4.4.cmml">=</mo><mrow id="S6.SS2.10.3.p1.17.m4.4.4.5" xref="S6.SS2.10.3.p1.17.m4.4.4.5.cmml"><mi id="S6.SS2.10.3.p1.17.m4.4.4.5.2" xref="S6.SS2.10.3.p1.17.m4.4.4.5.2.cmml">ν</mi><mo id="S6.SS2.10.3.p1.17.m4.4.4.5.1" xref="S6.SS2.10.3.p1.17.m4.4.4.5.1.cmml">⁢</mo><mrow id="S6.SS2.10.3.p1.17.m4.4.4.5.3.2" xref="S6.SS2.10.3.p1.17.m4.1.1.cmml"><mo id="S6.SS2.10.3.p1.17.m4.4.4.5.3.2.1" stretchy="false" xref="S6.SS2.10.3.p1.17.m4.1.1.cmml">(</mo><mover accent="true" id="S6.SS2.10.3.p1.17.m4.1.1" xref="S6.SS2.10.3.p1.17.m4.1.1.cmml"><mi id="S6.SS2.10.3.p1.17.m4.1.1.2" xref="S6.SS2.10.3.p1.17.m4.1.1.2.cmml">a</mi><mo id="S6.SS2.10.3.p1.17.m4.1.1.1" xref="S6.SS2.10.3.p1.17.m4.1.1.1.cmml">¯</mo></mover><mo id="S6.SS2.10.3.p1.17.m4.4.4.5.3.2.2" stretchy="false" xref="S6.SS2.10.3.p1.17.m4.1.1.cmml">)</mo></mrow></mrow><mo id="S6.SS2.10.3.p1.17.m4.4.4.6" xref="S6.SS2.10.3.p1.17.m4.4.4.6.cmml"><</mo><mrow id="S6.SS2.10.3.p1.17.m4.4.4.7" xref="S6.SS2.10.3.p1.17.m4.4.4.7.cmml"><mi id="S6.SS2.10.3.p1.17.m4.4.4.7.2" xref="S6.SS2.10.3.p1.17.m4.4.4.7.2.cmml">ν</mi><mo id="S6.SS2.10.3.p1.17.m4.4.4.7.1" xref="S6.SS2.10.3.p1.17.m4.4.4.7.1.cmml">⁢</mo><mrow id="S6.SS2.10.3.p1.17.m4.4.4.7.3.2" xref="S6.SS2.10.3.p1.17.m4.2.2.cmml"><mo id="S6.SS2.10.3.p1.17.m4.4.4.7.3.2.1" stretchy="false" xref="S6.SS2.10.3.p1.17.m4.2.2.cmml">(</mo><mover accent="true" id="S6.SS2.10.3.p1.17.m4.2.2" xref="S6.SS2.10.3.p1.17.m4.2.2.cmml"><mi id="S6.SS2.10.3.p1.17.m4.2.2.2" xref="S6.SS2.10.3.p1.17.m4.2.2.2.cmml">b</mi><mo id="S6.SS2.10.3.p1.17.m4.2.2.1" xref="S6.SS2.10.3.p1.17.m4.2.2.1.cmml">¯</mo></mover><mo id="S6.SS2.10.3.p1.17.m4.4.4.7.3.2.2" stretchy="false" xref="S6.SS2.10.3.p1.17.m4.2.2.cmml">)</mo></mrow></mrow><mo id="S6.SS2.10.3.p1.17.m4.4.4.8" xref="S6.SS2.10.3.p1.17.m4.4.4.8.cmml">=</mo><mrow id="S6.SS2.10.3.p1.17.m4.4.4.2" xref="S6.SS2.10.3.p1.17.m4.4.4.2.cmml"><mi id="S6.SS2.10.3.p1.17.m4.4.4.2.3" xref="S6.SS2.10.3.p1.17.m4.4.4.2.3.cmml">ν</mi><mo id="S6.SS2.10.3.p1.17.m4.4.4.2.2" xref="S6.SS2.10.3.p1.17.m4.4.4.2.2.cmml">⁢</mo><mrow id="S6.SS2.10.3.p1.17.m4.4.4.2.1.1" xref="S6.SS2.10.3.p1.17.m4.4.4.2.1.1.1.cmml"><mo id="S6.SS2.10.3.p1.17.m4.4.4.2.1.1.2" stretchy="false" xref="S6.SS2.10.3.p1.17.m4.4.4.2.1.1.1.cmml">(</mo><msub id="S6.SS2.10.3.p1.17.m4.4.4.2.1.1.1" xref="S6.SS2.10.3.p1.17.m4.4.4.2.1.1.1.cmml"><mi id="S6.SS2.10.3.p1.17.m4.4.4.2.1.1.1.2" xref="S6.SS2.10.3.p1.17.m4.4.4.2.1.1.1.2.cmml">b</mi><mi id="S6.SS2.10.3.p1.17.m4.4.4.2.1.1.1.3" xref="S6.SS2.10.3.p1.17.m4.4.4.2.1.1.1.3.cmml">m</mi></msub><mo id="S6.SS2.10.3.p1.17.m4.4.4.2.1.1.3" stretchy="false" xref="S6.SS2.10.3.p1.17.m4.4.4.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.10.3.p1.17.m4.4b"><apply id="S6.SS2.10.3.p1.17.m4.4.4.cmml" xref="S6.SS2.10.3.p1.17.m4.4.4"><and id="S6.SS2.10.3.p1.17.m4.4.4a.cmml" xref="S6.SS2.10.3.p1.17.m4.4.4"></and><apply id="S6.SS2.10.3.p1.17.m4.4.4b.cmml" xref="S6.SS2.10.3.p1.17.m4.4.4"><eq id="S6.SS2.10.3.p1.17.m4.4.4.4.cmml" xref="S6.SS2.10.3.p1.17.m4.4.4.4"></eq><apply id="S6.SS2.10.3.p1.17.m4.3.3.1.cmml" xref="S6.SS2.10.3.p1.17.m4.3.3.1"><times id="S6.SS2.10.3.p1.17.m4.3.3.1.2.cmml" xref="S6.SS2.10.3.p1.17.m4.3.3.1.2"></times><ci id="S6.SS2.10.3.p1.17.m4.3.3.1.3.cmml" xref="S6.SS2.10.3.p1.17.m4.3.3.1.3">𝜈</ci><apply id="S6.SS2.10.3.p1.17.m4.3.3.1.1.1.1.cmml" xref="S6.SS2.10.3.p1.17.m4.3.3.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.10.3.p1.17.m4.3.3.1.1.1.1.1.cmml" xref="S6.SS2.10.3.p1.17.m4.3.3.1.1.1">subscript</csymbol><ci id="S6.SS2.10.3.p1.17.m4.3.3.1.1.1.1.2.cmml" xref="S6.SS2.10.3.p1.17.m4.3.3.1.1.1.1.2">𝑎</ci><ci id="S6.SS2.10.3.p1.17.m4.3.3.1.1.1.1.3.cmml" xref="S6.SS2.10.3.p1.17.m4.3.3.1.1.1.1.3">𝑚</ci></apply></apply><apply id="S6.SS2.10.3.p1.17.m4.4.4.5.cmml" xref="S6.SS2.10.3.p1.17.m4.4.4.5"><times id="S6.SS2.10.3.p1.17.m4.4.4.5.1.cmml" xref="S6.SS2.10.3.p1.17.m4.4.4.5.1"></times><ci id="S6.SS2.10.3.p1.17.m4.4.4.5.2.cmml" xref="S6.SS2.10.3.p1.17.m4.4.4.5.2">𝜈</ci><apply id="S6.SS2.10.3.p1.17.m4.1.1.cmml" xref="S6.SS2.10.3.p1.17.m4.4.4.5.3.2"><ci id="S6.SS2.10.3.p1.17.m4.1.1.1.cmml" xref="S6.SS2.10.3.p1.17.m4.1.1.1">¯</ci><ci id="S6.SS2.10.3.p1.17.m4.1.1.2.cmml" xref="S6.SS2.10.3.p1.17.m4.1.1.2">𝑎</ci></apply></apply></apply><apply id="S6.SS2.10.3.p1.17.m4.4.4c.cmml" xref="S6.SS2.10.3.p1.17.m4.4.4"><lt id="S6.SS2.10.3.p1.17.m4.4.4.6.cmml" xref="S6.SS2.10.3.p1.17.m4.4.4.6"></lt><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.10.3.p1.17.m4.4.4.5.cmml" id="S6.SS2.10.3.p1.17.m4.4.4d.cmml" xref="S6.SS2.10.3.p1.17.m4.4.4"></share><apply id="S6.SS2.10.3.p1.17.m4.4.4.7.cmml" xref="S6.SS2.10.3.p1.17.m4.4.4.7"><times id="S6.SS2.10.3.p1.17.m4.4.4.7.1.cmml" xref="S6.SS2.10.3.p1.17.m4.4.4.7.1"></times><ci id="S6.SS2.10.3.p1.17.m4.4.4.7.2.cmml" xref="S6.SS2.10.3.p1.17.m4.4.4.7.2">𝜈</ci><apply id="S6.SS2.10.3.p1.17.m4.2.2.cmml" xref="S6.SS2.10.3.p1.17.m4.4.4.7.3.2"><ci id="S6.SS2.10.3.p1.17.m4.2.2.1.cmml" xref="S6.SS2.10.3.p1.17.m4.2.2.1">¯</ci><ci id="S6.SS2.10.3.p1.17.m4.2.2.2.cmml" xref="S6.SS2.10.3.p1.17.m4.2.2.2">𝑏</ci></apply></apply></apply><apply id="S6.SS2.10.3.p1.17.m4.4.4e.cmml" xref="S6.SS2.10.3.p1.17.m4.4.4"><eq id="S6.SS2.10.3.p1.17.m4.4.4.8.cmml" xref="S6.SS2.10.3.p1.17.m4.4.4.8"></eq><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.10.3.p1.17.m4.4.4.7.cmml" id="S6.SS2.10.3.p1.17.m4.4.4f.cmml" xref="S6.SS2.10.3.p1.17.m4.4.4"></share><apply id="S6.SS2.10.3.p1.17.m4.4.4.2.cmml" xref="S6.SS2.10.3.p1.17.m4.4.4.2"><times id="S6.SS2.10.3.p1.17.m4.4.4.2.2.cmml" xref="S6.SS2.10.3.p1.17.m4.4.4.2.2"></times><ci id="S6.SS2.10.3.p1.17.m4.4.4.2.3.cmml" xref="S6.SS2.10.3.p1.17.m4.4.4.2.3">𝜈</ci><apply id="S6.SS2.10.3.p1.17.m4.4.4.2.1.1.1.cmml" xref="S6.SS2.10.3.p1.17.m4.4.4.2.1.1"><csymbol cd="ambiguous" id="S6.SS2.10.3.p1.17.m4.4.4.2.1.1.1.1.cmml" xref="S6.SS2.10.3.p1.17.m4.4.4.2.1.1">subscript</csymbol><ci id="S6.SS2.10.3.p1.17.m4.4.4.2.1.1.1.2.cmml" xref="S6.SS2.10.3.p1.17.m4.4.4.2.1.1.1.2">𝑏</ci><ci id="S6.SS2.10.3.p1.17.m4.4.4.2.1.1.1.3.cmml" xref="S6.SS2.10.3.p1.17.m4.4.4.2.1.1.1.3">𝑚</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.10.3.p1.17.m4.4c">\nu(a_{m})=\nu(\bar{a})<\nu(\bar{b})=\nu(b_{m})</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.10.3.p1.17.m4.4d">italic_ν ( italic_a start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) = italic_ν ( over¯ start_ARG italic_a end_ARG ) < italic_ν ( over¯ start_ARG italic_b end_ARG ) = italic_ν ( italic_b start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT )</annotation></semantics></math>, thus <math alttext="\max(\operatorname{dom}(q_{\xi}))<\min(\operatorname{dom}(q_{\eta}))" class="ltx_Math" display="inline" id="S6.SS2.10.3.p1.18.m5.6"><semantics id="S6.SS2.10.3.p1.18.m5.6a"><mrow id="S6.SS2.10.3.p1.18.m5.6.6" xref="S6.SS2.10.3.p1.18.m5.6.6.cmml"><mrow id="S6.SS2.10.3.p1.18.m5.5.5.1.1" xref="S6.SS2.10.3.p1.18.m5.5.5.1.2.cmml"><mi id="S6.SS2.10.3.p1.18.m5.2.2" xref="S6.SS2.10.3.p1.18.m5.2.2.cmml">max</mi><mo id="S6.SS2.10.3.p1.18.m5.5.5.1.1a" xref="S6.SS2.10.3.p1.18.m5.5.5.1.2.cmml">⁡</mo><mrow id="S6.SS2.10.3.p1.18.m5.5.5.1.1.1" xref="S6.SS2.10.3.p1.18.m5.5.5.1.2.cmml"><mo id="S6.SS2.10.3.p1.18.m5.5.5.1.1.1.2" stretchy="false" xref="S6.SS2.10.3.p1.18.m5.5.5.1.2.cmml">(</mo><mrow id="S6.SS2.10.3.p1.18.m5.5.5.1.1.1.1.1" xref="S6.SS2.10.3.p1.18.m5.5.5.1.1.1.1.2.cmml"><mi id="S6.SS2.10.3.p1.18.m5.1.1" xref="S6.SS2.10.3.p1.18.m5.1.1.cmml">dom</mi><mo id="S6.SS2.10.3.p1.18.m5.5.5.1.1.1.1.1a" xref="S6.SS2.10.3.p1.18.m5.5.5.1.1.1.1.2.cmml">⁡</mo><mrow id="S6.SS2.10.3.p1.18.m5.5.5.1.1.1.1.1.1" xref="S6.SS2.10.3.p1.18.m5.5.5.1.1.1.1.2.cmml"><mo id="S6.SS2.10.3.p1.18.m5.5.5.1.1.1.1.1.1.2" stretchy="false" xref="S6.SS2.10.3.p1.18.m5.5.5.1.1.1.1.2.cmml">(</mo><msub id="S6.SS2.10.3.p1.18.m5.5.5.1.1.1.1.1.1.1" xref="S6.SS2.10.3.p1.18.m5.5.5.1.1.1.1.1.1.1.cmml"><mi id="S6.SS2.10.3.p1.18.m5.5.5.1.1.1.1.1.1.1.2" xref="S6.SS2.10.3.p1.18.m5.5.5.1.1.1.1.1.1.1.2.cmml">q</mi><mi id="S6.SS2.10.3.p1.18.m5.5.5.1.1.1.1.1.1.1.3" xref="S6.SS2.10.3.p1.18.m5.5.5.1.1.1.1.1.1.1.3.cmml">ξ</mi></msub><mo id="S6.SS2.10.3.p1.18.m5.5.5.1.1.1.1.1.1.3" stretchy="false" xref="S6.SS2.10.3.p1.18.m5.5.5.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S6.SS2.10.3.p1.18.m5.5.5.1.1.1.3" stretchy="false" xref="S6.SS2.10.3.p1.18.m5.5.5.1.2.cmml">)</mo></mrow></mrow><mo id="S6.SS2.10.3.p1.18.m5.6.6.3" xref="S6.SS2.10.3.p1.18.m5.6.6.3.cmml"><</mo><mrow id="S6.SS2.10.3.p1.18.m5.6.6.2.1" xref="S6.SS2.10.3.p1.18.m5.6.6.2.2.cmml"><mi id="S6.SS2.10.3.p1.18.m5.4.4" xref="S6.SS2.10.3.p1.18.m5.4.4.cmml">min</mi><mo id="S6.SS2.10.3.p1.18.m5.6.6.2.1a" xref="S6.SS2.10.3.p1.18.m5.6.6.2.2.cmml">⁡</mo><mrow id="S6.SS2.10.3.p1.18.m5.6.6.2.1.1" xref="S6.SS2.10.3.p1.18.m5.6.6.2.2.cmml"><mo id="S6.SS2.10.3.p1.18.m5.6.6.2.1.1.2" stretchy="false" xref="S6.SS2.10.3.p1.18.m5.6.6.2.2.cmml">(</mo><mrow id="S6.SS2.10.3.p1.18.m5.6.6.2.1.1.1.1" xref="S6.SS2.10.3.p1.18.m5.6.6.2.1.1.1.2.cmml"><mi id="S6.SS2.10.3.p1.18.m5.3.3" xref="S6.SS2.10.3.p1.18.m5.3.3.cmml">dom</mi><mo id="S6.SS2.10.3.p1.18.m5.6.6.2.1.1.1.1a" xref="S6.SS2.10.3.p1.18.m5.6.6.2.1.1.1.2.cmml">⁡</mo><mrow id="S6.SS2.10.3.p1.18.m5.6.6.2.1.1.1.1.1" xref="S6.SS2.10.3.p1.18.m5.6.6.2.1.1.1.2.cmml"><mo id="S6.SS2.10.3.p1.18.m5.6.6.2.1.1.1.1.1.2" stretchy="false" xref="S6.SS2.10.3.p1.18.m5.6.6.2.1.1.1.2.cmml">(</mo><msub id="S6.SS2.10.3.p1.18.m5.6.6.2.1.1.1.1.1.1" xref="S6.SS2.10.3.p1.18.m5.6.6.2.1.1.1.1.1.1.cmml"><mi id="S6.SS2.10.3.p1.18.m5.6.6.2.1.1.1.1.1.1.2" xref="S6.SS2.10.3.p1.18.m5.6.6.2.1.1.1.1.1.1.2.cmml">q</mi><mi id="S6.SS2.10.3.p1.18.m5.6.6.2.1.1.1.1.1.1.3" xref="S6.SS2.10.3.p1.18.m5.6.6.2.1.1.1.1.1.1.3.cmml">η</mi></msub><mo id="S6.SS2.10.3.p1.18.m5.6.6.2.1.1.1.1.1.3" stretchy="false" xref="S6.SS2.10.3.p1.18.m5.6.6.2.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S6.SS2.10.3.p1.18.m5.6.6.2.1.1.3" stretchy="false" xref="S6.SS2.10.3.p1.18.m5.6.6.2.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.10.3.p1.18.m5.6b"><apply id="S6.SS2.10.3.p1.18.m5.6.6.cmml" xref="S6.SS2.10.3.p1.18.m5.6.6"><lt id="S6.SS2.10.3.p1.18.m5.6.6.3.cmml" xref="S6.SS2.10.3.p1.18.m5.6.6.3"></lt><apply id="S6.SS2.10.3.p1.18.m5.5.5.1.2.cmml" xref="S6.SS2.10.3.p1.18.m5.5.5.1.1"><max id="S6.SS2.10.3.p1.18.m5.2.2.cmml" xref="S6.SS2.10.3.p1.18.m5.2.2"></max><apply id="S6.SS2.10.3.p1.18.m5.5.5.1.1.1.1.2.cmml" xref="S6.SS2.10.3.p1.18.m5.5.5.1.1.1.1.1"><ci id="S6.SS2.10.3.p1.18.m5.1.1.cmml" xref="S6.SS2.10.3.p1.18.m5.1.1">dom</ci><apply id="S6.SS2.10.3.p1.18.m5.5.5.1.1.1.1.1.1.1.cmml" xref="S6.SS2.10.3.p1.18.m5.5.5.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.10.3.p1.18.m5.5.5.1.1.1.1.1.1.1.1.cmml" xref="S6.SS2.10.3.p1.18.m5.5.5.1.1.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.10.3.p1.18.m5.5.5.1.1.1.1.1.1.1.2.cmml" xref="S6.SS2.10.3.p1.18.m5.5.5.1.1.1.1.1.1.1.2">𝑞</ci><ci id="S6.SS2.10.3.p1.18.m5.5.5.1.1.1.1.1.1.1.3.cmml" xref="S6.SS2.10.3.p1.18.m5.5.5.1.1.1.1.1.1.1.3">𝜉</ci></apply></apply></apply><apply id="S6.SS2.10.3.p1.18.m5.6.6.2.2.cmml" xref="S6.SS2.10.3.p1.18.m5.6.6.2.1"><min id="S6.SS2.10.3.p1.18.m5.4.4.cmml" xref="S6.SS2.10.3.p1.18.m5.4.4"></min><apply id="S6.SS2.10.3.p1.18.m5.6.6.2.1.1.1.2.cmml" xref="S6.SS2.10.3.p1.18.m5.6.6.2.1.1.1.1"><ci id="S6.SS2.10.3.p1.18.m5.3.3.cmml" xref="S6.SS2.10.3.p1.18.m5.3.3">dom</ci><apply id="S6.SS2.10.3.p1.18.m5.6.6.2.1.1.1.1.1.1.cmml" xref="S6.SS2.10.3.p1.18.m5.6.6.2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.10.3.p1.18.m5.6.6.2.1.1.1.1.1.1.1.cmml" xref="S6.SS2.10.3.p1.18.m5.6.6.2.1.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.10.3.p1.18.m5.6.6.2.1.1.1.1.1.1.2.cmml" xref="S6.SS2.10.3.p1.18.m5.6.6.2.1.1.1.1.1.1.2">𝑞</ci><ci id="S6.SS2.10.3.p1.18.m5.6.6.2.1.1.1.1.1.1.3.cmml" xref="S6.SS2.10.3.p1.18.m5.6.6.2.1.1.1.1.1.1.3">𝜂</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.10.3.p1.18.m5.6c">\max(\operatorname{dom}(q_{\xi}))<\min(\operatorname{dom}(q_{\eta}))</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.10.3.p1.18.m5.6d">roman_max ( roman_dom ( italic_q start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT ) ) < roman_min ( roman_dom ( italic_q start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT ) )</annotation></semantics></math>. So letting <math alttext="\Gamma" class="ltx_Math" display="inline" id="S6.SS2.10.3.p1.19.m6.1"><semantics id="S6.SS2.10.3.p1.19.m6.1a"><mi id="S6.SS2.10.3.p1.19.m6.1.1" mathvariant="normal" xref="S6.SS2.10.3.p1.19.m6.1.1.cmml">Γ</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.10.3.p1.19.m6.1b"><ci id="S6.SS2.10.3.p1.19.m6.1.1.cmml" xref="S6.SS2.10.3.p1.19.m6.1.1">Γ</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.10.3.p1.19.m6.1c">\Gamma</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.10.3.p1.19.m6.1d">roman_Γ</annotation></semantics></math> be a club of ordinals closed under <math alttext="\alpha\mapsto\min\{\xi:\operatorname{dom}(q_{\xi})\cap\alpha=\varnothing\}" class="ltx_Math" display="inline" id="S6.SS2.10.3.p1.20.m7.3"><semantics id="S6.SS2.10.3.p1.20.m7.3a"><mrow id="S6.SS2.10.3.p1.20.m7.3.3" xref="S6.SS2.10.3.p1.20.m7.3.3.cmml"><mi id="S6.SS2.10.3.p1.20.m7.3.3.3" xref="S6.SS2.10.3.p1.20.m7.3.3.3.cmml">α</mi><mo id="S6.SS2.10.3.p1.20.m7.3.3.2" stretchy="false" xref="S6.SS2.10.3.p1.20.m7.3.3.2.cmml">↦</mo><mrow id="S6.SS2.10.3.p1.20.m7.3.3.1.1" xref="S6.SS2.10.3.p1.20.m7.3.3.1.2.cmml"><mi id="S6.SS2.10.3.p1.20.m7.2.2" xref="S6.SS2.10.3.p1.20.m7.2.2.cmml">min</mi><mo id="S6.SS2.10.3.p1.20.m7.3.3.1.1a" xref="S6.SS2.10.3.p1.20.m7.3.3.1.2.cmml">⁡</mo><mrow id="S6.SS2.10.3.p1.20.m7.3.3.1.1.1" xref="S6.SS2.10.3.p1.20.m7.3.3.1.2.cmml"><mo id="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.2" stretchy="false" xref="S6.SS2.10.3.p1.20.m7.3.3.1.2.cmml">{</mo><mrow id="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1" xref="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.cmml"><mi id="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.3" xref="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.3.cmml">ξ</mi><mo id="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.2" lspace="0.278em" rspace="0.278em" xref="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.2.cmml">:</mo><mrow id="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.1" xref="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.1.cmml"><mrow id="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.1.1" xref="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.1.1.cmml"><mrow id="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.1.1.1.1" xref="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.1.1.1.2.cmml"><mi id="S6.SS2.10.3.p1.20.m7.1.1" xref="S6.SS2.10.3.p1.20.m7.1.1.cmml">dom</mi><mo id="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.1.1.1.1a" xref="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.1.1.1.2.cmml">⁡</mo><mrow id="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.1.1.1.1.1" xref="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.1.1.1.2.cmml"><mo id="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.1.1.1.2.cmml">(</mo><msub id="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.1.1.1.1.1.1" xref="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.1.1.1.1.1.1.2" xref="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.1.1.1.1.1.1.2.cmml">q</mi><mi id="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.1.1.1.1.1.1.3" xref="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.1.1.1.1.1.1.3.cmml">ξ</mi></msub><mo id="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.1.1.2" xref="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.1.1.2.cmml">∩</mo><mi id="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.1.1.3" xref="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.1.1.3.cmml">α</mi></mrow><mo id="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.1.2" xref="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.1.2.cmml">=</mo><mi id="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.1.3" mathvariant="normal" xref="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.1.3.cmml">∅</mi></mrow></mrow><mo id="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.3" stretchy="false" xref="S6.SS2.10.3.p1.20.m7.3.3.1.2.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.10.3.p1.20.m7.3b"><apply id="S6.SS2.10.3.p1.20.m7.3.3.cmml" xref="S6.SS2.10.3.p1.20.m7.3.3"><csymbol cd="latexml" id="S6.SS2.10.3.p1.20.m7.3.3.2.cmml" xref="S6.SS2.10.3.p1.20.m7.3.3.2">maps-to</csymbol><ci id="S6.SS2.10.3.p1.20.m7.3.3.3.cmml" xref="S6.SS2.10.3.p1.20.m7.3.3.3">𝛼</ci><apply id="S6.SS2.10.3.p1.20.m7.3.3.1.2.cmml" xref="S6.SS2.10.3.p1.20.m7.3.3.1.1"><min id="S6.SS2.10.3.p1.20.m7.2.2.cmml" xref="S6.SS2.10.3.p1.20.m7.2.2"></min><apply id="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.cmml" xref="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1"><ci id="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.2.cmml" xref="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.2">:</ci><ci id="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.3.cmml" xref="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.3">𝜉</ci><apply id="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.1.cmml" xref="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.1"><eq id="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.1.2.cmml" xref="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.1.2"></eq><apply id="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.1.1.cmml" xref="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.1.1"><intersect id="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.1.1.2.cmml" xref="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.1.1.2"></intersect><apply id="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.1.1.1.2.cmml" xref="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.1.1.1.1"><ci id="S6.SS2.10.3.p1.20.m7.1.1.cmml" xref="S6.SS2.10.3.p1.20.m7.1.1">dom</ci><apply id="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.1.1.1.1.1.1.cmml" xref="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.1.1.1.1.1.1.2.cmml" xref="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.1.1.1.1.1.1.2">𝑞</ci><ci id="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.1.1.1.1.1.1.3.cmml" xref="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.1.1.1.1.1.1.3">𝜉</ci></apply></apply><ci id="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.1.1.3.cmml" xref="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.1.1.3">𝛼</ci></apply><emptyset id="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.1.3.cmml" xref="S6.SS2.10.3.p1.20.m7.3.3.1.1.1.1.1.3"></emptyset></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.10.3.p1.20.m7.3c">\alpha\mapsto\min\{\xi:\operatorname{dom}(q_{\xi})\cap\alpha=\varnothing\}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.10.3.p1.20.m7.3d">italic_α ↦ roman_min { italic_ξ : roman_dom ( italic_q start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT ) ∩ italic_α = ∅ }</annotation></semantics></math> we easily obtain (a). (b) and (c) are obtained exactly as in <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.Thmtheorem8" title="Lemma 6.8. ‣ 6.1. Moore’s forcing ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">6.8</span></a>. We needed <math alttext="\Gamma" class="ltx_Math" display="inline" id="S6.SS2.10.3.p1.21.m8.1"><semantics id="S6.SS2.10.3.p1.21.m8.1a"><mi id="S6.SS2.10.3.p1.21.m8.1.1" mathvariant="normal" xref="S6.SS2.10.3.p1.21.m8.1.1.cmml">Γ</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.10.3.p1.21.m8.1b"><ci id="S6.SS2.10.3.p1.21.m8.1.1.cmml" xref="S6.SS2.10.3.p1.21.m8.1.1">Γ</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.10.3.p1.21.m8.1c">\Gamma</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.10.3.p1.21.m8.1d">roman_Γ</annotation></semantics></math> to be at least stationary, since (b) requires the use of Fodor’s lemma over <math alttext="\Gamma" class="ltx_Math" display="inline" id="S6.SS2.10.3.p1.22.m9.1"><semantics id="S6.SS2.10.3.p1.22.m9.1a"><mi id="S6.SS2.10.3.p1.22.m9.1.1" mathvariant="normal" xref="S6.SS2.10.3.p1.22.m9.1.1.cmml">Γ</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.10.3.p1.22.m9.1b"><ci id="S6.SS2.10.3.p1.22.m9.1.1.cmml" xref="S6.SS2.10.3.p1.22.m9.1.1">Γ</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.10.3.p1.22.m9.1c">\Gamma</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.10.3.p1.22.m9.1d">roman_Γ</annotation></semantics></math>.</p> </div> <figure class="ltx_figure" id="S6.F1"> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 1. </span>Branching pattern</figcaption><svg class="ltx_picture ltx_centering" height="970.42" id="S6.F1.pic1" overflow="visible" version="1.1" width="340.56"><g fill="#000000" stroke="#000000" transform="translate(0,970.42) matrix(1 0 0 -1 0 0) translate(-22.55,0) translate(0,1045.38)"><g stroke-linecap="round" stroke-linejoin="round" stroke-width="0.869pt"><path d="M 97.03 -296.03 C 107.5 -262.54 82.47 -227.65 67.97 -199.51 C 60.65 -185.29 53.21 -168.3 50.85 -152.29 C 49.47 -142.92 51.89 -134.09 51.89 -124.79" style="fill:none"></path></g><g stroke-linecap="round" stroke-linejoin="round" stroke-width="0.869pt"><path d="M 98.07 -266.97 C 98.07 -267.85 97.6 -265.26 97.55 -264.37 C 97.23 -258.94 97.42 -257.05 98.59 -251.92 C 101.06 -241.02 106.42 -233.71 114.16 -225.98 C 134.97 -205.16 145.02 -186.33 154.63 -158.52 C 157.66 -149.74 159.65 -142.32 159.82 -133.09 C 159.83 -132.32 161.71 -101.07 159.3 -96.25" style="fill:none"></path></g><g stroke-linecap="round" stroke-linejoin="round" stroke-width="0.869pt"><path d="M 51.63 -156.7 C 51.63 -153.54 42.57 -147.11 39.69 -143.21 C 36.1 -138.34 31.26 -132.94 29.32 -127.13 C 28 -123.19 29.32 -118.69 29.32 -114.67" style="fill:none"></path></g><g stroke-linecap="round" stroke-linejoin="round" stroke-width="0.869pt"><path d="M 57.6 -176.68 C 55.02 -174.11 56.93 -169.42 57.08 -165.78 C 57.27 -161.2 61.82 -158.53 63.82 -154.89 C 66.58 -149.87 68.63 -141.94 68.49 -136.21 C 68.3 -128.09 64.86 -119.27 64.86 -111.3" style="fill:none"></path></g><g stroke-linecap="round" stroke-linejoin="round" stroke-width="0.869pt"><path d="M 131.8 -205.22 C 131.8 -204.41 131.15 -204.58 130.76 -204.18 C 128.68 -202.1 127.8 -199.09 126.61 -196.4 C 126.03 -195.1 125.57 -189.07 125.57 -188.1 C 125.57 -180.6 129.39 -173.4 129.2 -165.78 C 129.1 -161.77 128.07 -157.84 127.65 -153.85 C 127.13 -148.94 127.79 -144.98 125.05 -140.88" style="fill:none"></path></g><g stroke-linecap="round" stroke-linejoin="round" stroke-width="0.869pt"><path d="M 128.42 -172.27 C 128.42 -173.27 129.11 -169.2 129.46 -168.64 C 130.68 -166.66 132.34 -164.94 134.13 -163.45 C 141.02 -157.71 143.87 -142.06 137.25 -135.43" style="fill:none"></path></g><g stroke-linecap="round" stroke-linejoin="round" stroke-width="0.869pt"><path d="M 126.61 -179.79 C 126.09 -179.79 125.97 -176.76 124.01 -175.64 C 118.85 -172.7 115.91 -170.14 114.16 -163.71 C 113.31 -160.63 115.82 -158.54 115.19 -155.41" style="fill:none"></path></g><g stroke-dasharray="3.0pt,1.5pt" stroke-dashoffset="0.0pt" stroke-width="0.4pt"><path d="M 57.08 -247.51 L 139.06 -247.25" style="fill:none"></path></g><g stroke-dasharray="3.0pt,1.5pt" stroke-dashoffset="0.0pt" stroke-width="0.4pt"><path d="M 41.51 -196.92 L 154.63 -196.92" style="fill:none"></path></g><g stroke-dasharray="3.0pt,1.5pt" stroke-dashoffset="0.0pt" stroke-width="0.4pt"><path d="M 28.89 -184.46 L 172.62 -184.98" style="fill:none"></path></g><g stroke-linecap="round" stroke-linejoin="round" stroke-width="0.869pt"><path d="M 296.8 -292.91 C 296.8 -289.02 298.75 -285.87 298.88 -282.02 C 299.21 -272 299.2 -261.97 295.77 -252.44 C 289.32 -234.53 279.73 -223.37 269.3 -208.85 C 261.62 -198.15 256.8 -186.33 251.14 -174.61 C 249.29 -170.78 246.78 -167.23 245.43 -163.19" style="fill:none"></path></g><g stroke-linecap="round" stroke-linejoin="round" stroke-width="0.869pt"><path d="M 299.05 -273.8 C 297.15 -273.8 299.19 -268.6 300.09 -265.5 C 302.81 -256.17 310.14 -250.12 317.73 -244.74 C 328.88 -236.85 340.27 -227.84 347.31 -215.69 C 350.79 -209.67 347.91 -205.15 350.42 -200.12" style="fill:none"></path></g><g stroke-linecap="round" stroke-linejoin="round" stroke-width="0.869pt"><path d="M 341.95 -222.77 C 337.82 -216.59 314.2 -212.65 326.38 -200.46" style="fill:none"></path></g><g stroke-linecap="round" stroke-linejoin="round" stroke-width="0.869pt"><path d="M 346.1 -217.67 C 353 -217.67 356.48 -211.62 356.48 -204.7 C 356.48 -200.23 352.91 -190.91 357.51 -188.62" style="fill:none"></path></g><g stroke-dasharray="3.0pt,1.5pt" stroke-dashoffset="0.0pt" stroke-width="0.4pt"><path d="M 261.69 -254.25 L 335.37 -254.25" style="fill:none"></path></g><g stroke-dasharray="3.0pt,1.5pt" stroke-dashoffset="0.0pt" stroke-width="0.4pt"><path d="M 264.46 -286.25 L 333.64 -286.25" style="fill:none"></path></g><g stroke-dasharray="3.0pt,1.5pt" stroke-dashoffset="0.0pt" stroke-width="0.4pt"><path d="M 250.28 -232.98 L 351.46 -232.98" style="fill:none"></path></g><g fill="#000000" stroke="#000000" stroke-linecap="round" stroke-linejoin="round" stroke-opacity="1" stroke-width="0.869pt"><path d="M 272.93 -213.39 C 272.93 -214.25 272.92 -212.91 273.19 -212.35 C 275.58 -207.58 277.84 -205.21 278.9 -200.42 C 280.24 -194.4 280.4 -187.22 278.9 -181.22 C 278.42 -179.31 276.59 -170.84 272.42 -170.84" style="fill:none"></path></g><g fill="#000000" stroke="#000000" stroke-linecap="round" stroke-linejoin="round" stroke-opacity="1" stroke-width="0.869pt"><path d="M 280.2 -196.01 C 284.42 -192.64 289.66 -186.64 288.76 -180.7 C 288.32 -177.82 287.45 -175.02 286.94 -172.14 C 286.77 -171.14 286.76 -168.77 288.24 -168.77" style="fill:none"></path></g><g fill="#000000" stroke="#000000" stroke-linecap="round" stroke-linejoin="round" stroke-opacity="1" stroke-width="0.869pt"><path d="M 293.17 -245.56 C 293.17 -246.26 295.07 -245.26 297.32 -241.93 C 301.38 -235.96 300.89 -233.74 301.73 -225.07 C 302.17 -220.52 299.95 -215.86 299.14 -211.58 C 297.54 -203.14 297.71 -194.66 298.36 -186.15" style="fill:none"></path></g><g stroke-width="0.4pt"><path d="M 22.83 -99.97" style="fill:none"></path><g fill="#000000" stroke="#000000" transform="matrix(1.0 0.0 0.0 1.0 24.15 -376.53)"><g class="ltx_tikzmatrix" transform="matrix(1 0 0 -1 0 270.925)"><g class="ltx_tikzmatrix_row" transform="matrix(1 0 0 1 0 272.56)"><g class="ltx_tikzmatrix_col ltx_nopad_l ltx_nopad_r" transform="matrix(1 0 0 -1 0 0)"><foreignobject height="8.65" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="9.78"><math alttext="a_{l}" class="ltx_Math" display="inline" id="S6.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1"><semantics id="S6.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1a"><msub id="S6.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1" xref="S6.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml"><mi id="S6.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2" xref="S6.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2.cmml">a</mi><mi id="S6.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3" xref="S6.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3.cmml">l</mi></msub><annotation-xml encoding="MathML-Content" id="S6.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1b"><apply id="S6.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml" xref="S6.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1"><csymbol cd="ambiguous" id="S6.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.1.cmml" xref="S6.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1">subscript</csymbol><ci id="S6.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2.cmml" xref="S6.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2">𝑎</ci><ci id="S6.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3.cmml" xref="S6.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3">𝑙</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1c">a_{l}</annotation><annotation encoding="application/x-llamapun" id="S6.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1d">italic_a start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT</annotation></semantics></math></foreignobject></g></g></g></g><path d="M 43.07 -106.02" style="fill:none"></path><g fill="#000000" stroke="#000000" transform="matrix(1.0 0.0 0.0 1.0 44.38 -402.37)"><g class="ltx_tikzmatrix" transform="matrix(1 0 0 -1 0 288.88)"><g class="ltx_tikzmatrix_row" transform="matrix(1 0 0 1 0 292.34)"><g class="ltx_tikzmatrix_col ltx_nopad_l ltx_nopad_r" transform="matrix(1 0 0 -1 0 0)"><foreignobject height="12.3" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="8.72"><math alttext="\xi_{i}" class="ltx_Math" display="inline" id="S6.F1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1"><semantics id="S6.F1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1a"><msub id="S6.F1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1" xref="S6.F1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml"><mi id="S6.F1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2" xref="S6.F1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2.cmml">ξ</mi><mi id="S6.F1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3" xref="S6.F1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S6.F1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1b"><apply id="S6.F1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml" xref="S6.F1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1"><csymbol cd="ambiguous" id="S6.F1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.1.cmml" xref="S6.F1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1">subscript</csymbol><ci id="S6.F1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2.cmml" xref="S6.F1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2">𝜉</ci><ci id="S6.F1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3.cmml" xref="S6.F1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.F1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1c">\xi_{i}</annotation><annotation encoding="application/x-llamapun" id="S6.F1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1d">italic_ξ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math></foreignobject></g></g></g></g><path d="M 58.12 -99.28" style="fill:none"></path><g fill="#000000" stroke="#000000" transform="matrix(1.0 0.0 0.0 1.0 59.43 -372.96)"><g class="ltx_tikzmatrix" transform="matrix(1 0 0 -1 0 268.555)"><g class="ltx_tikzmatrix_row" transform="matrix(1 0 0 1 0 270.7)"><g class="ltx_tikzmatrix_col ltx_nopad_l ltx_nopad_r" transform="matrix(1 0 0 -1 0 0)"><foreignobject height="7.63" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="11.03"><math alttext="a_{r}" class="ltx_Math" display="inline" id="S6.F1.pic1.3.3.3.3.3.3.3.3.3.3.3.3.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1"><semantics id="S6.F1.pic1.3.3.3.3.3.3.3.3.3.3.3.3.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1a"><msub id="S6.F1.pic1.3.3.3.3.3.3.3.3.3.3.3.3.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1" xref="S6.F1.pic1.3.3.3.3.3.3.3.3.3.3.3.3.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml"><mi id="S6.F1.pic1.3.3.3.3.3.3.3.3.3.3.3.3.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2" xref="S6.F1.pic1.3.3.3.3.3.3.3.3.3.3.3.3.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2.cmml">a</mi><mi id="S6.F1.pic1.3.3.3.3.3.3.3.3.3.3.3.3.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3" xref="S6.F1.pic1.3.3.3.3.3.3.3.3.3.3.3.3.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3.cmml">r</mi></msub><annotation-xml encoding="MathML-Content" id="S6.F1.pic1.3.3.3.3.3.3.3.3.3.3.3.3.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1b"><apply id="S6.F1.pic1.3.3.3.3.3.3.3.3.3.3.3.3.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml" xref="S6.F1.pic1.3.3.3.3.3.3.3.3.3.3.3.3.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1"><csymbol cd="ambiguous" id="S6.F1.pic1.3.3.3.3.3.3.3.3.3.3.3.3.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.1.cmml" xref="S6.F1.pic1.3.3.3.3.3.3.3.3.3.3.3.3.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1">subscript</csymbol><ci id="S6.F1.pic1.3.3.3.3.3.3.3.3.3.3.3.3.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2.cmml" xref="S6.F1.pic1.3.3.3.3.3.3.3.3.3.3.3.3.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2">𝑎</ci><ci id="S6.F1.pic1.3.3.3.3.3.3.3.3.3.3.3.3.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3.cmml" xref="S6.F1.pic1.3.3.3.3.3.3.3.3.3.3.3.3.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3">𝑟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.F1.pic1.3.3.3.3.3.3.3.3.3.3.3.3.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1c">a_{r}</annotation><annotation encoding="application/x-llamapun" id="S6.F1.pic1.3.3.3.3.3.3.3.3.3.3.3.3.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1d">italic_a start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT</annotation></semantics></math></foreignobject></g></g></g></g><path d="M 109.31 -139.75" style="fill:none"></path><g fill="#000000" stroke="#000000" transform="matrix(1.0 0.0 0.0 1.0 110.63 -526.04)"><g class="ltx_tikzmatrix" transform="matrix(1 0 0 -1 0 378.82)"><g class="ltx_tikzmatrix_row" transform="matrix(1 0 0 1 0 382.28)"><g class="ltx_tikzmatrix_col ltx_nopad_l ltx_nopad_r" transform="matrix(1 0 0 -1 0 0)"><foreignobject height="12.3" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="8.4"><math alttext="b_{l}" class="ltx_Math" display="inline" id="S6.F1.pic1.4.4.4.4.4.4.4.4.4.4.4.4.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1"><semantics id="S6.F1.pic1.4.4.4.4.4.4.4.4.4.4.4.4.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1a"><msub id="S6.F1.pic1.4.4.4.4.4.4.4.4.4.4.4.4.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1" xref="S6.F1.pic1.4.4.4.4.4.4.4.4.4.4.4.4.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml"><mi id="S6.F1.pic1.4.4.4.4.4.4.4.4.4.4.4.4.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2" xref="S6.F1.pic1.4.4.4.4.4.4.4.4.4.4.4.4.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2.cmml">b</mi><mi id="S6.F1.pic1.4.4.4.4.4.4.4.4.4.4.4.4.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3" xref="S6.F1.pic1.4.4.4.4.4.4.4.4.4.4.4.4.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3.cmml">l</mi></msub><annotation-xml encoding="MathML-Content" id="S6.F1.pic1.4.4.4.4.4.4.4.4.4.4.4.4.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1b"><apply id="S6.F1.pic1.4.4.4.4.4.4.4.4.4.4.4.4.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml" xref="S6.F1.pic1.4.4.4.4.4.4.4.4.4.4.4.4.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1"><csymbol cd="ambiguous" id="S6.F1.pic1.4.4.4.4.4.4.4.4.4.4.4.4.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.1.cmml" xref="S6.F1.pic1.4.4.4.4.4.4.4.4.4.4.4.4.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1">subscript</csymbol><ci id="S6.F1.pic1.4.4.4.4.4.4.4.4.4.4.4.4.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2.cmml" xref="S6.F1.pic1.4.4.4.4.4.4.4.4.4.4.4.4.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2">𝑏</ci><ci id="S6.F1.pic1.4.4.4.4.4.4.4.4.4.4.4.4.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3.cmml" xref="S6.F1.pic1.4.4.4.4.4.4.4.4.4.4.4.4.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3">𝑙</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.F1.pic1.4.4.4.4.4.4.4.4.4.4.4.4.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1c">b_{l}</annotation><annotation encoding="application/x-llamapun" id="S6.F1.pic1.4.4.4.4.4.4.4.4.4.4.4.4.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1d">italic_b start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT</annotation></semantics></math></foreignobject></g></g></g></g><path d="M 116.23 -126.44" style="fill:none"></path><g fill="#000000" stroke="#000000" transform="matrix(1.0 0.0 0.0 1.0 117.55 -474.94)"><g class="ltx_tikzmatrix" transform="matrix(1 0 0 -1 0 342.18)"><g class="ltx_tikzmatrix_row" transform="matrix(1 0 0 1 0 343.13)"><g class="ltx_tikzmatrix_col ltx_nopad_l ltx_nopad_r" transform="matrix(1 0 0 -1 0 0)"><foreignobject height="10.02" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="10.51"><math alttext="\eta_{j}" class="ltx_Math" display="inline" id="S6.F1.pic1.5.5.5.5.5.5.5.5.5.5.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1"><semantics id="S6.F1.pic1.5.5.5.5.5.5.5.5.5.5.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1a"><msub id="S6.F1.pic1.5.5.5.5.5.5.5.5.5.5.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1" xref="S6.F1.pic1.5.5.5.5.5.5.5.5.5.5.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml"><mi id="S6.F1.pic1.5.5.5.5.5.5.5.5.5.5.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2" xref="S6.F1.pic1.5.5.5.5.5.5.5.5.5.5.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2.cmml">η</mi><mi id="S6.F1.pic1.5.5.5.5.5.5.5.5.5.5.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3" xref="S6.F1.pic1.5.5.5.5.5.5.5.5.5.5.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S6.F1.pic1.5.5.5.5.5.5.5.5.5.5.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1b"><apply id="S6.F1.pic1.5.5.5.5.5.5.5.5.5.5.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml" xref="S6.F1.pic1.5.5.5.5.5.5.5.5.5.5.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1"><csymbol cd="ambiguous" id="S6.F1.pic1.5.5.5.5.5.5.5.5.5.5.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.1.cmml" xref="S6.F1.pic1.5.5.5.5.5.5.5.5.5.5.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1">subscript</csymbol><ci id="S6.F1.pic1.5.5.5.5.5.5.5.5.5.5.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2.cmml" xref="S6.F1.pic1.5.5.5.5.5.5.5.5.5.5.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2">𝜂</ci><ci id="S6.F1.pic1.5.5.5.5.5.5.5.5.5.5.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3.cmml" xref="S6.F1.pic1.5.5.5.5.5.5.5.5.5.5.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.F1.pic1.5.5.5.5.5.5.5.5.5.5.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1c">\eta_{j}</annotation><annotation encoding="application/x-llamapun" id="S6.F1.pic1.5.5.5.5.5.5.5.5.5.5.5.5.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1d">italic_η start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math></foreignobject></g></g></g></g><path d="M 133.87 -119.52" style="fill:none"></path><g fill="#000000" stroke="#000000" transform="matrix(1.0 0.0 0.0 1.0 135.19 -450.81)"><g class="ltx_tikzmatrix" transform="matrix(1 0 0 -1 0 324.34)"><g class="ltx_tikzmatrix_row" transform="matrix(1 0 0 1 0 328.32)"><g class="ltx_tikzmatrix_col ltx_nopad_l ltx_nopad_r" transform="matrix(1 0 0 -1 0 0)"><foreignobject height="11.28" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="9.65"><math alttext="b_{r}" class="ltx_Math" display="inline" id="S6.F1.pic1.6.6.6.6.6.6.6.6.6.6.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1"><semantics id="S6.F1.pic1.6.6.6.6.6.6.6.6.6.6.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1a"><msub id="S6.F1.pic1.6.6.6.6.6.6.6.6.6.6.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1" xref="S6.F1.pic1.6.6.6.6.6.6.6.6.6.6.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml"><mi id="S6.F1.pic1.6.6.6.6.6.6.6.6.6.6.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2" xref="S6.F1.pic1.6.6.6.6.6.6.6.6.6.6.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2.cmml">b</mi><mi id="S6.F1.pic1.6.6.6.6.6.6.6.6.6.6.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3" xref="S6.F1.pic1.6.6.6.6.6.6.6.6.6.6.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3.cmml">r</mi></msub><annotation-xml encoding="MathML-Content" id="S6.F1.pic1.6.6.6.6.6.6.6.6.6.6.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1b"><apply id="S6.F1.pic1.6.6.6.6.6.6.6.6.6.6.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml" xref="S6.F1.pic1.6.6.6.6.6.6.6.6.6.6.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1"><csymbol cd="ambiguous" id="S6.F1.pic1.6.6.6.6.6.6.6.6.6.6.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.1.cmml" xref="S6.F1.pic1.6.6.6.6.6.6.6.6.6.6.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1">subscript</csymbol><ci id="S6.F1.pic1.6.6.6.6.6.6.6.6.6.6.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2.cmml" xref="S6.F1.pic1.6.6.6.6.6.6.6.6.6.6.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2">𝑏</ci><ci id="S6.F1.pic1.6.6.6.6.6.6.6.6.6.6.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3.cmml" xref="S6.F1.pic1.6.6.6.6.6.6.6.6.6.6.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3">𝑟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.F1.pic1.6.6.6.6.6.6.6.6.6.6.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1c">b_{r}</annotation><annotation encoding="application/x-llamapun" id="S6.F1.pic1.6.6.6.6.6.6.6.6.6.6.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1d">italic_b start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT</annotation></semantics></math></foreignobject></g></g></g></g><path d="M 152.9 -75.24" style="fill:none"></path><g fill="#000000" stroke="#000000" transform="matrix(1.0 0.0 0.0 1.0 154.22 -290.86)"><g class="ltx_tikzmatrix" transform="matrix(1 0 0 -1 0 207.475)"><g class="ltx_tikzmatrix_row" transform="matrix(1 0 0 1 0 210.25)"><g class="ltx_tikzmatrix_col ltx_nopad_l ltx_nopad_r" transform="matrix(1 0 0 -1 0 0)"><foreignobject height="13.67" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="9.69"><math alttext="\xi_{j}" class="ltx_Math" display="inline" id="S6.F1.pic1.7.7.7.7.7.7.7.7.7.7.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1"><semantics id="S6.F1.pic1.7.7.7.7.7.7.7.7.7.7.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1a"><msub id="S6.F1.pic1.7.7.7.7.7.7.7.7.7.7.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1" xref="S6.F1.pic1.7.7.7.7.7.7.7.7.7.7.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml"><mi id="S6.F1.pic1.7.7.7.7.7.7.7.7.7.7.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2" xref="S6.F1.pic1.7.7.7.7.7.7.7.7.7.7.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2.cmml">ξ</mi><mi id="S6.F1.pic1.7.7.7.7.7.7.7.7.7.7.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3" xref="S6.F1.pic1.7.7.7.7.7.7.7.7.7.7.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S6.F1.pic1.7.7.7.7.7.7.7.7.7.7.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1b"><apply id="S6.F1.pic1.7.7.7.7.7.7.7.7.7.7.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml" xref="S6.F1.pic1.7.7.7.7.7.7.7.7.7.7.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1"><csymbol cd="ambiguous" id="S6.F1.pic1.7.7.7.7.7.7.7.7.7.7.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.1.cmml" xref="S6.F1.pic1.7.7.7.7.7.7.7.7.7.7.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1">subscript</csymbol><ci id="S6.F1.pic1.7.7.7.7.7.7.7.7.7.7.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2.cmml" xref="S6.F1.pic1.7.7.7.7.7.7.7.7.7.7.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2">𝜉</ci><ci id="S6.F1.pic1.7.7.7.7.7.7.7.7.7.7.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3.cmml" xref="S6.F1.pic1.7.7.7.7.7.7.7.7.7.7.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.F1.pic1.7.7.7.7.7.7.7.7.7.7.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1c">\xi_{j}</annotation><annotation encoding="application/x-llamapun" id="S6.F1.pic1.7.7.7.7.7.7.7.7.7.7.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1d">italic_ξ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math></foreignobject></g></g></g></g><path d="M 142.87 -243.01" style="fill:none"></path><g fill="#000000" stroke="#000000" transform="matrix(1.0 0.0 0.0 1.0 144.18 -901)"><g class="ltx_tikzmatrix" transform="matrix(1 0 0 -1 0 652.355)"><g class="ltx_tikzmatrix_row" transform="matrix(1 0 0 1 0 653.99)"><g class="ltx_tikzmatrix_col ltx_nopad_l ltx_nopad_r" transform="matrix(1 0 0 -1 0 0)"><foreignobject height="8.65" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="7.16"><math alttext="\gamma" class="ltx_Math" display="inline" id="S6.F1.pic1.8.8.8.8.8.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1"><semantics id="S6.F1.pic1.8.8.8.8.8.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1a"><mi id="S6.F1.pic1.8.8.8.8.8.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1" xref="S6.F1.pic1.8.8.8.8.8.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml">γ</mi><annotation-xml encoding="MathML-Content" id="S6.F1.pic1.8.8.8.8.8.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1b"><ci id="S6.F1.pic1.8.8.8.8.8.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml" xref="S6.F1.pic1.8.8.8.8.8.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1">𝛾</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.F1.pic1.8.8.8.8.8.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1c">\gamma</annotation><annotation encoding="application/x-llamapun" id="S6.F1.pic1.8.8.8.8.8.8.8.8.8.8.8.8.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1d">italic_γ</annotation></semantics></math></foreignobject></g></g></g></g><path d="M 155.67 -190.78" style="fill:none"></path><g fill="#000000" stroke="#000000" transform="matrix(1.0 0.0 0.0 1.0 156.98 -713.14)"><g class="ltx_tikzmatrix" transform="matrix(1 0 0 -1 0 514.89)"><g class="ltx_tikzmatrix_row" transform="matrix(1 0 0 1 0 518.35)"><g class="ltx_tikzmatrix_col ltx_nopad_l ltx_nopad_r" transform="matrix(1 0 0 -1 0 0)"><foreignobject height="12.3" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="6.05"><math alttext="\xi" class="ltx_Math" display="inline" id="S6.F1.pic1.9.9.9.9.9.9.9.9.9.9.9.9.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1"><semantics id="S6.F1.pic1.9.9.9.9.9.9.9.9.9.9.9.9.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1a"><mi id="S6.F1.pic1.9.9.9.9.9.9.9.9.9.9.9.9.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1" xref="S6.F1.pic1.9.9.9.9.9.9.9.9.9.9.9.9.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml">ξ</mi><annotation-xml encoding="MathML-Content" id="S6.F1.pic1.9.9.9.9.9.9.9.9.9.9.9.9.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1b"><ci id="S6.F1.pic1.9.9.9.9.9.9.9.9.9.9.9.9.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml" xref="S6.F1.pic1.9.9.9.9.9.9.9.9.9.9.9.9.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1">𝜉</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.F1.pic1.9.9.9.9.9.9.9.9.9.9.9.9.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1c">\xi</annotation><annotation encoding="application/x-llamapun" id="S6.F1.pic1.9.9.9.9.9.9.9.9.9.9.9.9.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1d">italic_ξ</annotation></semantics></math></foreignobject></g></g></g></g><path d="M 174.69 -179.71" style="fill:none"></path><g fill="#000000" stroke="#000000" transform="matrix(1.0 0.0 0.0 1.0 176.01 -668.89)"><g class="ltx_tikzmatrix" transform="matrix(1 0 0 -1 0 483.535)"><g class="ltx_tikzmatrix_row" transform="matrix(1 0 0 1 0 485.18)"><g class="ltx_tikzmatrix_col ltx_nopad_l ltx_nopad_r" transform="matrix(1 0 0 -1 0 0)"><foreignobject height="8.65" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="6.87"><math alttext="\eta" class="ltx_Math" display="inline" id="S6.F1.pic1.10.10.10.10.10.10.10.10.10.10.10.10.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1"><semantics id="S6.F1.pic1.10.10.10.10.10.10.10.10.10.10.10.10.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1a"><mi id="S6.F1.pic1.10.10.10.10.10.10.10.10.10.10.10.10.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1" xref="S6.F1.pic1.10.10.10.10.10.10.10.10.10.10.10.10.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S6.F1.pic1.10.10.10.10.10.10.10.10.10.10.10.10.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1b"><ci id="S6.F1.pic1.10.10.10.10.10.10.10.10.10.10.10.10.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml" xref="S6.F1.pic1.10.10.10.10.10.10.10.10.10.10.10.10.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.F1.pic1.10.10.10.10.10.10.10.10.10.10.10.10.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S6.F1.pic1.10.10.10.10.10.10.10.10.10.10.10.10.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1d">italic_η</annotation></semantics></math></foreignobject></g></g></g></g><path d="M 238.34 -150.48" style="fill:none"></path><g fill="#000000" stroke="#000000" transform="matrix(1.0 0.0 0.0 1.0 239.66 -561.71)"><g class="ltx_tikzmatrix" transform="matrix(1 0 0 -1 0 405.595)"><g class="ltx_tikzmatrix_row" transform="matrix(1 0 0 1 0 407.23)"><g class="ltx_tikzmatrix_col ltx_nopad_l ltx_nopad_r" transform="matrix(1 0 0 -1 0 0)"><foreignobject height="8.65" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="9.78"><math alttext="a_{l}" class="ltx_Math" display="inline" id="S6.F1.pic1.11.11.11.11.11.11.11.11.11.11.11.11.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1"><semantics id="S6.F1.pic1.11.11.11.11.11.11.11.11.11.11.11.11.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1a"><msub id="S6.F1.pic1.11.11.11.11.11.11.11.11.11.11.11.11.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1" xref="S6.F1.pic1.11.11.11.11.11.11.11.11.11.11.11.11.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml"><mi id="S6.F1.pic1.11.11.11.11.11.11.11.11.11.11.11.11.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2" xref="S6.F1.pic1.11.11.11.11.11.11.11.11.11.11.11.11.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2.cmml">a</mi><mi id="S6.F1.pic1.11.11.11.11.11.11.11.11.11.11.11.11.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3" xref="S6.F1.pic1.11.11.11.11.11.11.11.11.11.11.11.11.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3.cmml">l</mi></msub><annotation-xml encoding="MathML-Content" id="S6.F1.pic1.11.11.11.11.11.11.11.11.11.11.11.11.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1b"><apply id="S6.F1.pic1.11.11.11.11.11.11.11.11.11.11.11.11.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml" xref="S6.F1.pic1.11.11.11.11.11.11.11.11.11.11.11.11.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1"><csymbol cd="ambiguous" id="S6.F1.pic1.11.11.11.11.11.11.11.11.11.11.11.11.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.1.cmml" xref="S6.F1.pic1.11.11.11.11.11.11.11.11.11.11.11.11.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1">subscript</csymbol><ci id="S6.F1.pic1.11.11.11.11.11.11.11.11.11.11.11.11.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2.cmml" xref="S6.F1.pic1.11.11.11.11.11.11.11.11.11.11.11.11.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2">𝑎</ci><ci id="S6.F1.pic1.11.11.11.11.11.11.11.11.11.11.11.11.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3.cmml" xref="S6.F1.pic1.11.11.11.11.11.11.11.11.11.11.11.11.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3">𝑙</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.F1.pic1.11.11.11.11.11.11.11.11.11.11.11.11.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1c">a_{l}</annotation><annotation encoding="application/x-llamapun" id="S6.F1.pic1.11.11.11.11.11.11.11.11.11.11.11.11.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1d">italic_a start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT</annotation></semantics></math></foreignobject></g></g></g></g><path d="M 262.39 -147.88" style="fill:none"></path><g fill="#000000" stroke="#000000" transform="matrix(1.0 0.0 0.0 1.0 263.7 -555.85)"><g class="ltx_tikzmatrix" transform="matrix(1 0 0 -1 0 400.5)"><g class="ltx_tikzmatrix_row" transform="matrix(1 0 0 1 0 403.96)"><g class="ltx_tikzmatrix_col ltx_nopad_l ltx_nopad_r" transform="matrix(1 0 0 -1 0 0)"><foreignobject height="12.3" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="8.72"><math alttext="\xi_{i}" class="ltx_Math" display="inline" id="S6.F1.pic1.12.12.12.12.12.12.12.12.12.12.12.12.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1"><semantics id="S6.F1.pic1.12.12.12.12.12.12.12.12.12.12.12.12.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1a"><msub id="S6.F1.pic1.12.12.12.12.12.12.12.12.12.12.12.12.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1" xref="S6.F1.pic1.12.12.12.12.12.12.12.12.12.12.12.12.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml"><mi id="S6.F1.pic1.12.12.12.12.12.12.12.12.12.12.12.12.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2" xref="S6.F1.pic1.12.12.12.12.12.12.12.12.12.12.12.12.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2.cmml">ξ</mi><mi id="S6.F1.pic1.12.12.12.12.12.12.12.12.12.12.12.12.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3" xref="S6.F1.pic1.12.12.12.12.12.12.12.12.12.12.12.12.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S6.F1.pic1.12.12.12.12.12.12.12.12.12.12.12.12.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1b"><apply id="S6.F1.pic1.12.12.12.12.12.12.12.12.12.12.12.12.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml" xref="S6.F1.pic1.12.12.12.12.12.12.12.12.12.12.12.12.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1"><csymbol cd="ambiguous" id="S6.F1.pic1.12.12.12.12.12.12.12.12.12.12.12.12.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.1.cmml" xref="S6.F1.pic1.12.12.12.12.12.12.12.12.12.12.12.12.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1">subscript</csymbol><ci id="S6.F1.pic1.12.12.12.12.12.12.12.12.12.12.12.12.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2.cmml" xref="S6.F1.pic1.12.12.12.12.12.12.12.12.12.12.12.12.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2">𝜉</ci><ci id="S6.F1.pic1.12.12.12.12.12.12.12.12.12.12.12.12.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3.cmml" xref="S6.F1.pic1.12.12.12.12.12.12.12.12.12.12.12.12.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.F1.pic1.12.12.12.12.12.12.12.12.12.12.12.12.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1c">\xi_{i}</annotation><annotation encoding="application/x-llamapun" id="S6.F1.pic1.12.12.12.12.12.12.12.12.12.12.12.12.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1d">italic_ξ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math></foreignobject></g></g></g></g><path d="M 282.71 -156.19" style="fill:none"></path><g fill="#000000" stroke="#000000" transform="matrix(1.0 0.0 0.0 1.0 284.02 -581.62)"><g class="ltx_tikzmatrix" transform="matrix(1 0 0 -1 0 420.305)"><g class="ltx_tikzmatrix_row" transform="matrix(1 0 0 1 0 422.45)"><g class="ltx_tikzmatrix_col ltx_nopad_l ltx_nopad_r" transform="matrix(1 0 0 -1 0 0)"><foreignobject height="7.63" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="11.03"><math alttext="a_{r}" class="ltx_Math" display="inline" id="S6.F1.pic1.13.13.13.13.13.13.13.13.13.13.13.13.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1"><semantics id="S6.F1.pic1.13.13.13.13.13.13.13.13.13.13.13.13.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1a"><msub id="S6.F1.pic1.13.13.13.13.13.13.13.13.13.13.13.13.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1" xref="S6.F1.pic1.13.13.13.13.13.13.13.13.13.13.13.13.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml"><mi id="S6.F1.pic1.13.13.13.13.13.13.13.13.13.13.13.13.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2" xref="S6.F1.pic1.13.13.13.13.13.13.13.13.13.13.13.13.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2.cmml">a</mi><mi id="S6.F1.pic1.13.13.13.13.13.13.13.13.13.13.13.13.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3" xref="S6.F1.pic1.13.13.13.13.13.13.13.13.13.13.13.13.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3.cmml">r</mi></msub><annotation-xml encoding="MathML-Content" id="S6.F1.pic1.13.13.13.13.13.13.13.13.13.13.13.13.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1b"><apply id="S6.F1.pic1.13.13.13.13.13.13.13.13.13.13.13.13.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml" xref="S6.F1.pic1.13.13.13.13.13.13.13.13.13.13.13.13.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1"><csymbol cd="ambiguous" id="S6.F1.pic1.13.13.13.13.13.13.13.13.13.13.13.13.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.1.cmml" xref="S6.F1.pic1.13.13.13.13.13.13.13.13.13.13.13.13.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1">subscript</csymbol><ci id="S6.F1.pic1.13.13.13.13.13.13.13.13.13.13.13.13.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2.cmml" xref="S6.F1.pic1.13.13.13.13.13.13.13.13.13.13.13.13.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2">𝑎</ci><ci id="S6.F1.pic1.13.13.13.13.13.13.13.13.13.13.13.13.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3.cmml" xref="S6.F1.pic1.13.13.13.13.13.13.13.13.13.13.13.13.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3">𝑟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.F1.pic1.13.13.13.13.13.13.13.13.13.13.13.13.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1c">a_{r}</annotation><annotation encoding="application/x-llamapun" id="S6.F1.pic1.13.13.13.13.13.13.13.13.13.13.13.13.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1d">italic_a start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT</annotation></semantics></math></foreignobject></g></g></g></g><path d="M 320.67 -179.71" style="fill:none"></path><g fill="#000000" stroke="#000000" transform="matrix(1.0 0.0 0.0 1.0 321.99 -672.54)"><g class="ltx_tikzmatrix" transform="matrix(1 0 0 -1 0 485.37)"><g class="ltx_tikzmatrix_row" transform="matrix(1 0 0 1 0 488.83)"><g class="ltx_tikzmatrix_col ltx_nopad_l ltx_nopad_r" transform="matrix(1 0 0 -1 0 0)"><foreignobject height="12.3" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="8.4"><math alttext="b_{l}" class="ltx_Math" display="inline" id="S6.F1.pic1.14.14.14.14.14.14.14.14.14.14.14.14.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1"><semantics id="S6.F1.pic1.14.14.14.14.14.14.14.14.14.14.14.14.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1a"><msub id="S6.F1.pic1.14.14.14.14.14.14.14.14.14.14.14.14.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1" xref="S6.F1.pic1.14.14.14.14.14.14.14.14.14.14.14.14.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml"><mi id="S6.F1.pic1.14.14.14.14.14.14.14.14.14.14.14.14.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2" xref="S6.F1.pic1.14.14.14.14.14.14.14.14.14.14.14.14.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2.cmml">b</mi><mi id="S6.F1.pic1.14.14.14.14.14.14.14.14.14.14.14.14.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3" xref="S6.F1.pic1.14.14.14.14.14.14.14.14.14.14.14.14.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3.cmml">l</mi></msub><annotation-xml encoding="MathML-Content" id="S6.F1.pic1.14.14.14.14.14.14.14.14.14.14.14.14.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1b"><apply id="S6.F1.pic1.14.14.14.14.14.14.14.14.14.14.14.14.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml" xref="S6.F1.pic1.14.14.14.14.14.14.14.14.14.14.14.14.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1"><csymbol cd="ambiguous" id="S6.F1.pic1.14.14.14.14.14.14.14.14.14.14.14.14.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.1.cmml" xref="S6.F1.pic1.14.14.14.14.14.14.14.14.14.14.14.14.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1">subscript</csymbol><ci id="S6.F1.pic1.14.14.14.14.14.14.14.14.14.14.14.14.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2.cmml" xref="S6.F1.pic1.14.14.14.14.14.14.14.14.14.14.14.14.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2">𝑏</ci><ci id="S6.F1.pic1.14.14.14.14.14.14.14.14.14.14.14.14.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3.cmml" xref="S6.F1.pic1.14.14.14.14.14.14.14.14.14.14.14.14.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3">𝑙</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.F1.pic1.14.14.14.14.14.14.14.14.14.14.14.14.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1c">b_{l}</annotation><annotation encoding="application/x-llamapun" id="S6.F1.pic1.14.14.14.14.14.14.14.14.14.14.14.14.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1d">italic_b start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT</annotation></semantics></math></foreignobject></g></g></g></g><path d="M 337.45 -182.13" style="fill:none"></path><g fill="#000000" stroke="#000000" transform="matrix(1.0 0.0 0.0 1.0 338.76 -679.14)"><g class="ltx_tikzmatrix" transform="matrix(1 0 0 -1 0 490.69)"><g class="ltx_tikzmatrix_row" transform="matrix(1 0 0 1 0 491.64)"><g class="ltx_tikzmatrix_col ltx_nopad_l ltx_nopad_r" transform="matrix(1 0 0 -1 0 0)"><foreignobject height="10.02" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="10.51"><math alttext="\eta_{j}" class="ltx_Math" display="inline" id="S6.F1.pic1.15.15.15.15.15.15.15.15.15.15.15.15.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1"><semantics id="S6.F1.pic1.15.15.15.15.15.15.15.15.15.15.15.15.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1a"><msub id="S6.F1.pic1.15.15.15.15.15.15.15.15.15.15.15.15.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1" xref="S6.F1.pic1.15.15.15.15.15.15.15.15.15.15.15.15.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml"><mi id="S6.F1.pic1.15.15.15.15.15.15.15.15.15.15.15.15.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2" xref="S6.F1.pic1.15.15.15.15.15.15.15.15.15.15.15.15.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2.cmml">η</mi><mi id="S6.F1.pic1.15.15.15.15.15.15.15.15.15.15.15.15.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3" xref="S6.F1.pic1.15.15.15.15.15.15.15.15.15.15.15.15.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S6.F1.pic1.15.15.15.15.15.15.15.15.15.15.15.15.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1b"><apply id="S6.F1.pic1.15.15.15.15.15.15.15.15.15.15.15.15.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml" xref="S6.F1.pic1.15.15.15.15.15.15.15.15.15.15.15.15.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1"><csymbol cd="ambiguous" id="S6.F1.pic1.15.15.15.15.15.15.15.15.15.15.15.15.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.1.cmml" xref="S6.F1.pic1.15.15.15.15.15.15.15.15.15.15.15.15.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1">subscript</csymbol><ci id="S6.F1.pic1.15.15.15.15.15.15.15.15.15.15.15.15.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2.cmml" xref="S6.F1.pic1.15.15.15.15.15.15.15.15.15.15.15.15.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2">𝜂</ci><ci id="S6.F1.pic1.15.15.15.15.15.15.15.15.15.15.15.15.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3.cmml" xref="S6.F1.pic1.15.15.15.15.15.15.15.15.15.15.15.15.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.F1.pic1.15.15.15.15.15.15.15.15.15.15.15.15.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1c">\eta_{j}</annotation><annotation encoding="application/x-llamapun" id="S6.F1.pic1.15.15.15.15.15.15.15.15.15.15.15.15.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1d">italic_η start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math></foreignobject></g></g></g></g><path d="M 351.12 -170.02" style="fill:none"></path><g fill="#000000" stroke="#000000" transform="matrix(1.0 0.0 0.0 1.0 352.43 -636.01)"><g class="ltx_tikzmatrix" transform="matrix(1 0 0 -1 0 459.03)"><g class="ltx_tikzmatrix_row" transform="matrix(1 0 0 1 0 463.01)"><g class="ltx_tikzmatrix_col ltx_nopad_l ltx_nopad_r" transform="matrix(1 0 0 -1 0 0)"><foreignobject height="11.28" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="9.65"><math alttext="b_{r}" class="ltx_Math" display="inline" id="S6.F1.pic1.16.16.16.16.16.16.16.16.16.16.16.16.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1"><semantics id="S6.F1.pic1.16.16.16.16.16.16.16.16.16.16.16.16.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1a"><msub id="S6.F1.pic1.16.16.16.16.16.16.16.16.16.16.16.16.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1" xref="S6.F1.pic1.16.16.16.16.16.16.16.16.16.16.16.16.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml"><mi id="S6.F1.pic1.16.16.16.16.16.16.16.16.16.16.16.16.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2" xref="S6.F1.pic1.16.16.16.16.16.16.16.16.16.16.16.16.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2.cmml">b</mi><mi id="S6.F1.pic1.16.16.16.16.16.16.16.16.16.16.16.16.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3" xref="S6.F1.pic1.16.16.16.16.16.16.16.16.16.16.16.16.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3.cmml">r</mi></msub><annotation-xml encoding="MathML-Content" id="S6.F1.pic1.16.16.16.16.16.16.16.16.16.16.16.16.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1b"><apply id="S6.F1.pic1.16.16.16.16.16.16.16.16.16.16.16.16.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml" xref="S6.F1.pic1.16.16.16.16.16.16.16.16.16.16.16.16.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1"><csymbol cd="ambiguous" id="S6.F1.pic1.16.16.16.16.16.16.16.16.16.16.16.16.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.1.cmml" xref="S6.F1.pic1.16.16.16.16.16.16.16.16.16.16.16.16.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1">subscript</csymbol><ci id="S6.F1.pic1.16.16.16.16.16.16.16.16.16.16.16.16.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2.cmml" xref="S6.F1.pic1.16.16.16.16.16.16.16.16.16.16.16.16.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2">𝑏</ci><ci id="S6.F1.pic1.16.16.16.16.16.16.16.16.16.16.16.16.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3.cmml" xref="S6.F1.pic1.16.16.16.16.16.16.16.16.16.16.16.16.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3">𝑟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.F1.pic1.16.16.16.16.16.16.16.16.16.16.16.16.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1c">b_{r}</annotation><annotation encoding="application/x-llamapun" id="S6.F1.pic1.16.16.16.16.16.16.16.16.16.16.16.16.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1d">italic_b start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT</annotation></semantics></math></foreignobject></g></g></g></g><path d="M 352.5 -227.1" style="fill:none"></path><g fill="#000000" stroke="#000000" transform="matrix(1.0 0.0 0.0 1.0 353.81 -842.67)"><g class="ltx_tikzmatrix" transform="matrix(1 0 0 -1 0 609.925)"><g class="ltx_tikzmatrix_row" transform="matrix(1 0 0 1 0 611.56)"><g class="ltx_tikzmatrix_col ltx_nopad_l ltx_nopad_r" transform="matrix(1 0 0 -1 0 0)"><foreignobject height="8.65" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="6.87"><math alttext="\eta" class="ltx_Math" display="inline" id="S6.F1.pic1.17.17.17.17.17.17.17.17.17.17.17.17.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1"><semantics id="S6.F1.pic1.17.17.17.17.17.17.17.17.17.17.17.17.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1a"><mi id="S6.F1.pic1.17.17.17.17.17.17.17.17.17.17.17.17.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1" xref="S6.F1.pic1.17.17.17.17.17.17.17.17.17.17.17.17.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S6.F1.pic1.17.17.17.17.17.17.17.17.17.17.17.17.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1b"><ci id="S6.F1.pic1.17.17.17.17.17.17.17.17.17.17.17.17.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml" xref="S6.F1.pic1.17.17.17.17.17.17.17.17.17.17.17.17.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.F1.pic1.17.17.17.17.17.17.17.17.17.17.17.17.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S6.F1.pic1.17.17.17.17.17.17.17.17.17.17.17.17.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1d">italic_η</annotation></semantics></math></foreignobject></g></g></g></g><path d="M 337.8 -249.07" style="fill:none"></path><g fill="#000000" stroke="#000000" transform="matrix(1.0 0.0 0.0 1.0 339.11 -926.86)"><g class="ltx_tikzmatrix" transform="matrix(1 0 0 -1 0 670.33)"><g class="ltx_tikzmatrix_row" transform="matrix(1 0 0 1 0 673.79)"><g class="ltx_tikzmatrix_col ltx_nopad_l ltx_nopad_r" transform="matrix(1 0 0 -1 0 0)"><foreignobject height="12.3" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="6.05"><math alttext="\xi" class="ltx_Math" display="inline" id="S6.F1.pic1.18.18.18.18.18.18.18.18.18.18.18.18.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1"><semantics id="S6.F1.pic1.18.18.18.18.18.18.18.18.18.18.18.18.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1a"><mi id="S6.F1.pic1.18.18.18.18.18.18.18.18.18.18.18.18.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1" xref="S6.F1.pic1.18.18.18.18.18.18.18.18.18.18.18.18.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml">ξ</mi><annotation-xml encoding="MathML-Content" id="S6.F1.pic1.18.18.18.18.18.18.18.18.18.18.18.18.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1b"><ci id="S6.F1.pic1.18.18.18.18.18.18.18.18.18.18.18.18.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml" xref="S6.F1.pic1.18.18.18.18.18.18.18.18.18.18.18.18.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1">𝜉</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.F1.pic1.18.18.18.18.18.18.18.18.18.18.18.18.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1c">\xi</annotation><annotation encoding="application/x-llamapun" id="S6.F1.pic1.18.18.18.18.18.18.18.18.18.18.18.18.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1d">italic_ξ</annotation></semantics></math></foreignobject></g></g></g></g><path d="M 335.2 -282.1" style="fill:none"></path><g fill="#000000" stroke="#000000" transform="matrix(1.0 0.0 0.0 1.0 336.52 -1044.34)"><g class="ltx_tikzmatrix" transform="matrix(1 0 0 -1 0 756.595)"><g class="ltx_tikzmatrix_row" transform="matrix(1 0 0 1 0 758.24)"><g class="ltx_tikzmatrix_col ltx_nopad_l ltx_nopad_r" transform="matrix(1 0 0 -1 0 0)"><foreignobject height="8.65" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="7.16"><math alttext="\gamma" class="ltx_Math" display="inline" id="S6.F1.pic1.19.19.19.19.19.19.19.19.19.19.19.19.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1"><semantics id="S6.F1.pic1.19.19.19.19.19.19.19.19.19.19.19.19.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1a"><mi id="S6.F1.pic1.19.19.19.19.19.19.19.19.19.19.19.19.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1" xref="S6.F1.pic1.19.19.19.19.19.19.19.19.19.19.19.19.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml">γ</mi><annotation-xml encoding="MathML-Content" id="S6.F1.pic1.19.19.19.19.19.19.19.19.19.19.19.19.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1b"><ci id="S6.F1.pic1.19.19.19.19.19.19.19.19.19.19.19.19.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml" xref="S6.F1.pic1.19.19.19.19.19.19.19.19.19.19.19.19.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1">𝛾</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.F1.pic1.19.19.19.19.19.19.19.19.19.19.19.19.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1c">\gamma</annotation><annotation encoding="application/x-llamapun" id="S6.F1.pic1.19.19.19.19.19.19.19.19.19.19.19.19.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1d">italic_γ</annotation></semantics></math></foreignobject></g></g></g></g><path d="M 294.21 -166.3" style="fill:none"></path><g fill="#000000" stroke="#000000" transform="matrix(1.0 0.0 0.0 1.0 295.52 -624.76)"><g class="ltx_tikzmatrix" transform="matrix(1 0 0 -1 0 450.315)"><g class="ltx_tikzmatrix_row" transform="matrix(1 0 0 1 0 453.09)"><g class="ltx_tikzmatrix_col ltx_nopad_l ltx_nopad_r" transform="matrix(1 0 0 -1 0 0)"><foreignobject height="13.67" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="9.69"><math alttext="\xi_{j}" class="ltx_Math" display="inline" id="S6.F1.pic1.20.20.20.20.20.20.20.20.20.20.20.20.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1"><semantics id="S6.F1.pic1.20.20.20.20.20.20.20.20.20.20.20.20.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1a"><msub id="S6.F1.pic1.20.20.20.20.20.20.20.20.20.20.20.20.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1" xref="S6.F1.pic1.20.20.20.20.20.20.20.20.20.20.20.20.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml"><mi id="S6.F1.pic1.20.20.20.20.20.20.20.20.20.20.20.20.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2" xref="S6.F1.pic1.20.20.20.20.20.20.20.20.20.20.20.20.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2.cmml">ξ</mi><mi id="S6.F1.pic1.20.20.20.20.20.20.20.20.20.20.20.20.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3" xref="S6.F1.pic1.20.20.20.20.20.20.20.20.20.20.20.20.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S6.F1.pic1.20.20.20.20.20.20.20.20.20.20.20.20.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1b"><apply id="S6.F1.pic1.20.20.20.20.20.20.20.20.20.20.20.20.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml" xref="S6.F1.pic1.20.20.20.20.20.20.20.20.20.20.20.20.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1"><csymbol cd="ambiguous" id="S6.F1.pic1.20.20.20.20.20.20.20.20.20.20.20.20.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.1.cmml" xref="S6.F1.pic1.20.20.20.20.20.20.20.20.20.20.20.20.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1">subscript</csymbol><ci id="S6.F1.pic1.20.20.20.20.20.20.20.20.20.20.20.20.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2.cmml" xref="S6.F1.pic1.20.20.20.20.20.20.20.20.20.20.20.20.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2">𝜉</ci><ci id="S6.F1.pic1.20.20.20.20.20.20.20.20.20.20.20.20.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3.cmml" xref="S6.F1.pic1.20.20.20.20.20.20.20.20.20.20.20.20.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.F1.pic1.20.20.20.20.20.20.20.20.20.20.20.20.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1c">\xi_{j}</annotation><annotation encoding="application/x-llamapun" id="S6.F1.pic1.20.20.20.20.20.20.20.20.20.20.20.20.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1d">italic_ξ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math></foreignobject></g></g></g></g></g></g></svg> </figure> <div class="ltx_para" id="S6.SS2.11.4.p2"> <p class="ltx_p" id="S6.SS2.11.4.p2.12">Now let <math alttext="\Gamma" class="ltx_Math" display="inline" id="S6.SS2.11.4.p2.1.m1.1"><semantics id="S6.SS2.11.4.p2.1.m1.1a"><mi id="S6.SS2.11.4.p2.1.m1.1.1" mathvariant="normal" xref="S6.SS2.11.4.p2.1.m1.1.1.cmml">Γ</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.11.4.p2.1.m1.1b"><ci id="S6.SS2.11.4.p2.1.m1.1.1.cmml" xref="S6.SS2.11.4.p2.1.m1.1.1">Γ</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.11.4.p2.1.m1.1c">\Gamma</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.11.4.p2.1.m1.1d">roman_Γ</annotation></semantics></math> satisfy (a), (b) and (c). We claim that <math alttext="H^{\prime}:=\{p_{\xi}:\xi\in\Gamma\}" class="ltx_Math" display="inline" id="S6.SS2.11.4.p2.2.m2.2"><semantics id="S6.SS2.11.4.p2.2.m2.2a"><mrow id="S6.SS2.11.4.p2.2.m2.2.2" xref="S6.SS2.11.4.p2.2.m2.2.2.cmml"><msup id="S6.SS2.11.4.p2.2.m2.2.2.4" xref="S6.SS2.11.4.p2.2.m2.2.2.4.cmml"><mi id="S6.SS2.11.4.p2.2.m2.2.2.4.2" xref="S6.SS2.11.4.p2.2.m2.2.2.4.2.cmml">H</mi><mo id="S6.SS2.11.4.p2.2.m2.2.2.4.3" xref="S6.SS2.11.4.p2.2.m2.2.2.4.3.cmml">′</mo></msup><mo id="S6.SS2.11.4.p2.2.m2.2.2.3" lspace="0.278em" rspace="0.278em" xref="S6.SS2.11.4.p2.2.m2.2.2.3.cmml">:=</mo><mrow id="S6.SS2.11.4.p2.2.m2.2.2.2.2" xref="S6.SS2.11.4.p2.2.m2.2.2.2.3.cmml"><mo id="S6.SS2.11.4.p2.2.m2.2.2.2.2.3" stretchy="false" xref="S6.SS2.11.4.p2.2.m2.2.2.2.3.1.cmml">{</mo><msub id="S6.SS2.11.4.p2.2.m2.1.1.1.1.1" xref="S6.SS2.11.4.p2.2.m2.1.1.1.1.1.cmml"><mi id="S6.SS2.11.4.p2.2.m2.1.1.1.1.1.2" xref="S6.SS2.11.4.p2.2.m2.1.1.1.1.1.2.cmml">p</mi><mi id="S6.SS2.11.4.p2.2.m2.1.1.1.1.1.3" xref="S6.SS2.11.4.p2.2.m2.1.1.1.1.1.3.cmml">ξ</mi></msub><mo id="S6.SS2.11.4.p2.2.m2.2.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S6.SS2.11.4.p2.2.m2.2.2.2.3.1.cmml">:</mo><mrow id="S6.SS2.11.4.p2.2.m2.2.2.2.2.2" xref="S6.SS2.11.4.p2.2.m2.2.2.2.2.2.cmml"><mi id="S6.SS2.11.4.p2.2.m2.2.2.2.2.2.2" xref="S6.SS2.11.4.p2.2.m2.2.2.2.2.2.2.cmml">ξ</mi><mo id="S6.SS2.11.4.p2.2.m2.2.2.2.2.2.1" xref="S6.SS2.11.4.p2.2.m2.2.2.2.2.2.1.cmml">∈</mo><mi id="S6.SS2.11.4.p2.2.m2.2.2.2.2.2.3" mathvariant="normal" xref="S6.SS2.11.4.p2.2.m2.2.2.2.2.2.3.cmml">Γ</mi></mrow><mo id="S6.SS2.11.4.p2.2.m2.2.2.2.2.5" stretchy="false" xref="S6.SS2.11.4.p2.2.m2.2.2.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.11.4.p2.2.m2.2b"><apply id="S6.SS2.11.4.p2.2.m2.2.2.cmml" xref="S6.SS2.11.4.p2.2.m2.2.2"><csymbol cd="latexml" id="S6.SS2.11.4.p2.2.m2.2.2.3.cmml" xref="S6.SS2.11.4.p2.2.m2.2.2.3">assign</csymbol><apply id="S6.SS2.11.4.p2.2.m2.2.2.4.cmml" xref="S6.SS2.11.4.p2.2.m2.2.2.4"><csymbol cd="ambiguous" id="S6.SS2.11.4.p2.2.m2.2.2.4.1.cmml" xref="S6.SS2.11.4.p2.2.m2.2.2.4">superscript</csymbol><ci id="S6.SS2.11.4.p2.2.m2.2.2.4.2.cmml" xref="S6.SS2.11.4.p2.2.m2.2.2.4.2">𝐻</ci><ci id="S6.SS2.11.4.p2.2.m2.2.2.4.3.cmml" xref="S6.SS2.11.4.p2.2.m2.2.2.4.3">′</ci></apply><apply id="S6.SS2.11.4.p2.2.m2.2.2.2.3.cmml" xref="S6.SS2.11.4.p2.2.m2.2.2.2.2"><csymbol cd="latexml" id="S6.SS2.11.4.p2.2.m2.2.2.2.3.1.cmml" xref="S6.SS2.11.4.p2.2.m2.2.2.2.2.3">conditional-set</csymbol><apply id="S6.SS2.11.4.p2.2.m2.1.1.1.1.1.cmml" xref="S6.SS2.11.4.p2.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.11.4.p2.2.m2.1.1.1.1.1.1.cmml" xref="S6.SS2.11.4.p2.2.m2.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.11.4.p2.2.m2.1.1.1.1.1.2.cmml" xref="S6.SS2.11.4.p2.2.m2.1.1.1.1.1.2">𝑝</ci><ci id="S6.SS2.11.4.p2.2.m2.1.1.1.1.1.3.cmml" xref="S6.SS2.11.4.p2.2.m2.1.1.1.1.1.3">𝜉</ci></apply><apply id="S6.SS2.11.4.p2.2.m2.2.2.2.2.2.cmml" xref="S6.SS2.11.4.p2.2.m2.2.2.2.2.2"><in id="S6.SS2.11.4.p2.2.m2.2.2.2.2.2.1.cmml" xref="S6.SS2.11.4.p2.2.m2.2.2.2.2.2.1"></in><ci id="S6.SS2.11.4.p2.2.m2.2.2.2.2.2.2.cmml" xref="S6.SS2.11.4.p2.2.m2.2.2.2.2.2.2">𝜉</ci><ci id="S6.SS2.11.4.p2.2.m2.2.2.2.2.2.3.cmml" xref="S6.SS2.11.4.p2.2.m2.2.2.2.2.2.3">Γ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.11.4.p2.2.m2.2c">H^{\prime}:=\{p_{\xi}:\xi\in\Gamma\}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.11.4.p2.2.m2.2d">italic_H start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT := { italic_p start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT : italic_ξ ∈ roman_Γ }</annotation></semantics></math> works. For this fix <math alttext="\xi<\eta" class="ltx_Math" display="inline" id="S6.SS2.11.4.p2.3.m3.1"><semantics id="S6.SS2.11.4.p2.3.m3.1a"><mrow id="S6.SS2.11.4.p2.3.m3.1.1" xref="S6.SS2.11.4.p2.3.m3.1.1.cmml"><mi id="S6.SS2.11.4.p2.3.m3.1.1.2" xref="S6.SS2.11.4.p2.3.m3.1.1.2.cmml">ξ</mi><mo id="S6.SS2.11.4.p2.3.m3.1.1.1" xref="S6.SS2.11.4.p2.3.m3.1.1.1.cmml"><</mo><mi id="S6.SS2.11.4.p2.3.m3.1.1.3" xref="S6.SS2.11.4.p2.3.m3.1.1.3.cmml">η</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.11.4.p2.3.m3.1b"><apply id="S6.SS2.11.4.p2.3.m3.1.1.cmml" xref="S6.SS2.11.4.p2.3.m3.1.1"><lt id="S6.SS2.11.4.p2.3.m3.1.1.1.cmml" xref="S6.SS2.11.4.p2.3.m3.1.1.1"></lt><ci id="S6.SS2.11.4.p2.3.m3.1.1.2.cmml" xref="S6.SS2.11.4.p2.3.m3.1.1.2">𝜉</ci><ci id="S6.SS2.11.4.p2.3.m3.1.1.3.cmml" xref="S6.SS2.11.4.p2.3.m3.1.1.3">𝜂</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.11.4.p2.3.m3.1c">\xi<\eta</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.11.4.p2.3.m3.1d">italic_ξ < italic_η</annotation></semantics></math> in <math alttext="\Gamma" class="ltx_Math" display="inline" id="S6.SS2.11.4.p2.4.m4.1"><semantics id="S6.SS2.11.4.p2.4.m4.1a"><mi id="S6.SS2.11.4.p2.4.m4.1.1" mathvariant="normal" xref="S6.SS2.11.4.p2.4.m4.1.1.cmml">Γ</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.11.4.p2.4.m4.1b"><ci id="S6.SS2.11.4.p2.4.m4.1.1.cmml" xref="S6.SS2.11.4.p2.4.m4.1.1">Γ</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.11.4.p2.4.m4.1c">\Gamma</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.11.4.p2.4.m4.1d">roman_Γ</annotation></semantics></math>, and let <math alttext="\bar{a}\in\operatorname{dom}(p_{\xi})" class="ltx_Math" display="inline" id="S6.SS2.11.4.p2.5.m5.2"><semantics id="S6.SS2.11.4.p2.5.m5.2a"><mrow id="S6.SS2.11.4.p2.5.m5.2.2" xref="S6.SS2.11.4.p2.5.m5.2.2.cmml"><mover accent="true" id="S6.SS2.11.4.p2.5.m5.2.2.3" xref="S6.SS2.11.4.p2.5.m5.2.2.3.cmml"><mi id="S6.SS2.11.4.p2.5.m5.2.2.3.2" xref="S6.SS2.11.4.p2.5.m5.2.2.3.2.cmml">a</mi><mo id="S6.SS2.11.4.p2.5.m5.2.2.3.1" xref="S6.SS2.11.4.p2.5.m5.2.2.3.1.cmml">¯</mo></mover><mo id="S6.SS2.11.4.p2.5.m5.2.2.2" xref="S6.SS2.11.4.p2.5.m5.2.2.2.cmml">∈</mo><mrow id="S6.SS2.11.4.p2.5.m5.2.2.1.1" xref="S6.SS2.11.4.p2.5.m5.2.2.1.2.cmml"><mi id="S6.SS2.11.4.p2.5.m5.1.1" xref="S6.SS2.11.4.p2.5.m5.1.1.cmml">dom</mi><mo id="S6.SS2.11.4.p2.5.m5.2.2.1.1a" xref="S6.SS2.11.4.p2.5.m5.2.2.1.2.cmml">⁡</mo><mrow id="S6.SS2.11.4.p2.5.m5.2.2.1.1.1" xref="S6.SS2.11.4.p2.5.m5.2.2.1.2.cmml"><mo id="S6.SS2.11.4.p2.5.m5.2.2.1.1.1.2" stretchy="false" xref="S6.SS2.11.4.p2.5.m5.2.2.1.2.cmml">(</mo><msub id="S6.SS2.11.4.p2.5.m5.2.2.1.1.1.1" xref="S6.SS2.11.4.p2.5.m5.2.2.1.1.1.1.cmml"><mi id="S6.SS2.11.4.p2.5.m5.2.2.1.1.1.1.2" xref="S6.SS2.11.4.p2.5.m5.2.2.1.1.1.1.2.cmml">p</mi><mi id="S6.SS2.11.4.p2.5.m5.2.2.1.1.1.1.3" xref="S6.SS2.11.4.p2.5.m5.2.2.1.1.1.1.3.cmml">ξ</mi></msub><mo id="S6.SS2.11.4.p2.5.m5.2.2.1.1.1.3" stretchy="false" xref="S6.SS2.11.4.p2.5.m5.2.2.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.11.4.p2.5.m5.2b"><apply id="S6.SS2.11.4.p2.5.m5.2.2.cmml" xref="S6.SS2.11.4.p2.5.m5.2.2"><in id="S6.SS2.11.4.p2.5.m5.2.2.2.cmml" xref="S6.SS2.11.4.p2.5.m5.2.2.2"></in><apply id="S6.SS2.11.4.p2.5.m5.2.2.3.cmml" xref="S6.SS2.11.4.p2.5.m5.2.2.3"><ci id="S6.SS2.11.4.p2.5.m5.2.2.3.1.cmml" xref="S6.SS2.11.4.p2.5.m5.2.2.3.1">¯</ci><ci id="S6.SS2.11.4.p2.5.m5.2.2.3.2.cmml" xref="S6.SS2.11.4.p2.5.m5.2.2.3.2">𝑎</ci></apply><apply id="S6.SS2.11.4.p2.5.m5.2.2.1.2.cmml" xref="S6.SS2.11.4.p2.5.m5.2.2.1.1"><ci id="S6.SS2.11.4.p2.5.m5.1.1.cmml" xref="S6.SS2.11.4.p2.5.m5.1.1">dom</ci><apply id="S6.SS2.11.4.p2.5.m5.2.2.1.1.1.1.cmml" xref="S6.SS2.11.4.p2.5.m5.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.11.4.p2.5.m5.2.2.1.1.1.1.1.cmml" xref="S6.SS2.11.4.p2.5.m5.2.2.1.1.1.1">subscript</csymbol><ci id="S6.SS2.11.4.p2.5.m5.2.2.1.1.1.1.2.cmml" xref="S6.SS2.11.4.p2.5.m5.2.2.1.1.1.1.2">𝑝</ci><ci id="S6.SS2.11.4.p2.5.m5.2.2.1.1.1.1.3.cmml" xref="S6.SS2.11.4.p2.5.m5.2.2.1.1.1.1.3">𝜉</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.11.4.p2.5.m5.2c">\bar{a}\in\operatorname{dom}(p_{\xi})</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.11.4.p2.5.m5.2d">over¯ start_ARG italic_a end_ARG ∈ roman_dom ( italic_p start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT )</annotation></semantics></math> and <math alttext="\bar{b}\in\operatorname{dom}(p_{\eta})" class="ltx_Math" display="inline" id="S6.SS2.11.4.p2.6.m6.2"><semantics id="S6.SS2.11.4.p2.6.m6.2a"><mrow id="S6.SS2.11.4.p2.6.m6.2.2" xref="S6.SS2.11.4.p2.6.m6.2.2.cmml"><mover accent="true" id="S6.SS2.11.4.p2.6.m6.2.2.3" xref="S6.SS2.11.4.p2.6.m6.2.2.3.cmml"><mi id="S6.SS2.11.4.p2.6.m6.2.2.3.2" xref="S6.SS2.11.4.p2.6.m6.2.2.3.2.cmml">b</mi><mo id="S6.SS2.11.4.p2.6.m6.2.2.3.1" xref="S6.SS2.11.4.p2.6.m6.2.2.3.1.cmml">¯</mo></mover><mo id="S6.SS2.11.4.p2.6.m6.2.2.2" xref="S6.SS2.11.4.p2.6.m6.2.2.2.cmml">∈</mo><mrow id="S6.SS2.11.4.p2.6.m6.2.2.1.1" xref="S6.SS2.11.4.p2.6.m6.2.2.1.2.cmml"><mi id="S6.SS2.11.4.p2.6.m6.1.1" xref="S6.SS2.11.4.p2.6.m6.1.1.cmml">dom</mi><mo id="S6.SS2.11.4.p2.6.m6.2.2.1.1a" xref="S6.SS2.11.4.p2.6.m6.2.2.1.2.cmml">⁡</mo><mrow id="S6.SS2.11.4.p2.6.m6.2.2.1.1.1" xref="S6.SS2.11.4.p2.6.m6.2.2.1.2.cmml"><mo id="S6.SS2.11.4.p2.6.m6.2.2.1.1.1.2" stretchy="false" xref="S6.SS2.11.4.p2.6.m6.2.2.1.2.cmml">(</mo><msub id="S6.SS2.11.4.p2.6.m6.2.2.1.1.1.1" xref="S6.SS2.11.4.p2.6.m6.2.2.1.1.1.1.cmml"><mi id="S6.SS2.11.4.p2.6.m6.2.2.1.1.1.1.2" xref="S6.SS2.11.4.p2.6.m6.2.2.1.1.1.1.2.cmml">p</mi><mi id="S6.SS2.11.4.p2.6.m6.2.2.1.1.1.1.3" xref="S6.SS2.11.4.p2.6.m6.2.2.1.1.1.1.3.cmml">η</mi></msub><mo id="S6.SS2.11.4.p2.6.m6.2.2.1.1.1.3" stretchy="false" xref="S6.SS2.11.4.p2.6.m6.2.2.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.11.4.p2.6.m6.2b"><apply id="S6.SS2.11.4.p2.6.m6.2.2.cmml" xref="S6.SS2.11.4.p2.6.m6.2.2"><in id="S6.SS2.11.4.p2.6.m6.2.2.2.cmml" xref="S6.SS2.11.4.p2.6.m6.2.2.2"></in><apply id="S6.SS2.11.4.p2.6.m6.2.2.3.cmml" xref="S6.SS2.11.4.p2.6.m6.2.2.3"><ci id="S6.SS2.11.4.p2.6.m6.2.2.3.1.cmml" xref="S6.SS2.11.4.p2.6.m6.2.2.3.1">¯</ci><ci id="S6.SS2.11.4.p2.6.m6.2.2.3.2.cmml" xref="S6.SS2.11.4.p2.6.m6.2.2.3.2">𝑏</ci></apply><apply id="S6.SS2.11.4.p2.6.m6.2.2.1.2.cmml" xref="S6.SS2.11.4.p2.6.m6.2.2.1.1"><ci id="S6.SS2.11.4.p2.6.m6.1.1.cmml" xref="S6.SS2.11.4.p2.6.m6.1.1">dom</ci><apply id="S6.SS2.11.4.p2.6.m6.2.2.1.1.1.1.cmml" xref="S6.SS2.11.4.p2.6.m6.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.11.4.p2.6.m6.2.2.1.1.1.1.1.cmml" xref="S6.SS2.11.4.p2.6.m6.2.2.1.1.1.1">subscript</csymbol><ci id="S6.SS2.11.4.p2.6.m6.2.2.1.1.1.1.2.cmml" xref="S6.SS2.11.4.p2.6.m6.2.2.1.1.1.1.2">𝑝</ci><ci id="S6.SS2.11.4.p2.6.m6.2.2.1.1.1.1.3.cmml" xref="S6.SS2.11.4.p2.6.m6.2.2.1.1.1.1.3">𝜂</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.11.4.p2.6.m6.2c">\bar{b}\in\operatorname{dom}(p_{\eta})</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.11.4.p2.6.m6.2d">over¯ start_ARG italic_b end_ARG ∈ roman_dom ( italic_p start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT )</annotation></semantics></math>. We prove that <math alttext="[a_{l},a_{r}]\cap[b_{l},b_{r}]=\varnothing" class="ltx_Math" display="inline" id="S6.SS2.11.4.p2.7.m7.4"><semantics id="S6.SS2.11.4.p2.7.m7.4a"><mrow id="S6.SS2.11.4.p2.7.m7.4.4" xref="S6.SS2.11.4.p2.7.m7.4.4.cmml"><mrow id="S6.SS2.11.4.p2.7.m7.4.4.4" xref="S6.SS2.11.4.p2.7.m7.4.4.4.cmml"><mrow id="S6.SS2.11.4.p2.7.m7.2.2.2.2.2" xref="S6.SS2.11.4.p2.7.m7.2.2.2.2.3.cmml"><mo id="S6.SS2.11.4.p2.7.m7.2.2.2.2.2.3" stretchy="false" xref="S6.SS2.11.4.p2.7.m7.2.2.2.2.3.cmml">[</mo><msub id="S6.SS2.11.4.p2.7.m7.1.1.1.1.1.1" xref="S6.SS2.11.4.p2.7.m7.1.1.1.1.1.1.cmml"><mi id="S6.SS2.11.4.p2.7.m7.1.1.1.1.1.1.2" xref="S6.SS2.11.4.p2.7.m7.1.1.1.1.1.1.2.cmml">a</mi><mi id="S6.SS2.11.4.p2.7.m7.1.1.1.1.1.1.3" xref="S6.SS2.11.4.p2.7.m7.1.1.1.1.1.1.3.cmml">l</mi></msub><mo id="S6.SS2.11.4.p2.7.m7.2.2.2.2.2.4" xref="S6.SS2.11.4.p2.7.m7.2.2.2.2.3.cmml">,</mo><msub id="S6.SS2.11.4.p2.7.m7.2.2.2.2.2.2" xref="S6.SS2.11.4.p2.7.m7.2.2.2.2.2.2.cmml"><mi id="S6.SS2.11.4.p2.7.m7.2.2.2.2.2.2.2" xref="S6.SS2.11.4.p2.7.m7.2.2.2.2.2.2.2.cmml">a</mi><mi id="S6.SS2.11.4.p2.7.m7.2.2.2.2.2.2.3" xref="S6.SS2.11.4.p2.7.m7.2.2.2.2.2.2.3.cmml">r</mi></msub><mo id="S6.SS2.11.4.p2.7.m7.2.2.2.2.2.5" stretchy="false" xref="S6.SS2.11.4.p2.7.m7.2.2.2.2.3.cmml">]</mo></mrow><mo id="S6.SS2.11.4.p2.7.m7.4.4.4.5" xref="S6.SS2.11.4.p2.7.m7.4.4.4.5.cmml">∩</mo><mrow id="S6.SS2.11.4.p2.7.m7.4.4.4.4.2" xref="S6.SS2.11.4.p2.7.m7.4.4.4.4.3.cmml"><mo id="S6.SS2.11.4.p2.7.m7.4.4.4.4.2.3" stretchy="false" xref="S6.SS2.11.4.p2.7.m7.4.4.4.4.3.cmml">[</mo><msub id="S6.SS2.11.4.p2.7.m7.3.3.3.3.1.1" xref="S6.SS2.11.4.p2.7.m7.3.3.3.3.1.1.cmml"><mi id="S6.SS2.11.4.p2.7.m7.3.3.3.3.1.1.2" xref="S6.SS2.11.4.p2.7.m7.3.3.3.3.1.1.2.cmml">b</mi><mi id="S6.SS2.11.4.p2.7.m7.3.3.3.3.1.1.3" xref="S6.SS2.11.4.p2.7.m7.3.3.3.3.1.1.3.cmml">l</mi></msub><mo id="S6.SS2.11.4.p2.7.m7.4.4.4.4.2.4" xref="S6.SS2.11.4.p2.7.m7.4.4.4.4.3.cmml">,</mo><msub id="S6.SS2.11.4.p2.7.m7.4.4.4.4.2.2" xref="S6.SS2.11.4.p2.7.m7.4.4.4.4.2.2.cmml"><mi id="S6.SS2.11.4.p2.7.m7.4.4.4.4.2.2.2" xref="S6.SS2.11.4.p2.7.m7.4.4.4.4.2.2.2.cmml">b</mi><mi id="S6.SS2.11.4.p2.7.m7.4.4.4.4.2.2.3" xref="S6.SS2.11.4.p2.7.m7.4.4.4.4.2.2.3.cmml">r</mi></msub><mo id="S6.SS2.11.4.p2.7.m7.4.4.4.4.2.5" stretchy="false" xref="S6.SS2.11.4.p2.7.m7.4.4.4.4.3.cmml">]</mo></mrow></mrow><mo id="S6.SS2.11.4.p2.7.m7.4.4.5" xref="S6.SS2.11.4.p2.7.m7.4.4.5.cmml">=</mo><mi id="S6.SS2.11.4.p2.7.m7.4.4.6" mathvariant="normal" xref="S6.SS2.11.4.p2.7.m7.4.4.6.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.11.4.p2.7.m7.4b"><apply id="S6.SS2.11.4.p2.7.m7.4.4.cmml" xref="S6.SS2.11.4.p2.7.m7.4.4"><eq id="S6.SS2.11.4.p2.7.m7.4.4.5.cmml" xref="S6.SS2.11.4.p2.7.m7.4.4.5"></eq><apply id="S6.SS2.11.4.p2.7.m7.4.4.4.cmml" xref="S6.SS2.11.4.p2.7.m7.4.4.4"><intersect id="S6.SS2.11.4.p2.7.m7.4.4.4.5.cmml" xref="S6.SS2.11.4.p2.7.m7.4.4.4.5"></intersect><interval closure="closed" id="S6.SS2.11.4.p2.7.m7.2.2.2.2.3.cmml" xref="S6.SS2.11.4.p2.7.m7.2.2.2.2.2"><apply id="S6.SS2.11.4.p2.7.m7.1.1.1.1.1.1.cmml" xref="S6.SS2.11.4.p2.7.m7.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.11.4.p2.7.m7.1.1.1.1.1.1.1.cmml" xref="S6.SS2.11.4.p2.7.m7.1.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.11.4.p2.7.m7.1.1.1.1.1.1.2.cmml" xref="S6.SS2.11.4.p2.7.m7.1.1.1.1.1.1.2">𝑎</ci><ci id="S6.SS2.11.4.p2.7.m7.1.1.1.1.1.1.3.cmml" xref="S6.SS2.11.4.p2.7.m7.1.1.1.1.1.1.3">𝑙</ci></apply><apply id="S6.SS2.11.4.p2.7.m7.2.2.2.2.2.2.cmml" xref="S6.SS2.11.4.p2.7.m7.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.11.4.p2.7.m7.2.2.2.2.2.2.1.cmml" xref="S6.SS2.11.4.p2.7.m7.2.2.2.2.2.2">subscript</csymbol><ci id="S6.SS2.11.4.p2.7.m7.2.2.2.2.2.2.2.cmml" xref="S6.SS2.11.4.p2.7.m7.2.2.2.2.2.2.2">𝑎</ci><ci id="S6.SS2.11.4.p2.7.m7.2.2.2.2.2.2.3.cmml" xref="S6.SS2.11.4.p2.7.m7.2.2.2.2.2.2.3">𝑟</ci></apply></interval><interval closure="closed" id="S6.SS2.11.4.p2.7.m7.4.4.4.4.3.cmml" xref="S6.SS2.11.4.p2.7.m7.4.4.4.4.2"><apply id="S6.SS2.11.4.p2.7.m7.3.3.3.3.1.1.cmml" xref="S6.SS2.11.4.p2.7.m7.3.3.3.3.1.1"><csymbol cd="ambiguous" id="S6.SS2.11.4.p2.7.m7.3.3.3.3.1.1.1.cmml" xref="S6.SS2.11.4.p2.7.m7.3.3.3.3.1.1">subscript</csymbol><ci id="S6.SS2.11.4.p2.7.m7.3.3.3.3.1.1.2.cmml" xref="S6.SS2.11.4.p2.7.m7.3.3.3.3.1.1.2">𝑏</ci><ci id="S6.SS2.11.4.p2.7.m7.3.3.3.3.1.1.3.cmml" xref="S6.SS2.11.4.p2.7.m7.3.3.3.3.1.1.3">𝑙</ci></apply><apply id="S6.SS2.11.4.p2.7.m7.4.4.4.4.2.2.cmml" xref="S6.SS2.11.4.p2.7.m7.4.4.4.4.2.2"><csymbol cd="ambiguous" id="S6.SS2.11.4.p2.7.m7.4.4.4.4.2.2.1.cmml" xref="S6.SS2.11.4.p2.7.m7.4.4.4.4.2.2">subscript</csymbol><ci id="S6.SS2.11.4.p2.7.m7.4.4.4.4.2.2.2.cmml" xref="S6.SS2.11.4.p2.7.m7.4.4.4.4.2.2.2">𝑏</ci><ci id="S6.SS2.11.4.p2.7.m7.4.4.4.4.2.2.3.cmml" xref="S6.SS2.11.4.p2.7.m7.4.4.4.4.2.2.3">𝑟</ci></apply></interval></apply><emptyset id="S6.SS2.11.4.p2.7.m7.4.4.6.cmml" xref="S6.SS2.11.4.p2.7.m7.4.4.6"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.11.4.p2.7.m7.4c">[a_{l},a_{r}]\cap[b_{l},b_{r}]=\varnothing</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.11.4.p2.7.m7.4d">[ italic_a start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ] ∩ [ italic_b start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ] = ∅</annotation></semantics></math>. By definition of <math alttext="q_{\xi}" class="ltx_Math" display="inline" id="S6.SS2.11.4.p2.8.m8.1"><semantics id="S6.SS2.11.4.p2.8.m8.1a"><msub id="S6.SS2.11.4.p2.8.m8.1.1" xref="S6.SS2.11.4.p2.8.m8.1.1.cmml"><mi id="S6.SS2.11.4.p2.8.m8.1.1.2" xref="S6.SS2.11.4.p2.8.m8.1.1.2.cmml">q</mi><mi id="S6.SS2.11.4.p2.8.m8.1.1.3" xref="S6.SS2.11.4.p2.8.m8.1.1.3.cmml">ξ</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.11.4.p2.8.m8.1b"><apply id="S6.SS2.11.4.p2.8.m8.1.1.cmml" xref="S6.SS2.11.4.p2.8.m8.1.1"><csymbol cd="ambiguous" id="S6.SS2.11.4.p2.8.m8.1.1.1.cmml" xref="S6.SS2.11.4.p2.8.m8.1.1">subscript</csymbol><ci id="S6.SS2.11.4.p2.8.m8.1.1.2.cmml" xref="S6.SS2.11.4.p2.8.m8.1.1.2">𝑞</ci><ci id="S6.SS2.11.4.p2.8.m8.1.1.3.cmml" xref="S6.SS2.11.4.p2.8.m8.1.1.3">𝜉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.11.4.p2.8.m8.1c">q_{\xi}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.11.4.p2.8.m8.1d">italic_q start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT</annotation></semantics></math>, there are <math alttext="i,j<n" class="ltx_Math" display="inline" id="S6.SS2.11.4.p2.9.m9.2"><semantics id="S6.SS2.11.4.p2.9.m9.2a"><mrow id="S6.SS2.11.4.p2.9.m9.2.3" xref="S6.SS2.11.4.p2.9.m9.2.3.cmml"><mrow id="S6.SS2.11.4.p2.9.m9.2.3.2.2" xref="S6.SS2.11.4.p2.9.m9.2.3.2.1.cmml"><mi id="S6.SS2.11.4.p2.9.m9.1.1" xref="S6.SS2.11.4.p2.9.m9.1.1.cmml">i</mi><mo id="S6.SS2.11.4.p2.9.m9.2.3.2.2.1" xref="S6.SS2.11.4.p2.9.m9.2.3.2.1.cmml">,</mo><mi id="S6.SS2.11.4.p2.9.m9.2.2" xref="S6.SS2.11.4.p2.9.m9.2.2.cmml">j</mi></mrow><mo id="S6.SS2.11.4.p2.9.m9.2.3.1" xref="S6.SS2.11.4.p2.9.m9.2.3.1.cmml"><</mo><mi id="S6.SS2.11.4.p2.9.m9.2.3.3" xref="S6.SS2.11.4.p2.9.m9.2.3.3.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.11.4.p2.9.m9.2b"><apply id="S6.SS2.11.4.p2.9.m9.2.3.cmml" xref="S6.SS2.11.4.p2.9.m9.2.3"><lt id="S6.SS2.11.4.p2.9.m9.2.3.1.cmml" xref="S6.SS2.11.4.p2.9.m9.2.3.1"></lt><list id="S6.SS2.11.4.p2.9.m9.2.3.2.1.cmml" xref="S6.SS2.11.4.p2.9.m9.2.3.2.2"><ci id="S6.SS2.11.4.p2.9.m9.1.1.cmml" xref="S6.SS2.11.4.p2.9.m9.1.1">𝑖</ci><ci id="S6.SS2.11.4.p2.9.m9.2.2.cmml" xref="S6.SS2.11.4.p2.9.m9.2.2">𝑗</ci></list><ci id="S6.SS2.11.4.p2.9.m9.2.3.3.cmml" xref="S6.SS2.11.4.p2.9.m9.2.3.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.11.4.p2.9.m9.2c">i,j<n</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.11.4.p2.9.m9.2d">italic_i , italic_j < italic_n</annotation></semantics></math> such that <math alttext="\xi_{i}=a_{m}" class="ltx_Math" display="inline" id="S6.SS2.11.4.p2.10.m10.1"><semantics id="S6.SS2.11.4.p2.10.m10.1a"><mrow id="S6.SS2.11.4.p2.10.m10.1.1" xref="S6.SS2.11.4.p2.10.m10.1.1.cmml"><msub id="S6.SS2.11.4.p2.10.m10.1.1.2" xref="S6.SS2.11.4.p2.10.m10.1.1.2.cmml"><mi id="S6.SS2.11.4.p2.10.m10.1.1.2.2" xref="S6.SS2.11.4.p2.10.m10.1.1.2.2.cmml">ξ</mi><mi id="S6.SS2.11.4.p2.10.m10.1.1.2.3" xref="S6.SS2.11.4.p2.10.m10.1.1.2.3.cmml">i</mi></msub><mo id="S6.SS2.11.4.p2.10.m10.1.1.1" xref="S6.SS2.11.4.p2.10.m10.1.1.1.cmml">=</mo><msub id="S6.SS2.11.4.p2.10.m10.1.1.3" xref="S6.SS2.11.4.p2.10.m10.1.1.3.cmml"><mi id="S6.SS2.11.4.p2.10.m10.1.1.3.2" xref="S6.SS2.11.4.p2.10.m10.1.1.3.2.cmml">a</mi><mi id="S6.SS2.11.4.p2.10.m10.1.1.3.3" xref="S6.SS2.11.4.p2.10.m10.1.1.3.3.cmml">m</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.11.4.p2.10.m10.1b"><apply id="S6.SS2.11.4.p2.10.m10.1.1.cmml" xref="S6.SS2.11.4.p2.10.m10.1.1"><eq id="S6.SS2.11.4.p2.10.m10.1.1.1.cmml" xref="S6.SS2.11.4.p2.10.m10.1.1.1"></eq><apply id="S6.SS2.11.4.p2.10.m10.1.1.2.cmml" xref="S6.SS2.11.4.p2.10.m10.1.1.2"><csymbol cd="ambiguous" id="S6.SS2.11.4.p2.10.m10.1.1.2.1.cmml" xref="S6.SS2.11.4.p2.10.m10.1.1.2">subscript</csymbol><ci id="S6.SS2.11.4.p2.10.m10.1.1.2.2.cmml" xref="S6.SS2.11.4.p2.10.m10.1.1.2.2">𝜉</ci><ci id="S6.SS2.11.4.p2.10.m10.1.1.2.3.cmml" xref="S6.SS2.11.4.p2.10.m10.1.1.2.3">𝑖</ci></apply><apply id="S6.SS2.11.4.p2.10.m10.1.1.3.cmml" xref="S6.SS2.11.4.p2.10.m10.1.1.3"><csymbol cd="ambiguous" id="S6.SS2.11.4.p2.10.m10.1.1.3.1.cmml" xref="S6.SS2.11.4.p2.10.m10.1.1.3">subscript</csymbol><ci id="S6.SS2.11.4.p2.10.m10.1.1.3.2.cmml" xref="S6.SS2.11.4.p2.10.m10.1.1.3.2">𝑎</ci><ci id="S6.SS2.11.4.p2.10.m10.1.1.3.3.cmml" xref="S6.SS2.11.4.p2.10.m10.1.1.3.3">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.11.4.p2.10.m10.1c">\xi_{i}=a_{m}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.11.4.p2.10.m10.1d">italic_ξ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_a start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\eta_{j}=b_{m}" class="ltx_Math" display="inline" id="S6.SS2.11.4.p2.11.m11.1"><semantics id="S6.SS2.11.4.p2.11.m11.1a"><mrow id="S6.SS2.11.4.p2.11.m11.1.1" xref="S6.SS2.11.4.p2.11.m11.1.1.cmml"><msub id="S6.SS2.11.4.p2.11.m11.1.1.2" xref="S6.SS2.11.4.p2.11.m11.1.1.2.cmml"><mi id="S6.SS2.11.4.p2.11.m11.1.1.2.2" xref="S6.SS2.11.4.p2.11.m11.1.1.2.2.cmml">η</mi><mi id="S6.SS2.11.4.p2.11.m11.1.1.2.3" xref="S6.SS2.11.4.p2.11.m11.1.1.2.3.cmml">j</mi></msub><mo id="S6.SS2.11.4.p2.11.m11.1.1.1" xref="S6.SS2.11.4.p2.11.m11.1.1.1.cmml">=</mo><msub id="S6.SS2.11.4.p2.11.m11.1.1.3" xref="S6.SS2.11.4.p2.11.m11.1.1.3.cmml"><mi id="S6.SS2.11.4.p2.11.m11.1.1.3.2" xref="S6.SS2.11.4.p2.11.m11.1.1.3.2.cmml">b</mi><mi id="S6.SS2.11.4.p2.11.m11.1.1.3.3" xref="S6.SS2.11.4.p2.11.m11.1.1.3.3.cmml">m</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.11.4.p2.11.m11.1b"><apply id="S6.SS2.11.4.p2.11.m11.1.1.cmml" xref="S6.SS2.11.4.p2.11.m11.1.1"><eq id="S6.SS2.11.4.p2.11.m11.1.1.1.cmml" xref="S6.SS2.11.4.p2.11.m11.1.1.1"></eq><apply id="S6.SS2.11.4.p2.11.m11.1.1.2.cmml" xref="S6.SS2.11.4.p2.11.m11.1.1.2"><csymbol cd="ambiguous" id="S6.SS2.11.4.p2.11.m11.1.1.2.1.cmml" xref="S6.SS2.11.4.p2.11.m11.1.1.2">subscript</csymbol><ci id="S6.SS2.11.4.p2.11.m11.1.1.2.2.cmml" xref="S6.SS2.11.4.p2.11.m11.1.1.2.2">𝜂</ci><ci id="S6.SS2.11.4.p2.11.m11.1.1.2.3.cmml" xref="S6.SS2.11.4.p2.11.m11.1.1.2.3">𝑗</ci></apply><apply id="S6.SS2.11.4.p2.11.m11.1.1.3.cmml" xref="S6.SS2.11.4.p2.11.m11.1.1.3"><csymbol cd="ambiguous" id="S6.SS2.11.4.p2.11.m11.1.1.3.1.cmml" xref="S6.SS2.11.4.p2.11.m11.1.1.3">subscript</csymbol><ci id="S6.SS2.11.4.p2.11.m11.1.1.3.2.cmml" xref="S6.SS2.11.4.p2.11.m11.1.1.3.2">𝑏</ci><ci id="S6.SS2.11.4.p2.11.m11.1.1.3.3.cmml" xref="S6.SS2.11.4.p2.11.m11.1.1.3.3">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.11.4.p2.11.m11.1c">\eta_{j}=b_{m}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.11.4.p2.11.m11.1d">italic_η start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = italic_b start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math>. We claim that <math alttext="\Delta_{A}(\xi_{i},\eta_{j})<\xi" class="ltx_Math" display="inline" id="S6.SS2.11.4.p2.12.m12.2"><semantics id="S6.SS2.11.4.p2.12.m12.2a"><mrow id="S6.SS2.11.4.p2.12.m12.2.2" xref="S6.SS2.11.4.p2.12.m12.2.2.cmml"><mrow id="S6.SS2.11.4.p2.12.m12.2.2.2" xref="S6.SS2.11.4.p2.12.m12.2.2.2.cmml"><msub id="S6.SS2.11.4.p2.12.m12.2.2.2.4" xref="S6.SS2.11.4.p2.12.m12.2.2.2.4.cmml"><mi id="S6.SS2.11.4.p2.12.m12.2.2.2.4.2" mathvariant="normal" xref="S6.SS2.11.4.p2.12.m12.2.2.2.4.2.cmml">Δ</mi><mi id="S6.SS2.11.4.p2.12.m12.2.2.2.4.3" xref="S6.SS2.11.4.p2.12.m12.2.2.2.4.3.cmml">A</mi></msub><mo id="S6.SS2.11.4.p2.12.m12.2.2.2.3" xref="S6.SS2.11.4.p2.12.m12.2.2.2.3.cmml">⁢</mo><mrow id="S6.SS2.11.4.p2.12.m12.2.2.2.2.2" xref="S6.SS2.11.4.p2.12.m12.2.2.2.2.3.cmml"><mo id="S6.SS2.11.4.p2.12.m12.2.2.2.2.2.3" stretchy="false" xref="S6.SS2.11.4.p2.12.m12.2.2.2.2.3.cmml">(</mo><msub id="S6.SS2.11.4.p2.12.m12.1.1.1.1.1.1" xref="S6.SS2.11.4.p2.12.m12.1.1.1.1.1.1.cmml"><mi id="S6.SS2.11.4.p2.12.m12.1.1.1.1.1.1.2" xref="S6.SS2.11.4.p2.12.m12.1.1.1.1.1.1.2.cmml">ξ</mi><mi id="S6.SS2.11.4.p2.12.m12.1.1.1.1.1.1.3" xref="S6.SS2.11.4.p2.12.m12.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S6.SS2.11.4.p2.12.m12.2.2.2.2.2.4" xref="S6.SS2.11.4.p2.12.m12.2.2.2.2.3.cmml">,</mo><msub id="S6.SS2.11.4.p2.12.m12.2.2.2.2.2.2" xref="S6.SS2.11.4.p2.12.m12.2.2.2.2.2.2.cmml"><mi id="S6.SS2.11.4.p2.12.m12.2.2.2.2.2.2.2" xref="S6.SS2.11.4.p2.12.m12.2.2.2.2.2.2.2.cmml">η</mi><mi id="S6.SS2.11.4.p2.12.m12.2.2.2.2.2.2.3" xref="S6.SS2.11.4.p2.12.m12.2.2.2.2.2.2.3.cmml">j</mi></msub><mo id="S6.SS2.11.4.p2.12.m12.2.2.2.2.2.5" stretchy="false" xref="S6.SS2.11.4.p2.12.m12.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.SS2.11.4.p2.12.m12.2.2.3" xref="S6.SS2.11.4.p2.12.m12.2.2.3.cmml"><</mo><mi id="S6.SS2.11.4.p2.12.m12.2.2.4" xref="S6.SS2.11.4.p2.12.m12.2.2.4.cmml">ξ</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.11.4.p2.12.m12.2b"><apply id="S6.SS2.11.4.p2.12.m12.2.2.cmml" xref="S6.SS2.11.4.p2.12.m12.2.2"><lt id="S6.SS2.11.4.p2.12.m12.2.2.3.cmml" xref="S6.SS2.11.4.p2.12.m12.2.2.3"></lt><apply id="S6.SS2.11.4.p2.12.m12.2.2.2.cmml" xref="S6.SS2.11.4.p2.12.m12.2.2.2"><times id="S6.SS2.11.4.p2.12.m12.2.2.2.3.cmml" xref="S6.SS2.11.4.p2.12.m12.2.2.2.3"></times><apply id="S6.SS2.11.4.p2.12.m12.2.2.2.4.cmml" xref="S6.SS2.11.4.p2.12.m12.2.2.2.4"><csymbol cd="ambiguous" id="S6.SS2.11.4.p2.12.m12.2.2.2.4.1.cmml" xref="S6.SS2.11.4.p2.12.m12.2.2.2.4">subscript</csymbol><ci id="S6.SS2.11.4.p2.12.m12.2.2.2.4.2.cmml" xref="S6.SS2.11.4.p2.12.m12.2.2.2.4.2">Δ</ci><ci id="S6.SS2.11.4.p2.12.m12.2.2.2.4.3.cmml" xref="S6.SS2.11.4.p2.12.m12.2.2.2.4.3">𝐴</ci></apply><interval closure="open" id="S6.SS2.11.4.p2.12.m12.2.2.2.2.3.cmml" xref="S6.SS2.11.4.p2.12.m12.2.2.2.2.2"><apply id="S6.SS2.11.4.p2.12.m12.1.1.1.1.1.1.cmml" xref="S6.SS2.11.4.p2.12.m12.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.11.4.p2.12.m12.1.1.1.1.1.1.1.cmml" xref="S6.SS2.11.4.p2.12.m12.1.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.11.4.p2.12.m12.1.1.1.1.1.1.2.cmml" xref="S6.SS2.11.4.p2.12.m12.1.1.1.1.1.1.2">𝜉</ci><ci id="S6.SS2.11.4.p2.12.m12.1.1.1.1.1.1.3.cmml" xref="S6.SS2.11.4.p2.12.m12.1.1.1.1.1.1.3">𝑖</ci></apply><apply id="S6.SS2.11.4.p2.12.m12.2.2.2.2.2.2.cmml" xref="S6.SS2.11.4.p2.12.m12.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.11.4.p2.12.m12.2.2.2.2.2.2.1.cmml" xref="S6.SS2.11.4.p2.12.m12.2.2.2.2.2.2">subscript</csymbol><ci id="S6.SS2.11.4.p2.12.m12.2.2.2.2.2.2.2.cmml" xref="S6.SS2.11.4.p2.12.m12.2.2.2.2.2.2.2">𝜂</ci><ci id="S6.SS2.11.4.p2.12.m12.2.2.2.2.2.2.3.cmml" xref="S6.SS2.11.4.p2.12.m12.2.2.2.2.2.2.3">𝑗</ci></apply></interval></apply><ci id="S6.SS2.11.4.p2.12.m12.2.2.4.cmml" xref="S6.SS2.11.4.p2.12.m12.2.2.4">𝜉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.11.4.p2.12.m12.2c">\Delta_{A}(\xi_{i},\eta_{j})<\xi</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.11.4.p2.12.m12.2d">roman_Δ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ξ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_η start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) < italic_ξ</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S6.SS2.12.5.p3"> <p class="ltx_p" id="S6.SS2.12.5.p3.10">If <math alttext="i=j" class="ltx_Math" display="inline" id="S6.SS2.12.5.p3.1.m1.1"><semantics id="S6.SS2.12.5.p3.1.m1.1a"><mrow id="S6.SS2.12.5.p3.1.m1.1.1" xref="S6.SS2.12.5.p3.1.m1.1.1.cmml"><mi id="S6.SS2.12.5.p3.1.m1.1.1.2" xref="S6.SS2.12.5.p3.1.m1.1.1.2.cmml">i</mi><mo id="S6.SS2.12.5.p3.1.m1.1.1.1" xref="S6.SS2.12.5.p3.1.m1.1.1.1.cmml">=</mo><mi id="S6.SS2.12.5.p3.1.m1.1.1.3" xref="S6.SS2.12.5.p3.1.m1.1.1.3.cmml">j</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.12.5.p3.1.m1.1b"><apply id="S6.SS2.12.5.p3.1.m1.1.1.cmml" xref="S6.SS2.12.5.p3.1.m1.1.1"><eq id="S6.SS2.12.5.p3.1.m1.1.1.1.cmml" xref="S6.SS2.12.5.p3.1.m1.1.1.1"></eq><ci id="S6.SS2.12.5.p3.1.m1.1.1.2.cmml" xref="S6.SS2.12.5.p3.1.m1.1.1.2">𝑖</ci><ci id="S6.SS2.12.5.p3.1.m1.1.1.3.cmml" xref="S6.SS2.12.5.p3.1.m1.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.12.5.p3.1.m1.1c">i=j</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.12.5.p3.1.m1.1d">italic_i = italic_j</annotation></semantics></math> this follows directly from (c). So assume <math alttext="i\neq j" class="ltx_Math" display="inline" id="S6.SS2.12.5.p3.2.m2.1"><semantics id="S6.SS2.12.5.p3.2.m2.1a"><mrow id="S6.SS2.12.5.p3.2.m2.1.1" xref="S6.SS2.12.5.p3.2.m2.1.1.cmml"><mi id="S6.SS2.12.5.p3.2.m2.1.1.2" xref="S6.SS2.12.5.p3.2.m2.1.1.2.cmml">i</mi><mo id="S6.SS2.12.5.p3.2.m2.1.1.1" xref="S6.SS2.12.5.p3.2.m2.1.1.1.cmml">≠</mo><mi id="S6.SS2.12.5.p3.2.m2.1.1.3" xref="S6.SS2.12.5.p3.2.m2.1.1.3.cmml">j</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.12.5.p3.2.m2.1b"><apply id="S6.SS2.12.5.p3.2.m2.1.1.cmml" xref="S6.SS2.12.5.p3.2.m2.1.1"><neq id="S6.SS2.12.5.p3.2.m2.1.1.1.cmml" xref="S6.SS2.12.5.p3.2.m2.1.1.1"></neq><ci id="S6.SS2.12.5.p3.2.m2.1.1.2.cmml" xref="S6.SS2.12.5.p3.2.m2.1.1.2">𝑖</ci><ci id="S6.SS2.12.5.p3.2.m2.1.1.3.cmml" xref="S6.SS2.12.5.p3.2.m2.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.12.5.p3.2.m2.1c">i\neq j</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.12.5.p3.2.m2.1d">italic_i ≠ italic_j</annotation></semantics></math>. There are two cases depending on whether <math alttext="\Delta_{A}(\xi_{i},\xi_{j})<\xi" class="ltx_Math" display="inline" id="S6.SS2.12.5.p3.3.m3.2"><semantics id="S6.SS2.12.5.p3.3.m3.2a"><mrow id="S6.SS2.12.5.p3.3.m3.2.2" xref="S6.SS2.12.5.p3.3.m3.2.2.cmml"><mrow id="S6.SS2.12.5.p3.3.m3.2.2.2" xref="S6.SS2.12.5.p3.3.m3.2.2.2.cmml"><msub id="S6.SS2.12.5.p3.3.m3.2.2.2.4" xref="S6.SS2.12.5.p3.3.m3.2.2.2.4.cmml"><mi id="S6.SS2.12.5.p3.3.m3.2.2.2.4.2" mathvariant="normal" xref="S6.SS2.12.5.p3.3.m3.2.2.2.4.2.cmml">Δ</mi><mi id="S6.SS2.12.5.p3.3.m3.2.2.2.4.3" xref="S6.SS2.12.5.p3.3.m3.2.2.2.4.3.cmml">A</mi></msub><mo id="S6.SS2.12.5.p3.3.m3.2.2.2.3" xref="S6.SS2.12.5.p3.3.m3.2.2.2.3.cmml">⁢</mo><mrow id="S6.SS2.12.5.p3.3.m3.2.2.2.2.2" xref="S6.SS2.12.5.p3.3.m3.2.2.2.2.3.cmml"><mo id="S6.SS2.12.5.p3.3.m3.2.2.2.2.2.3" stretchy="false" xref="S6.SS2.12.5.p3.3.m3.2.2.2.2.3.cmml">(</mo><msub id="S6.SS2.12.5.p3.3.m3.1.1.1.1.1.1" xref="S6.SS2.12.5.p3.3.m3.1.1.1.1.1.1.cmml"><mi id="S6.SS2.12.5.p3.3.m3.1.1.1.1.1.1.2" xref="S6.SS2.12.5.p3.3.m3.1.1.1.1.1.1.2.cmml">ξ</mi><mi id="S6.SS2.12.5.p3.3.m3.1.1.1.1.1.1.3" xref="S6.SS2.12.5.p3.3.m3.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S6.SS2.12.5.p3.3.m3.2.2.2.2.2.4" xref="S6.SS2.12.5.p3.3.m3.2.2.2.2.3.cmml">,</mo><msub id="S6.SS2.12.5.p3.3.m3.2.2.2.2.2.2" xref="S6.SS2.12.5.p3.3.m3.2.2.2.2.2.2.cmml"><mi id="S6.SS2.12.5.p3.3.m3.2.2.2.2.2.2.2" xref="S6.SS2.12.5.p3.3.m3.2.2.2.2.2.2.2.cmml">ξ</mi><mi id="S6.SS2.12.5.p3.3.m3.2.2.2.2.2.2.3" xref="S6.SS2.12.5.p3.3.m3.2.2.2.2.2.2.3.cmml">j</mi></msub><mo id="S6.SS2.12.5.p3.3.m3.2.2.2.2.2.5" stretchy="false" xref="S6.SS2.12.5.p3.3.m3.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.SS2.12.5.p3.3.m3.2.2.3" xref="S6.SS2.12.5.p3.3.m3.2.2.3.cmml"><</mo><mi id="S6.SS2.12.5.p3.3.m3.2.2.4" xref="S6.SS2.12.5.p3.3.m3.2.2.4.cmml">ξ</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.12.5.p3.3.m3.2b"><apply id="S6.SS2.12.5.p3.3.m3.2.2.cmml" xref="S6.SS2.12.5.p3.3.m3.2.2"><lt id="S6.SS2.12.5.p3.3.m3.2.2.3.cmml" xref="S6.SS2.12.5.p3.3.m3.2.2.3"></lt><apply id="S6.SS2.12.5.p3.3.m3.2.2.2.cmml" xref="S6.SS2.12.5.p3.3.m3.2.2.2"><times id="S6.SS2.12.5.p3.3.m3.2.2.2.3.cmml" xref="S6.SS2.12.5.p3.3.m3.2.2.2.3"></times><apply id="S6.SS2.12.5.p3.3.m3.2.2.2.4.cmml" xref="S6.SS2.12.5.p3.3.m3.2.2.2.4"><csymbol cd="ambiguous" id="S6.SS2.12.5.p3.3.m3.2.2.2.4.1.cmml" xref="S6.SS2.12.5.p3.3.m3.2.2.2.4">subscript</csymbol><ci id="S6.SS2.12.5.p3.3.m3.2.2.2.4.2.cmml" xref="S6.SS2.12.5.p3.3.m3.2.2.2.4.2">Δ</ci><ci id="S6.SS2.12.5.p3.3.m3.2.2.2.4.3.cmml" xref="S6.SS2.12.5.p3.3.m3.2.2.2.4.3">𝐴</ci></apply><interval closure="open" id="S6.SS2.12.5.p3.3.m3.2.2.2.2.3.cmml" xref="S6.SS2.12.5.p3.3.m3.2.2.2.2.2"><apply id="S6.SS2.12.5.p3.3.m3.1.1.1.1.1.1.cmml" xref="S6.SS2.12.5.p3.3.m3.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.12.5.p3.3.m3.1.1.1.1.1.1.1.cmml" xref="S6.SS2.12.5.p3.3.m3.1.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.12.5.p3.3.m3.1.1.1.1.1.1.2.cmml" xref="S6.SS2.12.5.p3.3.m3.1.1.1.1.1.1.2">𝜉</ci><ci id="S6.SS2.12.5.p3.3.m3.1.1.1.1.1.1.3.cmml" xref="S6.SS2.12.5.p3.3.m3.1.1.1.1.1.1.3">𝑖</ci></apply><apply id="S6.SS2.12.5.p3.3.m3.2.2.2.2.2.2.cmml" xref="S6.SS2.12.5.p3.3.m3.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.12.5.p3.3.m3.2.2.2.2.2.2.1.cmml" xref="S6.SS2.12.5.p3.3.m3.2.2.2.2.2.2">subscript</csymbol><ci id="S6.SS2.12.5.p3.3.m3.2.2.2.2.2.2.2.cmml" xref="S6.SS2.12.5.p3.3.m3.2.2.2.2.2.2.2">𝜉</ci><ci id="S6.SS2.12.5.p3.3.m3.2.2.2.2.2.2.3.cmml" xref="S6.SS2.12.5.p3.3.m3.2.2.2.2.2.2.3">𝑗</ci></apply></interval></apply><ci id="S6.SS2.12.5.p3.3.m3.2.2.4.cmml" xref="S6.SS2.12.5.p3.3.m3.2.2.4">𝜉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.12.5.p3.3.m3.2c">\Delta_{A}(\xi_{i},\xi_{j})<\xi</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.12.5.p3.3.m3.2d">roman_Δ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ξ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_ξ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) < italic_ξ</annotation></semantics></math> or not. If <math alttext="\Delta_{A}(\xi_{i},\xi_{j})<\xi" class="ltx_Math" display="inline" id="S6.SS2.12.5.p3.4.m4.2"><semantics id="S6.SS2.12.5.p3.4.m4.2a"><mrow id="S6.SS2.12.5.p3.4.m4.2.2" xref="S6.SS2.12.5.p3.4.m4.2.2.cmml"><mrow id="S6.SS2.12.5.p3.4.m4.2.2.2" xref="S6.SS2.12.5.p3.4.m4.2.2.2.cmml"><msub id="S6.SS2.12.5.p3.4.m4.2.2.2.4" xref="S6.SS2.12.5.p3.4.m4.2.2.2.4.cmml"><mi id="S6.SS2.12.5.p3.4.m4.2.2.2.4.2" mathvariant="normal" xref="S6.SS2.12.5.p3.4.m4.2.2.2.4.2.cmml">Δ</mi><mi id="S6.SS2.12.5.p3.4.m4.2.2.2.4.3" xref="S6.SS2.12.5.p3.4.m4.2.2.2.4.3.cmml">A</mi></msub><mo id="S6.SS2.12.5.p3.4.m4.2.2.2.3" xref="S6.SS2.12.5.p3.4.m4.2.2.2.3.cmml">⁢</mo><mrow id="S6.SS2.12.5.p3.4.m4.2.2.2.2.2" xref="S6.SS2.12.5.p3.4.m4.2.2.2.2.3.cmml"><mo id="S6.SS2.12.5.p3.4.m4.2.2.2.2.2.3" stretchy="false" xref="S6.SS2.12.5.p3.4.m4.2.2.2.2.3.cmml">(</mo><msub id="S6.SS2.12.5.p3.4.m4.1.1.1.1.1.1" xref="S6.SS2.12.5.p3.4.m4.1.1.1.1.1.1.cmml"><mi id="S6.SS2.12.5.p3.4.m4.1.1.1.1.1.1.2" xref="S6.SS2.12.5.p3.4.m4.1.1.1.1.1.1.2.cmml">ξ</mi><mi id="S6.SS2.12.5.p3.4.m4.1.1.1.1.1.1.3" xref="S6.SS2.12.5.p3.4.m4.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S6.SS2.12.5.p3.4.m4.2.2.2.2.2.4" xref="S6.SS2.12.5.p3.4.m4.2.2.2.2.3.cmml">,</mo><msub id="S6.SS2.12.5.p3.4.m4.2.2.2.2.2.2" xref="S6.SS2.12.5.p3.4.m4.2.2.2.2.2.2.cmml"><mi id="S6.SS2.12.5.p3.4.m4.2.2.2.2.2.2.2" xref="S6.SS2.12.5.p3.4.m4.2.2.2.2.2.2.2.cmml">ξ</mi><mi id="S6.SS2.12.5.p3.4.m4.2.2.2.2.2.2.3" xref="S6.SS2.12.5.p3.4.m4.2.2.2.2.2.2.3.cmml">j</mi></msub><mo id="S6.SS2.12.5.p3.4.m4.2.2.2.2.2.5" stretchy="false" xref="S6.SS2.12.5.p3.4.m4.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.SS2.12.5.p3.4.m4.2.2.3" xref="S6.SS2.12.5.p3.4.m4.2.2.3.cmml"><</mo><mi id="S6.SS2.12.5.p3.4.m4.2.2.4" xref="S6.SS2.12.5.p3.4.m4.2.2.4.cmml">ξ</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.12.5.p3.4.m4.2b"><apply id="S6.SS2.12.5.p3.4.m4.2.2.cmml" xref="S6.SS2.12.5.p3.4.m4.2.2"><lt id="S6.SS2.12.5.p3.4.m4.2.2.3.cmml" xref="S6.SS2.12.5.p3.4.m4.2.2.3"></lt><apply id="S6.SS2.12.5.p3.4.m4.2.2.2.cmml" xref="S6.SS2.12.5.p3.4.m4.2.2.2"><times id="S6.SS2.12.5.p3.4.m4.2.2.2.3.cmml" xref="S6.SS2.12.5.p3.4.m4.2.2.2.3"></times><apply id="S6.SS2.12.5.p3.4.m4.2.2.2.4.cmml" xref="S6.SS2.12.5.p3.4.m4.2.2.2.4"><csymbol cd="ambiguous" id="S6.SS2.12.5.p3.4.m4.2.2.2.4.1.cmml" xref="S6.SS2.12.5.p3.4.m4.2.2.2.4">subscript</csymbol><ci id="S6.SS2.12.5.p3.4.m4.2.2.2.4.2.cmml" xref="S6.SS2.12.5.p3.4.m4.2.2.2.4.2">Δ</ci><ci id="S6.SS2.12.5.p3.4.m4.2.2.2.4.3.cmml" xref="S6.SS2.12.5.p3.4.m4.2.2.2.4.3">𝐴</ci></apply><interval closure="open" id="S6.SS2.12.5.p3.4.m4.2.2.2.2.3.cmml" xref="S6.SS2.12.5.p3.4.m4.2.2.2.2.2"><apply id="S6.SS2.12.5.p3.4.m4.1.1.1.1.1.1.cmml" xref="S6.SS2.12.5.p3.4.m4.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.12.5.p3.4.m4.1.1.1.1.1.1.1.cmml" xref="S6.SS2.12.5.p3.4.m4.1.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.12.5.p3.4.m4.1.1.1.1.1.1.2.cmml" xref="S6.SS2.12.5.p3.4.m4.1.1.1.1.1.1.2">𝜉</ci><ci id="S6.SS2.12.5.p3.4.m4.1.1.1.1.1.1.3.cmml" xref="S6.SS2.12.5.p3.4.m4.1.1.1.1.1.1.3">𝑖</ci></apply><apply id="S6.SS2.12.5.p3.4.m4.2.2.2.2.2.2.cmml" xref="S6.SS2.12.5.p3.4.m4.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.12.5.p3.4.m4.2.2.2.2.2.2.1.cmml" xref="S6.SS2.12.5.p3.4.m4.2.2.2.2.2.2">subscript</csymbol><ci id="S6.SS2.12.5.p3.4.m4.2.2.2.2.2.2.2.cmml" xref="S6.SS2.12.5.p3.4.m4.2.2.2.2.2.2.2">𝜉</ci><ci id="S6.SS2.12.5.p3.4.m4.2.2.2.2.2.2.3.cmml" xref="S6.SS2.12.5.p3.4.m4.2.2.2.2.2.2.3">𝑗</ci></apply></interval></apply><ci id="S6.SS2.12.5.p3.4.m4.2.2.4.cmml" xref="S6.SS2.12.5.p3.4.m4.2.2.4">𝜉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.12.5.p3.4.m4.2c">\Delta_{A}(\xi_{i},\xi_{j})<\xi</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.12.5.p3.4.m4.2d">roman_Δ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ξ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_ξ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) < italic_ξ</annotation></semantics></math> by (b) we have that <math alttext="\Delta_{A}(\xi_{i},\xi_{j})<\gamma" class="ltx_Math" display="inline" id="S6.SS2.12.5.p3.5.m5.2"><semantics id="S6.SS2.12.5.p3.5.m5.2a"><mrow id="S6.SS2.12.5.p3.5.m5.2.2" xref="S6.SS2.12.5.p3.5.m5.2.2.cmml"><mrow id="S6.SS2.12.5.p3.5.m5.2.2.2" xref="S6.SS2.12.5.p3.5.m5.2.2.2.cmml"><msub id="S6.SS2.12.5.p3.5.m5.2.2.2.4" xref="S6.SS2.12.5.p3.5.m5.2.2.2.4.cmml"><mi id="S6.SS2.12.5.p3.5.m5.2.2.2.4.2" mathvariant="normal" xref="S6.SS2.12.5.p3.5.m5.2.2.2.4.2.cmml">Δ</mi><mi id="S6.SS2.12.5.p3.5.m5.2.2.2.4.3" xref="S6.SS2.12.5.p3.5.m5.2.2.2.4.3.cmml">A</mi></msub><mo id="S6.SS2.12.5.p3.5.m5.2.2.2.3" xref="S6.SS2.12.5.p3.5.m5.2.2.2.3.cmml">⁢</mo><mrow id="S6.SS2.12.5.p3.5.m5.2.2.2.2.2" xref="S6.SS2.12.5.p3.5.m5.2.2.2.2.3.cmml"><mo id="S6.SS2.12.5.p3.5.m5.2.2.2.2.2.3" stretchy="false" xref="S6.SS2.12.5.p3.5.m5.2.2.2.2.3.cmml">(</mo><msub id="S6.SS2.12.5.p3.5.m5.1.1.1.1.1.1" xref="S6.SS2.12.5.p3.5.m5.1.1.1.1.1.1.cmml"><mi id="S6.SS2.12.5.p3.5.m5.1.1.1.1.1.1.2" xref="S6.SS2.12.5.p3.5.m5.1.1.1.1.1.1.2.cmml">ξ</mi><mi id="S6.SS2.12.5.p3.5.m5.1.1.1.1.1.1.3" xref="S6.SS2.12.5.p3.5.m5.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S6.SS2.12.5.p3.5.m5.2.2.2.2.2.4" xref="S6.SS2.12.5.p3.5.m5.2.2.2.2.3.cmml">,</mo><msub id="S6.SS2.12.5.p3.5.m5.2.2.2.2.2.2" xref="S6.SS2.12.5.p3.5.m5.2.2.2.2.2.2.cmml"><mi id="S6.SS2.12.5.p3.5.m5.2.2.2.2.2.2.2" xref="S6.SS2.12.5.p3.5.m5.2.2.2.2.2.2.2.cmml">ξ</mi><mi id="S6.SS2.12.5.p3.5.m5.2.2.2.2.2.2.3" xref="S6.SS2.12.5.p3.5.m5.2.2.2.2.2.2.3.cmml">j</mi></msub><mo id="S6.SS2.12.5.p3.5.m5.2.2.2.2.2.5" stretchy="false" xref="S6.SS2.12.5.p3.5.m5.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.SS2.12.5.p3.5.m5.2.2.3" xref="S6.SS2.12.5.p3.5.m5.2.2.3.cmml"><</mo><mi id="S6.SS2.12.5.p3.5.m5.2.2.4" xref="S6.SS2.12.5.p3.5.m5.2.2.4.cmml">γ</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.12.5.p3.5.m5.2b"><apply id="S6.SS2.12.5.p3.5.m5.2.2.cmml" xref="S6.SS2.12.5.p3.5.m5.2.2"><lt id="S6.SS2.12.5.p3.5.m5.2.2.3.cmml" xref="S6.SS2.12.5.p3.5.m5.2.2.3"></lt><apply id="S6.SS2.12.5.p3.5.m5.2.2.2.cmml" xref="S6.SS2.12.5.p3.5.m5.2.2.2"><times id="S6.SS2.12.5.p3.5.m5.2.2.2.3.cmml" xref="S6.SS2.12.5.p3.5.m5.2.2.2.3"></times><apply id="S6.SS2.12.5.p3.5.m5.2.2.2.4.cmml" xref="S6.SS2.12.5.p3.5.m5.2.2.2.4"><csymbol cd="ambiguous" id="S6.SS2.12.5.p3.5.m5.2.2.2.4.1.cmml" xref="S6.SS2.12.5.p3.5.m5.2.2.2.4">subscript</csymbol><ci id="S6.SS2.12.5.p3.5.m5.2.2.2.4.2.cmml" xref="S6.SS2.12.5.p3.5.m5.2.2.2.4.2">Δ</ci><ci id="S6.SS2.12.5.p3.5.m5.2.2.2.4.3.cmml" xref="S6.SS2.12.5.p3.5.m5.2.2.2.4.3">𝐴</ci></apply><interval closure="open" id="S6.SS2.12.5.p3.5.m5.2.2.2.2.3.cmml" xref="S6.SS2.12.5.p3.5.m5.2.2.2.2.2"><apply id="S6.SS2.12.5.p3.5.m5.1.1.1.1.1.1.cmml" xref="S6.SS2.12.5.p3.5.m5.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.12.5.p3.5.m5.1.1.1.1.1.1.1.cmml" xref="S6.SS2.12.5.p3.5.m5.1.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.12.5.p3.5.m5.1.1.1.1.1.1.2.cmml" xref="S6.SS2.12.5.p3.5.m5.1.1.1.1.1.1.2">𝜉</ci><ci id="S6.SS2.12.5.p3.5.m5.1.1.1.1.1.1.3.cmml" xref="S6.SS2.12.5.p3.5.m5.1.1.1.1.1.1.3">𝑖</ci></apply><apply id="S6.SS2.12.5.p3.5.m5.2.2.2.2.2.2.cmml" xref="S6.SS2.12.5.p3.5.m5.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.12.5.p3.5.m5.2.2.2.2.2.2.1.cmml" xref="S6.SS2.12.5.p3.5.m5.2.2.2.2.2.2">subscript</csymbol><ci id="S6.SS2.12.5.p3.5.m5.2.2.2.2.2.2.2.cmml" xref="S6.SS2.12.5.p3.5.m5.2.2.2.2.2.2.2">𝜉</ci><ci id="S6.SS2.12.5.p3.5.m5.2.2.2.2.2.2.3.cmml" xref="S6.SS2.12.5.p3.5.m5.2.2.2.2.2.2.3">𝑗</ci></apply></interval></apply><ci id="S6.SS2.12.5.p3.5.m5.2.2.4.cmml" xref="S6.SS2.12.5.p3.5.m5.2.2.4">𝛾</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.12.5.p3.5.m5.2c">\Delta_{A}(\xi_{i},\xi_{j})<\gamma</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.12.5.p3.5.m5.2d">roman_Δ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ξ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_ξ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) < italic_γ</annotation></semantics></math>, and by (c) that <math alttext="\gamma<\Delta_{A}(\xi_{j},\eta_{j})" class="ltx_Math" display="inline" id="S6.SS2.12.5.p3.6.m6.2"><semantics id="S6.SS2.12.5.p3.6.m6.2a"><mrow id="S6.SS2.12.5.p3.6.m6.2.2" xref="S6.SS2.12.5.p3.6.m6.2.2.cmml"><mi id="S6.SS2.12.5.p3.6.m6.2.2.4" xref="S6.SS2.12.5.p3.6.m6.2.2.4.cmml">γ</mi><mo id="S6.SS2.12.5.p3.6.m6.2.2.3" xref="S6.SS2.12.5.p3.6.m6.2.2.3.cmml"><</mo><mrow id="S6.SS2.12.5.p3.6.m6.2.2.2" xref="S6.SS2.12.5.p3.6.m6.2.2.2.cmml"><msub id="S6.SS2.12.5.p3.6.m6.2.2.2.4" xref="S6.SS2.12.5.p3.6.m6.2.2.2.4.cmml"><mi id="S6.SS2.12.5.p3.6.m6.2.2.2.4.2" mathvariant="normal" xref="S6.SS2.12.5.p3.6.m6.2.2.2.4.2.cmml">Δ</mi><mi id="S6.SS2.12.5.p3.6.m6.2.2.2.4.3" xref="S6.SS2.12.5.p3.6.m6.2.2.2.4.3.cmml">A</mi></msub><mo id="S6.SS2.12.5.p3.6.m6.2.2.2.3" xref="S6.SS2.12.5.p3.6.m6.2.2.2.3.cmml">⁢</mo><mrow id="S6.SS2.12.5.p3.6.m6.2.2.2.2.2" xref="S6.SS2.12.5.p3.6.m6.2.2.2.2.3.cmml"><mo id="S6.SS2.12.5.p3.6.m6.2.2.2.2.2.3" stretchy="false" xref="S6.SS2.12.5.p3.6.m6.2.2.2.2.3.cmml">(</mo><msub id="S6.SS2.12.5.p3.6.m6.1.1.1.1.1.1" xref="S6.SS2.12.5.p3.6.m6.1.1.1.1.1.1.cmml"><mi id="S6.SS2.12.5.p3.6.m6.1.1.1.1.1.1.2" xref="S6.SS2.12.5.p3.6.m6.1.1.1.1.1.1.2.cmml">ξ</mi><mi id="S6.SS2.12.5.p3.6.m6.1.1.1.1.1.1.3" xref="S6.SS2.12.5.p3.6.m6.1.1.1.1.1.1.3.cmml">j</mi></msub><mo id="S6.SS2.12.5.p3.6.m6.2.2.2.2.2.4" xref="S6.SS2.12.5.p3.6.m6.2.2.2.2.3.cmml">,</mo><msub id="S6.SS2.12.5.p3.6.m6.2.2.2.2.2.2" xref="S6.SS2.12.5.p3.6.m6.2.2.2.2.2.2.cmml"><mi id="S6.SS2.12.5.p3.6.m6.2.2.2.2.2.2.2" xref="S6.SS2.12.5.p3.6.m6.2.2.2.2.2.2.2.cmml">η</mi><mi id="S6.SS2.12.5.p3.6.m6.2.2.2.2.2.2.3" xref="S6.SS2.12.5.p3.6.m6.2.2.2.2.2.2.3.cmml">j</mi></msub><mo id="S6.SS2.12.5.p3.6.m6.2.2.2.2.2.5" stretchy="false" xref="S6.SS2.12.5.p3.6.m6.2.2.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.12.5.p3.6.m6.2b"><apply id="S6.SS2.12.5.p3.6.m6.2.2.cmml" xref="S6.SS2.12.5.p3.6.m6.2.2"><lt id="S6.SS2.12.5.p3.6.m6.2.2.3.cmml" xref="S6.SS2.12.5.p3.6.m6.2.2.3"></lt><ci id="S6.SS2.12.5.p3.6.m6.2.2.4.cmml" xref="S6.SS2.12.5.p3.6.m6.2.2.4">𝛾</ci><apply id="S6.SS2.12.5.p3.6.m6.2.2.2.cmml" xref="S6.SS2.12.5.p3.6.m6.2.2.2"><times id="S6.SS2.12.5.p3.6.m6.2.2.2.3.cmml" xref="S6.SS2.12.5.p3.6.m6.2.2.2.3"></times><apply id="S6.SS2.12.5.p3.6.m6.2.2.2.4.cmml" xref="S6.SS2.12.5.p3.6.m6.2.2.2.4"><csymbol cd="ambiguous" id="S6.SS2.12.5.p3.6.m6.2.2.2.4.1.cmml" xref="S6.SS2.12.5.p3.6.m6.2.2.2.4">subscript</csymbol><ci id="S6.SS2.12.5.p3.6.m6.2.2.2.4.2.cmml" xref="S6.SS2.12.5.p3.6.m6.2.2.2.4.2">Δ</ci><ci id="S6.SS2.12.5.p3.6.m6.2.2.2.4.3.cmml" xref="S6.SS2.12.5.p3.6.m6.2.2.2.4.3">𝐴</ci></apply><interval closure="open" id="S6.SS2.12.5.p3.6.m6.2.2.2.2.3.cmml" xref="S6.SS2.12.5.p3.6.m6.2.2.2.2.2"><apply id="S6.SS2.12.5.p3.6.m6.1.1.1.1.1.1.cmml" xref="S6.SS2.12.5.p3.6.m6.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.12.5.p3.6.m6.1.1.1.1.1.1.1.cmml" xref="S6.SS2.12.5.p3.6.m6.1.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.12.5.p3.6.m6.1.1.1.1.1.1.2.cmml" xref="S6.SS2.12.5.p3.6.m6.1.1.1.1.1.1.2">𝜉</ci><ci id="S6.SS2.12.5.p3.6.m6.1.1.1.1.1.1.3.cmml" xref="S6.SS2.12.5.p3.6.m6.1.1.1.1.1.1.3">𝑗</ci></apply><apply id="S6.SS2.12.5.p3.6.m6.2.2.2.2.2.2.cmml" xref="S6.SS2.12.5.p3.6.m6.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.12.5.p3.6.m6.2.2.2.2.2.2.1.cmml" xref="S6.SS2.12.5.p3.6.m6.2.2.2.2.2.2">subscript</csymbol><ci id="S6.SS2.12.5.p3.6.m6.2.2.2.2.2.2.2.cmml" xref="S6.SS2.12.5.p3.6.m6.2.2.2.2.2.2.2">𝜂</ci><ci id="S6.SS2.12.5.p3.6.m6.2.2.2.2.2.2.3.cmml" xref="S6.SS2.12.5.p3.6.m6.2.2.2.2.2.2.3">𝑗</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.12.5.p3.6.m6.2c">\gamma<\Delta_{A}(\xi_{j},\eta_{j})</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.12.5.p3.6.m6.2d">italic_γ < roman_Δ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ξ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT , italic_η start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT )</annotation></semantics></math>, then <math alttext="\Delta_{A}(\xi_{i},\eta_{j})<\gamma<\xi" class="ltx_Math" display="inline" id="S6.SS2.12.5.p3.7.m7.2"><semantics id="S6.SS2.12.5.p3.7.m7.2a"><mrow id="S6.SS2.12.5.p3.7.m7.2.2" xref="S6.SS2.12.5.p3.7.m7.2.2.cmml"><mrow id="S6.SS2.12.5.p3.7.m7.2.2.2" xref="S6.SS2.12.5.p3.7.m7.2.2.2.cmml"><msub id="S6.SS2.12.5.p3.7.m7.2.2.2.4" xref="S6.SS2.12.5.p3.7.m7.2.2.2.4.cmml"><mi id="S6.SS2.12.5.p3.7.m7.2.2.2.4.2" mathvariant="normal" xref="S6.SS2.12.5.p3.7.m7.2.2.2.4.2.cmml">Δ</mi><mi id="S6.SS2.12.5.p3.7.m7.2.2.2.4.3" xref="S6.SS2.12.5.p3.7.m7.2.2.2.4.3.cmml">A</mi></msub><mo id="S6.SS2.12.5.p3.7.m7.2.2.2.3" xref="S6.SS2.12.5.p3.7.m7.2.2.2.3.cmml">⁢</mo><mrow id="S6.SS2.12.5.p3.7.m7.2.2.2.2.2" xref="S6.SS2.12.5.p3.7.m7.2.2.2.2.3.cmml"><mo id="S6.SS2.12.5.p3.7.m7.2.2.2.2.2.3" stretchy="false" xref="S6.SS2.12.5.p3.7.m7.2.2.2.2.3.cmml">(</mo><msub id="S6.SS2.12.5.p3.7.m7.1.1.1.1.1.1" xref="S6.SS2.12.5.p3.7.m7.1.1.1.1.1.1.cmml"><mi id="S6.SS2.12.5.p3.7.m7.1.1.1.1.1.1.2" xref="S6.SS2.12.5.p3.7.m7.1.1.1.1.1.1.2.cmml">ξ</mi><mi id="S6.SS2.12.5.p3.7.m7.1.1.1.1.1.1.3" xref="S6.SS2.12.5.p3.7.m7.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S6.SS2.12.5.p3.7.m7.2.2.2.2.2.4" xref="S6.SS2.12.5.p3.7.m7.2.2.2.2.3.cmml">,</mo><msub id="S6.SS2.12.5.p3.7.m7.2.2.2.2.2.2" xref="S6.SS2.12.5.p3.7.m7.2.2.2.2.2.2.cmml"><mi id="S6.SS2.12.5.p3.7.m7.2.2.2.2.2.2.2" xref="S6.SS2.12.5.p3.7.m7.2.2.2.2.2.2.2.cmml">η</mi><mi id="S6.SS2.12.5.p3.7.m7.2.2.2.2.2.2.3" xref="S6.SS2.12.5.p3.7.m7.2.2.2.2.2.2.3.cmml">j</mi></msub><mo id="S6.SS2.12.5.p3.7.m7.2.2.2.2.2.5" stretchy="false" xref="S6.SS2.12.5.p3.7.m7.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.SS2.12.5.p3.7.m7.2.2.4" xref="S6.SS2.12.5.p3.7.m7.2.2.4.cmml"><</mo><mi id="S6.SS2.12.5.p3.7.m7.2.2.5" xref="S6.SS2.12.5.p3.7.m7.2.2.5.cmml">γ</mi><mo id="S6.SS2.12.5.p3.7.m7.2.2.6" xref="S6.SS2.12.5.p3.7.m7.2.2.6.cmml"><</mo><mi id="S6.SS2.12.5.p3.7.m7.2.2.7" xref="S6.SS2.12.5.p3.7.m7.2.2.7.cmml">ξ</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.12.5.p3.7.m7.2b"><apply id="S6.SS2.12.5.p3.7.m7.2.2.cmml" xref="S6.SS2.12.5.p3.7.m7.2.2"><and id="S6.SS2.12.5.p3.7.m7.2.2a.cmml" xref="S6.SS2.12.5.p3.7.m7.2.2"></and><apply id="S6.SS2.12.5.p3.7.m7.2.2b.cmml" xref="S6.SS2.12.5.p3.7.m7.2.2"><lt id="S6.SS2.12.5.p3.7.m7.2.2.4.cmml" xref="S6.SS2.12.5.p3.7.m7.2.2.4"></lt><apply id="S6.SS2.12.5.p3.7.m7.2.2.2.cmml" xref="S6.SS2.12.5.p3.7.m7.2.2.2"><times id="S6.SS2.12.5.p3.7.m7.2.2.2.3.cmml" xref="S6.SS2.12.5.p3.7.m7.2.2.2.3"></times><apply id="S6.SS2.12.5.p3.7.m7.2.2.2.4.cmml" xref="S6.SS2.12.5.p3.7.m7.2.2.2.4"><csymbol cd="ambiguous" id="S6.SS2.12.5.p3.7.m7.2.2.2.4.1.cmml" xref="S6.SS2.12.5.p3.7.m7.2.2.2.4">subscript</csymbol><ci id="S6.SS2.12.5.p3.7.m7.2.2.2.4.2.cmml" xref="S6.SS2.12.5.p3.7.m7.2.2.2.4.2">Δ</ci><ci id="S6.SS2.12.5.p3.7.m7.2.2.2.4.3.cmml" xref="S6.SS2.12.5.p3.7.m7.2.2.2.4.3">𝐴</ci></apply><interval closure="open" id="S6.SS2.12.5.p3.7.m7.2.2.2.2.3.cmml" xref="S6.SS2.12.5.p3.7.m7.2.2.2.2.2"><apply id="S6.SS2.12.5.p3.7.m7.1.1.1.1.1.1.cmml" xref="S6.SS2.12.5.p3.7.m7.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.12.5.p3.7.m7.1.1.1.1.1.1.1.cmml" xref="S6.SS2.12.5.p3.7.m7.1.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.12.5.p3.7.m7.1.1.1.1.1.1.2.cmml" xref="S6.SS2.12.5.p3.7.m7.1.1.1.1.1.1.2">𝜉</ci><ci id="S6.SS2.12.5.p3.7.m7.1.1.1.1.1.1.3.cmml" xref="S6.SS2.12.5.p3.7.m7.1.1.1.1.1.1.3">𝑖</ci></apply><apply id="S6.SS2.12.5.p3.7.m7.2.2.2.2.2.2.cmml" xref="S6.SS2.12.5.p3.7.m7.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.12.5.p3.7.m7.2.2.2.2.2.2.1.cmml" xref="S6.SS2.12.5.p3.7.m7.2.2.2.2.2.2">subscript</csymbol><ci id="S6.SS2.12.5.p3.7.m7.2.2.2.2.2.2.2.cmml" xref="S6.SS2.12.5.p3.7.m7.2.2.2.2.2.2.2">𝜂</ci><ci id="S6.SS2.12.5.p3.7.m7.2.2.2.2.2.2.3.cmml" xref="S6.SS2.12.5.p3.7.m7.2.2.2.2.2.2.3">𝑗</ci></apply></interval></apply><ci id="S6.SS2.12.5.p3.7.m7.2.2.5.cmml" xref="S6.SS2.12.5.p3.7.m7.2.2.5">𝛾</ci></apply><apply id="S6.SS2.12.5.p3.7.m7.2.2c.cmml" xref="S6.SS2.12.5.p3.7.m7.2.2"><lt id="S6.SS2.12.5.p3.7.m7.2.2.6.cmml" xref="S6.SS2.12.5.p3.7.m7.2.2.6"></lt><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.12.5.p3.7.m7.2.2.5.cmml" id="S6.SS2.12.5.p3.7.m7.2.2d.cmml" xref="S6.SS2.12.5.p3.7.m7.2.2"></share><ci id="S6.SS2.12.5.p3.7.m7.2.2.7.cmml" xref="S6.SS2.12.5.p3.7.m7.2.2.7">𝜉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.12.5.p3.7.m7.2c">\Delta_{A}(\xi_{i},\eta_{j})<\gamma<\xi</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.12.5.p3.7.m7.2d">roman_Δ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ξ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_η start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) < italic_γ < italic_ξ</annotation></semantics></math> follows. If <math alttext="\Delta_{A}(\xi_{i},\xi_{j})\geq\xi" class="ltx_Math" display="inline" id="S6.SS2.12.5.p3.8.m8.2"><semantics id="S6.SS2.12.5.p3.8.m8.2a"><mrow id="S6.SS2.12.5.p3.8.m8.2.2" xref="S6.SS2.12.5.p3.8.m8.2.2.cmml"><mrow id="S6.SS2.12.5.p3.8.m8.2.2.2" xref="S6.SS2.12.5.p3.8.m8.2.2.2.cmml"><msub id="S6.SS2.12.5.p3.8.m8.2.2.2.4" xref="S6.SS2.12.5.p3.8.m8.2.2.2.4.cmml"><mi id="S6.SS2.12.5.p3.8.m8.2.2.2.4.2" mathvariant="normal" xref="S6.SS2.12.5.p3.8.m8.2.2.2.4.2.cmml">Δ</mi><mi id="S6.SS2.12.5.p3.8.m8.2.2.2.4.3" xref="S6.SS2.12.5.p3.8.m8.2.2.2.4.3.cmml">A</mi></msub><mo id="S6.SS2.12.5.p3.8.m8.2.2.2.3" xref="S6.SS2.12.5.p3.8.m8.2.2.2.3.cmml">⁢</mo><mrow id="S6.SS2.12.5.p3.8.m8.2.2.2.2.2" xref="S6.SS2.12.5.p3.8.m8.2.2.2.2.3.cmml"><mo id="S6.SS2.12.5.p3.8.m8.2.2.2.2.2.3" stretchy="false" xref="S6.SS2.12.5.p3.8.m8.2.2.2.2.3.cmml">(</mo><msub id="S6.SS2.12.5.p3.8.m8.1.1.1.1.1.1" xref="S6.SS2.12.5.p3.8.m8.1.1.1.1.1.1.cmml"><mi id="S6.SS2.12.5.p3.8.m8.1.1.1.1.1.1.2" xref="S6.SS2.12.5.p3.8.m8.1.1.1.1.1.1.2.cmml">ξ</mi><mi id="S6.SS2.12.5.p3.8.m8.1.1.1.1.1.1.3" xref="S6.SS2.12.5.p3.8.m8.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S6.SS2.12.5.p3.8.m8.2.2.2.2.2.4" xref="S6.SS2.12.5.p3.8.m8.2.2.2.2.3.cmml">,</mo><msub id="S6.SS2.12.5.p3.8.m8.2.2.2.2.2.2" xref="S6.SS2.12.5.p3.8.m8.2.2.2.2.2.2.cmml"><mi id="S6.SS2.12.5.p3.8.m8.2.2.2.2.2.2.2" xref="S6.SS2.12.5.p3.8.m8.2.2.2.2.2.2.2.cmml">ξ</mi><mi id="S6.SS2.12.5.p3.8.m8.2.2.2.2.2.2.3" xref="S6.SS2.12.5.p3.8.m8.2.2.2.2.2.2.3.cmml">j</mi></msub><mo id="S6.SS2.12.5.p3.8.m8.2.2.2.2.2.5" stretchy="false" xref="S6.SS2.12.5.p3.8.m8.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.SS2.12.5.p3.8.m8.2.2.3" xref="S6.SS2.12.5.p3.8.m8.2.2.3.cmml">≥</mo><mi id="S6.SS2.12.5.p3.8.m8.2.2.4" xref="S6.SS2.12.5.p3.8.m8.2.2.4.cmml">ξ</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.12.5.p3.8.m8.2b"><apply id="S6.SS2.12.5.p3.8.m8.2.2.cmml" xref="S6.SS2.12.5.p3.8.m8.2.2"><geq id="S6.SS2.12.5.p3.8.m8.2.2.3.cmml" xref="S6.SS2.12.5.p3.8.m8.2.2.3"></geq><apply id="S6.SS2.12.5.p3.8.m8.2.2.2.cmml" xref="S6.SS2.12.5.p3.8.m8.2.2.2"><times id="S6.SS2.12.5.p3.8.m8.2.2.2.3.cmml" xref="S6.SS2.12.5.p3.8.m8.2.2.2.3"></times><apply id="S6.SS2.12.5.p3.8.m8.2.2.2.4.cmml" xref="S6.SS2.12.5.p3.8.m8.2.2.2.4"><csymbol cd="ambiguous" id="S6.SS2.12.5.p3.8.m8.2.2.2.4.1.cmml" xref="S6.SS2.12.5.p3.8.m8.2.2.2.4">subscript</csymbol><ci id="S6.SS2.12.5.p3.8.m8.2.2.2.4.2.cmml" xref="S6.SS2.12.5.p3.8.m8.2.2.2.4.2">Δ</ci><ci id="S6.SS2.12.5.p3.8.m8.2.2.2.4.3.cmml" xref="S6.SS2.12.5.p3.8.m8.2.2.2.4.3">𝐴</ci></apply><interval closure="open" id="S6.SS2.12.5.p3.8.m8.2.2.2.2.3.cmml" xref="S6.SS2.12.5.p3.8.m8.2.2.2.2.2"><apply id="S6.SS2.12.5.p3.8.m8.1.1.1.1.1.1.cmml" xref="S6.SS2.12.5.p3.8.m8.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.12.5.p3.8.m8.1.1.1.1.1.1.1.cmml" xref="S6.SS2.12.5.p3.8.m8.1.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.12.5.p3.8.m8.1.1.1.1.1.1.2.cmml" xref="S6.SS2.12.5.p3.8.m8.1.1.1.1.1.1.2">𝜉</ci><ci id="S6.SS2.12.5.p3.8.m8.1.1.1.1.1.1.3.cmml" xref="S6.SS2.12.5.p3.8.m8.1.1.1.1.1.1.3">𝑖</ci></apply><apply id="S6.SS2.12.5.p3.8.m8.2.2.2.2.2.2.cmml" xref="S6.SS2.12.5.p3.8.m8.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.12.5.p3.8.m8.2.2.2.2.2.2.1.cmml" xref="S6.SS2.12.5.p3.8.m8.2.2.2.2.2.2">subscript</csymbol><ci id="S6.SS2.12.5.p3.8.m8.2.2.2.2.2.2.2.cmml" xref="S6.SS2.12.5.p3.8.m8.2.2.2.2.2.2.2">𝜉</ci><ci id="S6.SS2.12.5.p3.8.m8.2.2.2.2.2.2.3.cmml" xref="S6.SS2.12.5.p3.8.m8.2.2.2.2.2.2.3">𝑗</ci></apply></interval></apply><ci id="S6.SS2.12.5.p3.8.m8.2.2.4.cmml" xref="S6.SS2.12.5.p3.8.m8.2.2.4">𝜉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.12.5.p3.8.m8.2c">\Delta_{A}(\xi_{i},\xi_{j})\geq\xi</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.12.5.p3.8.m8.2d">roman_Δ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ξ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_ξ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) ≥ italic_ξ</annotation></semantics></math>, then by (c) we have <math alttext="\Delta_{A}(\xi_{j},\eta_{j})<\xi" class="ltx_Math" display="inline" id="S6.SS2.12.5.p3.9.m9.2"><semantics id="S6.SS2.12.5.p3.9.m9.2a"><mrow id="S6.SS2.12.5.p3.9.m9.2.2" xref="S6.SS2.12.5.p3.9.m9.2.2.cmml"><mrow id="S6.SS2.12.5.p3.9.m9.2.2.2" xref="S6.SS2.12.5.p3.9.m9.2.2.2.cmml"><msub id="S6.SS2.12.5.p3.9.m9.2.2.2.4" xref="S6.SS2.12.5.p3.9.m9.2.2.2.4.cmml"><mi id="S6.SS2.12.5.p3.9.m9.2.2.2.4.2" mathvariant="normal" xref="S6.SS2.12.5.p3.9.m9.2.2.2.4.2.cmml">Δ</mi><mi id="S6.SS2.12.5.p3.9.m9.2.2.2.4.3" xref="S6.SS2.12.5.p3.9.m9.2.2.2.4.3.cmml">A</mi></msub><mo id="S6.SS2.12.5.p3.9.m9.2.2.2.3" xref="S6.SS2.12.5.p3.9.m9.2.2.2.3.cmml">⁢</mo><mrow id="S6.SS2.12.5.p3.9.m9.2.2.2.2.2" xref="S6.SS2.12.5.p3.9.m9.2.2.2.2.3.cmml"><mo id="S6.SS2.12.5.p3.9.m9.2.2.2.2.2.3" stretchy="false" xref="S6.SS2.12.5.p3.9.m9.2.2.2.2.3.cmml">(</mo><msub id="S6.SS2.12.5.p3.9.m9.1.1.1.1.1.1" xref="S6.SS2.12.5.p3.9.m9.1.1.1.1.1.1.cmml"><mi id="S6.SS2.12.5.p3.9.m9.1.1.1.1.1.1.2" xref="S6.SS2.12.5.p3.9.m9.1.1.1.1.1.1.2.cmml">ξ</mi><mi id="S6.SS2.12.5.p3.9.m9.1.1.1.1.1.1.3" xref="S6.SS2.12.5.p3.9.m9.1.1.1.1.1.1.3.cmml">j</mi></msub><mo id="S6.SS2.12.5.p3.9.m9.2.2.2.2.2.4" xref="S6.SS2.12.5.p3.9.m9.2.2.2.2.3.cmml">,</mo><msub id="S6.SS2.12.5.p3.9.m9.2.2.2.2.2.2" xref="S6.SS2.12.5.p3.9.m9.2.2.2.2.2.2.cmml"><mi id="S6.SS2.12.5.p3.9.m9.2.2.2.2.2.2.2" xref="S6.SS2.12.5.p3.9.m9.2.2.2.2.2.2.2.cmml">η</mi><mi id="S6.SS2.12.5.p3.9.m9.2.2.2.2.2.2.3" xref="S6.SS2.12.5.p3.9.m9.2.2.2.2.2.2.3.cmml">j</mi></msub><mo id="S6.SS2.12.5.p3.9.m9.2.2.2.2.2.5" stretchy="false" xref="S6.SS2.12.5.p3.9.m9.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.SS2.12.5.p3.9.m9.2.2.3" xref="S6.SS2.12.5.p3.9.m9.2.2.3.cmml"><</mo><mi id="S6.SS2.12.5.p3.9.m9.2.2.4" xref="S6.SS2.12.5.p3.9.m9.2.2.4.cmml">ξ</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.12.5.p3.9.m9.2b"><apply id="S6.SS2.12.5.p3.9.m9.2.2.cmml" xref="S6.SS2.12.5.p3.9.m9.2.2"><lt id="S6.SS2.12.5.p3.9.m9.2.2.3.cmml" xref="S6.SS2.12.5.p3.9.m9.2.2.3"></lt><apply id="S6.SS2.12.5.p3.9.m9.2.2.2.cmml" xref="S6.SS2.12.5.p3.9.m9.2.2.2"><times id="S6.SS2.12.5.p3.9.m9.2.2.2.3.cmml" xref="S6.SS2.12.5.p3.9.m9.2.2.2.3"></times><apply id="S6.SS2.12.5.p3.9.m9.2.2.2.4.cmml" xref="S6.SS2.12.5.p3.9.m9.2.2.2.4"><csymbol cd="ambiguous" id="S6.SS2.12.5.p3.9.m9.2.2.2.4.1.cmml" xref="S6.SS2.12.5.p3.9.m9.2.2.2.4">subscript</csymbol><ci id="S6.SS2.12.5.p3.9.m9.2.2.2.4.2.cmml" xref="S6.SS2.12.5.p3.9.m9.2.2.2.4.2">Δ</ci><ci id="S6.SS2.12.5.p3.9.m9.2.2.2.4.3.cmml" xref="S6.SS2.12.5.p3.9.m9.2.2.2.4.3">𝐴</ci></apply><interval closure="open" id="S6.SS2.12.5.p3.9.m9.2.2.2.2.3.cmml" xref="S6.SS2.12.5.p3.9.m9.2.2.2.2.2"><apply id="S6.SS2.12.5.p3.9.m9.1.1.1.1.1.1.cmml" xref="S6.SS2.12.5.p3.9.m9.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.12.5.p3.9.m9.1.1.1.1.1.1.1.cmml" xref="S6.SS2.12.5.p3.9.m9.1.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.12.5.p3.9.m9.1.1.1.1.1.1.2.cmml" xref="S6.SS2.12.5.p3.9.m9.1.1.1.1.1.1.2">𝜉</ci><ci id="S6.SS2.12.5.p3.9.m9.1.1.1.1.1.1.3.cmml" xref="S6.SS2.12.5.p3.9.m9.1.1.1.1.1.1.3">𝑗</ci></apply><apply id="S6.SS2.12.5.p3.9.m9.2.2.2.2.2.2.cmml" xref="S6.SS2.12.5.p3.9.m9.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.12.5.p3.9.m9.2.2.2.2.2.2.1.cmml" xref="S6.SS2.12.5.p3.9.m9.2.2.2.2.2.2">subscript</csymbol><ci id="S6.SS2.12.5.p3.9.m9.2.2.2.2.2.2.2.cmml" xref="S6.SS2.12.5.p3.9.m9.2.2.2.2.2.2.2">𝜂</ci><ci id="S6.SS2.12.5.p3.9.m9.2.2.2.2.2.2.3.cmml" xref="S6.SS2.12.5.p3.9.m9.2.2.2.2.2.2.3">𝑗</ci></apply></interval></apply><ci id="S6.SS2.12.5.p3.9.m9.2.2.4.cmml" xref="S6.SS2.12.5.p3.9.m9.2.2.4">𝜉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.12.5.p3.9.m9.2c">\Delta_{A}(\xi_{j},\eta_{j})<\xi</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.12.5.p3.9.m9.2d">roman_Δ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ξ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT , italic_η start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) < italic_ξ</annotation></semantics></math>, it follows that <math alttext="\Delta_{A}(\xi_{i},\eta_{j})<\xi" class="ltx_Math" display="inline" id="S6.SS2.12.5.p3.10.m10.2"><semantics id="S6.SS2.12.5.p3.10.m10.2a"><mrow id="S6.SS2.12.5.p3.10.m10.2.2" xref="S6.SS2.12.5.p3.10.m10.2.2.cmml"><mrow id="S6.SS2.12.5.p3.10.m10.2.2.2" xref="S6.SS2.12.5.p3.10.m10.2.2.2.cmml"><msub id="S6.SS2.12.5.p3.10.m10.2.2.2.4" xref="S6.SS2.12.5.p3.10.m10.2.2.2.4.cmml"><mi id="S6.SS2.12.5.p3.10.m10.2.2.2.4.2" mathvariant="normal" xref="S6.SS2.12.5.p3.10.m10.2.2.2.4.2.cmml">Δ</mi><mi id="S6.SS2.12.5.p3.10.m10.2.2.2.4.3" xref="S6.SS2.12.5.p3.10.m10.2.2.2.4.3.cmml">A</mi></msub><mo id="S6.SS2.12.5.p3.10.m10.2.2.2.3" xref="S6.SS2.12.5.p3.10.m10.2.2.2.3.cmml">⁢</mo><mrow id="S6.SS2.12.5.p3.10.m10.2.2.2.2.2" xref="S6.SS2.12.5.p3.10.m10.2.2.2.2.3.cmml"><mo id="S6.SS2.12.5.p3.10.m10.2.2.2.2.2.3" stretchy="false" xref="S6.SS2.12.5.p3.10.m10.2.2.2.2.3.cmml">(</mo><msub id="S6.SS2.12.5.p3.10.m10.1.1.1.1.1.1" xref="S6.SS2.12.5.p3.10.m10.1.1.1.1.1.1.cmml"><mi id="S6.SS2.12.5.p3.10.m10.1.1.1.1.1.1.2" xref="S6.SS2.12.5.p3.10.m10.1.1.1.1.1.1.2.cmml">ξ</mi><mi id="S6.SS2.12.5.p3.10.m10.1.1.1.1.1.1.3" xref="S6.SS2.12.5.p3.10.m10.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S6.SS2.12.5.p3.10.m10.2.2.2.2.2.4" xref="S6.SS2.12.5.p3.10.m10.2.2.2.2.3.cmml">,</mo><msub id="S6.SS2.12.5.p3.10.m10.2.2.2.2.2.2" xref="S6.SS2.12.5.p3.10.m10.2.2.2.2.2.2.cmml"><mi id="S6.SS2.12.5.p3.10.m10.2.2.2.2.2.2.2" xref="S6.SS2.12.5.p3.10.m10.2.2.2.2.2.2.2.cmml">η</mi><mi id="S6.SS2.12.5.p3.10.m10.2.2.2.2.2.2.3" xref="S6.SS2.12.5.p3.10.m10.2.2.2.2.2.2.3.cmml">j</mi></msub><mo id="S6.SS2.12.5.p3.10.m10.2.2.2.2.2.5" stretchy="false" xref="S6.SS2.12.5.p3.10.m10.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.SS2.12.5.p3.10.m10.2.2.3" xref="S6.SS2.12.5.p3.10.m10.2.2.3.cmml"><</mo><mi id="S6.SS2.12.5.p3.10.m10.2.2.4" xref="S6.SS2.12.5.p3.10.m10.2.2.4.cmml">ξ</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.12.5.p3.10.m10.2b"><apply id="S6.SS2.12.5.p3.10.m10.2.2.cmml" xref="S6.SS2.12.5.p3.10.m10.2.2"><lt id="S6.SS2.12.5.p3.10.m10.2.2.3.cmml" xref="S6.SS2.12.5.p3.10.m10.2.2.3"></lt><apply id="S6.SS2.12.5.p3.10.m10.2.2.2.cmml" xref="S6.SS2.12.5.p3.10.m10.2.2.2"><times id="S6.SS2.12.5.p3.10.m10.2.2.2.3.cmml" xref="S6.SS2.12.5.p3.10.m10.2.2.2.3"></times><apply id="S6.SS2.12.5.p3.10.m10.2.2.2.4.cmml" xref="S6.SS2.12.5.p3.10.m10.2.2.2.4"><csymbol cd="ambiguous" id="S6.SS2.12.5.p3.10.m10.2.2.2.4.1.cmml" xref="S6.SS2.12.5.p3.10.m10.2.2.2.4">subscript</csymbol><ci id="S6.SS2.12.5.p3.10.m10.2.2.2.4.2.cmml" xref="S6.SS2.12.5.p3.10.m10.2.2.2.4.2">Δ</ci><ci id="S6.SS2.12.5.p3.10.m10.2.2.2.4.3.cmml" xref="S6.SS2.12.5.p3.10.m10.2.2.2.4.3">𝐴</ci></apply><interval closure="open" id="S6.SS2.12.5.p3.10.m10.2.2.2.2.3.cmml" xref="S6.SS2.12.5.p3.10.m10.2.2.2.2.2"><apply id="S6.SS2.12.5.p3.10.m10.1.1.1.1.1.1.cmml" xref="S6.SS2.12.5.p3.10.m10.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.12.5.p3.10.m10.1.1.1.1.1.1.1.cmml" xref="S6.SS2.12.5.p3.10.m10.1.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.12.5.p3.10.m10.1.1.1.1.1.1.2.cmml" xref="S6.SS2.12.5.p3.10.m10.1.1.1.1.1.1.2">𝜉</ci><ci id="S6.SS2.12.5.p3.10.m10.1.1.1.1.1.1.3.cmml" xref="S6.SS2.12.5.p3.10.m10.1.1.1.1.1.1.3">𝑖</ci></apply><apply id="S6.SS2.12.5.p3.10.m10.2.2.2.2.2.2.cmml" xref="S6.SS2.12.5.p3.10.m10.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.12.5.p3.10.m10.2.2.2.2.2.2.1.cmml" xref="S6.SS2.12.5.p3.10.m10.2.2.2.2.2.2">subscript</csymbol><ci id="S6.SS2.12.5.p3.10.m10.2.2.2.2.2.2.2.cmml" xref="S6.SS2.12.5.p3.10.m10.2.2.2.2.2.2.2">𝜂</ci><ci id="S6.SS2.12.5.p3.10.m10.2.2.2.2.2.2.3.cmml" xref="S6.SS2.12.5.p3.10.m10.2.2.2.2.2.2.3">𝑗</ci></apply></interval></apply><ci id="S6.SS2.12.5.p3.10.m10.2.2.4.cmml" xref="S6.SS2.12.5.p3.10.m10.2.2.4">𝜉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.12.5.p3.10.m10.2c">\Delta_{A}(\xi_{i},\eta_{j})<\xi</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.12.5.p3.10.m10.2d">roman_Δ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ξ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_η start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) < italic_ξ</annotation></semantics></math>. See <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.F1" title="In Proof. ‣ Proof. ‣ 6.2. Forcing epimorphisms ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Fig.</span> <span class="ltx_text ltx_ref_tag">1</span></a>.</p> </div> <div class="ltx_para" id="S6.SS2.13.6.p4"> <p class="ltx_p" id="S6.SS2.13.6.p4.8">Now suppose that <math alttext="[a_{l},a_{r}]\cap[b_{l},b_{r}]\neq\varnothing" class="ltx_Math" display="inline" id="S6.SS2.13.6.p4.1.m1.4"><semantics id="S6.SS2.13.6.p4.1.m1.4a"><mrow id="S6.SS2.13.6.p4.1.m1.4.4" xref="S6.SS2.13.6.p4.1.m1.4.4.cmml"><mrow id="S6.SS2.13.6.p4.1.m1.4.4.4" xref="S6.SS2.13.6.p4.1.m1.4.4.4.cmml"><mrow id="S6.SS2.13.6.p4.1.m1.2.2.2.2.2" xref="S6.SS2.13.6.p4.1.m1.2.2.2.2.3.cmml"><mo id="S6.SS2.13.6.p4.1.m1.2.2.2.2.2.3" stretchy="false" xref="S6.SS2.13.6.p4.1.m1.2.2.2.2.3.cmml">[</mo><msub id="S6.SS2.13.6.p4.1.m1.1.1.1.1.1.1" xref="S6.SS2.13.6.p4.1.m1.1.1.1.1.1.1.cmml"><mi id="S6.SS2.13.6.p4.1.m1.1.1.1.1.1.1.2" xref="S6.SS2.13.6.p4.1.m1.1.1.1.1.1.1.2.cmml">a</mi><mi id="S6.SS2.13.6.p4.1.m1.1.1.1.1.1.1.3" xref="S6.SS2.13.6.p4.1.m1.1.1.1.1.1.1.3.cmml">l</mi></msub><mo id="S6.SS2.13.6.p4.1.m1.2.2.2.2.2.4" xref="S6.SS2.13.6.p4.1.m1.2.2.2.2.3.cmml">,</mo><msub id="S6.SS2.13.6.p4.1.m1.2.2.2.2.2.2" xref="S6.SS2.13.6.p4.1.m1.2.2.2.2.2.2.cmml"><mi id="S6.SS2.13.6.p4.1.m1.2.2.2.2.2.2.2" xref="S6.SS2.13.6.p4.1.m1.2.2.2.2.2.2.2.cmml">a</mi><mi id="S6.SS2.13.6.p4.1.m1.2.2.2.2.2.2.3" xref="S6.SS2.13.6.p4.1.m1.2.2.2.2.2.2.3.cmml">r</mi></msub><mo id="S6.SS2.13.6.p4.1.m1.2.2.2.2.2.5" stretchy="false" xref="S6.SS2.13.6.p4.1.m1.2.2.2.2.3.cmml">]</mo></mrow><mo id="S6.SS2.13.6.p4.1.m1.4.4.4.5" xref="S6.SS2.13.6.p4.1.m1.4.4.4.5.cmml">∩</mo><mrow id="S6.SS2.13.6.p4.1.m1.4.4.4.4.2" xref="S6.SS2.13.6.p4.1.m1.4.4.4.4.3.cmml"><mo id="S6.SS2.13.6.p4.1.m1.4.4.4.4.2.3" stretchy="false" xref="S6.SS2.13.6.p4.1.m1.4.4.4.4.3.cmml">[</mo><msub id="S6.SS2.13.6.p4.1.m1.3.3.3.3.1.1" xref="S6.SS2.13.6.p4.1.m1.3.3.3.3.1.1.cmml"><mi id="S6.SS2.13.6.p4.1.m1.3.3.3.3.1.1.2" xref="S6.SS2.13.6.p4.1.m1.3.3.3.3.1.1.2.cmml">b</mi><mi id="S6.SS2.13.6.p4.1.m1.3.3.3.3.1.1.3" xref="S6.SS2.13.6.p4.1.m1.3.3.3.3.1.1.3.cmml">l</mi></msub><mo id="S6.SS2.13.6.p4.1.m1.4.4.4.4.2.4" xref="S6.SS2.13.6.p4.1.m1.4.4.4.4.3.cmml">,</mo><msub id="S6.SS2.13.6.p4.1.m1.4.4.4.4.2.2" xref="S6.SS2.13.6.p4.1.m1.4.4.4.4.2.2.cmml"><mi id="S6.SS2.13.6.p4.1.m1.4.4.4.4.2.2.2" xref="S6.SS2.13.6.p4.1.m1.4.4.4.4.2.2.2.cmml">b</mi><mi id="S6.SS2.13.6.p4.1.m1.4.4.4.4.2.2.3" xref="S6.SS2.13.6.p4.1.m1.4.4.4.4.2.2.3.cmml">r</mi></msub><mo id="S6.SS2.13.6.p4.1.m1.4.4.4.4.2.5" stretchy="false" xref="S6.SS2.13.6.p4.1.m1.4.4.4.4.3.cmml">]</mo></mrow></mrow><mo id="S6.SS2.13.6.p4.1.m1.4.4.5" xref="S6.SS2.13.6.p4.1.m1.4.4.5.cmml">≠</mo><mi id="S6.SS2.13.6.p4.1.m1.4.4.6" mathvariant="normal" xref="S6.SS2.13.6.p4.1.m1.4.4.6.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.13.6.p4.1.m1.4b"><apply id="S6.SS2.13.6.p4.1.m1.4.4.cmml" xref="S6.SS2.13.6.p4.1.m1.4.4"><neq id="S6.SS2.13.6.p4.1.m1.4.4.5.cmml" xref="S6.SS2.13.6.p4.1.m1.4.4.5"></neq><apply id="S6.SS2.13.6.p4.1.m1.4.4.4.cmml" xref="S6.SS2.13.6.p4.1.m1.4.4.4"><intersect id="S6.SS2.13.6.p4.1.m1.4.4.4.5.cmml" xref="S6.SS2.13.6.p4.1.m1.4.4.4.5"></intersect><interval closure="closed" id="S6.SS2.13.6.p4.1.m1.2.2.2.2.3.cmml" xref="S6.SS2.13.6.p4.1.m1.2.2.2.2.2"><apply id="S6.SS2.13.6.p4.1.m1.1.1.1.1.1.1.cmml" xref="S6.SS2.13.6.p4.1.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.13.6.p4.1.m1.1.1.1.1.1.1.1.cmml" xref="S6.SS2.13.6.p4.1.m1.1.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.13.6.p4.1.m1.1.1.1.1.1.1.2.cmml" xref="S6.SS2.13.6.p4.1.m1.1.1.1.1.1.1.2">𝑎</ci><ci id="S6.SS2.13.6.p4.1.m1.1.1.1.1.1.1.3.cmml" xref="S6.SS2.13.6.p4.1.m1.1.1.1.1.1.1.3">𝑙</ci></apply><apply id="S6.SS2.13.6.p4.1.m1.2.2.2.2.2.2.cmml" xref="S6.SS2.13.6.p4.1.m1.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.13.6.p4.1.m1.2.2.2.2.2.2.1.cmml" xref="S6.SS2.13.6.p4.1.m1.2.2.2.2.2.2">subscript</csymbol><ci id="S6.SS2.13.6.p4.1.m1.2.2.2.2.2.2.2.cmml" xref="S6.SS2.13.6.p4.1.m1.2.2.2.2.2.2.2">𝑎</ci><ci id="S6.SS2.13.6.p4.1.m1.2.2.2.2.2.2.3.cmml" xref="S6.SS2.13.6.p4.1.m1.2.2.2.2.2.2.3">𝑟</ci></apply></interval><interval closure="closed" id="S6.SS2.13.6.p4.1.m1.4.4.4.4.3.cmml" xref="S6.SS2.13.6.p4.1.m1.4.4.4.4.2"><apply id="S6.SS2.13.6.p4.1.m1.3.3.3.3.1.1.cmml" xref="S6.SS2.13.6.p4.1.m1.3.3.3.3.1.1"><csymbol cd="ambiguous" id="S6.SS2.13.6.p4.1.m1.3.3.3.3.1.1.1.cmml" xref="S6.SS2.13.6.p4.1.m1.3.3.3.3.1.1">subscript</csymbol><ci id="S6.SS2.13.6.p4.1.m1.3.3.3.3.1.1.2.cmml" xref="S6.SS2.13.6.p4.1.m1.3.3.3.3.1.1.2">𝑏</ci><ci id="S6.SS2.13.6.p4.1.m1.3.3.3.3.1.1.3.cmml" xref="S6.SS2.13.6.p4.1.m1.3.3.3.3.1.1.3">𝑙</ci></apply><apply id="S6.SS2.13.6.p4.1.m1.4.4.4.4.2.2.cmml" xref="S6.SS2.13.6.p4.1.m1.4.4.4.4.2.2"><csymbol cd="ambiguous" id="S6.SS2.13.6.p4.1.m1.4.4.4.4.2.2.1.cmml" xref="S6.SS2.13.6.p4.1.m1.4.4.4.4.2.2">subscript</csymbol><ci id="S6.SS2.13.6.p4.1.m1.4.4.4.4.2.2.2.cmml" xref="S6.SS2.13.6.p4.1.m1.4.4.4.4.2.2.2">𝑏</ci><ci id="S6.SS2.13.6.p4.1.m1.4.4.4.4.2.2.3.cmml" xref="S6.SS2.13.6.p4.1.m1.4.4.4.4.2.2.3">𝑟</ci></apply></interval></apply><emptyset id="S6.SS2.13.6.p4.1.m1.4.4.6.cmml" xref="S6.SS2.13.6.p4.1.m1.4.4.6"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.13.6.p4.1.m1.4c">[a_{l},a_{r}]\cap[b_{l},b_{r}]\neq\varnothing</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.13.6.p4.1.m1.4d">[ italic_a start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ] ∩ [ italic_b start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ] ≠ ∅</annotation></semantics></math> and assume that <math alttext="\xi_{i}<_{A}\eta_{j}" class="ltx_Math" display="inline" id="S6.SS2.13.6.p4.2.m2.1"><semantics id="S6.SS2.13.6.p4.2.m2.1a"><mrow id="S6.SS2.13.6.p4.2.m2.1.1" xref="S6.SS2.13.6.p4.2.m2.1.1.cmml"><msub id="S6.SS2.13.6.p4.2.m2.1.1.2" xref="S6.SS2.13.6.p4.2.m2.1.1.2.cmml"><mi id="S6.SS2.13.6.p4.2.m2.1.1.2.2" xref="S6.SS2.13.6.p4.2.m2.1.1.2.2.cmml">ξ</mi><mi id="S6.SS2.13.6.p4.2.m2.1.1.2.3" xref="S6.SS2.13.6.p4.2.m2.1.1.2.3.cmml">i</mi></msub><msub id="S6.SS2.13.6.p4.2.m2.1.1.1" xref="S6.SS2.13.6.p4.2.m2.1.1.1.cmml"><mo id="S6.SS2.13.6.p4.2.m2.1.1.1.2" xref="S6.SS2.13.6.p4.2.m2.1.1.1.2.cmml"><</mo><mi id="S6.SS2.13.6.p4.2.m2.1.1.1.3" xref="S6.SS2.13.6.p4.2.m2.1.1.1.3.cmml">A</mi></msub><msub id="S6.SS2.13.6.p4.2.m2.1.1.3" xref="S6.SS2.13.6.p4.2.m2.1.1.3.cmml"><mi id="S6.SS2.13.6.p4.2.m2.1.1.3.2" xref="S6.SS2.13.6.p4.2.m2.1.1.3.2.cmml">η</mi><mi id="S6.SS2.13.6.p4.2.m2.1.1.3.3" xref="S6.SS2.13.6.p4.2.m2.1.1.3.3.cmml">j</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.13.6.p4.2.m2.1b"><apply id="S6.SS2.13.6.p4.2.m2.1.1.cmml" xref="S6.SS2.13.6.p4.2.m2.1.1"><apply id="S6.SS2.13.6.p4.2.m2.1.1.1.cmml" xref="S6.SS2.13.6.p4.2.m2.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.13.6.p4.2.m2.1.1.1.1.cmml" xref="S6.SS2.13.6.p4.2.m2.1.1.1">subscript</csymbol><lt id="S6.SS2.13.6.p4.2.m2.1.1.1.2.cmml" xref="S6.SS2.13.6.p4.2.m2.1.1.1.2"></lt><ci id="S6.SS2.13.6.p4.2.m2.1.1.1.3.cmml" xref="S6.SS2.13.6.p4.2.m2.1.1.1.3">𝐴</ci></apply><apply id="S6.SS2.13.6.p4.2.m2.1.1.2.cmml" xref="S6.SS2.13.6.p4.2.m2.1.1.2"><csymbol cd="ambiguous" id="S6.SS2.13.6.p4.2.m2.1.1.2.1.cmml" xref="S6.SS2.13.6.p4.2.m2.1.1.2">subscript</csymbol><ci id="S6.SS2.13.6.p4.2.m2.1.1.2.2.cmml" xref="S6.SS2.13.6.p4.2.m2.1.1.2.2">𝜉</ci><ci id="S6.SS2.13.6.p4.2.m2.1.1.2.3.cmml" xref="S6.SS2.13.6.p4.2.m2.1.1.2.3">𝑖</ci></apply><apply id="S6.SS2.13.6.p4.2.m2.1.1.3.cmml" xref="S6.SS2.13.6.p4.2.m2.1.1.3"><csymbol cd="ambiguous" id="S6.SS2.13.6.p4.2.m2.1.1.3.1.cmml" xref="S6.SS2.13.6.p4.2.m2.1.1.3">subscript</csymbol><ci id="S6.SS2.13.6.p4.2.m2.1.1.3.2.cmml" xref="S6.SS2.13.6.p4.2.m2.1.1.3.2">𝜂</ci><ci id="S6.SS2.13.6.p4.2.m2.1.1.3.3.cmml" xref="S6.SS2.13.6.p4.2.m2.1.1.3.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.13.6.p4.2.m2.1c">\xi_{i}<_{A}\eta_{j}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.13.6.p4.2.m2.1d">italic_ξ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT < start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>, the other case is analogous. Then, <math alttext="c:=\Delta_{A}(\xi_{i},\eta_{j})\in{[\xi_{i},\xi_{j}]}_{A}\subseteq[a_{l},a_{r}% ]\cup[b_{l},b_{r}]" class="ltx_Math" display="inline" id="S6.SS2.13.6.p4.3.m3.8"><semantics id="S6.SS2.13.6.p4.3.m3.8a"><mrow id="S6.SS2.13.6.p4.3.m3.8.8" xref="S6.SS2.13.6.p4.3.m3.8.8.cmml"><mi id="S6.SS2.13.6.p4.3.m3.8.8.10" xref="S6.SS2.13.6.p4.3.m3.8.8.10.cmml">c</mi><mo id="S6.SS2.13.6.p4.3.m3.8.8.11" lspace="0.278em" rspace="0.278em" xref="S6.SS2.13.6.p4.3.m3.8.8.11.cmml">:=</mo><mrow id="S6.SS2.13.6.p4.3.m3.2.2.2" xref="S6.SS2.13.6.p4.3.m3.2.2.2.cmml"><msub id="S6.SS2.13.6.p4.3.m3.2.2.2.4" xref="S6.SS2.13.6.p4.3.m3.2.2.2.4.cmml"><mi id="S6.SS2.13.6.p4.3.m3.2.2.2.4.2" mathvariant="normal" xref="S6.SS2.13.6.p4.3.m3.2.2.2.4.2.cmml">Δ</mi><mi id="S6.SS2.13.6.p4.3.m3.2.2.2.4.3" xref="S6.SS2.13.6.p4.3.m3.2.2.2.4.3.cmml">A</mi></msub><mo id="S6.SS2.13.6.p4.3.m3.2.2.2.3" xref="S6.SS2.13.6.p4.3.m3.2.2.2.3.cmml">⁢</mo><mrow id="S6.SS2.13.6.p4.3.m3.2.2.2.2.2" xref="S6.SS2.13.6.p4.3.m3.2.2.2.2.3.cmml"><mo id="S6.SS2.13.6.p4.3.m3.2.2.2.2.2.3" stretchy="false" xref="S6.SS2.13.6.p4.3.m3.2.2.2.2.3.cmml">(</mo><msub id="S6.SS2.13.6.p4.3.m3.1.1.1.1.1.1" xref="S6.SS2.13.6.p4.3.m3.1.1.1.1.1.1.cmml"><mi id="S6.SS2.13.6.p4.3.m3.1.1.1.1.1.1.2" xref="S6.SS2.13.6.p4.3.m3.1.1.1.1.1.1.2.cmml">ξ</mi><mi id="S6.SS2.13.6.p4.3.m3.1.1.1.1.1.1.3" xref="S6.SS2.13.6.p4.3.m3.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S6.SS2.13.6.p4.3.m3.2.2.2.2.2.4" xref="S6.SS2.13.6.p4.3.m3.2.2.2.2.3.cmml">,</mo><msub id="S6.SS2.13.6.p4.3.m3.2.2.2.2.2.2" xref="S6.SS2.13.6.p4.3.m3.2.2.2.2.2.2.cmml"><mi id="S6.SS2.13.6.p4.3.m3.2.2.2.2.2.2.2" xref="S6.SS2.13.6.p4.3.m3.2.2.2.2.2.2.2.cmml">η</mi><mi id="S6.SS2.13.6.p4.3.m3.2.2.2.2.2.2.3" xref="S6.SS2.13.6.p4.3.m3.2.2.2.2.2.2.3.cmml">j</mi></msub><mo id="S6.SS2.13.6.p4.3.m3.2.2.2.2.2.5" stretchy="false" xref="S6.SS2.13.6.p4.3.m3.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.SS2.13.6.p4.3.m3.8.8.12" xref="S6.SS2.13.6.p4.3.m3.8.8.12.cmml">∈</mo><msub id="S6.SS2.13.6.p4.3.m3.4.4.4" xref="S6.SS2.13.6.p4.3.m3.4.4.4.cmml"><mrow id="S6.SS2.13.6.p4.3.m3.4.4.4.2.2" xref="S6.SS2.13.6.p4.3.m3.4.4.4.2.3.cmml"><mo id="S6.SS2.13.6.p4.3.m3.4.4.4.2.2.3" stretchy="false" xref="S6.SS2.13.6.p4.3.m3.4.4.4.2.3.cmml">[</mo><msub id="S6.SS2.13.6.p4.3.m3.3.3.3.1.1.1" xref="S6.SS2.13.6.p4.3.m3.3.3.3.1.1.1.cmml"><mi id="S6.SS2.13.6.p4.3.m3.3.3.3.1.1.1.2" xref="S6.SS2.13.6.p4.3.m3.3.3.3.1.1.1.2.cmml">ξ</mi><mi id="S6.SS2.13.6.p4.3.m3.3.3.3.1.1.1.3" xref="S6.SS2.13.6.p4.3.m3.3.3.3.1.1.1.3.cmml">i</mi></msub><mo id="S6.SS2.13.6.p4.3.m3.4.4.4.2.2.4" xref="S6.SS2.13.6.p4.3.m3.4.4.4.2.3.cmml">,</mo><msub id="S6.SS2.13.6.p4.3.m3.4.4.4.2.2.2" xref="S6.SS2.13.6.p4.3.m3.4.4.4.2.2.2.cmml"><mi id="S6.SS2.13.6.p4.3.m3.4.4.4.2.2.2.2" xref="S6.SS2.13.6.p4.3.m3.4.4.4.2.2.2.2.cmml">ξ</mi><mi id="S6.SS2.13.6.p4.3.m3.4.4.4.2.2.2.3" xref="S6.SS2.13.6.p4.3.m3.4.4.4.2.2.2.3.cmml">j</mi></msub><mo id="S6.SS2.13.6.p4.3.m3.4.4.4.2.2.5" stretchy="false" xref="S6.SS2.13.6.p4.3.m3.4.4.4.2.3.cmml">]</mo></mrow><mi id="S6.SS2.13.6.p4.3.m3.4.4.4.4" xref="S6.SS2.13.6.p4.3.m3.4.4.4.4.cmml">A</mi></msub><mo id="S6.SS2.13.6.p4.3.m3.8.8.13" xref="S6.SS2.13.6.p4.3.m3.8.8.13.cmml">⊆</mo><mrow id="S6.SS2.13.6.p4.3.m3.8.8.8" xref="S6.SS2.13.6.p4.3.m3.8.8.8.cmml"><mrow id="S6.SS2.13.6.p4.3.m3.6.6.6.2.2" xref="S6.SS2.13.6.p4.3.m3.6.6.6.2.3.cmml"><mo id="S6.SS2.13.6.p4.3.m3.6.6.6.2.2.3" stretchy="false" xref="S6.SS2.13.6.p4.3.m3.6.6.6.2.3.cmml">[</mo><msub id="S6.SS2.13.6.p4.3.m3.5.5.5.1.1.1" xref="S6.SS2.13.6.p4.3.m3.5.5.5.1.1.1.cmml"><mi id="S6.SS2.13.6.p4.3.m3.5.5.5.1.1.1.2" xref="S6.SS2.13.6.p4.3.m3.5.5.5.1.1.1.2.cmml">a</mi><mi id="S6.SS2.13.6.p4.3.m3.5.5.5.1.1.1.3" xref="S6.SS2.13.6.p4.3.m3.5.5.5.1.1.1.3.cmml">l</mi></msub><mo id="S6.SS2.13.6.p4.3.m3.6.6.6.2.2.4" xref="S6.SS2.13.6.p4.3.m3.6.6.6.2.3.cmml">,</mo><msub id="S6.SS2.13.6.p4.3.m3.6.6.6.2.2.2" xref="S6.SS2.13.6.p4.3.m3.6.6.6.2.2.2.cmml"><mi id="S6.SS2.13.6.p4.3.m3.6.6.6.2.2.2.2" xref="S6.SS2.13.6.p4.3.m3.6.6.6.2.2.2.2.cmml">a</mi><mi id="S6.SS2.13.6.p4.3.m3.6.6.6.2.2.2.3" xref="S6.SS2.13.6.p4.3.m3.6.6.6.2.2.2.3.cmml">r</mi></msub><mo id="S6.SS2.13.6.p4.3.m3.6.6.6.2.2.5" stretchy="false" xref="S6.SS2.13.6.p4.3.m3.6.6.6.2.3.cmml">]</mo></mrow><mo id="S6.SS2.13.6.p4.3.m3.8.8.8.5" xref="S6.SS2.13.6.p4.3.m3.8.8.8.5.cmml">∪</mo><mrow id="S6.SS2.13.6.p4.3.m3.8.8.8.4.2" xref="S6.SS2.13.6.p4.3.m3.8.8.8.4.3.cmml"><mo id="S6.SS2.13.6.p4.3.m3.8.8.8.4.2.3" stretchy="false" xref="S6.SS2.13.6.p4.3.m3.8.8.8.4.3.cmml">[</mo><msub id="S6.SS2.13.6.p4.3.m3.7.7.7.3.1.1" xref="S6.SS2.13.6.p4.3.m3.7.7.7.3.1.1.cmml"><mi id="S6.SS2.13.6.p4.3.m3.7.7.7.3.1.1.2" xref="S6.SS2.13.6.p4.3.m3.7.7.7.3.1.1.2.cmml">b</mi><mi id="S6.SS2.13.6.p4.3.m3.7.7.7.3.1.1.3" xref="S6.SS2.13.6.p4.3.m3.7.7.7.3.1.1.3.cmml">l</mi></msub><mo id="S6.SS2.13.6.p4.3.m3.8.8.8.4.2.4" xref="S6.SS2.13.6.p4.3.m3.8.8.8.4.3.cmml">,</mo><msub id="S6.SS2.13.6.p4.3.m3.8.8.8.4.2.2" xref="S6.SS2.13.6.p4.3.m3.8.8.8.4.2.2.cmml"><mi id="S6.SS2.13.6.p4.3.m3.8.8.8.4.2.2.2" xref="S6.SS2.13.6.p4.3.m3.8.8.8.4.2.2.2.cmml">b</mi><mi id="S6.SS2.13.6.p4.3.m3.8.8.8.4.2.2.3" xref="S6.SS2.13.6.p4.3.m3.8.8.8.4.2.2.3.cmml">r</mi></msub><mo id="S6.SS2.13.6.p4.3.m3.8.8.8.4.2.5" stretchy="false" xref="S6.SS2.13.6.p4.3.m3.8.8.8.4.3.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.13.6.p4.3.m3.8b"><apply id="S6.SS2.13.6.p4.3.m3.8.8.cmml" xref="S6.SS2.13.6.p4.3.m3.8.8"><and id="S6.SS2.13.6.p4.3.m3.8.8a.cmml" xref="S6.SS2.13.6.p4.3.m3.8.8"></and><apply id="S6.SS2.13.6.p4.3.m3.8.8b.cmml" xref="S6.SS2.13.6.p4.3.m3.8.8"><csymbol cd="latexml" id="S6.SS2.13.6.p4.3.m3.8.8.11.cmml" xref="S6.SS2.13.6.p4.3.m3.8.8.11">assign</csymbol><ci id="S6.SS2.13.6.p4.3.m3.8.8.10.cmml" xref="S6.SS2.13.6.p4.3.m3.8.8.10">𝑐</ci><apply id="S6.SS2.13.6.p4.3.m3.2.2.2.cmml" xref="S6.SS2.13.6.p4.3.m3.2.2.2"><times id="S6.SS2.13.6.p4.3.m3.2.2.2.3.cmml" xref="S6.SS2.13.6.p4.3.m3.2.2.2.3"></times><apply id="S6.SS2.13.6.p4.3.m3.2.2.2.4.cmml" xref="S6.SS2.13.6.p4.3.m3.2.2.2.4"><csymbol cd="ambiguous" id="S6.SS2.13.6.p4.3.m3.2.2.2.4.1.cmml" xref="S6.SS2.13.6.p4.3.m3.2.2.2.4">subscript</csymbol><ci id="S6.SS2.13.6.p4.3.m3.2.2.2.4.2.cmml" xref="S6.SS2.13.6.p4.3.m3.2.2.2.4.2">Δ</ci><ci id="S6.SS2.13.6.p4.3.m3.2.2.2.4.3.cmml" xref="S6.SS2.13.6.p4.3.m3.2.2.2.4.3">𝐴</ci></apply><interval closure="open" id="S6.SS2.13.6.p4.3.m3.2.2.2.2.3.cmml" xref="S6.SS2.13.6.p4.3.m3.2.2.2.2.2"><apply id="S6.SS2.13.6.p4.3.m3.1.1.1.1.1.1.cmml" xref="S6.SS2.13.6.p4.3.m3.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.13.6.p4.3.m3.1.1.1.1.1.1.1.cmml" xref="S6.SS2.13.6.p4.3.m3.1.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.13.6.p4.3.m3.1.1.1.1.1.1.2.cmml" xref="S6.SS2.13.6.p4.3.m3.1.1.1.1.1.1.2">𝜉</ci><ci id="S6.SS2.13.6.p4.3.m3.1.1.1.1.1.1.3.cmml" xref="S6.SS2.13.6.p4.3.m3.1.1.1.1.1.1.3">𝑖</ci></apply><apply id="S6.SS2.13.6.p4.3.m3.2.2.2.2.2.2.cmml" xref="S6.SS2.13.6.p4.3.m3.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.13.6.p4.3.m3.2.2.2.2.2.2.1.cmml" xref="S6.SS2.13.6.p4.3.m3.2.2.2.2.2.2">subscript</csymbol><ci id="S6.SS2.13.6.p4.3.m3.2.2.2.2.2.2.2.cmml" xref="S6.SS2.13.6.p4.3.m3.2.2.2.2.2.2.2">𝜂</ci><ci id="S6.SS2.13.6.p4.3.m3.2.2.2.2.2.2.3.cmml" xref="S6.SS2.13.6.p4.3.m3.2.2.2.2.2.2.3">𝑗</ci></apply></interval></apply></apply><apply id="S6.SS2.13.6.p4.3.m3.8.8c.cmml" xref="S6.SS2.13.6.p4.3.m3.8.8"><in id="S6.SS2.13.6.p4.3.m3.8.8.12.cmml" xref="S6.SS2.13.6.p4.3.m3.8.8.12"></in><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.13.6.p4.3.m3.2.2.2.cmml" id="S6.SS2.13.6.p4.3.m3.8.8d.cmml" xref="S6.SS2.13.6.p4.3.m3.8.8"></share><apply id="S6.SS2.13.6.p4.3.m3.4.4.4.cmml" xref="S6.SS2.13.6.p4.3.m3.4.4.4"><csymbol cd="ambiguous" id="S6.SS2.13.6.p4.3.m3.4.4.4.3.cmml" xref="S6.SS2.13.6.p4.3.m3.4.4.4">subscript</csymbol><interval closure="closed" id="S6.SS2.13.6.p4.3.m3.4.4.4.2.3.cmml" xref="S6.SS2.13.6.p4.3.m3.4.4.4.2.2"><apply id="S6.SS2.13.6.p4.3.m3.3.3.3.1.1.1.cmml" xref="S6.SS2.13.6.p4.3.m3.3.3.3.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.13.6.p4.3.m3.3.3.3.1.1.1.1.cmml" xref="S6.SS2.13.6.p4.3.m3.3.3.3.1.1.1">subscript</csymbol><ci id="S6.SS2.13.6.p4.3.m3.3.3.3.1.1.1.2.cmml" xref="S6.SS2.13.6.p4.3.m3.3.3.3.1.1.1.2">𝜉</ci><ci id="S6.SS2.13.6.p4.3.m3.3.3.3.1.1.1.3.cmml" xref="S6.SS2.13.6.p4.3.m3.3.3.3.1.1.1.3">𝑖</ci></apply><apply id="S6.SS2.13.6.p4.3.m3.4.4.4.2.2.2.cmml" xref="S6.SS2.13.6.p4.3.m3.4.4.4.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.13.6.p4.3.m3.4.4.4.2.2.2.1.cmml" xref="S6.SS2.13.6.p4.3.m3.4.4.4.2.2.2">subscript</csymbol><ci id="S6.SS2.13.6.p4.3.m3.4.4.4.2.2.2.2.cmml" xref="S6.SS2.13.6.p4.3.m3.4.4.4.2.2.2.2">𝜉</ci><ci id="S6.SS2.13.6.p4.3.m3.4.4.4.2.2.2.3.cmml" xref="S6.SS2.13.6.p4.3.m3.4.4.4.2.2.2.3">𝑗</ci></apply></interval><ci id="S6.SS2.13.6.p4.3.m3.4.4.4.4.cmml" xref="S6.SS2.13.6.p4.3.m3.4.4.4.4">𝐴</ci></apply></apply><apply id="S6.SS2.13.6.p4.3.m3.8.8e.cmml" xref="S6.SS2.13.6.p4.3.m3.8.8"><subset id="S6.SS2.13.6.p4.3.m3.8.8.13.cmml" xref="S6.SS2.13.6.p4.3.m3.8.8.13"></subset><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.13.6.p4.3.m3.4.4.4.cmml" id="S6.SS2.13.6.p4.3.m3.8.8f.cmml" xref="S6.SS2.13.6.p4.3.m3.8.8"></share><apply id="S6.SS2.13.6.p4.3.m3.8.8.8.cmml" xref="S6.SS2.13.6.p4.3.m3.8.8.8"><union id="S6.SS2.13.6.p4.3.m3.8.8.8.5.cmml" xref="S6.SS2.13.6.p4.3.m3.8.8.8.5"></union><interval closure="closed" id="S6.SS2.13.6.p4.3.m3.6.6.6.2.3.cmml" xref="S6.SS2.13.6.p4.3.m3.6.6.6.2.2"><apply id="S6.SS2.13.6.p4.3.m3.5.5.5.1.1.1.cmml" xref="S6.SS2.13.6.p4.3.m3.5.5.5.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.13.6.p4.3.m3.5.5.5.1.1.1.1.cmml" xref="S6.SS2.13.6.p4.3.m3.5.5.5.1.1.1">subscript</csymbol><ci id="S6.SS2.13.6.p4.3.m3.5.5.5.1.1.1.2.cmml" xref="S6.SS2.13.6.p4.3.m3.5.5.5.1.1.1.2">𝑎</ci><ci id="S6.SS2.13.6.p4.3.m3.5.5.5.1.1.1.3.cmml" xref="S6.SS2.13.6.p4.3.m3.5.5.5.1.1.1.3">𝑙</ci></apply><apply id="S6.SS2.13.6.p4.3.m3.6.6.6.2.2.2.cmml" xref="S6.SS2.13.6.p4.3.m3.6.6.6.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.13.6.p4.3.m3.6.6.6.2.2.2.1.cmml" xref="S6.SS2.13.6.p4.3.m3.6.6.6.2.2.2">subscript</csymbol><ci id="S6.SS2.13.6.p4.3.m3.6.6.6.2.2.2.2.cmml" xref="S6.SS2.13.6.p4.3.m3.6.6.6.2.2.2.2">𝑎</ci><ci id="S6.SS2.13.6.p4.3.m3.6.6.6.2.2.2.3.cmml" xref="S6.SS2.13.6.p4.3.m3.6.6.6.2.2.2.3">𝑟</ci></apply></interval><interval closure="closed" id="S6.SS2.13.6.p4.3.m3.8.8.8.4.3.cmml" xref="S6.SS2.13.6.p4.3.m3.8.8.8.4.2"><apply id="S6.SS2.13.6.p4.3.m3.7.7.7.3.1.1.cmml" xref="S6.SS2.13.6.p4.3.m3.7.7.7.3.1.1"><csymbol cd="ambiguous" id="S6.SS2.13.6.p4.3.m3.7.7.7.3.1.1.1.cmml" xref="S6.SS2.13.6.p4.3.m3.7.7.7.3.1.1">subscript</csymbol><ci id="S6.SS2.13.6.p4.3.m3.7.7.7.3.1.1.2.cmml" xref="S6.SS2.13.6.p4.3.m3.7.7.7.3.1.1.2">𝑏</ci><ci id="S6.SS2.13.6.p4.3.m3.7.7.7.3.1.1.3.cmml" xref="S6.SS2.13.6.p4.3.m3.7.7.7.3.1.1.3">𝑙</ci></apply><apply id="S6.SS2.13.6.p4.3.m3.8.8.8.4.2.2.cmml" xref="S6.SS2.13.6.p4.3.m3.8.8.8.4.2.2"><csymbol cd="ambiguous" id="S6.SS2.13.6.p4.3.m3.8.8.8.4.2.2.1.cmml" xref="S6.SS2.13.6.p4.3.m3.8.8.8.4.2.2">subscript</csymbol><ci id="S6.SS2.13.6.p4.3.m3.8.8.8.4.2.2.2.cmml" xref="S6.SS2.13.6.p4.3.m3.8.8.8.4.2.2.2">𝑏</ci><ci id="S6.SS2.13.6.p4.3.m3.8.8.8.4.2.2.3.cmml" xref="S6.SS2.13.6.p4.3.m3.8.8.8.4.2.2.3">𝑟</ci></apply></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.13.6.p4.3.m3.8c">c:=\Delta_{A}(\xi_{i},\eta_{j})\in{[\xi_{i},\xi_{j}]}_{A}\subseteq[a_{l},a_{r}% ]\cup[b_{l},b_{r}]</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.13.6.p4.3.m3.8d">italic_c := roman_Δ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ξ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_η start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) ∈ [ italic_ξ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_ξ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ] start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⊆ [ italic_a start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ] ∪ [ italic_b start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ]</annotation></semantics></math>, so <math alttext="c\in[a_{l},a_{r}]" class="ltx_Math" display="inline" id="S6.SS2.13.6.p4.4.m4.2"><semantics id="S6.SS2.13.6.p4.4.m4.2a"><mrow id="S6.SS2.13.6.p4.4.m4.2.2" xref="S6.SS2.13.6.p4.4.m4.2.2.cmml"><mi id="S6.SS2.13.6.p4.4.m4.2.2.4" xref="S6.SS2.13.6.p4.4.m4.2.2.4.cmml">c</mi><mo id="S6.SS2.13.6.p4.4.m4.2.2.3" xref="S6.SS2.13.6.p4.4.m4.2.2.3.cmml">∈</mo><mrow id="S6.SS2.13.6.p4.4.m4.2.2.2.2" xref="S6.SS2.13.6.p4.4.m4.2.2.2.3.cmml"><mo id="S6.SS2.13.6.p4.4.m4.2.2.2.2.3" stretchy="false" xref="S6.SS2.13.6.p4.4.m4.2.2.2.3.cmml">[</mo><msub id="S6.SS2.13.6.p4.4.m4.1.1.1.1.1" xref="S6.SS2.13.6.p4.4.m4.1.1.1.1.1.cmml"><mi id="S6.SS2.13.6.p4.4.m4.1.1.1.1.1.2" xref="S6.SS2.13.6.p4.4.m4.1.1.1.1.1.2.cmml">a</mi><mi id="S6.SS2.13.6.p4.4.m4.1.1.1.1.1.3" xref="S6.SS2.13.6.p4.4.m4.1.1.1.1.1.3.cmml">l</mi></msub><mo id="S6.SS2.13.6.p4.4.m4.2.2.2.2.4" xref="S6.SS2.13.6.p4.4.m4.2.2.2.3.cmml">,</mo><msub id="S6.SS2.13.6.p4.4.m4.2.2.2.2.2" xref="S6.SS2.13.6.p4.4.m4.2.2.2.2.2.cmml"><mi id="S6.SS2.13.6.p4.4.m4.2.2.2.2.2.2" xref="S6.SS2.13.6.p4.4.m4.2.2.2.2.2.2.cmml">a</mi><mi id="S6.SS2.13.6.p4.4.m4.2.2.2.2.2.3" xref="S6.SS2.13.6.p4.4.m4.2.2.2.2.2.3.cmml">r</mi></msub><mo id="S6.SS2.13.6.p4.4.m4.2.2.2.2.5" stretchy="false" xref="S6.SS2.13.6.p4.4.m4.2.2.2.3.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.13.6.p4.4.m4.2b"><apply id="S6.SS2.13.6.p4.4.m4.2.2.cmml" xref="S6.SS2.13.6.p4.4.m4.2.2"><in id="S6.SS2.13.6.p4.4.m4.2.2.3.cmml" xref="S6.SS2.13.6.p4.4.m4.2.2.3"></in><ci id="S6.SS2.13.6.p4.4.m4.2.2.4.cmml" xref="S6.SS2.13.6.p4.4.m4.2.2.4">𝑐</ci><interval closure="closed" id="S6.SS2.13.6.p4.4.m4.2.2.2.3.cmml" xref="S6.SS2.13.6.p4.4.m4.2.2.2.2"><apply id="S6.SS2.13.6.p4.4.m4.1.1.1.1.1.cmml" xref="S6.SS2.13.6.p4.4.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.13.6.p4.4.m4.1.1.1.1.1.1.cmml" xref="S6.SS2.13.6.p4.4.m4.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.13.6.p4.4.m4.1.1.1.1.1.2.cmml" xref="S6.SS2.13.6.p4.4.m4.1.1.1.1.1.2">𝑎</ci><ci id="S6.SS2.13.6.p4.4.m4.1.1.1.1.1.3.cmml" xref="S6.SS2.13.6.p4.4.m4.1.1.1.1.1.3">𝑙</ci></apply><apply id="S6.SS2.13.6.p4.4.m4.2.2.2.2.2.cmml" xref="S6.SS2.13.6.p4.4.m4.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.13.6.p4.4.m4.2.2.2.2.2.1.cmml" xref="S6.SS2.13.6.p4.4.m4.2.2.2.2.2">subscript</csymbol><ci id="S6.SS2.13.6.p4.4.m4.2.2.2.2.2.2.cmml" xref="S6.SS2.13.6.p4.4.m4.2.2.2.2.2.2">𝑎</ci><ci id="S6.SS2.13.6.p4.4.m4.2.2.2.2.2.3.cmml" xref="S6.SS2.13.6.p4.4.m4.2.2.2.2.2.3">𝑟</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.13.6.p4.4.m4.2c">c\in[a_{l},a_{r}]</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.13.6.p4.4.m4.2d">italic_c ∈ [ italic_a start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ]</annotation></semantics></math> or <math alttext="c\in[b_{l},b_{r}]" class="ltx_Math" display="inline" id="S6.SS2.13.6.p4.5.m5.2"><semantics id="S6.SS2.13.6.p4.5.m5.2a"><mrow id="S6.SS2.13.6.p4.5.m5.2.2" xref="S6.SS2.13.6.p4.5.m5.2.2.cmml"><mi id="S6.SS2.13.6.p4.5.m5.2.2.4" xref="S6.SS2.13.6.p4.5.m5.2.2.4.cmml">c</mi><mo id="S6.SS2.13.6.p4.5.m5.2.2.3" xref="S6.SS2.13.6.p4.5.m5.2.2.3.cmml">∈</mo><mrow id="S6.SS2.13.6.p4.5.m5.2.2.2.2" xref="S6.SS2.13.6.p4.5.m5.2.2.2.3.cmml"><mo id="S6.SS2.13.6.p4.5.m5.2.2.2.2.3" stretchy="false" xref="S6.SS2.13.6.p4.5.m5.2.2.2.3.cmml">[</mo><msub id="S6.SS2.13.6.p4.5.m5.1.1.1.1.1" xref="S6.SS2.13.6.p4.5.m5.1.1.1.1.1.cmml"><mi id="S6.SS2.13.6.p4.5.m5.1.1.1.1.1.2" xref="S6.SS2.13.6.p4.5.m5.1.1.1.1.1.2.cmml">b</mi><mi id="S6.SS2.13.6.p4.5.m5.1.1.1.1.1.3" xref="S6.SS2.13.6.p4.5.m5.1.1.1.1.1.3.cmml">l</mi></msub><mo id="S6.SS2.13.6.p4.5.m5.2.2.2.2.4" xref="S6.SS2.13.6.p4.5.m5.2.2.2.3.cmml">,</mo><msub id="S6.SS2.13.6.p4.5.m5.2.2.2.2.2" xref="S6.SS2.13.6.p4.5.m5.2.2.2.2.2.cmml"><mi id="S6.SS2.13.6.p4.5.m5.2.2.2.2.2.2" xref="S6.SS2.13.6.p4.5.m5.2.2.2.2.2.2.cmml">b</mi><mi id="S6.SS2.13.6.p4.5.m5.2.2.2.2.2.3" xref="S6.SS2.13.6.p4.5.m5.2.2.2.2.2.3.cmml">r</mi></msub><mo id="S6.SS2.13.6.p4.5.m5.2.2.2.2.5" stretchy="false" xref="S6.SS2.13.6.p4.5.m5.2.2.2.3.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.13.6.p4.5.m5.2b"><apply id="S6.SS2.13.6.p4.5.m5.2.2.cmml" xref="S6.SS2.13.6.p4.5.m5.2.2"><in id="S6.SS2.13.6.p4.5.m5.2.2.3.cmml" xref="S6.SS2.13.6.p4.5.m5.2.2.3"></in><ci id="S6.SS2.13.6.p4.5.m5.2.2.4.cmml" xref="S6.SS2.13.6.p4.5.m5.2.2.4">𝑐</ci><interval closure="closed" id="S6.SS2.13.6.p4.5.m5.2.2.2.3.cmml" xref="S6.SS2.13.6.p4.5.m5.2.2.2.2"><apply id="S6.SS2.13.6.p4.5.m5.1.1.1.1.1.cmml" xref="S6.SS2.13.6.p4.5.m5.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.13.6.p4.5.m5.1.1.1.1.1.1.cmml" xref="S6.SS2.13.6.p4.5.m5.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.13.6.p4.5.m5.1.1.1.1.1.2.cmml" xref="S6.SS2.13.6.p4.5.m5.1.1.1.1.1.2">𝑏</ci><ci id="S6.SS2.13.6.p4.5.m5.1.1.1.1.1.3.cmml" xref="S6.SS2.13.6.p4.5.m5.1.1.1.1.1.3">𝑙</ci></apply><apply id="S6.SS2.13.6.p4.5.m5.2.2.2.2.2.cmml" xref="S6.SS2.13.6.p4.5.m5.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.13.6.p4.5.m5.2.2.2.2.2.1.cmml" xref="S6.SS2.13.6.p4.5.m5.2.2.2.2.2">subscript</csymbol><ci id="S6.SS2.13.6.p4.5.m5.2.2.2.2.2.2.cmml" xref="S6.SS2.13.6.p4.5.m5.2.2.2.2.2.2">𝑏</ci><ci id="S6.SS2.13.6.p4.5.m5.2.2.2.2.2.3.cmml" xref="S6.SS2.13.6.p4.5.m5.2.2.2.2.2.3">𝑟</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.13.6.p4.5.m5.2c">c\in[b_{l},b_{r}]</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.13.6.p4.5.m5.2d">italic_c ∈ [ italic_b start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ]</annotation></semantics></math> (or both). Either way we arrive at a contradiction since <math alttext="c=\Delta_{A}(\xi_{i},\eta_{j})<\xi" class="ltx_Math" display="inline" id="S6.SS2.13.6.p4.6.m6.2"><semantics id="S6.SS2.13.6.p4.6.m6.2a"><mrow id="S6.SS2.13.6.p4.6.m6.2.2" xref="S6.SS2.13.6.p4.6.m6.2.2.cmml"><mi id="S6.SS2.13.6.p4.6.m6.2.2.4" xref="S6.SS2.13.6.p4.6.m6.2.2.4.cmml">c</mi><mo id="S6.SS2.13.6.p4.6.m6.2.2.5" xref="S6.SS2.13.6.p4.6.m6.2.2.5.cmml">=</mo><mrow id="S6.SS2.13.6.p4.6.m6.2.2.2" xref="S6.SS2.13.6.p4.6.m6.2.2.2.cmml"><msub id="S6.SS2.13.6.p4.6.m6.2.2.2.4" xref="S6.SS2.13.6.p4.6.m6.2.2.2.4.cmml"><mi id="S6.SS2.13.6.p4.6.m6.2.2.2.4.2" mathvariant="normal" xref="S6.SS2.13.6.p4.6.m6.2.2.2.4.2.cmml">Δ</mi><mi id="S6.SS2.13.6.p4.6.m6.2.2.2.4.3" xref="S6.SS2.13.6.p4.6.m6.2.2.2.4.3.cmml">A</mi></msub><mo id="S6.SS2.13.6.p4.6.m6.2.2.2.3" xref="S6.SS2.13.6.p4.6.m6.2.2.2.3.cmml">⁢</mo><mrow id="S6.SS2.13.6.p4.6.m6.2.2.2.2.2" xref="S6.SS2.13.6.p4.6.m6.2.2.2.2.3.cmml"><mo id="S6.SS2.13.6.p4.6.m6.2.2.2.2.2.3" stretchy="false" xref="S6.SS2.13.6.p4.6.m6.2.2.2.2.3.cmml">(</mo><msub id="S6.SS2.13.6.p4.6.m6.1.1.1.1.1.1" xref="S6.SS2.13.6.p4.6.m6.1.1.1.1.1.1.cmml"><mi id="S6.SS2.13.6.p4.6.m6.1.1.1.1.1.1.2" xref="S6.SS2.13.6.p4.6.m6.1.1.1.1.1.1.2.cmml">ξ</mi><mi id="S6.SS2.13.6.p4.6.m6.1.1.1.1.1.1.3" xref="S6.SS2.13.6.p4.6.m6.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S6.SS2.13.6.p4.6.m6.2.2.2.2.2.4" xref="S6.SS2.13.6.p4.6.m6.2.2.2.2.3.cmml">,</mo><msub id="S6.SS2.13.6.p4.6.m6.2.2.2.2.2.2" xref="S6.SS2.13.6.p4.6.m6.2.2.2.2.2.2.cmml"><mi id="S6.SS2.13.6.p4.6.m6.2.2.2.2.2.2.2" xref="S6.SS2.13.6.p4.6.m6.2.2.2.2.2.2.2.cmml">η</mi><mi id="S6.SS2.13.6.p4.6.m6.2.2.2.2.2.2.3" xref="S6.SS2.13.6.p4.6.m6.2.2.2.2.2.2.3.cmml">j</mi></msub><mo id="S6.SS2.13.6.p4.6.m6.2.2.2.2.2.5" stretchy="false" xref="S6.SS2.13.6.p4.6.m6.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.SS2.13.6.p4.6.m6.2.2.6" xref="S6.SS2.13.6.p4.6.m6.2.2.6.cmml"><</mo><mi id="S6.SS2.13.6.p4.6.m6.2.2.7" xref="S6.SS2.13.6.p4.6.m6.2.2.7.cmml">ξ</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.13.6.p4.6.m6.2b"><apply id="S6.SS2.13.6.p4.6.m6.2.2.cmml" xref="S6.SS2.13.6.p4.6.m6.2.2"><and id="S6.SS2.13.6.p4.6.m6.2.2a.cmml" xref="S6.SS2.13.6.p4.6.m6.2.2"></and><apply id="S6.SS2.13.6.p4.6.m6.2.2b.cmml" xref="S6.SS2.13.6.p4.6.m6.2.2"><eq id="S6.SS2.13.6.p4.6.m6.2.2.5.cmml" xref="S6.SS2.13.6.p4.6.m6.2.2.5"></eq><ci id="S6.SS2.13.6.p4.6.m6.2.2.4.cmml" xref="S6.SS2.13.6.p4.6.m6.2.2.4">𝑐</ci><apply id="S6.SS2.13.6.p4.6.m6.2.2.2.cmml" xref="S6.SS2.13.6.p4.6.m6.2.2.2"><times id="S6.SS2.13.6.p4.6.m6.2.2.2.3.cmml" xref="S6.SS2.13.6.p4.6.m6.2.2.2.3"></times><apply id="S6.SS2.13.6.p4.6.m6.2.2.2.4.cmml" xref="S6.SS2.13.6.p4.6.m6.2.2.2.4"><csymbol cd="ambiguous" id="S6.SS2.13.6.p4.6.m6.2.2.2.4.1.cmml" xref="S6.SS2.13.6.p4.6.m6.2.2.2.4">subscript</csymbol><ci id="S6.SS2.13.6.p4.6.m6.2.2.2.4.2.cmml" xref="S6.SS2.13.6.p4.6.m6.2.2.2.4.2">Δ</ci><ci id="S6.SS2.13.6.p4.6.m6.2.2.2.4.3.cmml" xref="S6.SS2.13.6.p4.6.m6.2.2.2.4.3">𝐴</ci></apply><interval closure="open" id="S6.SS2.13.6.p4.6.m6.2.2.2.2.3.cmml" xref="S6.SS2.13.6.p4.6.m6.2.2.2.2.2"><apply id="S6.SS2.13.6.p4.6.m6.1.1.1.1.1.1.cmml" xref="S6.SS2.13.6.p4.6.m6.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.13.6.p4.6.m6.1.1.1.1.1.1.1.cmml" xref="S6.SS2.13.6.p4.6.m6.1.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.13.6.p4.6.m6.1.1.1.1.1.1.2.cmml" xref="S6.SS2.13.6.p4.6.m6.1.1.1.1.1.1.2">𝜉</ci><ci id="S6.SS2.13.6.p4.6.m6.1.1.1.1.1.1.3.cmml" xref="S6.SS2.13.6.p4.6.m6.1.1.1.1.1.1.3">𝑖</ci></apply><apply id="S6.SS2.13.6.p4.6.m6.2.2.2.2.2.2.cmml" xref="S6.SS2.13.6.p4.6.m6.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.13.6.p4.6.m6.2.2.2.2.2.2.1.cmml" xref="S6.SS2.13.6.p4.6.m6.2.2.2.2.2.2">subscript</csymbol><ci id="S6.SS2.13.6.p4.6.m6.2.2.2.2.2.2.2.cmml" xref="S6.SS2.13.6.p4.6.m6.2.2.2.2.2.2.2">𝜂</ci><ci id="S6.SS2.13.6.p4.6.m6.2.2.2.2.2.2.3.cmml" xref="S6.SS2.13.6.p4.6.m6.2.2.2.2.2.2.3">𝑗</ci></apply></interval></apply></apply><apply id="S6.SS2.13.6.p4.6.m6.2.2c.cmml" xref="S6.SS2.13.6.p4.6.m6.2.2"><lt id="S6.SS2.13.6.p4.6.m6.2.2.6.cmml" xref="S6.SS2.13.6.p4.6.m6.2.2.6"></lt><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.13.6.p4.6.m6.2.2.2.cmml" id="S6.SS2.13.6.p4.6.m6.2.2d.cmml" xref="S6.SS2.13.6.p4.6.m6.2.2"></share><ci id="S6.SS2.13.6.p4.6.m6.2.2.7.cmml" xref="S6.SS2.13.6.p4.6.m6.2.2.7">𝜉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.13.6.p4.6.m6.2c">c=\Delta_{A}(\xi_{i},\eta_{j})<\xi</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.13.6.p4.6.m6.2d">italic_c = roman_Δ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ξ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_η start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) < italic_ξ</annotation></semantics></math>, and (a) implies that <math alttext="\Delta_{A}(a_{l},a_{r})=a_{m}\geq\xi" class="ltx_Math" display="inline" id="S6.SS2.13.6.p4.7.m7.2"><semantics id="S6.SS2.13.6.p4.7.m7.2a"><mrow id="S6.SS2.13.6.p4.7.m7.2.2" xref="S6.SS2.13.6.p4.7.m7.2.2.cmml"><mrow id="S6.SS2.13.6.p4.7.m7.2.2.2" xref="S6.SS2.13.6.p4.7.m7.2.2.2.cmml"><msub id="S6.SS2.13.6.p4.7.m7.2.2.2.4" xref="S6.SS2.13.6.p4.7.m7.2.2.2.4.cmml"><mi id="S6.SS2.13.6.p4.7.m7.2.2.2.4.2" mathvariant="normal" xref="S6.SS2.13.6.p4.7.m7.2.2.2.4.2.cmml">Δ</mi><mi id="S6.SS2.13.6.p4.7.m7.2.2.2.4.3" xref="S6.SS2.13.6.p4.7.m7.2.2.2.4.3.cmml">A</mi></msub><mo id="S6.SS2.13.6.p4.7.m7.2.2.2.3" xref="S6.SS2.13.6.p4.7.m7.2.2.2.3.cmml">⁢</mo><mrow id="S6.SS2.13.6.p4.7.m7.2.2.2.2.2" xref="S6.SS2.13.6.p4.7.m7.2.2.2.2.3.cmml"><mo id="S6.SS2.13.6.p4.7.m7.2.2.2.2.2.3" stretchy="false" xref="S6.SS2.13.6.p4.7.m7.2.2.2.2.3.cmml">(</mo><msub id="S6.SS2.13.6.p4.7.m7.1.1.1.1.1.1" xref="S6.SS2.13.6.p4.7.m7.1.1.1.1.1.1.cmml"><mi id="S6.SS2.13.6.p4.7.m7.1.1.1.1.1.1.2" xref="S6.SS2.13.6.p4.7.m7.1.1.1.1.1.1.2.cmml">a</mi><mi id="S6.SS2.13.6.p4.7.m7.1.1.1.1.1.1.3" xref="S6.SS2.13.6.p4.7.m7.1.1.1.1.1.1.3.cmml">l</mi></msub><mo id="S6.SS2.13.6.p4.7.m7.2.2.2.2.2.4" xref="S6.SS2.13.6.p4.7.m7.2.2.2.2.3.cmml">,</mo><msub id="S6.SS2.13.6.p4.7.m7.2.2.2.2.2.2" xref="S6.SS2.13.6.p4.7.m7.2.2.2.2.2.2.cmml"><mi id="S6.SS2.13.6.p4.7.m7.2.2.2.2.2.2.2" xref="S6.SS2.13.6.p4.7.m7.2.2.2.2.2.2.2.cmml">a</mi><mi id="S6.SS2.13.6.p4.7.m7.2.2.2.2.2.2.3" xref="S6.SS2.13.6.p4.7.m7.2.2.2.2.2.2.3.cmml">r</mi></msub><mo id="S6.SS2.13.6.p4.7.m7.2.2.2.2.2.5" stretchy="false" xref="S6.SS2.13.6.p4.7.m7.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.SS2.13.6.p4.7.m7.2.2.4" xref="S6.SS2.13.6.p4.7.m7.2.2.4.cmml">=</mo><msub id="S6.SS2.13.6.p4.7.m7.2.2.5" xref="S6.SS2.13.6.p4.7.m7.2.2.5.cmml"><mi id="S6.SS2.13.6.p4.7.m7.2.2.5.2" xref="S6.SS2.13.6.p4.7.m7.2.2.5.2.cmml">a</mi><mi id="S6.SS2.13.6.p4.7.m7.2.2.5.3" xref="S6.SS2.13.6.p4.7.m7.2.2.5.3.cmml">m</mi></msub><mo id="S6.SS2.13.6.p4.7.m7.2.2.6" xref="S6.SS2.13.6.p4.7.m7.2.2.6.cmml">≥</mo><mi id="S6.SS2.13.6.p4.7.m7.2.2.7" xref="S6.SS2.13.6.p4.7.m7.2.2.7.cmml">ξ</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.13.6.p4.7.m7.2b"><apply id="S6.SS2.13.6.p4.7.m7.2.2.cmml" xref="S6.SS2.13.6.p4.7.m7.2.2"><and id="S6.SS2.13.6.p4.7.m7.2.2a.cmml" xref="S6.SS2.13.6.p4.7.m7.2.2"></and><apply id="S6.SS2.13.6.p4.7.m7.2.2b.cmml" xref="S6.SS2.13.6.p4.7.m7.2.2"><eq id="S6.SS2.13.6.p4.7.m7.2.2.4.cmml" xref="S6.SS2.13.6.p4.7.m7.2.2.4"></eq><apply id="S6.SS2.13.6.p4.7.m7.2.2.2.cmml" xref="S6.SS2.13.6.p4.7.m7.2.2.2"><times id="S6.SS2.13.6.p4.7.m7.2.2.2.3.cmml" xref="S6.SS2.13.6.p4.7.m7.2.2.2.3"></times><apply id="S6.SS2.13.6.p4.7.m7.2.2.2.4.cmml" xref="S6.SS2.13.6.p4.7.m7.2.2.2.4"><csymbol cd="ambiguous" id="S6.SS2.13.6.p4.7.m7.2.2.2.4.1.cmml" xref="S6.SS2.13.6.p4.7.m7.2.2.2.4">subscript</csymbol><ci id="S6.SS2.13.6.p4.7.m7.2.2.2.4.2.cmml" xref="S6.SS2.13.6.p4.7.m7.2.2.2.4.2">Δ</ci><ci id="S6.SS2.13.6.p4.7.m7.2.2.2.4.3.cmml" xref="S6.SS2.13.6.p4.7.m7.2.2.2.4.3">𝐴</ci></apply><interval closure="open" id="S6.SS2.13.6.p4.7.m7.2.2.2.2.3.cmml" xref="S6.SS2.13.6.p4.7.m7.2.2.2.2.2"><apply id="S6.SS2.13.6.p4.7.m7.1.1.1.1.1.1.cmml" xref="S6.SS2.13.6.p4.7.m7.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.13.6.p4.7.m7.1.1.1.1.1.1.1.cmml" xref="S6.SS2.13.6.p4.7.m7.1.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.13.6.p4.7.m7.1.1.1.1.1.1.2.cmml" xref="S6.SS2.13.6.p4.7.m7.1.1.1.1.1.1.2">𝑎</ci><ci id="S6.SS2.13.6.p4.7.m7.1.1.1.1.1.1.3.cmml" xref="S6.SS2.13.6.p4.7.m7.1.1.1.1.1.1.3">𝑙</ci></apply><apply id="S6.SS2.13.6.p4.7.m7.2.2.2.2.2.2.cmml" xref="S6.SS2.13.6.p4.7.m7.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.13.6.p4.7.m7.2.2.2.2.2.2.1.cmml" xref="S6.SS2.13.6.p4.7.m7.2.2.2.2.2.2">subscript</csymbol><ci id="S6.SS2.13.6.p4.7.m7.2.2.2.2.2.2.2.cmml" xref="S6.SS2.13.6.p4.7.m7.2.2.2.2.2.2.2">𝑎</ci><ci id="S6.SS2.13.6.p4.7.m7.2.2.2.2.2.2.3.cmml" xref="S6.SS2.13.6.p4.7.m7.2.2.2.2.2.2.3">𝑟</ci></apply></interval></apply><apply id="S6.SS2.13.6.p4.7.m7.2.2.5.cmml" xref="S6.SS2.13.6.p4.7.m7.2.2.5"><csymbol cd="ambiguous" id="S6.SS2.13.6.p4.7.m7.2.2.5.1.cmml" xref="S6.SS2.13.6.p4.7.m7.2.2.5">subscript</csymbol><ci id="S6.SS2.13.6.p4.7.m7.2.2.5.2.cmml" xref="S6.SS2.13.6.p4.7.m7.2.2.5.2">𝑎</ci><ci id="S6.SS2.13.6.p4.7.m7.2.2.5.3.cmml" xref="S6.SS2.13.6.p4.7.m7.2.2.5.3">𝑚</ci></apply></apply><apply id="S6.SS2.13.6.p4.7.m7.2.2c.cmml" xref="S6.SS2.13.6.p4.7.m7.2.2"><geq id="S6.SS2.13.6.p4.7.m7.2.2.6.cmml" xref="S6.SS2.13.6.p4.7.m7.2.2.6"></geq><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.13.6.p4.7.m7.2.2.5.cmml" id="S6.SS2.13.6.p4.7.m7.2.2d.cmml" xref="S6.SS2.13.6.p4.7.m7.2.2"></share><ci id="S6.SS2.13.6.p4.7.m7.2.2.7.cmml" xref="S6.SS2.13.6.p4.7.m7.2.2.7">𝜉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.13.6.p4.7.m7.2c">\Delta_{A}(a_{l},a_{r})=a_{m}\geq\xi</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.13.6.p4.7.m7.2d">roman_Δ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) = italic_a start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ≥ italic_ξ</annotation></semantics></math> and <math alttext="\Delta_{A}(b_{l},b_{r})=b_{m}\geq\eta>\xi" class="ltx_Math" display="inline" id="S6.SS2.13.6.p4.8.m8.2"><semantics id="S6.SS2.13.6.p4.8.m8.2a"><mrow id="S6.SS2.13.6.p4.8.m8.2.2" xref="S6.SS2.13.6.p4.8.m8.2.2.cmml"><mrow id="S6.SS2.13.6.p4.8.m8.2.2.2" xref="S6.SS2.13.6.p4.8.m8.2.2.2.cmml"><msub id="S6.SS2.13.6.p4.8.m8.2.2.2.4" xref="S6.SS2.13.6.p4.8.m8.2.2.2.4.cmml"><mi id="S6.SS2.13.6.p4.8.m8.2.2.2.4.2" mathvariant="normal" xref="S6.SS2.13.6.p4.8.m8.2.2.2.4.2.cmml">Δ</mi><mi id="S6.SS2.13.6.p4.8.m8.2.2.2.4.3" xref="S6.SS2.13.6.p4.8.m8.2.2.2.4.3.cmml">A</mi></msub><mo id="S6.SS2.13.6.p4.8.m8.2.2.2.3" xref="S6.SS2.13.6.p4.8.m8.2.2.2.3.cmml">⁢</mo><mrow id="S6.SS2.13.6.p4.8.m8.2.2.2.2.2" xref="S6.SS2.13.6.p4.8.m8.2.2.2.2.3.cmml"><mo id="S6.SS2.13.6.p4.8.m8.2.2.2.2.2.3" stretchy="false" xref="S6.SS2.13.6.p4.8.m8.2.2.2.2.3.cmml">(</mo><msub id="S6.SS2.13.6.p4.8.m8.1.1.1.1.1.1" xref="S6.SS2.13.6.p4.8.m8.1.1.1.1.1.1.cmml"><mi id="S6.SS2.13.6.p4.8.m8.1.1.1.1.1.1.2" xref="S6.SS2.13.6.p4.8.m8.1.1.1.1.1.1.2.cmml">b</mi><mi id="S6.SS2.13.6.p4.8.m8.1.1.1.1.1.1.3" xref="S6.SS2.13.6.p4.8.m8.1.1.1.1.1.1.3.cmml">l</mi></msub><mo id="S6.SS2.13.6.p4.8.m8.2.2.2.2.2.4" xref="S6.SS2.13.6.p4.8.m8.2.2.2.2.3.cmml">,</mo><msub id="S6.SS2.13.6.p4.8.m8.2.2.2.2.2.2" xref="S6.SS2.13.6.p4.8.m8.2.2.2.2.2.2.cmml"><mi id="S6.SS2.13.6.p4.8.m8.2.2.2.2.2.2.2" xref="S6.SS2.13.6.p4.8.m8.2.2.2.2.2.2.2.cmml">b</mi><mi id="S6.SS2.13.6.p4.8.m8.2.2.2.2.2.2.3" xref="S6.SS2.13.6.p4.8.m8.2.2.2.2.2.2.3.cmml">r</mi></msub><mo id="S6.SS2.13.6.p4.8.m8.2.2.2.2.2.5" stretchy="false" xref="S6.SS2.13.6.p4.8.m8.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.SS2.13.6.p4.8.m8.2.2.4" xref="S6.SS2.13.6.p4.8.m8.2.2.4.cmml">=</mo><msub id="S6.SS2.13.6.p4.8.m8.2.2.5" xref="S6.SS2.13.6.p4.8.m8.2.2.5.cmml"><mi id="S6.SS2.13.6.p4.8.m8.2.2.5.2" xref="S6.SS2.13.6.p4.8.m8.2.2.5.2.cmml">b</mi><mi id="S6.SS2.13.6.p4.8.m8.2.2.5.3" xref="S6.SS2.13.6.p4.8.m8.2.2.5.3.cmml">m</mi></msub><mo id="S6.SS2.13.6.p4.8.m8.2.2.6" xref="S6.SS2.13.6.p4.8.m8.2.2.6.cmml">≥</mo><mi id="S6.SS2.13.6.p4.8.m8.2.2.7" xref="S6.SS2.13.6.p4.8.m8.2.2.7.cmml">η</mi><mo id="S6.SS2.13.6.p4.8.m8.2.2.8" xref="S6.SS2.13.6.p4.8.m8.2.2.8.cmml">></mo><mi id="S6.SS2.13.6.p4.8.m8.2.2.9" xref="S6.SS2.13.6.p4.8.m8.2.2.9.cmml">ξ</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.13.6.p4.8.m8.2b"><apply id="S6.SS2.13.6.p4.8.m8.2.2.cmml" xref="S6.SS2.13.6.p4.8.m8.2.2"><and id="S6.SS2.13.6.p4.8.m8.2.2a.cmml" xref="S6.SS2.13.6.p4.8.m8.2.2"></and><apply id="S6.SS2.13.6.p4.8.m8.2.2b.cmml" xref="S6.SS2.13.6.p4.8.m8.2.2"><eq id="S6.SS2.13.6.p4.8.m8.2.2.4.cmml" xref="S6.SS2.13.6.p4.8.m8.2.2.4"></eq><apply id="S6.SS2.13.6.p4.8.m8.2.2.2.cmml" xref="S6.SS2.13.6.p4.8.m8.2.2.2"><times id="S6.SS2.13.6.p4.8.m8.2.2.2.3.cmml" xref="S6.SS2.13.6.p4.8.m8.2.2.2.3"></times><apply id="S6.SS2.13.6.p4.8.m8.2.2.2.4.cmml" xref="S6.SS2.13.6.p4.8.m8.2.2.2.4"><csymbol cd="ambiguous" id="S6.SS2.13.6.p4.8.m8.2.2.2.4.1.cmml" xref="S6.SS2.13.6.p4.8.m8.2.2.2.4">subscript</csymbol><ci id="S6.SS2.13.6.p4.8.m8.2.2.2.4.2.cmml" xref="S6.SS2.13.6.p4.8.m8.2.2.2.4.2">Δ</ci><ci id="S6.SS2.13.6.p4.8.m8.2.2.2.4.3.cmml" xref="S6.SS2.13.6.p4.8.m8.2.2.2.4.3">𝐴</ci></apply><interval closure="open" id="S6.SS2.13.6.p4.8.m8.2.2.2.2.3.cmml" xref="S6.SS2.13.6.p4.8.m8.2.2.2.2.2"><apply id="S6.SS2.13.6.p4.8.m8.1.1.1.1.1.1.cmml" xref="S6.SS2.13.6.p4.8.m8.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.13.6.p4.8.m8.1.1.1.1.1.1.1.cmml" xref="S6.SS2.13.6.p4.8.m8.1.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.13.6.p4.8.m8.1.1.1.1.1.1.2.cmml" xref="S6.SS2.13.6.p4.8.m8.1.1.1.1.1.1.2">𝑏</ci><ci id="S6.SS2.13.6.p4.8.m8.1.1.1.1.1.1.3.cmml" xref="S6.SS2.13.6.p4.8.m8.1.1.1.1.1.1.3">𝑙</ci></apply><apply id="S6.SS2.13.6.p4.8.m8.2.2.2.2.2.2.cmml" xref="S6.SS2.13.6.p4.8.m8.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.13.6.p4.8.m8.2.2.2.2.2.2.1.cmml" xref="S6.SS2.13.6.p4.8.m8.2.2.2.2.2.2">subscript</csymbol><ci id="S6.SS2.13.6.p4.8.m8.2.2.2.2.2.2.2.cmml" xref="S6.SS2.13.6.p4.8.m8.2.2.2.2.2.2.2">𝑏</ci><ci id="S6.SS2.13.6.p4.8.m8.2.2.2.2.2.2.3.cmml" xref="S6.SS2.13.6.p4.8.m8.2.2.2.2.2.2.3">𝑟</ci></apply></interval></apply><apply id="S6.SS2.13.6.p4.8.m8.2.2.5.cmml" xref="S6.SS2.13.6.p4.8.m8.2.2.5"><csymbol cd="ambiguous" id="S6.SS2.13.6.p4.8.m8.2.2.5.1.cmml" xref="S6.SS2.13.6.p4.8.m8.2.2.5">subscript</csymbol><ci id="S6.SS2.13.6.p4.8.m8.2.2.5.2.cmml" xref="S6.SS2.13.6.p4.8.m8.2.2.5.2">𝑏</ci><ci id="S6.SS2.13.6.p4.8.m8.2.2.5.3.cmml" xref="S6.SS2.13.6.p4.8.m8.2.2.5.3">𝑚</ci></apply></apply><apply id="S6.SS2.13.6.p4.8.m8.2.2c.cmml" xref="S6.SS2.13.6.p4.8.m8.2.2"><geq id="S6.SS2.13.6.p4.8.m8.2.2.6.cmml" xref="S6.SS2.13.6.p4.8.m8.2.2.6"></geq><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.13.6.p4.8.m8.2.2.5.cmml" id="S6.SS2.13.6.p4.8.m8.2.2d.cmml" xref="S6.SS2.13.6.p4.8.m8.2.2"></share><ci id="S6.SS2.13.6.p4.8.m8.2.2.7.cmml" xref="S6.SS2.13.6.p4.8.m8.2.2.7">𝜂</ci></apply><apply id="S6.SS2.13.6.p4.8.m8.2.2e.cmml" xref="S6.SS2.13.6.p4.8.m8.2.2"><gt id="S6.SS2.13.6.p4.8.m8.2.2.8.cmml" xref="S6.SS2.13.6.p4.8.m8.2.2.8"></gt><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.13.6.p4.8.m8.2.2.7.cmml" id="S6.SS2.13.6.p4.8.m8.2.2f.cmml" xref="S6.SS2.13.6.p4.8.m8.2.2"></share><ci id="S6.SS2.13.6.p4.8.m8.2.2.9.cmml" xref="S6.SS2.13.6.p4.8.m8.2.2.9">𝜉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.13.6.p4.8.m8.2c">\Delta_{A}(b_{l},b_{r})=b_{m}\geq\eta>\xi</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.13.6.p4.8.m8.2d">roman_Δ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_b start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) = italic_b start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ≥ italic_η > italic_ξ</annotation></semantics></math>. ∎</p> </div> </div> <div class="ltx_para" id="S6.SS2.14.p3"> <p class="ltx_p" id="S6.SS2.14.p3.1">∎</p> </div> </div> <div class="ltx_para" id="S6.SS2.p8"> <p class="ltx_p" id="S6.SS2.p8.1">We dedicate the rest of this section to show that the <math alttext="E" class="ltx_Math" display="inline" id="S6.SS2.p8.1.m1.1"><semantics id="S6.SS2.p8.1.m1.1a"><mi id="S6.SS2.p8.1.m1.1.1" xref="S6.SS2.p8.1.m1.1.1.cmml">E</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.p8.1.m1.1b"><ci id="S6.SS2.p8.1.m1.1.1.cmml" xref="S6.SS2.p8.1.m1.1.1">𝐸</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p8.1.m1.1c">E</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p8.1.m1.1d">italic_E</annotation></semantics></math> from <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.Thmtheorem10" title="Lemma 6.10. ‣ 6.2. Forcing epimorphisms ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">6.10</span></a> makes all the needed sets dense. For this we only use properties (2) and (3) (in fact (3) follows from (2)).</p> </div> <div class="ltx_para" id="S6.SS2.p9"> <p class="ltx_p" id="S6.SS2.p9.3">Say that <math alttext="\bar{a},\bar{b}\in\operatorname{dom}(p)" class="ltx_Math" display="inline" id="S6.SS2.p9.1.m1.4"><semantics id="S6.SS2.p9.1.m1.4a"><mrow id="S6.SS2.p9.1.m1.4.5" xref="S6.SS2.p9.1.m1.4.5.cmml"><mrow id="S6.SS2.p9.1.m1.4.5.2.2" xref="S6.SS2.p9.1.m1.4.5.2.1.cmml"><mover accent="true" id="S6.SS2.p9.1.m1.3.3" xref="S6.SS2.p9.1.m1.3.3.cmml"><mi id="S6.SS2.p9.1.m1.3.3.2" xref="S6.SS2.p9.1.m1.3.3.2.cmml">a</mi><mo id="S6.SS2.p9.1.m1.3.3.1" xref="S6.SS2.p9.1.m1.3.3.1.cmml">¯</mo></mover><mo id="S6.SS2.p9.1.m1.4.5.2.2.1" xref="S6.SS2.p9.1.m1.4.5.2.1.cmml">,</mo><mover accent="true" id="S6.SS2.p9.1.m1.4.4" xref="S6.SS2.p9.1.m1.4.4.cmml"><mi id="S6.SS2.p9.1.m1.4.4.2" xref="S6.SS2.p9.1.m1.4.4.2.cmml">b</mi><mo id="S6.SS2.p9.1.m1.4.4.1" xref="S6.SS2.p9.1.m1.4.4.1.cmml">¯</mo></mover></mrow><mo id="S6.SS2.p9.1.m1.4.5.1" xref="S6.SS2.p9.1.m1.4.5.1.cmml">∈</mo><mrow id="S6.SS2.p9.1.m1.4.5.3.2" xref="S6.SS2.p9.1.m1.4.5.3.1.cmml"><mi id="S6.SS2.p9.1.m1.1.1" xref="S6.SS2.p9.1.m1.1.1.cmml">dom</mi><mo id="S6.SS2.p9.1.m1.4.5.3.2a" xref="S6.SS2.p9.1.m1.4.5.3.1.cmml">⁡</mo><mrow id="S6.SS2.p9.1.m1.4.5.3.2.1" xref="S6.SS2.p9.1.m1.4.5.3.1.cmml"><mo id="S6.SS2.p9.1.m1.4.5.3.2.1.1" stretchy="false" xref="S6.SS2.p9.1.m1.4.5.3.1.cmml">(</mo><mi id="S6.SS2.p9.1.m1.2.2" xref="S6.SS2.p9.1.m1.2.2.cmml">p</mi><mo id="S6.SS2.p9.1.m1.4.5.3.2.1.2" stretchy="false" xref="S6.SS2.p9.1.m1.4.5.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p9.1.m1.4b"><apply id="S6.SS2.p9.1.m1.4.5.cmml" xref="S6.SS2.p9.1.m1.4.5"><in id="S6.SS2.p9.1.m1.4.5.1.cmml" xref="S6.SS2.p9.1.m1.4.5.1"></in><list id="S6.SS2.p9.1.m1.4.5.2.1.cmml" xref="S6.SS2.p9.1.m1.4.5.2.2"><apply id="S6.SS2.p9.1.m1.3.3.cmml" xref="S6.SS2.p9.1.m1.3.3"><ci id="S6.SS2.p9.1.m1.3.3.1.cmml" xref="S6.SS2.p9.1.m1.3.3.1">¯</ci><ci id="S6.SS2.p9.1.m1.3.3.2.cmml" xref="S6.SS2.p9.1.m1.3.3.2">𝑎</ci></apply><apply id="S6.SS2.p9.1.m1.4.4.cmml" xref="S6.SS2.p9.1.m1.4.4"><ci id="S6.SS2.p9.1.m1.4.4.1.cmml" xref="S6.SS2.p9.1.m1.4.4.1">¯</ci><ci id="S6.SS2.p9.1.m1.4.4.2.cmml" xref="S6.SS2.p9.1.m1.4.4.2">𝑏</ci></apply></list><apply id="S6.SS2.p9.1.m1.4.5.3.1.cmml" xref="S6.SS2.p9.1.m1.4.5.3.2"><ci id="S6.SS2.p9.1.m1.1.1.cmml" xref="S6.SS2.p9.1.m1.1.1">dom</ci><ci id="S6.SS2.p9.1.m1.2.2.cmml" xref="S6.SS2.p9.1.m1.2.2">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p9.1.m1.4c">\bar{a},\bar{b}\in\operatorname{dom}(p)</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p9.1.m1.4d">over¯ start_ARG italic_a end_ARG , over¯ start_ARG italic_b end_ARG ∈ roman_dom ( italic_p )</annotation></semantics></math> are <em class="ltx_emph ltx_font_italic" id="S6.SS2.p9.3.1">neighbors</em> if there is no <math alttext="\bar{c}\in\operatorname{dom}(p)" class="ltx_Math" display="inline" id="S6.SS2.p9.2.m2.2"><semantics id="S6.SS2.p9.2.m2.2a"><mrow id="S6.SS2.p9.2.m2.2.3" xref="S6.SS2.p9.2.m2.2.3.cmml"><mover accent="true" id="S6.SS2.p9.2.m2.2.3.2" xref="S6.SS2.p9.2.m2.2.3.2.cmml"><mi id="S6.SS2.p9.2.m2.2.3.2.2" xref="S6.SS2.p9.2.m2.2.3.2.2.cmml">c</mi><mo id="S6.SS2.p9.2.m2.2.3.2.1" xref="S6.SS2.p9.2.m2.2.3.2.1.cmml">¯</mo></mover><mo id="S6.SS2.p9.2.m2.2.3.1" xref="S6.SS2.p9.2.m2.2.3.1.cmml">∈</mo><mrow id="S6.SS2.p9.2.m2.2.3.3.2" xref="S6.SS2.p9.2.m2.2.3.3.1.cmml"><mi id="S6.SS2.p9.2.m2.1.1" xref="S6.SS2.p9.2.m2.1.1.cmml">dom</mi><mo id="S6.SS2.p9.2.m2.2.3.3.2a" xref="S6.SS2.p9.2.m2.2.3.3.1.cmml">⁡</mo><mrow id="S6.SS2.p9.2.m2.2.3.3.2.1" xref="S6.SS2.p9.2.m2.2.3.3.1.cmml"><mo id="S6.SS2.p9.2.m2.2.3.3.2.1.1" stretchy="false" xref="S6.SS2.p9.2.m2.2.3.3.1.cmml">(</mo><mi id="S6.SS2.p9.2.m2.2.2" xref="S6.SS2.p9.2.m2.2.2.cmml">p</mi><mo id="S6.SS2.p9.2.m2.2.3.3.2.1.2" stretchy="false" xref="S6.SS2.p9.2.m2.2.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p9.2.m2.2b"><apply id="S6.SS2.p9.2.m2.2.3.cmml" xref="S6.SS2.p9.2.m2.2.3"><in id="S6.SS2.p9.2.m2.2.3.1.cmml" xref="S6.SS2.p9.2.m2.2.3.1"></in><apply id="S6.SS2.p9.2.m2.2.3.2.cmml" xref="S6.SS2.p9.2.m2.2.3.2"><ci id="S6.SS2.p9.2.m2.2.3.2.1.cmml" xref="S6.SS2.p9.2.m2.2.3.2.1">¯</ci><ci id="S6.SS2.p9.2.m2.2.3.2.2.cmml" xref="S6.SS2.p9.2.m2.2.3.2.2">𝑐</ci></apply><apply id="S6.SS2.p9.2.m2.2.3.3.1.cmml" xref="S6.SS2.p9.2.m2.2.3.3.2"><ci id="S6.SS2.p9.2.m2.1.1.cmml" xref="S6.SS2.p9.2.m2.1.1">dom</ci><ci id="S6.SS2.p9.2.m2.2.2.cmml" xref="S6.SS2.p9.2.m2.2.2">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p9.2.m2.2c">\bar{c}\in\operatorname{dom}(p)</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p9.2.m2.2d">over¯ start_ARG italic_c end_ARG ∈ roman_dom ( italic_p )</annotation></semantics></math> strictly between them. When checking that new conditions are in <math alttext="P_{E}" class="ltx_Math" display="inline" id="S6.SS2.p9.3.m3.1"><semantics id="S6.SS2.p9.3.m3.1a"><msub id="S6.SS2.p9.3.m3.1.1" xref="S6.SS2.p9.3.m3.1.1.cmml"><mi id="S6.SS2.p9.3.m3.1.1.2" xref="S6.SS2.p9.3.m3.1.1.2.cmml">P</mi><mi id="S6.SS2.p9.3.m3.1.1.3" xref="S6.SS2.p9.3.m3.1.1.3.cmml">E</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.p9.3.m3.1b"><apply id="S6.SS2.p9.3.m3.1.1.cmml" xref="S6.SS2.p9.3.m3.1.1"><csymbol cd="ambiguous" id="S6.SS2.p9.3.m3.1.1.1.cmml" xref="S6.SS2.p9.3.m3.1.1">subscript</csymbol><ci id="S6.SS2.p9.3.m3.1.1.2.cmml" xref="S6.SS2.p9.3.m3.1.1.2">𝑃</ci><ci id="S6.SS2.p9.3.m3.1.1.3.cmml" xref="S6.SS2.p9.3.m3.1.1.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p9.3.m3.1c">P_{E}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p9.3.m3.1d">italic_P start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT</annotation></semantics></math> it will be useful to have the following.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S6.Thmtheorem15"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem15.1.1.1">Lemma 6.15</span></span><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem15.2.2">.</span> </h6> <div class="ltx_para" id="S6.Thmtheorem15.p1"> <p class="ltx_p" id="S6.Thmtheorem15.p1.5">Let <math alttext="p\in P" class="ltx_Math" display="inline" id="S6.Thmtheorem15.p1.1.m1.1"><semantics id="S6.Thmtheorem15.p1.1.m1.1a"><mrow id="S6.Thmtheorem15.p1.1.m1.1.1" xref="S6.Thmtheorem15.p1.1.m1.1.1.cmml"><mi id="S6.Thmtheorem15.p1.1.m1.1.1.2" xref="S6.Thmtheorem15.p1.1.m1.1.1.2.cmml">p</mi><mo id="S6.Thmtheorem15.p1.1.m1.1.1.1" xref="S6.Thmtheorem15.p1.1.m1.1.1.1.cmml">∈</mo><mi id="S6.Thmtheorem15.p1.1.m1.1.1.3" xref="S6.Thmtheorem15.p1.1.m1.1.1.3.cmml">P</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem15.p1.1.m1.1b"><apply id="S6.Thmtheorem15.p1.1.m1.1.1.cmml" xref="S6.Thmtheorem15.p1.1.m1.1.1"><in id="S6.Thmtheorem15.p1.1.m1.1.1.1.cmml" xref="S6.Thmtheorem15.p1.1.m1.1.1.1"></in><ci id="S6.Thmtheorem15.p1.1.m1.1.1.2.cmml" xref="S6.Thmtheorem15.p1.1.m1.1.1.2">𝑝</ci><ci id="S6.Thmtheorem15.p1.1.m1.1.1.3.cmml" xref="S6.Thmtheorem15.p1.1.m1.1.1.3">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem15.p1.1.m1.1c">p\in P</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem15.p1.1.m1.1d">italic_p ∈ italic_P</annotation></semantics></math> satisfy <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.Thmtheorem9" title="Definition 6.9. ‣ 6.2. Forcing epimorphisms ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">6.9</span></a> <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.I4.i1" title="Item (i) ‣ Definition 6.9. ‣ 6.2. Forcing epimorphisms ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">(i)</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.Thmtheorem9" title="Definition 6.9. ‣ 6.2. Forcing epimorphisms ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">6.9</span></a> <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.I4.i3" title="Item (iii) ‣ Definition 6.9. ‣ 6.2. Forcing epimorphisms ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">(iii)</span></a>. If for all neighbors <math alttext="\bar{a}<_{b}\bar{b}" class="ltx_Math" display="inline" id="S6.Thmtheorem15.p1.2.m2.1"><semantics id="S6.Thmtheorem15.p1.2.m2.1a"><mrow id="S6.Thmtheorem15.p1.2.m2.1.1" xref="S6.Thmtheorem15.p1.2.m2.1.1.cmml"><mover accent="true" id="S6.Thmtheorem15.p1.2.m2.1.1.2" xref="S6.Thmtheorem15.p1.2.m2.1.1.2.cmml"><mi id="S6.Thmtheorem15.p1.2.m2.1.1.2.2" xref="S6.Thmtheorem15.p1.2.m2.1.1.2.2.cmml">a</mi><mo id="S6.Thmtheorem15.p1.2.m2.1.1.2.1" xref="S6.Thmtheorem15.p1.2.m2.1.1.2.1.cmml">¯</mo></mover><msub id="S6.Thmtheorem15.p1.2.m2.1.1.1" xref="S6.Thmtheorem15.p1.2.m2.1.1.1.cmml"><mo id="S6.Thmtheorem15.p1.2.m2.1.1.1.2" xref="S6.Thmtheorem15.p1.2.m2.1.1.1.2.cmml"><</mo><mi id="S6.Thmtheorem15.p1.2.m2.1.1.1.3" xref="S6.Thmtheorem15.p1.2.m2.1.1.1.3.cmml">b</mi></msub><mover accent="true" id="S6.Thmtheorem15.p1.2.m2.1.1.3" xref="S6.Thmtheorem15.p1.2.m2.1.1.3.cmml"><mi id="S6.Thmtheorem15.p1.2.m2.1.1.3.2" xref="S6.Thmtheorem15.p1.2.m2.1.1.3.2.cmml">b</mi><mo id="S6.Thmtheorem15.p1.2.m2.1.1.3.1" xref="S6.Thmtheorem15.p1.2.m2.1.1.3.1.cmml">¯</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem15.p1.2.m2.1b"><apply id="S6.Thmtheorem15.p1.2.m2.1.1.cmml" xref="S6.Thmtheorem15.p1.2.m2.1.1"><apply id="S6.Thmtheorem15.p1.2.m2.1.1.1.cmml" xref="S6.Thmtheorem15.p1.2.m2.1.1.1"><csymbol cd="ambiguous" id="S6.Thmtheorem15.p1.2.m2.1.1.1.1.cmml" xref="S6.Thmtheorem15.p1.2.m2.1.1.1">subscript</csymbol><lt id="S6.Thmtheorem15.p1.2.m2.1.1.1.2.cmml" xref="S6.Thmtheorem15.p1.2.m2.1.1.1.2"></lt><ci id="S6.Thmtheorem15.p1.2.m2.1.1.1.3.cmml" xref="S6.Thmtheorem15.p1.2.m2.1.1.1.3">𝑏</ci></apply><apply id="S6.Thmtheorem15.p1.2.m2.1.1.2.cmml" xref="S6.Thmtheorem15.p1.2.m2.1.1.2"><ci id="S6.Thmtheorem15.p1.2.m2.1.1.2.1.cmml" xref="S6.Thmtheorem15.p1.2.m2.1.1.2.1">¯</ci><ci id="S6.Thmtheorem15.p1.2.m2.1.1.2.2.cmml" xref="S6.Thmtheorem15.p1.2.m2.1.1.2.2">𝑎</ci></apply><apply id="S6.Thmtheorem15.p1.2.m2.1.1.3.cmml" xref="S6.Thmtheorem15.p1.2.m2.1.1.3"><ci id="S6.Thmtheorem15.p1.2.m2.1.1.3.1.cmml" xref="S6.Thmtheorem15.p1.2.m2.1.1.3.1">¯</ci><ci id="S6.Thmtheorem15.p1.2.m2.1.1.3.2.cmml" xref="S6.Thmtheorem15.p1.2.m2.1.1.3.2">𝑏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem15.p1.2.m2.1c">\bar{a}<_{b}\bar{b}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem15.p1.2.m2.1d">over¯ start_ARG italic_a end_ARG < start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT over¯ start_ARG italic_b end_ARG</annotation></semantics></math> in <math alttext="\operatorname{dom}(p)" class="ltx_Math" display="inline" id="S6.Thmtheorem15.p1.3.m3.2"><semantics id="S6.Thmtheorem15.p1.3.m3.2a"><mrow id="S6.Thmtheorem15.p1.3.m3.2.3.2" xref="S6.Thmtheorem15.p1.3.m3.2.3.1.cmml"><mi id="S6.Thmtheorem15.p1.3.m3.1.1" xref="S6.Thmtheorem15.p1.3.m3.1.1.cmml">dom</mi><mo id="S6.Thmtheorem15.p1.3.m3.2.3.2a" xref="S6.Thmtheorem15.p1.3.m3.2.3.1.cmml">⁡</mo><mrow id="S6.Thmtheorem15.p1.3.m3.2.3.2.1" xref="S6.Thmtheorem15.p1.3.m3.2.3.1.cmml"><mo id="S6.Thmtheorem15.p1.3.m3.2.3.2.1.1" stretchy="false" xref="S6.Thmtheorem15.p1.3.m3.2.3.1.cmml">(</mo><mi id="S6.Thmtheorem15.p1.3.m3.2.2" xref="S6.Thmtheorem15.p1.3.m3.2.2.cmml">p</mi><mo id="S6.Thmtheorem15.p1.3.m3.2.3.2.1.2" stretchy="false" xref="S6.Thmtheorem15.p1.3.m3.2.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem15.p1.3.m3.2b"><apply id="S6.Thmtheorem15.p1.3.m3.2.3.1.cmml" xref="S6.Thmtheorem15.p1.3.m3.2.3.2"><ci id="S6.Thmtheorem15.p1.3.m3.1.1.cmml" xref="S6.Thmtheorem15.p1.3.m3.1.1">dom</ci><ci id="S6.Thmtheorem15.p1.3.m3.2.2.cmml" xref="S6.Thmtheorem15.p1.3.m3.2.2">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem15.p1.3.m3.2c">\operatorname{dom}(p)</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem15.p1.3.m3.2d">roman_dom ( italic_p )</annotation></semantics></math>, <math alttext="\nu(a_{r},b_{l})=\nu(p(\bar{a}),p(\bar{b}))" class="ltx_Math" display="inline" id="S6.Thmtheorem15.p1.4.m4.6"><semantics id="S6.Thmtheorem15.p1.4.m4.6a"><mrow id="S6.Thmtheorem15.p1.4.m4.6.6" xref="S6.Thmtheorem15.p1.4.m4.6.6.cmml"><mrow id="S6.Thmtheorem15.p1.4.m4.4.4.2" xref="S6.Thmtheorem15.p1.4.m4.4.4.2.cmml"><mi id="S6.Thmtheorem15.p1.4.m4.4.4.2.4" xref="S6.Thmtheorem15.p1.4.m4.4.4.2.4.cmml">ν</mi><mo id="S6.Thmtheorem15.p1.4.m4.4.4.2.3" xref="S6.Thmtheorem15.p1.4.m4.4.4.2.3.cmml">⁢</mo><mrow id="S6.Thmtheorem15.p1.4.m4.4.4.2.2.2" xref="S6.Thmtheorem15.p1.4.m4.4.4.2.2.3.cmml"><mo id="S6.Thmtheorem15.p1.4.m4.4.4.2.2.2.3" stretchy="false" xref="S6.Thmtheorem15.p1.4.m4.4.4.2.2.3.cmml">(</mo><msub id="S6.Thmtheorem15.p1.4.m4.3.3.1.1.1.1" xref="S6.Thmtheorem15.p1.4.m4.3.3.1.1.1.1.cmml"><mi id="S6.Thmtheorem15.p1.4.m4.3.3.1.1.1.1.2" xref="S6.Thmtheorem15.p1.4.m4.3.3.1.1.1.1.2.cmml">a</mi><mi id="S6.Thmtheorem15.p1.4.m4.3.3.1.1.1.1.3" xref="S6.Thmtheorem15.p1.4.m4.3.3.1.1.1.1.3.cmml">r</mi></msub><mo id="S6.Thmtheorem15.p1.4.m4.4.4.2.2.2.4" xref="S6.Thmtheorem15.p1.4.m4.4.4.2.2.3.cmml">,</mo><msub id="S6.Thmtheorem15.p1.4.m4.4.4.2.2.2.2" xref="S6.Thmtheorem15.p1.4.m4.4.4.2.2.2.2.cmml"><mi id="S6.Thmtheorem15.p1.4.m4.4.4.2.2.2.2.2" xref="S6.Thmtheorem15.p1.4.m4.4.4.2.2.2.2.2.cmml">b</mi><mi id="S6.Thmtheorem15.p1.4.m4.4.4.2.2.2.2.3" xref="S6.Thmtheorem15.p1.4.m4.4.4.2.2.2.2.3.cmml">l</mi></msub><mo id="S6.Thmtheorem15.p1.4.m4.4.4.2.2.2.5" stretchy="false" xref="S6.Thmtheorem15.p1.4.m4.4.4.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.Thmtheorem15.p1.4.m4.6.6.5" xref="S6.Thmtheorem15.p1.4.m4.6.6.5.cmml">=</mo><mrow id="S6.Thmtheorem15.p1.4.m4.6.6.4" xref="S6.Thmtheorem15.p1.4.m4.6.6.4.cmml"><mi id="S6.Thmtheorem15.p1.4.m4.6.6.4.4" xref="S6.Thmtheorem15.p1.4.m4.6.6.4.4.cmml">ν</mi><mo id="S6.Thmtheorem15.p1.4.m4.6.6.4.3" xref="S6.Thmtheorem15.p1.4.m4.6.6.4.3.cmml">⁢</mo><mrow id="S6.Thmtheorem15.p1.4.m4.6.6.4.2.2" xref="S6.Thmtheorem15.p1.4.m4.6.6.4.2.3.cmml"><mo id="S6.Thmtheorem15.p1.4.m4.6.6.4.2.2.3" stretchy="false" xref="S6.Thmtheorem15.p1.4.m4.6.6.4.2.3.cmml">(</mo><mrow id="S6.Thmtheorem15.p1.4.m4.5.5.3.1.1.1" xref="S6.Thmtheorem15.p1.4.m4.5.5.3.1.1.1.cmml"><mi id="S6.Thmtheorem15.p1.4.m4.5.5.3.1.1.1.2" xref="S6.Thmtheorem15.p1.4.m4.5.5.3.1.1.1.2.cmml">p</mi><mo id="S6.Thmtheorem15.p1.4.m4.5.5.3.1.1.1.1" xref="S6.Thmtheorem15.p1.4.m4.5.5.3.1.1.1.1.cmml">⁢</mo><mrow id="S6.Thmtheorem15.p1.4.m4.5.5.3.1.1.1.3.2" xref="S6.Thmtheorem15.p1.4.m4.1.1.cmml"><mo id="S6.Thmtheorem15.p1.4.m4.5.5.3.1.1.1.3.2.1" stretchy="false" xref="S6.Thmtheorem15.p1.4.m4.1.1.cmml">(</mo><mover accent="true" id="S6.Thmtheorem15.p1.4.m4.1.1" xref="S6.Thmtheorem15.p1.4.m4.1.1.cmml"><mi id="S6.Thmtheorem15.p1.4.m4.1.1.2" xref="S6.Thmtheorem15.p1.4.m4.1.1.2.cmml">a</mi><mo id="S6.Thmtheorem15.p1.4.m4.1.1.1" xref="S6.Thmtheorem15.p1.4.m4.1.1.1.cmml">¯</mo></mover><mo id="S6.Thmtheorem15.p1.4.m4.5.5.3.1.1.1.3.2.2" stretchy="false" xref="S6.Thmtheorem15.p1.4.m4.1.1.cmml">)</mo></mrow></mrow><mo id="S6.Thmtheorem15.p1.4.m4.6.6.4.2.2.4" xref="S6.Thmtheorem15.p1.4.m4.6.6.4.2.3.cmml">,</mo><mrow id="S6.Thmtheorem15.p1.4.m4.6.6.4.2.2.2" xref="S6.Thmtheorem15.p1.4.m4.6.6.4.2.2.2.cmml"><mi id="S6.Thmtheorem15.p1.4.m4.6.6.4.2.2.2.2" xref="S6.Thmtheorem15.p1.4.m4.6.6.4.2.2.2.2.cmml">p</mi><mo id="S6.Thmtheorem15.p1.4.m4.6.6.4.2.2.2.1" xref="S6.Thmtheorem15.p1.4.m4.6.6.4.2.2.2.1.cmml">⁢</mo><mrow id="S6.Thmtheorem15.p1.4.m4.6.6.4.2.2.2.3.2" xref="S6.Thmtheorem15.p1.4.m4.2.2.cmml"><mo id="S6.Thmtheorem15.p1.4.m4.6.6.4.2.2.2.3.2.1" stretchy="false" xref="S6.Thmtheorem15.p1.4.m4.2.2.cmml">(</mo><mover accent="true" id="S6.Thmtheorem15.p1.4.m4.2.2" xref="S6.Thmtheorem15.p1.4.m4.2.2.cmml"><mi id="S6.Thmtheorem15.p1.4.m4.2.2.2" xref="S6.Thmtheorem15.p1.4.m4.2.2.2.cmml">b</mi><mo id="S6.Thmtheorem15.p1.4.m4.2.2.1" xref="S6.Thmtheorem15.p1.4.m4.2.2.1.cmml">¯</mo></mover><mo id="S6.Thmtheorem15.p1.4.m4.6.6.4.2.2.2.3.2.2" stretchy="false" xref="S6.Thmtheorem15.p1.4.m4.2.2.cmml">)</mo></mrow></mrow><mo id="S6.Thmtheorem15.p1.4.m4.6.6.4.2.2.5" stretchy="false" xref="S6.Thmtheorem15.p1.4.m4.6.6.4.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem15.p1.4.m4.6b"><apply id="S6.Thmtheorem15.p1.4.m4.6.6.cmml" xref="S6.Thmtheorem15.p1.4.m4.6.6"><eq id="S6.Thmtheorem15.p1.4.m4.6.6.5.cmml" xref="S6.Thmtheorem15.p1.4.m4.6.6.5"></eq><apply id="S6.Thmtheorem15.p1.4.m4.4.4.2.cmml" xref="S6.Thmtheorem15.p1.4.m4.4.4.2"><times id="S6.Thmtheorem15.p1.4.m4.4.4.2.3.cmml" xref="S6.Thmtheorem15.p1.4.m4.4.4.2.3"></times><ci id="S6.Thmtheorem15.p1.4.m4.4.4.2.4.cmml" xref="S6.Thmtheorem15.p1.4.m4.4.4.2.4">𝜈</ci><interval closure="open" id="S6.Thmtheorem15.p1.4.m4.4.4.2.2.3.cmml" xref="S6.Thmtheorem15.p1.4.m4.4.4.2.2.2"><apply id="S6.Thmtheorem15.p1.4.m4.3.3.1.1.1.1.cmml" xref="S6.Thmtheorem15.p1.4.m4.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S6.Thmtheorem15.p1.4.m4.3.3.1.1.1.1.1.cmml" xref="S6.Thmtheorem15.p1.4.m4.3.3.1.1.1.1">subscript</csymbol><ci id="S6.Thmtheorem15.p1.4.m4.3.3.1.1.1.1.2.cmml" xref="S6.Thmtheorem15.p1.4.m4.3.3.1.1.1.1.2">𝑎</ci><ci id="S6.Thmtheorem15.p1.4.m4.3.3.1.1.1.1.3.cmml" xref="S6.Thmtheorem15.p1.4.m4.3.3.1.1.1.1.3">𝑟</ci></apply><apply id="S6.Thmtheorem15.p1.4.m4.4.4.2.2.2.2.cmml" xref="S6.Thmtheorem15.p1.4.m4.4.4.2.2.2.2"><csymbol cd="ambiguous" id="S6.Thmtheorem15.p1.4.m4.4.4.2.2.2.2.1.cmml" xref="S6.Thmtheorem15.p1.4.m4.4.4.2.2.2.2">subscript</csymbol><ci id="S6.Thmtheorem15.p1.4.m4.4.4.2.2.2.2.2.cmml" xref="S6.Thmtheorem15.p1.4.m4.4.4.2.2.2.2.2">𝑏</ci><ci id="S6.Thmtheorem15.p1.4.m4.4.4.2.2.2.2.3.cmml" xref="S6.Thmtheorem15.p1.4.m4.4.4.2.2.2.2.3">𝑙</ci></apply></interval></apply><apply id="S6.Thmtheorem15.p1.4.m4.6.6.4.cmml" xref="S6.Thmtheorem15.p1.4.m4.6.6.4"><times id="S6.Thmtheorem15.p1.4.m4.6.6.4.3.cmml" xref="S6.Thmtheorem15.p1.4.m4.6.6.4.3"></times><ci id="S6.Thmtheorem15.p1.4.m4.6.6.4.4.cmml" xref="S6.Thmtheorem15.p1.4.m4.6.6.4.4">𝜈</ci><interval closure="open" id="S6.Thmtheorem15.p1.4.m4.6.6.4.2.3.cmml" xref="S6.Thmtheorem15.p1.4.m4.6.6.4.2.2"><apply id="S6.Thmtheorem15.p1.4.m4.5.5.3.1.1.1.cmml" xref="S6.Thmtheorem15.p1.4.m4.5.5.3.1.1.1"><times id="S6.Thmtheorem15.p1.4.m4.5.5.3.1.1.1.1.cmml" xref="S6.Thmtheorem15.p1.4.m4.5.5.3.1.1.1.1"></times><ci id="S6.Thmtheorem15.p1.4.m4.5.5.3.1.1.1.2.cmml" xref="S6.Thmtheorem15.p1.4.m4.5.5.3.1.1.1.2">𝑝</ci><apply id="S6.Thmtheorem15.p1.4.m4.1.1.cmml" xref="S6.Thmtheorem15.p1.4.m4.5.5.3.1.1.1.3.2"><ci id="S6.Thmtheorem15.p1.4.m4.1.1.1.cmml" xref="S6.Thmtheorem15.p1.4.m4.1.1.1">¯</ci><ci id="S6.Thmtheorem15.p1.4.m4.1.1.2.cmml" xref="S6.Thmtheorem15.p1.4.m4.1.1.2">𝑎</ci></apply></apply><apply id="S6.Thmtheorem15.p1.4.m4.6.6.4.2.2.2.cmml" xref="S6.Thmtheorem15.p1.4.m4.6.6.4.2.2.2"><times id="S6.Thmtheorem15.p1.4.m4.6.6.4.2.2.2.1.cmml" xref="S6.Thmtheorem15.p1.4.m4.6.6.4.2.2.2.1"></times><ci id="S6.Thmtheorem15.p1.4.m4.6.6.4.2.2.2.2.cmml" xref="S6.Thmtheorem15.p1.4.m4.6.6.4.2.2.2.2">𝑝</ci><apply id="S6.Thmtheorem15.p1.4.m4.2.2.cmml" xref="S6.Thmtheorem15.p1.4.m4.6.6.4.2.2.2.3.2"><ci id="S6.Thmtheorem15.p1.4.m4.2.2.1.cmml" xref="S6.Thmtheorem15.p1.4.m4.2.2.1">¯</ci><ci id="S6.Thmtheorem15.p1.4.m4.2.2.2.cmml" xref="S6.Thmtheorem15.p1.4.m4.2.2.2">𝑏</ci></apply></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem15.p1.4.m4.6c">\nu(a_{r},b_{l})=\nu(p(\bar{a}),p(\bar{b}))</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem15.p1.4.m4.6d">italic_ν ( italic_a start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ) = italic_ν ( italic_p ( over¯ start_ARG italic_a end_ARG ) , italic_p ( over¯ start_ARG italic_b end_ARG ) )</annotation></semantics></math>, then <math alttext="p\in P_{E}" class="ltx_Math" display="inline" id="S6.Thmtheorem15.p1.5.m5.1"><semantics id="S6.Thmtheorem15.p1.5.m5.1a"><mrow id="S6.Thmtheorem15.p1.5.m5.1.1" xref="S6.Thmtheorem15.p1.5.m5.1.1.cmml"><mi id="S6.Thmtheorem15.p1.5.m5.1.1.2" xref="S6.Thmtheorem15.p1.5.m5.1.1.2.cmml">p</mi><mo id="S6.Thmtheorem15.p1.5.m5.1.1.1" xref="S6.Thmtheorem15.p1.5.m5.1.1.1.cmml">∈</mo><msub id="S6.Thmtheorem15.p1.5.m5.1.1.3" xref="S6.Thmtheorem15.p1.5.m5.1.1.3.cmml"><mi id="S6.Thmtheorem15.p1.5.m5.1.1.3.2" xref="S6.Thmtheorem15.p1.5.m5.1.1.3.2.cmml">P</mi><mi id="S6.Thmtheorem15.p1.5.m5.1.1.3.3" xref="S6.Thmtheorem15.p1.5.m5.1.1.3.3.cmml">E</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem15.p1.5.m5.1b"><apply id="S6.Thmtheorem15.p1.5.m5.1.1.cmml" xref="S6.Thmtheorem15.p1.5.m5.1.1"><in id="S6.Thmtheorem15.p1.5.m5.1.1.1.cmml" xref="S6.Thmtheorem15.p1.5.m5.1.1.1"></in><ci id="S6.Thmtheorem15.p1.5.m5.1.1.2.cmml" xref="S6.Thmtheorem15.p1.5.m5.1.1.2">𝑝</ci><apply id="S6.Thmtheorem15.p1.5.m5.1.1.3.cmml" xref="S6.Thmtheorem15.p1.5.m5.1.1.3"><csymbol cd="ambiguous" id="S6.Thmtheorem15.p1.5.m5.1.1.3.1.cmml" xref="S6.Thmtheorem15.p1.5.m5.1.1.3">subscript</csymbol><ci id="S6.Thmtheorem15.p1.5.m5.1.1.3.2.cmml" xref="S6.Thmtheorem15.p1.5.m5.1.1.3.2">𝑃</ci><ci id="S6.Thmtheorem15.p1.5.m5.1.1.3.3.cmml" xref="S6.Thmtheorem15.p1.5.m5.1.1.3.3">𝐸</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem15.p1.5.m5.1c">p\in P_{E}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem15.p1.5.m5.1d">italic_p ∈ italic_P start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> </div> <div class="ltx_proof" id="S6.SS2.16"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S6.SS2.15.p1"> <p class="ltx_p" id="S6.SS2.15.p1.7">It is enough to prove that if <math alttext="\bar{a}<_{b}\bar{b}<_{b}\bar{c}" class="ltx_Math" display="inline" id="S6.SS2.15.p1.1.m1.1"><semantics id="S6.SS2.15.p1.1.m1.1a"><mrow id="S6.SS2.15.p1.1.m1.1.1" xref="S6.SS2.15.p1.1.m1.1.1.cmml"><mover accent="true" id="S6.SS2.15.p1.1.m1.1.1.2" xref="S6.SS2.15.p1.1.m1.1.1.2.cmml"><mi id="S6.SS2.15.p1.1.m1.1.1.2.2" xref="S6.SS2.15.p1.1.m1.1.1.2.2.cmml">a</mi><mo id="S6.SS2.15.p1.1.m1.1.1.2.1" xref="S6.SS2.15.p1.1.m1.1.1.2.1.cmml">¯</mo></mover><msub id="S6.SS2.15.p1.1.m1.1.1.3" xref="S6.SS2.15.p1.1.m1.1.1.3.cmml"><mo id="S6.SS2.15.p1.1.m1.1.1.3.2" xref="S6.SS2.15.p1.1.m1.1.1.3.2.cmml"><</mo><mi id="S6.SS2.15.p1.1.m1.1.1.3.3" xref="S6.SS2.15.p1.1.m1.1.1.3.3.cmml">b</mi></msub><mover accent="true" id="S6.SS2.15.p1.1.m1.1.1.4" xref="S6.SS2.15.p1.1.m1.1.1.4.cmml"><mi id="S6.SS2.15.p1.1.m1.1.1.4.2" xref="S6.SS2.15.p1.1.m1.1.1.4.2.cmml">b</mi><mo id="S6.SS2.15.p1.1.m1.1.1.4.1" xref="S6.SS2.15.p1.1.m1.1.1.4.1.cmml">¯</mo></mover><msub id="S6.SS2.15.p1.1.m1.1.1.5" xref="S6.SS2.15.p1.1.m1.1.1.5.cmml"><mo id="S6.SS2.15.p1.1.m1.1.1.5.2" xref="S6.SS2.15.p1.1.m1.1.1.5.2.cmml"><</mo><mi id="S6.SS2.15.p1.1.m1.1.1.5.3" xref="S6.SS2.15.p1.1.m1.1.1.5.3.cmml">b</mi></msub><mover accent="true" id="S6.SS2.15.p1.1.m1.1.1.6" xref="S6.SS2.15.p1.1.m1.1.1.6.cmml"><mi id="S6.SS2.15.p1.1.m1.1.1.6.2" xref="S6.SS2.15.p1.1.m1.1.1.6.2.cmml">c</mi><mo id="S6.SS2.15.p1.1.m1.1.1.6.1" xref="S6.SS2.15.p1.1.m1.1.1.6.1.cmml">¯</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.15.p1.1.m1.1b"><apply id="S6.SS2.15.p1.1.m1.1.1.cmml" xref="S6.SS2.15.p1.1.m1.1.1"><and id="S6.SS2.15.p1.1.m1.1.1a.cmml" xref="S6.SS2.15.p1.1.m1.1.1"></and><apply id="S6.SS2.15.p1.1.m1.1.1b.cmml" xref="S6.SS2.15.p1.1.m1.1.1"><apply id="S6.SS2.15.p1.1.m1.1.1.3.cmml" xref="S6.SS2.15.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S6.SS2.15.p1.1.m1.1.1.3.1.cmml" xref="S6.SS2.15.p1.1.m1.1.1.3">subscript</csymbol><lt id="S6.SS2.15.p1.1.m1.1.1.3.2.cmml" xref="S6.SS2.15.p1.1.m1.1.1.3.2"></lt><ci id="S6.SS2.15.p1.1.m1.1.1.3.3.cmml" xref="S6.SS2.15.p1.1.m1.1.1.3.3">𝑏</ci></apply><apply id="S6.SS2.15.p1.1.m1.1.1.2.cmml" xref="S6.SS2.15.p1.1.m1.1.1.2"><ci id="S6.SS2.15.p1.1.m1.1.1.2.1.cmml" xref="S6.SS2.15.p1.1.m1.1.1.2.1">¯</ci><ci id="S6.SS2.15.p1.1.m1.1.1.2.2.cmml" xref="S6.SS2.15.p1.1.m1.1.1.2.2">𝑎</ci></apply><apply id="S6.SS2.15.p1.1.m1.1.1.4.cmml" xref="S6.SS2.15.p1.1.m1.1.1.4"><ci id="S6.SS2.15.p1.1.m1.1.1.4.1.cmml" xref="S6.SS2.15.p1.1.m1.1.1.4.1">¯</ci><ci id="S6.SS2.15.p1.1.m1.1.1.4.2.cmml" xref="S6.SS2.15.p1.1.m1.1.1.4.2">𝑏</ci></apply></apply><apply id="S6.SS2.15.p1.1.m1.1.1c.cmml" xref="S6.SS2.15.p1.1.m1.1.1"><apply id="S6.SS2.15.p1.1.m1.1.1.5.cmml" xref="S6.SS2.15.p1.1.m1.1.1.5"><csymbol cd="ambiguous" id="S6.SS2.15.p1.1.m1.1.1.5.1.cmml" xref="S6.SS2.15.p1.1.m1.1.1.5">subscript</csymbol><lt id="S6.SS2.15.p1.1.m1.1.1.5.2.cmml" xref="S6.SS2.15.p1.1.m1.1.1.5.2"></lt><ci id="S6.SS2.15.p1.1.m1.1.1.5.3.cmml" xref="S6.SS2.15.p1.1.m1.1.1.5.3">𝑏</ci></apply><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.15.p1.1.m1.1.1.4.cmml" id="S6.SS2.15.p1.1.m1.1.1d.cmml" xref="S6.SS2.15.p1.1.m1.1.1"></share><apply id="S6.SS2.15.p1.1.m1.1.1.6.cmml" xref="S6.SS2.15.p1.1.m1.1.1.6"><ci id="S6.SS2.15.p1.1.m1.1.1.6.1.cmml" xref="S6.SS2.15.p1.1.m1.1.1.6.1">¯</ci><ci id="S6.SS2.15.p1.1.m1.1.1.6.2.cmml" xref="S6.SS2.15.p1.1.m1.1.1.6.2">𝑐</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.15.p1.1.m1.1c">\bar{a}<_{b}\bar{b}<_{b}\bar{c}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.15.p1.1.m1.1d">over¯ start_ARG italic_a end_ARG < start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT over¯ start_ARG italic_b end_ARG < start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT over¯ start_ARG italic_c end_ARG</annotation></semantics></math> are in <math alttext="\operatorname{dom}(p)" class="ltx_Math" display="inline" id="S6.SS2.15.p1.2.m2.2"><semantics id="S6.SS2.15.p1.2.m2.2a"><mrow id="S6.SS2.15.p1.2.m2.2.3.2" xref="S6.SS2.15.p1.2.m2.2.3.1.cmml"><mi id="S6.SS2.15.p1.2.m2.1.1" xref="S6.SS2.15.p1.2.m2.1.1.cmml">dom</mi><mo id="S6.SS2.15.p1.2.m2.2.3.2a" xref="S6.SS2.15.p1.2.m2.2.3.1.cmml">⁡</mo><mrow id="S6.SS2.15.p1.2.m2.2.3.2.1" xref="S6.SS2.15.p1.2.m2.2.3.1.cmml"><mo id="S6.SS2.15.p1.2.m2.2.3.2.1.1" stretchy="false" xref="S6.SS2.15.p1.2.m2.2.3.1.cmml">(</mo><mi id="S6.SS2.15.p1.2.m2.2.2" xref="S6.SS2.15.p1.2.m2.2.2.cmml">p</mi><mo id="S6.SS2.15.p1.2.m2.2.3.2.1.2" stretchy="false" xref="S6.SS2.15.p1.2.m2.2.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.15.p1.2.m2.2b"><apply id="S6.SS2.15.p1.2.m2.2.3.1.cmml" xref="S6.SS2.15.p1.2.m2.2.3.2"><ci id="S6.SS2.15.p1.2.m2.1.1.cmml" xref="S6.SS2.15.p1.2.m2.1.1">dom</ci><ci id="S6.SS2.15.p1.2.m2.2.2.cmml" xref="S6.SS2.15.p1.2.m2.2.2">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.15.p1.2.m2.2c">\operatorname{dom}(p)</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.15.p1.2.m2.2d">roman_dom ( italic_p )</annotation></semantics></math> and <math alttext="\bar{a}" class="ltx_Math" display="inline" id="S6.SS2.15.p1.3.m3.1"><semantics id="S6.SS2.15.p1.3.m3.1a"><mover accent="true" id="S6.SS2.15.p1.3.m3.1.1" xref="S6.SS2.15.p1.3.m3.1.1.cmml"><mi id="S6.SS2.15.p1.3.m3.1.1.2" xref="S6.SS2.15.p1.3.m3.1.1.2.cmml">a</mi><mo id="S6.SS2.15.p1.3.m3.1.1.1" xref="S6.SS2.15.p1.3.m3.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S6.SS2.15.p1.3.m3.1b"><apply id="S6.SS2.15.p1.3.m3.1.1.cmml" xref="S6.SS2.15.p1.3.m3.1.1"><ci id="S6.SS2.15.p1.3.m3.1.1.1.cmml" xref="S6.SS2.15.p1.3.m3.1.1.1">¯</ci><ci id="S6.SS2.15.p1.3.m3.1.1.2.cmml" xref="S6.SS2.15.p1.3.m3.1.1.2">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.15.p1.3.m3.1c">\bar{a}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.15.p1.3.m3.1d">over¯ start_ARG italic_a end_ARG</annotation></semantics></math>,<math alttext="\bar{c}" class="ltx_Math" display="inline" id="S6.SS2.15.p1.4.m4.1"><semantics id="S6.SS2.15.p1.4.m4.1a"><mover accent="true" id="S6.SS2.15.p1.4.m4.1.1" xref="S6.SS2.15.p1.4.m4.1.1.cmml"><mi id="S6.SS2.15.p1.4.m4.1.1.2" xref="S6.SS2.15.p1.4.m4.1.1.2.cmml">c</mi><mo id="S6.SS2.15.p1.4.m4.1.1.1" xref="S6.SS2.15.p1.4.m4.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S6.SS2.15.p1.4.m4.1b"><apply id="S6.SS2.15.p1.4.m4.1.1.cmml" xref="S6.SS2.15.p1.4.m4.1.1"><ci id="S6.SS2.15.p1.4.m4.1.1.1.cmml" xref="S6.SS2.15.p1.4.m4.1.1.1">¯</ci><ci id="S6.SS2.15.p1.4.m4.1.1.2.cmml" xref="S6.SS2.15.p1.4.m4.1.1.2">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.15.p1.4.m4.1c">\bar{c}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.15.p1.4.m4.1d">over¯ start_ARG italic_c end_ARG</annotation></semantics></math> are neighbors of <math alttext="\bar{b}" class="ltx_Math" display="inline" id="S6.SS2.15.p1.5.m5.1"><semantics id="S6.SS2.15.p1.5.m5.1a"><mover accent="true" id="S6.SS2.15.p1.5.m5.1.1" xref="S6.SS2.15.p1.5.m5.1.1.cmml"><mi id="S6.SS2.15.p1.5.m5.1.1.2" xref="S6.SS2.15.p1.5.m5.1.1.2.cmml">b</mi><mo id="S6.SS2.15.p1.5.m5.1.1.1" xref="S6.SS2.15.p1.5.m5.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S6.SS2.15.p1.5.m5.1b"><apply id="S6.SS2.15.p1.5.m5.1.1.cmml" xref="S6.SS2.15.p1.5.m5.1.1"><ci id="S6.SS2.15.p1.5.m5.1.1.1.cmml" xref="S6.SS2.15.p1.5.m5.1.1.1">¯</ci><ci id="S6.SS2.15.p1.5.m5.1.1.2.cmml" xref="S6.SS2.15.p1.5.m5.1.1.2">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.15.p1.5.m5.1c">\bar{b}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.15.p1.5.m5.1d">over¯ start_ARG italic_b end_ARG</annotation></semantics></math>, then <math alttext="\nu(a_{r},c_{l})=\nu(p(\bar{a}),p(\bar{c}))" class="ltx_Math" display="inline" id="S6.SS2.15.p1.6.m6.6"><semantics id="S6.SS2.15.p1.6.m6.6a"><mrow id="S6.SS2.15.p1.6.m6.6.6" xref="S6.SS2.15.p1.6.m6.6.6.cmml"><mrow id="S6.SS2.15.p1.6.m6.4.4.2" xref="S6.SS2.15.p1.6.m6.4.4.2.cmml"><mi id="S6.SS2.15.p1.6.m6.4.4.2.4" xref="S6.SS2.15.p1.6.m6.4.4.2.4.cmml">ν</mi><mo id="S6.SS2.15.p1.6.m6.4.4.2.3" xref="S6.SS2.15.p1.6.m6.4.4.2.3.cmml">⁢</mo><mrow id="S6.SS2.15.p1.6.m6.4.4.2.2.2" xref="S6.SS2.15.p1.6.m6.4.4.2.2.3.cmml"><mo id="S6.SS2.15.p1.6.m6.4.4.2.2.2.3" stretchy="false" xref="S6.SS2.15.p1.6.m6.4.4.2.2.3.cmml">(</mo><msub id="S6.SS2.15.p1.6.m6.3.3.1.1.1.1" xref="S6.SS2.15.p1.6.m6.3.3.1.1.1.1.cmml"><mi id="S6.SS2.15.p1.6.m6.3.3.1.1.1.1.2" xref="S6.SS2.15.p1.6.m6.3.3.1.1.1.1.2.cmml">a</mi><mi id="S6.SS2.15.p1.6.m6.3.3.1.1.1.1.3" xref="S6.SS2.15.p1.6.m6.3.3.1.1.1.1.3.cmml">r</mi></msub><mo id="S6.SS2.15.p1.6.m6.4.4.2.2.2.4" xref="S6.SS2.15.p1.6.m6.4.4.2.2.3.cmml">,</mo><msub id="S6.SS2.15.p1.6.m6.4.4.2.2.2.2" xref="S6.SS2.15.p1.6.m6.4.4.2.2.2.2.cmml"><mi id="S6.SS2.15.p1.6.m6.4.4.2.2.2.2.2" xref="S6.SS2.15.p1.6.m6.4.4.2.2.2.2.2.cmml">c</mi><mi id="S6.SS2.15.p1.6.m6.4.4.2.2.2.2.3" xref="S6.SS2.15.p1.6.m6.4.4.2.2.2.2.3.cmml">l</mi></msub><mo id="S6.SS2.15.p1.6.m6.4.4.2.2.2.5" stretchy="false" xref="S6.SS2.15.p1.6.m6.4.4.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.SS2.15.p1.6.m6.6.6.5" xref="S6.SS2.15.p1.6.m6.6.6.5.cmml">=</mo><mrow id="S6.SS2.15.p1.6.m6.6.6.4" xref="S6.SS2.15.p1.6.m6.6.6.4.cmml"><mi id="S6.SS2.15.p1.6.m6.6.6.4.4" xref="S6.SS2.15.p1.6.m6.6.6.4.4.cmml">ν</mi><mo id="S6.SS2.15.p1.6.m6.6.6.4.3" xref="S6.SS2.15.p1.6.m6.6.6.4.3.cmml">⁢</mo><mrow id="S6.SS2.15.p1.6.m6.6.6.4.2.2" xref="S6.SS2.15.p1.6.m6.6.6.4.2.3.cmml"><mo id="S6.SS2.15.p1.6.m6.6.6.4.2.2.3" stretchy="false" xref="S6.SS2.15.p1.6.m6.6.6.4.2.3.cmml">(</mo><mrow id="S6.SS2.15.p1.6.m6.5.5.3.1.1.1" xref="S6.SS2.15.p1.6.m6.5.5.3.1.1.1.cmml"><mi id="S6.SS2.15.p1.6.m6.5.5.3.1.1.1.2" xref="S6.SS2.15.p1.6.m6.5.5.3.1.1.1.2.cmml">p</mi><mo id="S6.SS2.15.p1.6.m6.5.5.3.1.1.1.1" xref="S6.SS2.15.p1.6.m6.5.5.3.1.1.1.1.cmml">⁢</mo><mrow id="S6.SS2.15.p1.6.m6.5.5.3.1.1.1.3.2" xref="S6.SS2.15.p1.6.m6.1.1.cmml"><mo id="S6.SS2.15.p1.6.m6.5.5.3.1.1.1.3.2.1" stretchy="false" xref="S6.SS2.15.p1.6.m6.1.1.cmml">(</mo><mover accent="true" id="S6.SS2.15.p1.6.m6.1.1" xref="S6.SS2.15.p1.6.m6.1.1.cmml"><mi id="S6.SS2.15.p1.6.m6.1.1.2" xref="S6.SS2.15.p1.6.m6.1.1.2.cmml">a</mi><mo id="S6.SS2.15.p1.6.m6.1.1.1" xref="S6.SS2.15.p1.6.m6.1.1.1.cmml">¯</mo></mover><mo id="S6.SS2.15.p1.6.m6.5.5.3.1.1.1.3.2.2" stretchy="false" xref="S6.SS2.15.p1.6.m6.1.1.cmml">)</mo></mrow></mrow><mo id="S6.SS2.15.p1.6.m6.6.6.4.2.2.4" xref="S6.SS2.15.p1.6.m6.6.6.4.2.3.cmml">,</mo><mrow id="S6.SS2.15.p1.6.m6.6.6.4.2.2.2" xref="S6.SS2.15.p1.6.m6.6.6.4.2.2.2.cmml"><mi id="S6.SS2.15.p1.6.m6.6.6.4.2.2.2.2" xref="S6.SS2.15.p1.6.m6.6.6.4.2.2.2.2.cmml">p</mi><mo id="S6.SS2.15.p1.6.m6.6.6.4.2.2.2.1" xref="S6.SS2.15.p1.6.m6.6.6.4.2.2.2.1.cmml">⁢</mo><mrow id="S6.SS2.15.p1.6.m6.6.6.4.2.2.2.3.2" xref="S6.SS2.15.p1.6.m6.2.2.cmml"><mo id="S6.SS2.15.p1.6.m6.6.6.4.2.2.2.3.2.1" stretchy="false" xref="S6.SS2.15.p1.6.m6.2.2.cmml">(</mo><mover accent="true" id="S6.SS2.15.p1.6.m6.2.2" xref="S6.SS2.15.p1.6.m6.2.2.cmml"><mi id="S6.SS2.15.p1.6.m6.2.2.2" xref="S6.SS2.15.p1.6.m6.2.2.2.cmml">c</mi><mo id="S6.SS2.15.p1.6.m6.2.2.1" xref="S6.SS2.15.p1.6.m6.2.2.1.cmml">¯</mo></mover><mo id="S6.SS2.15.p1.6.m6.6.6.4.2.2.2.3.2.2" stretchy="false" xref="S6.SS2.15.p1.6.m6.2.2.cmml">)</mo></mrow></mrow><mo id="S6.SS2.15.p1.6.m6.6.6.4.2.2.5" stretchy="false" xref="S6.SS2.15.p1.6.m6.6.6.4.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.15.p1.6.m6.6b"><apply id="S6.SS2.15.p1.6.m6.6.6.cmml" xref="S6.SS2.15.p1.6.m6.6.6"><eq id="S6.SS2.15.p1.6.m6.6.6.5.cmml" xref="S6.SS2.15.p1.6.m6.6.6.5"></eq><apply id="S6.SS2.15.p1.6.m6.4.4.2.cmml" xref="S6.SS2.15.p1.6.m6.4.4.2"><times id="S6.SS2.15.p1.6.m6.4.4.2.3.cmml" xref="S6.SS2.15.p1.6.m6.4.4.2.3"></times><ci id="S6.SS2.15.p1.6.m6.4.4.2.4.cmml" xref="S6.SS2.15.p1.6.m6.4.4.2.4">𝜈</ci><interval closure="open" id="S6.SS2.15.p1.6.m6.4.4.2.2.3.cmml" xref="S6.SS2.15.p1.6.m6.4.4.2.2.2"><apply id="S6.SS2.15.p1.6.m6.3.3.1.1.1.1.cmml" xref="S6.SS2.15.p1.6.m6.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.15.p1.6.m6.3.3.1.1.1.1.1.cmml" xref="S6.SS2.15.p1.6.m6.3.3.1.1.1.1">subscript</csymbol><ci id="S6.SS2.15.p1.6.m6.3.3.1.1.1.1.2.cmml" xref="S6.SS2.15.p1.6.m6.3.3.1.1.1.1.2">𝑎</ci><ci id="S6.SS2.15.p1.6.m6.3.3.1.1.1.1.3.cmml" xref="S6.SS2.15.p1.6.m6.3.3.1.1.1.1.3">𝑟</ci></apply><apply id="S6.SS2.15.p1.6.m6.4.4.2.2.2.2.cmml" xref="S6.SS2.15.p1.6.m6.4.4.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.15.p1.6.m6.4.4.2.2.2.2.1.cmml" xref="S6.SS2.15.p1.6.m6.4.4.2.2.2.2">subscript</csymbol><ci id="S6.SS2.15.p1.6.m6.4.4.2.2.2.2.2.cmml" xref="S6.SS2.15.p1.6.m6.4.4.2.2.2.2.2">𝑐</ci><ci id="S6.SS2.15.p1.6.m6.4.4.2.2.2.2.3.cmml" xref="S6.SS2.15.p1.6.m6.4.4.2.2.2.2.3">𝑙</ci></apply></interval></apply><apply id="S6.SS2.15.p1.6.m6.6.6.4.cmml" xref="S6.SS2.15.p1.6.m6.6.6.4"><times id="S6.SS2.15.p1.6.m6.6.6.4.3.cmml" xref="S6.SS2.15.p1.6.m6.6.6.4.3"></times><ci id="S6.SS2.15.p1.6.m6.6.6.4.4.cmml" xref="S6.SS2.15.p1.6.m6.6.6.4.4">𝜈</ci><interval closure="open" id="S6.SS2.15.p1.6.m6.6.6.4.2.3.cmml" xref="S6.SS2.15.p1.6.m6.6.6.4.2.2"><apply id="S6.SS2.15.p1.6.m6.5.5.3.1.1.1.cmml" xref="S6.SS2.15.p1.6.m6.5.5.3.1.1.1"><times id="S6.SS2.15.p1.6.m6.5.5.3.1.1.1.1.cmml" xref="S6.SS2.15.p1.6.m6.5.5.3.1.1.1.1"></times><ci id="S6.SS2.15.p1.6.m6.5.5.3.1.1.1.2.cmml" xref="S6.SS2.15.p1.6.m6.5.5.3.1.1.1.2">𝑝</ci><apply id="S6.SS2.15.p1.6.m6.1.1.cmml" xref="S6.SS2.15.p1.6.m6.5.5.3.1.1.1.3.2"><ci id="S6.SS2.15.p1.6.m6.1.1.1.cmml" xref="S6.SS2.15.p1.6.m6.1.1.1">¯</ci><ci id="S6.SS2.15.p1.6.m6.1.1.2.cmml" xref="S6.SS2.15.p1.6.m6.1.1.2">𝑎</ci></apply></apply><apply id="S6.SS2.15.p1.6.m6.6.6.4.2.2.2.cmml" xref="S6.SS2.15.p1.6.m6.6.6.4.2.2.2"><times id="S6.SS2.15.p1.6.m6.6.6.4.2.2.2.1.cmml" xref="S6.SS2.15.p1.6.m6.6.6.4.2.2.2.1"></times><ci id="S6.SS2.15.p1.6.m6.6.6.4.2.2.2.2.cmml" xref="S6.SS2.15.p1.6.m6.6.6.4.2.2.2.2">𝑝</ci><apply id="S6.SS2.15.p1.6.m6.2.2.cmml" xref="S6.SS2.15.p1.6.m6.6.6.4.2.2.2.3.2"><ci id="S6.SS2.15.p1.6.m6.2.2.1.cmml" xref="S6.SS2.15.p1.6.m6.2.2.1">¯</ci><ci id="S6.SS2.15.p1.6.m6.2.2.2.cmml" xref="S6.SS2.15.p1.6.m6.2.2.2">𝑐</ci></apply></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.15.p1.6.m6.6c">\nu(a_{r},c_{l})=\nu(p(\bar{a}),p(\bar{c}))</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.15.p1.6.m6.6d">italic_ν ( italic_a start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT , italic_c start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ) = italic_ν ( italic_p ( over¯ start_ARG italic_a end_ARG ) , italic_p ( over¯ start_ARG italic_c end_ARG ) )</annotation></semantics></math>. By <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.Thmtheorem11" title="Lemma 6.11. ‣ 6.2. Forcing epimorphisms ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">6.11</span></a> (b), it is enough to prove that <math alttext="\nu(a_{m},c_{m})=\nu(p(\bar{a}),p(\bar{b}))" class="ltx_Math" display="inline" id="S6.SS2.15.p1.7.m7.6"><semantics id="S6.SS2.15.p1.7.m7.6a"><mrow id="S6.SS2.15.p1.7.m7.6.6" xref="S6.SS2.15.p1.7.m7.6.6.cmml"><mrow id="S6.SS2.15.p1.7.m7.4.4.2" xref="S6.SS2.15.p1.7.m7.4.4.2.cmml"><mi id="S6.SS2.15.p1.7.m7.4.4.2.4" xref="S6.SS2.15.p1.7.m7.4.4.2.4.cmml">ν</mi><mo id="S6.SS2.15.p1.7.m7.4.4.2.3" xref="S6.SS2.15.p1.7.m7.4.4.2.3.cmml">⁢</mo><mrow id="S6.SS2.15.p1.7.m7.4.4.2.2.2" xref="S6.SS2.15.p1.7.m7.4.4.2.2.3.cmml"><mo id="S6.SS2.15.p1.7.m7.4.4.2.2.2.3" stretchy="false" xref="S6.SS2.15.p1.7.m7.4.4.2.2.3.cmml">(</mo><msub id="S6.SS2.15.p1.7.m7.3.3.1.1.1.1" xref="S6.SS2.15.p1.7.m7.3.3.1.1.1.1.cmml"><mi id="S6.SS2.15.p1.7.m7.3.3.1.1.1.1.2" xref="S6.SS2.15.p1.7.m7.3.3.1.1.1.1.2.cmml">a</mi><mi id="S6.SS2.15.p1.7.m7.3.3.1.1.1.1.3" xref="S6.SS2.15.p1.7.m7.3.3.1.1.1.1.3.cmml">m</mi></msub><mo id="S6.SS2.15.p1.7.m7.4.4.2.2.2.4" xref="S6.SS2.15.p1.7.m7.4.4.2.2.3.cmml">,</mo><msub id="S6.SS2.15.p1.7.m7.4.4.2.2.2.2" xref="S6.SS2.15.p1.7.m7.4.4.2.2.2.2.cmml"><mi id="S6.SS2.15.p1.7.m7.4.4.2.2.2.2.2" xref="S6.SS2.15.p1.7.m7.4.4.2.2.2.2.2.cmml">c</mi><mi id="S6.SS2.15.p1.7.m7.4.4.2.2.2.2.3" xref="S6.SS2.15.p1.7.m7.4.4.2.2.2.2.3.cmml">m</mi></msub><mo id="S6.SS2.15.p1.7.m7.4.4.2.2.2.5" stretchy="false" xref="S6.SS2.15.p1.7.m7.4.4.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.SS2.15.p1.7.m7.6.6.5" xref="S6.SS2.15.p1.7.m7.6.6.5.cmml">=</mo><mrow id="S6.SS2.15.p1.7.m7.6.6.4" xref="S6.SS2.15.p1.7.m7.6.6.4.cmml"><mi id="S6.SS2.15.p1.7.m7.6.6.4.4" xref="S6.SS2.15.p1.7.m7.6.6.4.4.cmml">ν</mi><mo id="S6.SS2.15.p1.7.m7.6.6.4.3" xref="S6.SS2.15.p1.7.m7.6.6.4.3.cmml">⁢</mo><mrow id="S6.SS2.15.p1.7.m7.6.6.4.2.2" xref="S6.SS2.15.p1.7.m7.6.6.4.2.3.cmml"><mo id="S6.SS2.15.p1.7.m7.6.6.4.2.2.3" stretchy="false" xref="S6.SS2.15.p1.7.m7.6.6.4.2.3.cmml">(</mo><mrow id="S6.SS2.15.p1.7.m7.5.5.3.1.1.1" xref="S6.SS2.15.p1.7.m7.5.5.3.1.1.1.cmml"><mi id="S6.SS2.15.p1.7.m7.5.5.3.1.1.1.2" xref="S6.SS2.15.p1.7.m7.5.5.3.1.1.1.2.cmml">p</mi><mo id="S6.SS2.15.p1.7.m7.5.5.3.1.1.1.1" xref="S6.SS2.15.p1.7.m7.5.5.3.1.1.1.1.cmml">⁢</mo><mrow id="S6.SS2.15.p1.7.m7.5.5.3.1.1.1.3.2" xref="S6.SS2.15.p1.7.m7.1.1.cmml"><mo id="S6.SS2.15.p1.7.m7.5.5.3.1.1.1.3.2.1" stretchy="false" xref="S6.SS2.15.p1.7.m7.1.1.cmml">(</mo><mover accent="true" id="S6.SS2.15.p1.7.m7.1.1" xref="S6.SS2.15.p1.7.m7.1.1.cmml"><mi id="S6.SS2.15.p1.7.m7.1.1.2" xref="S6.SS2.15.p1.7.m7.1.1.2.cmml">a</mi><mo id="S6.SS2.15.p1.7.m7.1.1.1" xref="S6.SS2.15.p1.7.m7.1.1.1.cmml">¯</mo></mover><mo id="S6.SS2.15.p1.7.m7.5.5.3.1.1.1.3.2.2" stretchy="false" xref="S6.SS2.15.p1.7.m7.1.1.cmml">)</mo></mrow></mrow><mo id="S6.SS2.15.p1.7.m7.6.6.4.2.2.4" xref="S6.SS2.15.p1.7.m7.6.6.4.2.3.cmml">,</mo><mrow id="S6.SS2.15.p1.7.m7.6.6.4.2.2.2" xref="S6.SS2.15.p1.7.m7.6.6.4.2.2.2.cmml"><mi id="S6.SS2.15.p1.7.m7.6.6.4.2.2.2.2" xref="S6.SS2.15.p1.7.m7.6.6.4.2.2.2.2.cmml">p</mi><mo id="S6.SS2.15.p1.7.m7.6.6.4.2.2.2.1" xref="S6.SS2.15.p1.7.m7.6.6.4.2.2.2.1.cmml">⁢</mo><mrow id="S6.SS2.15.p1.7.m7.6.6.4.2.2.2.3.2" xref="S6.SS2.15.p1.7.m7.2.2.cmml"><mo id="S6.SS2.15.p1.7.m7.6.6.4.2.2.2.3.2.1" stretchy="false" xref="S6.SS2.15.p1.7.m7.2.2.cmml">(</mo><mover accent="true" id="S6.SS2.15.p1.7.m7.2.2" xref="S6.SS2.15.p1.7.m7.2.2.cmml"><mi id="S6.SS2.15.p1.7.m7.2.2.2" xref="S6.SS2.15.p1.7.m7.2.2.2.cmml">b</mi><mo id="S6.SS2.15.p1.7.m7.2.2.1" xref="S6.SS2.15.p1.7.m7.2.2.1.cmml">¯</mo></mover><mo id="S6.SS2.15.p1.7.m7.6.6.4.2.2.2.3.2.2" stretchy="false" xref="S6.SS2.15.p1.7.m7.2.2.cmml">)</mo></mrow></mrow><mo id="S6.SS2.15.p1.7.m7.6.6.4.2.2.5" stretchy="false" xref="S6.SS2.15.p1.7.m7.6.6.4.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.15.p1.7.m7.6b"><apply id="S6.SS2.15.p1.7.m7.6.6.cmml" xref="S6.SS2.15.p1.7.m7.6.6"><eq id="S6.SS2.15.p1.7.m7.6.6.5.cmml" xref="S6.SS2.15.p1.7.m7.6.6.5"></eq><apply id="S6.SS2.15.p1.7.m7.4.4.2.cmml" xref="S6.SS2.15.p1.7.m7.4.4.2"><times id="S6.SS2.15.p1.7.m7.4.4.2.3.cmml" xref="S6.SS2.15.p1.7.m7.4.4.2.3"></times><ci id="S6.SS2.15.p1.7.m7.4.4.2.4.cmml" xref="S6.SS2.15.p1.7.m7.4.4.2.4">𝜈</ci><interval closure="open" id="S6.SS2.15.p1.7.m7.4.4.2.2.3.cmml" xref="S6.SS2.15.p1.7.m7.4.4.2.2.2"><apply id="S6.SS2.15.p1.7.m7.3.3.1.1.1.1.cmml" xref="S6.SS2.15.p1.7.m7.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.15.p1.7.m7.3.3.1.1.1.1.1.cmml" xref="S6.SS2.15.p1.7.m7.3.3.1.1.1.1">subscript</csymbol><ci id="S6.SS2.15.p1.7.m7.3.3.1.1.1.1.2.cmml" xref="S6.SS2.15.p1.7.m7.3.3.1.1.1.1.2">𝑎</ci><ci id="S6.SS2.15.p1.7.m7.3.3.1.1.1.1.3.cmml" xref="S6.SS2.15.p1.7.m7.3.3.1.1.1.1.3">𝑚</ci></apply><apply id="S6.SS2.15.p1.7.m7.4.4.2.2.2.2.cmml" xref="S6.SS2.15.p1.7.m7.4.4.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.15.p1.7.m7.4.4.2.2.2.2.1.cmml" xref="S6.SS2.15.p1.7.m7.4.4.2.2.2.2">subscript</csymbol><ci id="S6.SS2.15.p1.7.m7.4.4.2.2.2.2.2.cmml" xref="S6.SS2.15.p1.7.m7.4.4.2.2.2.2.2">𝑐</ci><ci id="S6.SS2.15.p1.7.m7.4.4.2.2.2.2.3.cmml" xref="S6.SS2.15.p1.7.m7.4.4.2.2.2.2.3">𝑚</ci></apply></interval></apply><apply id="S6.SS2.15.p1.7.m7.6.6.4.cmml" xref="S6.SS2.15.p1.7.m7.6.6.4"><times id="S6.SS2.15.p1.7.m7.6.6.4.3.cmml" xref="S6.SS2.15.p1.7.m7.6.6.4.3"></times><ci id="S6.SS2.15.p1.7.m7.6.6.4.4.cmml" xref="S6.SS2.15.p1.7.m7.6.6.4.4">𝜈</ci><interval closure="open" id="S6.SS2.15.p1.7.m7.6.6.4.2.3.cmml" xref="S6.SS2.15.p1.7.m7.6.6.4.2.2"><apply id="S6.SS2.15.p1.7.m7.5.5.3.1.1.1.cmml" xref="S6.SS2.15.p1.7.m7.5.5.3.1.1.1"><times id="S6.SS2.15.p1.7.m7.5.5.3.1.1.1.1.cmml" xref="S6.SS2.15.p1.7.m7.5.5.3.1.1.1.1"></times><ci id="S6.SS2.15.p1.7.m7.5.5.3.1.1.1.2.cmml" xref="S6.SS2.15.p1.7.m7.5.5.3.1.1.1.2">𝑝</ci><apply id="S6.SS2.15.p1.7.m7.1.1.cmml" xref="S6.SS2.15.p1.7.m7.5.5.3.1.1.1.3.2"><ci id="S6.SS2.15.p1.7.m7.1.1.1.cmml" xref="S6.SS2.15.p1.7.m7.1.1.1">¯</ci><ci id="S6.SS2.15.p1.7.m7.1.1.2.cmml" xref="S6.SS2.15.p1.7.m7.1.1.2">𝑎</ci></apply></apply><apply id="S6.SS2.15.p1.7.m7.6.6.4.2.2.2.cmml" xref="S6.SS2.15.p1.7.m7.6.6.4.2.2.2"><times id="S6.SS2.15.p1.7.m7.6.6.4.2.2.2.1.cmml" xref="S6.SS2.15.p1.7.m7.6.6.4.2.2.2.1"></times><ci id="S6.SS2.15.p1.7.m7.6.6.4.2.2.2.2.cmml" xref="S6.SS2.15.p1.7.m7.6.6.4.2.2.2.2">𝑝</ci><apply id="S6.SS2.15.p1.7.m7.2.2.cmml" xref="S6.SS2.15.p1.7.m7.6.6.4.2.2.2.3.2"><ci id="S6.SS2.15.p1.7.m7.2.2.1.cmml" xref="S6.SS2.15.p1.7.m7.2.2.1">¯</ci><ci id="S6.SS2.15.p1.7.m7.2.2.2.cmml" xref="S6.SS2.15.p1.7.m7.2.2.2">𝑏</ci></apply></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.15.p1.7.m7.6c">\nu(a_{m},c_{m})=\nu(p(\bar{a}),p(\bar{b}))</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.15.p1.7.m7.6d">italic_ν ( italic_a start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT , italic_c start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) = italic_ν ( italic_p ( over¯ start_ARG italic_a end_ARG ) , italic_p ( over¯ start_ARG italic_b end_ARG ) )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S6.SS2.16.p2"> <p class="ltx_p" id="S6.SS2.16.p2.11">Since <math alttext="p(\bar{a})<_{X}p(\bar{b})<_{X}p(\bar{c})" class="ltx_Math" display="inline" id="S6.SS2.16.p2.1.m1.3"><semantics id="S6.SS2.16.p2.1.m1.3a"><mrow id="S6.SS2.16.p2.1.m1.3.4" xref="S6.SS2.16.p2.1.m1.3.4.cmml"><mrow id="S6.SS2.16.p2.1.m1.3.4.2" xref="S6.SS2.16.p2.1.m1.3.4.2.cmml"><mi id="S6.SS2.16.p2.1.m1.3.4.2.2" xref="S6.SS2.16.p2.1.m1.3.4.2.2.cmml">p</mi><mo id="S6.SS2.16.p2.1.m1.3.4.2.1" xref="S6.SS2.16.p2.1.m1.3.4.2.1.cmml">⁢</mo><mrow id="S6.SS2.16.p2.1.m1.3.4.2.3.2" xref="S6.SS2.16.p2.1.m1.1.1.cmml"><mo id="S6.SS2.16.p2.1.m1.3.4.2.3.2.1" stretchy="false" xref="S6.SS2.16.p2.1.m1.1.1.cmml">(</mo><mover accent="true" id="S6.SS2.16.p2.1.m1.1.1" xref="S6.SS2.16.p2.1.m1.1.1.cmml"><mi id="S6.SS2.16.p2.1.m1.1.1.2" xref="S6.SS2.16.p2.1.m1.1.1.2.cmml">a</mi><mo id="S6.SS2.16.p2.1.m1.1.1.1" xref="S6.SS2.16.p2.1.m1.1.1.1.cmml">¯</mo></mover><mo id="S6.SS2.16.p2.1.m1.3.4.2.3.2.2" stretchy="false" xref="S6.SS2.16.p2.1.m1.1.1.cmml">)</mo></mrow></mrow><msub id="S6.SS2.16.p2.1.m1.3.4.3" xref="S6.SS2.16.p2.1.m1.3.4.3.cmml"><mo id="S6.SS2.16.p2.1.m1.3.4.3.2" xref="S6.SS2.16.p2.1.m1.3.4.3.2.cmml"><</mo><mi id="S6.SS2.16.p2.1.m1.3.4.3.3" xref="S6.SS2.16.p2.1.m1.3.4.3.3.cmml">X</mi></msub><mrow id="S6.SS2.16.p2.1.m1.3.4.4" xref="S6.SS2.16.p2.1.m1.3.4.4.cmml"><mi id="S6.SS2.16.p2.1.m1.3.4.4.2" xref="S6.SS2.16.p2.1.m1.3.4.4.2.cmml">p</mi><mo id="S6.SS2.16.p2.1.m1.3.4.4.1" xref="S6.SS2.16.p2.1.m1.3.4.4.1.cmml">⁢</mo><mrow id="S6.SS2.16.p2.1.m1.3.4.4.3.2" xref="S6.SS2.16.p2.1.m1.2.2.cmml"><mo id="S6.SS2.16.p2.1.m1.3.4.4.3.2.1" stretchy="false" xref="S6.SS2.16.p2.1.m1.2.2.cmml">(</mo><mover accent="true" id="S6.SS2.16.p2.1.m1.2.2" xref="S6.SS2.16.p2.1.m1.2.2.cmml"><mi id="S6.SS2.16.p2.1.m1.2.2.2" xref="S6.SS2.16.p2.1.m1.2.2.2.cmml">b</mi><mo id="S6.SS2.16.p2.1.m1.2.2.1" xref="S6.SS2.16.p2.1.m1.2.2.1.cmml">¯</mo></mover><mo id="S6.SS2.16.p2.1.m1.3.4.4.3.2.2" stretchy="false" xref="S6.SS2.16.p2.1.m1.2.2.cmml">)</mo></mrow></mrow><msub id="S6.SS2.16.p2.1.m1.3.4.5" xref="S6.SS2.16.p2.1.m1.3.4.5.cmml"><mo id="S6.SS2.16.p2.1.m1.3.4.5.2" xref="S6.SS2.16.p2.1.m1.3.4.5.2.cmml"><</mo><mi id="S6.SS2.16.p2.1.m1.3.4.5.3" xref="S6.SS2.16.p2.1.m1.3.4.5.3.cmml">X</mi></msub><mrow id="S6.SS2.16.p2.1.m1.3.4.6" xref="S6.SS2.16.p2.1.m1.3.4.6.cmml"><mi id="S6.SS2.16.p2.1.m1.3.4.6.2" xref="S6.SS2.16.p2.1.m1.3.4.6.2.cmml">p</mi><mo id="S6.SS2.16.p2.1.m1.3.4.6.1" xref="S6.SS2.16.p2.1.m1.3.4.6.1.cmml">⁢</mo><mrow id="S6.SS2.16.p2.1.m1.3.4.6.3.2" xref="S6.SS2.16.p2.1.m1.3.3.cmml"><mo id="S6.SS2.16.p2.1.m1.3.4.6.3.2.1" stretchy="false" xref="S6.SS2.16.p2.1.m1.3.3.cmml">(</mo><mover accent="true" id="S6.SS2.16.p2.1.m1.3.3" xref="S6.SS2.16.p2.1.m1.3.3.cmml"><mi id="S6.SS2.16.p2.1.m1.3.3.2" xref="S6.SS2.16.p2.1.m1.3.3.2.cmml">c</mi><mo id="S6.SS2.16.p2.1.m1.3.3.1" xref="S6.SS2.16.p2.1.m1.3.3.1.cmml">¯</mo></mover><mo id="S6.SS2.16.p2.1.m1.3.4.6.3.2.2" stretchy="false" xref="S6.SS2.16.p2.1.m1.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.16.p2.1.m1.3b"><apply id="S6.SS2.16.p2.1.m1.3.4.cmml" xref="S6.SS2.16.p2.1.m1.3.4"><and id="S6.SS2.16.p2.1.m1.3.4a.cmml" xref="S6.SS2.16.p2.1.m1.3.4"></and><apply id="S6.SS2.16.p2.1.m1.3.4b.cmml" xref="S6.SS2.16.p2.1.m1.3.4"><apply id="S6.SS2.16.p2.1.m1.3.4.3.cmml" xref="S6.SS2.16.p2.1.m1.3.4.3"><csymbol cd="ambiguous" id="S6.SS2.16.p2.1.m1.3.4.3.1.cmml" xref="S6.SS2.16.p2.1.m1.3.4.3">subscript</csymbol><lt id="S6.SS2.16.p2.1.m1.3.4.3.2.cmml" xref="S6.SS2.16.p2.1.m1.3.4.3.2"></lt><ci id="S6.SS2.16.p2.1.m1.3.4.3.3.cmml" xref="S6.SS2.16.p2.1.m1.3.4.3.3">𝑋</ci></apply><apply id="S6.SS2.16.p2.1.m1.3.4.2.cmml" xref="S6.SS2.16.p2.1.m1.3.4.2"><times id="S6.SS2.16.p2.1.m1.3.4.2.1.cmml" xref="S6.SS2.16.p2.1.m1.3.4.2.1"></times><ci id="S6.SS2.16.p2.1.m1.3.4.2.2.cmml" xref="S6.SS2.16.p2.1.m1.3.4.2.2">𝑝</ci><apply id="S6.SS2.16.p2.1.m1.1.1.cmml" xref="S6.SS2.16.p2.1.m1.3.4.2.3.2"><ci id="S6.SS2.16.p2.1.m1.1.1.1.cmml" xref="S6.SS2.16.p2.1.m1.1.1.1">¯</ci><ci id="S6.SS2.16.p2.1.m1.1.1.2.cmml" xref="S6.SS2.16.p2.1.m1.1.1.2">𝑎</ci></apply></apply><apply id="S6.SS2.16.p2.1.m1.3.4.4.cmml" xref="S6.SS2.16.p2.1.m1.3.4.4"><times id="S6.SS2.16.p2.1.m1.3.4.4.1.cmml" xref="S6.SS2.16.p2.1.m1.3.4.4.1"></times><ci id="S6.SS2.16.p2.1.m1.3.4.4.2.cmml" xref="S6.SS2.16.p2.1.m1.3.4.4.2">𝑝</ci><apply id="S6.SS2.16.p2.1.m1.2.2.cmml" xref="S6.SS2.16.p2.1.m1.3.4.4.3.2"><ci id="S6.SS2.16.p2.1.m1.2.2.1.cmml" xref="S6.SS2.16.p2.1.m1.2.2.1">¯</ci><ci id="S6.SS2.16.p2.1.m1.2.2.2.cmml" xref="S6.SS2.16.p2.1.m1.2.2.2">𝑏</ci></apply></apply></apply><apply id="S6.SS2.16.p2.1.m1.3.4c.cmml" xref="S6.SS2.16.p2.1.m1.3.4"><apply id="S6.SS2.16.p2.1.m1.3.4.5.cmml" xref="S6.SS2.16.p2.1.m1.3.4.5"><csymbol cd="ambiguous" id="S6.SS2.16.p2.1.m1.3.4.5.1.cmml" xref="S6.SS2.16.p2.1.m1.3.4.5">subscript</csymbol><lt id="S6.SS2.16.p2.1.m1.3.4.5.2.cmml" xref="S6.SS2.16.p2.1.m1.3.4.5.2"></lt><ci id="S6.SS2.16.p2.1.m1.3.4.5.3.cmml" xref="S6.SS2.16.p2.1.m1.3.4.5.3">𝑋</ci></apply><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.16.p2.1.m1.3.4.4.cmml" id="S6.SS2.16.p2.1.m1.3.4d.cmml" xref="S6.SS2.16.p2.1.m1.3.4"></share><apply id="S6.SS2.16.p2.1.m1.3.4.6.cmml" xref="S6.SS2.16.p2.1.m1.3.4.6"><times id="S6.SS2.16.p2.1.m1.3.4.6.1.cmml" xref="S6.SS2.16.p2.1.m1.3.4.6.1"></times><ci id="S6.SS2.16.p2.1.m1.3.4.6.2.cmml" xref="S6.SS2.16.p2.1.m1.3.4.6.2">𝑝</ci><apply id="S6.SS2.16.p2.1.m1.3.3.cmml" xref="S6.SS2.16.p2.1.m1.3.4.6.3.2"><ci id="S6.SS2.16.p2.1.m1.3.3.1.cmml" xref="S6.SS2.16.p2.1.m1.3.3.1">¯</ci><ci id="S6.SS2.16.p2.1.m1.3.3.2.cmml" xref="S6.SS2.16.p2.1.m1.3.3.2">𝑐</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.16.p2.1.m1.3c">p(\bar{a})<_{X}p(\bar{b})<_{X}p(\bar{c})</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.16.p2.1.m1.3d">italic_p ( over¯ start_ARG italic_a end_ARG ) < start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT italic_p ( over¯ start_ARG italic_b end_ARG ) < start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT italic_p ( over¯ start_ARG italic_c end_ARG )</annotation></semantics></math> either <math alttext="\Delta(p(\bar{a}),p(\bar{c}))=\Delta(p(\bar{a}),p(\bar{b}))" class="ltx_Math" display="inline" id="S6.SS2.16.p2.2.m2.8"><semantics id="S6.SS2.16.p2.2.m2.8a"><mrow id="S6.SS2.16.p2.2.m2.8.8" xref="S6.SS2.16.p2.2.m2.8.8.cmml"><mrow id="S6.SS2.16.p2.2.m2.6.6.2" xref="S6.SS2.16.p2.2.m2.6.6.2.cmml"><mi id="S6.SS2.16.p2.2.m2.6.6.2.4" mathvariant="normal" xref="S6.SS2.16.p2.2.m2.6.6.2.4.cmml">Δ</mi><mo id="S6.SS2.16.p2.2.m2.6.6.2.3" xref="S6.SS2.16.p2.2.m2.6.6.2.3.cmml">⁢</mo><mrow id="S6.SS2.16.p2.2.m2.6.6.2.2.2" xref="S6.SS2.16.p2.2.m2.6.6.2.2.3.cmml"><mo id="S6.SS2.16.p2.2.m2.6.6.2.2.2.3" stretchy="false" xref="S6.SS2.16.p2.2.m2.6.6.2.2.3.cmml">(</mo><mrow id="S6.SS2.16.p2.2.m2.5.5.1.1.1.1" xref="S6.SS2.16.p2.2.m2.5.5.1.1.1.1.cmml"><mi id="S6.SS2.16.p2.2.m2.5.5.1.1.1.1.2" xref="S6.SS2.16.p2.2.m2.5.5.1.1.1.1.2.cmml">p</mi><mo id="S6.SS2.16.p2.2.m2.5.5.1.1.1.1.1" xref="S6.SS2.16.p2.2.m2.5.5.1.1.1.1.1.cmml">⁢</mo><mrow id="S6.SS2.16.p2.2.m2.5.5.1.1.1.1.3.2" xref="S6.SS2.16.p2.2.m2.1.1.cmml"><mo id="S6.SS2.16.p2.2.m2.5.5.1.1.1.1.3.2.1" stretchy="false" xref="S6.SS2.16.p2.2.m2.1.1.cmml">(</mo><mover accent="true" id="S6.SS2.16.p2.2.m2.1.1" xref="S6.SS2.16.p2.2.m2.1.1.cmml"><mi id="S6.SS2.16.p2.2.m2.1.1.2" xref="S6.SS2.16.p2.2.m2.1.1.2.cmml">a</mi><mo id="S6.SS2.16.p2.2.m2.1.1.1" xref="S6.SS2.16.p2.2.m2.1.1.1.cmml">¯</mo></mover><mo id="S6.SS2.16.p2.2.m2.5.5.1.1.1.1.3.2.2" stretchy="false" xref="S6.SS2.16.p2.2.m2.1.1.cmml">)</mo></mrow></mrow><mo id="S6.SS2.16.p2.2.m2.6.6.2.2.2.4" xref="S6.SS2.16.p2.2.m2.6.6.2.2.3.cmml">,</mo><mrow id="S6.SS2.16.p2.2.m2.6.6.2.2.2.2" xref="S6.SS2.16.p2.2.m2.6.6.2.2.2.2.cmml"><mi id="S6.SS2.16.p2.2.m2.6.6.2.2.2.2.2" xref="S6.SS2.16.p2.2.m2.6.6.2.2.2.2.2.cmml">p</mi><mo id="S6.SS2.16.p2.2.m2.6.6.2.2.2.2.1" xref="S6.SS2.16.p2.2.m2.6.6.2.2.2.2.1.cmml">⁢</mo><mrow id="S6.SS2.16.p2.2.m2.6.6.2.2.2.2.3.2" xref="S6.SS2.16.p2.2.m2.2.2.cmml"><mo id="S6.SS2.16.p2.2.m2.6.6.2.2.2.2.3.2.1" stretchy="false" xref="S6.SS2.16.p2.2.m2.2.2.cmml">(</mo><mover accent="true" id="S6.SS2.16.p2.2.m2.2.2" xref="S6.SS2.16.p2.2.m2.2.2.cmml"><mi id="S6.SS2.16.p2.2.m2.2.2.2" xref="S6.SS2.16.p2.2.m2.2.2.2.cmml">c</mi><mo id="S6.SS2.16.p2.2.m2.2.2.1" xref="S6.SS2.16.p2.2.m2.2.2.1.cmml">¯</mo></mover><mo id="S6.SS2.16.p2.2.m2.6.6.2.2.2.2.3.2.2" stretchy="false" xref="S6.SS2.16.p2.2.m2.2.2.cmml">)</mo></mrow></mrow><mo id="S6.SS2.16.p2.2.m2.6.6.2.2.2.5" stretchy="false" xref="S6.SS2.16.p2.2.m2.6.6.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.SS2.16.p2.2.m2.8.8.5" xref="S6.SS2.16.p2.2.m2.8.8.5.cmml">=</mo><mrow id="S6.SS2.16.p2.2.m2.8.8.4" xref="S6.SS2.16.p2.2.m2.8.8.4.cmml"><mi id="S6.SS2.16.p2.2.m2.8.8.4.4" mathvariant="normal" xref="S6.SS2.16.p2.2.m2.8.8.4.4.cmml">Δ</mi><mo id="S6.SS2.16.p2.2.m2.8.8.4.3" xref="S6.SS2.16.p2.2.m2.8.8.4.3.cmml">⁢</mo><mrow id="S6.SS2.16.p2.2.m2.8.8.4.2.2" xref="S6.SS2.16.p2.2.m2.8.8.4.2.3.cmml"><mo id="S6.SS2.16.p2.2.m2.8.8.4.2.2.3" stretchy="false" xref="S6.SS2.16.p2.2.m2.8.8.4.2.3.cmml">(</mo><mrow id="S6.SS2.16.p2.2.m2.7.7.3.1.1.1" xref="S6.SS2.16.p2.2.m2.7.7.3.1.1.1.cmml"><mi id="S6.SS2.16.p2.2.m2.7.7.3.1.1.1.2" xref="S6.SS2.16.p2.2.m2.7.7.3.1.1.1.2.cmml">p</mi><mo id="S6.SS2.16.p2.2.m2.7.7.3.1.1.1.1" xref="S6.SS2.16.p2.2.m2.7.7.3.1.1.1.1.cmml">⁢</mo><mrow id="S6.SS2.16.p2.2.m2.7.7.3.1.1.1.3.2" xref="S6.SS2.16.p2.2.m2.3.3.cmml"><mo id="S6.SS2.16.p2.2.m2.7.7.3.1.1.1.3.2.1" stretchy="false" xref="S6.SS2.16.p2.2.m2.3.3.cmml">(</mo><mover accent="true" id="S6.SS2.16.p2.2.m2.3.3" xref="S6.SS2.16.p2.2.m2.3.3.cmml"><mi id="S6.SS2.16.p2.2.m2.3.3.2" xref="S6.SS2.16.p2.2.m2.3.3.2.cmml">a</mi><mo id="S6.SS2.16.p2.2.m2.3.3.1" xref="S6.SS2.16.p2.2.m2.3.3.1.cmml">¯</mo></mover><mo id="S6.SS2.16.p2.2.m2.7.7.3.1.1.1.3.2.2" stretchy="false" xref="S6.SS2.16.p2.2.m2.3.3.cmml">)</mo></mrow></mrow><mo id="S6.SS2.16.p2.2.m2.8.8.4.2.2.4" xref="S6.SS2.16.p2.2.m2.8.8.4.2.3.cmml">,</mo><mrow id="S6.SS2.16.p2.2.m2.8.8.4.2.2.2" xref="S6.SS2.16.p2.2.m2.8.8.4.2.2.2.cmml"><mi id="S6.SS2.16.p2.2.m2.8.8.4.2.2.2.2" xref="S6.SS2.16.p2.2.m2.8.8.4.2.2.2.2.cmml">p</mi><mo id="S6.SS2.16.p2.2.m2.8.8.4.2.2.2.1" xref="S6.SS2.16.p2.2.m2.8.8.4.2.2.2.1.cmml">⁢</mo><mrow id="S6.SS2.16.p2.2.m2.8.8.4.2.2.2.3.2" xref="S6.SS2.16.p2.2.m2.4.4.cmml"><mo id="S6.SS2.16.p2.2.m2.8.8.4.2.2.2.3.2.1" stretchy="false" xref="S6.SS2.16.p2.2.m2.4.4.cmml">(</mo><mover accent="true" id="S6.SS2.16.p2.2.m2.4.4" xref="S6.SS2.16.p2.2.m2.4.4.cmml"><mi id="S6.SS2.16.p2.2.m2.4.4.2" xref="S6.SS2.16.p2.2.m2.4.4.2.cmml">b</mi><mo id="S6.SS2.16.p2.2.m2.4.4.1" xref="S6.SS2.16.p2.2.m2.4.4.1.cmml">¯</mo></mover><mo id="S6.SS2.16.p2.2.m2.8.8.4.2.2.2.3.2.2" stretchy="false" xref="S6.SS2.16.p2.2.m2.4.4.cmml">)</mo></mrow></mrow><mo id="S6.SS2.16.p2.2.m2.8.8.4.2.2.5" stretchy="false" xref="S6.SS2.16.p2.2.m2.8.8.4.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.16.p2.2.m2.8b"><apply id="S6.SS2.16.p2.2.m2.8.8.cmml" xref="S6.SS2.16.p2.2.m2.8.8"><eq id="S6.SS2.16.p2.2.m2.8.8.5.cmml" xref="S6.SS2.16.p2.2.m2.8.8.5"></eq><apply id="S6.SS2.16.p2.2.m2.6.6.2.cmml" xref="S6.SS2.16.p2.2.m2.6.6.2"><times id="S6.SS2.16.p2.2.m2.6.6.2.3.cmml" xref="S6.SS2.16.p2.2.m2.6.6.2.3"></times><ci id="S6.SS2.16.p2.2.m2.6.6.2.4.cmml" xref="S6.SS2.16.p2.2.m2.6.6.2.4">Δ</ci><interval closure="open" id="S6.SS2.16.p2.2.m2.6.6.2.2.3.cmml" xref="S6.SS2.16.p2.2.m2.6.6.2.2.2"><apply id="S6.SS2.16.p2.2.m2.5.5.1.1.1.1.cmml" xref="S6.SS2.16.p2.2.m2.5.5.1.1.1.1"><times id="S6.SS2.16.p2.2.m2.5.5.1.1.1.1.1.cmml" xref="S6.SS2.16.p2.2.m2.5.5.1.1.1.1.1"></times><ci id="S6.SS2.16.p2.2.m2.5.5.1.1.1.1.2.cmml" xref="S6.SS2.16.p2.2.m2.5.5.1.1.1.1.2">𝑝</ci><apply id="S6.SS2.16.p2.2.m2.1.1.cmml" xref="S6.SS2.16.p2.2.m2.5.5.1.1.1.1.3.2"><ci id="S6.SS2.16.p2.2.m2.1.1.1.cmml" xref="S6.SS2.16.p2.2.m2.1.1.1">¯</ci><ci id="S6.SS2.16.p2.2.m2.1.1.2.cmml" xref="S6.SS2.16.p2.2.m2.1.1.2">𝑎</ci></apply></apply><apply id="S6.SS2.16.p2.2.m2.6.6.2.2.2.2.cmml" xref="S6.SS2.16.p2.2.m2.6.6.2.2.2.2"><times id="S6.SS2.16.p2.2.m2.6.6.2.2.2.2.1.cmml" xref="S6.SS2.16.p2.2.m2.6.6.2.2.2.2.1"></times><ci id="S6.SS2.16.p2.2.m2.6.6.2.2.2.2.2.cmml" xref="S6.SS2.16.p2.2.m2.6.6.2.2.2.2.2">𝑝</ci><apply id="S6.SS2.16.p2.2.m2.2.2.cmml" xref="S6.SS2.16.p2.2.m2.6.6.2.2.2.2.3.2"><ci id="S6.SS2.16.p2.2.m2.2.2.1.cmml" xref="S6.SS2.16.p2.2.m2.2.2.1">¯</ci><ci id="S6.SS2.16.p2.2.m2.2.2.2.cmml" xref="S6.SS2.16.p2.2.m2.2.2.2">𝑐</ci></apply></apply></interval></apply><apply id="S6.SS2.16.p2.2.m2.8.8.4.cmml" xref="S6.SS2.16.p2.2.m2.8.8.4"><times id="S6.SS2.16.p2.2.m2.8.8.4.3.cmml" xref="S6.SS2.16.p2.2.m2.8.8.4.3"></times><ci id="S6.SS2.16.p2.2.m2.8.8.4.4.cmml" xref="S6.SS2.16.p2.2.m2.8.8.4.4">Δ</ci><interval closure="open" id="S6.SS2.16.p2.2.m2.8.8.4.2.3.cmml" xref="S6.SS2.16.p2.2.m2.8.8.4.2.2"><apply id="S6.SS2.16.p2.2.m2.7.7.3.1.1.1.cmml" xref="S6.SS2.16.p2.2.m2.7.7.3.1.1.1"><times id="S6.SS2.16.p2.2.m2.7.7.3.1.1.1.1.cmml" xref="S6.SS2.16.p2.2.m2.7.7.3.1.1.1.1"></times><ci id="S6.SS2.16.p2.2.m2.7.7.3.1.1.1.2.cmml" xref="S6.SS2.16.p2.2.m2.7.7.3.1.1.1.2">𝑝</ci><apply id="S6.SS2.16.p2.2.m2.3.3.cmml" xref="S6.SS2.16.p2.2.m2.7.7.3.1.1.1.3.2"><ci id="S6.SS2.16.p2.2.m2.3.3.1.cmml" xref="S6.SS2.16.p2.2.m2.3.3.1">¯</ci><ci id="S6.SS2.16.p2.2.m2.3.3.2.cmml" xref="S6.SS2.16.p2.2.m2.3.3.2">𝑎</ci></apply></apply><apply id="S6.SS2.16.p2.2.m2.8.8.4.2.2.2.cmml" xref="S6.SS2.16.p2.2.m2.8.8.4.2.2.2"><times id="S6.SS2.16.p2.2.m2.8.8.4.2.2.2.1.cmml" xref="S6.SS2.16.p2.2.m2.8.8.4.2.2.2.1"></times><ci id="S6.SS2.16.p2.2.m2.8.8.4.2.2.2.2.cmml" xref="S6.SS2.16.p2.2.m2.8.8.4.2.2.2.2">𝑝</ci><apply id="S6.SS2.16.p2.2.m2.4.4.cmml" xref="S6.SS2.16.p2.2.m2.8.8.4.2.2.2.3.2"><ci id="S6.SS2.16.p2.2.m2.4.4.1.cmml" xref="S6.SS2.16.p2.2.m2.4.4.1">¯</ci><ci id="S6.SS2.16.p2.2.m2.4.4.2.cmml" xref="S6.SS2.16.p2.2.m2.4.4.2">𝑏</ci></apply></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.16.p2.2.m2.8c">\Delta(p(\bar{a}),p(\bar{c}))=\Delta(p(\bar{a}),p(\bar{b}))</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.16.p2.2.m2.8d">roman_Δ ( italic_p ( over¯ start_ARG italic_a end_ARG ) , italic_p ( over¯ start_ARG italic_c end_ARG ) ) = roman_Δ ( italic_p ( over¯ start_ARG italic_a end_ARG ) , italic_p ( over¯ start_ARG italic_b end_ARG ) )</annotation></semantics></math> or <math alttext="\Delta(p(\bar{a}),p(\bar{c}))=\Delta(p(\bar{b}),p(\bar{c}))" class="ltx_Math" display="inline" id="S6.SS2.16.p2.3.m3.8"><semantics id="S6.SS2.16.p2.3.m3.8a"><mrow id="S6.SS2.16.p2.3.m3.8.8" xref="S6.SS2.16.p2.3.m3.8.8.cmml"><mrow id="S6.SS2.16.p2.3.m3.6.6.2" xref="S6.SS2.16.p2.3.m3.6.6.2.cmml"><mi id="S6.SS2.16.p2.3.m3.6.6.2.4" mathvariant="normal" xref="S6.SS2.16.p2.3.m3.6.6.2.4.cmml">Δ</mi><mo id="S6.SS2.16.p2.3.m3.6.6.2.3" xref="S6.SS2.16.p2.3.m3.6.6.2.3.cmml">⁢</mo><mrow id="S6.SS2.16.p2.3.m3.6.6.2.2.2" xref="S6.SS2.16.p2.3.m3.6.6.2.2.3.cmml"><mo id="S6.SS2.16.p2.3.m3.6.6.2.2.2.3" stretchy="false" xref="S6.SS2.16.p2.3.m3.6.6.2.2.3.cmml">(</mo><mrow id="S6.SS2.16.p2.3.m3.5.5.1.1.1.1" xref="S6.SS2.16.p2.3.m3.5.5.1.1.1.1.cmml"><mi id="S6.SS2.16.p2.3.m3.5.5.1.1.1.1.2" xref="S6.SS2.16.p2.3.m3.5.5.1.1.1.1.2.cmml">p</mi><mo id="S6.SS2.16.p2.3.m3.5.5.1.1.1.1.1" xref="S6.SS2.16.p2.3.m3.5.5.1.1.1.1.1.cmml">⁢</mo><mrow id="S6.SS2.16.p2.3.m3.5.5.1.1.1.1.3.2" xref="S6.SS2.16.p2.3.m3.1.1.cmml"><mo id="S6.SS2.16.p2.3.m3.5.5.1.1.1.1.3.2.1" stretchy="false" xref="S6.SS2.16.p2.3.m3.1.1.cmml">(</mo><mover accent="true" id="S6.SS2.16.p2.3.m3.1.1" xref="S6.SS2.16.p2.3.m3.1.1.cmml"><mi id="S6.SS2.16.p2.3.m3.1.1.2" xref="S6.SS2.16.p2.3.m3.1.1.2.cmml">a</mi><mo id="S6.SS2.16.p2.3.m3.1.1.1" xref="S6.SS2.16.p2.3.m3.1.1.1.cmml">¯</mo></mover><mo id="S6.SS2.16.p2.3.m3.5.5.1.1.1.1.3.2.2" stretchy="false" xref="S6.SS2.16.p2.3.m3.1.1.cmml">)</mo></mrow></mrow><mo id="S6.SS2.16.p2.3.m3.6.6.2.2.2.4" xref="S6.SS2.16.p2.3.m3.6.6.2.2.3.cmml">,</mo><mrow id="S6.SS2.16.p2.3.m3.6.6.2.2.2.2" xref="S6.SS2.16.p2.3.m3.6.6.2.2.2.2.cmml"><mi id="S6.SS2.16.p2.3.m3.6.6.2.2.2.2.2" xref="S6.SS2.16.p2.3.m3.6.6.2.2.2.2.2.cmml">p</mi><mo id="S6.SS2.16.p2.3.m3.6.6.2.2.2.2.1" xref="S6.SS2.16.p2.3.m3.6.6.2.2.2.2.1.cmml">⁢</mo><mrow id="S6.SS2.16.p2.3.m3.6.6.2.2.2.2.3.2" xref="S6.SS2.16.p2.3.m3.2.2.cmml"><mo id="S6.SS2.16.p2.3.m3.6.6.2.2.2.2.3.2.1" stretchy="false" xref="S6.SS2.16.p2.3.m3.2.2.cmml">(</mo><mover accent="true" id="S6.SS2.16.p2.3.m3.2.2" xref="S6.SS2.16.p2.3.m3.2.2.cmml"><mi id="S6.SS2.16.p2.3.m3.2.2.2" xref="S6.SS2.16.p2.3.m3.2.2.2.cmml">c</mi><mo id="S6.SS2.16.p2.3.m3.2.2.1" xref="S6.SS2.16.p2.3.m3.2.2.1.cmml">¯</mo></mover><mo id="S6.SS2.16.p2.3.m3.6.6.2.2.2.2.3.2.2" stretchy="false" xref="S6.SS2.16.p2.3.m3.2.2.cmml">)</mo></mrow></mrow><mo id="S6.SS2.16.p2.3.m3.6.6.2.2.2.5" stretchy="false" xref="S6.SS2.16.p2.3.m3.6.6.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.SS2.16.p2.3.m3.8.8.5" xref="S6.SS2.16.p2.3.m3.8.8.5.cmml">=</mo><mrow id="S6.SS2.16.p2.3.m3.8.8.4" xref="S6.SS2.16.p2.3.m3.8.8.4.cmml"><mi id="S6.SS2.16.p2.3.m3.8.8.4.4" mathvariant="normal" xref="S6.SS2.16.p2.3.m3.8.8.4.4.cmml">Δ</mi><mo id="S6.SS2.16.p2.3.m3.8.8.4.3" xref="S6.SS2.16.p2.3.m3.8.8.4.3.cmml">⁢</mo><mrow id="S6.SS2.16.p2.3.m3.8.8.4.2.2" xref="S6.SS2.16.p2.3.m3.8.8.4.2.3.cmml"><mo id="S6.SS2.16.p2.3.m3.8.8.4.2.2.3" stretchy="false" xref="S6.SS2.16.p2.3.m3.8.8.4.2.3.cmml">(</mo><mrow id="S6.SS2.16.p2.3.m3.7.7.3.1.1.1" xref="S6.SS2.16.p2.3.m3.7.7.3.1.1.1.cmml"><mi id="S6.SS2.16.p2.3.m3.7.7.3.1.1.1.2" xref="S6.SS2.16.p2.3.m3.7.7.3.1.1.1.2.cmml">p</mi><mo id="S6.SS2.16.p2.3.m3.7.7.3.1.1.1.1" xref="S6.SS2.16.p2.3.m3.7.7.3.1.1.1.1.cmml">⁢</mo><mrow id="S6.SS2.16.p2.3.m3.7.7.3.1.1.1.3.2" xref="S6.SS2.16.p2.3.m3.3.3.cmml"><mo id="S6.SS2.16.p2.3.m3.7.7.3.1.1.1.3.2.1" stretchy="false" xref="S6.SS2.16.p2.3.m3.3.3.cmml">(</mo><mover accent="true" id="S6.SS2.16.p2.3.m3.3.3" xref="S6.SS2.16.p2.3.m3.3.3.cmml"><mi id="S6.SS2.16.p2.3.m3.3.3.2" xref="S6.SS2.16.p2.3.m3.3.3.2.cmml">b</mi><mo id="S6.SS2.16.p2.3.m3.3.3.1" xref="S6.SS2.16.p2.3.m3.3.3.1.cmml">¯</mo></mover><mo id="S6.SS2.16.p2.3.m3.7.7.3.1.1.1.3.2.2" stretchy="false" xref="S6.SS2.16.p2.3.m3.3.3.cmml">)</mo></mrow></mrow><mo id="S6.SS2.16.p2.3.m3.8.8.4.2.2.4" xref="S6.SS2.16.p2.3.m3.8.8.4.2.3.cmml">,</mo><mrow id="S6.SS2.16.p2.3.m3.8.8.4.2.2.2" xref="S6.SS2.16.p2.3.m3.8.8.4.2.2.2.cmml"><mi id="S6.SS2.16.p2.3.m3.8.8.4.2.2.2.2" xref="S6.SS2.16.p2.3.m3.8.8.4.2.2.2.2.cmml">p</mi><mo id="S6.SS2.16.p2.3.m3.8.8.4.2.2.2.1" xref="S6.SS2.16.p2.3.m3.8.8.4.2.2.2.1.cmml">⁢</mo><mrow id="S6.SS2.16.p2.3.m3.8.8.4.2.2.2.3.2" xref="S6.SS2.16.p2.3.m3.4.4.cmml"><mo id="S6.SS2.16.p2.3.m3.8.8.4.2.2.2.3.2.1" stretchy="false" xref="S6.SS2.16.p2.3.m3.4.4.cmml">(</mo><mover accent="true" id="S6.SS2.16.p2.3.m3.4.4" xref="S6.SS2.16.p2.3.m3.4.4.cmml"><mi id="S6.SS2.16.p2.3.m3.4.4.2" xref="S6.SS2.16.p2.3.m3.4.4.2.cmml">c</mi><mo id="S6.SS2.16.p2.3.m3.4.4.1" xref="S6.SS2.16.p2.3.m3.4.4.1.cmml">¯</mo></mover><mo id="S6.SS2.16.p2.3.m3.8.8.4.2.2.2.3.2.2" stretchy="false" xref="S6.SS2.16.p2.3.m3.4.4.cmml">)</mo></mrow></mrow><mo id="S6.SS2.16.p2.3.m3.8.8.4.2.2.5" stretchy="false" xref="S6.SS2.16.p2.3.m3.8.8.4.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.16.p2.3.m3.8b"><apply id="S6.SS2.16.p2.3.m3.8.8.cmml" xref="S6.SS2.16.p2.3.m3.8.8"><eq id="S6.SS2.16.p2.3.m3.8.8.5.cmml" xref="S6.SS2.16.p2.3.m3.8.8.5"></eq><apply id="S6.SS2.16.p2.3.m3.6.6.2.cmml" xref="S6.SS2.16.p2.3.m3.6.6.2"><times id="S6.SS2.16.p2.3.m3.6.6.2.3.cmml" xref="S6.SS2.16.p2.3.m3.6.6.2.3"></times><ci id="S6.SS2.16.p2.3.m3.6.6.2.4.cmml" xref="S6.SS2.16.p2.3.m3.6.6.2.4">Δ</ci><interval closure="open" id="S6.SS2.16.p2.3.m3.6.6.2.2.3.cmml" xref="S6.SS2.16.p2.3.m3.6.6.2.2.2"><apply id="S6.SS2.16.p2.3.m3.5.5.1.1.1.1.cmml" xref="S6.SS2.16.p2.3.m3.5.5.1.1.1.1"><times id="S6.SS2.16.p2.3.m3.5.5.1.1.1.1.1.cmml" xref="S6.SS2.16.p2.3.m3.5.5.1.1.1.1.1"></times><ci id="S6.SS2.16.p2.3.m3.5.5.1.1.1.1.2.cmml" xref="S6.SS2.16.p2.3.m3.5.5.1.1.1.1.2">𝑝</ci><apply id="S6.SS2.16.p2.3.m3.1.1.cmml" xref="S6.SS2.16.p2.3.m3.5.5.1.1.1.1.3.2"><ci id="S6.SS2.16.p2.3.m3.1.1.1.cmml" xref="S6.SS2.16.p2.3.m3.1.1.1">¯</ci><ci id="S6.SS2.16.p2.3.m3.1.1.2.cmml" xref="S6.SS2.16.p2.3.m3.1.1.2">𝑎</ci></apply></apply><apply id="S6.SS2.16.p2.3.m3.6.6.2.2.2.2.cmml" xref="S6.SS2.16.p2.3.m3.6.6.2.2.2.2"><times id="S6.SS2.16.p2.3.m3.6.6.2.2.2.2.1.cmml" xref="S6.SS2.16.p2.3.m3.6.6.2.2.2.2.1"></times><ci id="S6.SS2.16.p2.3.m3.6.6.2.2.2.2.2.cmml" xref="S6.SS2.16.p2.3.m3.6.6.2.2.2.2.2">𝑝</ci><apply id="S6.SS2.16.p2.3.m3.2.2.cmml" xref="S6.SS2.16.p2.3.m3.6.6.2.2.2.2.3.2"><ci id="S6.SS2.16.p2.3.m3.2.2.1.cmml" xref="S6.SS2.16.p2.3.m3.2.2.1">¯</ci><ci id="S6.SS2.16.p2.3.m3.2.2.2.cmml" xref="S6.SS2.16.p2.3.m3.2.2.2">𝑐</ci></apply></apply></interval></apply><apply id="S6.SS2.16.p2.3.m3.8.8.4.cmml" xref="S6.SS2.16.p2.3.m3.8.8.4"><times id="S6.SS2.16.p2.3.m3.8.8.4.3.cmml" xref="S6.SS2.16.p2.3.m3.8.8.4.3"></times><ci id="S6.SS2.16.p2.3.m3.8.8.4.4.cmml" xref="S6.SS2.16.p2.3.m3.8.8.4.4">Δ</ci><interval closure="open" id="S6.SS2.16.p2.3.m3.8.8.4.2.3.cmml" xref="S6.SS2.16.p2.3.m3.8.8.4.2.2"><apply id="S6.SS2.16.p2.3.m3.7.7.3.1.1.1.cmml" xref="S6.SS2.16.p2.3.m3.7.7.3.1.1.1"><times id="S6.SS2.16.p2.3.m3.7.7.3.1.1.1.1.cmml" xref="S6.SS2.16.p2.3.m3.7.7.3.1.1.1.1"></times><ci id="S6.SS2.16.p2.3.m3.7.7.3.1.1.1.2.cmml" xref="S6.SS2.16.p2.3.m3.7.7.3.1.1.1.2">𝑝</ci><apply id="S6.SS2.16.p2.3.m3.3.3.cmml" xref="S6.SS2.16.p2.3.m3.7.7.3.1.1.1.3.2"><ci id="S6.SS2.16.p2.3.m3.3.3.1.cmml" xref="S6.SS2.16.p2.3.m3.3.3.1">¯</ci><ci id="S6.SS2.16.p2.3.m3.3.3.2.cmml" xref="S6.SS2.16.p2.3.m3.3.3.2">𝑏</ci></apply></apply><apply id="S6.SS2.16.p2.3.m3.8.8.4.2.2.2.cmml" xref="S6.SS2.16.p2.3.m3.8.8.4.2.2.2"><times id="S6.SS2.16.p2.3.m3.8.8.4.2.2.2.1.cmml" xref="S6.SS2.16.p2.3.m3.8.8.4.2.2.2.1"></times><ci id="S6.SS2.16.p2.3.m3.8.8.4.2.2.2.2.cmml" xref="S6.SS2.16.p2.3.m3.8.8.4.2.2.2.2">𝑝</ci><apply id="S6.SS2.16.p2.3.m3.4.4.cmml" xref="S6.SS2.16.p2.3.m3.8.8.4.2.2.2.3.2"><ci id="S6.SS2.16.p2.3.m3.4.4.1.cmml" xref="S6.SS2.16.p2.3.m3.4.4.1">¯</ci><ci id="S6.SS2.16.p2.3.m3.4.4.2.cmml" xref="S6.SS2.16.p2.3.m3.4.4.2">𝑐</ci></apply></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.16.p2.3.m3.8c">\Delta(p(\bar{a}),p(\bar{c}))=\Delta(p(\bar{b}),p(\bar{c}))</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.16.p2.3.m3.8d">roman_Δ ( italic_p ( over¯ start_ARG italic_a end_ARG ) , italic_p ( over¯ start_ARG italic_c end_ARG ) ) = roman_Δ ( italic_p ( over¯ start_ARG italic_b end_ARG ) , italic_p ( over¯ start_ARG italic_c end_ARG ) )</annotation></semantics></math>. Assume the second, the other case is symmetric. So <math alttext="\nu:=\nu(p(\bar{a}),p(\bar{c}))=\nu(p(\bar{b}),p(\bar{c}))" class="ltx_Math" display="inline" id="S6.SS2.16.p2.4.m4.8"><semantics id="S6.SS2.16.p2.4.m4.8a"><mrow id="S6.SS2.16.p2.4.m4.8.8" xref="S6.SS2.16.p2.4.m4.8.8.cmml"><mi id="S6.SS2.16.p2.4.m4.8.8.6" xref="S6.SS2.16.p2.4.m4.8.8.6.cmml">ν</mi><mo id="S6.SS2.16.p2.4.m4.8.8.7" lspace="0.278em" rspace="0.278em" xref="S6.SS2.16.p2.4.m4.8.8.7.cmml">:=</mo><mrow id="S6.SS2.16.p2.4.m4.6.6.2" xref="S6.SS2.16.p2.4.m4.6.6.2.cmml"><mi id="S6.SS2.16.p2.4.m4.6.6.2.4" xref="S6.SS2.16.p2.4.m4.6.6.2.4.cmml">ν</mi><mo id="S6.SS2.16.p2.4.m4.6.6.2.3" xref="S6.SS2.16.p2.4.m4.6.6.2.3.cmml">⁢</mo><mrow id="S6.SS2.16.p2.4.m4.6.6.2.2.2" xref="S6.SS2.16.p2.4.m4.6.6.2.2.3.cmml"><mo id="S6.SS2.16.p2.4.m4.6.6.2.2.2.3" stretchy="false" xref="S6.SS2.16.p2.4.m4.6.6.2.2.3.cmml">(</mo><mrow id="S6.SS2.16.p2.4.m4.5.5.1.1.1.1" xref="S6.SS2.16.p2.4.m4.5.5.1.1.1.1.cmml"><mi id="S6.SS2.16.p2.4.m4.5.5.1.1.1.1.2" xref="S6.SS2.16.p2.4.m4.5.5.1.1.1.1.2.cmml">p</mi><mo id="S6.SS2.16.p2.4.m4.5.5.1.1.1.1.1" xref="S6.SS2.16.p2.4.m4.5.5.1.1.1.1.1.cmml">⁢</mo><mrow id="S6.SS2.16.p2.4.m4.5.5.1.1.1.1.3.2" xref="S6.SS2.16.p2.4.m4.1.1.cmml"><mo id="S6.SS2.16.p2.4.m4.5.5.1.1.1.1.3.2.1" stretchy="false" xref="S6.SS2.16.p2.4.m4.1.1.cmml">(</mo><mover accent="true" id="S6.SS2.16.p2.4.m4.1.1" xref="S6.SS2.16.p2.4.m4.1.1.cmml"><mi id="S6.SS2.16.p2.4.m4.1.1.2" xref="S6.SS2.16.p2.4.m4.1.1.2.cmml">a</mi><mo id="S6.SS2.16.p2.4.m4.1.1.1" xref="S6.SS2.16.p2.4.m4.1.1.1.cmml">¯</mo></mover><mo id="S6.SS2.16.p2.4.m4.5.5.1.1.1.1.3.2.2" stretchy="false" xref="S6.SS2.16.p2.4.m4.1.1.cmml">)</mo></mrow></mrow><mo id="S6.SS2.16.p2.4.m4.6.6.2.2.2.4" xref="S6.SS2.16.p2.4.m4.6.6.2.2.3.cmml">,</mo><mrow id="S6.SS2.16.p2.4.m4.6.6.2.2.2.2" xref="S6.SS2.16.p2.4.m4.6.6.2.2.2.2.cmml"><mi id="S6.SS2.16.p2.4.m4.6.6.2.2.2.2.2" xref="S6.SS2.16.p2.4.m4.6.6.2.2.2.2.2.cmml">p</mi><mo id="S6.SS2.16.p2.4.m4.6.6.2.2.2.2.1" xref="S6.SS2.16.p2.4.m4.6.6.2.2.2.2.1.cmml">⁢</mo><mrow id="S6.SS2.16.p2.4.m4.6.6.2.2.2.2.3.2" xref="S6.SS2.16.p2.4.m4.2.2.cmml"><mo id="S6.SS2.16.p2.4.m4.6.6.2.2.2.2.3.2.1" stretchy="false" xref="S6.SS2.16.p2.4.m4.2.2.cmml">(</mo><mover accent="true" id="S6.SS2.16.p2.4.m4.2.2" xref="S6.SS2.16.p2.4.m4.2.2.cmml"><mi id="S6.SS2.16.p2.4.m4.2.2.2" xref="S6.SS2.16.p2.4.m4.2.2.2.cmml">c</mi><mo id="S6.SS2.16.p2.4.m4.2.2.1" xref="S6.SS2.16.p2.4.m4.2.2.1.cmml">¯</mo></mover><mo id="S6.SS2.16.p2.4.m4.6.6.2.2.2.2.3.2.2" stretchy="false" xref="S6.SS2.16.p2.4.m4.2.2.cmml">)</mo></mrow></mrow><mo id="S6.SS2.16.p2.4.m4.6.6.2.2.2.5" stretchy="false" xref="S6.SS2.16.p2.4.m4.6.6.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.SS2.16.p2.4.m4.8.8.8" xref="S6.SS2.16.p2.4.m4.8.8.8.cmml">=</mo><mrow id="S6.SS2.16.p2.4.m4.8.8.4" xref="S6.SS2.16.p2.4.m4.8.8.4.cmml"><mi id="S6.SS2.16.p2.4.m4.8.8.4.4" xref="S6.SS2.16.p2.4.m4.8.8.4.4.cmml">ν</mi><mo id="S6.SS2.16.p2.4.m4.8.8.4.3" xref="S6.SS2.16.p2.4.m4.8.8.4.3.cmml">⁢</mo><mrow id="S6.SS2.16.p2.4.m4.8.8.4.2.2" xref="S6.SS2.16.p2.4.m4.8.8.4.2.3.cmml"><mo id="S6.SS2.16.p2.4.m4.8.8.4.2.2.3" stretchy="false" xref="S6.SS2.16.p2.4.m4.8.8.4.2.3.cmml">(</mo><mrow id="S6.SS2.16.p2.4.m4.7.7.3.1.1.1" xref="S6.SS2.16.p2.4.m4.7.7.3.1.1.1.cmml"><mi id="S6.SS2.16.p2.4.m4.7.7.3.1.1.1.2" xref="S6.SS2.16.p2.4.m4.7.7.3.1.1.1.2.cmml">p</mi><mo id="S6.SS2.16.p2.4.m4.7.7.3.1.1.1.1" xref="S6.SS2.16.p2.4.m4.7.7.3.1.1.1.1.cmml">⁢</mo><mrow id="S6.SS2.16.p2.4.m4.7.7.3.1.1.1.3.2" xref="S6.SS2.16.p2.4.m4.3.3.cmml"><mo id="S6.SS2.16.p2.4.m4.7.7.3.1.1.1.3.2.1" stretchy="false" xref="S6.SS2.16.p2.4.m4.3.3.cmml">(</mo><mover accent="true" id="S6.SS2.16.p2.4.m4.3.3" xref="S6.SS2.16.p2.4.m4.3.3.cmml"><mi id="S6.SS2.16.p2.4.m4.3.3.2" xref="S6.SS2.16.p2.4.m4.3.3.2.cmml">b</mi><mo id="S6.SS2.16.p2.4.m4.3.3.1" xref="S6.SS2.16.p2.4.m4.3.3.1.cmml">¯</mo></mover><mo id="S6.SS2.16.p2.4.m4.7.7.3.1.1.1.3.2.2" stretchy="false" xref="S6.SS2.16.p2.4.m4.3.3.cmml">)</mo></mrow></mrow><mo id="S6.SS2.16.p2.4.m4.8.8.4.2.2.4" xref="S6.SS2.16.p2.4.m4.8.8.4.2.3.cmml">,</mo><mrow id="S6.SS2.16.p2.4.m4.8.8.4.2.2.2" xref="S6.SS2.16.p2.4.m4.8.8.4.2.2.2.cmml"><mi id="S6.SS2.16.p2.4.m4.8.8.4.2.2.2.2" xref="S6.SS2.16.p2.4.m4.8.8.4.2.2.2.2.cmml">p</mi><mo id="S6.SS2.16.p2.4.m4.8.8.4.2.2.2.1" xref="S6.SS2.16.p2.4.m4.8.8.4.2.2.2.1.cmml">⁢</mo><mrow id="S6.SS2.16.p2.4.m4.8.8.4.2.2.2.3.2" xref="S6.SS2.16.p2.4.m4.4.4.cmml"><mo id="S6.SS2.16.p2.4.m4.8.8.4.2.2.2.3.2.1" stretchy="false" xref="S6.SS2.16.p2.4.m4.4.4.cmml">(</mo><mover accent="true" id="S6.SS2.16.p2.4.m4.4.4" xref="S6.SS2.16.p2.4.m4.4.4.cmml"><mi id="S6.SS2.16.p2.4.m4.4.4.2" xref="S6.SS2.16.p2.4.m4.4.4.2.cmml">c</mi><mo id="S6.SS2.16.p2.4.m4.4.4.1" xref="S6.SS2.16.p2.4.m4.4.4.1.cmml">¯</mo></mover><mo id="S6.SS2.16.p2.4.m4.8.8.4.2.2.2.3.2.2" stretchy="false" xref="S6.SS2.16.p2.4.m4.4.4.cmml">)</mo></mrow></mrow><mo id="S6.SS2.16.p2.4.m4.8.8.4.2.2.5" stretchy="false" xref="S6.SS2.16.p2.4.m4.8.8.4.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.16.p2.4.m4.8b"><apply id="S6.SS2.16.p2.4.m4.8.8.cmml" xref="S6.SS2.16.p2.4.m4.8.8"><and id="S6.SS2.16.p2.4.m4.8.8a.cmml" xref="S6.SS2.16.p2.4.m4.8.8"></and><apply id="S6.SS2.16.p2.4.m4.8.8b.cmml" xref="S6.SS2.16.p2.4.m4.8.8"><csymbol cd="latexml" id="S6.SS2.16.p2.4.m4.8.8.7.cmml" xref="S6.SS2.16.p2.4.m4.8.8.7">assign</csymbol><ci id="S6.SS2.16.p2.4.m4.8.8.6.cmml" xref="S6.SS2.16.p2.4.m4.8.8.6">𝜈</ci><apply id="S6.SS2.16.p2.4.m4.6.6.2.cmml" xref="S6.SS2.16.p2.4.m4.6.6.2"><times id="S6.SS2.16.p2.4.m4.6.6.2.3.cmml" xref="S6.SS2.16.p2.4.m4.6.6.2.3"></times><ci id="S6.SS2.16.p2.4.m4.6.6.2.4.cmml" xref="S6.SS2.16.p2.4.m4.6.6.2.4">𝜈</ci><interval closure="open" id="S6.SS2.16.p2.4.m4.6.6.2.2.3.cmml" xref="S6.SS2.16.p2.4.m4.6.6.2.2.2"><apply id="S6.SS2.16.p2.4.m4.5.5.1.1.1.1.cmml" xref="S6.SS2.16.p2.4.m4.5.5.1.1.1.1"><times id="S6.SS2.16.p2.4.m4.5.5.1.1.1.1.1.cmml" xref="S6.SS2.16.p2.4.m4.5.5.1.1.1.1.1"></times><ci id="S6.SS2.16.p2.4.m4.5.5.1.1.1.1.2.cmml" xref="S6.SS2.16.p2.4.m4.5.5.1.1.1.1.2">𝑝</ci><apply id="S6.SS2.16.p2.4.m4.1.1.cmml" xref="S6.SS2.16.p2.4.m4.5.5.1.1.1.1.3.2"><ci id="S6.SS2.16.p2.4.m4.1.1.1.cmml" xref="S6.SS2.16.p2.4.m4.1.1.1">¯</ci><ci id="S6.SS2.16.p2.4.m4.1.1.2.cmml" xref="S6.SS2.16.p2.4.m4.1.1.2">𝑎</ci></apply></apply><apply id="S6.SS2.16.p2.4.m4.6.6.2.2.2.2.cmml" xref="S6.SS2.16.p2.4.m4.6.6.2.2.2.2"><times id="S6.SS2.16.p2.4.m4.6.6.2.2.2.2.1.cmml" xref="S6.SS2.16.p2.4.m4.6.6.2.2.2.2.1"></times><ci id="S6.SS2.16.p2.4.m4.6.6.2.2.2.2.2.cmml" xref="S6.SS2.16.p2.4.m4.6.6.2.2.2.2.2">𝑝</ci><apply id="S6.SS2.16.p2.4.m4.2.2.cmml" xref="S6.SS2.16.p2.4.m4.6.6.2.2.2.2.3.2"><ci id="S6.SS2.16.p2.4.m4.2.2.1.cmml" xref="S6.SS2.16.p2.4.m4.2.2.1">¯</ci><ci id="S6.SS2.16.p2.4.m4.2.2.2.cmml" xref="S6.SS2.16.p2.4.m4.2.2.2">𝑐</ci></apply></apply></interval></apply></apply><apply id="S6.SS2.16.p2.4.m4.8.8c.cmml" xref="S6.SS2.16.p2.4.m4.8.8"><eq id="S6.SS2.16.p2.4.m4.8.8.8.cmml" xref="S6.SS2.16.p2.4.m4.8.8.8"></eq><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.16.p2.4.m4.6.6.2.cmml" id="S6.SS2.16.p2.4.m4.8.8d.cmml" xref="S6.SS2.16.p2.4.m4.8.8"></share><apply id="S6.SS2.16.p2.4.m4.8.8.4.cmml" xref="S6.SS2.16.p2.4.m4.8.8.4"><times id="S6.SS2.16.p2.4.m4.8.8.4.3.cmml" xref="S6.SS2.16.p2.4.m4.8.8.4.3"></times><ci id="S6.SS2.16.p2.4.m4.8.8.4.4.cmml" xref="S6.SS2.16.p2.4.m4.8.8.4.4">𝜈</ci><interval closure="open" id="S6.SS2.16.p2.4.m4.8.8.4.2.3.cmml" xref="S6.SS2.16.p2.4.m4.8.8.4.2.2"><apply id="S6.SS2.16.p2.4.m4.7.7.3.1.1.1.cmml" xref="S6.SS2.16.p2.4.m4.7.7.3.1.1.1"><times id="S6.SS2.16.p2.4.m4.7.7.3.1.1.1.1.cmml" xref="S6.SS2.16.p2.4.m4.7.7.3.1.1.1.1"></times><ci id="S6.SS2.16.p2.4.m4.7.7.3.1.1.1.2.cmml" xref="S6.SS2.16.p2.4.m4.7.7.3.1.1.1.2">𝑝</ci><apply id="S6.SS2.16.p2.4.m4.3.3.cmml" xref="S6.SS2.16.p2.4.m4.7.7.3.1.1.1.3.2"><ci id="S6.SS2.16.p2.4.m4.3.3.1.cmml" xref="S6.SS2.16.p2.4.m4.3.3.1">¯</ci><ci id="S6.SS2.16.p2.4.m4.3.3.2.cmml" xref="S6.SS2.16.p2.4.m4.3.3.2">𝑏</ci></apply></apply><apply id="S6.SS2.16.p2.4.m4.8.8.4.2.2.2.cmml" xref="S6.SS2.16.p2.4.m4.8.8.4.2.2.2"><times id="S6.SS2.16.p2.4.m4.8.8.4.2.2.2.1.cmml" xref="S6.SS2.16.p2.4.m4.8.8.4.2.2.2.1"></times><ci id="S6.SS2.16.p2.4.m4.8.8.4.2.2.2.2.cmml" xref="S6.SS2.16.p2.4.m4.8.8.4.2.2.2.2">𝑝</ci><apply id="S6.SS2.16.p2.4.m4.4.4.cmml" xref="S6.SS2.16.p2.4.m4.8.8.4.2.2.2.3.2"><ci id="S6.SS2.16.p2.4.m4.4.4.1.cmml" xref="S6.SS2.16.p2.4.m4.4.4.1">¯</ci><ci id="S6.SS2.16.p2.4.m4.4.4.2.cmml" xref="S6.SS2.16.p2.4.m4.4.4.2">𝑐</ci></apply></apply></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.16.p2.4.m4.8c">\nu:=\nu(p(\bar{a}),p(\bar{c}))=\nu(p(\bar{b}),p(\bar{c}))</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.16.p2.4.m4.8d">italic_ν := italic_ν ( italic_p ( over¯ start_ARG italic_a end_ARG ) , italic_p ( over¯ start_ARG italic_c end_ARG ) ) = italic_ν ( italic_p ( over¯ start_ARG italic_b end_ARG ) , italic_p ( over¯ start_ARG italic_c end_ARG ) )</annotation></semantics></math>. Since <math alttext="p" class="ltx_Math" display="inline" id="S6.SS2.16.p2.5.m5.1"><semantics id="S6.SS2.16.p2.5.m5.1a"><mi id="S6.SS2.16.p2.5.m5.1.1" xref="S6.SS2.16.p2.5.m5.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.16.p2.5.m5.1b"><ci id="S6.SS2.16.p2.5.m5.1.1.cmml" xref="S6.SS2.16.p2.5.m5.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.16.p2.5.m5.1c">p</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.16.p2.5.m5.1d">italic_p</annotation></semantics></math> satisfies <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.Thmtheorem9" title="Definition 6.9. ‣ 6.2. Forcing epimorphisms ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">6.9</span></a> <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.I4.i2" title="Item (ii) ‣ Definition 6.9. ‣ 6.2. Forcing epimorphisms ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">(ii)</span></a> for neighbors, <math alttext="\nu=\nu(b_{m},c_{m})" class="ltx_Math" display="inline" id="S6.SS2.16.p2.6.m6.2"><semantics id="S6.SS2.16.p2.6.m6.2a"><mrow id="S6.SS2.16.p2.6.m6.2.2" xref="S6.SS2.16.p2.6.m6.2.2.cmml"><mi id="S6.SS2.16.p2.6.m6.2.2.4" xref="S6.SS2.16.p2.6.m6.2.2.4.cmml">ν</mi><mo id="S6.SS2.16.p2.6.m6.2.2.3" xref="S6.SS2.16.p2.6.m6.2.2.3.cmml">=</mo><mrow id="S6.SS2.16.p2.6.m6.2.2.2" xref="S6.SS2.16.p2.6.m6.2.2.2.cmml"><mi id="S6.SS2.16.p2.6.m6.2.2.2.4" xref="S6.SS2.16.p2.6.m6.2.2.2.4.cmml">ν</mi><mo id="S6.SS2.16.p2.6.m6.2.2.2.3" xref="S6.SS2.16.p2.6.m6.2.2.2.3.cmml">⁢</mo><mrow id="S6.SS2.16.p2.6.m6.2.2.2.2.2" xref="S6.SS2.16.p2.6.m6.2.2.2.2.3.cmml"><mo id="S6.SS2.16.p2.6.m6.2.2.2.2.2.3" stretchy="false" xref="S6.SS2.16.p2.6.m6.2.2.2.2.3.cmml">(</mo><msub id="S6.SS2.16.p2.6.m6.1.1.1.1.1.1" xref="S6.SS2.16.p2.6.m6.1.1.1.1.1.1.cmml"><mi id="S6.SS2.16.p2.6.m6.1.1.1.1.1.1.2" xref="S6.SS2.16.p2.6.m6.1.1.1.1.1.1.2.cmml">b</mi><mi id="S6.SS2.16.p2.6.m6.1.1.1.1.1.1.3" xref="S6.SS2.16.p2.6.m6.1.1.1.1.1.1.3.cmml">m</mi></msub><mo id="S6.SS2.16.p2.6.m6.2.2.2.2.2.4" xref="S6.SS2.16.p2.6.m6.2.2.2.2.3.cmml">,</mo><msub id="S6.SS2.16.p2.6.m6.2.2.2.2.2.2" xref="S6.SS2.16.p2.6.m6.2.2.2.2.2.2.cmml"><mi id="S6.SS2.16.p2.6.m6.2.2.2.2.2.2.2" xref="S6.SS2.16.p2.6.m6.2.2.2.2.2.2.2.cmml">c</mi><mi id="S6.SS2.16.p2.6.m6.2.2.2.2.2.2.3" xref="S6.SS2.16.p2.6.m6.2.2.2.2.2.2.3.cmml">m</mi></msub><mo id="S6.SS2.16.p2.6.m6.2.2.2.2.2.5" stretchy="false" xref="S6.SS2.16.p2.6.m6.2.2.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.16.p2.6.m6.2b"><apply id="S6.SS2.16.p2.6.m6.2.2.cmml" xref="S6.SS2.16.p2.6.m6.2.2"><eq id="S6.SS2.16.p2.6.m6.2.2.3.cmml" xref="S6.SS2.16.p2.6.m6.2.2.3"></eq><ci id="S6.SS2.16.p2.6.m6.2.2.4.cmml" xref="S6.SS2.16.p2.6.m6.2.2.4">𝜈</ci><apply id="S6.SS2.16.p2.6.m6.2.2.2.cmml" xref="S6.SS2.16.p2.6.m6.2.2.2"><times id="S6.SS2.16.p2.6.m6.2.2.2.3.cmml" xref="S6.SS2.16.p2.6.m6.2.2.2.3"></times><ci id="S6.SS2.16.p2.6.m6.2.2.2.4.cmml" xref="S6.SS2.16.p2.6.m6.2.2.2.4">𝜈</ci><interval closure="open" id="S6.SS2.16.p2.6.m6.2.2.2.2.3.cmml" xref="S6.SS2.16.p2.6.m6.2.2.2.2.2"><apply id="S6.SS2.16.p2.6.m6.1.1.1.1.1.1.cmml" xref="S6.SS2.16.p2.6.m6.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.16.p2.6.m6.1.1.1.1.1.1.1.cmml" xref="S6.SS2.16.p2.6.m6.1.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.16.p2.6.m6.1.1.1.1.1.1.2.cmml" xref="S6.SS2.16.p2.6.m6.1.1.1.1.1.1.2">𝑏</ci><ci id="S6.SS2.16.p2.6.m6.1.1.1.1.1.1.3.cmml" xref="S6.SS2.16.p2.6.m6.1.1.1.1.1.1.3">𝑚</ci></apply><apply id="S6.SS2.16.p2.6.m6.2.2.2.2.2.2.cmml" xref="S6.SS2.16.p2.6.m6.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.16.p2.6.m6.2.2.2.2.2.2.1.cmml" xref="S6.SS2.16.p2.6.m6.2.2.2.2.2.2">subscript</csymbol><ci id="S6.SS2.16.p2.6.m6.2.2.2.2.2.2.2.cmml" xref="S6.SS2.16.p2.6.m6.2.2.2.2.2.2.2">𝑐</ci><ci id="S6.SS2.16.p2.6.m6.2.2.2.2.2.2.3.cmml" xref="S6.SS2.16.p2.6.m6.2.2.2.2.2.2.3">𝑚</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.16.p2.6.m6.2c">\nu=\nu(b_{m},c_{m})</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.16.p2.6.m6.2d">italic_ν = italic_ν ( italic_b start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT , italic_c start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT )</annotation></semantics></math> and <math alttext="\mu:=\nu(a_{m},b_{m})=\nu(p(\bar{a}),p(\bar{b}))" class="ltx_Math" display="inline" id="S6.SS2.16.p2.7.m7.6"><semantics id="S6.SS2.16.p2.7.m7.6a"><mrow id="S6.SS2.16.p2.7.m7.6.6" xref="S6.SS2.16.p2.7.m7.6.6.cmml"><mi id="S6.SS2.16.p2.7.m7.6.6.6" xref="S6.SS2.16.p2.7.m7.6.6.6.cmml">μ</mi><mo id="S6.SS2.16.p2.7.m7.6.6.7" lspace="0.278em" rspace="0.278em" xref="S6.SS2.16.p2.7.m7.6.6.7.cmml">:=</mo><mrow id="S6.SS2.16.p2.7.m7.4.4.2" xref="S6.SS2.16.p2.7.m7.4.4.2.cmml"><mi id="S6.SS2.16.p2.7.m7.4.4.2.4" xref="S6.SS2.16.p2.7.m7.4.4.2.4.cmml">ν</mi><mo id="S6.SS2.16.p2.7.m7.4.4.2.3" xref="S6.SS2.16.p2.7.m7.4.4.2.3.cmml">⁢</mo><mrow id="S6.SS2.16.p2.7.m7.4.4.2.2.2" xref="S6.SS2.16.p2.7.m7.4.4.2.2.3.cmml"><mo id="S6.SS2.16.p2.7.m7.4.4.2.2.2.3" stretchy="false" xref="S6.SS2.16.p2.7.m7.4.4.2.2.3.cmml">(</mo><msub id="S6.SS2.16.p2.7.m7.3.3.1.1.1.1" xref="S6.SS2.16.p2.7.m7.3.3.1.1.1.1.cmml"><mi id="S6.SS2.16.p2.7.m7.3.3.1.1.1.1.2" xref="S6.SS2.16.p2.7.m7.3.3.1.1.1.1.2.cmml">a</mi><mi id="S6.SS2.16.p2.7.m7.3.3.1.1.1.1.3" xref="S6.SS2.16.p2.7.m7.3.3.1.1.1.1.3.cmml">m</mi></msub><mo id="S6.SS2.16.p2.7.m7.4.4.2.2.2.4" xref="S6.SS2.16.p2.7.m7.4.4.2.2.3.cmml">,</mo><msub id="S6.SS2.16.p2.7.m7.4.4.2.2.2.2" xref="S6.SS2.16.p2.7.m7.4.4.2.2.2.2.cmml"><mi id="S6.SS2.16.p2.7.m7.4.4.2.2.2.2.2" xref="S6.SS2.16.p2.7.m7.4.4.2.2.2.2.2.cmml">b</mi><mi id="S6.SS2.16.p2.7.m7.4.4.2.2.2.2.3" xref="S6.SS2.16.p2.7.m7.4.4.2.2.2.2.3.cmml">m</mi></msub><mo id="S6.SS2.16.p2.7.m7.4.4.2.2.2.5" stretchy="false" xref="S6.SS2.16.p2.7.m7.4.4.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.SS2.16.p2.7.m7.6.6.8" xref="S6.SS2.16.p2.7.m7.6.6.8.cmml">=</mo><mrow id="S6.SS2.16.p2.7.m7.6.6.4" xref="S6.SS2.16.p2.7.m7.6.6.4.cmml"><mi id="S6.SS2.16.p2.7.m7.6.6.4.4" xref="S6.SS2.16.p2.7.m7.6.6.4.4.cmml">ν</mi><mo id="S6.SS2.16.p2.7.m7.6.6.4.3" xref="S6.SS2.16.p2.7.m7.6.6.4.3.cmml">⁢</mo><mrow id="S6.SS2.16.p2.7.m7.6.6.4.2.2" xref="S6.SS2.16.p2.7.m7.6.6.4.2.3.cmml"><mo id="S6.SS2.16.p2.7.m7.6.6.4.2.2.3" stretchy="false" xref="S6.SS2.16.p2.7.m7.6.6.4.2.3.cmml">(</mo><mrow id="S6.SS2.16.p2.7.m7.5.5.3.1.1.1" xref="S6.SS2.16.p2.7.m7.5.5.3.1.1.1.cmml"><mi id="S6.SS2.16.p2.7.m7.5.5.3.1.1.1.2" xref="S6.SS2.16.p2.7.m7.5.5.3.1.1.1.2.cmml">p</mi><mo id="S6.SS2.16.p2.7.m7.5.5.3.1.1.1.1" xref="S6.SS2.16.p2.7.m7.5.5.3.1.1.1.1.cmml">⁢</mo><mrow id="S6.SS2.16.p2.7.m7.5.5.3.1.1.1.3.2" xref="S6.SS2.16.p2.7.m7.1.1.cmml"><mo id="S6.SS2.16.p2.7.m7.5.5.3.1.1.1.3.2.1" stretchy="false" xref="S6.SS2.16.p2.7.m7.1.1.cmml">(</mo><mover accent="true" id="S6.SS2.16.p2.7.m7.1.1" xref="S6.SS2.16.p2.7.m7.1.1.cmml"><mi id="S6.SS2.16.p2.7.m7.1.1.2" xref="S6.SS2.16.p2.7.m7.1.1.2.cmml">a</mi><mo id="S6.SS2.16.p2.7.m7.1.1.1" xref="S6.SS2.16.p2.7.m7.1.1.1.cmml">¯</mo></mover><mo id="S6.SS2.16.p2.7.m7.5.5.3.1.1.1.3.2.2" stretchy="false" xref="S6.SS2.16.p2.7.m7.1.1.cmml">)</mo></mrow></mrow><mo id="S6.SS2.16.p2.7.m7.6.6.4.2.2.4" xref="S6.SS2.16.p2.7.m7.6.6.4.2.3.cmml">,</mo><mrow id="S6.SS2.16.p2.7.m7.6.6.4.2.2.2" xref="S6.SS2.16.p2.7.m7.6.6.4.2.2.2.cmml"><mi id="S6.SS2.16.p2.7.m7.6.6.4.2.2.2.2" xref="S6.SS2.16.p2.7.m7.6.6.4.2.2.2.2.cmml">p</mi><mo id="S6.SS2.16.p2.7.m7.6.6.4.2.2.2.1" xref="S6.SS2.16.p2.7.m7.6.6.4.2.2.2.1.cmml">⁢</mo><mrow id="S6.SS2.16.p2.7.m7.6.6.4.2.2.2.3.2" xref="S6.SS2.16.p2.7.m7.2.2.cmml"><mo id="S6.SS2.16.p2.7.m7.6.6.4.2.2.2.3.2.1" stretchy="false" xref="S6.SS2.16.p2.7.m7.2.2.cmml">(</mo><mover accent="true" id="S6.SS2.16.p2.7.m7.2.2" xref="S6.SS2.16.p2.7.m7.2.2.cmml"><mi id="S6.SS2.16.p2.7.m7.2.2.2" xref="S6.SS2.16.p2.7.m7.2.2.2.cmml">b</mi><mo id="S6.SS2.16.p2.7.m7.2.2.1" xref="S6.SS2.16.p2.7.m7.2.2.1.cmml">¯</mo></mover><mo id="S6.SS2.16.p2.7.m7.6.6.4.2.2.2.3.2.2" stretchy="false" xref="S6.SS2.16.p2.7.m7.2.2.cmml">)</mo></mrow></mrow><mo id="S6.SS2.16.p2.7.m7.6.6.4.2.2.5" stretchy="false" xref="S6.SS2.16.p2.7.m7.6.6.4.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.16.p2.7.m7.6b"><apply id="S6.SS2.16.p2.7.m7.6.6.cmml" xref="S6.SS2.16.p2.7.m7.6.6"><and id="S6.SS2.16.p2.7.m7.6.6a.cmml" xref="S6.SS2.16.p2.7.m7.6.6"></and><apply id="S6.SS2.16.p2.7.m7.6.6b.cmml" xref="S6.SS2.16.p2.7.m7.6.6"><csymbol cd="latexml" id="S6.SS2.16.p2.7.m7.6.6.7.cmml" xref="S6.SS2.16.p2.7.m7.6.6.7">assign</csymbol><ci id="S6.SS2.16.p2.7.m7.6.6.6.cmml" xref="S6.SS2.16.p2.7.m7.6.6.6">𝜇</ci><apply id="S6.SS2.16.p2.7.m7.4.4.2.cmml" xref="S6.SS2.16.p2.7.m7.4.4.2"><times id="S6.SS2.16.p2.7.m7.4.4.2.3.cmml" xref="S6.SS2.16.p2.7.m7.4.4.2.3"></times><ci id="S6.SS2.16.p2.7.m7.4.4.2.4.cmml" xref="S6.SS2.16.p2.7.m7.4.4.2.4">𝜈</ci><interval closure="open" id="S6.SS2.16.p2.7.m7.4.4.2.2.3.cmml" xref="S6.SS2.16.p2.7.m7.4.4.2.2.2"><apply id="S6.SS2.16.p2.7.m7.3.3.1.1.1.1.cmml" xref="S6.SS2.16.p2.7.m7.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.16.p2.7.m7.3.3.1.1.1.1.1.cmml" xref="S6.SS2.16.p2.7.m7.3.3.1.1.1.1">subscript</csymbol><ci id="S6.SS2.16.p2.7.m7.3.3.1.1.1.1.2.cmml" xref="S6.SS2.16.p2.7.m7.3.3.1.1.1.1.2">𝑎</ci><ci id="S6.SS2.16.p2.7.m7.3.3.1.1.1.1.3.cmml" xref="S6.SS2.16.p2.7.m7.3.3.1.1.1.1.3">𝑚</ci></apply><apply id="S6.SS2.16.p2.7.m7.4.4.2.2.2.2.cmml" xref="S6.SS2.16.p2.7.m7.4.4.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.16.p2.7.m7.4.4.2.2.2.2.1.cmml" xref="S6.SS2.16.p2.7.m7.4.4.2.2.2.2">subscript</csymbol><ci id="S6.SS2.16.p2.7.m7.4.4.2.2.2.2.2.cmml" xref="S6.SS2.16.p2.7.m7.4.4.2.2.2.2.2">𝑏</ci><ci id="S6.SS2.16.p2.7.m7.4.4.2.2.2.2.3.cmml" xref="S6.SS2.16.p2.7.m7.4.4.2.2.2.2.3">𝑚</ci></apply></interval></apply></apply><apply id="S6.SS2.16.p2.7.m7.6.6c.cmml" xref="S6.SS2.16.p2.7.m7.6.6"><eq id="S6.SS2.16.p2.7.m7.6.6.8.cmml" xref="S6.SS2.16.p2.7.m7.6.6.8"></eq><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.16.p2.7.m7.4.4.2.cmml" id="S6.SS2.16.p2.7.m7.6.6d.cmml" xref="S6.SS2.16.p2.7.m7.6.6"></share><apply id="S6.SS2.16.p2.7.m7.6.6.4.cmml" xref="S6.SS2.16.p2.7.m7.6.6.4"><times id="S6.SS2.16.p2.7.m7.6.6.4.3.cmml" xref="S6.SS2.16.p2.7.m7.6.6.4.3"></times><ci id="S6.SS2.16.p2.7.m7.6.6.4.4.cmml" xref="S6.SS2.16.p2.7.m7.6.6.4.4">𝜈</ci><interval closure="open" id="S6.SS2.16.p2.7.m7.6.6.4.2.3.cmml" xref="S6.SS2.16.p2.7.m7.6.6.4.2.2"><apply id="S6.SS2.16.p2.7.m7.5.5.3.1.1.1.cmml" xref="S6.SS2.16.p2.7.m7.5.5.3.1.1.1"><times id="S6.SS2.16.p2.7.m7.5.5.3.1.1.1.1.cmml" xref="S6.SS2.16.p2.7.m7.5.5.3.1.1.1.1"></times><ci id="S6.SS2.16.p2.7.m7.5.5.3.1.1.1.2.cmml" xref="S6.SS2.16.p2.7.m7.5.5.3.1.1.1.2">𝑝</ci><apply id="S6.SS2.16.p2.7.m7.1.1.cmml" xref="S6.SS2.16.p2.7.m7.5.5.3.1.1.1.3.2"><ci id="S6.SS2.16.p2.7.m7.1.1.1.cmml" xref="S6.SS2.16.p2.7.m7.1.1.1">¯</ci><ci id="S6.SS2.16.p2.7.m7.1.1.2.cmml" xref="S6.SS2.16.p2.7.m7.1.1.2">𝑎</ci></apply></apply><apply id="S6.SS2.16.p2.7.m7.6.6.4.2.2.2.cmml" xref="S6.SS2.16.p2.7.m7.6.6.4.2.2.2"><times id="S6.SS2.16.p2.7.m7.6.6.4.2.2.2.1.cmml" xref="S6.SS2.16.p2.7.m7.6.6.4.2.2.2.1"></times><ci id="S6.SS2.16.p2.7.m7.6.6.4.2.2.2.2.cmml" xref="S6.SS2.16.p2.7.m7.6.6.4.2.2.2.2">𝑝</ci><apply id="S6.SS2.16.p2.7.m7.2.2.cmml" xref="S6.SS2.16.p2.7.m7.6.6.4.2.2.2.3.2"><ci id="S6.SS2.16.p2.7.m7.2.2.1.cmml" xref="S6.SS2.16.p2.7.m7.2.2.1">¯</ci><ci id="S6.SS2.16.p2.7.m7.2.2.2.cmml" xref="S6.SS2.16.p2.7.m7.2.2.2">𝑏</ci></apply></apply></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.16.p2.7.m7.6c">\mu:=\nu(a_{m},b_{m})=\nu(p(\bar{a}),p(\bar{b}))</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.16.p2.7.m7.6d">italic_μ := italic_ν ( italic_a start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) = italic_ν ( italic_p ( over¯ start_ARG italic_a end_ARG ) , italic_p ( over¯ start_ARG italic_b end_ARG ) )</annotation></semantics></math>. Clearly <math alttext="\nu\leq\mu" class="ltx_Math" display="inline" id="S6.SS2.16.p2.8.m8.1"><semantics id="S6.SS2.16.p2.8.m8.1a"><mrow id="S6.SS2.16.p2.8.m8.1.1" xref="S6.SS2.16.p2.8.m8.1.1.cmml"><mi id="S6.SS2.16.p2.8.m8.1.1.2" xref="S6.SS2.16.p2.8.m8.1.1.2.cmml">ν</mi><mo id="S6.SS2.16.p2.8.m8.1.1.1" xref="S6.SS2.16.p2.8.m8.1.1.1.cmml">≤</mo><mi id="S6.SS2.16.p2.8.m8.1.1.3" xref="S6.SS2.16.p2.8.m8.1.1.3.cmml">μ</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.16.p2.8.m8.1b"><apply id="S6.SS2.16.p2.8.m8.1.1.cmml" xref="S6.SS2.16.p2.8.m8.1.1"><leq id="S6.SS2.16.p2.8.m8.1.1.1.cmml" xref="S6.SS2.16.p2.8.m8.1.1.1"></leq><ci id="S6.SS2.16.p2.8.m8.1.1.2.cmml" xref="S6.SS2.16.p2.8.m8.1.1.2">𝜈</ci><ci id="S6.SS2.16.p2.8.m8.1.1.3.cmml" xref="S6.SS2.16.p2.8.m8.1.1.3">𝜇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.16.p2.8.m8.1c">\nu\leq\mu</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.16.p2.8.m8.1d">italic_ν ≤ italic_μ</annotation></semantics></math>, and if equality holds we are done. So assume <math alttext="\nu<\mu" class="ltx_Math" display="inline" id="S6.SS2.16.p2.9.m9.1"><semantics id="S6.SS2.16.p2.9.m9.1a"><mrow id="S6.SS2.16.p2.9.m9.1.1" xref="S6.SS2.16.p2.9.m9.1.1.cmml"><mi id="S6.SS2.16.p2.9.m9.1.1.2" xref="S6.SS2.16.p2.9.m9.1.1.2.cmml">ν</mi><mo id="S6.SS2.16.p2.9.m9.1.1.1" xref="S6.SS2.16.p2.9.m9.1.1.1.cmml"><</mo><mi id="S6.SS2.16.p2.9.m9.1.1.3" xref="S6.SS2.16.p2.9.m9.1.1.3.cmml">μ</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.16.p2.9.m9.1b"><apply id="S6.SS2.16.p2.9.m9.1.1.cmml" xref="S6.SS2.16.p2.9.m9.1.1"><lt id="S6.SS2.16.p2.9.m9.1.1.1.cmml" xref="S6.SS2.16.p2.9.m9.1.1.1"></lt><ci id="S6.SS2.16.p2.9.m9.1.1.2.cmml" xref="S6.SS2.16.p2.9.m9.1.1.2">𝜈</ci><ci id="S6.SS2.16.p2.9.m9.1.1.3.cmml" xref="S6.SS2.16.p2.9.m9.1.1.3">𝜇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.16.p2.9.m9.1c">\nu<\mu</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.16.p2.9.m9.1d">italic_ν < italic_μ</annotation></semantics></math>. This implies that <math alttext="\Delta(b_{m},c_{m})=\Delta(a_{m},c_{m})" class="ltx_Math" display="inline" id="S6.SS2.16.p2.10.m10.4"><semantics id="S6.SS2.16.p2.10.m10.4a"><mrow id="S6.SS2.16.p2.10.m10.4.4" xref="S6.SS2.16.p2.10.m10.4.4.cmml"><mrow id="S6.SS2.16.p2.10.m10.2.2.2" xref="S6.SS2.16.p2.10.m10.2.2.2.cmml"><mi id="S6.SS2.16.p2.10.m10.2.2.2.4" mathvariant="normal" xref="S6.SS2.16.p2.10.m10.2.2.2.4.cmml">Δ</mi><mo id="S6.SS2.16.p2.10.m10.2.2.2.3" xref="S6.SS2.16.p2.10.m10.2.2.2.3.cmml">⁢</mo><mrow id="S6.SS2.16.p2.10.m10.2.2.2.2.2" xref="S6.SS2.16.p2.10.m10.2.2.2.2.3.cmml"><mo id="S6.SS2.16.p2.10.m10.2.2.2.2.2.3" stretchy="false" xref="S6.SS2.16.p2.10.m10.2.2.2.2.3.cmml">(</mo><msub id="S6.SS2.16.p2.10.m10.1.1.1.1.1.1" xref="S6.SS2.16.p2.10.m10.1.1.1.1.1.1.cmml"><mi id="S6.SS2.16.p2.10.m10.1.1.1.1.1.1.2" xref="S6.SS2.16.p2.10.m10.1.1.1.1.1.1.2.cmml">b</mi><mi id="S6.SS2.16.p2.10.m10.1.1.1.1.1.1.3" xref="S6.SS2.16.p2.10.m10.1.1.1.1.1.1.3.cmml">m</mi></msub><mo id="S6.SS2.16.p2.10.m10.2.2.2.2.2.4" xref="S6.SS2.16.p2.10.m10.2.2.2.2.3.cmml">,</mo><msub id="S6.SS2.16.p2.10.m10.2.2.2.2.2.2" xref="S6.SS2.16.p2.10.m10.2.2.2.2.2.2.cmml"><mi id="S6.SS2.16.p2.10.m10.2.2.2.2.2.2.2" xref="S6.SS2.16.p2.10.m10.2.2.2.2.2.2.2.cmml">c</mi><mi id="S6.SS2.16.p2.10.m10.2.2.2.2.2.2.3" xref="S6.SS2.16.p2.10.m10.2.2.2.2.2.2.3.cmml">m</mi></msub><mo id="S6.SS2.16.p2.10.m10.2.2.2.2.2.5" stretchy="false" xref="S6.SS2.16.p2.10.m10.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.SS2.16.p2.10.m10.4.4.5" xref="S6.SS2.16.p2.10.m10.4.4.5.cmml">=</mo><mrow id="S6.SS2.16.p2.10.m10.4.4.4" xref="S6.SS2.16.p2.10.m10.4.4.4.cmml"><mi id="S6.SS2.16.p2.10.m10.4.4.4.4" mathvariant="normal" xref="S6.SS2.16.p2.10.m10.4.4.4.4.cmml">Δ</mi><mo id="S6.SS2.16.p2.10.m10.4.4.4.3" xref="S6.SS2.16.p2.10.m10.4.4.4.3.cmml">⁢</mo><mrow id="S6.SS2.16.p2.10.m10.4.4.4.2.2" xref="S6.SS2.16.p2.10.m10.4.4.4.2.3.cmml"><mo id="S6.SS2.16.p2.10.m10.4.4.4.2.2.3" stretchy="false" xref="S6.SS2.16.p2.10.m10.4.4.4.2.3.cmml">(</mo><msub id="S6.SS2.16.p2.10.m10.3.3.3.1.1.1" xref="S6.SS2.16.p2.10.m10.3.3.3.1.1.1.cmml"><mi id="S6.SS2.16.p2.10.m10.3.3.3.1.1.1.2" xref="S6.SS2.16.p2.10.m10.3.3.3.1.1.1.2.cmml">a</mi><mi id="S6.SS2.16.p2.10.m10.3.3.3.1.1.1.3" xref="S6.SS2.16.p2.10.m10.3.3.3.1.1.1.3.cmml">m</mi></msub><mo id="S6.SS2.16.p2.10.m10.4.4.4.2.2.4" xref="S6.SS2.16.p2.10.m10.4.4.4.2.3.cmml">,</mo><msub id="S6.SS2.16.p2.10.m10.4.4.4.2.2.2" xref="S6.SS2.16.p2.10.m10.4.4.4.2.2.2.cmml"><mi id="S6.SS2.16.p2.10.m10.4.4.4.2.2.2.2" xref="S6.SS2.16.p2.10.m10.4.4.4.2.2.2.2.cmml">c</mi><mi id="S6.SS2.16.p2.10.m10.4.4.4.2.2.2.3" xref="S6.SS2.16.p2.10.m10.4.4.4.2.2.2.3.cmml">m</mi></msub><mo id="S6.SS2.16.p2.10.m10.4.4.4.2.2.5" stretchy="false" xref="S6.SS2.16.p2.10.m10.4.4.4.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.16.p2.10.m10.4b"><apply id="S6.SS2.16.p2.10.m10.4.4.cmml" xref="S6.SS2.16.p2.10.m10.4.4"><eq id="S6.SS2.16.p2.10.m10.4.4.5.cmml" xref="S6.SS2.16.p2.10.m10.4.4.5"></eq><apply id="S6.SS2.16.p2.10.m10.2.2.2.cmml" xref="S6.SS2.16.p2.10.m10.2.2.2"><times id="S6.SS2.16.p2.10.m10.2.2.2.3.cmml" xref="S6.SS2.16.p2.10.m10.2.2.2.3"></times><ci id="S6.SS2.16.p2.10.m10.2.2.2.4.cmml" xref="S6.SS2.16.p2.10.m10.2.2.2.4">Δ</ci><interval closure="open" id="S6.SS2.16.p2.10.m10.2.2.2.2.3.cmml" xref="S6.SS2.16.p2.10.m10.2.2.2.2.2"><apply id="S6.SS2.16.p2.10.m10.1.1.1.1.1.1.cmml" xref="S6.SS2.16.p2.10.m10.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.16.p2.10.m10.1.1.1.1.1.1.1.cmml" xref="S6.SS2.16.p2.10.m10.1.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.16.p2.10.m10.1.1.1.1.1.1.2.cmml" xref="S6.SS2.16.p2.10.m10.1.1.1.1.1.1.2">𝑏</ci><ci id="S6.SS2.16.p2.10.m10.1.1.1.1.1.1.3.cmml" xref="S6.SS2.16.p2.10.m10.1.1.1.1.1.1.3">𝑚</ci></apply><apply id="S6.SS2.16.p2.10.m10.2.2.2.2.2.2.cmml" xref="S6.SS2.16.p2.10.m10.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.16.p2.10.m10.2.2.2.2.2.2.1.cmml" xref="S6.SS2.16.p2.10.m10.2.2.2.2.2.2">subscript</csymbol><ci id="S6.SS2.16.p2.10.m10.2.2.2.2.2.2.2.cmml" xref="S6.SS2.16.p2.10.m10.2.2.2.2.2.2.2">𝑐</ci><ci id="S6.SS2.16.p2.10.m10.2.2.2.2.2.2.3.cmml" xref="S6.SS2.16.p2.10.m10.2.2.2.2.2.2.3">𝑚</ci></apply></interval></apply><apply id="S6.SS2.16.p2.10.m10.4.4.4.cmml" xref="S6.SS2.16.p2.10.m10.4.4.4"><times id="S6.SS2.16.p2.10.m10.4.4.4.3.cmml" xref="S6.SS2.16.p2.10.m10.4.4.4.3"></times><ci id="S6.SS2.16.p2.10.m10.4.4.4.4.cmml" xref="S6.SS2.16.p2.10.m10.4.4.4.4">Δ</ci><interval closure="open" id="S6.SS2.16.p2.10.m10.4.4.4.2.3.cmml" xref="S6.SS2.16.p2.10.m10.4.4.4.2.2"><apply id="S6.SS2.16.p2.10.m10.3.3.3.1.1.1.cmml" xref="S6.SS2.16.p2.10.m10.3.3.3.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.16.p2.10.m10.3.3.3.1.1.1.1.cmml" xref="S6.SS2.16.p2.10.m10.3.3.3.1.1.1">subscript</csymbol><ci id="S6.SS2.16.p2.10.m10.3.3.3.1.1.1.2.cmml" xref="S6.SS2.16.p2.10.m10.3.3.3.1.1.1.2">𝑎</ci><ci id="S6.SS2.16.p2.10.m10.3.3.3.1.1.1.3.cmml" xref="S6.SS2.16.p2.10.m10.3.3.3.1.1.1.3">𝑚</ci></apply><apply id="S6.SS2.16.p2.10.m10.4.4.4.2.2.2.cmml" xref="S6.SS2.16.p2.10.m10.4.4.4.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.16.p2.10.m10.4.4.4.2.2.2.1.cmml" xref="S6.SS2.16.p2.10.m10.4.4.4.2.2.2">subscript</csymbol><ci id="S6.SS2.16.p2.10.m10.4.4.4.2.2.2.2.cmml" xref="S6.SS2.16.p2.10.m10.4.4.4.2.2.2.2">𝑐</ci><ci id="S6.SS2.16.p2.10.m10.4.4.4.2.2.2.3.cmml" xref="S6.SS2.16.p2.10.m10.4.4.4.2.2.2.3">𝑚</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.16.p2.10.m10.4c">\Delta(b_{m},c_{m})=\Delta(a_{m},c_{m})</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.16.p2.10.m10.4d">roman_Δ ( italic_b start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT , italic_c start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) = roman_Δ ( italic_a start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT , italic_c start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT )</annotation></semantics></math>, and then <math alttext="\nu(a_{m},c_{m})=\nu(b_{m},c_{m})=\nu=\nu(p(\bar{b}),p(\bar{c}))=\nu(p(\bar{a}% ),p(\bar{c}))" class="ltx_Math" display="inline" id="S6.SS2.16.p2.11.m11.12"><semantics id="S6.SS2.16.p2.11.m11.12a"><mrow id="S6.SS2.16.p2.11.m11.12.12" xref="S6.SS2.16.p2.11.m11.12.12.cmml"><mrow id="S6.SS2.16.p2.11.m11.6.6.2" xref="S6.SS2.16.p2.11.m11.6.6.2.cmml"><mi id="S6.SS2.16.p2.11.m11.6.6.2.4" xref="S6.SS2.16.p2.11.m11.6.6.2.4.cmml">ν</mi><mo id="S6.SS2.16.p2.11.m11.6.6.2.3" xref="S6.SS2.16.p2.11.m11.6.6.2.3.cmml">⁢</mo><mrow id="S6.SS2.16.p2.11.m11.6.6.2.2.2" xref="S6.SS2.16.p2.11.m11.6.6.2.2.3.cmml"><mo id="S6.SS2.16.p2.11.m11.6.6.2.2.2.3" stretchy="false" xref="S6.SS2.16.p2.11.m11.6.6.2.2.3.cmml">(</mo><msub id="S6.SS2.16.p2.11.m11.5.5.1.1.1.1" xref="S6.SS2.16.p2.11.m11.5.5.1.1.1.1.cmml"><mi id="S6.SS2.16.p2.11.m11.5.5.1.1.1.1.2" xref="S6.SS2.16.p2.11.m11.5.5.1.1.1.1.2.cmml">a</mi><mi id="S6.SS2.16.p2.11.m11.5.5.1.1.1.1.3" xref="S6.SS2.16.p2.11.m11.5.5.1.1.1.1.3.cmml">m</mi></msub><mo id="S6.SS2.16.p2.11.m11.6.6.2.2.2.4" xref="S6.SS2.16.p2.11.m11.6.6.2.2.3.cmml">,</mo><msub id="S6.SS2.16.p2.11.m11.6.6.2.2.2.2" xref="S6.SS2.16.p2.11.m11.6.6.2.2.2.2.cmml"><mi id="S6.SS2.16.p2.11.m11.6.6.2.2.2.2.2" xref="S6.SS2.16.p2.11.m11.6.6.2.2.2.2.2.cmml">c</mi><mi id="S6.SS2.16.p2.11.m11.6.6.2.2.2.2.3" xref="S6.SS2.16.p2.11.m11.6.6.2.2.2.2.3.cmml">m</mi></msub><mo id="S6.SS2.16.p2.11.m11.6.6.2.2.2.5" stretchy="false" xref="S6.SS2.16.p2.11.m11.6.6.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.SS2.16.p2.11.m11.12.12.10" xref="S6.SS2.16.p2.11.m11.12.12.10.cmml">=</mo><mrow id="S6.SS2.16.p2.11.m11.8.8.4" xref="S6.SS2.16.p2.11.m11.8.8.4.cmml"><mi id="S6.SS2.16.p2.11.m11.8.8.4.4" xref="S6.SS2.16.p2.11.m11.8.8.4.4.cmml">ν</mi><mo id="S6.SS2.16.p2.11.m11.8.8.4.3" xref="S6.SS2.16.p2.11.m11.8.8.4.3.cmml">⁢</mo><mrow id="S6.SS2.16.p2.11.m11.8.8.4.2.2" xref="S6.SS2.16.p2.11.m11.8.8.4.2.3.cmml"><mo id="S6.SS2.16.p2.11.m11.8.8.4.2.2.3" stretchy="false" xref="S6.SS2.16.p2.11.m11.8.8.4.2.3.cmml">(</mo><msub id="S6.SS2.16.p2.11.m11.7.7.3.1.1.1" xref="S6.SS2.16.p2.11.m11.7.7.3.1.1.1.cmml"><mi id="S6.SS2.16.p2.11.m11.7.7.3.1.1.1.2" xref="S6.SS2.16.p2.11.m11.7.7.3.1.1.1.2.cmml">b</mi><mi id="S6.SS2.16.p2.11.m11.7.7.3.1.1.1.3" xref="S6.SS2.16.p2.11.m11.7.7.3.1.1.1.3.cmml">m</mi></msub><mo id="S6.SS2.16.p2.11.m11.8.8.4.2.2.4" xref="S6.SS2.16.p2.11.m11.8.8.4.2.3.cmml">,</mo><msub id="S6.SS2.16.p2.11.m11.8.8.4.2.2.2" xref="S6.SS2.16.p2.11.m11.8.8.4.2.2.2.cmml"><mi id="S6.SS2.16.p2.11.m11.8.8.4.2.2.2.2" xref="S6.SS2.16.p2.11.m11.8.8.4.2.2.2.2.cmml">c</mi><mi id="S6.SS2.16.p2.11.m11.8.8.4.2.2.2.3" xref="S6.SS2.16.p2.11.m11.8.8.4.2.2.2.3.cmml">m</mi></msub><mo id="S6.SS2.16.p2.11.m11.8.8.4.2.2.5" stretchy="false" xref="S6.SS2.16.p2.11.m11.8.8.4.2.3.cmml">)</mo></mrow></mrow><mo id="S6.SS2.16.p2.11.m11.12.12.11" xref="S6.SS2.16.p2.11.m11.12.12.11.cmml">=</mo><mi id="S6.SS2.16.p2.11.m11.12.12.12" xref="S6.SS2.16.p2.11.m11.12.12.12.cmml">ν</mi><mo id="S6.SS2.16.p2.11.m11.12.12.13" xref="S6.SS2.16.p2.11.m11.12.12.13.cmml">=</mo><mrow id="S6.SS2.16.p2.11.m11.10.10.6" xref="S6.SS2.16.p2.11.m11.10.10.6.cmml"><mi id="S6.SS2.16.p2.11.m11.10.10.6.4" xref="S6.SS2.16.p2.11.m11.10.10.6.4.cmml">ν</mi><mo id="S6.SS2.16.p2.11.m11.10.10.6.3" xref="S6.SS2.16.p2.11.m11.10.10.6.3.cmml">⁢</mo><mrow id="S6.SS2.16.p2.11.m11.10.10.6.2.2" xref="S6.SS2.16.p2.11.m11.10.10.6.2.3.cmml"><mo id="S6.SS2.16.p2.11.m11.10.10.6.2.2.3" stretchy="false" xref="S6.SS2.16.p2.11.m11.10.10.6.2.3.cmml">(</mo><mrow id="S6.SS2.16.p2.11.m11.9.9.5.1.1.1" xref="S6.SS2.16.p2.11.m11.9.9.5.1.1.1.cmml"><mi id="S6.SS2.16.p2.11.m11.9.9.5.1.1.1.2" xref="S6.SS2.16.p2.11.m11.9.9.5.1.1.1.2.cmml">p</mi><mo id="S6.SS2.16.p2.11.m11.9.9.5.1.1.1.1" xref="S6.SS2.16.p2.11.m11.9.9.5.1.1.1.1.cmml">⁢</mo><mrow id="S6.SS2.16.p2.11.m11.9.9.5.1.1.1.3.2" xref="S6.SS2.16.p2.11.m11.1.1.cmml"><mo id="S6.SS2.16.p2.11.m11.9.9.5.1.1.1.3.2.1" stretchy="false" xref="S6.SS2.16.p2.11.m11.1.1.cmml">(</mo><mover accent="true" id="S6.SS2.16.p2.11.m11.1.1" xref="S6.SS2.16.p2.11.m11.1.1.cmml"><mi id="S6.SS2.16.p2.11.m11.1.1.2" xref="S6.SS2.16.p2.11.m11.1.1.2.cmml">b</mi><mo id="S6.SS2.16.p2.11.m11.1.1.1" xref="S6.SS2.16.p2.11.m11.1.1.1.cmml">¯</mo></mover><mo id="S6.SS2.16.p2.11.m11.9.9.5.1.1.1.3.2.2" stretchy="false" xref="S6.SS2.16.p2.11.m11.1.1.cmml">)</mo></mrow></mrow><mo id="S6.SS2.16.p2.11.m11.10.10.6.2.2.4" xref="S6.SS2.16.p2.11.m11.10.10.6.2.3.cmml">,</mo><mrow id="S6.SS2.16.p2.11.m11.10.10.6.2.2.2" xref="S6.SS2.16.p2.11.m11.10.10.6.2.2.2.cmml"><mi id="S6.SS2.16.p2.11.m11.10.10.6.2.2.2.2" xref="S6.SS2.16.p2.11.m11.10.10.6.2.2.2.2.cmml">p</mi><mo id="S6.SS2.16.p2.11.m11.10.10.6.2.2.2.1" xref="S6.SS2.16.p2.11.m11.10.10.6.2.2.2.1.cmml">⁢</mo><mrow id="S6.SS2.16.p2.11.m11.10.10.6.2.2.2.3.2" xref="S6.SS2.16.p2.11.m11.2.2.cmml"><mo id="S6.SS2.16.p2.11.m11.10.10.6.2.2.2.3.2.1" stretchy="false" xref="S6.SS2.16.p2.11.m11.2.2.cmml">(</mo><mover accent="true" id="S6.SS2.16.p2.11.m11.2.2" xref="S6.SS2.16.p2.11.m11.2.2.cmml"><mi id="S6.SS2.16.p2.11.m11.2.2.2" xref="S6.SS2.16.p2.11.m11.2.2.2.cmml">c</mi><mo id="S6.SS2.16.p2.11.m11.2.2.1" xref="S6.SS2.16.p2.11.m11.2.2.1.cmml">¯</mo></mover><mo id="S6.SS2.16.p2.11.m11.10.10.6.2.2.2.3.2.2" stretchy="false" xref="S6.SS2.16.p2.11.m11.2.2.cmml">)</mo></mrow></mrow><mo id="S6.SS2.16.p2.11.m11.10.10.6.2.2.5" stretchy="false" xref="S6.SS2.16.p2.11.m11.10.10.6.2.3.cmml">)</mo></mrow></mrow><mo id="S6.SS2.16.p2.11.m11.12.12.14" xref="S6.SS2.16.p2.11.m11.12.12.14.cmml">=</mo><mrow id="S6.SS2.16.p2.11.m11.12.12.8" xref="S6.SS2.16.p2.11.m11.12.12.8.cmml"><mi id="S6.SS2.16.p2.11.m11.12.12.8.4" xref="S6.SS2.16.p2.11.m11.12.12.8.4.cmml">ν</mi><mo id="S6.SS2.16.p2.11.m11.12.12.8.3" xref="S6.SS2.16.p2.11.m11.12.12.8.3.cmml">⁢</mo><mrow id="S6.SS2.16.p2.11.m11.12.12.8.2.2" xref="S6.SS2.16.p2.11.m11.12.12.8.2.3.cmml"><mo id="S6.SS2.16.p2.11.m11.12.12.8.2.2.3" stretchy="false" xref="S6.SS2.16.p2.11.m11.12.12.8.2.3.cmml">(</mo><mrow id="S6.SS2.16.p2.11.m11.11.11.7.1.1.1" xref="S6.SS2.16.p2.11.m11.11.11.7.1.1.1.cmml"><mi id="S6.SS2.16.p2.11.m11.11.11.7.1.1.1.2" xref="S6.SS2.16.p2.11.m11.11.11.7.1.1.1.2.cmml">p</mi><mo id="S6.SS2.16.p2.11.m11.11.11.7.1.1.1.1" xref="S6.SS2.16.p2.11.m11.11.11.7.1.1.1.1.cmml">⁢</mo><mrow id="S6.SS2.16.p2.11.m11.11.11.7.1.1.1.3.2" xref="S6.SS2.16.p2.11.m11.3.3.cmml"><mo id="S6.SS2.16.p2.11.m11.11.11.7.1.1.1.3.2.1" stretchy="false" xref="S6.SS2.16.p2.11.m11.3.3.cmml">(</mo><mover accent="true" id="S6.SS2.16.p2.11.m11.3.3" xref="S6.SS2.16.p2.11.m11.3.3.cmml"><mi id="S6.SS2.16.p2.11.m11.3.3.2" xref="S6.SS2.16.p2.11.m11.3.3.2.cmml">a</mi><mo id="S6.SS2.16.p2.11.m11.3.3.1" xref="S6.SS2.16.p2.11.m11.3.3.1.cmml">¯</mo></mover><mo id="S6.SS2.16.p2.11.m11.11.11.7.1.1.1.3.2.2" stretchy="false" xref="S6.SS2.16.p2.11.m11.3.3.cmml">)</mo></mrow></mrow><mo id="S6.SS2.16.p2.11.m11.12.12.8.2.2.4" xref="S6.SS2.16.p2.11.m11.12.12.8.2.3.cmml">,</mo><mrow id="S6.SS2.16.p2.11.m11.12.12.8.2.2.2" xref="S6.SS2.16.p2.11.m11.12.12.8.2.2.2.cmml"><mi id="S6.SS2.16.p2.11.m11.12.12.8.2.2.2.2" xref="S6.SS2.16.p2.11.m11.12.12.8.2.2.2.2.cmml">p</mi><mo id="S6.SS2.16.p2.11.m11.12.12.8.2.2.2.1" xref="S6.SS2.16.p2.11.m11.12.12.8.2.2.2.1.cmml">⁢</mo><mrow id="S6.SS2.16.p2.11.m11.12.12.8.2.2.2.3.2" xref="S6.SS2.16.p2.11.m11.4.4.cmml"><mo id="S6.SS2.16.p2.11.m11.12.12.8.2.2.2.3.2.1" stretchy="false" xref="S6.SS2.16.p2.11.m11.4.4.cmml">(</mo><mover accent="true" id="S6.SS2.16.p2.11.m11.4.4" xref="S6.SS2.16.p2.11.m11.4.4.cmml"><mi id="S6.SS2.16.p2.11.m11.4.4.2" xref="S6.SS2.16.p2.11.m11.4.4.2.cmml">c</mi><mo id="S6.SS2.16.p2.11.m11.4.4.1" xref="S6.SS2.16.p2.11.m11.4.4.1.cmml">¯</mo></mover><mo id="S6.SS2.16.p2.11.m11.12.12.8.2.2.2.3.2.2" stretchy="false" xref="S6.SS2.16.p2.11.m11.4.4.cmml">)</mo></mrow></mrow><mo id="S6.SS2.16.p2.11.m11.12.12.8.2.2.5" stretchy="false" xref="S6.SS2.16.p2.11.m11.12.12.8.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.16.p2.11.m11.12b"><apply id="S6.SS2.16.p2.11.m11.12.12.cmml" xref="S6.SS2.16.p2.11.m11.12.12"><and id="S6.SS2.16.p2.11.m11.12.12a.cmml" xref="S6.SS2.16.p2.11.m11.12.12"></and><apply id="S6.SS2.16.p2.11.m11.12.12b.cmml" xref="S6.SS2.16.p2.11.m11.12.12"><eq id="S6.SS2.16.p2.11.m11.12.12.10.cmml" xref="S6.SS2.16.p2.11.m11.12.12.10"></eq><apply id="S6.SS2.16.p2.11.m11.6.6.2.cmml" xref="S6.SS2.16.p2.11.m11.6.6.2"><times id="S6.SS2.16.p2.11.m11.6.6.2.3.cmml" xref="S6.SS2.16.p2.11.m11.6.6.2.3"></times><ci id="S6.SS2.16.p2.11.m11.6.6.2.4.cmml" xref="S6.SS2.16.p2.11.m11.6.6.2.4">𝜈</ci><interval closure="open" id="S6.SS2.16.p2.11.m11.6.6.2.2.3.cmml" xref="S6.SS2.16.p2.11.m11.6.6.2.2.2"><apply id="S6.SS2.16.p2.11.m11.5.5.1.1.1.1.cmml" xref="S6.SS2.16.p2.11.m11.5.5.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.16.p2.11.m11.5.5.1.1.1.1.1.cmml" xref="S6.SS2.16.p2.11.m11.5.5.1.1.1.1">subscript</csymbol><ci id="S6.SS2.16.p2.11.m11.5.5.1.1.1.1.2.cmml" xref="S6.SS2.16.p2.11.m11.5.5.1.1.1.1.2">𝑎</ci><ci id="S6.SS2.16.p2.11.m11.5.5.1.1.1.1.3.cmml" xref="S6.SS2.16.p2.11.m11.5.5.1.1.1.1.3">𝑚</ci></apply><apply id="S6.SS2.16.p2.11.m11.6.6.2.2.2.2.cmml" xref="S6.SS2.16.p2.11.m11.6.6.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.16.p2.11.m11.6.6.2.2.2.2.1.cmml" xref="S6.SS2.16.p2.11.m11.6.6.2.2.2.2">subscript</csymbol><ci id="S6.SS2.16.p2.11.m11.6.6.2.2.2.2.2.cmml" xref="S6.SS2.16.p2.11.m11.6.6.2.2.2.2.2">𝑐</ci><ci id="S6.SS2.16.p2.11.m11.6.6.2.2.2.2.3.cmml" xref="S6.SS2.16.p2.11.m11.6.6.2.2.2.2.3">𝑚</ci></apply></interval></apply><apply id="S6.SS2.16.p2.11.m11.8.8.4.cmml" xref="S6.SS2.16.p2.11.m11.8.8.4"><times id="S6.SS2.16.p2.11.m11.8.8.4.3.cmml" xref="S6.SS2.16.p2.11.m11.8.8.4.3"></times><ci id="S6.SS2.16.p2.11.m11.8.8.4.4.cmml" xref="S6.SS2.16.p2.11.m11.8.8.4.4">𝜈</ci><interval closure="open" id="S6.SS2.16.p2.11.m11.8.8.4.2.3.cmml" xref="S6.SS2.16.p2.11.m11.8.8.4.2.2"><apply id="S6.SS2.16.p2.11.m11.7.7.3.1.1.1.cmml" xref="S6.SS2.16.p2.11.m11.7.7.3.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.16.p2.11.m11.7.7.3.1.1.1.1.cmml" xref="S6.SS2.16.p2.11.m11.7.7.3.1.1.1">subscript</csymbol><ci id="S6.SS2.16.p2.11.m11.7.7.3.1.1.1.2.cmml" xref="S6.SS2.16.p2.11.m11.7.7.3.1.1.1.2">𝑏</ci><ci id="S6.SS2.16.p2.11.m11.7.7.3.1.1.1.3.cmml" xref="S6.SS2.16.p2.11.m11.7.7.3.1.1.1.3">𝑚</ci></apply><apply id="S6.SS2.16.p2.11.m11.8.8.4.2.2.2.cmml" xref="S6.SS2.16.p2.11.m11.8.8.4.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.16.p2.11.m11.8.8.4.2.2.2.1.cmml" xref="S6.SS2.16.p2.11.m11.8.8.4.2.2.2">subscript</csymbol><ci id="S6.SS2.16.p2.11.m11.8.8.4.2.2.2.2.cmml" xref="S6.SS2.16.p2.11.m11.8.8.4.2.2.2.2">𝑐</ci><ci id="S6.SS2.16.p2.11.m11.8.8.4.2.2.2.3.cmml" xref="S6.SS2.16.p2.11.m11.8.8.4.2.2.2.3">𝑚</ci></apply></interval></apply></apply><apply id="S6.SS2.16.p2.11.m11.12.12c.cmml" xref="S6.SS2.16.p2.11.m11.12.12"><eq id="S6.SS2.16.p2.11.m11.12.12.11.cmml" xref="S6.SS2.16.p2.11.m11.12.12.11"></eq><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.16.p2.11.m11.8.8.4.cmml" id="S6.SS2.16.p2.11.m11.12.12d.cmml" xref="S6.SS2.16.p2.11.m11.12.12"></share><ci id="S6.SS2.16.p2.11.m11.12.12.12.cmml" xref="S6.SS2.16.p2.11.m11.12.12.12">𝜈</ci></apply><apply id="S6.SS2.16.p2.11.m11.12.12e.cmml" xref="S6.SS2.16.p2.11.m11.12.12"><eq id="S6.SS2.16.p2.11.m11.12.12.13.cmml" xref="S6.SS2.16.p2.11.m11.12.12.13"></eq><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.16.p2.11.m11.12.12.12.cmml" id="S6.SS2.16.p2.11.m11.12.12f.cmml" xref="S6.SS2.16.p2.11.m11.12.12"></share><apply id="S6.SS2.16.p2.11.m11.10.10.6.cmml" xref="S6.SS2.16.p2.11.m11.10.10.6"><times id="S6.SS2.16.p2.11.m11.10.10.6.3.cmml" xref="S6.SS2.16.p2.11.m11.10.10.6.3"></times><ci id="S6.SS2.16.p2.11.m11.10.10.6.4.cmml" xref="S6.SS2.16.p2.11.m11.10.10.6.4">𝜈</ci><interval closure="open" id="S6.SS2.16.p2.11.m11.10.10.6.2.3.cmml" xref="S6.SS2.16.p2.11.m11.10.10.6.2.2"><apply id="S6.SS2.16.p2.11.m11.9.9.5.1.1.1.cmml" xref="S6.SS2.16.p2.11.m11.9.9.5.1.1.1"><times id="S6.SS2.16.p2.11.m11.9.9.5.1.1.1.1.cmml" xref="S6.SS2.16.p2.11.m11.9.9.5.1.1.1.1"></times><ci id="S6.SS2.16.p2.11.m11.9.9.5.1.1.1.2.cmml" xref="S6.SS2.16.p2.11.m11.9.9.5.1.1.1.2">𝑝</ci><apply id="S6.SS2.16.p2.11.m11.1.1.cmml" xref="S6.SS2.16.p2.11.m11.9.9.5.1.1.1.3.2"><ci id="S6.SS2.16.p2.11.m11.1.1.1.cmml" xref="S6.SS2.16.p2.11.m11.1.1.1">¯</ci><ci id="S6.SS2.16.p2.11.m11.1.1.2.cmml" xref="S6.SS2.16.p2.11.m11.1.1.2">𝑏</ci></apply></apply><apply id="S6.SS2.16.p2.11.m11.10.10.6.2.2.2.cmml" xref="S6.SS2.16.p2.11.m11.10.10.6.2.2.2"><times id="S6.SS2.16.p2.11.m11.10.10.6.2.2.2.1.cmml" xref="S6.SS2.16.p2.11.m11.10.10.6.2.2.2.1"></times><ci id="S6.SS2.16.p2.11.m11.10.10.6.2.2.2.2.cmml" xref="S6.SS2.16.p2.11.m11.10.10.6.2.2.2.2">𝑝</ci><apply id="S6.SS2.16.p2.11.m11.2.2.cmml" xref="S6.SS2.16.p2.11.m11.10.10.6.2.2.2.3.2"><ci id="S6.SS2.16.p2.11.m11.2.2.1.cmml" xref="S6.SS2.16.p2.11.m11.2.2.1">¯</ci><ci id="S6.SS2.16.p2.11.m11.2.2.2.cmml" xref="S6.SS2.16.p2.11.m11.2.2.2">𝑐</ci></apply></apply></interval></apply></apply><apply id="S6.SS2.16.p2.11.m11.12.12g.cmml" xref="S6.SS2.16.p2.11.m11.12.12"><eq id="S6.SS2.16.p2.11.m11.12.12.14.cmml" xref="S6.SS2.16.p2.11.m11.12.12.14"></eq><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.16.p2.11.m11.10.10.6.cmml" id="S6.SS2.16.p2.11.m11.12.12h.cmml" xref="S6.SS2.16.p2.11.m11.12.12"></share><apply id="S6.SS2.16.p2.11.m11.12.12.8.cmml" xref="S6.SS2.16.p2.11.m11.12.12.8"><times id="S6.SS2.16.p2.11.m11.12.12.8.3.cmml" xref="S6.SS2.16.p2.11.m11.12.12.8.3"></times><ci id="S6.SS2.16.p2.11.m11.12.12.8.4.cmml" xref="S6.SS2.16.p2.11.m11.12.12.8.4">𝜈</ci><interval closure="open" id="S6.SS2.16.p2.11.m11.12.12.8.2.3.cmml" xref="S6.SS2.16.p2.11.m11.12.12.8.2.2"><apply id="S6.SS2.16.p2.11.m11.11.11.7.1.1.1.cmml" xref="S6.SS2.16.p2.11.m11.11.11.7.1.1.1"><times id="S6.SS2.16.p2.11.m11.11.11.7.1.1.1.1.cmml" xref="S6.SS2.16.p2.11.m11.11.11.7.1.1.1.1"></times><ci id="S6.SS2.16.p2.11.m11.11.11.7.1.1.1.2.cmml" xref="S6.SS2.16.p2.11.m11.11.11.7.1.1.1.2">𝑝</ci><apply id="S6.SS2.16.p2.11.m11.3.3.cmml" xref="S6.SS2.16.p2.11.m11.11.11.7.1.1.1.3.2"><ci id="S6.SS2.16.p2.11.m11.3.3.1.cmml" xref="S6.SS2.16.p2.11.m11.3.3.1">¯</ci><ci id="S6.SS2.16.p2.11.m11.3.3.2.cmml" xref="S6.SS2.16.p2.11.m11.3.3.2">𝑎</ci></apply></apply><apply id="S6.SS2.16.p2.11.m11.12.12.8.2.2.2.cmml" xref="S6.SS2.16.p2.11.m11.12.12.8.2.2.2"><times id="S6.SS2.16.p2.11.m11.12.12.8.2.2.2.1.cmml" xref="S6.SS2.16.p2.11.m11.12.12.8.2.2.2.1"></times><ci id="S6.SS2.16.p2.11.m11.12.12.8.2.2.2.2.cmml" xref="S6.SS2.16.p2.11.m11.12.12.8.2.2.2.2">𝑝</ci><apply id="S6.SS2.16.p2.11.m11.4.4.cmml" xref="S6.SS2.16.p2.11.m11.12.12.8.2.2.2.3.2"><ci id="S6.SS2.16.p2.11.m11.4.4.1.cmml" xref="S6.SS2.16.p2.11.m11.4.4.1">¯</ci><ci id="S6.SS2.16.p2.11.m11.4.4.2.cmml" xref="S6.SS2.16.p2.11.m11.4.4.2">𝑐</ci></apply></apply></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.16.p2.11.m11.12c">\nu(a_{m},c_{m})=\nu(b_{m},c_{m})=\nu=\nu(p(\bar{b}),p(\bar{c}))=\nu(p(\bar{a}% ),p(\bar{c}))</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.16.p2.11.m11.12d">italic_ν ( italic_a start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT , italic_c start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) = italic_ν ( italic_b start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT , italic_c start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) = italic_ν = italic_ν ( italic_p ( over¯ start_ARG italic_b end_ARG ) , italic_p ( over¯ start_ARG italic_c end_ARG ) ) = italic_ν ( italic_p ( over¯ start_ARG italic_a end_ARG ) , italic_p ( over¯ start_ARG italic_c end_ARG ) )</annotation></semantics></math>. ∎</p> </div> </div> <div class="ltx_para" id="S6.SS2.p10"> <p class="ltx_p" id="S6.SS2.p10.1">The following summarizes some properties of that follow from the elementarity of <math alttext="E" class="ltx_Math" display="inline" id="S6.SS2.p10.1.m1.1"><semantics id="S6.SS2.p10.1.m1.1a"><mi id="S6.SS2.p10.1.m1.1.1" xref="S6.SS2.p10.1.m1.1.1.cmml">E</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.p10.1.m1.1b"><ci id="S6.SS2.p10.1.m1.1.1.cmml" xref="S6.SS2.p10.1.m1.1.1">𝐸</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p10.1.m1.1c">E</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p10.1.m1.1d">italic_E</annotation></semantics></math>, that will be frequently used without mention. Their proofs are standard, we give a proof of the last one as an example.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S6.Thmtheorem16"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem16.1.1.1">Lemma 6.16</span></span><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem16.2.2">.</span> </h6> <div class="ltx_para" id="S6.Thmtheorem16.p1"> <ol class="ltx_enumerate" id="S6.I8"> <li class="ltx_item" id="S6.I8.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(a)</span> <div class="ltx_para" id="S6.I8.i1.p1"> <p class="ltx_p" id="S6.I8.i1.p1.2">For all <math alttext="\nu\in E" class="ltx_Math" display="inline" id="S6.I8.i1.p1.1.m1.1"><semantics id="S6.I8.i1.p1.1.m1.1a"><mrow id="S6.I8.i1.p1.1.m1.1.1" xref="S6.I8.i1.p1.1.m1.1.1.cmml"><mi id="S6.I8.i1.p1.1.m1.1.1.2" xref="S6.I8.i1.p1.1.m1.1.1.2.cmml">ν</mi><mo id="S6.I8.i1.p1.1.m1.1.1.1" xref="S6.I8.i1.p1.1.m1.1.1.1.cmml">∈</mo><mi id="S6.I8.i1.p1.1.m1.1.1.3" xref="S6.I8.i1.p1.1.m1.1.1.3.cmml">E</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.I8.i1.p1.1.m1.1b"><apply id="S6.I8.i1.p1.1.m1.1.1.cmml" xref="S6.I8.i1.p1.1.m1.1.1"><in id="S6.I8.i1.p1.1.m1.1.1.1.cmml" xref="S6.I8.i1.p1.1.m1.1.1.1"></in><ci id="S6.I8.i1.p1.1.m1.1.1.2.cmml" xref="S6.I8.i1.p1.1.m1.1.1.2">𝜈</ci><ci id="S6.I8.i1.p1.1.m1.1.1.3.cmml" xref="S6.I8.i1.p1.1.m1.1.1.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I8.i1.p1.1.m1.1c">\nu\in E</annotation><annotation encoding="application/x-llamapun" id="S6.I8.i1.p1.1.m1.1d">italic_ν ∈ italic_E</annotation></semantics></math>, <math alttext="D_{\nu}=\nu" class="ltx_Math" display="inline" id="S6.I8.i1.p1.2.m2.1"><semantics id="S6.I8.i1.p1.2.m2.1a"><mrow id="S6.I8.i1.p1.2.m2.1.1" xref="S6.I8.i1.p1.2.m2.1.1.cmml"><msub id="S6.I8.i1.p1.2.m2.1.1.2" xref="S6.I8.i1.p1.2.m2.1.1.2.cmml"><mi id="S6.I8.i1.p1.2.m2.1.1.2.2" xref="S6.I8.i1.p1.2.m2.1.1.2.2.cmml">D</mi><mi id="S6.I8.i1.p1.2.m2.1.1.2.3" xref="S6.I8.i1.p1.2.m2.1.1.2.3.cmml">ν</mi></msub><mo id="S6.I8.i1.p1.2.m2.1.1.1" xref="S6.I8.i1.p1.2.m2.1.1.1.cmml">=</mo><mi id="S6.I8.i1.p1.2.m2.1.1.3" xref="S6.I8.i1.p1.2.m2.1.1.3.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.I8.i1.p1.2.m2.1b"><apply id="S6.I8.i1.p1.2.m2.1.1.cmml" xref="S6.I8.i1.p1.2.m2.1.1"><eq id="S6.I8.i1.p1.2.m2.1.1.1.cmml" xref="S6.I8.i1.p1.2.m2.1.1.1"></eq><apply id="S6.I8.i1.p1.2.m2.1.1.2.cmml" xref="S6.I8.i1.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="S6.I8.i1.p1.2.m2.1.1.2.1.cmml" xref="S6.I8.i1.p1.2.m2.1.1.2">subscript</csymbol><ci id="S6.I8.i1.p1.2.m2.1.1.2.2.cmml" xref="S6.I8.i1.p1.2.m2.1.1.2.2">𝐷</ci><ci id="S6.I8.i1.p1.2.m2.1.1.2.3.cmml" xref="S6.I8.i1.p1.2.m2.1.1.2.3">𝜈</ci></apply><ci id="S6.I8.i1.p1.2.m2.1.1.3.cmml" xref="S6.I8.i1.p1.2.m2.1.1.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I8.i1.p1.2.m2.1c">D_{\nu}=\nu</annotation><annotation encoding="application/x-llamapun" id="S6.I8.i1.p1.2.m2.1d">italic_D start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT = italic_ν</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S6.I8.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(b)</span> <div class="ltx_para" id="S6.I8.i2.p1"> <p class="ltx_p" id="S6.I8.i2.p1.1">Every <math alttext="\nu\in E" class="ltx_Math" display="inline" id="S6.I8.i2.p1.1.m1.1"><semantics id="S6.I8.i2.p1.1.m1.1a"><mrow id="S6.I8.i2.p1.1.m1.1.1" xref="S6.I8.i2.p1.1.m1.1.1.cmml"><mi id="S6.I8.i2.p1.1.m1.1.1.2" xref="S6.I8.i2.p1.1.m1.1.1.2.cmml">ν</mi><mo id="S6.I8.i2.p1.1.m1.1.1.1" xref="S6.I8.i2.p1.1.m1.1.1.1.cmml">∈</mo><mi id="S6.I8.i2.p1.1.m1.1.1.3" xref="S6.I8.i2.p1.1.m1.1.1.3.cmml">E</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.I8.i2.p1.1.m1.1b"><apply id="S6.I8.i2.p1.1.m1.1.1.cmml" xref="S6.I8.i2.p1.1.m1.1.1"><in id="S6.I8.i2.p1.1.m1.1.1.1.cmml" xref="S6.I8.i2.p1.1.m1.1.1.1"></in><ci id="S6.I8.i2.p1.1.m1.1.1.2.cmml" xref="S6.I8.i2.p1.1.m1.1.1.2">𝜈</ci><ci id="S6.I8.i2.p1.1.m1.1.1.3.cmml" xref="S6.I8.i2.p1.1.m1.1.1.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I8.i2.p1.1.m1.1c">\nu\in E</annotation><annotation encoding="application/x-llamapun" id="S6.I8.i2.p1.1.m1.1d">italic_ν ∈ italic_E</annotation></semantics></math> approximates all its elements (recall <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S5.Thmtheorem1" title="Definition 5.1. ‣ 5. An infinite antichain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Definition</span> <span class="ltx_text ltx_ref_tag">5.1</span></a>).</p> </div> </li> <li class="ltx_item" id="S6.I8.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(c)</span> <div class="ltx_para" id="S6.I8.i3.p1"> <p class="ltx_p" id="S6.I8.i3.p1.5">For all <math alttext="a\neq b" class="ltx_Math" display="inline" id="S6.I8.i3.p1.1.m1.1"><semantics id="S6.I8.i3.p1.1.m1.1a"><mrow id="S6.I8.i3.p1.1.m1.1.1" xref="S6.I8.i3.p1.1.m1.1.1.cmml"><mi id="S6.I8.i3.p1.1.m1.1.1.2" xref="S6.I8.i3.p1.1.m1.1.1.2.cmml">a</mi><mo id="S6.I8.i3.p1.1.m1.1.1.1" xref="S6.I8.i3.p1.1.m1.1.1.1.cmml">≠</mo><mi id="S6.I8.i3.p1.1.m1.1.1.3" xref="S6.I8.i3.p1.1.m1.1.1.3.cmml">b</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.I8.i3.p1.1.m1.1b"><apply id="S6.I8.i3.p1.1.m1.1.1.cmml" xref="S6.I8.i3.p1.1.m1.1.1"><neq id="S6.I8.i3.p1.1.m1.1.1.1.cmml" xref="S6.I8.i3.p1.1.m1.1.1.1"></neq><ci id="S6.I8.i3.p1.1.m1.1.1.2.cmml" xref="S6.I8.i3.p1.1.m1.1.1.2">𝑎</ci><ci id="S6.I8.i3.p1.1.m1.1.1.3.cmml" xref="S6.I8.i3.p1.1.m1.1.1.3">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I8.i3.p1.1.m1.1c">a\neq b</annotation><annotation encoding="application/x-llamapun" id="S6.I8.i3.p1.1.m1.1d">italic_a ≠ italic_b</annotation></semantics></math> and <math alttext="\nu\in E" class="ltx_Math" display="inline" id="S6.I8.i3.p1.2.m2.1"><semantics id="S6.I8.i3.p1.2.m2.1a"><mrow id="S6.I8.i3.p1.2.m2.1.1" xref="S6.I8.i3.p1.2.m2.1.1.cmml"><mi id="S6.I8.i3.p1.2.m2.1.1.2" xref="S6.I8.i3.p1.2.m2.1.1.2.cmml">ν</mi><mo id="S6.I8.i3.p1.2.m2.1.1.1" xref="S6.I8.i3.p1.2.m2.1.1.1.cmml">∈</mo><mi id="S6.I8.i3.p1.2.m2.1.1.3" xref="S6.I8.i3.p1.2.m2.1.1.3.cmml">E</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.I8.i3.p1.2.m2.1b"><apply id="S6.I8.i3.p1.2.m2.1.1.cmml" xref="S6.I8.i3.p1.2.m2.1.1"><in id="S6.I8.i3.p1.2.m2.1.1.1.cmml" xref="S6.I8.i3.p1.2.m2.1.1.1"></in><ci id="S6.I8.i3.p1.2.m2.1.1.2.cmml" xref="S6.I8.i3.p1.2.m2.1.1.2">𝜈</ci><ci id="S6.I8.i3.p1.2.m2.1.1.3.cmml" xref="S6.I8.i3.p1.2.m2.1.1.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I8.i3.p1.2.m2.1c">\nu\in E</annotation><annotation encoding="application/x-llamapun" id="S6.I8.i3.p1.2.m2.1d">italic_ν ∈ italic_E</annotation></semantics></math>, <math alttext="\Delta(a,b)\geq\nu" class="ltx_Math" display="inline" id="S6.I8.i3.p1.3.m3.2"><semantics id="S6.I8.i3.p1.3.m3.2a"><mrow id="S6.I8.i3.p1.3.m3.2.3" xref="S6.I8.i3.p1.3.m3.2.3.cmml"><mrow id="S6.I8.i3.p1.3.m3.2.3.2" xref="S6.I8.i3.p1.3.m3.2.3.2.cmml"><mi id="S6.I8.i3.p1.3.m3.2.3.2.2" mathvariant="normal" xref="S6.I8.i3.p1.3.m3.2.3.2.2.cmml">Δ</mi><mo id="S6.I8.i3.p1.3.m3.2.3.2.1" xref="S6.I8.i3.p1.3.m3.2.3.2.1.cmml">⁢</mo><mrow id="S6.I8.i3.p1.3.m3.2.3.2.3.2" xref="S6.I8.i3.p1.3.m3.2.3.2.3.1.cmml"><mo id="S6.I8.i3.p1.3.m3.2.3.2.3.2.1" stretchy="false" xref="S6.I8.i3.p1.3.m3.2.3.2.3.1.cmml">(</mo><mi id="S6.I8.i3.p1.3.m3.1.1" xref="S6.I8.i3.p1.3.m3.1.1.cmml">a</mi><mo id="S6.I8.i3.p1.3.m3.2.3.2.3.2.2" xref="S6.I8.i3.p1.3.m3.2.3.2.3.1.cmml">,</mo><mi id="S6.I8.i3.p1.3.m3.2.2" xref="S6.I8.i3.p1.3.m3.2.2.cmml">b</mi><mo id="S6.I8.i3.p1.3.m3.2.3.2.3.2.3" stretchy="false" xref="S6.I8.i3.p1.3.m3.2.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="S6.I8.i3.p1.3.m3.2.3.1" xref="S6.I8.i3.p1.3.m3.2.3.1.cmml">≥</mo><mi id="S6.I8.i3.p1.3.m3.2.3.3" xref="S6.I8.i3.p1.3.m3.2.3.3.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.I8.i3.p1.3.m3.2b"><apply id="S6.I8.i3.p1.3.m3.2.3.cmml" xref="S6.I8.i3.p1.3.m3.2.3"><geq id="S6.I8.i3.p1.3.m3.2.3.1.cmml" xref="S6.I8.i3.p1.3.m3.2.3.1"></geq><apply id="S6.I8.i3.p1.3.m3.2.3.2.cmml" xref="S6.I8.i3.p1.3.m3.2.3.2"><times id="S6.I8.i3.p1.3.m3.2.3.2.1.cmml" xref="S6.I8.i3.p1.3.m3.2.3.2.1"></times><ci id="S6.I8.i3.p1.3.m3.2.3.2.2.cmml" xref="S6.I8.i3.p1.3.m3.2.3.2.2">Δ</ci><interval closure="open" id="S6.I8.i3.p1.3.m3.2.3.2.3.1.cmml" xref="S6.I8.i3.p1.3.m3.2.3.2.3.2"><ci id="S6.I8.i3.p1.3.m3.1.1.cmml" xref="S6.I8.i3.p1.3.m3.1.1">𝑎</ci><ci id="S6.I8.i3.p1.3.m3.2.2.cmml" xref="S6.I8.i3.p1.3.m3.2.2">𝑏</ci></interval></apply><ci id="S6.I8.i3.p1.3.m3.2.3.3.cmml" xref="S6.I8.i3.p1.3.m3.2.3.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I8.i3.p1.3.m3.2c">\Delta(a,b)\geq\nu</annotation><annotation encoding="application/x-llamapun" id="S6.I8.i3.p1.3.m3.2d">roman_Δ ( italic_a , italic_b ) ≥ italic_ν</annotation></semantics></math> iff <math alttext="a,b\geq\nu" class="ltx_Math" display="inline" id="S6.I8.i3.p1.4.m4.2"><semantics id="S6.I8.i3.p1.4.m4.2a"><mrow id="S6.I8.i3.p1.4.m4.2.3" xref="S6.I8.i3.p1.4.m4.2.3.cmml"><mrow id="S6.I8.i3.p1.4.m4.2.3.2.2" xref="S6.I8.i3.p1.4.m4.2.3.2.1.cmml"><mi id="S6.I8.i3.p1.4.m4.1.1" xref="S6.I8.i3.p1.4.m4.1.1.cmml">a</mi><mo id="S6.I8.i3.p1.4.m4.2.3.2.2.1" xref="S6.I8.i3.p1.4.m4.2.3.2.1.cmml">,</mo><mi id="S6.I8.i3.p1.4.m4.2.2" xref="S6.I8.i3.p1.4.m4.2.2.cmml">b</mi></mrow><mo id="S6.I8.i3.p1.4.m4.2.3.1" xref="S6.I8.i3.p1.4.m4.2.3.1.cmml">≥</mo><mi id="S6.I8.i3.p1.4.m4.2.3.3" xref="S6.I8.i3.p1.4.m4.2.3.3.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.I8.i3.p1.4.m4.2b"><apply id="S6.I8.i3.p1.4.m4.2.3.cmml" xref="S6.I8.i3.p1.4.m4.2.3"><geq id="S6.I8.i3.p1.4.m4.2.3.1.cmml" xref="S6.I8.i3.p1.4.m4.2.3.1"></geq><list id="S6.I8.i3.p1.4.m4.2.3.2.1.cmml" xref="S6.I8.i3.p1.4.m4.2.3.2.2"><ci id="S6.I8.i3.p1.4.m4.1.1.cmml" xref="S6.I8.i3.p1.4.m4.1.1">𝑎</ci><ci id="S6.I8.i3.p1.4.m4.2.2.cmml" xref="S6.I8.i3.p1.4.m4.2.2">𝑏</ci></list><ci id="S6.I8.i3.p1.4.m4.2.3.3.cmml" xref="S6.I8.i3.p1.4.m4.2.3.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I8.i3.p1.4.m4.2c">a,b\geq\nu</annotation><annotation encoding="application/x-llamapun" id="S6.I8.i3.p1.4.m4.2d">italic_a , italic_b ≥ italic_ν</annotation></semantics></math> and they are in the same complementary interval of <math alttext="A\setminus\nu" class="ltx_Math" display="inline" id="S6.I8.i3.p1.5.m5.1"><semantics id="S6.I8.i3.p1.5.m5.1a"><mrow id="S6.I8.i3.p1.5.m5.1.1" xref="S6.I8.i3.p1.5.m5.1.1.cmml"><mi id="S6.I8.i3.p1.5.m5.1.1.2" xref="S6.I8.i3.p1.5.m5.1.1.2.cmml">A</mi><mo id="S6.I8.i3.p1.5.m5.1.1.1" xref="S6.I8.i3.p1.5.m5.1.1.1.cmml">∖</mo><mi id="S6.I8.i3.p1.5.m5.1.1.3" xref="S6.I8.i3.p1.5.m5.1.1.3.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.I8.i3.p1.5.m5.1b"><apply id="S6.I8.i3.p1.5.m5.1.1.cmml" xref="S6.I8.i3.p1.5.m5.1.1"><setdiff id="S6.I8.i3.p1.5.m5.1.1.1.cmml" xref="S6.I8.i3.p1.5.m5.1.1.1"></setdiff><ci id="S6.I8.i3.p1.5.m5.1.1.2.cmml" xref="S6.I8.i3.p1.5.m5.1.1.2">𝐴</ci><ci id="S6.I8.i3.p1.5.m5.1.1.3.cmml" xref="S6.I8.i3.p1.5.m5.1.1.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I8.i3.p1.5.m5.1c">A\setminus\nu</annotation><annotation encoding="application/x-llamapun" id="S6.I8.i3.p1.5.m5.1d">italic_A ∖ italic_ν</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S6.I8.i4" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(d)</span> <div class="ltx_para" id="S6.I8.i4.p1"> <p class="ltx_p" id="S6.I8.i4.p1.4">If <math alttext="\nu\in E" class="ltx_Math" display="inline" id="S6.I8.i4.p1.1.m1.1"><semantics id="S6.I8.i4.p1.1.m1.1a"><mrow id="S6.I8.i4.p1.1.m1.1.1" xref="S6.I8.i4.p1.1.m1.1.1.cmml"><mi id="S6.I8.i4.p1.1.m1.1.1.2" xref="S6.I8.i4.p1.1.m1.1.1.2.cmml">ν</mi><mo id="S6.I8.i4.p1.1.m1.1.1.1" xref="S6.I8.i4.p1.1.m1.1.1.1.cmml">∈</mo><mi id="S6.I8.i4.p1.1.m1.1.1.3" xref="S6.I8.i4.p1.1.m1.1.1.3.cmml">E</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.I8.i4.p1.1.m1.1b"><apply id="S6.I8.i4.p1.1.m1.1.1.cmml" xref="S6.I8.i4.p1.1.m1.1.1"><in id="S6.I8.i4.p1.1.m1.1.1.1.cmml" xref="S6.I8.i4.p1.1.m1.1.1.1"></in><ci id="S6.I8.i4.p1.1.m1.1.1.2.cmml" xref="S6.I8.i4.p1.1.m1.1.1.2">𝜈</ci><ci id="S6.I8.i4.p1.1.m1.1.1.3.cmml" xref="S6.I8.i4.p1.1.m1.1.1.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I8.i4.p1.1.m1.1c">\nu\in E</annotation><annotation encoding="application/x-llamapun" id="S6.I8.i4.p1.1.m1.1d">italic_ν ∈ italic_E</annotation></semantics></math> and <math alttext="a" class="ltx_Math" display="inline" id="S6.I8.i4.p1.2.m2.1"><semantics id="S6.I8.i4.p1.2.m2.1a"><mi id="S6.I8.i4.p1.2.m2.1.1" xref="S6.I8.i4.p1.2.m2.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S6.I8.i4.p1.2.m2.1b"><ci id="S6.I8.i4.p1.2.m2.1.1.cmml" xref="S6.I8.i4.p1.2.m2.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.I8.i4.p1.2.m2.1c">a</annotation><annotation encoding="application/x-llamapun" id="S6.I8.i4.p1.2.m2.1d">italic_a</annotation></semantics></math> is an endpoint of some complementary interval of <math alttext="A\setminus\nu" class="ltx_Math" display="inline" id="S6.I8.i4.p1.3.m3.1"><semantics id="S6.I8.i4.p1.3.m3.1a"><mrow id="S6.I8.i4.p1.3.m3.1.1" xref="S6.I8.i4.p1.3.m3.1.1.cmml"><mi id="S6.I8.i4.p1.3.m3.1.1.2" xref="S6.I8.i4.p1.3.m3.1.1.2.cmml">A</mi><mo id="S6.I8.i4.p1.3.m3.1.1.1" xref="S6.I8.i4.p1.3.m3.1.1.1.cmml">∖</mo><mi id="S6.I8.i4.p1.3.m3.1.1.3" xref="S6.I8.i4.p1.3.m3.1.1.3.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.I8.i4.p1.3.m3.1b"><apply id="S6.I8.i4.p1.3.m3.1.1.cmml" xref="S6.I8.i4.p1.3.m3.1.1"><setdiff id="S6.I8.i4.p1.3.m3.1.1.1.cmml" xref="S6.I8.i4.p1.3.m3.1.1.1"></setdiff><ci id="S6.I8.i4.p1.3.m3.1.1.2.cmml" xref="S6.I8.i4.p1.3.m3.1.1.2">𝐴</ci><ci id="S6.I8.i4.p1.3.m3.1.1.3.cmml" xref="S6.I8.i4.p1.3.m3.1.1.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I8.i4.p1.3.m3.1c">A\setminus\nu</annotation><annotation encoding="application/x-llamapun" id="S6.I8.i4.p1.3.m3.1d">italic_A ∖ italic_ν</annotation></semantics></math>, then <math alttext="\nu(a)=\nu" class="ltx_Math" display="inline" id="S6.I8.i4.p1.4.m4.1"><semantics id="S6.I8.i4.p1.4.m4.1a"><mrow id="S6.I8.i4.p1.4.m4.1.2" xref="S6.I8.i4.p1.4.m4.1.2.cmml"><mrow id="S6.I8.i4.p1.4.m4.1.2.2" xref="S6.I8.i4.p1.4.m4.1.2.2.cmml"><mi id="S6.I8.i4.p1.4.m4.1.2.2.2" xref="S6.I8.i4.p1.4.m4.1.2.2.2.cmml">ν</mi><mo id="S6.I8.i4.p1.4.m4.1.2.2.1" xref="S6.I8.i4.p1.4.m4.1.2.2.1.cmml">⁢</mo><mrow id="S6.I8.i4.p1.4.m4.1.2.2.3.2" xref="S6.I8.i4.p1.4.m4.1.2.2.cmml"><mo id="S6.I8.i4.p1.4.m4.1.2.2.3.2.1" stretchy="false" xref="S6.I8.i4.p1.4.m4.1.2.2.cmml">(</mo><mi id="S6.I8.i4.p1.4.m4.1.1" xref="S6.I8.i4.p1.4.m4.1.1.cmml">a</mi><mo id="S6.I8.i4.p1.4.m4.1.2.2.3.2.2" stretchy="false" xref="S6.I8.i4.p1.4.m4.1.2.2.cmml">)</mo></mrow></mrow><mo id="S6.I8.i4.p1.4.m4.1.2.1" xref="S6.I8.i4.p1.4.m4.1.2.1.cmml">=</mo><mi id="S6.I8.i4.p1.4.m4.1.2.3" xref="S6.I8.i4.p1.4.m4.1.2.3.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.I8.i4.p1.4.m4.1b"><apply id="S6.I8.i4.p1.4.m4.1.2.cmml" xref="S6.I8.i4.p1.4.m4.1.2"><eq id="S6.I8.i4.p1.4.m4.1.2.1.cmml" xref="S6.I8.i4.p1.4.m4.1.2.1"></eq><apply id="S6.I8.i4.p1.4.m4.1.2.2.cmml" xref="S6.I8.i4.p1.4.m4.1.2.2"><times id="S6.I8.i4.p1.4.m4.1.2.2.1.cmml" xref="S6.I8.i4.p1.4.m4.1.2.2.1"></times><ci id="S6.I8.i4.p1.4.m4.1.2.2.2.cmml" xref="S6.I8.i4.p1.4.m4.1.2.2.2">𝜈</ci><ci id="S6.I8.i4.p1.4.m4.1.1.cmml" xref="S6.I8.i4.p1.4.m4.1.1">𝑎</ci></apply><ci id="S6.I8.i4.p1.4.m4.1.2.3.cmml" xref="S6.I8.i4.p1.4.m4.1.2.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I8.i4.p1.4.m4.1c">\nu(a)=\nu</annotation><annotation encoding="application/x-llamapun" id="S6.I8.i4.p1.4.m4.1d">italic_ν ( italic_a ) = italic_ν</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S6.I8.i5" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(e)</span> <div class="ltx_para" id="S6.I8.i5.p1"> <p class="ltx_p" id="S6.I8.i5.p1.3">If <math alttext="\nu\in E\cup\{0\}" class="ltx_Math" display="inline" id="S6.I8.i5.p1.1.m1.1"><semantics id="S6.I8.i5.p1.1.m1.1a"><mrow id="S6.I8.i5.p1.1.m1.1.2" xref="S6.I8.i5.p1.1.m1.1.2.cmml"><mi id="S6.I8.i5.p1.1.m1.1.2.2" xref="S6.I8.i5.p1.1.m1.1.2.2.cmml">ν</mi><mo id="S6.I8.i5.p1.1.m1.1.2.1" xref="S6.I8.i5.p1.1.m1.1.2.1.cmml">∈</mo><mrow id="S6.I8.i5.p1.1.m1.1.2.3" xref="S6.I8.i5.p1.1.m1.1.2.3.cmml"><mi id="S6.I8.i5.p1.1.m1.1.2.3.2" xref="S6.I8.i5.p1.1.m1.1.2.3.2.cmml">E</mi><mo id="S6.I8.i5.p1.1.m1.1.2.3.1" xref="S6.I8.i5.p1.1.m1.1.2.3.1.cmml">∪</mo><mrow id="S6.I8.i5.p1.1.m1.1.2.3.3.2" xref="S6.I8.i5.p1.1.m1.1.2.3.3.1.cmml"><mo id="S6.I8.i5.p1.1.m1.1.2.3.3.2.1" stretchy="false" xref="S6.I8.i5.p1.1.m1.1.2.3.3.1.cmml">{</mo><mn id="S6.I8.i5.p1.1.m1.1.1" xref="S6.I8.i5.p1.1.m1.1.1.cmml">0</mn><mo id="S6.I8.i5.p1.1.m1.1.2.3.3.2.2" stretchy="false" xref="S6.I8.i5.p1.1.m1.1.2.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.I8.i5.p1.1.m1.1b"><apply id="S6.I8.i5.p1.1.m1.1.2.cmml" xref="S6.I8.i5.p1.1.m1.1.2"><in id="S6.I8.i5.p1.1.m1.1.2.1.cmml" xref="S6.I8.i5.p1.1.m1.1.2.1"></in><ci id="S6.I8.i5.p1.1.m1.1.2.2.cmml" xref="S6.I8.i5.p1.1.m1.1.2.2">𝜈</ci><apply id="S6.I8.i5.p1.1.m1.1.2.3.cmml" xref="S6.I8.i5.p1.1.m1.1.2.3"><union id="S6.I8.i5.p1.1.m1.1.2.3.1.cmml" xref="S6.I8.i5.p1.1.m1.1.2.3.1"></union><ci id="S6.I8.i5.p1.1.m1.1.2.3.2.cmml" xref="S6.I8.i5.p1.1.m1.1.2.3.2">𝐸</ci><set id="S6.I8.i5.p1.1.m1.1.2.3.3.1.cmml" xref="S6.I8.i5.p1.1.m1.1.2.3.3.2"><cn id="S6.I8.i5.p1.1.m1.1.1.cmml" type="integer" xref="S6.I8.i5.p1.1.m1.1.1">0</cn></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I8.i5.p1.1.m1.1c">\nu\in E\cup\{0\}</annotation><annotation encoding="application/x-llamapun" id="S6.I8.i5.p1.1.m1.1d">italic_ν ∈ italic_E ∪ { 0 }</annotation></semantics></math>, then <math alttext="\left[\nu,\nu^{+}\right[" class="ltx_Math" display="inline" id="S6.I8.i5.p1.2.m2.2"><semantics id="S6.I8.i5.p1.2.m2.2a"><mrow id="S6.I8.i5.p1.2.m2.2.2.1" xref="S6.I8.i5.p1.2.m2.2.2.2.cmml"><mo id="S6.I8.i5.p1.2.m2.2.2.1.2" xref="S6.I8.i5.p1.2.m2.2.2.2.cmml">[</mo><mi id="S6.I8.i5.p1.2.m2.1.1" xref="S6.I8.i5.p1.2.m2.1.1.cmml">ν</mi><mo id="S6.I8.i5.p1.2.m2.2.2.1.3" xref="S6.I8.i5.p1.2.m2.2.2.2.cmml">,</mo><msup id="S6.I8.i5.p1.2.m2.2.2.1.1" xref="S6.I8.i5.p1.2.m2.2.2.1.1.cmml"><mi id="S6.I8.i5.p1.2.m2.2.2.1.1.2" xref="S6.I8.i5.p1.2.m2.2.2.1.1.2.cmml">ν</mi><mo id="S6.I8.i5.p1.2.m2.2.2.1.1.3" xref="S6.I8.i5.p1.2.m2.2.2.1.1.3.cmml">+</mo></msup><mo id="S6.I8.i5.p1.2.m2.2.2.1.4" lspace="0em" stretchy="true" xref="S6.I8.i5.p1.2.m2.2.2.2.cmml">[</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.I8.i5.p1.2.m2.2b"><list id="S6.I8.i5.p1.2.m2.2.2.2.cmml" xref="S6.I8.i5.p1.2.m2.2.2.1"><ci id="S6.I8.i5.p1.2.m2.1.1.cmml" xref="S6.I8.i5.p1.2.m2.1.1">𝜈</ci><apply id="S6.I8.i5.p1.2.m2.2.2.1.1.cmml" xref="S6.I8.i5.p1.2.m2.2.2.1.1"><csymbol cd="ambiguous" id="S6.I8.i5.p1.2.m2.2.2.1.1.1.cmml" xref="S6.I8.i5.p1.2.m2.2.2.1.1">superscript</csymbol><ci id="S6.I8.i5.p1.2.m2.2.2.1.1.2.cmml" xref="S6.I8.i5.p1.2.m2.2.2.1.1.2">𝜈</ci><plus id="S6.I8.i5.p1.2.m2.2.2.1.1.3.cmml" xref="S6.I8.i5.p1.2.m2.2.2.1.1.3"></plus></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S6.I8.i5.p1.2.m2.2c">\left[\nu,\nu^{+}\right[</annotation><annotation encoding="application/x-llamapun" id="S6.I8.i5.p1.2.m2.2d">[ italic_ν , italic_ν start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT [</annotation></semantics></math> is cofinal and coinitial in every complementary interval of <math alttext="A\setminus\nu" class="ltx_Math" display="inline" id="S6.I8.i5.p1.3.m3.1"><semantics id="S6.I8.i5.p1.3.m3.1a"><mrow id="S6.I8.i5.p1.3.m3.1.1" xref="S6.I8.i5.p1.3.m3.1.1.cmml"><mi id="S6.I8.i5.p1.3.m3.1.1.2" xref="S6.I8.i5.p1.3.m3.1.1.2.cmml">A</mi><mo id="S6.I8.i5.p1.3.m3.1.1.1" xref="S6.I8.i5.p1.3.m3.1.1.1.cmml">∖</mo><mi id="S6.I8.i5.p1.3.m3.1.1.3" xref="S6.I8.i5.p1.3.m3.1.1.3.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.I8.i5.p1.3.m3.1b"><apply id="S6.I8.i5.p1.3.m3.1.1.cmml" xref="S6.I8.i5.p1.3.m3.1.1"><setdiff id="S6.I8.i5.p1.3.m3.1.1.1.cmml" xref="S6.I8.i5.p1.3.m3.1.1.1"></setdiff><ci id="S6.I8.i5.p1.3.m3.1.1.2.cmml" xref="S6.I8.i5.p1.3.m3.1.1.2">𝐴</ci><ci id="S6.I8.i5.p1.3.m3.1.1.3.cmml" xref="S6.I8.i5.p1.3.m3.1.1.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I8.i5.p1.3.m3.1c">A\setminus\nu</annotation><annotation encoding="application/x-llamapun" id="S6.I8.i5.p1.3.m3.1d">italic_A ∖ italic_ν</annotation></semantics></math>.</p> </div> </li> </ol> </div> </div> <div class="ltx_proof" id="S6.SS2.17"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S6.SS2.17.p1"> <p class="ltx_p" id="S6.SS2.17.p1.14">We prove the last one, the others are similar. Let <math alttext="N\prec H(\omega_{2})" class="ltx_Math" display="inline" id="S6.SS2.17.p1.1.m1.1"><semantics id="S6.SS2.17.p1.1.m1.1a"><mrow id="S6.SS2.17.p1.1.m1.1.1" xref="S6.SS2.17.p1.1.m1.1.1.cmml"><mi id="S6.SS2.17.p1.1.m1.1.1.3" xref="S6.SS2.17.p1.1.m1.1.1.3.cmml">N</mi><mo id="S6.SS2.17.p1.1.m1.1.1.2" xref="S6.SS2.17.p1.1.m1.1.1.2.cmml">≺</mo><mrow id="S6.SS2.17.p1.1.m1.1.1.1" xref="S6.SS2.17.p1.1.m1.1.1.1.cmml"><mi id="S6.SS2.17.p1.1.m1.1.1.1.3" xref="S6.SS2.17.p1.1.m1.1.1.1.3.cmml">H</mi><mo id="S6.SS2.17.p1.1.m1.1.1.1.2" xref="S6.SS2.17.p1.1.m1.1.1.1.2.cmml">⁢</mo><mrow id="S6.SS2.17.p1.1.m1.1.1.1.1.1" xref="S6.SS2.17.p1.1.m1.1.1.1.1.1.1.cmml"><mo id="S6.SS2.17.p1.1.m1.1.1.1.1.1.2" stretchy="false" xref="S6.SS2.17.p1.1.m1.1.1.1.1.1.1.cmml">(</mo><msub id="S6.SS2.17.p1.1.m1.1.1.1.1.1.1" xref="S6.SS2.17.p1.1.m1.1.1.1.1.1.1.cmml"><mi id="S6.SS2.17.p1.1.m1.1.1.1.1.1.1.2" xref="S6.SS2.17.p1.1.m1.1.1.1.1.1.1.2.cmml">ω</mi><mn id="S6.SS2.17.p1.1.m1.1.1.1.1.1.1.3" xref="S6.SS2.17.p1.1.m1.1.1.1.1.1.1.3.cmml">2</mn></msub><mo id="S6.SS2.17.p1.1.m1.1.1.1.1.1.3" stretchy="false" xref="S6.SS2.17.p1.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.17.p1.1.m1.1b"><apply id="S6.SS2.17.p1.1.m1.1.1.cmml" xref="S6.SS2.17.p1.1.m1.1.1"><csymbol cd="latexml" id="S6.SS2.17.p1.1.m1.1.1.2.cmml" xref="S6.SS2.17.p1.1.m1.1.1.2">precedes</csymbol><ci id="S6.SS2.17.p1.1.m1.1.1.3.cmml" xref="S6.SS2.17.p1.1.m1.1.1.3">𝑁</ci><apply id="S6.SS2.17.p1.1.m1.1.1.1.cmml" xref="S6.SS2.17.p1.1.m1.1.1.1"><times id="S6.SS2.17.p1.1.m1.1.1.1.2.cmml" xref="S6.SS2.17.p1.1.m1.1.1.1.2"></times><ci id="S6.SS2.17.p1.1.m1.1.1.1.3.cmml" xref="S6.SS2.17.p1.1.m1.1.1.1.3">𝐻</ci><apply id="S6.SS2.17.p1.1.m1.1.1.1.1.1.1.cmml" xref="S6.SS2.17.p1.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.17.p1.1.m1.1.1.1.1.1.1.1.cmml" xref="S6.SS2.17.p1.1.m1.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.17.p1.1.m1.1.1.1.1.1.1.2.cmml" xref="S6.SS2.17.p1.1.m1.1.1.1.1.1.1.2">𝜔</ci><cn id="S6.SS2.17.p1.1.m1.1.1.1.1.1.1.3.cmml" type="integer" xref="S6.SS2.17.p1.1.m1.1.1.1.1.1.1.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.17.p1.1.m1.1c">N\prec H(\omega_{2})</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.17.p1.1.m1.1d">italic_N ≺ italic_H ( italic_ω start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT )</annotation></semantics></math> witness that <math alttext="\nu^{+}" class="ltx_Math" display="inline" id="S6.SS2.17.p1.2.m2.1"><semantics id="S6.SS2.17.p1.2.m2.1a"><msup id="S6.SS2.17.p1.2.m2.1.1" xref="S6.SS2.17.p1.2.m2.1.1.cmml"><mi id="S6.SS2.17.p1.2.m2.1.1.2" xref="S6.SS2.17.p1.2.m2.1.1.2.cmml">ν</mi><mo id="S6.SS2.17.p1.2.m2.1.1.3" xref="S6.SS2.17.p1.2.m2.1.1.3.cmml">+</mo></msup><annotation-xml encoding="MathML-Content" id="S6.SS2.17.p1.2.m2.1b"><apply id="S6.SS2.17.p1.2.m2.1.1.cmml" xref="S6.SS2.17.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S6.SS2.17.p1.2.m2.1.1.1.cmml" xref="S6.SS2.17.p1.2.m2.1.1">superscript</csymbol><ci id="S6.SS2.17.p1.2.m2.1.1.2.cmml" xref="S6.SS2.17.p1.2.m2.1.1.2">𝜈</ci><plus id="S6.SS2.17.p1.2.m2.1.1.3.cmml" xref="S6.SS2.17.p1.2.m2.1.1.3"></plus></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.17.p1.2.m2.1c">\nu^{+}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.17.p1.2.m2.1d">italic_ν start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> is elementary for <math alttext="A" class="ltx_Math" display="inline" id="S6.SS2.17.p1.3.m3.1"><semantics id="S6.SS2.17.p1.3.m3.1a"><mi id="S6.SS2.17.p1.3.m3.1.1" xref="S6.SS2.17.p1.3.m3.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.17.p1.3.m3.1b"><ci id="S6.SS2.17.p1.3.m3.1.1.cmml" xref="S6.SS2.17.p1.3.m3.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.17.p1.3.m3.1c">A</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.17.p1.3.m3.1d">italic_A</annotation></semantics></math>. Observe that <math alttext="H(\omega_{2})" class="ltx_Math" display="inline" id="S6.SS2.17.p1.4.m4.1"><semantics id="S6.SS2.17.p1.4.m4.1a"><mrow id="S6.SS2.17.p1.4.m4.1.1" xref="S6.SS2.17.p1.4.m4.1.1.cmml"><mi id="S6.SS2.17.p1.4.m4.1.1.3" xref="S6.SS2.17.p1.4.m4.1.1.3.cmml">H</mi><mo id="S6.SS2.17.p1.4.m4.1.1.2" xref="S6.SS2.17.p1.4.m4.1.1.2.cmml">⁢</mo><mrow id="S6.SS2.17.p1.4.m4.1.1.1.1" xref="S6.SS2.17.p1.4.m4.1.1.1.1.1.cmml"><mo id="S6.SS2.17.p1.4.m4.1.1.1.1.2" stretchy="false" xref="S6.SS2.17.p1.4.m4.1.1.1.1.1.cmml">(</mo><msub id="S6.SS2.17.p1.4.m4.1.1.1.1.1" xref="S6.SS2.17.p1.4.m4.1.1.1.1.1.cmml"><mi id="S6.SS2.17.p1.4.m4.1.1.1.1.1.2" xref="S6.SS2.17.p1.4.m4.1.1.1.1.1.2.cmml">ω</mi><mn id="S6.SS2.17.p1.4.m4.1.1.1.1.1.3" xref="S6.SS2.17.p1.4.m4.1.1.1.1.1.3.cmml">2</mn></msub><mo id="S6.SS2.17.p1.4.m4.1.1.1.1.3" stretchy="false" xref="S6.SS2.17.p1.4.m4.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.17.p1.4.m4.1b"><apply id="S6.SS2.17.p1.4.m4.1.1.cmml" xref="S6.SS2.17.p1.4.m4.1.1"><times id="S6.SS2.17.p1.4.m4.1.1.2.cmml" xref="S6.SS2.17.p1.4.m4.1.1.2"></times><ci id="S6.SS2.17.p1.4.m4.1.1.3.cmml" xref="S6.SS2.17.p1.4.m4.1.1.3">𝐻</ci><apply id="S6.SS2.17.p1.4.m4.1.1.1.1.1.cmml" xref="S6.SS2.17.p1.4.m4.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.17.p1.4.m4.1.1.1.1.1.1.cmml" xref="S6.SS2.17.p1.4.m4.1.1.1.1">subscript</csymbol><ci id="S6.SS2.17.p1.4.m4.1.1.1.1.1.2.cmml" xref="S6.SS2.17.p1.4.m4.1.1.1.1.1.2">𝜔</ci><cn id="S6.SS2.17.p1.4.m4.1.1.1.1.1.3.cmml" type="integer" xref="S6.SS2.17.p1.4.m4.1.1.1.1.1.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.17.p1.4.m4.1c">H(\omega_{2})</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.17.p1.4.m4.1d">italic_H ( italic_ω start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT )</annotation></semantics></math> models that there is a countable <math alttext="X\subseteq A" class="ltx_Math" display="inline" id="S6.SS2.17.p1.5.m5.1"><semantics id="S6.SS2.17.p1.5.m5.1a"><mrow id="S6.SS2.17.p1.5.m5.1.1" xref="S6.SS2.17.p1.5.m5.1.1.cmml"><mi id="S6.SS2.17.p1.5.m5.1.1.2" xref="S6.SS2.17.p1.5.m5.1.1.2.cmml">X</mi><mo id="S6.SS2.17.p1.5.m5.1.1.1" xref="S6.SS2.17.p1.5.m5.1.1.1.cmml">⊆</mo><mi id="S6.SS2.17.p1.5.m5.1.1.3" xref="S6.SS2.17.p1.5.m5.1.1.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.17.p1.5.m5.1b"><apply id="S6.SS2.17.p1.5.m5.1.1.cmml" xref="S6.SS2.17.p1.5.m5.1.1"><subset id="S6.SS2.17.p1.5.m5.1.1.1.cmml" xref="S6.SS2.17.p1.5.m5.1.1.1"></subset><ci id="S6.SS2.17.p1.5.m5.1.1.2.cmml" xref="S6.SS2.17.p1.5.m5.1.1.2">𝑋</ci><ci id="S6.SS2.17.p1.5.m5.1.1.3.cmml" xref="S6.SS2.17.p1.5.m5.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.17.p1.5.m5.1c">X\subseteq A</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.17.p1.5.m5.1d">italic_X ⊆ italic_A</annotation></semantics></math> such that <math alttext="X" class="ltx_Math" display="inline" id="S6.SS2.17.p1.6.m6.1"><semantics id="S6.SS2.17.p1.6.m6.1a"><mi id="S6.SS2.17.p1.6.m6.1.1" xref="S6.SS2.17.p1.6.m6.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.17.p1.6.m6.1b"><ci id="S6.SS2.17.p1.6.m6.1.1.cmml" xref="S6.SS2.17.p1.6.m6.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.17.p1.6.m6.1c">X</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.17.p1.6.m6.1d">italic_X</annotation></semantics></math> is cofinal and coinitial in every complementary interval of <math alttext="A\setminus\nu" class="ltx_Math" display="inline" id="S6.SS2.17.p1.7.m7.1"><semantics id="S6.SS2.17.p1.7.m7.1a"><mrow id="S6.SS2.17.p1.7.m7.1.1" xref="S6.SS2.17.p1.7.m7.1.1.cmml"><mi id="S6.SS2.17.p1.7.m7.1.1.2" xref="S6.SS2.17.p1.7.m7.1.1.2.cmml">A</mi><mo id="S6.SS2.17.p1.7.m7.1.1.1" xref="S6.SS2.17.p1.7.m7.1.1.1.cmml">∖</mo><mi id="S6.SS2.17.p1.7.m7.1.1.3" xref="S6.SS2.17.p1.7.m7.1.1.3.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.17.p1.7.m7.1b"><apply id="S6.SS2.17.p1.7.m7.1.1.cmml" xref="S6.SS2.17.p1.7.m7.1.1"><setdiff id="S6.SS2.17.p1.7.m7.1.1.1.cmml" xref="S6.SS2.17.p1.7.m7.1.1.1"></setdiff><ci id="S6.SS2.17.p1.7.m7.1.1.2.cmml" xref="S6.SS2.17.p1.7.m7.1.1.2">𝐴</ci><ci id="S6.SS2.17.p1.7.m7.1.1.3.cmml" xref="S6.SS2.17.p1.7.m7.1.1.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.17.p1.7.m7.1c">A\setminus\nu</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.17.p1.7.m7.1d">italic_A ∖ italic_ν</annotation></semantics></math>, this is because there are countably many such intervals. Then, there must be such a set <math alttext="X" class="ltx_Math" display="inline" id="S6.SS2.17.p1.8.m8.1"><semantics id="S6.SS2.17.p1.8.m8.1a"><mi id="S6.SS2.17.p1.8.m8.1.1" xref="S6.SS2.17.p1.8.m8.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.17.p1.8.m8.1b"><ci id="S6.SS2.17.p1.8.m8.1.1.cmml" xref="S6.SS2.17.p1.8.m8.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.17.p1.8.m8.1c">X</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.17.p1.8.m8.1d">italic_X</annotation></semantics></math> inside <math alttext="N" class="ltx_Math" display="inline" id="S6.SS2.17.p1.9.m9.1"><semantics id="S6.SS2.17.p1.9.m9.1a"><mi id="S6.SS2.17.p1.9.m9.1.1" xref="S6.SS2.17.p1.9.m9.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.17.p1.9.m9.1b"><ci id="S6.SS2.17.p1.9.m9.1.1.cmml" xref="S6.SS2.17.p1.9.m9.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.17.p1.9.m9.1c">N</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.17.p1.9.m9.1d">italic_N</annotation></semantics></math>, and since <math alttext="X" class="ltx_Math" display="inline" id="S6.SS2.17.p1.10.m10.1"><semantics id="S6.SS2.17.p1.10.m10.1a"><mi id="S6.SS2.17.p1.10.m10.1.1" xref="S6.SS2.17.p1.10.m10.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.17.p1.10.m10.1b"><ci id="S6.SS2.17.p1.10.m10.1.1.cmml" xref="S6.SS2.17.p1.10.m10.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.17.p1.10.m10.1c">X</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.17.p1.10.m10.1d">italic_X</annotation></semantics></math> is countable, <math alttext="X\subseteq N" class="ltx_Math" display="inline" id="S6.SS2.17.p1.11.m11.1"><semantics id="S6.SS2.17.p1.11.m11.1a"><mrow id="S6.SS2.17.p1.11.m11.1.1" xref="S6.SS2.17.p1.11.m11.1.1.cmml"><mi id="S6.SS2.17.p1.11.m11.1.1.2" xref="S6.SS2.17.p1.11.m11.1.1.2.cmml">X</mi><mo id="S6.SS2.17.p1.11.m11.1.1.1" xref="S6.SS2.17.p1.11.m11.1.1.1.cmml">⊆</mo><mi id="S6.SS2.17.p1.11.m11.1.1.3" xref="S6.SS2.17.p1.11.m11.1.1.3.cmml">N</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.17.p1.11.m11.1b"><apply id="S6.SS2.17.p1.11.m11.1.1.cmml" xref="S6.SS2.17.p1.11.m11.1.1"><subset id="S6.SS2.17.p1.11.m11.1.1.1.cmml" xref="S6.SS2.17.p1.11.m11.1.1.1"></subset><ci id="S6.SS2.17.p1.11.m11.1.1.2.cmml" xref="S6.SS2.17.p1.11.m11.1.1.2">𝑋</ci><ci id="S6.SS2.17.p1.11.m11.1.1.3.cmml" xref="S6.SS2.17.p1.11.m11.1.1.3">𝑁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.17.p1.11.m11.1c">X\subseteq N</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.17.p1.11.m11.1d">italic_X ⊆ italic_N</annotation></semantics></math>. Since <math alttext="N\cap\omega_{1}=\nu^{+}" class="ltx_Math" display="inline" id="S6.SS2.17.p1.12.m12.1"><semantics id="S6.SS2.17.p1.12.m12.1a"><mrow id="S6.SS2.17.p1.12.m12.1.1" xref="S6.SS2.17.p1.12.m12.1.1.cmml"><mrow id="S6.SS2.17.p1.12.m12.1.1.2" xref="S6.SS2.17.p1.12.m12.1.1.2.cmml"><mi id="S6.SS2.17.p1.12.m12.1.1.2.2" xref="S6.SS2.17.p1.12.m12.1.1.2.2.cmml">N</mi><mo id="S6.SS2.17.p1.12.m12.1.1.2.1" xref="S6.SS2.17.p1.12.m12.1.1.2.1.cmml">∩</mo><msub id="S6.SS2.17.p1.12.m12.1.1.2.3" xref="S6.SS2.17.p1.12.m12.1.1.2.3.cmml"><mi id="S6.SS2.17.p1.12.m12.1.1.2.3.2" xref="S6.SS2.17.p1.12.m12.1.1.2.3.2.cmml">ω</mi><mn id="S6.SS2.17.p1.12.m12.1.1.2.3.3" xref="S6.SS2.17.p1.12.m12.1.1.2.3.3.cmml">1</mn></msub></mrow><mo id="S6.SS2.17.p1.12.m12.1.1.1" xref="S6.SS2.17.p1.12.m12.1.1.1.cmml">=</mo><msup id="S6.SS2.17.p1.12.m12.1.1.3" xref="S6.SS2.17.p1.12.m12.1.1.3.cmml"><mi id="S6.SS2.17.p1.12.m12.1.1.3.2" xref="S6.SS2.17.p1.12.m12.1.1.3.2.cmml">ν</mi><mo id="S6.SS2.17.p1.12.m12.1.1.3.3" xref="S6.SS2.17.p1.12.m12.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.17.p1.12.m12.1b"><apply id="S6.SS2.17.p1.12.m12.1.1.cmml" xref="S6.SS2.17.p1.12.m12.1.1"><eq id="S6.SS2.17.p1.12.m12.1.1.1.cmml" xref="S6.SS2.17.p1.12.m12.1.1.1"></eq><apply id="S6.SS2.17.p1.12.m12.1.1.2.cmml" xref="S6.SS2.17.p1.12.m12.1.1.2"><intersect id="S6.SS2.17.p1.12.m12.1.1.2.1.cmml" xref="S6.SS2.17.p1.12.m12.1.1.2.1"></intersect><ci id="S6.SS2.17.p1.12.m12.1.1.2.2.cmml" xref="S6.SS2.17.p1.12.m12.1.1.2.2">𝑁</ci><apply id="S6.SS2.17.p1.12.m12.1.1.2.3.cmml" xref="S6.SS2.17.p1.12.m12.1.1.2.3"><csymbol cd="ambiguous" id="S6.SS2.17.p1.12.m12.1.1.2.3.1.cmml" xref="S6.SS2.17.p1.12.m12.1.1.2.3">subscript</csymbol><ci id="S6.SS2.17.p1.12.m12.1.1.2.3.2.cmml" xref="S6.SS2.17.p1.12.m12.1.1.2.3.2">𝜔</ci><cn id="S6.SS2.17.p1.12.m12.1.1.2.3.3.cmml" type="integer" xref="S6.SS2.17.p1.12.m12.1.1.2.3.3">1</cn></apply></apply><apply id="S6.SS2.17.p1.12.m12.1.1.3.cmml" xref="S6.SS2.17.p1.12.m12.1.1.3"><csymbol cd="ambiguous" id="S6.SS2.17.p1.12.m12.1.1.3.1.cmml" xref="S6.SS2.17.p1.12.m12.1.1.3">superscript</csymbol><ci id="S6.SS2.17.p1.12.m12.1.1.3.2.cmml" xref="S6.SS2.17.p1.12.m12.1.1.3.2">𝜈</ci><plus id="S6.SS2.17.p1.12.m12.1.1.3.3.cmml" xref="S6.SS2.17.p1.12.m12.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.17.p1.12.m12.1c">N\cap\omega_{1}=\nu^{+}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.17.p1.12.m12.1d">italic_N ∩ italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_ν start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math>, this implies that <math alttext="X\subseteq\nu^{+}" class="ltx_Math" display="inline" id="S6.SS2.17.p1.13.m13.1"><semantics id="S6.SS2.17.p1.13.m13.1a"><mrow id="S6.SS2.17.p1.13.m13.1.1" xref="S6.SS2.17.p1.13.m13.1.1.cmml"><mi id="S6.SS2.17.p1.13.m13.1.1.2" xref="S6.SS2.17.p1.13.m13.1.1.2.cmml">X</mi><mo id="S6.SS2.17.p1.13.m13.1.1.1" xref="S6.SS2.17.p1.13.m13.1.1.1.cmml">⊆</mo><msup id="S6.SS2.17.p1.13.m13.1.1.3" xref="S6.SS2.17.p1.13.m13.1.1.3.cmml"><mi id="S6.SS2.17.p1.13.m13.1.1.3.2" xref="S6.SS2.17.p1.13.m13.1.1.3.2.cmml">ν</mi><mo id="S6.SS2.17.p1.13.m13.1.1.3.3" xref="S6.SS2.17.p1.13.m13.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.17.p1.13.m13.1b"><apply id="S6.SS2.17.p1.13.m13.1.1.cmml" xref="S6.SS2.17.p1.13.m13.1.1"><subset id="S6.SS2.17.p1.13.m13.1.1.1.cmml" xref="S6.SS2.17.p1.13.m13.1.1.1"></subset><ci id="S6.SS2.17.p1.13.m13.1.1.2.cmml" xref="S6.SS2.17.p1.13.m13.1.1.2">𝑋</ci><apply id="S6.SS2.17.p1.13.m13.1.1.3.cmml" xref="S6.SS2.17.p1.13.m13.1.1.3"><csymbol cd="ambiguous" id="S6.SS2.17.p1.13.m13.1.1.3.1.cmml" xref="S6.SS2.17.p1.13.m13.1.1.3">superscript</csymbol><ci id="S6.SS2.17.p1.13.m13.1.1.3.2.cmml" xref="S6.SS2.17.p1.13.m13.1.1.3.2">𝜈</ci><plus id="S6.SS2.17.p1.13.m13.1.1.3.3.cmml" xref="S6.SS2.17.p1.13.m13.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.17.p1.13.m13.1c">X\subseteq\nu^{+}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.17.p1.13.m13.1d">italic_X ⊆ italic_ν start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math>. That <math alttext="X\cap\nu=\varnothing" class="ltx_Math" display="inline" id="S6.SS2.17.p1.14.m14.1"><semantics id="S6.SS2.17.p1.14.m14.1a"><mrow id="S6.SS2.17.p1.14.m14.1.1" xref="S6.SS2.17.p1.14.m14.1.1.cmml"><mrow id="S6.SS2.17.p1.14.m14.1.1.2" xref="S6.SS2.17.p1.14.m14.1.1.2.cmml"><mi id="S6.SS2.17.p1.14.m14.1.1.2.2" xref="S6.SS2.17.p1.14.m14.1.1.2.2.cmml">X</mi><mo id="S6.SS2.17.p1.14.m14.1.1.2.1" xref="S6.SS2.17.p1.14.m14.1.1.2.1.cmml">∩</mo><mi id="S6.SS2.17.p1.14.m14.1.1.2.3" xref="S6.SS2.17.p1.14.m14.1.1.2.3.cmml">ν</mi></mrow><mo id="S6.SS2.17.p1.14.m14.1.1.1" xref="S6.SS2.17.p1.14.m14.1.1.1.cmml">=</mo><mi id="S6.SS2.17.p1.14.m14.1.1.3" mathvariant="normal" xref="S6.SS2.17.p1.14.m14.1.1.3.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.17.p1.14.m14.1b"><apply id="S6.SS2.17.p1.14.m14.1.1.cmml" xref="S6.SS2.17.p1.14.m14.1.1"><eq id="S6.SS2.17.p1.14.m14.1.1.1.cmml" xref="S6.SS2.17.p1.14.m14.1.1.1"></eq><apply id="S6.SS2.17.p1.14.m14.1.1.2.cmml" xref="S6.SS2.17.p1.14.m14.1.1.2"><intersect id="S6.SS2.17.p1.14.m14.1.1.2.1.cmml" xref="S6.SS2.17.p1.14.m14.1.1.2.1"></intersect><ci id="S6.SS2.17.p1.14.m14.1.1.2.2.cmml" xref="S6.SS2.17.p1.14.m14.1.1.2.2">𝑋</ci><ci id="S6.SS2.17.p1.14.m14.1.1.2.3.cmml" xref="S6.SS2.17.p1.14.m14.1.1.2.3">𝜈</ci></apply><emptyset id="S6.SS2.17.p1.14.m14.1.1.3.cmml" xref="S6.SS2.17.p1.14.m14.1.1.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.17.p1.14.m14.1c">X\cap\nu=\varnothing</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.17.p1.14.m14.1d">italic_X ∩ italic_ν = ∅</annotation></semantics></math> is by definition. ∎</p> </div> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S6.Thmtheorem17"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem17.1.1.1">Lemma 6.17</span></span><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem17.2.2">.</span> </h6> <div class="ltx_para" id="S6.Thmtheorem17.p1"> <p class="ltx_p" id="S6.Thmtheorem17.p1.5">Let <math alttext="a,b\in A" class="ltx_Math" display="inline" id="S6.Thmtheorem17.p1.1.m1.2"><semantics id="S6.Thmtheorem17.p1.1.m1.2a"><mrow id="S6.Thmtheorem17.p1.1.m1.2.3" xref="S6.Thmtheorem17.p1.1.m1.2.3.cmml"><mrow id="S6.Thmtheorem17.p1.1.m1.2.3.2.2" xref="S6.Thmtheorem17.p1.1.m1.2.3.2.1.cmml"><mi id="S6.Thmtheorem17.p1.1.m1.1.1" xref="S6.Thmtheorem17.p1.1.m1.1.1.cmml">a</mi><mo id="S6.Thmtheorem17.p1.1.m1.2.3.2.2.1" xref="S6.Thmtheorem17.p1.1.m1.2.3.2.1.cmml">,</mo><mi id="S6.Thmtheorem17.p1.1.m1.2.2" xref="S6.Thmtheorem17.p1.1.m1.2.2.cmml">b</mi></mrow><mo id="S6.Thmtheorem17.p1.1.m1.2.3.1" xref="S6.Thmtheorem17.p1.1.m1.2.3.1.cmml">∈</mo><mi id="S6.Thmtheorem17.p1.1.m1.2.3.3" xref="S6.Thmtheorem17.p1.1.m1.2.3.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem17.p1.1.m1.2b"><apply id="S6.Thmtheorem17.p1.1.m1.2.3.cmml" xref="S6.Thmtheorem17.p1.1.m1.2.3"><in id="S6.Thmtheorem17.p1.1.m1.2.3.1.cmml" xref="S6.Thmtheorem17.p1.1.m1.2.3.1"></in><list id="S6.Thmtheorem17.p1.1.m1.2.3.2.1.cmml" xref="S6.Thmtheorem17.p1.1.m1.2.3.2.2"><ci id="S6.Thmtheorem17.p1.1.m1.1.1.cmml" xref="S6.Thmtheorem17.p1.1.m1.1.1">𝑎</ci><ci id="S6.Thmtheorem17.p1.1.m1.2.2.cmml" xref="S6.Thmtheorem17.p1.1.m1.2.2">𝑏</ci></list><ci id="S6.Thmtheorem17.p1.1.m1.2.3.3.cmml" xref="S6.Thmtheorem17.p1.1.m1.2.3.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem17.p1.1.m1.2c">a,b\in A</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem17.p1.1.m1.2d">italic_a , italic_b ∈ italic_A</annotation></semantics></math> and <math alttext="\mu\in E\cup\{0\}" class="ltx_Math" display="inline" id="S6.Thmtheorem17.p1.2.m2.1"><semantics id="S6.Thmtheorem17.p1.2.m2.1a"><mrow id="S6.Thmtheorem17.p1.2.m2.1.2" xref="S6.Thmtheorem17.p1.2.m2.1.2.cmml"><mi id="S6.Thmtheorem17.p1.2.m2.1.2.2" xref="S6.Thmtheorem17.p1.2.m2.1.2.2.cmml">μ</mi><mo id="S6.Thmtheorem17.p1.2.m2.1.2.1" xref="S6.Thmtheorem17.p1.2.m2.1.2.1.cmml">∈</mo><mrow id="S6.Thmtheorem17.p1.2.m2.1.2.3" xref="S6.Thmtheorem17.p1.2.m2.1.2.3.cmml"><mi id="S6.Thmtheorem17.p1.2.m2.1.2.3.2" xref="S6.Thmtheorem17.p1.2.m2.1.2.3.2.cmml">E</mi><mo id="S6.Thmtheorem17.p1.2.m2.1.2.3.1" xref="S6.Thmtheorem17.p1.2.m2.1.2.3.1.cmml">∪</mo><mrow id="S6.Thmtheorem17.p1.2.m2.1.2.3.3.2" xref="S6.Thmtheorem17.p1.2.m2.1.2.3.3.1.cmml"><mo id="S6.Thmtheorem17.p1.2.m2.1.2.3.3.2.1" stretchy="false" xref="S6.Thmtheorem17.p1.2.m2.1.2.3.3.1.cmml">{</mo><mn id="S6.Thmtheorem17.p1.2.m2.1.1" xref="S6.Thmtheorem17.p1.2.m2.1.1.cmml">0</mn><mo id="S6.Thmtheorem17.p1.2.m2.1.2.3.3.2.2" stretchy="false" xref="S6.Thmtheorem17.p1.2.m2.1.2.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem17.p1.2.m2.1b"><apply id="S6.Thmtheorem17.p1.2.m2.1.2.cmml" xref="S6.Thmtheorem17.p1.2.m2.1.2"><in id="S6.Thmtheorem17.p1.2.m2.1.2.1.cmml" xref="S6.Thmtheorem17.p1.2.m2.1.2.1"></in><ci id="S6.Thmtheorem17.p1.2.m2.1.2.2.cmml" xref="S6.Thmtheorem17.p1.2.m2.1.2.2">𝜇</ci><apply id="S6.Thmtheorem17.p1.2.m2.1.2.3.cmml" xref="S6.Thmtheorem17.p1.2.m2.1.2.3"><union id="S6.Thmtheorem17.p1.2.m2.1.2.3.1.cmml" xref="S6.Thmtheorem17.p1.2.m2.1.2.3.1"></union><ci id="S6.Thmtheorem17.p1.2.m2.1.2.3.2.cmml" xref="S6.Thmtheorem17.p1.2.m2.1.2.3.2">𝐸</ci><set id="S6.Thmtheorem17.p1.2.m2.1.2.3.3.1.cmml" xref="S6.Thmtheorem17.p1.2.m2.1.2.3.3.2"><cn id="S6.Thmtheorem17.p1.2.m2.1.1.cmml" type="integer" xref="S6.Thmtheorem17.p1.2.m2.1.1">0</cn></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem17.p1.2.m2.1c">\mu\in E\cup\{0\}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem17.p1.2.m2.1d">italic_μ ∈ italic_E ∪ { 0 }</annotation></semantics></math> be such that <math alttext="a<_{A}b" class="ltx_Math" display="inline" id="S6.Thmtheorem17.p1.3.m3.1"><semantics id="S6.Thmtheorem17.p1.3.m3.1a"><mrow id="S6.Thmtheorem17.p1.3.m3.1.1" xref="S6.Thmtheorem17.p1.3.m3.1.1.cmml"><mi id="S6.Thmtheorem17.p1.3.m3.1.1.2" xref="S6.Thmtheorem17.p1.3.m3.1.1.2.cmml">a</mi><msub id="S6.Thmtheorem17.p1.3.m3.1.1.1" xref="S6.Thmtheorem17.p1.3.m3.1.1.1.cmml"><mo id="S6.Thmtheorem17.p1.3.m3.1.1.1.2" xref="S6.Thmtheorem17.p1.3.m3.1.1.1.2.cmml"><</mo><mi id="S6.Thmtheorem17.p1.3.m3.1.1.1.3" xref="S6.Thmtheorem17.p1.3.m3.1.1.1.3.cmml">A</mi></msub><mi id="S6.Thmtheorem17.p1.3.m3.1.1.3" xref="S6.Thmtheorem17.p1.3.m3.1.1.3.cmml">b</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem17.p1.3.m3.1b"><apply id="S6.Thmtheorem17.p1.3.m3.1.1.cmml" xref="S6.Thmtheorem17.p1.3.m3.1.1"><apply id="S6.Thmtheorem17.p1.3.m3.1.1.1.cmml" xref="S6.Thmtheorem17.p1.3.m3.1.1.1"><csymbol cd="ambiguous" id="S6.Thmtheorem17.p1.3.m3.1.1.1.1.cmml" xref="S6.Thmtheorem17.p1.3.m3.1.1.1">subscript</csymbol><lt id="S6.Thmtheorem17.p1.3.m3.1.1.1.2.cmml" xref="S6.Thmtheorem17.p1.3.m3.1.1.1.2"></lt><ci id="S6.Thmtheorem17.p1.3.m3.1.1.1.3.cmml" xref="S6.Thmtheorem17.p1.3.m3.1.1.1.3">𝐴</ci></apply><ci id="S6.Thmtheorem17.p1.3.m3.1.1.2.cmml" xref="S6.Thmtheorem17.p1.3.m3.1.1.2">𝑎</ci><ci id="S6.Thmtheorem17.p1.3.m3.1.1.3.cmml" xref="S6.Thmtheorem17.p1.3.m3.1.1.3">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem17.p1.3.m3.1c">a<_{A}b</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem17.p1.3.m3.1d">italic_a < start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_b</annotation></semantics></math>, and <math alttext="\nu(a,b)\leq\mu" class="ltx_Math" display="inline" id="S6.Thmtheorem17.p1.4.m4.2"><semantics id="S6.Thmtheorem17.p1.4.m4.2a"><mrow id="S6.Thmtheorem17.p1.4.m4.2.3" xref="S6.Thmtheorem17.p1.4.m4.2.3.cmml"><mrow id="S6.Thmtheorem17.p1.4.m4.2.3.2" xref="S6.Thmtheorem17.p1.4.m4.2.3.2.cmml"><mi id="S6.Thmtheorem17.p1.4.m4.2.3.2.2" xref="S6.Thmtheorem17.p1.4.m4.2.3.2.2.cmml">ν</mi><mo id="S6.Thmtheorem17.p1.4.m4.2.3.2.1" xref="S6.Thmtheorem17.p1.4.m4.2.3.2.1.cmml">⁢</mo><mrow id="S6.Thmtheorem17.p1.4.m4.2.3.2.3.2" xref="S6.Thmtheorem17.p1.4.m4.2.3.2.3.1.cmml"><mo id="S6.Thmtheorem17.p1.4.m4.2.3.2.3.2.1" stretchy="false" xref="S6.Thmtheorem17.p1.4.m4.2.3.2.3.1.cmml">(</mo><mi id="S6.Thmtheorem17.p1.4.m4.1.1" xref="S6.Thmtheorem17.p1.4.m4.1.1.cmml">a</mi><mo id="S6.Thmtheorem17.p1.4.m4.2.3.2.3.2.2" xref="S6.Thmtheorem17.p1.4.m4.2.3.2.3.1.cmml">,</mo><mi id="S6.Thmtheorem17.p1.4.m4.2.2" xref="S6.Thmtheorem17.p1.4.m4.2.2.cmml">b</mi><mo id="S6.Thmtheorem17.p1.4.m4.2.3.2.3.2.3" stretchy="false" xref="S6.Thmtheorem17.p1.4.m4.2.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="S6.Thmtheorem17.p1.4.m4.2.3.1" xref="S6.Thmtheorem17.p1.4.m4.2.3.1.cmml">≤</mo><mi id="S6.Thmtheorem17.p1.4.m4.2.3.3" xref="S6.Thmtheorem17.p1.4.m4.2.3.3.cmml">μ</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem17.p1.4.m4.2b"><apply id="S6.Thmtheorem17.p1.4.m4.2.3.cmml" xref="S6.Thmtheorem17.p1.4.m4.2.3"><leq id="S6.Thmtheorem17.p1.4.m4.2.3.1.cmml" xref="S6.Thmtheorem17.p1.4.m4.2.3.1"></leq><apply id="S6.Thmtheorem17.p1.4.m4.2.3.2.cmml" xref="S6.Thmtheorem17.p1.4.m4.2.3.2"><times id="S6.Thmtheorem17.p1.4.m4.2.3.2.1.cmml" xref="S6.Thmtheorem17.p1.4.m4.2.3.2.1"></times><ci id="S6.Thmtheorem17.p1.4.m4.2.3.2.2.cmml" xref="S6.Thmtheorem17.p1.4.m4.2.3.2.2">𝜈</ci><interval closure="open" id="S6.Thmtheorem17.p1.4.m4.2.3.2.3.1.cmml" xref="S6.Thmtheorem17.p1.4.m4.2.3.2.3.2"><ci id="S6.Thmtheorem17.p1.4.m4.1.1.cmml" xref="S6.Thmtheorem17.p1.4.m4.1.1">𝑎</ci><ci id="S6.Thmtheorem17.p1.4.m4.2.2.cmml" xref="S6.Thmtheorem17.p1.4.m4.2.2">𝑏</ci></interval></apply><ci id="S6.Thmtheorem17.p1.4.m4.2.3.3.cmml" xref="S6.Thmtheorem17.p1.4.m4.2.3.3">𝜇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem17.p1.4.m4.2c">\nu(a,b)\leq\mu</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem17.p1.4.m4.2d">italic_ν ( italic_a , italic_b ) ≤ italic_μ</annotation></semantics></math>. Then there are <math alttext="a^{\prime},b^{\prime}\in A" class="ltx_Math" display="inline" id="S6.Thmtheorem17.p1.5.m5.2"><semantics id="S6.Thmtheorem17.p1.5.m5.2a"><mrow id="S6.Thmtheorem17.p1.5.m5.2.2" xref="S6.Thmtheorem17.p1.5.m5.2.2.cmml"><mrow id="S6.Thmtheorem17.p1.5.m5.2.2.2.2" xref="S6.Thmtheorem17.p1.5.m5.2.2.2.3.cmml"><msup id="S6.Thmtheorem17.p1.5.m5.1.1.1.1.1" xref="S6.Thmtheorem17.p1.5.m5.1.1.1.1.1.cmml"><mi id="S6.Thmtheorem17.p1.5.m5.1.1.1.1.1.2" xref="S6.Thmtheorem17.p1.5.m5.1.1.1.1.1.2.cmml">a</mi><mo id="S6.Thmtheorem17.p1.5.m5.1.1.1.1.1.3" xref="S6.Thmtheorem17.p1.5.m5.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S6.Thmtheorem17.p1.5.m5.2.2.2.2.3" xref="S6.Thmtheorem17.p1.5.m5.2.2.2.3.cmml">,</mo><msup id="S6.Thmtheorem17.p1.5.m5.2.2.2.2.2" xref="S6.Thmtheorem17.p1.5.m5.2.2.2.2.2.cmml"><mi id="S6.Thmtheorem17.p1.5.m5.2.2.2.2.2.2" xref="S6.Thmtheorem17.p1.5.m5.2.2.2.2.2.2.cmml">b</mi><mo id="S6.Thmtheorem17.p1.5.m5.2.2.2.2.2.3" xref="S6.Thmtheorem17.p1.5.m5.2.2.2.2.2.3.cmml">′</mo></msup></mrow><mo id="S6.Thmtheorem17.p1.5.m5.2.2.3" xref="S6.Thmtheorem17.p1.5.m5.2.2.3.cmml">∈</mo><mi id="S6.Thmtheorem17.p1.5.m5.2.2.4" xref="S6.Thmtheorem17.p1.5.m5.2.2.4.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem17.p1.5.m5.2b"><apply id="S6.Thmtheorem17.p1.5.m5.2.2.cmml" xref="S6.Thmtheorem17.p1.5.m5.2.2"><in id="S6.Thmtheorem17.p1.5.m5.2.2.3.cmml" xref="S6.Thmtheorem17.p1.5.m5.2.2.3"></in><list id="S6.Thmtheorem17.p1.5.m5.2.2.2.3.cmml" xref="S6.Thmtheorem17.p1.5.m5.2.2.2.2"><apply id="S6.Thmtheorem17.p1.5.m5.1.1.1.1.1.cmml" xref="S6.Thmtheorem17.p1.5.m5.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.Thmtheorem17.p1.5.m5.1.1.1.1.1.1.cmml" xref="S6.Thmtheorem17.p1.5.m5.1.1.1.1.1">superscript</csymbol><ci id="S6.Thmtheorem17.p1.5.m5.1.1.1.1.1.2.cmml" xref="S6.Thmtheorem17.p1.5.m5.1.1.1.1.1.2">𝑎</ci><ci id="S6.Thmtheorem17.p1.5.m5.1.1.1.1.1.3.cmml" xref="S6.Thmtheorem17.p1.5.m5.1.1.1.1.1.3">′</ci></apply><apply id="S6.Thmtheorem17.p1.5.m5.2.2.2.2.2.cmml" xref="S6.Thmtheorem17.p1.5.m5.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.Thmtheorem17.p1.5.m5.2.2.2.2.2.1.cmml" xref="S6.Thmtheorem17.p1.5.m5.2.2.2.2.2">superscript</csymbol><ci id="S6.Thmtheorem17.p1.5.m5.2.2.2.2.2.2.cmml" xref="S6.Thmtheorem17.p1.5.m5.2.2.2.2.2.2">𝑏</ci><ci id="S6.Thmtheorem17.p1.5.m5.2.2.2.2.2.3.cmml" xref="S6.Thmtheorem17.p1.5.m5.2.2.2.2.2.3">′</ci></apply></list><ci id="S6.Thmtheorem17.p1.5.m5.2.2.4.cmml" xref="S6.Thmtheorem17.p1.5.m5.2.2.4">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem17.p1.5.m5.2c">a^{\prime},b^{\prime}\in A</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem17.p1.5.m5.2d">italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_A</annotation></semantics></math> such that:</p> <ol class="ltx_enumerate" id="S6.I9"> <li class="ltx_item" id="S6.I9.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(a)</span> <div class="ltx_para" id="S6.I9.i1.p1"> <p class="ltx_p" id="S6.I9.i1.p1.1"><math alttext="\nu(a^{\prime})=\nu(b^{\prime})=\nu(a^{\prime},b^{\prime})=\mu" class="ltx_Math" display="inline" id="S6.I9.i1.p1.1.m1.4"><semantics id="S6.I9.i1.p1.1.m1.4a"><mrow id="S6.I9.i1.p1.1.m1.4.4" xref="S6.I9.i1.p1.1.m1.4.4.cmml"><mrow id="S6.I9.i1.p1.1.m1.1.1.1" xref="S6.I9.i1.p1.1.m1.1.1.1.cmml"><mi id="S6.I9.i1.p1.1.m1.1.1.1.3" xref="S6.I9.i1.p1.1.m1.1.1.1.3.cmml">ν</mi><mo id="S6.I9.i1.p1.1.m1.1.1.1.2" xref="S6.I9.i1.p1.1.m1.1.1.1.2.cmml">⁢</mo><mrow id="S6.I9.i1.p1.1.m1.1.1.1.1.1" xref="S6.I9.i1.p1.1.m1.1.1.1.1.1.1.cmml"><mo id="S6.I9.i1.p1.1.m1.1.1.1.1.1.2" stretchy="false" xref="S6.I9.i1.p1.1.m1.1.1.1.1.1.1.cmml">(</mo><msup id="S6.I9.i1.p1.1.m1.1.1.1.1.1.1" xref="S6.I9.i1.p1.1.m1.1.1.1.1.1.1.cmml"><mi id="S6.I9.i1.p1.1.m1.1.1.1.1.1.1.2" xref="S6.I9.i1.p1.1.m1.1.1.1.1.1.1.2.cmml">a</mi><mo id="S6.I9.i1.p1.1.m1.1.1.1.1.1.1.3" xref="S6.I9.i1.p1.1.m1.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S6.I9.i1.p1.1.m1.1.1.1.1.1.3" stretchy="false" xref="S6.I9.i1.p1.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.I9.i1.p1.1.m1.4.4.6" xref="S6.I9.i1.p1.1.m1.4.4.6.cmml">=</mo><mrow id="S6.I9.i1.p1.1.m1.2.2.2" xref="S6.I9.i1.p1.1.m1.2.2.2.cmml"><mi id="S6.I9.i1.p1.1.m1.2.2.2.3" xref="S6.I9.i1.p1.1.m1.2.2.2.3.cmml">ν</mi><mo id="S6.I9.i1.p1.1.m1.2.2.2.2" xref="S6.I9.i1.p1.1.m1.2.2.2.2.cmml">⁢</mo><mrow id="S6.I9.i1.p1.1.m1.2.2.2.1.1" xref="S6.I9.i1.p1.1.m1.2.2.2.1.1.1.cmml"><mo id="S6.I9.i1.p1.1.m1.2.2.2.1.1.2" stretchy="false" xref="S6.I9.i1.p1.1.m1.2.2.2.1.1.1.cmml">(</mo><msup id="S6.I9.i1.p1.1.m1.2.2.2.1.1.1" xref="S6.I9.i1.p1.1.m1.2.2.2.1.1.1.cmml"><mi id="S6.I9.i1.p1.1.m1.2.2.2.1.1.1.2" xref="S6.I9.i1.p1.1.m1.2.2.2.1.1.1.2.cmml">b</mi><mo id="S6.I9.i1.p1.1.m1.2.2.2.1.1.1.3" xref="S6.I9.i1.p1.1.m1.2.2.2.1.1.1.3.cmml">′</mo></msup><mo id="S6.I9.i1.p1.1.m1.2.2.2.1.1.3" stretchy="false" xref="S6.I9.i1.p1.1.m1.2.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.I9.i1.p1.1.m1.4.4.7" xref="S6.I9.i1.p1.1.m1.4.4.7.cmml">=</mo><mrow id="S6.I9.i1.p1.1.m1.4.4.4" xref="S6.I9.i1.p1.1.m1.4.4.4.cmml"><mi id="S6.I9.i1.p1.1.m1.4.4.4.4" xref="S6.I9.i1.p1.1.m1.4.4.4.4.cmml">ν</mi><mo id="S6.I9.i1.p1.1.m1.4.4.4.3" xref="S6.I9.i1.p1.1.m1.4.4.4.3.cmml">⁢</mo><mrow id="S6.I9.i1.p1.1.m1.4.4.4.2.2" xref="S6.I9.i1.p1.1.m1.4.4.4.2.3.cmml"><mo id="S6.I9.i1.p1.1.m1.4.4.4.2.2.3" stretchy="false" xref="S6.I9.i1.p1.1.m1.4.4.4.2.3.cmml">(</mo><msup id="S6.I9.i1.p1.1.m1.3.3.3.1.1.1" xref="S6.I9.i1.p1.1.m1.3.3.3.1.1.1.cmml"><mi id="S6.I9.i1.p1.1.m1.3.3.3.1.1.1.2" xref="S6.I9.i1.p1.1.m1.3.3.3.1.1.1.2.cmml">a</mi><mo id="S6.I9.i1.p1.1.m1.3.3.3.1.1.1.3" xref="S6.I9.i1.p1.1.m1.3.3.3.1.1.1.3.cmml">′</mo></msup><mo id="S6.I9.i1.p1.1.m1.4.4.4.2.2.4" xref="S6.I9.i1.p1.1.m1.4.4.4.2.3.cmml">,</mo><msup id="S6.I9.i1.p1.1.m1.4.4.4.2.2.2" xref="S6.I9.i1.p1.1.m1.4.4.4.2.2.2.cmml"><mi id="S6.I9.i1.p1.1.m1.4.4.4.2.2.2.2" xref="S6.I9.i1.p1.1.m1.4.4.4.2.2.2.2.cmml">b</mi><mo id="S6.I9.i1.p1.1.m1.4.4.4.2.2.2.3" xref="S6.I9.i1.p1.1.m1.4.4.4.2.2.2.3.cmml">′</mo></msup><mo id="S6.I9.i1.p1.1.m1.4.4.4.2.2.5" stretchy="false" xref="S6.I9.i1.p1.1.m1.4.4.4.2.3.cmml">)</mo></mrow></mrow><mo id="S6.I9.i1.p1.1.m1.4.4.8" xref="S6.I9.i1.p1.1.m1.4.4.8.cmml">=</mo><mi id="S6.I9.i1.p1.1.m1.4.4.9" xref="S6.I9.i1.p1.1.m1.4.4.9.cmml">μ</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.I9.i1.p1.1.m1.4b"><apply id="S6.I9.i1.p1.1.m1.4.4.cmml" xref="S6.I9.i1.p1.1.m1.4.4"><and id="S6.I9.i1.p1.1.m1.4.4a.cmml" xref="S6.I9.i1.p1.1.m1.4.4"></and><apply id="S6.I9.i1.p1.1.m1.4.4b.cmml" xref="S6.I9.i1.p1.1.m1.4.4"><eq id="S6.I9.i1.p1.1.m1.4.4.6.cmml" xref="S6.I9.i1.p1.1.m1.4.4.6"></eq><apply id="S6.I9.i1.p1.1.m1.1.1.1.cmml" xref="S6.I9.i1.p1.1.m1.1.1.1"><times id="S6.I9.i1.p1.1.m1.1.1.1.2.cmml" xref="S6.I9.i1.p1.1.m1.1.1.1.2"></times><ci id="S6.I9.i1.p1.1.m1.1.1.1.3.cmml" xref="S6.I9.i1.p1.1.m1.1.1.1.3">𝜈</ci><apply id="S6.I9.i1.p1.1.m1.1.1.1.1.1.1.cmml" xref="S6.I9.i1.p1.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.I9.i1.p1.1.m1.1.1.1.1.1.1.1.cmml" xref="S6.I9.i1.p1.1.m1.1.1.1.1.1">superscript</csymbol><ci id="S6.I9.i1.p1.1.m1.1.1.1.1.1.1.2.cmml" xref="S6.I9.i1.p1.1.m1.1.1.1.1.1.1.2">𝑎</ci><ci id="S6.I9.i1.p1.1.m1.1.1.1.1.1.1.3.cmml" xref="S6.I9.i1.p1.1.m1.1.1.1.1.1.1.3">′</ci></apply></apply><apply id="S6.I9.i1.p1.1.m1.2.2.2.cmml" xref="S6.I9.i1.p1.1.m1.2.2.2"><times id="S6.I9.i1.p1.1.m1.2.2.2.2.cmml" xref="S6.I9.i1.p1.1.m1.2.2.2.2"></times><ci id="S6.I9.i1.p1.1.m1.2.2.2.3.cmml" xref="S6.I9.i1.p1.1.m1.2.2.2.3">𝜈</ci><apply id="S6.I9.i1.p1.1.m1.2.2.2.1.1.1.cmml" xref="S6.I9.i1.p1.1.m1.2.2.2.1.1"><csymbol cd="ambiguous" id="S6.I9.i1.p1.1.m1.2.2.2.1.1.1.1.cmml" xref="S6.I9.i1.p1.1.m1.2.2.2.1.1">superscript</csymbol><ci id="S6.I9.i1.p1.1.m1.2.2.2.1.1.1.2.cmml" xref="S6.I9.i1.p1.1.m1.2.2.2.1.1.1.2">𝑏</ci><ci id="S6.I9.i1.p1.1.m1.2.2.2.1.1.1.3.cmml" xref="S6.I9.i1.p1.1.m1.2.2.2.1.1.1.3">′</ci></apply></apply></apply><apply id="S6.I9.i1.p1.1.m1.4.4c.cmml" xref="S6.I9.i1.p1.1.m1.4.4"><eq id="S6.I9.i1.p1.1.m1.4.4.7.cmml" xref="S6.I9.i1.p1.1.m1.4.4.7"></eq><share href="https://arxiv.org/html/2503.13728v1#S6.I9.i1.p1.1.m1.2.2.2.cmml" id="S6.I9.i1.p1.1.m1.4.4d.cmml" xref="S6.I9.i1.p1.1.m1.4.4"></share><apply id="S6.I9.i1.p1.1.m1.4.4.4.cmml" xref="S6.I9.i1.p1.1.m1.4.4.4"><times id="S6.I9.i1.p1.1.m1.4.4.4.3.cmml" xref="S6.I9.i1.p1.1.m1.4.4.4.3"></times><ci id="S6.I9.i1.p1.1.m1.4.4.4.4.cmml" xref="S6.I9.i1.p1.1.m1.4.4.4.4">𝜈</ci><interval closure="open" id="S6.I9.i1.p1.1.m1.4.4.4.2.3.cmml" xref="S6.I9.i1.p1.1.m1.4.4.4.2.2"><apply id="S6.I9.i1.p1.1.m1.3.3.3.1.1.1.cmml" xref="S6.I9.i1.p1.1.m1.3.3.3.1.1.1"><csymbol cd="ambiguous" id="S6.I9.i1.p1.1.m1.3.3.3.1.1.1.1.cmml" xref="S6.I9.i1.p1.1.m1.3.3.3.1.1.1">superscript</csymbol><ci id="S6.I9.i1.p1.1.m1.3.3.3.1.1.1.2.cmml" xref="S6.I9.i1.p1.1.m1.3.3.3.1.1.1.2">𝑎</ci><ci id="S6.I9.i1.p1.1.m1.3.3.3.1.1.1.3.cmml" xref="S6.I9.i1.p1.1.m1.3.3.3.1.1.1.3">′</ci></apply><apply id="S6.I9.i1.p1.1.m1.4.4.4.2.2.2.cmml" xref="S6.I9.i1.p1.1.m1.4.4.4.2.2.2"><csymbol cd="ambiguous" id="S6.I9.i1.p1.1.m1.4.4.4.2.2.2.1.cmml" xref="S6.I9.i1.p1.1.m1.4.4.4.2.2.2">superscript</csymbol><ci id="S6.I9.i1.p1.1.m1.4.4.4.2.2.2.2.cmml" xref="S6.I9.i1.p1.1.m1.4.4.4.2.2.2.2">𝑏</ci><ci id="S6.I9.i1.p1.1.m1.4.4.4.2.2.2.3.cmml" xref="S6.I9.i1.p1.1.m1.4.4.4.2.2.2.3">′</ci></apply></interval></apply></apply><apply id="S6.I9.i1.p1.1.m1.4.4e.cmml" xref="S6.I9.i1.p1.1.m1.4.4"><eq id="S6.I9.i1.p1.1.m1.4.4.8.cmml" xref="S6.I9.i1.p1.1.m1.4.4.8"></eq><share href="https://arxiv.org/html/2503.13728v1#S6.I9.i1.p1.1.m1.4.4.4.cmml" id="S6.I9.i1.p1.1.m1.4.4f.cmml" xref="S6.I9.i1.p1.1.m1.4.4"></share><ci id="S6.I9.i1.p1.1.m1.4.4.9.cmml" xref="S6.I9.i1.p1.1.m1.4.4.9">𝜇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I9.i1.p1.1.m1.4c">\nu(a^{\prime})=\nu(b^{\prime})=\nu(a^{\prime},b^{\prime})=\mu</annotation><annotation encoding="application/x-llamapun" id="S6.I9.i1.p1.1.m1.4d">italic_ν ( italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = italic_ν ( italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = italic_ν ( italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = italic_μ</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S6.I9.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(b)</span> <div class="ltx_para" id="S6.I9.i2.p1"> <p class="ltx_p" id="S6.I9.i2.p1.1"><math alttext="a<_{A}a^{\prime}<_{A}b^{\prime}<_{A}b" class="ltx_Math" display="inline" id="S6.I9.i2.p1.1.m1.1"><semantics id="S6.I9.i2.p1.1.m1.1a"><mrow id="S6.I9.i2.p1.1.m1.1.1" xref="S6.I9.i2.p1.1.m1.1.1.cmml"><mi id="S6.I9.i2.p1.1.m1.1.1.2" xref="S6.I9.i2.p1.1.m1.1.1.2.cmml">a</mi><msub id="S6.I9.i2.p1.1.m1.1.1.3" xref="S6.I9.i2.p1.1.m1.1.1.3.cmml"><mo id="S6.I9.i2.p1.1.m1.1.1.3.2" xref="S6.I9.i2.p1.1.m1.1.1.3.2.cmml"><</mo><mi id="S6.I9.i2.p1.1.m1.1.1.3.3" xref="S6.I9.i2.p1.1.m1.1.1.3.3.cmml">A</mi></msub><msup id="S6.I9.i2.p1.1.m1.1.1.4" xref="S6.I9.i2.p1.1.m1.1.1.4.cmml"><mi id="S6.I9.i2.p1.1.m1.1.1.4.2" xref="S6.I9.i2.p1.1.m1.1.1.4.2.cmml">a</mi><mo id="S6.I9.i2.p1.1.m1.1.1.4.3" xref="S6.I9.i2.p1.1.m1.1.1.4.3.cmml">′</mo></msup><msub id="S6.I9.i2.p1.1.m1.1.1.5" xref="S6.I9.i2.p1.1.m1.1.1.5.cmml"><mo id="S6.I9.i2.p1.1.m1.1.1.5.2" xref="S6.I9.i2.p1.1.m1.1.1.5.2.cmml"><</mo><mi id="S6.I9.i2.p1.1.m1.1.1.5.3" xref="S6.I9.i2.p1.1.m1.1.1.5.3.cmml">A</mi></msub><msup id="S6.I9.i2.p1.1.m1.1.1.6" xref="S6.I9.i2.p1.1.m1.1.1.6.cmml"><mi id="S6.I9.i2.p1.1.m1.1.1.6.2" xref="S6.I9.i2.p1.1.m1.1.1.6.2.cmml">b</mi><mo id="S6.I9.i2.p1.1.m1.1.1.6.3" xref="S6.I9.i2.p1.1.m1.1.1.6.3.cmml">′</mo></msup><msub id="S6.I9.i2.p1.1.m1.1.1.7" xref="S6.I9.i2.p1.1.m1.1.1.7.cmml"><mo id="S6.I9.i2.p1.1.m1.1.1.7.2" xref="S6.I9.i2.p1.1.m1.1.1.7.2.cmml"><</mo><mi id="S6.I9.i2.p1.1.m1.1.1.7.3" xref="S6.I9.i2.p1.1.m1.1.1.7.3.cmml">A</mi></msub><mi id="S6.I9.i2.p1.1.m1.1.1.8" xref="S6.I9.i2.p1.1.m1.1.1.8.cmml">b</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.I9.i2.p1.1.m1.1b"><apply id="S6.I9.i2.p1.1.m1.1.1.cmml" xref="S6.I9.i2.p1.1.m1.1.1"><and id="S6.I9.i2.p1.1.m1.1.1a.cmml" xref="S6.I9.i2.p1.1.m1.1.1"></and><apply id="S6.I9.i2.p1.1.m1.1.1b.cmml" xref="S6.I9.i2.p1.1.m1.1.1"><apply id="S6.I9.i2.p1.1.m1.1.1.3.cmml" xref="S6.I9.i2.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S6.I9.i2.p1.1.m1.1.1.3.1.cmml" xref="S6.I9.i2.p1.1.m1.1.1.3">subscript</csymbol><lt id="S6.I9.i2.p1.1.m1.1.1.3.2.cmml" xref="S6.I9.i2.p1.1.m1.1.1.3.2"></lt><ci id="S6.I9.i2.p1.1.m1.1.1.3.3.cmml" xref="S6.I9.i2.p1.1.m1.1.1.3.3">𝐴</ci></apply><ci id="S6.I9.i2.p1.1.m1.1.1.2.cmml" xref="S6.I9.i2.p1.1.m1.1.1.2">𝑎</ci><apply id="S6.I9.i2.p1.1.m1.1.1.4.cmml" xref="S6.I9.i2.p1.1.m1.1.1.4"><csymbol cd="ambiguous" id="S6.I9.i2.p1.1.m1.1.1.4.1.cmml" xref="S6.I9.i2.p1.1.m1.1.1.4">superscript</csymbol><ci id="S6.I9.i2.p1.1.m1.1.1.4.2.cmml" xref="S6.I9.i2.p1.1.m1.1.1.4.2">𝑎</ci><ci id="S6.I9.i2.p1.1.m1.1.1.4.3.cmml" xref="S6.I9.i2.p1.1.m1.1.1.4.3">′</ci></apply></apply><apply id="S6.I9.i2.p1.1.m1.1.1c.cmml" xref="S6.I9.i2.p1.1.m1.1.1"><apply id="S6.I9.i2.p1.1.m1.1.1.5.cmml" xref="S6.I9.i2.p1.1.m1.1.1.5"><csymbol cd="ambiguous" id="S6.I9.i2.p1.1.m1.1.1.5.1.cmml" xref="S6.I9.i2.p1.1.m1.1.1.5">subscript</csymbol><lt id="S6.I9.i2.p1.1.m1.1.1.5.2.cmml" xref="S6.I9.i2.p1.1.m1.1.1.5.2"></lt><ci id="S6.I9.i2.p1.1.m1.1.1.5.3.cmml" xref="S6.I9.i2.p1.1.m1.1.1.5.3">𝐴</ci></apply><share href="https://arxiv.org/html/2503.13728v1#S6.I9.i2.p1.1.m1.1.1.4.cmml" id="S6.I9.i2.p1.1.m1.1.1d.cmml" xref="S6.I9.i2.p1.1.m1.1.1"></share><apply id="S6.I9.i2.p1.1.m1.1.1.6.cmml" xref="S6.I9.i2.p1.1.m1.1.1.6"><csymbol cd="ambiguous" id="S6.I9.i2.p1.1.m1.1.1.6.1.cmml" xref="S6.I9.i2.p1.1.m1.1.1.6">superscript</csymbol><ci id="S6.I9.i2.p1.1.m1.1.1.6.2.cmml" xref="S6.I9.i2.p1.1.m1.1.1.6.2">𝑏</ci><ci id="S6.I9.i2.p1.1.m1.1.1.6.3.cmml" xref="S6.I9.i2.p1.1.m1.1.1.6.3">′</ci></apply></apply><apply id="S6.I9.i2.p1.1.m1.1.1e.cmml" xref="S6.I9.i2.p1.1.m1.1.1"><apply id="S6.I9.i2.p1.1.m1.1.1.7.cmml" xref="S6.I9.i2.p1.1.m1.1.1.7"><csymbol cd="ambiguous" id="S6.I9.i2.p1.1.m1.1.1.7.1.cmml" xref="S6.I9.i2.p1.1.m1.1.1.7">subscript</csymbol><lt id="S6.I9.i2.p1.1.m1.1.1.7.2.cmml" xref="S6.I9.i2.p1.1.m1.1.1.7.2"></lt><ci id="S6.I9.i2.p1.1.m1.1.1.7.3.cmml" xref="S6.I9.i2.p1.1.m1.1.1.7.3">𝐴</ci></apply><share href="https://arxiv.org/html/2503.13728v1#S6.I9.i2.p1.1.m1.1.1.6.cmml" id="S6.I9.i2.p1.1.m1.1.1f.cmml" xref="S6.I9.i2.p1.1.m1.1.1"></share><ci id="S6.I9.i2.p1.1.m1.1.1.8.cmml" xref="S6.I9.i2.p1.1.m1.1.1.8">𝑏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I9.i2.p1.1.m1.1c">a<_{A}a^{\prime}<_{A}b^{\prime}<_{A}b</annotation><annotation encoding="application/x-llamapun" id="S6.I9.i2.p1.1.m1.1d">italic_a < start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT < start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT < start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_b</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S6.I9.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(c)</span> <div class="ltx_para" id="S6.I9.i3.p1"> <p class="ltx_p" id="S6.I9.i3.p1.3"><math alttext="a^{\prime}" class="ltx_Math" display="inline" id="S6.I9.i3.p1.1.m1.1"><semantics id="S6.I9.i3.p1.1.m1.1a"><msup id="S6.I9.i3.p1.1.m1.1.1" xref="S6.I9.i3.p1.1.m1.1.1.cmml"><mi id="S6.I9.i3.p1.1.m1.1.1.2" xref="S6.I9.i3.p1.1.m1.1.1.2.cmml">a</mi><mo id="S6.I9.i3.p1.1.m1.1.1.3" xref="S6.I9.i3.p1.1.m1.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S6.I9.i3.p1.1.m1.1b"><apply id="S6.I9.i3.p1.1.m1.1.1.cmml" xref="S6.I9.i3.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S6.I9.i3.p1.1.m1.1.1.1.cmml" xref="S6.I9.i3.p1.1.m1.1.1">superscript</csymbol><ci id="S6.I9.i3.p1.1.m1.1.1.2.cmml" xref="S6.I9.i3.p1.1.m1.1.1.2">𝑎</ci><ci id="S6.I9.i3.p1.1.m1.1.1.3.cmml" xref="S6.I9.i3.p1.1.m1.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I9.i3.p1.1.m1.1c">a^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S6.I9.i3.p1.1.m1.1d">italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="b^{\prime}" class="ltx_Math" display="inline" id="S6.I9.i3.p1.2.m2.1"><semantics id="S6.I9.i3.p1.2.m2.1a"><msup id="S6.I9.i3.p1.2.m2.1.1" xref="S6.I9.i3.p1.2.m2.1.1.cmml"><mi id="S6.I9.i3.p1.2.m2.1.1.2" xref="S6.I9.i3.p1.2.m2.1.1.2.cmml">b</mi><mo id="S6.I9.i3.p1.2.m2.1.1.3" xref="S6.I9.i3.p1.2.m2.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S6.I9.i3.p1.2.m2.1b"><apply id="S6.I9.i3.p1.2.m2.1.1.cmml" xref="S6.I9.i3.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S6.I9.i3.p1.2.m2.1.1.1.cmml" xref="S6.I9.i3.p1.2.m2.1.1">superscript</csymbol><ci id="S6.I9.i3.p1.2.m2.1.1.2.cmml" xref="S6.I9.i3.p1.2.m2.1.1.2">𝑏</ci><ci id="S6.I9.i3.p1.2.m2.1.1.3.cmml" xref="S6.I9.i3.p1.2.m2.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I9.i3.p1.2.m2.1c">b^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S6.I9.i3.p1.2.m2.1d">italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> are not endpoints of their (by (a) they are in the same one) complementary interval of <math alttext="A\setminus\mu" class="ltx_Math" display="inline" id="S6.I9.i3.p1.3.m3.1"><semantics id="S6.I9.i3.p1.3.m3.1a"><mrow id="S6.I9.i3.p1.3.m3.1.1" xref="S6.I9.i3.p1.3.m3.1.1.cmml"><mi id="S6.I9.i3.p1.3.m3.1.1.2" xref="S6.I9.i3.p1.3.m3.1.1.2.cmml">A</mi><mo id="S6.I9.i3.p1.3.m3.1.1.1" xref="S6.I9.i3.p1.3.m3.1.1.1.cmml">∖</mo><mi id="S6.I9.i3.p1.3.m3.1.1.3" xref="S6.I9.i3.p1.3.m3.1.1.3.cmml">μ</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.I9.i3.p1.3.m3.1b"><apply id="S6.I9.i3.p1.3.m3.1.1.cmml" xref="S6.I9.i3.p1.3.m3.1.1"><setdiff id="S6.I9.i3.p1.3.m3.1.1.1.cmml" xref="S6.I9.i3.p1.3.m3.1.1.1"></setdiff><ci id="S6.I9.i3.p1.3.m3.1.1.2.cmml" xref="S6.I9.i3.p1.3.m3.1.1.2">𝐴</ci><ci id="S6.I9.i3.p1.3.m3.1.1.3.cmml" xref="S6.I9.i3.p1.3.m3.1.1.3">𝜇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I9.i3.p1.3.m3.1c">A\setminus\mu</annotation><annotation encoding="application/x-llamapun" id="S6.I9.i3.p1.3.m3.1d">italic_A ∖ italic_μ</annotation></semantics></math>.</p> </div> </li> </ol> </div> </div> <div class="ltx_proof" id="S6.SS2.19"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S6.SS2.18.p1"> <p class="ltx_p" id="S6.SS2.18.p1.14">We first find <math alttext="c^{\prime},c^{\prime\prime}<\mu^{+}" class="ltx_Math" display="inline" id="S6.SS2.18.p1.1.m1.2"><semantics id="S6.SS2.18.p1.1.m1.2a"><mrow id="S6.SS2.18.p1.1.m1.2.2" xref="S6.SS2.18.p1.1.m1.2.2.cmml"><mrow id="S6.SS2.18.p1.1.m1.2.2.2.2" xref="S6.SS2.18.p1.1.m1.2.2.2.3.cmml"><msup id="S6.SS2.18.p1.1.m1.1.1.1.1.1" xref="S6.SS2.18.p1.1.m1.1.1.1.1.1.cmml"><mi id="S6.SS2.18.p1.1.m1.1.1.1.1.1.2" xref="S6.SS2.18.p1.1.m1.1.1.1.1.1.2.cmml">c</mi><mo id="S6.SS2.18.p1.1.m1.1.1.1.1.1.3" xref="S6.SS2.18.p1.1.m1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S6.SS2.18.p1.1.m1.2.2.2.2.3" xref="S6.SS2.18.p1.1.m1.2.2.2.3.cmml">,</mo><msup id="S6.SS2.18.p1.1.m1.2.2.2.2.2" xref="S6.SS2.18.p1.1.m1.2.2.2.2.2.cmml"><mi id="S6.SS2.18.p1.1.m1.2.2.2.2.2.2" xref="S6.SS2.18.p1.1.m1.2.2.2.2.2.2.cmml">c</mi><mo id="S6.SS2.18.p1.1.m1.2.2.2.2.2.3" xref="S6.SS2.18.p1.1.m1.2.2.2.2.2.3.cmml">′′</mo></msup></mrow><mo id="S6.SS2.18.p1.1.m1.2.2.3" xref="S6.SS2.18.p1.1.m1.2.2.3.cmml"><</mo><msup id="S6.SS2.18.p1.1.m1.2.2.4" xref="S6.SS2.18.p1.1.m1.2.2.4.cmml"><mi id="S6.SS2.18.p1.1.m1.2.2.4.2" xref="S6.SS2.18.p1.1.m1.2.2.4.2.cmml">μ</mi><mo id="S6.SS2.18.p1.1.m1.2.2.4.3" xref="S6.SS2.18.p1.1.m1.2.2.4.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.18.p1.1.m1.2b"><apply id="S6.SS2.18.p1.1.m1.2.2.cmml" xref="S6.SS2.18.p1.1.m1.2.2"><lt id="S6.SS2.18.p1.1.m1.2.2.3.cmml" xref="S6.SS2.18.p1.1.m1.2.2.3"></lt><list id="S6.SS2.18.p1.1.m1.2.2.2.3.cmml" xref="S6.SS2.18.p1.1.m1.2.2.2.2"><apply id="S6.SS2.18.p1.1.m1.1.1.1.1.1.cmml" xref="S6.SS2.18.p1.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.18.p1.1.m1.1.1.1.1.1.1.cmml" xref="S6.SS2.18.p1.1.m1.1.1.1.1.1">superscript</csymbol><ci id="S6.SS2.18.p1.1.m1.1.1.1.1.1.2.cmml" xref="S6.SS2.18.p1.1.m1.1.1.1.1.1.2">𝑐</ci><ci id="S6.SS2.18.p1.1.m1.1.1.1.1.1.3.cmml" xref="S6.SS2.18.p1.1.m1.1.1.1.1.1.3">′</ci></apply><apply id="S6.SS2.18.p1.1.m1.2.2.2.2.2.cmml" xref="S6.SS2.18.p1.1.m1.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.18.p1.1.m1.2.2.2.2.2.1.cmml" xref="S6.SS2.18.p1.1.m1.2.2.2.2.2">superscript</csymbol><ci id="S6.SS2.18.p1.1.m1.2.2.2.2.2.2.cmml" xref="S6.SS2.18.p1.1.m1.2.2.2.2.2.2">𝑐</ci><ci id="S6.SS2.18.p1.1.m1.2.2.2.2.2.3.cmml" xref="S6.SS2.18.p1.1.m1.2.2.2.2.2.3">′′</ci></apply></list><apply id="S6.SS2.18.p1.1.m1.2.2.4.cmml" xref="S6.SS2.18.p1.1.m1.2.2.4"><csymbol cd="ambiguous" id="S6.SS2.18.p1.1.m1.2.2.4.1.cmml" xref="S6.SS2.18.p1.1.m1.2.2.4">superscript</csymbol><ci id="S6.SS2.18.p1.1.m1.2.2.4.2.cmml" xref="S6.SS2.18.p1.1.m1.2.2.4.2">𝜇</ci><plus id="S6.SS2.18.p1.1.m1.2.2.4.3.cmml" xref="S6.SS2.18.p1.1.m1.2.2.4.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.18.p1.1.m1.2c">c^{\prime},c^{\prime\prime}<\mu^{+}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.18.p1.1.m1.2d">italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_c start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT < italic_μ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> such that <math alttext="a<_{A}c^{\prime}<_{A}c^{\prime\prime}<_{A}b" class="ltx_Math" display="inline" id="S6.SS2.18.p1.2.m2.1"><semantics id="S6.SS2.18.p1.2.m2.1a"><mrow id="S6.SS2.18.p1.2.m2.1.1" xref="S6.SS2.18.p1.2.m2.1.1.cmml"><mi id="S6.SS2.18.p1.2.m2.1.1.2" xref="S6.SS2.18.p1.2.m2.1.1.2.cmml">a</mi><msub id="S6.SS2.18.p1.2.m2.1.1.3" xref="S6.SS2.18.p1.2.m2.1.1.3.cmml"><mo id="S6.SS2.18.p1.2.m2.1.1.3.2" xref="S6.SS2.18.p1.2.m2.1.1.3.2.cmml"><</mo><mi id="S6.SS2.18.p1.2.m2.1.1.3.3" xref="S6.SS2.18.p1.2.m2.1.1.3.3.cmml">A</mi></msub><msup id="S6.SS2.18.p1.2.m2.1.1.4" xref="S6.SS2.18.p1.2.m2.1.1.4.cmml"><mi id="S6.SS2.18.p1.2.m2.1.1.4.2" xref="S6.SS2.18.p1.2.m2.1.1.4.2.cmml">c</mi><mo id="S6.SS2.18.p1.2.m2.1.1.4.3" xref="S6.SS2.18.p1.2.m2.1.1.4.3.cmml">′</mo></msup><msub id="S6.SS2.18.p1.2.m2.1.1.5" xref="S6.SS2.18.p1.2.m2.1.1.5.cmml"><mo id="S6.SS2.18.p1.2.m2.1.1.5.2" xref="S6.SS2.18.p1.2.m2.1.1.5.2.cmml"><</mo><mi id="S6.SS2.18.p1.2.m2.1.1.5.3" xref="S6.SS2.18.p1.2.m2.1.1.5.3.cmml">A</mi></msub><msup id="S6.SS2.18.p1.2.m2.1.1.6" xref="S6.SS2.18.p1.2.m2.1.1.6.cmml"><mi id="S6.SS2.18.p1.2.m2.1.1.6.2" xref="S6.SS2.18.p1.2.m2.1.1.6.2.cmml">c</mi><mo id="S6.SS2.18.p1.2.m2.1.1.6.3" xref="S6.SS2.18.p1.2.m2.1.1.6.3.cmml">′′</mo></msup><msub id="S6.SS2.18.p1.2.m2.1.1.7" xref="S6.SS2.18.p1.2.m2.1.1.7.cmml"><mo id="S6.SS2.18.p1.2.m2.1.1.7.2" xref="S6.SS2.18.p1.2.m2.1.1.7.2.cmml"><</mo><mi id="S6.SS2.18.p1.2.m2.1.1.7.3" xref="S6.SS2.18.p1.2.m2.1.1.7.3.cmml">A</mi></msub><mi id="S6.SS2.18.p1.2.m2.1.1.8" xref="S6.SS2.18.p1.2.m2.1.1.8.cmml">b</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.18.p1.2.m2.1b"><apply id="S6.SS2.18.p1.2.m2.1.1.cmml" xref="S6.SS2.18.p1.2.m2.1.1"><and id="S6.SS2.18.p1.2.m2.1.1a.cmml" xref="S6.SS2.18.p1.2.m2.1.1"></and><apply id="S6.SS2.18.p1.2.m2.1.1b.cmml" xref="S6.SS2.18.p1.2.m2.1.1"><apply id="S6.SS2.18.p1.2.m2.1.1.3.cmml" xref="S6.SS2.18.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S6.SS2.18.p1.2.m2.1.1.3.1.cmml" xref="S6.SS2.18.p1.2.m2.1.1.3">subscript</csymbol><lt id="S6.SS2.18.p1.2.m2.1.1.3.2.cmml" xref="S6.SS2.18.p1.2.m2.1.1.3.2"></lt><ci id="S6.SS2.18.p1.2.m2.1.1.3.3.cmml" xref="S6.SS2.18.p1.2.m2.1.1.3.3">𝐴</ci></apply><ci id="S6.SS2.18.p1.2.m2.1.1.2.cmml" xref="S6.SS2.18.p1.2.m2.1.1.2">𝑎</ci><apply id="S6.SS2.18.p1.2.m2.1.1.4.cmml" xref="S6.SS2.18.p1.2.m2.1.1.4"><csymbol cd="ambiguous" id="S6.SS2.18.p1.2.m2.1.1.4.1.cmml" xref="S6.SS2.18.p1.2.m2.1.1.4">superscript</csymbol><ci id="S6.SS2.18.p1.2.m2.1.1.4.2.cmml" xref="S6.SS2.18.p1.2.m2.1.1.4.2">𝑐</ci><ci id="S6.SS2.18.p1.2.m2.1.1.4.3.cmml" xref="S6.SS2.18.p1.2.m2.1.1.4.3">′</ci></apply></apply><apply id="S6.SS2.18.p1.2.m2.1.1c.cmml" xref="S6.SS2.18.p1.2.m2.1.1"><apply id="S6.SS2.18.p1.2.m2.1.1.5.cmml" xref="S6.SS2.18.p1.2.m2.1.1.5"><csymbol cd="ambiguous" id="S6.SS2.18.p1.2.m2.1.1.5.1.cmml" xref="S6.SS2.18.p1.2.m2.1.1.5">subscript</csymbol><lt id="S6.SS2.18.p1.2.m2.1.1.5.2.cmml" xref="S6.SS2.18.p1.2.m2.1.1.5.2"></lt><ci id="S6.SS2.18.p1.2.m2.1.1.5.3.cmml" xref="S6.SS2.18.p1.2.m2.1.1.5.3">𝐴</ci></apply><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.18.p1.2.m2.1.1.4.cmml" id="S6.SS2.18.p1.2.m2.1.1d.cmml" xref="S6.SS2.18.p1.2.m2.1.1"></share><apply id="S6.SS2.18.p1.2.m2.1.1.6.cmml" xref="S6.SS2.18.p1.2.m2.1.1.6"><csymbol cd="ambiguous" id="S6.SS2.18.p1.2.m2.1.1.6.1.cmml" xref="S6.SS2.18.p1.2.m2.1.1.6">superscript</csymbol><ci id="S6.SS2.18.p1.2.m2.1.1.6.2.cmml" xref="S6.SS2.18.p1.2.m2.1.1.6.2">𝑐</ci><ci id="S6.SS2.18.p1.2.m2.1.1.6.3.cmml" xref="S6.SS2.18.p1.2.m2.1.1.6.3">′′</ci></apply></apply><apply id="S6.SS2.18.p1.2.m2.1.1e.cmml" xref="S6.SS2.18.p1.2.m2.1.1"><apply id="S6.SS2.18.p1.2.m2.1.1.7.cmml" xref="S6.SS2.18.p1.2.m2.1.1.7"><csymbol cd="ambiguous" id="S6.SS2.18.p1.2.m2.1.1.7.1.cmml" xref="S6.SS2.18.p1.2.m2.1.1.7">subscript</csymbol><lt id="S6.SS2.18.p1.2.m2.1.1.7.2.cmml" xref="S6.SS2.18.p1.2.m2.1.1.7.2"></lt><ci id="S6.SS2.18.p1.2.m2.1.1.7.3.cmml" xref="S6.SS2.18.p1.2.m2.1.1.7.3">𝐴</ci></apply><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.18.p1.2.m2.1.1.6.cmml" id="S6.SS2.18.p1.2.m2.1.1f.cmml" xref="S6.SS2.18.p1.2.m2.1.1"></share><ci id="S6.SS2.18.p1.2.m2.1.1.8.cmml" xref="S6.SS2.18.p1.2.m2.1.1.8">𝑏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.18.p1.2.m2.1c">a<_{A}c^{\prime}<_{A}c^{\prime\prime}<_{A}b</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.18.p1.2.m2.1d">italic_a < start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT < start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_c start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT < start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_b</annotation></semantics></math>. Let <math alttext="c:=\Delta(a,b)" class="ltx_Math" display="inline" id="S6.SS2.18.p1.3.m3.2"><semantics id="S6.SS2.18.p1.3.m3.2a"><mrow id="S6.SS2.18.p1.3.m3.2.3" xref="S6.SS2.18.p1.3.m3.2.3.cmml"><mi id="S6.SS2.18.p1.3.m3.2.3.2" xref="S6.SS2.18.p1.3.m3.2.3.2.cmml">c</mi><mo id="S6.SS2.18.p1.3.m3.2.3.1" lspace="0.278em" rspace="0.278em" xref="S6.SS2.18.p1.3.m3.2.3.1.cmml">:=</mo><mrow id="S6.SS2.18.p1.3.m3.2.3.3" xref="S6.SS2.18.p1.3.m3.2.3.3.cmml"><mi id="S6.SS2.18.p1.3.m3.2.3.3.2" mathvariant="normal" xref="S6.SS2.18.p1.3.m3.2.3.3.2.cmml">Δ</mi><mo id="S6.SS2.18.p1.3.m3.2.3.3.1" xref="S6.SS2.18.p1.3.m3.2.3.3.1.cmml">⁢</mo><mrow id="S6.SS2.18.p1.3.m3.2.3.3.3.2" xref="S6.SS2.18.p1.3.m3.2.3.3.3.1.cmml"><mo id="S6.SS2.18.p1.3.m3.2.3.3.3.2.1" stretchy="false" xref="S6.SS2.18.p1.3.m3.2.3.3.3.1.cmml">(</mo><mi id="S6.SS2.18.p1.3.m3.1.1" xref="S6.SS2.18.p1.3.m3.1.1.cmml">a</mi><mo id="S6.SS2.18.p1.3.m3.2.3.3.3.2.2" xref="S6.SS2.18.p1.3.m3.2.3.3.3.1.cmml">,</mo><mi id="S6.SS2.18.p1.3.m3.2.2" xref="S6.SS2.18.p1.3.m3.2.2.cmml">b</mi><mo id="S6.SS2.18.p1.3.m3.2.3.3.3.2.3" stretchy="false" xref="S6.SS2.18.p1.3.m3.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.18.p1.3.m3.2b"><apply id="S6.SS2.18.p1.3.m3.2.3.cmml" xref="S6.SS2.18.p1.3.m3.2.3"><csymbol cd="latexml" id="S6.SS2.18.p1.3.m3.2.3.1.cmml" xref="S6.SS2.18.p1.3.m3.2.3.1">assign</csymbol><ci id="S6.SS2.18.p1.3.m3.2.3.2.cmml" xref="S6.SS2.18.p1.3.m3.2.3.2">𝑐</ci><apply id="S6.SS2.18.p1.3.m3.2.3.3.cmml" xref="S6.SS2.18.p1.3.m3.2.3.3"><times id="S6.SS2.18.p1.3.m3.2.3.3.1.cmml" xref="S6.SS2.18.p1.3.m3.2.3.3.1"></times><ci id="S6.SS2.18.p1.3.m3.2.3.3.2.cmml" xref="S6.SS2.18.p1.3.m3.2.3.3.2">Δ</ci><interval closure="open" id="S6.SS2.18.p1.3.m3.2.3.3.3.1.cmml" xref="S6.SS2.18.p1.3.m3.2.3.3.3.2"><ci id="S6.SS2.18.p1.3.m3.1.1.cmml" xref="S6.SS2.18.p1.3.m3.1.1">𝑎</ci><ci id="S6.SS2.18.p1.3.m3.2.2.cmml" xref="S6.SS2.18.p1.3.m3.2.2">𝑏</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.18.p1.3.m3.2c">c:=\Delta(a,b)</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.18.p1.3.m3.2d">italic_c := roman_Δ ( italic_a , italic_b )</annotation></semantics></math>, and observe that <math alttext="a\leq_{A}c<_{A}b" class="ltx_Math" display="inline" id="S6.SS2.18.p1.4.m4.1"><semantics id="S6.SS2.18.p1.4.m4.1a"><mrow id="S6.SS2.18.p1.4.m4.1.1" xref="S6.SS2.18.p1.4.m4.1.1.cmml"><mi id="S6.SS2.18.p1.4.m4.1.1.2" xref="S6.SS2.18.p1.4.m4.1.1.2.cmml">a</mi><msub id="S6.SS2.18.p1.4.m4.1.1.3" xref="S6.SS2.18.p1.4.m4.1.1.3.cmml"><mo id="S6.SS2.18.p1.4.m4.1.1.3.2" xref="S6.SS2.18.p1.4.m4.1.1.3.2.cmml">≤</mo><mi id="S6.SS2.18.p1.4.m4.1.1.3.3" xref="S6.SS2.18.p1.4.m4.1.1.3.3.cmml">A</mi></msub><mi id="S6.SS2.18.p1.4.m4.1.1.4" xref="S6.SS2.18.p1.4.m4.1.1.4.cmml">c</mi><msub id="S6.SS2.18.p1.4.m4.1.1.5" xref="S6.SS2.18.p1.4.m4.1.1.5.cmml"><mo id="S6.SS2.18.p1.4.m4.1.1.5.2" xref="S6.SS2.18.p1.4.m4.1.1.5.2.cmml"><</mo><mi id="S6.SS2.18.p1.4.m4.1.1.5.3" xref="S6.SS2.18.p1.4.m4.1.1.5.3.cmml">A</mi></msub><mi id="S6.SS2.18.p1.4.m4.1.1.6" xref="S6.SS2.18.p1.4.m4.1.1.6.cmml">b</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.18.p1.4.m4.1b"><apply id="S6.SS2.18.p1.4.m4.1.1.cmml" xref="S6.SS2.18.p1.4.m4.1.1"><and id="S6.SS2.18.p1.4.m4.1.1a.cmml" xref="S6.SS2.18.p1.4.m4.1.1"></and><apply id="S6.SS2.18.p1.4.m4.1.1b.cmml" xref="S6.SS2.18.p1.4.m4.1.1"><apply id="S6.SS2.18.p1.4.m4.1.1.3.cmml" xref="S6.SS2.18.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="S6.SS2.18.p1.4.m4.1.1.3.1.cmml" xref="S6.SS2.18.p1.4.m4.1.1.3">subscript</csymbol><leq id="S6.SS2.18.p1.4.m4.1.1.3.2.cmml" xref="S6.SS2.18.p1.4.m4.1.1.3.2"></leq><ci id="S6.SS2.18.p1.4.m4.1.1.3.3.cmml" xref="S6.SS2.18.p1.4.m4.1.1.3.3">𝐴</ci></apply><ci id="S6.SS2.18.p1.4.m4.1.1.2.cmml" xref="S6.SS2.18.p1.4.m4.1.1.2">𝑎</ci><ci id="S6.SS2.18.p1.4.m4.1.1.4.cmml" xref="S6.SS2.18.p1.4.m4.1.1.4">𝑐</ci></apply><apply id="S6.SS2.18.p1.4.m4.1.1c.cmml" xref="S6.SS2.18.p1.4.m4.1.1"><apply id="S6.SS2.18.p1.4.m4.1.1.5.cmml" xref="S6.SS2.18.p1.4.m4.1.1.5"><csymbol cd="ambiguous" id="S6.SS2.18.p1.4.m4.1.1.5.1.cmml" xref="S6.SS2.18.p1.4.m4.1.1.5">subscript</csymbol><lt id="S6.SS2.18.p1.4.m4.1.1.5.2.cmml" xref="S6.SS2.18.p1.4.m4.1.1.5.2"></lt><ci id="S6.SS2.18.p1.4.m4.1.1.5.3.cmml" xref="S6.SS2.18.p1.4.m4.1.1.5.3">𝐴</ci></apply><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.18.p1.4.m4.1.1.4.cmml" id="S6.SS2.18.p1.4.m4.1.1d.cmml" xref="S6.SS2.18.p1.4.m4.1.1"></share><ci id="S6.SS2.18.p1.4.m4.1.1.6.cmml" xref="S6.SS2.18.p1.4.m4.1.1.6">𝑏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.18.p1.4.m4.1c">a\leq_{A}c<_{A}b</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.18.p1.4.m4.1d">italic_a ≤ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_c < start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_b</annotation></semantics></math> and <math alttext="\nu(c)=\nu(a,b)" class="ltx_Math" display="inline" id="S6.SS2.18.p1.5.m5.3"><semantics id="S6.SS2.18.p1.5.m5.3a"><mrow id="S6.SS2.18.p1.5.m5.3.4" xref="S6.SS2.18.p1.5.m5.3.4.cmml"><mrow id="S6.SS2.18.p1.5.m5.3.4.2" xref="S6.SS2.18.p1.5.m5.3.4.2.cmml"><mi id="S6.SS2.18.p1.5.m5.3.4.2.2" xref="S6.SS2.18.p1.5.m5.3.4.2.2.cmml">ν</mi><mo id="S6.SS2.18.p1.5.m5.3.4.2.1" xref="S6.SS2.18.p1.5.m5.3.4.2.1.cmml">⁢</mo><mrow id="S6.SS2.18.p1.5.m5.3.4.2.3.2" xref="S6.SS2.18.p1.5.m5.3.4.2.cmml"><mo id="S6.SS2.18.p1.5.m5.3.4.2.3.2.1" stretchy="false" xref="S6.SS2.18.p1.5.m5.3.4.2.cmml">(</mo><mi id="S6.SS2.18.p1.5.m5.1.1" xref="S6.SS2.18.p1.5.m5.1.1.cmml">c</mi><mo id="S6.SS2.18.p1.5.m5.3.4.2.3.2.2" stretchy="false" xref="S6.SS2.18.p1.5.m5.3.4.2.cmml">)</mo></mrow></mrow><mo id="S6.SS2.18.p1.5.m5.3.4.1" xref="S6.SS2.18.p1.5.m5.3.4.1.cmml">=</mo><mrow id="S6.SS2.18.p1.5.m5.3.4.3" xref="S6.SS2.18.p1.5.m5.3.4.3.cmml"><mi id="S6.SS2.18.p1.5.m5.3.4.3.2" xref="S6.SS2.18.p1.5.m5.3.4.3.2.cmml">ν</mi><mo id="S6.SS2.18.p1.5.m5.3.4.3.1" xref="S6.SS2.18.p1.5.m5.3.4.3.1.cmml">⁢</mo><mrow id="S6.SS2.18.p1.5.m5.3.4.3.3.2" xref="S6.SS2.18.p1.5.m5.3.4.3.3.1.cmml"><mo id="S6.SS2.18.p1.5.m5.3.4.3.3.2.1" stretchy="false" xref="S6.SS2.18.p1.5.m5.3.4.3.3.1.cmml">(</mo><mi id="S6.SS2.18.p1.5.m5.2.2" xref="S6.SS2.18.p1.5.m5.2.2.cmml">a</mi><mo id="S6.SS2.18.p1.5.m5.3.4.3.3.2.2" xref="S6.SS2.18.p1.5.m5.3.4.3.3.1.cmml">,</mo><mi id="S6.SS2.18.p1.5.m5.3.3" xref="S6.SS2.18.p1.5.m5.3.3.cmml">b</mi><mo id="S6.SS2.18.p1.5.m5.3.4.3.3.2.3" stretchy="false" xref="S6.SS2.18.p1.5.m5.3.4.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.18.p1.5.m5.3b"><apply id="S6.SS2.18.p1.5.m5.3.4.cmml" xref="S6.SS2.18.p1.5.m5.3.4"><eq id="S6.SS2.18.p1.5.m5.3.4.1.cmml" xref="S6.SS2.18.p1.5.m5.3.4.1"></eq><apply id="S6.SS2.18.p1.5.m5.3.4.2.cmml" xref="S6.SS2.18.p1.5.m5.3.4.2"><times id="S6.SS2.18.p1.5.m5.3.4.2.1.cmml" xref="S6.SS2.18.p1.5.m5.3.4.2.1"></times><ci id="S6.SS2.18.p1.5.m5.3.4.2.2.cmml" xref="S6.SS2.18.p1.5.m5.3.4.2.2">𝜈</ci><ci id="S6.SS2.18.p1.5.m5.1.1.cmml" xref="S6.SS2.18.p1.5.m5.1.1">𝑐</ci></apply><apply id="S6.SS2.18.p1.5.m5.3.4.3.cmml" xref="S6.SS2.18.p1.5.m5.3.4.3"><times id="S6.SS2.18.p1.5.m5.3.4.3.1.cmml" xref="S6.SS2.18.p1.5.m5.3.4.3.1"></times><ci id="S6.SS2.18.p1.5.m5.3.4.3.2.cmml" xref="S6.SS2.18.p1.5.m5.3.4.3.2">𝜈</ci><interval closure="open" id="S6.SS2.18.p1.5.m5.3.4.3.3.1.cmml" xref="S6.SS2.18.p1.5.m5.3.4.3.3.2"><ci id="S6.SS2.18.p1.5.m5.2.2.cmml" xref="S6.SS2.18.p1.5.m5.2.2">𝑎</ci><ci id="S6.SS2.18.p1.5.m5.3.3.cmml" xref="S6.SS2.18.p1.5.m5.3.3">𝑏</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.18.p1.5.m5.3c">\nu(c)=\nu(a,b)</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.18.p1.5.m5.3d">italic_ν ( italic_c ) = italic_ν ( italic_a , italic_b )</annotation></semantics></math> hold. Using the <math alttext="\aleph_{1}" class="ltx_Math" display="inline" id="S6.SS2.18.p1.6.m6.1"><semantics id="S6.SS2.18.p1.6.m6.1a"><msub id="S6.SS2.18.p1.6.m6.1.1" xref="S6.SS2.18.p1.6.m6.1.1.cmml"><mi id="S6.SS2.18.p1.6.m6.1.1.2" mathvariant="normal" xref="S6.SS2.18.p1.6.m6.1.1.2.cmml">ℵ</mi><mn id="S6.SS2.18.p1.6.m6.1.1.3" xref="S6.SS2.18.p1.6.m6.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.18.p1.6.m6.1b"><apply id="S6.SS2.18.p1.6.m6.1.1.cmml" xref="S6.SS2.18.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S6.SS2.18.p1.6.m6.1.1.1.cmml" xref="S6.SS2.18.p1.6.m6.1.1">subscript</csymbol><ci id="S6.SS2.18.p1.6.m6.1.1.2.cmml" xref="S6.SS2.18.p1.6.m6.1.1.2">ℵ</ci><cn id="S6.SS2.18.p1.6.m6.1.1.3.cmml" type="integer" xref="S6.SS2.18.p1.6.m6.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.18.p1.6.m6.1c">\aleph_{1}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.18.p1.6.m6.1d">roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-density of <math alttext="A" class="ltx_Math" display="inline" id="S6.SS2.18.p1.7.m7.1"><semantics id="S6.SS2.18.p1.7.m7.1a"><mi id="S6.SS2.18.p1.7.m7.1.1" xref="S6.SS2.18.p1.7.m7.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.18.p1.7.m7.1b"><ci id="S6.SS2.18.p1.7.m7.1.1.cmml" xref="S6.SS2.18.p1.7.m7.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.18.p1.7.m7.1c">A</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.18.p1.7.m7.1d">italic_A</annotation></semantics></math>, and <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.Thmtheorem16" title="Lemma 6.16. ‣ 6.2. Forcing epimorphisms ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">6.16</span></a> (b) for <math alttext="\nu(a,b)^{+}" class="ltx_Math" display="inline" id="S6.SS2.18.p1.8.m8.2"><semantics id="S6.SS2.18.p1.8.m8.2a"><mrow id="S6.SS2.18.p1.8.m8.2.3" xref="S6.SS2.18.p1.8.m8.2.3.cmml"><mi id="S6.SS2.18.p1.8.m8.2.3.2" xref="S6.SS2.18.p1.8.m8.2.3.2.cmml">ν</mi><mo id="S6.SS2.18.p1.8.m8.2.3.1" xref="S6.SS2.18.p1.8.m8.2.3.1.cmml">⁢</mo><msup id="S6.SS2.18.p1.8.m8.2.3.3" xref="S6.SS2.18.p1.8.m8.2.3.3.cmml"><mrow id="S6.SS2.18.p1.8.m8.2.3.3.2.2" xref="S6.SS2.18.p1.8.m8.2.3.3.2.1.cmml"><mo id="S6.SS2.18.p1.8.m8.2.3.3.2.2.1" stretchy="false" xref="S6.SS2.18.p1.8.m8.2.3.3.2.1.cmml">(</mo><mi id="S6.SS2.18.p1.8.m8.1.1" xref="S6.SS2.18.p1.8.m8.1.1.cmml">a</mi><mo id="S6.SS2.18.p1.8.m8.2.3.3.2.2.2" xref="S6.SS2.18.p1.8.m8.2.3.3.2.1.cmml">,</mo><mi id="S6.SS2.18.p1.8.m8.2.2" xref="S6.SS2.18.p1.8.m8.2.2.cmml">b</mi><mo id="S6.SS2.18.p1.8.m8.2.3.3.2.2.3" stretchy="false" xref="S6.SS2.18.p1.8.m8.2.3.3.2.1.cmml">)</mo></mrow><mo id="S6.SS2.18.p1.8.m8.2.3.3.3" xref="S6.SS2.18.p1.8.m8.2.3.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.18.p1.8.m8.2b"><apply id="S6.SS2.18.p1.8.m8.2.3.cmml" xref="S6.SS2.18.p1.8.m8.2.3"><times id="S6.SS2.18.p1.8.m8.2.3.1.cmml" xref="S6.SS2.18.p1.8.m8.2.3.1"></times><ci id="S6.SS2.18.p1.8.m8.2.3.2.cmml" xref="S6.SS2.18.p1.8.m8.2.3.2">𝜈</ci><apply id="S6.SS2.18.p1.8.m8.2.3.3.cmml" xref="S6.SS2.18.p1.8.m8.2.3.3"><csymbol cd="ambiguous" id="S6.SS2.18.p1.8.m8.2.3.3.1.cmml" xref="S6.SS2.18.p1.8.m8.2.3.3">superscript</csymbol><interval closure="open" id="S6.SS2.18.p1.8.m8.2.3.3.2.1.cmml" xref="S6.SS2.18.p1.8.m8.2.3.3.2.2"><ci id="S6.SS2.18.p1.8.m8.1.1.cmml" xref="S6.SS2.18.p1.8.m8.1.1">𝑎</ci><ci id="S6.SS2.18.p1.8.m8.2.2.cmml" xref="S6.SS2.18.p1.8.m8.2.2">𝑏</ci></interval><plus id="S6.SS2.18.p1.8.m8.2.3.3.3.cmml" xref="S6.SS2.18.p1.8.m8.2.3.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.18.p1.8.m8.2c">\nu(a,b)^{+}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.18.p1.8.m8.2d">italic_ν ( italic_a , italic_b ) start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> together with the fact that <math alttext="\nu(c)<\nu(a,b)^{+}" class="ltx_Math" display="inline" id="S6.SS2.18.p1.9.m9.3"><semantics id="S6.SS2.18.p1.9.m9.3a"><mrow id="S6.SS2.18.p1.9.m9.3.4" xref="S6.SS2.18.p1.9.m9.3.4.cmml"><mrow id="S6.SS2.18.p1.9.m9.3.4.2" xref="S6.SS2.18.p1.9.m9.3.4.2.cmml"><mi id="S6.SS2.18.p1.9.m9.3.4.2.2" xref="S6.SS2.18.p1.9.m9.3.4.2.2.cmml">ν</mi><mo id="S6.SS2.18.p1.9.m9.3.4.2.1" xref="S6.SS2.18.p1.9.m9.3.4.2.1.cmml">⁢</mo><mrow id="S6.SS2.18.p1.9.m9.3.4.2.3.2" xref="S6.SS2.18.p1.9.m9.3.4.2.cmml"><mo id="S6.SS2.18.p1.9.m9.3.4.2.3.2.1" stretchy="false" xref="S6.SS2.18.p1.9.m9.3.4.2.cmml">(</mo><mi id="S6.SS2.18.p1.9.m9.1.1" xref="S6.SS2.18.p1.9.m9.1.1.cmml">c</mi><mo id="S6.SS2.18.p1.9.m9.3.4.2.3.2.2" stretchy="false" xref="S6.SS2.18.p1.9.m9.3.4.2.cmml">)</mo></mrow></mrow><mo id="S6.SS2.18.p1.9.m9.3.4.1" xref="S6.SS2.18.p1.9.m9.3.4.1.cmml"><</mo><mrow id="S6.SS2.18.p1.9.m9.3.4.3" xref="S6.SS2.18.p1.9.m9.3.4.3.cmml"><mi id="S6.SS2.18.p1.9.m9.3.4.3.2" xref="S6.SS2.18.p1.9.m9.3.4.3.2.cmml">ν</mi><mo id="S6.SS2.18.p1.9.m9.3.4.3.1" xref="S6.SS2.18.p1.9.m9.3.4.3.1.cmml">⁢</mo><msup id="S6.SS2.18.p1.9.m9.3.4.3.3" xref="S6.SS2.18.p1.9.m9.3.4.3.3.cmml"><mrow id="S6.SS2.18.p1.9.m9.3.4.3.3.2.2" xref="S6.SS2.18.p1.9.m9.3.4.3.3.2.1.cmml"><mo id="S6.SS2.18.p1.9.m9.3.4.3.3.2.2.1" stretchy="false" xref="S6.SS2.18.p1.9.m9.3.4.3.3.2.1.cmml">(</mo><mi id="S6.SS2.18.p1.9.m9.2.2" xref="S6.SS2.18.p1.9.m9.2.2.cmml">a</mi><mo id="S6.SS2.18.p1.9.m9.3.4.3.3.2.2.2" xref="S6.SS2.18.p1.9.m9.3.4.3.3.2.1.cmml">,</mo><mi id="S6.SS2.18.p1.9.m9.3.3" xref="S6.SS2.18.p1.9.m9.3.3.cmml">b</mi><mo id="S6.SS2.18.p1.9.m9.3.4.3.3.2.2.3" stretchy="false" xref="S6.SS2.18.p1.9.m9.3.4.3.3.2.1.cmml">)</mo></mrow><mo id="S6.SS2.18.p1.9.m9.3.4.3.3.3" xref="S6.SS2.18.p1.9.m9.3.4.3.3.3.cmml">+</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.18.p1.9.m9.3b"><apply id="S6.SS2.18.p1.9.m9.3.4.cmml" xref="S6.SS2.18.p1.9.m9.3.4"><lt id="S6.SS2.18.p1.9.m9.3.4.1.cmml" xref="S6.SS2.18.p1.9.m9.3.4.1"></lt><apply id="S6.SS2.18.p1.9.m9.3.4.2.cmml" xref="S6.SS2.18.p1.9.m9.3.4.2"><times id="S6.SS2.18.p1.9.m9.3.4.2.1.cmml" xref="S6.SS2.18.p1.9.m9.3.4.2.1"></times><ci id="S6.SS2.18.p1.9.m9.3.4.2.2.cmml" xref="S6.SS2.18.p1.9.m9.3.4.2.2">𝜈</ci><ci id="S6.SS2.18.p1.9.m9.1.1.cmml" xref="S6.SS2.18.p1.9.m9.1.1">𝑐</ci></apply><apply id="S6.SS2.18.p1.9.m9.3.4.3.cmml" xref="S6.SS2.18.p1.9.m9.3.4.3"><times id="S6.SS2.18.p1.9.m9.3.4.3.1.cmml" xref="S6.SS2.18.p1.9.m9.3.4.3.1"></times><ci id="S6.SS2.18.p1.9.m9.3.4.3.2.cmml" xref="S6.SS2.18.p1.9.m9.3.4.3.2">𝜈</ci><apply id="S6.SS2.18.p1.9.m9.3.4.3.3.cmml" xref="S6.SS2.18.p1.9.m9.3.4.3.3"><csymbol cd="ambiguous" id="S6.SS2.18.p1.9.m9.3.4.3.3.1.cmml" xref="S6.SS2.18.p1.9.m9.3.4.3.3">superscript</csymbol><interval closure="open" id="S6.SS2.18.p1.9.m9.3.4.3.3.2.1.cmml" xref="S6.SS2.18.p1.9.m9.3.4.3.3.2.2"><ci id="S6.SS2.18.p1.9.m9.2.2.cmml" xref="S6.SS2.18.p1.9.m9.2.2">𝑎</ci><ci id="S6.SS2.18.p1.9.m9.3.3.cmml" xref="S6.SS2.18.p1.9.m9.3.3">𝑏</ci></interval><plus id="S6.SS2.18.p1.9.m9.3.4.3.3.3.cmml" xref="S6.SS2.18.p1.9.m9.3.4.3.3.3"></plus></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.18.p1.9.m9.3c">\nu(c)<\nu(a,b)^{+}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.18.p1.9.m9.3d">italic_ν ( italic_c ) < italic_ν ( italic_a , italic_b ) start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math>, approximating <math alttext="c" class="ltx_Math" display="inline" id="S6.SS2.18.p1.10.m10.1"><semantics id="S6.SS2.18.p1.10.m10.1a"><mi id="S6.SS2.18.p1.10.m10.1.1" xref="S6.SS2.18.p1.10.m10.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.18.p1.10.m10.1b"><ci id="S6.SS2.18.p1.10.m10.1.1.cmml" xref="S6.SS2.18.p1.10.m10.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.18.p1.10.m10.1c">c</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.18.p1.10.m10.1d">italic_c</annotation></semantics></math> from the right one finds <math alttext="c^{\prime},c^{\prime\prime}\in A" class="ltx_Math" display="inline" id="S6.SS2.18.p1.11.m11.2"><semantics id="S6.SS2.18.p1.11.m11.2a"><mrow id="S6.SS2.18.p1.11.m11.2.2" xref="S6.SS2.18.p1.11.m11.2.2.cmml"><mrow id="S6.SS2.18.p1.11.m11.2.2.2.2" xref="S6.SS2.18.p1.11.m11.2.2.2.3.cmml"><msup id="S6.SS2.18.p1.11.m11.1.1.1.1.1" xref="S6.SS2.18.p1.11.m11.1.1.1.1.1.cmml"><mi id="S6.SS2.18.p1.11.m11.1.1.1.1.1.2" xref="S6.SS2.18.p1.11.m11.1.1.1.1.1.2.cmml">c</mi><mo id="S6.SS2.18.p1.11.m11.1.1.1.1.1.3" xref="S6.SS2.18.p1.11.m11.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S6.SS2.18.p1.11.m11.2.2.2.2.3" xref="S6.SS2.18.p1.11.m11.2.2.2.3.cmml">,</mo><msup id="S6.SS2.18.p1.11.m11.2.2.2.2.2" xref="S6.SS2.18.p1.11.m11.2.2.2.2.2.cmml"><mi id="S6.SS2.18.p1.11.m11.2.2.2.2.2.2" xref="S6.SS2.18.p1.11.m11.2.2.2.2.2.2.cmml">c</mi><mo id="S6.SS2.18.p1.11.m11.2.2.2.2.2.3" xref="S6.SS2.18.p1.11.m11.2.2.2.2.2.3.cmml">′′</mo></msup></mrow><mo id="S6.SS2.18.p1.11.m11.2.2.3" xref="S6.SS2.18.p1.11.m11.2.2.3.cmml">∈</mo><mi id="S6.SS2.18.p1.11.m11.2.2.4" xref="S6.SS2.18.p1.11.m11.2.2.4.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.18.p1.11.m11.2b"><apply id="S6.SS2.18.p1.11.m11.2.2.cmml" xref="S6.SS2.18.p1.11.m11.2.2"><in id="S6.SS2.18.p1.11.m11.2.2.3.cmml" xref="S6.SS2.18.p1.11.m11.2.2.3"></in><list id="S6.SS2.18.p1.11.m11.2.2.2.3.cmml" xref="S6.SS2.18.p1.11.m11.2.2.2.2"><apply id="S6.SS2.18.p1.11.m11.1.1.1.1.1.cmml" xref="S6.SS2.18.p1.11.m11.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.18.p1.11.m11.1.1.1.1.1.1.cmml" xref="S6.SS2.18.p1.11.m11.1.1.1.1.1">superscript</csymbol><ci id="S6.SS2.18.p1.11.m11.1.1.1.1.1.2.cmml" xref="S6.SS2.18.p1.11.m11.1.1.1.1.1.2">𝑐</ci><ci id="S6.SS2.18.p1.11.m11.1.1.1.1.1.3.cmml" xref="S6.SS2.18.p1.11.m11.1.1.1.1.1.3">′</ci></apply><apply id="S6.SS2.18.p1.11.m11.2.2.2.2.2.cmml" xref="S6.SS2.18.p1.11.m11.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.18.p1.11.m11.2.2.2.2.2.1.cmml" xref="S6.SS2.18.p1.11.m11.2.2.2.2.2">superscript</csymbol><ci id="S6.SS2.18.p1.11.m11.2.2.2.2.2.2.cmml" xref="S6.SS2.18.p1.11.m11.2.2.2.2.2.2">𝑐</ci><ci id="S6.SS2.18.p1.11.m11.2.2.2.2.2.3.cmml" xref="S6.SS2.18.p1.11.m11.2.2.2.2.2.3">′′</ci></apply></list><ci id="S6.SS2.18.p1.11.m11.2.2.4.cmml" xref="S6.SS2.18.p1.11.m11.2.2.4">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.18.p1.11.m11.2c">c^{\prime},c^{\prime\prime}\in A</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.18.p1.11.m11.2d">italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_c start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ∈ italic_A</annotation></semantics></math> such that <math alttext="c<_{A}c^{\prime}<_{A}c^{\prime\prime}<_{A}b" class="ltx_Math" display="inline" id="S6.SS2.18.p1.12.m12.1"><semantics id="S6.SS2.18.p1.12.m12.1a"><mrow id="S6.SS2.18.p1.12.m12.1.1" xref="S6.SS2.18.p1.12.m12.1.1.cmml"><mi id="S6.SS2.18.p1.12.m12.1.1.2" xref="S6.SS2.18.p1.12.m12.1.1.2.cmml">c</mi><msub id="S6.SS2.18.p1.12.m12.1.1.3" xref="S6.SS2.18.p1.12.m12.1.1.3.cmml"><mo id="S6.SS2.18.p1.12.m12.1.1.3.2" xref="S6.SS2.18.p1.12.m12.1.1.3.2.cmml"><</mo><mi id="S6.SS2.18.p1.12.m12.1.1.3.3" xref="S6.SS2.18.p1.12.m12.1.1.3.3.cmml">A</mi></msub><msup id="S6.SS2.18.p1.12.m12.1.1.4" xref="S6.SS2.18.p1.12.m12.1.1.4.cmml"><mi id="S6.SS2.18.p1.12.m12.1.1.4.2" xref="S6.SS2.18.p1.12.m12.1.1.4.2.cmml">c</mi><mo id="S6.SS2.18.p1.12.m12.1.1.4.3" xref="S6.SS2.18.p1.12.m12.1.1.4.3.cmml">′</mo></msup><msub id="S6.SS2.18.p1.12.m12.1.1.5" xref="S6.SS2.18.p1.12.m12.1.1.5.cmml"><mo id="S6.SS2.18.p1.12.m12.1.1.5.2" xref="S6.SS2.18.p1.12.m12.1.1.5.2.cmml"><</mo><mi id="S6.SS2.18.p1.12.m12.1.1.5.3" xref="S6.SS2.18.p1.12.m12.1.1.5.3.cmml">A</mi></msub><msup id="S6.SS2.18.p1.12.m12.1.1.6" xref="S6.SS2.18.p1.12.m12.1.1.6.cmml"><mi id="S6.SS2.18.p1.12.m12.1.1.6.2" xref="S6.SS2.18.p1.12.m12.1.1.6.2.cmml">c</mi><mo id="S6.SS2.18.p1.12.m12.1.1.6.3" xref="S6.SS2.18.p1.12.m12.1.1.6.3.cmml">′′</mo></msup><msub id="S6.SS2.18.p1.12.m12.1.1.7" xref="S6.SS2.18.p1.12.m12.1.1.7.cmml"><mo id="S6.SS2.18.p1.12.m12.1.1.7.2" xref="S6.SS2.18.p1.12.m12.1.1.7.2.cmml"><</mo><mi id="S6.SS2.18.p1.12.m12.1.1.7.3" xref="S6.SS2.18.p1.12.m12.1.1.7.3.cmml">A</mi></msub><mi id="S6.SS2.18.p1.12.m12.1.1.8" xref="S6.SS2.18.p1.12.m12.1.1.8.cmml">b</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.18.p1.12.m12.1b"><apply id="S6.SS2.18.p1.12.m12.1.1.cmml" xref="S6.SS2.18.p1.12.m12.1.1"><and id="S6.SS2.18.p1.12.m12.1.1a.cmml" xref="S6.SS2.18.p1.12.m12.1.1"></and><apply id="S6.SS2.18.p1.12.m12.1.1b.cmml" xref="S6.SS2.18.p1.12.m12.1.1"><apply id="S6.SS2.18.p1.12.m12.1.1.3.cmml" xref="S6.SS2.18.p1.12.m12.1.1.3"><csymbol cd="ambiguous" id="S6.SS2.18.p1.12.m12.1.1.3.1.cmml" xref="S6.SS2.18.p1.12.m12.1.1.3">subscript</csymbol><lt id="S6.SS2.18.p1.12.m12.1.1.3.2.cmml" xref="S6.SS2.18.p1.12.m12.1.1.3.2"></lt><ci id="S6.SS2.18.p1.12.m12.1.1.3.3.cmml" xref="S6.SS2.18.p1.12.m12.1.1.3.3">𝐴</ci></apply><ci id="S6.SS2.18.p1.12.m12.1.1.2.cmml" xref="S6.SS2.18.p1.12.m12.1.1.2">𝑐</ci><apply id="S6.SS2.18.p1.12.m12.1.1.4.cmml" xref="S6.SS2.18.p1.12.m12.1.1.4"><csymbol cd="ambiguous" id="S6.SS2.18.p1.12.m12.1.1.4.1.cmml" xref="S6.SS2.18.p1.12.m12.1.1.4">superscript</csymbol><ci id="S6.SS2.18.p1.12.m12.1.1.4.2.cmml" xref="S6.SS2.18.p1.12.m12.1.1.4.2">𝑐</ci><ci id="S6.SS2.18.p1.12.m12.1.1.4.3.cmml" xref="S6.SS2.18.p1.12.m12.1.1.4.3">′</ci></apply></apply><apply id="S6.SS2.18.p1.12.m12.1.1c.cmml" xref="S6.SS2.18.p1.12.m12.1.1"><apply id="S6.SS2.18.p1.12.m12.1.1.5.cmml" xref="S6.SS2.18.p1.12.m12.1.1.5"><csymbol cd="ambiguous" id="S6.SS2.18.p1.12.m12.1.1.5.1.cmml" xref="S6.SS2.18.p1.12.m12.1.1.5">subscript</csymbol><lt id="S6.SS2.18.p1.12.m12.1.1.5.2.cmml" xref="S6.SS2.18.p1.12.m12.1.1.5.2"></lt><ci id="S6.SS2.18.p1.12.m12.1.1.5.3.cmml" xref="S6.SS2.18.p1.12.m12.1.1.5.3">𝐴</ci></apply><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.18.p1.12.m12.1.1.4.cmml" id="S6.SS2.18.p1.12.m12.1.1d.cmml" xref="S6.SS2.18.p1.12.m12.1.1"></share><apply id="S6.SS2.18.p1.12.m12.1.1.6.cmml" xref="S6.SS2.18.p1.12.m12.1.1.6"><csymbol cd="ambiguous" id="S6.SS2.18.p1.12.m12.1.1.6.1.cmml" xref="S6.SS2.18.p1.12.m12.1.1.6">superscript</csymbol><ci id="S6.SS2.18.p1.12.m12.1.1.6.2.cmml" xref="S6.SS2.18.p1.12.m12.1.1.6.2">𝑐</ci><ci id="S6.SS2.18.p1.12.m12.1.1.6.3.cmml" xref="S6.SS2.18.p1.12.m12.1.1.6.3">′′</ci></apply></apply><apply id="S6.SS2.18.p1.12.m12.1.1e.cmml" xref="S6.SS2.18.p1.12.m12.1.1"><apply id="S6.SS2.18.p1.12.m12.1.1.7.cmml" xref="S6.SS2.18.p1.12.m12.1.1.7"><csymbol cd="ambiguous" id="S6.SS2.18.p1.12.m12.1.1.7.1.cmml" xref="S6.SS2.18.p1.12.m12.1.1.7">subscript</csymbol><lt id="S6.SS2.18.p1.12.m12.1.1.7.2.cmml" xref="S6.SS2.18.p1.12.m12.1.1.7.2"></lt><ci id="S6.SS2.18.p1.12.m12.1.1.7.3.cmml" xref="S6.SS2.18.p1.12.m12.1.1.7.3">𝐴</ci></apply><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.18.p1.12.m12.1.1.6.cmml" id="S6.SS2.18.p1.12.m12.1.1f.cmml" xref="S6.SS2.18.p1.12.m12.1.1"></share><ci id="S6.SS2.18.p1.12.m12.1.1.8.cmml" xref="S6.SS2.18.p1.12.m12.1.1.8">𝑏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.18.p1.12.m12.1c">c<_{A}c^{\prime}<_{A}c^{\prime\prime}<_{A}b</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.18.p1.12.m12.1d">italic_c < start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT < start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_c start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT < start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_b</annotation></semantics></math> and <math alttext="\nu(c^{\prime})=\nu(c^{\prime\prime})=\nu(a,b)\leq\mu" class="ltx_Math" display="inline" id="S6.SS2.18.p1.13.m13.4"><semantics id="S6.SS2.18.p1.13.m13.4a"><mrow id="S6.SS2.18.p1.13.m13.4.4" xref="S6.SS2.18.p1.13.m13.4.4.cmml"><mrow id="S6.SS2.18.p1.13.m13.3.3.1" xref="S6.SS2.18.p1.13.m13.3.3.1.cmml"><mi id="S6.SS2.18.p1.13.m13.3.3.1.3" xref="S6.SS2.18.p1.13.m13.3.3.1.3.cmml">ν</mi><mo id="S6.SS2.18.p1.13.m13.3.3.1.2" xref="S6.SS2.18.p1.13.m13.3.3.1.2.cmml">⁢</mo><mrow id="S6.SS2.18.p1.13.m13.3.3.1.1.1" xref="S6.SS2.18.p1.13.m13.3.3.1.1.1.1.cmml"><mo id="S6.SS2.18.p1.13.m13.3.3.1.1.1.2" stretchy="false" xref="S6.SS2.18.p1.13.m13.3.3.1.1.1.1.cmml">(</mo><msup id="S6.SS2.18.p1.13.m13.3.3.1.1.1.1" xref="S6.SS2.18.p1.13.m13.3.3.1.1.1.1.cmml"><mi id="S6.SS2.18.p1.13.m13.3.3.1.1.1.1.2" xref="S6.SS2.18.p1.13.m13.3.3.1.1.1.1.2.cmml">c</mi><mo id="S6.SS2.18.p1.13.m13.3.3.1.1.1.1.3" xref="S6.SS2.18.p1.13.m13.3.3.1.1.1.1.3.cmml">′</mo></msup><mo id="S6.SS2.18.p1.13.m13.3.3.1.1.1.3" stretchy="false" xref="S6.SS2.18.p1.13.m13.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.SS2.18.p1.13.m13.4.4.4" xref="S6.SS2.18.p1.13.m13.4.4.4.cmml">=</mo><mrow id="S6.SS2.18.p1.13.m13.4.4.2" xref="S6.SS2.18.p1.13.m13.4.4.2.cmml"><mi id="S6.SS2.18.p1.13.m13.4.4.2.3" xref="S6.SS2.18.p1.13.m13.4.4.2.3.cmml">ν</mi><mo id="S6.SS2.18.p1.13.m13.4.4.2.2" xref="S6.SS2.18.p1.13.m13.4.4.2.2.cmml">⁢</mo><mrow id="S6.SS2.18.p1.13.m13.4.4.2.1.1" xref="S6.SS2.18.p1.13.m13.4.4.2.1.1.1.cmml"><mo id="S6.SS2.18.p1.13.m13.4.4.2.1.1.2" stretchy="false" xref="S6.SS2.18.p1.13.m13.4.4.2.1.1.1.cmml">(</mo><msup id="S6.SS2.18.p1.13.m13.4.4.2.1.1.1" xref="S6.SS2.18.p1.13.m13.4.4.2.1.1.1.cmml"><mi id="S6.SS2.18.p1.13.m13.4.4.2.1.1.1.2" xref="S6.SS2.18.p1.13.m13.4.4.2.1.1.1.2.cmml">c</mi><mo id="S6.SS2.18.p1.13.m13.4.4.2.1.1.1.3" xref="S6.SS2.18.p1.13.m13.4.4.2.1.1.1.3.cmml">′′</mo></msup><mo id="S6.SS2.18.p1.13.m13.4.4.2.1.1.3" stretchy="false" xref="S6.SS2.18.p1.13.m13.4.4.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.SS2.18.p1.13.m13.4.4.5" xref="S6.SS2.18.p1.13.m13.4.4.5.cmml">=</mo><mrow id="S6.SS2.18.p1.13.m13.4.4.6" xref="S6.SS2.18.p1.13.m13.4.4.6.cmml"><mi id="S6.SS2.18.p1.13.m13.4.4.6.2" xref="S6.SS2.18.p1.13.m13.4.4.6.2.cmml">ν</mi><mo id="S6.SS2.18.p1.13.m13.4.4.6.1" xref="S6.SS2.18.p1.13.m13.4.4.6.1.cmml">⁢</mo><mrow id="S6.SS2.18.p1.13.m13.4.4.6.3.2" xref="S6.SS2.18.p1.13.m13.4.4.6.3.1.cmml"><mo id="S6.SS2.18.p1.13.m13.4.4.6.3.2.1" stretchy="false" xref="S6.SS2.18.p1.13.m13.4.4.6.3.1.cmml">(</mo><mi id="S6.SS2.18.p1.13.m13.1.1" xref="S6.SS2.18.p1.13.m13.1.1.cmml">a</mi><mo id="S6.SS2.18.p1.13.m13.4.4.6.3.2.2" xref="S6.SS2.18.p1.13.m13.4.4.6.3.1.cmml">,</mo><mi id="S6.SS2.18.p1.13.m13.2.2" xref="S6.SS2.18.p1.13.m13.2.2.cmml">b</mi><mo id="S6.SS2.18.p1.13.m13.4.4.6.3.2.3" stretchy="false" xref="S6.SS2.18.p1.13.m13.4.4.6.3.1.cmml">)</mo></mrow></mrow><mo id="S6.SS2.18.p1.13.m13.4.4.7" xref="S6.SS2.18.p1.13.m13.4.4.7.cmml">≤</mo><mi id="S6.SS2.18.p1.13.m13.4.4.8" xref="S6.SS2.18.p1.13.m13.4.4.8.cmml">μ</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.18.p1.13.m13.4b"><apply id="S6.SS2.18.p1.13.m13.4.4.cmml" xref="S6.SS2.18.p1.13.m13.4.4"><and id="S6.SS2.18.p1.13.m13.4.4a.cmml" xref="S6.SS2.18.p1.13.m13.4.4"></and><apply id="S6.SS2.18.p1.13.m13.4.4b.cmml" xref="S6.SS2.18.p1.13.m13.4.4"><eq id="S6.SS2.18.p1.13.m13.4.4.4.cmml" xref="S6.SS2.18.p1.13.m13.4.4.4"></eq><apply id="S6.SS2.18.p1.13.m13.3.3.1.cmml" xref="S6.SS2.18.p1.13.m13.3.3.1"><times id="S6.SS2.18.p1.13.m13.3.3.1.2.cmml" xref="S6.SS2.18.p1.13.m13.3.3.1.2"></times><ci id="S6.SS2.18.p1.13.m13.3.3.1.3.cmml" xref="S6.SS2.18.p1.13.m13.3.3.1.3">𝜈</ci><apply id="S6.SS2.18.p1.13.m13.3.3.1.1.1.1.cmml" xref="S6.SS2.18.p1.13.m13.3.3.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.18.p1.13.m13.3.3.1.1.1.1.1.cmml" xref="S6.SS2.18.p1.13.m13.3.3.1.1.1">superscript</csymbol><ci id="S6.SS2.18.p1.13.m13.3.3.1.1.1.1.2.cmml" xref="S6.SS2.18.p1.13.m13.3.3.1.1.1.1.2">𝑐</ci><ci id="S6.SS2.18.p1.13.m13.3.3.1.1.1.1.3.cmml" xref="S6.SS2.18.p1.13.m13.3.3.1.1.1.1.3">′</ci></apply></apply><apply id="S6.SS2.18.p1.13.m13.4.4.2.cmml" xref="S6.SS2.18.p1.13.m13.4.4.2"><times id="S6.SS2.18.p1.13.m13.4.4.2.2.cmml" xref="S6.SS2.18.p1.13.m13.4.4.2.2"></times><ci id="S6.SS2.18.p1.13.m13.4.4.2.3.cmml" xref="S6.SS2.18.p1.13.m13.4.4.2.3">𝜈</ci><apply id="S6.SS2.18.p1.13.m13.4.4.2.1.1.1.cmml" xref="S6.SS2.18.p1.13.m13.4.4.2.1.1"><csymbol cd="ambiguous" id="S6.SS2.18.p1.13.m13.4.4.2.1.1.1.1.cmml" xref="S6.SS2.18.p1.13.m13.4.4.2.1.1">superscript</csymbol><ci id="S6.SS2.18.p1.13.m13.4.4.2.1.1.1.2.cmml" xref="S6.SS2.18.p1.13.m13.4.4.2.1.1.1.2">𝑐</ci><ci id="S6.SS2.18.p1.13.m13.4.4.2.1.1.1.3.cmml" xref="S6.SS2.18.p1.13.m13.4.4.2.1.1.1.3">′′</ci></apply></apply></apply><apply id="S6.SS2.18.p1.13.m13.4.4c.cmml" xref="S6.SS2.18.p1.13.m13.4.4"><eq id="S6.SS2.18.p1.13.m13.4.4.5.cmml" xref="S6.SS2.18.p1.13.m13.4.4.5"></eq><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.18.p1.13.m13.4.4.2.cmml" id="S6.SS2.18.p1.13.m13.4.4d.cmml" xref="S6.SS2.18.p1.13.m13.4.4"></share><apply id="S6.SS2.18.p1.13.m13.4.4.6.cmml" xref="S6.SS2.18.p1.13.m13.4.4.6"><times id="S6.SS2.18.p1.13.m13.4.4.6.1.cmml" xref="S6.SS2.18.p1.13.m13.4.4.6.1"></times><ci id="S6.SS2.18.p1.13.m13.4.4.6.2.cmml" xref="S6.SS2.18.p1.13.m13.4.4.6.2">𝜈</ci><interval closure="open" id="S6.SS2.18.p1.13.m13.4.4.6.3.1.cmml" xref="S6.SS2.18.p1.13.m13.4.4.6.3.2"><ci id="S6.SS2.18.p1.13.m13.1.1.cmml" xref="S6.SS2.18.p1.13.m13.1.1">𝑎</ci><ci id="S6.SS2.18.p1.13.m13.2.2.cmml" xref="S6.SS2.18.p1.13.m13.2.2">𝑏</ci></interval></apply></apply><apply id="S6.SS2.18.p1.13.m13.4.4e.cmml" xref="S6.SS2.18.p1.13.m13.4.4"><leq id="S6.SS2.18.p1.13.m13.4.4.7.cmml" xref="S6.SS2.18.p1.13.m13.4.4.7"></leq><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.18.p1.13.m13.4.4.6.cmml" id="S6.SS2.18.p1.13.m13.4.4f.cmml" xref="S6.SS2.18.p1.13.m13.4.4"></share><ci id="S6.SS2.18.p1.13.m13.4.4.8.cmml" xref="S6.SS2.18.p1.13.m13.4.4.8">𝜇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.18.p1.13.m13.4c">\nu(c^{\prime})=\nu(c^{\prime\prime})=\nu(a,b)\leq\mu</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.18.p1.13.m13.4d">italic_ν ( italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = italic_ν ( italic_c start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) = italic_ν ( italic_a , italic_b ) ≤ italic_μ</annotation></semantics></math>, thus <math alttext="c^{\prime},c^{\prime\prime}<\mu^{+}" class="ltx_Math" display="inline" id="S6.SS2.18.p1.14.m14.2"><semantics id="S6.SS2.18.p1.14.m14.2a"><mrow id="S6.SS2.18.p1.14.m14.2.2" xref="S6.SS2.18.p1.14.m14.2.2.cmml"><mrow id="S6.SS2.18.p1.14.m14.2.2.2.2" xref="S6.SS2.18.p1.14.m14.2.2.2.3.cmml"><msup id="S6.SS2.18.p1.14.m14.1.1.1.1.1" xref="S6.SS2.18.p1.14.m14.1.1.1.1.1.cmml"><mi id="S6.SS2.18.p1.14.m14.1.1.1.1.1.2" xref="S6.SS2.18.p1.14.m14.1.1.1.1.1.2.cmml">c</mi><mo id="S6.SS2.18.p1.14.m14.1.1.1.1.1.3" xref="S6.SS2.18.p1.14.m14.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S6.SS2.18.p1.14.m14.2.2.2.2.3" xref="S6.SS2.18.p1.14.m14.2.2.2.3.cmml">,</mo><msup id="S6.SS2.18.p1.14.m14.2.2.2.2.2" xref="S6.SS2.18.p1.14.m14.2.2.2.2.2.cmml"><mi id="S6.SS2.18.p1.14.m14.2.2.2.2.2.2" xref="S6.SS2.18.p1.14.m14.2.2.2.2.2.2.cmml">c</mi><mo id="S6.SS2.18.p1.14.m14.2.2.2.2.2.3" xref="S6.SS2.18.p1.14.m14.2.2.2.2.2.3.cmml">′′</mo></msup></mrow><mo id="S6.SS2.18.p1.14.m14.2.2.3" xref="S6.SS2.18.p1.14.m14.2.2.3.cmml"><</mo><msup id="S6.SS2.18.p1.14.m14.2.2.4" xref="S6.SS2.18.p1.14.m14.2.2.4.cmml"><mi id="S6.SS2.18.p1.14.m14.2.2.4.2" xref="S6.SS2.18.p1.14.m14.2.2.4.2.cmml">μ</mi><mo id="S6.SS2.18.p1.14.m14.2.2.4.3" xref="S6.SS2.18.p1.14.m14.2.2.4.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.18.p1.14.m14.2b"><apply id="S6.SS2.18.p1.14.m14.2.2.cmml" xref="S6.SS2.18.p1.14.m14.2.2"><lt id="S6.SS2.18.p1.14.m14.2.2.3.cmml" xref="S6.SS2.18.p1.14.m14.2.2.3"></lt><list id="S6.SS2.18.p1.14.m14.2.2.2.3.cmml" xref="S6.SS2.18.p1.14.m14.2.2.2.2"><apply id="S6.SS2.18.p1.14.m14.1.1.1.1.1.cmml" xref="S6.SS2.18.p1.14.m14.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.18.p1.14.m14.1.1.1.1.1.1.cmml" xref="S6.SS2.18.p1.14.m14.1.1.1.1.1">superscript</csymbol><ci id="S6.SS2.18.p1.14.m14.1.1.1.1.1.2.cmml" xref="S6.SS2.18.p1.14.m14.1.1.1.1.1.2">𝑐</ci><ci id="S6.SS2.18.p1.14.m14.1.1.1.1.1.3.cmml" xref="S6.SS2.18.p1.14.m14.1.1.1.1.1.3">′</ci></apply><apply id="S6.SS2.18.p1.14.m14.2.2.2.2.2.cmml" xref="S6.SS2.18.p1.14.m14.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.18.p1.14.m14.2.2.2.2.2.1.cmml" xref="S6.SS2.18.p1.14.m14.2.2.2.2.2">superscript</csymbol><ci id="S6.SS2.18.p1.14.m14.2.2.2.2.2.2.cmml" xref="S6.SS2.18.p1.14.m14.2.2.2.2.2.2">𝑐</ci><ci id="S6.SS2.18.p1.14.m14.2.2.2.2.2.3.cmml" xref="S6.SS2.18.p1.14.m14.2.2.2.2.2.3">′′</ci></apply></list><apply id="S6.SS2.18.p1.14.m14.2.2.4.cmml" xref="S6.SS2.18.p1.14.m14.2.2.4"><csymbol cd="ambiguous" id="S6.SS2.18.p1.14.m14.2.2.4.1.cmml" xref="S6.SS2.18.p1.14.m14.2.2.4">superscript</csymbol><ci id="S6.SS2.18.p1.14.m14.2.2.4.2.cmml" xref="S6.SS2.18.p1.14.m14.2.2.4.2">𝜇</ci><plus id="S6.SS2.18.p1.14.m14.2.2.4.3.cmml" xref="S6.SS2.18.p1.14.m14.2.2.4.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.18.p1.14.m14.2c">c^{\prime},c^{\prime\prime}<\mu^{+}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.18.p1.14.m14.2d">italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_c start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT < italic_μ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S6.SS2.19.p2"> <p class="ltx_p" id="S6.SS2.19.p2.12">Now by <math alttext="\aleph_{1}" class="ltx_Math" display="inline" id="S6.SS2.19.p2.1.m1.1"><semantics id="S6.SS2.19.p2.1.m1.1a"><msub id="S6.SS2.19.p2.1.m1.1.1" xref="S6.SS2.19.p2.1.m1.1.1.cmml"><mi id="S6.SS2.19.p2.1.m1.1.1.2" mathvariant="normal" xref="S6.SS2.19.p2.1.m1.1.1.2.cmml">ℵ</mi><mn id="S6.SS2.19.p2.1.m1.1.1.3" xref="S6.SS2.19.p2.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.19.p2.1.m1.1b"><apply id="S6.SS2.19.p2.1.m1.1.1.cmml" xref="S6.SS2.19.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S6.SS2.19.p2.1.m1.1.1.1.cmml" xref="S6.SS2.19.p2.1.m1.1.1">subscript</csymbol><ci id="S6.SS2.19.p2.1.m1.1.1.2.cmml" xref="S6.SS2.19.p2.1.m1.1.1.2">ℵ</ci><cn id="S6.SS2.19.p2.1.m1.1.1.3.cmml" type="integer" xref="S6.SS2.19.p2.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.19.p2.1.m1.1c">\aleph_{1}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.19.p2.1.m1.1d">roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-density of <math alttext="A" class="ltx_Math" display="inline" id="S6.SS2.19.p2.2.m2.1"><semantics id="S6.SS2.19.p2.2.m2.1a"><mi id="S6.SS2.19.p2.2.m2.1.1" xref="S6.SS2.19.p2.2.m2.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.19.p2.2.m2.1b"><ci id="S6.SS2.19.p2.2.m2.1.1.cmml" xref="S6.SS2.19.p2.2.m2.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.19.p2.2.m2.1c">A</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.19.p2.2.m2.1d">italic_A</annotation></semantics></math>, there must be an open interval <math alttext="I" class="ltx_Math" display="inline" id="S6.SS2.19.p2.3.m3.1"><semantics id="S6.SS2.19.p2.3.m3.1a"><mi id="S6.SS2.19.p2.3.m3.1.1" xref="S6.SS2.19.p2.3.m3.1.1.cmml">I</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.19.p2.3.m3.1b"><ci id="S6.SS2.19.p2.3.m3.1.1.cmml" xref="S6.SS2.19.p2.3.m3.1.1">𝐼</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.19.p2.3.m3.1c">I</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.19.p2.3.m3.1d">italic_I</annotation></semantics></math> of <math alttext="A" class="ltx_Math" display="inline" id="S6.SS2.19.p2.4.m4.1"><semantics id="S6.SS2.19.p2.4.m4.1a"><mi id="S6.SS2.19.p2.4.m4.1.1" xref="S6.SS2.19.p2.4.m4.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.19.p2.4.m4.1b"><ci id="S6.SS2.19.p2.4.m4.1.1.cmml" xref="S6.SS2.19.p2.4.m4.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.19.p2.4.m4.1c">A</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.19.p2.4.m4.1d">italic_A</annotation></semantics></math> such that <math alttext="I\subseteq[c^{\prime},c^{\prime\prime}]_{A}" class="ltx_Math" display="inline" id="S6.SS2.19.p2.5.m5.2"><semantics id="S6.SS2.19.p2.5.m5.2a"><mrow id="S6.SS2.19.p2.5.m5.2.2" xref="S6.SS2.19.p2.5.m5.2.2.cmml"><mi id="S6.SS2.19.p2.5.m5.2.2.4" xref="S6.SS2.19.p2.5.m5.2.2.4.cmml">I</mi><mo id="S6.SS2.19.p2.5.m5.2.2.3" xref="S6.SS2.19.p2.5.m5.2.2.3.cmml">⊆</mo><msub id="S6.SS2.19.p2.5.m5.2.2.2" xref="S6.SS2.19.p2.5.m5.2.2.2.cmml"><mrow id="S6.SS2.19.p2.5.m5.2.2.2.2.2" xref="S6.SS2.19.p2.5.m5.2.2.2.2.3.cmml"><mo id="S6.SS2.19.p2.5.m5.2.2.2.2.2.3" stretchy="false" xref="S6.SS2.19.p2.5.m5.2.2.2.2.3.cmml">[</mo><msup id="S6.SS2.19.p2.5.m5.1.1.1.1.1.1" xref="S6.SS2.19.p2.5.m5.1.1.1.1.1.1.cmml"><mi id="S6.SS2.19.p2.5.m5.1.1.1.1.1.1.2" xref="S6.SS2.19.p2.5.m5.1.1.1.1.1.1.2.cmml">c</mi><mo id="S6.SS2.19.p2.5.m5.1.1.1.1.1.1.3" xref="S6.SS2.19.p2.5.m5.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S6.SS2.19.p2.5.m5.2.2.2.2.2.4" xref="S6.SS2.19.p2.5.m5.2.2.2.2.3.cmml">,</mo><msup id="S6.SS2.19.p2.5.m5.2.2.2.2.2.2" xref="S6.SS2.19.p2.5.m5.2.2.2.2.2.2.cmml"><mi id="S6.SS2.19.p2.5.m5.2.2.2.2.2.2.2" xref="S6.SS2.19.p2.5.m5.2.2.2.2.2.2.2.cmml">c</mi><mo id="S6.SS2.19.p2.5.m5.2.2.2.2.2.2.3" xref="S6.SS2.19.p2.5.m5.2.2.2.2.2.2.3.cmml">′′</mo></msup><mo id="S6.SS2.19.p2.5.m5.2.2.2.2.2.5" stretchy="false" xref="S6.SS2.19.p2.5.m5.2.2.2.2.3.cmml">]</mo></mrow><mi id="S6.SS2.19.p2.5.m5.2.2.2.4" xref="S6.SS2.19.p2.5.m5.2.2.2.4.cmml">A</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.19.p2.5.m5.2b"><apply id="S6.SS2.19.p2.5.m5.2.2.cmml" xref="S6.SS2.19.p2.5.m5.2.2"><subset id="S6.SS2.19.p2.5.m5.2.2.3.cmml" xref="S6.SS2.19.p2.5.m5.2.2.3"></subset><ci id="S6.SS2.19.p2.5.m5.2.2.4.cmml" xref="S6.SS2.19.p2.5.m5.2.2.4">𝐼</ci><apply id="S6.SS2.19.p2.5.m5.2.2.2.cmml" xref="S6.SS2.19.p2.5.m5.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.19.p2.5.m5.2.2.2.3.cmml" xref="S6.SS2.19.p2.5.m5.2.2.2">subscript</csymbol><interval closure="closed" id="S6.SS2.19.p2.5.m5.2.2.2.2.3.cmml" xref="S6.SS2.19.p2.5.m5.2.2.2.2.2"><apply id="S6.SS2.19.p2.5.m5.1.1.1.1.1.1.cmml" xref="S6.SS2.19.p2.5.m5.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.19.p2.5.m5.1.1.1.1.1.1.1.cmml" xref="S6.SS2.19.p2.5.m5.1.1.1.1.1.1">superscript</csymbol><ci id="S6.SS2.19.p2.5.m5.1.1.1.1.1.1.2.cmml" xref="S6.SS2.19.p2.5.m5.1.1.1.1.1.1.2">𝑐</ci><ci id="S6.SS2.19.p2.5.m5.1.1.1.1.1.1.3.cmml" xref="S6.SS2.19.p2.5.m5.1.1.1.1.1.1.3">′</ci></apply><apply id="S6.SS2.19.p2.5.m5.2.2.2.2.2.2.cmml" xref="S6.SS2.19.p2.5.m5.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.19.p2.5.m5.2.2.2.2.2.2.1.cmml" xref="S6.SS2.19.p2.5.m5.2.2.2.2.2.2">superscript</csymbol><ci id="S6.SS2.19.p2.5.m5.2.2.2.2.2.2.2.cmml" xref="S6.SS2.19.p2.5.m5.2.2.2.2.2.2.2">𝑐</ci><ci id="S6.SS2.19.p2.5.m5.2.2.2.2.2.2.3.cmml" xref="S6.SS2.19.p2.5.m5.2.2.2.2.2.2.3">′′</ci></apply></interval><ci id="S6.SS2.19.p2.5.m5.2.2.2.4.cmml" xref="S6.SS2.19.p2.5.m5.2.2.2.4">𝐴</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.19.p2.5.m5.2c">I\subseteq[c^{\prime},c^{\prime\prime}]_{A}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.19.p2.5.m5.2d">italic_I ⊆ [ italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_c start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ] start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="I\cap\mu=\varnothing" class="ltx_Math" display="inline" id="S6.SS2.19.p2.6.m6.1"><semantics id="S6.SS2.19.p2.6.m6.1a"><mrow id="S6.SS2.19.p2.6.m6.1.1" xref="S6.SS2.19.p2.6.m6.1.1.cmml"><mrow id="S6.SS2.19.p2.6.m6.1.1.2" xref="S6.SS2.19.p2.6.m6.1.1.2.cmml"><mi id="S6.SS2.19.p2.6.m6.1.1.2.2" xref="S6.SS2.19.p2.6.m6.1.1.2.2.cmml">I</mi><mo id="S6.SS2.19.p2.6.m6.1.1.2.1" xref="S6.SS2.19.p2.6.m6.1.1.2.1.cmml">∩</mo><mi id="S6.SS2.19.p2.6.m6.1.1.2.3" xref="S6.SS2.19.p2.6.m6.1.1.2.3.cmml">μ</mi></mrow><mo id="S6.SS2.19.p2.6.m6.1.1.1" xref="S6.SS2.19.p2.6.m6.1.1.1.cmml">=</mo><mi id="S6.SS2.19.p2.6.m6.1.1.3" mathvariant="normal" xref="S6.SS2.19.p2.6.m6.1.1.3.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.19.p2.6.m6.1b"><apply id="S6.SS2.19.p2.6.m6.1.1.cmml" xref="S6.SS2.19.p2.6.m6.1.1"><eq id="S6.SS2.19.p2.6.m6.1.1.1.cmml" xref="S6.SS2.19.p2.6.m6.1.1.1"></eq><apply id="S6.SS2.19.p2.6.m6.1.1.2.cmml" xref="S6.SS2.19.p2.6.m6.1.1.2"><intersect id="S6.SS2.19.p2.6.m6.1.1.2.1.cmml" xref="S6.SS2.19.p2.6.m6.1.1.2.1"></intersect><ci id="S6.SS2.19.p2.6.m6.1.1.2.2.cmml" xref="S6.SS2.19.p2.6.m6.1.1.2.2">𝐼</ci><ci id="S6.SS2.19.p2.6.m6.1.1.2.3.cmml" xref="S6.SS2.19.p2.6.m6.1.1.2.3">𝜇</ci></apply><emptyset id="S6.SS2.19.p2.6.m6.1.1.3.cmml" xref="S6.SS2.19.p2.6.m6.1.1.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.19.p2.6.m6.1c">I\cap\mu=\varnothing</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.19.p2.6.m6.1d">italic_I ∩ italic_μ = ∅</annotation></semantics></math>, i.e., it is contained in a complementary interval of <math alttext="A\setminus\mu" class="ltx_Math" display="inline" id="S6.SS2.19.p2.7.m7.1"><semantics id="S6.SS2.19.p2.7.m7.1a"><mrow id="S6.SS2.19.p2.7.m7.1.1" xref="S6.SS2.19.p2.7.m7.1.1.cmml"><mi id="S6.SS2.19.p2.7.m7.1.1.2" xref="S6.SS2.19.p2.7.m7.1.1.2.cmml">A</mi><mo id="S6.SS2.19.p2.7.m7.1.1.1" xref="S6.SS2.19.p2.7.m7.1.1.1.cmml">∖</mo><mi id="S6.SS2.19.p2.7.m7.1.1.3" xref="S6.SS2.19.p2.7.m7.1.1.3.cmml">μ</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.19.p2.7.m7.1b"><apply id="S6.SS2.19.p2.7.m7.1.1.cmml" xref="S6.SS2.19.p2.7.m7.1.1"><setdiff id="S6.SS2.19.p2.7.m7.1.1.1.cmml" xref="S6.SS2.19.p2.7.m7.1.1.1"></setdiff><ci id="S6.SS2.19.p2.7.m7.1.1.2.cmml" xref="S6.SS2.19.p2.7.m7.1.1.2">𝐴</ci><ci id="S6.SS2.19.p2.7.m7.1.1.3.cmml" xref="S6.SS2.19.p2.7.m7.1.1.3">𝜇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.19.p2.7.m7.1c">A\setminus\mu</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.19.p2.7.m7.1d">italic_A ∖ italic_μ</annotation></semantics></math>, then any <math alttext="a^{\prime},b^{\prime}\in I" class="ltx_Math" display="inline" id="S6.SS2.19.p2.8.m8.2"><semantics id="S6.SS2.19.p2.8.m8.2a"><mrow id="S6.SS2.19.p2.8.m8.2.2" xref="S6.SS2.19.p2.8.m8.2.2.cmml"><mrow id="S6.SS2.19.p2.8.m8.2.2.2.2" xref="S6.SS2.19.p2.8.m8.2.2.2.3.cmml"><msup id="S6.SS2.19.p2.8.m8.1.1.1.1.1" xref="S6.SS2.19.p2.8.m8.1.1.1.1.1.cmml"><mi id="S6.SS2.19.p2.8.m8.1.1.1.1.1.2" xref="S6.SS2.19.p2.8.m8.1.1.1.1.1.2.cmml">a</mi><mo id="S6.SS2.19.p2.8.m8.1.1.1.1.1.3" xref="S6.SS2.19.p2.8.m8.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S6.SS2.19.p2.8.m8.2.2.2.2.3" xref="S6.SS2.19.p2.8.m8.2.2.2.3.cmml">,</mo><msup id="S6.SS2.19.p2.8.m8.2.2.2.2.2" xref="S6.SS2.19.p2.8.m8.2.2.2.2.2.cmml"><mi id="S6.SS2.19.p2.8.m8.2.2.2.2.2.2" xref="S6.SS2.19.p2.8.m8.2.2.2.2.2.2.cmml">b</mi><mo id="S6.SS2.19.p2.8.m8.2.2.2.2.2.3" xref="S6.SS2.19.p2.8.m8.2.2.2.2.2.3.cmml">′</mo></msup></mrow><mo id="S6.SS2.19.p2.8.m8.2.2.3" xref="S6.SS2.19.p2.8.m8.2.2.3.cmml">∈</mo><mi id="S6.SS2.19.p2.8.m8.2.2.4" xref="S6.SS2.19.p2.8.m8.2.2.4.cmml">I</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.19.p2.8.m8.2b"><apply id="S6.SS2.19.p2.8.m8.2.2.cmml" xref="S6.SS2.19.p2.8.m8.2.2"><in id="S6.SS2.19.p2.8.m8.2.2.3.cmml" xref="S6.SS2.19.p2.8.m8.2.2.3"></in><list id="S6.SS2.19.p2.8.m8.2.2.2.3.cmml" xref="S6.SS2.19.p2.8.m8.2.2.2.2"><apply id="S6.SS2.19.p2.8.m8.1.1.1.1.1.cmml" xref="S6.SS2.19.p2.8.m8.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.19.p2.8.m8.1.1.1.1.1.1.cmml" xref="S6.SS2.19.p2.8.m8.1.1.1.1.1">superscript</csymbol><ci id="S6.SS2.19.p2.8.m8.1.1.1.1.1.2.cmml" xref="S6.SS2.19.p2.8.m8.1.1.1.1.1.2">𝑎</ci><ci id="S6.SS2.19.p2.8.m8.1.1.1.1.1.3.cmml" xref="S6.SS2.19.p2.8.m8.1.1.1.1.1.3">′</ci></apply><apply id="S6.SS2.19.p2.8.m8.2.2.2.2.2.cmml" xref="S6.SS2.19.p2.8.m8.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.19.p2.8.m8.2.2.2.2.2.1.cmml" xref="S6.SS2.19.p2.8.m8.2.2.2.2.2">superscript</csymbol><ci id="S6.SS2.19.p2.8.m8.2.2.2.2.2.2.cmml" xref="S6.SS2.19.p2.8.m8.2.2.2.2.2.2">𝑏</ci><ci id="S6.SS2.19.p2.8.m8.2.2.2.2.2.3.cmml" xref="S6.SS2.19.p2.8.m8.2.2.2.2.2.3">′</ci></apply></list><ci id="S6.SS2.19.p2.8.m8.2.2.4.cmml" xref="S6.SS2.19.p2.8.m8.2.2.4">𝐼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.19.p2.8.m8.2c">a^{\prime},b^{\prime}\in I</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.19.p2.8.m8.2d">italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_I</annotation></semantics></math> satisfy (b), (c) and <math alttext="\mu\leq\nu(a^{\prime},b^{\prime})\leq\nu(a^{\prime}),\nu(b^{\prime})" class="ltx_Math" display="inline" id="S6.SS2.19.p2.9.m9.2"><semantics id="S6.SS2.19.p2.9.m9.2a"><mrow id="S6.SS2.19.p2.9.m9.2.2.2" xref="S6.SS2.19.p2.9.m9.2.2.3.cmml"><mrow id="S6.SS2.19.p2.9.m9.1.1.1.1" xref="S6.SS2.19.p2.9.m9.1.1.1.1.cmml"><mi id="S6.SS2.19.p2.9.m9.1.1.1.1.5" xref="S6.SS2.19.p2.9.m9.1.1.1.1.5.cmml">μ</mi><mo id="S6.SS2.19.p2.9.m9.1.1.1.1.6" xref="S6.SS2.19.p2.9.m9.1.1.1.1.6.cmml">≤</mo><mrow id="S6.SS2.19.p2.9.m9.1.1.1.1.2" xref="S6.SS2.19.p2.9.m9.1.1.1.1.2.cmml"><mi id="S6.SS2.19.p2.9.m9.1.1.1.1.2.4" xref="S6.SS2.19.p2.9.m9.1.1.1.1.2.4.cmml">ν</mi><mo id="S6.SS2.19.p2.9.m9.1.1.1.1.2.3" xref="S6.SS2.19.p2.9.m9.1.1.1.1.2.3.cmml">⁢</mo><mrow id="S6.SS2.19.p2.9.m9.1.1.1.1.2.2.2" xref="S6.SS2.19.p2.9.m9.1.1.1.1.2.2.3.cmml"><mo id="S6.SS2.19.p2.9.m9.1.1.1.1.2.2.2.3" stretchy="false" xref="S6.SS2.19.p2.9.m9.1.1.1.1.2.2.3.cmml">(</mo><msup id="S6.SS2.19.p2.9.m9.1.1.1.1.1.1.1.1" xref="S6.SS2.19.p2.9.m9.1.1.1.1.1.1.1.1.cmml"><mi id="S6.SS2.19.p2.9.m9.1.1.1.1.1.1.1.1.2" xref="S6.SS2.19.p2.9.m9.1.1.1.1.1.1.1.1.2.cmml">a</mi><mo id="S6.SS2.19.p2.9.m9.1.1.1.1.1.1.1.1.3" xref="S6.SS2.19.p2.9.m9.1.1.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S6.SS2.19.p2.9.m9.1.1.1.1.2.2.2.4" xref="S6.SS2.19.p2.9.m9.1.1.1.1.2.2.3.cmml">,</mo><msup id="S6.SS2.19.p2.9.m9.1.1.1.1.2.2.2.2" xref="S6.SS2.19.p2.9.m9.1.1.1.1.2.2.2.2.cmml"><mi id="S6.SS2.19.p2.9.m9.1.1.1.1.2.2.2.2.2" xref="S6.SS2.19.p2.9.m9.1.1.1.1.2.2.2.2.2.cmml">b</mi><mo id="S6.SS2.19.p2.9.m9.1.1.1.1.2.2.2.2.3" xref="S6.SS2.19.p2.9.m9.1.1.1.1.2.2.2.2.3.cmml">′</mo></msup><mo id="S6.SS2.19.p2.9.m9.1.1.1.1.2.2.2.5" stretchy="false" xref="S6.SS2.19.p2.9.m9.1.1.1.1.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.SS2.19.p2.9.m9.1.1.1.1.7" xref="S6.SS2.19.p2.9.m9.1.1.1.1.7.cmml">≤</mo><mrow id="S6.SS2.19.p2.9.m9.1.1.1.1.3" xref="S6.SS2.19.p2.9.m9.1.1.1.1.3.cmml"><mi id="S6.SS2.19.p2.9.m9.1.1.1.1.3.3" xref="S6.SS2.19.p2.9.m9.1.1.1.1.3.3.cmml">ν</mi><mo id="S6.SS2.19.p2.9.m9.1.1.1.1.3.2" xref="S6.SS2.19.p2.9.m9.1.1.1.1.3.2.cmml">⁢</mo><mrow id="S6.SS2.19.p2.9.m9.1.1.1.1.3.1.1" xref="S6.SS2.19.p2.9.m9.1.1.1.1.3.1.1.1.cmml"><mo id="S6.SS2.19.p2.9.m9.1.1.1.1.3.1.1.2" stretchy="false" xref="S6.SS2.19.p2.9.m9.1.1.1.1.3.1.1.1.cmml">(</mo><msup id="S6.SS2.19.p2.9.m9.1.1.1.1.3.1.1.1" xref="S6.SS2.19.p2.9.m9.1.1.1.1.3.1.1.1.cmml"><mi id="S6.SS2.19.p2.9.m9.1.1.1.1.3.1.1.1.2" xref="S6.SS2.19.p2.9.m9.1.1.1.1.3.1.1.1.2.cmml">a</mi><mo id="S6.SS2.19.p2.9.m9.1.1.1.1.3.1.1.1.3" xref="S6.SS2.19.p2.9.m9.1.1.1.1.3.1.1.1.3.cmml">′</mo></msup><mo id="S6.SS2.19.p2.9.m9.1.1.1.1.3.1.1.3" stretchy="false" xref="S6.SS2.19.p2.9.m9.1.1.1.1.3.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S6.SS2.19.p2.9.m9.2.2.2.3" xref="S6.SS2.19.p2.9.m9.2.2.3a.cmml">,</mo><mrow id="S6.SS2.19.p2.9.m9.2.2.2.2" xref="S6.SS2.19.p2.9.m9.2.2.2.2.cmml"><mi id="S6.SS2.19.p2.9.m9.2.2.2.2.3" xref="S6.SS2.19.p2.9.m9.2.2.2.2.3.cmml">ν</mi><mo id="S6.SS2.19.p2.9.m9.2.2.2.2.2" xref="S6.SS2.19.p2.9.m9.2.2.2.2.2.cmml">⁢</mo><mrow id="S6.SS2.19.p2.9.m9.2.2.2.2.1.1" xref="S6.SS2.19.p2.9.m9.2.2.2.2.1.1.1.cmml"><mo id="S6.SS2.19.p2.9.m9.2.2.2.2.1.1.2" stretchy="false" xref="S6.SS2.19.p2.9.m9.2.2.2.2.1.1.1.cmml">(</mo><msup id="S6.SS2.19.p2.9.m9.2.2.2.2.1.1.1" xref="S6.SS2.19.p2.9.m9.2.2.2.2.1.1.1.cmml"><mi id="S6.SS2.19.p2.9.m9.2.2.2.2.1.1.1.2" xref="S6.SS2.19.p2.9.m9.2.2.2.2.1.1.1.2.cmml">b</mi><mo id="S6.SS2.19.p2.9.m9.2.2.2.2.1.1.1.3" xref="S6.SS2.19.p2.9.m9.2.2.2.2.1.1.1.3.cmml">′</mo></msup><mo id="S6.SS2.19.p2.9.m9.2.2.2.2.1.1.3" stretchy="false" xref="S6.SS2.19.p2.9.m9.2.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.19.p2.9.m9.2b"><apply id="S6.SS2.19.p2.9.m9.2.2.3.cmml" xref="S6.SS2.19.p2.9.m9.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.19.p2.9.m9.2.2.3a.cmml" xref="S6.SS2.19.p2.9.m9.2.2.2.3">formulae-sequence</csymbol><apply id="S6.SS2.19.p2.9.m9.1.1.1.1.cmml" xref="S6.SS2.19.p2.9.m9.1.1.1.1"><and id="S6.SS2.19.p2.9.m9.1.1.1.1a.cmml" xref="S6.SS2.19.p2.9.m9.1.1.1.1"></and><apply id="S6.SS2.19.p2.9.m9.1.1.1.1b.cmml" xref="S6.SS2.19.p2.9.m9.1.1.1.1"><leq id="S6.SS2.19.p2.9.m9.1.1.1.1.6.cmml" xref="S6.SS2.19.p2.9.m9.1.1.1.1.6"></leq><ci id="S6.SS2.19.p2.9.m9.1.1.1.1.5.cmml" xref="S6.SS2.19.p2.9.m9.1.1.1.1.5">𝜇</ci><apply id="S6.SS2.19.p2.9.m9.1.1.1.1.2.cmml" xref="S6.SS2.19.p2.9.m9.1.1.1.1.2"><times id="S6.SS2.19.p2.9.m9.1.1.1.1.2.3.cmml" xref="S6.SS2.19.p2.9.m9.1.1.1.1.2.3"></times><ci id="S6.SS2.19.p2.9.m9.1.1.1.1.2.4.cmml" xref="S6.SS2.19.p2.9.m9.1.1.1.1.2.4">𝜈</ci><interval closure="open" id="S6.SS2.19.p2.9.m9.1.1.1.1.2.2.3.cmml" xref="S6.SS2.19.p2.9.m9.1.1.1.1.2.2.2"><apply id="S6.SS2.19.p2.9.m9.1.1.1.1.1.1.1.1.cmml" xref="S6.SS2.19.p2.9.m9.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.19.p2.9.m9.1.1.1.1.1.1.1.1.1.cmml" xref="S6.SS2.19.p2.9.m9.1.1.1.1.1.1.1.1">superscript</csymbol><ci id="S6.SS2.19.p2.9.m9.1.1.1.1.1.1.1.1.2.cmml" xref="S6.SS2.19.p2.9.m9.1.1.1.1.1.1.1.1.2">𝑎</ci><ci id="S6.SS2.19.p2.9.m9.1.1.1.1.1.1.1.1.3.cmml" xref="S6.SS2.19.p2.9.m9.1.1.1.1.1.1.1.1.3">′</ci></apply><apply id="S6.SS2.19.p2.9.m9.1.1.1.1.2.2.2.2.cmml" xref="S6.SS2.19.p2.9.m9.1.1.1.1.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.19.p2.9.m9.1.1.1.1.2.2.2.2.1.cmml" xref="S6.SS2.19.p2.9.m9.1.1.1.1.2.2.2.2">superscript</csymbol><ci id="S6.SS2.19.p2.9.m9.1.1.1.1.2.2.2.2.2.cmml" xref="S6.SS2.19.p2.9.m9.1.1.1.1.2.2.2.2.2">𝑏</ci><ci id="S6.SS2.19.p2.9.m9.1.1.1.1.2.2.2.2.3.cmml" xref="S6.SS2.19.p2.9.m9.1.1.1.1.2.2.2.2.3">′</ci></apply></interval></apply></apply><apply id="S6.SS2.19.p2.9.m9.1.1.1.1c.cmml" xref="S6.SS2.19.p2.9.m9.1.1.1.1"><leq id="S6.SS2.19.p2.9.m9.1.1.1.1.7.cmml" xref="S6.SS2.19.p2.9.m9.1.1.1.1.7"></leq><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.19.p2.9.m9.1.1.1.1.2.cmml" id="S6.SS2.19.p2.9.m9.1.1.1.1d.cmml" xref="S6.SS2.19.p2.9.m9.1.1.1.1"></share><apply id="S6.SS2.19.p2.9.m9.1.1.1.1.3.cmml" xref="S6.SS2.19.p2.9.m9.1.1.1.1.3"><times id="S6.SS2.19.p2.9.m9.1.1.1.1.3.2.cmml" xref="S6.SS2.19.p2.9.m9.1.1.1.1.3.2"></times><ci id="S6.SS2.19.p2.9.m9.1.1.1.1.3.3.cmml" xref="S6.SS2.19.p2.9.m9.1.1.1.1.3.3">𝜈</ci><apply id="S6.SS2.19.p2.9.m9.1.1.1.1.3.1.1.1.cmml" xref="S6.SS2.19.p2.9.m9.1.1.1.1.3.1.1"><csymbol cd="ambiguous" id="S6.SS2.19.p2.9.m9.1.1.1.1.3.1.1.1.1.cmml" xref="S6.SS2.19.p2.9.m9.1.1.1.1.3.1.1">superscript</csymbol><ci id="S6.SS2.19.p2.9.m9.1.1.1.1.3.1.1.1.2.cmml" xref="S6.SS2.19.p2.9.m9.1.1.1.1.3.1.1.1.2">𝑎</ci><ci id="S6.SS2.19.p2.9.m9.1.1.1.1.3.1.1.1.3.cmml" xref="S6.SS2.19.p2.9.m9.1.1.1.1.3.1.1.1.3">′</ci></apply></apply></apply></apply><apply id="S6.SS2.19.p2.9.m9.2.2.2.2.cmml" xref="S6.SS2.19.p2.9.m9.2.2.2.2"><times id="S6.SS2.19.p2.9.m9.2.2.2.2.2.cmml" xref="S6.SS2.19.p2.9.m9.2.2.2.2.2"></times><ci id="S6.SS2.19.p2.9.m9.2.2.2.2.3.cmml" xref="S6.SS2.19.p2.9.m9.2.2.2.2.3">𝜈</ci><apply id="S6.SS2.19.p2.9.m9.2.2.2.2.1.1.1.cmml" xref="S6.SS2.19.p2.9.m9.2.2.2.2.1.1"><csymbol cd="ambiguous" id="S6.SS2.19.p2.9.m9.2.2.2.2.1.1.1.1.cmml" xref="S6.SS2.19.p2.9.m9.2.2.2.2.1.1">superscript</csymbol><ci id="S6.SS2.19.p2.9.m9.2.2.2.2.1.1.1.2.cmml" xref="S6.SS2.19.p2.9.m9.2.2.2.2.1.1.1.2">𝑏</ci><ci id="S6.SS2.19.p2.9.m9.2.2.2.2.1.1.1.3.cmml" xref="S6.SS2.19.p2.9.m9.2.2.2.2.1.1.1.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.19.p2.9.m9.2c">\mu\leq\nu(a^{\prime},b^{\prime})\leq\nu(a^{\prime}),\nu(b^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.19.p2.9.m9.2d">italic_μ ≤ italic_ν ( italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ≤ italic_ν ( italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) , italic_ν ( italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math>. Applying the above inside a model witnessing the elementarity of <math alttext="\mu^{+}" class="ltx_Math" display="inline" id="S6.SS2.19.p2.10.m10.1"><semantics id="S6.SS2.19.p2.10.m10.1a"><msup id="S6.SS2.19.p2.10.m10.1.1" xref="S6.SS2.19.p2.10.m10.1.1.cmml"><mi id="S6.SS2.19.p2.10.m10.1.1.2" xref="S6.SS2.19.p2.10.m10.1.1.2.cmml">μ</mi><mo id="S6.SS2.19.p2.10.m10.1.1.3" xref="S6.SS2.19.p2.10.m10.1.1.3.cmml">+</mo></msup><annotation-xml encoding="MathML-Content" id="S6.SS2.19.p2.10.m10.1b"><apply id="S6.SS2.19.p2.10.m10.1.1.cmml" xref="S6.SS2.19.p2.10.m10.1.1"><csymbol cd="ambiguous" id="S6.SS2.19.p2.10.m10.1.1.1.cmml" xref="S6.SS2.19.p2.10.m10.1.1">superscript</csymbol><ci id="S6.SS2.19.p2.10.m10.1.1.2.cmml" xref="S6.SS2.19.p2.10.m10.1.1.2">𝜇</ci><plus id="S6.SS2.19.p2.10.m10.1.1.3.cmml" xref="S6.SS2.19.p2.10.m10.1.1.3"></plus></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.19.p2.10.m10.1c">\mu^{+}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.19.p2.10.m10.1d">italic_μ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math>, and using the fact that <math alttext="c^{\prime},c^{\prime\prime}<\mu^{+}" class="ltx_Math" display="inline" id="S6.SS2.19.p2.11.m11.2"><semantics id="S6.SS2.19.p2.11.m11.2a"><mrow id="S6.SS2.19.p2.11.m11.2.2" xref="S6.SS2.19.p2.11.m11.2.2.cmml"><mrow id="S6.SS2.19.p2.11.m11.2.2.2.2" xref="S6.SS2.19.p2.11.m11.2.2.2.3.cmml"><msup id="S6.SS2.19.p2.11.m11.1.1.1.1.1" xref="S6.SS2.19.p2.11.m11.1.1.1.1.1.cmml"><mi id="S6.SS2.19.p2.11.m11.1.1.1.1.1.2" xref="S6.SS2.19.p2.11.m11.1.1.1.1.1.2.cmml">c</mi><mo id="S6.SS2.19.p2.11.m11.1.1.1.1.1.3" xref="S6.SS2.19.p2.11.m11.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S6.SS2.19.p2.11.m11.2.2.2.2.3" xref="S6.SS2.19.p2.11.m11.2.2.2.3.cmml">,</mo><msup id="S6.SS2.19.p2.11.m11.2.2.2.2.2" xref="S6.SS2.19.p2.11.m11.2.2.2.2.2.cmml"><mi id="S6.SS2.19.p2.11.m11.2.2.2.2.2.2" xref="S6.SS2.19.p2.11.m11.2.2.2.2.2.2.cmml">c</mi><mo id="S6.SS2.19.p2.11.m11.2.2.2.2.2.3" xref="S6.SS2.19.p2.11.m11.2.2.2.2.2.3.cmml">′′</mo></msup></mrow><mo id="S6.SS2.19.p2.11.m11.2.2.3" xref="S6.SS2.19.p2.11.m11.2.2.3.cmml"><</mo><msup id="S6.SS2.19.p2.11.m11.2.2.4" xref="S6.SS2.19.p2.11.m11.2.2.4.cmml"><mi id="S6.SS2.19.p2.11.m11.2.2.4.2" xref="S6.SS2.19.p2.11.m11.2.2.4.2.cmml">μ</mi><mo id="S6.SS2.19.p2.11.m11.2.2.4.3" xref="S6.SS2.19.p2.11.m11.2.2.4.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.19.p2.11.m11.2b"><apply id="S6.SS2.19.p2.11.m11.2.2.cmml" xref="S6.SS2.19.p2.11.m11.2.2"><lt id="S6.SS2.19.p2.11.m11.2.2.3.cmml" xref="S6.SS2.19.p2.11.m11.2.2.3"></lt><list id="S6.SS2.19.p2.11.m11.2.2.2.3.cmml" xref="S6.SS2.19.p2.11.m11.2.2.2.2"><apply id="S6.SS2.19.p2.11.m11.1.1.1.1.1.cmml" xref="S6.SS2.19.p2.11.m11.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.19.p2.11.m11.1.1.1.1.1.1.cmml" xref="S6.SS2.19.p2.11.m11.1.1.1.1.1">superscript</csymbol><ci id="S6.SS2.19.p2.11.m11.1.1.1.1.1.2.cmml" xref="S6.SS2.19.p2.11.m11.1.1.1.1.1.2">𝑐</ci><ci id="S6.SS2.19.p2.11.m11.1.1.1.1.1.3.cmml" xref="S6.SS2.19.p2.11.m11.1.1.1.1.1.3">′</ci></apply><apply id="S6.SS2.19.p2.11.m11.2.2.2.2.2.cmml" xref="S6.SS2.19.p2.11.m11.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.19.p2.11.m11.2.2.2.2.2.1.cmml" xref="S6.SS2.19.p2.11.m11.2.2.2.2.2">superscript</csymbol><ci id="S6.SS2.19.p2.11.m11.2.2.2.2.2.2.cmml" xref="S6.SS2.19.p2.11.m11.2.2.2.2.2.2">𝑐</ci><ci id="S6.SS2.19.p2.11.m11.2.2.2.2.2.3.cmml" xref="S6.SS2.19.p2.11.m11.2.2.2.2.2.3">′′</ci></apply></list><apply id="S6.SS2.19.p2.11.m11.2.2.4.cmml" xref="S6.SS2.19.p2.11.m11.2.2.4"><csymbol cd="ambiguous" id="S6.SS2.19.p2.11.m11.2.2.4.1.cmml" xref="S6.SS2.19.p2.11.m11.2.2.4">superscript</csymbol><ci id="S6.SS2.19.p2.11.m11.2.2.4.2.cmml" xref="S6.SS2.19.p2.11.m11.2.2.4.2">𝜇</ci><plus id="S6.SS2.19.p2.11.m11.2.2.4.3.cmml" xref="S6.SS2.19.p2.11.m11.2.2.4.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.19.p2.11.m11.2c">c^{\prime},c^{\prime\prime}<\mu^{+}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.19.p2.11.m11.2d">italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_c start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT < italic_μ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math>, one also achieves that <math alttext="\nu(a^{\prime}),\nu(b^{\prime})<\mu^{+}" class="ltx_Math" display="inline" id="S6.SS2.19.p2.12.m12.2"><semantics id="S6.SS2.19.p2.12.m12.2a"><mrow id="S6.SS2.19.p2.12.m12.2.2" xref="S6.SS2.19.p2.12.m12.2.2.cmml"><mrow id="S6.SS2.19.p2.12.m12.2.2.2.2" xref="S6.SS2.19.p2.12.m12.2.2.2.3.cmml"><mrow id="S6.SS2.19.p2.12.m12.1.1.1.1.1" xref="S6.SS2.19.p2.12.m12.1.1.1.1.1.cmml"><mi id="S6.SS2.19.p2.12.m12.1.1.1.1.1.3" xref="S6.SS2.19.p2.12.m12.1.1.1.1.1.3.cmml">ν</mi><mo id="S6.SS2.19.p2.12.m12.1.1.1.1.1.2" xref="S6.SS2.19.p2.12.m12.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S6.SS2.19.p2.12.m12.1.1.1.1.1.1.1" xref="S6.SS2.19.p2.12.m12.1.1.1.1.1.1.1.1.cmml"><mo id="S6.SS2.19.p2.12.m12.1.1.1.1.1.1.1.2" stretchy="false" xref="S6.SS2.19.p2.12.m12.1.1.1.1.1.1.1.1.cmml">(</mo><msup id="S6.SS2.19.p2.12.m12.1.1.1.1.1.1.1.1" xref="S6.SS2.19.p2.12.m12.1.1.1.1.1.1.1.1.cmml"><mi id="S6.SS2.19.p2.12.m12.1.1.1.1.1.1.1.1.2" xref="S6.SS2.19.p2.12.m12.1.1.1.1.1.1.1.1.2.cmml">a</mi><mo id="S6.SS2.19.p2.12.m12.1.1.1.1.1.1.1.1.3" xref="S6.SS2.19.p2.12.m12.1.1.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S6.SS2.19.p2.12.m12.1.1.1.1.1.1.1.3" stretchy="false" xref="S6.SS2.19.p2.12.m12.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.SS2.19.p2.12.m12.2.2.2.2.3" xref="S6.SS2.19.p2.12.m12.2.2.2.3.cmml">,</mo><mrow id="S6.SS2.19.p2.12.m12.2.2.2.2.2" xref="S6.SS2.19.p2.12.m12.2.2.2.2.2.cmml"><mi id="S6.SS2.19.p2.12.m12.2.2.2.2.2.3" xref="S6.SS2.19.p2.12.m12.2.2.2.2.2.3.cmml">ν</mi><mo id="S6.SS2.19.p2.12.m12.2.2.2.2.2.2" xref="S6.SS2.19.p2.12.m12.2.2.2.2.2.2.cmml">⁢</mo><mrow id="S6.SS2.19.p2.12.m12.2.2.2.2.2.1.1" xref="S6.SS2.19.p2.12.m12.2.2.2.2.2.1.1.1.cmml"><mo id="S6.SS2.19.p2.12.m12.2.2.2.2.2.1.1.2" stretchy="false" xref="S6.SS2.19.p2.12.m12.2.2.2.2.2.1.1.1.cmml">(</mo><msup id="S6.SS2.19.p2.12.m12.2.2.2.2.2.1.1.1" xref="S6.SS2.19.p2.12.m12.2.2.2.2.2.1.1.1.cmml"><mi id="S6.SS2.19.p2.12.m12.2.2.2.2.2.1.1.1.2" xref="S6.SS2.19.p2.12.m12.2.2.2.2.2.1.1.1.2.cmml">b</mi><mo id="S6.SS2.19.p2.12.m12.2.2.2.2.2.1.1.1.3" xref="S6.SS2.19.p2.12.m12.2.2.2.2.2.1.1.1.3.cmml">′</mo></msup><mo id="S6.SS2.19.p2.12.m12.2.2.2.2.2.1.1.3" stretchy="false" xref="S6.SS2.19.p2.12.m12.2.2.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S6.SS2.19.p2.12.m12.2.2.3" xref="S6.SS2.19.p2.12.m12.2.2.3.cmml"><</mo><msup id="S6.SS2.19.p2.12.m12.2.2.4" xref="S6.SS2.19.p2.12.m12.2.2.4.cmml"><mi id="S6.SS2.19.p2.12.m12.2.2.4.2" xref="S6.SS2.19.p2.12.m12.2.2.4.2.cmml">μ</mi><mo id="S6.SS2.19.p2.12.m12.2.2.4.3" xref="S6.SS2.19.p2.12.m12.2.2.4.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.19.p2.12.m12.2b"><apply id="S6.SS2.19.p2.12.m12.2.2.cmml" xref="S6.SS2.19.p2.12.m12.2.2"><lt id="S6.SS2.19.p2.12.m12.2.2.3.cmml" xref="S6.SS2.19.p2.12.m12.2.2.3"></lt><list id="S6.SS2.19.p2.12.m12.2.2.2.3.cmml" xref="S6.SS2.19.p2.12.m12.2.2.2.2"><apply id="S6.SS2.19.p2.12.m12.1.1.1.1.1.cmml" xref="S6.SS2.19.p2.12.m12.1.1.1.1.1"><times id="S6.SS2.19.p2.12.m12.1.1.1.1.1.2.cmml" xref="S6.SS2.19.p2.12.m12.1.1.1.1.1.2"></times><ci id="S6.SS2.19.p2.12.m12.1.1.1.1.1.3.cmml" xref="S6.SS2.19.p2.12.m12.1.1.1.1.1.3">𝜈</ci><apply id="S6.SS2.19.p2.12.m12.1.1.1.1.1.1.1.1.cmml" xref="S6.SS2.19.p2.12.m12.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.19.p2.12.m12.1.1.1.1.1.1.1.1.1.cmml" xref="S6.SS2.19.p2.12.m12.1.1.1.1.1.1.1">superscript</csymbol><ci id="S6.SS2.19.p2.12.m12.1.1.1.1.1.1.1.1.2.cmml" xref="S6.SS2.19.p2.12.m12.1.1.1.1.1.1.1.1.2">𝑎</ci><ci id="S6.SS2.19.p2.12.m12.1.1.1.1.1.1.1.1.3.cmml" xref="S6.SS2.19.p2.12.m12.1.1.1.1.1.1.1.1.3">′</ci></apply></apply><apply id="S6.SS2.19.p2.12.m12.2.2.2.2.2.cmml" xref="S6.SS2.19.p2.12.m12.2.2.2.2.2"><times id="S6.SS2.19.p2.12.m12.2.2.2.2.2.2.cmml" xref="S6.SS2.19.p2.12.m12.2.2.2.2.2.2"></times><ci id="S6.SS2.19.p2.12.m12.2.2.2.2.2.3.cmml" xref="S6.SS2.19.p2.12.m12.2.2.2.2.2.3">𝜈</ci><apply id="S6.SS2.19.p2.12.m12.2.2.2.2.2.1.1.1.cmml" xref="S6.SS2.19.p2.12.m12.2.2.2.2.2.1.1"><csymbol cd="ambiguous" id="S6.SS2.19.p2.12.m12.2.2.2.2.2.1.1.1.1.cmml" xref="S6.SS2.19.p2.12.m12.2.2.2.2.2.1.1">superscript</csymbol><ci id="S6.SS2.19.p2.12.m12.2.2.2.2.2.1.1.1.2.cmml" xref="S6.SS2.19.p2.12.m12.2.2.2.2.2.1.1.1.2">𝑏</ci><ci id="S6.SS2.19.p2.12.m12.2.2.2.2.2.1.1.1.3.cmml" xref="S6.SS2.19.p2.12.m12.2.2.2.2.2.1.1.1.3">′</ci></apply></apply></list><apply id="S6.SS2.19.p2.12.m12.2.2.4.cmml" xref="S6.SS2.19.p2.12.m12.2.2.4"><csymbol cd="ambiguous" id="S6.SS2.19.p2.12.m12.2.2.4.1.cmml" xref="S6.SS2.19.p2.12.m12.2.2.4">superscript</csymbol><ci id="S6.SS2.19.p2.12.m12.2.2.4.2.cmml" xref="S6.SS2.19.p2.12.m12.2.2.4.2">𝜇</ci><plus id="S6.SS2.19.p2.12.m12.2.2.4.3.cmml" xref="S6.SS2.19.p2.12.m12.2.2.4.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.19.p2.12.m12.2c">\nu(a^{\prime}),\nu(b^{\prime})<\mu^{+}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.19.p2.12.m12.2d">italic_ν ( italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) , italic_ν ( italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) < italic_μ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math>. ∎</p> </div> </div> <div class="ltx_para" id="S6.SS2.p11"> <p class="ltx_p" id="S6.SS2.p11.1">We now proceed to prove the relevant densities. As explained before <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.Thmtheorem10" title="Lemma 6.10. ‣ 6.2. Forcing epimorphisms ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">6.10</span></a>, this together with the ccc finishes the proof of <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.Thmtheorem2" title="Theorem 6.2. ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">6.2</span></a>. We will prove the first of these lemmas giving all the necessary details and tricks, in the second we give less details but the tricks are the same.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S6.Thmtheorem18"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem18.1.1.1">Lemma 6.18</span></span><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem18.2.2">.</span> </h6> <div class="ltx_para" id="S6.Thmtheorem18.p1"> <p class="ltx_p" id="S6.Thmtheorem18.p1.2">For every <math alttext="x\in X" class="ltx_Math" display="inline" id="S6.Thmtheorem18.p1.1.m1.1"><semantics id="S6.Thmtheorem18.p1.1.m1.1a"><mrow id="S6.Thmtheorem18.p1.1.m1.1.1" xref="S6.Thmtheorem18.p1.1.m1.1.1.cmml"><mi id="S6.Thmtheorem18.p1.1.m1.1.1.2" xref="S6.Thmtheorem18.p1.1.m1.1.1.2.cmml">x</mi><mo id="S6.Thmtheorem18.p1.1.m1.1.1.1" xref="S6.Thmtheorem18.p1.1.m1.1.1.1.cmml">∈</mo><mi id="S6.Thmtheorem18.p1.1.m1.1.1.3" xref="S6.Thmtheorem18.p1.1.m1.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem18.p1.1.m1.1b"><apply id="S6.Thmtheorem18.p1.1.m1.1.1.cmml" xref="S6.Thmtheorem18.p1.1.m1.1.1"><in id="S6.Thmtheorem18.p1.1.m1.1.1.1.cmml" xref="S6.Thmtheorem18.p1.1.m1.1.1.1"></in><ci id="S6.Thmtheorem18.p1.1.m1.1.1.2.cmml" xref="S6.Thmtheorem18.p1.1.m1.1.1.2">𝑥</ci><ci id="S6.Thmtheorem18.p1.1.m1.1.1.3.cmml" xref="S6.Thmtheorem18.p1.1.m1.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem18.p1.1.m1.1c">x\in X</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem18.p1.1.m1.1d">italic_x ∈ italic_X</annotation></semantics></math>, <math alttext="\{p\in P_{E}:x\in\operatorname{ran}(f_{p})\}" class="ltx_Math" display="inline" id="S6.Thmtheorem18.p1.2.m2.3"><semantics id="S6.Thmtheorem18.p1.2.m2.3a"><mrow id="S6.Thmtheorem18.p1.2.m2.3.3.2" xref="S6.Thmtheorem18.p1.2.m2.3.3.3.cmml"><mo id="S6.Thmtheorem18.p1.2.m2.3.3.2.3" stretchy="false" xref="S6.Thmtheorem18.p1.2.m2.3.3.3.1.cmml">{</mo><mrow id="S6.Thmtheorem18.p1.2.m2.2.2.1.1" xref="S6.Thmtheorem18.p1.2.m2.2.2.1.1.cmml"><mi id="S6.Thmtheorem18.p1.2.m2.2.2.1.1.2" xref="S6.Thmtheorem18.p1.2.m2.2.2.1.1.2.cmml">p</mi><mo id="S6.Thmtheorem18.p1.2.m2.2.2.1.1.1" xref="S6.Thmtheorem18.p1.2.m2.2.2.1.1.1.cmml">∈</mo><msub id="S6.Thmtheorem18.p1.2.m2.2.2.1.1.3" xref="S6.Thmtheorem18.p1.2.m2.2.2.1.1.3.cmml"><mi id="S6.Thmtheorem18.p1.2.m2.2.2.1.1.3.2" xref="S6.Thmtheorem18.p1.2.m2.2.2.1.1.3.2.cmml">P</mi><mi id="S6.Thmtheorem18.p1.2.m2.2.2.1.1.3.3" xref="S6.Thmtheorem18.p1.2.m2.2.2.1.1.3.3.cmml">E</mi></msub></mrow><mo id="S6.Thmtheorem18.p1.2.m2.3.3.2.4" lspace="0.278em" rspace="0.278em" xref="S6.Thmtheorem18.p1.2.m2.3.3.3.1.cmml">:</mo><mrow id="S6.Thmtheorem18.p1.2.m2.3.3.2.2" xref="S6.Thmtheorem18.p1.2.m2.3.3.2.2.cmml"><mi id="S6.Thmtheorem18.p1.2.m2.3.3.2.2.3" xref="S6.Thmtheorem18.p1.2.m2.3.3.2.2.3.cmml">x</mi><mo id="S6.Thmtheorem18.p1.2.m2.3.3.2.2.2" xref="S6.Thmtheorem18.p1.2.m2.3.3.2.2.2.cmml">∈</mo><mrow id="S6.Thmtheorem18.p1.2.m2.3.3.2.2.1.1" xref="S6.Thmtheorem18.p1.2.m2.3.3.2.2.1.2.cmml"><mi id="S6.Thmtheorem18.p1.2.m2.1.1" xref="S6.Thmtheorem18.p1.2.m2.1.1.cmml">ran</mi><mo id="S6.Thmtheorem18.p1.2.m2.3.3.2.2.1.1a" xref="S6.Thmtheorem18.p1.2.m2.3.3.2.2.1.2.cmml">⁡</mo><mrow id="S6.Thmtheorem18.p1.2.m2.3.3.2.2.1.1.1" xref="S6.Thmtheorem18.p1.2.m2.3.3.2.2.1.2.cmml"><mo id="S6.Thmtheorem18.p1.2.m2.3.3.2.2.1.1.1.2" stretchy="false" xref="S6.Thmtheorem18.p1.2.m2.3.3.2.2.1.2.cmml">(</mo><msub id="S6.Thmtheorem18.p1.2.m2.3.3.2.2.1.1.1.1" xref="S6.Thmtheorem18.p1.2.m2.3.3.2.2.1.1.1.1.cmml"><mi id="S6.Thmtheorem18.p1.2.m2.3.3.2.2.1.1.1.1.2" xref="S6.Thmtheorem18.p1.2.m2.3.3.2.2.1.1.1.1.2.cmml">f</mi><mi id="S6.Thmtheorem18.p1.2.m2.3.3.2.2.1.1.1.1.3" xref="S6.Thmtheorem18.p1.2.m2.3.3.2.2.1.1.1.1.3.cmml">p</mi></msub><mo id="S6.Thmtheorem18.p1.2.m2.3.3.2.2.1.1.1.3" stretchy="false" xref="S6.Thmtheorem18.p1.2.m2.3.3.2.2.1.2.cmml">)</mo></mrow></mrow></mrow><mo id="S6.Thmtheorem18.p1.2.m2.3.3.2.5" stretchy="false" xref="S6.Thmtheorem18.p1.2.m2.3.3.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem18.p1.2.m2.3b"><apply id="S6.Thmtheorem18.p1.2.m2.3.3.3.cmml" xref="S6.Thmtheorem18.p1.2.m2.3.3.2"><csymbol cd="latexml" id="S6.Thmtheorem18.p1.2.m2.3.3.3.1.cmml" xref="S6.Thmtheorem18.p1.2.m2.3.3.2.3">conditional-set</csymbol><apply id="S6.Thmtheorem18.p1.2.m2.2.2.1.1.cmml" xref="S6.Thmtheorem18.p1.2.m2.2.2.1.1"><in id="S6.Thmtheorem18.p1.2.m2.2.2.1.1.1.cmml" xref="S6.Thmtheorem18.p1.2.m2.2.2.1.1.1"></in><ci id="S6.Thmtheorem18.p1.2.m2.2.2.1.1.2.cmml" xref="S6.Thmtheorem18.p1.2.m2.2.2.1.1.2">𝑝</ci><apply id="S6.Thmtheorem18.p1.2.m2.2.2.1.1.3.cmml" xref="S6.Thmtheorem18.p1.2.m2.2.2.1.1.3"><csymbol cd="ambiguous" id="S6.Thmtheorem18.p1.2.m2.2.2.1.1.3.1.cmml" xref="S6.Thmtheorem18.p1.2.m2.2.2.1.1.3">subscript</csymbol><ci id="S6.Thmtheorem18.p1.2.m2.2.2.1.1.3.2.cmml" xref="S6.Thmtheorem18.p1.2.m2.2.2.1.1.3.2">𝑃</ci><ci id="S6.Thmtheorem18.p1.2.m2.2.2.1.1.3.3.cmml" xref="S6.Thmtheorem18.p1.2.m2.2.2.1.1.3.3">𝐸</ci></apply></apply><apply id="S6.Thmtheorem18.p1.2.m2.3.3.2.2.cmml" xref="S6.Thmtheorem18.p1.2.m2.3.3.2.2"><in id="S6.Thmtheorem18.p1.2.m2.3.3.2.2.2.cmml" xref="S6.Thmtheorem18.p1.2.m2.3.3.2.2.2"></in><ci id="S6.Thmtheorem18.p1.2.m2.3.3.2.2.3.cmml" xref="S6.Thmtheorem18.p1.2.m2.3.3.2.2.3">𝑥</ci><apply id="S6.Thmtheorem18.p1.2.m2.3.3.2.2.1.2.cmml" xref="S6.Thmtheorem18.p1.2.m2.3.3.2.2.1.1"><ci id="S6.Thmtheorem18.p1.2.m2.1.1.cmml" xref="S6.Thmtheorem18.p1.2.m2.1.1">ran</ci><apply id="S6.Thmtheorem18.p1.2.m2.3.3.2.2.1.1.1.1.cmml" xref="S6.Thmtheorem18.p1.2.m2.3.3.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S6.Thmtheorem18.p1.2.m2.3.3.2.2.1.1.1.1.1.cmml" xref="S6.Thmtheorem18.p1.2.m2.3.3.2.2.1.1.1.1">subscript</csymbol><ci id="S6.Thmtheorem18.p1.2.m2.3.3.2.2.1.1.1.1.2.cmml" xref="S6.Thmtheorem18.p1.2.m2.3.3.2.2.1.1.1.1.2">𝑓</ci><ci id="S6.Thmtheorem18.p1.2.m2.3.3.2.2.1.1.1.1.3.cmml" xref="S6.Thmtheorem18.p1.2.m2.3.3.2.2.1.1.1.1.3">𝑝</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem18.p1.2.m2.3c">\{p\in P_{E}:x\in\operatorname{ran}(f_{p})\}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem18.p1.2.m2.3d">{ italic_p ∈ italic_P start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT : italic_x ∈ roman_ran ( italic_f start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ) }</annotation></semantics></math> is dense.</p> </div> </div> <div class="ltx_proof" id="S6.SS2.24"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S6.SS2.20.p1"> <p class="ltx_p" id="S6.SS2.20.p1.15">Assume <math alttext="p\in P_{E}" class="ltx_Math" display="inline" id="S6.SS2.20.p1.1.m1.1"><semantics id="S6.SS2.20.p1.1.m1.1a"><mrow id="S6.SS2.20.p1.1.m1.1.1" xref="S6.SS2.20.p1.1.m1.1.1.cmml"><mi id="S6.SS2.20.p1.1.m1.1.1.2" xref="S6.SS2.20.p1.1.m1.1.1.2.cmml">p</mi><mo id="S6.SS2.20.p1.1.m1.1.1.1" xref="S6.SS2.20.p1.1.m1.1.1.1.cmml">∈</mo><msub id="S6.SS2.20.p1.1.m1.1.1.3" xref="S6.SS2.20.p1.1.m1.1.1.3.cmml"><mi id="S6.SS2.20.p1.1.m1.1.1.3.2" xref="S6.SS2.20.p1.1.m1.1.1.3.2.cmml">P</mi><mi id="S6.SS2.20.p1.1.m1.1.1.3.3" xref="S6.SS2.20.p1.1.m1.1.1.3.3.cmml">E</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.20.p1.1.m1.1b"><apply id="S6.SS2.20.p1.1.m1.1.1.cmml" xref="S6.SS2.20.p1.1.m1.1.1"><in id="S6.SS2.20.p1.1.m1.1.1.1.cmml" xref="S6.SS2.20.p1.1.m1.1.1.1"></in><ci id="S6.SS2.20.p1.1.m1.1.1.2.cmml" xref="S6.SS2.20.p1.1.m1.1.1.2">𝑝</ci><apply id="S6.SS2.20.p1.1.m1.1.1.3.cmml" xref="S6.SS2.20.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S6.SS2.20.p1.1.m1.1.1.3.1.cmml" xref="S6.SS2.20.p1.1.m1.1.1.3">subscript</csymbol><ci id="S6.SS2.20.p1.1.m1.1.1.3.2.cmml" xref="S6.SS2.20.p1.1.m1.1.1.3.2">𝑃</ci><ci id="S6.SS2.20.p1.1.m1.1.1.3.3.cmml" xref="S6.SS2.20.p1.1.m1.1.1.3.3">𝐸</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.20.p1.1.m1.1c">p\in P_{E}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.20.p1.1.m1.1d">italic_p ∈ italic_P start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="z\notin\operatorname{ran}(p)" class="ltx_Math" display="inline" id="S6.SS2.20.p1.2.m2.2"><semantics id="S6.SS2.20.p1.2.m2.2a"><mrow id="S6.SS2.20.p1.2.m2.2.3" xref="S6.SS2.20.p1.2.m2.2.3.cmml"><mi id="S6.SS2.20.p1.2.m2.2.3.2" xref="S6.SS2.20.p1.2.m2.2.3.2.cmml">z</mi><mo id="S6.SS2.20.p1.2.m2.2.3.1" xref="S6.SS2.20.p1.2.m2.2.3.1.cmml">∉</mo><mrow id="S6.SS2.20.p1.2.m2.2.3.3.2" xref="S6.SS2.20.p1.2.m2.2.3.3.1.cmml"><mi id="S6.SS2.20.p1.2.m2.1.1" xref="S6.SS2.20.p1.2.m2.1.1.cmml">ran</mi><mo id="S6.SS2.20.p1.2.m2.2.3.3.2a" xref="S6.SS2.20.p1.2.m2.2.3.3.1.cmml">⁡</mo><mrow id="S6.SS2.20.p1.2.m2.2.3.3.2.1" xref="S6.SS2.20.p1.2.m2.2.3.3.1.cmml"><mo id="S6.SS2.20.p1.2.m2.2.3.3.2.1.1" stretchy="false" xref="S6.SS2.20.p1.2.m2.2.3.3.1.cmml">(</mo><mi id="S6.SS2.20.p1.2.m2.2.2" xref="S6.SS2.20.p1.2.m2.2.2.cmml">p</mi><mo id="S6.SS2.20.p1.2.m2.2.3.3.2.1.2" stretchy="false" xref="S6.SS2.20.p1.2.m2.2.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.20.p1.2.m2.2b"><apply id="S6.SS2.20.p1.2.m2.2.3.cmml" xref="S6.SS2.20.p1.2.m2.2.3"><notin id="S6.SS2.20.p1.2.m2.2.3.1.cmml" xref="S6.SS2.20.p1.2.m2.2.3.1"></notin><ci id="S6.SS2.20.p1.2.m2.2.3.2.cmml" xref="S6.SS2.20.p1.2.m2.2.3.2">𝑧</ci><apply id="S6.SS2.20.p1.2.m2.2.3.3.1.cmml" xref="S6.SS2.20.p1.2.m2.2.3.3.2"><ci id="S6.SS2.20.p1.2.m2.1.1.cmml" xref="S6.SS2.20.p1.2.m2.1.1">ran</ci><ci id="S6.SS2.20.p1.2.m2.2.2.cmml" xref="S6.SS2.20.p1.2.m2.2.2">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.20.p1.2.m2.2c">z\notin\operatorname{ran}(p)</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.20.p1.2.m2.2d">italic_z ∉ roman_ran ( italic_p )</annotation></semantics></math>. Let <math alttext="x<_{X}y" class="ltx_Math" display="inline" id="S6.SS2.20.p1.3.m3.1"><semantics id="S6.SS2.20.p1.3.m3.1a"><mrow id="S6.SS2.20.p1.3.m3.1.1" xref="S6.SS2.20.p1.3.m3.1.1.cmml"><mi id="S6.SS2.20.p1.3.m3.1.1.2" xref="S6.SS2.20.p1.3.m3.1.1.2.cmml">x</mi><msub id="S6.SS2.20.p1.3.m3.1.1.1" xref="S6.SS2.20.p1.3.m3.1.1.1.cmml"><mo id="S6.SS2.20.p1.3.m3.1.1.1.2" xref="S6.SS2.20.p1.3.m3.1.1.1.2.cmml"><</mo><mi id="S6.SS2.20.p1.3.m3.1.1.1.3" xref="S6.SS2.20.p1.3.m3.1.1.1.3.cmml">X</mi></msub><mi id="S6.SS2.20.p1.3.m3.1.1.3" xref="S6.SS2.20.p1.3.m3.1.1.3.cmml">y</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.20.p1.3.m3.1b"><apply id="S6.SS2.20.p1.3.m3.1.1.cmml" xref="S6.SS2.20.p1.3.m3.1.1"><apply id="S6.SS2.20.p1.3.m3.1.1.1.cmml" xref="S6.SS2.20.p1.3.m3.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.20.p1.3.m3.1.1.1.1.cmml" xref="S6.SS2.20.p1.3.m3.1.1.1">subscript</csymbol><lt id="S6.SS2.20.p1.3.m3.1.1.1.2.cmml" xref="S6.SS2.20.p1.3.m3.1.1.1.2"></lt><ci id="S6.SS2.20.p1.3.m3.1.1.1.3.cmml" xref="S6.SS2.20.p1.3.m3.1.1.1.3">𝑋</ci></apply><ci id="S6.SS2.20.p1.3.m3.1.1.2.cmml" xref="S6.SS2.20.p1.3.m3.1.1.2">𝑥</ci><ci id="S6.SS2.20.p1.3.m3.1.1.3.cmml" xref="S6.SS2.20.p1.3.m3.1.1.3">𝑦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.20.p1.3.m3.1c">x<_{X}y</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.20.p1.3.m3.1d">italic_x < start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT italic_y</annotation></semantics></math> be in <math alttext="\operatorname{dom}(p)" class="ltx_Math" display="inline" id="S6.SS2.20.p1.4.m4.2"><semantics id="S6.SS2.20.p1.4.m4.2a"><mrow id="S6.SS2.20.p1.4.m4.2.3.2" xref="S6.SS2.20.p1.4.m4.2.3.1.cmml"><mi id="S6.SS2.20.p1.4.m4.1.1" xref="S6.SS2.20.p1.4.m4.1.1.cmml">dom</mi><mo id="S6.SS2.20.p1.4.m4.2.3.2a" xref="S6.SS2.20.p1.4.m4.2.3.1.cmml">⁡</mo><mrow id="S6.SS2.20.p1.4.m4.2.3.2.1" xref="S6.SS2.20.p1.4.m4.2.3.1.cmml"><mo id="S6.SS2.20.p1.4.m4.2.3.2.1.1" stretchy="false" xref="S6.SS2.20.p1.4.m4.2.3.1.cmml">(</mo><mi id="S6.SS2.20.p1.4.m4.2.2" xref="S6.SS2.20.p1.4.m4.2.2.cmml">p</mi><mo id="S6.SS2.20.p1.4.m4.2.3.2.1.2" stretchy="false" xref="S6.SS2.20.p1.4.m4.2.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.20.p1.4.m4.2b"><apply id="S6.SS2.20.p1.4.m4.2.3.1.cmml" xref="S6.SS2.20.p1.4.m4.2.3.2"><ci id="S6.SS2.20.p1.4.m4.1.1.cmml" xref="S6.SS2.20.p1.4.m4.1.1">dom</ci><ci id="S6.SS2.20.p1.4.m4.2.2.cmml" xref="S6.SS2.20.p1.4.m4.2.2">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.20.p1.4.m4.2c">\operatorname{dom}(p)</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.20.p1.4.m4.2d">roman_dom ( italic_p )</annotation></semantics></math> and neighbors to <math alttext="z" class="ltx_Math" display="inline" id="S6.SS2.20.p1.5.m5.1"><semantics id="S6.SS2.20.p1.5.m5.1a"><mi id="S6.SS2.20.p1.5.m5.1.1" xref="S6.SS2.20.p1.5.m5.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.20.p1.5.m5.1b"><ci id="S6.SS2.20.p1.5.m5.1.1.cmml" xref="S6.SS2.20.p1.5.m5.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.20.p1.5.m5.1c">z</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.20.p1.5.m5.1d">italic_z</annotation></semantics></math>. The case when <math alttext="x" class="ltx_Math" display="inline" id="S6.SS2.20.p1.6.m6.1"><semantics id="S6.SS2.20.p1.6.m6.1a"><mi id="S6.SS2.20.p1.6.m6.1.1" xref="S6.SS2.20.p1.6.m6.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.20.p1.6.m6.1b"><ci id="S6.SS2.20.p1.6.m6.1.1.cmml" xref="S6.SS2.20.p1.6.m6.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.20.p1.6.m6.1c">x</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.20.p1.6.m6.1d">italic_x</annotation></semantics></math> and/or <math alttext="y" class="ltx_Math" display="inline" id="S6.SS2.20.p1.7.m7.1"><semantics id="S6.SS2.20.p1.7.m7.1a"><mi id="S6.SS2.20.p1.7.m7.1.1" xref="S6.SS2.20.p1.7.m7.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.20.p1.7.m7.1b"><ci id="S6.SS2.20.p1.7.m7.1.1.cmml" xref="S6.SS2.20.p1.7.m7.1.1">𝑦</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.20.p1.7.m7.1c">y</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.20.p1.7.m7.1d">italic_y</annotation></semantics></math> do not exist is similar. Because the arguments are symmetric, we may assume that <math alttext="\Delta(x,z)<\Delta(z,y)" class="ltx_Math" display="inline" id="S6.SS2.20.p1.8.m8.4"><semantics id="S6.SS2.20.p1.8.m8.4a"><mrow id="S6.SS2.20.p1.8.m8.4.5" xref="S6.SS2.20.p1.8.m8.4.5.cmml"><mrow id="S6.SS2.20.p1.8.m8.4.5.2" xref="S6.SS2.20.p1.8.m8.4.5.2.cmml"><mi id="S6.SS2.20.p1.8.m8.4.5.2.2" mathvariant="normal" xref="S6.SS2.20.p1.8.m8.4.5.2.2.cmml">Δ</mi><mo id="S6.SS2.20.p1.8.m8.4.5.2.1" xref="S6.SS2.20.p1.8.m8.4.5.2.1.cmml">⁢</mo><mrow id="S6.SS2.20.p1.8.m8.4.5.2.3.2" xref="S6.SS2.20.p1.8.m8.4.5.2.3.1.cmml"><mo id="S6.SS2.20.p1.8.m8.4.5.2.3.2.1" stretchy="false" xref="S6.SS2.20.p1.8.m8.4.5.2.3.1.cmml">(</mo><mi id="S6.SS2.20.p1.8.m8.1.1" xref="S6.SS2.20.p1.8.m8.1.1.cmml">x</mi><mo id="S6.SS2.20.p1.8.m8.4.5.2.3.2.2" xref="S6.SS2.20.p1.8.m8.4.5.2.3.1.cmml">,</mo><mi id="S6.SS2.20.p1.8.m8.2.2" xref="S6.SS2.20.p1.8.m8.2.2.cmml">z</mi><mo id="S6.SS2.20.p1.8.m8.4.5.2.3.2.3" stretchy="false" xref="S6.SS2.20.p1.8.m8.4.5.2.3.1.cmml">)</mo></mrow></mrow><mo id="S6.SS2.20.p1.8.m8.4.5.1" xref="S6.SS2.20.p1.8.m8.4.5.1.cmml"><</mo><mrow id="S6.SS2.20.p1.8.m8.4.5.3" xref="S6.SS2.20.p1.8.m8.4.5.3.cmml"><mi id="S6.SS2.20.p1.8.m8.4.5.3.2" mathvariant="normal" xref="S6.SS2.20.p1.8.m8.4.5.3.2.cmml">Δ</mi><mo id="S6.SS2.20.p1.8.m8.4.5.3.1" xref="S6.SS2.20.p1.8.m8.4.5.3.1.cmml">⁢</mo><mrow id="S6.SS2.20.p1.8.m8.4.5.3.3.2" xref="S6.SS2.20.p1.8.m8.4.5.3.3.1.cmml"><mo id="S6.SS2.20.p1.8.m8.4.5.3.3.2.1" stretchy="false" xref="S6.SS2.20.p1.8.m8.4.5.3.3.1.cmml">(</mo><mi id="S6.SS2.20.p1.8.m8.3.3" xref="S6.SS2.20.p1.8.m8.3.3.cmml">z</mi><mo id="S6.SS2.20.p1.8.m8.4.5.3.3.2.2" xref="S6.SS2.20.p1.8.m8.4.5.3.3.1.cmml">,</mo><mi id="S6.SS2.20.p1.8.m8.4.4" xref="S6.SS2.20.p1.8.m8.4.4.cmml">y</mi><mo id="S6.SS2.20.p1.8.m8.4.5.3.3.2.3" stretchy="false" xref="S6.SS2.20.p1.8.m8.4.5.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.20.p1.8.m8.4b"><apply id="S6.SS2.20.p1.8.m8.4.5.cmml" xref="S6.SS2.20.p1.8.m8.4.5"><lt id="S6.SS2.20.p1.8.m8.4.5.1.cmml" xref="S6.SS2.20.p1.8.m8.4.5.1"></lt><apply id="S6.SS2.20.p1.8.m8.4.5.2.cmml" xref="S6.SS2.20.p1.8.m8.4.5.2"><times id="S6.SS2.20.p1.8.m8.4.5.2.1.cmml" xref="S6.SS2.20.p1.8.m8.4.5.2.1"></times><ci id="S6.SS2.20.p1.8.m8.4.5.2.2.cmml" xref="S6.SS2.20.p1.8.m8.4.5.2.2">Δ</ci><interval closure="open" id="S6.SS2.20.p1.8.m8.4.5.2.3.1.cmml" xref="S6.SS2.20.p1.8.m8.4.5.2.3.2"><ci id="S6.SS2.20.p1.8.m8.1.1.cmml" xref="S6.SS2.20.p1.8.m8.1.1">𝑥</ci><ci id="S6.SS2.20.p1.8.m8.2.2.cmml" xref="S6.SS2.20.p1.8.m8.2.2">𝑧</ci></interval></apply><apply id="S6.SS2.20.p1.8.m8.4.5.3.cmml" xref="S6.SS2.20.p1.8.m8.4.5.3"><times id="S6.SS2.20.p1.8.m8.4.5.3.1.cmml" xref="S6.SS2.20.p1.8.m8.4.5.3.1"></times><ci id="S6.SS2.20.p1.8.m8.4.5.3.2.cmml" xref="S6.SS2.20.p1.8.m8.4.5.3.2">Δ</ci><interval closure="open" id="S6.SS2.20.p1.8.m8.4.5.3.3.1.cmml" xref="S6.SS2.20.p1.8.m8.4.5.3.3.2"><ci id="S6.SS2.20.p1.8.m8.3.3.cmml" xref="S6.SS2.20.p1.8.m8.3.3">𝑧</ci><ci id="S6.SS2.20.p1.8.m8.4.4.cmml" xref="S6.SS2.20.p1.8.m8.4.4">𝑦</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.20.p1.8.m8.4c">\Delta(x,z)<\Delta(z,y)</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.20.p1.8.m8.4d">roman_Δ ( italic_x , italic_z ) < roman_Δ ( italic_z , italic_y )</annotation></semantics></math>. Let <math alttext="\nu:=\nu(z,y)" class="ltx_Math" display="inline" id="S6.SS2.20.p1.9.m9.2"><semantics id="S6.SS2.20.p1.9.m9.2a"><mrow id="S6.SS2.20.p1.9.m9.2.3" xref="S6.SS2.20.p1.9.m9.2.3.cmml"><mi id="S6.SS2.20.p1.9.m9.2.3.2" xref="S6.SS2.20.p1.9.m9.2.3.2.cmml">ν</mi><mo id="S6.SS2.20.p1.9.m9.2.3.1" lspace="0.278em" rspace="0.278em" xref="S6.SS2.20.p1.9.m9.2.3.1.cmml">:=</mo><mrow id="S6.SS2.20.p1.9.m9.2.3.3" xref="S6.SS2.20.p1.9.m9.2.3.3.cmml"><mi id="S6.SS2.20.p1.9.m9.2.3.3.2" xref="S6.SS2.20.p1.9.m9.2.3.3.2.cmml">ν</mi><mo id="S6.SS2.20.p1.9.m9.2.3.3.1" xref="S6.SS2.20.p1.9.m9.2.3.3.1.cmml">⁢</mo><mrow id="S6.SS2.20.p1.9.m9.2.3.3.3.2" xref="S6.SS2.20.p1.9.m9.2.3.3.3.1.cmml"><mo id="S6.SS2.20.p1.9.m9.2.3.3.3.2.1" stretchy="false" xref="S6.SS2.20.p1.9.m9.2.3.3.3.1.cmml">(</mo><mi id="S6.SS2.20.p1.9.m9.1.1" xref="S6.SS2.20.p1.9.m9.1.1.cmml">z</mi><mo id="S6.SS2.20.p1.9.m9.2.3.3.3.2.2" xref="S6.SS2.20.p1.9.m9.2.3.3.3.1.cmml">,</mo><mi id="S6.SS2.20.p1.9.m9.2.2" xref="S6.SS2.20.p1.9.m9.2.2.cmml">y</mi><mo id="S6.SS2.20.p1.9.m9.2.3.3.3.2.3" stretchy="false" xref="S6.SS2.20.p1.9.m9.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.20.p1.9.m9.2b"><apply id="S6.SS2.20.p1.9.m9.2.3.cmml" xref="S6.SS2.20.p1.9.m9.2.3"><csymbol cd="latexml" id="S6.SS2.20.p1.9.m9.2.3.1.cmml" xref="S6.SS2.20.p1.9.m9.2.3.1">assign</csymbol><ci id="S6.SS2.20.p1.9.m9.2.3.2.cmml" xref="S6.SS2.20.p1.9.m9.2.3.2">𝜈</ci><apply id="S6.SS2.20.p1.9.m9.2.3.3.cmml" xref="S6.SS2.20.p1.9.m9.2.3.3"><times id="S6.SS2.20.p1.9.m9.2.3.3.1.cmml" xref="S6.SS2.20.p1.9.m9.2.3.3.1"></times><ci id="S6.SS2.20.p1.9.m9.2.3.3.2.cmml" xref="S6.SS2.20.p1.9.m9.2.3.3.2">𝜈</ci><interval closure="open" id="S6.SS2.20.p1.9.m9.2.3.3.3.1.cmml" xref="S6.SS2.20.p1.9.m9.2.3.3.3.2"><ci id="S6.SS2.20.p1.9.m9.1.1.cmml" xref="S6.SS2.20.p1.9.m9.1.1">𝑧</ci><ci id="S6.SS2.20.p1.9.m9.2.2.cmml" xref="S6.SS2.20.p1.9.m9.2.2">𝑦</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.20.p1.9.m9.2c">\nu:=\nu(z,y)</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.20.p1.9.m9.2d">italic_ν := italic_ν ( italic_z , italic_y )</annotation></semantics></math>, <math alttext="\bar{a}:=p^{-1}(x)" class="ltx_Math" display="inline" id="S6.SS2.20.p1.10.m10.1"><semantics id="S6.SS2.20.p1.10.m10.1a"><mrow id="S6.SS2.20.p1.10.m10.1.2" xref="S6.SS2.20.p1.10.m10.1.2.cmml"><mover accent="true" id="S6.SS2.20.p1.10.m10.1.2.2" xref="S6.SS2.20.p1.10.m10.1.2.2.cmml"><mi id="S6.SS2.20.p1.10.m10.1.2.2.2" xref="S6.SS2.20.p1.10.m10.1.2.2.2.cmml">a</mi><mo id="S6.SS2.20.p1.10.m10.1.2.2.1" xref="S6.SS2.20.p1.10.m10.1.2.2.1.cmml">¯</mo></mover><mo id="S6.SS2.20.p1.10.m10.1.2.1" lspace="0.278em" rspace="0.278em" xref="S6.SS2.20.p1.10.m10.1.2.1.cmml">:=</mo><mrow id="S6.SS2.20.p1.10.m10.1.2.3" xref="S6.SS2.20.p1.10.m10.1.2.3.cmml"><msup id="S6.SS2.20.p1.10.m10.1.2.3.2" xref="S6.SS2.20.p1.10.m10.1.2.3.2.cmml"><mi id="S6.SS2.20.p1.10.m10.1.2.3.2.2" xref="S6.SS2.20.p1.10.m10.1.2.3.2.2.cmml">p</mi><mrow id="S6.SS2.20.p1.10.m10.1.2.3.2.3" xref="S6.SS2.20.p1.10.m10.1.2.3.2.3.cmml"><mo id="S6.SS2.20.p1.10.m10.1.2.3.2.3a" xref="S6.SS2.20.p1.10.m10.1.2.3.2.3.cmml">−</mo><mn id="S6.SS2.20.p1.10.m10.1.2.3.2.3.2" xref="S6.SS2.20.p1.10.m10.1.2.3.2.3.2.cmml">1</mn></mrow></msup><mo id="S6.SS2.20.p1.10.m10.1.2.3.1" xref="S6.SS2.20.p1.10.m10.1.2.3.1.cmml">⁢</mo><mrow id="S6.SS2.20.p1.10.m10.1.2.3.3.2" xref="S6.SS2.20.p1.10.m10.1.2.3.cmml"><mo id="S6.SS2.20.p1.10.m10.1.2.3.3.2.1" stretchy="false" xref="S6.SS2.20.p1.10.m10.1.2.3.cmml">(</mo><mi id="S6.SS2.20.p1.10.m10.1.1" xref="S6.SS2.20.p1.10.m10.1.1.cmml">x</mi><mo id="S6.SS2.20.p1.10.m10.1.2.3.3.2.2" stretchy="false" xref="S6.SS2.20.p1.10.m10.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.20.p1.10.m10.1b"><apply id="S6.SS2.20.p1.10.m10.1.2.cmml" xref="S6.SS2.20.p1.10.m10.1.2"><csymbol cd="latexml" id="S6.SS2.20.p1.10.m10.1.2.1.cmml" xref="S6.SS2.20.p1.10.m10.1.2.1">assign</csymbol><apply id="S6.SS2.20.p1.10.m10.1.2.2.cmml" xref="S6.SS2.20.p1.10.m10.1.2.2"><ci id="S6.SS2.20.p1.10.m10.1.2.2.1.cmml" xref="S6.SS2.20.p1.10.m10.1.2.2.1">¯</ci><ci id="S6.SS2.20.p1.10.m10.1.2.2.2.cmml" xref="S6.SS2.20.p1.10.m10.1.2.2.2">𝑎</ci></apply><apply id="S6.SS2.20.p1.10.m10.1.2.3.cmml" xref="S6.SS2.20.p1.10.m10.1.2.3"><times id="S6.SS2.20.p1.10.m10.1.2.3.1.cmml" xref="S6.SS2.20.p1.10.m10.1.2.3.1"></times><apply id="S6.SS2.20.p1.10.m10.1.2.3.2.cmml" xref="S6.SS2.20.p1.10.m10.1.2.3.2"><csymbol cd="ambiguous" id="S6.SS2.20.p1.10.m10.1.2.3.2.1.cmml" xref="S6.SS2.20.p1.10.m10.1.2.3.2">superscript</csymbol><ci id="S6.SS2.20.p1.10.m10.1.2.3.2.2.cmml" xref="S6.SS2.20.p1.10.m10.1.2.3.2.2">𝑝</ci><apply id="S6.SS2.20.p1.10.m10.1.2.3.2.3.cmml" xref="S6.SS2.20.p1.10.m10.1.2.3.2.3"><minus id="S6.SS2.20.p1.10.m10.1.2.3.2.3.1.cmml" xref="S6.SS2.20.p1.10.m10.1.2.3.2.3"></minus><cn id="S6.SS2.20.p1.10.m10.1.2.3.2.3.2.cmml" type="integer" xref="S6.SS2.20.p1.10.m10.1.2.3.2.3.2">1</cn></apply></apply><ci id="S6.SS2.20.p1.10.m10.1.1.cmml" xref="S6.SS2.20.p1.10.m10.1.1">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.20.p1.10.m10.1c">\bar{a}:=p^{-1}(x)</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.20.p1.10.m10.1d">over¯ start_ARG italic_a end_ARG := italic_p start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_x )</annotation></semantics></math> and <math alttext="\bar{b}:=p^{-1}(y)" class="ltx_Math" display="inline" id="S6.SS2.20.p1.11.m11.1"><semantics id="S6.SS2.20.p1.11.m11.1a"><mrow id="S6.SS2.20.p1.11.m11.1.2" xref="S6.SS2.20.p1.11.m11.1.2.cmml"><mover accent="true" id="S6.SS2.20.p1.11.m11.1.2.2" xref="S6.SS2.20.p1.11.m11.1.2.2.cmml"><mi id="S6.SS2.20.p1.11.m11.1.2.2.2" xref="S6.SS2.20.p1.11.m11.1.2.2.2.cmml">b</mi><mo id="S6.SS2.20.p1.11.m11.1.2.2.1" xref="S6.SS2.20.p1.11.m11.1.2.2.1.cmml">¯</mo></mover><mo id="S6.SS2.20.p1.11.m11.1.2.1" lspace="0.278em" rspace="0.278em" xref="S6.SS2.20.p1.11.m11.1.2.1.cmml">:=</mo><mrow id="S6.SS2.20.p1.11.m11.1.2.3" xref="S6.SS2.20.p1.11.m11.1.2.3.cmml"><msup id="S6.SS2.20.p1.11.m11.1.2.3.2" xref="S6.SS2.20.p1.11.m11.1.2.3.2.cmml"><mi id="S6.SS2.20.p1.11.m11.1.2.3.2.2" xref="S6.SS2.20.p1.11.m11.1.2.3.2.2.cmml">p</mi><mrow id="S6.SS2.20.p1.11.m11.1.2.3.2.3" xref="S6.SS2.20.p1.11.m11.1.2.3.2.3.cmml"><mo id="S6.SS2.20.p1.11.m11.1.2.3.2.3a" xref="S6.SS2.20.p1.11.m11.1.2.3.2.3.cmml">−</mo><mn id="S6.SS2.20.p1.11.m11.1.2.3.2.3.2" xref="S6.SS2.20.p1.11.m11.1.2.3.2.3.2.cmml">1</mn></mrow></msup><mo id="S6.SS2.20.p1.11.m11.1.2.3.1" xref="S6.SS2.20.p1.11.m11.1.2.3.1.cmml">⁢</mo><mrow id="S6.SS2.20.p1.11.m11.1.2.3.3.2" xref="S6.SS2.20.p1.11.m11.1.2.3.cmml"><mo id="S6.SS2.20.p1.11.m11.1.2.3.3.2.1" stretchy="false" xref="S6.SS2.20.p1.11.m11.1.2.3.cmml">(</mo><mi id="S6.SS2.20.p1.11.m11.1.1" xref="S6.SS2.20.p1.11.m11.1.1.cmml">y</mi><mo id="S6.SS2.20.p1.11.m11.1.2.3.3.2.2" stretchy="false" xref="S6.SS2.20.p1.11.m11.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.20.p1.11.m11.1b"><apply id="S6.SS2.20.p1.11.m11.1.2.cmml" xref="S6.SS2.20.p1.11.m11.1.2"><csymbol cd="latexml" id="S6.SS2.20.p1.11.m11.1.2.1.cmml" xref="S6.SS2.20.p1.11.m11.1.2.1">assign</csymbol><apply id="S6.SS2.20.p1.11.m11.1.2.2.cmml" xref="S6.SS2.20.p1.11.m11.1.2.2"><ci id="S6.SS2.20.p1.11.m11.1.2.2.1.cmml" xref="S6.SS2.20.p1.11.m11.1.2.2.1">¯</ci><ci id="S6.SS2.20.p1.11.m11.1.2.2.2.cmml" xref="S6.SS2.20.p1.11.m11.1.2.2.2">𝑏</ci></apply><apply id="S6.SS2.20.p1.11.m11.1.2.3.cmml" xref="S6.SS2.20.p1.11.m11.1.2.3"><times id="S6.SS2.20.p1.11.m11.1.2.3.1.cmml" xref="S6.SS2.20.p1.11.m11.1.2.3.1"></times><apply id="S6.SS2.20.p1.11.m11.1.2.3.2.cmml" xref="S6.SS2.20.p1.11.m11.1.2.3.2"><csymbol cd="ambiguous" id="S6.SS2.20.p1.11.m11.1.2.3.2.1.cmml" xref="S6.SS2.20.p1.11.m11.1.2.3.2">superscript</csymbol><ci id="S6.SS2.20.p1.11.m11.1.2.3.2.2.cmml" xref="S6.SS2.20.p1.11.m11.1.2.3.2.2">𝑝</ci><apply id="S6.SS2.20.p1.11.m11.1.2.3.2.3.cmml" xref="S6.SS2.20.p1.11.m11.1.2.3.2.3"><minus id="S6.SS2.20.p1.11.m11.1.2.3.2.3.1.cmml" xref="S6.SS2.20.p1.11.m11.1.2.3.2.3"></minus><cn id="S6.SS2.20.p1.11.m11.1.2.3.2.3.2.cmml" type="integer" xref="S6.SS2.20.p1.11.m11.1.2.3.2.3.2">1</cn></apply></apply><ci id="S6.SS2.20.p1.11.m11.1.1.cmml" xref="S6.SS2.20.p1.11.m11.1.1">𝑦</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.20.p1.11.m11.1c">\bar{b}:=p^{-1}(y)</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.20.p1.11.m11.1d">over¯ start_ARG italic_b end_ARG := italic_p start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_y )</annotation></semantics></math>. Note that by <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.Thmtheorem11" title="Lemma 6.11. ‣ 6.2. Forcing epimorphisms ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">6.11</span></a> (a), <math alttext="\nu\leq\nu(y)" class="ltx_Math" display="inline" id="S6.SS2.20.p1.12.m12.1"><semantics id="S6.SS2.20.p1.12.m12.1a"><mrow id="S6.SS2.20.p1.12.m12.1.2" xref="S6.SS2.20.p1.12.m12.1.2.cmml"><mi id="S6.SS2.20.p1.12.m12.1.2.2" xref="S6.SS2.20.p1.12.m12.1.2.2.cmml">ν</mi><mo id="S6.SS2.20.p1.12.m12.1.2.1" xref="S6.SS2.20.p1.12.m12.1.2.1.cmml">≤</mo><mrow id="S6.SS2.20.p1.12.m12.1.2.3" xref="S6.SS2.20.p1.12.m12.1.2.3.cmml"><mi id="S6.SS2.20.p1.12.m12.1.2.3.2" xref="S6.SS2.20.p1.12.m12.1.2.3.2.cmml">ν</mi><mo id="S6.SS2.20.p1.12.m12.1.2.3.1" xref="S6.SS2.20.p1.12.m12.1.2.3.1.cmml">⁢</mo><mrow id="S6.SS2.20.p1.12.m12.1.2.3.3.2" xref="S6.SS2.20.p1.12.m12.1.2.3.cmml"><mo id="S6.SS2.20.p1.12.m12.1.2.3.3.2.1" stretchy="false" xref="S6.SS2.20.p1.12.m12.1.2.3.cmml">(</mo><mi id="S6.SS2.20.p1.12.m12.1.1" xref="S6.SS2.20.p1.12.m12.1.1.cmml">y</mi><mo id="S6.SS2.20.p1.12.m12.1.2.3.3.2.2" stretchy="false" xref="S6.SS2.20.p1.12.m12.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.20.p1.12.m12.1b"><apply id="S6.SS2.20.p1.12.m12.1.2.cmml" xref="S6.SS2.20.p1.12.m12.1.2"><leq id="S6.SS2.20.p1.12.m12.1.2.1.cmml" xref="S6.SS2.20.p1.12.m12.1.2.1"></leq><ci id="S6.SS2.20.p1.12.m12.1.2.2.cmml" xref="S6.SS2.20.p1.12.m12.1.2.2">𝜈</ci><apply id="S6.SS2.20.p1.12.m12.1.2.3.cmml" xref="S6.SS2.20.p1.12.m12.1.2.3"><times id="S6.SS2.20.p1.12.m12.1.2.3.1.cmml" xref="S6.SS2.20.p1.12.m12.1.2.3.1"></times><ci id="S6.SS2.20.p1.12.m12.1.2.3.2.cmml" xref="S6.SS2.20.p1.12.m12.1.2.3.2">𝜈</ci><ci id="S6.SS2.20.p1.12.m12.1.1.cmml" xref="S6.SS2.20.p1.12.m12.1.1">𝑦</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.20.p1.12.m12.1c">\nu\leq\nu(y)</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.20.p1.12.m12.1d">italic_ν ≤ italic_ν ( italic_y )</annotation></semantics></math>, and by definition of <math alttext="P_{E}" class="ltx_Math" display="inline" id="S6.SS2.20.p1.13.m13.1"><semantics id="S6.SS2.20.p1.13.m13.1a"><msub id="S6.SS2.20.p1.13.m13.1.1" xref="S6.SS2.20.p1.13.m13.1.1.cmml"><mi id="S6.SS2.20.p1.13.m13.1.1.2" xref="S6.SS2.20.p1.13.m13.1.1.2.cmml">P</mi><mi id="S6.SS2.20.p1.13.m13.1.1.3" xref="S6.SS2.20.p1.13.m13.1.1.3.cmml">E</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.20.p1.13.m13.1b"><apply id="S6.SS2.20.p1.13.m13.1.1.cmml" xref="S6.SS2.20.p1.13.m13.1.1"><csymbol cd="ambiguous" id="S6.SS2.20.p1.13.m13.1.1.1.cmml" xref="S6.SS2.20.p1.13.m13.1.1">subscript</csymbol><ci id="S6.SS2.20.p1.13.m13.1.1.2.cmml" xref="S6.SS2.20.p1.13.m13.1.1.2">𝑃</ci><ci id="S6.SS2.20.p1.13.m13.1.1.3.cmml" xref="S6.SS2.20.p1.13.m13.1.1.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.20.p1.13.m13.1c">P_{E}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.20.p1.13.m13.1d">italic_P start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\nu(y)=\nu(b_{l})" class="ltx_Math" display="inline" id="S6.SS2.20.p1.14.m14.2"><semantics id="S6.SS2.20.p1.14.m14.2a"><mrow id="S6.SS2.20.p1.14.m14.2.2" xref="S6.SS2.20.p1.14.m14.2.2.cmml"><mrow id="S6.SS2.20.p1.14.m14.2.2.3" xref="S6.SS2.20.p1.14.m14.2.2.3.cmml"><mi id="S6.SS2.20.p1.14.m14.2.2.3.2" xref="S6.SS2.20.p1.14.m14.2.2.3.2.cmml">ν</mi><mo id="S6.SS2.20.p1.14.m14.2.2.3.1" xref="S6.SS2.20.p1.14.m14.2.2.3.1.cmml">⁢</mo><mrow id="S6.SS2.20.p1.14.m14.2.2.3.3.2" xref="S6.SS2.20.p1.14.m14.2.2.3.cmml"><mo id="S6.SS2.20.p1.14.m14.2.2.3.3.2.1" stretchy="false" xref="S6.SS2.20.p1.14.m14.2.2.3.cmml">(</mo><mi id="S6.SS2.20.p1.14.m14.1.1" xref="S6.SS2.20.p1.14.m14.1.1.cmml">y</mi><mo id="S6.SS2.20.p1.14.m14.2.2.3.3.2.2" stretchy="false" xref="S6.SS2.20.p1.14.m14.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.SS2.20.p1.14.m14.2.2.2" xref="S6.SS2.20.p1.14.m14.2.2.2.cmml">=</mo><mrow id="S6.SS2.20.p1.14.m14.2.2.1" xref="S6.SS2.20.p1.14.m14.2.2.1.cmml"><mi id="S6.SS2.20.p1.14.m14.2.2.1.3" xref="S6.SS2.20.p1.14.m14.2.2.1.3.cmml">ν</mi><mo id="S6.SS2.20.p1.14.m14.2.2.1.2" xref="S6.SS2.20.p1.14.m14.2.2.1.2.cmml">⁢</mo><mrow id="S6.SS2.20.p1.14.m14.2.2.1.1.1" xref="S6.SS2.20.p1.14.m14.2.2.1.1.1.1.cmml"><mo id="S6.SS2.20.p1.14.m14.2.2.1.1.1.2" stretchy="false" xref="S6.SS2.20.p1.14.m14.2.2.1.1.1.1.cmml">(</mo><msub id="S6.SS2.20.p1.14.m14.2.2.1.1.1.1" xref="S6.SS2.20.p1.14.m14.2.2.1.1.1.1.cmml"><mi id="S6.SS2.20.p1.14.m14.2.2.1.1.1.1.2" xref="S6.SS2.20.p1.14.m14.2.2.1.1.1.1.2.cmml">b</mi><mi id="S6.SS2.20.p1.14.m14.2.2.1.1.1.1.3" xref="S6.SS2.20.p1.14.m14.2.2.1.1.1.1.3.cmml">l</mi></msub><mo id="S6.SS2.20.p1.14.m14.2.2.1.1.1.3" stretchy="false" xref="S6.SS2.20.p1.14.m14.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.20.p1.14.m14.2b"><apply id="S6.SS2.20.p1.14.m14.2.2.cmml" xref="S6.SS2.20.p1.14.m14.2.2"><eq id="S6.SS2.20.p1.14.m14.2.2.2.cmml" xref="S6.SS2.20.p1.14.m14.2.2.2"></eq><apply id="S6.SS2.20.p1.14.m14.2.2.3.cmml" xref="S6.SS2.20.p1.14.m14.2.2.3"><times id="S6.SS2.20.p1.14.m14.2.2.3.1.cmml" xref="S6.SS2.20.p1.14.m14.2.2.3.1"></times><ci id="S6.SS2.20.p1.14.m14.2.2.3.2.cmml" xref="S6.SS2.20.p1.14.m14.2.2.3.2">𝜈</ci><ci id="S6.SS2.20.p1.14.m14.1.1.cmml" xref="S6.SS2.20.p1.14.m14.1.1">𝑦</ci></apply><apply id="S6.SS2.20.p1.14.m14.2.2.1.cmml" xref="S6.SS2.20.p1.14.m14.2.2.1"><times id="S6.SS2.20.p1.14.m14.2.2.1.2.cmml" xref="S6.SS2.20.p1.14.m14.2.2.1.2"></times><ci id="S6.SS2.20.p1.14.m14.2.2.1.3.cmml" xref="S6.SS2.20.p1.14.m14.2.2.1.3">𝜈</ci><apply id="S6.SS2.20.p1.14.m14.2.2.1.1.1.1.cmml" xref="S6.SS2.20.p1.14.m14.2.2.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.20.p1.14.m14.2.2.1.1.1.1.1.cmml" xref="S6.SS2.20.p1.14.m14.2.2.1.1.1">subscript</csymbol><ci id="S6.SS2.20.p1.14.m14.2.2.1.1.1.1.2.cmml" xref="S6.SS2.20.p1.14.m14.2.2.1.1.1.1.2">𝑏</ci><ci id="S6.SS2.20.p1.14.m14.2.2.1.1.1.1.3.cmml" xref="S6.SS2.20.p1.14.m14.2.2.1.1.1.1.3">𝑙</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.20.p1.14.m14.2c">\nu(y)=\nu(b_{l})</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.20.p1.14.m14.2d">italic_ν ( italic_y ) = italic_ν ( italic_b start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT )</annotation></semantics></math>. Observe also that <math alttext="\Delta(x,z)=\Delta(x,y)" class="ltx_Math" display="inline" id="S6.SS2.20.p1.15.m15.4"><semantics id="S6.SS2.20.p1.15.m15.4a"><mrow id="S6.SS2.20.p1.15.m15.4.5" xref="S6.SS2.20.p1.15.m15.4.5.cmml"><mrow id="S6.SS2.20.p1.15.m15.4.5.2" xref="S6.SS2.20.p1.15.m15.4.5.2.cmml"><mi id="S6.SS2.20.p1.15.m15.4.5.2.2" mathvariant="normal" xref="S6.SS2.20.p1.15.m15.4.5.2.2.cmml">Δ</mi><mo id="S6.SS2.20.p1.15.m15.4.5.2.1" xref="S6.SS2.20.p1.15.m15.4.5.2.1.cmml">⁢</mo><mrow id="S6.SS2.20.p1.15.m15.4.5.2.3.2" xref="S6.SS2.20.p1.15.m15.4.5.2.3.1.cmml"><mo id="S6.SS2.20.p1.15.m15.4.5.2.3.2.1" stretchy="false" xref="S6.SS2.20.p1.15.m15.4.5.2.3.1.cmml">(</mo><mi id="S6.SS2.20.p1.15.m15.1.1" xref="S6.SS2.20.p1.15.m15.1.1.cmml">x</mi><mo id="S6.SS2.20.p1.15.m15.4.5.2.3.2.2" xref="S6.SS2.20.p1.15.m15.4.5.2.3.1.cmml">,</mo><mi id="S6.SS2.20.p1.15.m15.2.2" xref="S6.SS2.20.p1.15.m15.2.2.cmml">z</mi><mo id="S6.SS2.20.p1.15.m15.4.5.2.3.2.3" stretchy="false" xref="S6.SS2.20.p1.15.m15.4.5.2.3.1.cmml">)</mo></mrow></mrow><mo id="S6.SS2.20.p1.15.m15.4.5.1" xref="S6.SS2.20.p1.15.m15.4.5.1.cmml">=</mo><mrow id="S6.SS2.20.p1.15.m15.4.5.3" xref="S6.SS2.20.p1.15.m15.4.5.3.cmml"><mi id="S6.SS2.20.p1.15.m15.4.5.3.2" mathvariant="normal" xref="S6.SS2.20.p1.15.m15.4.5.3.2.cmml">Δ</mi><mo id="S6.SS2.20.p1.15.m15.4.5.3.1" xref="S6.SS2.20.p1.15.m15.4.5.3.1.cmml">⁢</mo><mrow id="S6.SS2.20.p1.15.m15.4.5.3.3.2" xref="S6.SS2.20.p1.15.m15.4.5.3.3.1.cmml"><mo id="S6.SS2.20.p1.15.m15.4.5.3.3.2.1" stretchy="false" xref="S6.SS2.20.p1.15.m15.4.5.3.3.1.cmml">(</mo><mi id="S6.SS2.20.p1.15.m15.3.3" xref="S6.SS2.20.p1.15.m15.3.3.cmml">x</mi><mo id="S6.SS2.20.p1.15.m15.4.5.3.3.2.2" xref="S6.SS2.20.p1.15.m15.4.5.3.3.1.cmml">,</mo><mi id="S6.SS2.20.p1.15.m15.4.4" xref="S6.SS2.20.p1.15.m15.4.4.cmml">y</mi><mo id="S6.SS2.20.p1.15.m15.4.5.3.3.2.3" stretchy="false" xref="S6.SS2.20.p1.15.m15.4.5.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.20.p1.15.m15.4b"><apply id="S6.SS2.20.p1.15.m15.4.5.cmml" xref="S6.SS2.20.p1.15.m15.4.5"><eq id="S6.SS2.20.p1.15.m15.4.5.1.cmml" xref="S6.SS2.20.p1.15.m15.4.5.1"></eq><apply id="S6.SS2.20.p1.15.m15.4.5.2.cmml" xref="S6.SS2.20.p1.15.m15.4.5.2"><times id="S6.SS2.20.p1.15.m15.4.5.2.1.cmml" xref="S6.SS2.20.p1.15.m15.4.5.2.1"></times><ci id="S6.SS2.20.p1.15.m15.4.5.2.2.cmml" xref="S6.SS2.20.p1.15.m15.4.5.2.2">Δ</ci><interval closure="open" id="S6.SS2.20.p1.15.m15.4.5.2.3.1.cmml" xref="S6.SS2.20.p1.15.m15.4.5.2.3.2"><ci id="S6.SS2.20.p1.15.m15.1.1.cmml" xref="S6.SS2.20.p1.15.m15.1.1">𝑥</ci><ci id="S6.SS2.20.p1.15.m15.2.2.cmml" xref="S6.SS2.20.p1.15.m15.2.2">𝑧</ci></interval></apply><apply id="S6.SS2.20.p1.15.m15.4.5.3.cmml" xref="S6.SS2.20.p1.15.m15.4.5.3"><times id="S6.SS2.20.p1.15.m15.4.5.3.1.cmml" xref="S6.SS2.20.p1.15.m15.4.5.3.1"></times><ci id="S6.SS2.20.p1.15.m15.4.5.3.2.cmml" xref="S6.SS2.20.p1.15.m15.4.5.3.2">Δ</ci><interval closure="open" id="S6.SS2.20.p1.15.m15.4.5.3.3.1.cmml" xref="S6.SS2.20.p1.15.m15.4.5.3.3.2"><ci id="S6.SS2.20.p1.15.m15.3.3.cmml" xref="S6.SS2.20.p1.15.m15.3.3">𝑥</ci><ci id="S6.SS2.20.p1.15.m15.4.4.cmml" xref="S6.SS2.20.p1.15.m15.4.4">𝑦</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.20.p1.15.m15.4c">\Delta(x,z)=\Delta(x,y)</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.20.p1.15.m15.4d">roman_Δ ( italic_x , italic_z ) = roman_Δ ( italic_x , italic_y )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S6.SS2.21.p2"> <p class="ltx_p" id="S6.SS2.21.p2.9">First we claim that <math alttext="b_{l}" class="ltx_Math" display="inline" id="S6.SS2.21.p2.1.m1.1"><semantics id="S6.SS2.21.p2.1.m1.1a"><msub id="S6.SS2.21.p2.1.m1.1.1" xref="S6.SS2.21.p2.1.m1.1.1.cmml"><mi id="S6.SS2.21.p2.1.m1.1.1.2" xref="S6.SS2.21.p2.1.m1.1.1.2.cmml">b</mi><mi id="S6.SS2.21.p2.1.m1.1.1.3" xref="S6.SS2.21.p2.1.m1.1.1.3.cmml">l</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.21.p2.1.m1.1b"><apply id="S6.SS2.21.p2.1.m1.1.1.cmml" xref="S6.SS2.21.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S6.SS2.21.p2.1.m1.1.1.1.cmml" xref="S6.SS2.21.p2.1.m1.1.1">subscript</csymbol><ci id="S6.SS2.21.p2.1.m1.1.1.2.cmml" xref="S6.SS2.21.p2.1.m1.1.1.2">𝑏</ci><ci id="S6.SS2.21.p2.1.m1.1.1.3.cmml" xref="S6.SS2.21.p2.1.m1.1.1.3">𝑙</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.21.p2.1.m1.1c">b_{l}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.21.p2.1.m1.1d">italic_b start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT</annotation></semantics></math> is not the left endpoint of its complementary interval of <math alttext="A\setminus\nu" class="ltx_Math" display="inline" id="S6.SS2.21.p2.2.m2.1"><semantics id="S6.SS2.21.p2.2.m2.1a"><mrow id="S6.SS2.21.p2.2.m2.1.1" xref="S6.SS2.21.p2.2.m2.1.1.cmml"><mi id="S6.SS2.21.p2.2.m2.1.1.2" xref="S6.SS2.21.p2.2.m2.1.1.2.cmml">A</mi><mo id="S6.SS2.21.p2.2.m2.1.1.1" xref="S6.SS2.21.p2.2.m2.1.1.1.cmml">∖</mo><mi id="S6.SS2.21.p2.2.m2.1.1.3" xref="S6.SS2.21.p2.2.m2.1.1.3.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.21.p2.2.m2.1b"><apply id="S6.SS2.21.p2.2.m2.1.1.cmml" xref="S6.SS2.21.p2.2.m2.1.1"><setdiff id="S6.SS2.21.p2.2.m2.1.1.1.cmml" xref="S6.SS2.21.p2.2.m2.1.1.1"></setdiff><ci id="S6.SS2.21.p2.2.m2.1.1.2.cmml" xref="S6.SS2.21.p2.2.m2.1.1.2">𝐴</ci><ci id="S6.SS2.21.p2.2.m2.1.1.3.cmml" xref="S6.SS2.21.p2.2.m2.1.1.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.21.p2.2.m2.1c">A\setminus\nu</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.21.p2.2.m2.1d">italic_A ∖ italic_ν</annotation></semantics></math>. If <math alttext="\nu<\nu(y)" class="ltx_Math" display="inline" id="S6.SS2.21.p2.3.m3.1"><semantics id="S6.SS2.21.p2.3.m3.1a"><mrow id="S6.SS2.21.p2.3.m3.1.2" xref="S6.SS2.21.p2.3.m3.1.2.cmml"><mi id="S6.SS2.21.p2.3.m3.1.2.2" xref="S6.SS2.21.p2.3.m3.1.2.2.cmml">ν</mi><mo id="S6.SS2.21.p2.3.m3.1.2.1" xref="S6.SS2.21.p2.3.m3.1.2.1.cmml"><</mo><mrow id="S6.SS2.21.p2.3.m3.1.2.3" xref="S6.SS2.21.p2.3.m3.1.2.3.cmml"><mi id="S6.SS2.21.p2.3.m3.1.2.3.2" xref="S6.SS2.21.p2.3.m3.1.2.3.2.cmml">ν</mi><mo id="S6.SS2.21.p2.3.m3.1.2.3.1" xref="S6.SS2.21.p2.3.m3.1.2.3.1.cmml">⁢</mo><mrow id="S6.SS2.21.p2.3.m3.1.2.3.3.2" xref="S6.SS2.21.p2.3.m3.1.2.3.cmml"><mo id="S6.SS2.21.p2.3.m3.1.2.3.3.2.1" stretchy="false" xref="S6.SS2.21.p2.3.m3.1.2.3.cmml">(</mo><mi id="S6.SS2.21.p2.3.m3.1.1" xref="S6.SS2.21.p2.3.m3.1.1.cmml">y</mi><mo id="S6.SS2.21.p2.3.m3.1.2.3.3.2.2" stretchy="false" xref="S6.SS2.21.p2.3.m3.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.21.p2.3.m3.1b"><apply id="S6.SS2.21.p2.3.m3.1.2.cmml" xref="S6.SS2.21.p2.3.m3.1.2"><lt id="S6.SS2.21.p2.3.m3.1.2.1.cmml" xref="S6.SS2.21.p2.3.m3.1.2.1"></lt><ci id="S6.SS2.21.p2.3.m3.1.2.2.cmml" xref="S6.SS2.21.p2.3.m3.1.2.2">𝜈</ci><apply id="S6.SS2.21.p2.3.m3.1.2.3.cmml" xref="S6.SS2.21.p2.3.m3.1.2.3"><times id="S6.SS2.21.p2.3.m3.1.2.3.1.cmml" xref="S6.SS2.21.p2.3.m3.1.2.3.1"></times><ci id="S6.SS2.21.p2.3.m3.1.2.3.2.cmml" xref="S6.SS2.21.p2.3.m3.1.2.3.2">𝜈</ci><ci id="S6.SS2.21.p2.3.m3.1.1.cmml" xref="S6.SS2.21.p2.3.m3.1.1">𝑦</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.21.p2.3.m3.1c">\nu<\nu(y)</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.21.p2.3.m3.1d">italic_ν < italic_ν ( italic_y )</annotation></semantics></math> then this follows from <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.Thmtheorem16" title="Lemma 6.16. ‣ 6.2. Forcing epimorphisms ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">6.16</span></a> (d) and the fact that <math alttext="\nu(y)=\nu(b_{l})" class="ltx_Math" display="inline" id="S6.SS2.21.p2.4.m4.2"><semantics id="S6.SS2.21.p2.4.m4.2a"><mrow id="S6.SS2.21.p2.4.m4.2.2" xref="S6.SS2.21.p2.4.m4.2.2.cmml"><mrow id="S6.SS2.21.p2.4.m4.2.2.3" xref="S6.SS2.21.p2.4.m4.2.2.3.cmml"><mi id="S6.SS2.21.p2.4.m4.2.2.3.2" xref="S6.SS2.21.p2.4.m4.2.2.3.2.cmml">ν</mi><mo id="S6.SS2.21.p2.4.m4.2.2.3.1" xref="S6.SS2.21.p2.4.m4.2.2.3.1.cmml">⁢</mo><mrow id="S6.SS2.21.p2.4.m4.2.2.3.3.2" xref="S6.SS2.21.p2.4.m4.2.2.3.cmml"><mo id="S6.SS2.21.p2.4.m4.2.2.3.3.2.1" stretchy="false" xref="S6.SS2.21.p2.4.m4.2.2.3.cmml">(</mo><mi id="S6.SS2.21.p2.4.m4.1.1" xref="S6.SS2.21.p2.4.m4.1.1.cmml">y</mi><mo id="S6.SS2.21.p2.4.m4.2.2.3.3.2.2" stretchy="false" xref="S6.SS2.21.p2.4.m4.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.SS2.21.p2.4.m4.2.2.2" xref="S6.SS2.21.p2.4.m4.2.2.2.cmml">=</mo><mrow id="S6.SS2.21.p2.4.m4.2.2.1" xref="S6.SS2.21.p2.4.m4.2.2.1.cmml"><mi id="S6.SS2.21.p2.4.m4.2.2.1.3" xref="S6.SS2.21.p2.4.m4.2.2.1.3.cmml">ν</mi><mo id="S6.SS2.21.p2.4.m4.2.2.1.2" xref="S6.SS2.21.p2.4.m4.2.2.1.2.cmml">⁢</mo><mrow id="S6.SS2.21.p2.4.m4.2.2.1.1.1" xref="S6.SS2.21.p2.4.m4.2.2.1.1.1.1.cmml"><mo id="S6.SS2.21.p2.4.m4.2.2.1.1.1.2" stretchy="false" xref="S6.SS2.21.p2.4.m4.2.2.1.1.1.1.cmml">(</mo><msub id="S6.SS2.21.p2.4.m4.2.2.1.1.1.1" xref="S6.SS2.21.p2.4.m4.2.2.1.1.1.1.cmml"><mi id="S6.SS2.21.p2.4.m4.2.2.1.1.1.1.2" xref="S6.SS2.21.p2.4.m4.2.2.1.1.1.1.2.cmml">b</mi><mi id="S6.SS2.21.p2.4.m4.2.2.1.1.1.1.3" xref="S6.SS2.21.p2.4.m4.2.2.1.1.1.1.3.cmml">l</mi></msub><mo id="S6.SS2.21.p2.4.m4.2.2.1.1.1.3" stretchy="false" xref="S6.SS2.21.p2.4.m4.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.21.p2.4.m4.2b"><apply id="S6.SS2.21.p2.4.m4.2.2.cmml" xref="S6.SS2.21.p2.4.m4.2.2"><eq id="S6.SS2.21.p2.4.m4.2.2.2.cmml" xref="S6.SS2.21.p2.4.m4.2.2.2"></eq><apply id="S6.SS2.21.p2.4.m4.2.2.3.cmml" xref="S6.SS2.21.p2.4.m4.2.2.3"><times id="S6.SS2.21.p2.4.m4.2.2.3.1.cmml" xref="S6.SS2.21.p2.4.m4.2.2.3.1"></times><ci id="S6.SS2.21.p2.4.m4.2.2.3.2.cmml" xref="S6.SS2.21.p2.4.m4.2.2.3.2">𝜈</ci><ci id="S6.SS2.21.p2.4.m4.1.1.cmml" xref="S6.SS2.21.p2.4.m4.1.1">𝑦</ci></apply><apply id="S6.SS2.21.p2.4.m4.2.2.1.cmml" xref="S6.SS2.21.p2.4.m4.2.2.1"><times id="S6.SS2.21.p2.4.m4.2.2.1.2.cmml" xref="S6.SS2.21.p2.4.m4.2.2.1.2"></times><ci id="S6.SS2.21.p2.4.m4.2.2.1.3.cmml" xref="S6.SS2.21.p2.4.m4.2.2.1.3">𝜈</ci><apply id="S6.SS2.21.p2.4.m4.2.2.1.1.1.1.cmml" xref="S6.SS2.21.p2.4.m4.2.2.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.21.p2.4.m4.2.2.1.1.1.1.1.cmml" xref="S6.SS2.21.p2.4.m4.2.2.1.1.1">subscript</csymbol><ci id="S6.SS2.21.p2.4.m4.2.2.1.1.1.1.2.cmml" xref="S6.SS2.21.p2.4.m4.2.2.1.1.1.1.2">𝑏</ci><ci id="S6.SS2.21.p2.4.m4.2.2.1.1.1.1.3.cmml" xref="S6.SS2.21.p2.4.m4.2.2.1.1.1.1.3">𝑙</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.21.p2.4.m4.2c">\nu(y)=\nu(b_{l})</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.21.p2.4.m4.2d">italic_ν ( italic_y ) = italic_ν ( italic_b start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT )</annotation></semantics></math>. If <math alttext="\nu=\nu(y)" class="ltx_Math" display="inline" id="S6.SS2.21.p2.5.m5.1"><semantics id="S6.SS2.21.p2.5.m5.1a"><mrow id="S6.SS2.21.p2.5.m5.1.2" xref="S6.SS2.21.p2.5.m5.1.2.cmml"><mi id="S6.SS2.21.p2.5.m5.1.2.2" xref="S6.SS2.21.p2.5.m5.1.2.2.cmml">ν</mi><mo id="S6.SS2.21.p2.5.m5.1.2.1" xref="S6.SS2.21.p2.5.m5.1.2.1.cmml">=</mo><mrow id="S6.SS2.21.p2.5.m5.1.2.3" xref="S6.SS2.21.p2.5.m5.1.2.3.cmml"><mi id="S6.SS2.21.p2.5.m5.1.2.3.2" xref="S6.SS2.21.p2.5.m5.1.2.3.2.cmml">ν</mi><mo id="S6.SS2.21.p2.5.m5.1.2.3.1" xref="S6.SS2.21.p2.5.m5.1.2.3.1.cmml">⁢</mo><mrow id="S6.SS2.21.p2.5.m5.1.2.3.3.2" xref="S6.SS2.21.p2.5.m5.1.2.3.cmml"><mo id="S6.SS2.21.p2.5.m5.1.2.3.3.2.1" stretchy="false" xref="S6.SS2.21.p2.5.m5.1.2.3.cmml">(</mo><mi id="S6.SS2.21.p2.5.m5.1.1" xref="S6.SS2.21.p2.5.m5.1.1.cmml">y</mi><mo id="S6.SS2.21.p2.5.m5.1.2.3.3.2.2" stretchy="false" xref="S6.SS2.21.p2.5.m5.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.21.p2.5.m5.1b"><apply id="S6.SS2.21.p2.5.m5.1.2.cmml" xref="S6.SS2.21.p2.5.m5.1.2"><eq id="S6.SS2.21.p2.5.m5.1.2.1.cmml" xref="S6.SS2.21.p2.5.m5.1.2.1"></eq><ci id="S6.SS2.21.p2.5.m5.1.2.2.cmml" xref="S6.SS2.21.p2.5.m5.1.2.2">𝜈</ci><apply id="S6.SS2.21.p2.5.m5.1.2.3.cmml" xref="S6.SS2.21.p2.5.m5.1.2.3"><times id="S6.SS2.21.p2.5.m5.1.2.3.1.cmml" xref="S6.SS2.21.p2.5.m5.1.2.3.1"></times><ci id="S6.SS2.21.p2.5.m5.1.2.3.2.cmml" xref="S6.SS2.21.p2.5.m5.1.2.3.2">𝜈</ci><ci id="S6.SS2.21.p2.5.m5.1.1.cmml" xref="S6.SS2.21.p2.5.m5.1.1">𝑦</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.21.p2.5.m5.1c">\nu=\nu(y)</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.21.p2.5.m5.1d">italic_ν = italic_ν ( italic_y )</annotation></semantics></math>, then it follows from <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.Thmtheorem9" title="Definition 6.9. ‣ 6.2. Forcing epimorphisms ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">6.9</span></a> <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.I4.i3" title="Item (iii) ‣ Definition 6.9. ‣ 6.2. Forcing epimorphisms ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">(iii)</span></a> and the fact that <math alttext="y" class="ltx_Math" display="inline" id="S6.SS2.21.p2.6.m6.1"><semantics id="S6.SS2.21.p2.6.m6.1a"><mi id="S6.SS2.21.p2.6.m6.1.1" xref="S6.SS2.21.p2.6.m6.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.21.p2.6.m6.1b"><ci id="S6.SS2.21.p2.6.m6.1.1.cmml" xref="S6.SS2.21.p2.6.m6.1.1">𝑦</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.21.p2.6.m6.1c">y</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.21.p2.6.m6.1d">italic_y</annotation></semantics></math> is not the left endpoint of its complementary interval of <math alttext="X\setminus\nu(\bar{b})" class="ltx_Math" display="inline" id="S6.SS2.21.p2.7.m7.1"><semantics id="S6.SS2.21.p2.7.m7.1a"><mrow id="S6.SS2.21.p2.7.m7.1.2" xref="S6.SS2.21.p2.7.m7.1.2.cmml"><mi id="S6.SS2.21.p2.7.m7.1.2.2" xref="S6.SS2.21.p2.7.m7.1.2.2.cmml">X</mi><mo id="S6.SS2.21.p2.7.m7.1.2.1" xref="S6.SS2.21.p2.7.m7.1.2.1.cmml">∖</mo><mrow id="S6.SS2.21.p2.7.m7.1.2.3" xref="S6.SS2.21.p2.7.m7.1.2.3.cmml"><mi id="S6.SS2.21.p2.7.m7.1.2.3.2" xref="S6.SS2.21.p2.7.m7.1.2.3.2.cmml">ν</mi><mo id="S6.SS2.21.p2.7.m7.1.2.3.1" xref="S6.SS2.21.p2.7.m7.1.2.3.1.cmml">⁢</mo><mrow id="S6.SS2.21.p2.7.m7.1.2.3.3.2" xref="S6.SS2.21.p2.7.m7.1.1.cmml"><mo id="S6.SS2.21.p2.7.m7.1.2.3.3.2.1" stretchy="false" xref="S6.SS2.21.p2.7.m7.1.1.cmml">(</mo><mover accent="true" id="S6.SS2.21.p2.7.m7.1.1" xref="S6.SS2.21.p2.7.m7.1.1.cmml"><mi id="S6.SS2.21.p2.7.m7.1.1.2" xref="S6.SS2.21.p2.7.m7.1.1.2.cmml">b</mi><mo id="S6.SS2.21.p2.7.m7.1.1.1" xref="S6.SS2.21.p2.7.m7.1.1.1.cmml">¯</mo></mover><mo id="S6.SS2.21.p2.7.m7.1.2.3.3.2.2" stretchy="false" xref="S6.SS2.21.p2.7.m7.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.21.p2.7.m7.1b"><apply id="S6.SS2.21.p2.7.m7.1.2.cmml" xref="S6.SS2.21.p2.7.m7.1.2"><setdiff id="S6.SS2.21.p2.7.m7.1.2.1.cmml" xref="S6.SS2.21.p2.7.m7.1.2.1"></setdiff><ci id="S6.SS2.21.p2.7.m7.1.2.2.cmml" xref="S6.SS2.21.p2.7.m7.1.2.2">𝑋</ci><apply id="S6.SS2.21.p2.7.m7.1.2.3.cmml" xref="S6.SS2.21.p2.7.m7.1.2.3"><times id="S6.SS2.21.p2.7.m7.1.2.3.1.cmml" xref="S6.SS2.21.p2.7.m7.1.2.3.1"></times><ci id="S6.SS2.21.p2.7.m7.1.2.3.2.cmml" xref="S6.SS2.21.p2.7.m7.1.2.3.2">𝜈</ci><apply id="S6.SS2.21.p2.7.m7.1.1.cmml" xref="S6.SS2.21.p2.7.m7.1.2.3.3.2"><ci id="S6.SS2.21.p2.7.m7.1.1.1.cmml" xref="S6.SS2.21.p2.7.m7.1.1.1">¯</ci><ci id="S6.SS2.21.p2.7.m7.1.1.2.cmml" xref="S6.SS2.21.p2.7.m7.1.1.2">𝑏</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.21.p2.7.m7.1c">X\setminus\nu(\bar{b})</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.21.p2.7.m7.1d">italic_X ∖ italic_ν ( over¯ start_ARG italic_b end_ARG )</annotation></semantics></math>, since <math alttext="z<_{X}y" class="ltx_Math" display="inline" id="S6.SS2.21.p2.8.m8.1"><semantics id="S6.SS2.21.p2.8.m8.1a"><mrow id="S6.SS2.21.p2.8.m8.1.1" xref="S6.SS2.21.p2.8.m8.1.1.cmml"><mi id="S6.SS2.21.p2.8.m8.1.1.2" xref="S6.SS2.21.p2.8.m8.1.1.2.cmml">z</mi><msub id="S6.SS2.21.p2.8.m8.1.1.1" xref="S6.SS2.21.p2.8.m8.1.1.1.cmml"><mo id="S6.SS2.21.p2.8.m8.1.1.1.2" xref="S6.SS2.21.p2.8.m8.1.1.1.2.cmml"><</mo><mi id="S6.SS2.21.p2.8.m8.1.1.1.3" xref="S6.SS2.21.p2.8.m8.1.1.1.3.cmml">X</mi></msub><mi id="S6.SS2.21.p2.8.m8.1.1.3" xref="S6.SS2.21.p2.8.m8.1.1.3.cmml">y</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.21.p2.8.m8.1b"><apply id="S6.SS2.21.p2.8.m8.1.1.cmml" xref="S6.SS2.21.p2.8.m8.1.1"><apply id="S6.SS2.21.p2.8.m8.1.1.1.cmml" xref="S6.SS2.21.p2.8.m8.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.21.p2.8.m8.1.1.1.1.cmml" xref="S6.SS2.21.p2.8.m8.1.1.1">subscript</csymbol><lt id="S6.SS2.21.p2.8.m8.1.1.1.2.cmml" xref="S6.SS2.21.p2.8.m8.1.1.1.2"></lt><ci id="S6.SS2.21.p2.8.m8.1.1.1.3.cmml" xref="S6.SS2.21.p2.8.m8.1.1.1.3">𝑋</ci></apply><ci id="S6.SS2.21.p2.8.m8.1.1.2.cmml" xref="S6.SS2.21.p2.8.m8.1.1.2">𝑧</ci><ci id="S6.SS2.21.p2.8.m8.1.1.3.cmml" xref="S6.SS2.21.p2.8.m8.1.1.3">𝑦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.21.p2.8.m8.1c">z<_{X}y</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.21.p2.8.m8.1d">italic_z < start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT italic_y</annotation></semantics></math> and <math alttext="\Delta(z,y)=\nu=\nu(y)" class="ltx_Math" display="inline" id="S6.SS2.21.p2.9.m9.3"><semantics id="S6.SS2.21.p2.9.m9.3a"><mrow id="S6.SS2.21.p2.9.m9.3.4" xref="S6.SS2.21.p2.9.m9.3.4.cmml"><mrow id="S6.SS2.21.p2.9.m9.3.4.2" xref="S6.SS2.21.p2.9.m9.3.4.2.cmml"><mi id="S6.SS2.21.p2.9.m9.3.4.2.2" mathvariant="normal" xref="S6.SS2.21.p2.9.m9.3.4.2.2.cmml">Δ</mi><mo id="S6.SS2.21.p2.9.m9.3.4.2.1" xref="S6.SS2.21.p2.9.m9.3.4.2.1.cmml">⁢</mo><mrow id="S6.SS2.21.p2.9.m9.3.4.2.3.2" xref="S6.SS2.21.p2.9.m9.3.4.2.3.1.cmml"><mo id="S6.SS2.21.p2.9.m9.3.4.2.3.2.1" stretchy="false" xref="S6.SS2.21.p2.9.m9.3.4.2.3.1.cmml">(</mo><mi id="S6.SS2.21.p2.9.m9.1.1" xref="S6.SS2.21.p2.9.m9.1.1.cmml">z</mi><mo id="S6.SS2.21.p2.9.m9.3.4.2.3.2.2" xref="S6.SS2.21.p2.9.m9.3.4.2.3.1.cmml">,</mo><mi id="S6.SS2.21.p2.9.m9.2.2" xref="S6.SS2.21.p2.9.m9.2.2.cmml">y</mi><mo id="S6.SS2.21.p2.9.m9.3.4.2.3.2.3" stretchy="false" xref="S6.SS2.21.p2.9.m9.3.4.2.3.1.cmml">)</mo></mrow></mrow><mo id="S6.SS2.21.p2.9.m9.3.4.3" xref="S6.SS2.21.p2.9.m9.3.4.3.cmml">=</mo><mi id="S6.SS2.21.p2.9.m9.3.4.4" xref="S6.SS2.21.p2.9.m9.3.4.4.cmml">ν</mi><mo id="S6.SS2.21.p2.9.m9.3.4.5" xref="S6.SS2.21.p2.9.m9.3.4.5.cmml">=</mo><mrow id="S6.SS2.21.p2.9.m9.3.4.6" xref="S6.SS2.21.p2.9.m9.3.4.6.cmml"><mi id="S6.SS2.21.p2.9.m9.3.4.6.2" xref="S6.SS2.21.p2.9.m9.3.4.6.2.cmml">ν</mi><mo id="S6.SS2.21.p2.9.m9.3.4.6.1" xref="S6.SS2.21.p2.9.m9.3.4.6.1.cmml">⁢</mo><mrow id="S6.SS2.21.p2.9.m9.3.4.6.3.2" xref="S6.SS2.21.p2.9.m9.3.4.6.cmml"><mo id="S6.SS2.21.p2.9.m9.3.4.6.3.2.1" stretchy="false" xref="S6.SS2.21.p2.9.m9.3.4.6.cmml">(</mo><mi id="S6.SS2.21.p2.9.m9.3.3" xref="S6.SS2.21.p2.9.m9.3.3.cmml">y</mi><mo id="S6.SS2.21.p2.9.m9.3.4.6.3.2.2" stretchy="false" xref="S6.SS2.21.p2.9.m9.3.4.6.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.21.p2.9.m9.3b"><apply id="S6.SS2.21.p2.9.m9.3.4.cmml" xref="S6.SS2.21.p2.9.m9.3.4"><and id="S6.SS2.21.p2.9.m9.3.4a.cmml" xref="S6.SS2.21.p2.9.m9.3.4"></and><apply id="S6.SS2.21.p2.9.m9.3.4b.cmml" xref="S6.SS2.21.p2.9.m9.3.4"><eq id="S6.SS2.21.p2.9.m9.3.4.3.cmml" xref="S6.SS2.21.p2.9.m9.3.4.3"></eq><apply id="S6.SS2.21.p2.9.m9.3.4.2.cmml" xref="S6.SS2.21.p2.9.m9.3.4.2"><times id="S6.SS2.21.p2.9.m9.3.4.2.1.cmml" xref="S6.SS2.21.p2.9.m9.3.4.2.1"></times><ci id="S6.SS2.21.p2.9.m9.3.4.2.2.cmml" xref="S6.SS2.21.p2.9.m9.3.4.2.2">Δ</ci><interval closure="open" id="S6.SS2.21.p2.9.m9.3.4.2.3.1.cmml" xref="S6.SS2.21.p2.9.m9.3.4.2.3.2"><ci id="S6.SS2.21.p2.9.m9.1.1.cmml" xref="S6.SS2.21.p2.9.m9.1.1">𝑧</ci><ci id="S6.SS2.21.p2.9.m9.2.2.cmml" xref="S6.SS2.21.p2.9.m9.2.2">𝑦</ci></interval></apply><ci id="S6.SS2.21.p2.9.m9.3.4.4.cmml" xref="S6.SS2.21.p2.9.m9.3.4.4">𝜈</ci></apply><apply id="S6.SS2.21.p2.9.m9.3.4c.cmml" xref="S6.SS2.21.p2.9.m9.3.4"><eq id="S6.SS2.21.p2.9.m9.3.4.5.cmml" xref="S6.SS2.21.p2.9.m9.3.4.5"></eq><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.21.p2.9.m9.3.4.4.cmml" id="S6.SS2.21.p2.9.m9.3.4d.cmml" xref="S6.SS2.21.p2.9.m9.3.4"></share><apply id="S6.SS2.21.p2.9.m9.3.4.6.cmml" xref="S6.SS2.21.p2.9.m9.3.4.6"><times id="S6.SS2.21.p2.9.m9.3.4.6.1.cmml" xref="S6.SS2.21.p2.9.m9.3.4.6.1"></times><ci id="S6.SS2.21.p2.9.m9.3.4.6.2.cmml" xref="S6.SS2.21.p2.9.m9.3.4.6.2">𝜈</ci><ci id="S6.SS2.21.p2.9.m9.3.3.cmml" xref="S6.SS2.21.p2.9.m9.3.3">𝑦</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.21.p2.9.m9.3c">\Delta(z,y)=\nu=\nu(y)</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.21.p2.9.m9.3d">roman_Δ ( italic_z , italic_y ) = italic_ν = italic_ν ( italic_y )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S6.SS2.22.p3"> <p class="ltx_p" id="S6.SS2.22.p3.16">Now we claim that there is <math alttext="c^{\prime}" class="ltx_Math" display="inline" id="S6.SS2.22.p3.1.m1.1"><semantics id="S6.SS2.22.p3.1.m1.1a"><msup id="S6.SS2.22.p3.1.m1.1.1" xref="S6.SS2.22.p3.1.m1.1.1.cmml"><mi id="S6.SS2.22.p3.1.m1.1.1.2" xref="S6.SS2.22.p3.1.m1.1.1.2.cmml">c</mi><mo id="S6.SS2.22.p3.1.m1.1.1.3" xref="S6.SS2.22.p3.1.m1.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S6.SS2.22.p3.1.m1.1b"><apply id="S6.SS2.22.p3.1.m1.1.1.cmml" xref="S6.SS2.22.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S6.SS2.22.p3.1.m1.1.1.1.cmml" xref="S6.SS2.22.p3.1.m1.1.1">superscript</csymbol><ci id="S6.SS2.22.p3.1.m1.1.1.2.cmml" xref="S6.SS2.22.p3.1.m1.1.1.2">𝑐</ci><ci id="S6.SS2.22.p3.1.m1.1.1.3.cmml" xref="S6.SS2.22.p3.1.m1.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.22.p3.1.m1.1c">c^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.22.p3.1.m1.1d">italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> such that <math alttext="\nu(c^{\prime})=\nu" class="ltx_Math" display="inline" id="S6.SS2.22.p3.2.m2.1"><semantics id="S6.SS2.22.p3.2.m2.1a"><mrow id="S6.SS2.22.p3.2.m2.1.1" xref="S6.SS2.22.p3.2.m2.1.1.cmml"><mrow id="S6.SS2.22.p3.2.m2.1.1.1" xref="S6.SS2.22.p3.2.m2.1.1.1.cmml"><mi id="S6.SS2.22.p3.2.m2.1.1.1.3" xref="S6.SS2.22.p3.2.m2.1.1.1.3.cmml">ν</mi><mo id="S6.SS2.22.p3.2.m2.1.1.1.2" xref="S6.SS2.22.p3.2.m2.1.1.1.2.cmml">⁢</mo><mrow id="S6.SS2.22.p3.2.m2.1.1.1.1.1" xref="S6.SS2.22.p3.2.m2.1.1.1.1.1.1.cmml"><mo id="S6.SS2.22.p3.2.m2.1.1.1.1.1.2" stretchy="false" xref="S6.SS2.22.p3.2.m2.1.1.1.1.1.1.cmml">(</mo><msup id="S6.SS2.22.p3.2.m2.1.1.1.1.1.1" xref="S6.SS2.22.p3.2.m2.1.1.1.1.1.1.cmml"><mi id="S6.SS2.22.p3.2.m2.1.1.1.1.1.1.2" xref="S6.SS2.22.p3.2.m2.1.1.1.1.1.1.2.cmml">c</mi><mo id="S6.SS2.22.p3.2.m2.1.1.1.1.1.1.3" xref="S6.SS2.22.p3.2.m2.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S6.SS2.22.p3.2.m2.1.1.1.1.1.3" stretchy="false" xref="S6.SS2.22.p3.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.SS2.22.p3.2.m2.1.1.2" xref="S6.SS2.22.p3.2.m2.1.1.2.cmml">=</mo><mi id="S6.SS2.22.p3.2.m2.1.1.3" xref="S6.SS2.22.p3.2.m2.1.1.3.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.22.p3.2.m2.1b"><apply id="S6.SS2.22.p3.2.m2.1.1.cmml" xref="S6.SS2.22.p3.2.m2.1.1"><eq id="S6.SS2.22.p3.2.m2.1.1.2.cmml" xref="S6.SS2.22.p3.2.m2.1.1.2"></eq><apply id="S6.SS2.22.p3.2.m2.1.1.1.cmml" xref="S6.SS2.22.p3.2.m2.1.1.1"><times id="S6.SS2.22.p3.2.m2.1.1.1.2.cmml" xref="S6.SS2.22.p3.2.m2.1.1.1.2"></times><ci id="S6.SS2.22.p3.2.m2.1.1.1.3.cmml" xref="S6.SS2.22.p3.2.m2.1.1.1.3">𝜈</ci><apply id="S6.SS2.22.p3.2.m2.1.1.1.1.1.1.cmml" xref="S6.SS2.22.p3.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.22.p3.2.m2.1.1.1.1.1.1.1.cmml" xref="S6.SS2.22.p3.2.m2.1.1.1.1.1">superscript</csymbol><ci id="S6.SS2.22.p3.2.m2.1.1.1.1.1.1.2.cmml" xref="S6.SS2.22.p3.2.m2.1.1.1.1.1.1.2">𝑐</ci><ci id="S6.SS2.22.p3.2.m2.1.1.1.1.1.1.3.cmml" xref="S6.SS2.22.p3.2.m2.1.1.1.1.1.1.3">′</ci></apply></apply><ci id="S6.SS2.22.p3.2.m2.1.1.3.cmml" xref="S6.SS2.22.p3.2.m2.1.1.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.22.p3.2.m2.1c">\nu(c^{\prime})=\nu</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.22.p3.2.m2.1d">italic_ν ( italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = italic_ν</annotation></semantics></math>, <math alttext="a_{l}<_{A}c^{\prime}<_{A}b_{l}" class="ltx_Math" display="inline" id="S6.SS2.22.p3.3.m3.1"><semantics id="S6.SS2.22.p3.3.m3.1a"><mrow id="S6.SS2.22.p3.3.m3.1.1" xref="S6.SS2.22.p3.3.m3.1.1.cmml"><msub id="S6.SS2.22.p3.3.m3.1.1.2" xref="S6.SS2.22.p3.3.m3.1.1.2.cmml"><mi id="S6.SS2.22.p3.3.m3.1.1.2.2" xref="S6.SS2.22.p3.3.m3.1.1.2.2.cmml">a</mi><mi id="S6.SS2.22.p3.3.m3.1.1.2.3" xref="S6.SS2.22.p3.3.m3.1.1.2.3.cmml">l</mi></msub><msub id="S6.SS2.22.p3.3.m3.1.1.3" xref="S6.SS2.22.p3.3.m3.1.1.3.cmml"><mo id="S6.SS2.22.p3.3.m3.1.1.3.2" xref="S6.SS2.22.p3.3.m3.1.1.3.2.cmml"><</mo><mi id="S6.SS2.22.p3.3.m3.1.1.3.3" xref="S6.SS2.22.p3.3.m3.1.1.3.3.cmml">A</mi></msub><msup id="S6.SS2.22.p3.3.m3.1.1.4" xref="S6.SS2.22.p3.3.m3.1.1.4.cmml"><mi id="S6.SS2.22.p3.3.m3.1.1.4.2" xref="S6.SS2.22.p3.3.m3.1.1.4.2.cmml">c</mi><mo id="S6.SS2.22.p3.3.m3.1.1.4.3" xref="S6.SS2.22.p3.3.m3.1.1.4.3.cmml">′</mo></msup><msub id="S6.SS2.22.p3.3.m3.1.1.5" xref="S6.SS2.22.p3.3.m3.1.1.5.cmml"><mo id="S6.SS2.22.p3.3.m3.1.1.5.2" xref="S6.SS2.22.p3.3.m3.1.1.5.2.cmml"><</mo><mi id="S6.SS2.22.p3.3.m3.1.1.5.3" xref="S6.SS2.22.p3.3.m3.1.1.5.3.cmml">A</mi></msub><msub id="S6.SS2.22.p3.3.m3.1.1.6" xref="S6.SS2.22.p3.3.m3.1.1.6.cmml"><mi id="S6.SS2.22.p3.3.m3.1.1.6.2" xref="S6.SS2.22.p3.3.m3.1.1.6.2.cmml">b</mi><mi id="S6.SS2.22.p3.3.m3.1.1.6.3" xref="S6.SS2.22.p3.3.m3.1.1.6.3.cmml">l</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.22.p3.3.m3.1b"><apply id="S6.SS2.22.p3.3.m3.1.1.cmml" xref="S6.SS2.22.p3.3.m3.1.1"><and id="S6.SS2.22.p3.3.m3.1.1a.cmml" xref="S6.SS2.22.p3.3.m3.1.1"></and><apply id="S6.SS2.22.p3.3.m3.1.1b.cmml" xref="S6.SS2.22.p3.3.m3.1.1"><apply id="S6.SS2.22.p3.3.m3.1.1.3.cmml" xref="S6.SS2.22.p3.3.m3.1.1.3"><csymbol cd="ambiguous" id="S6.SS2.22.p3.3.m3.1.1.3.1.cmml" xref="S6.SS2.22.p3.3.m3.1.1.3">subscript</csymbol><lt id="S6.SS2.22.p3.3.m3.1.1.3.2.cmml" xref="S6.SS2.22.p3.3.m3.1.1.3.2"></lt><ci id="S6.SS2.22.p3.3.m3.1.1.3.3.cmml" xref="S6.SS2.22.p3.3.m3.1.1.3.3">𝐴</ci></apply><apply id="S6.SS2.22.p3.3.m3.1.1.2.cmml" xref="S6.SS2.22.p3.3.m3.1.1.2"><csymbol cd="ambiguous" id="S6.SS2.22.p3.3.m3.1.1.2.1.cmml" xref="S6.SS2.22.p3.3.m3.1.1.2">subscript</csymbol><ci id="S6.SS2.22.p3.3.m3.1.1.2.2.cmml" xref="S6.SS2.22.p3.3.m3.1.1.2.2">𝑎</ci><ci id="S6.SS2.22.p3.3.m3.1.1.2.3.cmml" xref="S6.SS2.22.p3.3.m3.1.1.2.3">𝑙</ci></apply><apply id="S6.SS2.22.p3.3.m3.1.1.4.cmml" xref="S6.SS2.22.p3.3.m3.1.1.4"><csymbol cd="ambiguous" id="S6.SS2.22.p3.3.m3.1.1.4.1.cmml" xref="S6.SS2.22.p3.3.m3.1.1.4">superscript</csymbol><ci id="S6.SS2.22.p3.3.m3.1.1.4.2.cmml" xref="S6.SS2.22.p3.3.m3.1.1.4.2">𝑐</ci><ci id="S6.SS2.22.p3.3.m3.1.1.4.3.cmml" xref="S6.SS2.22.p3.3.m3.1.1.4.3">′</ci></apply></apply><apply id="S6.SS2.22.p3.3.m3.1.1c.cmml" xref="S6.SS2.22.p3.3.m3.1.1"><apply id="S6.SS2.22.p3.3.m3.1.1.5.cmml" xref="S6.SS2.22.p3.3.m3.1.1.5"><csymbol cd="ambiguous" id="S6.SS2.22.p3.3.m3.1.1.5.1.cmml" xref="S6.SS2.22.p3.3.m3.1.1.5">subscript</csymbol><lt id="S6.SS2.22.p3.3.m3.1.1.5.2.cmml" xref="S6.SS2.22.p3.3.m3.1.1.5.2"></lt><ci id="S6.SS2.22.p3.3.m3.1.1.5.3.cmml" xref="S6.SS2.22.p3.3.m3.1.1.5.3">𝐴</ci></apply><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.22.p3.3.m3.1.1.4.cmml" id="S6.SS2.22.p3.3.m3.1.1d.cmml" xref="S6.SS2.22.p3.3.m3.1.1"></share><apply id="S6.SS2.22.p3.3.m3.1.1.6.cmml" xref="S6.SS2.22.p3.3.m3.1.1.6"><csymbol cd="ambiguous" id="S6.SS2.22.p3.3.m3.1.1.6.1.cmml" xref="S6.SS2.22.p3.3.m3.1.1.6">subscript</csymbol><ci id="S6.SS2.22.p3.3.m3.1.1.6.2.cmml" xref="S6.SS2.22.p3.3.m3.1.1.6.2">𝑏</ci><ci id="S6.SS2.22.p3.3.m3.1.1.6.3.cmml" xref="S6.SS2.22.p3.3.m3.1.1.6.3">𝑙</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.22.p3.3.m3.1c">a_{l}<_{A}c^{\prime}<_{A}b_{l}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.22.p3.3.m3.1d">italic_a start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT < start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT < start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT</annotation></semantics></math>, and such that <math alttext="c^{\prime}" class="ltx_Math" display="inline" id="S6.SS2.22.p3.4.m4.1"><semantics id="S6.SS2.22.p3.4.m4.1a"><msup id="S6.SS2.22.p3.4.m4.1.1" xref="S6.SS2.22.p3.4.m4.1.1.cmml"><mi id="S6.SS2.22.p3.4.m4.1.1.2" xref="S6.SS2.22.p3.4.m4.1.1.2.cmml">c</mi><mo id="S6.SS2.22.p3.4.m4.1.1.3" xref="S6.SS2.22.p3.4.m4.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S6.SS2.22.p3.4.m4.1b"><apply id="S6.SS2.22.p3.4.m4.1.1.cmml" xref="S6.SS2.22.p3.4.m4.1.1"><csymbol cd="ambiguous" id="S6.SS2.22.p3.4.m4.1.1.1.cmml" xref="S6.SS2.22.p3.4.m4.1.1">superscript</csymbol><ci id="S6.SS2.22.p3.4.m4.1.1.2.cmml" xref="S6.SS2.22.p3.4.m4.1.1.2">𝑐</ci><ci id="S6.SS2.22.p3.4.m4.1.1.3.cmml" xref="S6.SS2.22.p3.4.m4.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.22.p3.4.m4.1c">c^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.22.p3.4.m4.1d">italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> is in the same complementary interval of <math alttext="A\setminus\nu" class="ltx_Math" display="inline" id="S6.SS2.22.p3.5.m5.1"><semantics id="S6.SS2.22.p3.5.m5.1a"><mrow id="S6.SS2.22.p3.5.m5.1.1" xref="S6.SS2.22.p3.5.m5.1.1.cmml"><mi id="S6.SS2.22.p3.5.m5.1.1.2" xref="S6.SS2.22.p3.5.m5.1.1.2.cmml">A</mi><mo id="S6.SS2.22.p3.5.m5.1.1.1" xref="S6.SS2.22.p3.5.m5.1.1.1.cmml">∖</mo><mi id="S6.SS2.22.p3.5.m5.1.1.3" xref="S6.SS2.22.p3.5.m5.1.1.3.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.22.p3.5.m5.1b"><apply id="S6.SS2.22.p3.5.m5.1.1.cmml" xref="S6.SS2.22.p3.5.m5.1.1"><setdiff id="S6.SS2.22.p3.5.m5.1.1.1.cmml" xref="S6.SS2.22.p3.5.m5.1.1.1"></setdiff><ci id="S6.SS2.22.p3.5.m5.1.1.2.cmml" xref="S6.SS2.22.p3.5.m5.1.1.2">𝐴</ci><ci id="S6.SS2.22.p3.5.m5.1.1.3.cmml" xref="S6.SS2.22.p3.5.m5.1.1.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.22.p3.5.m5.1c">A\setminus\nu</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.22.p3.5.m5.1d">italic_A ∖ italic_ν</annotation></semantics></math> in which <math alttext="b_{l}" class="ltx_Math" display="inline" id="S6.SS2.22.p3.6.m6.1"><semantics id="S6.SS2.22.p3.6.m6.1a"><msub id="S6.SS2.22.p3.6.m6.1.1" xref="S6.SS2.22.p3.6.m6.1.1.cmml"><mi id="S6.SS2.22.p3.6.m6.1.1.2" xref="S6.SS2.22.p3.6.m6.1.1.2.cmml">b</mi><mi id="S6.SS2.22.p3.6.m6.1.1.3" xref="S6.SS2.22.p3.6.m6.1.1.3.cmml">l</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.22.p3.6.m6.1b"><apply id="S6.SS2.22.p3.6.m6.1.1.cmml" xref="S6.SS2.22.p3.6.m6.1.1"><csymbol cd="ambiguous" id="S6.SS2.22.p3.6.m6.1.1.1.cmml" xref="S6.SS2.22.p3.6.m6.1.1">subscript</csymbol><ci id="S6.SS2.22.p3.6.m6.1.1.2.cmml" xref="S6.SS2.22.p3.6.m6.1.1.2">𝑏</ci><ci id="S6.SS2.22.p3.6.m6.1.1.3.cmml" xref="S6.SS2.22.p3.6.m6.1.1.3">𝑙</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.22.p3.6.m6.1c">b_{l}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.22.p3.6.m6.1d">italic_b start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT</annotation></semantics></math> is. There are two cases, if <math alttext="\nu(a_{r},b_{l})<\nu" class="ltx_Math" display="inline" id="S6.SS2.22.p3.7.m7.2"><semantics id="S6.SS2.22.p3.7.m7.2a"><mrow id="S6.SS2.22.p3.7.m7.2.2" xref="S6.SS2.22.p3.7.m7.2.2.cmml"><mrow id="S6.SS2.22.p3.7.m7.2.2.2" xref="S6.SS2.22.p3.7.m7.2.2.2.cmml"><mi id="S6.SS2.22.p3.7.m7.2.2.2.4" xref="S6.SS2.22.p3.7.m7.2.2.2.4.cmml">ν</mi><mo id="S6.SS2.22.p3.7.m7.2.2.2.3" xref="S6.SS2.22.p3.7.m7.2.2.2.3.cmml">⁢</mo><mrow id="S6.SS2.22.p3.7.m7.2.2.2.2.2" xref="S6.SS2.22.p3.7.m7.2.2.2.2.3.cmml"><mo id="S6.SS2.22.p3.7.m7.2.2.2.2.2.3" stretchy="false" xref="S6.SS2.22.p3.7.m7.2.2.2.2.3.cmml">(</mo><msub id="S6.SS2.22.p3.7.m7.1.1.1.1.1.1" xref="S6.SS2.22.p3.7.m7.1.1.1.1.1.1.cmml"><mi id="S6.SS2.22.p3.7.m7.1.1.1.1.1.1.2" xref="S6.SS2.22.p3.7.m7.1.1.1.1.1.1.2.cmml">a</mi><mi id="S6.SS2.22.p3.7.m7.1.1.1.1.1.1.3" xref="S6.SS2.22.p3.7.m7.1.1.1.1.1.1.3.cmml">r</mi></msub><mo id="S6.SS2.22.p3.7.m7.2.2.2.2.2.4" xref="S6.SS2.22.p3.7.m7.2.2.2.2.3.cmml">,</mo><msub id="S6.SS2.22.p3.7.m7.2.2.2.2.2.2" xref="S6.SS2.22.p3.7.m7.2.2.2.2.2.2.cmml"><mi id="S6.SS2.22.p3.7.m7.2.2.2.2.2.2.2" xref="S6.SS2.22.p3.7.m7.2.2.2.2.2.2.2.cmml">b</mi><mi id="S6.SS2.22.p3.7.m7.2.2.2.2.2.2.3" xref="S6.SS2.22.p3.7.m7.2.2.2.2.2.2.3.cmml">l</mi></msub><mo id="S6.SS2.22.p3.7.m7.2.2.2.2.2.5" stretchy="false" xref="S6.SS2.22.p3.7.m7.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.SS2.22.p3.7.m7.2.2.3" xref="S6.SS2.22.p3.7.m7.2.2.3.cmml"><</mo><mi id="S6.SS2.22.p3.7.m7.2.2.4" xref="S6.SS2.22.p3.7.m7.2.2.4.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.22.p3.7.m7.2b"><apply id="S6.SS2.22.p3.7.m7.2.2.cmml" xref="S6.SS2.22.p3.7.m7.2.2"><lt id="S6.SS2.22.p3.7.m7.2.2.3.cmml" xref="S6.SS2.22.p3.7.m7.2.2.3"></lt><apply id="S6.SS2.22.p3.7.m7.2.2.2.cmml" xref="S6.SS2.22.p3.7.m7.2.2.2"><times id="S6.SS2.22.p3.7.m7.2.2.2.3.cmml" xref="S6.SS2.22.p3.7.m7.2.2.2.3"></times><ci id="S6.SS2.22.p3.7.m7.2.2.2.4.cmml" xref="S6.SS2.22.p3.7.m7.2.2.2.4">𝜈</ci><interval closure="open" id="S6.SS2.22.p3.7.m7.2.2.2.2.3.cmml" xref="S6.SS2.22.p3.7.m7.2.2.2.2.2"><apply id="S6.SS2.22.p3.7.m7.1.1.1.1.1.1.cmml" xref="S6.SS2.22.p3.7.m7.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.22.p3.7.m7.1.1.1.1.1.1.1.cmml" xref="S6.SS2.22.p3.7.m7.1.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.22.p3.7.m7.1.1.1.1.1.1.2.cmml" xref="S6.SS2.22.p3.7.m7.1.1.1.1.1.1.2">𝑎</ci><ci id="S6.SS2.22.p3.7.m7.1.1.1.1.1.1.3.cmml" xref="S6.SS2.22.p3.7.m7.1.1.1.1.1.1.3">𝑟</ci></apply><apply id="S6.SS2.22.p3.7.m7.2.2.2.2.2.2.cmml" xref="S6.SS2.22.p3.7.m7.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.22.p3.7.m7.2.2.2.2.2.2.1.cmml" xref="S6.SS2.22.p3.7.m7.2.2.2.2.2.2">subscript</csymbol><ci id="S6.SS2.22.p3.7.m7.2.2.2.2.2.2.2.cmml" xref="S6.SS2.22.p3.7.m7.2.2.2.2.2.2.2">𝑏</ci><ci id="S6.SS2.22.p3.7.m7.2.2.2.2.2.2.3.cmml" xref="S6.SS2.22.p3.7.m7.2.2.2.2.2.2.3">𝑙</ci></apply></interval></apply><ci id="S6.SS2.22.p3.7.m7.2.2.4.cmml" xref="S6.SS2.22.p3.7.m7.2.2.4">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.22.p3.7.m7.2c">\nu(a_{r},b_{l})<\nu</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.22.p3.7.m7.2d">italic_ν ( italic_a start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ) < italic_ν</annotation></semantics></math> (that is if they are not in the same complementary interval of <math alttext="A\setminus\nu" class="ltx_Math" display="inline" id="S6.SS2.22.p3.8.m8.1"><semantics id="S6.SS2.22.p3.8.m8.1a"><mrow id="S6.SS2.22.p3.8.m8.1.1" xref="S6.SS2.22.p3.8.m8.1.1.cmml"><mi id="S6.SS2.22.p3.8.m8.1.1.2" xref="S6.SS2.22.p3.8.m8.1.1.2.cmml">A</mi><mo id="S6.SS2.22.p3.8.m8.1.1.1" xref="S6.SS2.22.p3.8.m8.1.1.1.cmml">∖</mo><mi id="S6.SS2.22.p3.8.m8.1.1.3" xref="S6.SS2.22.p3.8.m8.1.1.3.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.22.p3.8.m8.1b"><apply id="S6.SS2.22.p3.8.m8.1.1.cmml" xref="S6.SS2.22.p3.8.m8.1.1"><setdiff id="S6.SS2.22.p3.8.m8.1.1.1.cmml" xref="S6.SS2.22.p3.8.m8.1.1.1"></setdiff><ci id="S6.SS2.22.p3.8.m8.1.1.2.cmml" xref="S6.SS2.22.p3.8.m8.1.1.2">𝐴</ci><ci id="S6.SS2.22.p3.8.m8.1.1.3.cmml" xref="S6.SS2.22.p3.8.m8.1.1.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.22.p3.8.m8.1c">A\setminus\nu</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.22.p3.8.m8.1d">italic_A ∖ italic_ν</annotation></semantics></math>) then use <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.Thmtheorem16" title="Lemma 6.16. ‣ 6.2. Forcing epimorphisms ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">6.16</span></a> (e) and the fact that <math alttext="b_{l}" class="ltx_Math" display="inline" id="S6.SS2.22.p3.9.m9.1"><semantics id="S6.SS2.22.p3.9.m9.1a"><msub id="S6.SS2.22.p3.9.m9.1.1" xref="S6.SS2.22.p3.9.m9.1.1.cmml"><mi id="S6.SS2.22.p3.9.m9.1.1.2" xref="S6.SS2.22.p3.9.m9.1.1.2.cmml">b</mi><mi id="S6.SS2.22.p3.9.m9.1.1.3" xref="S6.SS2.22.p3.9.m9.1.1.3.cmml">l</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.22.p3.9.m9.1b"><apply id="S6.SS2.22.p3.9.m9.1.1.cmml" xref="S6.SS2.22.p3.9.m9.1.1"><csymbol cd="ambiguous" id="S6.SS2.22.p3.9.m9.1.1.1.cmml" xref="S6.SS2.22.p3.9.m9.1.1">subscript</csymbol><ci id="S6.SS2.22.p3.9.m9.1.1.2.cmml" xref="S6.SS2.22.p3.9.m9.1.1.2">𝑏</ci><ci id="S6.SS2.22.p3.9.m9.1.1.3.cmml" xref="S6.SS2.22.p3.9.m9.1.1.3">𝑙</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.22.p3.9.m9.1c">b_{l}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.22.p3.9.m9.1d">italic_b start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT</annotation></semantics></math> is not the left endpoint of its complementary interval in <math alttext="A\setminus\nu" class="ltx_Math" display="inline" id="S6.SS2.22.p3.10.m10.1"><semantics id="S6.SS2.22.p3.10.m10.1a"><mrow id="S6.SS2.22.p3.10.m10.1.1" xref="S6.SS2.22.p3.10.m10.1.1.cmml"><mi id="S6.SS2.22.p3.10.m10.1.1.2" xref="S6.SS2.22.p3.10.m10.1.1.2.cmml">A</mi><mo id="S6.SS2.22.p3.10.m10.1.1.1" xref="S6.SS2.22.p3.10.m10.1.1.1.cmml">∖</mo><mi id="S6.SS2.22.p3.10.m10.1.1.3" xref="S6.SS2.22.p3.10.m10.1.1.3.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.22.p3.10.m10.1b"><apply id="S6.SS2.22.p3.10.m10.1.1.cmml" xref="S6.SS2.22.p3.10.m10.1.1"><setdiff id="S6.SS2.22.p3.10.m10.1.1.1.cmml" xref="S6.SS2.22.p3.10.m10.1.1.1"></setdiff><ci id="S6.SS2.22.p3.10.m10.1.1.2.cmml" xref="S6.SS2.22.p3.10.m10.1.1.2">𝐴</ci><ci id="S6.SS2.22.p3.10.m10.1.1.3.cmml" xref="S6.SS2.22.p3.10.m10.1.1.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.22.p3.10.m10.1c">A\setminus\nu</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.22.p3.10.m10.1d">italic_A ∖ italic_ν</annotation></semantics></math>. If <math alttext="\nu(a_{r},b_{l})=\nu" class="ltx_Math" display="inline" id="S6.SS2.22.p3.11.m11.2"><semantics id="S6.SS2.22.p3.11.m11.2a"><mrow id="S6.SS2.22.p3.11.m11.2.2" xref="S6.SS2.22.p3.11.m11.2.2.cmml"><mrow id="S6.SS2.22.p3.11.m11.2.2.2" xref="S6.SS2.22.p3.11.m11.2.2.2.cmml"><mi id="S6.SS2.22.p3.11.m11.2.2.2.4" xref="S6.SS2.22.p3.11.m11.2.2.2.4.cmml">ν</mi><mo id="S6.SS2.22.p3.11.m11.2.2.2.3" xref="S6.SS2.22.p3.11.m11.2.2.2.3.cmml">⁢</mo><mrow id="S6.SS2.22.p3.11.m11.2.2.2.2.2" xref="S6.SS2.22.p3.11.m11.2.2.2.2.3.cmml"><mo id="S6.SS2.22.p3.11.m11.2.2.2.2.2.3" stretchy="false" xref="S6.SS2.22.p3.11.m11.2.2.2.2.3.cmml">(</mo><msub id="S6.SS2.22.p3.11.m11.1.1.1.1.1.1" xref="S6.SS2.22.p3.11.m11.1.1.1.1.1.1.cmml"><mi id="S6.SS2.22.p3.11.m11.1.1.1.1.1.1.2" xref="S6.SS2.22.p3.11.m11.1.1.1.1.1.1.2.cmml">a</mi><mi id="S6.SS2.22.p3.11.m11.1.1.1.1.1.1.3" xref="S6.SS2.22.p3.11.m11.1.1.1.1.1.1.3.cmml">r</mi></msub><mo id="S6.SS2.22.p3.11.m11.2.2.2.2.2.4" xref="S6.SS2.22.p3.11.m11.2.2.2.2.3.cmml">,</mo><msub id="S6.SS2.22.p3.11.m11.2.2.2.2.2.2" xref="S6.SS2.22.p3.11.m11.2.2.2.2.2.2.cmml"><mi id="S6.SS2.22.p3.11.m11.2.2.2.2.2.2.2" xref="S6.SS2.22.p3.11.m11.2.2.2.2.2.2.2.cmml">b</mi><mi id="S6.SS2.22.p3.11.m11.2.2.2.2.2.2.3" xref="S6.SS2.22.p3.11.m11.2.2.2.2.2.2.3.cmml">l</mi></msub><mo id="S6.SS2.22.p3.11.m11.2.2.2.2.2.5" stretchy="false" xref="S6.SS2.22.p3.11.m11.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.SS2.22.p3.11.m11.2.2.3" xref="S6.SS2.22.p3.11.m11.2.2.3.cmml">=</mo><mi id="S6.SS2.22.p3.11.m11.2.2.4" xref="S6.SS2.22.p3.11.m11.2.2.4.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.22.p3.11.m11.2b"><apply id="S6.SS2.22.p3.11.m11.2.2.cmml" xref="S6.SS2.22.p3.11.m11.2.2"><eq id="S6.SS2.22.p3.11.m11.2.2.3.cmml" xref="S6.SS2.22.p3.11.m11.2.2.3"></eq><apply id="S6.SS2.22.p3.11.m11.2.2.2.cmml" xref="S6.SS2.22.p3.11.m11.2.2.2"><times id="S6.SS2.22.p3.11.m11.2.2.2.3.cmml" xref="S6.SS2.22.p3.11.m11.2.2.2.3"></times><ci id="S6.SS2.22.p3.11.m11.2.2.2.4.cmml" xref="S6.SS2.22.p3.11.m11.2.2.2.4">𝜈</ci><interval closure="open" id="S6.SS2.22.p3.11.m11.2.2.2.2.3.cmml" xref="S6.SS2.22.p3.11.m11.2.2.2.2.2"><apply id="S6.SS2.22.p3.11.m11.1.1.1.1.1.1.cmml" xref="S6.SS2.22.p3.11.m11.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.22.p3.11.m11.1.1.1.1.1.1.1.cmml" xref="S6.SS2.22.p3.11.m11.1.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.22.p3.11.m11.1.1.1.1.1.1.2.cmml" xref="S6.SS2.22.p3.11.m11.1.1.1.1.1.1.2">𝑎</ci><ci id="S6.SS2.22.p3.11.m11.1.1.1.1.1.1.3.cmml" xref="S6.SS2.22.p3.11.m11.1.1.1.1.1.1.3">𝑟</ci></apply><apply id="S6.SS2.22.p3.11.m11.2.2.2.2.2.2.cmml" xref="S6.SS2.22.p3.11.m11.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.22.p3.11.m11.2.2.2.2.2.2.1.cmml" xref="S6.SS2.22.p3.11.m11.2.2.2.2.2.2">subscript</csymbol><ci id="S6.SS2.22.p3.11.m11.2.2.2.2.2.2.2.cmml" xref="S6.SS2.22.p3.11.m11.2.2.2.2.2.2.2">𝑏</ci><ci id="S6.SS2.22.p3.11.m11.2.2.2.2.2.2.3.cmml" xref="S6.SS2.22.p3.11.m11.2.2.2.2.2.2.3">𝑙</ci></apply></interval></apply><ci id="S6.SS2.22.p3.11.m11.2.2.4.cmml" xref="S6.SS2.22.p3.11.m11.2.2.4">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.22.p3.11.m11.2c">\nu(a_{r},b_{l})=\nu</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.22.p3.11.m11.2d">italic_ν ( italic_a start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ) = italic_ν</annotation></semantics></math>, then <math alttext="\Delta(a_{r},b_{l})<\nu^{+}" class="ltx_Math" display="inline" id="S6.SS2.22.p3.12.m12.2"><semantics id="S6.SS2.22.p3.12.m12.2a"><mrow id="S6.SS2.22.p3.12.m12.2.2" xref="S6.SS2.22.p3.12.m12.2.2.cmml"><mrow id="S6.SS2.22.p3.12.m12.2.2.2" xref="S6.SS2.22.p3.12.m12.2.2.2.cmml"><mi id="S6.SS2.22.p3.12.m12.2.2.2.4" mathvariant="normal" xref="S6.SS2.22.p3.12.m12.2.2.2.4.cmml">Δ</mi><mo id="S6.SS2.22.p3.12.m12.2.2.2.3" xref="S6.SS2.22.p3.12.m12.2.2.2.3.cmml">⁢</mo><mrow id="S6.SS2.22.p3.12.m12.2.2.2.2.2" xref="S6.SS2.22.p3.12.m12.2.2.2.2.3.cmml"><mo id="S6.SS2.22.p3.12.m12.2.2.2.2.2.3" stretchy="false" xref="S6.SS2.22.p3.12.m12.2.2.2.2.3.cmml">(</mo><msub id="S6.SS2.22.p3.12.m12.1.1.1.1.1.1" xref="S6.SS2.22.p3.12.m12.1.1.1.1.1.1.cmml"><mi id="S6.SS2.22.p3.12.m12.1.1.1.1.1.1.2" xref="S6.SS2.22.p3.12.m12.1.1.1.1.1.1.2.cmml">a</mi><mi id="S6.SS2.22.p3.12.m12.1.1.1.1.1.1.3" xref="S6.SS2.22.p3.12.m12.1.1.1.1.1.1.3.cmml">r</mi></msub><mo id="S6.SS2.22.p3.12.m12.2.2.2.2.2.4" xref="S6.SS2.22.p3.12.m12.2.2.2.2.3.cmml">,</mo><msub id="S6.SS2.22.p3.12.m12.2.2.2.2.2.2" xref="S6.SS2.22.p3.12.m12.2.2.2.2.2.2.cmml"><mi id="S6.SS2.22.p3.12.m12.2.2.2.2.2.2.2" xref="S6.SS2.22.p3.12.m12.2.2.2.2.2.2.2.cmml">b</mi><mi id="S6.SS2.22.p3.12.m12.2.2.2.2.2.2.3" xref="S6.SS2.22.p3.12.m12.2.2.2.2.2.2.3.cmml">l</mi></msub><mo id="S6.SS2.22.p3.12.m12.2.2.2.2.2.5" stretchy="false" xref="S6.SS2.22.p3.12.m12.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.SS2.22.p3.12.m12.2.2.3" xref="S6.SS2.22.p3.12.m12.2.2.3.cmml"><</mo><msup id="S6.SS2.22.p3.12.m12.2.2.4" xref="S6.SS2.22.p3.12.m12.2.2.4.cmml"><mi id="S6.SS2.22.p3.12.m12.2.2.4.2" xref="S6.SS2.22.p3.12.m12.2.2.4.2.cmml">ν</mi><mo id="S6.SS2.22.p3.12.m12.2.2.4.3" xref="S6.SS2.22.p3.12.m12.2.2.4.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.22.p3.12.m12.2b"><apply id="S6.SS2.22.p3.12.m12.2.2.cmml" xref="S6.SS2.22.p3.12.m12.2.2"><lt id="S6.SS2.22.p3.12.m12.2.2.3.cmml" xref="S6.SS2.22.p3.12.m12.2.2.3"></lt><apply id="S6.SS2.22.p3.12.m12.2.2.2.cmml" xref="S6.SS2.22.p3.12.m12.2.2.2"><times id="S6.SS2.22.p3.12.m12.2.2.2.3.cmml" xref="S6.SS2.22.p3.12.m12.2.2.2.3"></times><ci id="S6.SS2.22.p3.12.m12.2.2.2.4.cmml" xref="S6.SS2.22.p3.12.m12.2.2.2.4">Δ</ci><interval closure="open" id="S6.SS2.22.p3.12.m12.2.2.2.2.3.cmml" xref="S6.SS2.22.p3.12.m12.2.2.2.2.2"><apply id="S6.SS2.22.p3.12.m12.1.1.1.1.1.1.cmml" xref="S6.SS2.22.p3.12.m12.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.22.p3.12.m12.1.1.1.1.1.1.1.cmml" xref="S6.SS2.22.p3.12.m12.1.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.22.p3.12.m12.1.1.1.1.1.1.2.cmml" xref="S6.SS2.22.p3.12.m12.1.1.1.1.1.1.2">𝑎</ci><ci id="S6.SS2.22.p3.12.m12.1.1.1.1.1.1.3.cmml" xref="S6.SS2.22.p3.12.m12.1.1.1.1.1.1.3">𝑟</ci></apply><apply id="S6.SS2.22.p3.12.m12.2.2.2.2.2.2.cmml" xref="S6.SS2.22.p3.12.m12.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.22.p3.12.m12.2.2.2.2.2.2.1.cmml" xref="S6.SS2.22.p3.12.m12.2.2.2.2.2.2">subscript</csymbol><ci id="S6.SS2.22.p3.12.m12.2.2.2.2.2.2.2.cmml" xref="S6.SS2.22.p3.12.m12.2.2.2.2.2.2.2">𝑏</ci><ci id="S6.SS2.22.p3.12.m12.2.2.2.2.2.2.3.cmml" xref="S6.SS2.22.p3.12.m12.2.2.2.2.2.2.3">𝑙</ci></apply></interval></apply><apply id="S6.SS2.22.p3.12.m12.2.2.4.cmml" xref="S6.SS2.22.p3.12.m12.2.2.4"><csymbol cd="ambiguous" id="S6.SS2.22.p3.12.m12.2.2.4.1.cmml" xref="S6.SS2.22.p3.12.m12.2.2.4">superscript</csymbol><ci id="S6.SS2.22.p3.12.m12.2.2.4.2.cmml" xref="S6.SS2.22.p3.12.m12.2.2.4.2">𝜈</ci><plus id="S6.SS2.22.p3.12.m12.2.2.4.3.cmml" xref="S6.SS2.22.p3.12.m12.2.2.4.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.22.p3.12.m12.2c">\Delta(a_{r},b_{l})<\nu^{+}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.22.p3.12.m12.2d">roman_Δ ( italic_a start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ) < italic_ν start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="a_{l}\leq_{A}\Delta(a_{r},b_{l})<_{A}b_{r}" class="ltx_Math" display="inline" id="S6.SS2.22.p3.13.m13.2"><semantics id="S6.SS2.22.p3.13.m13.2a"><mrow id="S6.SS2.22.p3.13.m13.2.2" xref="S6.SS2.22.p3.13.m13.2.2.cmml"><msub id="S6.SS2.22.p3.13.m13.2.2.4" xref="S6.SS2.22.p3.13.m13.2.2.4.cmml"><mi id="S6.SS2.22.p3.13.m13.2.2.4.2" xref="S6.SS2.22.p3.13.m13.2.2.4.2.cmml">a</mi><mi id="S6.SS2.22.p3.13.m13.2.2.4.3" xref="S6.SS2.22.p3.13.m13.2.2.4.3.cmml">l</mi></msub><msub id="S6.SS2.22.p3.13.m13.2.2.5" xref="S6.SS2.22.p3.13.m13.2.2.5.cmml"><mo id="S6.SS2.22.p3.13.m13.2.2.5.2" xref="S6.SS2.22.p3.13.m13.2.2.5.2.cmml">≤</mo><mi id="S6.SS2.22.p3.13.m13.2.2.5.3" xref="S6.SS2.22.p3.13.m13.2.2.5.3.cmml">A</mi></msub><mrow id="S6.SS2.22.p3.13.m13.2.2.2" xref="S6.SS2.22.p3.13.m13.2.2.2.cmml"><mi id="S6.SS2.22.p3.13.m13.2.2.2.4" mathvariant="normal" xref="S6.SS2.22.p3.13.m13.2.2.2.4.cmml">Δ</mi><mo id="S6.SS2.22.p3.13.m13.2.2.2.3" xref="S6.SS2.22.p3.13.m13.2.2.2.3.cmml">⁢</mo><mrow id="S6.SS2.22.p3.13.m13.2.2.2.2.2" xref="S6.SS2.22.p3.13.m13.2.2.2.2.3.cmml"><mo id="S6.SS2.22.p3.13.m13.2.2.2.2.2.3" stretchy="false" xref="S6.SS2.22.p3.13.m13.2.2.2.2.3.cmml">(</mo><msub id="S6.SS2.22.p3.13.m13.1.1.1.1.1.1" xref="S6.SS2.22.p3.13.m13.1.1.1.1.1.1.cmml"><mi id="S6.SS2.22.p3.13.m13.1.1.1.1.1.1.2" xref="S6.SS2.22.p3.13.m13.1.1.1.1.1.1.2.cmml">a</mi><mi id="S6.SS2.22.p3.13.m13.1.1.1.1.1.1.3" xref="S6.SS2.22.p3.13.m13.1.1.1.1.1.1.3.cmml">r</mi></msub><mo id="S6.SS2.22.p3.13.m13.2.2.2.2.2.4" xref="S6.SS2.22.p3.13.m13.2.2.2.2.3.cmml">,</mo><msub id="S6.SS2.22.p3.13.m13.2.2.2.2.2.2" xref="S6.SS2.22.p3.13.m13.2.2.2.2.2.2.cmml"><mi id="S6.SS2.22.p3.13.m13.2.2.2.2.2.2.2" xref="S6.SS2.22.p3.13.m13.2.2.2.2.2.2.2.cmml">b</mi><mi id="S6.SS2.22.p3.13.m13.2.2.2.2.2.2.3" xref="S6.SS2.22.p3.13.m13.2.2.2.2.2.2.3.cmml">l</mi></msub><mo id="S6.SS2.22.p3.13.m13.2.2.2.2.2.5" stretchy="false" xref="S6.SS2.22.p3.13.m13.2.2.2.2.3.cmml">)</mo></mrow></mrow><msub id="S6.SS2.22.p3.13.m13.2.2.6" xref="S6.SS2.22.p3.13.m13.2.2.6.cmml"><mo id="S6.SS2.22.p3.13.m13.2.2.6.2" xref="S6.SS2.22.p3.13.m13.2.2.6.2.cmml"><</mo><mi id="S6.SS2.22.p3.13.m13.2.2.6.3" xref="S6.SS2.22.p3.13.m13.2.2.6.3.cmml">A</mi></msub><msub id="S6.SS2.22.p3.13.m13.2.2.7" xref="S6.SS2.22.p3.13.m13.2.2.7.cmml"><mi id="S6.SS2.22.p3.13.m13.2.2.7.2" xref="S6.SS2.22.p3.13.m13.2.2.7.2.cmml">b</mi><mi id="S6.SS2.22.p3.13.m13.2.2.7.3" xref="S6.SS2.22.p3.13.m13.2.2.7.3.cmml">r</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.22.p3.13.m13.2b"><apply id="S6.SS2.22.p3.13.m13.2.2.cmml" xref="S6.SS2.22.p3.13.m13.2.2"><and id="S6.SS2.22.p3.13.m13.2.2a.cmml" xref="S6.SS2.22.p3.13.m13.2.2"></and><apply id="S6.SS2.22.p3.13.m13.2.2b.cmml" xref="S6.SS2.22.p3.13.m13.2.2"><apply id="S6.SS2.22.p3.13.m13.2.2.5.cmml" xref="S6.SS2.22.p3.13.m13.2.2.5"><csymbol cd="ambiguous" id="S6.SS2.22.p3.13.m13.2.2.5.1.cmml" xref="S6.SS2.22.p3.13.m13.2.2.5">subscript</csymbol><leq id="S6.SS2.22.p3.13.m13.2.2.5.2.cmml" xref="S6.SS2.22.p3.13.m13.2.2.5.2"></leq><ci id="S6.SS2.22.p3.13.m13.2.2.5.3.cmml" xref="S6.SS2.22.p3.13.m13.2.2.5.3">𝐴</ci></apply><apply id="S6.SS2.22.p3.13.m13.2.2.4.cmml" xref="S6.SS2.22.p3.13.m13.2.2.4"><csymbol cd="ambiguous" id="S6.SS2.22.p3.13.m13.2.2.4.1.cmml" xref="S6.SS2.22.p3.13.m13.2.2.4">subscript</csymbol><ci id="S6.SS2.22.p3.13.m13.2.2.4.2.cmml" xref="S6.SS2.22.p3.13.m13.2.2.4.2">𝑎</ci><ci id="S6.SS2.22.p3.13.m13.2.2.4.3.cmml" xref="S6.SS2.22.p3.13.m13.2.2.4.3">𝑙</ci></apply><apply id="S6.SS2.22.p3.13.m13.2.2.2.cmml" xref="S6.SS2.22.p3.13.m13.2.2.2"><times id="S6.SS2.22.p3.13.m13.2.2.2.3.cmml" xref="S6.SS2.22.p3.13.m13.2.2.2.3"></times><ci id="S6.SS2.22.p3.13.m13.2.2.2.4.cmml" xref="S6.SS2.22.p3.13.m13.2.2.2.4">Δ</ci><interval closure="open" id="S6.SS2.22.p3.13.m13.2.2.2.2.3.cmml" xref="S6.SS2.22.p3.13.m13.2.2.2.2.2"><apply id="S6.SS2.22.p3.13.m13.1.1.1.1.1.1.cmml" xref="S6.SS2.22.p3.13.m13.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.22.p3.13.m13.1.1.1.1.1.1.1.cmml" xref="S6.SS2.22.p3.13.m13.1.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.22.p3.13.m13.1.1.1.1.1.1.2.cmml" xref="S6.SS2.22.p3.13.m13.1.1.1.1.1.1.2">𝑎</ci><ci id="S6.SS2.22.p3.13.m13.1.1.1.1.1.1.3.cmml" xref="S6.SS2.22.p3.13.m13.1.1.1.1.1.1.3">𝑟</ci></apply><apply id="S6.SS2.22.p3.13.m13.2.2.2.2.2.2.cmml" xref="S6.SS2.22.p3.13.m13.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.22.p3.13.m13.2.2.2.2.2.2.1.cmml" xref="S6.SS2.22.p3.13.m13.2.2.2.2.2.2">subscript</csymbol><ci id="S6.SS2.22.p3.13.m13.2.2.2.2.2.2.2.cmml" xref="S6.SS2.22.p3.13.m13.2.2.2.2.2.2.2">𝑏</ci><ci id="S6.SS2.22.p3.13.m13.2.2.2.2.2.2.3.cmml" xref="S6.SS2.22.p3.13.m13.2.2.2.2.2.2.3">𝑙</ci></apply></interval></apply></apply><apply id="S6.SS2.22.p3.13.m13.2.2c.cmml" xref="S6.SS2.22.p3.13.m13.2.2"><apply id="S6.SS2.22.p3.13.m13.2.2.6.cmml" xref="S6.SS2.22.p3.13.m13.2.2.6"><csymbol cd="ambiguous" id="S6.SS2.22.p3.13.m13.2.2.6.1.cmml" xref="S6.SS2.22.p3.13.m13.2.2.6">subscript</csymbol><lt id="S6.SS2.22.p3.13.m13.2.2.6.2.cmml" xref="S6.SS2.22.p3.13.m13.2.2.6.2"></lt><ci id="S6.SS2.22.p3.13.m13.2.2.6.3.cmml" xref="S6.SS2.22.p3.13.m13.2.2.6.3">𝐴</ci></apply><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.22.p3.13.m13.2.2.2.cmml" id="S6.SS2.22.p3.13.m13.2.2d.cmml" xref="S6.SS2.22.p3.13.m13.2.2"></share><apply id="S6.SS2.22.p3.13.m13.2.2.7.cmml" xref="S6.SS2.22.p3.13.m13.2.2.7"><csymbol cd="ambiguous" id="S6.SS2.22.p3.13.m13.2.2.7.1.cmml" xref="S6.SS2.22.p3.13.m13.2.2.7">subscript</csymbol><ci id="S6.SS2.22.p3.13.m13.2.2.7.2.cmml" xref="S6.SS2.22.p3.13.m13.2.2.7.2">𝑏</ci><ci id="S6.SS2.22.p3.13.m13.2.2.7.3.cmml" xref="S6.SS2.22.p3.13.m13.2.2.7.3">𝑟</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.22.p3.13.m13.2c">a_{l}\leq_{A}\Delta(a_{r},b_{l})<_{A}b_{r}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.22.p3.13.m13.2d">italic_a start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ≤ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT roman_Δ ( italic_a start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ) < start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT</annotation></semantics></math>, and then one can use <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.Thmtheorem16" title="Lemma 6.16. ‣ 6.2. Forcing epimorphisms ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">6.16</span></a> (b) and <math alttext="\aleph_{1}" class="ltx_Math" display="inline" id="S6.SS2.22.p3.14.m14.1"><semantics id="S6.SS2.22.p3.14.m14.1a"><msub id="S6.SS2.22.p3.14.m14.1.1" xref="S6.SS2.22.p3.14.m14.1.1.cmml"><mi id="S6.SS2.22.p3.14.m14.1.1.2" mathvariant="normal" xref="S6.SS2.22.p3.14.m14.1.1.2.cmml">ℵ</mi><mn id="S6.SS2.22.p3.14.m14.1.1.3" xref="S6.SS2.22.p3.14.m14.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.22.p3.14.m14.1b"><apply id="S6.SS2.22.p3.14.m14.1.1.cmml" xref="S6.SS2.22.p3.14.m14.1.1"><csymbol cd="ambiguous" id="S6.SS2.22.p3.14.m14.1.1.1.cmml" xref="S6.SS2.22.p3.14.m14.1.1">subscript</csymbol><ci id="S6.SS2.22.p3.14.m14.1.1.2.cmml" xref="S6.SS2.22.p3.14.m14.1.1.2">ℵ</ci><cn id="S6.SS2.22.p3.14.m14.1.1.3.cmml" type="integer" xref="S6.SS2.22.p3.14.m14.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.22.p3.14.m14.1c">\aleph_{1}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.22.p3.14.m14.1d">roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-density to approximate <math alttext="\Delta(a_{r},b_{l})" class="ltx_Math" display="inline" id="S6.SS2.22.p3.15.m15.2"><semantics id="S6.SS2.22.p3.15.m15.2a"><mrow id="S6.SS2.22.p3.15.m15.2.2" xref="S6.SS2.22.p3.15.m15.2.2.cmml"><mi id="S6.SS2.22.p3.15.m15.2.2.4" mathvariant="normal" xref="S6.SS2.22.p3.15.m15.2.2.4.cmml">Δ</mi><mo id="S6.SS2.22.p3.15.m15.2.2.3" xref="S6.SS2.22.p3.15.m15.2.2.3.cmml">⁢</mo><mrow id="S6.SS2.22.p3.15.m15.2.2.2.2" xref="S6.SS2.22.p3.15.m15.2.2.2.3.cmml"><mo id="S6.SS2.22.p3.15.m15.2.2.2.2.3" stretchy="false" xref="S6.SS2.22.p3.15.m15.2.2.2.3.cmml">(</mo><msub id="S6.SS2.22.p3.15.m15.1.1.1.1.1" xref="S6.SS2.22.p3.15.m15.1.1.1.1.1.cmml"><mi id="S6.SS2.22.p3.15.m15.1.1.1.1.1.2" xref="S6.SS2.22.p3.15.m15.1.1.1.1.1.2.cmml">a</mi><mi id="S6.SS2.22.p3.15.m15.1.1.1.1.1.3" xref="S6.SS2.22.p3.15.m15.1.1.1.1.1.3.cmml">r</mi></msub><mo id="S6.SS2.22.p3.15.m15.2.2.2.2.4" xref="S6.SS2.22.p3.15.m15.2.2.2.3.cmml">,</mo><msub id="S6.SS2.22.p3.15.m15.2.2.2.2.2" xref="S6.SS2.22.p3.15.m15.2.2.2.2.2.cmml"><mi id="S6.SS2.22.p3.15.m15.2.2.2.2.2.2" xref="S6.SS2.22.p3.15.m15.2.2.2.2.2.2.cmml">b</mi><mi id="S6.SS2.22.p3.15.m15.2.2.2.2.2.3" xref="S6.SS2.22.p3.15.m15.2.2.2.2.2.3.cmml">l</mi></msub><mo id="S6.SS2.22.p3.15.m15.2.2.2.2.5" stretchy="false" xref="S6.SS2.22.p3.15.m15.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.22.p3.15.m15.2b"><apply id="S6.SS2.22.p3.15.m15.2.2.cmml" xref="S6.SS2.22.p3.15.m15.2.2"><times id="S6.SS2.22.p3.15.m15.2.2.3.cmml" xref="S6.SS2.22.p3.15.m15.2.2.3"></times><ci id="S6.SS2.22.p3.15.m15.2.2.4.cmml" xref="S6.SS2.22.p3.15.m15.2.2.4">Δ</ci><interval closure="open" id="S6.SS2.22.p3.15.m15.2.2.2.3.cmml" xref="S6.SS2.22.p3.15.m15.2.2.2.2"><apply id="S6.SS2.22.p3.15.m15.1.1.1.1.1.cmml" xref="S6.SS2.22.p3.15.m15.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.22.p3.15.m15.1.1.1.1.1.1.cmml" xref="S6.SS2.22.p3.15.m15.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.22.p3.15.m15.1.1.1.1.1.2.cmml" xref="S6.SS2.22.p3.15.m15.1.1.1.1.1.2">𝑎</ci><ci id="S6.SS2.22.p3.15.m15.1.1.1.1.1.3.cmml" xref="S6.SS2.22.p3.15.m15.1.1.1.1.1.3">𝑟</ci></apply><apply id="S6.SS2.22.p3.15.m15.2.2.2.2.2.cmml" xref="S6.SS2.22.p3.15.m15.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.22.p3.15.m15.2.2.2.2.2.1.cmml" xref="S6.SS2.22.p3.15.m15.2.2.2.2.2">subscript</csymbol><ci id="S6.SS2.22.p3.15.m15.2.2.2.2.2.2.cmml" xref="S6.SS2.22.p3.15.m15.2.2.2.2.2.2">𝑏</ci><ci id="S6.SS2.22.p3.15.m15.2.2.2.2.2.3.cmml" xref="S6.SS2.22.p3.15.m15.2.2.2.2.2.3">𝑙</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.22.p3.15.m15.2c">\Delta(a_{r},b_{l})</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.22.p3.15.m15.2d">roman_Δ ( italic_a start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT )</annotation></semantics></math> from the right, thus finding <math alttext="c^{\prime}" class="ltx_Math" display="inline" id="S6.SS2.22.p3.16.m16.1"><semantics id="S6.SS2.22.p3.16.m16.1a"><msup id="S6.SS2.22.p3.16.m16.1.1" xref="S6.SS2.22.p3.16.m16.1.1.cmml"><mi id="S6.SS2.22.p3.16.m16.1.1.2" xref="S6.SS2.22.p3.16.m16.1.1.2.cmml">c</mi><mo id="S6.SS2.22.p3.16.m16.1.1.3" xref="S6.SS2.22.p3.16.m16.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S6.SS2.22.p3.16.m16.1b"><apply id="S6.SS2.22.p3.16.m16.1.1.cmml" xref="S6.SS2.22.p3.16.m16.1.1"><csymbol cd="ambiguous" id="S6.SS2.22.p3.16.m16.1.1.1.cmml" xref="S6.SS2.22.p3.16.m16.1.1">superscript</csymbol><ci id="S6.SS2.22.p3.16.m16.1.1.2.cmml" xref="S6.SS2.22.p3.16.m16.1.1.2">𝑐</ci><ci id="S6.SS2.22.p3.16.m16.1.1.3.cmml" xref="S6.SS2.22.p3.16.m16.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.22.p3.16.m16.1c">c^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.22.p3.16.m16.1d">italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S6.SS2.23.p4"> <p class="ltx_p" id="S6.SS2.23.p4.16">Repeating the above argument, and using the <math alttext="\aleph_{1}" class="ltx_Math" display="inline" id="S6.SS2.23.p4.1.m1.1"><semantics id="S6.SS2.23.p4.1.m1.1a"><msub id="S6.SS2.23.p4.1.m1.1.1" xref="S6.SS2.23.p4.1.m1.1.1.cmml"><mi id="S6.SS2.23.p4.1.m1.1.1.2" mathvariant="normal" xref="S6.SS2.23.p4.1.m1.1.1.2.cmml">ℵ</mi><mn id="S6.SS2.23.p4.1.m1.1.1.3" xref="S6.SS2.23.p4.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.23.p4.1.m1.1b"><apply id="S6.SS2.23.p4.1.m1.1.1.cmml" xref="S6.SS2.23.p4.1.m1.1.1"><csymbol cd="ambiguous" id="S6.SS2.23.p4.1.m1.1.1.1.cmml" xref="S6.SS2.23.p4.1.m1.1.1">subscript</csymbol><ci id="S6.SS2.23.p4.1.m1.1.1.2.cmml" xref="S6.SS2.23.p4.1.m1.1.1.2">ℵ</ci><cn id="S6.SS2.23.p4.1.m1.1.1.3.cmml" type="integer" xref="S6.SS2.23.p4.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.23.p4.1.m1.1c">\aleph_{1}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.23.p4.1.m1.1d">roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-density of <math alttext="A" class="ltx_Math" display="inline" id="S6.SS2.23.p4.2.m2.1"><semantics id="S6.SS2.23.p4.2.m2.1a"><mi id="S6.SS2.23.p4.2.m2.1.1" xref="S6.SS2.23.p4.2.m2.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.23.p4.2.m2.1b"><ci id="S6.SS2.23.p4.2.m2.1.1.cmml" xref="S6.SS2.23.p4.2.m2.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.23.p4.2.m2.1c">A</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.23.p4.2.m2.1d">italic_A</annotation></semantics></math>, we find <math alttext="c^{\prime\prime}" class="ltx_Math" display="inline" id="S6.SS2.23.p4.3.m3.1"><semantics id="S6.SS2.23.p4.3.m3.1a"><msup id="S6.SS2.23.p4.3.m3.1.1" xref="S6.SS2.23.p4.3.m3.1.1.cmml"><mi id="S6.SS2.23.p4.3.m3.1.1.2" xref="S6.SS2.23.p4.3.m3.1.1.2.cmml">c</mi><mo id="S6.SS2.23.p4.3.m3.1.1.3" xref="S6.SS2.23.p4.3.m3.1.1.3.cmml">′′</mo></msup><annotation-xml encoding="MathML-Content" id="S6.SS2.23.p4.3.m3.1b"><apply id="S6.SS2.23.p4.3.m3.1.1.cmml" xref="S6.SS2.23.p4.3.m3.1.1"><csymbol cd="ambiguous" id="S6.SS2.23.p4.3.m3.1.1.1.cmml" xref="S6.SS2.23.p4.3.m3.1.1">superscript</csymbol><ci id="S6.SS2.23.p4.3.m3.1.1.2.cmml" xref="S6.SS2.23.p4.3.m3.1.1.2">𝑐</ci><ci id="S6.SS2.23.p4.3.m3.1.1.3.cmml" xref="S6.SS2.23.p4.3.m3.1.1.3">′′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.23.p4.3.m3.1c">c^{\prime\prime}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.23.p4.3.m3.1d">italic_c start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT</annotation></semantics></math> such that <math alttext="\nu(c^{\prime\prime})=\nu" class="ltx_Math" display="inline" id="S6.SS2.23.p4.4.m4.1"><semantics id="S6.SS2.23.p4.4.m4.1a"><mrow id="S6.SS2.23.p4.4.m4.1.1" xref="S6.SS2.23.p4.4.m4.1.1.cmml"><mrow id="S6.SS2.23.p4.4.m4.1.1.1" xref="S6.SS2.23.p4.4.m4.1.1.1.cmml"><mi id="S6.SS2.23.p4.4.m4.1.1.1.3" xref="S6.SS2.23.p4.4.m4.1.1.1.3.cmml">ν</mi><mo id="S6.SS2.23.p4.4.m4.1.1.1.2" xref="S6.SS2.23.p4.4.m4.1.1.1.2.cmml">⁢</mo><mrow id="S6.SS2.23.p4.4.m4.1.1.1.1.1" xref="S6.SS2.23.p4.4.m4.1.1.1.1.1.1.cmml"><mo id="S6.SS2.23.p4.4.m4.1.1.1.1.1.2" stretchy="false" xref="S6.SS2.23.p4.4.m4.1.1.1.1.1.1.cmml">(</mo><msup id="S6.SS2.23.p4.4.m4.1.1.1.1.1.1" xref="S6.SS2.23.p4.4.m4.1.1.1.1.1.1.cmml"><mi id="S6.SS2.23.p4.4.m4.1.1.1.1.1.1.2" xref="S6.SS2.23.p4.4.m4.1.1.1.1.1.1.2.cmml">c</mi><mo id="S6.SS2.23.p4.4.m4.1.1.1.1.1.1.3" xref="S6.SS2.23.p4.4.m4.1.1.1.1.1.1.3.cmml">′′</mo></msup><mo id="S6.SS2.23.p4.4.m4.1.1.1.1.1.3" stretchy="false" xref="S6.SS2.23.p4.4.m4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.SS2.23.p4.4.m4.1.1.2" xref="S6.SS2.23.p4.4.m4.1.1.2.cmml">=</mo><mi id="S6.SS2.23.p4.4.m4.1.1.3" xref="S6.SS2.23.p4.4.m4.1.1.3.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.23.p4.4.m4.1b"><apply id="S6.SS2.23.p4.4.m4.1.1.cmml" xref="S6.SS2.23.p4.4.m4.1.1"><eq id="S6.SS2.23.p4.4.m4.1.1.2.cmml" xref="S6.SS2.23.p4.4.m4.1.1.2"></eq><apply id="S6.SS2.23.p4.4.m4.1.1.1.cmml" xref="S6.SS2.23.p4.4.m4.1.1.1"><times id="S6.SS2.23.p4.4.m4.1.1.1.2.cmml" xref="S6.SS2.23.p4.4.m4.1.1.1.2"></times><ci id="S6.SS2.23.p4.4.m4.1.1.1.3.cmml" xref="S6.SS2.23.p4.4.m4.1.1.1.3">𝜈</ci><apply id="S6.SS2.23.p4.4.m4.1.1.1.1.1.1.cmml" xref="S6.SS2.23.p4.4.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.23.p4.4.m4.1.1.1.1.1.1.1.cmml" xref="S6.SS2.23.p4.4.m4.1.1.1.1.1">superscript</csymbol><ci id="S6.SS2.23.p4.4.m4.1.1.1.1.1.1.2.cmml" xref="S6.SS2.23.p4.4.m4.1.1.1.1.1.1.2">𝑐</ci><ci id="S6.SS2.23.p4.4.m4.1.1.1.1.1.1.3.cmml" xref="S6.SS2.23.p4.4.m4.1.1.1.1.1.1.3">′′</ci></apply></apply><ci id="S6.SS2.23.p4.4.m4.1.1.3.cmml" xref="S6.SS2.23.p4.4.m4.1.1.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.23.p4.4.m4.1c">\nu(c^{\prime\prime})=\nu</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.23.p4.4.m4.1d">italic_ν ( italic_c start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) = italic_ν</annotation></semantics></math> and <math alttext="c^{\prime}<_{A}c^{\prime\prime}<_{A}b_{l}" class="ltx_Math" display="inline" id="S6.SS2.23.p4.5.m5.1"><semantics id="S6.SS2.23.p4.5.m5.1a"><mrow id="S6.SS2.23.p4.5.m5.1.1" xref="S6.SS2.23.p4.5.m5.1.1.cmml"><msup id="S6.SS2.23.p4.5.m5.1.1.2" xref="S6.SS2.23.p4.5.m5.1.1.2.cmml"><mi id="S6.SS2.23.p4.5.m5.1.1.2.2" xref="S6.SS2.23.p4.5.m5.1.1.2.2.cmml">c</mi><mo id="S6.SS2.23.p4.5.m5.1.1.2.3" xref="S6.SS2.23.p4.5.m5.1.1.2.3.cmml">′</mo></msup><msub id="S6.SS2.23.p4.5.m5.1.1.3" xref="S6.SS2.23.p4.5.m5.1.1.3.cmml"><mo id="S6.SS2.23.p4.5.m5.1.1.3.2" xref="S6.SS2.23.p4.5.m5.1.1.3.2.cmml"><</mo><mi id="S6.SS2.23.p4.5.m5.1.1.3.3" xref="S6.SS2.23.p4.5.m5.1.1.3.3.cmml">A</mi></msub><msup id="S6.SS2.23.p4.5.m5.1.1.4" xref="S6.SS2.23.p4.5.m5.1.1.4.cmml"><mi id="S6.SS2.23.p4.5.m5.1.1.4.2" xref="S6.SS2.23.p4.5.m5.1.1.4.2.cmml">c</mi><mo id="S6.SS2.23.p4.5.m5.1.1.4.3" xref="S6.SS2.23.p4.5.m5.1.1.4.3.cmml">′′</mo></msup><msub id="S6.SS2.23.p4.5.m5.1.1.5" xref="S6.SS2.23.p4.5.m5.1.1.5.cmml"><mo id="S6.SS2.23.p4.5.m5.1.1.5.2" xref="S6.SS2.23.p4.5.m5.1.1.5.2.cmml"><</mo><mi id="S6.SS2.23.p4.5.m5.1.1.5.3" xref="S6.SS2.23.p4.5.m5.1.1.5.3.cmml">A</mi></msub><msub id="S6.SS2.23.p4.5.m5.1.1.6" xref="S6.SS2.23.p4.5.m5.1.1.6.cmml"><mi id="S6.SS2.23.p4.5.m5.1.1.6.2" xref="S6.SS2.23.p4.5.m5.1.1.6.2.cmml">b</mi><mi id="S6.SS2.23.p4.5.m5.1.1.6.3" xref="S6.SS2.23.p4.5.m5.1.1.6.3.cmml">l</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.23.p4.5.m5.1b"><apply id="S6.SS2.23.p4.5.m5.1.1.cmml" xref="S6.SS2.23.p4.5.m5.1.1"><and id="S6.SS2.23.p4.5.m5.1.1a.cmml" xref="S6.SS2.23.p4.5.m5.1.1"></and><apply id="S6.SS2.23.p4.5.m5.1.1b.cmml" xref="S6.SS2.23.p4.5.m5.1.1"><apply id="S6.SS2.23.p4.5.m5.1.1.3.cmml" xref="S6.SS2.23.p4.5.m5.1.1.3"><csymbol cd="ambiguous" id="S6.SS2.23.p4.5.m5.1.1.3.1.cmml" xref="S6.SS2.23.p4.5.m5.1.1.3">subscript</csymbol><lt id="S6.SS2.23.p4.5.m5.1.1.3.2.cmml" xref="S6.SS2.23.p4.5.m5.1.1.3.2"></lt><ci id="S6.SS2.23.p4.5.m5.1.1.3.3.cmml" xref="S6.SS2.23.p4.5.m5.1.1.3.3">𝐴</ci></apply><apply id="S6.SS2.23.p4.5.m5.1.1.2.cmml" xref="S6.SS2.23.p4.5.m5.1.1.2"><csymbol cd="ambiguous" id="S6.SS2.23.p4.5.m5.1.1.2.1.cmml" xref="S6.SS2.23.p4.5.m5.1.1.2">superscript</csymbol><ci id="S6.SS2.23.p4.5.m5.1.1.2.2.cmml" xref="S6.SS2.23.p4.5.m5.1.1.2.2">𝑐</ci><ci id="S6.SS2.23.p4.5.m5.1.1.2.3.cmml" xref="S6.SS2.23.p4.5.m5.1.1.2.3">′</ci></apply><apply id="S6.SS2.23.p4.5.m5.1.1.4.cmml" xref="S6.SS2.23.p4.5.m5.1.1.4"><csymbol cd="ambiguous" id="S6.SS2.23.p4.5.m5.1.1.4.1.cmml" xref="S6.SS2.23.p4.5.m5.1.1.4">superscript</csymbol><ci id="S6.SS2.23.p4.5.m5.1.1.4.2.cmml" xref="S6.SS2.23.p4.5.m5.1.1.4.2">𝑐</ci><ci id="S6.SS2.23.p4.5.m5.1.1.4.3.cmml" xref="S6.SS2.23.p4.5.m5.1.1.4.3">′′</ci></apply></apply><apply id="S6.SS2.23.p4.5.m5.1.1c.cmml" xref="S6.SS2.23.p4.5.m5.1.1"><apply id="S6.SS2.23.p4.5.m5.1.1.5.cmml" xref="S6.SS2.23.p4.5.m5.1.1.5"><csymbol cd="ambiguous" id="S6.SS2.23.p4.5.m5.1.1.5.1.cmml" xref="S6.SS2.23.p4.5.m5.1.1.5">subscript</csymbol><lt id="S6.SS2.23.p4.5.m5.1.1.5.2.cmml" xref="S6.SS2.23.p4.5.m5.1.1.5.2"></lt><ci id="S6.SS2.23.p4.5.m5.1.1.5.3.cmml" xref="S6.SS2.23.p4.5.m5.1.1.5.3">𝐴</ci></apply><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.23.p4.5.m5.1.1.4.cmml" id="S6.SS2.23.p4.5.m5.1.1d.cmml" xref="S6.SS2.23.p4.5.m5.1.1"></share><apply id="S6.SS2.23.p4.5.m5.1.1.6.cmml" xref="S6.SS2.23.p4.5.m5.1.1.6"><csymbol cd="ambiguous" id="S6.SS2.23.p4.5.m5.1.1.6.1.cmml" xref="S6.SS2.23.p4.5.m5.1.1.6">subscript</csymbol><ci id="S6.SS2.23.p4.5.m5.1.1.6.2.cmml" xref="S6.SS2.23.p4.5.m5.1.1.6.2">𝑏</ci><ci id="S6.SS2.23.p4.5.m5.1.1.6.3.cmml" xref="S6.SS2.23.p4.5.m5.1.1.6.3">𝑙</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.23.p4.5.m5.1c">c^{\prime}<_{A}c^{\prime\prime}<_{A}b_{l}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.23.p4.5.m5.1d">italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT < start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_c start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT < start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT</annotation></semantics></math>. Now note that <math alttext="\nu\leq\nu(z)" class="ltx_Math" display="inline" id="S6.SS2.23.p4.6.m6.1"><semantics id="S6.SS2.23.p4.6.m6.1a"><mrow id="S6.SS2.23.p4.6.m6.1.2" xref="S6.SS2.23.p4.6.m6.1.2.cmml"><mi id="S6.SS2.23.p4.6.m6.1.2.2" xref="S6.SS2.23.p4.6.m6.1.2.2.cmml">ν</mi><mo id="S6.SS2.23.p4.6.m6.1.2.1" xref="S6.SS2.23.p4.6.m6.1.2.1.cmml">≤</mo><mrow id="S6.SS2.23.p4.6.m6.1.2.3" xref="S6.SS2.23.p4.6.m6.1.2.3.cmml"><mi id="S6.SS2.23.p4.6.m6.1.2.3.2" xref="S6.SS2.23.p4.6.m6.1.2.3.2.cmml">ν</mi><mo id="S6.SS2.23.p4.6.m6.1.2.3.1" xref="S6.SS2.23.p4.6.m6.1.2.3.1.cmml">⁢</mo><mrow id="S6.SS2.23.p4.6.m6.1.2.3.3.2" xref="S6.SS2.23.p4.6.m6.1.2.3.cmml"><mo id="S6.SS2.23.p4.6.m6.1.2.3.3.2.1" stretchy="false" xref="S6.SS2.23.p4.6.m6.1.2.3.cmml">(</mo><mi id="S6.SS2.23.p4.6.m6.1.1" xref="S6.SS2.23.p4.6.m6.1.1.cmml">z</mi><mo id="S6.SS2.23.p4.6.m6.1.2.3.3.2.2" stretchy="false" xref="S6.SS2.23.p4.6.m6.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.23.p4.6.m6.1b"><apply id="S6.SS2.23.p4.6.m6.1.2.cmml" xref="S6.SS2.23.p4.6.m6.1.2"><leq id="S6.SS2.23.p4.6.m6.1.2.1.cmml" xref="S6.SS2.23.p4.6.m6.1.2.1"></leq><ci id="S6.SS2.23.p4.6.m6.1.2.2.cmml" xref="S6.SS2.23.p4.6.m6.1.2.2">𝜈</ci><apply id="S6.SS2.23.p4.6.m6.1.2.3.cmml" xref="S6.SS2.23.p4.6.m6.1.2.3"><times id="S6.SS2.23.p4.6.m6.1.2.3.1.cmml" xref="S6.SS2.23.p4.6.m6.1.2.3.1"></times><ci id="S6.SS2.23.p4.6.m6.1.2.3.2.cmml" xref="S6.SS2.23.p4.6.m6.1.2.3.2">𝜈</ci><ci id="S6.SS2.23.p4.6.m6.1.1.cmml" xref="S6.SS2.23.p4.6.m6.1.1">𝑧</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.23.p4.6.m6.1c">\nu\leq\nu(z)</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.23.p4.6.m6.1d">italic_ν ≤ italic_ν ( italic_z )</annotation></semantics></math>, thus using <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.Thmtheorem17" title="Lemma 6.17. ‣ 6.2. Forcing epimorphisms ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">6.17</span></a> one finds <math alttext="c_{l}" class="ltx_Math" display="inline" id="S6.SS2.23.p4.7.m7.1"><semantics id="S6.SS2.23.p4.7.m7.1a"><msub id="S6.SS2.23.p4.7.m7.1.1" xref="S6.SS2.23.p4.7.m7.1.1.cmml"><mi id="S6.SS2.23.p4.7.m7.1.1.2" xref="S6.SS2.23.p4.7.m7.1.1.2.cmml">c</mi><mi id="S6.SS2.23.p4.7.m7.1.1.3" xref="S6.SS2.23.p4.7.m7.1.1.3.cmml">l</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.23.p4.7.m7.1b"><apply id="S6.SS2.23.p4.7.m7.1.1.cmml" xref="S6.SS2.23.p4.7.m7.1.1"><csymbol cd="ambiguous" id="S6.SS2.23.p4.7.m7.1.1.1.cmml" xref="S6.SS2.23.p4.7.m7.1.1">subscript</csymbol><ci id="S6.SS2.23.p4.7.m7.1.1.2.cmml" xref="S6.SS2.23.p4.7.m7.1.1.2">𝑐</ci><ci id="S6.SS2.23.p4.7.m7.1.1.3.cmml" xref="S6.SS2.23.p4.7.m7.1.1.3">𝑙</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.23.p4.7.m7.1c">c_{l}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.23.p4.7.m7.1d">italic_c start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="c_{r}" class="ltx_Math" display="inline" id="S6.SS2.23.p4.8.m8.1"><semantics id="S6.SS2.23.p4.8.m8.1a"><msub id="S6.SS2.23.p4.8.m8.1.1" xref="S6.SS2.23.p4.8.m8.1.1.cmml"><mi id="S6.SS2.23.p4.8.m8.1.1.2" xref="S6.SS2.23.p4.8.m8.1.1.2.cmml">c</mi><mi id="S6.SS2.23.p4.8.m8.1.1.3" xref="S6.SS2.23.p4.8.m8.1.1.3.cmml">r</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.23.p4.8.m8.1b"><apply id="S6.SS2.23.p4.8.m8.1.1.cmml" xref="S6.SS2.23.p4.8.m8.1.1"><csymbol cd="ambiguous" id="S6.SS2.23.p4.8.m8.1.1.1.cmml" xref="S6.SS2.23.p4.8.m8.1.1">subscript</csymbol><ci id="S6.SS2.23.p4.8.m8.1.1.2.cmml" xref="S6.SS2.23.p4.8.m8.1.1.2">𝑐</ci><ci id="S6.SS2.23.p4.8.m8.1.1.3.cmml" xref="S6.SS2.23.p4.8.m8.1.1.3">𝑟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.23.p4.8.m8.1c">c_{r}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.23.p4.8.m8.1d">italic_c start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="\nu(c_{l})=\nu(c_{r})=\nu(z)" class="ltx_Math" display="inline" id="S6.SS2.23.p4.9.m9.3"><semantics id="S6.SS2.23.p4.9.m9.3a"><mrow id="S6.SS2.23.p4.9.m9.3.3" xref="S6.SS2.23.p4.9.m9.3.3.cmml"><mrow id="S6.SS2.23.p4.9.m9.2.2.1" xref="S6.SS2.23.p4.9.m9.2.2.1.cmml"><mi id="S6.SS2.23.p4.9.m9.2.2.1.3" xref="S6.SS2.23.p4.9.m9.2.2.1.3.cmml">ν</mi><mo id="S6.SS2.23.p4.9.m9.2.2.1.2" xref="S6.SS2.23.p4.9.m9.2.2.1.2.cmml">⁢</mo><mrow id="S6.SS2.23.p4.9.m9.2.2.1.1.1" xref="S6.SS2.23.p4.9.m9.2.2.1.1.1.1.cmml"><mo id="S6.SS2.23.p4.9.m9.2.2.1.1.1.2" stretchy="false" xref="S6.SS2.23.p4.9.m9.2.2.1.1.1.1.cmml">(</mo><msub id="S6.SS2.23.p4.9.m9.2.2.1.1.1.1" xref="S6.SS2.23.p4.9.m9.2.2.1.1.1.1.cmml"><mi id="S6.SS2.23.p4.9.m9.2.2.1.1.1.1.2" xref="S6.SS2.23.p4.9.m9.2.2.1.1.1.1.2.cmml">c</mi><mi id="S6.SS2.23.p4.9.m9.2.2.1.1.1.1.3" xref="S6.SS2.23.p4.9.m9.2.2.1.1.1.1.3.cmml">l</mi></msub><mo id="S6.SS2.23.p4.9.m9.2.2.1.1.1.3" stretchy="false" xref="S6.SS2.23.p4.9.m9.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.SS2.23.p4.9.m9.3.3.4" xref="S6.SS2.23.p4.9.m9.3.3.4.cmml">=</mo><mrow id="S6.SS2.23.p4.9.m9.3.3.2" xref="S6.SS2.23.p4.9.m9.3.3.2.cmml"><mi id="S6.SS2.23.p4.9.m9.3.3.2.3" xref="S6.SS2.23.p4.9.m9.3.3.2.3.cmml">ν</mi><mo id="S6.SS2.23.p4.9.m9.3.3.2.2" xref="S6.SS2.23.p4.9.m9.3.3.2.2.cmml">⁢</mo><mrow id="S6.SS2.23.p4.9.m9.3.3.2.1.1" xref="S6.SS2.23.p4.9.m9.3.3.2.1.1.1.cmml"><mo id="S6.SS2.23.p4.9.m9.3.3.2.1.1.2" stretchy="false" xref="S6.SS2.23.p4.9.m9.3.3.2.1.1.1.cmml">(</mo><msub id="S6.SS2.23.p4.9.m9.3.3.2.1.1.1" xref="S6.SS2.23.p4.9.m9.3.3.2.1.1.1.cmml"><mi id="S6.SS2.23.p4.9.m9.3.3.2.1.1.1.2" xref="S6.SS2.23.p4.9.m9.3.3.2.1.1.1.2.cmml">c</mi><mi id="S6.SS2.23.p4.9.m9.3.3.2.1.1.1.3" xref="S6.SS2.23.p4.9.m9.3.3.2.1.1.1.3.cmml">r</mi></msub><mo id="S6.SS2.23.p4.9.m9.3.3.2.1.1.3" stretchy="false" xref="S6.SS2.23.p4.9.m9.3.3.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.SS2.23.p4.9.m9.3.3.5" xref="S6.SS2.23.p4.9.m9.3.3.5.cmml">=</mo><mrow id="S6.SS2.23.p4.9.m9.3.3.6" xref="S6.SS2.23.p4.9.m9.3.3.6.cmml"><mi id="S6.SS2.23.p4.9.m9.3.3.6.2" xref="S6.SS2.23.p4.9.m9.3.3.6.2.cmml">ν</mi><mo id="S6.SS2.23.p4.9.m9.3.3.6.1" xref="S6.SS2.23.p4.9.m9.3.3.6.1.cmml">⁢</mo><mrow id="S6.SS2.23.p4.9.m9.3.3.6.3.2" xref="S6.SS2.23.p4.9.m9.3.3.6.cmml"><mo id="S6.SS2.23.p4.9.m9.3.3.6.3.2.1" stretchy="false" xref="S6.SS2.23.p4.9.m9.3.3.6.cmml">(</mo><mi id="S6.SS2.23.p4.9.m9.1.1" xref="S6.SS2.23.p4.9.m9.1.1.cmml">z</mi><mo id="S6.SS2.23.p4.9.m9.3.3.6.3.2.2" stretchy="false" xref="S6.SS2.23.p4.9.m9.3.3.6.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.23.p4.9.m9.3b"><apply id="S6.SS2.23.p4.9.m9.3.3.cmml" xref="S6.SS2.23.p4.9.m9.3.3"><and id="S6.SS2.23.p4.9.m9.3.3a.cmml" xref="S6.SS2.23.p4.9.m9.3.3"></and><apply id="S6.SS2.23.p4.9.m9.3.3b.cmml" xref="S6.SS2.23.p4.9.m9.3.3"><eq id="S6.SS2.23.p4.9.m9.3.3.4.cmml" xref="S6.SS2.23.p4.9.m9.3.3.4"></eq><apply id="S6.SS2.23.p4.9.m9.2.2.1.cmml" xref="S6.SS2.23.p4.9.m9.2.2.1"><times id="S6.SS2.23.p4.9.m9.2.2.1.2.cmml" xref="S6.SS2.23.p4.9.m9.2.2.1.2"></times><ci id="S6.SS2.23.p4.9.m9.2.2.1.3.cmml" xref="S6.SS2.23.p4.9.m9.2.2.1.3">𝜈</ci><apply id="S6.SS2.23.p4.9.m9.2.2.1.1.1.1.cmml" xref="S6.SS2.23.p4.9.m9.2.2.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.23.p4.9.m9.2.2.1.1.1.1.1.cmml" xref="S6.SS2.23.p4.9.m9.2.2.1.1.1">subscript</csymbol><ci id="S6.SS2.23.p4.9.m9.2.2.1.1.1.1.2.cmml" xref="S6.SS2.23.p4.9.m9.2.2.1.1.1.1.2">𝑐</ci><ci id="S6.SS2.23.p4.9.m9.2.2.1.1.1.1.3.cmml" xref="S6.SS2.23.p4.9.m9.2.2.1.1.1.1.3">𝑙</ci></apply></apply><apply id="S6.SS2.23.p4.9.m9.3.3.2.cmml" xref="S6.SS2.23.p4.9.m9.3.3.2"><times id="S6.SS2.23.p4.9.m9.3.3.2.2.cmml" xref="S6.SS2.23.p4.9.m9.3.3.2.2"></times><ci id="S6.SS2.23.p4.9.m9.3.3.2.3.cmml" xref="S6.SS2.23.p4.9.m9.3.3.2.3">𝜈</ci><apply id="S6.SS2.23.p4.9.m9.3.3.2.1.1.1.cmml" xref="S6.SS2.23.p4.9.m9.3.3.2.1.1"><csymbol cd="ambiguous" id="S6.SS2.23.p4.9.m9.3.3.2.1.1.1.1.cmml" xref="S6.SS2.23.p4.9.m9.3.3.2.1.1">subscript</csymbol><ci id="S6.SS2.23.p4.9.m9.3.3.2.1.1.1.2.cmml" xref="S6.SS2.23.p4.9.m9.3.3.2.1.1.1.2">𝑐</ci><ci id="S6.SS2.23.p4.9.m9.3.3.2.1.1.1.3.cmml" xref="S6.SS2.23.p4.9.m9.3.3.2.1.1.1.3">𝑟</ci></apply></apply></apply><apply id="S6.SS2.23.p4.9.m9.3.3c.cmml" xref="S6.SS2.23.p4.9.m9.3.3"><eq id="S6.SS2.23.p4.9.m9.3.3.5.cmml" xref="S6.SS2.23.p4.9.m9.3.3.5"></eq><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.23.p4.9.m9.3.3.2.cmml" id="S6.SS2.23.p4.9.m9.3.3d.cmml" xref="S6.SS2.23.p4.9.m9.3.3"></share><apply id="S6.SS2.23.p4.9.m9.3.3.6.cmml" xref="S6.SS2.23.p4.9.m9.3.3.6"><times id="S6.SS2.23.p4.9.m9.3.3.6.1.cmml" xref="S6.SS2.23.p4.9.m9.3.3.6.1"></times><ci id="S6.SS2.23.p4.9.m9.3.3.6.2.cmml" xref="S6.SS2.23.p4.9.m9.3.3.6.2">𝜈</ci><ci id="S6.SS2.23.p4.9.m9.1.1.cmml" xref="S6.SS2.23.p4.9.m9.1.1">𝑧</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.23.p4.9.m9.3c">\nu(c_{l})=\nu(c_{r})=\nu(z)</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.23.p4.9.m9.3d">italic_ν ( italic_c start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ) = italic_ν ( italic_c start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) = italic_ν ( italic_z )</annotation></semantics></math>, <math alttext="c^{\prime}<_{A}c_{l}<_{A}c_{r}<_{A}c^{\prime\prime}" class="ltx_Math" display="inline" id="S6.SS2.23.p4.10.m10.1"><semantics id="S6.SS2.23.p4.10.m10.1a"><mrow id="S6.SS2.23.p4.10.m10.1.1" xref="S6.SS2.23.p4.10.m10.1.1.cmml"><msup id="S6.SS2.23.p4.10.m10.1.1.2" xref="S6.SS2.23.p4.10.m10.1.1.2.cmml"><mi id="S6.SS2.23.p4.10.m10.1.1.2.2" xref="S6.SS2.23.p4.10.m10.1.1.2.2.cmml">c</mi><mo id="S6.SS2.23.p4.10.m10.1.1.2.3" xref="S6.SS2.23.p4.10.m10.1.1.2.3.cmml">′</mo></msup><msub id="S6.SS2.23.p4.10.m10.1.1.3" xref="S6.SS2.23.p4.10.m10.1.1.3.cmml"><mo id="S6.SS2.23.p4.10.m10.1.1.3.2" xref="S6.SS2.23.p4.10.m10.1.1.3.2.cmml"><</mo><mi id="S6.SS2.23.p4.10.m10.1.1.3.3" xref="S6.SS2.23.p4.10.m10.1.1.3.3.cmml">A</mi></msub><msub id="S6.SS2.23.p4.10.m10.1.1.4" xref="S6.SS2.23.p4.10.m10.1.1.4.cmml"><mi id="S6.SS2.23.p4.10.m10.1.1.4.2" xref="S6.SS2.23.p4.10.m10.1.1.4.2.cmml">c</mi><mi id="S6.SS2.23.p4.10.m10.1.1.4.3" xref="S6.SS2.23.p4.10.m10.1.1.4.3.cmml">l</mi></msub><msub id="S6.SS2.23.p4.10.m10.1.1.5" xref="S6.SS2.23.p4.10.m10.1.1.5.cmml"><mo id="S6.SS2.23.p4.10.m10.1.1.5.2" xref="S6.SS2.23.p4.10.m10.1.1.5.2.cmml"><</mo><mi id="S6.SS2.23.p4.10.m10.1.1.5.3" xref="S6.SS2.23.p4.10.m10.1.1.5.3.cmml">A</mi></msub><msub id="S6.SS2.23.p4.10.m10.1.1.6" xref="S6.SS2.23.p4.10.m10.1.1.6.cmml"><mi id="S6.SS2.23.p4.10.m10.1.1.6.2" xref="S6.SS2.23.p4.10.m10.1.1.6.2.cmml">c</mi><mi id="S6.SS2.23.p4.10.m10.1.1.6.3" xref="S6.SS2.23.p4.10.m10.1.1.6.3.cmml">r</mi></msub><msub id="S6.SS2.23.p4.10.m10.1.1.7" xref="S6.SS2.23.p4.10.m10.1.1.7.cmml"><mo id="S6.SS2.23.p4.10.m10.1.1.7.2" xref="S6.SS2.23.p4.10.m10.1.1.7.2.cmml"><</mo><mi id="S6.SS2.23.p4.10.m10.1.1.7.3" xref="S6.SS2.23.p4.10.m10.1.1.7.3.cmml">A</mi></msub><msup id="S6.SS2.23.p4.10.m10.1.1.8" xref="S6.SS2.23.p4.10.m10.1.1.8.cmml"><mi id="S6.SS2.23.p4.10.m10.1.1.8.2" xref="S6.SS2.23.p4.10.m10.1.1.8.2.cmml">c</mi><mo id="S6.SS2.23.p4.10.m10.1.1.8.3" xref="S6.SS2.23.p4.10.m10.1.1.8.3.cmml">′′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.23.p4.10.m10.1b"><apply id="S6.SS2.23.p4.10.m10.1.1.cmml" xref="S6.SS2.23.p4.10.m10.1.1"><and id="S6.SS2.23.p4.10.m10.1.1a.cmml" xref="S6.SS2.23.p4.10.m10.1.1"></and><apply id="S6.SS2.23.p4.10.m10.1.1b.cmml" xref="S6.SS2.23.p4.10.m10.1.1"><apply id="S6.SS2.23.p4.10.m10.1.1.3.cmml" xref="S6.SS2.23.p4.10.m10.1.1.3"><csymbol cd="ambiguous" id="S6.SS2.23.p4.10.m10.1.1.3.1.cmml" xref="S6.SS2.23.p4.10.m10.1.1.3">subscript</csymbol><lt id="S6.SS2.23.p4.10.m10.1.1.3.2.cmml" xref="S6.SS2.23.p4.10.m10.1.1.3.2"></lt><ci id="S6.SS2.23.p4.10.m10.1.1.3.3.cmml" xref="S6.SS2.23.p4.10.m10.1.1.3.3">𝐴</ci></apply><apply id="S6.SS2.23.p4.10.m10.1.1.2.cmml" xref="S6.SS2.23.p4.10.m10.1.1.2"><csymbol cd="ambiguous" id="S6.SS2.23.p4.10.m10.1.1.2.1.cmml" xref="S6.SS2.23.p4.10.m10.1.1.2">superscript</csymbol><ci id="S6.SS2.23.p4.10.m10.1.1.2.2.cmml" xref="S6.SS2.23.p4.10.m10.1.1.2.2">𝑐</ci><ci id="S6.SS2.23.p4.10.m10.1.1.2.3.cmml" xref="S6.SS2.23.p4.10.m10.1.1.2.3">′</ci></apply><apply id="S6.SS2.23.p4.10.m10.1.1.4.cmml" xref="S6.SS2.23.p4.10.m10.1.1.4"><csymbol cd="ambiguous" id="S6.SS2.23.p4.10.m10.1.1.4.1.cmml" xref="S6.SS2.23.p4.10.m10.1.1.4">subscript</csymbol><ci id="S6.SS2.23.p4.10.m10.1.1.4.2.cmml" xref="S6.SS2.23.p4.10.m10.1.1.4.2">𝑐</ci><ci id="S6.SS2.23.p4.10.m10.1.1.4.3.cmml" xref="S6.SS2.23.p4.10.m10.1.1.4.3">𝑙</ci></apply></apply><apply id="S6.SS2.23.p4.10.m10.1.1c.cmml" xref="S6.SS2.23.p4.10.m10.1.1"><apply id="S6.SS2.23.p4.10.m10.1.1.5.cmml" xref="S6.SS2.23.p4.10.m10.1.1.5"><csymbol cd="ambiguous" id="S6.SS2.23.p4.10.m10.1.1.5.1.cmml" xref="S6.SS2.23.p4.10.m10.1.1.5">subscript</csymbol><lt id="S6.SS2.23.p4.10.m10.1.1.5.2.cmml" xref="S6.SS2.23.p4.10.m10.1.1.5.2"></lt><ci id="S6.SS2.23.p4.10.m10.1.1.5.3.cmml" xref="S6.SS2.23.p4.10.m10.1.1.5.3">𝐴</ci></apply><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.23.p4.10.m10.1.1.4.cmml" id="S6.SS2.23.p4.10.m10.1.1d.cmml" xref="S6.SS2.23.p4.10.m10.1.1"></share><apply id="S6.SS2.23.p4.10.m10.1.1.6.cmml" xref="S6.SS2.23.p4.10.m10.1.1.6"><csymbol cd="ambiguous" id="S6.SS2.23.p4.10.m10.1.1.6.1.cmml" xref="S6.SS2.23.p4.10.m10.1.1.6">subscript</csymbol><ci id="S6.SS2.23.p4.10.m10.1.1.6.2.cmml" xref="S6.SS2.23.p4.10.m10.1.1.6.2">𝑐</ci><ci id="S6.SS2.23.p4.10.m10.1.1.6.3.cmml" xref="S6.SS2.23.p4.10.m10.1.1.6.3">𝑟</ci></apply></apply><apply id="S6.SS2.23.p4.10.m10.1.1e.cmml" xref="S6.SS2.23.p4.10.m10.1.1"><apply id="S6.SS2.23.p4.10.m10.1.1.7.cmml" xref="S6.SS2.23.p4.10.m10.1.1.7"><csymbol cd="ambiguous" id="S6.SS2.23.p4.10.m10.1.1.7.1.cmml" xref="S6.SS2.23.p4.10.m10.1.1.7">subscript</csymbol><lt id="S6.SS2.23.p4.10.m10.1.1.7.2.cmml" xref="S6.SS2.23.p4.10.m10.1.1.7.2"></lt><ci id="S6.SS2.23.p4.10.m10.1.1.7.3.cmml" xref="S6.SS2.23.p4.10.m10.1.1.7.3">𝐴</ci></apply><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.23.p4.10.m10.1.1.6.cmml" id="S6.SS2.23.p4.10.m10.1.1f.cmml" xref="S6.SS2.23.p4.10.m10.1.1"></share><apply id="S6.SS2.23.p4.10.m10.1.1.8.cmml" xref="S6.SS2.23.p4.10.m10.1.1.8"><csymbol cd="ambiguous" id="S6.SS2.23.p4.10.m10.1.1.8.1.cmml" xref="S6.SS2.23.p4.10.m10.1.1.8">superscript</csymbol><ci id="S6.SS2.23.p4.10.m10.1.1.8.2.cmml" xref="S6.SS2.23.p4.10.m10.1.1.8.2">𝑐</ci><ci id="S6.SS2.23.p4.10.m10.1.1.8.3.cmml" xref="S6.SS2.23.p4.10.m10.1.1.8.3">′′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.23.p4.10.m10.1c">c^{\prime}<_{A}c_{l}<_{A}c_{r}<_{A}c^{\prime\prime}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.23.p4.10.m10.1d">italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT < start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_c start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT < start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_c start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT < start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_c start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="c_{l}" class="ltx_Math" display="inline" id="S6.SS2.23.p4.11.m11.1"><semantics id="S6.SS2.23.p4.11.m11.1a"><msub id="S6.SS2.23.p4.11.m11.1.1" xref="S6.SS2.23.p4.11.m11.1.1.cmml"><mi id="S6.SS2.23.p4.11.m11.1.1.2" xref="S6.SS2.23.p4.11.m11.1.1.2.cmml">c</mi><mi id="S6.SS2.23.p4.11.m11.1.1.3" xref="S6.SS2.23.p4.11.m11.1.1.3.cmml">l</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.23.p4.11.m11.1b"><apply id="S6.SS2.23.p4.11.m11.1.1.cmml" xref="S6.SS2.23.p4.11.m11.1.1"><csymbol cd="ambiguous" id="S6.SS2.23.p4.11.m11.1.1.1.cmml" xref="S6.SS2.23.p4.11.m11.1.1">subscript</csymbol><ci id="S6.SS2.23.p4.11.m11.1.1.2.cmml" xref="S6.SS2.23.p4.11.m11.1.1.2">𝑐</ci><ci id="S6.SS2.23.p4.11.m11.1.1.3.cmml" xref="S6.SS2.23.p4.11.m11.1.1.3">𝑙</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.23.p4.11.m11.1c">c_{l}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.23.p4.11.m11.1d">italic_c start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="c_{r}" class="ltx_Math" display="inline" id="S6.SS2.23.p4.12.m12.1"><semantics id="S6.SS2.23.p4.12.m12.1a"><msub id="S6.SS2.23.p4.12.m12.1.1" xref="S6.SS2.23.p4.12.m12.1.1.cmml"><mi id="S6.SS2.23.p4.12.m12.1.1.2" xref="S6.SS2.23.p4.12.m12.1.1.2.cmml">c</mi><mi id="S6.SS2.23.p4.12.m12.1.1.3" xref="S6.SS2.23.p4.12.m12.1.1.3.cmml">r</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.23.p4.12.m12.1b"><apply id="S6.SS2.23.p4.12.m12.1.1.cmml" xref="S6.SS2.23.p4.12.m12.1.1"><csymbol cd="ambiguous" id="S6.SS2.23.p4.12.m12.1.1.1.cmml" xref="S6.SS2.23.p4.12.m12.1.1">subscript</csymbol><ci id="S6.SS2.23.p4.12.m12.1.1.2.cmml" xref="S6.SS2.23.p4.12.m12.1.1.2">𝑐</ci><ci id="S6.SS2.23.p4.12.m12.1.1.3.cmml" xref="S6.SS2.23.p4.12.m12.1.1.3">𝑟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.23.p4.12.m12.1c">c_{r}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.23.p4.12.m12.1d">italic_c start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT</annotation></semantics></math> are in the same complementary interval of <math alttext="A\setminus\nu(z)" class="ltx_Math" display="inline" id="S6.SS2.23.p4.13.m13.1"><semantics id="S6.SS2.23.p4.13.m13.1a"><mrow id="S6.SS2.23.p4.13.m13.1.2" xref="S6.SS2.23.p4.13.m13.1.2.cmml"><mi id="S6.SS2.23.p4.13.m13.1.2.2" xref="S6.SS2.23.p4.13.m13.1.2.2.cmml">A</mi><mo id="S6.SS2.23.p4.13.m13.1.2.1" xref="S6.SS2.23.p4.13.m13.1.2.1.cmml">∖</mo><mrow id="S6.SS2.23.p4.13.m13.1.2.3" xref="S6.SS2.23.p4.13.m13.1.2.3.cmml"><mi id="S6.SS2.23.p4.13.m13.1.2.3.2" xref="S6.SS2.23.p4.13.m13.1.2.3.2.cmml">ν</mi><mo id="S6.SS2.23.p4.13.m13.1.2.3.1" xref="S6.SS2.23.p4.13.m13.1.2.3.1.cmml">⁢</mo><mrow id="S6.SS2.23.p4.13.m13.1.2.3.3.2" xref="S6.SS2.23.p4.13.m13.1.2.3.cmml"><mo id="S6.SS2.23.p4.13.m13.1.2.3.3.2.1" stretchy="false" xref="S6.SS2.23.p4.13.m13.1.2.3.cmml">(</mo><mi id="S6.SS2.23.p4.13.m13.1.1" xref="S6.SS2.23.p4.13.m13.1.1.cmml">z</mi><mo id="S6.SS2.23.p4.13.m13.1.2.3.3.2.2" stretchy="false" xref="S6.SS2.23.p4.13.m13.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.23.p4.13.m13.1b"><apply id="S6.SS2.23.p4.13.m13.1.2.cmml" xref="S6.SS2.23.p4.13.m13.1.2"><setdiff id="S6.SS2.23.p4.13.m13.1.2.1.cmml" xref="S6.SS2.23.p4.13.m13.1.2.1"></setdiff><ci id="S6.SS2.23.p4.13.m13.1.2.2.cmml" xref="S6.SS2.23.p4.13.m13.1.2.2">𝐴</ci><apply id="S6.SS2.23.p4.13.m13.1.2.3.cmml" xref="S6.SS2.23.p4.13.m13.1.2.3"><times id="S6.SS2.23.p4.13.m13.1.2.3.1.cmml" xref="S6.SS2.23.p4.13.m13.1.2.3.1"></times><ci id="S6.SS2.23.p4.13.m13.1.2.3.2.cmml" xref="S6.SS2.23.p4.13.m13.1.2.3.2">𝜈</ci><ci id="S6.SS2.23.p4.13.m13.1.1.cmml" xref="S6.SS2.23.p4.13.m13.1.1">𝑧</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.23.p4.13.m13.1c">A\setminus\nu(z)</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.23.p4.13.m13.1d">italic_A ∖ italic_ν ( italic_z )</annotation></semantics></math> (that is <math alttext="\Delta(c_{l},c_{r})\geq\nu(z)" class="ltx_Math" display="inline" id="S6.SS2.23.p4.14.m14.3"><semantics id="S6.SS2.23.p4.14.m14.3a"><mrow id="S6.SS2.23.p4.14.m14.3.3" xref="S6.SS2.23.p4.14.m14.3.3.cmml"><mrow id="S6.SS2.23.p4.14.m14.3.3.2" xref="S6.SS2.23.p4.14.m14.3.3.2.cmml"><mi id="S6.SS2.23.p4.14.m14.3.3.2.4" mathvariant="normal" xref="S6.SS2.23.p4.14.m14.3.3.2.4.cmml">Δ</mi><mo id="S6.SS2.23.p4.14.m14.3.3.2.3" xref="S6.SS2.23.p4.14.m14.3.3.2.3.cmml">⁢</mo><mrow id="S6.SS2.23.p4.14.m14.3.3.2.2.2" xref="S6.SS2.23.p4.14.m14.3.3.2.2.3.cmml"><mo id="S6.SS2.23.p4.14.m14.3.3.2.2.2.3" stretchy="false" xref="S6.SS2.23.p4.14.m14.3.3.2.2.3.cmml">(</mo><msub id="S6.SS2.23.p4.14.m14.2.2.1.1.1.1" xref="S6.SS2.23.p4.14.m14.2.2.1.1.1.1.cmml"><mi id="S6.SS2.23.p4.14.m14.2.2.1.1.1.1.2" xref="S6.SS2.23.p4.14.m14.2.2.1.1.1.1.2.cmml">c</mi><mi id="S6.SS2.23.p4.14.m14.2.2.1.1.1.1.3" xref="S6.SS2.23.p4.14.m14.2.2.1.1.1.1.3.cmml">l</mi></msub><mo id="S6.SS2.23.p4.14.m14.3.3.2.2.2.4" xref="S6.SS2.23.p4.14.m14.3.3.2.2.3.cmml">,</mo><msub id="S6.SS2.23.p4.14.m14.3.3.2.2.2.2" xref="S6.SS2.23.p4.14.m14.3.3.2.2.2.2.cmml"><mi id="S6.SS2.23.p4.14.m14.3.3.2.2.2.2.2" xref="S6.SS2.23.p4.14.m14.3.3.2.2.2.2.2.cmml">c</mi><mi id="S6.SS2.23.p4.14.m14.3.3.2.2.2.2.3" xref="S6.SS2.23.p4.14.m14.3.3.2.2.2.2.3.cmml">r</mi></msub><mo id="S6.SS2.23.p4.14.m14.3.3.2.2.2.5" stretchy="false" xref="S6.SS2.23.p4.14.m14.3.3.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.SS2.23.p4.14.m14.3.3.3" xref="S6.SS2.23.p4.14.m14.3.3.3.cmml">≥</mo><mrow id="S6.SS2.23.p4.14.m14.3.3.4" xref="S6.SS2.23.p4.14.m14.3.3.4.cmml"><mi id="S6.SS2.23.p4.14.m14.3.3.4.2" xref="S6.SS2.23.p4.14.m14.3.3.4.2.cmml">ν</mi><mo id="S6.SS2.23.p4.14.m14.3.3.4.1" xref="S6.SS2.23.p4.14.m14.3.3.4.1.cmml">⁢</mo><mrow id="S6.SS2.23.p4.14.m14.3.3.4.3.2" xref="S6.SS2.23.p4.14.m14.3.3.4.cmml"><mo id="S6.SS2.23.p4.14.m14.3.3.4.3.2.1" stretchy="false" xref="S6.SS2.23.p4.14.m14.3.3.4.cmml">(</mo><mi id="S6.SS2.23.p4.14.m14.1.1" xref="S6.SS2.23.p4.14.m14.1.1.cmml">z</mi><mo id="S6.SS2.23.p4.14.m14.3.3.4.3.2.2" stretchy="false" xref="S6.SS2.23.p4.14.m14.3.3.4.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.23.p4.14.m14.3b"><apply id="S6.SS2.23.p4.14.m14.3.3.cmml" xref="S6.SS2.23.p4.14.m14.3.3"><geq id="S6.SS2.23.p4.14.m14.3.3.3.cmml" xref="S6.SS2.23.p4.14.m14.3.3.3"></geq><apply id="S6.SS2.23.p4.14.m14.3.3.2.cmml" xref="S6.SS2.23.p4.14.m14.3.3.2"><times id="S6.SS2.23.p4.14.m14.3.3.2.3.cmml" xref="S6.SS2.23.p4.14.m14.3.3.2.3"></times><ci id="S6.SS2.23.p4.14.m14.3.3.2.4.cmml" xref="S6.SS2.23.p4.14.m14.3.3.2.4">Δ</ci><interval closure="open" id="S6.SS2.23.p4.14.m14.3.3.2.2.3.cmml" xref="S6.SS2.23.p4.14.m14.3.3.2.2.2"><apply id="S6.SS2.23.p4.14.m14.2.2.1.1.1.1.cmml" xref="S6.SS2.23.p4.14.m14.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.23.p4.14.m14.2.2.1.1.1.1.1.cmml" xref="S6.SS2.23.p4.14.m14.2.2.1.1.1.1">subscript</csymbol><ci id="S6.SS2.23.p4.14.m14.2.2.1.1.1.1.2.cmml" xref="S6.SS2.23.p4.14.m14.2.2.1.1.1.1.2">𝑐</ci><ci id="S6.SS2.23.p4.14.m14.2.2.1.1.1.1.3.cmml" xref="S6.SS2.23.p4.14.m14.2.2.1.1.1.1.3">𝑙</ci></apply><apply id="S6.SS2.23.p4.14.m14.3.3.2.2.2.2.cmml" xref="S6.SS2.23.p4.14.m14.3.3.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.23.p4.14.m14.3.3.2.2.2.2.1.cmml" xref="S6.SS2.23.p4.14.m14.3.3.2.2.2.2">subscript</csymbol><ci id="S6.SS2.23.p4.14.m14.3.3.2.2.2.2.2.cmml" xref="S6.SS2.23.p4.14.m14.3.3.2.2.2.2.2">𝑐</ci><ci id="S6.SS2.23.p4.14.m14.3.3.2.2.2.2.3.cmml" xref="S6.SS2.23.p4.14.m14.3.3.2.2.2.2.3">𝑟</ci></apply></interval></apply><apply id="S6.SS2.23.p4.14.m14.3.3.4.cmml" xref="S6.SS2.23.p4.14.m14.3.3.4"><times id="S6.SS2.23.p4.14.m14.3.3.4.1.cmml" xref="S6.SS2.23.p4.14.m14.3.3.4.1"></times><ci id="S6.SS2.23.p4.14.m14.3.3.4.2.cmml" xref="S6.SS2.23.p4.14.m14.3.3.4.2">𝜈</ci><ci id="S6.SS2.23.p4.14.m14.1.1.cmml" xref="S6.SS2.23.p4.14.m14.1.1">𝑧</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.23.p4.14.m14.3c">\Delta(c_{l},c_{r})\geq\nu(z)</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.23.p4.14.m14.3d">roman_Δ ( italic_c start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT , italic_c start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) ≥ italic_ν ( italic_z )</annotation></semantics></math>) and are not endpoints of it. We claim that <math alttext="p^{\prime}:=p\cup\{((c_{l},c_{r}),z)\}" class="ltx_Math" display="inline" id="S6.SS2.23.p4.15.m15.2"><semantics id="S6.SS2.23.p4.15.m15.2a"><mrow id="S6.SS2.23.p4.15.m15.2.2" xref="S6.SS2.23.p4.15.m15.2.2.cmml"><msup id="S6.SS2.23.p4.15.m15.2.2.3" xref="S6.SS2.23.p4.15.m15.2.2.3.cmml"><mi id="S6.SS2.23.p4.15.m15.2.2.3.2" xref="S6.SS2.23.p4.15.m15.2.2.3.2.cmml">p</mi><mo id="S6.SS2.23.p4.15.m15.2.2.3.3" xref="S6.SS2.23.p4.15.m15.2.2.3.3.cmml">′</mo></msup><mo id="S6.SS2.23.p4.15.m15.2.2.2" lspace="0.278em" rspace="0.278em" xref="S6.SS2.23.p4.15.m15.2.2.2.cmml">:=</mo><mrow id="S6.SS2.23.p4.15.m15.2.2.1" xref="S6.SS2.23.p4.15.m15.2.2.1.cmml"><mi id="S6.SS2.23.p4.15.m15.2.2.1.3" xref="S6.SS2.23.p4.15.m15.2.2.1.3.cmml">p</mi><mo id="S6.SS2.23.p4.15.m15.2.2.1.2" xref="S6.SS2.23.p4.15.m15.2.2.1.2.cmml">∪</mo><mrow id="S6.SS2.23.p4.15.m15.2.2.1.1.1" xref="S6.SS2.23.p4.15.m15.2.2.1.1.2.cmml"><mo id="S6.SS2.23.p4.15.m15.2.2.1.1.1.2" stretchy="false" xref="S6.SS2.23.p4.15.m15.2.2.1.1.2.cmml">{</mo><mrow id="S6.SS2.23.p4.15.m15.2.2.1.1.1.1.1" xref="S6.SS2.23.p4.15.m15.2.2.1.1.1.1.2.cmml"><mo id="S6.SS2.23.p4.15.m15.2.2.1.1.1.1.1.2" stretchy="false" xref="S6.SS2.23.p4.15.m15.2.2.1.1.1.1.2.cmml">(</mo><mrow id="S6.SS2.23.p4.15.m15.2.2.1.1.1.1.1.1.2" xref="S6.SS2.23.p4.15.m15.2.2.1.1.1.1.1.1.3.cmml"><mo id="S6.SS2.23.p4.15.m15.2.2.1.1.1.1.1.1.2.3" stretchy="false" xref="S6.SS2.23.p4.15.m15.2.2.1.1.1.1.1.1.3.cmml">(</mo><msub id="S6.SS2.23.p4.15.m15.2.2.1.1.1.1.1.1.1.1" xref="S6.SS2.23.p4.15.m15.2.2.1.1.1.1.1.1.1.1.cmml"><mi id="S6.SS2.23.p4.15.m15.2.2.1.1.1.1.1.1.1.1.2" xref="S6.SS2.23.p4.15.m15.2.2.1.1.1.1.1.1.1.1.2.cmml">c</mi><mi id="S6.SS2.23.p4.15.m15.2.2.1.1.1.1.1.1.1.1.3" xref="S6.SS2.23.p4.15.m15.2.2.1.1.1.1.1.1.1.1.3.cmml">l</mi></msub><mo id="S6.SS2.23.p4.15.m15.2.2.1.1.1.1.1.1.2.4" xref="S6.SS2.23.p4.15.m15.2.2.1.1.1.1.1.1.3.cmml">,</mo><msub id="S6.SS2.23.p4.15.m15.2.2.1.1.1.1.1.1.2.2" xref="S6.SS2.23.p4.15.m15.2.2.1.1.1.1.1.1.2.2.cmml"><mi id="S6.SS2.23.p4.15.m15.2.2.1.1.1.1.1.1.2.2.2" xref="S6.SS2.23.p4.15.m15.2.2.1.1.1.1.1.1.2.2.2.cmml">c</mi><mi id="S6.SS2.23.p4.15.m15.2.2.1.1.1.1.1.1.2.2.3" xref="S6.SS2.23.p4.15.m15.2.2.1.1.1.1.1.1.2.2.3.cmml">r</mi></msub><mo id="S6.SS2.23.p4.15.m15.2.2.1.1.1.1.1.1.2.5" stretchy="false" xref="S6.SS2.23.p4.15.m15.2.2.1.1.1.1.1.1.3.cmml">)</mo></mrow><mo id="S6.SS2.23.p4.15.m15.2.2.1.1.1.1.1.3" xref="S6.SS2.23.p4.15.m15.2.2.1.1.1.1.2.cmml">,</mo><mi id="S6.SS2.23.p4.15.m15.1.1" xref="S6.SS2.23.p4.15.m15.1.1.cmml">z</mi><mo id="S6.SS2.23.p4.15.m15.2.2.1.1.1.1.1.4" stretchy="false" xref="S6.SS2.23.p4.15.m15.2.2.1.1.1.1.2.cmml">)</mo></mrow><mo id="S6.SS2.23.p4.15.m15.2.2.1.1.1.3" stretchy="false" xref="S6.SS2.23.p4.15.m15.2.2.1.1.2.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.23.p4.15.m15.2b"><apply id="S6.SS2.23.p4.15.m15.2.2.cmml" xref="S6.SS2.23.p4.15.m15.2.2"><csymbol cd="latexml" id="S6.SS2.23.p4.15.m15.2.2.2.cmml" xref="S6.SS2.23.p4.15.m15.2.2.2">assign</csymbol><apply id="S6.SS2.23.p4.15.m15.2.2.3.cmml" xref="S6.SS2.23.p4.15.m15.2.2.3"><csymbol cd="ambiguous" id="S6.SS2.23.p4.15.m15.2.2.3.1.cmml" xref="S6.SS2.23.p4.15.m15.2.2.3">superscript</csymbol><ci id="S6.SS2.23.p4.15.m15.2.2.3.2.cmml" xref="S6.SS2.23.p4.15.m15.2.2.3.2">𝑝</ci><ci id="S6.SS2.23.p4.15.m15.2.2.3.3.cmml" xref="S6.SS2.23.p4.15.m15.2.2.3.3">′</ci></apply><apply id="S6.SS2.23.p4.15.m15.2.2.1.cmml" xref="S6.SS2.23.p4.15.m15.2.2.1"><union id="S6.SS2.23.p4.15.m15.2.2.1.2.cmml" xref="S6.SS2.23.p4.15.m15.2.2.1.2"></union><ci id="S6.SS2.23.p4.15.m15.2.2.1.3.cmml" xref="S6.SS2.23.p4.15.m15.2.2.1.3">𝑝</ci><set id="S6.SS2.23.p4.15.m15.2.2.1.1.2.cmml" xref="S6.SS2.23.p4.15.m15.2.2.1.1.1"><interval closure="open" id="S6.SS2.23.p4.15.m15.2.2.1.1.1.1.2.cmml" xref="S6.SS2.23.p4.15.m15.2.2.1.1.1.1.1"><interval closure="open" id="S6.SS2.23.p4.15.m15.2.2.1.1.1.1.1.1.3.cmml" xref="S6.SS2.23.p4.15.m15.2.2.1.1.1.1.1.1.2"><apply id="S6.SS2.23.p4.15.m15.2.2.1.1.1.1.1.1.1.1.cmml" xref="S6.SS2.23.p4.15.m15.2.2.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.23.p4.15.m15.2.2.1.1.1.1.1.1.1.1.1.cmml" xref="S6.SS2.23.p4.15.m15.2.2.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.23.p4.15.m15.2.2.1.1.1.1.1.1.1.1.2.cmml" xref="S6.SS2.23.p4.15.m15.2.2.1.1.1.1.1.1.1.1.2">𝑐</ci><ci id="S6.SS2.23.p4.15.m15.2.2.1.1.1.1.1.1.1.1.3.cmml" xref="S6.SS2.23.p4.15.m15.2.2.1.1.1.1.1.1.1.1.3">𝑙</ci></apply><apply id="S6.SS2.23.p4.15.m15.2.2.1.1.1.1.1.1.2.2.cmml" xref="S6.SS2.23.p4.15.m15.2.2.1.1.1.1.1.1.2.2"><csymbol cd="ambiguous" id="S6.SS2.23.p4.15.m15.2.2.1.1.1.1.1.1.2.2.1.cmml" xref="S6.SS2.23.p4.15.m15.2.2.1.1.1.1.1.1.2.2">subscript</csymbol><ci id="S6.SS2.23.p4.15.m15.2.2.1.1.1.1.1.1.2.2.2.cmml" xref="S6.SS2.23.p4.15.m15.2.2.1.1.1.1.1.1.2.2.2">𝑐</ci><ci id="S6.SS2.23.p4.15.m15.2.2.1.1.1.1.1.1.2.2.3.cmml" xref="S6.SS2.23.p4.15.m15.2.2.1.1.1.1.1.1.2.2.3">𝑟</ci></apply></interval><ci id="S6.SS2.23.p4.15.m15.1.1.cmml" xref="S6.SS2.23.p4.15.m15.1.1">𝑧</ci></interval></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.23.p4.15.m15.2c">p^{\prime}:=p\cup\{((c_{l},c_{r}),z)\}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.23.p4.15.m15.2d">italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT := italic_p ∪ { ( ( italic_c start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT , italic_c start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) , italic_z ) }</annotation></semantics></math> is in <math alttext="P_{E}" class="ltx_Math" display="inline" id="S6.SS2.23.p4.16.m16.1"><semantics id="S6.SS2.23.p4.16.m16.1a"><msub id="S6.SS2.23.p4.16.m16.1.1" xref="S6.SS2.23.p4.16.m16.1.1.cmml"><mi id="S6.SS2.23.p4.16.m16.1.1.2" xref="S6.SS2.23.p4.16.m16.1.1.2.cmml">P</mi><mi id="S6.SS2.23.p4.16.m16.1.1.3" xref="S6.SS2.23.p4.16.m16.1.1.3.cmml">E</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.23.p4.16.m16.1b"><apply id="S6.SS2.23.p4.16.m16.1.1.cmml" xref="S6.SS2.23.p4.16.m16.1.1"><csymbol cd="ambiguous" id="S6.SS2.23.p4.16.m16.1.1.1.cmml" xref="S6.SS2.23.p4.16.m16.1.1">subscript</csymbol><ci id="S6.SS2.23.p4.16.m16.1.1.2.cmml" xref="S6.SS2.23.p4.16.m16.1.1.2">𝑃</ci><ci id="S6.SS2.23.p4.16.m16.1.1.3.cmml" xref="S6.SS2.23.p4.16.m16.1.1.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.23.p4.16.m16.1c">P_{E}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.23.p4.16.m16.1d">italic_P start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S6.SS2.24.p5"> <p class="ltx_p" id="S6.SS2.24.p5.13">By construction <math alttext="[c_{l},c_{r}]" class="ltx_Math" display="inline" id="S6.SS2.24.p5.1.m1.2"><semantics id="S6.SS2.24.p5.1.m1.2a"><mrow id="S6.SS2.24.p5.1.m1.2.2.2" xref="S6.SS2.24.p5.1.m1.2.2.3.cmml"><mo id="S6.SS2.24.p5.1.m1.2.2.2.3" stretchy="false" xref="S6.SS2.24.p5.1.m1.2.2.3.cmml">[</mo><msub id="S6.SS2.24.p5.1.m1.1.1.1.1" xref="S6.SS2.24.p5.1.m1.1.1.1.1.cmml"><mi id="S6.SS2.24.p5.1.m1.1.1.1.1.2" xref="S6.SS2.24.p5.1.m1.1.1.1.1.2.cmml">c</mi><mi id="S6.SS2.24.p5.1.m1.1.1.1.1.3" xref="S6.SS2.24.p5.1.m1.1.1.1.1.3.cmml">l</mi></msub><mo id="S6.SS2.24.p5.1.m1.2.2.2.4" xref="S6.SS2.24.p5.1.m1.2.2.3.cmml">,</mo><msub id="S6.SS2.24.p5.1.m1.2.2.2.2" xref="S6.SS2.24.p5.1.m1.2.2.2.2.cmml"><mi id="S6.SS2.24.p5.1.m1.2.2.2.2.2" xref="S6.SS2.24.p5.1.m1.2.2.2.2.2.cmml">c</mi><mi id="S6.SS2.24.p5.1.m1.2.2.2.2.3" xref="S6.SS2.24.p5.1.m1.2.2.2.2.3.cmml">r</mi></msub><mo id="S6.SS2.24.p5.1.m1.2.2.2.5" stretchy="false" xref="S6.SS2.24.p5.1.m1.2.2.3.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.24.p5.1.m1.2b"><interval closure="closed" id="S6.SS2.24.p5.1.m1.2.2.3.cmml" xref="S6.SS2.24.p5.1.m1.2.2.2"><apply id="S6.SS2.24.p5.1.m1.1.1.1.1.cmml" xref="S6.SS2.24.p5.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.24.p5.1.m1.1.1.1.1.1.cmml" xref="S6.SS2.24.p5.1.m1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.24.p5.1.m1.1.1.1.1.2.cmml" xref="S6.SS2.24.p5.1.m1.1.1.1.1.2">𝑐</ci><ci id="S6.SS2.24.p5.1.m1.1.1.1.1.3.cmml" xref="S6.SS2.24.p5.1.m1.1.1.1.1.3">𝑙</ci></apply><apply id="S6.SS2.24.p5.1.m1.2.2.2.2.cmml" xref="S6.SS2.24.p5.1.m1.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.24.p5.1.m1.2.2.2.2.1.cmml" xref="S6.SS2.24.p5.1.m1.2.2.2.2">subscript</csymbol><ci id="S6.SS2.24.p5.1.m1.2.2.2.2.2.cmml" xref="S6.SS2.24.p5.1.m1.2.2.2.2.2">𝑐</ci><ci id="S6.SS2.24.p5.1.m1.2.2.2.2.3.cmml" xref="S6.SS2.24.p5.1.m1.2.2.2.2.3">𝑟</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.24.p5.1.m1.2c">[c_{l},c_{r}]</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.24.p5.1.m1.2d">[ italic_c start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT , italic_c start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ]</annotation></semantics></math> is disjoint from <math alttext="\operatorname{dom}(f_{p})" class="ltx_Math" display="inline" id="S6.SS2.24.p5.2.m2.2"><semantics id="S6.SS2.24.p5.2.m2.2a"><mrow id="S6.SS2.24.p5.2.m2.2.2.1" xref="S6.SS2.24.p5.2.m2.2.2.2.cmml"><mi id="S6.SS2.24.p5.2.m2.1.1" xref="S6.SS2.24.p5.2.m2.1.1.cmml">dom</mi><mo id="S6.SS2.24.p5.2.m2.2.2.1a" xref="S6.SS2.24.p5.2.m2.2.2.2.cmml">⁡</mo><mrow id="S6.SS2.24.p5.2.m2.2.2.1.1" xref="S6.SS2.24.p5.2.m2.2.2.2.cmml"><mo id="S6.SS2.24.p5.2.m2.2.2.1.1.2" stretchy="false" xref="S6.SS2.24.p5.2.m2.2.2.2.cmml">(</mo><msub id="S6.SS2.24.p5.2.m2.2.2.1.1.1" xref="S6.SS2.24.p5.2.m2.2.2.1.1.1.cmml"><mi id="S6.SS2.24.p5.2.m2.2.2.1.1.1.2" xref="S6.SS2.24.p5.2.m2.2.2.1.1.1.2.cmml">f</mi><mi id="S6.SS2.24.p5.2.m2.2.2.1.1.1.3" xref="S6.SS2.24.p5.2.m2.2.2.1.1.1.3.cmml">p</mi></msub><mo id="S6.SS2.24.p5.2.m2.2.2.1.1.3" stretchy="false" xref="S6.SS2.24.p5.2.m2.2.2.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.24.p5.2.m2.2b"><apply id="S6.SS2.24.p5.2.m2.2.2.2.cmml" xref="S6.SS2.24.p5.2.m2.2.2.1"><ci id="S6.SS2.24.p5.2.m2.1.1.cmml" xref="S6.SS2.24.p5.2.m2.1.1">dom</ci><apply id="S6.SS2.24.p5.2.m2.2.2.1.1.1.cmml" xref="S6.SS2.24.p5.2.m2.2.2.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.24.p5.2.m2.2.2.1.1.1.1.cmml" xref="S6.SS2.24.p5.2.m2.2.2.1.1.1">subscript</csymbol><ci id="S6.SS2.24.p5.2.m2.2.2.1.1.1.2.cmml" xref="S6.SS2.24.p5.2.m2.2.2.1.1.1.2">𝑓</ci><ci id="S6.SS2.24.p5.2.m2.2.2.1.1.1.3.cmml" xref="S6.SS2.24.p5.2.m2.2.2.1.1.1.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.24.p5.2.m2.2c">\operatorname{dom}(f_{p})</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.24.p5.2.m2.2d">roman_dom ( italic_f start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT )</annotation></semantics></math>, and <math alttext="p^{\prime}" class="ltx_Math" display="inline" id="S6.SS2.24.p5.3.m3.1"><semantics id="S6.SS2.24.p5.3.m3.1a"><msup id="S6.SS2.24.p5.3.m3.1.1" xref="S6.SS2.24.p5.3.m3.1.1.cmml"><mi id="S6.SS2.24.p5.3.m3.1.1.2" xref="S6.SS2.24.p5.3.m3.1.1.2.cmml">p</mi><mo id="S6.SS2.24.p5.3.m3.1.1.3" xref="S6.SS2.24.p5.3.m3.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S6.SS2.24.p5.3.m3.1b"><apply id="S6.SS2.24.p5.3.m3.1.1.cmml" xref="S6.SS2.24.p5.3.m3.1.1"><csymbol cd="ambiguous" id="S6.SS2.24.p5.3.m3.1.1.1.cmml" xref="S6.SS2.24.p5.3.m3.1.1">superscript</csymbol><ci id="S6.SS2.24.p5.3.m3.1.1.2.cmml" xref="S6.SS2.24.p5.3.m3.1.1.2">𝑝</ci><ci id="S6.SS2.24.p5.3.m3.1.1.3.cmml" xref="S6.SS2.24.p5.3.m3.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.24.p5.3.m3.1c">p^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.24.p5.3.m3.1d">italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> is increasing. Condition <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.Thmtheorem9" title="Definition 6.9. ‣ 6.2. Forcing epimorphisms ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">6.9</span></a> <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.I4.i1" title="Item (i) ‣ Definition 6.9. ‣ 6.2. Forcing epimorphisms ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">(i)</span></a> follows from the fact that <math alttext="\nu(c_{l})=\nu(c_{r})=\nu(z)" class="ltx_Math" display="inline" id="S6.SS2.24.p5.4.m4.3"><semantics id="S6.SS2.24.p5.4.m4.3a"><mrow id="S6.SS2.24.p5.4.m4.3.3" xref="S6.SS2.24.p5.4.m4.3.3.cmml"><mrow id="S6.SS2.24.p5.4.m4.2.2.1" xref="S6.SS2.24.p5.4.m4.2.2.1.cmml"><mi id="S6.SS2.24.p5.4.m4.2.2.1.3" xref="S6.SS2.24.p5.4.m4.2.2.1.3.cmml">ν</mi><mo id="S6.SS2.24.p5.4.m4.2.2.1.2" xref="S6.SS2.24.p5.4.m4.2.2.1.2.cmml">⁢</mo><mrow id="S6.SS2.24.p5.4.m4.2.2.1.1.1" xref="S6.SS2.24.p5.4.m4.2.2.1.1.1.1.cmml"><mo id="S6.SS2.24.p5.4.m4.2.2.1.1.1.2" stretchy="false" xref="S6.SS2.24.p5.4.m4.2.2.1.1.1.1.cmml">(</mo><msub id="S6.SS2.24.p5.4.m4.2.2.1.1.1.1" xref="S6.SS2.24.p5.4.m4.2.2.1.1.1.1.cmml"><mi id="S6.SS2.24.p5.4.m4.2.2.1.1.1.1.2" xref="S6.SS2.24.p5.4.m4.2.2.1.1.1.1.2.cmml">c</mi><mi id="S6.SS2.24.p5.4.m4.2.2.1.1.1.1.3" xref="S6.SS2.24.p5.4.m4.2.2.1.1.1.1.3.cmml">l</mi></msub><mo id="S6.SS2.24.p5.4.m4.2.2.1.1.1.3" stretchy="false" xref="S6.SS2.24.p5.4.m4.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.SS2.24.p5.4.m4.3.3.4" xref="S6.SS2.24.p5.4.m4.3.3.4.cmml">=</mo><mrow id="S6.SS2.24.p5.4.m4.3.3.2" xref="S6.SS2.24.p5.4.m4.3.3.2.cmml"><mi id="S6.SS2.24.p5.4.m4.3.3.2.3" xref="S6.SS2.24.p5.4.m4.3.3.2.3.cmml">ν</mi><mo id="S6.SS2.24.p5.4.m4.3.3.2.2" xref="S6.SS2.24.p5.4.m4.3.3.2.2.cmml">⁢</mo><mrow id="S6.SS2.24.p5.4.m4.3.3.2.1.1" xref="S6.SS2.24.p5.4.m4.3.3.2.1.1.1.cmml"><mo id="S6.SS2.24.p5.4.m4.3.3.2.1.1.2" stretchy="false" xref="S6.SS2.24.p5.4.m4.3.3.2.1.1.1.cmml">(</mo><msub id="S6.SS2.24.p5.4.m4.3.3.2.1.1.1" xref="S6.SS2.24.p5.4.m4.3.3.2.1.1.1.cmml"><mi id="S6.SS2.24.p5.4.m4.3.3.2.1.1.1.2" xref="S6.SS2.24.p5.4.m4.3.3.2.1.1.1.2.cmml">c</mi><mi id="S6.SS2.24.p5.4.m4.3.3.2.1.1.1.3" xref="S6.SS2.24.p5.4.m4.3.3.2.1.1.1.3.cmml">r</mi></msub><mo id="S6.SS2.24.p5.4.m4.3.3.2.1.1.3" stretchy="false" xref="S6.SS2.24.p5.4.m4.3.3.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.SS2.24.p5.4.m4.3.3.5" xref="S6.SS2.24.p5.4.m4.3.3.5.cmml">=</mo><mrow id="S6.SS2.24.p5.4.m4.3.3.6" xref="S6.SS2.24.p5.4.m4.3.3.6.cmml"><mi id="S6.SS2.24.p5.4.m4.3.3.6.2" xref="S6.SS2.24.p5.4.m4.3.3.6.2.cmml">ν</mi><mo id="S6.SS2.24.p5.4.m4.3.3.6.1" xref="S6.SS2.24.p5.4.m4.3.3.6.1.cmml">⁢</mo><mrow id="S6.SS2.24.p5.4.m4.3.3.6.3.2" xref="S6.SS2.24.p5.4.m4.3.3.6.cmml"><mo id="S6.SS2.24.p5.4.m4.3.3.6.3.2.1" stretchy="false" xref="S6.SS2.24.p5.4.m4.3.3.6.cmml">(</mo><mi id="S6.SS2.24.p5.4.m4.1.1" xref="S6.SS2.24.p5.4.m4.1.1.cmml">z</mi><mo id="S6.SS2.24.p5.4.m4.3.3.6.3.2.2" stretchy="false" xref="S6.SS2.24.p5.4.m4.3.3.6.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.24.p5.4.m4.3b"><apply id="S6.SS2.24.p5.4.m4.3.3.cmml" xref="S6.SS2.24.p5.4.m4.3.3"><and id="S6.SS2.24.p5.4.m4.3.3a.cmml" xref="S6.SS2.24.p5.4.m4.3.3"></and><apply id="S6.SS2.24.p5.4.m4.3.3b.cmml" xref="S6.SS2.24.p5.4.m4.3.3"><eq id="S6.SS2.24.p5.4.m4.3.3.4.cmml" xref="S6.SS2.24.p5.4.m4.3.3.4"></eq><apply id="S6.SS2.24.p5.4.m4.2.2.1.cmml" xref="S6.SS2.24.p5.4.m4.2.2.1"><times id="S6.SS2.24.p5.4.m4.2.2.1.2.cmml" xref="S6.SS2.24.p5.4.m4.2.2.1.2"></times><ci id="S6.SS2.24.p5.4.m4.2.2.1.3.cmml" xref="S6.SS2.24.p5.4.m4.2.2.1.3">𝜈</ci><apply id="S6.SS2.24.p5.4.m4.2.2.1.1.1.1.cmml" xref="S6.SS2.24.p5.4.m4.2.2.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.24.p5.4.m4.2.2.1.1.1.1.1.cmml" xref="S6.SS2.24.p5.4.m4.2.2.1.1.1">subscript</csymbol><ci id="S6.SS2.24.p5.4.m4.2.2.1.1.1.1.2.cmml" xref="S6.SS2.24.p5.4.m4.2.2.1.1.1.1.2">𝑐</ci><ci id="S6.SS2.24.p5.4.m4.2.2.1.1.1.1.3.cmml" xref="S6.SS2.24.p5.4.m4.2.2.1.1.1.1.3">𝑙</ci></apply></apply><apply id="S6.SS2.24.p5.4.m4.3.3.2.cmml" xref="S6.SS2.24.p5.4.m4.3.3.2"><times id="S6.SS2.24.p5.4.m4.3.3.2.2.cmml" xref="S6.SS2.24.p5.4.m4.3.3.2.2"></times><ci id="S6.SS2.24.p5.4.m4.3.3.2.3.cmml" xref="S6.SS2.24.p5.4.m4.3.3.2.3">𝜈</ci><apply id="S6.SS2.24.p5.4.m4.3.3.2.1.1.1.cmml" xref="S6.SS2.24.p5.4.m4.3.3.2.1.1"><csymbol cd="ambiguous" id="S6.SS2.24.p5.4.m4.3.3.2.1.1.1.1.cmml" xref="S6.SS2.24.p5.4.m4.3.3.2.1.1">subscript</csymbol><ci id="S6.SS2.24.p5.4.m4.3.3.2.1.1.1.2.cmml" xref="S6.SS2.24.p5.4.m4.3.3.2.1.1.1.2">𝑐</ci><ci id="S6.SS2.24.p5.4.m4.3.3.2.1.1.1.3.cmml" xref="S6.SS2.24.p5.4.m4.3.3.2.1.1.1.3">𝑟</ci></apply></apply></apply><apply id="S6.SS2.24.p5.4.m4.3.3c.cmml" xref="S6.SS2.24.p5.4.m4.3.3"><eq id="S6.SS2.24.p5.4.m4.3.3.5.cmml" xref="S6.SS2.24.p5.4.m4.3.3.5"></eq><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.24.p5.4.m4.3.3.2.cmml" id="S6.SS2.24.p5.4.m4.3.3d.cmml" xref="S6.SS2.24.p5.4.m4.3.3"></share><apply id="S6.SS2.24.p5.4.m4.3.3.6.cmml" xref="S6.SS2.24.p5.4.m4.3.3.6"><times id="S6.SS2.24.p5.4.m4.3.3.6.1.cmml" xref="S6.SS2.24.p5.4.m4.3.3.6.1"></times><ci id="S6.SS2.24.p5.4.m4.3.3.6.2.cmml" xref="S6.SS2.24.p5.4.m4.3.3.6.2">𝜈</ci><ci id="S6.SS2.24.p5.4.m4.1.1.cmml" xref="S6.SS2.24.p5.4.m4.1.1">𝑧</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.24.p5.4.m4.3c">\nu(c_{l})=\nu(c_{r})=\nu(z)</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.24.p5.4.m4.3d">italic_ν ( italic_c start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ) = italic_ν ( italic_c start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) = italic_ν ( italic_z )</annotation></semantics></math> and <math alttext="\nu(c_{l},c_{r})=\nu(z)" class="ltx_Math" display="inline" id="S6.SS2.24.p5.5.m5.3"><semantics id="S6.SS2.24.p5.5.m5.3a"><mrow id="S6.SS2.24.p5.5.m5.3.3" xref="S6.SS2.24.p5.5.m5.3.3.cmml"><mrow id="S6.SS2.24.p5.5.m5.3.3.2" xref="S6.SS2.24.p5.5.m5.3.3.2.cmml"><mi id="S6.SS2.24.p5.5.m5.3.3.2.4" xref="S6.SS2.24.p5.5.m5.3.3.2.4.cmml">ν</mi><mo id="S6.SS2.24.p5.5.m5.3.3.2.3" xref="S6.SS2.24.p5.5.m5.3.3.2.3.cmml">⁢</mo><mrow id="S6.SS2.24.p5.5.m5.3.3.2.2.2" xref="S6.SS2.24.p5.5.m5.3.3.2.2.3.cmml"><mo id="S6.SS2.24.p5.5.m5.3.3.2.2.2.3" stretchy="false" xref="S6.SS2.24.p5.5.m5.3.3.2.2.3.cmml">(</mo><msub id="S6.SS2.24.p5.5.m5.2.2.1.1.1.1" xref="S6.SS2.24.p5.5.m5.2.2.1.1.1.1.cmml"><mi id="S6.SS2.24.p5.5.m5.2.2.1.1.1.1.2" xref="S6.SS2.24.p5.5.m5.2.2.1.1.1.1.2.cmml">c</mi><mi id="S6.SS2.24.p5.5.m5.2.2.1.1.1.1.3" xref="S6.SS2.24.p5.5.m5.2.2.1.1.1.1.3.cmml">l</mi></msub><mo id="S6.SS2.24.p5.5.m5.3.3.2.2.2.4" xref="S6.SS2.24.p5.5.m5.3.3.2.2.3.cmml">,</mo><msub id="S6.SS2.24.p5.5.m5.3.3.2.2.2.2" xref="S6.SS2.24.p5.5.m5.3.3.2.2.2.2.cmml"><mi id="S6.SS2.24.p5.5.m5.3.3.2.2.2.2.2" xref="S6.SS2.24.p5.5.m5.3.3.2.2.2.2.2.cmml">c</mi><mi id="S6.SS2.24.p5.5.m5.3.3.2.2.2.2.3" xref="S6.SS2.24.p5.5.m5.3.3.2.2.2.2.3.cmml">r</mi></msub><mo id="S6.SS2.24.p5.5.m5.3.3.2.2.2.5" stretchy="false" xref="S6.SS2.24.p5.5.m5.3.3.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.SS2.24.p5.5.m5.3.3.3" xref="S6.SS2.24.p5.5.m5.3.3.3.cmml">=</mo><mrow id="S6.SS2.24.p5.5.m5.3.3.4" xref="S6.SS2.24.p5.5.m5.3.3.4.cmml"><mi id="S6.SS2.24.p5.5.m5.3.3.4.2" xref="S6.SS2.24.p5.5.m5.3.3.4.2.cmml">ν</mi><mo id="S6.SS2.24.p5.5.m5.3.3.4.1" xref="S6.SS2.24.p5.5.m5.3.3.4.1.cmml">⁢</mo><mrow id="S6.SS2.24.p5.5.m5.3.3.4.3.2" xref="S6.SS2.24.p5.5.m5.3.3.4.cmml"><mo id="S6.SS2.24.p5.5.m5.3.3.4.3.2.1" stretchy="false" xref="S6.SS2.24.p5.5.m5.3.3.4.cmml">(</mo><mi id="S6.SS2.24.p5.5.m5.1.1" xref="S6.SS2.24.p5.5.m5.1.1.cmml">z</mi><mo id="S6.SS2.24.p5.5.m5.3.3.4.3.2.2" stretchy="false" xref="S6.SS2.24.p5.5.m5.3.3.4.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.24.p5.5.m5.3b"><apply id="S6.SS2.24.p5.5.m5.3.3.cmml" xref="S6.SS2.24.p5.5.m5.3.3"><eq id="S6.SS2.24.p5.5.m5.3.3.3.cmml" xref="S6.SS2.24.p5.5.m5.3.3.3"></eq><apply id="S6.SS2.24.p5.5.m5.3.3.2.cmml" xref="S6.SS2.24.p5.5.m5.3.3.2"><times id="S6.SS2.24.p5.5.m5.3.3.2.3.cmml" xref="S6.SS2.24.p5.5.m5.3.3.2.3"></times><ci id="S6.SS2.24.p5.5.m5.3.3.2.4.cmml" xref="S6.SS2.24.p5.5.m5.3.3.2.4">𝜈</ci><interval closure="open" id="S6.SS2.24.p5.5.m5.3.3.2.2.3.cmml" xref="S6.SS2.24.p5.5.m5.3.3.2.2.2"><apply id="S6.SS2.24.p5.5.m5.2.2.1.1.1.1.cmml" xref="S6.SS2.24.p5.5.m5.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.24.p5.5.m5.2.2.1.1.1.1.1.cmml" xref="S6.SS2.24.p5.5.m5.2.2.1.1.1.1">subscript</csymbol><ci id="S6.SS2.24.p5.5.m5.2.2.1.1.1.1.2.cmml" xref="S6.SS2.24.p5.5.m5.2.2.1.1.1.1.2">𝑐</ci><ci id="S6.SS2.24.p5.5.m5.2.2.1.1.1.1.3.cmml" xref="S6.SS2.24.p5.5.m5.2.2.1.1.1.1.3">𝑙</ci></apply><apply id="S6.SS2.24.p5.5.m5.3.3.2.2.2.2.cmml" xref="S6.SS2.24.p5.5.m5.3.3.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.24.p5.5.m5.3.3.2.2.2.2.1.cmml" xref="S6.SS2.24.p5.5.m5.3.3.2.2.2.2">subscript</csymbol><ci id="S6.SS2.24.p5.5.m5.3.3.2.2.2.2.2.cmml" xref="S6.SS2.24.p5.5.m5.3.3.2.2.2.2.2">𝑐</ci><ci id="S6.SS2.24.p5.5.m5.3.3.2.2.2.2.3.cmml" xref="S6.SS2.24.p5.5.m5.3.3.2.2.2.2.3">𝑟</ci></apply></interval></apply><apply id="S6.SS2.24.p5.5.m5.3.3.4.cmml" xref="S6.SS2.24.p5.5.m5.3.3.4"><times id="S6.SS2.24.p5.5.m5.3.3.4.1.cmml" xref="S6.SS2.24.p5.5.m5.3.3.4.1"></times><ci id="S6.SS2.24.p5.5.m5.3.3.4.2.cmml" xref="S6.SS2.24.p5.5.m5.3.3.4.2">𝜈</ci><ci id="S6.SS2.24.p5.5.m5.1.1.cmml" xref="S6.SS2.24.p5.5.m5.1.1">𝑧</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.24.p5.5.m5.3c">\nu(c_{l},c_{r})=\nu(z)</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.24.p5.5.m5.3d">italic_ν ( italic_c start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT , italic_c start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) = italic_ν ( italic_z )</annotation></semantics></math> because then <math alttext="c_{l}" class="ltx_Math" display="inline" id="S6.SS2.24.p5.6.m6.1"><semantics id="S6.SS2.24.p5.6.m6.1a"><msub id="S6.SS2.24.p5.6.m6.1.1" xref="S6.SS2.24.p5.6.m6.1.1.cmml"><mi id="S6.SS2.24.p5.6.m6.1.1.2" xref="S6.SS2.24.p5.6.m6.1.1.2.cmml">c</mi><mi id="S6.SS2.24.p5.6.m6.1.1.3" xref="S6.SS2.24.p5.6.m6.1.1.3.cmml">l</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.24.p5.6.m6.1b"><apply id="S6.SS2.24.p5.6.m6.1.1.cmml" xref="S6.SS2.24.p5.6.m6.1.1"><csymbol cd="ambiguous" id="S6.SS2.24.p5.6.m6.1.1.1.cmml" xref="S6.SS2.24.p5.6.m6.1.1">subscript</csymbol><ci id="S6.SS2.24.p5.6.m6.1.1.2.cmml" xref="S6.SS2.24.p5.6.m6.1.1.2">𝑐</ci><ci id="S6.SS2.24.p5.6.m6.1.1.3.cmml" xref="S6.SS2.24.p5.6.m6.1.1.3">𝑙</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.24.p5.6.m6.1c">c_{l}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.24.p5.6.m6.1d">italic_c start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="c_{r}" class="ltx_Math" display="inline" id="S6.SS2.24.p5.7.m7.1"><semantics id="S6.SS2.24.p5.7.m7.1a"><msub id="S6.SS2.24.p5.7.m7.1.1" xref="S6.SS2.24.p5.7.m7.1.1.cmml"><mi id="S6.SS2.24.p5.7.m7.1.1.2" xref="S6.SS2.24.p5.7.m7.1.1.2.cmml">c</mi><mi id="S6.SS2.24.p5.7.m7.1.1.3" xref="S6.SS2.24.p5.7.m7.1.1.3.cmml">r</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.24.p5.7.m7.1b"><apply id="S6.SS2.24.p5.7.m7.1.1.cmml" xref="S6.SS2.24.p5.7.m7.1.1"><csymbol cd="ambiguous" id="S6.SS2.24.p5.7.m7.1.1.1.cmml" xref="S6.SS2.24.p5.7.m7.1.1">subscript</csymbol><ci id="S6.SS2.24.p5.7.m7.1.1.2.cmml" xref="S6.SS2.24.p5.7.m7.1.1.2">𝑐</ci><ci id="S6.SS2.24.p5.7.m7.1.1.3.cmml" xref="S6.SS2.24.p5.7.m7.1.1.3">𝑟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.24.p5.7.m7.1c">c_{r}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.24.p5.7.m7.1d">italic_c start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT</annotation></semantics></math> are in the same complementary interval of <math alttext="A\setminus\nu(z)" class="ltx_Math" display="inline" id="S6.SS2.24.p5.8.m8.1"><semantics id="S6.SS2.24.p5.8.m8.1a"><mrow id="S6.SS2.24.p5.8.m8.1.2" xref="S6.SS2.24.p5.8.m8.1.2.cmml"><mi id="S6.SS2.24.p5.8.m8.1.2.2" xref="S6.SS2.24.p5.8.m8.1.2.2.cmml">A</mi><mo id="S6.SS2.24.p5.8.m8.1.2.1" xref="S6.SS2.24.p5.8.m8.1.2.1.cmml">∖</mo><mrow id="S6.SS2.24.p5.8.m8.1.2.3" xref="S6.SS2.24.p5.8.m8.1.2.3.cmml"><mi id="S6.SS2.24.p5.8.m8.1.2.3.2" xref="S6.SS2.24.p5.8.m8.1.2.3.2.cmml">ν</mi><mo id="S6.SS2.24.p5.8.m8.1.2.3.1" xref="S6.SS2.24.p5.8.m8.1.2.3.1.cmml">⁢</mo><mrow id="S6.SS2.24.p5.8.m8.1.2.3.3.2" xref="S6.SS2.24.p5.8.m8.1.2.3.cmml"><mo id="S6.SS2.24.p5.8.m8.1.2.3.3.2.1" stretchy="false" xref="S6.SS2.24.p5.8.m8.1.2.3.cmml">(</mo><mi id="S6.SS2.24.p5.8.m8.1.1" xref="S6.SS2.24.p5.8.m8.1.1.cmml">z</mi><mo id="S6.SS2.24.p5.8.m8.1.2.3.3.2.2" stretchy="false" xref="S6.SS2.24.p5.8.m8.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.24.p5.8.m8.1b"><apply id="S6.SS2.24.p5.8.m8.1.2.cmml" xref="S6.SS2.24.p5.8.m8.1.2"><setdiff id="S6.SS2.24.p5.8.m8.1.2.1.cmml" xref="S6.SS2.24.p5.8.m8.1.2.1"></setdiff><ci id="S6.SS2.24.p5.8.m8.1.2.2.cmml" xref="S6.SS2.24.p5.8.m8.1.2.2">𝐴</ci><apply id="S6.SS2.24.p5.8.m8.1.2.3.cmml" xref="S6.SS2.24.p5.8.m8.1.2.3"><times id="S6.SS2.24.p5.8.m8.1.2.3.1.cmml" xref="S6.SS2.24.p5.8.m8.1.2.3.1"></times><ci id="S6.SS2.24.p5.8.m8.1.2.3.2.cmml" xref="S6.SS2.24.p5.8.m8.1.2.3.2">𝜈</ci><ci id="S6.SS2.24.p5.8.m8.1.1.cmml" xref="S6.SS2.24.p5.8.m8.1.1">𝑧</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.24.p5.8.m8.1c">A\setminus\nu(z)</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.24.p5.8.m8.1d">italic_A ∖ italic_ν ( italic_z )</annotation></semantics></math>. <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.Thmtheorem9" title="Definition 6.9. ‣ 6.2. Forcing epimorphisms ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">6.9</span></a> <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.I4.i3" title="Item (iii) ‣ Definition 6.9. ‣ 6.2. Forcing epimorphisms ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">(iii)</span></a> follows because we chose <math alttext="c_{l}" class="ltx_Math" display="inline" id="S6.SS2.24.p5.9.m9.1"><semantics id="S6.SS2.24.p5.9.m9.1a"><msub id="S6.SS2.24.p5.9.m9.1.1" xref="S6.SS2.24.p5.9.m9.1.1.cmml"><mi id="S6.SS2.24.p5.9.m9.1.1.2" xref="S6.SS2.24.p5.9.m9.1.1.2.cmml">c</mi><mi id="S6.SS2.24.p5.9.m9.1.1.3" xref="S6.SS2.24.p5.9.m9.1.1.3.cmml">l</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.24.p5.9.m9.1b"><apply id="S6.SS2.24.p5.9.m9.1.1.cmml" xref="S6.SS2.24.p5.9.m9.1.1"><csymbol cd="ambiguous" id="S6.SS2.24.p5.9.m9.1.1.1.cmml" xref="S6.SS2.24.p5.9.m9.1.1">subscript</csymbol><ci id="S6.SS2.24.p5.9.m9.1.1.2.cmml" xref="S6.SS2.24.p5.9.m9.1.1.2">𝑐</ci><ci id="S6.SS2.24.p5.9.m9.1.1.3.cmml" xref="S6.SS2.24.p5.9.m9.1.1.3">𝑙</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.24.p5.9.m9.1c">c_{l}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.24.p5.9.m9.1d">italic_c start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="c_{r}" class="ltx_Math" display="inline" id="S6.SS2.24.p5.10.m10.1"><semantics id="S6.SS2.24.p5.10.m10.1a"><msub id="S6.SS2.24.p5.10.m10.1.1" xref="S6.SS2.24.p5.10.m10.1.1.cmml"><mi id="S6.SS2.24.p5.10.m10.1.1.2" xref="S6.SS2.24.p5.10.m10.1.1.2.cmml">c</mi><mi id="S6.SS2.24.p5.10.m10.1.1.3" xref="S6.SS2.24.p5.10.m10.1.1.3.cmml">r</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.24.p5.10.m10.1b"><apply id="S6.SS2.24.p5.10.m10.1.1.cmml" xref="S6.SS2.24.p5.10.m10.1.1"><csymbol cd="ambiguous" id="S6.SS2.24.p5.10.m10.1.1.1.cmml" xref="S6.SS2.24.p5.10.m10.1.1">subscript</csymbol><ci id="S6.SS2.24.p5.10.m10.1.1.2.cmml" xref="S6.SS2.24.p5.10.m10.1.1.2">𝑐</ci><ci id="S6.SS2.24.p5.10.m10.1.1.3.cmml" xref="S6.SS2.24.p5.10.m10.1.1.3">𝑟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.24.p5.10.m10.1c">c_{r}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.24.p5.10.m10.1d">italic_c start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT</annotation></semantics></math> to not be endpoints of <math alttext="I" class="ltx_Math" display="inline" id="S6.SS2.24.p5.11.m11.1"><semantics id="S6.SS2.24.p5.11.m11.1a"><mi id="S6.SS2.24.p5.11.m11.1.1" xref="S6.SS2.24.p5.11.m11.1.1.cmml">I</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.24.p5.11.m11.1b"><ci id="S6.SS2.24.p5.11.m11.1.1.cmml" xref="S6.SS2.24.p5.11.m11.1.1">𝐼</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.24.p5.11.m11.1c">I</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.24.p5.11.m11.1d">italic_I</annotation></semantics></math>. Using <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.Thmtheorem15" title="Lemma 6.15. ‣ 6.2. Forcing epimorphisms ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">6.15</span></a> we see that <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.Thmtheorem9" title="Definition 6.9. ‣ 6.2. Forcing epimorphisms ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">6.9</span></a> <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.I4.i2" title="Item (ii) ‣ Definition 6.9. ‣ 6.2. Forcing epimorphisms ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">(ii)</span></a> holds because by construction <math alttext="\Delta(a_{r},c_{l})=\Delta(a_{r},b_{l})=\Delta(x,z)" class="ltx_Math" display="inline" id="S6.SS2.24.p5.12.m12.6"><semantics id="S6.SS2.24.p5.12.m12.6a"><mrow id="S6.SS2.24.p5.12.m12.6.6" xref="S6.SS2.24.p5.12.m12.6.6.cmml"><mrow id="S6.SS2.24.p5.12.m12.4.4.2" xref="S6.SS2.24.p5.12.m12.4.4.2.cmml"><mi id="S6.SS2.24.p5.12.m12.4.4.2.4" mathvariant="normal" xref="S6.SS2.24.p5.12.m12.4.4.2.4.cmml">Δ</mi><mo id="S6.SS2.24.p5.12.m12.4.4.2.3" xref="S6.SS2.24.p5.12.m12.4.4.2.3.cmml">⁢</mo><mrow id="S6.SS2.24.p5.12.m12.4.4.2.2.2" xref="S6.SS2.24.p5.12.m12.4.4.2.2.3.cmml"><mo id="S6.SS2.24.p5.12.m12.4.4.2.2.2.3" stretchy="false" xref="S6.SS2.24.p5.12.m12.4.4.2.2.3.cmml">(</mo><msub id="S6.SS2.24.p5.12.m12.3.3.1.1.1.1" xref="S6.SS2.24.p5.12.m12.3.3.1.1.1.1.cmml"><mi id="S6.SS2.24.p5.12.m12.3.3.1.1.1.1.2" xref="S6.SS2.24.p5.12.m12.3.3.1.1.1.1.2.cmml">a</mi><mi id="S6.SS2.24.p5.12.m12.3.3.1.1.1.1.3" xref="S6.SS2.24.p5.12.m12.3.3.1.1.1.1.3.cmml">r</mi></msub><mo id="S6.SS2.24.p5.12.m12.4.4.2.2.2.4" xref="S6.SS2.24.p5.12.m12.4.4.2.2.3.cmml">,</mo><msub id="S6.SS2.24.p5.12.m12.4.4.2.2.2.2" xref="S6.SS2.24.p5.12.m12.4.4.2.2.2.2.cmml"><mi id="S6.SS2.24.p5.12.m12.4.4.2.2.2.2.2" xref="S6.SS2.24.p5.12.m12.4.4.2.2.2.2.2.cmml">c</mi><mi id="S6.SS2.24.p5.12.m12.4.4.2.2.2.2.3" xref="S6.SS2.24.p5.12.m12.4.4.2.2.2.2.3.cmml">l</mi></msub><mo id="S6.SS2.24.p5.12.m12.4.4.2.2.2.5" stretchy="false" xref="S6.SS2.24.p5.12.m12.4.4.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.SS2.24.p5.12.m12.6.6.6" xref="S6.SS2.24.p5.12.m12.6.6.6.cmml">=</mo><mrow id="S6.SS2.24.p5.12.m12.6.6.4" xref="S6.SS2.24.p5.12.m12.6.6.4.cmml"><mi id="S6.SS2.24.p5.12.m12.6.6.4.4" mathvariant="normal" xref="S6.SS2.24.p5.12.m12.6.6.4.4.cmml">Δ</mi><mo id="S6.SS2.24.p5.12.m12.6.6.4.3" xref="S6.SS2.24.p5.12.m12.6.6.4.3.cmml">⁢</mo><mrow id="S6.SS2.24.p5.12.m12.6.6.4.2.2" xref="S6.SS2.24.p5.12.m12.6.6.4.2.3.cmml"><mo id="S6.SS2.24.p5.12.m12.6.6.4.2.2.3" stretchy="false" xref="S6.SS2.24.p5.12.m12.6.6.4.2.3.cmml">(</mo><msub id="S6.SS2.24.p5.12.m12.5.5.3.1.1.1" xref="S6.SS2.24.p5.12.m12.5.5.3.1.1.1.cmml"><mi id="S6.SS2.24.p5.12.m12.5.5.3.1.1.1.2" xref="S6.SS2.24.p5.12.m12.5.5.3.1.1.1.2.cmml">a</mi><mi id="S6.SS2.24.p5.12.m12.5.5.3.1.1.1.3" xref="S6.SS2.24.p5.12.m12.5.5.3.1.1.1.3.cmml">r</mi></msub><mo id="S6.SS2.24.p5.12.m12.6.6.4.2.2.4" xref="S6.SS2.24.p5.12.m12.6.6.4.2.3.cmml">,</mo><msub id="S6.SS2.24.p5.12.m12.6.6.4.2.2.2" xref="S6.SS2.24.p5.12.m12.6.6.4.2.2.2.cmml"><mi id="S6.SS2.24.p5.12.m12.6.6.4.2.2.2.2" xref="S6.SS2.24.p5.12.m12.6.6.4.2.2.2.2.cmml">b</mi><mi id="S6.SS2.24.p5.12.m12.6.6.4.2.2.2.3" xref="S6.SS2.24.p5.12.m12.6.6.4.2.2.2.3.cmml">l</mi></msub><mo id="S6.SS2.24.p5.12.m12.6.6.4.2.2.5" stretchy="false" xref="S6.SS2.24.p5.12.m12.6.6.4.2.3.cmml">)</mo></mrow></mrow><mo id="S6.SS2.24.p5.12.m12.6.6.7" xref="S6.SS2.24.p5.12.m12.6.6.7.cmml">=</mo><mrow id="S6.SS2.24.p5.12.m12.6.6.8" xref="S6.SS2.24.p5.12.m12.6.6.8.cmml"><mi id="S6.SS2.24.p5.12.m12.6.6.8.2" mathvariant="normal" xref="S6.SS2.24.p5.12.m12.6.6.8.2.cmml">Δ</mi><mo id="S6.SS2.24.p5.12.m12.6.6.8.1" xref="S6.SS2.24.p5.12.m12.6.6.8.1.cmml">⁢</mo><mrow id="S6.SS2.24.p5.12.m12.6.6.8.3.2" xref="S6.SS2.24.p5.12.m12.6.6.8.3.1.cmml"><mo id="S6.SS2.24.p5.12.m12.6.6.8.3.2.1" stretchy="false" xref="S6.SS2.24.p5.12.m12.6.6.8.3.1.cmml">(</mo><mi id="S6.SS2.24.p5.12.m12.1.1" xref="S6.SS2.24.p5.12.m12.1.1.cmml">x</mi><mo id="S6.SS2.24.p5.12.m12.6.6.8.3.2.2" xref="S6.SS2.24.p5.12.m12.6.6.8.3.1.cmml">,</mo><mi id="S6.SS2.24.p5.12.m12.2.2" xref="S6.SS2.24.p5.12.m12.2.2.cmml">z</mi><mo id="S6.SS2.24.p5.12.m12.6.6.8.3.2.3" stretchy="false" xref="S6.SS2.24.p5.12.m12.6.6.8.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.24.p5.12.m12.6b"><apply id="S6.SS2.24.p5.12.m12.6.6.cmml" xref="S6.SS2.24.p5.12.m12.6.6"><and id="S6.SS2.24.p5.12.m12.6.6a.cmml" xref="S6.SS2.24.p5.12.m12.6.6"></and><apply id="S6.SS2.24.p5.12.m12.6.6b.cmml" xref="S6.SS2.24.p5.12.m12.6.6"><eq id="S6.SS2.24.p5.12.m12.6.6.6.cmml" xref="S6.SS2.24.p5.12.m12.6.6.6"></eq><apply id="S6.SS2.24.p5.12.m12.4.4.2.cmml" xref="S6.SS2.24.p5.12.m12.4.4.2"><times id="S6.SS2.24.p5.12.m12.4.4.2.3.cmml" xref="S6.SS2.24.p5.12.m12.4.4.2.3"></times><ci id="S6.SS2.24.p5.12.m12.4.4.2.4.cmml" xref="S6.SS2.24.p5.12.m12.4.4.2.4">Δ</ci><interval closure="open" id="S6.SS2.24.p5.12.m12.4.4.2.2.3.cmml" xref="S6.SS2.24.p5.12.m12.4.4.2.2.2"><apply id="S6.SS2.24.p5.12.m12.3.3.1.1.1.1.cmml" xref="S6.SS2.24.p5.12.m12.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.24.p5.12.m12.3.3.1.1.1.1.1.cmml" xref="S6.SS2.24.p5.12.m12.3.3.1.1.1.1">subscript</csymbol><ci id="S6.SS2.24.p5.12.m12.3.3.1.1.1.1.2.cmml" xref="S6.SS2.24.p5.12.m12.3.3.1.1.1.1.2">𝑎</ci><ci id="S6.SS2.24.p5.12.m12.3.3.1.1.1.1.3.cmml" xref="S6.SS2.24.p5.12.m12.3.3.1.1.1.1.3">𝑟</ci></apply><apply id="S6.SS2.24.p5.12.m12.4.4.2.2.2.2.cmml" xref="S6.SS2.24.p5.12.m12.4.4.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.24.p5.12.m12.4.4.2.2.2.2.1.cmml" xref="S6.SS2.24.p5.12.m12.4.4.2.2.2.2">subscript</csymbol><ci id="S6.SS2.24.p5.12.m12.4.4.2.2.2.2.2.cmml" xref="S6.SS2.24.p5.12.m12.4.4.2.2.2.2.2">𝑐</ci><ci id="S6.SS2.24.p5.12.m12.4.4.2.2.2.2.3.cmml" xref="S6.SS2.24.p5.12.m12.4.4.2.2.2.2.3">𝑙</ci></apply></interval></apply><apply id="S6.SS2.24.p5.12.m12.6.6.4.cmml" xref="S6.SS2.24.p5.12.m12.6.6.4"><times id="S6.SS2.24.p5.12.m12.6.6.4.3.cmml" xref="S6.SS2.24.p5.12.m12.6.6.4.3"></times><ci id="S6.SS2.24.p5.12.m12.6.6.4.4.cmml" xref="S6.SS2.24.p5.12.m12.6.6.4.4">Δ</ci><interval closure="open" id="S6.SS2.24.p5.12.m12.6.6.4.2.3.cmml" xref="S6.SS2.24.p5.12.m12.6.6.4.2.2"><apply id="S6.SS2.24.p5.12.m12.5.5.3.1.1.1.cmml" xref="S6.SS2.24.p5.12.m12.5.5.3.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.24.p5.12.m12.5.5.3.1.1.1.1.cmml" xref="S6.SS2.24.p5.12.m12.5.5.3.1.1.1">subscript</csymbol><ci id="S6.SS2.24.p5.12.m12.5.5.3.1.1.1.2.cmml" xref="S6.SS2.24.p5.12.m12.5.5.3.1.1.1.2">𝑎</ci><ci id="S6.SS2.24.p5.12.m12.5.5.3.1.1.1.3.cmml" xref="S6.SS2.24.p5.12.m12.5.5.3.1.1.1.3">𝑟</ci></apply><apply id="S6.SS2.24.p5.12.m12.6.6.4.2.2.2.cmml" xref="S6.SS2.24.p5.12.m12.6.6.4.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.24.p5.12.m12.6.6.4.2.2.2.1.cmml" xref="S6.SS2.24.p5.12.m12.6.6.4.2.2.2">subscript</csymbol><ci id="S6.SS2.24.p5.12.m12.6.6.4.2.2.2.2.cmml" xref="S6.SS2.24.p5.12.m12.6.6.4.2.2.2.2">𝑏</ci><ci id="S6.SS2.24.p5.12.m12.6.6.4.2.2.2.3.cmml" xref="S6.SS2.24.p5.12.m12.6.6.4.2.2.2.3">𝑙</ci></apply></interval></apply></apply><apply id="S6.SS2.24.p5.12.m12.6.6c.cmml" xref="S6.SS2.24.p5.12.m12.6.6"><eq id="S6.SS2.24.p5.12.m12.6.6.7.cmml" xref="S6.SS2.24.p5.12.m12.6.6.7"></eq><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.24.p5.12.m12.6.6.4.cmml" id="S6.SS2.24.p5.12.m12.6.6d.cmml" xref="S6.SS2.24.p5.12.m12.6.6"></share><apply id="S6.SS2.24.p5.12.m12.6.6.8.cmml" xref="S6.SS2.24.p5.12.m12.6.6.8"><times id="S6.SS2.24.p5.12.m12.6.6.8.1.cmml" xref="S6.SS2.24.p5.12.m12.6.6.8.1"></times><ci id="S6.SS2.24.p5.12.m12.6.6.8.2.cmml" xref="S6.SS2.24.p5.12.m12.6.6.8.2">Δ</ci><interval closure="open" id="S6.SS2.24.p5.12.m12.6.6.8.3.1.cmml" xref="S6.SS2.24.p5.12.m12.6.6.8.3.2"><ci id="S6.SS2.24.p5.12.m12.1.1.cmml" xref="S6.SS2.24.p5.12.m12.1.1">𝑥</ci><ci id="S6.SS2.24.p5.12.m12.2.2.cmml" xref="S6.SS2.24.p5.12.m12.2.2">𝑧</ci></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.24.p5.12.m12.6c">\Delta(a_{r},c_{l})=\Delta(a_{r},b_{l})=\Delta(x,z)</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.24.p5.12.m12.6d">roman_Δ ( italic_a start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT , italic_c start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ) = roman_Δ ( italic_a start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ) = roman_Δ ( italic_x , italic_z )</annotation></semantics></math>, and <math alttext="\nu(c_{r},b_{l})=\nu=\nu(z,y)" class="ltx_Math" display="inline" id="S6.SS2.24.p5.13.m13.4"><semantics id="S6.SS2.24.p5.13.m13.4a"><mrow id="S6.SS2.24.p5.13.m13.4.4" xref="S6.SS2.24.p5.13.m13.4.4.cmml"><mrow id="S6.SS2.24.p5.13.m13.4.4.2" xref="S6.SS2.24.p5.13.m13.4.4.2.cmml"><mi id="S6.SS2.24.p5.13.m13.4.4.2.4" xref="S6.SS2.24.p5.13.m13.4.4.2.4.cmml">ν</mi><mo id="S6.SS2.24.p5.13.m13.4.4.2.3" xref="S6.SS2.24.p5.13.m13.4.4.2.3.cmml">⁢</mo><mrow id="S6.SS2.24.p5.13.m13.4.4.2.2.2" xref="S6.SS2.24.p5.13.m13.4.4.2.2.3.cmml"><mo id="S6.SS2.24.p5.13.m13.4.4.2.2.2.3" stretchy="false" xref="S6.SS2.24.p5.13.m13.4.4.2.2.3.cmml">(</mo><msub id="S6.SS2.24.p5.13.m13.3.3.1.1.1.1" xref="S6.SS2.24.p5.13.m13.3.3.1.1.1.1.cmml"><mi id="S6.SS2.24.p5.13.m13.3.3.1.1.1.1.2" xref="S6.SS2.24.p5.13.m13.3.3.1.1.1.1.2.cmml">c</mi><mi id="S6.SS2.24.p5.13.m13.3.3.1.1.1.1.3" xref="S6.SS2.24.p5.13.m13.3.3.1.1.1.1.3.cmml">r</mi></msub><mo id="S6.SS2.24.p5.13.m13.4.4.2.2.2.4" xref="S6.SS2.24.p5.13.m13.4.4.2.2.3.cmml">,</mo><msub id="S6.SS2.24.p5.13.m13.4.4.2.2.2.2" xref="S6.SS2.24.p5.13.m13.4.4.2.2.2.2.cmml"><mi id="S6.SS2.24.p5.13.m13.4.4.2.2.2.2.2" xref="S6.SS2.24.p5.13.m13.4.4.2.2.2.2.2.cmml">b</mi><mi id="S6.SS2.24.p5.13.m13.4.4.2.2.2.2.3" xref="S6.SS2.24.p5.13.m13.4.4.2.2.2.2.3.cmml">l</mi></msub><mo id="S6.SS2.24.p5.13.m13.4.4.2.2.2.5" stretchy="false" xref="S6.SS2.24.p5.13.m13.4.4.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.SS2.24.p5.13.m13.4.4.4" xref="S6.SS2.24.p5.13.m13.4.4.4.cmml">=</mo><mi id="S6.SS2.24.p5.13.m13.4.4.5" xref="S6.SS2.24.p5.13.m13.4.4.5.cmml">ν</mi><mo id="S6.SS2.24.p5.13.m13.4.4.6" xref="S6.SS2.24.p5.13.m13.4.4.6.cmml">=</mo><mrow id="S6.SS2.24.p5.13.m13.4.4.7" xref="S6.SS2.24.p5.13.m13.4.4.7.cmml"><mi id="S6.SS2.24.p5.13.m13.4.4.7.2" xref="S6.SS2.24.p5.13.m13.4.4.7.2.cmml">ν</mi><mo id="S6.SS2.24.p5.13.m13.4.4.7.1" xref="S6.SS2.24.p5.13.m13.4.4.7.1.cmml">⁢</mo><mrow id="S6.SS2.24.p5.13.m13.4.4.7.3.2" xref="S6.SS2.24.p5.13.m13.4.4.7.3.1.cmml"><mo id="S6.SS2.24.p5.13.m13.4.4.7.3.2.1" stretchy="false" xref="S6.SS2.24.p5.13.m13.4.4.7.3.1.cmml">(</mo><mi id="S6.SS2.24.p5.13.m13.1.1" xref="S6.SS2.24.p5.13.m13.1.1.cmml">z</mi><mo id="S6.SS2.24.p5.13.m13.4.4.7.3.2.2" xref="S6.SS2.24.p5.13.m13.4.4.7.3.1.cmml">,</mo><mi id="S6.SS2.24.p5.13.m13.2.2" xref="S6.SS2.24.p5.13.m13.2.2.cmml">y</mi><mo id="S6.SS2.24.p5.13.m13.4.4.7.3.2.3" stretchy="false" xref="S6.SS2.24.p5.13.m13.4.4.7.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.24.p5.13.m13.4b"><apply id="S6.SS2.24.p5.13.m13.4.4.cmml" xref="S6.SS2.24.p5.13.m13.4.4"><and id="S6.SS2.24.p5.13.m13.4.4a.cmml" xref="S6.SS2.24.p5.13.m13.4.4"></and><apply id="S6.SS2.24.p5.13.m13.4.4b.cmml" xref="S6.SS2.24.p5.13.m13.4.4"><eq id="S6.SS2.24.p5.13.m13.4.4.4.cmml" xref="S6.SS2.24.p5.13.m13.4.4.4"></eq><apply id="S6.SS2.24.p5.13.m13.4.4.2.cmml" xref="S6.SS2.24.p5.13.m13.4.4.2"><times id="S6.SS2.24.p5.13.m13.4.4.2.3.cmml" xref="S6.SS2.24.p5.13.m13.4.4.2.3"></times><ci id="S6.SS2.24.p5.13.m13.4.4.2.4.cmml" xref="S6.SS2.24.p5.13.m13.4.4.2.4">𝜈</ci><interval closure="open" id="S6.SS2.24.p5.13.m13.4.4.2.2.3.cmml" xref="S6.SS2.24.p5.13.m13.4.4.2.2.2"><apply id="S6.SS2.24.p5.13.m13.3.3.1.1.1.1.cmml" xref="S6.SS2.24.p5.13.m13.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.24.p5.13.m13.3.3.1.1.1.1.1.cmml" xref="S6.SS2.24.p5.13.m13.3.3.1.1.1.1">subscript</csymbol><ci id="S6.SS2.24.p5.13.m13.3.3.1.1.1.1.2.cmml" xref="S6.SS2.24.p5.13.m13.3.3.1.1.1.1.2">𝑐</ci><ci id="S6.SS2.24.p5.13.m13.3.3.1.1.1.1.3.cmml" xref="S6.SS2.24.p5.13.m13.3.3.1.1.1.1.3">𝑟</ci></apply><apply id="S6.SS2.24.p5.13.m13.4.4.2.2.2.2.cmml" xref="S6.SS2.24.p5.13.m13.4.4.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.24.p5.13.m13.4.4.2.2.2.2.1.cmml" xref="S6.SS2.24.p5.13.m13.4.4.2.2.2.2">subscript</csymbol><ci id="S6.SS2.24.p5.13.m13.4.4.2.2.2.2.2.cmml" xref="S6.SS2.24.p5.13.m13.4.4.2.2.2.2.2">𝑏</ci><ci id="S6.SS2.24.p5.13.m13.4.4.2.2.2.2.3.cmml" xref="S6.SS2.24.p5.13.m13.4.4.2.2.2.2.3">𝑙</ci></apply></interval></apply><ci id="S6.SS2.24.p5.13.m13.4.4.5.cmml" xref="S6.SS2.24.p5.13.m13.4.4.5">𝜈</ci></apply><apply id="S6.SS2.24.p5.13.m13.4.4c.cmml" xref="S6.SS2.24.p5.13.m13.4.4"><eq id="S6.SS2.24.p5.13.m13.4.4.6.cmml" xref="S6.SS2.24.p5.13.m13.4.4.6"></eq><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.24.p5.13.m13.4.4.5.cmml" id="S6.SS2.24.p5.13.m13.4.4d.cmml" xref="S6.SS2.24.p5.13.m13.4.4"></share><apply id="S6.SS2.24.p5.13.m13.4.4.7.cmml" xref="S6.SS2.24.p5.13.m13.4.4.7"><times id="S6.SS2.24.p5.13.m13.4.4.7.1.cmml" xref="S6.SS2.24.p5.13.m13.4.4.7.1"></times><ci id="S6.SS2.24.p5.13.m13.4.4.7.2.cmml" xref="S6.SS2.24.p5.13.m13.4.4.7.2">𝜈</ci><interval closure="open" id="S6.SS2.24.p5.13.m13.4.4.7.3.1.cmml" xref="S6.SS2.24.p5.13.m13.4.4.7.3.2"><ci id="S6.SS2.24.p5.13.m13.1.1.cmml" xref="S6.SS2.24.p5.13.m13.1.1">𝑧</ci><ci id="S6.SS2.24.p5.13.m13.2.2.cmml" xref="S6.SS2.24.p5.13.m13.2.2">𝑦</ci></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.24.p5.13.m13.4c">\nu(c_{r},b_{l})=\nu=\nu(z,y)</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.24.p5.13.m13.4d">italic_ν ( italic_c start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ) = italic_ν = italic_ν ( italic_z , italic_y )</annotation></semantics></math>. ∎</p> </div> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S6.Thmtheorem19"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem19.1.1.1">Lemma 6.19</span></span><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem19.2.2">.</span> </h6> <div class="ltx_para" id="S6.Thmtheorem19.p1"> <p class="ltx_p" id="S6.Thmtheorem19.p1.2">For every <math alttext="a\in A" class="ltx_Math" display="inline" id="S6.Thmtheorem19.p1.1.m1.1"><semantics id="S6.Thmtheorem19.p1.1.m1.1a"><mrow id="S6.Thmtheorem19.p1.1.m1.1.1" xref="S6.Thmtheorem19.p1.1.m1.1.1.cmml"><mi id="S6.Thmtheorem19.p1.1.m1.1.1.2" xref="S6.Thmtheorem19.p1.1.m1.1.1.2.cmml">a</mi><mo id="S6.Thmtheorem19.p1.1.m1.1.1.1" xref="S6.Thmtheorem19.p1.1.m1.1.1.1.cmml">∈</mo><mi id="S6.Thmtheorem19.p1.1.m1.1.1.3" xref="S6.Thmtheorem19.p1.1.m1.1.1.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem19.p1.1.m1.1b"><apply id="S6.Thmtheorem19.p1.1.m1.1.1.cmml" xref="S6.Thmtheorem19.p1.1.m1.1.1"><in id="S6.Thmtheorem19.p1.1.m1.1.1.1.cmml" xref="S6.Thmtheorem19.p1.1.m1.1.1.1"></in><ci id="S6.Thmtheorem19.p1.1.m1.1.1.2.cmml" xref="S6.Thmtheorem19.p1.1.m1.1.1.2">𝑎</ci><ci id="S6.Thmtheorem19.p1.1.m1.1.1.3.cmml" xref="S6.Thmtheorem19.p1.1.m1.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem19.p1.1.m1.1c">a\in A</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem19.p1.1.m1.1d">italic_a ∈ italic_A</annotation></semantics></math>, <math alttext="\{p\in P_{E}:a\in\operatorname{dom}(f_{p})\}" class="ltx_Math" display="inline" id="S6.Thmtheorem19.p1.2.m2.3"><semantics id="S6.Thmtheorem19.p1.2.m2.3a"><mrow id="S6.Thmtheorem19.p1.2.m2.3.3.2" xref="S6.Thmtheorem19.p1.2.m2.3.3.3.cmml"><mo id="S6.Thmtheorem19.p1.2.m2.3.3.2.3" stretchy="false" xref="S6.Thmtheorem19.p1.2.m2.3.3.3.1.cmml">{</mo><mrow id="S6.Thmtheorem19.p1.2.m2.2.2.1.1" xref="S6.Thmtheorem19.p1.2.m2.2.2.1.1.cmml"><mi id="S6.Thmtheorem19.p1.2.m2.2.2.1.1.2" xref="S6.Thmtheorem19.p1.2.m2.2.2.1.1.2.cmml">p</mi><mo id="S6.Thmtheorem19.p1.2.m2.2.2.1.1.1" xref="S6.Thmtheorem19.p1.2.m2.2.2.1.1.1.cmml">∈</mo><msub id="S6.Thmtheorem19.p1.2.m2.2.2.1.1.3" xref="S6.Thmtheorem19.p1.2.m2.2.2.1.1.3.cmml"><mi id="S6.Thmtheorem19.p1.2.m2.2.2.1.1.3.2" xref="S6.Thmtheorem19.p1.2.m2.2.2.1.1.3.2.cmml">P</mi><mi id="S6.Thmtheorem19.p1.2.m2.2.2.1.1.3.3" xref="S6.Thmtheorem19.p1.2.m2.2.2.1.1.3.3.cmml">E</mi></msub></mrow><mo id="S6.Thmtheorem19.p1.2.m2.3.3.2.4" lspace="0.278em" rspace="0.278em" xref="S6.Thmtheorem19.p1.2.m2.3.3.3.1.cmml">:</mo><mrow id="S6.Thmtheorem19.p1.2.m2.3.3.2.2" xref="S6.Thmtheorem19.p1.2.m2.3.3.2.2.cmml"><mi id="S6.Thmtheorem19.p1.2.m2.3.3.2.2.3" xref="S6.Thmtheorem19.p1.2.m2.3.3.2.2.3.cmml">a</mi><mo id="S6.Thmtheorem19.p1.2.m2.3.3.2.2.2" xref="S6.Thmtheorem19.p1.2.m2.3.3.2.2.2.cmml">∈</mo><mrow id="S6.Thmtheorem19.p1.2.m2.3.3.2.2.1.1" xref="S6.Thmtheorem19.p1.2.m2.3.3.2.2.1.2.cmml"><mi id="S6.Thmtheorem19.p1.2.m2.1.1" xref="S6.Thmtheorem19.p1.2.m2.1.1.cmml">dom</mi><mo id="S6.Thmtheorem19.p1.2.m2.3.3.2.2.1.1a" xref="S6.Thmtheorem19.p1.2.m2.3.3.2.2.1.2.cmml">⁡</mo><mrow id="S6.Thmtheorem19.p1.2.m2.3.3.2.2.1.1.1" xref="S6.Thmtheorem19.p1.2.m2.3.3.2.2.1.2.cmml"><mo id="S6.Thmtheorem19.p1.2.m2.3.3.2.2.1.1.1.2" stretchy="false" xref="S6.Thmtheorem19.p1.2.m2.3.3.2.2.1.2.cmml">(</mo><msub id="S6.Thmtheorem19.p1.2.m2.3.3.2.2.1.1.1.1" xref="S6.Thmtheorem19.p1.2.m2.3.3.2.2.1.1.1.1.cmml"><mi id="S6.Thmtheorem19.p1.2.m2.3.3.2.2.1.1.1.1.2" xref="S6.Thmtheorem19.p1.2.m2.3.3.2.2.1.1.1.1.2.cmml">f</mi><mi id="S6.Thmtheorem19.p1.2.m2.3.3.2.2.1.1.1.1.3" xref="S6.Thmtheorem19.p1.2.m2.3.3.2.2.1.1.1.1.3.cmml">p</mi></msub><mo id="S6.Thmtheorem19.p1.2.m2.3.3.2.2.1.1.1.3" stretchy="false" xref="S6.Thmtheorem19.p1.2.m2.3.3.2.2.1.2.cmml">)</mo></mrow></mrow></mrow><mo id="S6.Thmtheorem19.p1.2.m2.3.3.2.5" stretchy="false" xref="S6.Thmtheorem19.p1.2.m2.3.3.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem19.p1.2.m2.3b"><apply id="S6.Thmtheorem19.p1.2.m2.3.3.3.cmml" xref="S6.Thmtheorem19.p1.2.m2.3.3.2"><csymbol cd="latexml" id="S6.Thmtheorem19.p1.2.m2.3.3.3.1.cmml" xref="S6.Thmtheorem19.p1.2.m2.3.3.2.3">conditional-set</csymbol><apply id="S6.Thmtheorem19.p1.2.m2.2.2.1.1.cmml" xref="S6.Thmtheorem19.p1.2.m2.2.2.1.1"><in id="S6.Thmtheorem19.p1.2.m2.2.2.1.1.1.cmml" xref="S6.Thmtheorem19.p1.2.m2.2.2.1.1.1"></in><ci id="S6.Thmtheorem19.p1.2.m2.2.2.1.1.2.cmml" xref="S6.Thmtheorem19.p1.2.m2.2.2.1.1.2">𝑝</ci><apply id="S6.Thmtheorem19.p1.2.m2.2.2.1.1.3.cmml" xref="S6.Thmtheorem19.p1.2.m2.2.2.1.1.3"><csymbol cd="ambiguous" id="S6.Thmtheorem19.p1.2.m2.2.2.1.1.3.1.cmml" xref="S6.Thmtheorem19.p1.2.m2.2.2.1.1.3">subscript</csymbol><ci id="S6.Thmtheorem19.p1.2.m2.2.2.1.1.3.2.cmml" xref="S6.Thmtheorem19.p1.2.m2.2.2.1.1.3.2">𝑃</ci><ci id="S6.Thmtheorem19.p1.2.m2.2.2.1.1.3.3.cmml" xref="S6.Thmtheorem19.p1.2.m2.2.2.1.1.3.3">𝐸</ci></apply></apply><apply id="S6.Thmtheorem19.p1.2.m2.3.3.2.2.cmml" xref="S6.Thmtheorem19.p1.2.m2.3.3.2.2"><in id="S6.Thmtheorem19.p1.2.m2.3.3.2.2.2.cmml" xref="S6.Thmtheorem19.p1.2.m2.3.3.2.2.2"></in><ci id="S6.Thmtheorem19.p1.2.m2.3.3.2.2.3.cmml" xref="S6.Thmtheorem19.p1.2.m2.3.3.2.2.3">𝑎</ci><apply id="S6.Thmtheorem19.p1.2.m2.3.3.2.2.1.2.cmml" xref="S6.Thmtheorem19.p1.2.m2.3.3.2.2.1.1"><ci id="S6.Thmtheorem19.p1.2.m2.1.1.cmml" xref="S6.Thmtheorem19.p1.2.m2.1.1">dom</ci><apply id="S6.Thmtheorem19.p1.2.m2.3.3.2.2.1.1.1.1.cmml" xref="S6.Thmtheorem19.p1.2.m2.3.3.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S6.Thmtheorem19.p1.2.m2.3.3.2.2.1.1.1.1.1.cmml" xref="S6.Thmtheorem19.p1.2.m2.3.3.2.2.1.1.1.1">subscript</csymbol><ci id="S6.Thmtheorem19.p1.2.m2.3.3.2.2.1.1.1.1.2.cmml" xref="S6.Thmtheorem19.p1.2.m2.3.3.2.2.1.1.1.1.2">𝑓</ci><ci id="S6.Thmtheorem19.p1.2.m2.3.3.2.2.1.1.1.1.3.cmml" xref="S6.Thmtheorem19.p1.2.m2.3.3.2.2.1.1.1.1.3">𝑝</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem19.p1.2.m2.3c">\{p\in P_{E}:a\in\operatorname{dom}(f_{p})\}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem19.p1.2.m2.3d">{ italic_p ∈ italic_P start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT : italic_a ∈ roman_dom ( italic_f start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ) }</annotation></semantics></math> is dense.</p> </div> </div> <div class="ltx_proof" id="S6.SS2.30"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S6.SS2.25.p1"> <p class="ltx_p" id="S6.SS2.25.p1.11">Take <math alttext="p\in P_{E}" class="ltx_Math" display="inline" id="S6.SS2.25.p1.1.m1.1"><semantics id="S6.SS2.25.p1.1.m1.1a"><mrow id="S6.SS2.25.p1.1.m1.1.1" xref="S6.SS2.25.p1.1.m1.1.1.cmml"><mi id="S6.SS2.25.p1.1.m1.1.1.2" xref="S6.SS2.25.p1.1.m1.1.1.2.cmml">p</mi><mo id="S6.SS2.25.p1.1.m1.1.1.1" xref="S6.SS2.25.p1.1.m1.1.1.1.cmml">∈</mo><msub id="S6.SS2.25.p1.1.m1.1.1.3" xref="S6.SS2.25.p1.1.m1.1.1.3.cmml"><mi id="S6.SS2.25.p1.1.m1.1.1.3.2" xref="S6.SS2.25.p1.1.m1.1.1.3.2.cmml">P</mi><mi id="S6.SS2.25.p1.1.m1.1.1.3.3" xref="S6.SS2.25.p1.1.m1.1.1.3.3.cmml">E</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.25.p1.1.m1.1b"><apply id="S6.SS2.25.p1.1.m1.1.1.cmml" xref="S6.SS2.25.p1.1.m1.1.1"><in id="S6.SS2.25.p1.1.m1.1.1.1.cmml" xref="S6.SS2.25.p1.1.m1.1.1.1"></in><ci id="S6.SS2.25.p1.1.m1.1.1.2.cmml" xref="S6.SS2.25.p1.1.m1.1.1.2">𝑝</ci><apply id="S6.SS2.25.p1.1.m1.1.1.3.cmml" xref="S6.SS2.25.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S6.SS2.25.p1.1.m1.1.1.3.1.cmml" xref="S6.SS2.25.p1.1.m1.1.1.3">subscript</csymbol><ci id="S6.SS2.25.p1.1.m1.1.1.3.2.cmml" xref="S6.SS2.25.p1.1.m1.1.1.3.2">𝑃</ci><ci id="S6.SS2.25.p1.1.m1.1.1.3.3.cmml" xref="S6.SS2.25.p1.1.m1.1.1.3.3">𝐸</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.25.p1.1.m1.1c">p\in P_{E}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.25.p1.1.m1.1d">italic_p ∈ italic_P start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="c\notin\operatorname{dom}(f_{p})" class="ltx_Math" display="inline" id="S6.SS2.25.p1.2.m2.2"><semantics id="S6.SS2.25.p1.2.m2.2a"><mrow id="S6.SS2.25.p1.2.m2.2.2" xref="S6.SS2.25.p1.2.m2.2.2.cmml"><mi id="S6.SS2.25.p1.2.m2.2.2.3" xref="S6.SS2.25.p1.2.m2.2.2.3.cmml">c</mi><mo id="S6.SS2.25.p1.2.m2.2.2.2" xref="S6.SS2.25.p1.2.m2.2.2.2.cmml">∉</mo><mrow id="S6.SS2.25.p1.2.m2.2.2.1.1" xref="S6.SS2.25.p1.2.m2.2.2.1.2.cmml"><mi id="S6.SS2.25.p1.2.m2.1.1" xref="S6.SS2.25.p1.2.m2.1.1.cmml">dom</mi><mo id="S6.SS2.25.p1.2.m2.2.2.1.1a" xref="S6.SS2.25.p1.2.m2.2.2.1.2.cmml">⁡</mo><mrow id="S6.SS2.25.p1.2.m2.2.2.1.1.1" xref="S6.SS2.25.p1.2.m2.2.2.1.2.cmml"><mo id="S6.SS2.25.p1.2.m2.2.2.1.1.1.2" stretchy="false" xref="S6.SS2.25.p1.2.m2.2.2.1.2.cmml">(</mo><msub id="S6.SS2.25.p1.2.m2.2.2.1.1.1.1" xref="S6.SS2.25.p1.2.m2.2.2.1.1.1.1.cmml"><mi id="S6.SS2.25.p1.2.m2.2.2.1.1.1.1.2" xref="S6.SS2.25.p1.2.m2.2.2.1.1.1.1.2.cmml">f</mi><mi id="S6.SS2.25.p1.2.m2.2.2.1.1.1.1.3" xref="S6.SS2.25.p1.2.m2.2.2.1.1.1.1.3.cmml">p</mi></msub><mo id="S6.SS2.25.p1.2.m2.2.2.1.1.1.3" stretchy="false" xref="S6.SS2.25.p1.2.m2.2.2.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.25.p1.2.m2.2b"><apply id="S6.SS2.25.p1.2.m2.2.2.cmml" xref="S6.SS2.25.p1.2.m2.2.2"><notin id="S6.SS2.25.p1.2.m2.2.2.2.cmml" xref="S6.SS2.25.p1.2.m2.2.2.2"></notin><ci id="S6.SS2.25.p1.2.m2.2.2.3.cmml" xref="S6.SS2.25.p1.2.m2.2.2.3">𝑐</ci><apply id="S6.SS2.25.p1.2.m2.2.2.1.2.cmml" xref="S6.SS2.25.p1.2.m2.2.2.1.1"><ci id="S6.SS2.25.p1.2.m2.1.1.cmml" xref="S6.SS2.25.p1.2.m2.1.1">dom</ci><apply id="S6.SS2.25.p1.2.m2.2.2.1.1.1.1.cmml" xref="S6.SS2.25.p1.2.m2.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.25.p1.2.m2.2.2.1.1.1.1.1.cmml" xref="S6.SS2.25.p1.2.m2.2.2.1.1.1.1">subscript</csymbol><ci id="S6.SS2.25.p1.2.m2.2.2.1.1.1.1.2.cmml" xref="S6.SS2.25.p1.2.m2.2.2.1.1.1.1.2">𝑓</ci><ci id="S6.SS2.25.p1.2.m2.2.2.1.1.1.1.3.cmml" xref="S6.SS2.25.p1.2.m2.2.2.1.1.1.1.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.25.p1.2.m2.2c">c\notin\operatorname{dom}(f_{p})</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.25.p1.2.m2.2d">italic_c ∉ roman_dom ( italic_f start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT )</annotation></semantics></math>. Then let <math alttext="\bar{a}<\bar{b}" class="ltx_Math" display="inline" id="S6.SS2.25.p1.3.m3.1"><semantics id="S6.SS2.25.p1.3.m3.1a"><mrow id="S6.SS2.25.p1.3.m3.1.1" xref="S6.SS2.25.p1.3.m3.1.1.cmml"><mover accent="true" id="S6.SS2.25.p1.3.m3.1.1.2" xref="S6.SS2.25.p1.3.m3.1.1.2.cmml"><mi id="S6.SS2.25.p1.3.m3.1.1.2.2" xref="S6.SS2.25.p1.3.m3.1.1.2.2.cmml">a</mi><mo id="S6.SS2.25.p1.3.m3.1.1.2.1" xref="S6.SS2.25.p1.3.m3.1.1.2.1.cmml">¯</mo></mover><mo id="S6.SS2.25.p1.3.m3.1.1.1" xref="S6.SS2.25.p1.3.m3.1.1.1.cmml"><</mo><mover accent="true" id="S6.SS2.25.p1.3.m3.1.1.3" xref="S6.SS2.25.p1.3.m3.1.1.3.cmml"><mi id="S6.SS2.25.p1.3.m3.1.1.3.2" xref="S6.SS2.25.p1.3.m3.1.1.3.2.cmml">b</mi><mo id="S6.SS2.25.p1.3.m3.1.1.3.1" xref="S6.SS2.25.p1.3.m3.1.1.3.1.cmml">¯</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.25.p1.3.m3.1b"><apply id="S6.SS2.25.p1.3.m3.1.1.cmml" xref="S6.SS2.25.p1.3.m3.1.1"><lt id="S6.SS2.25.p1.3.m3.1.1.1.cmml" xref="S6.SS2.25.p1.3.m3.1.1.1"></lt><apply id="S6.SS2.25.p1.3.m3.1.1.2.cmml" xref="S6.SS2.25.p1.3.m3.1.1.2"><ci id="S6.SS2.25.p1.3.m3.1.1.2.1.cmml" xref="S6.SS2.25.p1.3.m3.1.1.2.1">¯</ci><ci id="S6.SS2.25.p1.3.m3.1.1.2.2.cmml" xref="S6.SS2.25.p1.3.m3.1.1.2.2">𝑎</ci></apply><apply id="S6.SS2.25.p1.3.m3.1.1.3.cmml" xref="S6.SS2.25.p1.3.m3.1.1.3"><ci id="S6.SS2.25.p1.3.m3.1.1.3.1.cmml" xref="S6.SS2.25.p1.3.m3.1.1.3.1">¯</ci><ci id="S6.SS2.25.p1.3.m3.1.1.3.2.cmml" xref="S6.SS2.25.p1.3.m3.1.1.3.2">𝑏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.25.p1.3.m3.1c">\bar{a}<\bar{b}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.25.p1.3.m3.1d">over¯ start_ARG italic_a end_ARG < over¯ start_ARG italic_b end_ARG</annotation></semantics></math> in <math alttext="\operatorname{dom}(p)" class="ltx_Math" display="inline" id="S6.SS2.25.p1.4.m4.2"><semantics id="S6.SS2.25.p1.4.m4.2a"><mrow id="S6.SS2.25.p1.4.m4.2.3.2" xref="S6.SS2.25.p1.4.m4.2.3.1.cmml"><mi id="S6.SS2.25.p1.4.m4.1.1" xref="S6.SS2.25.p1.4.m4.1.1.cmml">dom</mi><mo id="S6.SS2.25.p1.4.m4.2.3.2a" xref="S6.SS2.25.p1.4.m4.2.3.1.cmml">⁡</mo><mrow id="S6.SS2.25.p1.4.m4.2.3.2.1" xref="S6.SS2.25.p1.4.m4.2.3.1.cmml"><mo id="S6.SS2.25.p1.4.m4.2.3.2.1.1" stretchy="false" xref="S6.SS2.25.p1.4.m4.2.3.1.cmml">(</mo><mi id="S6.SS2.25.p1.4.m4.2.2" xref="S6.SS2.25.p1.4.m4.2.2.cmml">p</mi><mo id="S6.SS2.25.p1.4.m4.2.3.2.1.2" stretchy="false" xref="S6.SS2.25.p1.4.m4.2.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.25.p1.4.m4.2b"><apply id="S6.SS2.25.p1.4.m4.2.3.1.cmml" xref="S6.SS2.25.p1.4.m4.2.3.2"><ci id="S6.SS2.25.p1.4.m4.1.1.cmml" xref="S6.SS2.25.p1.4.m4.1.1">dom</ci><ci id="S6.SS2.25.p1.4.m4.2.2.cmml" xref="S6.SS2.25.p1.4.m4.2.2">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.25.p1.4.m4.2c">\operatorname{dom}(p)</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.25.p1.4.m4.2d">roman_dom ( italic_p )</annotation></semantics></math> be neighbors of <math alttext="c" class="ltx_Math" display="inline" id="S6.SS2.25.p1.5.m5.1"><semantics id="S6.SS2.25.p1.5.m5.1a"><mi id="S6.SS2.25.p1.5.m5.1.1" xref="S6.SS2.25.p1.5.m5.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.25.p1.5.m5.1b"><ci id="S6.SS2.25.p1.5.m5.1.1.cmml" xref="S6.SS2.25.p1.5.m5.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.25.p1.5.m5.1c">c</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.25.p1.5.m5.1d">italic_c</annotation></semantics></math>. The case when there is no such <math alttext="\bar{a}" class="ltx_Math" display="inline" id="S6.SS2.25.p1.6.m6.1"><semantics id="S6.SS2.25.p1.6.m6.1a"><mover accent="true" id="S6.SS2.25.p1.6.m6.1.1" xref="S6.SS2.25.p1.6.m6.1.1.cmml"><mi id="S6.SS2.25.p1.6.m6.1.1.2" xref="S6.SS2.25.p1.6.m6.1.1.2.cmml">a</mi><mo id="S6.SS2.25.p1.6.m6.1.1.1" xref="S6.SS2.25.p1.6.m6.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S6.SS2.25.p1.6.m6.1b"><apply id="S6.SS2.25.p1.6.m6.1.1.cmml" xref="S6.SS2.25.p1.6.m6.1.1"><ci id="S6.SS2.25.p1.6.m6.1.1.1.cmml" xref="S6.SS2.25.p1.6.m6.1.1.1">¯</ci><ci id="S6.SS2.25.p1.6.m6.1.1.2.cmml" xref="S6.SS2.25.p1.6.m6.1.1.2">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.25.p1.6.m6.1c">\bar{a}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.25.p1.6.m6.1d">over¯ start_ARG italic_a end_ARG</annotation></semantics></math> and/or <math alttext="\bar{b}" class="ltx_Math" display="inline" id="S6.SS2.25.p1.7.m7.1"><semantics id="S6.SS2.25.p1.7.m7.1a"><mover accent="true" id="S6.SS2.25.p1.7.m7.1.1" xref="S6.SS2.25.p1.7.m7.1.1.cmml"><mi id="S6.SS2.25.p1.7.m7.1.1.2" xref="S6.SS2.25.p1.7.m7.1.1.2.cmml">b</mi><mo id="S6.SS2.25.p1.7.m7.1.1.1" xref="S6.SS2.25.p1.7.m7.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S6.SS2.25.p1.7.m7.1b"><apply id="S6.SS2.25.p1.7.m7.1.1.cmml" xref="S6.SS2.25.p1.7.m7.1.1"><ci id="S6.SS2.25.p1.7.m7.1.1.1.cmml" xref="S6.SS2.25.p1.7.m7.1.1.1">¯</ci><ci id="S6.SS2.25.p1.7.m7.1.1.2.cmml" xref="S6.SS2.25.p1.7.m7.1.1.2">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.25.p1.7.m7.1c">\bar{b}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.25.p1.7.m7.1d">over¯ start_ARG italic_b end_ARG</annotation></semantics></math> is similar. Because the arguments are symmetric, we may assume that <math alttext="\Delta(a_{r},c)<\Delta(c,b_{l})" class="ltx_Math" display="inline" id="S6.SS2.25.p1.8.m8.4"><semantics id="S6.SS2.25.p1.8.m8.4a"><mrow id="S6.SS2.25.p1.8.m8.4.4" xref="S6.SS2.25.p1.8.m8.4.4.cmml"><mrow id="S6.SS2.25.p1.8.m8.3.3.1" xref="S6.SS2.25.p1.8.m8.3.3.1.cmml"><mi id="S6.SS2.25.p1.8.m8.3.3.1.3" mathvariant="normal" xref="S6.SS2.25.p1.8.m8.3.3.1.3.cmml">Δ</mi><mo id="S6.SS2.25.p1.8.m8.3.3.1.2" xref="S6.SS2.25.p1.8.m8.3.3.1.2.cmml">⁢</mo><mrow id="S6.SS2.25.p1.8.m8.3.3.1.1.1" xref="S6.SS2.25.p1.8.m8.3.3.1.1.2.cmml"><mo id="S6.SS2.25.p1.8.m8.3.3.1.1.1.2" stretchy="false" xref="S6.SS2.25.p1.8.m8.3.3.1.1.2.cmml">(</mo><msub id="S6.SS2.25.p1.8.m8.3.3.1.1.1.1" xref="S6.SS2.25.p1.8.m8.3.3.1.1.1.1.cmml"><mi id="S6.SS2.25.p1.8.m8.3.3.1.1.1.1.2" xref="S6.SS2.25.p1.8.m8.3.3.1.1.1.1.2.cmml">a</mi><mi id="S6.SS2.25.p1.8.m8.3.3.1.1.1.1.3" xref="S6.SS2.25.p1.8.m8.3.3.1.1.1.1.3.cmml">r</mi></msub><mo id="S6.SS2.25.p1.8.m8.3.3.1.1.1.3" xref="S6.SS2.25.p1.8.m8.3.3.1.1.2.cmml">,</mo><mi id="S6.SS2.25.p1.8.m8.1.1" xref="S6.SS2.25.p1.8.m8.1.1.cmml">c</mi><mo id="S6.SS2.25.p1.8.m8.3.3.1.1.1.4" stretchy="false" xref="S6.SS2.25.p1.8.m8.3.3.1.1.2.cmml">)</mo></mrow></mrow><mo id="S6.SS2.25.p1.8.m8.4.4.3" xref="S6.SS2.25.p1.8.m8.4.4.3.cmml"><</mo><mrow id="S6.SS2.25.p1.8.m8.4.4.2" xref="S6.SS2.25.p1.8.m8.4.4.2.cmml"><mi id="S6.SS2.25.p1.8.m8.4.4.2.3" mathvariant="normal" xref="S6.SS2.25.p1.8.m8.4.4.2.3.cmml">Δ</mi><mo id="S6.SS2.25.p1.8.m8.4.4.2.2" xref="S6.SS2.25.p1.8.m8.4.4.2.2.cmml">⁢</mo><mrow id="S6.SS2.25.p1.8.m8.4.4.2.1.1" xref="S6.SS2.25.p1.8.m8.4.4.2.1.2.cmml"><mo id="S6.SS2.25.p1.8.m8.4.4.2.1.1.2" stretchy="false" xref="S6.SS2.25.p1.8.m8.4.4.2.1.2.cmml">(</mo><mi id="S6.SS2.25.p1.8.m8.2.2" xref="S6.SS2.25.p1.8.m8.2.2.cmml">c</mi><mo id="S6.SS2.25.p1.8.m8.4.4.2.1.1.3" xref="S6.SS2.25.p1.8.m8.4.4.2.1.2.cmml">,</mo><msub id="S6.SS2.25.p1.8.m8.4.4.2.1.1.1" xref="S6.SS2.25.p1.8.m8.4.4.2.1.1.1.cmml"><mi id="S6.SS2.25.p1.8.m8.4.4.2.1.1.1.2" xref="S6.SS2.25.p1.8.m8.4.4.2.1.1.1.2.cmml">b</mi><mi id="S6.SS2.25.p1.8.m8.4.4.2.1.1.1.3" xref="S6.SS2.25.p1.8.m8.4.4.2.1.1.1.3.cmml">l</mi></msub><mo id="S6.SS2.25.p1.8.m8.4.4.2.1.1.4" stretchy="false" xref="S6.SS2.25.p1.8.m8.4.4.2.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.25.p1.8.m8.4b"><apply id="S6.SS2.25.p1.8.m8.4.4.cmml" xref="S6.SS2.25.p1.8.m8.4.4"><lt id="S6.SS2.25.p1.8.m8.4.4.3.cmml" xref="S6.SS2.25.p1.8.m8.4.4.3"></lt><apply id="S6.SS2.25.p1.8.m8.3.3.1.cmml" xref="S6.SS2.25.p1.8.m8.3.3.1"><times id="S6.SS2.25.p1.8.m8.3.3.1.2.cmml" xref="S6.SS2.25.p1.8.m8.3.3.1.2"></times><ci id="S6.SS2.25.p1.8.m8.3.3.1.3.cmml" xref="S6.SS2.25.p1.8.m8.3.3.1.3">Δ</ci><interval closure="open" id="S6.SS2.25.p1.8.m8.3.3.1.1.2.cmml" xref="S6.SS2.25.p1.8.m8.3.3.1.1.1"><apply id="S6.SS2.25.p1.8.m8.3.3.1.1.1.1.cmml" xref="S6.SS2.25.p1.8.m8.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.25.p1.8.m8.3.3.1.1.1.1.1.cmml" xref="S6.SS2.25.p1.8.m8.3.3.1.1.1.1">subscript</csymbol><ci id="S6.SS2.25.p1.8.m8.3.3.1.1.1.1.2.cmml" xref="S6.SS2.25.p1.8.m8.3.3.1.1.1.1.2">𝑎</ci><ci id="S6.SS2.25.p1.8.m8.3.3.1.1.1.1.3.cmml" xref="S6.SS2.25.p1.8.m8.3.3.1.1.1.1.3">𝑟</ci></apply><ci id="S6.SS2.25.p1.8.m8.1.1.cmml" xref="S6.SS2.25.p1.8.m8.1.1">𝑐</ci></interval></apply><apply id="S6.SS2.25.p1.8.m8.4.4.2.cmml" xref="S6.SS2.25.p1.8.m8.4.4.2"><times id="S6.SS2.25.p1.8.m8.4.4.2.2.cmml" xref="S6.SS2.25.p1.8.m8.4.4.2.2"></times><ci id="S6.SS2.25.p1.8.m8.4.4.2.3.cmml" xref="S6.SS2.25.p1.8.m8.4.4.2.3">Δ</ci><interval closure="open" id="S6.SS2.25.p1.8.m8.4.4.2.1.2.cmml" xref="S6.SS2.25.p1.8.m8.4.4.2.1.1"><ci id="S6.SS2.25.p1.8.m8.2.2.cmml" xref="S6.SS2.25.p1.8.m8.2.2">𝑐</ci><apply id="S6.SS2.25.p1.8.m8.4.4.2.1.1.1.cmml" xref="S6.SS2.25.p1.8.m8.4.4.2.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.25.p1.8.m8.4.4.2.1.1.1.1.cmml" xref="S6.SS2.25.p1.8.m8.4.4.2.1.1.1">subscript</csymbol><ci id="S6.SS2.25.p1.8.m8.4.4.2.1.1.1.2.cmml" xref="S6.SS2.25.p1.8.m8.4.4.2.1.1.1.2">𝑏</ci><ci id="S6.SS2.25.p1.8.m8.4.4.2.1.1.1.3.cmml" xref="S6.SS2.25.p1.8.m8.4.4.2.1.1.1.3">𝑙</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.25.p1.8.m8.4c">\Delta(a_{r},c)<\Delta(c,b_{l})</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.25.p1.8.m8.4d">roman_Δ ( italic_a start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT , italic_c ) < roman_Δ ( italic_c , italic_b start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT )</annotation></semantics></math>. Let <math alttext="\nu:=\nu(c,b_{l})" class="ltx_Math" display="inline" id="S6.SS2.25.p1.9.m9.2"><semantics id="S6.SS2.25.p1.9.m9.2a"><mrow id="S6.SS2.25.p1.9.m9.2.2" xref="S6.SS2.25.p1.9.m9.2.2.cmml"><mi id="S6.SS2.25.p1.9.m9.2.2.3" xref="S6.SS2.25.p1.9.m9.2.2.3.cmml">ν</mi><mo id="S6.SS2.25.p1.9.m9.2.2.2" lspace="0.278em" rspace="0.278em" xref="S6.SS2.25.p1.9.m9.2.2.2.cmml">:=</mo><mrow id="S6.SS2.25.p1.9.m9.2.2.1" xref="S6.SS2.25.p1.9.m9.2.2.1.cmml"><mi id="S6.SS2.25.p1.9.m9.2.2.1.3" xref="S6.SS2.25.p1.9.m9.2.2.1.3.cmml">ν</mi><mo id="S6.SS2.25.p1.9.m9.2.2.1.2" xref="S6.SS2.25.p1.9.m9.2.2.1.2.cmml">⁢</mo><mrow id="S6.SS2.25.p1.9.m9.2.2.1.1.1" xref="S6.SS2.25.p1.9.m9.2.2.1.1.2.cmml"><mo id="S6.SS2.25.p1.9.m9.2.2.1.1.1.2" stretchy="false" xref="S6.SS2.25.p1.9.m9.2.2.1.1.2.cmml">(</mo><mi id="S6.SS2.25.p1.9.m9.1.1" xref="S6.SS2.25.p1.9.m9.1.1.cmml">c</mi><mo id="S6.SS2.25.p1.9.m9.2.2.1.1.1.3" xref="S6.SS2.25.p1.9.m9.2.2.1.1.2.cmml">,</mo><msub id="S6.SS2.25.p1.9.m9.2.2.1.1.1.1" xref="S6.SS2.25.p1.9.m9.2.2.1.1.1.1.cmml"><mi id="S6.SS2.25.p1.9.m9.2.2.1.1.1.1.2" xref="S6.SS2.25.p1.9.m9.2.2.1.1.1.1.2.cmml">b</mi><mi id="S6.SS2.25.p1.9.m9.2.2.1.1.1.1.3" xref="S6.SS2.25.p1.9.m9.2.2.1.1.1.1.3.cmml">l</mi></msub><mo id="S6.SS2.25.p1.9.m9.2.2.1.1.1.4" stretchy="false" xref="S6.SS2.25.p1.9.m9.2.2.1.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.25.p1.9.m9.2b"><apply id="S6.SS2.25.p1.9.m9.2.2.cmml" xref="S6.SS2.25.p1.9.m9.2.2"><csymbol cd="latexml" id="S6.SS2.25.p1.9.m9.2.2.2.cmml" xref="S6.SS2.25.p1.9.m9.2.2.2">assign</csymbol><ci id="S6.SS2.25.p1.9.m9.2.2.3.cmml" xref="S6.SS2.25.p1.9.m9.2.2.3">𝜈</ci><apply id="S6.SS2.25.p1.9.m9.2.2.1.cmml" xref="S6.SS2.25.p1.9.m9.2.2.1"><times id="S6.SS2.25.p1.9.m9.2.2.1.2.cmml" xref="S6.SS2.25.p1.9.m9.2.2.1.2"></times><ci id="S6.SS2.25.p1.9.m9.2.2.1.3.cmml" xref="S6.SS2.25.p1.9.m9.2.2.1.3">𝜈</ci><interval closure="open" id="S6.SS2.25.p1.9.m9.2.2.1.1.2.cmml" xref="S6.SS2.25.p1.9.m9.2.2.1.1.1"><ci id="S6.SS2.25.p1.9.m9.1.1.cmml" xref="S6.SS2.25.p1.9.m9.1.1">𝑐</ci><apply id="S6.SS2.25.p1.9.m9.2.2.1.1.1.1.cmml" xref="S6.SS2.25.p1.9.m9.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.25.p1.9.m9.2.2.1.1.1.1.1.cmml" xref="S6.SS2.25.p1.9.m9.2.2.1.1.1.1">subscript</csymbol><ci id="S6.SS2.25.p1.9.m9.2.2.1.1.1.1.2.cmml" xref="S6.SS2.25.p1.9.m9.2.2.1.1.1.1.2">𝑏</ci><ci id="S6.SS2.25.p1.9.m9.2.2.1.1.1.1.3.cmml" xref="S6.SS2.25.p1.9.m9.2.2.1.1.1.1.3">𝑙</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.25.p1.9.m9.2c">\nu:=\nu(c,b_{l})</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.25.p1.9.m9.2d">italic_ν := italic_ν ( italic_c , italic_b start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT )</annotation></semantics></math>, <math alttext="x:=p(\bar{a})" class="ltx_Math" display="inline" id="S6.SS2.25.p1.10.m10.1"><semantics id="S6.SS2.25.p1.10.m10.1a"><mrow id="S6.SS2.25.p1.10.m10.1.2" xref="S6.SS2.25.p1.10.m10.1.2.cmml"><mi id="S6.SS2.25.p1.10.m10.1.2.2" xref="S6.SS2.25.p1.10.m10.1.2.2.cmml">x</mi><mo id="S6.SS2.25.p1.10.m10.1.2.1" lspace="0.278em" rspace="0.278em" xref="S6.SS2.25.p1.10.m10.1.2.1.cmml">:=</mo><mrow id="S6.SS2.25.p1.10.m10.1.2.3" xref="S6.SS2.25.p1.10.m10.1.2.3.cmml"><mi id="S6.SS2.25.p1.10.m10.1.2.3.2" xref="S6.SS2.25.p1.10.m10.1.2.3.2.cmml">p</mi><mo id="S6.SS2.25.p1.10.m10.1.2.3.1" xref="S6.SS2.25.p1.10.m10.1.2.3.1.cmml">⁢</mo><mrow id="S6.SS2.25.p1.10.m10.1.2.3.3.2" xref="S6.SS2.25.p1.10.m10.1.1.cmml"><mo id="S6.SS2.25.p1.10.m10.1.2.3.3.2.1" stretchy="false" xref="S6.SS2.25.p1.10.m10.1.1.cmml">(</mo><mover accent="true" id="S6.SS2.25.p1.10.m10.1.1" xref="S6.SS2.25.p1.10.m10.1.1.cmml"><mi id="S6.SS2.25.p1.10.m10.1.1.2" xref="S6.SS2.25.p1.10.m10.1.1.2.cmml">a</mi><mo id="S6.SS2.25.p1.10.m10.1.1.1" xref="S6.SS2.25.p1.10.m10.1.1.1.cmml">¯</mo></mover><mo id="S6.SS2.25.p1.10.m10.1.2.3.3.2.2" stretchy="false" xref="S6.SS2.25.p1.10.m10.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.25.p1.10.m10.1b"><apply id="S6.SS2.25.p1.10.m10.1.2.cmml" xref="S6.SS2.25.p1.10.m10.1.2"><csymbol cd="latexml" id="S6.SS2.25.p1.10.m10.1.2.1.cmml" xref="S6.SS2.25.p1.10.m10.1.2.1">assign</csymbol><ci id="S6.SS2.25.p1.10.m10.1.2.2.cmml" xref="S6.SS2.25.p1.10.m10.1.2.2">𝑥</ci><apply id="S6.SS2.25.p1.10.m10.1.2.3.cmml" xref="S6.SS2.25.p1.10.m10.1.2.3"><times id="S6.SS2.25.p1.10.m10.1.2.3.1.cmml" xref="S6.SS2.25.p1.10.m10.1.2.3.1"></times><ci id="S6.SS2.25.p1.10.m10.1.2.3.2.cmml" xref="S6.SS2.25.p1.10.m10.1.2.3.2">𝑝</ci><apply id="S6.SS2.25.p1.10.m10.1.1.cmml" xref="S6.SS2.25.p1.10.m10.1.2.3.3.2"><ci id="S6.SS2.25.p1.10.m10.1.1.1.cmml" xref="S6.SS2.25.p1.10.m10.1.1.1">¯</ci><ci id="S6.SS2.25.p1.10.m10.1.1.2.cmml" xref="S6.SS2.25.p1.10.m10.1.1.2">𝑎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.25.p1.10.m10.1c">x:=p(\bar{a})</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.25.p1.10.m10.1d">italic_x := italic_p ( over¯ start_ARG italic_a end_ARG )</annotation></semantics></math> and and <math alttext="y:=p(\bar{y})" class="ltx_Math" display="inline" id="S6.SS2.25.p1.11.m11.1"><semantics id="S6.SS2.25.p1.11.m11.1a"><mrow id="S6.SS2.25.p1.11.m11.1.2" xref="S6.SS2.25.p1.11.m11.1.2.cmml"><mi id="S6.SS2.25.p1.11.m11.1.2.2" xref="S6.SS2.25.p1.11.m11.1.2.2.cmml">y</mi><mo id="S6.SS2.25.p1.11.m11.1.2.1" lspace="0.278em" rspace="0.278em" xref="S6.SS2.25.p1.11.m11.1.2.1.cmml">:=</mo><mrow id="S6.SS2.25.p1.11.m11.1.2.3" xref="S6.SS2.25.p1.11.m11.1.2.3.cmml"><mi id="S6.SS2.25.p1.11.m11.1.2.3.2" xref="S6.SS2.25.p1.11.m11.1.2.3.2.cmml">p</mi><mo id="S6.SS2.25.p1.11.m11.1.2.3.1" xref="S6.SS2.25.p1.11.m11.1.2.3.1.cmml">⁢</mo><mrow id="S6.SS2.25.p1.11.m11.1.2.3.3.2" xref="S6.SS2.25.p1.11.m11.1.1.cmml"><mo id="S6.SS2.25.p1.11.m11.1.2.3.3.2.1" stretchy="false" xref="S6.SS2.25.p1.11.m11.1.1.cmml">(</mo><mover accent="true" id="S6.SS2.25.p1.11.m11.1.1" xref="S6.SS2.25.p1.11.m11.1.1.cmml"><mi id="S6.SS2.25.p1.11.m11.1.1.2" xref="S6.SS2.25.p1.11.m11.1.1.2.cmml">y</mi><mo id="S6.SS2.25.p1.11.m11.1.1.1" xref="S6.SS2.25.p1.11.m11.1.1.1.cmml">¯</mo></mover><mo id="S6.SS2.25.p1.11.m11.1.2.3.3.2.2" stretchy="false" xref="S6.SS2.25.p1.11.m11.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.25.p1.11.m11.1b"><apply id="S6.SS2.25.p1.11.m11.1.2.cmml" xref="S6.SS2.25.p1.11.m11.1.2"><csymbol cd="latexml" id="S6.SS2.25.p1.11.m11.1.2.1.cmml" xref="S6.SS2.25.p1.11.m11.1.2.1">assign</csymbol><ci id="S6.SS2.25.p1.11.m11.1.2.2.cmml" xref="S6.SS2.25.p1.11.m11.1.2.2">𝑦</ci><apply id="S6.SS2.25.p1.11.m11.1.2.3.cmml" xref="S6.SS2.25.p1.11.m11.1.2.3"><times id="S6.SS2.25.p1.11.m11.1.2.3.1.cmml" xref="S6.SS2.25.p1.11.m11.1.2.3.1"></times><ci id="S6.SS2.25.p1.11.m11.1.2.3.2.cmml" xref="S6.SS2.25.p1.11.m11.1.2.3.2">𝑝</ci><apply id="S6.SS2.25.p1.11.m11.1.1.cmml" xref="S6.SS2.25.p1.11.m11.1.2.3.3.2"><ci id="S6.SS2.25.p1.11.m11.1.1.1.cmml" xref="S6.SS2.25.p1.11.m11.1.1.1">¯</ci><ci id="S6.SS2.25.p1.11.m11.1.1.2.cmml" xref="S6.SS2.25.p1.11.m11.1.1.2">𝑦</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.25.p1.11.m11.1c">y:=p(\bar{y})</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.25.p1.11.m11.1d">italic_y := italic_p ( over¯ start_ARG italic_y end_ARG )</annotation></semantics></math>. Then</p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex13"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\nu(x,y)=\nu(a_{r},b_{l})\leq\nu\leq\nu(b_{l})=\nu(y)" class="ltx_Math" display="block" id="S6.Ex13.m1.6"><semantics id="S6.Ex13.m1.6a"><mrow id="S6.Ex13.m1.6.6" xref="S6.Ex13.m1.6.6.cmml"><mrow id="S6.Ex13.m1.6.6.5" xref="S6.Ex13.m1.6.6.5.cmml"><mi id="S6.Ex13.m1.6.6.5.2" xref="S6.Ex13.m1.6.6.5.2.cmml">ν</mi><mo id="S6.Ex13.m1.6.6.5.1" xref="S6.Ex13.m1.6.6.5.1.cmml">⁢</mo><mrow id="S6.Ex13.m1.6.6.5.3.2" xref="S6.Ex13.m1.6.6.5.3.1.cmml"><mo id="S6.Ex13.m1.6.6.5.3.2.1" stretchy="false" xref="S6.Ex13.m1.6.6.5.3.1.cmml">(</mo><mi id="S6.Ex13.m1.1.1" xref="S6.Ex13.m1.1.1.cmml">x</mi><mo id="S6.Ex13.m1.6.6.5.3.2.2" xref="S6.Ex13.m1.6.6.5.3.1.cmml">,</mo><mi id="S6.Ex13.m1.2.2" xref="S6.Ex13.m1.2.2.cmml">y</mi><mo id="S6.Ex13.m1.6.6.5.3.2.3" stretchy="false" xref="S6.Ex13.m1.6.6.5.3.1.cmml">)</mo></mrow></mrow><mo id="S6.Ex13.m1.6.6.6" xref="S6.Ex13.m1.6.6.6.cmml">=</mo><mrow id="S6.Ex13.m1.5.5.2" xref="S6.Ex13.m1.5.5.2.cmml"><mi id="S6.Ex13.m1.5.5.2.4" xref="S6.Ex13.m1.5.5.2.4.cmml">ν</mi><mo id="S6.Ex13.m1.5.5.2.3" xref="S6.Ex13.m1.5.5.2.3.cmml">⁢</mo><mrow id="S6.Ex13.m1.5.5.2.2.2" xref="S6.Ex13.m1.5.5.2.2.3.cmml"><mo id="S6.Ex13.m1.5.5.2.2.2.3" stretchy="false" xref="S6.Ex13.m1.5.5.2.2.3.cmml">(</mo><msub id="S6.Ex13.m1.4.4.1.1.1.1" xref="S6.Ex13.m1.4.4.1.1.1.1.cmml"><mi id="S6.Ex13.m1.4.4.1.1.1.1.2" xref="S6.Ex13.m1.4.4.1.1.1.1.2.cmml">a</mi><mi id="S6.Ex13.m1.4.4.1.1.1.1.3" xref="S6.Ex13.m1.4.4.1.1.1.1.3.cmml">r</mi></msub><mo id="S6.Ex13.m1.5.5.2.2.2.4" xref="S6.Ex13.m1.5.5.2.2.3.cmml">,</mo><msub id="S6.Ex13.m1.5.5.2.2.2.2" xref="S6.Ex13.m1.5.5.2.2.2.2.cmml"><mi id="S6.Ex13.m1.5.5.2.2.2.2.2" xref="S6.Ex13.m1.5.5.2.2.2.2.2.cmml">b</mi><mi id="S6.Ex13.m1.5.5.2.2.2.2.3" xref="S6.Ex13.m1.5.5.2.2.2.2.3.cmml">l</mi></msub><mo id="S6.Ex13.m1.5.5.2.2.2.5" stretchy="false" xref="S6.Ex13.m1.5.5.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.Ex13.m1.6.6.7" xref="S6.Ex13.m1.6.6.7.cmml">≤</mo><mi id="S6.Ex13.m1.6.6.8" xref="S6.Ex13.m1.6.6.8.cmml">ν</mi><mo id="S6.Ex13.m1.6.6.9" xref="S6.Ex13.m1.6.6.9.cmml">≤</mo><mrow id="S6.Ex13.m1.6.6.3" xref="S6.Ex13.m1.6.6.3.cmml"><mi id="S6.Ex13.m1.6.6.3.3" xref="S6.Ex13.m1.6.6.3.3.cmml">ν</mi><mo id="S6.Ex13.m1.6.6.3.2" xref="S6.Ex13.m1.6.6.3.2.cmml">⁢</mo><mrow id="S6.Ex13.m1.6.6.3.1.1" xref="S6.Ex13.m1.6.6.3.1.1.1.cmml"><mo id="S6.Ex13.m1.6.6.3.1.1.2" stretchy="false" xref="S6.Ex13.m1.6.6.3.1.1.1.cmml">(</mo><msub id="S6.Ex13.m1.6.6.3.1.1.1" xref="S6.Ex13.m1.6.6.3.1.1.1.cmml"><mi id="S6.Ex13.m1.6.6.3.1.1.1.2" xref="S6.Ex13.m1.6.6.3.1.1.1.2.cmml">b</mi><mi id="S6.Ex13.m1.6.6.3.1.1.1.3" xref="S6.Ex13.m1.6.6.3.1.1.1.3.cmml">l</mi></msub><mo id="S6.Ex13.m1.6.6.3.1.1.3" stretchy="false" xref="S6.Ex13.m1.6.6.3.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.Ex13.m1.6.6.10" xref="S6.Ex13.m1.6.6.10.cmml">=</mo><mrow id="S6.Ex13.m1.6.6.11" xref="S6.Ex13.m1.6.6.11.cmml"><mi id="S6.Ex13.m1.6.6.11.2" xref="S6.Ex13.m1.6.6.11.2.cmml">ν</mi><mo id="S6.Ex13.m1.6.6.11.1" xref="S6.Ex13.m1.6.6.11.1.cmml">⁢</mo><mrow id="S6.Ex13.m1.6.6.11.3.2" xref="S6.Ex13.m1.6.6.11.cmml"><mo id="S6.Ex13.m1.6.6.11.3.2.1" stretchy="false" xref="S6.Ex13.m1.6.6.11.cmml">(</mo><mi id="S6.Ex13.m1.3.3" xref="S6.Ex13.m1.3.3.cmml">y</mi><mo id="S6.Ex13.m1.6.6.11.3.2.2" stretchy="false" xref="S6.Ex13.m1.6.6.11.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Ex13.m1.6b"><apply id="S6.Ex13.m1.6.6.cmml" xref="S6.Ex13.m1.6.6"><and id="S6.Ex13.m1.6.6a.cmml" xref="S6.Ex13.m1.6.6"></and><apply id="S6.Ex13.m1.6.6b.cmml" xref="S6.Ex13.m1.6.6"><eq id="S6.Ex13.m1.6.6.6.cmml" xref="S6.Ex13.m1.6.6.6"></eq><apply id="S6.Ex13.m1.6.6.5.cmml" xref="S6.Ex13.m1.6.6.5"><times id="S6.Ex13.m1.6.6.5.1.cmml" xref="S6.Ex13.m1.6.6.5.1"></times><ci id="S6.Ex13.m1.6.6.5.2.cmml" xref="S6.Ex13.m1.6.6.5.2">𝜈</ci><interval closure="open" id="S6.Ex13.m1.6.6.5.3.1.cmml" xref="S6.Ex13.m1.6.6.5.3.2"><ci id="S6.Ex13.m1.1.1.cmml" xref="S6.Ex13.m1.1.1">𝑥</ci><ci id="S6.Ex13.m1.2.2.cmml" xref="S6.Ex13.m1.2.2">𝑦</ci></interval></apply><apply id="S6.Ex13.m1.5.5.2.cmml" xref="S6.Ex13.m1.5.5.2"><times id="S6.Ex13.m1.5.5.2.3.cmml" xref="S6.Ex13.m1.5.5.2.3"></times><ci id="S6.Ex13.m1.5.5.2.4.cmml" xref="S6.Ex13.m1.5.5.2.4">𝜈</ci><interval closure="open" id="S6.Ex13.m1.5.5.2.2.3.cmml" xref="S6.Ex13.m1.5.5.2.2.2"><apply id="S6.Ex13.m1.4.4.1.1.1.1.cmml" xref="S6.Ex13.m1.4.4.1.1.1.1"><csymbol cd="ambiguous" id="S6.Ex13.m1.4.4.1.1.1.1.1.cmml" xref="S6.Ex13.m1.4.4.1.1.1.1">subscript</csymbol><ci id="S6.Ex13.m1.4.4.1.1.1.1.2.cmml" xref="S6.Ex13.m1.4.4.1.1.1.1.2">𝑎</ci><ci id="S6.Ex13.m1.4.4.1.1.1.1.3.cmml" xref="S6.Ex13.m1.4.4.1.1.1.1.3">𝑟</ci></apply><apply id="S6.Ex13.m1.5.5.2.2.2.2.cmml" xref="S6.Ex13.m1.5.5.2.2.2.2"><csymbol cd="ambiguous" id="S6.Ex13.m1.5.5.2.2.2.2.1.cmml" xref="S6.Ex13.m1.5.5.2.2.2.2">subscript</csymbol><ci id="S6.Ex13.m1.5.5.2.2.2.2.2.cmml" xref="S6.Ex13.m1.5.5.2.2.2.2.2">𝑏</ci><ci id="S6.Ex13.m1.5.5.2.2.2.2.3.cmml" xref="S6.Ex13.m1.5.5.2.2.2.2.3">𝑙</ci></apply></interval></apply></apply><apply id="S6.Ex13.m1.6.6c.cmml" xref="S6.Ex13.m1.6.6"><leq id="S6.Ex13.m1.6.6.7.cmml" xref="S6.Ex13.m1.6.6.7"></leq><share href="https://arxiv.org/html/2503.13728v1#S6.Ex13.m1.5.5.2.cmml" id="S6.Ex13.m1.6.6d.cmml" xref="S6.Ex13.m1.6.6"></share><ci id="S6.Ex13.m1.6.6.8.cmml" xref="S6.Ex13.m1.6.6.8">𝜈</ci></apply><apply id="S6.Ex13.m1.6.6e.cmml" xref="S6.Ex13.m1.6.6"><leq id="S6.Ex13.m1.6.6.9.cmml" xref="S6.Ex13.m1.6.6.9"></leq><share href="https://arxiv.org/html/2503.13728v1#S6.Ex13.m1.6.6.8.cmml" id="S6.Ex13.m1.6.6f.cmml" xref="S6.Ex13.m1.6.6"></share><apply id="S6.Ex13.m1.6.6.3.cmml" xref="S6.Ex13.m1.6.6.3"><times id="S6.Ex13.m1.6.6.3.2.cmml" xref="S6.Ex13.m1.6.6.3.2"></times><ci id="S6.Ex13.m1.6.6.3.3.cmml" xref="S6.Ex13.m1.6.6.3.3">𝜈</ci><apply id="S6.Ex13.m1.6.6.3.1.1.1.cmml" xref="S6.Ex13.m1.6.6.3.1.1"><csymbol cd="ambiguous" id="S6.Ex13.m1.6.6.3.1.1.1.1.cmml" xref="S6.Ex13.m1.6.6.3.1.1">subscript</csymbol><ci id="S6.Ex13.m1.6.6.3.1.1.1.2.cmml" xref="S6.Ex13.m1.6.6.3.1.1.1.2">𝑏</ci><ci id="S6.Ex13.m1.6.6.3.1.1.1.3.cmml" xref="S6.Ex13.m1.6.6.3.1.1.1.3">𝑙</ci></apply></apply></apply><apply id="S6.Ex13.m1.6.6g.cmml" xref="S6.Ex13.m1.6.6"><eq id="S6.Ex13.m1.6.6.10.cmml" xref="S6.Ex13.m1.6.6.10"></eq><share href="https://arxiv.org/html/2503.13728v1#S6.Ex13.m1.6.6.3.cmml" id="S6.Ex13.m1.6.6h.cmml" xref="S6.Ex13.m1.6.6"></share><apply id="S6.Ex13.m1.6.6.11.cmml" xref="S6.Ex13.m1.6.6.11"><times id="S6.Ex13.m1.6.6.11.1.cmml" xref="S6.Ex13.m1.6.6.11.1"></times><ci id="S6.Ex13.m1.6.6.11.2.cmml" xref="S6.Ex13.m1.6.6.11.2">𝜈</ci><ci id="S6.Ex13.m1.3.3.cmml" xref="S6.Ex13.m1.3.3">𝑦</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex13.m1.6c">\nu(x,y)=\nu(a_{r},b_{l})\leq\nu\leq\nu(b_{l})=\nu(y)</annotation><annotation encoding="application/x-llamapun" id="S6.Ex13.m1.6d">italic_ν ( italic_x , italic_y ) = italic_ν ( italic_a start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ) ≤ italic_ν ≤ italic_ν ( italic_b start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ) = italic_ν ( italic_y )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S6.SS2.26.p2"> <p class="ltx_p" id="S6.SS2.26.p2.12"><span class="ltx_text ltx_framed ltx_framed_underline" id="S6.SS2.26.p2.12.1">Case 1</span>: <math alttext="\nu=\nu(a_{r},b_{l})" class="ltx_Math" display="inline" id="S6.SS2.26.p2.1.m1.2"><semantics id="S6.SS2.26.p2.1.m1.2a"><mrow id="S6.SS2.26.p2.1.m1.2.2" xref="S6.SS2.26.p2.1.m1.2.2.cmml"><mi id="S6.SS2.26.p2.1.m1.2.2.4" xref="S6.SS2.26.p2.1.m1.2.2.4.cmml">ν</mi><mo id="S6.SS2.26.p2.1.m1.2.2.3" xref="S6.SS2.26.p2.1.m1.2.2.3.cmml">=</mo><mrow id="S6.SS2.26.p2.1.m1.2.2.2" xref="S6.SS2.26.p2.1.m1.2.2.2.cmml"><mi id="S6.SS2.26.p2.1.m1.2.2.2.4" xref="S6.SS2.26.p2.1.m1.2.2.2.4.cmml">ν</mi><mo id="S6.SS2.26.p2.1.m1.2.2.2.3" xref="S6.SS2.26.p2.1.m1.2.2.2.3.cmml">⁢</mo><mrow id="S6.SS2.26.p2.1.m1.2.2.2.2.2" xref="S6.SS2.26.p2.1.m1.2.2.2.2.3.cmml"><mo id="S6.SS2.26.p2.1.m1.2.2.2.2.2.3" stretchy="false" xref="S6.SS2.26.p2.1.m1.2.2.2.2.3.cmml">(</mo><msub id="S6.SS2.26.p2.1.m1.1.1.1.1.1.1" xref="S6.SS2.26.p2.1.m1.1.1.1.1.1.1.cmml"><mi id="S6.SS2.26.p2.1.m1.1.1.1.1.1.1.2" xref="S6.SS2.26.p2.1.m1.1.1.1.1.1.1.2.cmml">a</mi><mi id="S6.SS2.26.p2.1.m1.1.1.1.1.1.1.3" xref="S6.SS2.26.p2.1.m1.1.1.1.1.1.1.3.cmml">r</mi></msub><mo id="S6.SS2.26.p2.1.m1.2.2.2.2.2.4" xref="S6.SS2.26.p2.1.m1.2.2.2.2.3.cmml">,</mo><msub id="S6.SS2.26.p2.1.m1.2.2.2.2.2.2" xref="S6.SS2.26.p2.1.m1.2.2.2.2.2.2.cmml"><mi id="S6.SS2.26.p2.1.m1.2.2.2.2.2.2.2" xref="S6.SS2.26.p2.1.m1.2.2.2.2.2.2.2.cmml">b</mi><mi id="S6.SS2.26.p2.1.m1.2.2.2.2.2.2.3" xref="S6.SS2.26.p2.1.m1.2.2.2.2.2.2.3.cmml">l</mi></msub><mo id="S6.SS2.26.p2.1.m1.2.2.2.2.2.5" stretchy="false" xref="S6.SS2.26.p2.1.m1.2.2.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.26.p2.1.m1.2b"><apply id="S6.SS2.26.p2.1.m1.2.2.cmml" xref="S6.SS2.26.p2.1.m1.2.2"><eq id="S6.SS2.26.p2.1.m1.2.2.3.cmml" xref="S6.SS2.26.p2.1.m1.2.2.3"></eq><ci id="S6.SS2.26.p2.1.m1.2.2.4.cmml" xref="S6.SS2.26.p2.1.m1.2.2.4">𝜈</ci><apply id="S6.SS2.26.p2.1.m1.2.2.2.cmml" xref="S6.SS2.26.p2.1.m1.2.2.2"><times id="S6.SS2.26.p2.1.m1.2.2.2.3.cmml" xref="S6.SS2.26.p2.1.m1.2.2.2.3"></times><ci id="S6.SS2.26.p2.1.m1.2.2.2.4.cmml" xref="S6.SS2.26.p2.1.m1.2.2.2.4">𝜈</ci><interval closure="open" id="S6.SS2.26.p2.1.m1.2.2.2.2.3.cmml" xref="S6.SS2.26.p2.1.m1.2.2.2.2.2"><apply id="S6.SS2.26.p2.1.m1.1.1.1.1.1.1.cmml" xref="S6.SS2.26.p2.1.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.26.p2.1.m1.1.1.1.1.1.1.1.cmml" xref="S6.SS2.26.p2.1.m1.1.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.26.p2.1.m1.1.1.1.1.1.1.2.cmml" xref="S6.SS2.26.p2.1.m1.1.1.1.1.1.1.2">𝑎</ci><ci id="S6.SS2.26.p2.1.m1.1.1.1.1.1.1.3.cmml" xref="S6.SS2.26.p2.1.m1.1.1.1.1.1.1.3">𝑟</ci></apply><apply id="S6.SS2.26.p2.1.m1.2.2.2.2.2.2.cmml" xref="S6.SS2.26.p2.1.m1.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.26.p2.1.m1.2.2.2.2.2.2.1.cmml" xref="S6.SS2.26.p2.1.m1.2.2.2.2.2.2">subscript</csymbol><ci id="S6.SS2.26.p2.1.m1.2.2.2.2.2.2.2.cmml" xref="S6.SS2.26.p2.1.m1.2.2.2.2.2.2.2">𝑏</ci><ci id="S6.SS2.26.p2.1.m1.2.2.2.2.2.2.3.cmml" xref="S6.SS2.26.p2.1.m1.2.2.2.2.2.2.3">𝑙</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.26.p2.1.m1.2c">\nu=\nu(a_{r},b_{l})</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.26.p2.1.m1.2d">italic_ν = italic_ν ( italic_a start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT )</annotation></semantics></math>. Note that then <math alttext="\nu=\nu(x,y)" class="ltx_Math" display="inline" id="S6.SS2.26.p2.2.m2.2"><semantics id="S6.SS2.26.p2.2.m2.2a"><mrow id="S6.SS2.26.p2.2.m2.2.3" xref="S6.SS2.26.p2.2.m2.2.3.cmml"><mi id="S6.SS2.26.p2.2.m2.2.3.2" xref="S6.SS2.26.p2.2.m2.2.3.2.cmml">ν</mi><mo id="S6.SS2.26.p2.2.m2.2.3.1" xref="S6.SS2.26.p2.2.m2.2.3.1.cmml">=</mo><mrow id="S6.SS2.26.p2.2.m2.2.3.3" xref="S6.SS2.26.p2.2.m2.2.3.3.cmml"><mi id="S6.SS2.26.p2.2.m2.2.3.3.2" xref="S6.SS2.26.p2.2.m2.2.3.3.2.cmml">ν</mi><mo id="S6.SS2.26.p2.2.m2.2.3.3.1" xref="S6.SS2.26.p2.2.m2.2.3.3.1.cmml">⁢</mo><mrow id="S6.SS2.26.p2.2.m2.2.3.3.3.2" xref="S6.SS2.26.p2.2.m2.2.3.3.3.1.cmml"><mo id="S6.SS2.26.p2.2.m2.2.3.3.3.2.1" stretchy="false" xref="S6.SS2.26.p2.2.m2.2.3.3.3.1.cmml">(</mo><mi id="S6.SS2.26.p2.2.m2.1.1" xref="S6.SS2.26.p2.2.m2.1.1.cmml">x</mi><mo id="S6.SS2.26.p2.2.m2.2.3.3.3.2.2" xref="S6.SS2.26.p2.2.m2.2.3.3.3.1.cmml">,</mo><mi id="S6.SS2.26.p2.2.m2.2.2" xref="S6.SS2.26.p2.2.m2.2.2.cmml">y</mi><mo id="S6.SS2.26.p2.2.m2.2.3.3.3.2.3" stretchy="false" xref="S6.SS2.26.p2.2.m2.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.26.p2.2.m2.2b"><apply id="S6.SS2.26.p2.2.m2.2.3.cmml" xref="S6.SS2.26.p2.2.m2.2.3"><eq id="S6.SS2.26.p2.2.m2.2.3.1.cmml" xref="S6.SS2.26.p2.2.m2.2.3.1"></eq><ci id="S6.SS2.26.p2.2.m2.2.3.2.cmml" xref="S6.SS2.26.p2.2.m2.2.3.2">𝜈</ci><apply id="S6.SS2.26.p2.2.m2.2.3.3.cmml" xref="S6.SS2.26.p2.2.m2.2.3.3"><times id="S6.SS2.26.p2.2.m2.2.3.3.1.cmml" xref="S6.SS2.26.p2.2.m2.2.3.3.1"></times><ci id="S6.SS2.26.p2.2.m2.2.3.3.2.cmml" xref="S6.SS2.26.p2.2.m2.2.3.3.2">𝜈</ci><interval closure="open" id="S6.SS2.26.p2.2.m2.2.3.3.3.1.cmml" xref="S6.SS2.26.p2.2.m2.2.3.3.3.2"><ci id="S6.SS2.26.p2.2.m2.1.1.cmml" xref="S6.SS2.26.p2.2.m2.1.1">𝑥</ci><ci id="S6.SS2.26.p2.2.m2.2.2.cmml" xref="S6.SS2.26.p2.2.m2.2.2">𝑦</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.26.p2.2.m2.2c">\nu=\nu(x,y)</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.26.p2.2.m2.2d">italic_ν = italic_ν ( italic_x , italic_y )</annotation></semantics></math> and that in this case <math alttext="c" class="ltx_Math" display="inline" id="S6.SS2.26.p2.3.m3.1"><semantics id="S6.SS2.26.p2.3.m3.1a"><mi id="S6.SS2.26.p2.3.m3.1.1" xref="S6.SS2.26.p2.3.m3.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.26.p2.3.m3.1b"><ci id="S6.SS2.26.p2.3.m3.1.1.cmml" xref="S6.SS2.26.p2.3.m3.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.26.p2.3.m3.1c">c</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.26.p2.3.m3.1d">italic_c</annotation></semantics></math> is not an endpoint of its complementary interval of <math alttext="A\setminus\nu" class="ltx_Math" display="inline" id="S6.SS2.26.p2.4.m4.1"><semantics id="S6.SS2.26.p2.4.m4.1a"><mrow id="S6.SS2.26.p2.4.m4.1.1" xref="S6.SS2.26.p2.4.m4.1.1.cmml"><mi id="S6.SS2.26.p2.4.m4.1.1.2" xref="S6.SS2.26.p2.4.m4.1.1.2.cmml">A</mi><mo id="S6.SS2.26.p2.4.m4.1.1.1" xref="S6.SS2.26.p2.4.m4.1.1.1.cmml">∖</mo><mi id="S6.SS2.26.p2.4.m4.1.1.3" xref="S6.SS2.26.p2.4.m4.1.1.3.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.26.p2.4.m4.1b"><apply id="S6.SS2.26.p2.4.m4.1.1.cmml" xref="S6.SS2.26.p2.4.m4.1.1"><setdiff id="S6.SS2.26.p2.4.m4.1.1.1.cmml" xref="S6.SS2.26.p2.4.m4.1.1.1"></setdiff><ci id="S6.SS2.26.p2.4.m4.1.1.2.cmml" xref="S6.SS2.26.p2.4.m4.1.1.2">𝐴</ci><ci id="S6.SS2.26.p2.4.m4.1.1.3.cmml" xref="S6.SS2.26.p2.4.m4.1.1.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.26.p2.4.m4.1c">A\setminus\nu</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.26.p2.4.m4.1d">italic_A ∖ italic_ν</annotation></semantics></math>. Using elementarity and <math alttext="\aleph_{1}" class="ltx_Math" display="inline" id="S6.SS2.26.p2.5.m5.1"><semantics id="S6.SS2.26.p2.5.m5.1a"><msub id="S6.SS2.26.p2.5.m5.1.1" xref="S6.SS2.26.p2.5.m5.1.1.cmml"><mi id="S6.SS2.26.p2.5.m5.1.1.2" mathvariant="normal" xref="S6.SS2.26.p2.5.m5.1.1.2.cmml">ℵ</mi><mn id="S6.SS2.26.p2.5.m5.1.1.3" xref="S6.SS2.26.p2.5.m5.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.26.p2.5.m5.1b"><apply id="S6.SS2.26.p2.5.m5.1.1.cmml" xref="S6.SS2.26.p2.5.m5.1.1"><csymbol cd="ambiguous" id="S6.SS2.26.p2.5.m5.1.1.1.cmml" xref="S6.SS2.26.p2.5.m5.1.1">subscript</csymbol><ci id="S6.SS2.26.p2.5.m5.1.1.2.cmml" xref="S6.SS2.26.p2.5.m5.1.1.2">ℵ</ci><cn id="S6.SS2.26.p2.5.m5.1.1.3.cmml" type="integer" xref="S6.SS2.26.p2.5.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.26.p2.5.m5.1c">\aleph_{1}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.26.p2.5.m5.1d">roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-density one finds <math alttext="a_{r}<_{A}c_{l}<_{A}c<_{A}c_{r}<_{A}b_{l}" class="ltx_Math" display="inline" id="S6.SS2.26.p2.6.m6.1"><semantics id="S6.SS2.26.p2.6.m6.1a"><mrow id="S6.SS2.26.p2.6.m6.1.1" xref="S6.SS2.26.p2.6.m6.1.1.cmml"><msub id="S6.SS2.26.p2.6.m6.1.1.2" xref="S6.SS2.26.p2.6.m6.1.1.2.cmml"><mi id="S6.SS2.26.p2.6.m6.1.1.2.2" xref="S6.SS2.26.p2.6.m6.1.1.2.2.cmml">a</mi><mi id="S6.SS2.26.p2.6.m6.1.1.2.3" xref="S6.SS2.26.p2.6.m6.1.1.2.3.cmml">r</mi></msub><msub id="S6.SS2.26.p2.6.m6.1.1.3" xref="S6.SS2.26.p2.6.m6.1.1.3.cmml"><mo id="S6.SS2.26.p2.6.m6.1.1.3.2" xref="S6.SS2.26.p2.6.m6.1.1.3.2.cmml"><</mo><mi id="S6.SS2.26.p2.6.m6.1.1.3.3" xref="S6.SS2.26.p2.6.m6.1.1.3.3.cmml">A</mi></msub><msub id="S6.SS2.26.p2.6.m6.1.1.4" xref="S6.SS2.26.p2.6.m6.1.1.4.cmml"><mi id="S6.SS2.26.p2.6.m6.1.1.4.2" xref="S6.SS2.26.p2.6.m6.1.1.4.2.cmml">c</mi><mi id="S6.SS2.26.p2.6.m6.1.1.4.3" xref="S6.SS2.26.p2.6.m6.1.1.4.3.cmml">l</mi></msub><msub id="S6.SS2.26.p2.6.m6.1.1.5" xref="S6.SS2.26.p2.6.m6.1.1.5.cmml"><mo id="S6.SS2.26.p2.6.m6.1.1.5.2" xref="S6.SS2.26.p2.6.m6.1.1.5.2.cmml"><</mo><mi id="S6.SS2.26.p2.6.m6.1.1.5.3" xref="S6.SS2.26.p2.6.m6.1.1.5.3.cmml">A</mi></msub><mi id="S6.SS2.26.p2.6.m6.1.1.6" xref="S6.SS2.26.p2.6.m6.1.1.6.cmml">c</mi><msub id="S6.SS2.26.p2.6.m6.1.1.7" xref="S6.SS2.26.p2.6.m6.1.1.7.cmml"><mo id="S6.SS2.26.p2.6.m6.1.1.7.2" xref="S6.SS2.26.p2.6.m6.1.1.7.2.cmml"><</mo><mi id="S6.SS2.26.p2.6.m6.1.1.7.3" xref="S6.SS2.26.p2.6.m6.1.1.7.3.cmml">A</mi></msub><msub id="S6.SS2.26.p2.6.m6.1.1.8" xref="S6.SS2.26.p2.6.m6.1.1.8.cmml"><mi id="S6.SS2.26.p2.6.m6.1.1.8.2" xref="S6.SS2.26.p2.6.m6.1.1.8.2.cmml">c</mi><mi id="S6.SS2.26.p2.6.m6.1.1.8.3" xref="S6.SS2.26.p2.6.m6.1.1.8.3.cmml">r</mi></msub><msub id="S6.SS2.26.p2.6.m6.1.1.9" xref="S6.SS2.26.p2.6.m6.1.1.9.cmml"><mo id="S6.SS2.26.p2.6.m6.1.1.9.2" xref="S6.SS2.26.p2.6.m6.1.1.9.2.cmml"><</mo><mi id="S6.SS2.26.p2.6.m6.1.1.9.3" xref="S6.SS2.26.p2.6.m6.1.1.9.3.cmml">A</mi></msub><msub id="S6.SS2.26.p2.6.m6.1.1.10" xref="S6.SS2.26.p2.6.m6.1.1.10.cmml"><mi id="S6.SS2.26.p2.6.m6.1.1.10.2" xref="S6.SS2.26.p2.6.m6.1.1.10.2.cmml">b</mi><mi id="S6.SS2.26.p2.6.m6.1.1.10.3" xref="S6.SS2.26.p2.6.m6.1.1.10.3.cmml">l</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.26.p2.6.m6.1b"><apply id="S6.SS2.26.p2.6.m6.1.1.cmml" xref="S6.SS2.26.p2.6.m6.1.1"><and id="S6.SS2.26.p2.6.m6.1.1a.cmml" xref="S6.SS2.26.p2.6.m6.1.1"></and><apply id="S6.SS2.26.p2.6.m6.1.1b.cmml" xref="S6.SS2.26.p2.6.m6.1.1"><apply id="S6.SS2.26.p2.6.m6.1.1.3.cmml" xref="S6.SS2.26.p2.6.m6.1.1.3"><csymbol cd="ambiguous" id="S6.SS2.26.p2.6.m6.1.1.3.1.cmml" xref="S6.SS2.26.p2.6.m6.1.1.3">subscript</csymbol><lt id="S6.SS2.26.p2.6.m6.1.1.3.2.cmml" xref="S6.SS2.26.p2.6.m6.1.1.3.2"></lt><ci id="S6.SS2.26.p2.6.m6.1.1.3.3.cmml" xref="S6.SS2.26.p2.6.m6.1.1.3.3">𝐴</ci></apply><apply id="S6.SS2.26.p2.6.m6.1.1.2.cmml" xref="S6.SS2.26.p2.6.m6.1.1.2"><csymbol cd="ambiguous" id="S6.SS2.26.p2.6.m6.1.1.2.1.cmml" xref="S6.SS2.26.p2.6.m6.1.1.2">subscript</csymbol><ci id="S6.SS2.26.p2.6.m6.1.1.2.2.cmml" xref="S6.SS2.26.p2.6.m6.1.1.2.2">𝑎</ci><ci id="S6.SS2.26.p2.6.m6.1.1.2.3.cmml" xref="S6.SS2.26.p2.6.m6.1.1.2.3">𝑟</ci></apply><apply id="S6.SS2.26.p2.6.m6.1.1.4.cmml" xref="S6.SS2.26.p2.6.m6.1.1.4"><csymbol cd="ambiguous" id="S6.SS2.26.p2.6.m6.1.1.4.1.cmml" xref="S6.SS2.26.p2.6.m6.1.1.4">subscript</csymbol><ci id="S6.SS2.26.p2.6.m6.1.1.4.2.cmml" xref="S6.SS2.26.p2.6.m6.1.1.4.2">𝑐</ci><ci id="S6.SS2.26.p2.6.m6.1.1.4.3.cmml" xref="S6.SS2.26.p2.6.m6.1.1.4.3">𝑙</ci></apply></apply><apply id="S6.SS2.26.p2.6.m6.1.1c.cmml" xref="S6.SS2.26.p2.6.m6.1.1"><apply id="S6.SS2.26.p2.6.m6.1.1.5.cmml" xref="S6.SS2.26.p2.6.m6.1.1.5"><csymbol cd="ambiguous" id="S6.SS2.26.p2.6.m6.1.1.5.1.cmml" xref="S6.SS2.26.p2.6.m6.1.1.5">subscript</csymbol><lt id="S6.SS2.26.p2.6.m6.1.1.5.2.cmml" xref="S6.SS2.26.p2.6.m6.1.1.5.2"></lt><ci id="S6.SS2.26.p2.6.m6.1.1.5.3.cmml" xref="S6.SS2.26.p2.6.m6.1.1.5.3">𝐴</ci></apply><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.26.p2.6.m6.1.1.4.cmml" id="S6.SS2.26.p2.6.m6.1.1d.cmml" xref="S6.SS2.26.p2.6.m6.1.1"></share><ci id="S6.SS2.26.p2.6.m6.1.1.6.cmml" xref="S6.SS2.26.p2.6.m6.1.1.6">𝑐</ci></apply><apply id="S6.SS2.26.p2.6.m6.1.1e.cmml" xref="S6.SS2.26.p2.6.m6.1.1"><apply id="S6.SS2.26.p2.6.m6.1.1.7.cmml" xref="S6.SS2.26.p2.6.m6.1.1.7"><csymbol cd="ambiguous" id="S6.SS2.26.p2.6.m6.1.1.7.1.cmml" xref="S6.SS2.26.p2.6.m6.1.1.7">subscript</csymbol><lt id="S6.SS2.26.p2.6.m6.1.1.7.2.cmml" xref="S6.SS2.26.p2.6.m6.1.1.7.2"></lt><ci id="S6.SS2.26.p2.6.m6.1.1.7.3.cmml" xref="S6.SS2.26.p2.6.m6.1.1.7.3">𝐴</ci></apply><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.26.p2.6.m6.1.1.6.cmml" id="S6.SS2.26.p2.6.m6.1.1f.cmml" xref="S6.SS2.26.p2.6.m6.1.1"></share><apply id="S6.SS2.26.p2.6.m6.1.1.8.cmml" xref="S6.SS2.26.p2.6.m6.1.1.8"><csymbol cd="ambiguous" id="S6.SS2.26.p2.6.m6.1.1.8.1.cmml" xref="S6.SS2.26.p2.6.m6.1.1.8">subscript</csymbol><ci id="S6.SS2.26.p2.6.m6.1.1.8.2.cmml" xref="S6.SS2.26.p2.6.m6.1.1.8.2">𝑐</ci><ci id="S6.SS2.26.p2.6.m6.1.1.8.3.cmml" xref="S6.SS2.26.p2.6.m6.1.1.8.3">𝑟</ci></apply></apply><apply id="S6.SS2.26.p2.6.m6.1.1g.cmml" xref="S6.SS2.26.p2.6.m6.1.1"><apply id="S6.SS2.26.p2.6.m6.1.1.9.cmml" xref="S6.SS2.26.p2.6.m6.1.1.9"><csymbol cd="ambiguous" id="S6.SS2.26.p2.6.m6.1.1.9.1.cmml" xref="S6.SS2.26.p2.6.m6.1.1.9">subscript</csymbol><lt id="S6.SS2.26.p2.6.m6.1.1.9.2.cmml" xref="S6.SS2.26.p2.6.m6.1.1.9.2"></lt><ci id="S6.SS2.26.p2.6.m6.1.1.9.3.cmml" xref="S6.SS2.26.p2.6.m6.1.1.9.3">𝐴</ci></apply><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.26.p2.6.m6.1.1.8.cmml" id="S6.SS2.26.p2.6.m6.1.1h.cmml" xref="S6.SS2.26.p2.6.m6.1.1"></share><apply id="S6.SS2.26.p2.6.m6.1.1.10.cmml" xref="S6.SS2.26.p2.6.m6.1.1.10"><csymbol cd="ambiguous" id="S6.SS2.26.p2.6.m6.1.1.10.1.cmml" xref="S6.SS2.26.p2.6.m6.1.1.10">subscript</csymbol><ci id="S6.SS2.26.p2.6.m6.1.1.10.2.cmml" xref="S6.SS2.26.p2.6.m6.1.1.10.2">𝑏</ci><ci id="S6.SS2.26.p2.6.m6.1.1.10.3.cmml" xref="S6.SS2.26.p2.6.m6.1.1.10.3">𝑙</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.26.p2.6.m6.1c">a_{r}<_{A}c_{l}<_{A}c<_{A}c_{r}<_{A}b_{l}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.26.p2.6.m6.1d">italic_a start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT < start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_c start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT < start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_c < start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_c start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT < start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="\nu(c_{l})=\nu(c_{r})=\nu" class="ltx_Math" display="inline" id="S6.SS2.26.p2.7.m7.2"><semantics id="S6.SS2.26.p2.7.m7.2a"><mrow id="S6.SS2.26.p2.7.m7.2.2" xref="S6.SS2.26.p2.7.m7.2.2.cmml"><mrow id="S6.SS2.26.p2.7.m7.1.1.1" xref="S6.SS2.26.p2.7.m7.1.1.1.cmml"><mi id="S6.SS2.26.p2.7.m7.1.1.1.3" xref="S6.SS2.26.p2.7.m7.1.1.1.3.cmml">ν</mi><mo id="S6.SS2.26.p2.7.m7.1.1.1.2" xref="S6.SS2.26.p2.7.m7.1.1.1.2.cmml">⁢</mo><mrow id="S6.SS2.26.p2.7.m7.1.1.1.1.1" xref="S6.SS2.26.p2.7.m7.1.1.1.1.1.1.cmml"><mo id="S6.SS2.26.p2.7.m7.1.1.1.1.1.2" stretchy="false" xref="S6.SS2.26.p2.7.m7.1.1.1.1.1.1.cmml">(</mo><msub id="S6.SS2.26.p2.7.m7.1.1.1.1.1.1" xref="S6.SS2.26.p2.7.m7.1.1.1.1.1.1.cmml"><mi id="S6.SS2.26.p2.7.m7.1.1.1.1.1.1.2" xref="S6.SS2.26.p2.7.m7.1.1.1.1.1.1.2.cmml">c</mi><mi id="S6.SS2.26.p2.7.m7.1.1.1.1.1.1.3" xref="S6.SS2.26.p2.7.m7.1.1.1.1.1.1.3.cmml">l</mi></msub><mo id="S6.SS2.26.p2.7.m7.1.1.1.1.1.3" stretchy="false" xref="S6.SS2.26.p2.7.m7.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.SS2.26.p2.7.m7.2.2.4" xref="S6.SS2.26.p2.7.m7.2.2.4.cmml">=</mo><mrow id="S6.SS2.26.p2.7.m7.2.2.2" xref="S6.SS2.26.p2.7.m7.2.2.2.cmml"><mi id="S6.SS2.26.p2.7.m7.2.2.2.3" xref="S6.SS2.26.p2.7.m7.2.2.2.3.cmml">ν</mi><mo id="S6.SS2.26.p2.7.m7.2.2.2.2" xref="S6.SS2.26.p2.7.m7.2.2.2.2.cmml">⁢</mo><mrow id="S6.SS2.26.p2.7.m7.2.2.2.1.1" xref="S6.SS2.26.p2.7.m7.2.2.2.1.1.1.cmml"><mo id="S6.SS2.26.p2.7.m7.2.2.2.1.1.2" stretchy="false" xref="S6.SS2.26.p2.7.m7.2.2.2.1.1.1.cmml">(</mo><msub id="S6.SS2.26.p2.7.m7.2.2.2.1.1.1" xref="S6.SS2.26.p2.7.m7.2.2.2.1.1.1.cmml"><mi id="S6.SS2.26.p2.7.m7.2.2.2.1.1.1.2" xref="S6.SS2.26.p2.7.m7.2.2.2.1.1.1.2.cmml">c</mi><mi id="S6.SS2.26.p2.7.m7.2.2.2.1.1.1.3" xref="S6.SS2.26.p2.7.m7.2.2.2.1.1.1.3.cmml">r</mi></msub><mo id="S6.SS2.26.p2.7.m7.2.2.2.1.1.3" stretchy="false" xref="S6.SS2.26.p2.7.m7.2.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.SS2.26.p2.7.m7.2.2.5" xref="S6.SS2.26.p2.7.m7.2.2.5.cmml">=</mo><mi id="S6.SS2.26.p2.7.m7.2.2.6" xref="S6.SS2.26.p2.7.m7.2.2.6.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.26.p2.7.m7.2b"><apply id="S6.SS2.26.p2.7.m7.2.2.cmml" xref="S6.SS2.26.p2.7.m7.2.2"><and id="S6.SS2.26.p2.7.m7.2.2a.cmml" xref="S6.SS2.26.p2.7.m7.2.2"></and><apply id="S6.SS2.26.p2.7.m7.2.2b.cmml" xref="S6.SS2.26.p2.7.m7.2.2"><eq id="S6.SS2.26.p2.7.m7.2.2.4.cmml" xref="S6.SS2.26.p2.7.m7.2.2.4"></eq><apply id="S6.SS2.26.p2.7.m7.1.1.1.cmml" xref="S6.SS2.26.p2.7.m7.1.1.1"><times id="S6.SS2.26.p2.7.m7.1.1.1.2.cmml" xref="S6.SS2.26.p2.7.m7.1.1.1.2"></times><ci id="S6.SS2.26.p2.7.m7.1.1.1.3.cmml" xref="S6.SS2.26.p2.7.m7.1.1.1.3">𝜈</ci><apply id="S6.SS2.26.p2.7.m7.1.1.1.1.1.1.cmml" xref="S6.SS2.26.p2.7.m7.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.26.p2.7.m7.1.1.1.1.1.1.1.cmml" xref="S6.SS2.26.p2.7.m7.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.26.p2.7.m7.1.1.1.1.1.1.2.cmml" xref="S6.SS2.26.p2.7.m7.1.1.1.1.1.1.2">𝑐</ci><ci id="S6.SS2.26.p2.7.m7.1.1.1.1.1.1.3.cmml" xref="S6.SS2.26.p2.7.m7.1.1.1.1.1.1.3">𝑙</ci></apply></apply><apply id="S6.SS2.26.p2.7.m7.2.2.2.cmml" xref="S6.SS2.26.p2.7.m7.2.2.2"><times id="S6.SS2.26.p2.7.m7.2.2.2.2.cmml" xref="S6.SS2.26.p2.7.m7.2.2.2.2"></times><ci id="S6.SS2.26.p2.7.m7.2.2.2.3.cmml" xref="S6.SS2.26.p2.7.m7.2.2.2.3">𝜈</ci><apply id="S6.SS2.26.p2.7.m7.2.2.2.1.1.1.cmml" xref="S6.SS2.26.p2.7.m7.2.2.2.1.1"><csymbol cd="ambiguous" id="S6.SS2.26.p2.7.m7.2.2.2.1.1.1.1.cmml" xref="S6.SS2.26.p2.7.m7.2.2.2.1.1">subscript</csymbol><ci id="S6.SS2.26.p2.7.m7.2.2.2.1.1.1.2.cmml" xref="S6.SS2.26.p2.7.m7.2.2.2.1.1.1.2">𝑐</ci><ci id="S6.SS2.26.p2.7.m7.2.2.2.1.1.1.3.cmml" xref="S6.SS2.26.p2.7.m7.2.2.2.1.1.1.3">𝑟</ci></apply></apply></apply><apply id="S6.SS2.26.p2.7.m7.2.2c.cmml" xref="S6.SS2.26.p2.7.m7.2.2"><eq id="S6.SS2.26.p2.7.m7.2.2.5.cmml" xref="S6.SS2.26.p2.7.m7.2.2.5"></eq><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.26.p2.7.m7.2.2.2.cmml" id="S6.SS2.26.p2.7.m7.2.2d.cmml" xref="S6.SS2.26.p2.7.m7.2.2"></share><ci id="S6.SS2.26.p2.7.m7.2.2.6.cmml" xref="S6.SS2.26.p2.7.m7.2.2.6">𝜈</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.26.p2.7.m7.2c">\nu(c_{l})=\nu(c_{r})=\nu</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.26.p2.7.m7.2d">italic_ν ( italic_c start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ) = italic_ν ( italic_c start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) = italic_ν</annotation></semantics></math>, and <math alttext="x<_{X}z<_{X}y" class="ltx_Math" display="inline" id="S6.SS2.26.p2.8.m8.1"><semantics id="S6.SS2.26.p2.8.m8.1a"><mrow id="S6.SS2.26.p2.8.m8.1.1" xref="S6.SS2.26.p2.8.m8.1.1.cmml"><mi id="S6.SS2.26.p2.8.m8.1.1.2" xref="S6.SS2.26.p2.8.m8.1.1.2.cmml">x</mi><msub id="S6.SS2.26.p2.8.m8.1.1.3" xref="S6.SS2.26.p2.8.m8.1.1.3.cmml"><mo id="S6.SS2.26.p2.8.m8.1.1.3.2" xref="S6.SS2.26.p2.8.m8.1.1.3.2.cmml"><</mo><mi id="S6.SS2.26.p2.8.m8.1.1.3.3" xref="S6.SS2.26.p2.8.m8.1.1.3.3.cmml">X</mi></msub><mi id="S6.SS2.26.p2.8.m8.1.1.4" xref="S6.SS2.26.p2.8.m8.1.1.4.cmml">z</mi><msub id="S6.SS2.26.p2.8.m8.1.1.5" xref="S6.SS2.26.p2.8.m8.1.1.5.cmml"><mo id="S6.SS2.26.p2.8.m8.1.1.5.2" xref="S6.SS2.26.p2.8.m8.1.1.5.2.cmml"><</mo><mi id="S6.SS2.26.p2.8.m8.1.1.5.3" xref="S6.SS2.26.p2.8.m8.1.1.5.3.cmml">X</mi></msub><mi id="S6.SS2.26.p2.8.m8.1.1.6" xref="S6.SS2.26.p2.8.m8.1.1.6.cmml">y</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.26.p2.8.m8.1b"><apply id="S6.SS2.26.p2.8.m8.1.1.cmml" xref="S6.SS2.26.p2.8.m8.1.1"><and id="S6.SS2.26.p2.8.m8.1.1a.cmml" xref="S6.SS2.26.p2.8.m8.1.1"></and><apply id="S6.SS2.26.p2.8.m8.1.1b.cmml" xref="S6.SS2.26.p2.8.m8.1.1"><apply id="S6.SS2.26.p2.8.m8.1.1.3.cmml" xref="S6.SS2.26.p2.8.m8.1.1.3"><csymbol cd="ambiguous" id="S6.SS2.26.p2.8.m8.1.1.3.1.cmml" xref="S6.SS2.26.p2.8.m8.1.1.3">subscript</csymbol><lt id="S6.SS2.26.p2.8.m8.1.1.3.2.cmml" xref="S6.SS2.26.p2.8.m8.1.1.3.2"></lt><ci id="S6.SS2.26.p2.8.m8.1.1.3.3.cmml" xref="S6.SS2.26.p2.8.m8.1.1.3.3">𝑋</ci></apply><ci id="S6.SS2.26.p2.8.m8.1.1.2.cmml" xref="S6.SS2.26.p2.8.m8.1.1.2">𝑥</ci><ci id="S6.SS2.26.p2.8.m8.1.1.4.cmml" xref="S6.SS2.26.p2.8.m8.1.1.4">𝑧</ci></apply><apply id="S6.SS2.26.p2.8.m8.1.1c.cmml" xref="S6.SS2.26.p2.8.m8.1.1"><apply id="S6.SS2.26.p2.8.m8.1.1.5.cmml" xref="S6.SS2.26.p2.8.m8.1.1.5"><csymbol cd="ambiguous" id="S6.SS2.26.p2.8.m8.1.1.5.1.cmml" xref="S6.SS2.26.p2.8.m8.1.1.5">subscript</csymbol><lt id="S6.SS2.26.p2.8.m8.1.1.5.2.cmml" xref="S6.SS2.26.p2.8.m8.1.1.5.2"></lt><ci id="S6.SS2.26.p2.8.m8.1.1.5.3.cmml" xref="S6.SS2.26.p2.8.m8.1.1.5.3">𝑋</ci></apply><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.26.p2.8.m8.1.1.4.cmml" id="S6.SS2.26.p2.8.m8.1.1d.cmml" xref="S6.SS2.26.p2.8.m8.1.1"></share><ci id="S6.SS2.26.p2.8.m8.1.1.6.cmml" xref="S6.SS2.26.p2.8.m8.1.1.6">𝑦</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.26.p2.8.m8.1c">x<_{X}z<_{X}y</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.26.p2.8.m8.1d">italic_x < start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT italic_z < start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT italic_y</annotation></semantics></math> such that <math alttext="\nu(z)=\nu" class="ltx_Math" display="inline" id="S6.SS2.26.p2.9.m9.1"><semantics id="S6.SS2.26.p2.9.m9.1a"><mrow id="S6.SS2.26.p2.9.m9.1.2" xref="S6.SS2.26.p2.9.m9.1.2.cmml"><mrow id="S6.SS2.26.p2.9.m9.1.2.2" xref="S6.SS2.26.p2.9.m9.1.2.2.cmml"><mi id="S6.SS2.26.p2.9.m9.1.2.2.2" xref="S6.SS2.26.p2.9.m9.1.2.2.2.cmml">ν</mi><mo id="S6.SS2.26.p2.9.m9.1.2.2.1" xref="S6.SS2.26.p2.9.m9.1.2.2.1.cmml">⁢</mo><mrow id="S6.SS2.26.p2.9.m9.1.2.2.3.2" xref="S6.SS2.26.p2.9.m9.1.2.2.cmml"><mo id="S6.SS2.26.p2.9.m9.1.2.2.3.2.1" stretchy="false" xref="S6.SS2.26.p2.9.m9.1.2.2.cmml">(</mo><mi id="S6.SS2.26.p2.9.m9.1.1" xref="S6.SS2.26.p2.9.m9.1.1.cmml">z</mi><mo id="S6.SS2.26.p2.9.m9.1.2.2.3.2.2" stretchy="false" xref="S6.SS2.26.p2.9.m9.1.2.2.cmml">)</mo></mrow></mrow><mo id="S6.SS2.26.p2.9.m9.1.2.1" xref="S6.SS2.26.p2.9.m9.1.2.1.cmml">=</mo><mi id="S6.SS2.26.p2.9.m9.1.2.3" xref="S6.SS2.26.p2.9.m9.1.2.3.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.26.p2.9.m9.1b"><apply id="S6.SS2.26.p2.9.m9.1.2.cmml" xref="S6.SS2.26.p2.9.m9.1.2"><eq id="S6.SS2.26.p2.9.m9.1.2.1.cmml" xref="S6.SS2.26.p2.9.m9.1.2.1"></eq><apply id="S6.SS2.26.p2.9.m9.1.2.2.cmml" xref="S6.SS2.26.p2.9.m9.1.2.2"><times id="S6.SS2.26.p2.9.m9.1.2.2.1.cmml" xref="S6.SS2.26.p2.9.m9.1.2.2.1"></times><ci id="S6.SS2.26.p2.9.m9.1.2.2.2.cmml" xref="S6.SS2.26.p2.9.m9.1.2.2.2">𝜈</ci><ci id="S6.SS2.26.p2.9.m9.1.1.cmml" xref="S6.SS2.26.p2.9.m9.1.1">𝑧</ci></apply><ci id="S6.SS2.26.p2.9.m9.1.2.3.cmml" xref="S6.SS2.26.p2.9.m9.1.2.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.26.p2.9.m9.1c">\nu(z)=\nu</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.26.p2.9.m9.1d">italic_ν ( italic_z ) = italic_ν</annotation></semantics></math> (and thus also <math alttext="\Delta(x,z)=\Delta(x,y)=\nu" class="ltx_Math" display="inline" id="S6.SS2.26.p2.10.m10.4"><semantics id="S6.SS2.26.p2.10.m10.4a"><mrow id="S6.SS2.26.p2.10.m10.4.5" xref="S6.SS2.26.p2.10.m10.4.5.cmml"><mrow id="S6.SS2.26.p2.10.m10.4.5.2" xref="S6.SS2.26.p2.10.m10.4.5.2.cmml"><mi id="S6.SS2.26.p2.10.m10.4.5.2.2" mathvariant="normal" xref="S6.SS2.26.p2.10.m10.4.5.2.2.cmml">Δ</mi><mo id="S6.SS2.26.p2.10.m10.4.5.2.1" xref="S6.SS2.26.p2.10.m10.4.5.2.1.cmml">⁢</mo><mrow id="S6.SS2.26.p2.10.m10.4.5.2.3.2" xref="S6.SS2.26.p2.10.m10.4.5.2.3.1.cmml"><mo id="S6.SS2.26.p2.10.m10.4.5.2.3.2.1" stretchy="false" xref="S6.SS2.26.p2.10.m10.4.5.2.3.1.cmml">(</mo><mi id="S6.SS2.26.p2.10.m10.1.1" xref="S6.SS2.26.p2.10.m10.1.1.cmml">x</mi><mo id="S6.SS2.26.p2.10.m10.4.5.2.3.2.2" xref="S6.SS2.26.p2.10.m10.4.5.2.3.1.cmml">,</mo><mi id="S6.SS2.26.p2.10.m10.2.2" xref="S6.SS2.26.p2.10.m10.2.2.cmml">z</mi><mo id="S6.SS2.26.p2.10.m10.4.5.2.3.2.3" stretchy="false" xref="S6.SS2.26.p2.10.m10.4.5.2.3.1.cmml">)</mo></mrow></mrow><mo id="S6.SS2.26.p2.10.m10.4.5.3" xref="S6.SS2.26.p2.10.m10.4.5.3.cmml">=</mo><mrow id="S6.SS2.26.p2.10.m10.4.5.4" xref="S6.SS2.26.p2.10.m10.4.5.4.cmml"><mi id="S6.SS2.26.p2.10.m10.4.5.4.2" mathvariant="normal" xref="S6.SS2.26.p2.10.m10.4.5.4.2.cmml">Δ</mi><mo id="S6.SS2.26.p2.10.m10.4.5.4.1" xref="S6.SS2.26.p2.10.m10.4.5.4.1.cmml">⁢</mo><mrow id="S6.SS2.26.p2.10.m10.4.5.4.3.2" xref="S6.SS2.26.p2.10.m10.4.5.4.3.1.cmml"><mo id="S6.SS2.26.p2.10.m10.4.5.4.3.2.1" stretchy="false" xref="S6.SS2.26.p2.10.m10.4.5.4.3.1.cmml">(</mo><mi id="S6.SS2.26.p2.10.m10.3.3" xref="S6.SS2.26.p2.10.m10.3.3.cmml">x</mi><mo id="S6.SS2.26.p2.10.m10.4.5.4.3.2.2" xref="S6.SS2.26.p2.10.m10.4.5.4.3.1.cmml">,</mo><mi id="S6.SS2.26.p2.10.m10.4.4" xref="S6.SS2.26.p2.10.m10.4.4.cmml">y</mi><mo id="S6.SS2.26.p2.10.m10.4.5.4.3.2.3" stretchy="false" xref="S6.SS2.26.p2.10.m10.4.5.4.3.1.cmml">)</mo></mrow></mrow><mo id="S6.SS2.26.p2.10.m10.4.5.5" xref="S6.SS2.26.p2.10.m10.4.5.5.cmml">=</mo><mi id="S6.SS2.26.p2.10.m10.4.5.6" xref="S6.SS2.26.p2.10.m10.4.5.6.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.26.p2.10.m10.4b"><apply id="S6.SS2.26.p2.10.m10.4.5.cmml" xref="S6.SS2.26.p2.10.m10.4.5"><and id="S6.SS2.26.p2.10.m10.4.5a.cmml" xref="S6.SS2.26.p2.10.m10.4.5"></and><apply id="S6.SS2.26.p2.10.m10.4.5b.cmml" xref="S6.SS2.26.p2.10.m10.4.5"><eq id="S6.SS2.26.p2.10.m10.4.5.3.cmml" xref="S6.SS2.26.p2.10.m10.4.5.3"></eq><apply id="S6.SS2.26.p2.10.m10.4.5.2.cmml" xref="S6.SS2.26.p2.10.m10.4.5.2"><times id="S6.SS2.26.p2.10.m10.4.5.2.1.cmml" xref="S6.SS2.26.p2.10.m10.4.5.2.1"></times><ci id="S6.SS2.26.p2.10.m10.4.5.2.2.cmml" xref="S6.SS2.26.p2.10.m10.4.5.2.2">Δ</ci><interval closure="open" id="S6.SS2.26.p2.10.m10.4.5.2.3.1.cmml" xref="S6.SS2.26.p2.10.m10.4.5.2.3.2"><ci id="S6.SS2.26.p2.10.m10.1.1.cmml" xref="S6.SS2.26.p2.10.m10.1.1">𝑥</ci><ci id="S6.SS2.26.p2.10.m10.2.2.cmml" xref="S6.SS2.26.p2.10.m10.2.2">𝑧</ci></interval></apply><apply id="S6.SS2.26.p2.10.m10.4.5.4.cmml" xref="S6.SS2.26.p2.10.m10.4.5.4"><times id="S6.SS2.26.p2.10.m10.4.5.4.1.cmml" xref="S6.SS2.26.p2.10.m10.4.5.4.1"></times><ci id="S6.SS2.26.p2.10.m10.4.5.4.2.cmml" xref="S6.SS2.26.p2.10.m10.4.5.4.2">Δ</ci><interval closure="open" id="S6.SS2.26.p2.10.m10.4.5.4.3.1.cmml" xref="S6.SS2.26.p2.10.m10.4.5.4.3.2"><ci id="S6.SS2.26.p2.10.m10.3.3.cmml" xref="S6.SS2.26.p2.10.m10.3.3">𝑥</ci><ci id="S6.SS2.26.p2.10.m10.4.4.cmml" xref="S6.SS2.26.p2.10.m10.4.4">𝑦</ci></interval></apply></apply><apply id="S6.SS2.26.p2.10.m10.4.5c.cmml" xref="S6.SS2.26.p2.10.m10.4.5"><eq id="S6.SS2.26.p2.10.m10.4.5.5.cmml" xref="S6.SS2.26.p2.10.m10.4.5.5"></eq><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.26.p2.10.m10.4.5.4.cmml" id="S6.SS2.26.p2.10.m10.4.5d.cmml" xref="S6.SS2.26.p2.10.m10.4.5"></share><ci id="S6.SS2.26.p2.10.m10.4.5.6.cmml" xref="S6.SS2.26.p2.10.m10.4.5.6">𝜈</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.26.p2.10.m10.4c">\Delta(x,z)=\Delta(x,y)=\nu</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.26.p2.10.m10.4d">roman_Δ ( italic_x , italic_z ) = roman_Δ ( italic_x , italic_y ) = italic_ν</annotation></semantics></math>). Then <math alttext="p\cup\{((c_{l},c_{r}),z)\}" class="ltx_Math" display="inline" id="S6.SS2.26.p2.11.m11.2"><semantics id="S6.SS2.26.p2.11.m11.2a"><mrow id="S6.SS2.26.p2.11.m11.2.2" xref="S6.SS2.26.p2.11.m11.2.2.cmml"><mi id="S6.SS2.26.p2.11.m11.2.2.3" xref="S6.SS2.26.p2.11.m11.2.2.3.cmml">p</mi><mo id="S6.SS2.26.p2.11.m11.2.2.2" xref="S6.SS2.26.p2.11.m11.2.2.2.cmml">∪</mo><mrow id="S6.SS2.26.p2.11.m11.2.2.1.1" xref="S6.SS2.26.p2.11.m11.2.2.1.2.cmml"><mo id="S6.SS2.26.p2.11.m11.2.2.1.1.2" stretchy="false" xref="S6.SS2.26.p2.11.m11.2.2.1.2.cmml">{</mo><mrow id="S6.SS2.26.p2.11.m11.2.2.1.1.1.1" xref="S6.SS2.26.p2.11.m11.2.2.1.1.1.2.cmml"><mo id="S6.SS2.26.p2.11.m11.2.2.1.1.1.1.2" stretchy="false" xref="S6.SS2.26.p2.11.m11.2.2.1.1.1.2.cmml">(</mo><mrow id="S6.SS2.26.p2.11.m11.2.2.1.1.1.1.1.2" xref="S6.SS2.26.p2.11.m11.2.2.1.1.1.1.1.3.cmml"><mo id="S6.SS2.26.p2.11.m11.2.2.1.1.1.1.1.2.3" stretchy="false" xref="S6.SS2.26.p2.11.m11.2.2.1.1.1.1.1.3.cmml">(</mo><msub id="S6.SS2.26.p2.11.m11.2.2.1.1.1.1.1.1.1" xref="S6.SS2.26.p2.11.m11.2.2.1.1.1.1.1.1.1.cmml"><mi id="S6.SS2.26.p2.11.m11.2.2.1.1.1.1.1.1.1.2" xref="S6.SS2.26.p2.11.m11.2.2.1.1.1.1.1.1.1.2.cmml">c</mi><mi id="S6.SS2.26.p2.11.m11.2.2.1.1.1.1.1.1.1.3" xref="S6.SS2.26.p2.11.m11.2.2.1.1.1.1.1.1.1.3.cmml">l</mi></msub><mo id="S6.SS2.26.p2.11.m11.2.2.1.1.1.1.1.2.4" xref="S6.SS2.26.p2.11.m11.2.2.1.1.1.1.1.3.cmml">,</mo><msub id="S6.SS2.26.p2.11.m11.2.2.1.1.1.1.1.2.2" xref="S6.SS2.26.p2.11.m11.2.2.1.1.1.1.1.2.2.cmml"><mi id="S6.SS2.26.p2.11.m11.2.2.1.1.1.1.1.2.2.2" xref="S6.SS2.26.p2.11.m11.2.2.1.1.1.1.1.2.2.2.cmml">c</mi><mi id="S6.SS2.26.p2.11.m11.2.2.1.1.1.1.1.2.2.3" xref="S6.SS2.26.p2.11.m11.2.2.1.1.1.1.1.2.2.3.cmml">r</mi></msub><mo id="S6.SS2.26.p2.11.m11.2.2.1.1.1.1.1.2.5" stretchy="false" xref="S6.SS2.26.p2.11.m11.2.2.1.1.1.1.1.3.cmml">)</mo></mrow><mo id="S6.SS2.26.p2.11.m11.2.2.1.1.1.1.3" xref="S6.SS2.26.p2.11.m11.2.2.1.1.1.2.cmml">,</mo><mi id="S6.SS2.26.p2.11.m11.1.1" xref="S6.SS2.26.p2.11.m11.1.1.cmml">z</mi><mo id="S6.SS2.26.p2.11.m11.2.2.1.1.1.1.4" stretchy="false" xref="S6.SS2.26.p2.11.m11.2.2.1.1.1.2.cmml">)</mo></mrow><mo id="S6.SS2.26.p2.11.m11.2.2.1.1.3" stretchy="false" xref="S6.SS2.26.p2.11.m11.2.2.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.26.p2.11.m11.2b"><apply id="S6.SS2.26.p2.11.m11.2.2.cmml" xref="S6.SS2.26.p2.11.m11.2.2"><union id="S6.SS2.26.p2.11.m11.2.2.2.cmml" xref="S6.SS2.26.p2.11.m11.2.2.2"></union><ci id="S6.SS2.26.p2.11.m11.2.2.3.cmml" xref="S6.SS2.26.p2.11.m11.2.2.3">𝑝</ci><set id="S6.SS2.26.p2.11.m11.2.2.1.2.cmml" xref="S6.SS2.26.p2.11.m11.2.2.1.1"><interval closure="open" id="S6.SS2.26.p2.11.m11.2.2.1.1.1.2.cmml" xref="S6.SS2.26.p2.11.m11.2.2.1.1.1.1"><interval closure="open" id="S6.SS2.26.p2.11.m11.2.2.1.1.1.1.1.3.cmml" xref="S6.SS2.26.p2.11.m11.2.2.1.1.1.1.1.2"><apply id="S6.SS2.26.p2.11.m11.2.2.1.1.1.1.1.1.1.cmml" xref="S6.SS2.26.p2.11.m11.2.2.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.26.p2.11.m11.2.2.1.1.1.1.1.1.1.1.cmml" xref="S6.SS2.26.p2.11.m11.2.2.1.1.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.26.p2.11.m11.2.2.1.1.1.1.1.1.1.2.cmml" xref="S6.SS2.26.p2.11.m11.2.2.1.1.1.1.1.1.1.2">𝑐</ci><ci id="S6.SS2.26.p2.11.m11.2.2.1.1.1.1.1.1.1.3.cmml" xref="S6.SS2.26.p2.11.m11.2.2.1.1.1.1.1.1.1.3">𝑙</ci></apply><apply id="S6.SS2.26.p2.11.m11.2.2.1.1.1.1.1.2.2.cmml" xref="S6.SS2.26.p2.11.m11.2.2.1.1.1.1.1.2.2"><csymbol cd="ambiguous" id="S6.SS2.26.p2.11.m11.2.2.1.1.1.1.1.2.2.1.cmml" xref="S6.SS2.26.p2.11.m11.2.2.1.1.1.1.1.2.2">subscript</csymbol><ci id="S6.SS2.26.p2.11.m11.2.2.1.1.1.1.1.2.2.2.cmml" xref="S6.SS2.26.p2.11.m11.2.2.1.1.1.1.1.2.2.2">𝑐</ci><ci id="S6.SS2.26.p2.11.m11.2.2.1.1.1.1.1.2.2.3.cmml" xref="S6.SS2.26.p2.11.m11.2.2.1.1.1.1.1.2.2.3">𝑟</ci></apply></interval><ci id="S6.SS2.26.p2.11.m11.1.1.cmml" xref="S6.SS2.26.p2.11.m11.1.1">𝑧</ci></interval></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.26.p2.11.m11.2c">p\cup\{((c_{l},c_{r}),z)\}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.26.p2.11.m11.2d">italic_p ∪ { ( ( italic_c start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT , italic_c start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) , italic_z ) }</annotation></semantics></math> is in <math alttext="P_{E}" class="ltx_Math" display="inline" id="S6.SS2.26.p2.12.m12.1"><semantics id="S6.SS2.26.p2.12.m12.1a"><msub id="S6.SS2.26.p2.12.m12.1.1" xref="S6.SS2.26.p2.12.m12.1.1.cmml"><mi id="S6.SS2.26.p2.12.m12.1.1.2" xref="S6.SS2.26.p2.12.m12.1.1.2.cmml">P</mi><mi id="S6.SS2.26.p2.12.m12.1.1.3" xref="S6.SS2.26.p2.12.m12.1.1.3.cmml">E</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.26.p2.12.m12.1b"><apply id="S6.SS2.26.p2.12.m12.1.1.cmml" xref="S6.SS2.26.p2.12.m12.1.1"><csymbol cd="ambiguous" id="S6.SS2.26.p2.12.m12.1.1.1.cmml" xref="S6.SS2.26.p2.12.m12.1.1">subscript</csymbol><ci id="S6.SS2.26.p2.12.m12.1.1.2.cmml" xref="S6.SS2.26.p2.12.m12.1.1.2">𝑃</ci><ci id="S6.SS2.26.p2.12.m12.1.1.3.cmml" xref="S6.SS2.26.p2.12.m12.1.1.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.26.p2.12.m12.1c">P_{E}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.26.p2.12.m12.1d">italic_P start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S6.SS2.27.p3"> <p class="ltx_p" id="S6.SS2.27.p3.13"><span class="ltx_text ltx_framed ltx_framed_underline" id="S6.SS2.27.p3.13.1">Case 2</span>: <math alttext="\nu(a_{r},b_{l})<\nu" class="ltx_Math" display="inline" id="S6.SS2.27.p3.1.m1.2"><semantics id="S6.SS2.27.p3.1.m1.2a"><mrow id="S6.SS2.27.p3.1.m1.2.2" xref="S6.SS2.27.p3.1.m1.2.2.cmml"><mrow id="S6.SS2.27.p3.1.m1.2.2.2" xref="S6.SS2.27.p3.1.m1.2.2.2.cmml"><mi id="S6.SS2.27.p3.1.m1.2.2.2.4" xref="S6.SS2.27.p3.1.m1.2.2.2.4.cmml">ν</mi><mo id="S6.SS2.27.p3.1.m1.2.2.2.3" xref="S6.SS2.27.p3.1.m1.2.2.2.3.cmml">⁢</mo><mrow id="S6.SS2.27.p3.1.m1.2.2.2.2.2" xref="S6.SS2.27.p3.1.m1.2.2.2.2.3.cmml"><mo id="S6.SS2.27.p3.1.m1.2.2.2.2.2.3" stretchy="false" xref="S6.SS2.27.p3.1.m1.2.2.2.2.3.cmml">(</mo><msub id="S6.SS2.27.p3.1.m1.1.1.1.1.1.1" xref="S6.SS2.27.p3.1.m1.1.1.1.1.1.1.cmml"><mi id="S6.SS2.27.p3.1.m1.1.1.1.1.1.1.2" xref="S6.SS2.27.p3.1.m1.1.1.1.1.1.1.2.cmml">a</mi><mi id="S6.SS2.27.p3.1.m1.1.1.1.1.1.1.3" xref="S6.SS2.27.p3.1.m1.1.1.1.1.1.1.3.cmml">r</mi></msub><mo id="S6.SS2.27.p3.1.m1.2.2.2.2.2.4" xref="S6.SS2.27.p3.1.m1.2.2.2.2.3.cmml">,</mo><msub id="S6.SS2.27.p3.1.m1.2.2.2.2.2.2" xref="S6.SS2.27.p3.1.m1.2.2.2.2.2.2.cmml"><mi id="S6.SS2.27.p3.1.m1.2.2.2.2.2.2.2" xref="S6.SS2.27.p3.1.m1.2.2.2.2.2.2.2.cmml">b</mi><mi id="S6.SS2.27.p3.1.m1.2.2.2.2.2.2.3" xref="S6.SS2.27.p3.1.m1.2.2.2.2.2.2.3.cmml">l</mi></msub><mo id="S6.SS2.27.p3.1.m1.2.2.2.2.2.5" stretchy="false" xref="S6.SS2.27.p3.1.m1.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.SS2.27.p3.1.m1.2.2.3" xref="S6.SS2.27.p3.1.m1.2.2.3.cmml"><</mo><mi id="S6.SS2.27.p3.1.m1.2.2.4" xref="S6.SS2.27.p3.1.m1.2.2.4.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.27.p3.1.m1.2b"><apply id="S6.SS2.27.p3.1.m1.2.2.cmml" xref="S6.SS2.27.p3.1.m1.2.2"><lt id="S6.SS2.27.p3.1.m1.2.2.3.cmml" xref="S6.SS2.27.p3.1.m1.2.2.3"></lt><apply id="S6.SS2.27.p3.1.m1.2.2.2.cmml" xref="S6.SS2.27.p3.1.m1.2.2.2"><times id="S6.SS2.27.p3.1.m1.2.2.2.3.cmml" xref="S6.SS2.27.p3.1.m1.2.2.2.3"></times><ci id="S6.SS2.27.p3.1.m1.2.2.2.4.cmml" xref="S6.SS2.27.p3.1.m1.2.2.2.4">𝜈</ci><interval closure="open" id="S6.SS2.27.p3.1.m1.2.2.2.2.3.cmml" xref="S6.SS2.27.p3.1.m1.2.2.2.2.2"><apply id="S6.SS2.27.p3.1.m1.1.1.1.1.1.1.cmml" xref="S6.SS2.27.p3.1.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.27.p3.1.m1.1.1.1.1.1.1.1.cmml" xref="S6.SS2.27.p3.1.m1.1.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.27.p3.1.m1.1.1.1.1.1.1.2.cmml" xref="S6.SS2.27.p3.1.m1.1.1.1.1.1.1.2">𝑎</ci><ci id="S6.SS2.27.p3.1.m1.1.1.1.1.1.1.3.cmml" xref="S6.SS2.27.p3.1.m1.1.1.1.1.1.1.3">𝑟</ci></apply><apply id="S6.SS2.27.p3.1.m1.2.2.2.2.2.2.cmml" xref="S6.SS2.27.p3.1.m1.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.27.p3.1.m1.2.2.2.2.2.2.1.cmml" xref="S6.SS2.27.p3.1.m1.2.2.2.2.2.2">subscript</csymbol><ci id="S6.SS2.27.p3.1.m1.2.2.2.2.2.2.2.cmml" xref="S6.SS2.27.p3.1.m1.2.2.2.2.2.2.2">𝑏</ci><ci id="S6.SS2.27.p3.1.m1.2.2.2.2.2.2.3.cmml" xref="S6.SS2.27.p3.1.m1.2.2.2.2.2.2.3">𝑙</ci></apply></interval></apply><ci id="S6.SS2.27.p3.1.m1.2.2.4.cmml" xref="S6.SS2.27.p3.1.m1.2.2.4">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.27.p3.1.m1.2c">\nu(a_{r},b_{l})<\nu</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.27.p3.1.m1.2d">italic_ν ( italic_a start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ) < italic_ν</annotation></semantics></math> and <math alttext="y" class="ltx_Math" display="inline" id="S6.SS2.27.p3.2.m2.1"><semantics id="S6.SS2.27.p3.2.m2.1a"><mi id="S6.SS2.27.p3.2.m2.1.1" xref="S6.SS2.27.p3.2.m2.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.27.p3.2.m2.1b"><ci id="S6.SS2.27.p3.2.m2.1.1.cmml" xref="S6.SS2.27.p3.2.m2.1.1">𝑦</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.27.p3.2.m2.1c">y</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.27.p3.2.m2.1d">italic_y</annotation></semantics></math> is the left endpoint of its complementary interval in <math alttext="X\setminus\nu" class="ltx_Math" display="inline" id="S6.SS2.27.p3.3.m3.1"><semantics id="S6.SS2.27.p3.3.m3.1a"><mrow id="S6.SS2.27.p3.3.m3.1.1" xref="S6.SS2.27.p3.3.m3.1.1.cmml"><mi id="S6.SS2.27.p3.3.m3.1.1.2" xref="S6.SS2.27.p3.3.m3.1.1.2.cmml">X</mi><mo id="S6.SS2.27.p3.3.m3.1.1.1" xref="S6.SS2.27.p3.3.m3.1.1.1.cmml">∖</mo><mi id="S6.SS2.27.p3.3.m3.1.1.3" xref="S6.SS2.27.p3.3.m3.1.1.3.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.27.p3.3.m3.1b"><apply id="S6.SS2.27.p3.3.m3.1.1.cmml" xref="S6.SS2.27.p3.3.m3.1.1"><setdiff id="S6.SS2.27.p3.3.m3.1.1.1.cmml" xref="S6.SS2.27.p3.3.m3.1.1.1"></setdiff><ci id="S6.SS2.27.p3.3.m3.1.1.2.cmml" xref="S6.SS2.27.p3.3.m3.1.1.2">𝑋</ci><ci id="S6.SS2.27.p3.3.m3.1.1.3.cmml" xref="S6.SS2.27.p3.3.m3.1.1.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.27.p3.3.m3.1c">X\setminus\nu</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.27.p3.3.m3.1d">italic_X ∖ italic_ν</annotation></semantics></math>. Note that in this case it must be that <math alttext="\nu=\nu(y)=\nu(b_{l})" class="ltx_Math" display="inline" id="S6.SS2.27.p3.4.m4.2"><semantics id="S6.SS2.27.p3.4.m4.2a"><mrow id="S6.SS2.27.p3.4.m4.2.2" xref="S6.SS2.27.p3.4.m4.2.2.cmml"><mi id="S6.SS2.27.p3.4.m4.2.2.3" xref="S6.SS2.27.p3.4.m4.2.2.3.cmml">ν</mi><mo id="S6.SS2.27.p3.4.m4.2.2.4" xref="S6.SS2.27.p3.4.m4.2.2.4.cmml">=</mo><mrow id="S6.SS2.27.p3.4.m4.2.2.5" xref="S6.SS2.27.p3.4.m4.2.2.5.cmml"><mi id="S6.SS2.27.p3.4.m4.2.2.5.2" xref="S6.SS2.27.p3.4.m4.2.2.5.2.cmml">ν</mi><mo id="S6.SS2.27.p3.4.m4.2.2.5.1" xref="S6.SS2.27.p3.4.m4.2.2.5.1.cmml">⁢</mo><mrow id="S6.SS2.27.p3.4.m4.2.2.5.3.2" xref="S6.SS2.27.p3.4.m4.2.2.5.cmml"><mo id="S6.SS2.27.p3.4.m4.2.2.5.3.2.1" stretchy="false" xref="S6.SS2.27.p3.4.m4.2.2.5.cmml">(</mo><mi id="S6.SS2.27.p3.4.m4.1.1" xref="S6.SS2.27.p3.4.m4.1.1.cmml">y</mi><mo id="S6.SS2.27.p3.4.m4.2.2.5.3.2.2" stretchy="false" xref="S6.SS2.27.p3.4.m4.2.2.5.cmml">)</mo></mrow></mrow><mo id="S6.SS2.27.p3.4.m4.2.2.6" xref="S6.SS2.27.p3.4.m4.2.2.6.cmml">=</mo><mrow id="S6.SS2.27.p3.4.m4.2.2.1" xref="S6.SS2.27.p3.4.m4.2.2.1.cmml"><mi id="S6.SS2.27.p3.4.m4.2.2.1.3" xref="S6.SS2.27.p3.4.m4.2.2.1.3.cmml">ν</mi><mo id="S6.SS2.27.p3.4.m4.2.2.1.2" xref="S6.SS2.27.p3.4.m4.2.2.1.2.cmml">⁢</mo><mrow id="S6.SS2.27.p3.4.m4.2.2.1.1.1" xref="S6.SS2.27.p3.4.m4.2.2.1.1.1.1.cmml"><mo id="S6.SS2.27.p3.4.m4.2.2.1.1.1.2" stretchy="false" xref="S6.SS2.27.p3.4.m4.2.2.1.1.1.1.cmml">(</mo><msub id="S6.SS2.27.p3.4.m4.2.2.1.1.1.1" xref="S6.SS2.27.p3.4.m4.2.2.1.1.1.1.cmml"><mi id="S6.SS2.27.p3.4.m4.2.2.1.1.1.1.2" xref="S6.SS2.27.p3.4.m4.2.2.1.1.1.1.2.cmml">b</mi><mi id="S6.SS2.27.p3.4.m4.2.2.1.1.1.1.3" xref="S6.SS2.27.p3.4.m4.2.2.1.1.1.1.3.cmml">l</mi></msub><mo id="S6.SS2.27.p3.4.m4.2.2.1.1.1.3" stretchy="false" xref="S6.SS2.27.p3.4.m4.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.27.p3.4.m4.2b"><apply id="S6.SS2.27.p3.4.m4.2.2.cmml" xref="S6.SS2.27.p3.4.m4.2.2"><and id="S6.SS2.27.p3.4.m4.2.2a.cmml" xref="S6.SS2.27.p3.4.m4.2.2"></and><apply id="S6.SS2.27.p3.4.m4.2.2b.cmml" xref="S6.SS2.27.p3.4.m4.2.2"><eq id="S6.SS2.27.p3.4.m4.2.2.4.cmml" xref="S6.SS2.27.p3.4.m4.2.2.4"></eq><ci id="S6.SS2.27.p3.4.m4.2.2.3.cmml" xref="S6.SS2.27.p3.4.m4.2.2.3">𝜈</ci><apply id="S6.SS2.27.p3.4.m4.2.2.5.cmml" xref="S6.SS2.27.p3.4.m4.2.2.5"><times id="S6.SS2.27.p3.4.m4.2.2.5.1.cmml" xref="S6.SS2.27.p3.4.m4.2.2.5.1"></times><ci id="S6.SS2.27.p3.4.m4.2.2.5.2.cmml" xref="S6.SS2.27.p3.4.m4.2.2.5.2">𝜈</ci><ci id="S6.SS2.27.p3.4.m4.1.1.cmml" xref="S6.SS2.27.p3.4.m4.1.1">𝑦</ci></apply></apply><apply id="S6.SS2.27.p3.4.m4.2.2c.cmml" xref="S6.SS2.27.p3.4.m4.2.2"><eq id="S6.SS2.27.p3.4.m4.2.2.6.cmml" xref="S6.SS2.27.p3.4.m4.2.2.6"></eq><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.27.p3.4.m4.2.2.5.cmml" id="S6.SS2.27.p3.4.m4.2.2d.cmml" xref="S6.SS2.27.p3.4.m4.2.2"></share><apply id="S6.SS2.27.p3.4.m4.2.2.1.cmml" xref="S6.SS2.27.p3.4.m4.2.2.1"><times id="S6.SS2.27.p3.4.m4.2.2.1.2.cmml" xref="S6.SS2.27.p3.4.m4.2.2.1.2"></times><ci id="S6.SS2.27.p3.4.m4.2.2.1.3.cmml" xref="S6.SS2.27.p3.4.m4.2.2.1.3">𝜈</ci><apply id="S6.SS2.27.p3.4.m4.2.2.1.1.1.1.cmml" xref="S6.SS2.27.p3.4.m4.2.2.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.27.p3.4.m4.2.2.1.1.1.1.1.cmml" xref="S6.SS2.27.p3.4.m4.2.2.1.1.1">subscript</csymbol><ci id="S6.SS2.27.p3.4.m4.2.2.1.1.1.1.2.cmml" xref="S6.SS2.27.p3.4.m4.2.2.1.1.1.1.2">𝑏</ci><ci id="S6.SS2.27.p3.4.m4.2.2.1.1.1.1.3.cmml" xref="S6.SS2.27.p3.4.m4.2.2.1.1.1.1.3">𝑙</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.27.p3.4.m4.2c">\nu=\nu(y)=\nu(b_{l})</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.27.p3.4.m4.2d">italic_ν = italic_ν ( italic_y ) = italic_ν ( italic_b start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT )</annotation></semantics></math>. Using elementarity one finds <math alttext="c_{l}\leq_{A}c" class="ltx_Math" display="inline" id="S6.SS2.27.p3.5.m5.1"><semantics id="S6.SS2.27.p3.5.m5.1a"><mrow id="S6.SS2.27.p3.5.m5.1.1" xref="S6.SS2.27.p3.5.m5.1.1.cmml"><msub id="S6.SS2.27.p3.5.m5.1.1.2" xref="S6.SS2.27.p3.5.m5.1.1.2.cmml"><mi id="S6.SS2.27.p3.5.m5.1.1.2.2" xref="S6.SS2.27.p3.5.m5.1.1.2.2.cmml">c</mi><mi id="S6.SS2.27.p3.5.m5.1.1.2.3" xref="S6.SS2.27.p3.5.m5.1.1.2.3.cmml">l</mi></msub><msub id="S6.SS2.27.p3.5.m5.1.1.1" xref="S6.SS2.27.p3.5.m5.1.1.1.cmml"><mo id="S6.SS2.27.p3.5.m5.1.1.1.2" xref="S6.SS2.27.p3.5.m5.1.1.1.2.cmml">≤</mo><mi id="S6.SS2.27.p3.5.m5.1.1.1.3" xref="S6.SS2.27.p3.5.m5.1.1.1.3.cmml">A</mi></msub><mi id="S6.SS2.27.p3.5.m5.1.1.3" xref="S6.SS2.27.p3.5.m5.1.1.3.cmml">c</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.27.p3.5.m5.1b"><apply id="S6.SS2.27.p3.5.m5.1.1.cmml" xref="S6.SS2.27.p3.5.m5.1.1"><apply id="S6.SS2.27.p3.5.m5.1.1.1.cmml" xref="S6.SS2.27.p3.5.m5.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.27.p3.5.m5.1.1.1.1.cmml" xref="S6.SS2.27.p3.5.m5.1.1.1">subscript</csymbol><leq id="S6.SS2.27.p3.5.m5.1.1.1.2.cmml" xref="S6.SS2.27.p3.5.m5.1.1.1.2"></leq><ci id="S6.SS2.27.p3.5.m5.1.1.1.3.cmml" xref="S6.SS2.27.p3.5.m5.1.1.1.3">𝐴</ci></apply><apply id="S6.SS2.27.p3.5.m5.1.1.2.cmml" xref="S6.SS2.27.p3.5.m5.1.1.2"><csymbol cd="ambiguous" id="S6.SS2.27.p3.5.m5.1.1.2.1.cmml" xref="S6.SS2.27.p3.5.m5.1.1.2">subscript</csymbol><ci id="S6.SS2.27.p3.5.m5.1.1.2.2.cmml" xref="S6.SS2.27.p3.5.m5.1.1.2.2">𝑐</ci><ci id="S6.SS2.27.p3.5.m5.1.1.2.3.cmml" xref="S6.SS2.27.p3.5.m5.1.1.2.3">𝑙</ci></apply><ci id="S6.SS2.27.p3.5.m5.1.1.3.cmml" xref="S6.SS2.27.p3.5.m5.1.1.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.27.p3.5.m5.1c">c_{l}\leq_{A}c</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.27.p3.5.m5.1d">italic_c start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ≤ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_c</annotation></semantics></math> in the complementary interval of <math alttext="A\setminus\nu" class="ltx_Math" display="inline" id="S6.SS2.27.p3.6.m6.1"><semantics id="S6.SS2.27.p3.6.m6.1a"><mrow id="S6.SS2.27.p3.6.m6.1.1" xref="S6.SS2.27.p3.6.m6.1.1.cmml"><mi id="S6.SS2.27.p3.6.m6.1.1.2" xref="S6.SS2.27.p3.6.m6.1.1.2.cmml">A</mi><mo id="S6.SS2.27.p3.6.m6.1.1.1" xref="S6.SS2.27.p3.6.m6.1.1.1.cmml">∖</mo><mi id="S6.SS2.27.p3.6.m6.1.1.3" xref="S6.SS2.27.p3.6.m6.1.1.3.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.27.p3.6.m6.1b"><apply id="S6.SS2.27.p3.6.m6.1.1.cmml" xref="S6.SS2.27.p3.6.m6.1.1"><setdiff id="S6.SS2.27.p3.6.m6.1.1.1.cmml" xref="S6.SS2.27.p3.6.m6.1.1.1"></setdiff><ci id="S6.SS2.27.p3.6.m6.1.1.2.cmml" xref="S6.SS2.27.p3.6.m6.1.1.2">𝐴</ci><ci id="S6.SS2.27.p3.6.m6.1.1.3.cmml" xref="S6.SS2.27.p3.6.m6.1.1.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.27.p3.6.m6.1c">A\setminus\nu</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.27.p3.6.m6.1d">italic_A ∖ italic_ν</annotation></semantics></math> of <math alttext="b_{l}" class="ltx_Math" display="inline" id="S6.SS2.27.p3.7.m7.1"><semantics id="S6.SS2.27.p3.7.m7.1a"><msub id="S6.SS2.27.p3.7.m7.1.1" xref="S6.SS2.27.p3.7.m7.1.1.cmml"><mi id="S6.SS2.27.p3.7.m7.1.1.2" xref="S6.SS2.27.p3.7.m7.1.1.2.cmml">b</mi><mi id="S6.SS2.27.p3.7.m7.1.1.3" xref="S6.SS2.27.p3.7.m7.1.1.3.cmml">l</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.27.p3.7.m7.1b"><apply id="S6.SS2.27.p3.7.m7.1.1.cmml" xref="S6.SS2.27.p3.7.m7.1.1"><csymbol cd="ambiguous" id="S6.SS2.27.p3.7.m7.1.1.1.cmml" xref="S6.SS2.27.p3.7.m7.1.1">subscript</csymbol><ci id="S6.SS2.27.p3.7.m7.1.1.2.cmml" xref="S6.SS2.27.p3.7.m7.1.1.2">𝑏</ci><ci id="S6.SS2.27.p3.7.m7.1.1.3.cmml" xref="S6.SS2.27.p3.7.m7.1.1.3">𝑙</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.27.p3.7.m7.1c">b_{l}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.27.p3.7.m7.1d">italic_b start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="\nu(c_{l})=\nu" class="ltx_Math" display="inline" id="S6.SS2.27.p3.8.m8.1"><semantics id="S6.SS2.27.p3.8.m8.1a"><mrow id="S6.SS2.27.p3.8.m8.1.1" xref="S6.SS2.27.p3.8.m8.1.1.cmml"><mrow id="S6.SS2.27.p3.8.m8.1.1.1" xref="S6.SS2.27.p3.8.m8.1.1.1.cmml"><mi id="S6.SS2.27.p3.8.m8.1.1.1.3" xref="S6.SS2.27.p3.8.m8.1.1.1.3.cmml">ν</mi><mo id="S6.SS2.27.p3.8.m8.1.1.1.2" xref="S6.SS2.27.p3.8.m8.1.1.1.2.cmml">⁢</mo><mrow id="S6.SS2.27.p3.8.m8.1.1.1.1.1" xref="S6.SS2.27.p3.8.m8.1.1.1.1.1.1.cmml"><mo id="S6.SS2.27.p3.8.m8.1.1.1.1.1.2" stretchy="false" xref="S6.SS2.27.p3.8.m8.1.1.1.1.1.1.cmml">(</mo><msub id="S6.SS2.27.p3.8.m8.1.1.1.1.1.1" xref="S6.SS2.27.p3.8.m8.1.1.1.1.1.1.cmml"><mi id="S6.SS2.27.p3.8.m8.1.1.1.1.1.1.2" xref="S6.SS2.27.p3.8.m8.1.1.1.1.1.1.2.cmml">c</mi><mi id="S6.SS2.27.p3.8.m8.1.1.1.1.1.1.3" xref="S6.SS2.27.p3.8.m8.1.1.1.1.1.1.3.cmml">l</mi></msub><mo id="S6.SS2.27.p3.8.m8.1.1.1.1.1.3" stretchy="false" xref="S6.SS2.27.p3.8.m8.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.SS2.27.p3.8.m8.1.1.2" xref="S6.SS2.27.p3.8.m8.1.1.2.cmml">=</mo><mi id="S6.SS2.27.p3.8.m8.1.1.3" xref="S6.SS2.27.p3.8.m8.1.1.3.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.27.p3.8.m8.1b"><apply id="S6.SS2.27.p3.8.m8.1.1.cmml" xref="S6.SS2.27.p3.8.m8.1.1"><eq id="S6.SS2.27.p3.8.m8.1.1.2.cmml" xref="S6.SS2.27.p3.8.m8.1.1.2"></eq><apply id="S6.SS2.27.p3.8.m8.1.1.1.cmml" xref="S6.SS2.27.p3.8.m8.1.1.1"><times id="S6.SS2.27.p3.8.m8.1.1.1.2.cmml" xref="S6.SS2.27.p3.8.m8.1.1.1.2"></times><ci id="S6.SS2.27.p3.8.m8.1.1.1.3.cmml" xref="S6.SS2.27.p3.8.m8.1.1.1.3">𝜈</ci><apply id="S6.SS2.27.p3.8.m8.1.1.1.1.1.1.cmml" xref="S6.SS2.27.p3.8.m8.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.27.p3.8.m8.1.1.1.1.1.1.1.cmml" xref="S6.SS2.27.p3.8.m8.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.27.p3.8.m8.1.1.1.1.1.1.2.cmml" xref="S6.SS2.27.p3.8.m8.1.1.1.1.1.1.2">𝑐</ci><ci id="S6.SS2.27.p3.8.m8.1.1.1.1.1.1.3.cmml" xref="S6.SS2.27.p3.8.m8.1.1.1.1.1.1.3">𝑙</ci></apply></apply><ci id="S6.SS2.27.p3.8.m8.1.1.3.cmml" xref="S6.SS2.27.p3.8.m8.1.1.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.27.p3.8.m8.1c">\nu(c_{l})=\nu</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.27.p3.8.m8.1d">italic_ν ( italic_c start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ) = italic_ν</annotation></semantics></math>. We claim that <math alttext="p^{\prime}:=(p\setminus\{((b_{l},b_{r}),y)\})\cup\{((c_{l},b_{r}),y)\}" class="ltx_Math" display="inline" id="S6.SS2.27.p3.9.m9.4"><semantics id="S6.SS2.27.p3.9.m9.4a"><mrow id="S6.SS2.27.p3.9.m9.4.4" xref="S6.SS2.27.p3.9.m9.4.4.cmml"><msup id="S6.SS2.27.p3.9.m9.4.4.4" xref="S6.SS2.27.p3.9.m9.4.4.4.cmml"><mi id="S6.SS2.27.p3.9.m9.4.4.4.2" xref="S6.SS2.27.p3.9.m9.4.4.4.2.cmml">p</mi><mo id="S6.SS2.27.p3.9.m9.4.4.4.3" xref="S6.SS2.27.p3.9.m9.4.4.4.3.cmml">′</mo></msup><mo id="S6.SS2.27.p3.9.m9.4.4.3" lspace="0.278em" rspace="0.278em" xref="S6.SS2.27.p3.9.m9.4.4.3.cmml">:=</mo><mrow id="S6.SS2.27.p3.9.m9.4.4.2" xref="S6.SS2.27.p3.9.m9.4.4.2.cmml"><mrow id="S6.SS2.27.p3.9.m9.3.3.1.1.1" xref="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.cmml"><mo id="S6.SS2.27.p3.9.m9.3.3.1.1.1.2" stretchy="false" xref="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.cmml">(</mo><mrow id="S6.SS2.27.p3.9.m9.3.3.1.1.1.1" xref="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.cmml"><mi id="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.3" xref="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.3.cmml">p</mi><mo id="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.2" xref="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.2.cmml">∖</mo><mrow id="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.1.1" xref="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.1.2.cmml"><mo id="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.1.1.2" stretchy="false" xref="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.1.2.cmml">{</mo><mrow id="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.1.1.1.1" xref="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.1.1.1.2.cmml"><mo id="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.1.1.1.2.cmml">(</mo><mrow id="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.1.1.1.1.1.2" xref="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.1.1.1.1.1.3.cmml"><mo id="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.1.1.1.1.1.2.3" stretchy="false" xref="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.1.1.1.1.1.3.cmml">(</mo><msub id="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.1.1.1.1.1.1.1" xref="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.1.1.1.1.1.1.1.2" xref="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.1.1.1.1.1.1.1.2.cmml">b</mi><mi id="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.1.1.1.1.1.1.1.3" xref="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.1.1.1.1.1.1.1.3.cmml">l</mi></msub><mo id="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.1.1.1.1.1.2.4" xref="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.1.1.1.1.1.3.cmml">,</mo><msub id="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.1.1.1.1.1.2.2" xref="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.1.1.1.1.1.2.2.cmml"><mi id="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.1.1.1.1.1.2.2.2" xref="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.1.1.1.1.1.2.2.2.cmml">b</mi><mi id="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.1.1.1.1.1.2.2.3" xref="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.1.1.1.1.1.2.2.3.cmml">r</mi></msub><mo id="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.1.1.1.1.1.2.5" stretchy="false" xref="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.1.1.1.1.1.3.cmml">)</mo></mrow><mo id="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.1.1.1.1.3" xref="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.1.1.1.2.cmml">,</mo><mi id="S6.SS2.27.p3.9.m9.1.1" xref="S6.SS2.27.p3.9.m9.1.1.cmml">y</mi><mo id="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.1.1.1.1.4" stretchy="false" xref="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.1.1.1.2.cmml">)</mo></mrow><mo id="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.1.1.3" stretchy="false" xref="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.1.2.cmml">}</mo></mrow></mrow><mo id="S6.SS2.27.p3.9.m9.3.3.1.1.1.3" stretchy="false" xref="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.cmml">)</mo></mrow><mo id="S6.SS2.27.p3.9.m9.4.4.2.3" xref="S6.SS2.27.p3.9.m9.4.4.2.3.cmml">∪</mo><mrow id="S6.SS2.27.p3.9.m9.4.4.2.2.1" xref="S6.SS2.27.p3.9.m9.4.4.2.2.2.cmml"><mo id="S6.SS2.27.p3.9.m9.4.4.2.2.1.2" stretchy="false" xref="S6.SS2.27.p3.9.m9.4.4.2.2.2.cmml">{</mo><mrow id="S6.SS2.27.p3.9.m9.4.4.2.2.1.1.1" xref="S6.SS2.27.p3.9.m9.4.4.2.2.1.1.2.cmml"><mo id="S6.SS2.27.p3.9.m9.4.4.2.2.1.1.1.2" stretchy="false" xref="S6.SS2.27.p3.9.m9.4.4.2.2.1.1.2.cmml">(</mo><mrow id="S6.SS2.27.p3.9.m9.4.4.2.2.1.1.1.1.2" xref="S6.SS2.27.p3.9.m9.4.4.2.2.1.1.1.1.3.cmml"><mo id="S6.SS2.27.p3.9.m9.4.4.2.2.1.1.1.1.2.3" stretchy="false" xref="S6.SS2.27.p3.9.m9.4.4.2.2.1.1.1.1.3.cmml">(</mo><msub id="S6.SS2.27.p3.9.m9.4.4.2.2.1.1.1.1.1.1" xref="S6.SS2.27.p3.9.m9.4.4.2.2.1.1.1.1.1.1.cmml"><mi id="S6.SS2.27.p3.9.m9.4.4.2.2.1.1.1.1.1.1.2" xref="S6.SS2.27.p3.9.m9.4.4.2.2.1.1.1.1.1.1.2.cmml">c</mi><mi id="S6.SS2.27.p3.9.m9.4.4.2.2.1.1.1.1.1.1.3" xref="S6.SS2.27.p3.9.m9.4.4.2.2.1.1.1.1.1.1.3.cmml">l</mi></msub><mo id="S6.SS2.27.p3.9.m9.4.4.2.2.1.1.1.1.2.4" xref="S6.SS2.27.p3.9.m9.4.4.2.2.1.1.1.1.3.cmml">,</mo><msub id="S6.SS2.27.p3.9.m9.4.4.2.2.1.1.1.1.2.2" xref="S6.SS2.27.p3.9.m9.4.4.2.2.1.1.1.1.2.2.cmml"><mi id="S6.SS2.27.p3.9.m9.4.4.2.2.1.1.1.1.2.2.2" xref="S6.SS2.27.p3.9.m9.4.4.2.2.1.1.1.1.2.2.2.cmml">b</mi><mi id="S6.SS2.27.p3.9.m9.4.4.2.2.1.1.1.1.2.2.3" xref="S6.SS2.27.p3.9.m9.4.4.2.2.1.1.1.1.2.2.3.cmml">r</mi></msub><mo id="S6.SS2.27.p3.9.m9.4.4.2.2.1.1.1.1.2.5" stretchy="false" xref="S6.SS2.27.p3.9.m9.4.4.2.2.1.1.1.1.3.cmml">)</mo></mrow><mo id="S6.SS2.27.p3.9.m9.4.4.2.2.1.1.1.3" xref="S6.SS2.27.p3.9.m9.4.4.2.2.1.1.2.cmml">,</mo><mi id="S6.SS2.27.p3.9.m9.2.2" xref="S6.SS2.27.p3.9.m9.2.2.cmml">y</mi><mo id="S6.SS2.27.p3.9.m9.4.4.2.2.1.1.1.4" stretchy="false" xref="S6.SS2.27.p3.9.m9.4.4.2.2.1.1.2.cmml">)</mo></mrow><mo id="S6.SS2.27.p3.9.m9.4.4.2.2.1.3" stretchy="false" xref="S6.SS2.27.p3.9.m9.4.4.2.2.2.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.27.p3.9.m9.4b"><apply id="S6.SS2.27.p3.9.m9.4.4.cmml" xref="S6.SS2.27.p3.9.m9.4.4"><csymbol cd="latexml" id="S6.SS2.27.p3.9.m9.4.4.3.cmml" xref="S6.SS2.27.p3.9.m9.4.4.3">assign</csymbol><apply id="S6.SS2.27.p3.9.m9.4.4.4.cmml" xref="S6.SS2.27.p3.9.m9.4.4.4"><csymbol cd="ambiguous" id="S6.SS2.27.p3.9.m9.4.4.4.1.cmml" xref="S6.SS2.27.p3.9.m9.4.4.4">superscript</csymbol><ci id="S6.SS2.27.p3.9.m9.4.4.4.2.cmml" xref="S6.SS2.27.p3.9.m9.4.4.4.2">𝑝</ci><ci id="S6.SS2.27.p3.9.m9.4.4.4.3.cmml" xref="S6.SS2.27.p3.9.m9.4.4.4.3">′</ci></apply><apply id="S6.SS2.27.p3.9.m9.4.4.2.cmml" xref="S6.SS2.27.p3.9.m9.4.4.2"><union id="S6.SS2.27.p3.9.m9.4.4.2.3.cmml" xref="S6.SS2.27.p3.9.m9.4.4.2.3"></union><apply id="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.cmml" xref="S6.SS2.27.p3.9.m9.3.3.1.1.1"><setdiff id="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.2.cmml" xref="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.2"></setdiff><ci id="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.3.cmml" xref="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.3">𝑝</ci><set id="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.1.2.cmml" xref="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.1.1"><interval closure="open" id="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.1.1.1.2.cmml" xref="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.1.1.1.1"><interval closure="open" id="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.1.1.1.1.1.3.cmml" xref="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.1.1.1.1.1.2"><apply id="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.1.1.1.1.1.1.1.2.cmml" xref="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.1.1.1.1.1.1.1.2">𝑏</ci><ci id="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.1.1.1.1.1.1.1.3.cmml" xref="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.1.1.1.1.1.1.1.3">𝑙</ci></apply><apply id="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.1.1.1.1.1.2.2.cmml" xref="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.1.1.1.1.1.2.2"><csymbol cd="ambiguous" id="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.1.1.1.1.1.2.2.1.cmml" xref="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.1.1.1.1.1.2.2">subscript</csymbol><ci id="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.1.1.1.1.1.2.2.2.cmml" xref="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.1.1.1.1.1.2.2.2">𝑏</ci><ci id="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.1.1.1.1.1.2.2.3.cmml" xref="S6.SS2.27.p3.9.m9.3.3.1.1.1.1.1.1.1.1.1.2.2.3">𝑟</ci></apply></interval><ci id="S6.SS2.27.p3.9.m9.1.1.cmml" xref="S6.SS2.27.p3.9.m9.1.1">𝑦</ci></interval></set></apply><set id="S6.SS2.27.p3.9.m9.4.4.2.2.2.cmml" xref="S6.SS2.27.p3.9.m9.4.4.2.2.1"><interval closure="open" id="S6.SS2.27.p3.9.m9.4.4.2.2.1.1.2.cmml" xref="S6.SS2.27.p3.9.m9.4.4.2.2.1.1.1"><interval closure="open" id="S6.SS2.27.p3.9.m9.4.4.2.2.1.1.1.1.3.cmml" xref="S6.SS2.27.p3.9.m9.4.4.2.2.1.1.1.1.2"><apply id="S6.SS2.27.p3.9.m9.4.4.2.2.1.1.1.1.1.1.cmml" xref="S6.SS2.27.p3.9.m9.4.4.2.2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.27.p3.9.m9.4.4.2.2.1.1.1.1.1.1.1.cmml" xref="S6.SS2.27.p3.9.m9.4.4.2.2.1.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.27.p3.9.m9.4.4.2.2.1.1.1.1.1.1.2.cmml" xref="S6.SS2.27.p3.9.m9.4.4.2.2.1.1.1.1.1.1.2">𝑐</ci><ci id="S6.SS2.27.p3.9.m9.4.4.2.2.1.1.1.1.1.1.3.cmml" xref="S6.SS2.27.p3.9.m9.4.4.2.2.1.1.1.1.1.1.3">𝑙</ci></apply><apply id="S6.SS2.27.p3.9.m9.4.4.2.2.1.1.1.1.2.2.cmml" xref="S6.SS2.27.p3.9.m9.4.4.2.2.1.1.1.1.2.2"><csymbol cd="ambiguous" id="S6.SS2.27.p3.9.m9.4.4.2.2.1.1.1.1.2.2.1.cmml" xref="S6.SS2.27.p3.9.m9.4.4.2.2.1.1.1.1.2.2">subscript</csymbol><ci id="S6.SS2.27.p3.9.m9.4.4.2.2.1.1.1.1.2.2.2.cmml" xref="S6.SS2.27.p3.9.m9.4.4.2.2.1.1.1.1.2.2.2">𝑏</ci><ci id="S6.SS2.27.p3.9.m9.4.4.2.2.1.1.1.1.2.2.3.cmml" xref="S6.SS2.27.p3.9.m9.4.4.2.2.1.1.1.1.2.2.3">𝑟</ci></apply></interval><ci id="S6.SS2.27.p3.9.m9.2.2.cmml" xref="S6.SS2.27.p3.9.m9.2.2">𝑦</ci></interval></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.27.p3.9.m9.4c">p^{\prime}:=(p\setminus\{((b_{l},b_{r}),y)\})\cup\{((c_{l},b_{r}),y)\}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.27.p3.9.m9.4d">italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT := ( italic_p ∖ { ( ( italic_b start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) , italic_y ) } ) ∪ { ( ( italic_c start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) , italic_y ) }</annotation></semantics></math> is in <math alttext="P_{E}" class="ltx_Math" display="inline" id="S6.SS2.27.p3.10.m10.1"><semantics id="S6.SS2.27.p3.10.m10.1a"><msub id="S6.SS2.27.p3.10.m10.1.1" xref="S6.SS2.27.p3.10.m10.1.1.cmml"><mi id="S6.SS2.27.p3.10.m10.1.1.2" xref="S6.SS2.27.p3.10.m10.1.1.2.cmml">P</mi><mi id="S6.SS2.27.p3.10.m10.1.1.3" xref="S6.SS2.27.p3.10.m10.1.1.3.cmml">E</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.27.p3.10.m10.1b"><apply id="S6.SS2.27.p3.10.m10.1.1.cmml" xref="S6.SS2.27.p3.10.m10.1.1"><csymbol cd="ambiguous" id="S6.SS2.27.p3.10.m10.1.1.1.cmml" xref="S6.SS2.27.p3.10.m10.1.1">subscript</csymbol><ci id="S6.SS2.27.p3.10.m10.1.1.2.cmml" xref="S6.SS2.27.p3.10.m10.1.1.2">𝑃</ci><ci id="S6.SS2.27.p3.10.m10.1.1.3.cmml" xref="S6.SS2.27.p3.10.m10.1.1.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.27.p3.10.m10.1c">P_{E}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.27.p3.10.m10.1d">italic_P start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT</annotation></semantics></math>. <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.Thmtheorem9" title="Definition 6.9. ‣ 6.2. Forcing epimorphisms ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">6.9</span></a> <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.I4.i3" title="Item (iii) ‣ Definition 6.9. ‣ 6.2. Forcing epimorphisms ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">(iii)</span></a> is trivially satisfied since <math alttext="y" class="ltx_Math" display="inline" id="S6.SS2.27.p3.11.m11.1"><semantics id="S6.SS2.27.p3.11.m11.1a"><mi id="S6.SS2.27.p3.11.m11.1.1" xref="S6.SS2.27.p3.11.m11.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.27.p3.11.m11.1b"><ci id="S6.SS2.27.p3.11.m11.1.1.cmml" xref="S6.SS2.27.p3.11.m11.1.1">𝑦</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.27.p3.11.m11.1c">y</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.27.p3.11.m11.1d">italic_y</annotation></semantics></math> is the left endpoint of its complementary interval of <math alttext="A\setminus\nu" class="ltx_Math" display="inline" id="S6.SS2.27.p3.12.m12.1"><semantics id="S6.SS2.27.p3.12.m12.1a"><mrow id="S6.SS2.27.p3.12.m12.1.1" xref="S6.SS2.27.p3.12.m12.1.1.cmml"><mi id="S6.SS2.27.p3.12.m12.1.1.2" xref="S6.SS2.27.p3.12.m12.1.1.2.cmml">A</mi><mo id="S6.SS2.27.p3.12.m12.1.1.1" xref="S6.SS2.27.p3.12.m12.1.1.1.cmml">∖</mo><mi id="S6.SS2.27.p3.12.m12.1.1.3" xref="S6.SS2.27.p3.12.m12.1.1.3.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.27.p3.12.m12.1b"><apply id="S6.SS2.27.p3.12.m12.1.1.cmml" xref="S6.SS2.27.p3.12.m12.1.1"><setdiff id="S6.SS2.27.p3.12.m12.1.1.1.cmml" xref="S6.SS2.27.p3.12.m12.1.1.1"></setdiff><ci id="S6.SS2.27.p3.12.m12.1.1.2.cmml" xref="S6.SS2.27.p3.12.m12.1.1.2">𝐴</ci><ci id="S6.SS2.27.p3.12.m12.1.1.3.cmml" xref="S6.SS2.27.p3.12.m12.1.1.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.27.p3.12.m12.1c">A\setminus\nu</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.27.p3.12.m12.1d">italic_A ∖ italic_ν</annotation></semantics></math>, and noting that <math alttext="\nu=\nu(c_{l})=\nu(c_{l},b_{r})=\nu(b_{l},b_{r})" class="ltx_Math" display="inline" id="S6.SS2.27.p3.13.m13.5"><semantics id="S6.SS2.27.p3.13.m13.5a"><mrow id="S6.SS2.27.p3.13.m13.5.5" xref="S6.SS2.27.p3.13.m13.5.5.cmml"><mi id="S6.SS2.27.p3.13.m13.5.5.7" xref="S6.SS2.27.p3.13.m13.5.5.7.cmml">ν</mi><mo id="S6.SS2.27.p3.13.m13.5.5.8" xref="S6.SS2.27.p3.13.m13.5.5.8.cmml">=</mo><mrow id="S6.SS2.27.p3.13.m13.1.1.1" xref="S6.SS2.27.p3.13.m13.1.1.1.cmml"><mi id="S6.SS2.27.p3.13.m13.1.1.1.3" xref="S6.SS2.27.p3.13.m13.1.1.1.3.cmml">ν</mi><mo id="S6.SS2.27.p3.13.m13.1.1.1.2" xref="S6.SS2.27.p3.13.m13.1.1.1.2.cmml">⁢</mo><mrow id="S6.SS2.27.p3.13.m13.1.1.1.1.1" xref="S6.SS2.27.p3.13.m13.1.1.1.1.1.1.cmml"><mo id="S6.SS2.27.p3.13.m13.1.1.1.1.1.2" stretchy="false" xref="S6.SS2.27.p3.13.m13.1.1.1.1.1.1.cmml">(</mo><msub id="S6.SS2.27.p3.13.m13.1.1.1.1.1.1" xref="S6.SS2.27.p3.13.m13.1.1.1.1.1.1.cmml"><mi id="S6.SS2.27.p3.13.m13.1.1.1.1.1.1.2" xref="S6.SS2.27.p3.13.m13.1.1.1.1.1.1.2.cmml">c</mi><mi id="S6.SS2.27.p3.13.m13.1.1.1.1.1.1.3" xref="S6.SS2.27.p3.13.m13.1.1.1.1.1.1.3.cmml">l</mi></msub><mo id="S6.SS2.27.p3.13.m13.1.1.1.1.1.3" stretchy="false" xref="S6.SS2.27.p3.13.m13.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.SS2.27.p3.13.m13.5.5.9" xref="S6.SS2.27.p3.13.m13.5.5.9.cmml">=</mo><mrow id="S6.SS2.27.p3.13.m13.3.3.3" xref="S6.SS2.27.p3.13.m13.3.3.3.cmml"><mi id="S6.SS2.27.p3.13.m13.3.3.3.4" xref="S6.SS2.27.p3.13.m13.3.3.3.4.cmml">ν</mi><mo id="S6.SS2.27.p3.13.m13.3.3.3.3" xref="S6.SS2.27.p3.13.m13.3.3.3.3.cmml">⁢</mo><mrow id="S6.SS2.27.p3.13.m13.3.3.3.2.2" xref="S6.SS2.27.p3.13.m13.3.3.3.2.3.cmml"><mo id="S6.SS2.27.p3.13.m13.3.3.3.2.2.3" stretchy="false" xref="S6.SS2.27.p3.13.m13.3.3.3.2.3.cmml">(</mo><msub id="S6.SS2.27.p3.13.m13.2.2.2.1.1.1" xref="S6.SS2.27.p3.13.m13.2.2.2.1.1.1.cmml"><mi id="S6.SS2.27.p3.13.m13.2.2.2.1.1.1.2" xref="S6.SS2.27.p3.13.m13.2.2.2.1.1.1.2.cmml">c</mi><mi id="S6.SS2.27.p3.13.m13.2.2.2.1.1.1.3" xref="S6.SS2.27.p3.13.m13.2.2.2.1.1.1.3.cmml">l</mi></msub><mo id="S6.SS2.27.p3.13.m13.3.3.3.2.2.4" xref="S6.SS2.27.p3.13.m13.3.3.3.2.3.cmml">,</mo><msub id="S6.SS2.27.p3.13.m13.3.3.3.2.2.2" xref="S6.SS2.27.p3.13.m13.3.3.3.2.2.2.cmml"><mi id="S6.SS2.27.p3.13.m13.3.3.3.2.2.2.2" xref="S6.SS2.27.p3.13.m13.3.3.3.2.2.2.2.cmml">b</mi><mi id="S6.SS2.27.p3.13.m13.3.3.3.2.2.2.3" xref="S6.SS2.27.p3.13.m13.3.3.3.2.2.2.3.cmml">r</mi></msub><mo id="S6.SS2.27.p3.13.m13.3.3.3.2.2.5" stretchy="false" xref="S6.SS2.27.p3.13.m13.3.3.3.2.3.cmml">)</mo></mrow></mrow><mo id="S6.SS2.27.p3.13.m13.5.5.10" xref="S6.SS2.27.p3.13.m13.5.5.10.cmml">=</mo><mrow id="S6.SS2.27.p3.13.m13.5.5.5" xref="S6.SS2.27.p3.13.m13.5.5.5.cmml"><mi id="S6.SS2.27.p3.13.m13.5.5.5.4" xref="S6.SS2.27.p3.13.m13.5.5.5.4.cmml">ν</mi><mo id="S6.SS2.27.p3.13.m13.5.5.5.3" xref="S6.SS2.27.p3.13.m13.5.5.5.3.cmml">⁢</mo><mrow id="S6.SS2.27.p3.13.m13.5.5.5.2.2" xref="S6.SS2.27.p3.13.m13.5.5.5.2.3.cmml"><mo id="S6.SS2.27.p3.13.m13.5.5.5.2.2.3" stretchy="false" xref="S6.SS2.27.p3.13.m13.5.5.5.2.3.cmml">(</mo><msub id="S6.SS2.27.p3.13.m13.4.4.4.1.1.1" xref="S6.SS2.27.p3.13.m13.4.4.4.1.1.1.cmml"><mi id="S6.SS2.27.p3.13.m13.4.4.4.1.1.1.2" xref="S6.SS2.27.p3.13.m13.4.4.4.1.1.1.2.cmml">b</mi><mi id="S6.SS2.27.p3.13.m13.4.4.4.1.1.1.3" xref="S6.SS2.27.p3.13.m13.4.4.4.1.1.1.3.cmml">l</mi></msub><mo id="S6.SS2.27.p3.13.m13.5.5.5.2.2.4" xref="S6.SS2.27.p3.13.m13.5.5.5.2.3.cmml">,</mo><msub id="S6.SS2.27.p3.13.m13.5.5.5.2.2.2" xref="S6.SS2.27.p3.13.m13.5.5.5.2.2.2.cmml"><mi id="S6.SS2.27.p3.13.m13.5.5.5.2.2.2.2" xref="S6.SS2.27.p3.13.m13.5.5.5.2.2.2.2.cmml">b</mi><mi id="S6.SS2.27.p3.13.m13.5.5.5.2.2.2.3" xref="S6.SS2.27.p3.13.m13.5.5.5.2.2.2.3.cmml">r</mi></msub><mo id="S6.SS2.27.p3.13.m13.5.5.5.2.2.5" stretchy="false" xref="S6.SS2.27.p3.13.m13.5.5.5.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.27.p3.13.m13.5b"><apply id="S6.SS2.27.p3.13.m13.5.5.cmml" xref="S6.SS2.27.p3.13.m13.5.5"><and id="S6.SS2.27.p3.13.m13.5.5a.cmml" xref="S6.SS2.27.p3.13.m13.5.5"></and><apply id="S6.SS2.27.p3.13.m13.5.5b.cmml" xref="S6.SS2.27.p3.13.m13.5.5"><eq id="S6.SS2.27.p3.13.m13.5.5.8.cmml" xref="S6.SS2.27.p3.13.m13.5.5.8"></eq><ci id="S6.SS2.27.p3.13.m13.5.5.7.cmml" xref="S6.SS2.27.p3.13.m13.5.5.7">𝜈</ci><apply id="S6.SS2.27.p3.13.m13.1.1.1.cmml" xref="S6.SS2.27.p3.13.m13.1.1.1"><times id="S6.SS2.27.p3.13.m13.1.1.1.2.cmml" xref="S6.SS2.27.p3.13.m13.1.1.1.2"></times><ci id="S6.SS2.27.p3.13.m13.1.1.1.3.cmml" xref="S6.SS2.27.p3.13.m13.1.1.1.3">𝜈</ci><apply id="S6.SS2.27.p3.13.m13.1.1.1.1.1.1.cmml" xref="S6.SS2.27.p3.13.m13.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.27.p3.13.m13.1.1.1.1.1.1.1.cmml" xref="S6.SS2.27.p3.13.m13.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.27.p3.13.m13.1.1.1.1.1.1.2.cmml" xref="S6.SS2.27.p3.13.m13.1.1.1.1.1.1.2">𝑐</ci><ci id="S6.SS2.27.p3.13.m13.1.1.1.1.1.1.3.cmml" xref="S6.SS2.27.p3.13.m13.1.1.1.1.1.1.3">𝑙</ci></apply></apply></apply><apply id="S6.SS2.27.p3.13.m13.5.5c.cmml" xref="S6.SS2.27.p3.13.m13.5.5"><eq id="S6.SS2.27.p3.13.m13.5.5.9.cmml" xref="S6.SS2.27.p3.13.m13.5.5.9"></eq><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.27.p3.13.m13.1.1.1.cmml" id="S6.SS2.27.p3.13.m13.5.5d.cmml" xref="S6.SS2.27.p3.13.m13.5.5"></share><apply id="S6.SS2.27.p3.13.m13.3.3.3.cmml" xref="S6.SS2.27.p3.13.m13.3.3.3"><times id="S6.SS2.27.p3.13.m13.3.3.3.3.cmml" xref="S6.SS2.27.p3.13.m13.3.3.3.3"></times><ci id="S6.SS2.27.p3.13.m13.3.3.3.4.cmml" xref="S6.SS2.27.p3.13.m13.3.3.3.4">𝜈</ci><interval closure="open" id="S6.SS2.27.p3.13.m13.3.3.3.2.3.cmml" xref="S6.SS2.27.p3.13.m13.3.3.3.2.2"><apply id="S6.SS2.27.p3.13.m13.2.2.2.1.1.1.cmml" xref="S6.SS2.27.p3.13.m13.2.2.2.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.27.p3.13.m13.2.2.2.1.1.1.1.cmml" xref="S6.SS2.27.p3.13.m13.2.2.2.1.1.1">subscript</csymbol><ci id="S6.SS2.27.p3.13.m13.2.2.2.1.1.1.2.cmml" xref="S6.SS2.27.p3.13.m13.2.2.2.1.1.1.2">𝑐</ci><ci id="S6.SS2.27.p3.13.m13.2.2.2.1.1.1.3.cmml" xref="S6.SS2.27.p3.13.m13.2.2.2.1.1.1.3">𝑙</ci></apply><apply id="S6.SS2.27.p3.13.m13.3.3.3.2.2.2.cmml" xref="S6.SS2.27.p3.13.m13.3.3.3.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.27.p3.13.m13.3.3.3.2.2.2.1.cmml" xref="S6.SS2.27.p3.13.m13.3.3.3.2.2.2">subscript</csymbol><ci id="S6.SS2.27.p3.13.m13.3.3.3.2.2.2.2.cmml" xref="S6.SS2.27.p3.13.m13.3.3.3.2.2.2.2">𝑏</ci><ci id="S6.SS2.27.p3.13.m13.3.3.3.2.2.2.3.cmml" xref="S6.SS2.27.p3.13.m13.3.3.3.2.2.2.3">𝑟</ci></apply></interval></apply></apply><apply id="S6.SS2.27.p3.13.m13.5.5e.cmml" xref="S6.SS2.27.p3.13.m13.5.5"><eq id="S6.SS2.27.p3.13.m13.5.5.10.cmml" xref="S6.SS2.27.p3.13.m13.5.5.10"></eq><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.27.p3.13.m13.3.3.3.cmml" id="S6.SS2.27.p3.13.m13.5.5f.cmml" xref="S6.SS2.27.p3.13.m13.5.5"></share><apply id="S6.SS2.27.p3.13.m13.5.5.5.cmml" xref="S6.SS2.27.p3.13.m13.5.5.5"><times id="S6.SS2.27.p3.13.m13.5.5.5.3.cmml" xref="S6.SS2.27.p3.13.m13.5.5.5.3"></times><ci id="S6.SS2.27.p3.13.m13.5.5.5.4.cmml" xref="S6.SS2.27.p3.13.m13.5.5.5.4">𝜈</ci><interval closure="open" id="S6.SS2.27.p3.13.m13.5.5.5.2.3.cmml" xref="S6.SS2.27.p3.13.m13.5.5.5.2.2"><apply id="S6.SS2.27.p3.13.m13.4.4.4.1.1.1.cmml" xref="S6.SS2.27.p3.13.m13.4.4.4.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.27.p3.13.m13.4.4.4.1.1.1.1.cmml" xref="S6.SS2.27.p3.13.m13.4.4.4.1.1.1">subscript</csymbol><ci id="S6.SS2.27.p3.13.m13.4.4.4.1.1.1.2.cmml" xref="S6.SS2.27.p3.13.m13.4.4.4.1.1.1.2">𝑏</ci><ci id="S6.SS2.27.p3.13.m13.4.4.4.1.1.1.3.cmml" xref="S6.SS2.27.p3.13.m13.4.4.4.1.1.1.3">𝑙</ci></apply><apply id="S6.SS2.27.p3.13.m13.5.5.5.2.2.2.cmml" xref="S6.SS2.27.p3.13.m13.5.5.5.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.27.p3.13.m13.5.5.5.2.2.2.1.cmml" xref="S6.SS2.27.p3.13.m13.5.5.5.2.2.2">subscript</csymbol><ci id="S6.SS2.27.p3.13.m13.5.5.5.2.2.2.2.cmml" xref="S6.SS2.27.p3.13.m13.5.5.5.2.2.2.2">𝑏</ci><ci id="S6.SS2.27.p3.13.m13.5.5.5.2.2.2.3.cmml" xref="S6.SS2.27.p3.13.m13.5.5.5.2.2.2.3">𝑟</ci></apply></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.27.p3.13.m13.5c">\nu=\nu(c_{l})=\nu(c_{l},b_{r})=\nu(b_{l},b_{r})</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.27.p3.13.m13.5d">italic_ν = italic_ν ( italic_c start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ) = italic_ν ( italic_c start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) = italic_ν ( italic_b start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT )</annotation></semantics></math>, one sees that <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.Thmtheorem9" title="Definition 6.9. ‣ 6.2. Forcing epimorphisms ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">6.9</span></a> <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.I4.i1" title="Item (i) ‣ Definition 6.9. ‣ 6.2. Forcing epimorphisms ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">(i)</span></a> is satisfied, as well as <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.Thmtheorem9" title="Definition 6.9. ‣ 6.2. Forcing epimorphisms ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">6.9</span></a> <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.I4.i2" title="Item (ii) ‣ Definition 6.9. ‣ 6.2. Forcing epimorphisms ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">(ii)</span></a>.</p> </div> <div class="ltx_para" id="S6.SS2.28.p4"> <p class="ltx_p" id="S6.SS2.28.p4.3"><span class="ltx_text ltx_framed ltx_framed_underline" id="S6.SS2.28.p4.3.1">Case 3</span>: <math alttext="\nu(a_{r},b_{l})<\nu" class="ltx_Math" display="inline" id="S6.SS2.28.p4.1.m1.2"><semantics id="S6.SS2.28.p4.1.m1.2a"><mrow id="S6.SS2.28.p4.1.m1.2.2" xref="S6.SS2.28.p4.1.m1.2.2.cmml"><mrow id="S6.SS2.28.p4.1.m1.2.2.2" xref="S6.SS2.28.p4.1.m1.2.2.2.cmml"><mi id="S6.SS2.28.p4.1.m1.2.2.2.4" xref="S6.SS2.28.p4.1.m1.2.2.2.4.cmml">ν</mi><mo id="S6.SS2.28.p4.1.m1.2.2.2.3" xref="S6.SS2.28.p4.1.m1.2.2.2.3.cmml">⁢</mo><mrow id="S6.SS2.28.p4.1.m1.2.2.2.2.2" xref="S6.SS2.28.p4.1.m1.2.2.2.2.3.cmml"><mo id="S6.SS2.28.p4.1.m1.2.2.2.2.2.3" stretchy="false" xref="S6.SS2.28.p4.1.m1.2.2.2.2.3.cmml">(</mo><msub id="S6.SS2.28.p4.1.m1.1.1.1.1.1.1" xref="S6.SS2.28.p4.1.m1.1.1.1.1.1.1.cmml"><mi id="S6.SS2.28.p4.1.m1.1.1.1.1.1.1.2" xref="S6.SS2.28.p4.1.m1.1.1.1.1.1.1.2.cmml">a</mi><mi id="S6.SS2.28.p4.1.m1.1.1.1.1.1.1.3" xref="S6.SS2.28.p4.1.m1.1.1.1.1.1.1.3.cmml">r</mi></msub><mo id="S6.SS2.28.p4.1.m1.2.2.2.2.2.4" xref="S6.SS2.28.p4.1.m1.2.2.2.2.3.cmml">,</mo><msub id="S6.SS2.28.p4.1.m1.2.2.2.2.2.2" xref="S6.SS2.28.p4.1.m1.2.2.2.2.2.2.cmml"><mi id="S6.SS2.28.p4.1.m1.2.2.2.2.2.2.2" xref="S6.SS2.28.p4.1.m1.2.2.2.2.2.2.2.cmml">b</mi><mi id="S6.SS2.28.p4.1.m1.2.2.2.2.2.2.3" xref="S6.SS2.28.p4.1.m1.2.2.2.2.2.2.3.cmml">l</mi></msub><mo id="S6.SS2.28.p4.1.m1.2.2.2.2.2.5" stretchy="false" xref="S6.SS2.28.p4.1.m1.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.SS2.28.p4.1.m1.2.2.3" xref="S6.SS2.28.p4.1.m1.2.2.3.cmml"><</mo><mi id="S6.SS2.28.p4.1.m1.2.2.4" xref="S6.SS2.28.p4.1.m1.2.2.4.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.28.p4.1.m1.2b"><apply id="S6.SS2.28.p4.1.m1.2.2.cmml" xref="S6.SS2.28.p4.1.m1.2.2"><lt id="S6.SS2.28.p4.1.m1.2.2.3.cmml" xref="S6.SS2.28.p4.1.m1.2.2.3"></lt><apply id="S6.SS2.28.p4.1.m1.2.2.2.cmml" xref="S6.SS2.28.p4.1.m1.2.2.2"><times id="S6.SS2.28.p4.1.m1.2.2.2.3.cmml" xref="S6.SS2.28.p4.1.m1.2.2.2.3"></times><ci id="S6.SS2.28.p4.1.m1.2.2.2.4.cmml" xref="S6.SS2.28.p4.1.m1.2.2.2.4">𝜈</ci><interval closure="open" id="S6.SS2.28.p4.1.m1.2.2.2.2.3.cmml" xref="S6.SS2.28.p4.1.m1.2.2.2.2.2"><apply id="S6.SS2.28.p4.1.m1.1.1.1.1.1.1.cmml" xref="S6.SS2.28.p4.1.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.28.p4.1.m1.1.1.1.1.1.1.1.cmml" xref="S6.SS2.28.p4.1.m1.1.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.28.p4.1.m1.1.1.1.1.1.1.2.cmml" xref="S6.SS2.28.p4.1.m1.1.1.1.1.1.1.2">𝑎</ci><ci id="S6.SS2.28.p4.1.m1.1.1.1.1.1.1.3.cmml" xref="S6.SS2.28.p4.1.m1.1.1.1.1.1.1.3">𝑟</ci></apply><apply id="S6.SS2.28.p4.1.m1.2.2.2.2.2.2.cmml" xref="S6.SS2.28.p4.1.m1.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.28.p4.1.m1.2.2.2.2.2.2.1.cmml" xref="S6.SS2.28.p4.1.m1.2.2.2.2.2.2">subscript</csymbol><ci id="S6.SS2.28.p4.1.m1.2.2.2.2.2.2.2.cmml" xref="S6.SS2.28.p4.1.m1.2.2.2.2.2.2.2">𝑏</ci><ci id="S6.SS2.28.p4.1.m1.2.2.2.2.2.2.3.cmml" xref="S6.SS2.28.p4.1.m1.2.2.2.2.2.2.3">𝑙</ci></apply></interval></apply><ci id="S6.SS2.28.p4.1.m1.2.2.4.cmml" xref="S6.SS2.28.p4.1.m1.2.2.4">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.28.p4.1.m1.2c">\nu(a_{r},b_{l})<\nu</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.28.p4.1.m1.2d">italic_ν ( italic_a start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ) < italic_ν</annotation></semantics></math> and <math alttext="y" class="ltx_Math" display="inline" id="S6.SS2.28.p4.2.m2.1"><semantics id="S6.SS2.28.p4.2.m2.1a"><mi id="S6.SS2.28.p4.2.m2.1.1" xref="S6.SS2.28.p4.2.m2.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.28.p4.2.m2.1b"><ci id="S6.SS2.28.p4.2.m2.1.1.cmml" xref="S6.SS2.28.p4.2.m2.1.1">𝑦</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.28.p4.2.m2.1c">y</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.28.p4.2.m2.1d">italic_y</annotation></semantics></math> is not the left endpoint of its complementary interval of <math alttext="X\setminus\nu" class="ltx_Math" display="inline" id="S6.SS2.28.p4.3.m3.1"><semantics id="S6.SS2.28.p4.3.m3.1a"><mrow id="S6.SS2.28.p4.3.m3.1.1" xref="S6.SS2.28.p4.3.m3.1.1.cmml"><mi id="S6.SS2.28.p4.3.m3.1.1.2" xref="S6.SS2.28.p4.3.m3.1.1.2.cmml">X</mi><mo id="S6.SS2.28.p4.3.m3.1.1.1" xref="S6.SS2.28.p4.3.m3.1.1.1.cmml">∖</mo><mi id="S6.SS2.28.p4.3.m3.1.1.3" xref="S6.SS2.28.p4.3.m3.1.1.3.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.28.p4.3.m3.1b"><apply id="S6.SS2.28.p4.3.m3.1.1.cmml" xref="S6.SS2.28.p4.3.m3.1.1"><setdiff id="S6.SS2.28.p4.3.m3.1.1.1.cmml" xref="S6.SS2.28.p4.3.m3.1.1.1"></setdiff><ci id="S6.SS2.28.p4.3.m3.1.1.2.cmml" xref="S6.SS2.28.p4.3.m3.1.1.2">𝑋</ci><ci id="S6.SS2.28.p4.3.m3.1.1.3.cmml" xref="S6.SS2.28.p4.3.m3.1.1.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.28.p4.3.m3.1c">X\setminus\nu</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.28.p4.3.m3.1d">italic_X ∖ italic_ν</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S6.SS2.29.p5"> <p class="ltx_p" id="S6.SS2.29.p5.20"><span class="ltx_text ltx_framed ltx_framed_underline" id="S6.SS2.29.p5.20.1">Subcase 3.1</span>: <math alttext="c" class="ltx_Math" display="inline" id="S6.SS2.29.p5.1.m1.1"><semantics id="S6.SS2.29.p5.1.m1.1a"><mi id="S6.SS2.29.p5.1.m1.1.1" xref="S6.SS2.29.p5.1.m1.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.29.p5.1.m1.1b"><ci id="S6.SS2.29.p5.1.m1.1.1.cmml" xref="S6.SS2.29.p5.1.m1.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.29.p5.1.m1.1c">c</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.29.p5.1.m1.1d">italic_c</annotation></semantics></math> is the left endpoint of its complementary interval in <math alttext="A\setminus\nu" class="ltx_Math" display="inline" id="S6.SS2.29.p5.2.m2.1"><semantics id="S6.SS2.29.p5.2.m2.1a"><mrow id="S6.SS2.29.p5.2.m2.1.1" xref="S6.SS2.29.p5.2.m2.1.1.cmml"><mi id="S6.SS2.29.p5.2.m2.1.1.2" xref="S6.SS2.29.p5.2.m2.1.1.2.cmml">A</mi><mo id="S6.SS2.29.p5.2.m2.1.1.1" xref="S6.SS2.29.p5.2.m2.1.1.1.cmml">∖</mo><mi id="S6.SS2.29.p5.2.m2.1.1.3" xref="S6.SS2.29.p5.2.m2.1.1.3.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.29.p5.2.m2.1b"><apply id="S6.SS2.29.p5.2.m2.1.1.cmml" xref="S6.SS2.29.p5.2.m2.1.1"><setdiff id="S6.SS2.29.p5.2.m2.1.1.1.cmml" xref="S6.SS2.29.p5.2.m2.1.1.1"></setdiff><ci id="S6.SS2.29.p5.2.m2.1.1.2.cmml" xref="S6.SS2.29.p5.2.m2.1.1.2">𝐴</ci><ci id="S6.SS2.29.p5.2.m2.1.1.3.cmml" xref="S6.SS2.29.p5.2.m2.1.1.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.29.p5.2.m2.1c">A\setminus\nu</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.29.p5.2.m2.1d">italic_A ∖ italic_ν</annotation></semantics></math>. It is in this case where we need to use the hypothesis on <math alttext="\mathscr{L}" class="ltx_Math" display="inline" id="S6.SS2.29.p5.3.m3.1"><semantics id="S6.SS2.29.p5.3.m3.1a"><mi class="ltx_font_mathscript" id="S6.SS2.29.p5.3.m3.1.1" xref="S6.SS2.29.p5.3.m3.1.1.cmml">ℒ</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.29.p5.3.m3.1b"><ci id="S6.SS2.29.p5.3.m3.1.1.cmml" xref="S6.SS2.29.p5.3.m3.1.1">ℒ</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.29.p5.3.m3.1c">\mathscr{L}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.29.p5.3.m3.1d">script_L</annotation></semantics></math> and <math alttext="\mathscr{R}" class="ltx_Math" display="inline" id="S6.SS2.29.p5.4.m4.1"><semantics id="S6.SS2.29.p5.4.m4.1a"><mi class="ltx_font_mathscript" id="S6.SS2.29.p5.4.m4.1.1" xref="S6.SS2.29.p5.4.m4.1.1.cmml">ℛ</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.29.p5.4.m4.1b"><ci id="S6.SS2.29.p5.4.m4.1.1.cmml" xref="S6.SS2.29.p5.4.m4.1.1">ℛ</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.29.p5.4.m4.1c">\mathscr{R}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.29.p5.4.m4.1d">script_R</annotation></semantics></math>. Note that in this case we have <math alttext="\nu(c)=\nu" class="ltx_Math" display="inline" id="S6.SS2.29.p5.5.m5.1"><semantics id="S6.SS2.29.p5.5.m5.1a"><mrow id="S6.SS2.29.p5.5.m5.1.2" xref="S6.SS2.29.p5.5.m5.1.2.cmml"><mrow id="S6.SS2.29.p5.5.m5.1.2.2" xref="S6.SS2.29.p5.5.m5.1.2.2.cmml"><mi id="S6.SS2.29.p5.5.m5.1.2.2.2" xref="S6.SS2.29.p5.5.m5.1.2.2.2.cmml">ν</mi><mo id="S6.SS2.29.p5.5.m5.1.2.2.1" xref="S6.SS2.29.p5.5.m5.1.2.2.1.cmml">⁢</mo><mrow id="S6.SS2.29.p5.5.m5.1.2.2.3.2" xref="S6.SS2.29.p5.5.m5.1.2.2.cmml"><mo id="S6.SS2.29.p5.5.m5.1.2.2.3.2.1" stretchy="false" xref="S6.SS2.29.p5.5.m5.1.2.2.cmml">(</mo><mi id="S6.SS2.29.p5.5.m5.1.1" xref="S6.SS2.29.p5.5.m5.1.1.cmml">c</mi><mo id="S6.SS2.29.p5.5.m5.1.2.2.3.2.2" stretchy="false" xref="S6.SS2.29.p5.5.m5.1.2.2.cmml">)</mo></mrow></mrow><mo id="S6.SS2.29.p5.5.m5.1.2.1" xref="S6.SS2.29.p5.5.m5.1.2.1.cmml">=</mo><mi id="S6.SS2.29.p5.5.m5.1.2.3" xref="S6.SS2.29.p5.5.m5.1.2.3.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.29.p5.5.m5.1b"><apply id="S6.SS2.29.p5.5.m5.1.2.cmml" xref="S6.SS2.29.p5.5.m5.1.2"><eq id="S6.SS2.29.p5.5.m5.1.2.1.cmml" xref="S6.SS2.29.p5.5.m5.1.2.1"></eq><apply id="S6.SS2.29.p5.5.m5.1.2.2.cmml" xref="S6.SS2.29.p5.5.m5.1.2.2"><times id="S6.SS2.29.p5.5.m5.1.2.2.1.cmml" xref="S6.SS2.29.p5.5.m5.1.2.2.1"></times><ci id="S6.SS2.29.p5.5.m5.1.2.2.2.cmml" xref="S6.SS2.29.p5.5.m5.1.2.2.2">𝜈</ci><ci id="S6.SS2.29.p5.5.m5.1.1.cmml" xref="S6.SS2.29.p5.5.m5.1.1">𝑐</ci></apply><ci id="S6.SS2.29.p5.5.m5.1.2.3.cmml" xref="S6.SS2.29.p5.5.m5.1.2.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.29.p5.5.m5.1c">\nu(c)=\nu</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.29.p5.5.m5.1d">italic_ν ( italic_c ) = italic_ν</annotation></semantics></math>. By the hypotheses on <math alttext="A" class="ltx_Math" display="inline" id="S6.SS2.29.p5.6.m6.1"><semantics id="S6.SS2.29.p5.6.m6.1a"><mi id="S6.SS2.29.p5.6.m6.1.1" xref="S6.SS2.29.p5.6.m6.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.29.p5.6.m6.1b"><ci id="S6.SS2.29.p5.6.m6.1.1.cmml" xref="S6.SS2.29.p5.6.m6.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.29.p5.6.m6.1c">A</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.29.p5.6.m6.1d">italic_A</annotation></semantics></math> and <math alttext="X" class="ltx_Math" display="inline" id="S6.SS2.29.p5.7.m7.1"><semantics id="S6.SS2.29.p5.7.m7.1a"><mi id="S6.SS2.29.p5.7.m7.1.1" xref="S6.SS2.29.p5.7.m7.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.29.p5.7.m7.1b"><ci id="S6.SS2.29.p5.7.m7.1.1.cmml" xref="S6.SS2.29.p5.7.m7.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.29.p5.7.m7.1c">X</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.29.p5.7.m7.1d">italic_X</annotation></semantics></math>, and noting that by <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.Thmtheorem16" title="Lemma 6.16. ‣ 6.2. Forcing epimorphisms ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">6.16</span></a> (a) <math alttext="A\setminus D_{\nu}^{A}=A\setminus\nu" class="ltx_Math" display="inline" id="S6.SS2.29.p5.8.m8.1"><semantics id="S6.SS2.29.p5.8.m8.1a"><mrow id="S6.SS2.29.p5.8.m8.1.1" xref="S6.SS2.29.p5.8.m8.1.1.cmml"><mrow id="S6.SS2.29.p5.8.m8.1.1.2" xref="S6.SS2.29.p5.8.m8.1.1.2.cmml"><mi id="S6.SS2.29.p5.8.m8.1.1.2.2" xref="S6.SS2.29.p5.8.m8.1.1.2.2.cmml">A</mi><mo id="S6.SS2.29.p5.8.m8.1.1.2.1" xref="S6.SS2.29.p5.8.m8.1.1.2.1.cmml">∖</mo><msubsup id="S6.SS2.29.p5.8.m8.1.1.2.3" xref="S6.SS2.29.p5.8.m8.1.1.2.3.cmml"><mi id="S6.SS2.29.p5.8.m8.1.1.2.3.2.2" xref="S6.SS2.29.p5.8.m8.1.1.2.3.2.2.cmml">D</mi><mi id="S6.SS2.29.p5.8.m8.1.1.2.3.2.3" xref="S6.SS2.29.p5.8.m8.1.1.2.3.2.3.cmml">ν</mi><mi id="S6.SS2.29.p5.8.m8.1.1.2.3.3" xref="S6.SS2.29.p5.8.m8.1.1.2.3.3.cmml">A</mi></msubsup></mrow><mo id="S6.SS2.29.p5.8.m8.1.1.1" xref="S6.SS2.29.p5.8.m8.1.1.1.cmml">=</mo><mrow id="S6.SS2.29.p5.8.m8.1.1.3" xref="S6.SS2.29.p5.8.m8.1.1.3.cmml"><mi id="S6.SS2.29.p5.8.m8.1.1.3.2" xref="S6.SS2.29.p5.8.m8.1.1.3.2.cmml">A</mi><mo id="S6.SS2.29.p5.8.m8.1.1.3.1" xref="S6.SS2.29.p5.8.m8.1.1.3.1.cmml">∖</mo><mi id="S6.SS2.29.p5.8.m8.1.1.3.3" xref="S6.SS2.29.p5.8.m8.1.1.3.3.cmml">ν</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.29.p5.8.m8.1b"><apply id="S6.SS2.29.p5.8.m8.1.1.cmml" xref="S6.SS2.29.p5.8.m8.1.1"><eq id="S6.SS2.29.p5.8.m8.1.1.1.cmml" xref="S6.SS2.29.p5.8.m8.1.1.1"></eq><apply id="S6.SS2.29.p5.8.m8.1.1.2.cmml" xref="S6.SS2.29.p5.8.m8.1.1.2"><setdiff id="S6.SS2.29.p5.8.m8.1.1.2.1.cmml" xref="S6.SS2.29.p5.8.m8.1.1.2.1"></setdiff><ci id="S6.SS2.29.p5.8.m8.1.1.2.2.cmml" xref="S6.SS2.29.p5.8.m8.1.1.2.2">𝐴</ci><apply id="S6.SS2.29.p5.8.m8.1.1.2.3.cmml" xref="S6.SS2.29.p5.8.m8.1.1.2.3"><csymbol cd="ambiguous" id="S6.SS2.29.p5.8.m8.1.1.2.3.1.cmml" xref="S6.SS2.29.p5.8.m8.1.1.2.3">superscript</csymbol><apply id="S6.SS2.29.p5.8.m8.1.1.2.3.2.cmml" xref="S6.SS2.29.p5.8.m8.1.1.2.3"><csymbol cd="ambiguous" id="S6.SS2.29.p5.8.m8.1.1.2.3.2.1.cmml" xref="S6.SS2.29.p5.8.m8.1.1.2.3">subscript</csymbol><ci id="S6.SS2.29.p5.8.m8.1.1.2.3.2.2.cmml" xref="S6.SS2.29.p5.8.m8.1.1.2.3.2.2">𝐷</ci><ci id="S6.SS2.29.p5.8.m8.1.1.2.3.2.3.cmml" xref="S6.SS2.29.p5.8.m8.1.1.2.3.2.3">𝜈</ci></apply><ci id="S6.SS2.29.p5.8.m8.1.1.2.3.3.cmml" xref="S6.SS2.29.p5.8.m8.1.1.2.3.3">𝐴</ci></apply></apply><apply id="S6.SS2.29.p5.8.m8.1.1.3.cmml" xref="S6.SS2.29.p5.8.m8.1.1.3"><setdiff id="S6.SS2.29.p5.8.m8.1.1.3.1.cmml" xref="S6.SS2.29.p5.8.m8.1.1.3.1"></setdiff><ci id="S6.SS2.29.p5.8.m8.1.1.3.2.cmml" xref="S6.SS2.29.p5.8.m8.1.1.3.2">𝐴</ci><ci id="S6.SS2.29.p5.8.m8.1.1.3.3.cmml" xref="S6.SS2.29.p5.8.m8.1.1.3.3">𝜈</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.29.p5.8.m8.1c">A\setminus D_{\nu}^{A}=A\setminus\nu</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.29.p5.8.m8.1d">italic_A ∖ italic_D start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT = italic_A ∖ italic_ν</annotation></semantics></math>, we know that every complementary interval of <math alttext="X\setminus D_{\nu}^{X}=X\setminus\nu" class="ltx_Math" display="inline" id="S6.SS2.29.p5.9.m9.1"><semantics id="S6.SS2.29.p5.9.m9.1a"><mrow id="S6.SS2.29.p5.9.m9.1.1" xref="S6.SS2.29.p5.9.m9.1.1.cmml"><mrow id="S6.SS2.29.p5.9.m9.1.1.2" xref="S6.SS2.29.p5.9.m9.1.1.2.cmml"><mi id="S6.SS2.29.p5.9.m9.1.1.2.2" xref="S6.SS2.29.p5.9.m9.1.1.2.2.cmml">X</mi><mo id="S6.SS2.29.p5.9.m9.1.1.2.1" xref="S6.SS2.29.p5.9.m9.1.1.2.1.cmml">∖</mo><msubsup id="S6.SS2.29.p5.9.m9.1.1.2.3" xref="S6.SS2.29.p5.9.m9.1.1.2.3.cmml"><mi id="S6.SS2.29.p5.9.m9.1.1.2.3.2.2" xref="S6.SS2.29.p5.9.m9.1.1.2.3.2.2.cmml">D</mi><mi id="S6.SS2.29.p5.9.m9.1.1.2.3.2.3" xref="S6.SS2.29.p5.9.m9.1.1.2.3.2.3.cmml">ν</mi><mi id="S6.SS2.29.p5.9.m9.1.1.2.3.3" xref="S6.SS2.29.p5.9.m9.1.1.2.3.3.cmml">X</mi></msubsup></mrow><mo id="S6.SS2.29.p5.9.m9.1.1.1" xref="S6.SS2.29.p5.9.m9.1.1.1.cmml">=</mo><mrow id="S6.SS2.29.p5.9.m9.1.1.3" xref="S6.SS2.29.p5.9.m9.1.1.3.cmml"><mi id="S6.SS2.29.p5.9.m9.1.1.3.2" xref="S6.SS2.29.p5.9.m9.1.1.3.2.cmml">X</mi><mo id="S6.SS2.29.p5.9.m9.1.1.3.1" xref="S6.SS2.29.p5.9.m9.1.1.3.1.cmml">∖</mo><mi id="S6.SS2.29.p5.9.m9.1.1.3.3" xref="S6.SS2.29.p5.9.m9.1.1.3.3.cmml">ν</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.29.p5.9.m9.1b"><apply id="S6.SS2.29.p5.9.m9.1.1.cmml" xref="S6.SS2.29.p5.9.m9.1.1"><eq id="S6.SS2.29.p5.9.m9.1.1.1.cmml" xref="S6.SS2.29.p5.9.m9.1.1.1"></eq><apply id="S6.SS2.29.p5.9.m9.1.1.2.cmml" xref="S6.SS2.29.p5.9.m9.1.1.2"><setdiff id="S6.SS2.29.p5.9.m9.1.1.2.1.cmml" xref="S6.SS2.29.p5.9.m9.1.1.2.1"></setdiff><ci id="S6.SS2.29.p5.9.m9.1.1.2.2.cmml" xref="S6.SS2.29.p5.9.m9.1.1.2.2">𝑋</ci><apply id="S6.SS2.29.p5.9.m9.1.1.2.3.cmml" xref="S6.SS2.29.p5.9.m9.1.1.2.3"><csymbol cd="ambiguous" id="S6.SS2.29.p5.9.m9.1.1.2.3.1.cmml" xref="S6.SS2.29.p5.9.m9.1.1.2.3">superscript</csymbol><apply id="S6.SS2.29.p5.9.m9.1.1.2.3.2.cmml" xref="S6.SS2.29.p5.9.m9.1.1.2.3"><csymbol cd="ambiguous" id="S6.SS2.29.p5.9.m9.1.1.2.3.2.1.cmml" xref="S6.SS2.29.p5.9.m9.1.1.2.3">subscript</csymbol><ci id="S6.SS2.29.p5.9.m9.1.1.2.3.2.2.cmml" xref="S6.SS2.29.p5.9.m9.1.1.2.3.2.2">𝐷</ci><ci id="S6.SS2.29.p5.9.m9.1.1.2.3.2.3.cmml" xref="S6.SS2.29.p5.9.m9.1.1.2.3.2.3">𝜈</ci></apply><ci id="S6.SS2.29.p5.9.m9.1.1.2.3.3.cmml" xref="S6.SS2.29.p5.9.m9.1.1.2.3.3">𝑋</ci></apply></apply><apply id="S6.SS2.29.p5.9.m9.1.1.3.cmml" xref="S6.SS2.29.p5.9.m9.1.1.3"><setdiff id="S6.SS2.29.p5.9.m9.1.1.3.1.cmml" xref="S6.SS2.29.p5.9.m9.1.1.3.1"></setdiff><ci id="S6.SS2.29.p5.9.m9.1.1.3.2.cmml" xref="S6.SS2.29.p5.9.m9.1.1.3.2">𝑋</ci><ci id="S6.SS2.29.p5.9.m9.1.1.3.3.cmml" xref="S6.SS2.29.p5.9.m9.1.1.3.3">𝜈</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.29.p5.9.m9.1c">X\setminus D_{\nu}^{X}=X\setminus\nu</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.29.p5.9.m9.1d">italic_X ∖ italic_D start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT = italic_X ∖ italic_ν</annotation></semantics></math> has a left endpoint. Let <math alttext="z" class="ltx_Math" display="inline" id="S6.SS2.29.p5.10.m10.1"><semantics id="S6.SS2.29.p5.10.m10.1a"><mi id="S6.SS2.29.p5.10.m10.1.1" xref="S6.SS2.29.p5.10.m10.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.29.p5.10.m10.1b"><ci id="S6.SS2.29.p5.10.m10.1.1.cmml" xref="S6.SS2.29.p5.10.m10.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.29.p5.10.m10.1c">z</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.29.p5.10.m10.1d">italic_z</annotation></semantics></math> be the left endpoint of the complementary interval of <math alttext="X\setminus\nu" class="ltx_Math" display="inline" id="S6.SS2.29.p5.11.m11.1"><semantics id="S6.SS2.29.p5.11.m11.1a"><mrow id="S6.SS2.29.p5.11.m11.1.1" xref="S6.SS2.29.p5.11.m11.1.1.cmml"><mi id="S6.SS2.29.p5.11.m11.1.1.2" xref="S6.SS2.29.p5.11.m11.1.1.2.cmml">X</mi><mo id="S6.SS2.29.p5.11.m11.1.1.1" xref="S6.SS2.29.p5.11.m11.1.1.1.cmml">∖</mo><mi id="S6.SS2.29.p5.11.m11.1.1.3" xref="S6.SS2.29.p5.11.m11.1.1.3.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.29.p5.11.m11.1b"><apply id="S6.SS2.29.p5.11.m11.1.1.cmml" xref="S6.SS2.29.p5.11.m11.1.1"><setdiff id="S6.SS2.29.p5.11.m11.1.1.1.cmml" xref="S6.SS2.29.p5.11.m11.1.1.1"></setdiff><ci id="S6.SS2.29.p5.11.m11.1.1.2.cmml" xref="S6.SS2.29.p5.11.m11.1.1.2">𝑋</ci><ci id="S6.SS2.29.p5.11.m11.1.1.3.cmml" xref="S6.SS2.29.p5.11.m11.1.1.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.29.p5.11.m11.1c">X\setminus\nu</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.29.p5.11.m11.1d">italic_X ∖ italic_ν</annotation></semantics></math> in which <math alttext="y" class="ltx_Math" display="inline" id="S6.SS2.29.p5.12.m12.1"><semantics id="S6.SS2.29.p5.12.m12.1a"><mi id="S6.SS2.29.p5.12.m12.1.1" xref="S6.SS2.29.p5.12.m12.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.29.p5.12.m12.1b"><ci id="S6.SS2.29.p5.12.m12.1.1.cmml" xref="S6.SS2.29.p5.12.m12.1.1">𝑦</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.29.p5.12.m12.1c">y</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.29.p5.12.m12.1d">italic_y</annotation></semantics></math> is, and note that then <math alttext="x<_{X}z<_{X}y" class="ltx_Math" display="inline" id="S6.SS2.29.p5.13.m13.1"><semantics id="S6.SS2.29.p5.13.m13.1a"><mrow id="S6.SS2.29.p5.13.m13.1.1" xref="S6.SS2.29.p5.13.m13.1.1.cmml"><mi id="S6.SS2.29.p5.13.m13.1.1.2" xref="S6.SS2.29.p5.13.m13.1.1.2.cmml">x</mi><msub id="S6.SS2.29.p5.13.m13.1.1.3" xref="S6.SS2.29.p5.13.m13.1.1.3.cmml"><mo id="S6.SS2.29.p5.13.m13.1.1.3.2" xref="S6.SS2.29.p5.13.m13.1.1.3.2.cmml"><</mo><mi id="S6.SS2.29.p5.13.m13.1.1.3.3" xref="S6.SS2.29.p5.13.m13.1.1.3.3.cmml">X</mi></msub><mi id="S6.SS2.29.p5.13.m13.1.1.4" xref="S6.SS2.29.p5.13.m13.1.1.4.cmml">z</mi><msub id="S6.SS2.29.p5.13.m13.1.1.5" xref="S6.SS2.29.p5.13.m13.1.1.5.cmml"><mo id="S6.SS2.29.p5.13.m13.1.1.5.2" xref="S6.SS2.29.p5.13.m13.1.1.5.2.cmml"><</mo><mi id="S6.SS2.29.p5.13.m13.1.1.5.3" xref="S6.SS2.29.p5.13.m13.1.1.5.3.cmml">X</mi></msub><mi id="S6.SS2.29.p5.13.m13.1.1.6" xref="S6.SS2.29.p5.13.m13.1.1.6.cmml">y</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.29.p5.13.m13.1b"><apply id="S6.SS2.29.p5.13.m13.1.1.cmml" xref="S6.SS2.29.p5.13.m13.1.1"><and id="S6.SS2.29.p5.13.m13.1.1a.cmml" xref="S6.SS2.29.p5.13.m13.1.1"></and><apply id="S6.SS2.29.p5.13.m13.1.1b.cmml" xref="S6.SS2.29.p5.13.m13.1.1"><apply id="S6.SS2.29.p5.13.m13.1.1.3.cmml" xref="S6.SS2.29.p5.13.m13.1.1.3"><csymbol cd="ambiguous" id="S6.SS2.29.p5.13.m13.1.1.3.1.cmml" xref="S6.SS2.29.p5.13.m13.1.1.3">subscript</csymbol><lt id="S6.SS2.29.p5.13.m13.1.1.3.2.cmml" xref="S6.SS2.29.p5.13.m13.1.1.3.2"></lt><ci id="S6.SS2.29.p5.13.m13.1.1.3.3.cmml" xref="S6.SS2.29.p5.13.m13.1.1.3.3">𝑋</ci></apply><ci id="S6.SS2.29.p5.13.m13.1.1.2.cmml" xref="S6.SS2.29.p5.13.m13.1.1.2">𝑥</ci><ci id="S6.SS2.29.p5.13.m13.1.1.4.cmml" xref="S6.SS2.29.p5.13.m13.1.1.4">𝑧</ci></apply><apply id="S6.SS2.29.p5.13.m13.1.1c.cmml" xref="S6.SS2.29.p5.13.m13.1.1"><apply id="S6.SS2.29.p5.13.m13.1.1.5.cmml" xref="S6.SS2.29.p5.13.m13.1.1.5"><csymbol cd="ambiguous" id="S6.SS2.29.p5.13.m13.1.1.5.1.cmml" xref="S6.SS2.29.p5.13.m13.1.1.5">subscript</csymbol><lt id="S6.SS2.29.p5.13.m13.1.1.5.2.cmml" xref="S6.SS2.29.p5.13.m13.1.1.5.2"></lt><ci id="S6.SS2.29.p5.13.m13.1.1.5.3.cmml" xref="S6.SS2.29.p5.13.m13.1.1.5.3">𝑋</ci></apply><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.29.p5.13.m13.1.1.4.cmml" id="S6.SS2.29.p5.13.m13.1.1d.cmml" xref="S6.SS2.29.p5.13.m13.1.1"></share><ci id="S6.SS2.29.p5.13.m13.1.1.6.cmml" xref="S6.SS2.29.p5.13.m13.1.1.6">𝑦</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.29.p5.13.m13.1c">x<_{X}z<_{X}y</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.29.p5.13.m13.1d">italic_x < start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT italic_z < start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT italic_y</annotation></semantics></math> and <math alttext="\nu(z)=\nu" class="ltx_Math" display="inline" id="S6.SS2.29.p5.14.m14.1"><semantics id="S6.SS2.29.p5.14.m14.1a"><mrow id="S6.SS2.29.p5.14.m14.1.2" xref="S6.SS2.29.p5.14.m14.1.2.cmml"><mrow id="S6.SS2.29.p5.14.m14.1.2.2" xref="S6.SS2.29.p5.14.m14.1.2.2.cmml"><mi id="S6.SS2.29.p5.14.m14.1.2.2.2" xref="S6.SS2.29.p5.14.m14.1.2.2.2.cmml">ν</mi><mo id="S6.SS2.29.p5.14.m14.1.2.2.1" xref="S6.SS2.29.p5.14.m14.1.2.2.1.cmml">⁢</mo><mrow id="S6.SS2.29.p5.14.m14.1.2.2.3.2" xref="S6.SS2.29.p5.14.m14.1.2.2.cmml"><mo id="S6.SS2.29.p5.14.m14.1.2.2.3.2.1" stretchy="false" xref="S6.SS2.29.p5.14.m14.1.2.2.cmml">(</mo><mi id="S6.SS2.29.p5.14.m14.1.1" xref="S6.SS2.29.p5.14.m14.1.1.cmml">z</mi><mo id="S6.SS2.29.p5.14.m14.1.2.2.3.2.2" stretchy="false" xref="S6.SS2.29.p5.14.m14.1.2.2.cmml">)</mo></mrow></mrow><mo id="S6.SS2.29.p5.14.m14.1.2.1" xref="S6.SS2.29.p5.14.m14.1.2.1.cmml">=</mo><mi id="S6.SS2.29.p5.14.m14.1.2.3" xref="S6.SS2.29.p5.14.m14.1.2.3.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.29.p5.14.m14.1b"><apply id="S6.SS2.29.p5.14.m14.1.2.cmml" xref="S6.SS2.29.p5.14.m14.1.2"><eq id="S6.SS2.29.p5.14.m14.1.2.1.cmml" xref="S6.SS2.29.p5.14.m14.1.2.1"></eq><apply id="S6.SS2.29.p5.14.m14.1.2.2.cmml" xref="S6.SS2.29.p5.14.m14.1.2.2"><times id="S6.SS2.29.p5.14.m14.1.2.2.1.cmml" xref="S6.SS2.29.p5.14.m14.1.2.2.1"></times><ci id="S6.SS2.29.p5.14.m14.1.2.2.2.cmml" xref="S6.SS2.29.p5.14.m14.1.2.2.2">𝜈</ci><ci id="S6.SS2.29.p5.14.m14.1.1.cmml" xref="S6.SS2.29.p5.14.m14.1.1">𝑧</ci></apply><ci id="S6.SS2.29.p5.14.m14.1.2.3.cmml" xref="S6.SS2.29.p5.14.m14.1.2.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.29.p5.14.m14.1c">\nu(z)=\nu</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.29.p5.14.m14.1d">italic_ν ( italic_z ) = italic_ν</annotation></semantics></math> by <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.Thmtheorem16" title="Lemma 6.16. ‣ 6.2. Forcing epimorphisms ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">6.16</span></a> (d). As in Case 1 one finds <math alttext="c_{r}" class="ltx_Math" display="inline" id="S6.SS2.29.p5.15.m15.1"><semantics id="S6.SS2.29.p5.15.m15.1a"><msub id="S6.SS2.29.p5.15.m15.1.1" xref="S6.SS2.29.p5.15.m15.1.1.cmml"><mi id="S6.SS2.29.p5.15.m15.1.1.2" xref="S6.SS2.29.p5.15.m15.1.1.2.cmml">c</mi><mi id="S6.SS2.29.p5.15.m15.1.1.3" xref="S6.SS2.29.p5.15.m15.1.1.3.cmml">r</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.29.p5.15.m15.1b"><apply id="S6.SS2.29.p5.15.m15.1.1.cmml" xref="S6.SS2.29.p5.15.m15.1.1"><csymbol cd="ambiguous" id="S6.SS2.29.p5.15.m15.1.1.1.cmml" xref="S6.SS2.29.p5.15.m15.1.1">subscript</csymbol><ci id="S6.SS2.29.p5.15.m15.1.1.2.cmml" xref="S6.SS2.29.p5.15.m15.1.1.2">𝑐</ci><ci id="S6.SS2.29.p5.15.m15.1.1.3.cmml" xref="S6.SS2.29.p5.15.m15.1.1.3">𝑟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.29.p5.15.m15.1c">c_{r}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.29.p5.15.m15.1d">italic_c start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT</annotation></semantics></math> such <math alttext="\nu(c_{r})=\nu" class="ltx_Math" display="inline" id="S6.SS2.29.p5.16.m16.1"><semantics id="S6.SS2.29.p5.16.m16.1a"><mrow id="S6.SS2.29.p5.16.m16.1.1" xref="S6.SS2.29.p5.16.m16.1.1.cmml"><mrow id="S6.SS2.29.p5.16.m16.1.1.1" xref="S6.SS2.29.p5.16.m16.1.1.1.cmml"><mi id="S6.SS2.29.p5.16.m16.1.1.1.3" xref="S6.SS2.29.p5.16.m16.1.1.1.3.cmml">ν</mi><mo id="S6.SS2.29.p5.16.m16.1.1.1.2" xref="S6.SS2.29.p5.16.m16.1.1.1.2.cmml">⁢</mo><mrow id="S6.SS2.29.p5.16.m16.1.1.1.1.1" xref="S6.SS2.29.p5.16.m16.1.1.1.1.1.1.cmml"><mo id="S6.SS2.29.p5.16.m16.1.1.1.1.1.2" stretchy="false" xref="S6.SS2.29.p5.16.m16.1.1.1.1.1.1.cmml">(</mo><msub id="S6.SS2.29.p5.16.m16.1.1.1.1.1.1" xref="S6.SS2.29.p5.16.m16.1.1.1.1.1.1.cmml"><mi id="S6.SS2.29.p5.16.m16.1.1.1.1.1.1.2" xref="S6.SS2.29.p5.16.m16.1.1.1.1.1.1.2.cmml">c</mi><mi id="S6.SS2.29.p5.16.m16.1.1.1.1.1.1.3" xref="S6.SS2.29.p5.16.m16.1.1.1.1.1.1.3.cmml">r</mi></msub><mo id="S6.SS2.29.p5.16.m16.1.1.1.1.1.3" stretchy="false" xref="S6.SS2.29.p5.16.m16.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.SS2.29.p5.16.m16.1.1.2" xref="S6.SS2.29.p5.16.m16.1.1.2.cmml">=</mo><mi id="S6.SS2.29.p5.16.m16.1.1.3" xref="S6.SS2.29.p5.16.m16.1.1.3.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.29.p5.16.m16.1b"><apply id="S6.SS2.29.p5.16.m16.1.1.cmml" xref="S6.SS2.29.p5.16.m16.1.1"><eq id="S6.SS2.29.p5.16.m16.1.1.2.cmml" xref="S6.SS2.29.p5.16.m16.1.1.2"></eq><apply id="S6.SS2.29.p5.16.m16.1.1.1.cmml" xref="S6.SS2.29.p5.16.m16.1.1.1"><times id="S6.SS2.29.p5.16.m16.1.1.1.2.cmml" xref="S6.SS2.29.p5.16.m16.1.1.1.2"></times><ci id="S6.SS2.29.p5.16.m16.1.1.1.3.cmml" xref="S6.SS2.29.p5.16.m16.1.1.1.3">𝜈</ci><apply id="S6.SS2.29.p5.16.m16.1.1.1.1.1.1.cmml" xref="S6.SS2.29.p5.16.m16.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.29.p5.16.m16.1.1.1.1.1.1.1.cmml" xref="S6.SS2.29.p5.16.m16.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.29.p5.16.m16.1.1.1.1.1.1.2.cmml" xref="S6.SS2.29.p5.16.m16.1.1.1.1.1.1.2">𝑐</ci><ci id="S6.SS2.29.p5.16.m16.1.1.1.1.1.1.3.cmml" xref="S6.SS2.29.p5.16.m16.1.1.1.1.1.1.3">𝑟</ci></apply></apply><ci id="S6.SS2.29.p5.16.m16.1.1.3.cmml" xref="S6.SS2.29.p5.16.m16.1.1.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.29.p5.16.m16.1c">\nu(c_{r})=\nu</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.29.p5.16.m16.1d">italic_ν ( italic_c start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) = italic_ν</annotation></semantics></math>, <math alttext="c_{l}<_{A}c_{r}<_{A}b_{l}" class="ltx_Math" display="inline" id="S6.SS2.29.p5.17.m17.1"><semantics id="S6.SS2.29.p5.17.m17.1a"><mrow id="S6.SS2.29.p5.17.m17.1.1" xref="S6.SS2.29.p5.17.m17.1.1.cmml"><msub id="S6.SS2.29.p5.17.m17.1.1.2" xref="S6.SS2.29.p5.17.m17.1.1.2.cmml"><mi id="S6.SS2.29.p5.17.m17.1.1.2.2" xref="S6.SS2.29.p5.17.m17.1.1.2.2.cmml">c</mi><mi id="S6.SS2.29.p5.17.m17.1.1.2.3" xref="S6.SS2.29.p5.17.m17.1.1.2.3.cmml">l</mi></msub><msub id="S6.SS2.29.p5.17.m17.1.1.3" xref="S6.SS2.29.p5.17.m17.1.1.3.cmml"><mo id="S6.SS2.29.p5.17.m17.1.1.3.2" xref="S6.SS2.29.p5.17.m17.1.1.3.2.cmml"><</mo><mi id="S6.SS2.29.p5.17.m17.1.1.3.3" xref="S6.SS2.29.p5.17.m17.1.1.3.3.cmml">A</mi></msub><msub id="S6.SS2.29.p5.17.m17.1.1.4" xref="S6.SS2.29.p5.17.m17.1.1.4.cmml"><mi id="S6.SS2.29.p5.17.m17.1.1.4.2" xref="S6.SS2.29.p5.17.m17.1.1.4.2.cmml">c</mi><mi id="S6.SS2.29.p5.17.m17.1.1.4.3" xref="S6.SS2.29.p5.17.m17.1.1.4.3.cmml">r</mi></msub><msub id="S6.SS2.29.p5.17.m17.1.1.5" xref="S6.SS2.29.p5.17.m17.1.1.5.cmml"><mo id="S6.SS2.29.p5.17.m17.1.1.5.2" xref="S6.SS2.29.p5.17.m17.1.1.5.2.cmml"><</mo><mi id="S6.SS2.29.p5.17.m17.1.1.5.3" xref="S6.SS2.29.p5.17.m17.1.1.5.3.cmml">A</mi></msub><msub id="S6.SS2.29.p5.17.m17.1.1.6" xref="S6.SS2.29.p5.17.m17.1.1.6.cmml"><mi id="S6.SS2.29.p5.17.m17.1.1.6.2" xref="S6.SS2.29.p5.17.m17.1.1.6.2.cmml">b</mi><mi id="S6.SS2.29.p5.17.m17.1.1.6.3" xref="S6.SS2.29.p5.17.m17.1.1.6.3.cmml">l</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.29.p5.17.m17.1b"><apply id="S6.SS2.29.p5.17.m17.1.1.cmml" xref="S6.SS2.29.p5.17.m17.1.1"><and id="S6.SS2.29.p5.17.m17.1.1a.cmml" xref="S6.SS2.29.p5.17.m17.1.1"></and><apply id="S6.SS2.29.p5.17.m17.1.1b.cmml" xref="S6.SS2.29.p5.17.m17.1.1"><apply id="S6.SS2.29.p5.17.m17.1.1.3.cmml" xref="S6.SS2.29.p5.17.m17.1.1.3"><csymbol cd="ambiguous" id="S6.SS2.29.p5.17.m17.1.1.3.1.cmml" xref="S6.SS2.29.p5.17.m17.1.1.3">subscript</csymbol><lt id="S6.SS2.29.p5.17.m17.1.1.3.2.cmml" xref="S6.SS2.29.p5.17.m17.1.1.3.2"></lt><ci id="S6.SS2.29.p5.17.m17.1.1.3.3.cmml" xref="S6.SS2.29.p5.17.m17.1.1.3.3">𝐴</ci></apply><apply id="S6.SS2.29.p5.17.m17.1.1.2.cmml" xref="S6.SS2.29.p5.17.m17.1.1.2"><csymbol cd="ambiguous" id="S6.SS2.29.p5.17.m17.1.1.2.1.cmml" xref="S6.SS2.29.p5.17.m17.1.1.2">subscript</csymbol><ci id="S6.SS2.29.p5.17.m17.1.1.2.2.cmml" xref="S6.SS2.29.p5.17.m17.1.1.2.2">𝑐</ci><ci id="S6.SS2.29.p5.17.m17.1.1.2.3.cmml" xref="S6.SS2.29.p5.17.m17.1.1.2.3">𝑙</ci></apply><apply id="S6.SS2.29.p5.17.m17.1.1.4.cmml" xref="S6.SS2.29.p5.17.m17.1.1.4"><csymbol cd="ambiguous" id="S6.SS2.29.p5.17.m17.1.1.4.1.cmml" xref="S6.SS2.29.p5.17.m17.1.1.4">subscript</csymbol><ci id="S6.SS2.29.p5.17.m17.1.1.4.2.cmml" xref="S6.SS2.29.p5.17.m17.1.1.4.2">𝑐</ci><ci id="S6.SS2.29.p5.17.m17.1.1.4.3.cmml" xref="S6.SS2.29.p5.17.m17.1.1.4.3">𝑟</ci></apply></apply><apply id="S6.SS2.29.p5.17.m17.1.1c.cmml" xref="S6.SS2.29.p5.17.m17.1.1"><apply id="S6.SS2.29.p5.17.m17.1.1.5.cmml" xref="S6.SS2.29.p5.17.m17.1.1.5"><csymbol cd="ambiguous" id="S6.SS2.29.p5.17.m17.1.1.5.1.cmml" xref="S6.SS2.29.p5.17.m17.1.1.5">subscript</csymbol><lt id="S6.SS2.29.p5.17.m17.1.1.5.2.cmml" xref="S6.SS2.29.p5.17.m17.1.1.5.2"></lt><ci id="S6.SS2.29.p5.17.m17.1.1.5.3.cmml" xref="S6.SS2.29.p5.17.m17.1.1.5.3">𝐴</ci></apply><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.29.p5.17.m17.1.1.4.cmml" id="S6.SS2.29.p5.17.m17.1.1d.cmml" xref="S6.SS2.29.p5.17.m17.1.1"></share><apply id="S6.SS2.29.p5.17.m17.1.1.6.cmml" xref="S6.SS2.29.p5.17.m17.1.1.6"><csymbol cd="ambiguous" id="S6.SS2.29.p5.17.m17.1.1.6.1.cmml" xref="S6.SS2.29.p5.17.m17.1.1.6">subscript</csymbol><ci id="S6.SS2.29.p5.17.m17.1.1.6.2.cmml" xref="S6.SS2.29.p5.17.m17.1.1.6.2">𝑏</ci><ci id="S6.SS2.29.p5.17.m17.1.1.6.3.cmml" xref="S6.SS2.29.p5.17.m17.1.1.6.3">𝑙</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.29.p5.17.m17.1c">c_{l}<_{A}c_{r}<_{A}b_{l}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.29.p5.17.m17.1d">italic_c start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT < start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_c start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT < start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\nu(c_{l},c_{r})=\nu" class="ltx_Math" display="inline" id="S6.SS2.29.p5.18.m18.2"><semantics id="S6.SS2.29.p5.18.m18.2a"><mrow id="S6.SS2.29.p5.18.m18.2.2" xref="S6.SS2.29.p5.18.m18.2.2.cmml"><mrow id="S6.SS2.29.p5.18.m18.2.2.2" xref="S6.SS2.29.p5.18.m18.2.2.2.cmml"><mi id="S6.SS2.29.p5.18.m18.2.2.2.4" xref="S6.SS2.29.p5.18.m18.2.2.2.4.cmml">ν</mi><mo id="S6.SS2.29.p5.18.m18.2.2.2.3" xref="S6.SS2.29.p5.18.m18.2.2.2.3.cmml">⁢</mo><mrow id="S6.SS2.29.p5.18.m18.2.2.2.2.2" xref="S6.SS2.29.p5.18.m18.2.2.2.2.3.cmml"><mo id="S6.SS2.29.p5.18.m18.2.2.2.2.2.3" stretchy="false" xref="S6.SS2.29.p5.18.m18.2.2.2.2.3.cmml">(</mo><msub id="S6.SS2.29.p5.18.m18.1.1.1.1.1.1" xref="S6.SS2.29.p5.18.m18.1.1.1.1.1.1.cmml"><mi id="S6.SS2.29.p5.18.m18.1.1.1.1.1.1.2" xref="S6.SS2.29.p5.18.m18.1.1.1.1.1.1.2.cmml">c</mi><mi id="S6.SS2.29.p5.18.m18.1.1.1.1.1.1.3" xref="S6.SS2.29.p5.18.m18.1.1.1.1.1.1.3.cmml">l</mi></msub><mo id="S6.SS2.29.p5.18.m18.2.2.2.2.2.4" xref="S6.SS2.29.p5.18.m18.2.2.2.2.3.cmml">,</mo><msub id="S6.SS2.29.p5.18.m18.2.2.2.2.2.2" xref="S6.SS2.29.p5.18.m18.2.2.2.2.2.2.cmml"><mi id="S6.SS2.29.p5.18.m18.2.2.2.2.2.2.2" xref="S6.SS2.29.p5.18.m18.2.2.2.2.2.2.2.cmml">c</mi><mi id="S6.SS2.29.p5.18.m18.2.2.2.2.2.2.3" xref="S6.SS2.29.p5.18.m18.2.2.2.2.2.2.3.cmml">r</mi></msub><mo id="S6.SS2.29.p5.18.m18.2.2.2.2.2.5" stretchy="false" xref="S6.SS2.29.p5.18.m18.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.SS2.29.p5.18.m18.2.2.3" xref="S6.SS2.29.p5.18.m18.2.2.3.cmml">=</mo><mi id="S6.SS2.29.p5.18.m18.2.2.4" xref="S6.SS2.29.p5.18.m18.2.2.4.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.29.p5.18.m18.2b"><apply id="S6.SS2.29.p5.18.m18.2.2.cmml" xref="S6.SS2.29.p5.18.m18.2.2"><eq id="S6.SS2.29.p5.18.m18.2.2.3.cmml" xref="S6.SS2.29.p5.18.m18.2.2.3"></eq><apply id="S6.SS2.29.p5.18.m18.2.2.2.cmml" xref="S6.SS2.29.p5.18.m18.2.2.2"><times id="S6.SS2.29.p5.18.m18.2.2.2.3.cmml" xref="S6.SS2.29.p5.18.m18.2.2.2.3"></times><ci id="S6.SS2.29.p5.18.m18.2.2.2.4.cmml" xref="S6.SS2.29.p5.18.m18.2.2.2.4">𝜈</ci><interval closure="open" id="S6.SS2.29.p5.18.m18.2.2.2.2.3.cmml" xref="S6.SS2.29.p5.18.m18.2.2.2.2.2"><apply id="S6.SS2.29.p5.18.m18.1.1.1.1.1.1.cmml" xref="S6.SS2.29.p5.18.m18.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.29.p5.18.m18.1.1.1.1.1.1.1.cmml" xref="S6.SS2.29.p5.18.m18.1.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.29.p5.18.m18.1.1.1.1.1.1.2.cmml" xref="S6.SS2.29.p5.18.m18.1.1.1.1.1.1.2">𝑐</ci><ci id="S6.SS2.29.p5.18.m18.1.1.1.1.1.1.3.cmml" xref="S6.SS2.29.p5.18.m18.1.1.1.1.1.1.3">𝑙</ci></apply><apply id="S6.SS2.29.p5.18.m18.2.2.2.2.2.2.cmml" xref="S6.SS2.29.p5.18.m18.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.29.p5.18.m18.2.2.2.2.2.2.1.cmml" xref="S6.SS2.29.p5.18.m18.2.2.2.2.2.2">subscript</csymbol><ci id="S6.SS2.29.p5.18.m18.2.2.2.2.2.2.2.cmml" xref="S6.SS2.29.p5.18.m18.2.2.2.2.2.2.2">𝑐</ci><ci id="S6.SS2.29.p5.18.m18.2.2.2.2.2.2.3.cmml" xref="S6.SS2.29.p5.18.m18.2.2.2.2.2.2.3">𝑟</ci></apply></interval></apply><ci id="S6.SS2.29.p5.18.m18.2.2.4.cmml" xref="S6.SS2.29.p5.18.m18.2.2.4">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.29.p5.18.m18.2c">\nu(c_{l},c_{r})=\nu</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.29.p5.18.m18.2d">italic_ν ( italic_c start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT , italic_c start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) = italic_ν</annotation></semantics></math>. Then <math alttext="p\cup\{((c,c_{r}),z)\}" class="ltx_Math" display="inline" id="S6.SS2.29.p5.19.m19.3"><semantics id="S6.SS2.29.p5.19.m19.3a"><mrow id="S6.SS2.29.p5.19.m19.3.3" xref="S6.SS2.29.p5.19.m19.3.3.cmml"><mi id="S6.SS2.29.p5.19.m19.3.3.3" xref="S6.SS2.29.p5.19.m19.3.3.3.cmml">p</mi><mo id="S6.SS2.29.p5.19.m19.3.3.2" xref="S6.SS2.29.p5.19.m19.3.3.2.cmml">∪</mo><mrow id="S6.SS2.29.p5.19.m19.3.3.1.1" xref="S6.SS2.29.p5.19.m19.3.3.1.2.cmml"><mo id="S6.SS2.29.p5.19.m19.3.3.1.1.2" stretchy="false" xref="S6.SS2.29.p5.19.m19.3.3.1.2.cmml">{</mo><mrow id="S6.SS2.29.p5.19.m19.3.3.1.1.1.1" xref="S6.SS2.29.p5.19.m19.3.3.1.1.1.2.cmml"><mo id="S6.SS2.29.p5.19.m19.3.3.1.1.1.1.2" stretchy="false" xref="S6.SS2.29.p5.19.m19.3.3.1.1.1.2.cmml">(</mo><mrow id="S6.SS2.29.p5.19.m19.3.3.1.1.1.1.1.1" xref="S6.SS2.29.p5.19.m19.3.3.1.1.1.1.1.2.cmml"><mo id="S6.SS2.29.p5.19.m19.3.3.1.1.1.1.1.1.2" stretchy="false" xref="S6.SS2.29.p5.19.m19.3.3.1.1.1.1.1.2.cmml">(</mo><mi id="S6.SS2.29.p5.19.m19.1.1" xref="S6.SS2.29.p5.19.m19.1.1.cmml">c</mi><mo id="S6.SS2.29.p5.19.m19.3.3.1.1.1.1.1.1.3" xref="S6.SS2.29.p5.19.m19.3.3.1.1.1.1.1.2.cmml">,</mo><msub id="S6.SS2.29.p5.19.m19.3.3.1.1.1.1.1.1.1" xref="S6.SS2.29.p5.19.m19.3.3.1.1.1.1.1.1.1.cmml"><mi id="S6.SS2.29.p5.19.m19.3.3.1.1.1.1.1.1.1.2" xref="S6.SS2.29.p5.19.m19.3.3.1.1.1.1.1.1.1.2.cmml">c</mi><mi id="S6.SS2.29.p5.19.m19.3.3.1.1.1.1.1.1.1.3" xref="S6.SS2.29.p5.19.m19.3.3.1.1.1.1.1.1.1.3.cmml">r</mi></msub><mo id="S6.SS2.29.p5.19.m19.3.3.1.1.1.1.1.1.4" stretchy="false" xref="S6.SS2.29.p5.19.m19.3.3.1.1.1.1.1.2.cmml">)</mo></mrow><mo id="S6.SS2.29.p5.19.m19.3.3.1.1.1.1.3" xref="S6.SS2.29.p5.19.m19.3.3.1.1.1.2.cmml">,</mo><mi id="S6.SS2.29.p5.19.m19.2.2" xref="S6.SS2.29.p5.19.m19.2.2.cmml">z</mi><mo id="S6.SS2.29.p5.19.m19.3.3.1.1.1.1.4" stretchy="false" xref="S6.SS2.29.p5.19.m19.3.3.1.1.1.2.cmml">)</mo></mrow><mo id="S6.SS2.29.p5.19.m19.3.3.1.1.3" stretchy="false" xref="S6.SS2.29.p5.19.m19.3.3.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.29.p5.19.m19.3b"><apply id="S6.SS2.29.p5.19.m19.3.3.cmml" xref="S6.SS2.29.p5.19.m19.3.3"><union id="S6.SS2.29.p5.19.m19.3.3.2.cmml" xref="S6.SS2.29.p5.19.m19.3.3.2"></union><ci id="S6.SS2.29.p5.19.m19.3.3.3.cmml" xref="S6.SS2.29.p5.19.m19.3.3.3">𝑝</ci><set id="S6.SS2.29.p5.19.m19.3.3.1.2.cmml" xref="S6.SS2.29.p5.19.m19.3.3.1.1"><interval closure="open" id="S6.SS2.29.p5.19.m19.3.3.1.1.1.2.cmml" xref="S6.SS2.29.p5.19.m19.3.3.1.1.1.1"><interval closure="open" id="S6.SS2.29.p5.19.m19.3.3.1.1.1.1.1.2.cmml" xref="S6.SS2.29.p5.19.m19.3.3.1.1.1.1.1.1"><ci id="S6.SS2.29.p5.19.m19.1.1.cmml" xref="S6.SS2.29.p5.19.m19.1.1">𝑐</ci><apply id="S6.SS2.29.p5.19.m19.3.3.1.1.1.1.1.1.1.cmml" xref="S6.SS2.29.p5.19.m19.3.3.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.29.p5.19.m19.3.3.1.1.1.1.1.1.1.1.cmml" xref="S6.SS2.29.p5.19.m19.3.3.1.1.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.29.p5.19.m19.3.3.1.1.1.1.1.1.1.2.cmml" xref="S6.SS2.29.p5.19.m19.3.3.1.1.1.1.1.1.1.2">𝑐</ci><ci id="S6.SS2.29.p5.19.m19.3.3.1.1.1.1.1.1.1.3.cmml" xref="S6.SS2.29.p5.19.m19.3.3.1.1.1.1.1.1.1.3">𝑟</ci></apply></interval><ci id="S6.SS2.29.p5.19.m19.2.2.cmml" xref="S6.SS2.29.p5.19.m19.2.2">𝑧</ci></interval></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.29.p5.19.m19.3c">p\cup\{((c,c_{r}),z)\}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.29.p5.19.m19.3d">italic_p ∪ { ( ( italic_c , italic_c start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) , italic_z ) }</annotation></semantics></math> is in <math alttext="P_{E}" class="ltx_Math" display="inline" id="S6.SS2.29.p5.20.m20.1"><semantics id="S6.SS2.29.p5.20.m20.1a"><msub id="S6.SS2.29.p5.20.m20.1.1" xref="S6.SS2.29.p5.20.m20.1.1.cmml"><mi id="S6.SS2.29.p5.20.m20.1.1.2" xref="S6.SS2.29.p5.20.m20.1.1.2.cmml">P</mi><mi id="S6.SS2.29.p5.20.m20.1.1.3" xref="S6.SS2.29.p5.20.m20.1.1.3.cmml">E</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.29.p5.20.m20.1b"><apply id="S6.SS2.29.p5.20.m20.1.1.cmml" xref="S6.SS2.29.p5.20.m20.1.1"><csymbol cd="ambiguous" id="S6.SS2.29.p5.20.m20.1.1.1.cmml" xref="S6.SS2.29.p5.20.m20.1.1">subscript</csymbol><ci id="S6.SS2.29.p5.20.m20.1.1.2.cmml" xref="S6.SS2.29.p5.20.m20.1.1.2">𝑃</ci><ci id="S6.SS2.29.p5.20.m20.1.1.3.cmml" xref="S6.SS2.29.p5.20.m20.1.1.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.29.p5.20.m20.1c">P_{E}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.29.p5.20.m20.1d">italic_P start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S6.SS2.30.p6"> <p class="ltx_p" id="S6.SS2.30.p6.15"><span class="ltx_text ltx_framed ltx_framed_underline" id="S6.SS2.30.p6.15.1">Subcase 3.2</span>: <math alttext="c" class="ltx_Math" display="inline" id="S6.SS2.30.p6.1.m1.1"><semantics id="S6.SS2.30.p6.1.m1.1a"><mi id="S6.SS2.30.p6.1.m1.1.1" xref="S6.SS2.30.p6.1.m1.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.30.p6.1.m1.1b"><ci id="S6.SS2.30.p6.1.m1.1.1.cmml" xref="S6.SS2.30.p6.1.m1.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.30.p6.1.m1.1c">c</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.30.p6.1.m1.1d">italic_c</annotation></semantics></math> is not the left endpoint of its complementary interval in <math alttext="A\setminus\nu" class="ltx_Math" display="inline" id="S6.SS2.30.p6.2.m2.1"><semantics id="S6.SS2.30.p6.2.m2.1a"><mrow id="S6.SS2.30.p6.2.m2.1.1" xref="S6.SS2.30.p6.2.m2.1.1.cmml"><mi id="S6.SS2.30.p6.2.m2.1.1.2" xref="S6.SS2.30.p6.2.m2.1.1.2.cmml">A</mi><mo id="S6.SS2.30.p6.2.m2.1.1.1" xref="S6.SS2.30.p6.2.m2.1.1.1.cmml">∖</mo><mi id="S6.SS2.30.p6.2.m2.1.1.3" xref="S6.SS2.30.p6.2.m2.1.1.3.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.30.p6.2.m2.1b"><apply id="S6.SS2.30.p6.2.m2.1.1.cmml" xref="S6.SS2.30.p6.2.m2.1.1"><setdiff id="S6.SS2.30.p6.2.m2.1.1.1.cmml" xref="S6.SS2.30.p6.2.m2.1.1.1"></setdiff><ci id="S6.SS2.30.p6.2.m2.1.1.2.cmml" xref="S6.SS2.30.p6.2.m2.1.1.2">𝐴</ci><ci id="S6.SS2.30.p6.2.m2.1.1.3.cmml" xref="S6.SS2.30.p6.2.m2.1.1.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.30.p6.2.m2.1c">A\setminus\nu</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.30.p6.2.m2.1d">italic_A ∖ italic_ν</annotation></semantics></math>. Using elementarity and <math alttext="\aleph_{1}" class="ltx_Math" display="inline" id="S6.SS2.30.p6.3.m3.1"><semantics id="S6.SS2.30.p6.3.m3.1a"><msub id="S6.SS2.30.p6.3.m3.1.1" xref="S6.SS2.30.p6.3.m3.1.1.cmml"><mi id="S6.SS2.30.p6.3.m3.1.1.2" mathvariant="normal" xref="S6.SS2.30.p6.3.m3.1.1.2.cmml">ℵ</mi><mn id="S6.SS2.30.p6.3.m3.1.1.3" xref="S6.SS2.30.p6.3.m3.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.30.p6.3.m3.1b"><apply id="S6.SS2.30.p6.3.m3.1.1.cmml" xref="S6.SS2.30.p6.3.m3.1.1"><csymbol cd="ambiguous" id="S6.SS2.30.p6.3.m3.1.1.1.cmml" xref="S6.SS2.30.p6.3.m3.1.1">subscript</csymbol><ci id="S6.SS2.30.p6.3.m3.1.1.2.cmml" xref="S6.SS2.30.p6.3.m3.1.1.2">ℵ</ci><cn id="S6.SS2.30.p6.3.m3.1.1.3.cmml" type="integer" xref="S6.SS2.30.p6.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.30.p6.3.m3.1c">\aleph_{1}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.30.p6.3.m3.1d">roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-density let <math alttext="z" class="ltx_Math" display="inline" id="S6.SS2.30.p6.4.m4.1"><semantics id="S6.SS2.30.p6.4.m4.1a"><mi id="S6.SS2.30.p6.4.m4.1.1" xref="S6.SS2.30.p6.4.m4.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.30.p6.4.m4.1b"><ci id="S6.SS2.30.p6.4.m4.1.1.cmml" xref="S6.SS2.30.p6.4.m4.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.30.p6.4.m4.1c">z</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.30.p6.4.m4.1d">italic_z</annotation></semantics></math> be in the same complementary interval of <math alttext="X\setminus\nu" class="ltx_Math" display="inline" id="S6.SS2.30.p6.5.m5.1"><semantics id="S6.SS2.30.p6.5.m5.1a"><mrow id="S6.SS2.30.p6.5.m5.1.1" xref="S6.SS2.30.p6.5.m5.1.1.cmml"><mi id="S6.SS2.30.p6.5.m5.1.1.2" xref="S6.SS2.30.p6.5.m5.1.1.2.cmml">X</mi><mo id="S6.SS2.30.p6.5.m5.1.1.1" xref="S6.SS2.30.p6.5.m5.1.1.1.cmml">∖</mo><mi id="S6.SS2.30.p6.5.m5.1.1.3" xref="S6.SS2.30.p6.5.m5.1.1.3.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.30.p6.5.m5.1b"><apply id="S6.SS2.30.p6.5.m5.1.1.cmml" xref="S6.SS2.30.p6.5.m5.1.1"><setdiff id="S6.SS2.30.p6.5.m5.1.1.1.cmml" xref="S6.SS2.30.p6.5.m5.1.1.1"></setdiff><ci id="S6.SS2.30.p6.5.m5.1.1.2.cmml" xref="S6.SS2.30.p6.5.m5.1.1.2">𝑋</ci><ci id="S6.SS2.30.p6.5.m5.1.1.3.cmml" xref="S6.SS2.30.p6.5.m5.1.1.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.30.p6.5.m5.1c">X\setminus\nu</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.30.p6.5.m5.1d">italic_X ∖ italic_ν</annotation></semantics></math> of <math alttext="y" class="ltx_Math" display="inline" id="S6.SS2.30.p6.6.m6.1"><semantics id="S6.SS2.30.p6.6.m6.1a"><mi id="S6.SS2.30.p6.6.m6.1.1" xref="S6.SS2.30.p6.6.m6.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.30.p6.6.m6.1b"><ci id="S6.SS2.30.p6.6.m6.1.1.cmml" xref="S6.SS2.30.p6.6.m6.1.1">𝑦</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.30.p6.6.m6.1c">y</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.30.p6.6.m6.1d">italic_y</annotation></semantics></math>, and such that <math alttext="\nu(z)=\nu" class="ltx_Math" display="inline" id="S6.SS2.30.p6.7.m7.1"><semantics id="S6.SS2.30.p6.7.m7.1a"><mrow id="S6.SS2.30.p6.7.m7.1.2" xref="S6.SS2.30.p6.7.m7.1.2.cmml"><mrow id="S6.SS2.30.p6.7.m7.1.2.2" xref="S6.SS2.30.p6.7.m7.1.2.2.cmml"><mi id="S6.SS2.30.p6.7.m7.1.2.2.2" xref="S6.SS2.30.p6.7.m7.1.2.2.2.cmml">ν</mi><mo id="S6.SS2.30.p6.7.m7.1.2.2.1" xref="S6.SS2.30.p6.7.m7.1.2.2.1.cmml">⁢</mo><mrow id="S6.SS2.30.p6.7.m7.1.2.2.3.2" xref="S6.SS2.30.p6.7.m7.1.2.2.cmml"><mo id="S6.SS2.30.p6.7.m7.1.2.2.3.2.1" stretchy="false" xref="S6.SS2.30.p6.7.m7.1.2.2.cmml">(</mo><mi id="S6.SS2.30.p6.7.m7.1.1" xref="S6.SS2.30.p6.7.m7.1.1.cmml">z</mi><mo id="S6.SS2.30.p6.7.m7.1.2.2.3.2.2" stretchy="false" xref="S6.SS2.30.p6.7.m7.1.2.2.cmml">)</mo></mrow></mrow><mo id="S6.SS2.30.p6.7.m7.1.2.1" xref="S6.SS2.30.p6.7.m7.1.2.1.cmml">=</mo><mi id="S6.SS2.30.p6.7.m7.1.2.3" xref="S6.SS2.30.p6.7.m7.1.2.3.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.30.p6.7.m7.1b"><apply id="S6.SS2.30.p6.7.m7.1.2.cmml" xref="S6.SS2.30.p6.7.m7.1.2"><eq id="S6.SS2.30.p6.7.m7.1.2.1.cmml" xref="S6.SS2.30.p6.7.m7.1.2.1"></eq><apply id="S6.SS2.30.p6.7.m7.1.2.2.cmml" xref="S6.SS2.30.p6.7.m7.1.2.2"><times id="S6.SS2.30.p6.7.m7.1.2.2.1.cmml" xref="S6.SS2.30.p6.7.m7.1.2.2.1"></times><ci id="S6.SS2.30.p6.7.m7.1.2.2.2.cmml" xref="S6.SS2.30.p6.7.m7.1.2.2.2">𝜈</ci><ci id="S6.SS2.30.p6.7.m7.1.1.cmml" xref="S6.SS2.30.p6.7.m7.1.1">𝑧</ci></apply><ci id="S6.SS2.30.p6.7.m7.1.2.3.cmml" xref="S6.SS2.30.p6.7.m7.1.2.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.30.p6.7.m7.1c">\nu(z)=\nu</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.30.p6.7.m7.1d">italic_ν ( italic_z ) = italic_ν</annotation></semantics></math> and <math alttext="z<_{X}y" class="ltx_Math" display="inline" id="S6.SS2.30.p6.8.m8.1"><semantics id="S6.SS2.30.p6.8.m8.1a"><mrow id="S6.SS2.30.p6.8.m8.1.1" xref="S6.SS2.30.p6.8.m8.1.1.cmml"><mi id="S6.SS2.30.p6.8.m8.1.1.2" xref="S6.SS2.30.p6.8.m8.1.1.2.cmml">z</mi><msub id="S6.SS2.30.p6.8.m8.1.1.1" xref="S6.SS2.30.p6.8.m8.1.1.1.cmml"><mo id="S6.SS2.30.p6.8.m8.1.1.1.2" xref="S6.SS2.30.p6.8.m8.1.1.1.2.cmml"><</mo><mi id="S6.SS2.30.p6.8.m8.1.1.1.3" xref="S6.SS2.30.p6.8.m8.1.1.1.3.cmml">X</mi></msub><mi id="S6.SS2.30.p6.8.m8.1.1.3" xref="S6.SS2.30.p6.8.m8.1.1.3.cmml">y</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.30.p6.8.m8.1b"><apply id="S6.SS2.30.p6.8.m8.1.1.cmml" xref="S6.SS2.30.p6.8.m8.1.1"><apply id="S6.SS2.30.p6.8.m8.1.1.1.cmml" xref="S6.SS2.30.p6.8.m8.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.30.p6.8.m8.1.1.1.1.cmml" xref="S6.SS2.30.p6.8.m8.1.1.1">subscript</csymbol><lt id="S6.SS2.30.p6.8.m8.1.1.1.2.cmml" xref="S6.SS2.30.p6.8.m8.1.1.1.2"></lt><ci id="S6.SS2.30.p6.8.m8.1.1.1.3.cmml" xref="S6.SS2.30.p6.8.m8.1.1.1.3">𝑋</ci></apply><ci id="S6.SS2.30.p6.8.m8.1.1.2.cmml" xref="S6.SS2.30.p6.8.m8.1.1.2">𝑧</ci><ci id="S6.SS2.30.p6.8.m8.1.1.3.cmml" xref="S6.SS2.30.p6.8.m8.1.1.3">𝑦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.30.p6.8.m8.1c">z<_{X}y</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.30.p6.8.m8.1d">italic_z < start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT italic_y</annotation></semantics></math>. This can be done since we know that <math alttext="y" class="ltx_Math" display="inline" id="S6.SS2.30.p6.9.m9.1"><semantics id="S6.SS2.30.p6.9.m9.1a"><mi id="S6.SS2.30.p6.9.m9.1.1" xref="S6.SS2.30.p6.9.m9.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.30.p6.9.m9.1b"><ci id="S6.SS2.30.p6.9.m9.1.1.cmml" xref="S6.SS2.30.p6.9.m9.1.1">𝑦</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.30.p6.9.m9.1c">y</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.30.p6.9.m9.1d">italic_y</annotation></semantics></math> is not the left endpoint. As in Case 1 one finds <math alttext="c_{l}" class="ltx_Math" display="inline" id="S6.SS2.30.p6.10.m10.1"><semantics id="S6.SS2.30.p6.10.m10.1a"><msub id="S6.SS2.30.p6.10.m10.1.1" xref="S6.SS2.30.p6.10.m10.1.1.cmml"><mi id="S6.SS2.30.p6.10.m10.1.1.2" xref="S6.SS2.30.p6.10.m10.1.1.2.cmml">c</mi><mi id="S6.SS2.30.p6.10.m10.1.1.3" xref="S6.SS2.30.p6.10.m10.1.1.3.cmml">l</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.30.p6.10.m10.1b"><apply id="S6.SS2.30.p6.10.m10.1.1.cmml" xref="S6.SS2.30.p6.10.m10.1.1"><csymbol cd="ambiguous" id="S6.SS2.30.p6.10.m10.1.1.1.cmml" xref="S6.SS2.30.p6.10.m10.1.1">subscript</csymbol><ci id="S6.SS2.30.p6.10.m10.1.1.2.cmml" xref="S6.SS2.30.p6.10.m10.1.1.2">𝑐</ci><ci id="S6.SS2.30.p6.10.m10.1.1.3.cmml" xref="S6.SS2.30.p6.10.m10.1.1.3">𝑙</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.30.p6.10.m10.1c">c_{l}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.30.p6.10.m10.1d">italic_c start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="c_{r}" class="ltx_Math" display="inline" id="S6.SS2.30.p6.11.m11.1"><semantics id="S6.SS2.30.p6.11.m11.1a"><msub id="S6.SS2.30.p6.11.m11.1.1" xref="S6.SS2.30.p6.11.m11.1.1.cmml"><mi id="S6.SS2.30.p6.11.m11.1.1.2" xref="S6.SS2.30.p6.11.m11.1.1.2.cmml">c</mi><mi id="S6.SS2.30.p6.11.m11.1.1.3" xref="S6.SS2.30.p6.11.m11.1.1.3.cmml">r</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.30.p6.11.m11.1b"><apply id="S6.SS2.30.p6.11.m11.1.1.cmml" xref="S6.SS2.30.p6.11.m11.1.1"><csymbol cd="ambiguous" id="S6.SS2.30.p6.11.m11.1.1.1.cmml" xref="S6.SS2.30.p6.11.m11.1.1">subscript</csymbol><ci id="S6.SS2.30.p6.11.m11.1.1.2.cmml" xref="S6.SS2.30.p6.11.m11.1.1.2">𝑐</ci><ci id="S6.SS2.30.p6.11.m11.1.1.3.cmml" xref="S6.SS2.30.p6.11.m11.1.1.3">𝑟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.30.p6.11.m11.1c">c_{r}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.30.p6.11.m11.1d">italic_c start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="c_{l}<_{A}c<_{A}c_{r}" class="ltx_Math" display="inline" id="S6.SS2.30.p6.12.m12.1"><semantics id="S6.SS2.30.p6.12.m12.1a"><mrow id="S6.SS2.30.p6.12.m12.1.1" xref="S6.SS2.30.p6.12.m12.1.1.cmml"><msub id="S6.SS2.30.p6.12.m12.1.1.2" xref="S6.SS2.30.p6.12.m12.1.1.2.cmml"><mi id="S6.SS2.30.p6.12.m12.1.1.2.2" xref="S6.SS2.30.p6.12.m12.1.1.2.2.cmml">c</mi><mi id="S6.SS2.30.p6.12.m12.1.1.2.3" xref="S6.SS2.30.p6.12.m12.1.1.2.3.cmml">l</mi></msub><msub id="S6.SS2.30.p6.12.m12.1.1.3" xref="S6.SS2.30.p6.12.m12.1.1.3.cmml"><mo id="S6.SS2.30.p6.12.m12.1.1.3.2" xref="S6.SS2.30.p6.12.m12.1.1.3.2.cmml"><</mo><mi id="S6.SS2.30.p6.12.m12.1.1.3.3" xref="S6.SS2.30.p6.12.m12.1.1.3.3.cmml">A</mi></msub><mi id="S6.SS2.30.p6.12.m12.1.1.4" xref="S6.SS2.30.p6.12.m12.1.1.4.cmml">c</mi><msub id="S6.SS2.30.p6.12.m12.1.1.5" xref="S6.SS2.30.p6.12.m12.1.1.5.cmml"><mo id="S6.SS2.30.p6.12.m12.1.1.5.2" xref="S6.SS2.30.p6.12.m12.1.1.5.2.cmml"><</mo><mi id="S6.SS2.30.p6.12.m12.1.1.5.3" xref="S6.SS2.30.p6.12.m12.1.1.5.3.cmml">A</mi></msub><msub id="S6.SS2.30.p6.12.m12.1.1.6" xref="S6.SS2.30.p6.12.m12.1.1.6.cmml"><mi id="S6.SS2.30.p6.12.m12.1.1.6.2" xref="S6.SS2.30.p6.12.m12.1.1.6.2.cmml">c</mi><mi id="S6.SS2.30.p6.12.m12.1.1.6.3" xref="S6.SS2.30.p6.12.m12.1.1.6.3.cmml">r</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.30.p6.12.m12.1b"><apply id="S6.SS2.30.p6.12.m12.1.1.cmml" xref="S6.SS2.30.p6.12.m12.1.1"><and id="S6.SS2.30.p6.12.m12.1.1a.cmml" xref="S6.SS2.30.p6.12.m12.1.1"></and><apply id="S6.SS2.30.p6.12.m12.1.1b.cmml" xref="S6.SS2.30.p6.12.m12.1.1"><apply id="S6.SS2.30.p6.12.m12.1.1.3.cmml" xref="S6.SS2.30.p6.12.m12.1.1.3"><csymbol cd="ambiguous" id="S6.SS2.30.p6.12.m12.1.1.3.1.cmml" xref="S6.SS2.30.p6.12.m12.1.1.3">subscript</csymbol><lt id="S6.SS2.30.p6.12.m12.1.1.3.2.cmml" xref="S6.SS2.30.p6.12.m12.1.1.3.2"></lt><ci id="S6.SS2.30.p6.12.m12.1.1.3.3.cmml" xref="S6.SS2.30.p6.12.m12.1.1.3.3">𝐴</ci></apply><apply id="S6.SS2.30.p6.12.m12.1.1.2.cmml" xref="S6.SS2.30.p6.12.m12.1.1.2"><csymbol cd="ambiguous" id="S6.SS2.30.p6.12.m12.1.1.2.1.cmml" xref="S6.SS2.30.p6.12.m12.1.1.2">subscript</csymbol><ci id="S6.SS2.30.p6.12.m12.1.1.2.2.cmml" xref="S6.SS2.30.p6.12.m12.1.1.2.2">𝑐</ci><ci id="S6.SS2.30.p6.12.m12.1.1.2.3.cmml" xref="S6.SS2.30.p6.12.m12.1.1.2.3">𝑙</ci></apply><ci id="S6.SS2.30.p6.12.m12.1.1.4.cmml" xref="S6.SS2.30.p6.12.m12.1.1.4">𝑐</ci></apply><apply id="S6.SS2.30.p6.12.m12.1.1c.cmml" xref="S6.SS2.30.p6.12.m12.1.1"><apply id="S6.SS2.30.p6.12.m12.1.1.5.cmml" xref="S6.SS2.30.p6.12.m12.1.1.5"><csymbol cd="ambiguous" id="S6.SS2.30.p6.12.m12.1.1.5.1.cmml" xref="S6.SS2.30.p6.12.m12.1.1.5">subscript</csymbol><lt id="S6.SS2.30.p6.12.m12.1.1.5.2.cmml" xref="S6.SS2.30.p6.12.m12.1.1.5.2"></lt><ci id="S6.SS2.30.p6.12.m12.1.1.5.3.cmml" xref="S6.SS2.30.p6.12.m12.1.1.5.3">𝐴</ci></apply><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.30.p6.12.m12.1.1.4.cmml" id="S6.SS2.30.p6.12.m12.1.1d.cmml" xref="S6.SS2.30.p6.12.m12.1.1"></share><apply id="S6.SS2.30.p6.12.m12.1.1.6.cmml" xref="S6.SS2.30.p6.12.m12.1.1.6"><csymbol cd="ambiguous" id="S6.SS2.30.p6.12.m12.1.1.6.1.cmml" xref="S6.SS2.30.p6.12.m12.1.1.6">subscript</csymbol><ci id="S6.SS2.30.p6.12.m12.1.1.6.2.cmml" xref="S6.SS2.30.p6.12.m12.1.1.6.2">𝑐</ci><ci id="S6.SS2.30.p6.12.m12.1.1.6.3.cmml" xref="S6.SS2.30.p6.12.m12.1.1.6.3">𝑟</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.30.p6.12.m12.1c">c_{l}<_{A}c<_{A}c_{r}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.30.p6.12.m12.1d">italic_c start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT < start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_c < start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_c start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\nu(c_{l})=\nu(c_{r})=\nu" class="ltx_Math" display="inline" id="S6.SS2.30.p6.13.m13.2"><semantics id="S6.SS2.30.p6.13.m13.2a"><mrow id="S6.SS2.30.p6.13.m13.2.2" xref="S6.SS2.30.p6.13.m13.2.2.cmml"><mrow id="S6.SS2.30.p6.13.m13.1.1.1" xref="S6.SS2.30.p6.13.m13.1.1.1.cmml"><mi id="S6.SS2.30.p6.13.m13.1.1.1.3" xref="S6.SS2.30.p6.13.m13.1.1.1.3.cmml">ν</mi><mo id="S6.SS2.30.p6.13.m13.1.1.1.2" xref="S6.SS2.30.p6.13.m13.1.1.1.2.cmml">⁢</mo><mrow id="S6.SS2.30.p6.13.m13.1.1.1.1.1" xref="S6.SS2.30.p6.13.m13.1.1.1.1.1.1.cmml"><mo id="S6.SS2.30.p6.13.m13.1.1.1.1.1.2" stretchy="false" xref="S6.SS2.30.p6.13.m13.1.1.1.1.1.1.cmml">(</mo><msub id="S6.SS2.30.p6.13.m13.1.1.1.1.1.1" xref="S6.SS2.30.p6.13.m13.1.1.1.1.1.1.cmml"><mi id="S6.SS2.30.p6.13.m13.1.1.1.1.1.1.2" xref="S6.SS2.30.p6.13.m13.1.1.1.1.1.1.2.cmml">c</mi><mi id="S6.SS2.30.p6.13.m13.1.1.1.1.1.1.3" xref="S6.SS2.30.p6.13.m13.1.1.1.1.1.1.3.cmml">l</mi></msub><mo id="S6.SS2.30.p6.13.m13.1.1.1.1.1.3" stretchy="false" xref="S6.SS2.30.p6.13.m13.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.SS2.30.p6.13.m13.2.2.4" xref="S6.SS2.30.p6.13.m13.2.2.4.cmml">=</mo><mrow id="S6.SS2.30.p6.13.m13.2.2.2" xref="S6.SS2.30.p6.13.m13.2.2.2.cmml"><mi id="S6.SS2.30.p6.13.m13.2.2.2.3" xref="S6.SS2.30.p6.13.m13.2.2.2.3.cmml">ν</mi><mo id="S6.SS2.30.p6.13.m13.2.2.2.2" xref="S6.SS2.30.p6.13.m13.2.2.2.2.cmml">⁢</mo><mrow id="S6.SS2.30.p6.13.m13.2.2.2.1.1" xref="S6.SS2.30.p6.13.m13.2.2.2.1.1.1.cmml"><mo id="S6.SS2.30.p6.13.m13.2.2.2.1.1.2" stretchy="false" xref="S6.SS2.30.p6.13.m13.2.2.2.1.1.1.cmml">(</mo><msub id="S6.SS2.30.p6.13.m13.2.2.2.1.1.1" xref="S6.SS2.30.p6.13.m13.2.2.2.1.1.1.cmml"><mi id="S6.SS2.30.p6.13.m13.2.2.2.1.1.1.2" xref="S6.SS2.30.p6.13.m13.2.2.2.1.1.1.2.cmml">c</mi><mi id="S6.SS2.30.p6.13.m13.2.2.2.1.1.1.3" xref="S6.SS2.30.p6.13.m13.2.2.2.1.1.1.3.cmml">r</mi></msub><mo id="S6.SS2.30.p6.13.m13.2.2.2.1.1.3" stretchy="false" xref="S6.SS2.30.p6.13.m13.2.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.SS2.30.p6.13.m13.2.2.5" xref="S6.SS2.30.p6.13.m13.2.2.5.cmml">=</mo><mi id="S6.SS2.30.p6.13.m13.2.2.6" xref="S6.SS2.30.p6.13.m13.2.2.6.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.30.p6.13.m13.2b"><apply id="S6.SS2.30.p6.13.m13.2.2.cmml" xref="S6.SS2.30.p6.13.m13.2.2"><and id="S6.SS2.30.p6.13.m13.2.2a.cmml" xref="S6.SS2.30.p6.13.m13.2.2"></and><apply id="S6.SS2.30.p6.13.m13.2.2b.cmml" xref="S6.SS2.30.p6.13.m13.2.2"><eq id="S6.SS2.30.p6.13.m13.2.2.4.cmml" xref="S6.SS2.30.p6.13.m13.2.2.4"></eq><apply id="S6.SS2.30.p6.13.m13.1.1.1.cmml" xref="S6.SS2.30.p6.13.m13.1.1.1"><times id="S6.SS2.30.p6.13.m13.1.1.1.2.cmml" xref="S6.SS2.30.p6.13.m13.1.1.1.2"></times><ci id="S6.SS2.30.p6.13.m13.1.1.1.3.cmml" xref="S6.SS2.30.p6.13.m13.1.1.1.3">𝜈</ci><apply id="S6.SS2.30.p6.13.m13.1.1.1.1.1.1.cmml" xref="S6.SS2.30.p6.13.m13.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.30.p6.13.m13.1.1.1.1.1.1.1.cmml" xref="S6.SS2.30.p6.13.m13.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.30.p6.13.m13.1.1.1.1.1.1.2.cmml" xref="S6.SS2.30.p6.13.m13.1.1.1.1.1.1.2">𝑐</ci><ci id="S6.SS2.30.p6.13.m13.1.1.1.1.1.1.3.cmml" xref="S6.SS2.30.p6.13.m13.1.1.1.1.1.1.3">𝑙</ci></apply></apply><apply id="S6.SS2.30.p6.13.m13.2.2.2.cmml" xref="S6.SS2.30.p6.13.m13.2.2.2"><times id="S6.SS2.30.p6.13.m13.2.2.2.2.cmml" xref="S6.SS2.30.p6.13.m13.2.2.2.2"></times><ci id="S6.SS2.30.p6.13.m13.2.2.2.3.cmml" xref="S6.SS2.30.p6.13.m13.2.2.2.3">𝜈</ci><apply id="S6.SS2.30.p6.13.m13.2.2.2.1.1.1.cmml" xref="S6.SS2.30.p6.13.m13.2.2.2.1.1"><csymbol cd="ambiguous" id="S6.SS2.30.p6.13.m13.2.2.2.1.1.1.1.cmml" xref="S6.SS2.30.p6.13.m13.2.2.2.1.1">subscript</csymbol><ci id="S6.SS2.30.p6.13.m13.2.2.2.1.1.1.2.cmml" xref="S6.SS2.30.p6.13.m13.2.2.2.1.1.1.2">𝑐</ci><ci id="S6.SS2.30.p6.13.m13.2.2.2.1.1.1.3.cmml" xref="S6.SS2.30.p6.13.m13.2.2.2.1.1.1.3">𝑟</ci></apply></apply></apply><apply id="S6.SS2.30.p6.13.m13.2.2c.cmml" xref="S6.SS2.30.p6.13.m13.2.2"><eq id="S6.SS2.30.p6.13.m13.2.2.5.cmml" xref="S6.SS2.30.p6.13.m13.2.2.5"></eq><share href="https://arxiv.org/html/2503.13728v1#S6.SS2.30.p6.13.m13.2.2.2.cmml" id="S6.SS2.30.p6.13.m13.2.2d.cmml" xref="S6.SS2.30.p6.13.m13.2.2"></share><ci id="S6.SS2.30.p6.13.m13.2.2.6.cmml" xref="S6.SS2.30.p6.13.m13.2.2.6">𝜈</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.30.p6.13.m13.2c">\nu(c_{l})=\nu(c_{r})=\nu</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.30.p6.13.m13.2d">italic_ν ( italic_c start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT ) = italic_ν ( italic_c start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) = italic_ν</annotation></semantics></math> and <math alttext="\nu(c_{l},c_{r})=\nu" class="ltx_Math" display="inline" id="S6.SS2.30.p6.14.m14.2"><semantics id="S6.SS2.30.p6.14.m14.2a"><mrow id="S6.SS2.30.p6.14.m14.2.2" xref="S6.SS2.30.p6.14.m14.2.2.cmml"><mrow id="S6.SS2.30.p6.14.m14.2.2.2" xref="S6.SS2.30.p6.14.m14.2.2.2.cmml"><mi id="S6.SS2.30.p6.14.m14.2.2.2.4" xref="S6.SS2.30.p6.14.m14.2.2.2.4.cmml">ν</mi><mo id="S6.SS2.30.p6.14.m14.2.2.2.3" xref="S6.SS2.30.p6.14.m14.2.2.2.3.cmml">⁢</mo><mrow id="S6.SS2.30.p6.14.m14.2.2.2.2.2" xref="S6.SS2.30.p6.14.m14.2.2.2.2.3.cmml"><mo id="S6.SS2.30.p6.14.m14.2.2.2.2.2.3" stretchy="false" xref="S6.SS2.30.p6.14.m14.2.2.2.2.3.cmml">(</mo><msub id="S6.SS2.30.p6.14.m14.1.1.1.1.1.1" xref="S6.SS2.30.p6.14.m14.1.1.1.1.1.1.cmml"><mi id="S6.SS2.30.p6.14.m14.1.1.1.1.1.1.2" xref="S6.SS2.30.p6.14.m14.1.1.1.1.1.1.2.cmml">c</mi><mi id="S6.SS2.30.p6.14.m14.1.1.1.1.1.1.3" xref="S6.SS2.30.p6.14.m14.1.1.1.1.1.1.3.cmml">l</mi></msub><mo id="S6.SS2.30.p6.14.m14.2.2.2.2.2.4" xref="S6.SS2.30.p6.14.m14.2.2.2.2.3.cmml">,</mo><msub id="S6.SS2.30.p6.14.m14.2.2.2.2.2.2" xref="S6.SS2.30.p6.14.m14.2.2.2.2.2.2.cmml"><mi id="S6.SS2.30.p6.14.m14.2.2.2.2.2.2.2" xref="S6.SS2.30.p6.14.m14.2.2.2.2.2.2.2.cmml">c</mi><mi id="S6.SS2.30.p6.14.m14.2.2.2.2.2.2.3" xref="S6.SS2.30.p6.14.m14.2.2.2.2.2.2.3.cmml">r</mi></msub><mo id="S6.SS2.30.p6.14.m14.2.2.2.2.2.5" stretchy="false" xref="S6.SS2.30.p6.14.m14.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.SS2.30.p6.14.m14.2.2.3" xref="S6.SS2.30.p6.14.m14.2.2.3.cmml">=</mo><mi id="S6.SS2.30.p6.14.m14.2.2.4" xref="S6.SS2.30.p6.14.m14.2.2.4.cmml">ν</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.30.p6.14.m14.2b"><apply id="S6.SS2.30.p6.14.m14.2.2.cmml" xref="S6.SS2.30.p6.14.m14.2.2"><eq id="S6.SS2.30.p6.14.m14.2.2.3.cmml" xref="S6.SS2.30.p6.14.m14.2.2.3"></eq><apply id="S6.SS2.30.p6.14.m14.2.2.2.cmml" xref="S6.SS2.30.p6.14.m14.2.2.2"><times id="S6.SS2.30.p6.14.m14.2.2.2.3.cmml" xref="S6.SS2.30.p6.14.m14.2.2.2.3"></times><ci id="S6.SS2.30.p6.14.m14.2.2.2.4.cmml" xref="S6.SS2.30.p6.14.m14.2.2.2.4">𝜈</ci><interval closure="open" id="S6.SS2.30.p6.14.m14.2.2.2.2.3.cmml" xref="S6.SS2.30.p6.14.m14.2.2.2.2.2"><apply id="S6.SS2.30.p6.14.m14.1.1.1.1.1.1.cmml" xref="S6.SS2.30.p6.14.m14.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.30.p6.14.m14.1.1.1.1.1.1.1.cmml" xref="S6.SS2.30.p6.14.m14.1.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.30.p6.14.m14.1.1.1.1.1.1.2.cmml" xref="S6.SS2.30.p6.14.m14.1.1.1.1.1.1.2">𝑐</ci><ci id="S6.SS2.30.p6.14.m14.1.1.1.1.1.1.3.cmml" xref="S6.SS2.30.p6.14.m14.1.1.1.1.1.1.3">𝑙</ci></apply><apply id="S6.SS2.30.p6.14.m14.2.2.2.2.2.2.cmml" xref="S6.SS2.30.p6.14.m14.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.30.p6.14.m14.2.2.2.2.2.2.1.cmml" xref="S6.SS2.30.p6.14.m14.2.2.2.2.2.2">subscript</csymbol><ci id="S6.SS2.30.p6.14.m14.2.2.2.2.2.2.2.cmml" xref="S6.SS2.30.p6.14.m14.2.2.2.2.2.2.2">𝑐</ci><ci id="S6.SS2.30.p6.14.m14.2.2.2.2.2.2.3.cmml" xref="S6.SS2.30.p6.14.m14.2.2.2.2.2.2.3">𝑟</ci></apply></interval></apply><ci id="S6.SS2.30.p6.14.m14.2.2.4.cmml" xref="S6.SS2.30.p6.14.m14.2.2.4">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.30.p6.14.m14.2c">\nu(c_{l},c_{r})=\nu</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.30.p6.14.m14.2d">italic_ν ( italic_c start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT , italic_c start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) = italic_ν</annotation></semantics></math> and thus <math alttext="p\cup\{((c_{l},c_{r}),z)\}\in P_{E}" class="ltx_Math" display="inline" id="S6.SS2.30.p6.15.m15.2"><semantics id="S6.SS2.30.p6.15.m15.2a"><mrow id="S6.SS2.30.p6.15.m15.2.2" xref="S6.SS2.30.p6.15.m15.2.2.cmml"><mrow id="S6.SS2.30.p6.15.m15.2.2.1" xref="S6.SS2.30.p6.15.m15.2.2.1.cmml"><mi id="S6.SS2.30.p6.15.m15.2.2.1.3" xref="S6.SS2.30.p6.15.m15.2.2.1.3.cmml">p</mi><mo id="S6.SS2.30.p6.15.m15.2.2.1.2" xref="S6.SS2.30.p6.15.m15.2.2.1.2.cmml">∪</mo><mrow id="S6.SS2.30.p6.15.m15.2.2.1.1.1" xref="S6.SS2.30.p6.15.m15.2.2.1.1.2.cmml"><mo id="S6.SS2.30.p6.15.m15.2.2.1.1.1.2" stretchy="false" xref="S6.SS2.30.p6.15.m15.2.2.1.1.2.cmml">{</mo><mrow id="S6.SS2.30.p6.15.m15.2.2.1.1.1.1.1" xref="S6.SS2.30.p6.15.m15.2.2.1.1.1.1.2.cmml"><mo id="S6.SS2.30.p6.15.m15.2.2.1.1.1.1.1.2" stretchy="false" xref="S6.SS2.30.p6.15.m15.2.2.1.1.1.1.2.cmml">(</mo><mrow id="S6.SS2.30.p6.15.m15.2.2.1.1.1.1.1.1.2" xref="S6.SS2.30.p6.15.m15.2.2.1.1.1.1.1.1.3.cmml"><mo id="S6.SS2.30.p6.15.m15.2.2.1.1.1.1.1.1.2.3" stretchy="false" xref="S6.SS2.30.p6.15.m15.2.2.1.1.1.1.1.1.3.cmml">(</mo><msub id="S6.SS2.30.p6.15.m15.2.2.1.1.1.1.1.1.1.1" xref="S6.SS2.30.p6.15.m15.2.2.1.1.1.1.1.1.1.1.cmml"><mi id="S6.SS2.30.p6.15.m15.2.2.1.1.1.1.1.1.1.1.2" xref="S6.SS2.30.p6.15.m15.2.2.1.1.1.1.1.1.1.1.2.cmml">c</mi><mi id="S6.SS2.30.p6.15.m15.2.2.1.1.1.1.1.1.1.1.3" xref="S6.SS2.30.p6.15.m15.2.2.1.1.1.1.1.1.1.1.3.cmml">l</mi></msub><mo id="S6.SS2.30.p6.15.m15.2.2.1.1.1.1.1.1.2.4" xref="S6.SS2.30.p6.15.m15.2.2.1.1.1.1.1.1.3.cmml">,</mo><msub id="S6.SS2.30.p6.15.m15.2.2.1.1.1.1.1.1.2.2" xref="S6.SS2.30.p6.15.m15.2.2.1.1.1.1.1.1.2.2.cmml"><mi id="S6.SS2.30.p6.15.m15.2.2.1.1.1.1.1.1.2.2.2" xref="S6.SS2.30.p6.15.m15.2.2.1.1.1.1.1.1.2.2.2.cmml">c</mi><mi id="S6.SS2.30.p6.15.m15.2.2.1.1.1.1.1.1.2.2.3" xref="S6.SS2.30.p6.15.m15.2.2.1.1.1.1.1.1.2.2.3.cmml">r</mi></msub><mo id="S6.SS2.30.p6.15.m15.2.2.1.1.1.1.1.1.2.5" stretchy="false" xref="S6.SS2.30.p6.15.m15.2.2.1.1.1.1.1.1.3.cmml">)</mo></mrow><mo id="S6.SS2.30.p6.15.m15.2.2.1.1.1.1.1.3" xref="S6.SS2.30.p6.15.m15.2.2.1.1.1.1.2.cmml">,</mo><mi id="S6.SS2.30.p6.15.m15.1.1" xref="S6.SS2.30.p6.15.m15.1.1.cmml">z</mi><mo id="S6.SS2.30.p6.15.m15.2.2.1.1.1.1.1.4" stretchy="false" xref="S6.SS2.30.p6.15.m15.2.2.1.1.1.1.2.cmml">)</mo></mrow><mo id="S6.SS2.30.p6.15.m15.2.2.1.1.1.3" stretchy="false" xref="S6.SS2.30.p6.15.m15.2.2.1.1.2.cmml">}</mo></mrow></mrow><mo id="S6.SS2.30.p6.15.m15.2.2.2" xref="S6.SS2.30.p6.15.m15.2.2.2.cmml">∈</mo><msub id="S6.SS2.30.p6.15.m15.2.2.3" xref="S6.SS2.30.p6.15.m15.2.2.3.cmml"><mi id="S6.SS2.30.p6.15.m15.2.2.3.2" xref="S6.SS2.30.p6.15.m15.2.2.3.2.cmml">P</mi><mi id="S6.SS2.30.p6.15.m15.2.2.3.3" xref="S6.SS2.30.p6.15.m15.2.2.3.3.cmml">E</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.30.p6.15.m15.2b"><apply id="S6.SS2.30.p6.15.m15.2.2.cmml" xref="S6.SS2.30.p6.15.m15.2.2"><in id="S6.SS2.30.p6.15.m15.2.2.2.cmml" xref="S6.SS2.30.p6.15.m15.2.2.2"></in><apply id="S6.SS2.30.p6.15.m15.2.2.1.cmml" xref="S6.SS2.30.p6.15.m15.2.2.1"><union id="S6.SS2.30.p6.15.m15.2.2.1.2.cmml" xref="S6.SS2.30.p6.15.m15.2.2.1.2"></union><ci id="S6.SS2.30.p6.15.m15.2.2.1.3.cmml" xref="S6.SS2.30.p6.15.m15.2.2.1.3">𝑝</ci><set id="S6.SS2.30.p6.15.m15.2.2.1.1.2.cmml" xref="S6.SS2.30.p6.15.m15.2.2.1.1.1"><interval closure="open" id="S6.SS2.30.p6.15.m15.2.2.1.1.1.1.2.cmml" xref="S6.SS2.30.p6.15.m15.2.2.1.1.1.1.1"><interval closure="open" id="S6.SS2.30.p6.15.m15.2.2.1.1.1.1.1.1.3.cmml" xref="S6.SS2.30.p6.15.m15.2.2.1.1.1.1.1.1.2"><apply id="S6.SS2.30.p6.15.m15.2.2.1.1.1.1.1.1.1.1.cmml" xref="S6.SS2.30.p6.15.m15.2.2.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.30.p6.15.m15.2.2.1.1.1.1.1.1.1.1.1.cmml" xref="S6.SS2.30.p6.15.m15.2.2.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S6.SS2.30.p6.15.m15.2.2.1.1.1.1.1.1.1.1.2.cmml" xref="S6.SS2.30.p6.15.m15.2.2.1.1.1.1.1.1.1.1.2">𝑐</ci><ci id="S6.SS2.30.p6.15.m15.2.2.1.1.1.1.1.1.1.1.3.cmml" xref="S6.SS2.30.p6.15.m15.2.2.1.1.1.1.1.1.1.1.3">𝑙</ci></apply><apply id="S6.SS2.30.p6.15.m15.2.2.1.1.1.1.1.1.2.2.cmml" xref="S6.SS2.30.p6.15.m15.2.2.1.1.1.1.1.1.2.2"><csymbol cd="ambiguous" id="S6.SS2.30.p6.15.m15.2.2.1.1.1.1.1.1.2.2.1.cmml" xref="S6.SS2.30.p6.15.m15.2.2.1.1.1.1.1.1.2.2">subscript</csymbol><ci id="S6.SS2.30.p6.15.m15.2.2.1.1.1.1.1.1.2.2.2.cmml" xref="S6.SS2.30.p6.15.m15.2.2.1.1.1.1.1.1.2.2.2">𝑐</ci><ci id="S6.SS2.30.p6.15.m15.2.2.1.1.1.1.1.1.2.2.3.cmml" xref="S6.SS2.30.p6.15.m15.2.2.1.1.1.1.1.1.2.2.3">𝑟</ci></apply></interval><ci id="S6.SS2.30.p6.15.m15.1.1.cmml" xref="S6.SS2.30.p6.15.m15.1.1">𝑧</ci></interval></set></apply><apply id="S6.SS2.30.p6.15.m15.2.2.3.cmml" xref="S6.SS2.30.p6.15.m15.2.2.3"><csymbol cd="ambiguous" id="S6.SS2.30.p6.15.m15.2.2.3.1.cmml" xref="S6.SS2.30.p6.15.m15.2.2.3">subscript</csymbol><ci id="S6.SS2.30.p6.15.m15.2.2.3.2.cmml" xref="S6.SS2.30.p6.15.m15.2.2.3.2">𝑃</ci><ci id="S6.SS2.30.p6.15.m15.2.2.3.3.cmml" xref="S6.SS2.30.p6.15.m15.2.2.3.3">𝐸</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.30.p6.15.m15.2c">p\cup\{((c_{l},c_{r}),z)\}\in P_{E}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.30.p6.15.m15.2d">italic_p ∪ { ( ( italic_c start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT , italic_c start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) , italic_z ) } ∈ italic_P start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT</annotation></semantics></math>. ∎</p> </div> </div> </section> </section> <section class="ltx_section" id="S7"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">7. </span>A two element basis for the Aronszajn lines</h2> <div class="ltx_para" id="S7.p1"> <p class="ltx_p" id="S7.p1.1">Although we have shown that the analogy between countable linear orders and Aronszajn lines does not extend perfectly to <math alttext="\trianglelefteq" class="ltx_Math" display="inline" id="S7.p1.1.m1.1"><semantics id="S7.p1.1.m1.1a"><mi id="S7.p1.1.m1.1.1" mathvariant="normal" xref="S7.p1.1.m1.1.1.cmml">⊴</mi><annotation-xml encoding="MathML-Content" id="S7.p1.1.m1.1b"><ci id="S7.p1.1.m1.1.1.cmml" xref="S7.p1.1.m1.1.1">⊴</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.p1.1.m1.1c">\trianglelefteq</annotation><annotation encoding="application/x-llamapun" id="S7.p1.1.m1.1d">⊴</annotation></semantics></math>, we do have a positive result.</p> </div> <div class="ltx_para" id="S7.p2"> <p class="ltx_p" id="S7.p2.10">Recall that in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib4" title="">4</a>]</cite> it is shown that <math alttext="\omega" class="ltx_Math" display="inline" id="S7.p2.1.m1.1"><semantics id="S7.p2.1.m1.1a"><mi id="S7.p2.1.m1.1.1" xref="S7.p2.1.m1.1.1.cmml">ω</mi><annotation-xml encoding="MathML-Content" id="S7.p2.1.m1.1b"><ci id="S7.p2.1.m1.1.1.cmml" xref="S7.p2.1.m1.1.1">𝜔</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.p2.1.m1.1c">\omega</annotation><annotation encoding="application/x-llamapun" id="S7.p2.1.m1.1d">italic_ω</annotation></semantics></math>, <math alttext="\omega+1" class="ltx_Math" display="inline" id="S7.p2.2.m2.1"><semantics id="S7.p2.2.m2.1a"><mrow id="S7.p2.2.m2.1.1" xref="S7.p2.2.m2.1.1.cmml"><mi id="S7.p2.2.m2.1.1.2" xref="S7.p2.2.m2.1.1.2.cmml">ω</mi><mo id="S7.p2.2.m2.1.1.1" xref="S7.p2.2.m2.1.1.1.cmml">+</mo><mn id="S7.p2.2.m2.1.1.3" xref="S7.p2.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.p2.2.m2.1b"><apply id="S7.p2.2.m2.1.1.cmml" xref="S7.p2.2.m2.1.1"><plus id="S7.p2.2.m2.1.1.1.cmml" xref="S7.p2.2.m2.1.1.1"></plus><ci id="S7.p2.2.m2.1.1.2.cmml" xref="S7.p2.2.m2.1.1.2">𝜔</ci><cn id="S7.p2.2.m2.1.1.3.cmml" type="integer" xref="S7.p2.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.p2.2.m2.1c">\omega+1</annotation><annotation encoding="application/x-llamapun" id="S7.p2.2.m2.1d">italic_ω + 1</annotation></semantics></math> and <math alttext="\omega^{\star}" class="ltx_Math" display="inline" id="S7.p2.3.m3.1"><semantics id="S7.p2.3.m3.1a"><msup id="S7.p2.3.m3.1.1" xref="S7.p2.3.m3.1.1.cmml"><mi id="S7.p2.3.m3.1.1.2" xref="S7.p2.3.m3.1.1.2.cmml">ω</mi><mo id="S7.p2.3.m3.1.1.3" xref="S7.p2.3.m3.1.1.3.cmml">⋆</mo></msup><annotation-xml encoding="MathML-Content" id="S7.p2.3.m3.1b"><apply id="S7.p2.3.m3.1.1.cmml" xref="S7.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S7.p2.3.m3.1.1.1.cmml" xref="S7.p2.3.m3.1.1">superscript</csymbol><ci id="S7.p2.3.m3.1.1.2.cmml" xref="S7.p2.3.m3.1.1.2">𝜔</ci><ci id="S7.p2.3.m3.1.1.3.cmml" xref="S7.p2.3.m3.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.p2.3.m3.1c">\omega^{\star}</annotation><annotation encoding="application/x-llamapun" id="S7.p2.3.m3.1d">italic_ω start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="1+\omega^{\star}" class="ltx_Math" display="inline" id="S7.p2.4.m4.1"><semantics id="S7.p2.4.m4.1a"><mrow id="S7.p2.4.m4.1.1" xref="S7.p2.4.m4.1.1.cmml"><mn id="S7.p2.4.m4.1.1.2" xref="S7.p2.4.m4.1.1.2.cmml">1</mn><mo id="S7.p2.4.m4.1.1.1" xref="S7.p2.4.m4.1.1.1.cmml">+</mo><msup id="S7.p2.4.m4.1.1.3" xref="S7.p2.4.m4.1.1.3.cmml"><mi id="S7.p2.4.m4.1.1.3.2" xref="S7.p2.4.m4.1.1.3.2.cmml">ω</mi><mo id="S7.p2.4.m4.1.1.3.3" xref="S7.p2.4.m4.1.1.3.3.cmml">⋆</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S7.p2.4.m4.1b"><apply id="S7.p2.4.m4.1.1.cmml" xref="S7.p2.4.m4.1.1"><plus id="S7.p2.4.m4.1.1.1.cmml" xref="S7.p2.4.m4.1.1.1"></plus><cn id="S7.p2.4.m4.1.1.2.cmml" type="integer" xref="S7.p2.4.m4.1.1.2">1</cn><apply id="S7.p2.4.m4.1.1.3.cmml" xref="S7.p2.4.m4.1.1.3"><csymbol cd="ambiguous" id="S7.p2.4.m4.1.1.3.1.cmml" xref="S7.p2.4.m4.1.1.3">superscript</csymbol><ci id="S7.p2.4.m4.1.1.3.2.cmml" xref="S7.p2.4.m4.1.1.3.2">𝜔</ci><ci id="S7.p2.4.m4.1.1.3.3.cmml" xref="S7.p2.4.m4.1.1.3.3">⋆</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.p2.4.m4.1c">1+\omega^{\star}</annotation><annotation encoding="application/x-llamapun" id="S7.p2.4.m4.1d">1 + italic_ω start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math> form a <math alttext="\trianglelefteq" class="ltx_Math" display="inline" id="S7.p2.5.m5.1"><semantics id="S7.p2.5.m5.1a"><mi id="S7.p2.5.m5.1.1" mathvariant="normal" xref="S7.p2.5.m5.1.1.cmml">⊴</mi><annotation-xml encoding="MathML-Content" id="S7.p2.5.m5.1b"><ci id="S7.p2.5.m5.1.1.cmml" xref="S7.p2.5.m5.1.1">⊴</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.p2.5.m5.1c">\trianglelefteq</annotation><annotation encoding="application/x-llamapun" id="S7.p2.5.m5.1d">⊴</annotation></semantics></math>-basis for the countable linear orders. We do have an extension of this to the class of Countryman lines. Using the results of <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S4" title="4. Aronszajn line decompositions ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">4</span></a> let let <math alttext="C" class="ltx_Math" display="inline" id="S7.p2.6.m6.1"><semantics id="S7.p2.6.m6.1a"><mi id="S7.p2.6.m6.1.1" xref="S7.p2.6.m6.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S7.p2.6.m6.1b"><ci id="S7.p2.6.m6.1.1.cmml" xref="S7.p2.6.m6.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.p2.6.m6.1c">C</annotation><annotation encoding="application/x-llamapun" id="S7.p2.6.m6.1d">italic_C</annotation></semantics></math> be an <math alttext="\aleph_{1}" class="ltx_Math" display="inline" id="S7.p2.7.m7.1"><semantics id="S7.p2.7.m7.1a"><msub id="S7.p2.7.m7.1.1" xref="S7.p2.7.m7.1.1.cmml"><mi id="S7.p2.7.m7.1.1.2" mathvariant="normal" xref="S7.p2.7.m7.1.1.2.cmml">ℵ</mi><mn id="S7.p2.7.m7.1.1.3" xref="S7.p2.7.m7.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S7.p2.7.m7.1b"><apply id="S7.p2.7.m7.1.1.cmml" xref="S7.p2.7.m7.1.1"><csymbol cd="ambiguous" id="S7.p2.7.m7.1.1.1.cmml" xref="S7.p2.7.m7.1.1">subscript</csymbol><ci id="S7.p2.7.m7.1.1.2.cmml" xref="S7.p2.7.m7.1.1.2">ℵ</ci><cn id="S7.p2.7.m7.1.1.3.cmml" type="integer" xref="S7.p2.7.m7.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.p2.7.m7.1c">\aleph_{1}</annotation><annotation encoding="application/x-llamapun" id="S7.p2.7.m7.1d">roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-dense Countryman and <math alttext="D" class="ltx_Math" display="inline" id="S7.p2.8.m8.1"><semantics id="S7.p2.8.m8.1a"><mi id="S7.p2.8.m8.1.1" xref="S7.p2.8.m8.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S7.p2.8.m8.1b"><ci id="S7.p2.8.m8.1.1.cmml" xref="S7.p2.8.m8.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.p2.8.m8.1c">D</annotation><annotation encoding="application/x-llamapun" id="S7.p2.8.m8.1d">italic_D</annotation></semantics></math> a decomposition for <math alttext="C" class="ltx_Math" display="inline" id="S7.p2.9.m9.1"><semantics id="S7.p2.9.m9.1a"><mi id="S7.p2.9.m9.1.1" xref="S7.p2.9.m9.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S7.p2.9.m9.1b"><ci id="S7.p2.9.m9.1.1.cmml" xref="S7.p2.9.m9.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.p2.9.m9.1c">C</annotation><annotation encoding="application/x-llamapun" id="S7.p2.9.m9.1d">italic_C</annotation></semantics></math> such that <math alttext="\hat{\mathscr{L}}(C,D)=\hat{\mathscr{R}}(C,D)=\omega_{1}" class="ltx_Math" display="inline" id="S7.p2.10.m10.4"><semantics id="S7.p2.10.m10.4a"><mrow id="S7.p2.10.m10.4.5" xref="S7.p2.10.m10.4.5.cmml"><mrow id="S7.p2.10.m10.4.5.2" xref="S7.p2.10.m10.4.5.2.cmml"><mover accent="true" id="S7.p2.10.m10.4.5.2.2" xref="S7.p2.10.m10.4.5.2.2.cmml"><mi class="ltx_font_mathscript" id="S7.p2.10.m10.4.5.2.2.2" xref="S7.p2.10.m10.4.5.2.2.2.cmml">ℒ</mi><mo id="S7.p2.10.m10.4.5.2.2.1" xref="S7.p2.10.m10.4.5.2.2.1.cmml">^</mo></mover><mo id="S7.p2.10.m10.4.5.2.1" xref="S7.p2.10.m10.4.5.2.1.cmml">⁢</mo><mrow id="S7.p2.10.m10.4.5.2.3.2" xref="S7.p2.10.m10.4.5.2.3.1.cmml"><mo id="S7.p2.10.m10.4.5.2.3.2.1" stretchy="false" xref="S7.p2.10.m10.4.5.2.3.1.cmml">(</mo><mi id="S7.p2.10.m10.1.1" xref="S7.p2.10.m10.1.1.cmml">C</mi><mo id="S7.p2.10.m10.4.5.2.3.2.2" xref="S7.p2.10.m10.4.5.2.3.1.cmml">,</mo><mi id="S7.p2.10.m10.2.2" xref="S7.p2.10.m10.2.2.cmml">D</mi><mo id="S7.p2.10.m10.4.5.2.3.2.3" stretchy="false" xref="S7.p2.10.m10.4.5.2.3.1.cmml">)</mo></mrow></mrow><mo id="S7.p2.10.m10.4.5.3" xref="S7.p2.10.m10.4.5.3.cmml">=</mo><mrow id="S7.p2.10.m10.4.5.4" xref="S7.p2.10.m10.4.5.4.cmml"><mover accent="true" id="S7.p2.10.m10.4.5.4.2" xref="S7.p2.10.m10.4.5.4.2.cmml"><mi class="ltx_font_mathscript" id="S7.p2.10.m10.4.5.4.2.2" xref="S7.p2.10.m10.4.5.4.2.2.cmml">ℛ</mi><mo id="S7.p2.10.m10.4.5.4.2.1" xref="S7.p2.10.m10.4.5.4.2.1.cmml">^</mo></mover><mo id="S7.p2.10.m10.4.5.4.1" xref="S7.p2.10.m10.4.5.4.1.cmml">⁢</mo><mrow id="S7.p2.10.m10.4.5.4.3.2" xref="S7.p2.10.m10.4.5.4.3.1.cmml"><mo id="S7.p2.10.m10.4.5.4.3.2.1" stretchy="false" xref="S7.p2.10.m10.4.5.4.3.1.cmml">(</mo><mi id="S7.p2.10.m10.3.3" xref="S7.p2.10.m10.3.3.cmml">C</mi><mo id="S7.p2.10.m10.4.5.4.3.2.2" xref="S7.p2.10.m10.4.5.4.3.1.cmml">,</mo><mi id="S7.p2.10.m10.4.4" xref="S7.p2.10.m10.4.4.cmml">D</mi><mo id="S7.p2.10.m10.4.5.4.3.2.3" stretchy="false" xref="S7.p2.10.m10.4.5.4.3.1.cmml">)</mo></mrow></mrow><mo id="S7.p2.10.m10.4.5.5" xref="S7.p2.10.m10.4.5.5.cmml">=</mo><msub id="S7.p2.10.m10.4.5.6" xref="S7.p2.10.m10.4.5.6.cmml"><mi id="S7.p2.10.m10.4.5.6.2" xref="S7.p2.10.m10.4.5.6.2.cmml">ω</mi><mn id="S7.p2.10.m10.4.5.6.3" xref="S7.p2.10.m10.4.5.6.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.p2.10.m10.4b"><apply id="S7.p2.10.m10.4.5.cmml" xref="S7.p2.10.m10.4.5"><and id="S7.p2.10.m10.4.5a.cmml" xref="S7.p2.10.m10.4.5"></and><apply id="S7.p2.10.m10.4.5b.cmml" xref="S7.p2.10.m10.4.5"><eq id="S7.p2.10.m10.4.5.3.cmml" xref="S7.p2.10.m10.4.5.3"></eq><apply id="S7.p2.10.m10.4.5.2.cmml" xref="S7.p2.10.m10.4.5.2"><times id="S7.p2.10.m10.4.5.2.1.cmml" xref="S7.p2.10.m10.4.5.2.1"></times><apply id="S7.p2.10.m10.4.5.2.2.cmml" xref="S7.p2.10.m10.4.5.2.2"><ci id="S7.p2.10.m10.4.5.2.2.1.cmml" xref="S7.p2.10.m10.4.5.2.2.1">^</ci><ci id="S7.p2.10.m10.4.5.2.2.2.cmml" xref="S7.p2.10.m10.4.5.2.2.2">ℒ</ci></apply><interval closure="open" id="S7.p2.10.m10.4.5.2.3.1.cmml" xref="S7.p2.10.m10.4.5.2.3.2"><ci id="S7.p2.10.m10.1.1.cmml" xref="S7.p2.10.m10.1.1">𝐶</ci><ci id="S7.p2.10.m10.2.2.cmml" xref="S7.p2.10.m10.2.2">𝐷</ci></interval></apply><apply id="S7.p2.10.m10.4.5.4.cmml" xref="S7.p2.10.m10.4.5.4"><times id="S7.p2.10.m10.4.5.4.1.cmml" xref="S7.p2.10.m10.4.5.4.1"></times><apply id="S7.p2.10.m10.4.5.4.2.cmml" xref="S7.p2.10.m10.4.5.4.2"><ci id="S7.p2.10.m10.4.5.4.2.1.cmml" xref="S7.p2.10.m10.4.5.4.2.1">^</ci><ci id="S7.p2.10.m10.4.5.4.2.2.cmml" xref="S7.p2.10.m10.4.5.4.2.2">ℛ</ci></apply><interval closure="open" id="S7.p2.10.m10.4.5.4.3.1.cmml" xref="S7.p2.10.m10.4.5.4.3.2"><ci id="S7.p2.10.m10.3.3.cmml" xref="S7.p2.10.m10.3.3">𝐶</ci><ci id="S7.p2.10.m10.4.4.cmml" xref="S7.p2.10.m10.4.4">𝐷</ci></interval></apply></apply><apply id="S7.p2.10.m10.4.5c.cmml" xref="S7.p2.10.m10.4.5"><eq id="S7.p2.10.m10.4.5.5.cmml" xref="S7.p2.10.m10.4.5.5"></eq><share href="https://arxiv.org/html/2503.13728v1#S7.p2.10.m10.4.5.4.cmml" id="S7.p2.10.m10.4.5d.cmml" xref="S7.p2.10.m10.4.5"></share><apply id="S7.p2.10.m10.4.5.6.cmml" xref="S7.p2.10.m10.4.5.6"><csymbol cd="ambiguous" id="S7.p2.10.m10.4.5.6.1.cmml" xref="S7.p2.10.m10.4.5.6">subscript</csymbol><ci id="S7.p2.10.m10.4.5.6.2.cmml" xref="S7.p2.10.m10.4.5.6.2">𝜔</ci><cn id="S7.p2.10.m10.4.5.6.3.cmml" type="integer" xref="S7.p2.10.m10.4.5.6.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.p2.10.m10.4c">\hat{\mathscr{L}}(C,D)=\hat{\mathscr{R}}(C,D)=\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S7.p2.10.m10.4d">over^ start_ARG script_L end_ARG ( italic_C , italic_D ) = over^ start_ARG script_R end_ARG ( italic_C , italic_D ) = italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S7.Thmtheorem1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S7.Thmtheorem1.1.1.1">Lemma 7.1</span></span><span class="ltx_text ltx_font_bold" id="S7.Thmtheorem1.2.2">.</span> </h6> <div class="ltx_para" id="S7.Thmtheorem1.p1"> <p class="ltx_p" id="S7.Thmtheorem1.p1.4">Assume <math alttext="\mathsf{MA}_{\aleph_{1}}" class="ltx_Math" display="inline" id="S7.Thmtheorem1.p1.1.m1.1"><semantics id="S7.Thmtheorem1.p1.1.m1.1a"><msub id="S7.Thmtheorem1.p1.1.m1.1.1" xref="S7.Thmtheorem1.p1.1.m1.1.1.cmml"><mi id="S7.Thmtheorem1.p1.1.m1.1.1.2" xref="S7.Thmtheorem1.p1.1.m1.1.1.2.cmml">𝖬𝖠</mi><msub id="S7.Thmtheorem1.p1.1.m1.1.1.3" xref="S7.Thmtheorem1.p1.1.m1.1.1.3.cmml"><mi id="S7.Thmtheorem1.p1.1.m1.1.1.3.2" mathvariant="normal" xref="S7.Thmtheorem1.p1.1.m1.1.1.3.2.cmml">ℵ</mi><mn id="S7.Thmtheorem1.p1.1.m1.1.1.3.3" xref="S7.Thmtheorem1.p1.1.m1.1.1.3.3.cmml">1</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem1.p1.1.m1.1b"><apply id="S7.Thmtheorem1.p1.1.m1.1.1.cmml" xref="S7.Thmtheorem1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem1.p1.1.m1.1.1.1.cmml" xref="S7.Thmtheorem1.p1.1.m1.1.1">subscript</csymbol><ci id="S7.Thmtheorem1.p1.1.m1.1.1.2.cmml" xref="S7.Thmtheorem1.p1.1.m1.1.1.2">𝖬𝖠</ci><apply id="S7.Thmtheorem1.p1.1.m1.1.1.3.cmml" xref="S7.Thmtheorem1.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S7.Thmtheorem1.p1.1.m1.1.1.3.1.cmml" xref="S7.Thmtheorem1.p1.1.m1.1.1.3">subscript</csymbol><ci id="S7.Thmtheorem1.p1.1.m1.1.1.3.2.cmml" xref="S7.Thmtheorem1.p1.1.m1.1.1.3.2">ℵ</ci><cn id="S7.Thmtheorem1.p1.1.m1.1.1.3.3.cmml" type="integer" xref="S7.Thmtheorem1.p1.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem1.p1.1.m1.1c">\mathsf{MA}_{\aleph_{1}}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem1.p1.1.m1.1d">sansserif_MA start_POSTSUBSCRIPT roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. <math alttext="1+C+1" class="ltx_Math" display="inline" id="S7.Thmtheorem1.p1.2.m2.1"><semantics id="S7.Thmtheorem1.p1.2.m2.1a"><mrow id="S7.Thmtheorem1.p1.2.m2.1.1" xref="S7.Thmtheorem1.p1.2.m2.1.1.cmml"><mn id="S7.Thmtheorem1.p1.2.m2.1.1.2" xref="S7.Thmtheorem1.p1.2.m2.1.1.2.cmml">1</mn><mo id="S7.Thmtheorem1.p1.2.m2.1.1.1" xref="S7.Thmtheorem1.p1.2.m2.1.1.1.cmml">+</mo><mi id="S7.Thmtheorem1.p1.2.m2.1.1.3" xref="S7.Thmtheorem1.p1.2.m2.1.1.3.cmml">C</mi><mo id="S7.Thmtheorem1.p1.2.m2.1.1.1a" xref="S7.Thmtheorem1.p1.2.m2.1.1.1.cmml">+</mo><mn id="S7.Thmtheorem1.p1.2.m2.1.1.4" xref="S7.Thmtheorem1.p1.2.m2.1.1.4.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem1.p1.2.m2.1b"><apply id="S7.Thmtheorem1.p1.2.m2.1.1.cmml" xref="S7.Thmtheorem1.p1.2.m2.1.1"><plus id="S7.Thmtheorem1.p1.2.m2.1.1.1.cmml" xref="S7.Thmtheorem1.p1.2.m2.1.1.1"></plus><cn id="S7.Thmtheorem1.p1.2.m2.1.1.2.cmml" type="integer" xref="S7.Thmtheorem1.p1.2.m2.1.1.2">1</cn><ci id="S7.Thmtheorem1.p1.2.m2.1.1.3.cmml" xref="S7.Thmtheorem1.p1.2.m2.1.1.3">𝐶</ci><cn id="S7.Thmtheorem1.p1.2.m2.1.1.4.cmml" type="integer" xref="S7.Thmtheorem1.p1.2.m2.1.1.4">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem1.p1.2.m2.1c">1+C+1</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem1.p1.2.m2.1d">1 + italic_C + 1</annotation></semantics></math> and <math alttext="1+C^{\star}+1" class="ltx_Math" display="inline" id="S7.Thmtheorem1.p1.3.m3.1"><semantics id="S7.Thmtheorem1.p1.3.m3.1a"><mrow id="S7.Thmtheorem1.p1.3.m3.1.1" xref="S7.Thmtheorem1.p1.3.m3.1.1.cmml"><mn id="S7.Thmtheorem1.p1.3.m3.1.1.2" xref="S7.Thmtheorem1.p1.3.m3.1.1.2.cmml">1</mn><mo id="S7.Thmtheorem1.p1.3.m3.1.1.1" xref="S7.Thmtheorem1.p1.3.m3.1.1.1.cmml">+</mo><msup id="S7.Thmtheorem1.p1.3.m3.1.1.3" xref="S7.Thmtheorem1.p1.3.m3.1.1.3.cmml"><mi id="S7.Thmtheorem1.p1.3.m3.1.1.3.2" xref="S7.Thmtheorem1.p1.3.m3.1.1.3.2.cmml">C</mi><mo id="S7.Thmtheorem1.p1.3.m3.1.1.3.3" xref="S7.Thmtheorem1.p1.3.m3.1.1.3.3.cmml">⋆</mo></msup><mo id="S7.Thmtheorem1.p1.3.m3.1.1.1a" xref="S7.Thmtheorem1.p1.3.m3.1.1.1.cmml">+</mo><mn id="S7.Thmtheorem1.p1.3.m3.1.1.4" xref="S7.Thmtheorem1.p1.3.m3.1.1.4.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem1.p1.3.m3.1b"><apply id="S7.Thmtheorem1.p1.3.m3.1.1.cmml" xref="S7.Thmtheorem1.p1.3.m3.1.1"><plus id="S7.Thmtheorem1.p1.3.m3.1.1.1.cmml" xref="S7.Thmtheorem1.p1.3.m3.1.1.1"></plus><cn id="S7.Thmtheorem1.p1.3.m3.1.1.2.cmml" type="integer" xref="S7.Thmtheorem1.p1.3.m3.1.1.2">1</cn><apply id="S7.Thmtheorem1.p1.3.m3.1.1.3.cmml" xref="S7.Thmtheorem1.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S7.Thmtheorem1.p1.3.m3.1.1.3.1.cmml" xref="S7.Thmtheorem1.p1.3.m3.1.1.3">superscript</csymbol><ci id="S7.Thmtheorem1.p1.3.m3.1.1.3.2.cmml" xref="S7.Thmtheorem1.p1.3.m3.1.1.3.2">𝐶</ci><ci id="S7.Thmtheorem1.p1.3.m3.1.1.3.3.cmml" xref="S7.Thmtheorem1.p1.3.m3.1.1.3.3">⋆</ci></apply><cn id="S7.Thmtheorem1.p1.3.m3.1.1.4.cmml" type="integer" xref="S7.Thmtheorem1.p1.3.m3.1.1.4">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem1.p1.3.m3.1c">1+C^{\star}+1</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem1.p1.3.m3.1d">1 + italic_C start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT + 1</annotation></semantics></math> form a <math alttext="\trianglelefteq" class="ltx_Math" display="inline" id="S7.Thmtheorem1.p1.4.m4.1"><semantics id="S7.Thmtheorem1.p1.4.m4.1a"><mi id="S7.Thmtheorem1.p1.4.m4.1.1" mathvariant="normal" xref="S7.Thmtheorem1.p1.4.m4.1.1.cmml">⊴</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem1.p1.4.m4.1b"><ci id="S7.Thmtheorem1.p1.4.m4.1.1.cmml" xref="S7.Thmtheorem1.p1.4.m4.1.1">⊴</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem1.p1.4.m4.1c">\trianglelefteq</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem1.p1.4.m4.1d">⊴</annotation></semantics></math>-basis for the Countryman lines</p> </div> </div> <div class="ltx_proof" id="S7.1"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S7.1.p1"> <p class="ltx_p" id="S7.1.p1.8">It is enough to prove that if <math alttext="A\preceq C" class="ltx_Math" display="inline" id="S7.1.p1.1.m1.1"><semantics id="S7.1.p1.1.m1.1a"><mrow id="S7.1.p1.1.m1.1.1" xref="S7.1.p1.1.m1.1.1.cmml"><mi id="S7.1.p1.1.m1.1.1.2" xref="S7.1.p1.1.m1.1.1.2.cmml">A</mi><mo id="S7.1.p1.1.m1.1.1.1" xref="S7.1.p1.1.m1.1.1.1.cmml">⪯</mo><mi id="S7.1.p1.1.m1.1.1.3" xref="S7.1.p1.1.m1.1.1.3.cmml">C</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.1.p1.1.m1.1b"><apply id="S7.1.p1.1.m1.1.1.cmml" xref="S7.1.p1.1.m1.1.1"><csymbol cd="latexml" id="S7.1.p1.1.m1.1.1.1.cmml" xref="S7.1.p1.1.m1.1.1.1">precedes-or-equals</csymbol><ci id="S7.1.p1.1.m1.1.1.2.cmml" xref="S7.1.p1.1.m1.1.1.2">𝐴</ci><ci id="S7.1.p1.1.m1.1.1.3.cmml" xref="S7.1.p1.1.m1.1.1.3">𝐶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.1.p1.1.m1.1c">A\preceq C</annotation><annotation encoding="application/x-llamapun" id="S7.1.p1.1.m1.1d">italic_A ⪯ italic_C</annotation></semantics></math> is uncountable, then <math alttext="1+C+1\trianglelefteq A" class="ltx_Math" display="inline" id="S7.1.p1.2.m2.1"><semantics id="S7.1.p1.2.m2.1a"><mrow id="S7.1.p1.2.m2.1.1" xref="S7.1.p1.2.m2.1.1.cmml"><mn id="S7.1.p1.2.m2.1.1.2" xref="S7.1.p1.2.m2.1.1.2.cmml">1</mn><mo id="S7.1.p1.2.m2.1.1.1" xref="S7.1.p1.2.m2.1.1.1.cmml">+</mo><mi id="S7.1.p1.2.m2.1.1.3" xref="S7.1.p1.2.m2.1.1.3.cmml">C</mi><mo id="S7.1.p1.2.m2.1.1.1a" xref="S7.1.p1.2.m2.1.1.1.cmml">+</mo><mrow id="S7.1.p1.2.m2.1.1.4" xref="S7.1.p1.2.m2.1.1.4.cmml"><mn id="S7.1.p1.2.m2.1.1.4.2" xref="S7.1.p1.2.m2.1.1.4.2.cmml">1</mn><mo id="S7.1.p1.2.m2.1.1.4.1" xref="S7.1.p1.2.m2.1.1.4.1.cmml">⁢</mo><mi id="S7.1.p1.2.m2.1.1.4.3" mathvariant="normal" xref="S7.1.p1.2.m2.1.1.4.3.cmml">⊴</mi><mo id="S7.1.p1.2.m2.1.1.4.1a" xref="S7.1.p1.2.m2.1.1.4.1.cmml">⁢</mo><mi id="S7.1.p1.2.m2.1.1.4.4" xref="S7.1.p1.2.m2.1.1.4.4.cmml">A</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.1.p1.2.m2.1b"><apply id="S7.1.p1.2.m2.1.1.cmml" xref="S7.1.p1.2.m2.1.1"><plus id="S7.1.p1.2.m2.1.1.1.cmml" xref="S7.1.p1.2.m2.1.1.1"></plus><cn id="S7.1.p1.2.m2.1.1.2.cmml" type="integer" xref="S7.1.p1.2.m2.1.1.2">1</cn><ci id="S7.1.p1.2.m2.1.1.3.cmml" xref="S7.1.p1.2.m2.1.1.3">𝐶</ci><apply id="S7.1.p1.2.m2.1.1.4.cmml" xref="S7.1.p1.2.m2.1.1.4"><times id="S7.1.p1.2.m2.1.1.4.1.cmml" xref="S7.1.p1.2.m2.1.1.4.1"></times><cn id="S7.1.p1.2.m2.1.1.4.2.cmml" type="integer" xref="S7.1.p1.2.m2.1.1.4.2">1</cn><ci id="S7.1.p1.2.m2.1.1.4.3.cmml" xref="S7.1.p1.2.m2.1.1.4.3">⊴</ci><ci id="S7.1.p1.2.m2.1.1.4.4.cmml" xref="S7.1.p1.2.m2.1.1.4.4">𝐴</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.1.p1.2.m2.1c">1+C+1\trianglelefteq A</annotation><annotation encoding="application/x-llamapun" id="S7.1.p1.2.m2.1d">1 + italic_C + 1 ⊴ italic_A</annotation></semantics></math>. First observe that <math alttext="A\trianglerighteq 1+B+1" class="ltx_Math" display="inline" id="S7.1.p1.3.m3.1"><semantics id="S7.1.p1.3.m3.1a"><mrow id="S7.1.p1.3.m3.1.1" xref="S7.1.p1.3.m3.1.1.cmml"><mrow id="S7.1.p1.3.m3.1.1.2" xref="S7.1.p1.3.m3.1.1.2.cmml"><mi id="S7.1.p1.3.m3.1.1.2.2" xref="S7.1.p1.3.m3.1.1.2.2.cmml">A</mi><mo id="S7.1.p1.3.m3.1.1.2.1" xref="S7.1.p1.3.m3.1.1.2.1.cmml">⁢</mo><mi id="S7.1.p1.3.m3.1.1.2.3" mathvariant="normal" xref="S7.1.p1.3.m3.1.1.2.3.cmml">⊵</mi><mo id="S7.1.p1.3.m3.1.1.2.1a" xref="S7.1.p1.3.m3.1.1.2.1.cmml">⁢</mo><mn id="S7.1.p1.3.m3.1.1.2.4" xref="S7.1.p1.3.m3.1.1.2.4.cmml">1</mn></mrow><mo id="S7.1.p1.3.m3.1.1.1" xref="S7.1.p1.3.m3.1.1.1.cmml">+</mo><mi id="S7.1.p1.3.m3.1.1.3" xref="S7.1.p1.3.m3.1.1.3.cmml">B</mi><mo id="S7.1.p1.3.m3.1.1.1a" xref="S7.1.p1.3.m3.1.1.1.cmml">+</mo><mn id="S7.1.p1.3.m3.1.1.4" xref="S7.1.p1.3.m3.1.1.4.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.1.p1.3.m3.1b"><apply id="S7.1.p1.3.m3.1.1.cmml" xref="S7.1.p1.3.m3.1.1"><plus id="S7.1.p1.3.m3.1.1.1.cmml" xref="S7.1.p1.3.m3.1.1.1"></plus><apply id="S7.1.p1.3.m3.1.1.2.cmml" xref="S7.1.p1.3.m3.1.1.2"><times id="S7.1.p1.3.m3.1.1.2.1.cmml" xref="S7.1.p1.3.m3.1.1.2.1"></times><ci id="S7.1.p1.3.m3.1.1.2.2.cmml" xref="S7.1.p1.3.m3.1.1.2.2">𝐴</ci><ci id="S7.1.p1.3.m3.1.1.2.3.cmml" xref="S7.1.p1.3.m3.1.1.2.3">⊵</ci><cn id="S7.1.p1.3.m3.1.1.2.4.cmml" type="integer" xref="S7.1.p1.3.m3.1.1.2.4">1</cn></apply><ci id="S7.1.p1.3.m3.1.1.3.cmml" xref="S7.1.p1.3.m3.1.1.3">𝐵</ci><cn id="S7.1.p1.3.m3.1.1.4.cmml" type="integer" xref="S7.1.p1.3.m3.1.1.4">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.1.p1.3.m3.1c">A\trianglerighteq 1+B+1</annotation><annotation encoding="application/x-llamapun" id="S7.1.p1.3.m3.1d">italic_A ⊵ 1 + italic_B + 1</annotation></semantics></math> for some <math alttext="\aleph_{1}" class="ltx_Math" display="inline" id="S7.1.p1.4.m4.1"><semantics id="S7.1.p1.4.m4.1a"><msub id="S7.1.p1.4.m4.1.1" xref="S7.1.p1.4.m4.1.1.cmml"><mi id="S7.1.p1.4.m4.1.1.2" mathvariant="normal" xref="S7.1.p1.4.m4.1.1.2.cmml">ℵ</mi><mn id="S7.1.p1.4.m4.1.1.3" xref="S7.1.p1.4.m4.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S7.1.p1.4.m4.1b"><apply id="S7.1.p1.4.m4.1.1.cmml" xref="S7.1.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S7.1.p1.4.m4.1.1.1.cmml" xref="S7.1.p1.4.m4.1.1">subscript</csymbol><ci id="S7.1.p1.4.m4.1.1.2.cmml" xref="S7.1.p1.4.m4.1.1.2">ℵ</ci><cn id="S7.1.p1.4.m4.1.1.3.cmml" type="integer" xref="S7.1.p1.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.1.p1.4.m4.1c">\aleph_{1}</annotation><annotation encoding="application/x-llamapun" id="S7.1.p1.4.m4.1d">roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-dense Aronszajn line <math alttext="B" class="ltx_Math" display="inline" id="S7.1.p1.5.m5.1"><semantics id="S7.1.p1.5.m5.1a"><mi id="S7.1.p1.5.m5.1.1" xref="S7.1.p1.5.m5.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S7.1.p1.5.m5.1b"><ci id="S7.1.p1.5.m5.1.1.cmml" xref="S7.1.p1.5.m5.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.1.p1.5.m5.1c">B</annotation><annotation encoding="application/x-llamapun" id="S7.1.p1.5.m5.1d">italic_B</annotation></semantics></math>. This can be proven by taking the usual quotient on <math alttext="A" class="ltx_Math" display="inline" id="S7.1.p1.6.m6.1"><semantics id="S7.1.p1.6.m6.1a"><mi id="S7.1.p1.6.m6.1.1" xref="S7.1.p1.6.m6.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S7.1.p1.6.m6.1b"><ci id="S7.1.p1.6.m6.1.1.cmml" xref="S7.1.p1.6.m6.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.1.p1.6.m6.1c">A</annotation><annotation encoding="application/x-llamapun" id="S7.1.p1.6.m6.1d">italic_A</annotation></semantics></math> defined by <math alttext="a\sim b" class="ltx_Math" display="inline" id="S7.1.p1.7.m7.1"><semantics id="S7.1.p1.7.m7.1a"><mrow id="S7.1.p1.7.m7.1.1" xref="S7.1.p1.7.m7.1.1.cmml"><mi id="S7.1.p1.7.m7.1.1.2" xref="S7.1.p1.7.m7.1.1.2.cmml">a</mi><mo id="S7.1.p1.7.m7.1.1.1" xref="S7.1.p1.7.m7.1.1.1.cmml">∼</mo><mi id="S7.1.p1.7.m7.1.1.3" xref="S7.1.p1.7.m7.1.1.3.cmml">b</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.1.p1.7.m7.1b"><apply id="S7.1.p1.7.m7.1.1.cmml" xref="S7.1.p1.7.m7.1.1"><csymbol cd="latexml" id="S7.1.p1.7.m7.1.1.1.cmml" xref="S7.1.p1.7.m7.1.1.1">similar-to</csymbol><ci id="S7.1.p1.7.m7.1.1.2.cmml" xref="S7.1.p1.7.m7.1.1.2">𝑎</ci><ci id="S7.1.p1.7.m7.1.1.3.cmml" xref="S7.1.p1.7.m7.1.1.3">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.1.p1.7.m7.1c">a\sim b</annotation><annotation encoding="application/x-llamapun" id="S7.1.p1.7.m7.1d">italic_a ∼ italic_b</annotation></semantics></math> if there are countably many points between them. Then <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6.Thmtheorem2" title="Theorem 6.2. ‣ 6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">6.2</span></a> implies that <math alttext="B\trianglerighteq C" class="ltx_Math" display="inline" id="S7.1.p1.8.m8.1"><semantics id="S7.1.p1.8.m8.1a"><mrow id="S7.1.p1.8.m8.1.1" xref="S7.1.p1.8.m8.1.1.cmml"><mi id="S7.1.p1.8.m8.1.1.2" xref="S7.1.p1.8.m8.1.1.2.cmml">B</mi><mo id="S7.1.p1.8.m8.1.1.1" xref="S7.1.p1.8.m8.1.1.1.cmml">⁢</mo><mi id="S7.1.p1.8.m8.1.1.3" mathvariant="normal" xref="S7.1.p1.8.m8.1.1.3.cmml">⊵</mi><mo id="S7.1.p1.8.m8.1.1.1a" xref="S7.1.p1.8.m8.1.1.1.cmml">⁢</mo><mi id="S7.1.p1.8.m8.1.1.4" xref="S7.1.p1.8.m8.1.1.4.cmml">C</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.1.p1.8.m8.1b"><apply id="S7.1.p1.8.m8.1.1.cmml" xref="S7.1.p1.8.m8.1.1"><times id="S7.1.p1.8.m8.1.1.1.cmml" xref="S7.1.p1.8.m8.1.1.1"></times><ci id="S7.1.p1.8.m8.1.1.2.cmml" xref="S7.1.p1.8.m8.1.1.2">𝐵</ci><ci id="S7.1.p1.8.m8.1.1.3.cmml" xref="S7.1.p1.8.m8.1.1.3">⊵</ci><ci id="S7.1.p1.8.m8.1.1.4.cmml" xref="S7.1.p1.8.m8.1.1.4">𝐶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.1.p1.8.m8.1c">B\trianglerighteq C</annotation><annotation encoding="application/x-llamapun" id="S7.1.p1.8.m8.1d">italic_B ⊵ italic_C</annotation></semantics></math> which finishes the proof. ∎</p> </div> </div> <div class="ltx_para" id="S7.p3"> <p class="ltx_p" id="S7.p3.5">Under <math alttext="\mathsf{PFA}" class="ltx_Math" display="inline" id="S7.p3.1.m1.1"><semantics id="S7.p3.1.m1.1a"><mi id="S7.p3.1.m1.1.1" xref="S7.p3.1.m1.1.1.cmml">𝖯𝖥𝖠</mi><annotation-xml encoding="MathML-Content" id="S7.p3.1.m1.1b"><ci id="S7.p3.1.m1.1.1.cmml" xref="S7.p3.1.m1.1.1">𝖯𝖥𝖠</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.p3.1.m1.1c">\mathsf{PFA}</annotation><annotation encoding="application/x-llamapun" id="S7.p3.1.m1.1d">sansserif_PFA</annotation></semantics></math> we can say more. Recall that Moore proved <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib16" title="">16</a>, Theorem 1.3]</cite> that under <math alttext="\mathsf{PFA}" class="ltx_Math" display="inline" id="S7.p3.2.m2.1"><semantics id="S7.p3.2.m2.1a"><mi id="S7.p3.2.m2.1.1" xref="S7.p3.2.m2.1.1.cmml">𝖯𝖥𝖠</mi><annotation-xml encoding="MathML-Content" id="S7.p3.2.m2.1b"><ci id="S7.p3.2.m2.1.1.cmml" xref="S7.p3.2.m2.1.1">𝖯𝖥𝖠</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.p3.2.m2.1c">\mathsf{PFA}</annotation><annotation encoding="application/x-llamapun" id="S7.p3.2.m2.1d">sansserif_PFA</annotation></semantics></math> any non Countryman Aronszajn line contains a copy of <math alttext="L" class="ltx_Math" display="inline" id="S7.p3.3.m3.1"><semantics id="S7.p3.3.m3.1a"><mi id="S7.p3.3.m3.1.1" xref="S7.p3.3.m3.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S7.p3.3.m3.1b"><ci id="S7.p3.3.m3.1.1.cmml" xref="S7.p3.3.m3.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.p3.3.m3.1c">L</annotation><annotation encoding="application/x-llamapun" id="S7.p3.3.m3.1d">italic_L</annotation></semantics></math> and <math alttext="L^{*}" class="ltx_Math" display="inline" id="S7.p3.4.m4.1"><semantics id="S7.p3.4.m4.1a"><msup id="S7.p3.4.m4.1.1" xref="S7.p3.4.m4.1.1.cmml"><mi id="S7.p3.4.m4.1.1.2" xref="S7.p3.4.m4.1.1.2.cmml">L</mi><mo id="S7.p3.4.m4.1.1.3" xref="S7.p3.4.m4.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S7.p3.4.m4.1b"><apply id="S7.p3.4.m4.1.1.cmml" xref="S7.p3.4.m4.1.1"><csymbol cd="ambiguous" id="S7.p3.4.m4.1.1.1.cmml" xref="S7.p3.4.m4.1.1">superscript</csymbol><ci id="S7.p3.4.m4.1.1.2.cmml" xref="S7.p3.4.m4.1.1.2">𝐿</ci><times id="S7.p3.4.m4.1.1.3.cmml" xref="S7.p3.4.m4.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.p3.4.m4.1c">L^{*}</annotation><annotation encoding="application/x-llamapun" id="S7.p3.4.m4.1d">italic_L start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> for any given Countryman line <math alttext="L" class="ltx_Math" display="inline" id="S7.p3.5.m5.1"><semantics id="S7.p3.5.m5.1a"><mi id="S7.p3.5.m5.1.1" xref="S7.p3.5.m5.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S7.p3.5.m5.1b"><ci id="S7.p3.5.m5.1.1.cmml" xref="S7.p3.5.m5.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.p3.5.m5.1c">L</annotation><annotation encoding="application/x-llamapun" id="S7.p3.5.m5.1d">italic_L</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S7.Thmtheorem2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S7.Thmtheorem2.1.1.1">Theorem 7.2</span></span><span class="ltx_text ltx_font_bold" id="S7.Thmtheorem2.2.2">.</span> </h6> <div class="ltx_para" id="S7.Thmtheorem2.p1"> <p class="ltx_p" id="S7.Thmtheorem2.p1.4"><span class="ltx_text ltx_font_italic" id="S7.Thmtheorem2.p1.4.4">Assume <math alttext="\mathsf{PFA}" class="ltx_Math" display="inline" id="S7.Thmtheorem2.p1.1.1.m1.1"><semantics id="S7.Thmtheorem2.p1.1.1.m1.1a"><mi id="S7.Thmtheorem2.p1.1.1.m1.1.1" xref="S7.Thmtheorem2.p1.1.1.m1.1.1.cmml">𝖯𝖥𝖠</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem2.p1.1.1.m1.1b"><ci id="S7.Thmtheorem2.p1.1.1.m1.1.1.cmml" xref="S7.Thmtheorem2.p1.1.1.m1.1.1">𝖯𝖥𝖠</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem2.p1.1.1.m1.1c">\mathsf{PFA}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem2.p1.1.1.m1.1d">sansserif_PFA</annotation></semantics></math>. <math alttext="1+C+1" class="ltx_Math" display="inline" id="S7.Thmtheorem2.p1.2.2.m2.1"><semantics id="S7.Thmtheorem2.p1.2.2.m2.1a"><mrow id="S7.Thmtheorem2.p1.2.2.m2.1.1" xref="S7.Thmtheorem2.p1.2.2.m2.1.1.cmml"><mn id="S7.Thmtheorem2.p1.2.2.m2.1.1.2" xref="S7.Thmtheorem2.p1.2.2.m2.1.1.2.cmml">1</mn><mo id="S7.Thmtheorem2.p1.2.2.m2.1.1.1" xref="S7.Thmtheorem2.p1.2.2.m2.1.1.1.cmml">+</mo><mi id="S7.Thmtheorem2.p1.2.2.m2.1.1.3" xref="S7.Thmtheorem2.p1.2.2.m2.1.1.3.cmml">C</mi><mo id="S7.Thmtheorem2.p1.2.2.m2.1.1.1a" xref="S7.Thmtheorem2.p1.2.2.m2.1.1.1.cmml">+</mo><mn id="S7.Thmtheorem2.p1.2.2.m2.1.1.4" xref="S7.Thmtheorem2.p1.2.2.m2.1.1.4.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem2.p1.2.2.m2.1b"><apply id="S7.Thmtheorem2.p1.2.2.m2.1.1.cmml" xref="S7.Thmtheorem2.p1.2.2.m2.1.1"><plus id="S7.Thmtheorem2.p1.2.2.m2.1.1.1.cmml" xref="S7.Thmtheorem2.p1.2.2.m2.1.1.1"></plus><cn id="S7.Thmtheorem2.p1.2.2.m2.1.1.2.cmml" type="integer" xref="S7.Thmtheorem2.p1.2.2.m2.1.1.2">1</cn><ci id="S7.Thmtheorem2.p1.2.2.m2.1.1.3.cmml" xref="S7.Thmtheorem2.p1.2.2.m2.1.1.3">𝐶</ci><cn id="S7.Thmtheorem2.p1.2.2.m2.1.1.4.cmml" type="integer" xref="S7.Thmtheorem2.p1.2.2.m2.1.1.4">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem2.p1.2.2.m2.1c">1+C+1</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem2.p1.2.2.m2.1d">1 + italic_C + 1</annotation></semantics></math> and <math alttext="1+C^{\star}+1" class="ltx_Math" display="inline" id="S7.Thmtheorem2.p1.3.3.m3.1"><semantics id="S7.Thmtheorem2.p1.3.3.m3.1a"><mrow id="S7.Thmtheorem2.p1.3.3.m3.1.1" xref="S7.Thmtheorem2.p1.3.3.m3.1.1.cmml"><mn id="S7.Thmtheorem2.p1.3.3.m3.1.1.2" xref="S7.Thmtheorem2.p1.3.3.m3.1.1.2.cmml">1</mn><mo id="S7.Thmtheorem2.p1.3.3.m3.1.1.1" xref="S7.Thmtheorem2.p1.3.3.m3.1.1.1.cmml">+</mo><msup id="S7.Thmtheorem2.p1.3.3.m3.1.1.3" xref="S7.Thmtheorem2.p1.3.3.m3.1.1.3.cmml"><mi id="S7.Thmtheorem2.p1.3.3.m3.1.1.3.2" xref="S7.Thmtheorem2.p1.3.3.m3.1.1.3.2.cmml">C</mi><mo id="S7.Thmtheorem2.p1.3.3.m3.1.1.3.3" xref="S7.Thmtheorem2.p1.3.3.m3.1.1.3.3.cmml">⋆</mo></msup><mo id="S7.Thmtheorem2.p1.3.3.m3.1.1.1a" xref="S7.Thmtheorem2.p1.3.3.m3.1.1.1.cmml">+</mo><mn id="S7.Thmtheorem2.p1.3.3.m3.1.1.4" xref="S7.Thmtheorem2.p1.3.3.m3.1.1.4.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem2.p1.3.3.m3.1b"><apply id="S7.Thmtheorem2.p1.3.3.m3.1.1.cmml" xref="S7.Thmtheorem2.p1.3.3.m3.1.1"><plus id="S7.Thmtheorem2.p1.3.3.m3.1.1.1.cmml" xref="S7.Thmtheorem2.p1.3.3.m3.1.1.1"></plus><cn id="S7.Thmtheorem2.p1.3.3.m3.1.1.2.cmml" type="integer" xref="S7.Thmtheorem2.p1.3.3.m3.1.1.2">1</cn><apply id="S7.Thmtheorem2.p1.3.3.m3.1.1.3.cmml" xref="S7.Thmtheorem2.p1.3.3.m3.1.1.3"><csymbol cd="ambiguous" id="S7.Thmtheorem2.p1.3.3.m3.1.1.3.1.cmml" xref="S7.Thmtheorem2.p1.3.3.m3.1.1.3">superscript</csymbol><ci id="S7.Thmtheorem2.p1.3.3.m3.1.1.3.2.cmml" xref="S7.Thmtheorem2.p1.3.3.m3.1.1.3.2">𝐶</ci><ci id="S7.Thmtheorem2.p1.3.3.m3.1.1.3.3.cmml" xref="S7.Thmtheorem2.p1.3.3.m3.1.1.3.3">⋆</ci></apply><cn id="S7.Thmtheorem2.p1.3.3.m3.1.1.4.cmml" type="integer" xref="S7.Thmtheorem2.p1.3.3.m3.1.1.4">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem2.p1.3.3.m3.1c">1+C^{\star}+1</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem2.p1.3.3.m3.1d">1 + italic_C start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT + 1</annotation></semantics></math> form a <math alttext="\trianglelefteq" class="ltx_Math" display="inline" id="S7.Thmtheorem2.p1.4.4.m4.1"><semantics id="S7.Thmtheorem2.p1.4.4.m4.1a"><mi id="S7.Thmtheorem2.p1.4.4.m4.1.1" mathvariant="normal" xref="S7.Thmtheorem2.p1.4.4.m4.1.1.cmml">⊴</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem2.p1.4.4.m4.1b"><ci id="S7.Thmtheorem2.p1.4.4.m4.1.1.cmml" xref="S7.Thmtheorem2.p1.4.4.m4.1.1">⊴</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem2.p1.4.4.m4.1c">\trianglelefteq</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem2.p1.4.4.m4.1d">⊴</annotation></semantics></math>-basis for the Aronszajn lines.</span></p> </div> </div> <div class="ltx_proof" id="S7.4"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S7.2.p1"> <p class="ltx_p" id="S7.2.p1.2">Taking in to account <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S7.Thmtheorem1" title="Lemma 7.1. ‣ 7. A two element basis for the Aronszajn lines ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">7.1</span></a> it is enough to prove that if <math alttext="A" class="ltx_Math" display="inline" id="S7.2.p1.1.m1.1"><semantics id="S7.2.p1.1.m1.1a"><mi id="S7.2.p1.1.m1.1.1" xref="S7.2.p1.1.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S7.2.p1.1.m1.1b"><ci id="S7.2.p1.1.m1.1.1.cmml" xref="S7.2.p1.1.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.2.p1.1.m1.1c">A</annotation><annotation encoding="application/x-llamapun" id="S7.2.p1.1.m1.1d">italic_A</annotation></semantics></math> is any Aronszajn line, then <math alttext="A" class="ltx_Math" display="inline" id="S7.2.p1.2.m2.1"><semantics id="S7.2.p1.2.m2.1a"><mi id="S7.2.p1.2.m2.1.1" xref="S7.2.p1.2.m2.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S7.2.p1.2.m2.1b"><ci id="S7.2.p1.2.m2.1.1.cmml" xref="S7.2.p1.2.m2.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.2.p1.2.m2.1c">A</annotation><annotation encoding="application/x-llamapun" id="S7.2.p1.2.m2.1d">italic_A</annotation></semantics></math> surjects onto some Countryman line.</p> </div> <div class="ltx_para" id="S7.3.p2"> <p class="ltx_p" id="S7.3.p2.7">By <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S2.Thmtheorem9" title="Theorem 2.9. ‣ 2. Aronszajn and Countryman lines ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">2.9</span></a> there is <math alttext="B\subseteq A" class="ltx_Math" display="inline" id="S7.3.p2.1.m1.1"><semantics id="S7.3.p2.1.m1.1a"><mrow id="S7.3.p2.1.m1.1.1" xref="S7.3.p2.1.m1.1.1.cmml"><mi id="S7.3.p2.1.m1.1.1.2" xref="S7.3.p2.1.m1.1.1.2.cmml">B</mi><mo id="S7.3.p2.1.m1.1.1.1" xref="S7.3.p2.1.m1.1.1.1.cmml">⊆</mo><mi id="S7.3.p2.1.m1.1.1.3" xref="S7.3.p2.1.m1.1.1.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.3.p2.1.m1.1b"><apply id="S7.3.p2.1.m1.1.1.cmml" xref="S7.3.p2.1.m1.1.1"><subset id="S7.3.p2.1.m1.1.1.1.cmml" xref="S7.3.p2.1.m1.1.1.1"></subset><ci id="S7.3.p2.1.m1.1.1.2.cmml" xref="S7.3.p2.1.m1.1.1.2">𝐵</ci><ci id="S7.3.p2.1.m1.1.1.3.cmml" xref="S7.3.p2.1.m1.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.3.p2.1.m1.1c">B\subseteq A</annotation><annotation encoding="application/x-llamapun" id="S7.3.p2.1.m1.1d">italic_B ⊆ italic_A</annotation></semantics></math> that is an <math alttext="\aleph_{1}" class="ltx_Math" display="inline" id="S7.3.p2.2.m2.1"><semantics id="S7.3.p2.2.m2.1a"><msub id="S7.3.p2.2.m2.1.1" xref="S7.3.p2.2.m2.1.1.cmml"><mi id="S7.3.p2.2.m2.1.1.2" mathvariant="normal" xref="S7.3.p2.2.m2.1.1.2.cmml">ℵ</mi><mn id="S7.3.p2.2.m2.1.1.3" xref="S7.3.p2.2.m2.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S7.3.p2.2.m2.1b"><apply id="S7.3.p2.2.m2.1.1.cmml" xref="S7.3.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S7.3.p2.2.m2.1.1.1.cmml" xref="S7.3.p2.2.m2.1.1">subscript</csymbol><ci id="S7.3.p2.2.m2.1.1.2.cmml" xref="S7.3.p2.2.m2.1.1.2">ℵ</ci><cn id="S7.3.p2.2.m2.1.1.3.cmml" type="integer" xref="S7.3.p2.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.3.p2.2.m2.1c">\aleph_{1}</annotation><annotation encoding="application/x-llamapun" id="S7.3.p2.2.m2.1d">roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-dense Countryman line with the induced order. For <math alttext="a<_{A}b" class="ltx_Math" display="inline" id="S7.3.p2.3.m3.1"><semantics id="S7.3.p2.3.m3.1a"><mrow id="S7.3.p2.3.m3.1.1" xref="S7.3.p2.3.m3.1.1.cmml"><mi id="S7.3.p2.3.m3.1.1.2" xref="S7.3.p2.3.m3.1.1.2.cmml">a</mi><msub id="S7.3.p2.3.m3.1.1.1" xref="S7.3.p2.3.m3.1.1.1.cmml"><mo id="S7.3.p2.3.m3.1.1.1.2" xref="S7.3.p2.3.m3.1.1.1.2.cmml"><</mo><mi id="S7.3.p2.3.m3.1.1.1.3" xref="S7.3.p2.3.m3.1.1.1.3.cmml">A</mi></msub><mi id="S7.3.p2.3.m3.1.1.3" xref="S7.3.p2.3.m3.1.1.3.cmml">b</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.3.p2.3.m3.1b"><apply id="S7.3.p2.3.m3.1.1.cmml" xref="S7.3.p2.3.m3.1.1"><apply id="S7.3.p2.3.m3.1.1.1.cmml" xref="S7.3.p2.3.m3.1.1.1"><csymbol cd="ambiguous" id="S7.3.p2.3.m3.1.1.1.1.cmml" xref="S7.3.p2.3.m3.1.1.1">subscript</csymbol><lt id="S7.3.p2.3.m3.1.1.1.2.cmml" xref="S7.3.p2.3.m3.1.1.1.2"></lt><ci id="S7.3.p2.3.m3.1.1.1.3.cmml" xref="S7.3.p2.3.m3.1.1.1.3">𝐴</ci></apply><ci id="S7.3.p2.3.m3.1.1.2.cmml" xref="S7.3.p2.3.m3.1.1.2">𝑎</ci><ci id="S7.3.p2.3.m3.1.1.3.cmml" xref="S7.3.p2.3.m3.1.1.3">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.3.p2.3.m3.1c">a<_{A}b</annotation><annotation encoding="application/x-llamapun" id="S7.3.p2.3.m3.1d">italic_a < start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_b</annotation></semantics></math>, define <math alttext="a\sim b" class="ltx_Math" display="inline" id="S7.3.p2.4.m4.1"><semantics id="S7.3.p2.4.m4.1a"><mrow id="S7.3.p2.4.m4.1.1" xref="S7.3.p2.4.m4.1.1.cmml"><mi id="S7.3.p2.4.m4.1.1.2" xref="S7.3.p2.4.m4.1.1.2.cmml">a</mi><mo id="S7.3.p2.4.m4.1.1.1" xref="S7.3.p2.4.m4.1.1.1.cmml">∼</mo><mi id="S7.3.p2.4.m4.1.1.3" xref="S7.3.p2.4.m4.1.1.3.cmml">b</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.3.p2.4.m4.1b"><apply id="S7.3.p2.4.m4.1.1.cmml" xref="S7.3.p2.4.m4.1.1"><csymbol cd="latexml" id="S7.3.p2.4.m4.1.1.1.cmml" xref="S7.3.p2.4.m4.1.1.1">similar-to</csymbol><ci id="S7.3.p2.4.m4.1.1.2.cmml" xref="S7.3.p2.4.m4.1.1.2">𝑎</ci><ci id="S7.3.p2.4.m4.1.1.3.cmml" xref="S7.3.p2.4.m4.1.1.3">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.3.p2.4.m4.1c">a\sim b</annotation><annotation encoding="application/x-llamapun" id="S7.3.p2.4.m4.1d">italic_a ∼ italic_b</annotation></semantics></math> if <math alttext="[a,b]_{A}\cap B=\varnothing" class="ltx_Math" display="inline" id="S7.3.p2.5.m5.2"><semantics id="S7.3.p2.5.m5.2a"><mrow id="S7.3.p2.5.m5.2.3" xref="S7.3.p2.5.m5.2.3.cmml"><mrow id="S7.3.p2.5.m5.2.3.2" xref="S7.3.p2.5.m5.2.3.2.cmml"><msub id="S7.3.p2.5.m5.2.3.2.2" xref="S7.3.p2.5.m5.2.3.2.2.cmml"><mrow id="S7.3.p2.5.m5.2.3.2.2.2.2" xref="S7.3.p2.5.m5.2.3.2.2.2.1.cmml"><mo id="S7.3.p2.5.m5.2.3.2.2.2.2.1" stretchy="false" xref="S7.3.p2.5.m5.2.3.2.2.2.1.cmml">[</mo><mi id="S7.3.p2.5.m5.1.1" xref="S7.3.p2.5.m5.1.1.cmml">a</mi><mo id="S7.3.p2.5.m5.2.3.2.2.2.2.2" xref="S7.3.p2.5.m5.2.3.2.2.2.1.cmml">,</mo><mi id="S7.3.p2.5.m5.2.2" xref="S7.3.p2.5.m5.2.2.cmml">b</mi><mo id="S7.3.p2.5.m5.2.3.2.2.2.2.3" stretchy="false" xref="S7.3.p2.5.m5.2.3.2.2.2.1.cmml">]</mo></mrow><mi id="S7.3.p2.5.m5.2.3.2.2.3" xref="S7.3.p2.5.m5.2.3.2.2.3.cmml">A</mi></msub><mo id="S7.3.p2.5.m5.2.3.2.1" xref="S7.3.p2.5.m5.2.3.2.1.cmml">∩</mo><mi id="S7.3.p2.5.m5.2.3.2.3" xref="S7.3.p2.5.m5.2.3.2.3.cmml">B</mi></mrow><mo id="S7.3.p2.5.m5.2.3.1" xref="S7.3.p2.5.m5.2.3.1.cmml">=</mo><mi id="S7.3.p2.5.m5.2.3.3" mathvariant="normal" xref="S7.3.p2.5.m5.2.3.3.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.3.p2.5.m5.2b"><apply id="S7.3.p2.5.m5.2.3.cmml" xref="S7.3.p2.5.m5.2.3"><eq id="S7.3.p2.5.m5.2.3.1.cmml" xref="S7.3.p2.5.m5.2.3.1"></eq><apply id="S7.3.p2.5.m5.2.3.2.cmml" xref="S7.3.p2.5.m5.2.3.2"><intersect id="S7.3.p2.5.m5.2.3.2.1.cmml" xref="S7.3.p2.5.m5.2.3.2.1"></intersect><apply id="S7.3.p2.5.m5.2.3.2.2.cmml" xref="S7.3.p2.5.m5.2.3.2.2"><csymbol cd="ambiguous" id="S7.3.p2.5.m5.2.3.2.2.1.cmml" xref="S7.3.p2.5.m5.2.3.2.2">subscript</csymbol><interval closure="closed" id="S7.3.p2.5.m5.2.3.2.2.2.1.cmml" xref="S7.3.p2.5.m5.2.3.2.2.2.2"><ci id="S7.3.p2.5.m5.1.1.cmml" xref="S7.3.p2.5.m5.1.1">𝑎</ci><ci id="S7.3.p2.5.m5.2.2.cmml" xref="S7.3.p2.5.m5.2.2">𝑏</ci></interval><ci id="S7.3.p2.5.m5.2.3.2.2.3.cmml" xref="S7.3.p2.5.m5.2.3.2.2.3">𝐴</ci></apply><ci id="S7.3.p2.5.m5.2.3.2.3.cmml" xref="S7.3.p2.5.m5.2.3.2.3">𝐵</ci></apply><emptyset id="S7.3.p2.5.m5.2.3.3.cmml" xref="S7.3.p2.5.m5.2.3.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.3.p2.5.m5.2c">[a,b]_{A}\cap B=\varnothing</annotation><annotation encoding="application/x-llamapun" id="S7.3.p2.5.m5.2d">[ italic_a , italic_b ] start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ∩ italic_B = ∅</annotation></semantics></math>. We claim that <math alttext="A/\sim" class="ltx_math_unparsed" display="inline" id="S7.3.p2.6.m6.1"><semantics id="S7.3.p2.6.m6.1a"><mrow id="S7.3.p2.6.m6.1b"><mi id="S7.3.p2.6.m6.1.1">A</mi><mo id="S7.3.p2.6.m6.1.2" rspace="0em">/</mo><mo id="S7.3.p2.6.m6.1.3" lspace="0em">∼</mo></mrow><annotation encoding="application/x-tex" id="S7.3.p2.6.m6.1c">A/\sim</annotation><annotation encoding="application/x-llamapun" id="S7.3.p2.6.m6.1d">italic_A / ∼</annotation></semantics></math> is Countryman, this finishes the proof since clearly <math alttext="A\trianglerighteq A/\sim" class="ltx_math_unparsed" display="inline" id="S7.3.p2.7.m7.1"><semantics id="S7.3.p2.7.m7.1a"><mrow id="S7.3.p2.7.m7.1b"><mi id="S7.3.p2.7.m7.1.1">A</mi><mi id="S7.3.p2.7.m7.1.2" mathvariant="normal">⊵</mi><mi id="S7.3.p2.7.m7.1.3">A</mi><mo id="S7.3.p2.7.m7.1.4" rspace="0em">/</mo><mo id="S7.3.p2.7.m7.1.5" lspace="0em">∼</mo></mrow><annotation encoding="application/x-tex" id="S7.3.p2.7.m7.1c">A\trianglerighteq A/\sim</annotation><annotation encoding="application/x-llamapun" id="S7.3.p2.7.m7.1d">italic_A ⊵ italic_A / ∼</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S7.4.p3"> <p class="ltx_p" id="S7.4.p3.18">Suppose not, then by <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib16" title="">16</a>, Theorem 1.3]</cite> <math alttext="A/\sim" class="ltx_math_unparsed" display="inline" id="S7.4.p3.1.m1.1"><semantics id="S7.4.p3.1.m1.1a"><mrow id="S7.4.p3.1.m1.1b"><mi id="S7.4.p3.1.m1.1.1">A</mi><mo id="S7.4.p3.1.m1.1.2" rspace="0em">/</mo><mo id="S7.4.p3.1.m1.1.3" lspace="0em">∼</mo></mrow><annotation encoding="application/x-tex" id="S7.4.p3.1.m1.1c">A/\sim</annotation><annotation encoding="application/x-llamapun" id="S7.4.p3.1.m1.1d">italic_A / ∼</annotation></semantics></math> contains a copy of <math alttext="B^{\star}" class="ltx_Math" display="inline" id="S7.4.p3.2.m2.1"><semantics id="S7.4.p3.2.m2.1a"><msup id="S7.4.p3.2.m2.1.1" xref="S7.4.p3.2.m2.1.1.cmml"><mi id="S7.4.p3.2.m2.1.1.2" xref="S7.4.p3.2.m2.1.1.2.cmml">B</mi><mo id="S7.4.p3.2.m2.1.1.3" xref="S7.4.p3.2.m2.1.1.3.cmml">⋆</mo></msup><annotation-xml encoding="MathML-Content" id="S7.4.p3.2.m2.1b"><apply id="S7.4.p3.2.m2.1.1.cmml" xref="S7.4.p3.2.m2.1.1"><csymbol cd="ambiguous" id="S7.4.p3.2.m2.1.1.1.cmml" xref="S7.4.p3.2.m2.1.1">superscript</csymbol><ci id="S7.4.p3.2.m2.1.1.2.cmml" xref="S7.4.p3.2.m2.1.1.2">𝐵</ci><ci id="S7.4.p3.2.m2.1.1.3.cmml" xref="S7.4.p3.2.m2.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.4.p3.2.m2.1c">B^{\star}</annotation><annotation encoding="application/x-llamapun" id="S7.4.p3.2.m2.1d">italic_B start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math>, and thus there is <math alttext="f:B^{\star}\to A" class="ltx_Math" display="inline" id="S7.4.p3.3.m3.1"><semantics id="S7.4.p3.3.m3.1a"><mrow id="S7.4.p3.3.m3.1.1" xref="S7.4.p3.3.m3.1.1.cmml"><mi id="S7.4.p3.3.m3.1.1.2" xref="S7.4.p3.3.m3.1.1.2.cmml">f</mi><mo id="S7.4.p3.3.m3.1.1.1" lspace="0.278em" rspace="0.278em" xref="S7.4.p3.3.m3.1.1.1.cmml">:</mo><mrow id="S7.4.p3.3.m3.1.1.3" xref="S7.4.p3.3.m3.1.1.3.cmml"><msup id="S7.4.p3.3.m3.1.1.3.2" xref="S7.4.p3.3.m3.1.1.3.2.cmml"><mi id="S7.4.p3.3.m3.1.1.3.2.2" xref="S7.4.p3.3.m3.1.1.3.2.2.cmml">B</mi><mo id="S7.4.p3.3.m3.1.1.3.2.3" xref="S7.4.p3.3.m3.1.1.3.2.3.cmml">⋆</mo></msup><mo id="S7.4.p3.3.m3.1.1.3.1" stretchy="false" xref="S7.4.p3.3.m3.1.1.3.1.cmml">→</mo><mi id="S7.4.p3.3.m3.1.1.3.3" xref="S7.4.p3.3.m3.1.1.3.3.cmml">A</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.4.p3.3.m3.1b"><apply id="S7.4.p3.3.m3.1.1.cmml" xref="S7.4.p3.3.m3.1.1"><ci id="S7.4.p3.3.m3.1.1.1.cmml" xref="S7.4.p3.3.m3.1.1.1">:</ci><ci id="S7.4.p3.3.m3.1.1.2.cmml" xref="S7.4.p3.3.m3.1.1.2">𝑓</ci><apply id="S7.4.p3.3.m3.1.1.3.cmml" xref="S7.4.p3.3.m3.1.1.3"><ci id="S7.4.p3.3.m3.1.1.3.1.cmml" xref="S7.4.p3.3.m3.1.1.3.1">→</ci><apply id="S7.4.p3.3.m3.1.1.3.2.cmml" xref="S7.4.p3.3.m3.1.1.3.2"><csymbol cd="ambiguous" id="S7.4.p3.3.m3.1.1.3.2.1.cmml" xref="S7.4.p3.3.m3.1.1.3.2">superscript</csymbol><ci id="S7.4.p3.3.m3.1.1.3.2.2.cmml" xref="S7.4.p3.3.m3.1.1.3.2.2">𝐵</ci><ci id="S7.4.p3.3.m3.1.1.3.2.3.cmml" xref="S7.4.p3.3.m3.1.1.3.2.3">⋆</ci></apply><ci id="S7.4.p3.3.m3.1.1.3.3.cmml" xref="S7.4.p3.3.m3.1.1.3.3">𝐴</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.4.p3.3.m3.1c">f:B^{\star}\to A</annotation><annotation encoding="application/x-llamapun" id="S7.4.p3.3.m3.1d">italic_f : italic_B start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT → italic_A</annotation></semantics></math> such that for every <math alttext="x<_{B^{\star}}y" class="ltx_Math" display="inline" id="S7.4.p3.4.m4.1"><semantics id="S7.4.p3.4.m4.1a"><mrow id="S7.4.p3.4.m4.1.1" xref="S7.4.p3.4.m4.1.1.cmml"><mi id="S7.4.p3.4.m4.1.1.2" xref="S7.4.p3.4.m4.1.1.2.cmml">x</mi><msub id="S7.4.p3.4.m4.1.1.1" xref="S7.4.p3.4.m4.1.1.1.cmml"><mo id="S7.4.p3.4.m4.1.1.1.2" xref="S7.4.p3.4.m4.1.1.1.2.cmml"><</mo><msup id="S7.4.p3.4.m4.1.1.1.3" xref="S7.4.p3.4.m4.1.1.1.3.cmml"><mi id="S7.4.p3.4.m4.1.1.1.3.2" xref="S7.4.p3.4.m4.1.1.1.3.2.cmml">B</mi><mo id="S7.4.p3.4.m4.1.1.1.3.3" xref="S7.4.p3.4.m4.1.1.1.3.3.cmml">⋆</mo></msup></msub><mi id="S7.4.p3.4.m4.1.1.3" xref="S7.4.p3.4.m4.1.1.3.cmml">y</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.4.p3.4.m4.1b"><apply id="S7.4.p3.4.m4.1.1.cmml" xref="S7.4.p3.4.m4.1.1"><apply id="S7.4.p3.4.m4.1.1.1.cmml" xref="S7.4.p3.4.m4.1.1.1"><csymbol cd="ambiguous" id="S7.4.p3.4.m4.1.1.1.1.cmml" xref="S7.4.p3.4.m4.1.1.1">subscript</csymbol><lt id="S7.4.p3.4.m4.1.1.1.2.cmml" xref="S7.4.p3.4.m4.1.1.1.2"></lt><apply id="S7.4.p3.4.m4.1.1.1.3.cmml" xref="S7.4.p3.4.m4.1.1.1.3"><csymbol cd="ambiguous" id="S7.4.p3.4.m4.1.1.1.3.1.cmml" xref="S7.4.p3.4.m4.1.1.1.3">superscript</csymbol><ci id="S7.4.p3.4.m4.1.1.1.3.2.cmml" xref="S7.4.p3.4.m4.1.1.1.3.2">𝐵</ci><ci id="S7.4.p3.4.m4.1.1.1.3.3.cmml" xref="S7.4.p3.4.m4.1.1.1.3.3">⋆</ci></apply></apply><ci id="S7.4.p3.4.m4.1.1.2.cmml" xref="S7.4.p3.4.m4.1.1.2">𝑥</ci><ci id="S7.4.p3.4.m4.1.1.3.cmml" xref="S7.4.p3.4.m4.1.1.3">𝑦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.4.p3.4.m4.1c">x<_{B^{\star}}y</annotation><annotation encoding="application/x-llamapun" id="S7.4.p3.4.m4.1d">italic_x < start_POSTSUBSCRIPT italic_B start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_y</annotation></semantics></math>, <math alttext="f(x)<_{A}f(y)" class="ltx_Math" display="inline" id="S7.4.p3.5.m5.2"><semantics id="S7.4.p3.5.m5.2a"><mrow id="S7.4.p3.5.m5.2.3" xref="S7.4.p3.5.m5.2.3.cmml"><mrow id="S7.4.p3.5.m5.2.3.2" xref="S7.4.p3.5.m5.2.3.2.cmml"><mi id="S7.4.p3.5.m5.2.3.2.2" xref="S7.4.p3.5.m5.2.3.2.2.cmml">f</mi><mo id="S7.4.p3.5.m5.2.3.2.1" xref="S7.4.p3.5.m5.2.3.2.1.cmml">⁢</mo><mrow id="S7.4.p3.5.m5.2.3.2.3.2" xref="S7.4.p3.5.m5.2.3.2.cmml"><mo id="S7.4.p3.5.m5.2.3.2.3.2.1" stretchy="false" xref="S7.4.p3.5.m5.2.3.2.cmml">(</mo><mi id="S7.4.p3.5.m5.1.1" xref="S7.4.p3.5.m5.1.1.cmml">x</mi><mo id="S7.4.p3.5.m5.2.3.2.3.2.2" stretchy="false" xref="S7.4.p3.5.m5.2.3.2.cmml">)</mo></mrow></mrow><msub id="S7.4.p3.5.m5.2.3.1" xref="S7.4.p3.5.m5.2.3.1.cmml"><mo id="S7.4.p3.5.m5.2.3.1.2" xref="S7.4.p3.5.m5.2.3.1.2.cmml"><</mo><mi id="S7.4.p3.5.m5.2.3.1.3" xref="S7.4.p3.5.m5.2.3.1.3.cmml">A</mi></msub><mrow id="S7.4.p3.5.m5.2.3.3" xref="S7.4.p3.5.m5.2.3.3.cmml"><mi id="S7.4.p3.5.m5.2.3.3.2" xref="S7.4.p3.5.m5.2.3.3.2.cmml">f</mi><mo id="S7.4.p3.5.m5.2.3.3.1" xref="S7.4.p3.5.m5.2.3.3.1.cmml">⁢</mo><mrow id="S7.4.p3.5.m5.2.3.3.3.2" xref="S7.4.p3.5.m5.2.3.3.cmml"><mo id="S7.4.p3.5.m5.2.3.3.3.2.1" stretchy="false" xref="S7.4.p3.5.m5.2.3.3.cmml">(</mo><mi id="S7.4.p3.5.m5.2.2" xref="S7.4.p3.5.m5.2.2.cmml">y</mi><mo id="S7.4.p3.5.m5.2.3.3.3.2.2" stretchy="false" xref="S7.4.p3.5.m5.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.4.p3.5.m5.2b"><apply id="S7.4.p3.5.m5.2.3.cmml" xref="S7.4.p3.5.m5.2.3"><apply id="S7.4.p3.5.m5.2.3.1.cmml" xref="S7.4.p3.5.m5.2.3.1"><csymbol cd="ambiguous" id="S7.4.p3.5.m5.2.3.1.1.cmml" xref="S7.4.p3.5.m5.2.3.1">subscript</csymbol><lt id="S7.4.p3.5.m5.2.3.1.2.cmml" xref="S7.4.p3.5.m5.2.3.1.2"></lt><ci id="S7.4.p3.5.m5.2.3.1.3.cmml" xref="S7.4.p3.5.m5.2.3.1.3">𝐴</ci></apply><apply id="S7.4.p3.5.m5.2.3.2.cmml" xref="S7.4.p3.5.m5.2.3.2"><times id="S7.4.p3.5.m5.2.3.2.1.cmml" xref="S7.4.p3.5.m5.2.3.2.1"></times><ci id="S7.4.p3.5.m5.2.3.2.2.cmml" xref="S7.4.p3.5.m5.2.3.2.2">𝑓</ci><ci id="S7.4.p3.5.m5.1.1.cmml" xref="S7.4.p3.5.m5.1.1">𝑥</ci></apply><apply id="S7.4.p3.5.m5.2.3.3.cmml" xref="S7.4.p3.5.m5.2.3.3"><times id="S7.4.p3.5.m5.2.3.3.1.cmml" xref="S7.4.p3.5.m5.2.3.3.1"></times><ci id="S7.4.p3.5.m5.2.3.3.2.cmml" xref="S7.4.p3.5.m5.2.3.3.2">𝑓</ci><ci id="S7.4.p3.5.m5.2.2.cmml" xref="S7.4.p3.5.m5.2.2">𝑦</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.4.p3.5.m5.2c">f(x)<_{A}f(y)</annotation><annotation encoding="application/x-llamapun" id="S7.4.p3.5.m5.2d">italic_f ( italic_x ) < start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_f ( italic_y )</annotation></semantics></math> and <math alttext="f(a)\nsim f(b)" class="ltx_Math" display="inline" id="S7.4.p3.6.m6.2"><semantics id="S7.4.p3.6.m6.2a"><mrow id="S7.4.p3.6.m6.2.3" xref="S7.4.p3.6.m6.2.3.cmml"><mrow id="S7.4.p3.6.m6.2.3.2" xref="S7.4.p3.6.m6.2.3.2.cmml"><mi id="S7.4.p3.6.m6.2.3.2.2" xref="S7.4.p3.6.m6.2.3.2.2.cmml">f</mi><mo id="S7.4.p3.6.m6.2.3.2.1" xref="S7.4.p3.6.m6.2.3.2.1.cmml">⁢</mo><mrow id="S7.4.p3.6.m6.2.3.2.3.2" xref="S7.4.p3.6.m6.2.3.2.cmml"><mo id="S7.4.p3.6.m6.2.3.2.3.2.1" stretchy="false" xref="S7.4.p3.6.m6.2.3.2.cmml">(</mo><mi id="S7.4.p3.6.m6.1.1" xref="S7.4.p3.6.m6.1.1.cmml">a</mi><mo id="S7.4.p3.6.m6.2.3.2.3.2.2" stretchy="false" xref="S7.4.p3.6.m6.2.3.2.cmml">)</mo></mrow></mrow><mo id="S7.4.p3.6.m6.2.3.1" xref="S7.4.p3.6.m6.2.3.1.cmml">≁</mo><mrow id="S7.4.p3.6.m6.2.3.3" xref="S7.4.p3.6.m6.2.3.3.cmml"><mi id="S7.4.p3.6.m6.2.3.3.2" xref="S7.4.p3.6.m6.2.3.3.2.cmml">f</mi><mo id="S7.4.p3.6.m6.2.3.3.1" xref="S7.4.p3.6.m6.2.3.3.1.cmml">⁢</mo><mrow id="S7.4.p3.6.m6.2.3.3.3.2" xref="S7.4.p3.6.m6.2.3.3.cmml"><mo id="S7.4.p3.6.m6.2.3.3.3.2.1" stretchy="false" xref="S7.4.p3.6.m6.2.3.3.cmml">(</mo><mi id="S7.4.p3.6.m6.2.2" xref="S7.4.p3.6.m6.2.2.cmml">b</mi><mo id="S7.4.p3.6.m6.2.3.3.3.2.2" stretchy="false" xref="S7.4.p3.6.m6.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.4.p3.6.m6.2b"><apply id="S7.4.p3.6.m6.2.3.cmml" xref="S7.4.p3.6.m6.2.3"><csymbol cd="latexml" id="S7.4.p3.6.m6.2.3.1.cmml" xref="S7.4.p3.6.m6.2.3.1">not-similar-to</csymbol><apply id="S7.4.p3.6.m6.2.3.2.cmml" xref="S7.4.p3.6.m6.2.3.2"><times id="S7.4.p3.6.m6.2.3.2.1.cmml" xref="S7.4.p3.6.m6.2.3.2.1"></times><ci id="S7.4.p3.6.m6.2.3.2.2.cmml" xref="S7.4.p3.6.m6.2.3.2.2">𝑓</ci><ci id="S7.4.p3.6.m6.1.1.cmml" xref="S7.4.p3.6.m6.1.1">𝑎</ci></apply><apply id="S7.4.p3.6.m6.2.3.3.cmml" xref="S7.4.p3.6.m6.2.3.3"><times id="S7.4.p3.6.m6.2.3.3.1.cmml" xref="S7.4.p3.6.m6.2.3.3.1"></times><ci id="S7.4.p3.6.m6.2.3.3.2.cmml" xref="S7.4.p3.6.m6.2.3.3.2">𝑓</ci><ci id="S7.4.p3.6.m6.2.2.cmml" xref="S7.4.p3.6.m6.2.2">𝑏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.4.p3.6.m6.2c">f(a)\nsim f(b)</annotation><annotation encoding="application/x-llamapun" id="S7.4.p3.6.m6.2d">italic_f ( italic_a ) ≁ italic_f ( italic_b )</annotation></semantics></math>. Since <math alttext="B^{\star}" class="ltx_Math" display="inline" id="S7.4.p3.7.m7.1"><semantics id="S7.4.p3.7.m7.1a"><msup id="S7.4.p3.7.m7.1.1" xref="S7.4.p3.7.m7.1.1.cmml"><mi id="S7.4.p3.7.m7.1.1.2" xref="S7.4.p3.7.m7.1.1.2.cmml">B</mi><mo id="S7.4.p3.7.m7.1.1.3" xref="S7.4.p3.7.m7.1.1.3.cmml">⋆</mo></msup><annotation-xml encoding="MathML-Content" id="S7.4.p3.7.m7.1b"><apply id="S7.4.p3.7.m7.1.1.cmml" xref="S7.4.p3.7.m7.1.1"><csymbol cd="ambiguous" id="S7.4.p3.7.m7.1.1.1.cmml" xref="S7.4.p3.7.m7.1.1">superscript</csymbol><ci id="S7.4.p3.7.m7.1.1.2.cmml" xref="S7.4.p3.7.m7.1.1.2">𝐵</ci><ci id="S7.4.p3.7.m7.1.1.3.cmml" xref="S7.4.p3.7.m7.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.4.p3.7.m7.1c">B^{\star}</annotation><annotation encoding="application/x-llamapun" id="S7.4.p3.7.m7.1d">italic_B start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math> is <math alttext="\aleph_{1}" class="ltx_Math" display="inline" id="S7.4.p3.8.m8.1"><semantics id="S7.4.p3.8.m8.1a"><msub id="S7.4.p3.8.m8.1.1" xref="S7.4.p3.8.m8.1.1.cmml"><mi id="S7.4.p3.8.m8.1.1.2" mathvariant="normal" xref="S7.4.p3.8.m8.1.1.2.cmml">ℵ</mi><mn id="S7.4.p3.8.m8.1.1.3" xref="S7.4.p3.8.m8.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S7.4.p3.8.m8.1b"><apply id="S7.4.p3.8.m8.1.1.cmml" xref="S7.4.p3.8.m8.1.1"><csymbol cd="ambiguous" id="S7.4.p3.8.m8.1.1.1.cmml" xref="S7.4.p3.8.m8.1.1">subscript</csymbol><ci id="S7.4.p3.8.m8.1.1.2.cmml" xref="S7.4.p3.8.m8.1.1.2">ℵ</ci><cn id="S7.4.p3.8.m8.1.1.3.cmml" type="integer" xref="S7.4.p3.8.m8.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.4.p3.8.m8.1c">\aleph_{1}</annotation><annotation encoding="application/x-llamapun" id="S7.4.p3.8.m8.1d">roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-dense and not Suslin (use <math alttext="\mathsf{PFA}" class="ltx_Math" display="inline" id="S7.4.p3.9.m9.1"><semantics id="S7.4.p3.9.m9.1a"><mi id="S7.4.p3.9.m9.1.1" xref="S7.4.p3.9.m9.1.1.cmml">𝖯𝖥𝖠</mi><annotation-xml encoding="MathML-Content" id="S7.4.p3.9.m9.1b"><ci id="S7.4.p3.9.m9.1.1.cmml" xref="S7.4.p3.9.m9.1.1">𝖯𝖥𝖠</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.4.p3.9.m9.1c">\mathsf{PFA}</annotation><annotation encoding="application/x-llamapun" id="S7.4.p3.9.m9.1d">sansserif_PFA</annotation></semantics></math> or <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib14" title="">14</a>, Fact 3.9]</cite>), there is <math alttext="\{{[x_{\xi},y_{\xi}]}_{B^{\star}}:\xi<\omega_{1}\}" class="ltx_Math" display="inline" id="S7.4.p3.10.m10.2"><semantics id="S7.4.p3.10.m10.2a"><mrow id="S7.4.p3.10.m10.2.2.2" xref="S7.4.p3.10.m10.2.2.3.cmml"><mo id="S7.4.p3.10.m10.2.2.2.3" stretchy="false" xref="S7.4.p3.10.m10.2.2.3.1.cmml">{</mo><msub id="S7.4.p3.10.m10.1.1.1.1" xref="S7.4.p3.10.m10.1.1.1.1.cmml"><mrow id="S7.4.p3.10.m10.1.1.1.1.2.2" xref="S7.4.p3.10.m10.1.1.1.1.2.3.cmml"><mo id="S7.4.p3.10.m10.1.1.1.1.2.2.3" stretchy="false" xref="S7.4.p3.10.m10.1.1.1.1.2.3.cmml">[</mo><msub id="S7.4.p3.10.m10.1.1.1.1.1.1.1" xref="S7.4.p3.10.m10.1.1.1.1.1.1.1.cmml"><mi id="S7.4.p3.10.m10.1.1.1.1.1.1.1.2" xref="S7.4.p3.10.m10.1.1.1.1.1.1.1.2.cmml">x</mi><mi id="S7.4.p3.10.m10.1.1.1.1.1.1.1.3" xref="S7.4.p3.10.m10.1.1.1.1.1.1.1.3.cmml">ξ</mi></msub><mo id="S7.4.p3.10.m10.1.1.1.1.2.2.4" xref="S7.4.p3.10.m10.1.1.1.1.2.3.cmml">,</mo><msub id="S7.4.p3.10.m10.1.1.1.1.2.2.2" xref="S7.4.p3.10.m10.1.1.1.1.2.2.2.cmml"><mi id="S7.4.p3.10.m10.1.1.1.1.2.2.2.2" xref="S7.4.p3.10.m10.1.1.1.1.2.2.2.2.cmml">y</mi><mi id="S7.4.p3.10.m10.1.1.1.1.2.2.2.3" xref="S7.4.p3.10.m10.1.1.1.1.2.2.2.3.cmml">ξ</mi></msub><mo id="S7.4.p3.10.m10.1.1.1.1.2.2.5" rspace="0.278em" stretchy="false" xref="S7.4.p3.10.m10.1.1.1.1.2.3.cmml">]</mo></mrow><msup id="S7.4.p3.10.m10.1.1.1.1.4" xref="S7.4.p3.10.m10.1.1.1.1.4.cmml"><mi id="S7.4.p3.10.m10.1.1.1.1.4.2" xref="S7.4.p3.10.m10.1.1.1.1.4.2.cmml">B</mi><mo id="S7.4.p3.10.m10.1.1.1.1.4.3" xref="S7.4.p3.10.m10.1.1.1.1.4.3.cmml">⋆</mo></msup></msub><mo id="S7.4.p3.10.m10.2.2.2.4" rspace="0.278em" xref="S7.4.p3.10.m10.2.2.3.1.cmml">:</mo><mrow id="S7.4.p3.10.m10.2.2.2.2" xref="S7.4.p3.10.m10.2.2.2.2.cmml"><mi id="S7.4.p3.10.m10.2.2.2.2.2" xref="S7.4.p3.10.m10.2.2.2.2.2.cmml">ξ</mi><mo id="S7.4.p3.10.m10.2.2.2.2.1" xref="S7.4.p3.10.m10.2.2.2.2.1.cmml"><</mo><msub id="S7.4.p3.10.m10.2.2.2.2.3" xref="S7.4.p3.10.m10.2.2.2.2.3.cmml"><mi id="S7.4.p3.10.m10.2.2.2.2.3.2" xref="S7.4.p3.10.m10.2.2.2.2.3.2.cmml">ω</mi><mn id="S7.4.p3.10.m10.2.2.2.2.3.3" xref="S7.4.p3.10.m10.2.2.2.2.3.3.cmml">1</mn></msub></mrow><mo id="S7.4.p3.10.m10.2.2.2.5" stretchy="false" xref="S7.4.p3.10.m10.2.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.4.p3.10.m10.2b"><apply id="S7.4.p3.10.m10.2.2.3.cmml" xref="S7.4.p3.10.m10.2.2.2"><csymbol cd="latexml" id="S7.4.p3.10.m10.2.2.3.1.cmml" xref="S7.4.p3.10.m10.2.2.2.3">conditional-set</csymbol><apply id="S7.4.p3.10.m10.1.1.1.1.cmml" xref="S7.4.p3.10.m10.1.1.1.1"><csymbol cd="ambiguous" id="S7.4.p3.10.m10.1.1.1.1.3.cmml" xref="S7.4.p3.10.m10.1.1.1.1">subscript</csymbol><interval closure="closed" id="S7.4.p3.10.m10.1.1.1.1.2.3.cmml" xref="S7.4.p3.10.m10.1.1.1.1.2.2"><apply id="S7.4.p3.10.m10.1.1.1.1.1.1.1.cmml" xref="S7.4.p3.10.m10.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S7.4.p3.10.m10.1.1.1.1.1.1.1.1.cmml" xref="S7.4.p3.10.m10.1.1.1.1.1.1.1">subscript</csymbol><ci id="S7.4.p3.10.m10.1.1.1.1.1.1.1.2.cmml" xref="S7.4.p3.10.m10.1.1.1.1.1.1.1.2">𝑥</ci><ci id="S7.4.p3.10.m10.1.1.1.1.1.1.1.3.cmml" xref="S7.4.p3.10.m10.1.1.1.1.1.1.1.3">𝜉</ci></apply><apply id="S7.4.p3.10.m10.1.1.1.1.2.2.2.cmml" xref="S7.4.p3.10.m10.1.1.1.1.2.2.2"><csymbol cd="ambiguous" id="S7.4.p3.10.m10.1.1.1.1.2.2.2.1.cmml" xref="S7.4.p3.10.m10.1.1.1.1.2.2.2">subscript</csymbol><ci id="S7.4.p3.10.m10.1.1.1.1.2.2.2.2.cmml" xref="S7.4.p3.10.m10.1.1.1.1.2.2.2.2">𝑦</ci><ci id="S7.4.p3.10.m10.1.1.1.1.2.2.2.3.cmml" xref="S7.4.p3.10.m10.1.1.1.1.2.2.2.3">𝜉</ci></apply></interval><apply id="S7.4.p3.10.m10.1.1.1.1.4.cmml" xref="S7.4.p3.10.m10.1.1.1.1.4"><csymbol cd="ambiguous" id="S7.4.p3.10.m10.1.1.1.1.4.1.cmml" xref="S7.4.p3.10.m10.1.1.1.1.4">superscript</csymbol><ci id="S7.4.p3.10.m10.1.1.1.1.4.2.cmml" xref="S7.4.p3.10.m10.1.1.1.1.4.2">𝐵</ci><ci id="S7.4.p3.10.m10.1.1.1.1.4.3.cmml" xref="S7.4.p3.10.m10.1.1.1.1.4.3">⋆</ci></apply></apply><apply id="S7.4.p3.10.m10.2.2.2.2.cmml" xref="S7.4.p3.10.m10.2.2.2.2"><lt id="S7.4.p3.10.m10.2.2.2.2.1.cmml" xref="S7.4.p3.10.m10.2.2.2.2.1"></lt><ci id="S7.4.p3.10.m10.2.2.2.2.2.cmml" xref="S7.4.p3.10.m10.2.2.2.2.2">𝜉</ci><apply id="S7.4.p3.10.m10.2.2.2.2.3.cmml" xref="S7.4.p3.10.m10.2.2.2.2.3"><csymbol cd="ambiguous" id="S7.4.p3.10.m10.2.2.2.2.3.1.cmml" xref="S7.4.p3.10.m10.2.2.2.2.3">subscript</csymbol><ci id="S7.4.p3.10.m10.2.2.2.2.3.2.cmml" xref="S7.4.p3.10.m10.2.2.2.2.3.2">𝜔</ci><cn id="S7.4.p3.10.m10.2.2.2.2.3.3.cmml" type="integer" xref="S7.4.p3.10.m10.2.2.2.2.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.4.p3.10.m10.2c">\{{[x_{\xi},y_{\xi}]}_{B^{\star}}:\xi<\omega_{1}\}</annotation><annotation encoding="application/x-llamapun" id="S7.4.p3.10.m10.2d">{ [ italic_x start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT , italic_y start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT ] start_POSTSUBSCRIPT italic_B start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT : italic_ξ < italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT }</annotation></semantics></math> a collection of disjoint closed intervals of <math alttext="B^{\star}" class="ltx_Math" display="inline" id="S7.4.p3.11.m11.1"><semantics id="S7.4.p3.11.m11.1a"><msup id="S7.4.p3.11.m11.1.1" xref="S7.4.p3.11.m11.1.1.cmml"><mi id="S7.4.p3.11.m11.1.1.2" xref="S7.4.p3.11.m11.1.1.2.cmml">B</mi><mo id="S7.4.p3.11.m11.1.1.3" xref="S7.4.p3.11.m11.1.1.3.cmml">⋆</mo></msup><annotation-xml encoding="MathML-Content" id="S7.4.p3.11.m11.1b"><apply id="S7.4.p3.11.m11.1.1.cmml" xref="S7.4.p3.11.m11.1.1"><csymbol cd="ambiguous" id="S7.4.p3.11.m11.1.1.1.cmml" xref="S7.4.p3.11.m11.1.1">superscript</csymbol><ci id="S7.4.p3.11.m11.1.1.2.cmml" xref="S7.4.p3.11.m11.1.1.2">𝐵</ci><ci id="S7.4.p3.11.m11.1.1.3.cmml" xref="S7.4.p3.11.m11.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.4.p3.11.m11.1c">B^{\star}</annotation><annotation encoding="application/x-llamapun" id="S7.4.p3.11.m11.1d">italic_B start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math>. Now for each <math alttext="\xi<\omega_{1}" class="ltx_Math" display="inline" id="S7.4.p3.12.m12.1"><semantics id="S7.4.p3.12.m12.1a"><mrow id="S7.4.p3.12.m12.1.1" xref="S7.4.p3.12.m12.1.1.cmml"><mi id="S7.4.p3.12.m12.1.1.2" xref="S7.4.p3.12.m12.1.1.2.cmml">ξ</mi><mo id="S7.4.p3.12.m12.1.1.1" xref="S7.4.p3.12.m12.1.1.1.cmml"><</mo><msub id="S7.4.p3.12.m12.1.1.3" xref="S7.4.p3.12.m12.1.1.3.cmml"><mi id="S7.4.p3.12.m12.1.1.3.2" xref="S7.4.p3.12.m12.1.1.3.2.cmml">ω</mi><mn id="S7.4.p3.12.m12.1.1.3.3" xref="S7.4.p3.12.m12.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.4.p3.12.m12.1b"><apply id="S7.4.p3.12.m12.1.1.cmml" xref="S7.4.p3.12.m12.1.1"><lt id="S7.4.p3.12.m12.1.1.1.cmml" xref="S7.4.p3.12.m12.1.1.1"></lt><ci id="S7.4.p3.12.m12.1.1.2.cmml" xref="S7.4.p3.12.m12.1.1.2">𝜉</ci><apply id="S7.4.p3.12.m12.1.1.3.cmml" xref="S7.4.p3.12.m12.1.1.3"><csymbol cd="ambiguous" id="S7.4.p3.12.m12.1.1.3.1.cmml" xref="S7.4.p3.12.m12.1.1.3">subscript</csymbol><ci id="S7.4.p3.12.m12.1.1.3.2.cmml" xref="S7.4.p3.12.m12.1.1.3.2">𝜔</ci><cn id="S7.4.p3.12.m12.1.1.3.3.cmml" type="integer" xref="S7.4.p3.12.m12.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.4.p3.12.m12.1c">\xi<\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S7.4.p3.12.m12.1d">italic_ξ < italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="f(x_{\xi})<_{A}g(y_{\xi})" class="ltx_Math" display="inline" id="S7.4.p3.13.m13.2"><semantics id="S7.4.p3.13.m13.2a"><mrow id="S7.4.p3.13.m13.2.2" xref="S7.4.p3.13.m13.2.2.cmml"><mrow id="S7.4.p3.13.m13.1.1.1" xref="S7.4.p3.13.m13.1.1.1.cmml"><mi id="S7.4.p3.13.m13.1.1.1.3" xref="S7.4.p3.13.m13.1.1.1.3.cmml">f</mi><mo id="S7.4.p3.13.m13.1.1.1.2" xref="S7.4.p3.13.m13.1.1.1.2.cmml">⁢</mo><mrow id="S7.4.p3.13.m13.1.1.1.1.1" xref="S7.4.p3.13.m13.1.1.1.1.1.1.cmml"><mo id="S7.4.p3.13.m13.1.1.1.1.1.2" stretchy="false" xref="S7.4.p3.13.m13.1.1.1.1.1.1.cmml">(</mo><msub id="S7.4.p3.13.m13.1.1.1.1.1.1" xref="S7.4.p3.13.m13.1.1.1.1.1.1.cmml"><mi id="S7.4.p3.13.m13.1.1.1.1.1.1.2" xref="S7.4.p3.13.m13.1.1.1.1.1.1.2.cmml">x</mi><mi id="S7.4.p3.13.m13.1.1.1.1.1.1.3" xref="S7.4.p3.13.m13.1.1.1.1.1.1.3.cmml">ξ</mi></msub><mo id="S7.4.p3.13.m13.1.1.1.1.1.3" stretchy="false" xref="S7.4.p3.13.m13.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><msub id="S7.4.p3.13.m13.2.2.3" xref="S7.4.p3.13.m13.2.2.3.cmml"><mo id="S7.4.p3.13.m13.2.2.3.2" xref="S7.4.p3.13.m13.2.2.3.2.cmml"><</mo><mi id="S7.4.p3.13.m13.2.2.3.3" xref="S7.4.p3.13.m13.2.2.3.3.cmml">A</mi></msub><mrow id="S7.4.p3.13.m13.2.2.2" xref="S7.4.p3.13.m13.2.2.2.cmml"><mi id="S7.4.p3.13.m13.2.2.2.3" xref="S7.4.p3.13.m13.2.2.2.3.cmml">g</mi><mo id="S7.4.p3.13.m13.2.2.2.2" xref="S7.4.p3.13.m13.2.2.2.2.cmml">⁢</mo><mrow id="S7.4.p3.13.m13.2.2.2.1.1" xref="S7.4.p3.13.m13.2.2.2.1.1.1.cmml"><mo id="S7.4.p3.13.m13.2.2.2.1.1.2" stretchy="false" xref="S7.4.p3.13.m13.2.2.2.1.1.1.cmml">(</mo><msub id="S7.4.p3.13.m13.2.2.2.1.1.1" xref="S7.4.p3.13.m13.2.2.2.1.1.1.cmml"><mi id="S7.4.p3.13.m13.2.2.2.1.1.1.2" xref="S7.4.p3.13.m13.2.2.2.1.1.1.2.cmml">y</mi><mi id="S7.4.p3.13.m13.2.2.2.1.1.1.3" xref="S7.4.p3.13.m13.2.2.2.1.1.1.3.cmml">ξ</mi></msub><mo id="S7.4.p3.13.m13.2.2.2.1.1.3" stretchy="false" xref="S7.4.p3.13.m13.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.4.p3.13.m13.2b"><apply id="S7.4.p3.13.m13.2.2.cmml" xref="S7.4.p3.13.m13.2.2"><apply id="S7.4.p3.13.m13.2.2.3.cmml" xref="S7.4.p3.13.m13.2.2.3"><csymbol cd="ambiguous" id="S7.4.p3.13.m13.2.2.3.1.cmml" xref="S7.4.p3.13.m13.2.2.3">subscript</csymbol><lt id="S7.4.p3.13.m13.2.2.3.2.cmml" xref="S7.4.p3.13.m13.2.2.3.2"></lt><ci id="S7.4.p3.13.m13.2.2.3.3.cmml" xref="S7.4.p3.13.m13.2.2.3.3">𝐴</ci></apply><apply id="S7.4.p3.13.m13.1.1.1.cmml" xref="S7.4.p3.13.m13.1.1.1"><times id="S7.4.p3.13.m13.1.1.1.2.cmml" xref="S7.4.p3.13.m13.1.1.1.2"></times><ci id="S7.4.p3.13.m13.1.1.1.3.cmml" xref="S7.4.p3.13.m13.1.1.1.3">𝑓</ci><apply id="S7.4.p3.13.m13.1.1.1.1.1.1.cmml" xref="S7.4.p3.13.m13.1.1.1.1.1"><csymbol cd="ambiguous" id="S7.4.p3.13.m13.1.1.1.1.1.1.1.cmml" xref="S7.4.p3.13.m13.1.1.1.1.1">subscript</csymbol><ci id="S7.4.p3.13.m13.1.1.1.1.1.1.2.cmml" xref="S7.4.p3.13.m13.1.1.1.1.1.1.2">𝑥</ci><ci id="S7.4.p3.13.m13.1.1.1.1.1.1.3.cmml" xref="S7.4.p3.13.m13.1.1.1.1.1.1.3">𝜉</ci></apply></apply><apply id="S7.4.p3.13.m13.2.2.2.cmml" xref="S7.4.p3.13.m13.2.2.2"><times id="S7.4.p3.13.m13.2.2.2.2.cmml" xref="S7.4.p3.13.m13.2.2.2.2"></times><ci id="S7.4.p3.13.m13.2.2.2.3.cmml" xref="S7.4.p3.13.m13.2.2.2.3">𝑔</ci><apply id="S7.4.p3.13.m13.2.2.2.1.1.1.cmml" xref="S7.4.p3.13.m13.2.2.2.1.1"><csymbol cd="ambiguous" id="S7.4.p3.13.m13.2.2.2.1.1.1.1.cmml" xref="S7.4.p3.13.m13.2.2.2.1.1">subscript</csymbol><ci id="S7.4.p3.13.m13.2.2.2.1.1.1.2.cmml" xref="S7.4.p3.13.m13.2.2.2.1.1.1.2">𝑦</ci><ci id="S7.4.p3.13.m13.2.2.2.1.1.1.3.cmml" xref="S7.4.p3.13.m13.2.2.2.1.1.1.3">𝜉</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.4.p3.13.m13.2c">f(x_{\xi})<_{A}g(y_{\xi})</annotation><annotation encoding="application/x-llamapun" id="S7.4.p3.13.m13.2d">italic_f ( italic_x start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT ) < start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_g ( italic_y start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT )</annotation></semantics></math> and <math alttext="f(x_{\xi})\nsim g(y_{\xi})" class="ltx_Math" display="inline" id="S7.4.p3.14.m14.2"><semantics id="S7.4.p3.14.m14.2a"><mrow id="S7.4.p3.14.m14.2.2" xref="S7.4.p3.14.m14.2.2.cmml"><mrow id="S7.4.p3.14.m14.1.1.1" xref="S7.4.p3.14.m14.1.1.1.cmml"><mi id="S7.4.p3.14.m14.1.1.1.3" xref="S7.4.p3.14.m14.1.1.1.3.cmml">f</mi><mo id="S7.4.p3.14.m14.1.1.1.2" xref="S7.4.p3.14.m14.1.1.1.2.cmml">⁢</mo><mrow id="S7.4.p3.14.m14.1.1.1.1.1" xref="S7.4.p3.14.m14.1.1.1.1.1.1.cmml"><mo id="S7.4.p3.14.m14.1.1.1.1.1.2" stretchy="false" xref="S7.4.p3.14.m14.1.1.1.1.1.1.cmml">(</mo><msub id="S7.4.p3.14.m14.1.1.1.1.1.1" xref="S7.4.p3.14.m14.1.1.1.1.1.1.cmml"><mi id="S7.4.p3.14.m14.1.1.1.1.1.1.2" xref="S7.4.p3.14.m14.1.1.1.1.1.1.2.cmml">x</mi><mi id="S7.4.p3.14.m14.1.1.1.1.1.1.3" xref="S7.4.p3.14.m14.1.1.1.1.1.1.3.cmml">ξ</mi></msub><mo id="S7.4.p3.14.m14.1.1.1.1.1.3" stretchy="false" xref="S7.4.p3.14.m14.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S7.4.p3.14.m14.2.2.3" xref="S7.4.p3.14.m14.2.2.3.cmml">≁</mo><mrow id="S7.4.p3.14.m14.2.2.2" xref="S7.4.p3.14.m14.2.2.2.cmml"><mi id="S7.4.p3.14.m14.2.2.2.3" xref="S7.4.p3.14.m14.2.2.2.3.cmml">g</mi><mo id="S7.4.p3.14.m14.2.2.2.2" xref="S7.4.p3.14.m14.2.2.2.2.cmml">⁢</mo><mrow id="S7.4.p3.14.m14.2.2.2.1.1" xref="S7.4.p3.14.m14.2.2.2.1.1.1.cmml"><mo id="S7.4.p3.14.m14.2.2.2.1.1.2" stretchy="false" xref="S7.4.p3.14.m14.2.2.2.1.1.1.cmml">(</mo><msub id="S7.4.p3.14.m14.2.2.2.1.1.1" xref="S7.4.p3.14.m14.2.2.2.1.1.1.cmml"><mi id="S7.4.p3.14.m14.2.2.2.1.1.1.2" xref="S7.4.p3.14.m14.2.2.2.1.1.1.2.cmml">y</mi><mi id="S7.4.p3.14.m14.2.2.2.1.1.1.3" xref="S7.4.p3.14.m14.2.2.2.1.1.1.3.cmml">ξ</mi></msub><mo id="S7.4.p3.14.m14.2.2.2.1.1.3" stretchy="false" xref="S7.4.p3.14.m14.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.4.p3.14.m14.2b"><apply id="S7.4.p3.14.m14.2.2.cmml" xref="S7.4.p3.14.m14.2.2"><csymbol cd="latexml" id="S7.4.p3.14.m14.2.2.3.cmml" xref="S7.4.p3.14.m14.2.2.3">not-similar-to</csymbol><apply id="S7.4.p3.14.m14.1.1.1.cmml" xref="S7.4.p3.14.m14.1.1.1"><times id="S7.4.p3.14.m14.1.1.1.2.cmml" xref="S7.4.p3.14.m14.1.1.1.2"></times><ci id="S7.4.p3.14.m14.1.1.1.3.cmml" xref="S7.4.p3.14.m14.1.1.1.3">𝑓</ci><apply id="S7.4.p3.14.m14.1.1.1.1.1.1.cmml" xref="S7.4.p3.14.m14.1.1.1.1.1"><csymbol cd="ambiguous" id="S7.4.p3.14.m14.1.1.1.1.1.1.1.cmml" xref="S7.4.p3.14.m14.1.1.1.1.1">subscript</csymbol><ci id="S7.4.p3.14.m14.1.1.1.1.1.1.2.cmml" xref="S7.4.p3.14.m14.1.1.1.1.1.1.2">𝑥</ci><ci id="S7.4.p3.14.m14.1.1.1.1.1.1.3.cmml" xref="S7.4.p3.14.m14.1.1.1.1.1.1.3">𝜉</ci></apply></apply><apply id="S7.4.p3.14.m14.2.2.2.cmml" xref="S7.4.p3.14.m14.2.2.2"><times id="S7.4.p3.14.m14.2.2.2.2.cmml" xref="S7.4.p3.14.m14.2.2.2.2"></times><ci id="S7.4.p3.14.m14.2.2.2.3.cmml" xref="S7.4.p3.14.m14.2.2.2.3">𝑔</ci><apply id="S7.4.p3.14.m14.2.2.2.1.1.1.cmml" xref="S7.4.p3.14.m14.2.2.2.1.1"><csymbol cd="ambiguous" id="S7.4.p3.14.m14.2.2.2.1.1.1.1.cmml" xref="S7.4.p3.14.m14.2.2.2.1.1">subscript</csymbol><ci id="S7.4.p3.14.m14.2.2.2.1.1.1.2.cmml" xref="S7.4.p3.14.m14.2.2.2.1.1.1.2">𝑦</ci><ci id="S7.4.p3.14.m14.2.2.2.1.1.1.3.cmml" xref="S7.4.p3.14.m14.2.2.2.1.1.1.3">𝜉</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.4.p3.14.m14.2c">f(x_{\xi})\nsim g(y_{\xi})</annotation><annotation encoding="application/x-llamapun" id="S7.4.p3.14.m14.2d">italic_f ( italic_x start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT ) ≁ italic_g ( italic_y start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT )</annotation></semantics></math>, and thus there is some <math alttext="a_{\xi}\in B\cap{[f(x_{\xi}),f(y_{\xi})]}_{A}" class="ltx_Math" display="inline" id="S7.4.p3.15.m15.2"><semantics id="S7.4.p3.15.m15.2a"><mrow id="S7.4.p3.15.m15.2.2" xref="S7.4.p3.15.m15.2.2.cmml"><msub id="S7.4.p3.15.m15.2.2.4" xref="S7.4.p3.15.m15.2.2.4.cmml"><mi id="S7.4.p3.15.m15.2.2.4.2" xref="S7.4.p3.15.m15.2.2.4.2.cmml">a</mi><mi id="S7.4.p3.15.m15.2.2.4.3" xref="S7.4.p3.15.m15.2.2.4.3.cmml">ξ</mi></msub><mo id="S7.4.p3.15.m15.2.2.3" xref="S7.4.p3.15.m15.2.2.3.cmml">∈</mo><mrow id="S7.4.p3.15.m15.2.2.2" xref="S7.4.p3.15.m15.2.2.2.cmml"><mi id="S7.4.p3.15.m15.2.2.2.4" xref="S7.4.p3.15.m15.2.2.2.4.cmml">B</mi><mo id="S7.4.p3.15.m15.2.2.2.3" xref="S7.4.p3.15.m15.2.2.2.3.cmml">∩</mo><msub id="S7.4.p3.15.m15.2.2.2.2" xref="S7.4.p3.15.m15.2.2.2.2.cmml"><mrow id="S7.4.p3.15.m15.2.2.2.2.2.2" xref="S7.4.p3.15.m15.2.2.2.2.2.3.cmml"><mo id="S7.4.p3.15.m15.2.2.2.2.2.2.3" stretchy="false" xref="S7.4.p3.15.m15.2.2.2.2.2.3.cmml">[</mo><mrow id="S7.4.p3.15.m15.1.1.1.1.1.1.1" xref="S7.4.p3.15.m15.1.1.1.1.1.1.1.cmml"><mi id="S7.4.p3.15.m15.1.1.1.1.1.1.1.3" xref="S7.4.p3.15.m15.1.1.1.1.1.1.1.3.cmml">f</mi><mo id="S7.4.p3.15.m15.1.1.1.1.1.1.1.2" xref="S7.4.p3.15.m15.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S7.4.p3.15.m15.1.1.1.1.1.1.1.1.1" xref="S7.4.p3.15.m15.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S7.4.p3.15.m15.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S7.4.p3.15.m15.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S7.4.p3.15.m15.1.1.1.1.1.1.1.1.1.1" xref="S7.4.p3.15.m15.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="S7.4.p3.15.m15.1.1.1.1.1.1.1.1.1.1.2" xref="S7.4.p3.15.m15.1.1.1.1.1.1.1.1.1.1.2.cmml">x</mi><mi id="S7.4.p3.15.m15.1.1.1.1.1.1.1.1.1.1.3" xref="S7.4.p3.15.m15.1.1.1.1.1.1.1.1.1.1.3.cmml">ξ</mi></msub><mo id="S7.4.p3.15.m15.1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S7.4.p3.15.m15.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S7.4.p3.15.m15.2.2.2.2.2.2.4" xref="S7.4.p3.15.m15.2.2.2.2.2.3.cmml">,</mo><mrow id="S7.4.p3.15.m15.2.2.2.2.2.2.2" xref="S7.4.p3.15.m15.2.2.2.2.2.2.2.cmml"><mi id="S7.4.p3.15.m15.2.2.2.2.2.2.2.3" xref="S7.4.p3.15.m15.2.2.2.2.2.2.2.3.cmml">f</mi><mo id="S7.4.p3.15.m15.2.2.2.2.2.2.2.2" xref="S7.4.p3.15.m15.2.2.2.2.2.2.2.2.cmml">⁢</mo><mrow id="S7.4.p3.15.m15.2.2.2.2.2.2.2.1.1" xref="S7.4.p3.15.m15.2.2.2.2.2.2.2.1.1.1.cmml"><mo id="S7.4.p3.15.m15.2.2.2.2.2.2.2.1.1.2" stretchy="false" xref="S7.4.p3.15.m15.2.2.2.2.2.2.2.1.1.1.cmml">(</mo><msub id="S7.4.p3.15.m15.2.2.2.2.2.2.2.1.1.1" xref="S7.4.p3.15.m15.2.2.2.2.2.2.2.1.1.1.cmml"><mi id="S7.4.p3.15.m15.2.2.2.2.2.2.2.1.1.1.2" xref="S7.4.p3.15.m15.2.2.2.2.2.2.2.1.1.1.2.cmml">y</mi><mi id="S7.4.p3.15.m15.2.2.2.2.2.2.2.1.1.1.3" xref="S7.4.p3.15.m15.2.2.2.2.2.2.2.1.1.1.3.cmml">ξ</mi></msub><mo id="S7.4.p3.15.m15.2.2.2.2.2.2.2.1.1.3" stretchy="false" xref="S7.4.p3.15.m15.2.2.2.2.2.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S7.4.p3.15.m15.2.2.2.2.2.2.5" stretchy="false" xref="S7.4.p3.15.m15.2.2.2.2.2.3.cmml">]</mo></mrow><mi id="S7.4.p3.15.m15.2.2.2.2.4" xref="S7.4.p3.15.m15.2.2.2.2.4.cmml">A</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.4.p3.15.m15.2b"><apply id="S7.4.p3.15.m15.2.2.cmml" xref="S7.4.p3.15.m15.2.2"><in id="S7.4.p3.15.m15.2.2.3.cmml" xref="S7.4.p3.15.m15.2.2.3"></in><apply id="S7.4.p3.15.m15.2.2.4.cmml" xref="S7.4.p3.15.m15.2.2.4"><csymbol cd="ambiguous" id="S7.4.p3.15.m15.2.2.4.1.cmml" xref="S7.4.p3.15.m15.2.2.4">subscript</csymbol><ci id="S7.4.p3.15.m15.2.2.4.2.cmml" xref="S7.4.p3.15.m15.2.2.4.2">𝑎</ci><ci id="S7.4.p3.15.m15.2.2.4.3.cmml" xref="S7.4.p3.15.m15.2.2.4.3">𝜉</ci></apply><apply id="S7.4.p3.15.m15.2.2.2.cmml" xref="S7.4.p3.15.m15.2.2.2"><intersect id="S7.4.p3.15.m15.2.2.2.3.cmml" xref="S7.4.p3.15.m15.2.2.2.3"></intersect><ci id="S7.4.p3.15.m15.2.2.2.4.cmml" xref="S7.4.p3.15.m15.2.2.2.4">𝐵</ci><apply id="S7.4.p3.15.m15.2.2.2.2.cmml" xref="S7.4.p3.15.m15.2.2.2.2"><csymbol cd="ambiguous" id="S7.4.p3.15.m15.2.2.2.2.3.cmml" xref="S7.4.p3.15.m15.2.2.2.2">subscript</csymbol><interval closure="closed" id="S7.4.p3.15.m15.2.2.2.2.2.3.cmml" xref="S7.4.p3.15.m15.2.2.2.2.2.2"><apply id="S7.4.p3.15.m15.1.1.1.1.1.1.1.cmml" xref="S7.4.p3.15.m15.1.1.1.1.1.1.1"><times id="S7.4.p3.15.m15.1.1.1.1.1.1.1.2.cmml" xref="S7.4.p3.15.m15.1.1.1.1.1.1.1.2"></times><ci id="S7.4.p3.15.m15.1.1.1.1.1.1.1.3.cmml" xref="S7.4.p3.15.m15.1.1.1.1.1.1.1.3">𝑓</ci><apply id="S7.4.p3.15.m15.1.1.1.1.1.1.1.1.1.1.cmml" xref="S7.4.p3.15.m15.1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S7.4.p3.15.m15.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S7.4.p3.15.m15.1.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S7.4.p3.15.m15.1.1.1.1.1.1.1.1.1.1.2.cmml" xref="S7.4.p3.15.m15.1.1.1.1.1.1.1.1.1.1.2">𝑥</ci><ci id="S7.4.p3.15.m15.1.1.1.1.1.1.1.1.1.1.3.cmml" xref="S7.4.p3.15.m15.1.1.1.1.1.1.1.1.1.1.3">𝜉</ci></apply></apply><apply id="S7.4.p3.15.m15.2.2.2.2.2.2.2.cmml" xref="S7.4.p3.15.m15.2.2.2.2.2.2.2"><times id="S7.4.p3.15.m15.2.2.2.2.2.2.2.2.cmml" xref="S7.4.p3.15.m15.2.2.2.2.2.2.2.2"></times><ci id="S7.4.p3.15.m15.2.2.2.2.2.2.2.3.cmml" xref="S7.4.p3.15.m15.2.2.2.2.2.2.2.3">𝑓</ci><apply id="S7.4.p3.15.m15.2.2.2.2.2.2.2.1.1.1.cmml" xref="S7.4.p3.15.m15.2.2.2.2.2.2.2.1.1"><csymbol cd="ambiguous" id="S7.4.p3.15.m15.2.2.2.2.2.2.2.1.1.1.1.cmml" xref="S7.4.p3.15.m15.2.2.2.2.2.2.2.1.1">subscript</csymbol><ci id="S7.4.p3.15.m15.2.2.2.2.2.2.2.1.1.1.2.cmml" xref="S7.4.p3.15.m15.2.2.2.2.2.2.2.1.1.1.2">𝑦</ci><ci id="S7.4.p3.15.m15.2.2.2.2.2.2.2.1.1.1.3.cmml" xref="S7.4.p3.15.m15.2.2.2.2.2.2.2.1.1.1.3">𝜉</ci></apply></apply></interval><ci id="S7.4.p3.15.m15.2.2.2.2.4.cmml" xref="S7.4.p3.15.m15.2.2.2.2.4">𝐴</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.4.p3.15.m15.2c">a_{\xi}\in B\cap{[f(x_{\xi}),f(y_{\xi})]}_{A}</annotation><annotation encoding="application/x-llamapun" id="S7.4.p3.15.m15.2d">italic_a start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT ∈ italic_B ∩ [ italic_f ( italic_x start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT ) , italic_f ( italic_y start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT ) ] start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT</annotation></semantics></math>. Then <math alttext="x_{\xi}\mapsto a_{\xi}" class="ltx_Math" display="inline" id="S7.4.p3.16.m16.1"><semantics id="S7.4.p3.16.m16.1a"><mrow id="S7.4.p3.16.m16.1.1" xref="S7.4.p3.16.m16.1.1.cmml"><msub id="S7.4.p3.16.m16.1.1.2" xref="S7.4.p3.16.m16.1.1.2.cmml"><mi id="S7.4.p3.16.m16.1.1.2.2" xref="S7.4.p3.16.m16.1.1.2.2.cmml">x</mi><mi id="S7.4.p3.16.m16.1.1.2.3" xref="S7.4.p3.16.m16.1.1.2.3.cmml">ξ</mi></msub><mo id="S7.4.p3.16.m16.1.1.1" stretchy="false" xref="S7.4.p3.16.m16.1.1.1.cmml">↦</mo><msub id="S7.4.p3.16.m16.1.1.3" xref="S7.4.p3.16.m16.1.1.3.cmml"><mi id="S7.4.p3.16.m16.1.1.3.2" xref="S7.4.p3.16.m16.1.1.3.2.cmml">a</mi><mi id="S7.4.p3.16.m16.1.1.3.3" xref="S7.4.p3.16.m16.1.1.3.3.cmml">ξ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.4.p3.16.m16.1b"><apply id="S7.4.p3.16.m16.1.1.cmml" xref="S7.4.p3.16.m16.1.1"><csymbol cd="latexml" id="S7.4.p3.16.m16.1.1.1.cmml" xref="S7.4.p3.16.m16.1.1.1">maps-to</csymbol><apply id="S7.4.p3.16.m16.1.1.2.cmml" xref="S7.4.p3.16.m16.1.1.2"><csymbol cd="ambiguous" id="S7.4.p3.16.m16.1.1.2.1.cmml" xref="S7.4.p3.16.m16.1.1.2">subscript</csymbol><ci id="S7.4.p3.16.m16.1.1.2.2.cmml" xref="S7.4.p3.16.m16.1.1.2.2">𝑥</ci><ci id="S7.4.p3.16.m16.1.1.2.3.cmml" xref="S7.4.p3.16.m16.1.1.2.3">𝜉</ci></apply><apply id="S7.4.p3.16.m16.1.1.3.cmml" xref="S7.4.p3.16.m16.1.1.3"><csymbol cd="ambiguous" id="S7.4.p3.16.m16.1.1.3.1.cmml" xref="S7.4.p3.16.m16.1.1.3">subscript</csymbol><ci id="S7.4.p3.16.m16.1.1.3.2.cmml" xref="S7.4.p3.16.m16.1.1.3.2">𝑎</ci><ci id="S7.4.p3.16.m16.1.1.3.3.cmml" xref="S7.4.p3.16.m16.1.1.3.3">𝜉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.4.p3.16.m16.1c">x_{\xi}\mapsto a_{\xi}</annotation><annotation encoding="application/x-llamapun" id="S7.4.p3.16.m16.1d">italic_x start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT ↦ italic_a start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT</annotation></semantics></math> is an embedding from an uncountable suborder of <math alttext="B^{\star}" class="ltx_Math" display="inline" id="S7.4.p3.17.m17.1"><semantics id="S7.4.p3.17.m17.1a"><msup id="S7.4.p3.17.m17.1.1" xref="S7.4.p3.17.m17.1.1.cmml"><mi id="S7.4.p3.17.m17.1.1.2" xref="S7.4.p3.17.m17.1.1.2.cmml">B</mi><mo id="S7.4.p3.17.m17.1.1.3" xref="S7.4.p3.17.m17.1.1.3.cmml">⋆</mo></msup><annotation-xml encoding="MathML-Content" id="S7.4.p3.17.m17.1b"><apply id="S7.4.p3.17.m17.1.1.cmml" xref="S7.4.p3.17.m17.1.1"><csymbol cd="ambiguous" id="S7.4.p3.17.m17.1.1.1.cmml" xref="S7.4.p3.17.m17.1.1">superscript</csymbol><ci id="S7.4.p3.17.m17.1.1.2.cmml" xref="S7.4.p3.17.m17.1.1.2">𝐵</ci><ci id="S7.4.p3.17.m17.1.1.3.cmml" xref="S7.4.p3.17.m17.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.4.p3.17.m17.1c">B^{\star}</annotation><annotation encoding="application/x-llamapun" id="S7.4.p3.17.m17.1d">italic_B start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math> into <math alttext="B" class="ltx_Math" display="inline" id="S7.4.p3.18.m18.1"><semantics id="S7.4.p3.18.m18.1a"><mi id="S7.4.p3.18.m18.1.1" xref="S7.4.p3.18.m18.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S7.4.p3.18.m18.1b"><ci id="S7.4.p3.18.m18.1.1.cmml" xref="S7.4.p3.18.m18.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.4.p3.18.m18.1c">B</annotation><annotation encoding="application/x-llamapun" id="S7.4.p3.18.m18.1d">italic_B</annotation></semantics></math>, which contradicts <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S2.Thmtheorem6" title="Lemma 2.6. ‣ 2. Aronszajn and Countryman lines ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">2.6</span></a>. ∎</p> </div> </div> <section class="ltx_subsection" id="S7.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">7.1. </span>On a basis for all uncountable linear orders</h3> <div class="ltx_para" id="S7.SS1.p1"> <p class="ltx_p" id="S7.SS1.p1.9">The previous suggest the following natural question. Under <math alttext="\mathsf{PFA}" class="ltx_Math" display="inline" id="S7.SS1.p1.1.m1.1"><semantics id="S7.SS1.p1.1.m1.1a"><mi id="S7.SS1.p1.1.m1.1.1" xref="S7.SS1.p1.1.m1.1.1.cmml">𝖯𝖥𝖠</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.p1.1.m1.1b"><ci id="S7.SS1.p1.1.m1.1.1.cmml" xref="S7.SS1.p1.1.m1.1.1">𝖯𝖥𝖠</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p1.1.m1.1c">\mathsf{PFA}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p1.1.m1.1d">sansserif_PFA</annotation></semantics></math>, is there a finite <math alttext="\trianglelefteq" class="ltx_Math" display="inline" id="S7.SS1.p1.2.m2.1"><semantics id="S7.SS1.p1.2.m2.1a"><mi id="S7.SS1.p1.2.m2.1.1" mathvariant="normal" xref="S7.SS1.p1.2.m2.1.1.cmml">⊴</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.p1.2.m2.1b"><ci id="S7.SS1.p1.2.m2.1.1.cmml" xref="S7.SS1.p1.2.m2.1.1">⊴</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p1.2.m2.1c">\trianglelefteq</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p1.2.m2.1d">⊴</annotation></semantics></math>-basis for the uncountable linear orders? It is easily seen that if <math alttext="L" class="ltx_Math" display="inline" id="S7.SS1.p1.3.m3.1"><semantics id="S7.SS1.p1.3.m3.1a"><mi id="S7.SS1.p1.3.m3.1.1" xref="S7.SS1.p1.3.m3.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.p1.3.m3.1b"><ci id="S7.SS1.p1.3.m3.1.1.cmml" xref="S7.SS1.p1.3.m3.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p1.3.m3.1c">L</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p1.3.m3.1d">italic_L</annotation></semantics></math> is not short, then <math alttext="L" class="ltx_Math" display="inline" id="S7.SS1.p1.4.m4.1"><semantics id="S7.SS1.p1.4.m4.1a"><mi id="S7.SS1.p1.4.m4.1.1" xref="S7.SS1.p1.4.m4.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.p1.4.m4.1b"><ci id="S7.SS1.p1.4.m4.1.1.cmml" xref="S7.SS1.p1.4.m4.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p1.4.m4.1c">L</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p1.4.m4.1d">italic_L</annotation></semantics></math> surjects onto <math alttext="\omega_{1}" class="ltx_Math" display="inline" id="S7.SS1.p1.5.m5.1"><semantics id="S7.SS1.p1.5.m5.1a"><msub id="S7.SS1.p1.5.m5.1.1" xref="S7.SS1.p1.5.m5.1.1.cmml"><mi id="S7.SS1.p1.5.m5.1.1.2" xref="S7.SS1.p1.5.m5.1.1.2.cmml">ω</mi><mn id="S7.SS1.p1.5.m5.1.1.3" xref="S7.SS1.p1.5.m5.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S7.SS1.p1.5.m5.1b"><apply id="S7.SS1.p1.5.m5.1.1.cmml" xref="S7.SS1.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S7.SS1.p1.5.m5.1.1.1.cmml" xref="S7.SS1.p1.5.m5.1.1">subscript</csymbol><ci id="S7.SS1.p1.5.m5.1.1.2.cmml" xref="S7.SS1.p1.5.m5.1.1.2">𝜔</ci><cn id="S7.SS1.p1.5.m5.1.1.3.cmml" type="integer" xref="S7.SS1.p1.5.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p1.5.m5.1c">\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p1.5.m5.1d">italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\omega_{1}+1" class="ltx_Math" display="inline" id="S7.SS1.p1.6.m6.1"><semantics id="S7.SS1.p1.6.m6.1a"><mrow id="S7.SS1.p1.6.m6.1.1" xref="S7.SS1.p1.6.m6.1.1.cmml"><msub id="S7.SS1.p1.6.m6.1.1.2" xref="S7.SS1.p1.6.m6.1.1.2.cmml"><mi id="S7.SS1.p1.6.m6.1.1.2.2" xref="S7.SS1.p1.6.m6.1.1.2.2.cmml">ω</mi><mn id="S7.SS1.p1.6.m6.1.1.2.3" xref="S7.SS1.p1.6.m6.1.1.2.3.cmml">1</mn></msub><mo id="S7.SS1.p1.6.m6.1.1.1" xref="S7.SS1.p1.6.m6.1.1.1.cmml">+</mo><mn id="S7.SS1.p1.6.m6.1.1.3" xref="S7.SS1.p1.6.m6.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.p1.6.m6.1b"><apply id="S7.SS1.p1.6.m6.1.1.cmml" xref="S7.SS1.p1.6.m6.1.1"><plus id="S7.SS1.p1.6.m6.1.1.1.cmml" xref="S7.SS1.p1.6.m6.1.1.1"></plus><apply id="S7.SS1.p1.6.m6.1.1.2.cmml" xref="S7.SS1.p1.6.m6.1.1.2"><csymbol cd="ambiguous" id="S7.SS1.p1.6.m6.1.1.2.1.cmml" xref="S7.SS1.p1.6.m6.1.1.2">subscript</csymbol><ci id="S7.SS1.p1.6.m6.1.1.2.2.cmml" xref="S7.SS1.p1.6.m6.1.1.2.2">𝜔</ci><cn id="S7.SS1.p1.6.m6.1.1.2.3.cmml" type="integer" xref="S7.SS1.p1.6.m6.1.1.2.3">1</cn></apply><cn id="S7.SS1.p1.6.m6.1.1.3.cmml" type="integer" xref="S7.SS1.p1.6.m6.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p1.6.m6.1c">\omega_{1}+1</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p1.6.m6.1d">italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 1</annotation></semantics></math>, <math alttext="\omega_{1}^{\star}" class="ltx_Math" display="inline" id="S7.SS1.p1.7.m7.1"><semantics id="S7.SS1.p1.7.m7.1a"><msubsup id="S7.SS1.p1.7.m7.1.1" xref="S7.SS1.p1.7.m7.1.1.cmml"><mi id="S7.SS1.p1.7.m7.1.1.2.2" xref="S7.SS1.p1.7.m7.1.1.2.2.cmml">ω</mi><mn id="S7.SS1.p1.7.m7.1.1.2.3" xref="S7.SS1.p1.7.m7.1.1.2.3.cmml">1</mn><mo id="S7.SS1.p1.7.m7.1.1.3" xref="S7.SS1.p1.7.m7.1.1.3.cmml">⋆</mo></msubsup><annotation-xml encoding="MathML-Content" id="S7.SS1.p1.7.m7.1b"><apply id="S7.SS1.p1.7.m7.1.1.cmml" xref="S7.SS1.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S7.SS1.p1.7.m7.1.1.1.cmml" xref="S7.SS1.p1.7.m7.1.1">superscript</csymbol><apply id="S7.SS1.p1.7.m7.1.1.2.cmml" xref="S7.SS1.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S7.SS1.p1.7.m7.1.1.2.1.cmml" xref="S7.SS1.p1.7.m7.1.1">subscript</csymbol><ci id="S7.SS1.p1.7.m7.1.1.2.2.cmml" xref="S7.SS1.p1.7.m7.1.1.2.2">𝜔</ci><cn id="S7.SS1.p1.7.m7.1.1.2.3.cmml" type="integer" xref="S7.SS1.p1.7.m7.1.1.2.3">1</cn></apply><ci id="S7.SS1.p1.7.m7.1.1.3.cmml" xref="S7.SS1.p1.7.m7.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p1.7.m7.1c">\omega_{1}^{\star}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p1.7.m7.1d">italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math> or <math alttext="1+\omega_{1}^{\star}" class="ltx_Math" display="inline" id="S7.SS1.p1.8.m8.1"><semantics id="S7.SS1.p1.8.m8.1a"><mrow id="S7.SS1.p1.8.m8.1.1" xref="S7.SS1.p1.8.m8.1.1.cmml"><mn id="S7.SS1.p1.8.m8.1.1.2" xref="S7.SS1.p1.8.m8.1.1.2.cmml">1</mn><mo id="S7.SS1.p1.8.m8.1.1.1" xref="S7.SS1.p1.8.m8.1.1.1.cmml">+</mo><msubsup id="S7.SS1.p1.8.m8.1.1.3" xref="S7.SS1.p1.8.m8.1.1.3.cmml"><mi id="S7.SS1.p1.8.m8.1.1.3.2.2" xref="S7.SS1.p1.8.m8.1.1.3.2.2.cmml">ω</mi><mn id="S7.SS1.p1.8.m8.1.1.3.2.3" xref="S7.SS1.p1.8.m8.1.1.3.2.3.cmml">1</mn><mo id="S7.SS1.p1.8.m8.1.1.3.3" xref="S7.SS1.p1.8.m8.1.1.3.3.cmml">⋆</mo></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.p1.8.m8.1b"><apply id="S7.SS1.p1.8.m8.1.1.cmml" xref="S7.SS1.p1.8.m8.1.1"><plus id="S7.SS1.p1.8.m8.1.1.1.cmml" xref="S7.SS1.p1.8.m8.1.1.1"></plus><cn id="S7.SS1.p1.8.m8.1.1.2.cmml" type="integer" xref="S7.SS1.p1.8.m8.1.1.2">1</cn><apply id="S7.SS1.p1.8.m8.1.1.3.cmml" xref="S7.SS1.p1.8.m8.1.1.3"><csymbol cd="ambiguous" id="S7.SS1.p1.8.m8.1.1.3.1.cmml" xref="S7.SS1.p1.8.m8.1.1.3">superscript</csymbol><apply id="S7.SS1.p1.8.m8.1.1.3.2.cmml" xref="S7.SS1.p1.8.m8.1.1.3"><csymbol cd="ambiguous" id="S7.SS1.p1.8.m8.1.1.3.2.1.cmml" xref="S7.SS1.p1.8.m8.1.1.3">subscript</csymbol><ci id="S7.SS1.p1.8.m8.1.1.3.2.2.cmml" xref="S7.SS1.p1.8.m8.1.1.3.2.2">𝜔</ci><cn id="S7.SS1.p1.8.m8.1.1.3.2.3.cmml" type="integer" xref="S7.SS1.p1.8.m8.1.1.3.2.3">1</cn></apply><ci id="S7.SS1.p1.8.m8.1.1.3.3.cmml" xref="S7.SS1.p1.8.m8.1.1.3.3">⋆</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p1.8.m8.1c">1+\omega_{1}^{\star}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p1.8.m8.1d">1 + italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math>. However we will see that no matter what set theoretic hypotheses are in use, this cannot be extended to all uncountable linear orders. This will be done by proving (in <math alttext="\mathsf{ZFC}" class="ltx_Math" display="inline" id="S7.SS1.p1.9.m9.1"><semantics id="S7.SS1.p1.9.m9.1a"><mi id="S7.SS1.p1.9.m9.1.1" xref="S7.SS1.p1.9.m9.1.1.cmml">𝖹𝖥𝖢</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.p1.9.m9.1b"><ci id="S7.SS1.p1.9.m9.1.1.cmml" xref="S7.SS1.p1.9.m9.1.1">𝖹𝖥𝖢</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p1.9.m9.1c">\mathsf{ZFC}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p1.9.m9.1d">sansserif_ZFC</annotation></semantics></math>) the following theorem.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S7.Thmtheorem3"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S7.Thmtheorem3.1.1.1">Theorem 7.3</span></span><span class="ltx_text ltx_font_bold" id="S7.Thmtheorem3.2.2">.</span> </h6> <div class="ltx_para" id="S7.Thmtheorem3.p1"> <p class="ltx_p" id="S7.Thmtheorem3.p1.2"><span class="ltx_text ltx_font_italic" id="S7.Thmtheorem3.p1.2.2">Any <math alttext="\trianglelefteq" class="ltx_Math" display="inline" id="S7.Thmtheorem3.p1.1.1.m1.1"><semantics id="S7.Thmtheorem3.p1.1.1.m1.1a"><mi id="S7.Thmtheorem3.p1.1.1.m1.1.1" mathvariant="normal" xref="S7.Thmtheorem3.p1.1.1.m1.1.1.cmml">⊴</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem3.p1.1.1.m1.1b"><ci id="S7.Thmtheorem3.p1.1.1.m1.1.1.cmml" xref="S7.Thmtheorem3.p1.1.1.m1.1.1">⊴</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem3.p1.1.1.m1.1c">\trianglelefteq</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem3.p1.1.1.m1.1d">⊴</annotation></semantics></math>-basis for the uncountable real orders has at least <math alttext="\mathfrak{c}^{+}" class="ltx_Math" display="inline" id="S7.Thmtheorem3.p1.2.2.m2.1"><semantics id="S7.Thmtheorem3.p1.2.2.m2.1a"><msup id="S7.Thmtheorem3.p1.2.2.m2.1.1" xref="S7.Thmtheorem3.p1.2.2.m2.1.1.cmml"><mi id="S7.Thmtheorem3.p1.2.2.m2.1.1.2" xref="S7.Thmtheorem3.p1.2.2.m2.1.1.2.cmml">𝔠</mi><mo id="S7.Thmtheorem3.p1.2.2.m2.1.1.3" xref="S7.Thmtheorem3.p1.2.2.m2.1.1.3.cmml">+</mo></msup><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem3.p1.2.2.m2.1b"><apply id="S7.Thmtheorem3.p1.2.2.m2.1.1.cmml" xref="S7.Thmtheorem3.p1.2.2.m2.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem3.p1.2.2.m2.1.1.1.cmml" xref="S7.Thmtheorem3.p1.2.2.m2.1.1">superscript</csymbol><ci id="S7.Thmtheorem3.p1.2.2.m2.1.1.2.cmml" xref="S7.Thmtheorem3.p1.2.2.m2.1.1.2">𝔠</ci><plus id="S7.Thmtheorem3.p1.2.2.m2.1.1.3.cmml" xref="S7.Thmtheorem3.p1.2.2.m2.1.1.3"></plus></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem3.p1.2.2.m2.1c">\mathfrak{c}^{+}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem3.p1.2.2.m2.1d">fraktur_c start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> elements.</span></p> </div> </div> <div class="ltx_para" id="S7.SS1.p2"> <p class="ltx_p" id="S7.SS1.p2.14">Let <math alttext="\mathscr{F}" class="ltx_Math" display="inline" id="S7.SS1.p2.1.m1.1"><semantics id="S7.SS1.p2.1.m1.1a"><mi class="ltx_font_mathscript" id="S7.SS1.p2.1.m1.1.1" xref="S7.SS1.p2.1.m1.1.1.cmml">ℱ</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.p2.1.m1.1b"><ci id="S7.SS1.p2.1.m1.1.1.cmml" xref="S7.SS1.p2.1.m1.1.1">ℱ</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p2.1.m1.1c">\mathscr{F}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p2.1.m1.1d">script_F</annotation></semantics></math> be the family of epimorphisms between subsets of reals. That is, <math alttext="f\in\mathscr{F}" class="ltx_Math" display="inline" id="S7.SS1.p2.2.m2.1"><semantics id="S7.SS1.p2.2.m2.1a"><mrow id="S7.SS1.p2.2.m2.1.1" xref="S7.SS1.p2.2.m2.1.1.cmml"><mi id="S7.SS1.p2.2.m2.1.1.2" xref="S7.SS1.p2.2.m2.1.1.2.cmml">f</mi><mo id="S7.SS1.p2.2.m2.1.1.1" xref="S7.SS1.p2.2.m2.1.1.1.cmml">∈</mo><mi class="ltx_font_mathscript" id="S7.SS1.p2.2.m2.1.1.3" xref="S7.SS1.p2.2.m2.1.1.3.cmml">ℱ</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.p2.2.m2.1b"><apply id="S7.SS1.p2.2.m2.1.1.cmml" xref="S7.SS1.p2.2.m2.1.1"><in id="S7.SS1.p2.2.m2.1.1.1.cmml" xref="S7.SS1.p2.2.m2.1.1.1"></in><ci id="S7.SS1.p2.2.m2.1.1.2.cmml" xref="S7.SS1.p2.2.m2.1.1.2">𝑓</ci><ci id="S7.SS1.p2.2.m2.1.1.3.cmml" xref="S7.SS1.p2.2.m2.1.1.3">ℱ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p2.2.m2.1c">f\in\mathscr{F}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p2.2.m2.1d">italic_f ∈ script_F</annotation></semantics></math> iff <math alttext="\operatorname{dom}(f),\operatorname{ran}(f)\subseteq\mathbb{R}" class="ltx_Math" display="inline" id="S7.SS1.p2.3.m3.6"><semantics id="S7.SS1.p2.3.m3.6a"><mrow id="S7.SS1.p2.3.m3.6.6" xref="S7.SS1.p2.3.m3.6.6.cmml"><mrow id="S7.SS1.p2.3.m3.6.6.2.2" xref="S7.SS1.p2.3.m3.6.6.2.3.cmml"><mrow id="S7.SS1.p2.3.m3.5.5.1.1.1.2" xref="S7.SS1.p2.3.m3.5.5.1.1.1.1.cmml"><mi id="S7.SS1.p2.3.m3.1.1" xref="S7.SS1.p2.3.m3.1.1.cmml">dom</mi><mo id="S7.SS1.p2.3.m3.5.5.1.1.1.2a" xref="S7.SS1.p2.3.m3.5.5.1.1.1.1.cmml">⁡</mo><mrow id="S7.SS1.p2.3.m3.5.5.1.1.1.2.1" xref="S7.SS1.p2.3.m3.5.5.1.1.1.1.cmml"><mo id="S7.SS1.p2.3.m3.5.5.1.1.1.2.1.1" stretchy="false" xref="S7.SS1.p2.3.m3.5.5.1.1.1.1.cmml">(</mo><mi id="S7.SS1.p2.3.m3.2.2" xref="S7.SS1.p2.3.m3.2.2.cmml">f</mi><mo id="S7.SS1.p2.3.m3.5.5.1.1.1.2.1.2" stretchy="false" xref="S7.SS1.p2.3.m3.5.5.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S7.SS1.p2.3.m3.6.6.2.2.3" xref="S7.SS1.p2.3.m3.6.6.2.3.cmml">,</mo><mrow id="S7.SS1.p2.3.m3.6.6.2.2.2.2" xref="S7.SS1.p2.3.m3.6.6.2.2.2.1.cmml"><mi id="S7.SS1.p2.3.m3.3.3" xref="S7.SS1.p2.3.m3.3.3.cmml">ran</mi><mo id="S7.SS1.p2.3.m3.6.6.2.2.2.2a" xref="S7.SS1.p2.3.m3.6.6.2.2.2.1.cmml">⁡</mo><mrow id="S7.SS1.p2.3.m3.6.6.2.2.2.2.1" xref="S7.SS1.p2.3.m3.6.6.2.2.2.1.cmml"><mo id="S7.SS1.p2.3.m3.6.6.2.2.2.2.1.1" stretchy="false" xref="S7.SS1.p2.3.m3.6.6.2.2.2.1.cmml">(</mo><mi id="S7.SS1.p2.3.m3.4.4" xref="S7.SS1.p2.3.m3.4.4.cmml">f</mi><mo id="S7.SS1.p2.3.m3.6.6.2.2.2.2.1.2" stretchy="false" xref="S7.SS1.p2.3.m3.6.6.2.2.2.1.cmml">)</mo></mrow></mrow></mrow><mo id="S7.SS1.p2.3.m3.6.6.3" xref="S7.SS1.p2.3.m3.6.6.3.cmml">⊆</mo><mi id="S7.SS1.p2.3.m3.6.6.4" xref="S7.SS1.p2.3.m3.6.6.4.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.p2.3.m3.6b"><apply id="S7.SS1.p2.3.m3.6.6.cmml" xref="S7.SS1.p2.3.m3.6.6"><subset id="S7.SS1.p2.3.m3.6.6.3.cmml" xref="S7.SS1.p2.3.m3.6.6.3"></subset><list id="S7.SS1.p2.3.m3.6.6.2.3.cmml" xref="S7.SS1.p2.3.m3.6.6.2.2"><apply id="S7.SS1.p2.3.m3.5.5.1.1.1.1.cmml" xref="S7.SS1.p2.3.m3.5.5.1.1.1.2"><ci id="S7.SS1.p2.3.m3.1.1.cmml" xref="S7.SS1.p2.3.m3.1.1">dom</ci><ci id="S7.SS1.p2.3.m3.2.2.cmml" xref="S7.SS1.p2.3.m3.2.2">𝑓</ci></apply><apply id="S7.SS1.p2.3.m3.6.6.2.2.2.1.cmml" xref="S7.SS1.p2.3.m3.6.6.2.2.2.2"><ci id="S7.SS1.p2.3.m3.3.3.cmml" xref="S7.SS1.p2.3.m3.3.3">ran</ci><ci id="S7.SS1.p2.3.m3.4.4.cmml" xref="S7.SS1.p2.3.m3.4.4">𝑓</ci></apply></list><ci id="S7.SS1.p2.3.m3.6.6.4.cmml" xref="S7.SS1.p2.3.m3.6.6.4">ℝ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p2.3.m3.6c">\operatorname{dom}(f),\operatorname{ran}(f)\subseteq\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p2.3.m3.6d">roman_dom ( italic_f ) , roman_ran ( italic_f ) ⊆ blackboard_R</annotation></semantics></math>, and for all <math alttext="x<y" class="ltx_Math" display="inline" id="S7.SS1.p2.4.m4.1"><semantics id="S7.SS1.p2.4.m4.1a"><mrow id="S7.SS1.p2.4.m4.1.1" xref="S7.SS1.p2.4.m4.1.1.cmml"><mi id="S7.SS1.p2.4.m4.1.1.2" xref="S7.SS1.p2.4.m4.1.1.2.cmml">x</mi><mo id="S7.SS1.p2.4.m4.1.1.1" xref="S7.SS1.p2.4.m4.1.1.1.cmml"><</mo><mi id="S7.SS1.p2.4.m4.1.1.3" xref="S7.SS1.p2.4.m4.1.1.3.cmml">y</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.p2.4.m4.1b"><apply id="S7.SS1.p2.4.m4.1.1.cmml" xref="S7.SS1.p2.4.m4.1.1"><lt id="S7.SS1.p2.4.m4.1.1.1.cmml" xref="S7.SS1.p2.4.m4.1.1.1"></lt><ci id="S7.SS1.p2.4.m4.1.1.2.cmml" xref="S7.SS1.p2.4.m4.1.1.2">𝑥</ci><ci id="S7.SS1.p2.4.m4.1.1.3.cmml" xref="S7.SS1.p2.4.m4.1.1.3">𝑦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p2.4.m4.1c">x<y</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p2.4.m4.1d">italic_x < italic_y</annotation></semantics></math> in <math alttext="\operatorname{dom}(f)" class="ltx_Math" display="inline" id="S7.SS1.p2.5.m5.2"><semantics id="S7.SS1.p2.5.m5.2a"><mrow id="S7.SS1.p2.5.m5.2.3.2" xref="S7.SS1.p2.5.m5.2.3.1.cmml"><mi id="S7.SS1.p2.5.m5.1.1" xref="S7.SS1.p2.5.m5.1.1.cmml">dom</mi><mo id="S7.SS1.p2.5.m5.2.3.2a" xref="S7.SS1.p2.5.m5.2.3.1.cmml">⁡</mo><mrow id="S7.SS1.p2.5.m5.2.3.2.1" xref="S7.SS1.p2.5.m5.2.3.1.cmml"><mo id="S7.SS1.p2.5.m5.2.3.2.1.1" stretchy="false" xref="S7.SS1.p2.5.m5.2.3.1.cmml">(</mo><mi id="S7.SS1.p2.5.m5.2.2" xref="S7.SS1.p2.5.m5.2.2.cmml">f</mi><mo id="S7.SS1.p2.5.m5.2.3.2.1.2" stretchy="false" xref="S7.SS1.p2.5.m5.2.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.p2.5.m5.2b"><apply id="S7.SS1.p2.5.m5.2.3.1.cmml" xref="S7.SS1.p2.5.m5.2.3.2"><ci id="S7.SS1.p2.5.m5.1.1.cmml" xref="S7.SS1.p2.5.m5.1.1">dom</ci><ci id="S7.SS1.p2.5.m5.2.2.cmml" xref="S7.SS1.p2.5.m5.2.2">𝑓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p2.5.m5.2c">\operatorname{dom}(f)</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p2.5.m5.2d">roman_dom ( italic_f )</annotation></semantics></math>, <math alttext="f(x)\leq f(y)" class="ltx_Math" display="inline" id="S7.SS1.p2.6.m6.2"><semantics id="S7.SS1.p2.6.m6.2a"><mrow id="S7.SS1.p2.6.m6.2.3" xref="S7.SS1.p2.6.m6.2.3.cmml"><mrow id="S7.SS1.p2.6.m6.2.3.2" xref="S7.SS1.p2.6.m6.2.3.2.cmml"><mi id="S7.SS1.p2.6.m6.2.3.2.2" xref="S7.SS1.p2.6.m6.2.3.2.2.cmml">f</mi><mo id="S7.SS1.p2.6.m6.2.3.2.1" xref="S7.SS1.p2.6.m6.2.3.2.1.cmml">⁢</mo><mrow id="S7.SS1.p2.6.m6.2.3.2.3.2" xref="S7.SS1.p2.6.m6.2.3.2.cmml"><mo id="S7.SS1.p2.6.m6.2.3.2.3.2.1" stretchy="false" xref="S7.SS1.p2.6.m6.2.3.2.cmml">(</mo><mi id="S7.SS1.p2.6.m6.1.1" xref="S7.SS1.p2.6.m6.1.1.cmml">x</mi><mo id="S7.SS1.p2.6.m6.2.3.2.3.2.2" stretchy="false" xref="S7.SS1.p2.6.m6.2.3.2.cmml">)</mo></mrow></mrow><mo id="S7.SS1.p2.6.m6.2.3.1" xref="S7.SS1.p2.6.m6.2.3.1.cmml">≤</mo><mrow id="S7.SS1.p2.6.m6.2.3.3" xref="S7.SS1.p2.6.m6.2.3.3.cmml"><mi id="S7.SS1.p2.6.m6.2.3.3.2" xref="S7.SS1.p2.6.m6.2.3.3.2.cmml">f</mi><mo id="S7.SS1.p2.6.m6.2.3.3.1" xref="S7.SS1.p2.6.m6.2.3.3.1.cmml">⁢</mo><mrow id="S7.SS1.p2.6.m6.2.3.3.3.2" xref="S7.SS1.p2.6.m6.2.3.3.cmml"><mo id="S7.SS1.p2.6.m6.2.3.3.3.2.1" stretchy="false" xref="S7.SS1.p2.6.m6.2.3.3.cmml">(</mo><mi id="S7.SS1.p2.6.m6.2.2" xref="S7.SS1.p2.6.m6.2.2.cmml">y</mi><mo id="S7.SS1.p2.6.m6.2.3.3.3.2.2" stretchy="false" xref="S7.SS1.p2.6.m6.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.p2.6.m6.2b"><apply id="S7.SS1.p2.6.m6.2.3.cmml" xref="S7.SS1.p2.6.m6.2.3"><leq id="S7.SS1.p2.6.m6.2.3.1.cmml" xref="S7.SS1.p2.6.m6.2.3.1"></leq><apply id="S7.SS1.p2.6.m6.2.3.2.cmml" xref="S7.SS1.p2.6.m6.2.3.2"><times id="S7.SS1.p2.6.m6.2.3.2.1.cmml" xref="S7.SS1.p2.6.m6.2.3.2.1"></times><ci id="S7.SS1.p2.6.m6.2.3.2.2.cmml" xref="S7.SS1.p2.6.m6.2.3.2.2">𝑓</ci><ci id="S7.SS1.p2.6.m6.1.1.cmml" xref="S7.SS1.p2.6.m6.1.1">𝑥</ci></apply><apply id="S7.SS1.p2.6.m6.2.3.3.cmml" xref="S7.SS1.p2.6.m6.2.3.3"><times id="S7.SS1.p2.6.m6.2.3.3.1.cmml" xref="S7.SS1.p2.6.m6.2.3.3.1"></times><ci id="S7.SS1.p2.6.m6.2.3.3.2.cmml" xref="S7.SS1.p2.6.m6.2.3.3.2">𝑓</ci><ci id="S7.SS1.p2.6.m6.2.2.cmml" xref="S7.SS1.p2.6.m6.2.2">𝑦</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p2.6.m6.2c">f(x)\leq f(y)</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p2.6.m6.2d">italic_f ( italic_x ) ≤ italic_f ( italic_y )</annotation></semantics></math>. Following <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib3" title="">3</a>]</cite> we let <math alttext="\bar{f}" class="ltx_Math" display="inline" id="S7.SS1.p2.7.m7.1"><semantics id="S7.SS1.p2.7.m7.1a"><mover accent="true" id="S7.SS1.p2.7.m7.1.1" xref="S7.SS1.p2.7.m7.1.1.cmml"><mi id="S7.SS1.p2.7.m7.1.1.2" xref="S7.SS1.p2.7.m7.1.1.2.cmml">f</mi><mo id="S7.SS1.p2.7.m7.1.1.1" xref="S7.SS1.p2.7.m7.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S7.SS1.p2.7.m7.1b"><apply id="S7.SS1.p2.7.m7.1.1.cmml" xref="S7.SS1.p2.7.m7.1.1"><ci id="S7.SS1.p2.7.m7.1.1.1.cmml" xref="S7.SS1.p2.7.m7.1.1.1">¯</ci><ci id="S7.SS1.p2.7.m7.1.1.2.cmml" xref="S7.SS1.p2.7.m7.1.1.2">𝑓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p2.7.m7.1c">\bar{f}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p2.7.m7.1d">over¯ start_ARG italic_f end_ARG</annotation></semantics></math> be the set of <math alttext="(x,y)" class="ltx_Math" display="inline" id="S7.SS1.p2.8.m8.2"><semantics id="S7.SS1.p2.8.m8.2a"><mrow id="S7.SS1.p2.8.m8.2.3.2" xref="S7.SS1.p2.8.m8.2.3.1.cmml"><mo id="S7.SS1.p2.8.m8.2.3.2.1" stretchy="false" xref="S7.SS1.p2.8.m8.2.3.1.cmml">(</mo><mi id="S7.SS1.p2.8.m8.1.1" xref="S7.SS1.p2.8.m8.1.1.cmml">x</mi><mo id="S7.SS1.p2.8.m8.2.3.2.2" xref="S7.SS1.p2.8.m8.2.3.1.cmml">,</mo><mi id="S7.SS1.p2.8.m8.2.2" xref="S7.SS1.p2.8.m8.2.2.cmml">y</mi><mo id="S7.SS1.p2.8.m8.2.3.2.3" stretchy="false" xref="S7.SS1.p2.8.m8.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.p2.8.m8.2b"><interval closure="open" id="S7.SS1.p2.8.m8.2.3.1.cmml" xref="S7.SS1.p2.8.m8.2.3.2"><ci id="S7.SS1.p2.8.m8.1.1.cmml" xref="S7.SS1.p2.8.m8.1.1">𝑥</ci><ci id="S7.SS1.p2.8.m8.2.2.cmml" xref="S7.SS1.p2.8.m8.2.2">𝑦</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p2.8.m8.2c">(x,y)</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p2.8.m8.2d">( italic_x , italic_y )</annotation></semantics></math> such that for every <math alttext="\varepsilon>0" class="ltx_Math" display="inline" id="S7.SS1.p2.9.m9.1"><semantics id="S7.SS1.p2.9.m9.1a"><mrow id="S7.SS1.p2.9.m9.1.1" xref="S7.SS1.p2.9.m9.1.1.cmml"><mi id="S7.SS1.p2.9.m9.1.1.2" xref="S7.SS1.p2.9.m9.1.1.2.cmml">ε</mi><mo id="S7.SS1.p2.9.m9.1.1.1" xref="S7.SS1.p2.9.m9.1.1.1.cmml">></mo><mn id="S7.SS1.p2.9.m9.1.1.3" xref="S7.SS1.p2.9.m9.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.p2.9.m9.1b"><apply id="S7.SS1.p2.9.m9.1.1.cmml" xref="S7.SS1.p2.9.m9.1.1"><gt id="S7.SS1.p2.9.m9.1.1.1.cmml" xref="S7.SS1.p2.9.m9.1.1.1"></gt><ci id="S7.SS1.p2.9.m9.1.1.2.cmml" xref="S7.SS1.p2.9.m9.1.1.2">𝜀</ci><cn id="S7.SS1.p2.9.m9.1.1.3.cmml" type="integer" xref="S7.SS1.p2.9.m9.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p2.9.m9.1c">\varepsilon>0</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p2.9.m9.1d">italic_ε > 0</annotation></semantics></math>, there are <math alttext="x_{1},x_{2}\in\operatorname{dom}(f)" class="ltx_Math" display="inline" id="S7.SS1.p2.10.m10.4"><semantics id="S7.SS1.p2.10.m10.4a"><mrow id="S7.SS1.p2.10.m10.4.4" xref="S7.SS1.p2.10.m10.4.4.cmml"><mrow id="S7.SS1.p2.10.m10.4.4.2.2" xref="S7.SS1.p2.10.m10.4.4.2.3.cmml"><msub id="S7.SS1.p2.10.m10.3.3.1.1.1" xref="S7.SS1.p2.10.m10.3.3.1.1.1.cmml"><mi id="S7.SS1.p2.10.m10.3.3.1.1.1.2" xref="S7.SS1.p2.10.m10.3.3.1.1.1.2.cmml">x</mi><mn id="S7.SS1.p2.10.m10.3.3.1.1.1.3" xref="S7.SS1.p2.10.m10.3.3.1.1.1.3.cmml">1</mn></msub><mo id="S7.SS1.p2.10.m10.4.4.2.2.3" xref="S7.SS1.p2.10.m10.4.4.2.3.cmml">,</mo><msub id="S7.SS1.p2.10.m10.4.4.2.2.2" xref="S7.SS1.p2.10.m10.4.4.2.2.2.cmml"><mi id="S7.SS1.p2.10.m10.4.4.2.2.2.2" xref="S7.SS1.p2.10.m10.4.4.2.2.2.2.cmml">x</mi><mn id="S7.SS1.p2.10.m10.4.4.2.2.2.3" xref="S7.SS1.p2.10.m10.4.4.2.2.2.3.cmml">2</mn></msub></mrow><mo id="S7.SS1.p2.10.m10.4.4.3" xref="S7.SS1.p2.10.m10.4.4.3.cmml">∈</mo><mrow id="S7.SS1.p2.10.m10.4.4.4.2" xref="S7.SS1.p2.10.m10.4.4.4.1.cmml"><mi id="S7.SS1.p2.10.m10.1.1" xref="S7.SS1.p2.10.m10.1.1.cmml">dom</mi><mo id="S7.SS1.p2.10.m10.4.4.4.2a" xref="S7.SS1.p2.10.m10.4.4.4.1.cmml">⁡</mo><mrow id="S7.SS1.p2.10.m10.4.4.4.2.1" xref="S7.SS1.p2.10.m10.4.4.4.1.cmml"><mo id="S7.SS1.p2.10.m10.4.4.4.2.1.1" stretchy="false" xref="S7.SS1.p2.10.m10.4.4.4.1.cmml">(</mo><mi id="S7.SS1.p2.10.m10.2.2" xref="S7.SS1.p2.10.m10.2.2.cmml">f</mi><mo id="S7.SS1.p2.10.m10.4.4.4.2.1.2" stretchy="false" xref="S7.SS1.p2.10.m10.4.4.4.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.p2.10.m10.4b"><apply id="S7.SS1.p2.10.m10.4.4.cmml" xref="S7.SS1.p2.10.m10.4.4"><in id="S7.SS1.p2.10.m10.4.4.3.cmml" xref="S7.SS1.p2.10.m10.4.4.3"></in><list id="S7.SS1.p2.10.m10.4.4.2.3.cmml" xref="S7.SS1.p2.10.m10.4.4.2.2"><apply id="S7.SS1.p2.10.m10.3.3.1.1.1.cmml" xref="S7.SS1.p2.10.m10.3.3.1.1.1"><csymbol cd="ambiguous" id="S7.SS1.p2.10.m10.3.3.1.1.1.1.cmml" xref="S7.SS1.p2.10.m10.3.3.1.1.1">subscript</csymbol><ci id="S7.SS1.p2.10.m10.3.3.1.1.1.2.cmml" xref="S7.SS1.p2.10.m10.3.3.1.1.1.2">𝑥</ci><cn id="S7.SS1.p2.10.m10.3.3.1.1.1.3.cmml" type="integer" xref="S7.SS1.p2.10.m10.3.3.1.1.1.3">1</cn></apply><apply id="S7.SS1.p2.10.m10.4.4.2.2.2.cmml" xref="S7.SS1.p2.10.m10.4.4.2.2.2"><csymbol cd="ambiguous" id="S7.SS1.p2.10.m10.4.4.2.2.2.1.cmml" xref="S7.SS1.p2.10.m10.4.4.2.2.2">subscript</csymbol><ci id="S7.SS1.p2.10.m10.4.4.2.2.2.2.cmml" xref="S7.SS1.p2.10.m10.4.4.2.2.2.2">𝑥</ci><cn id="S7.SS1.p2.10.m10.4.4.2.2.2.3.cmml" type="integer" xref="S7.SS1.p2.10.m10.4.4.2.2.2.3">2</cn></apply></list><apply id="S7.SS1.p2.10.m10.4.4.4.1.cmml" xref="S7.SS1.p2.10.m10.4.4.4.2"><ci id="S7.SS1.p2.10.m10.1.1.cmml" xref="S7.SS1.p2.10.m10.1.1">dom</ci><ci id="S7.SS1.p2.10.m10.2.2.cmml" xref="S7.SS1.p2.10.m10.2.2">𝑓</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p2.10.m10.4c">x_{1},x_{2}\in\operatorname{dom}(f)</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p2.10.m10.4d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ roman_dom ( italic_f )</annotation></semantics></math> such that <math alttext="0\leq x-x_{1},x_{2}-x,y-f(x_{1}),f(x_{2})-y<\varepsilon" class="ltx_Math" display="inline" id="S7.SS1.p2.11.m11.2"><semantics id="S7.SS1.p2.11.m11.2a"><mrow id="S7.SS1.p2.11.m11.2.2.2" xref="S7.SS1.p2.11.m11.2.2.3.cmml"><mrow id="S7.SS1.p2.11.m11.1.1.1.1" xref="S7.SS1.p2.11.m11.1.1.1.1.cmml"><mn id="S7.SS1.p2.11.m11.1.1.1.1.5" xref="S7.SS1.p2.11.m11.1.1.1.1.5.cmml">0</mn><mo id="S7.SS1.p2.11.m11.1.1.1.1.4" xref="S7.SS1.p2.11.m11.1.1.1.1.4.cmml">≤</mo><mrow id="S7.SS1.p2.11.m11.1.1.1.1.3.3" xref="S7.SS1.p2.11.m11.1.1.1.1.3.4.cmml"><mrow id="S7.SS1.p2.11.m11.1.1.1.1.1.1.1" xref="S7.SS1.p2.11.m11.1.1.1.1.1.1.1.cmml"><mi id="S7.SS1.p2.11.m11.1.1.1.1.1.1.1.2" xref="S7.SS1.p2.11.m11.1.1.1.1.1.1.1.2.cmml">x</mi><mo id="S7.SS1.p2.11.m11.1.1.1.1.1.1.1.1" xref="S7.SS1.p2.11.m11.1.1.1.1.1.1.1.1.cmml">−</mo><msub id="S7.SS1.p2.11.m11.1.1.1.1.1.1.1.3" xref="S7.SS1.p2.11.m11.1.1.1.1.1.1.1.3.cmml"><mi id="S7.SS1.p2.11.m11.1.1.1.1.1.1.1.3.2" xref="S7.SS1.p2.11.m11.1.1.1.1.1.1.1.3.2.cmml">x</mi><mn id="S7.SS1.p2.11.m11.1.1.1.1.1.1.1.3.3" xref="S7.SS1.p2.11.m11.1.1.1.1.1.1.1.3.3.cmml">1</mn></msub></mrow><mo id="S7.SS1.p2.11.m11.1.1.1.1.3.3.4" xref="S7.SS1.p2.11.m11.1.1.1.1.3.4.cmml">,</mo><mrow id="S7.SS1.p2.11.m11.1.1.1.1.2.2.2" xref="S7.SS1.p2.11.m11.1.1.1.1.2.2.2.cmml"><msub id="S7.SS1.p2.11.m11.1.1.1.1.2.2.2.2" xref="S7.SS1.p2.11.m11.1.1.1.1.2.2.2.2.cmml"><mi id="S7.SS1.p2.11.m11.1.1.1.1.2.2.2.2.2" xref="S7.SS1.p2.11.m11.1.1.1.1.2.2.2.2.2.cmml">x</mi><mn id="S7.SS1.p2.11.m11.1.1.1.1.2.2.2.2.3" xref="S7.SS1.p2.11.m11.1.1.1.1.2.2.2.2.3.cmml">2</mn></msub><mo id="S7.SS1.p2.11.m11.1.1.1.1.2.2.2.1" xref="S7.SS1.p2.11.m11.1.1.1.1.2.2.2.1.cmml">−</mo><mi id="S7.SS1.p2.11.m11.1.1.1.1.2.2.2.3" xref="S7.SS1.p2.11.m11.1.1.1.1.2.2.2.3.cmml">x</mi></mrow><mo id="S7.SS1.p2.11.m11.1.1.1.1.3.3.5" xref="S7.SS1.p2.11.m11.1.1.1.1.3.4.cmml">,</mo><mrow id="S7.SS1.p2.11.m11.1.1.1.1.3.3.3" xref="S7.SS1.p2.11.m11.1.1.1.1.3.3.3.cmml"><mi id="S7.SS1.p2.11.m11.1.1.1.1.3.3.3.3" xref="S7.SS1.p2.11.m11.1.1.1.1.3.3.3.3.cmml">y</mi><mo id="S7.SS1.p2.11.m11.1.1.1.1.3.3.3.2" xref="S7.SS1.p2.11.m11.1.1.1.1.3.3.3.2.cmml">−</mo><mrow id="S7.SS1.p2.11.m11.1.1.1.1.3.3.3.1" xref="S7.SS1.p2.11.m11.1.1.1.1.3.3.3.1.cmml"><mi id="S7.SS1.p2.11.m11.1.1.1.1.3.3.3.1.3" xref="S7.SS1.p2.11.m11.1.1.1.1.3.3.3.1.3.cmml">f</mi><mo id="S7.SS1.p2.11.m11.1.1.1.1.3.3.3.1.2" xref="S7.SS1.p2.11.m11.1.1.1.1.3.3.3.1.2.cmml">⁢</mo><mrow id="S7.SS1.p2.11.m11.1.1.1.1.3.3.3.1.1.1" xref="S7.SS1.p2.11.m11.1.1.1.1.3.3.3.1.1.1.1.cmml"><mo id="S7.SS1.p2.11.m11.1.1.1.1.3.3.3.1.1.1.2" stretchy="false" xref="S7.SS1.p2.11.m11.1.1.1.1.3.3.3.1.1.1.1.cmml">(</mo><msub id="S7.SS1.p2.11.m11.1.1.1.1.3.3.3.1.1.1.1" xref="S7.SS1.p2.11.m11.1.1.1.1.3.3.3.1.1.1.1.cmml"><mi id="S7.SS1.p2.11.m11.1.1.1.1.3.3.3.1.1.1.1.2" xref="S7.SS1.p2.11.m11.1.1.1.1.3.3.3.1.1.1.1.2.cmml">x</mi><mn id="S7.SS1.p2.11.m11.1.1.1.1.3.3.3.1.1.1.1.3" xref="S7.SS1.p2.11.m11.1.1.1.1.3.3.3.1.1.1.1.3.cmml">1</mn></msub><mo id="S7.SS1.p2.11.m11.1.1.1.1.3.3.3.1.1.1.3" stretchy="false" xref="S7.SS1.p2.11.m11.1.1.1.1.3.3.3.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow></mrow><mo id="S7.SS1.p2.11.m11.2.2.2.3" xref="S7.SS1.p2.11.m11.2.2.3a.cmml">,</mo><mrow id="S7.SS1.p2.11.m11.2.2.2.2" xref="S7.SS1.p2.11.m11.2.2.2.2.cmml"><mrow id="S7.SS1.p2.11.m11.2.2.2.2.1" xref="S7.SS1.p2.11.m11.2.2.2.2.1.cmml"><mrow id="S7.SS1.p2.11.m11.2.2.2.2.1.1" xref="S7.SS1.p2.11.m11.2.2.2.2.1.1.cmml"><mi id="S7.SS1.p2.11.m11.2.2.2.2.1.1.3" xref="S7.SS1.p2.11.m11.2.2.2.2.1.1.3.cmml">f</mi><mo id="S7.SS1.p2.11.m11.2.2.2.2.1.1.2" xref="S7.SS1.p2.11.m11.2.2.2.2.1.1.2.cmml">⁢</mo><mrow id="S7.SS1.p2.11.m11.2.2.2.2.1.1.1.1" xref="S7.SS1.p2.11.m11.2.2.2.2.1.1.1.1.1.cmml"><mo id="S7.SS1.p2.11.m11.2.2.2.2.1.1.1.1.2" stretchy="false" xref="S7.SS1.p2.11.m11.2.2.2.2.1.1.1.1.1.cmml">(</mo><msub id="S7.SS1.p2.11.m11.2.2.2.2.1.1.1.1.1" xref="S7.SS1.p2.11.m11.2.2.2.2.1.1.1.1.1.cmml"><mi id="S7.SS1.p2.11.m11.2.2.2.2.1.1.1.1.1.2" xref="S7.SS1.p2.11.m11.2.2.2.2.1.1.1.1.1.2.cmml">x</mi><mn id="S7.SS1.p2.11.m11.2.2.2.2.1.1.1.1.1.3" xref="S7.SS1.p2.11.m11.2.2.2.2.1.1.1.1.1.3.cmml">2</mn></msub><mo id="S7.SS1.p2.11.m11.2.2.2.2.1.1.1.1.3" stretchy="false" xref="S7.SS1.p2.11.m11.2.2.2.2.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S7.SS1.p2.11.m11.2.2.2.2.1.2" xref="S7.SS1.p2.11.m11.2.2.2.2.1.2.cmml">−</mo><mi id="S7.SS1.p2.11.m11.2.2.2.2.1.3" xref="S7.SS1.p2.11.m11.2.2.2.2.1.3.cmml">y</mi></mrow><mo id="S7.SS1.p2.11.m11.2.2.2.2.2" xref="S7.SS1.p2.11.m11.2.2.2.2.2.cmml"><</mo><mi id="S7.SS1.p2.11.m11.2.2.2.2.3" xref="S7.SS1.p2.11.m11.2.2.2.2.3.cmml">ε</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.p2.11.m11.2b"><apply id="S7.SS1.p2.11.m11.2.2.3.cmml" xref="S7.SS1.p2.11.m11.2.2.2"><csymbol cd="ambiguous" id="S7.SS1.p2.11.m11.2.2.3a.cmml" xref="S7.SS1.p2.11.m11.2.2.2.3">formulae-sequence</csymbol><apply id="S7.SS1.p2.11.m11.1.1.1.1.cmml" xref="S7.SS1.p2.11.m11.1.1.1.1"><leq id="S7.SS1.p2.11.m11.1.1.1.1.4.cmml" xref="S7.SS1.p2.11.m11.1.1.1.1.4"></leq><cn id="S7.SS1.p2.11.m11.1.1.1.1.5.cmml" type="integer" xref="S7.SS1.p2.11.m11.1.1.1.1.5">0</cn><list id="S7.SS1.p2.11.m11.1.1.1.1.3.4.cmml" xref="S7.SS1.p2.11.m11.1.1.1.1.3.3"><apply id="S7.SS1.p2.11.m11.1.1.1.1.1.1.1.cmml" xref="S7.SS1.p2.11.m11.1.1.1.1.1.1.1"><minus id="S7.SS1.p2.11.m11.1.1.1.1.1.1.1.1.cmml" xref="S7.SS1.p2.11.m11.1.1.1.1.1.1.1.1"></minus><ci id="S7.SS1.p2.11.m11.1.1.1.1.1.1.1.2.cmml" xref="S7.SS1.p2.11.m11.1.1.1.1.1.1.1.2">𝑥</ci><apply id="S7.SS1.p2.11.m11.1.1.1.1.1.1.1.3.cmml" xref="S7.SS1.p2.11.m11.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S7.SS1.p2.11.m11.1.1.1.1.1.1.1.3.1.cmml" xref="S7.SS1.p2.11.m11.1.1.1.1.1.1.1.3">subscript</csymbol><ci id="S7.SS1.p2.11.m11.1.1.1.1.1.1.1.3.2.cmml" xref="S7.SS1.p2.11.m11.1.1.1.1.1.1.1.3.2">𝑥</ci><cn id="S7.SS1.p2.11.m11.1.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S7.SS1.p2.11.m11.1.1.1.1.1.1.1.3.3">1</cn></apply></apply><apply id="S7.SS1.p2.11.m11.1.1.1.1.2.2.2.cmml" xref="S7.SS1.p2.11.m11.1.1.1.1.2.2.2"><minus id="S7.SS1.p2.11.m11.1.1.1.1.2.2.2.1.cmml" xref="S7.SS1.p2.11.m11.1.1.1.1.2.2.2.1"></minus><apply id="S7.SS1.p2.11.m11.1.1.1.1.2.2.2.2.cmml" xref="S7.SS1.p2.11.m11.1.1.1.1.2.2.2.2"><csymbol cd="ambiguous" id="S7.SS1.p2.11.m11.1.1.1.1.2.2.2.2.1.cmml" xref="S7.SS1.p2.11.m11.1.1.1.1.2.2.2.2">subscript</csymbol><ci id="S7.SS1.p2.11.m11.1.1.1.1.2.2.2.2.2.cmml" xref="S7.SS1.p2.11.m11.1.1.1.1.2.2.2.2.2">𝑥</ci><cn id="S7.SS1.p2.11.m11.1.1.1.1.2.2.2.2.3.cmml" type="integer" xref="S7.SS1.p2.11.m11.1.1.1.1.2.2.2.2.3">2</cn></apply><ci id="S7.SS1.p2.11.m11.1.1.1.1.2.2.2.3.cmml" xref="S7.SS1.p2.11.m11.1.1.1.1.2.2.2.3">𝑥</ci></apply><apply id="S7.SS1.p2.11.m11.1.1.1.1.3.3.3.cmml" xref="S7.SS1.p2.11.m11.1.1.1.1.3.3.3"><minus id="S7.SS1.p2.11.m11.1.1.1.1.3.3.3.2.cmml" xref="S7.SS1.p2.11.m11.1.1.1.1.3.3.3.2"></minus><ci id="S7.SS1.p2.11.m11.1.1.1.1.3.3.3.3.cmml" xref="S7.SS1.p2.11.m11.1.1.1.1.3.3.3.3">𝑦</ci><apply id="S7.SS1.p2.11.m11.1.1.1.1.3.3.3.1.cmml" xref="S7.SS1.p2.11.m11.1.1.1.1.3.3.3.1"><times id="S7.SS1.p2.11.m11.1.1.1.1.3.3.3.1.2.cmml" xref="S7.SS1.p2.11.m11.1.1.1.1.3.3.3.1.2"></times><ci id="S7.SS1.p2.11.m11.1.1.1.1.3.3.3.1.3.cmml" xref="S7.SS1.p2.11.m11.1.1.1.1.3.3.3.1.3">𝑓</ci><apply id="S7.SS1.p2.11.m11.1.1.1.1.3.3.3.1.1.1.1.cmml" xref="S7.SS1.p2.11.m11.1.1.1.1.3.3.3.1.1.1"><csymbol cd="ambiguous" id="S7.SS1.p2.11.m11.1.1.1.1.3.3.3.1.1.1.1.1.cmml" xref="S7.SS1.p2.11.m11.1.1.1.1.3.3.3.1.1.1">subscript</csymbol><ci id="S7.SS1.p2.11.m11.1.1.1.1.3.3.3.1.1.1.1.2.cmml" xref="S7.SS1.p2.11.m11.1.1.1.1.3.3.3.1.1.1.1.2">𝑥</ci><cn id="S7.SS1.p2.11.m11.1.1.1.1.3.3.3.1.1.1.1.3.cmml" type="integer" xref="S7.SS1.p2.11.m11.1.1.1.1.3.3.3.1.1.1.1.3">1</cn></apply></apply></apply></list></apply><apply id="S7.SS1.p2.11.m11.2.2.2.2.cmml" xref="S7.SS1.p2.11.m11.2.2.2.2"><lt id="S7.SS1.p2.11.m11.2.2.2.2.2.cmml" xref="S7.SS1.p2.11.m11.2.2.2.2.2"></lt><apply id="S7.SS1.p2.11.m11.2.2.2.2.1.cmml" xref="S7.SS1.p2.11.m11.2.2.2.2.1"><minus id="S7.SS1.p2.11.m11.2.2.2.2.1.2.cmml" xref="S7.SS1.p2.11.m11.2.2.2.2.1.2"></minus><apply id="S7.SS1.p2.11.m11.2.2.2.2.1.1.cmml" xref="S7.SS1.p2.11.m11.2.2.2.2.1.1"><times id="S7.SS1.p2.11.m11.2.2.2.2.1.1.2.cmml" xref="S7.SS1.p2.11.m11.2.2.2.2.1.1.2"></times><ci id="S7.SS1.p2.11.m11.2.2.2.2.1.1.3.cmml" xref="S7.SS1.p2.11.m11.2.2.2.2.1.1.3">𝑓</ci><apply id="S7.SS1.p2.11.m11.2.2.2.2.1.1.1.1.1.cmml" xref="S7.SS1.p2.11.m11.2.2.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S7.SS1.p2.11.m11.2.2.2.2.1.1.1.1.1.1.cmml" xref="S7.SS1.p2.11.m11.2.2.2.2.1.1.1.1">subscript</csymbol><ci id="S7.SS1.p2.11.m11.2.2.2.2.1.1.1.1.1.2.cmml" xref="S7.SS1.p2.11.m11.2.2.2.2.1.1.1.1.1.2">𝑥</ci><cn id="S7.SS1.p2.11.m11.2.2.2.2.1.1.1.1.1.3.cmml" type="integer" xref="S7.SS1.p2.11.m11.2.2.2.2.1.1.1.1.1.3">2</cn></apply></apply><ci id="S7.SS1.p2.11.m11.2.2.2.2.1.3.cmml" xref="S7.SS1.p2.11.m11.2.2.2.2.1.3">𝑦</ci></apply><ci id="S7.SS1.p2.11.m11.2.2.2.2.3.cmml" xref="S7.SS1.p2.11.m11.2.2.2.2.3">𝜀</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p2.11.m11.2c">0\leq x-x_{1},x_{2}-x,y-f(x_{1}),f(x_{2})-y<\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p2.11.m11.2d">0 ≤ italic_x - italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT - italic_x , italic_y - italic_f ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) , italic_f ( italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) - italic_y < italic_ε</annotation></semantics></math>. It is clear that <math alttext="\bar{f}\in\mathscr{F}" class="ltx_Math" display="inline" id="S7.SS1.p2.12.m12.1"><semantics id="S7.SS1.p2.12.m12.1a"><mrow id="S7.SS1.p2.12.m12.1.1" xref="S7.SS1.p2.12.m12.1.1.cmml"><mover accent="true" id="S7.SS1.p2.12.m12.1.1.2" xref="S7.SS1.p2.12.m12.1.1.2.cmml"><mi id="S7.SS1.p2.12.m12.1.1.2.2" xref="S7.SS1.p2.12.m12.1.1.2.2.cmml">f</mi><mo id="S7.SS1.p2.12.m12.1.1.2.1" xref="S7.SS1.p2.12.m12.1.1.2.1.cmml">¯</mo></mover><mo id="S7.SS1.p2.12.m12.1.1.1" xref="S7.SS1.p2.12.m12.1.1.1.cmml">∈</mo><mi class="ltx_font_mathscript" id="S7.SS1.p2.12.m12.1.1.3" xref="S7.SS1.p2.12.m12.1.1.3.cmml">ℱ</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.p2.12.m12.1b"><apply id="S7.SS1.p2.12.m12.1.1.cmml" xref="S7.SS1.p2.12.m12.1.1"><in id="S7.SS1.p2.12.m12.1.1.1.cmml" xref="S7.SS1.p2.12.m12.1.1.1"></in><apply id="S7.SS1.p2.12.m12.1.1.2.cmml" xref="S7.SS1.p2.12.m12.1.1.2"><ci id="S7.SS1.p2.12.m12.1.1.2.1.cmml" xref="S7.SS1.p2.12.m12.1.1.2.1">¯</ci><ci id="S7.SS1.p2.12.m12.1.1.2.2.cmml" xref="S7.SS1.p2.12.m12.1.1.2.2">𝑓</ci></apply><ci id="S7.SS1.p2.12.m12.1.1.3.cmml" xref="S7.SS1.p2.12.m12.1.1.3">ℱ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p2.12.m12.1c">\bar{f}\in\mathscr{F}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p2.12.m12.1d">over¯ start_ARG italic_f end_ARG ∈ script_F</annotation></semantics></math> and <math alttext="f\subseteq\bar{f}" class="ltx_Math" display="inline" id="S7.SS1.p2.13.m13.1"><semantics id="S7.SS1.p2.13.m13.1a"><mrow id="S7.SS1.p2.13.m13.1.1" xref="S7.SS1.p2.13.m13.1.1.cmml"><mi id="S7.SS1.p2.13.m13.1.1.2" xref="S7.SS1.p2.13.m13.1.1.2.cmml">f</mi><mo id="S7.SS1.p2.13.m13.1.1.1" xref="S7.SS1.p2.13.m13.1.1.1.cmml">⊆</mo><mover accent="true" id="S7.SS1.p2.13.m13.1.1.3" xref="S7.SS1.p2.13.m13.1.1.3.cmml"><mi id="S7.SS1.p2.13.m13.1.1.3.2" xref="S7.SS1.p2.13.m13.1.1.3.2.cmml">f</mi><mo id="S7.SS1.p2.13.m13.1.1.3.1" xref="S7.SS1.p2.13.m13.1.1.3.1.cmml">¯</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.p2.13.m13.1b"><apply id="S7.SS1.p2.13.m13.1.1.cmml" xref="S7.SS1.p2.13.m13.1.1"><subset id="S7.SS1.p2.13.m13.1.1.1.cmml" xref="S7.SS1.p2.13.m13.1.1.1"></subset><ci id="S7.SS1.p2.13.m13.1.1.2.cmml" xref="S7.SS1.p2.13.m13.1.1.2">𝑓</ci><apply id="S7.SS1.p2.13.m13.1.1.3.cmml" xref="S7.SS1.p2.13.m13.1.1.3"><ci id="S7.SS1.p2.13.m13.1.1.3.1.cmml" xref="S7.SS1.p2.13.m13.1.1.3.1">¯</ci><ci id="S7.SS1.p2.13.m13.1.1.3.2.cmml" xref="S7.SS1.p2.13.m13.1.1.3.2">𝑓</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p2.13.m13.1c">f\subseteq\bar{f}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p2.13.m13.1d">italic_f ⊆ over¯ start_ARG italic_f end_ARG</annotation></semantics></math>. The following is essentially Lemma 2.2 of <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib3" title="">3</a>]</cite>, though he only deals with increasing functions, i.e., the <math alttext="f\in\mathscr{F}" class="ltx_Math" display="inline" id="S7.SS1.p2.14.m14.1"><semantics id="S7.SS1.p2.14.m14.1a"><mrow id="S7.SS1.p2.14.m14.1.1" xref="S7.SS1.p2.14.m14.1.1.cmml"><mi id="S7.SS1.p2.14.m14.1.1.2" xref="S7.SS1.p2.14.m14.1.1.2.cmml">f</mi><mo id="S7.SS1.p2.14.m14.1.1.1" xref="S7.SS1.p2.14.m14.1.1.1.cmml">∈</mo><mi class="ltx_font_mathscript" id="S7.SS1.p2.14.m14.1.1.3" xref="S7.SS1.p2.14.m14.1.1.3.cmml">ℱ</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.p2.14.m14.1b"><apply id="S7.SS1.p2.14.m14.1.1.cmml" xref="S7.SS1.p2.14.m14.1.1"><in id="S7.SS1.p2.14.m14.1.1.1.cmml" xref="S7.SS1.p2.14.m14.1.1.1"></in><ci id="S7.SS1.p2.14.m14.1.1.2.cmml" xref="S7.SS1.p2.14.m14.1.1.2">𝑓</ci><ci id="S7.SS1.p2.14.m14.1.1.3.cmml" xref="S7.SS1.p2.14.m14.1.1.3">ℱ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p2.14.m14.1c">f\in\mathscr{F}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p2.14.m14.1d">italic_f ∈ script_F</annotation></semantics></math> that are also injective. We sketch the proof because passing from increasing to montone does require a small tweak to Baumgartner’s proof.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S7.Thmtheorem4"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S7.Thmtheorem4.1.1.1">Lemma 7.4</span></span><span class="ltx_text ltx_font_bold" id="S7.Thmtheorem4.2.2">.</span> </h6> <div class="ltx_para" id="S7.Thmtheorem4.p1"> <p class="ltx_p" id="S7.Thmtheorem4.p1.5">For every <math alttext="f\in\mathscr{F}" class="ltx_Math" display="inline" id="S7.Thmtheorem4.p1.1.m1.1"><semantics id="S7.Thmtheorem4.p1.1.m1.1a"><mrow id="S7.Thmtheorem4.p1.1.m1.1.1" xref="S7.Thmtheorem4.p1.1.m1.1.1.cmml"><mi id="S7.Thmtheorem4.p1.1.m1.1.1.2" xref="S7.Thmtheorem4.p1.1.m1.1.1.2.cmml">f</mi><mo id="S7.Thmtheorem4.p1.1.m1.1.1.1" xref="S7.Thmtheorem4.p1.1.m1.1.1.1.cmml">∈</mo><mi class="ltx_font_mathscript" id="S7.Thmtheorem4.p1.1.m1.1.1.3" xref="S7.Thmtheorem4.p1.1.m1.1.1.3.cmml">ℱ</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem4.p1.1.m1.1b"><apply id="S7.Thmtheorem4.p1.1.m1.1.1.cmml" xref="S7.Thmtheorem4.p1.1.m1.1.1"><in id="S7.Thmtheorem4.p1.1.m1.1.1.1.cmml" xref="S7.Thmtheorem4.p1.1.m1.1.1.1"></in><ci id="S7.Thmtheorem4.p1.1.m1.1.1.2.cmml" xref="S7.Thmtheorem4.p1.1.m1.1.1.2">𝑓</ci><ci id="S7.Thmtheorem4.p1.1.m1.1.1.3.cmml" xref="S7.Thmtheorem4.p1.1.m1.1.1.3">ℱ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem4.p1.1.m1.1c">f\in\mathscr{F}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem4.p1.1.m1.1d">italic_f ∈ script_F</annotation></semantics></math> there is a countable <math alttext="g\subseteq f" class="ltx_Math" display="inline" id="S7.Thmtheorem4.p1.2.m2.1"><semantics id="S7.Thmtheorem4.p1.2.m2.1a"><mrow id="S7.Thmtheorem4.p1.2.m2.1.1" xref="S7.Thmtheorem4.p1.2.m2.1.1.cmml"><mi id="S7.Thmtheorem4.p1.2.m2.1.1.2" xref="S7.Thmtheorem4.p1.2.m2.1.1.2.cmml">g</mi><mo id="S7.Thmtheorem4.p1.2.m2.1.1.1" xref="S7.Thmtheorem4.p1.2.m2.1.1.1.cmml">⊆</mo><mi id="S7.Thmtheorem4.p1.2.m2.1.1.3" xref="S7.Thmtheorem4.p1.2.m2.1.1.3.cmml">f</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem4.p1.2.m2.1b"><apply id="S7.Thmtheorem4.p1.2.m2.1.1.cmml" xref="S7.Thmtheorem4.p1.2.m2.1.1"><subset id="S7.Thmtheorem4.p1.2.m2.1.1.1.cmml" xref="S7.Thmtheorem4.p1.2.m2.1.1.1"></subset><ci id="S7.Thmtheorem4.p1.2.m2.1.1.2.cmml" xref="S7.Thmtheorem4.p1.2.m2.1.1.2">𝑔</ci><ci id="S7.Thmtheorem4.p1.2.m2.1.1.3.cmml" xref="S7.Thmtheorem4.p1.2.m2.1.1.3">𝑓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem4.p1.2.m2.1c">g\subseteq f</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem4.p1.2.m2.1d">italic_g ⊆ italic_f</annotation></semantics></math> such that <math alttext="f\subseteq\bar{g}" class="ltx_Math" display="inline" id="S7.Thmtheorem4.p1.3.m3.1"><semantics id="S7.Thmtheorem4.p1.3.m3.1a"><mrow id="S7.Thmtheorem4.p1.3.m3.1.1" xref="S7.Thmtheorem4.p1.3.m3.1.1.cmml"><mi id="S7.Thmtheorem4.p1.3.m3.1.1.2" xref="S7.Thmtheorem4.p1.3.m3.1.1.2.cmml">f</mi><mo id="S7.Thmtheorem4.p1.3.m3.1.1.1" xref="S7.Thmtheorem4.p1.3.m3.1.1.1.cmml">⊆</mo><mover accent="true" id="S7.Thmtheorem4.p1.3.m3.1.1.3" xref="S7.Thmtheorem4.p1.3.m3.1.1.3.cmml"><mi id="S7.Thmtheorem4.p1.3.m3.1.1.3.2" xref="S7.Thmtheorem4.p1.3.m3.1.1.3.2.cmml">g</mi><mo id="S7.Thmtheorem4.p1.3.m3.1.1.3.1" xref="S7.Thmtheorem4.p1.3.m3.1.1.3.1.cmml">¯</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem4.p1.3.m3.1b"><apply id="S7.Thmtheorem4.p1.3.m3.1.1.cmml" xref="S7.Thmtheorem4.p1.3.m3.1.1"><subset id="S7.Thmtheorem4.p1.3.m3.1.1.1.cmml" xref="S7.Thmtheorem4.p1.3.m3.1.1.1"></subset><ci id="S7.Thmtheorem4.p1.3.m3.1.1.2.cmml" xref="S7.Thmtheorem4.p1.3.m3.1.1.2">𝑓</ci><apply id="S7.Thmtheorem4.p1.3.m3.1.1.3.cmml" xref="S7.Thmtheorem4.p1.3.m3.1.1.3"><ci id="S7.Thmtheorem4.p1.3.m3.1.1.3.1.cmml" xref="S7.Thmtheorem4.p1.3.m3.1.1.3.1">¯</ci><ci id="S7.Thmtheorem4.p1.3.m3.1.1.3.2.cmml" xref="S7.Thmtheorem4.p1.3.m3.1.1.3.2">𝑔</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem4.p1.3.m3.1c">f\subseteq\bar{g}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem4.p1.3.m3.1d">italic_f ⊆ over¯ start_ARG italic_g end_ARG</annotation></semantics></math>. Moreover, if <math alttext="f" class="ltx_Math" display="inline" id="S7.Thmtheorem4.p1.4.m4.1"><semantics id="S7.Thmtheorem4.p1.4.m4.1a"><mi id="S7.Thmtheorem4.p1.4.m4.1.1" xref="S7.Thmtheorem4.p1.4.m4.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem4.p1.4.m4.1b"><ci id="S7.Thmtheorem4.p1.4.m4.1.1.cmml" xref="S7.Thmtheorem4.p1.4.m4.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem4.p1.4.m4.1c">f</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem4.p1.4.m4.1d">italic_f</annotation></semantics></math> is injective then so is <math alttext="\bar{g}" class="ltx_Math" display="inline" id="S7.Thmtheorem4.p1.5.m5.1"><semantics id="S7.Thmtheorem4.p1.5.m5.1a"><mover accent="true" id="S7.Thmtheorem4.p1.5.m5.1.1" xref="S7.Thmtheorem4.p1.5.m5.1.1.cmml"><mi id="S7.Thmtheorem4.p1.5.m5.1.1.2" xref="S7.Thmtheorem4.p1.5.m5.1.1.2.cmml">g</mi><mo id="S7.Thmtheorem4.p1.5.m5.1.1.1" xref="S7.Thmtheorem4.p1.5.m5.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem4.p1.5.m5.1b"><apply id="S7.Thmtheorem4.p1.5.m5.1.1.cmml" xref="S7.Thmtheorem4.p1.5.m5.1.1"><ci id="S7.Thmtheorem4.p1.5.m5.1.1.1.cmml" xref="S7.Thmtheorem4.p1.5.m5.1.1.1">¯</ci><ci id="S7.Thmtheorem4.p1.5.m5.1.1.2.cmml" xref="S7.Thmtheorem4.p1.5.m5.1.1.2">𝑔</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem4.p1.5.m5.1c">\bar{g}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem4.p1.5.m5.1d">over¯ start_ARG italic_g end_ARG</annotation></semantics></math>.</p> </div> </div> <div class="ltx_proof" id="S7.SS1.1"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S7.SS1.1.p1"> <p class="ltx_p" id="S7.SS1.1.p1.22">Let <math alttext="A" class="ltx_Math" display="inline" id="S7.SS1.1.p1.1.m1.1"><semantics id="S7.SS1.1.p1.1.m1.1a"><mi id="S7.SS1.1.p1.1.m1.1.1" xref="S7.SS1.1.p1.1.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.1.p1.1.m1.1b"><ci id="S7.SS1.1.p1.1.m1.1.1.cmml" xref="S7.SS1.1.p1.1.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.1.p1.1.m1.1c">A</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.1.p1.1.m1.1d">italic_A</annotation></semantics></math> be a countable subset of <math alttext="\operatorname{dom}(f)" class="ltx_Math" display="inline" id="S7.SS1.1.p1.2.m2.2"><semantics id="S7.SS1.1.p1.2.m2.2a"><mrow id="S7.SS1.1.p1.2.m2.2.3.2" xref="S7.SS1.1.p1.2.m2.2.3.1.cmml"><mi id="S7.SS1.1.p1.2.m2.1.1" xref="S7.SS1.1.p1.2.m2.1.1.cmml">dom</mi><mo id="S7.SS1.1.p1.2.m2.2.3.2a" xref="S7.SS1.1.p1.2.m2.2.3.1.cmml">⁡</mo><mrow id="S7.SS1.1.p1.2.m2.2.3.2.1" xref="S7.SS1.1.p1.2.m2.2.3.1.cmml"><mo id="S7.SS1.1.p1.2.m2.2.3.2.1.1" stretchy="false" xref="S7.SS1.1.p1.2.m2.2.3.1.cmml">(</mo><mi id="S7.SS1.1.p1.2.m2.2.2" xref="S7.SS1.1.p1.2.m2.2.2.cmml">f</mi><mo id="S7.SS1.1.p1.2.m2.2.3.2.1.2" stretchy="false" xref="S7.SS1.1.p1.2.m2.2.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.1.p1.2.m2.2b"><apply id="S7.SS1.1.p1.2.m2.2.3.1.cmml" xref="S7.SS1.1.p1.2.m2.2.3.2"><ci id="S7.SS1.1.p1.2.m2.1.1.cmml" xref="S7.SS1.1.p1.2.m2.1.1">dom</ci><ci id="S7.SS1.1.p1.2.m2.2.2.cmml" xref="S7.SS1.1.p1.2.m2.2.2">𝑓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.1.p1.2.m2.2c">\operatorname{dom}(f)</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.1.p1.2.m2.2d">roman_dom ( italic_f )</annotation></semantics></math> such that every <math alttext="x\in\operatorname{dom}(f)" class="ltx_Math" display="inline" id="S7.SS1.1.p1.3.m3.2"><semantics id="S7.SS1.1.p1.3.m3.2a"><mrow id="S7.SS1.1.p1.3.m3.2.3" xref="S7.SS1.1.p1.3.m3.2.3.cmml"><mi id="S7.SS1.1.p1.3.m3.2.3.2" xref="S7.SS1.1.p1.3.m3.2.3.2.cmml">x</mi><mo id="S7.SS1.1.p1.3.m3.2.3.1" xref="S7.SS1.1.p1.3.m3.2.3.1.cmml">∈</mo><mrow id="S7.SS1.1.p1.3.m3.2.3.3.2" xref="S7.SS1.1.p1.3.m3.2.3.3.1.cmml"><mi id="S7.SS1.1.p1.3.m3.1.1" xref="S7.SS1.1.p1.3.m3.1.1.cmml">dom</mi><mo id="S7.SS1.1.p1.3.m3.2.3.3.2a" xref="S7.SS1.1.p1.3.m3.2.3.3.1.cmml">⁡</mo><mrow id="S7.SS1.1.p1.3.m3.2.3.3.2.1" xref="S7.SS1.1.p1.3.m3.2.3.3.1.cmml"><mo id="S7.SS1.1.p1.3.m3.2.3.3.2.1.1" stretchy="false" xref="S7.SS1.1.p1.3.m3.2.3.3.1.cmml">(</mo><mi id="S7.SS1.1.p1.3.m3.2.2" xref="S7.SS1.1.p1.3.m3.2.2.cmml">f</mi><mo id="S7.SS1.1.p1.3.m3.2.3.3.2.1.2" stretchy="false" xref="S7.SS1.1.p1.3.m3.2.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.1.p1.3.m3.2b"><apply id="S7.SS1.1.p1.3.m3.2.3.cmml" xref="S7.SS1.1.p1.3.m3.2.3"><in id="S7.SS1.1.p1.3.m3.2.3.1.cmml" xref="S7.SS1.1.p1.3.m3.2.3.1"></in><ci id="S7.SS1.1.p1.3.m3.2.3.2.cmml" xref="S7.SS1.1.p1.3.m3.2.3.2">𝑥</ci><apply id="S7.SS1.1.p1.3.m3.2.3.3.1.cmml" xref="S7.SS1.1.p1.3.m3.2.3.3.2"><ci id="S7.SS1.1.p1.3.m3.1.1.cmml" xref="S7.SS1.1.p1.3.m3.1.1">dom</ci><ci id="S7.SS1.1.p1.3.m3.2.2.cmml" xref="S7.SS1.1.p1.3.m3.2.2">𝑓</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.1.p1.3.m3.2c">x\in\operatorname{dom}(f)</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.1.p1.3.m3.2d">italic_x ∈ roman_dom ( italic_f )</annotation></semantics></math> is either in <math alttext="A" class="ltx_Math" display="inline" id="S7.SS1.1.p1.4.m4.1"><semantics id="S7.SS1.1.p1.4.m4.1a"><mi id="S7.SS1.1.p1.4.m4.1.1" xref="S7.SS1.1.p1.4.m4.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.1.p1.4.m4.1b"><ci id="S7.SS1.1.p1.4.m4.1.1.cmml" xref="S7.SS1.1.p1.4.m4.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.1.p1.4.m4.1c">A</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.1.p1.4.m4.1d">italic_A</annotation></semantics></math>, or is a limit point (in <math alttext="\mathbb{R}" class="ltx_Math" display="inline" id="S7.SS1.1.p1.5.m5.1"><semantics id="S7.SS1.1.p1.5.m5.1a"><mi id="S7.SS1.1.p1.5.m5.1.1" xref="S7.SS1.1.p1.5.m5.1.1.cmml">ℝ</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.1.p1.5.m5.1b"><ci id="S7.SS1.1.p1.5.m5.1.1.cmml" xref="S7.SS1.1.p1.5.m5.1.1">ℝ</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.1.p1.5.m5.1c">\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.1.p1.5.m5.1d">blackboard_R</annotation></semantics></math>) of both <math alttext="\{a\in A:a<x\}" class="ltx_Math" display="inline" id="S7.SS1.1.p1.6.m6.2"><semantics id="S7.SS1.1.p1.6.m6.2a"><mrow id="S7.SS1.1.p1.6.m6.2.2.2" xref="S7.SS1.1.p1.6.m6.2.2.3.cmml"><mo id="S7.SS1.1.p1.6.m6.2.2.2.3" stretchy="false" xref="S7.SS1.1.p1.6.m6.2.2.3.1.cmml">{</mo><mrow id="S7.SS1.1.p1.6.m6.1.1.1.1" xref="S7.SS1.1.p1.6.m6.1.1.1.1.cmml"><mi id="S7.SS1.1.p1.6.m6.1.1.1.1.2" xref="S7.SS1.1.p1.6.m6.1.1.1.1.2.cmml">a</mi><mo id="S7.SS1.1.p1.6.m6.1.1.1.1.1" xref="S7.SS1.1.p1.6.m6.1.1.1.1.1.cmml">∈</mo><mi id="S7.SS1.1.p1.6.m6.1.1.1.1.3" xref="S7.SS1.1.p1.6.m6.1.1.1.1.3.cmml">A</mi></mrow><mo id="S7.SS1.1.p1.6.m6.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S7.SS1.1.p1.6.m6.2.2.3.1.cmml">:</mo><mrow id="S7.SS1.1.p1.6.m6.2.2.2.2" xref="S7.SS1.1.p1.6.m6.2.2.2.2.cmml"><mi id="S7.SS1.1.p1.6.m6.2.2.2.2.2" xref="S7.SS1.1.p1.6.m6.2.2.2.2.2.cmml">a</mi><mo id="S7.SS1.1.p1.6.m6.2.2.2.2.1" xref="S7.SS1.1.p1.6.m6.2.2.2.2.1.cmml"><</mo><mi id="S7.SS1.1.p1.6.m6.2.2.2.2.3" xref="S7.SS1.1.p1.6.m6.2.2.2.2.3.cmml">x</mi></mrow><mo id="S7.SS1.1.p1.6.m6.2.2.2.5" stretchy="false" xref="S7.SS1.1.p1.6.m6.2.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.1.p1.6.m6.2b"><apply id="S7.SS1.1.p1.6.m6.2.2.3.cmml" xref="S7.SS1.1.p1.6.m6.2.2.2"><csymbol cd="latexml" id="S7.SS1.1.p1.6.m6.2.2.3.1.cmml" xref="S7.SS1.1.p1.6.m6.2.2.2.3">conditional-set</csymbol><apply id="S7.SS1.1.p1.6.m6.1.1.1.1.cmml" xref="S7.SS1.1.p1.6.m6.1.1.1.1"><in id="S7.SS1.1.p1.6.m6.1.1.1.1.1.cmml" xref="S7.SS1.1.p1.6.m6.1.1.1.1.1"></in><ci id="S7.SS1.1.p1.6.m6.1.1.1.1.2.cmml" xref="S7.SS1.1.p1.6.m6.1.1.1.1.2">𝑎</ci><ci id="S7.SS1.1.p1.6.m6.1.1.1.1.3.cmml" xref="S7.SS1.1.p1.6.m6.1.1.1.1.3">𝐴</ci></apply><apply id="S7.SS1.1.p1.6.m6.2.2.2.2.cmml" xref="S7.SS1.1.p1.6.m6.2.2.2.2"><lt id="S7.SS1.1.p1.6.m6.2.2.2.2.1.cmml" xref="S7.SS1.1.p1.6.m6.2.2.2.2.1"></lt><ci id="S7.SS1.1.p1.6.m6.2.2.2.2.2.cmml" xref="S7.SS1.1.p1.6.m6.2.2.2.2.2">𝑎</ci><ci id="S7.SS1.1.p1.6.m6.2.2.2.2.3.cmml" xref="S7.SS1.1.p1.6.m6.2.2.2.2.3">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.1.p1.6.m6.2c">\{a\in A:a<x\}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.1.p1.6.m6.2d">{ italic_a ∈ italic_A : italic_a < italic_x }</annotation></semantics></math> and <math alttext="\{a\in A:x<a\}" class="ltx_Math" display="inline" id="S7.SS1.1.p1.7.m7.2"><semantics id="S7.SS1.1.p1.7.m7.2a"><mrow id="S7.SS1.1.p1.7.m7.2.2.2" xref="S7.SS1.1.p1.7.m7.2.2.3.cmml"><mo id="S7.SS1.1.p1.7.m7.2.2.2.3" stretchy="false" xref="S7.SS1.1.p1.7.m7.2.2.3.1.cmml">{</mo><mrow id="S7.SS1.1.p1.7.m7.1.1.1.1" xref="S7.SS1.1.p1.7.m7.1.1.1.1.cmml"><mi id="S7.SS1.1.p1.7.m7.1.1.1.1.2" xref="S7.SS1.1.p1.7.m7.1.1.1.1.2.cmml">a</mi><mo id="S7.SS1.1.p1.7.m7.1.1.1.1.1" xref="S7.SS1.1.p1.7.m7.1.1.1.1.1.cmml">∈</mo><mi id="S7.SS1.1.p1.7.m7.1.1.1.1.3" xref="S7.SS1.1.p1.7.m7.1.1.1.1.3.cmml">A</mi></mrow><mo id="S7.SS1.1.p1.7.m7.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S7.SS1.1.p1.7.m7.2.2.3.1.cmml">:</mo><mrow id="S7.SS1.1.p1.7.m7.2.2.2.2" xref="S7.SS1.1.p1.7.m7.2.2.2.2.cmml"><mi id="S7.SS1.1.p1.7.m7.2.2.2.2.2" xref="S7.SS1.1.p1.7.m7.2.2.2.2.2.cmml">x</mi><mo id="S7.SS1.1.p1.7.m7.2.2.2.2.1" xref="S7.SS1.1.p1.7.m7.2.2.2.2.1.cmml"><</mo><mi id="S7.SS1.1.p1.7.m7.2.2.2.2.3" xref="S7.SS1.1.p1.7.m7.2.2.2.2.3.cmml">a</mi></mrow><mo id="S7.SS1.1.p1.7.m7.2.2.2.5" stretchy="false" xref="S7.SS1.1.p1.7.m7.2.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.1.p1.7.m7.2b"><apply id="S7.SS1.1.p1.7.m7.2.2.3.cmml" xref="S7.SS1.1.p1.7.m7.2.2.2"><csymbol cd="latexml" id="S7.SS1.1.p1.7.m7.2.2.3.1.cmml" xref="S7.SS1.1.p1.7.m7.2.2.2.3">conditional-set</csymbol><apply id="S7.SS1.1.p1.7.m7.1.1.1.1.cmml" xref="S7.SS1.1.p1.7.m7.1.1.1.1"><in id="S7.SS1.1.p1.7.m7.1.1.1.1.1.cmml" xref="S7.SS1.1.p1.7.m7.1.1.1.1.1"></in><ci id="S7.SS1.1.p1.7.m7.1.1.1.1.2.cmml" xref="S7.SS1.1.p1.7.m7.1.1.1.1.2">𝑎</ci><ci id="S7.SS1.1.p1.7.m7.1.1.1.1.3.cmml" xref="S7.SS1.1.p1.7.m7.1.1.1.1.3">𝐴</ci></apply><apply id="S7.SS1.1.p1.7.m7.2.2.2.2.cmml" xref="S7.SS1.1.p1.7.m7.2.2.2.2"><lt id="S7.SS1.1.p1.7.m7.2.2.2.2.1.cmml" xref="S7.SS1.1.p1.7.m7.2.2.2.2.1"></lt><ci id="S7.SS1.1.p1.7.m7.2.2.2.2.2.cmml" xref="S7.SS1.1.p1.7.m7.2.2.2.2.2">𝑥</ci><ci id="S7.SS1.1.p1.7.m7.2.2.2.2.3.cmml" xref="S7.SS1.1.p1.7.m7.2.2.2.2.3">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.1.p1.7.m7.2c">\{a\in A:x<a\}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.1.p1.7.m7.2d">{ italic_a ∈ italic_A : italic_x < italic_a }</annotation></semantics></math>. Let <math alttext="B\subseteq\operatorname{ran}(f)" class="ltx_Math" display="inline" id="S7.SS1.1.p1.8.m8.2"><semantics id="S7.SS1.1.p1.8.m8.2a"><mrow id="S7.SS1.1.p1.8.m8.2.3" xref="S7.SS1.1.p1.8.m8.2.3.cmml"><mi id="S7.SS1.1.p1.8.m8.2.3.2" xref="S7.SS1.1.p1.8.m8.2.3.2.cmml">B</mi><mo id="S7.SS1.1.p1.8.m8.2.3.1" xref="S7.SS1.1.p1.8.m8.2.3.1.cmml">⊆</mo><mrow id="S7.SS1.1.p1.8.m8.2.3.3.2" xref="S7.SS1.1.p1.8.m8.2.3.3.1.cmml"><mi id="S7.SS1.1.p1.8.m8.1.1" xref="S7.SS1.1.p1.8.m8.1.1.cmml">ran</mi><mo id="S7.SS1.1.p1.8.m8.2.3.3.2a" xref="S7.SS1.1.p1.8.m8.2.3.3.1.cmml">⁡</mo><mrow id="S7.SS1.1.p1.8.m8.2.3.3.2.1" xref="S7.SS1.1.p1.8.m8.2.3.3.1.cmml"><mo id="S7.SS1.1.p1.8.m8.2.3.3.2.1.1" stretchy="false" xref="S7.SS1.1.p1.8.m8.2.3.3.1.cmml">(</mo><mi id="S7.SS1.1.p1.8.m8.2.2" xref="S7.SS1.1.p1.8.m8.2.2.cmml">f</mi><mo id="S7.SS1.1.p1.8.m8.2.3.3.2.1.2" stretchy="false" xref="S7.SS1.1.p1.8.m8.2.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.1.p1.8.m8.2b"><apply id="S7.SS1.1.p1.8.m8.2.3.cmml" xref="S7.SS1.1.p1.8.m8.2.3"><subset id="S7.SS1.1.p1.8.m8.2.3.1.cmml" xref="S7.SS1.1.p1.8.m8.2.3.1"></subset><ci id="S7.SS1.1.p1.8.m8.2.3.2.cmml" xref="S7.SS1.1.p1.8.m8.2.3.2">𝐵</ci><apply id="S7.SS1.1.p1.8.m8.2.3.3.1.cmml" xref="S7.SS1.1.p1.8.m8.2.3.3.2"><ci id="S7.SS1.1.p1.8.m8.1.1.cmml" xref="S7.SS1.1.p1.8.m8.1.1">ran</ci><ci id="S7.SS1.1.p1.8.m8.2.2.cmml" xref="S7.SS1.1.p1.8.m8.2.2">𝑓</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.1.p1.8.m8.2c">B\subseteq\operatorname{ran}(f)</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.1.p1.8.m8.2d">italic_B ⊆ roman_ran ( italic_f )</annotation></semantics></math> be defined similarly. Now using the ccc of <math alttext="\mathbb{R}" class="ltx_Math" display="inline" id="S7.SS1.1.p1.9.m9.1"><semantics id="S7.SS1.1.p1.9.m9.1a"><mi id="S7.SS1.1.p1.9.m9.1.1" xref="S7.SS1.1.p1.9.m9.1.1.cmml">ℝ</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.1.p1.9.m9.1b"><ci id="S7.SS1.1.p1.9.m9.1.1.cmml" xref="S7.SS1.1.p1.9.m9.1.1">ℝ</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.1.p1.9.m9.1c">\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.1.p1.9.m9.1d">blackboard_R</annotation></semantics></math> and the fact that <math alttext="f" class="ltx_Math" display="inline" id="S7.SS1.1.p1.10.m10.1"><semantics id="S7.SS1.1.p1.10.m10.1a"><mi id="S7.SS1.1.p1.10.m10.1.1" xref="S7.SS1.1.p1.10.m10.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.1.p1.10.m10.1b"><ci id="S7.SS1.1.p1.10.m10.1.1.cmml" xref="S7.SS1.1.p1.10.m10.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.1.p1.10.m10.1c">f</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.1.p1.10.m10.1d">italic_f</annotation></semantics></math> is monotone, one sees that there are only countably many <math alttext="y\in\operatorname{ran}(f)" class="ltx_Math" display="inline" id="S7.SS1.1.p1.11.m11.2"><semantics id="S7.SS1.1.p1.11.m11.2a"><mrow id="S7.SS1.1.p1.11.m11.2.3" xref="S7.SS1.1.p1.11.m11.2.3.cmml"><mi id="S7.SS1.1.p1.11.m11.2.3.2" xref="S7.SS1.1.p1.11.m11.2.3.2.cmml">y</mi><mo id="S7.SS1.1.p1.11.m11.2.3.1" xref="S7.SS1.1.p1.11.m11.2.3.1.cmml">∈</mo><mrow id="S7.SS1.1.p1.11.m11.2.3.3.2" xref="S7.SS1.1.p1.11.m11.2.3.3.1.cmml"><mi id="S7.SS1.1.p1.11.m11.1.1" xref="S7.SS1.1.p1.11.m11.1.1.cmml">ran</mi><mo id="S7.SS1.1.p1.11.m11.2.3.3.2a" xref="S7.SS1.1.p1.11.m11.2.3.3.1.cmml">⁡</mo><mrow id="S7.SS1.1.p1.11.m11.2.3.3.2.1" xref="S7.SS1.1.p1.11.m11.2.3.3.1.cmml"><mo id="S7.SS1.1.p1.11.m11.2.3.3.2.1.1" stretchy="false" xref="S7.SS1.1.p1.11.m11.2.3.3.1.cmml">(</mo><mi id="S7.SS1.1.p1.11.m11.2.2" xref="S7.SS1.1.p1.11.m11.2.2.cmml">f</mi><mo id="S7.SS1.1.p1.11.m11.2.3.3.2.1.2" stretchy="false" xref="S7.SS1.1.p1.11.m11.2.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.1.p1.11.m11.2b"><apply id="S7.SS1.1.p1.11.m11.2.3.cmml" xref="S7.SS1.1.p1.11.m11.2.3"><in id="S7.SS1.1.p1.11.m11.2.3.1.cmml" xref="S7.SS1.1.p1.11.m11.2.3.1"></in><ci id="S7.SS1.1.p1.11.m11.2.3.2.cmml" xref="S7.SS1.1.p1.11.m11.2.3.2">𝑦</ci><apply id="S7.SS1.1.p1.11.m11.2.3.3.1.cmml" xref="S7.SS1.1.p1.11.m11.2.3.3.2"><ci id="S7.SS1.1.p1.11.m11.1.1.cmml" xref="S7.SS1.1.p1.11.m11.1.1">ran</ci><ci id="S7.SS1.1.p1.11.m11.2.2.cmml" xref="S7.SS1.1.p1.11.m11.2.2">𝑓</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.1.p1.11.m11.2c">y\in\operatorname{ran}(f)</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.1.p1.11.m11.2d">italic_y ∈ roman_ran ( italic_f )</annotation></semantics></math> such that <math alttext="f^{-1}(y)" class="ltx_Math" display="inline" id="S7.SS1.1.p1.12.m12.1"><semantics id="S7.SS1.1.p1.12.m12.1a"><mrow id="S7.SS1.1.p1.12.m12.1.2" xref="S7.SS1.1.p1.12.m12.1.2.cmml"><msup id="S7.SS1.1.p1.12.m12.1.2.2" xref="S7.SS1.1.p1.12.m12.1.2.2.cmml"><mi id="S7.SS1.1.p1.12.m12.1.2.2.2" xref="S7.SS1.1.p1.12.m12.1.2.2.2.cmml">f</mi><mrow id="S7.SS1.1.p1.12.m12.1.2.2.3" xref="S7.SS1.1.p1.12.m12.1.2.2.3.cmml"><mo id="S7.SS1.1.p1.12.m12.1.2.2.3a" xref="S7.SS1.1.p1.12.m12.1.2.2.3.cmml">−</mo><mn id="S7.SS1.1.p1.12.m12.1.2.2.3.2" xref="S7.SS1.1.p1.12.m12.1.2.2.3.2.cmml">1</mn></mrow></msup><mo id="S7.SS1.1.p1.12.m12.1.2.1" xref="S7.SS1.1.p1.12.m12.1.2.1.cmml">⁢</mo><mrow id="S7.SS1.1.p1.12.m12.1.2.3.2" xref="S7.SS1.1.p1.12.m12.1.2.cmml"><mo id="S7.SS1.1.p1.12.m12.1.2.3.2.1" stretchy="false" xref="S7.SS1.1.p1.12.m12.1.2.cmml">(</mo><mi id="S7.SS1.1.p1.12.m12.1.1" xref="S7.SS1.1.p1.12.m12.1.1.cmml">y</mi><mo id="S7.SS1.1.p1.12.m12.1.2.3.2.2" stretchy="false" xref="S7.SS1.1.p1.12.m12.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.1.p1.12.m12.1b"><apply id="S7.SS1.1.p1.12.m12.1.2.cmml" xref="S7.SS1.1.p1.12.m12.1.2"><times id="S7.SS1.1.p1.12.m12.1.2.1.cmml" xref="S7.SS1.1.p1.12.m12.1.2.1"></times><apply id="S7.SS1.1.p1.12.m12.1.2.2.cmml" xref="S7.SS1.1.p1.12.m12.1.2.2"><csymbol cd="ambiguous" id="S7.SS1.1.p1.12.m12.1.2.2.1.cmml" xref="S7.SS1.1.p1.12.m12.1.2.2">superscript</csymbol><ci id="S7.SS1.1.p1.12.m12.1.2.2.2.cmml" xref="S7.SS1.1.p1.12.m12.1.2.2.2">𝑓</ci><apply id="S7.SS1.1.p1.12.m12.1.2.2.3.cmml" xref="S7.SS1.1.p1.12.m12.1.2.2.3"><minus id="S7.SS1.1.p1.12.m12.1.2.2.3.1.cmml" xref="S7.SS1.1.p1.12.m12.1.2.2.3"></minus><cn id="S7.SS1.1.p1.12.m12.1.2.2.3.2.cmml" type="integer" xref="S7.SS1.1.p1.12.m12.1.2.2.3.2">1</cn></apply></apply><ci id="S7.SS1.1.p1.12.m12.1.1.cmml" xref="S7.SS1.1.p1.12.m12.1.1">𝑦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.1.p1.12.m12.1c">f^{-1}(y)</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.1.p1.12.m12.1d">italic_f start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_y )</annotation></semantics></math> has more then one element. Therefore, by enlarging <math alttext="A" class="ltx_Math" display="inline" id="S7.SS1.1.p1.13.m13.1"><semantics id="S7.SS1.1.p1.13.m13.1a"><mi id="S7.SS1.1.p1.13.m13.1.1" xref="S7.SS1.1.p1.13.m13.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.1.p1.13.m13.1b"><ci id="S7.SS1.1.p1.13.m13.1.1.cmml" xref="S7.SS1.1.p1.13.m13.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.1.p1.13.m13.1c">A</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.1.p1.13.m13.1d">italic_A</annotation></semantics></math> and <math alttext="B" class="ltx_Math" display="inline" id="S7.SS1.1.p1.14.m14.1"><semantics id="S7.SS1.1.p1.14.m14.1a"><mi id="S7.SS1.1.p1.14.m14.1.1" xref="S7.SS1.1.p1.14.m14.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.1.p1.14.m14.1b"><ci id="S7.SS1.1.p1.14.m14.1.1.cmml" xref="S7.SS1.1.p1.14.m14.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.1.p1.14.m14.1c">B</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.1.p1.14.m14.1d">italic_B</annotation></semantics></math> we may assume <math alttext="B" class="ltx_Math" display="inline" id="S7.SS1.1.p1.15.m15.1"><semantics id="S7.SS1.1.p1.15.m15.1a"><mi id="S7.SS1.1.p1.15.m15.1.1" xref="S7.SS1.1.p1.15.m15.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.1.p1.15.m15.1b"><ci id="S7.SS1.1.p1.15.m15.1.1.cmml" xref="S7.SS1.1.p1.15.m15.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.1.p1.15.m15.1c">B</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.1.p1.15.m15.1d">italic_B</annotation></semantics></math> contains every such <math alttext="y" class="ltx_Math" display="inline" id="S7.SS1.1.p1.16.m16.1"><semantics id="S7.SS1.1.p1.16.m16.1a"><mi id="S7.SS1.1.p1.16.m16.1.1" xref="S7.SS1.1.p1.16.m16.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.1.p1.16.m16.1b"><ci id="S7.SS1.1.p1.16.m16.1.1.cmml" xref="S7.SS1.1.p1.16.m16.1.1">𝑦</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.1.p1.16.m16.1c">y</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.1.p1.16.m16.1d">italic_y</annotation></semantics></math>, and <math alttext="A" class="ltx_Math" display="inline" id="S7.SS1.1.p1.17.m17.1"><semantics id="S7.SS1.1.p1.17.m17.1a"><mi id="S7.SS1.1.p1.17.m17.1.1" xref="S7.SS1.1.p1.17.m17.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.1.p1.17.m17.1b"><ci id="S7.SS1.1.p1.17.m17.1.1.cmml" xref="S7.SS1.1.p1.17.m17.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.1.p1.17.m17.1c">A</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.1.p1.17.m17.1d">italic_A</annotation></semantics></math> contains the endpoints of <math alttext="f^{-1}(y)" class="ltx_Math" display="inline" id="S7.SS1.1.p1.18.m18.1"><semantics id="S7.SS1.1.p1.18.m18.1a"><mrow id="S7.SS1.1.p1.18.m18.1.2" xref="S7.SS1.1.p1.18.m18.1.2.cmml"><msup id="S7.SS1.1.p1.18.m18.1.2.2" xref="S7.SS1.1.p1.18.m18.1.2.2.cmml"><mi id="S7.SS1.1.p1.18.m18.1.2.2.2" xref="S7.SS1.1.p1.18.m18.1.2.2.2.cmml">f</mi><mrow id="S7.SS1.1.p1.18.m18.1.2.2.3" xref="S7.SS1.1.p1.18.m18.1.2.2.3.cmml"><mo id="S7.SS1.1.p1.18.m18.1.2.2.3a" xref="S7.SS1.1.p1.18.m18.1.2.2.3.cmml">−</mo><mn id="S7.SS1.1.p1.18.m18.1.2.2.3.2" xref="S7.SS1.1.p1.18.m18.1.2.2.3.2.cmml">1</mn></mrow></msup><mo id="S7.SS1.1.p1.18.m18.1.2.1" xref="S7.SS1.1.p1.18.m18.1.2.1.cmml">⁢</mo><mrow id="S7.SS1.1.p1.18.m18.1.2.3.2" xref="S7.SS1.1.p1.18.m18.1.2.cmml"><mo id="S7.SS1.1.p1.18.m18.1.2.3.2.1" stretchy="false" xref="S7.SS1.1.p1.18.m18.1.2.cmml">(</mo><mi id="S7.SS1.1.p1.18.m18.1.1" xref="S7.SS1.1.p1.18.m18.1.1.cmml">y</mi><mo id="S7.SS1.1.p1.18.m18.1.2.3.2.2" stretchy="false" xref="S7.SS1.1.p1.18.m18.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.1.p1.18.m18.1b"><apply id="S7.SS1.1.p1.18.m18.1.2.cmml" xref="S7.SS1.1.p1.18.m18.1.2"><times id="S7.SS1.1.p1.18.m18.1.2.1.cmml" xref="S7.SS1.1.p1.18.m18.1.2.1"></times><apply id="S7.SS1.1.p1.18.m18.1.2.2.cmml" xref="S7.SS1.1.p1.18.m18.1.2.2"><csymbol cd="ambiguous" id="S7.SS1.1.p1.18.m18.1.2.2.1.cmml" xref="S7.SS1.1.p1.18.m18.1.2.2">superscript</csymbol><ci id="S7.SS1.1.p1.18.m18.1.2.2.2.cmml" xref="S7.SS1.1.p1.18.m18.1.2.2.2">𝑓</ci><apply id="S7.SS1.1.p1.18.m18.1.2.2.3.cmml" xref="S7.SS1.1.p1.18.m18.1.2.2.3"><minus id="S7.SS1.1.p1.18.m18.1.2.2.3.1.cmml" xref="S7.SS1.1.p1.18.m18.1.2.2.3"></minus><cn id="S7.SS1.1.p1.18.m18.1.2.2.3.2.cmml" type="integer" xref="S7.SS1.1.p1.18.m18.1.2.2.3.2">1</cn></apply></apply><ci id="S7.SS1.1.p1.18.m18.1.1.cmml" xref="S7.SS1.1.p1.18.m18.1.1">𝑦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.1.p1.18.m18.1c">f^{-1}(y)</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.1.p1.18.m18.1d">italic_f start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_y )</annotation></semantics></math> if it has any. Finally, enlarging <math alttext="A" class="ltx_Math" display="inline" id="S7.SS1.1.p1.19.m19.1"><semantics id="S7.SS1.1.p1.19.m19.1a"><mi id="S7.SS1.1.p1.19.m19.1.1" xref="S7.SS1.1.p1.19.m19.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.1.p1.19.m19.1b"><ci id="S7.SS1.1.p1.19.m19.1.1.cmml" xref="S7.SS1.1.p1.19.m19.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.1.p1.19.m19.1c">A</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.1.p1.19.m19.1d">italic_A</annotation></semantics></math> and <math alttext="B" class="ltx_Math" display="inline" id="S7.SS1.1.p1.20.m20.1"><semantics id="S7.SS1.1.p1.20.m20.1a"><mi id="S7.SS1.1.p1.20.m20.1.1" xref="S7.SS1.1.p1.20.m20.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.1.p1.20.m20.1b"><ci id="S7.SS1.1.p1.20.m20.1.1.cmml" xref="S7.SS1.1.p1.20.m20.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.1.p1.20.m20.1c">B</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.1.p1.20.m20.1d">italic_B</annotation></semantics></math> again we may assume that <math alttext="f(A)=B" class="ltx_Math" display="inline" id="S7.SS1.1.p1.21.m21.1"><semantics id="S7.SS1.1.p1.21.m21.1a"><mrow id="S7.SS1.1.p1.21.m21.1.2" xref="S7.SS1.1.p1.21.m21.1.2.cmml"><mrow id="S7.SS1.1.p1.21.m21.1.2.2" xref="S7.SS1.1.p1.21.m21.1.2.2.cmml"><mi id="S7.SS1.1.p1.21.m21.1.2.2.2" xref="S7.SS1.1.p1.21.m21.1.2.2.2.cmml">f</mi><mo id="S7.SS1.1.p1.21.m21.1.2.2.1" xref="S7.SS1.1.p1.21.m21.1.2.2.1.cmml">⁢</mo><mrow id="S7.SS1.1.p1.21.m21.1.2.2.3.2" xref="S7.SS1.1.p1.21.m21.1.2.2.cmml"><mo id="S7.SS1.1.p1.21.m21.1.2.2.3.2.1" stretchy="false" xref="S7.SS1.1.p1.21.m21.1.2.2.cmml">(</mo><mi id="S7.SS1.1.p1.21.m21.1.1" xref="S7.SS1.1.p1.21.m21.1.1.cmml">A</mi><mo id="S7.SS1.1.p1.21.m21.1.2.2.3.2.2" stretchy="false" xref="S7.SS1.1.p1.21.m21.1.2.2.cmml">)</mo></mrow></mrow><mo id="S7.SS1.1.p1.21.m21.1.2.1" xref="S7.SS1.1.p1.21.m21.1.2.1.cmml">=</mo><mi id="S7.SS1.1.p1.21.m21.1.2.3" xref="S7.SS1.1.p1.21.m21.1.2.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.1.p1.21.m21.1b"><apply id="S7.SS1.1.p1.21.m21.1.2.cmml" xref="S7.SS1.1.p1.21.m21.1.2"><eq id="S7.SS1.1.p1.21.m21.1.2.1.cmml" xref="S7.SS1.1.p1.21.m21.1.2.1"></eq><apply id="S7.SS1.1.p1.21.m21.1.2.2.cmml" xref="S7.SS1.1.p1.21.m21.1.2.2"><times id="S7.SS1.1.p1.21.m21.1.2.2.1.cmml" xref="S7.SS1.1.p1.21.m21.1.2.2.1"></times><ci id="S7.SS1.1.p1.21.m21.1.2.2.2.cmml" xref="S7.SS1.1.p1.21.m21.1.2.2.2">𝑓</ci><ci id="S7.SS1.1.p1.21.m21.1.1.cmml" xref="S7.SS1.1.p1.21.m21.1.1">𝐴</ci></apply><ci id="S7.SS1.1.p1.21.m21.1.2.3.cmml" xref="S7.SS1.1.p1.21.m21.1.2.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.1.p1.21.m21.1c">f(A)=B</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.1.p1.21.m21.1d">italic_f ( italic_A ) = italic_B</annotation></semantics></math>. We leave the reader to verify that then <math alttext="g:=f|_{A}" class="ltx_Math" display="inline" id="S7.SS1.1.p1.22.m22.2"><semantics id="S7.SS1.1.p1.22.m22.2a"><mrow id="S7.SS1.1.p1.22.m22.2.3" xref="S7.SS1.1.p1.22.m22.2.3.cmml"><mi id="S7.SS1.1.p1.22.m22.2.3.2" xref="S7.SS1.1.p1.22.m22.2.3.2.cmml">g</mi><mo id="S7.SS1.1.p1.22.m22.2.3.1" lspace="0.278em" rspace="0.278em" xref="S7.SS1.1.p1.22.m22.2.3.1.cmml">:=</mo><msub id="S7.SS1.1.p1.22.m22.2.3.3.2" xref="S7.SS1.1.p1.22.m22.2.3.3.1.cmml"><mrow id="S7.SS1.1.p1.22.m22.2.3.3.2.2" xref="S7.SS1.1.p1.22.m22.2.3.3.1.cmml"><mi id="S7.SS1.1.p1.22.m22.1.1" xref="S7.SS1.1.p1.22.m22.1.1.cmml">f</mi><mo id="S7.SS1.1.p1.22.m22.2.3.3.2.2.1" stretchy="false" xref="S7.SS1.1.p1.22.m22.2.3.3.1.1.cmml">|</mo></mrow><mi id="S7.SS1.1.p1.22.m22.2.2.1" xref="S7.SS1.1.p1.22.m22.2.2.1.cmml">A</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.1.p1.22.m22.2b"><apply id="S7.SS1.1.p1.22.m22.2.3.cmml" xref="S7.SS1.1.p1.22.m22.2.3"><csymbol cd="latexml" id="S7.SS1.1.p1.22.m22.2.3.1.cmml" xref="S7.SS1.1.p1.22.m22.2.3.1">assign</csymbol><ci id="S7.SS1.1.p1.22.m22.2.3.2.cmml" xref="S7.SS1.1.p1.22.m22.2.3.2">𝑔</ci><apply id="S7.SS1.1.p1.22.m22.2.3.3.1.cmml" xref="S7.SS1.1.p1.22.m22.2.3.3.2"><csymbol cd="latexml" id="S7.SS1.1.p1.22.m22.2.3.3.1.1.cmml" xref="S7.SS1.1.p1.22.m22.2.3.3.2.2.1">evaluated-at</csymbol><ci id="S7.SS1.1.p1.22.m22.1.1.cmml" xref="S7.SS1.1.p1.22.m22.1.1">𝑓</ci><ci id="S7.SS1.1.p1.22.m22.2.2.1.cmml" xref="S7.SS1.1.p1.22.m22.2.2.1">𝐴</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.1.p1.22.m22.2c">g:=f|_{A}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.1.p1.22.m22.2d">italic_g := italic_f | start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT</annotation></semantics></math> works. ∎</p> </div> </div> <div class="ltx_para" id="S7.SS1.p3"> <p class="ltx_p" id="S7.SS1.p3.10">Since there are only <math alttext="\mathfrak{c}" class="ltx_Math" display="inline" id="S7.SS1.p3.1.m1.1"><semantics id="S7.SS1.p3.1.m1.1a"><mi id="S7.SS1.p3.1.m1.1.1" xref="S7.SS1.p3.1.m1.1.1.cmml">𝔠</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.p3.1.m1.1b"><ci id="S7.SS1.p3.1.m1.1.1.cmml" xref="S7.SS1.p3.1.m1.1.1">𝔠</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p3.1.m1.1c">\mathfrak{c}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p3.1.m1.1d">fraktur_c</annotation></semantics></math> many countable elements in <math alttext="\mathscr{F}" class="ltx_Math" display="inline" id="S7.SS1.p3.2.m2.1"><semantics id="S7.SS1.p3.2.m2.1a"><mi class="ltx_font_mathscript" id="S7.SS1.p3.2.m2.1.1" xref="S7.SS1.p3.2.m2.1.1.cmml">ℱ</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.p3.2.m2.1b"><ci id="S7.SS1.p3.2.m2.1.1.cmml" xref="S7.SS1.p3.2.m2.1.1">ℱ</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p3.2.m2.1c">\mathscr{F}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p3.2.m2.1d">script_F</annotation></semantics></math>, the previous implies that there is a sequence <math alttext="\langle f_{\alpha}:\alpha<\mathfrak{c}\rangle" class="ltx_math_unparsed" display="inline" id="S7.SS1.p3.3.m3.1"><semantics id="S7.SS1.p3.3.m3.1a"><mrow id="S7.SS1.p3.3.m3.1b"><mo id="S7.SS1.p3.3.m3.1.2" stretchy="false">⟨</mo><msub id="S7.SS1.p3.3.m3.1.3"><mi id="S7.SS1.p3.3.m3.1.3.2">f</mi><mi id="S7.SS1.p3.3.m3.1.3.3">α</mi></msub><mo id="S7.SS1.p3.3.m3.1.4" lspace="0.278em" rspace="0.278em">:</mo><mi id="S7.SS1.p3.3.m3.1.5">α</mi><mo id="S7.SS1.p3.3.m3.1.6"><</mo><mi id="S7.SS1.p3.3.m3.1.1">𝔠</mi><mo id="S7.SS1.p3.3.m3.1.7" stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex" id="S7.SS1.p3.3.m3.1c">\langle f_{\alpha}:\alpha<\mathfrak{c}\rangle</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p3.3.m3.1d">⟨ italic_f start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT : italic_α < fraktur_c ⟩</annotation></semantics></math> of elements in <math alttext="\mathscr{F}" class="ltx_Math" display="inline" id="S7.SS1.p3.4.m4.1"><semantics id="S7.SS1.p3.4.m4.1a"><mi class="ltx_font_mathscript" id="S7.SS1.p3.4.m4.1.1" xref="S7.SS1.p3.4.m4.1.1.cmml">ℱ</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.p3.4.m4.1b"><ci id="S7.SS1.p3.4.m4.1.1.cmml" xref="S7.SS1.p3.4.m4.1.1">ℱ</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p3.4.m4.1c">\mathscr{F}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p3.4.m4.1d">script_F</annotation></semantics></math> such that for every <math alttext="f\in\mathscr{F}" class="ltx_Math" display="inline" id="S7.SS1.p3.5.m5.1"><semantics id="S7.SS1.p3.5.m5.1a"><mrow id="S7.SS1.p3.5.m5.1.1" xref="S7.SS1.p3.5.m5.1.1.cmml"><mi id="S7.SS1.p3.5.m5.1.1.2" xref="S7.SS1.p3.5.m5.1.1.2.cmml">f</mi><mo id="S7.SS1.p3.5.m5.1.1.1" xref="S7.SS1.p3.5.m5.1.1.1.cmml">∈</mo><mi class="ltx_font_mathscript" id="S7.SS1.p3.5.m5.1.1.3" xref="S7.SS1.p3.5.m5.1.1.3.cmml">ℱ</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.p3.5.m5.1b"><apply id="S7.SS1.p3.5.m5.1.1.cmml" xref="S7.SS1.p3.5.m5.1.1"><in id="S7.SS1.p3.5.m5.1.1.1.cmml" xref="S7.SS1.p3.5.m5.1.1.1"></in><ci id="S7.SS1.p3.5.m5.1.1.2.cmml" xref="S7.SS1.p3.5.m5.1.1.2">𝑓</ci><ci id="S7.SS1.p3.5.m5.1.1.3.cmml" xref="S7.SS1.p3.5.m5.1.1.3">ℱ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p3.5.m5.1c">f\in\mathscr{F}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p3.5.m5.1d">italic_f ∈ script_F</annotation></semantics></math>, there is <math alttext="\alpha" class="ltx_Math" display="inline" id="S7.SS1.p3.6.m6.1"><semantics id="S7.SS1.p3.6.m6.1a"><mi id="S7.SS1.p3.6.m6.1.1" xref="S7.SS1.p3.6.m6.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.p3.6.m6.1b"><ci id="S7.SS1.p3.6.m6.1.1.cmml" xref="S7.SS1.p3.6.m6.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p3.6.m6.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p3.6.m6.1d">italic_α</annotation></semantics></math> such that <math alttext="f\subseteq f_{\alpha}" class="ltx_Math" display="inline" id="S7.SS1.p3.7.m7.1"><semantics id="S7.SS1.p3.7.m7.1a"><mrow id="S7.SS1.p3.7.m7.1.1" xref="S7.SS1.p3.7.m7.1.1.cmml"><mi id="S7.SS1.p3.7.m7.1.1.2" xref="S7.SS1.p3.7.m7.1.1.2.cmml">f</mi><mo id="S7.SS1.p3.7.m7.1.1.1" xref="S7.SS1.p3.7.m7.1.1.1.cmml">⊆</mo><msub id="S7.SS1.p3.7.m7.1.1.3" xref="S7.SS1.p3.7.m7.1.1.3.cmml"><mi id="S7.SS1.p3.7.m7.1.1.3.2" xref="S7.SS1.p3.7.m7.1.1.3.2.cmml">f</mi><mi id="S7.SS1.p3.7.m7.1.1.3.3" xref="S7.SS1.p3.7.m7.1.1.3.3.cmml">α</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.p3.7.m7.1b"><apply id="S7.SS1.p3.7.m7.1.1.cmml" xref="S7.SS1.p3.7.m7.1.1"><subset id="S7.SS1.p3.7.m7.1.1.1.cmml" xref="S7.SS1.p3.7.m7.1.1.1"></subset><ci id="S7.SS1.p3.7.m7.1.1.2.cmml" xref="S7.SS1.p3.7.m7.1.1.2">𝑓</ci><apply id="S7.SS1.p3.7.m7.1.1.3.cmml" xref="S7.SS1.p3.7.m7.1.1.3"><csymbol cd="ambiguous" id="S7.SS1.p3.7.m7.1.1.3.1.cmml" xref="S7.SS1.p3.7.m7.1.1.3">subscript</csymbol><ci id="S7.SS1.p3.7.m7.1.1.3.2.cmml" xref="S7.SS1.p3.7.m7.1.1.3.2">𝑓</ci><ci id="S7.SS1.p3.7.m7.1.1.3.3.cmml" xref="S7.SS1.p3.7.m7.1.1.3.3">𝛼</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p3.7.m7.1c">f\subseteq f_{\alpha}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p3.7.m7.1d">italic_f ⊆ italic_f start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT</annotation></semantics></math>. Moreover, if <math alttext="f" class="ltx_Math" display="inline" id="S7.SS1.p3.8.m8.1"><semantics id="S7.SS1.p3.8.m8.1a"><mi id="S7.SS1.p3.8.m8.1.1" xref="S7.SS1.p3.8.m8.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.p3.8.m8.1b"><ci id="S7.SS1.p3.8.m8.1.1.cmml" xref="S7.SS1.p3.8.m8.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p3.8.m8.1c">f</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p3.8.m8.1d">italic_f</annotation></semantics></math> is injective then the <math alttext="\alpha" class="ltx_Math" display="inline" id="S7.SS1.p3.9.m9.1"><semantics id="S7.SS1.p3.9.m9.1a"><mi id="S7.SS1.p3.9.m9.1.1" xref="S7.SS1.p3.9.m9.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.p3.9.m9.1b"><ci id="S7.SS1.p3.9.m9.1.1.cmml" xref="S7.SS1.p3.9.m9.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p3.9.m9.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p3.9.m9.1d">italic_α</annotation></semantics></math> can be chosen so that <math alttext="f_{\alpha}" class="ltx_Math" display="inline" id="S7.SS1.p3.10.m10.1"><semantics id="S7.SS1.p3.10.m10.1a"><msub id="S7.SS1.p3.10.m10.1.1" xref="S7.SS1.p3.10.m10.1.1.cmml"><mi id="S7.SS1.p3.10.m10.1.1.2" xref="S7.SS1.p3.10.m10.1.1.2.cmml">f</mi><mi id="S7.SS1.p3.10.m10.1.1.3" xref="S7.SS1.p3.10.m10.1.1.3.cmml">α</mi></msub><annotation-xml encoding="MathML-Content" id="S7.SS1.p3.10.m10.1b"><apply id="S7.SS1.p3.10.m10.1.1.cmml" xref="S7.SS1.p3.10.m10.1.1"><csymbol cd="ambiguous" id="S7.SS1.p3.10.m10.1.1.1.cmml" xref="S7.SS1.p3.10.m10.1.1">subscript</csymbol><ci id="S7.SS1.p3.10.m10.1.1.2.cmml" xref="S7.SS1.p3.10.m10.1.1.2">𝑓</ci><ci id="S7.SS1.p3.10.m10.1.1.3.cmml" xref="S7.SS1.p3.10.m10.1.1.3">𝛼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p3.10.m10.1c">f_{\alpha}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p3.10.m10.1d">italic_f start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT</annotation></semantics></math> is also injective.</p> </div> <div class="ltx_para" id="S7.SS1.p4"> <p class="ltx_p" id="S7.SS1.p4.23">We now construct by recursion a sequence <math alttext="\langle A_{\alpha}:\alpha<\mathfrak{c}\rangle" class="ltx_math_unparsed" display="inline" id="S7.SS1.p4.1.m1.1"><semantics id="S7.SS1.p4.1.m1.1a"><mrow id="S7.SS1.p4.1.m1.1b"><mo id="S7.SS1.p4.1.m1.1.2" stretchy="false">⟨</mo><msub id="S7.SS1.p4.1.m1.1.3"><mi id="S7.SS1.p4.1.m1.1.3.2">A</mi><mi id="S7.SS1.p4.1.m1.1.3.3">α</mi></msub><mo id="S7.SS1.p4.1.m1.1.4" lspace="0.278em" rspace="0.278em">:</mo><mi id="S7.SS1.p4.1.m1.1.5">α</mi><mo id="S7.SS1.p4.1.m1.1.6"><</mo><mi id="S7.SS1.p4.1.m1.1.1">𝔠</mi><mo id="S7.SS1.p4.1.m1.1.7" stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex" id="S7.SS1.p4.1.m1.1c">\langle A_{\alpha}:\alpha<\mathfrak{c}\rangle</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p4.1.m1.1d">⟨ italic_A start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT : italic_α < fraktur_c ⟩</annotation></semantics></math> of pairwise disjoint subsets of <math alttext="\mathbb{R}" class="ltx_Math" display="inline" id="S7.SS1.p4.2.m2.1"><semantics id="S7.SS1.p4.2.m2.1a"><mi id="S7.SS1.p4.2.m2.1.1" xref="S7.SS1.p4.2.m2.1.1.cmml">ℝ</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.p4.2.m2.1b"><ci id="S7.SS1.p4.2.m2.1.1.cmml" xref="S7.SS1.p4.2.m2.1.1">ℝ</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p4.2.m2.1c">\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p4.2.m2.1d">blackboard_R</annotation></semantics></math>. The construction closely follows that of <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib3" title="">3</a>]</cite>, we only generalize it as to allow adding more then one (but not too many) elements at each step: At step <math alttext="\alpha" class="ltx_Math" display="inline" id="S7.SS1.p4.3.m3.1"><semantics id="S7.SS1.p4.3.m3.1a"><mi id="S7.SS1.p4.3.m3.1.1" xref="S7.SS1.p4.3.m3.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.p4.3.m3.1b"><ci id="S7.SS1.p4.3.m3.1.1.cmml" xref="S7.SS1.p4.3.m3.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p4.3.m3.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p4.3.m3.1d">italic_α</annotation></semantics></math>, assume that <math alttext="A_{\xi}" class="ltx_Math" display="inline" id="S7.SS1.p4.4.m4.1"><semantics id="S7.SS1.p4.4.m4.1a"><msub id="S7.SS1.p4.4.m4.1.1" xref="S7.SS1.p4.4.m4.1.1.cmml"><mi id="S7.SS1.p4.4.m4.1.1.2" xref="S7.SS1.p4.4.m4.1.1.2.cmml">A</mi><mi id="S7.SS1.p4.4.m4.1.1.3" xref="S7.SS1.p4.4.m4.1.1.3.cmml">ξ</mi></msub><annotation-xml encoding="MathML-Content" id="S7.SS1.p4.4.m4.1b"><apply id="S7.SS1.p4.4.m4.1.1.cmml" xref="S7.SS1.p4.4.m4.1.1"><csymbol cd="ambiguous" id="S7.SS1.p4.4.m4.1.1.1.cmml" xref="S7.SS1.p4.4.m4.1.1">subscript</csymbol><ci id="S7.SS1.p4.4.m4.1.1.2.cmml" xref="S7.SS1.p4.4.m4.1.1.2">𝐴</ci><ci id="S7.SS1.p4.4.m4.1.1.3.cmml" xref="S7.SS1.p4.4.m4.1.1.3">𝜉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p4.4.m4.1c">A_{\xi}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p4.4.m4.1d">italic_A start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT</annotation></semantics></math> has been defined for every <math alttext="\xi<\alpha" class="ltx_Math" display="inline" id="S7.SS1.p4.5.m5.1"><semantics id="S7.SS1.p4.5.m5.1a"><mrow id="S7.SS1.p4.5.m5.1.1" xref="S7.SS1.p4.5.m5.1.1.cmml"><mi id="S7.SS1.p4.5.m5.1.1.2" xref="S7.SS1.p4.5.m5.1.1.2.cmml">ξ</mi><mo id="S7.SS1.p4.5.m5.1.1.1" xref="S7.SS1.p4.5.m5.1.1.1.cmml"><</mo><mi id="S7.SS1.p4.5.m5.1.1.3" xref="S7.SS1.p4.5.m5.1.1.3.cmml">α</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.p4.5.m5.1b"><apply id="S7.SS1.p4.5.m5.1.1.cmml" xref="S7.SS1.p4.5.m5.1.1"><lt id="S7.SS1.p4.5.m5.1.1.1.cmml" xref="S7.SS1.p4.5.m5.1.1.1"></lt><ci id="S7.SS1.p4.5.m5.1.1.2.cmml" xref="S7.SS1.p4.5.m5.1.1.2">𝜉</ci><ci id="S7.SS1.p4.5.m5.1.1.3.cmml" xref="S7.SS1.p4.5.m5.1.1.3">𝛼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p4.5.m5.1c">\xi<\alpha</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p4.5.m5.1d">italic_ξ < italic_α</annotation></semantics></math>, that they are pairwise disjoint, and that for each <math alttext="\xi<\alpha" class="ltx_Math" display="inline" id="S7.SS1.p4.6.m6.1"><semantics id="S7.SS1.p4.6.m6.1a"><mrow id="S7.SS1.p4.6.m6.1.1" xref="S7.SS1.p4.6.m6.1.1.cmml"><mi id="S7.SS1.p4.6.m6.1.1.2" xref="S7.SS1.p4.6.m6.1.1.2.cmml">ξ</mi><mo id="S7.SS1.p4.6.m6.1.1.1" xref="S7.SS1.p4.6.m6.1.1.1.cmml"><</mo><mi id="S7.SS1.p4.6.m6.1.1.3" xref="S7.SS1.p4.6.m6.1.1.3.cmml">α</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.p4.6.m6.1b"><apply id="S7.SS1.p4.6.m6.1.1.cmml" xref="S7.SS1.p4.6.m6.1.1"><lt id="S7.SS1.p4.6.m6.1.1.1.cmml" xref="S7.SS1.p4.6.m6.1.1.1"></lt><ci id="S7.SS1.p4.6.m6.1.1.2.cmml" xref="S7.SS1.p4.6.m6.1.1.2">𝜉</ci><ci id="S7.SS1.p4.6.m6.1.1.3.cmml" xref="S7.SS1.p4.6.m6.1.1.3">𝛼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p4.6.m6.1c">\xi<\alpha</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p4.6.m6.1d">italic_ξ < italic_α</annotation></semantics></math>, <math alttext="1\leq|A_{\xi}|\leq\max(\aleph_{0},|\xi|)" class="ltx_Math" display="inline" id="S7.SS1.p4.7.m7.5"><semantics id="S7.SS1.p4.7.m7.5a"><mrow id="S7.SS1.p4.7.m7.5.5" xref="S7.SS1.p4.7.m7.5.5.cmml"><mn id="S7.SS1.p4.7.m7.5.5.5" xref="S7.SS1.p4.7.m7.5.5.5.cmml">1</mn><mo id="S7.SS1.p4.7.m7.5.5.6" xref="S7.SS1.p4.7.m7.5.5.6.cmml">≤</mo><mrow id="S7.SS1.p4.7.m7.3.3.1.1" xref="S7.SS1.p4.7.m7.3.3.1.2.cmml"><mo id="S7.SS1.p4.7.m7.3.3.1.1.2" stretchy="false" xref="S7.SS1.p4.7.m7.3.3.1.2.1.cmml">|</mo><msub id="S7.SS1.p4.7.m7.3.3.1.1.1" xref="S7.SS1.p4.7.m7.3.3.1.1.1.cmml"><mi id="S7.SS1.p4.7.m7.3.3.1.1.1.2" xref="S7.SS1.p4.7.m7.3.3.1.1.1.2.cmml">A</mi><mi id="S7.SS1.p4.7.m7.3.3.1.1.1.3" xref="S7.SS1.p4.7.m7.3.3.1.1.1.3.cmml">ξ</mi></msub><mo id="S7.SS1.p4.7.m7.3.3.1.1.3" stretchy="false" xref="S7.SS1.p4.7.m7.3.3.1.2.1.cmml">|</mo></mrow><mo id="S7.SS1.p4.7.m7.5.5.7" xref="S7.SS1.p4.7.m7.5.5.7.cmml">≤</mo><mrow id="S7.SS1.p4.7.m7.5.5.3.2" xref="S7.SS1.p4.7.m7.5.5.3.3.cmml"><mi id="S7.SS1.p4.7.m7.2.2" xref="S7.SS1.p4.7.m7.2.2.cmml">max</mi><mo id="S7.SS1.p4.7.m7.5.5.3.2a" xref="S7.SS1.p4.7.m7.5.5.3.3.cmml">⁡</mo><mrow id="S7.SS1.p4.7.m7.5.5.3.2.2" xref="S7.SS1.p4.7.m7.5.5.3.3.cmml"><mo id="S7.SS1.p4.7.m7.5.5.3.2.2.3" stretchy="false" xref="S7.SS1.p4.7.m7.5.5.3.3.cmml">(</mo><msub id="S7.SS1.p4.7.m7.4.4.2.1.1.1" xref="S7.SS1.p4.7.m7.4.4.2.1.1.1.cmml"><mi id="S7.SS1.p4.7.m7.4.4.2.1.1.1.2" mathvariant="normal" xref="S7.SS1.p4.7.m7.4.4.2.1.1.1.2.cmml">ℵ</mi><mn id="S7.SS1.p4.7.m7.4.4.2.1.1.1.3" xref="S7.SS1.p4.7.m7.4.4.2.1.1.1.3.cmml">0</mn></msub><mo id="S7.SS1.p4.7.m7.5.5.3.2.2.4" xref="S7.SS1.p4.7.m7.5.5.3.3.cmml">,</mo><mrow id="S7.SS1.p4.7.m7.5.5.3.2.2.2.2" xref="S7.SS1.p4.7.m7.5.5.3.2.2.2.1.cmml"><mo id="S7.SS1.p4.7.m7.5.5.3.2.2.2.2.1" stretchy="false" xref="S7.SS1.p4.7.m7.5.5.3.2.2.2.1.1.cmml">|</mo><mi id="S7.SS1.p4.7.m7.1.1" xref="S7.SS1.p4.7.m7.1.1.cmml">ξ</mi><mo id="S7.SS1.p4.7.m7.5.5.3.2.2.2.2.2" stretchy="false" xref="S7.SS1.p4.7.m7.5.5.3.2.2.2.1.1.cmml">|</mo></mrow><mo id="S7.SS1.p4.7.m7.5.5.3.2.2.5" stretchy="false" xref="S7.SS1.p4.7.m7.5.5.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.p4.7.m7.5b"><apply id="S7.SS1.p4.7.m7.5.5.cmml" xref="S7.SS1.p4.7.m7.5.5"><and id="S7.SS1.p4.7.m7.5.5a.cmml" xref="S7.SS1.p4.7.m7.5.5"></and><apply id="S7.SS1.p4.7.m7.5.5b.cmml" xref="S7.SS1.p4.7.m7.5.5"><leq id="S7.SS1.p4.7.m7.5.5.6.cmml" xref="S7.SS1.p4.7.m7.5.5.6"></leq><cn id="S7.SS1.p4.7.m7.5.5.5.cmml" type="integer" xref="S7.SS1.p4.7.m7.5.5.5">1</cn><apply id="S7.SS1.p4.7.m7.3.3.1.2.cmml" xref="S7.SS1.p4.7.m7.3.3.1.1"><abs id="S7.SS1.p4.7.m7.3.3.1.2.1.cmml" xref="S7.SS1.p4.7.m7.3.3.1.1.2"></abs><apply id="S7.SS1.p4.7.m7.3.3.1.1.1.cmml" xref="S7.SS1.p4.7.m7.3.3.1.1.1"><csymbol cd="ambiguous" id="S7.SS1.p4.7.m7.3.3.1.1.1.1.cmml" xref="S7.SS1.p4.7.m7.3.3.1.1.1">subscript</csymbol><ci id="S7.SS1.p4.7.m7.3.3.1.1.1.2.cmml" xref="S7.SS1.p4.7.m7.3.3.1.1.1.2">𝐴</ci><ci id="S7.SS1.p4.7.m7.3.3.1.1.1.3.cmml" xref="S7.SS1.p4.7.m7.3.3.1.1.1.3">𝜉</ci></apply></apply></apply><apply id="S7.SS1.p4.7.m7.5.5c.cmml" xref="S7.SS1.p4.7.m7.5.5"><leq id="S7.SS1.p4.7.m7.5.5.7.cmml" xref="S7.SS1.p4.7.m7.5.5.7"></leq><share href="https://arxiv.org/html/2503.13728v1#S7.SS1.p4.7.m7.3.3.1.cmml" id="S7.SS1.p4.7.m7.5.5d.cmml" xref="S7.SS1.p4.7.m7.5.5"></share><apply id="S7.SS1.p4.7.m7.5.5.3.3.cmml" xref="S7.SS1.p4.7.m7.5.5.3.2"><max id="S7.SS1.p4.7.m7.2.2.cmml" xref="S7.SS1.p4.7.m7.2.2"></max><apply id="S7.SS1.p4.7.m7.4.4.2.1.1.1.cmml" xref="S7.SS1.p4.7.m7.4.4.2.1.1.1"><csymbol cd="ambiguous" id="S7.SS1.p4.7.m7.4.4.2.1.1.1.1.cmml" xref="S7.SS1.p4.7.m7.4.4.2.1.1.1">subscript</csymbol><ci id="S7.SS1.p4.7.m7.4.4.2.1.1.1.2.cmml" xref="S7.SS1.p4.7.m7.4.4.2.1.1.1.2">ℵ</ci><cn id="S7.SS1.p4.7.m7.4.4.2.1.1.1.3.cmml" type="integer" xref="S7.SS1.p4.7.m7.4.4.2.1.1.1.3">0</cn></apply><apply id="S7.SS1.p4.7.m7.5.5.3.2.2.2.1.cmml" xref="S7.SS1.p4.7.m7.5.5.3.2.2.2.2"><abs id="S7.SS1.p4.7.m7.5.5.3.2.2.2.1.1.cmml" xref="S7.SS1.p4.7.m7.5.5.3.2.2.2.2.1"></abs><ci id="S7.SS1.p4.7.m7.1.1.cmml" xref="S7.SS1.p4.7.m7.1.1">𝜉</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p4.7.m7.5c">1\leq|A_{\xi}|\leq\max(\aleph_{0},|\xi|)</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p4.7.m7.5d">1 ≤ | italic_A start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT | ≤ roman_max ( roman_ℵ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , | italic_ξ | )</annotation></semantics></math>. Note then that <math alttext="U_{\alpha}:=\bigcup_{\xi<\alpha}A_{\alpha}" class="ltx_Math" display="inline" id="S7.SS1.p4.8.m8.1"><semantics id="S7.SS1.p4.8.m8.1a"><mrow id="S7.SS1.p4.8.m8.1.1" xref="S7.SS1.p4.8.m8.1.1.cmml"><msub id="S7.SS1.p4.8.m8.1.1.2" xref="S7.SS1.p4.8.m8.1.1.2.cmml"><mi id="S7.SS1.p4.8.m8.1.1.2.2" xref="S7.SS1.p4.8.m8.1.1.2.2.cmml">U</mi><mi id="S7.SS1.p4.8.m8.1.1.2.3" xref="S7.SS1.p4.8.m8.1.1.2.3.cmml">α</mi></msub><mo id="S7.SS1.p4.8.m8.1.1.1" lspace="0.278em" rspace="0.111em" xref="S7.SS1.p4.8.m8.1.1.1.cmml">:=</mo><mrow id="S7.SS1.p4.8.m8.1.1.3" xref="S7.SS1.p4.8.m8.1.1.3.cmml"><msub id="S7.SS1.p4.8.m8.1.1.3.1" xref="S7.SS1.p4.8.m8.1.1.3.1.cmml"><mo id="S7.SS1.p4.8.m8.1.1.3.1.2" xref="S7.SS1.p4.8.m8.1.1.3.1.2.cmml">⋃</mo><mrow id="S7.SS1.p4.8.m8.1.1.3.1.3" xref="S7.SS1.p4.8.m8.1.1.3.1.3.cmml"><mi id="S7.SS1.p4.8.m8.1.1.3.1.3.2" xref="S7.SS1.p4.8.m8.1.1.3.1.3.2.cmml">ξ</mi><mo id="S7.SS1.p4.8.m8.1.1.3.1.3.1" xref="S7.SS1.p4.8.m8.1.1.3.1.3.1.cmml"><</mo><mi id="S7.SS1.p4.8.m8.1.1.3.1.3.3" xref="S7.SS1.p4.8.m8.1.1.3.1.3.3.cmml">α</mi></mrow></msub><msub id="S7.SS1.p4.8.m8.1.1.3.2" xref="S7.SS1.p4.8.m8.1.1.3.2.cmml"><mi id="S7.SS1.p4.8.m8.1.1.3.2.2" xref="S7.SS1.p4.8.m8.1.1.3.2.2.cmml">A</mi><mi id="S7.SS1.p4.8.m8.1.1.3.2.3" xref="S7.SS1.p4.8.m8.1.1.3.2.3.cmml">α</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.p4.8.m8.1b"><apply id="S7.SS1.p4.8.m8.1.1.cmml" xref="S7.SS1.p4.8.m8.1.1"><csymbol cd="latexml" id="S7.SS1.p4.8.m8.1.1.1.cmml" xref="S7.SS1.p4.8.m8.1.1.1">assign</csymbol><apply id="S7.SS1.p4.8.m8.1.1.2.cmml" xref="S7.SS1.p4.8.m8.1.1.2"><csymbol cd="ambiguous" id="S7.SS1.p4.8.m8.1.1.2.1.cmml" xref="S7.SS1.p4.8.m8.1.1.2">subscript</csymbol><ci id="S7.SS1.p4.8.m8.1.1.2.2.cmml" xref="S7.SS1.p4.8.m8.1.1.2.2">𝑈</ci><ci id="S7.SS1.p4.8.m8.1.1.2.3.cmml" xref="S7.SS1.p4.8.m8.1.1.2.3">𝛼</ci></apply><apply id="S7.SS1.p4.8.m8.1.1.3.cmml" xref="S7.SS1.p4.8.m8.1.1.3"><apply id="S7.SS1.p4.8.m8.1.1.3.1.cmml" xref="S7.SS1.p4.8.m8.1.1.3.1"><csymbol cd="ambiguous" id="S7.SS1.p4.8.m8.1.1.3.1.1.cmml" xref="S7.SS1.p4.8.m8.1.1.3.1">subscript</csymbol><union id="S7.SS1.p4.8.m8.1.1.3.1.2.cmml" xref="S7.SS1.p4.8.m8.1.1.3.1.2"></union><apply id="S7.SS1.p4.8.m8.1.1.3.1.3.cmml" xref="S7.SS1.p4.8.m8.1.1.3.1.3"><lt id="S7.SS1.p4.8.m8.1.1.3.1.3.1.cmml" xref="S7.SS1.p4.8.m8.1.1.3.1.3.1"></lt><ci id="S7.SS1.p4.8.m8.1.1.3.1.3.2.cmml" xref="S7.SS1.p4.8.m8.1.1.3.1.3.2">𝜉</ci><ci id="S7.SS1.p4.8.m8.1.1.3.1.3.3.cmml" xref="S7.SS1.p4.8.m8.1.1.3.1.3.3">𝛼</ci></apply></apply><apply id="S7.SS1.p4.8.m8.1.1.3.2.cmml" xref="S7.SS1.p4.8.m8.1.1.3.2"><csymbol cd="ambiguous" id="S7.SS1.p4.8.m8.1.1.3.2.1.cmml" xref="S7.SS1.p4.8.m8.1.1.3.2">subscript</csymbol><ci id="S7.SS1.p4.8.m8.1.1.3.2.2.cmml" xref="S7.SS1.p4.8.m8.1.1.3.2.2">𝐴</ci><ci id="S7.SS1.p4.8.m8.1.1.3.2.3.cmml" xref="S7.SS1.p4.8.m8.1.1.3.2.3">𝛼</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p4.8.m8.1c">U_{\alpha}:=\bigcup_{\xi<\alpha}A_{\alpha}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p4.8.m8.1d">italic_U start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT := ⋃ start_POSTSUBSCRIPT italic_ξ < italic_α end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT</annotation></semantics></math> has cardinality less than <math alttext="\mathfrak{c}" class="ltx_Math" display="inline" id="S7.SS1.p4.9.m9.1"><semantics id="S7.SS1.p4.9.m9.1a"><mi id="S7.SS1.p4.9.m9.1.1" xref="S7.SS1.p4.9.m9.1.1.cmml">𝔠</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.p4.9.m9.1b"><ci id="S7.SS1.p4.9.m9.1.1.cmml" xref="S7.SS1.p4.9.m9.1.1">𝔠</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p4.9.m9.1c">\mathfrak{c}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p4.9.m9.1d">fraktur_c</annotation></semantics></math>. Let <math alttext="Z_{\alpha}" class="ltx_Math" display="inline" id="S7.SS1.p4.10.m10.1"><semantics id="S7.SS1.p4.10.m10.1a"><msub id="S7.SS1.p4.10.m10.1.1" xref="S7.SS1.p4.10.m10.1.1.cmml"><mi id="S7.SS1.p4.10.m10.1.1.2" xref="S7.SS1.p4.10.m10.1.1.2.cmml">Z</mi><mi id="S7.SS1.p4.10.m10.1.1.3" xref="S7.SS1.p4.10.m10.1.1.3.cmml">α</mi></msub><annotation-xml encoding="MathML-Content" id="S7.SS1.p4.10.m10.1b"><apply id="S7.SS1.p4.10.m10.1.1.cmml" xref="S7.SS1.p4.10.m10.1.1"><csymbol cd="ambiguous" id="S7.SS1.p4.10.m10.1.1.1.cmml" xref="S7.SS1.p4.10.m10.1.1">subscript</csymbol><ci id="S7.SS1.p4.10.m10.1.1.2.cmml" xref="S7.SS1.p4.10.m10.1.1.2">𝑍</ci><ci id="S7.SS1.p4.10.m10.1.1.3.cmml" xref="S7.SS1.p4.10.m10.1.1.3">𝛼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p4.10.m10.1c">Z_{\alpha}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p4.10.m10.1d">italic_Z start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT</annotation></semantics></math> be the closure of <math alttext="U_{\alpha}" class="ltx_Math" display="inline" id="S7.SS1.p4.11.m11.1"><semantics id="S7.SS1.p4.11.m11.1a"><msub id="S7.SS1.p4.11.m11.1.1" xref="S7.SS1.p4.11.m11.1.1.cmml"><mi id="S7.SS1.p4.11.m11.1.1.2" xref="S7.SS1.p4.11.m11.1.1.2.cmml">U</mi><mi id="S7.SS1.p4.11.m11.1.1.3" xref="S7.SS1.p4.11.m11.1.1.3.cmml">α</mi></msub><annotation-xml encoding="MathML-Content" id="S7.SS1.p4.11.m11.1b"><apply id="S7.SS1.p4.11.m11.1.1.cmml" xref="S7.SS1.p4.11.m11.1.1"><csymbol cd="ambiguous" id="S7.SS1.p4.11.m11.1.1.1.cmml" xref="S7.SS1.p4.11.m11.1.1">subscript</csymbol><ci id="S7.SS1.p4.11.m11.1.1.2.cmml" xref="S7.SS1.p4.11.m11.1.1.2">𝑈</ci><ci id="S7.SS1.p4.11.m11.1.1.3.cmml" xref="S7.SS1.p4.11.m11.1.1.3">𝛼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p4.11.m11.1c">U_{\alpha}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p4.11.m11.1d">italic_U start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT</annotation></semantics></math> under <math alttext="\mathrm{id}" class="ltx_Math" display="inline" id="S7.SS1.p4.12.m12.1"><semantics id="S7.SS1.p4.12.m12.1a"><mi id="S7.SS1.p4.12.m12.1.1" xref="S7.SS1.p4.12.m12.1.1.cmml">id</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.p4.12.m12.1b"><ci id="S7.SS1.p4.12.m12.1.1.cmml" xref="S7.SS1.p4.12.m12.1.1">id</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p4.12.m12.1c">\mathrm{id}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p4.12.m12.1d">roman_id</annotation></semantics></math>, and every <math alttext="f_{\xi}" class="ltx_Math" display="inline" id="S7.SS1.p4.13.m13.1"><semantics id="S7.SS1.p4.13.m13.1a"><msub id="S7.SS1.p4.13.m13.1.1" xref="S7.SS1.p4.13.m13.1.1.cmml"><mi id="S7.SS1.p4.13.m13.1.1.2" xref="S7.SS1.p4.13.m13.1.1.2.cmml">f</mi><mi id="S7.SS1.p4.13.m13.1.1.3" xref="S7.SS1.p4.13.m13.1.1.3.cmml">ξ</mi></msub><annotation-xml encoding="MathML-Content" id="S7.SS1.p4.13.m13.1b"><apply id="S7.SS1.p4.13.m13.1.1.cmml" xref="S7.SS1.p4.13.m13.1.1"><csymbol cd="ambiguous" id="S7.SS1.p4.13.m13.1.1.1.cmml" xref="S7.SS1.p4.13.m13.1.1">subscript</csymbol><ci id="S7.SS1.p4.13.m13.1.1.2.cmml" xref="S7.SS1.p4.13.m13.1.1.2">𝑓</ci><ci id="S7.SS1.p4.13.m13.1.1.3.cmml" xref="S7.SS1.p4.13.m13.1.1.3">𝜉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p4.13.m13.1c">f_{\xi}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p4.13.m13.1d">italic_f start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="f_{\xi}^{-1}" class="ltx_Math" display="inline" id="S7.SS1.p4.14.m14.1"><semantics id="S7.SS1.p4.14.m14.1a"><msubsup id="S7.SS1.p4.14.m14.1.1" xref="S7.SS1.p4.14.m14.1.1.cmml"><mi id="S7.SS1.p4.14.m14.1.1.2.2" xref="S7.SS1.p4.14.m14.1.1.2.2.cmml">f</mi><mi id="S7.SS1.p4.14.m14.1.1.2.3" xref="S7.SS1.p4.14.m14.1.1.2.3.cmml">ξ</mi><mrow id="S7.SS1.p4.14.m14.1.1.3" xref="S7.SS1.p4.14.m14.1.1.3.cmml"><mo id="S7.SS1.p4.14.m14.1.1.3a" xref="S7.SS1.p4.14.m14.1.1.3.cmml">−</mo><mn id="S7.SS1.p4.14.m14.1.1.3.2" xref="S7.SS1.p4.14.m14.1.1.3.2.cmml">1</mn></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S7.SS1.p4.14.m14.1b"><apply id="S7.SS1.p4.14.m14.1.1.cmml" xref="S7.SS1.p4.14.m14.1.1"><csymbol cd="ambiguous" id="S7.SS1.p4.14.m14.1.1.1.cmml" xref="S7.SS1.p4.14.m14.1.1">superscript</csymbol><apply id="S7.SS1.p4.14.m14.1.1.2.cmml" xref="S7.SS1.p4.14.m14.1.1"><csymbol cd="ambiguous" id="S7.SS1.p4.14.m14.1.1.2.1.cmml" xref="S7.SS1.p4.14.m14.1.1">subscript</csymbol><ci id="S7.SS1.p4.14.m14.1.1.2.2.cmml" xref="S7.SS1.p4.14.m14.1.1.2.2">𝑓</ci><ci id="S7.SS1.p4.14.m14.1.1.2.3.cmml" xref="S7.SS1.p4.14.m14.1.1.2.3">𝜉</ci></apply><apply id="S7.SS1.p4.14.m14.1.1.3.cmml" xref="S7.SS1.p4.14.m14.1.1.3"><minus id="S7.SS1.p4.14.m14.1.1.3.1.cmml" xref="S7.SS1.p4.14.m14.1.1.3"></minus><cn id="S7.SS1.p4.14.m14.1.1.3.2.cmml" type="integer" xref="S7.SS1.p4.14.m14.1.1.3.2">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p4.14.m14.1c">f_{\xi}^{-1}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p4.14.m14.1d">italic_f start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT</annotation></semantics></math> (when <math alttext="f_{\xi}" class="ltx_Math" display="inline" id="S7.SS1.p4.15.m15.1"><semantics id="S7.SS1.p4.15.m15.1a"><msub id="S7.SS1.p4.15.m15.1.1" xref="S7.SS1.p4.15.m15.1.1.cmml"><mi id="S7.SS1.p4.15.m15.1.1.2" xref="S7.SS1.p4.15.m15.1.1.2.cmml">f</mi><mi id="S7.SS1.p4.15.m15.1.1.3" xref="S7.SS1.p4.15.m15.1.1.3.cmml">ξ</mi></msub><annotation-xml encoding="MathML-Content" id="S7.SS1.p4.15.m15.1b"><apply id="S7.SS1.p4.15.m15.1.1.cmml" xref="S7.SS1.p4.15.m15.1.1"><csymbol cd="ambiguous" id="S7.SS1.p4.15.m15.1.1.1.cmml" xref="S7.SS1.p4.15.m15.1.1">subscript</csymbol><ci id="S7.SS1.p4.15.m15.1.1.2.cmml" xref="S7.SS1.p4.15.m15.1.1.2">𝑓</ci><ci id="S7.SS1.p4.15.m15.1.1.3.cmml" xref="S7.SS1.p4.15.m15.1.1.3">𝜉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p4.15.m15.1c">f_{\xi}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p4.15.m15.1d">italic_f start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT</annotation></semantics></math> is injective) for <math alttext="\xi<\alpha" class="ltx_Math" display="inline" id="S7.SS1.p4.16.m16.1"><semantics id="S7.SS1.p4.16.m16.1a"><mrow id="S7.SS1.p4.16.m16.1.1" xref="S7.SS1.p4.16.m16.1.1.cmml"><mi id="S7.SS1.p4.16.m16.1.1.2" xref="S7.SS1.p4.16.m16.1.1.2.cmml">ξ</mi><mo id="S7.SS1.p4.16.m16.1.1.1" xref="S7.SS1.p4.16.m16.1.1.1.cmml"><</mo><mi id="S7.SS1.p4.16.m16.1.1.3" xref="S7.SS1.p4.16.m16.1.1.3.cmml">α</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.p4.16.m16.1b"><apply id="S7.SS1.p4.16.m16.1.1.cmml" xref="S7.SS1.p4.16.m16.1.1"><lt id="S7.SS1.p4.16.m16.1.1.1.cmml" xref="S7.SS1.p4.16.m16.1.1.1"></lt><ci id="S7.SS1.p4.16.m16.1.1.2.cmml" xref="S7.SS1.p4.16.m16.1.1.2">𝜉</ci><ci id="S7.SS1.p4.16.m16.1.1.3.cmml" xref="S7.SS1.p4.16.m16.1.1.3">𝛼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p4.16.m16.1c">\xi<\alpha</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p4.16.m16.1d">italic_ξ < italic_α</annotation></semantics></math>. Then <math alttext="U_{\alpha}\subseteq Z_{\alpha}" class="ltx_Math" display="inline" id="S7.SS1.p4.17.m17.1"><semantics id="S7.SS1.p4.17.m17.1a"><mrow id="S7.SS1.p4.17.m17.1.1" xref="S7.SS1.p4.17.m17.1.1.cmml"><msub id="S7.SS1.p4.17.m17.1.1.2" xref="S7.SS1.p4.17.m17.1.1.2.cmml"><mi id="S7.SS1.p4.17.m17.1.1.2.2" xref="S7.SS1.p4.17.m17.1.1.2.2.cmml">U</mi><mi id="S7.SS1.p4.17.m17.1.1.2.3" xref="S7.SS1.p4.17.m17.1.1.2.3.cmml">α</mi></msub><mo id="S7.SS1.p4.17.m17.1.1.1" xref="S7.SS1.p4.17.m17.1.1.1.cmml">⊆</mo><msub id="S7.SS1.p4.17.m17.1.1.3" xref="S7.SS1.p4.17.m17.1.1.3.cmml"><mi id="S7.SS1.p4.17.m17.1.1.3.2" xref="S7.SS1.p4.17.m17.1.1.3.2.cmml">Z</mi><mi id="S7.SS1.p4.17.m17.1.1.3.3" xref="S7.SS1.p4.17.m17.1.1.3.3.cmml">α</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.p4.17.m17.1b"><apply id="S7.SS1.p4.17.m17.1.1.cmml" xref="S7.SS1.p4.17.m17.1.1"><subset id="S7.SS1.p4.17.m17.1.1.1.cmml" xref="S7.SS1.p4.17.m17.1.1.1"></subset><apply id="S7.SS1.p4.17.m17.1.1.2.cmml" xref="S7.SS1.p4.17.m17.1.1.2"><csymbol cd="ambiguous" id="S7.SS1.p4.17.m17.1.1.2.1.cmml" xref="S7.SS1.p4.17.m17.1.1.2">subscript</csymbol><ci id="S7.SS1.p4.17.m17.1.1.2.2.cmml" xref="S7.SS1.p4.17.m17.1.1.2.2">𝑈</ci><ci id="S7.SS1.p4.17.m17.1.1.2.3.cmml" xref="S7.SS1.p4.17.m17.1.1.2.3">𝛼</ci></apply><apply id="S7.SS1.p4.17.m17.1.1.3.cmml" xref="S7.SS1.p4.17.m17.1.1.3"><csymbol cd="ambiguous" id="S7.SS1.p4.17.m17.1.1.3.1.cmml" xref="S7.SS1.p4.17.m17.1.1.3">subscript</csymbol><ci id="S7.SS1.p4.17.m17.1.1.3.2.cmml" xref="S7.SS1.p4.17.m17.1.1.3.2">𝑍</ci><ci id="S7.SS1.p4.17.m17.1.1.3.3.cmml" xref="S7.SS1.p4.17.m17.1.1.3.3">𝛼</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p4.17.m17.1c">U_{\alpha}\subseteq Z_{\alpha}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p4.17.m17.1d">italic_U start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ⊆ italic_Z start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="|Z_{\alpha}|<\mathfrak{c}" class="ltx_Math" display="inline" id="S7.SS1.p4.18.m18.1"><semantics id="S7.SS1.p4.18.m18.1a"><mrow id="S7.SS1.p4.18.m18.1.1" xref="S7.SS1.p4.18.m18.1.1.cmml"><mrow id="S7.SS1.p4.18.m18.1.1.1.1" xref="S7.SS1.p4.18.m18.1.1.1.2.cmml"><mo id="S7.SS1.p4.18.m18.1.1.1.1.2" stretchy="false" xref="S7.SS1.p4.18.m18.1.1.1.2.1.cmml">|</mo><msub id="S7.SS1.p4.18.m18.1.1.1.1.1" xref="S7.SS1.p4.18.m18.1.1.1.1.1.cmml"><mi id="S7.SS1.p4.18.m18.1.1.1.1.1.2" xref="S7.SS1.p4.18.m18.1.1.1.1.1.2.cmml">Z</mi><mi id="S7.SS1.p4.18.m18.1.1.1.1.1.3" xref="S7.SS1.p4.18.m18.1.1.1.1.1.3.cmml">α</mi></msub><mo id="S7.SS1.p4.18.m18.1.1.1.1.3" stretchy="false" xref="S7.SS1.p4.18.m18.1.1.1.2.1.cmml">|</mo></mrow><mo id="S7.SS1.p4.18.m18.1.1.2" xref="S7.SS1.p4.18.m18.1.1.2.cmml"><</mo><mi id="S7.SS1.p4.18.m18.1.1.3" xref="S7.SS1.p4.18.m18.1.1.3.cmml">𝔠</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.p4.18.m18.1b"><apply id="S7.SS1.p4.18.m18.1.1.cmml" xref="S7.SS1.p4.18.m18.1.1"><lt id="S7.SS1.p4.18.m18.1.1.2.cmml" xref="S7.SS1.p4.18.m18.1.1.2"></lt><apply id="S7.SS1.p4.18.m18.1.1.1.2.cmml" xref="S7.SS1.p4.18.m18.1.1.1.1"><abs id="S7.SS1.p4.18.m18.1.1.1.2.1.cmml" xref="S7.SS1.p4.18.m18.1.1.1.1.2"></abs><apply id="S7.SS1.p4.18.m18.1.1.1.1.1.cmml" xref="S7.SS1.p4.18.m18.1.1.1.1.1"><csymbol cd="ambiguous" id="S7.SS1.p4.18.m18.1.1.1.1.1.1.cmml" xref="S7.SS1.p4.18.m18.1.1.1.1.1">subscript</csymbol><ci id="S7.SS1.p4.18.m18.1.1.1.1.1.2.cmml" xref="S7.SS1.p4.18.m18.1.1.1.1.1.2">𝑍</ci><ci id="S7.SS1.p4.18.m18.1.1.1.1.1.3.cmml" xref="S7.SS1.p4.18.m18.1.1.1.1.1.3">𝛼</ci></apply></apply><ci id="S7.SS1.p4.18.m18.1.1.3.cmml" xref="S7.SS1.p4.18.m18.1.1.3">𝔠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p4.18.m18.1c">|Z_{\alpha}|<\mathfrak{c}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p4.18.m18.1d">| italic_Z start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT | < fraktur_c</annotation></semantics></math>. Now we let <math alttext="A_{\alpha}" class="ltx_Math" display="inline" id="S7.SS1.p4.19.m19.1"><semantics id="S7.SS1.p4.19.m19.1a"><msub id="S7.SS1.p4.19.m19.1.1" xref="S7.SS1.p4.19.m19.1.1.cmml"><mi id="S7.SS1.p4.19.m19.1.1.2" xref="S7.SS1.p4.19.m19.1.1.2.cmml">A</mi><mi id="S7.SS1.p4.19.m19.1.1.3" xref="S7.SS1.p4.19.m19.1.1.3.cmml">α</mi></msub><annotation-xml encoding="MathML-Content" id="S7.SS1.p4.19.m19.1b"><apply id="S7.SS1.p4.19.m19.1.1.cmml" xref="S7.SS1.p4.19.m19.1.1"><csymbol cd="ambiguous" id="S7.SS1.p4.19.m19.1.1.1.cmml" xref="S7.SS1.p4.19.m19.1.1">subscript</csymbol><ci id="S7.SS1.p4.19.m19.1.1.2.cmml" xref="S7.SS1.p4.19.m19.1.1.2">𝐴</ci><ci id="S7.SS1.p4.19.m19.1.1.3.cmml" xref="S7.SS1.p4.19.m19.1.1.3">𝛼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p4.19.m19.1c">A_{\alpha}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p4.19.m19.1d">italic_A start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT</annotation></semantics></math> be any subset of <math alttext="\mathbb{R}" class="ltx_Math" display="inline" id="S7.SS1.p4.20.m20.1"><semantics id="S7.SS1.p4.20.m20.1a"><mi id="S7.SS1.p4.20.m20.1.1" xref="S7.SS1.p4.20.m20.1.1.cmml">ℝ</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.p4.20.m20.1b"><ci id="S7.SS1.p4.20.m20.1.1.cmml" xref="S7.SS1.p4.20.m20.1.1">ℝ</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p4.20.m20.1c">\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p4.20.m20.1d">blackboard_R</annotation></semantics></math> disjoint from <math alttext="Z_{\alpha}" class="ltx_Math" display="inline" id="S7.SS1.p4.21.m21.1"><semantics id="S7.SS1.p4.21.m21.1a"><msub id="S7.SS1.p4.21.m21.1.1" xref="S7.SS1.p4.21.m21.1.1.cmml"><mi id="S7.SS1.p4.21.m21.1.1.2" xref="S7.SS1.p4.21.m21.1.1.2.cmml">Z</mi><mi id="S7.SS1.p4.21.m21.1.1.3" xref="S7.SS1.p4.21.m21.1.1.3.cmml">α</mi></msub><annotation-xml encoding="MathML-Content" id="S7.SS1.p4.21.m21.1b"><apply id="S7.SS1.p4.21.m21.1.1.cmml" xref="S7.SS1.p4.21.m21.1.1"><csymbol cd="ambiguous" id="S7.SS1.p4.21.m21.1.1.1.cmml" xref="S7.SS1.p4.21.m21.1.1">subscript</csymbol><ci id="S7.SS1.p4.21.m21.1.1.2.cmml" xref="S7.SS1.p4.21.m21.1.1.2">𝑍</ci><ci id="S7.SS1.p4.21.m21.1.1.3.cmml" xref="S7.SS1.p4.21.m21.1.1.3">𝛼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p4.21.m21.1c">Z_{\alpha}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p4.21.m21.1d">italic_Z start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT</annotation></semantics></math>, and such that <math alttext="1\leq|A_{\alpha}|\leq\max(\aleph_{0},|\alpha|)" class="ltx_Math" display="inline" id="S7.SS1.p4.22.m22.5"><semantics id="S7.SS1.p4.22.m22.5a"><mrow id="S7.SS1.p4.22.m22.5.5" xref="S7.SS1.p4.22.m22.5.5.cmml"><mn id="S7.SS1.p4.22.m22.5.5.5" xref="S7.SS1.p4.22.m22.5.5.5.cmml">1</mn><mo id="S7.SS1.p4.22.m22.5.5.6" xref="S7.SS1.p4.22.m22.5.5.6.cmml">≤</mo><mrow id="S7.SS1.p4.22.m22.3.3.1.1" xref="S7.SS1.p4.22.m22.3.3.1.2.cmml"><mo id="S7.SS1.p4.22.m22.3.3.1.1.2" stretchy="false" xref="S7.SS1.p4.22.m22.3.3.1.2.1.cmml">|</mo><msub id="S7.SS1.p4.22.m22.3.3.1.1.1" xref="S7.SS1.p4.22.m22.3.3.1.1.1.cmml"><mi id="S7.SS1.p4.22.m22.3.3.1.1.1.2" xref="S7.SS1.p4.22.m22.3.3.1.1.1.2.cmml">A</mi><mi id="S7.SS1.p4.22.m22.3.3.1.1.1.3" xref="S7.SS1.p4.22.m22.3.3.1.1.1.3.cmml">α</mi></msub><mo id="S7.SS1.p4.22.m22.3.3.1.1.3" stretchy="false" xref="S7.SS1.p4.22.m22.3.3.1.2.1.cmml">|</mo></mrow><mo id="S7.SS1.p4.22.m22.5.5.7" xref="S7.SS1.p4.22.m22.5.5.7.cmml">≤</mo><mrow id="S7.SS1.p4.22.m22.5.5.3.2" xref="S7.SS1.p4.22.m22.5.5.3.3.cmml"><mi id="S7.SS1.p4.22.m22.2.2" xref="S7.SS1.p4.22.m22.2.2.cmml">max</mi><mo id="S7.SS1.p4.22.m22.5.5.3.2a" xref="S7.SS1.p4.22.m22.5.5.3.3.cmml">⁡</mo><mrow id="S7.SS1.p4.22.m22.5.5.3.2.2" xref="S7.SS1.p4.22.m22.5.5.3.3.cmml"><mo id="S7.SS1.p4.22.m22.5.5.3.2.2.3" stretchy="false" xref="S7.SS1.p4.22.m22.5.5.3.3.cmml">(</mo><msub id="S7.SS1.p4.22.m22.4.4.2.1.1.1" xref="S7.SS1.p4.22.m22.4.4.2.1.1.1.cmml"><mi id="S7.SS1.p4.22.m22.4.4.2.1.1.1.2" mathvariant="normal" xref="S7.SS1.p4.22.m22.4.4.2.1.1.1.2.cmml">ℵ</mi><mn id="S7.SS1.p4.22.m22.4.4.2.1.1.1.3" xref="S7.SS1.p4.22.m22.4.4.2.1.1.1.3.cmml">0</mn></msub><mo id="S7.SS1.p4.22.m22.5.5.3.2.2.4" xref="S7.SS1.p4.22.m22.5.5.3.3.cmml">,</mo><mrow id="S7.SS1.p4.22.m22.5.5.3.2.2.2.2" xref="S7.SS1.p4.22.m22.5.5.3.2.2.2.1.cmml"><mo id="S7.SS1.p4.22.m22.5.5.3.2.2.2.2.1" stretchy="false" xref="S7.SS1.p4.22.m22.5.5.3.2.2.2.1.1.cmml">|</mo><mi id="S7.SS1.p4.22.m22.1.1" xref="S7.SS1.p4.22.m22.1.1.cmml">α</mi><mo id="S7.SS1.p4.22.m22.5.5.3.2.2.2.2.2" stretchy="false" xref="S7.SS1.p4.22.m22.5.5.3.2.2.2.1.1.cmml">|</mo></mrow><mo id="S7.SS1.p4.22.m22.5.5.3.2.2.5" stretchy="false" xref="S7.SS1.p4.22.m22.5.5.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.p4.22.m22.5b"><apply id="S7.SS1.p4.22.m22.5.5.cmml" xref="S7.SS1.p4.22.m22.5.5"><and id="S7.SS1.p4.22.m22.5.5a.cmml" xref="S7.SS1.p4.22.m22.5.5"></and><apply id="S7.SS1.p4.22.m22.5.5b.cmml" xref="S7.SS1.p4.22.m22.5.5"><leq id="S7.SS1.p4.22.m22.5.5.6.cmml" xref="S7.SS1.p4.22.m22.5.5.6"></leq><cn id="S7.SS1.p4.22.m22.5.5.5.cmml" type="integer" xref="S7.SS1.p4.22.m22.5.5.5">1</cn><apply id="S7.SS1.p4.22.m22.3.3.1.2.cmml" xref="S7.SS1.p4.22.m22.3.3.1.1"><abs id="S7.SS1.p4.22.m22.3.3.1.2.1.cmml" xref="S7.SS1.p4.22.m22.3.3.1.1.2"></abs><apply id="S7.SS1.p4.22.m22.3.3.1.1.1.cmml" xref="S7.SS1.p4.22.m22.3.3.1.1.1"><csymbol cd="ambiguous" id="S7.SS1.p4.22.m22.3.3.1.1.1.1.cmml" xref="S7.SS1.p4.22.m22.3.3.1.1.1">subscript</csymbol><ci id="S7.SS1.p4.22.m22.3.3.1.1.1.2.cmml" xref="S7.SS1.p4.22.m22.3.3.1.1.1.2">𝐴</ci><ci id="S7.SS1.p4.22.m22.3.3.1.1.1.3.cmml" xref="S7.SS1.p4.22.m22.3.3.1.1.1.3">𝛼</ci></apply></apply></apply><apply id="S7.SS1.p4.22.m22.5.5c.cmml" xref="S7.SS1.p4.22.m22.5.5"><leq id="S7.SS1.p4.22.m22.5.5.7.cmml" xref="S7.SS1.p4.22.m22.5.5.7"></leq><share href="https://arxiv.org/html/2503.13728v1#S7.SS1.p4.22.m22.3.3.1.cmml" id="S7.SS1.p4.22.m22.5.5d.cmml" xref="S7.SS1.p4.22.m22.5.5"></share><apply id="S7.SS1.p4.22.m22.5.5.3.3.cmml" xref="S7.SS1.p4.22.m22.5.5.3.2"><max id="S7.SS1.p4.22.m22.2.2.cmml" xref="S7.SS1.p4.22.m22.2.2"></max><apply id="S7.SS1.p4.22.m22.4.4.2.1.1.1.cmml" xref="S7.SS1.p4.22.m22.4.4.2.1.1.1"><csymbol cd="ambiguous" id="S7.SS1.p4.22.m22.4.4.2.1.1.1.1.cmml" xref="S7.SS1.p4.22.m22.4.4.2.1.1.1">subscript</csymbol><ci id="S7.SS1.p4.22.m22.4.4.2.1.1.1.2.cmml" xref="S7.SS1.p4.22.m22.4.4.2.1.1.1.2">ℵ</ci><cn id="S7.SS1.p4.22.m22.4.4.2.1.1.1.3.cmml" type="integer" xref="S7.SS1.p4.22.m22.4.4.2.1.1.1.3">0</cn></apply><apply id="S7.SS1.p4.22.m22.5.5.3.2.2.2.1.cmml" xref="S7.SS1.p4.22.m22.5.5.3.2.2.2.2"><abs id="S7.SS1.p4.22.m22.5.5.3.2.2.2.1.1.cmml" xref="S7.SS1.p4.22.m22.5.5.3.2.2.2.2.1"></abs><ci id="S7.SS1.p4.22.m22.1.1.cmml" xref="S7.SS1.p4.22.m22.1.1">𝛼</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p4.22.m22.5c">1\leq|A_{\alpha}|\leq\max(\aleph_{0},|\alpha|)</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p4.22.m22.5d">1 ≤ | italic_A start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT | ≤ roman_max ( roman_ℵ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , | italic_α | )</annotation></semantics></math>. We later specify a particular way to chose the <math alttext="A_{\alpha}" class="ltx_Math" display="inline" id="S7.SS1.p4.23.m23.1"><semantics id="S7.SS1.p4.23.m23.1a"><msub id="S7.SS1.p4.23.m23.1.1" xref="S7.SS1.p4.23.m23.1.1.cmml"><mi id="S7.SS1.p4.23.m23.1.1.2" xref="S7.SS1.p4.23.m23.1.1.2.cmml">A</mi><mi id="S7.SS1.p4.23.m23.1.1.3" xref="S7.SS1.p4.23.m23.1.1.3.cmml">α</mi></msub><annotation-xml encoding="MathML-Content" id="S7.SS1.p4.23.m23.1b"><apply id="S7.SS1.p4.23.m23.1.1.cmml" xref="S7.SS1.p4.23.m23.1.1"><csymbol cd="ambiguous" id="S7.SS1.p4.23.m23.1.1.1.cmml" xref="S7.SS1.p4.23.m23.1.1">subscript</csymbol><ci id="S7.SS1.p4.23.m23.1.1.2.cmml" xref="S7.SS1.p4.23.m23.1.1.2">𝐴</ci><ci id="S7.SS1.p4.23.m23.1.1.3.cmml" xref="S7.SS1.p4.23.m23.1.1.3">𝛼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p4.23.m23.1c">A_{\alpha}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p4.23.m23.1d">italic_A start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT</annotation></semantics></math> as to achieve a particular property, but for the following two lemmas any choice satisfying the above works.</p> </div> <div class="ltx_para" id="S7.SS1.p5"> <p class="ltx_p" id="S7.SS1.p5.2">For <math alttext="X\subseteq\mathfrak{c}" class="ltx_Math" display="inline" id="S7.SS1.p5.1.m1.1"><semantics id="S7.SS1.p5.1.m1.1a"><mrow id="S7.SS1.p5.1.m1.1.1" xref="S7.SS1.p5.1.m1.1.1.cmml"><mi id="S7.SS1.p5.1.m1.1.1.2" xref="S7.SS1.p5.1.m1.1.1.2.cmml">X</mi><mo id="S7.SS1.p5.1.m1.1.1.1" xref="S7.SS1.p5.1.m1.1.1.1.cmml">⊆</mo><mi id="S7.SS1.p5.1.m1.1.1.3" xref="S7.SS1.p5.1.m1.1.1.3.cmml">𝔠</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.p5.1.m1.1b"><apply id="S7.SS1.p5.1.m1.1.1.cmml" xref="S7.SS1.p5.1.m1.1.1"><subset id="S7.SS1.p5.1.m1.1.1.1.cmml" xref="S7.SS1.p5.1.m1.1.1.1"></subset><ci id="S7.SS1.p5.1.m1.1.1.2.cmml" xref="S7.SS1.p5.1.m1.1.1.2">𝑋</ci><ci id="S7.SS1.p5.1.m1.1.1.3.cmml" xref="S7.SS1.p5.1.m1.1.1.3">𝔠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p5.1.m1.1c">X\subseteq\mathfrak{c}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p5.1.m1.1d">italic_X ⊆ fraktur_c</annotation></semantics></math>, let <math alttext="A(X):=\bigcup_{\alpha\in X}A_{\alpha}" class="ltx_Math" display="inline" id="S7.SS1.p5.2.m2.1"><semantics id="S7.SS1.p5.2.m2.1a"><mrow id="S7.SS1.p5.2.m2.1.2" xref="S7.SS1.p5.2.m2.1.2.cmml"><mrow id="S7.SS1.p5.2.m2.1.2.2" xref="S7.SS1.p5.2.m2.1.2.2.cmml"><mi id="S7.SS1.p5.2.m2.1.2.2.2" xref="S7.SS1.p5.2.m2.1.2.2.2.cmml">A</mi><mo id="S7.SS1.p5.2.m2.1.2.2.1" xref="S7.SS1.p5.2.m2.1.2.2.1.cmml">⁢</mo><mrow id="S7.SS1.p5.2.m2.1.2.2.3.2" xref="S7.SS1.p5.2.m2.1.2.2.cmml"><mo id="S7.SS1.p5.2.m2.1.2.2.3.2.1" stretchy="false" xref="S7.SS1.p5.2.m2.1.2.2.cmml">(</mo><mi id="S7.SS1.p5.2.m2.1.1" xref="S7.SS1.p5.2.m2.1.1.cmml">X</mi><mo id="S7.SS1.p5.2.m2.1.2.2.3.2.2" rspace="0.278em" stretchy="false" xref="S7.SS1.p5.2.m2.1.2.2.cmml">)</mo></mrow></mrow><mo id="S7.SS1.p5.2.m2.1.2.1" rspace="0.111em" xref="S7.SS1.p5.2.m2.1.2.1.cmml">:=</mo><mrow id="S7.SS1.p5.2.m2.1.2.3" xref="S7.SS1.p5.2.m2.1.2.3.cmml"><msub id="S7.SS1.p5.2.m2.1.2.3.1" xref="S7.SS1.p5.2.m2.1.2.3.1.cmml"><mo id="S7.SS1.p5.2.m2.1.2.3.1.2" xref="S7.SS1.p5.2.m2.1.2.3.1.2.cmml">⋃</mo><mrow id="S7.SS1.p5.2.m2.1.2.3.1.3" xref="S7.SS1.p5.2.m2.1.2.3.1.3.cmml"><mi id="S7.SS1.p5.2.m2.1.2.3.1.3.2" xref="S7.SS1.p5.2.m2.1.2.3.1.3.2.cmml">α</mi><mo id="S7.SS1.p5.2.m2.1.2.3.1.3.1" xref="S7.SS1.p5.2.m2.1.2.3.1.3.1.cmml">∈</mo><mi id="S7.SS1.p5.2.m2.1.2.3.1.3.3" xref="S7.SS1.p5.2.m2.1.2.3.1.3.3.cmml">X</mi></mrow></msub><msub id="S7.SS1.p5.2.m2.1.2.3.2" xref="S7.SS1.p5.2.m2.1.2.3.2.cmml"><mi id="S7.SS1.p5.2.m2.1.2.3.2.2" xref="S7.SS1.p5.2.m2.1.2.3.2.2.cmml">A</mi><mi id="S7.SS1.p5.2.m2.1.2.3.2.3" xref="S7.SS1.p5.2.m2.1.2.3.2.3.cmml">α</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.p5.2.m2.1b"><apply id="S7.SS1.p5.2.m2.1.2.cmml" xref="S7.SS1.p5.2.m2.1.2"><csymbol cd="latexml" id="S7.SS1.p5.2.m2.1.2.1.cmml" xref="S7.SS1.p5.2.m2.1.2.1">assign</csymbol><apply id="S7.SS1.p5.2.m2.1.2.2.cmml" xref="S7.SS1.p5.2.m2.1.2.2"><times id="S7.SS1.p5.2.m2.1.2.2.1.cmml" xref="S7.SS1.p5.2.m2.1.2.2.1"></times><ci id="S7.SS1.p5.2.m2.1.2.2.2.cmml" xref="S7.SS1.p5.2.m2.1.2.2.2">𝐴</ci><ci id="S7.SS1.p5.2.m2.1.1.cmml" xref="S7.SS1.p5.2.m2.1.1">𝑋</ci></apply><apply id="S7.SS1.p5.2.m2.1.2.3.cmml" xref="S7.SS1.p5.2.m2.1.2.3"><apply id="S7.SS1.p5.2.m2.1.2.3.1.cmml" xref="S7.SS1.p5.2.m2.1.2.3.1"><csymbol cd="ambiguous" id="S7.SS1.p5.2.m2.1.2.3.1.1.cmml" xref="S7.SS1.p5.2.m2.1.2.3.1">subscript</csymbol><union id="S7.SS1.p5.2.m2.1.2.3.1.2.cmml" xref="S7.SS1.p5.2.m2.1.2.3.1.2"></union><apply id="S7.SS1.p5.2.m2.1.2.3.1.3.cmml" xref="S7.SS1.p5.2.m2.1.2.3.1.3"><in id="S7.SS1.p5.2.m2.1.2.3.1.3.1.cmml" xref="S7.SS1.p5.2.m2.1.2.3.1.3.1"></in><ci id="S7.SS1.p5.2.m2.1.2.3.1.3.2.cmml" xref="S7.SS1.p5.2.m2.1.2.3.1.3.2">𝛼</ci><ci id="S7.SS1.p5.2.m2.1.2.3.1.3.3.cmml" xref="S7.SS1.p5.2.m2.1.2.3.1.3.3">𝑋</ci></apply></apply><apply id="S7.SS1.p5.2.m2.1.2.3.2.cmml" xref="S7.SS1.p5.2.m2.1.2.3.2"><csymbol cd="ambiguous" id="S7.SS1.p5.2.m2.1.2.3.2.1.cmml" xref="S7.SS1.p5.2.m2.1.2.3.2">subscript</csymbol><ci id="S7.SS1.p5.2.m2.1.2.3.2.2.cmml" xref="S7.SS1.p5.2.m2.1.2.3.2.2">𝐴</ci><ci id="S7.SS1.p5.2.m2.1.2.3.2.3.cmml" xref="S7.SS1.p5.2.m2.1.2.3.2.3">𝛼</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p5.2.m2.1c">A(X):=\bigcup_{\alpha\in X}A_{\alpha}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p5.2.m2.1d">italic_A ( italic_X ) := ⋃ start_POSTSUBSCRIPT italic_α ∈ italic_X end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT</annotation></semantics></math>. The following lemma (and its proof) is a straightforward modifications of Lemma 2.3 in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib3" title="">3</a>]</cite>.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S7.Thmtheorem5"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S7.Thmtheorem5.1.1.1">Lemma 7.5</span></span><span class="ltx_text ltx_font_bold" id="S7.Thmtheorem5.2.2">.</span> </h6> <div class="ltx_para" id="S7.Thmtheorem5.p1"> <p class="ltx_p" id="S7.Thmtheorem5.p1.4">Suppose that <math alttext="Y\subseteq\mathfrak{c}" class="ltx_Math" display="inline" id="S7.Thmtheorem5.p1.1.m1.1"><semantics id="S7.Thmtheorem5.p1.1.m1.1a"><mrow id="S7.Thmtheorem5.p1.1.m1.1.1" xref="S7.Thmtheorem5.p1.1.m1.1.1.cmml"><mi id="S7.Thmtheorem5.p1.1.m1.1.1.2" xref="S7.Thmtheorem5.p1.1.m1.1.1.2.cmml">Y</mi><mo id="S7.Thmtheorem5.p1.1.m1.1.1.1" xref="S7.Thmtheorem5.p1.1.m1.1.1.1.cmml">⊆</mo><mi id="S7.Thmtheorem5.p1.1.m1.1.1.3" xref="S7.Thmtheorem5.p1.1.m1.1.1.3.cmml">𝔠</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem5.p1.1.m1.1b"><apply id="S7.Thmtheorem5.p1.1.m1.1.1.cmml" xref="S7.Thmtheorem5.p1.1.m1.1.1"><subset id="S7.Thmtheorem5.p1.1.m1.1.1.1.cmml" xref="S7.Thmtheorem5.p1.1.m1.1.1.1"></subset><ci id="S7.Thmtheorem5.p1.1.m1.1.1.2.cmml" xref="S7.Thmtheorem5.p1.1.m1.1.1.2">𝑌</ci><ci id="S7.Thmtheorem5.p1.1.m1.1.1.3.cmml" xref="S7.Thmtheorem5.p1.1.m1.1.1.3">𝔠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem5.p1.1.m1.1c">Y\subseteq\mathfrak{c}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem5.p1.1.m1.1d">italic_Y ⊆ fraktur_c</annotation></semantics></math>, and that <math alttext="B\subseteq\mathbb{R}" class="ltx_Math" display="inline" id="S7.Thmtheorem5.p1.2.m2.1"><semantics id="S7.Thmtheorem5.p1.2.m2.1a"><mrow id="S7.Thmtheorem5.p1.2.m2.1.1" xref="S7.Thmtheorem5.p1.2.m2.1.1.cmml"><mi id="S7.Thmtheorem5.p1.2.m2.1.1.2" xref="S7.Thmtheorem5.p1.2.m2.1.1.2.cmml">B</mi><mo id="S7.Thmtheorem5.p1.2.m2.1.1.1" xref="S7.Thmtheorem5.p1.2.m2.1.1.1.cmml">⊆</mo><mi id="S7.Thmtheorem5.p1.2.m2.1.1.3" xref="S7.Thmtheorem5.p1.2.m2.1.1.3.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem5.p1.2.m2.1b"><apply id="S7.Thmtheorem5.p1.2.m2.1.1.cmml" xref="S7.Thmtheorem5.p1.2.m2.1.1"><subset id="S7.Thmtheorem5.p1.2.m2.1.1.1.cmml" xref="S7.Thmtheorem5.p1.2.m2.1.1.1"></subset><ci id="S7.Thmtheorem5.p1.2.m2.1.1.2.cmml" xref="S7.Thmtheorem5.p1.2.m2.1.1.2">𝐵</ci><ci id="S7.Thmtheorem5.p1.2.m2.1.1.3.cmml" xref="S7.Thmtheorem5.p1.2.m2.1.1.3">ℝ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem5.p1.2.m2.1c">B\subseteq\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem5.p1.2.m2.1d">italic_B ⊆ blackboard_R</annotation></semantics></math> is such that <math alttext="\{\alpha<\mathfrak{c}:B\cap A_{\alpha}\neq\varnothing\}\setminus Y" class="ltx_Math" display="inline" id="S7.Thmtheorem5.p1.3.m3.2"><semantics id="S7.Thmtheorem5.p1.3.m3.2a"><mrow id="S7.Thmtheorem5.p1.3.m3.2.2" xref="S7.Thmtheorem5.p1.3.m3.2.2.cmml"><mrow id="S7.Thmtheorem5.p1.3.m3.2.2.2.2" xref="S7.Thmtheorem5.p1.3.m3.2.2.2.3.cmml"><mo id="S7.Thmtheorem5.p1.3.m3.2.2.2.2.3" stretchy="false" xref="S7.Thmtheorem5.p1.3.m3.2.2.2.3.1.cmml">{</mo><mrow id="S7.Thmtheorem5.p1.3.m3.1.1.1.1.1" xref="S7.Thmtheorem5.p1.3.m3.1.1.1.1.1.cmml"><mi id="S7.Thmtheorem5.p1.3.m3.1.1.1.1.1.2" xref="S7.Thmtheorem5.p1.3.m3.1.1.1.1.1.2.cmml">α</mi><mo id="S7.Thmtheorem5.p1.3.m3.1.1.1.1.1.1" xref="S7.Thmtheorem5.p1.3.m3.1.1.1.1.1.1.cmml"><</mo><mi id="S7.Thmtheorem5.p1.3.m3.1.1.1.1.1.3" xref="S7.Thmtheorem5.p1.3.m3.1.1.1.1.1.3.cmml">𝔠</mi></mrow><mo id="S7.Thmtheorem5.p1.3.m3.2.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S7.Thmtheorem5.p1.3.m3.2.2.2.3.1.cmml">:</mo><mrow id="S7.Thmtheorem5.p1.3.m3.2.2.2.2.2" xref="S7.Thmtheorem5.p1.3.m3.2.2.2.2.2.cmml"><mrow id="S7.Thmtheorem5.p1.3.m3.2.2.2.2.2.2" xref="S7.Thmtheorem5.p1.3.m3.2.2.2.2.2.2.cmml"><mi id="S7.Thmtheorem5.p1.3.m3.2.2.2.2.2.2.2" xref="S7.Thmtheorem5.p1.3.m3.2.2.2.2.2.2.2.cmml">B</mi><mo id="S7.Thmtheorem5.p1.3.m3.2.2.2.2.2.2.1" xref="S7.Thmtheorem5.p1.3.m3.2.2.2.2.2.2.1.cmml">∩</mo><msub id="S7.Thmtheorem5.p1.3.m3.2.2.2.2.2.2.3" xref="S7.Thmtheorem5.p1.3.m3.2.2.2.2.2.2.3.cmml"><mi id="S7.Thmtheorem5.p1.3.m3.2.2.2.2.2.2.3.2" xref="S7.Thmtheorem5.p1.3.m3.2.2.2.2.2.2.3.2.cmml">A</mi><mi id="S7.Thmtheorem5.p1.3.m3.2.2.2.2.2.2.3.3" xref="S7.Thmtheorem5.p1.3.m3.2.2.2.2.2.2.3.3.cmml">α</mi></msub></mrow><mo id="S7.Thmtheorem5.p1.3.m3.2.2.2.2.2.1" xref="S7.Thmtheorem5.p1.3.m3.2.2.2.2.2.1.cmml">≠</mo><mi id="S7.Thmtheorem5.p1.3.m3.2.2.2.2.2.3" mathvariant="normal" xref="S7.Thmtheorem5.p1.3.m3.2.2.2.2.2.3.cmml">∅</mi></mrow><mo id="S7.Thmtheorem5.p1.3.m3.2.2.2.2.5" stretchy="false" xref="S7.Thmtheorem5.p1.3.m3.2.2.2.3.1.cmml">}</mo></mrow><mo id="S7.Thmtheorem5.p1.3.m3.2.2.3" xref="S7.Thmtheorem5.p1.3.m3.2.2.3.cmml">∖</mo><mi id="S7.Thmtheorem5.p1.3.m3.2.2.4" xref="S7.Thmtheorem5.p1.3.m3.2.2.4.cmml">Y</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem5.p1.3.m3.2b"><apply id="S7.Thmtheorem5.p1.3.m3.2.2.cmml" xref="S7.Thmtheorem5.p1.3.m3.2.2"><setdiff id="S7.Thmtheorem5.p1.3.m3.2.2.3.cmml" xref="S7.Thmtheorem5.p1.3.m3.2.2.3"></setdiff><apply id="S7.Thmtheorem5.p1.3.m3.2.2.2.3.cmml" xref="S7.Thmtheorem5.p1.3.m3.2.2.2.2"><csymbol cd="latexml" id="S7.Thmtheorem5.p1.3.m3.2.2.2.3.1.cmml" xref="S7.Thmtheorem5.p1.3.m3.2.2.2.2.3">conditional-set</csymbol><apply id="S7.Thmtheorem5.p1.3.m3.1.1.1.1.1.cmml" xref="S7.Thmtheorem5.p1.3.m3.1.1.1.1.1"><lt id="S7.Thmtheorem5.p1.3.m3.1.1.1.1.1.1.cmml" xref="S7.Thmtheorem5.p1.3.m3.1.1.1.1.1.1"></lt><ci id="S7.Thmtheorem5.p1.3.m3.1.1.1.1.1.2.cmml" xref="S7.Thmtheorem5.p1.3.m3.1.1.1.1.1.2">𝛼</ci><ci id="S7.Thmtheorem5.p1.3.m3.1.1.1.1.1.3.cmml" xref="S7.Thmtheorem5.p1.3.m3.1.1.1.1.1.3">𝔠</ci></apply><apply id="S7.Thmtheorem5.p1.3.m3.2.2.2.2.2.cmml" xref="S7.Thmtheorem5.p1.3.m3.2.2.2.2.2"><neq id="S7.Thmtheorem5.p1.3.m3.2.2.2.2.2.1.cmml" xref="S7.Thmtheorem5.p1.3.m3.2.2.2.2.2.1"></neq><apply id="S7.Thmtheorem5.p1.3.m3.2.2.2.2.2.2.cmml" xref="S7.Thmtheorem5.p1.3.m3.2.2.2.2.2.2"><intersect id="S7.Thmtheorem5.p1.3.m3.2.2.2.2.2.2.1.cmml" xref="S7.Thmtheorem5.p1.3.m3.2.2.2.2.2.2.1"></intersect><ci id="S7.Thmtheorem5.p1.3.m3.2.2.2.2.2.2.2.cmml" xref="S7.Thmtheorem5.p1.3.m3.2.2.2.2.2.2.2">𝐵</ci><apply id="S7.Thmtheorem5.p1.3.m3.2.2.2.2.2.2.3.cmml" xref="S7.Thmtheorem5.p1.3.m3.2.2.2.2.2.2.3"><csymbol cd="ambiguous" id="S7.Thmtheorem5.p1.3.m3.2.2.2.2.2.2.3.1.cmml" xref="S7.Thmtheorem5.p1.3.m3.2.2.2.2.2.2.3">subscript</csymbol><ci id="S7.Thmtheorem5.p1.3.m3.2.2.2.2.2.2.3.2.cmml" xref="S7.Thmtheorem5.p1.3.m3.2.2.2.2.2.2.3.2">𝐴</ci><ci id="S7.Thmtheorem5.p1.3.m3.2.2.2.2.2.2.3.3.cmml" xref="S7.Thmtheorem5.p1.3.m3.2.2.2.2.2.2.3.3">𝛼</ci></apply></apply><emptyset id="S7.Thmtheorem5.p1.3.m3.2.2.2.2.2.3.cmml" xref="S7.Thmtheorem5.p1.3.m3.2.2.2.2.2.3"></emptyset></apply></apply><ci id="S7.Thmtheorem5.p1.3.m3.2.2.4.cmml" xref="S7.Thmtheorem5.p1.3.m3.2.2.4">𝑌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem5.p1.3.m3.2c">\{\alpha<\mathfrak{c}:B\cap A_{\alpha}\neq\varnothing\}\setminus Y</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem5.p1.3.m3.2d">{ italic_α < fraktur_c : italic_B ∩ italic_A start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ≠ ∅ } ∖ italic_Y</annotation></semantics></math> is cofinal. Then <math alttext="B\npreceq A(Y)" class="ltx_Math" display="inline" id="S7.Thmtheorem5.p1.4.m4.1"><semantics id="S7.Thmtheorem5.p1.4.m4.1a"><mrow id="S7.Thmtheorem5.p1.4.m4.1.2" xref="S7.Thmtheorem5.p1.4.m4.1.2.cmml"><mi id="S7.Thmtheorem5.p1.4.m4.1.2.2" xref="S7.Thmtheorem5.p1.4.m4.1.2.2.cmml">B</mi><mo id="S7.Thmtheorem5.p1.4.m4.1.2.1" xref="S7.Thmtheorem5.p1.4.m4.1.2.1.cmml">⋠</mo><mrow id="S7.Thmtheorem5.p1.4.m4.1.2.3" xref="S7.Thmtheorem5.p1.4.m4.1.2.3.cmml"><mi id="S7.Thmtheorem5.p1.4.m4.1.2.3.2" xref="S7.Thmtheorem5.p1.4.m4.1.2.3.2.cmml">A</mi><mo id="S7.Thmtheorem5.p1.4.m4.1.2.3.1" xref="S7.Thmtheorem5.p1.4.m4.1.2.3.1.cmml">⁢</mo><mrow id="S7.Thmtheorem5.p1.4.m4.1.2.3.3.2" xref="S7.Thmtheorem5.p1.4.m4.1.2.3.cmml"><mo id="S7.Thmtheorem5.p1.4.m4.1.2.3.3.2.1" stretchy="false" xref="S7.Thmtheorem5.p1.4.m4.1.2.3.cmml">(</mo><mi id="S7.Thmtheorem5.p1.4.m4.1.1" xref="S7.Thmtheorem5.p1.4.m4.1.1.cmml">Y</mi><mo id="S7.Thmtheorem5.p1.4.m4.1.2.3.3.2.2" stretchy="false" xref="S7.Thmtheorem5.p1.4.m4.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem5.p1.4.m4.1b"><apply id="S7.Thmtheorem5.p1.4.m4.1.2.cmml" xref="S7.Thmtheorem5.p1.4.m4.1.2"><csymbol cd="latexml" id="S7.Thmtheorem5.p1.4.m4.1.2.1.cmml" xref="S7.Thmtheorem5.p1.4.m4.1.2.1">not-precedes-nor-equals</csymbol><ci id="S7.Thmtheorem5.p1.4.m4.1.2.2.cmml" xref="S7.Thmtheorem5.p1.4.m4.1.2.2">𝐵</ci><apply id="S7.Thmtheorem5.p1.4.m4.1.2.3.cmml" xref="S7.Thmtheorem5.p1.4.m4.1.2.3"><times id="S7.Thmtheorem5.p1.4.m4.1.2.3.1.cmml" xref="S7.Thmtheorem5.p1.4.m4.1.2.3.1"></times><ci id="S7.Thmtheorem5.p1.4.m4.1.2.3.2.cmml" xref="S7.Thmtheorem5.p1.4.m4.1.2.3.2">𝐴</ci><ci id="S7.Thmtheorem5.p1.4.m4.1.1.cmml" xref="S7.Thmtheorem5.p1.4.m4.1.1">𝑌</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem5.p1.4.m4.1c">B\npreceq A(Y)</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem5.p1.4.m4.1d">italic_B ⋠ italic_A ( italic_Y )</annotation></semantics></math>.</p> </div> </div> <div class="ltx_para" id="S7.SS1.p6"> <p class="ltx_p" id="S7.SS1.p6.7">In <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib3" title="">3</a>]</cite> (Theorem 1.2 (a)), it is proven that there is a family <math alttext="S" class="ltx_Math" display="inline" id="S7.SS1.p6.1.m1.1"><semantics id="S7.SS1.p6.1.m1.1a"><mi id="S7.SS1.p6.1.m1.1.1" xref="S7.SS1.p6.1.m1.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.p6.1.m1.1b"><ci id="S7.SS1.p6.1.m1.1.1.cmml" xref="S7.SS1.p6.1.m1.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p6.1.m1.1c">S</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p6.1.m1.1d">italic_S</annotation></semantics></math> of size <math alttext="\mathfrak{c}^{+}" class="ltx_Math" display="inline" id="S7.SS1.p6.2.m2.1"><semantics id="S7.SS1.p6.2.m2.1a"><msup id="S7.SS1.p6.2.m2.1.1" xref="S7.SS1.p6.2.m2.1.1.cmml"><mi id="S7.SS1.p6.2.m2.1.1.2" xref="S7.SS1.p6.2.m2.1.1.2.cmml">𝔠</mi><mo id="S7.SS1.p6.2.m2.1.1.3" xref="S7.SS1.p6.2.m2.1.1.3.cmml">+</mo></msup><annotation-xml encoding="MathML-Content" id="S7.SS1.p6.2.m2.1b"><apply id="S7.SS1.p6.2.m2.1.1.cmml" xref="S7.SS1.p6.2.m2.1.1"><csymbol cd="ambiguous" id="S7.SS1.p6.2.m2.1.1.1.cmml" xref="S7.SS1.p6.2.m2.1.1">superscript</csymbol><ci id="S7.SS1.p6.2.m2.1.1.2.cmml" xref="S7.SS1.p6.2.m2.1.1.2">𝔠</ci><plus id="S7.SS1.p6.2.m2.1.1.3.cmml" xref="S7.SS1.p6.2.m2.1.1.3"></plus></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p6.2.m2.1c">\mathfrak{c}^{+}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p6.2.m2.1d">fraktur_c start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> of subsets of <math alttext="\mathfrak{c}" class="ltx_Math" display="inline" id="S7.SS1.p6.3.m3.1"><semantics id="S7.SS1.p6.3.m3.1a"><mi id="S7.SS1.p6.3.m3.1.1" xref="S7.SS1.p6.3.m3.1.1.cmml">𝔠</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.p6.3.m3.1b"><ci id="S7.SS1.p6.3.m3.1.1.cmml" xref="S7.SS1.p6.3.m3.1.1">𝔠</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p6.3.m3.1c">\mathfrak{c}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p6.3.m3.1d">fraktur_c</annotation></semantics></math> such that for every <math alttext="X\neq Y" class="ltx_Math" display="inline" id="S7.SS1.p6.4.m4.1"><semantics id="S7.SS1.p6.4.m4.1a"><mrow id="S7.SS1.p6.4.m4.1.1" xref="S7.SS1.p6.4.m4.1.1.cmml"><mi id="S7.SS1.p6.4.m4.1.1.2" xref="S7.SS1.p6.4.m4.1.1.2.cmml">X</mi><mo id="S7.SS1.p6.4.m4.1.1.1" xref="S7.SS1.p6.4.m4.1.1.1.cmml">≠</mo><mi id="S7.SS1.p6.4.m4.1.1.3" xref="S7.SS1.p6.4.m4.1.1.3.cmml">Y</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.p6.4.m4.1b"><apply id="S7.SS1.p6.4.m4.1.1.cmml" xref="S7.SS1.p6.4.m4.1.1"><neq id="S7.SS1.p6.4.m4.1.1.1.cmml" xref="S7.SS1.p6.4.m4.1.1.1"></neq><ci id="S7.SS1.p6.4.m4.1.1.2.cmml" xref="S7.SS1.p6.4.m4.1.1.2">𝑋</ci><ci id="S7.SS1.p6.4.m4.1.1.3.cmml" xref="S7.SS1.p6.4.m4.1.1.3">𝑌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p6.4.m4.1c">X\neq Y</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p6.4.m4.1d">italic_X ≠ italic_Y</annotation></semantics></math> in <math alttext="S" class="ltx_Math" display="inline" id="S7.SS1.p6.5.m5.1"><semantics id="S7.SS1.p6.5.m5.1a"><mi id="S7.SS1.p6.5.m5.1.1" xref="S7.SS1.p6.5.m5.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.p6.5.m5.1b"><ci id="S7.SS1.p6.5.m5.1.1.cmml" xref="S7.SS1.p6.5.m5.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p6.5.m5.1c">S</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p6.5.m5.1d">italic_S</annotation></semantics></math>, <math alttext="|X|=|Y|=\mathfrak{c}" class="ltx_Math" display="inline" id="S7.SS1.p6.6.m6.2"><semantics id="S7.SS1.p6.6.m6.2a"><mrow id="S7.SS1.p6.6.m6.2.3" xref="S7.SS1.p6.6.m6.2.3.cmml"><mrow id="S7.SS1.p6.6.m6.2.3.2.2" xref="S7.SS1.p6.6.m6.2.3.2.1.cmml"><mo id="S7.SS1.p6.6.m6.2.3.2.2.1" stretchy="false" xref="S7.SS1.p6.6.m6.2.3.2.1.1.cmml">|</mo><mi id="S7.SS1.p6.6.m6.1.1" xref="S7.SS1.p6.6.m6.1.1.cmml">X</mi><mo id="S7.SS1.p6.6.m6.2.3.2.2.2" stretchy="false" xref="S7.SS1.p6.6.m6.2.3.2.1.1.cmml">|</mo></mrow><mo id="S7.SS1.p6.6.m6.2.3.3" xref="S7.SS1.p6.6.m6.2.3.3.cmml">=</mo><mrow id="S7.SS1.p6.6.m6.2.3.4.2" xref="S7.SS1.p6.6.m6.2.3.4.1.cmml"><mo id="S7.SS1.p6.6.m6.2.3.4.2.1" stretchy="false" xref="S7.SS1.p6.6.m6.2.3.4.1.1.cmml">|</mo><mi id="S7.SS1.p6.6.m6.2.2" xref="S7.SS1.p6.6.m6.2.2.cmml">Y</mi><mo id="S7.SS1.p6.6.m6.2.3.4.2.2" stretchy="false" xref="S7.SS1.p6.6.m6.2.3.4.1.1.cmml">|</mo></mrow><mo id="S7.SS1.p6.6.m6.2.3.5" xref="S7.SS1.p6.6.m6.2.3.5.cmml">=</mo><mi id="S7.SS1.p6.6.m6.2.3.6" xref="S7.SS1.p6.6.m6.2.3.6.cmml">𝔠</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.p6.6.m6.2b"><apply id="S7.SS1.p6.6.m6.2.3.cmml" xref="S7.SS1.p6.6.m6.2.3"><and id="S7.SS1.p6.6.m6.2.3a.cmml" xref="S7.SS1.p6.6.m6.2.3"></and><apply id="S7.SS1.p6.6.m6.2.3b.cmml" xref="S7.SS1.p6.6.m6.2.3"><eq id="S7.SS1.p6.6.m6.2.3.3.cmml" xref="S7.SS1.p6.6.m6.2.3.3"></eq><apply id="S7.SS1.p6.6.m6.2.3.2.1.cmml" xref="S7.SS1.p6.6.m6.2.3.2.2"><abs id="S7.SS1.p6.6.m6.2.3.2.1.1.cmml" xref="S7.SS1.p6.6.m6.2.3.2.2.1"></abs><ci id="S7.SS1.p6.6.m6.1.1.cmml" xref="S7.SS1.p6.6.m6.1.1">𝑋</ci></apply><apply id="S7.SS1.p6.6.m6.2.3.4.1.cmml" xref="S7.SS1.p6.6.m6.2.3.4.2"><abs id="S7.SS1.p6.6.m6.2.3.4.1.1.cmml" xref="S7.SS1.p6.6.m6.2.3.4.2.1"></abs><ci id="S7.SS1.p6.6.m6.2.2.cmml" xref="S7.SS1.p6.6.m6.2.2">𝑌</ci></apply></apply><apply id="S7.SS1.p6.6.m6.2.3c.cmml" xref="S7.SS1.p6.6.m6.2.3"><eq id="S7.SS1.p6.6.m6.2.3.5.cmml" xref="S7.SS1.p6.6.m6.2.3.5"></eq><share href="https://arxiv.org/html/2503.13728v1#S7.SS1.p6.6.m6.2.3.4.cmml" id="S7.SS1.p6.6.m6.2.3d.cmml" xref="S7.SS1.p6.6.m6.2.3"></share><ci id="S7.SS1.p6.6.m6.2.3.6.cmml" xref="S7.SS1.p6.6.m6.2.3.6">𝔠</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p6.6.m6.2c">|X|=|Y|=\mathfrak{c}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p6.6.m6.2d">| italic_X | = | italic_Y | = fraktur_c</annotation></semantics></math> and <math alttext="\sup(X\cap Y)<\mathfrak{c}" class="ltx_Math" display="inline" id="S7.SS1.p6.7.m7.1"><semantics id="S7.SS1.p6.7.m7.1a"><mrow id="S7.SS1.p6.7.m7.1.1" xref="S7.SS1.p6.7.m7.1.1.cmml"><mrow id="S7.SS1.p6.7.m7.1.1.1" xref="S7.SS1.p6.7.m7.1.1.1.cmml"><mo id="S7.SS1.p6.7.m7.1.1.1.2" rspace="0em" xref="S7.SS1.p6.7.m7.1.1.1.2.cmml">sup</mo><mrow id="S7.SS1.p6.7.m7.1.1.1.1.1" xref="S7.SS1.p6.7.m7.1.1.1.1.1.1.cmml"><mo id="S7.SS1.p6.7.m7.1.1.1.1.1.2" stretchy="false" xref="S7.SS1.p6.7.m7.1.1.1.1.1.1.cmml">(</mo><mrow id="S7.SS1.p6.7.m7.1.1.1.1.1.1" xref="S7.SS1.p6.7.m7.1.1.1.1.1.1.cmml"><mi id="S7.SS1.p6.7.m7.1.1.1.1.1.1.2" xref="S7.SS1.p6.7.m7.1.1.1.1.1.1.2.cmml">X</mi><mo id="S7.SS1.p6.7.m7.1.1.1.1.1.1.1" xref="S7.SS1.p6.7.m7.1.1.1.1.1.1.1.cmml">∩</mo><mi id="S7.SS1.p6.7.m7.1.1.1.1.1.1.3" xref="S7.SS1.p6.7.m7.1.1.1.1.1.1.3.cmml">Y</mi></mrow><mo id="S7.SS1.p6.7.m7.1.1.1.1.1.3" stretchy="false" xref="S7.SS1.p6.7.m7.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S7.SS1.p6.7.m7.1.1.2" xref="S7.SS1.p6.7.m7.1.1.2.cmml"><</mo><mi id="S7.SS1.p6.7.m7.1.1.3" xref="S7.SS1.p6.7.m7.1.1.3.cmml">𝔠</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.p6.7.m7.1b"><apply id="S7.SS1.p6.7.m7.1.1.cmml" xref="S7.SS1.p6.7.m7.1.1"><lt id="S7.SS1.p6.7.m7.1.1.2.cmml" xref="S7.SS1.p6.7.m7.1.1.2"></lt><apply id="S7.SS1.p6.7.m7.1.1.1.cmml" xref="S7.SS1.p6.7.m7.1.1.1"><csymbol cd="latexml" id="S7.SS1.p6.7.m7.1.1.1.2.cmml" xref="S7.SS1.p6.7.m7.1.1.1.2">supremum</csymbol><apply id="S7.SS1.p6.7.m7.1.1.1.1.1.1.cmml" xref="S7.SS1.p6.7.m7.1.1.1.1.1"><intersect id="S7.SS1.p6.7.m7.1.1.1.1.1.1.1.cmml" xref="S7.SS1.p6.7.m7.1.1.1.1.1.1.1"></intersect><ci id="S7.SS1.p6.7.m7.1.1.1.1.1.1.2.cmml" xref="S7.SS1.p6.7.m7.1.1.1.1.1.1.2">𝑋</ci><ci id="S7.SS1.p6.7.m7.1.1.1.1.1.1.3.cmml" xref="S7.SS1.p6.7.m7.1.1.1.1.1.1.3">𝑌</ci></apply></apply><ci id="S7.SS1.p6.7.m7.1.1.3.cmml" xref="S7.SS1.p6.7.m7.1.1.3">𝔠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p6.7.m7.1c">\sup(X\cap Y)<\mathfrak{c}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p6.7.m7.1d">roman_sup ( italic_X ∩ italic_Y ) < fraktur_c</annotation></semantics></math>. For the rest of this section we fix such a family. The following is Theorem 2.4 (b) of <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib3" title="">3</a>]</cite>. We give the proof because is not given there.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S7.Thmtheorem6"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S7.Thmtheorem6.1.1.1">Lemma 7.6</span></span><span class="ltx_text ltx_font_bold" id="S7.Thmtheorem6.2.2">.</span> </h6> <div class="ltx_para" id="S7.Thmtheorem6.p1"> <p class="ltx_p" id="S7.Thmtheorem6.p1.4">Let <math alttext="X\neq Y" class="ltx_Math" display="inline" id="S7.Thmtheorem6.p1.1.m1.1"><semantics id="S7.Thmtheorem6.p1.1.m1.1a"><mrow id="S7.Thmtheorem6.p1.1.m1.1.1" xref="S7.Thmtheorem6.p1.1.m1.1.1.cmml"><mi id="S7.Thmtheorem6.p1.1.m1.1.1.2" xref="S7.Thmtheorem6.p1.1.m1.1.1.2.cmml">X</mi><mo id="S7.Thmtheorem6.p1.1.m1.1.1.1" xref="S7.Thmtheorem6.p1.1.m1.1.1.1.cmml">≠</mo><mi id="S7.Thmtheorem6.p1.1.m1.1.1.3" xref="S7.Thmtheorem6.p1.1.m1.1.1.3.cmml">Y</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem6.p1.1.m1.1b"><apply id="S7.Thmtheorem6.p1.1.m1.1.1.cmml" xref="S7.Thmtheorem6.p1.1.m1.1.1"><neq id="S7.Thmtheorem6.p1.1.m1.1.1.1.cmml" xref="S7.Thmtheorem6.p1.1.m1.1.1.1"></neq><ci id="S7.Thmtheorem6.p1.1.m1.1.1.2.cmml" xref="S7.Thmtheorem6.p1.1.m1.1.1.2">𝑋</ci><ci id="S7.Thmtheorem6.p1.1.m1.1.1.3.cmml" xref="S7.Thmtheorem6.p1.1.m1.1.1.3">𝑌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem6.p1.1.m1.1c">X\neq Y</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem6.p1.1.m1.1d">italic_X ≠ italic_Y</annotation></semantics></math> be in <math alttext="S" class="ltx_Math" display="inline" id="S7.Thmtheorem6.p1.2.m2.1"><semantics id="S7.Thmtheorem6.p1.2.m2.1a"><mi id="S7.Thmtheorem6.p1.2.m2.1.1" xref="S7.Thmtheorem6.p1.2.m2.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem6.p1.2.m2.1b"><ci id="S7.Thmtheorem6.p1.2.m2.1.1.cmml" xref="S7.Thmtheorem6.p1.2.m2.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem6.p1.2.m2.1c">S</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem6.p1.2.m2.1d">italic_S</annotation></semantics></math>. If <math alttext="B\preceq A(X),A(Y)" class="ltx_Math" display="inline" id="S7.Thmtheorem6.p1.3.m3.4"><semantics id="S7.Thmtheorem6.p1.3.m3.4a"><mrow id="S7.Thmtheorem6.p1.3.m3.4.4" xref="S7.Thmtheorem6.p1.3.m3.4.4.cmml"><mi id="S7.Thmtheorem6.p1.3.m3.4.4.4" xref="S7.Thmtheorem6.p1.3.m3.4.4.4.cmml">B</mi><mo id="S7.Thmtheorem6.p1.3.m3.4.4.3" xref="S7.Thmtheorem6.p1.3.m3.4.4.3.cmml">⪯</mo><mrow id="S7.Thmtheorem6.p1.3.m3.4.4.2.2" xref="S7.Thmtheorem6.p1.3.m3.4.4.2.3.cmml"><mrow id="S7.Thmtheorem6.p1.3.m3.3.3.1.1.1" xref="S7.Thmtheorem6.p1.3.m3.3.3.1.1.1.cmml"><mi id="S7.Thmtheorem6.p1.3.m3.3.3.1.1.1.2" xref="S7.Thmtheorem6.p1.3.m3.3.3.1.1.1.2.cmml">A</mi><mo id="S7.Thmtheorem6.p1.3.m3.3.3.1.1.1.1" xref="S7.Thmtheorem6.p1.3.m3.3.3.1.1.1.1.cmml">⁢</mo><mrow id="S7.Thmtheorem6.p1.3.m3.3.3.1.1.1.3.2" xref="S7.Thmtheorem6.p1.3.m3.3.3.1.1.1.cmml"><mo id="S7.Thmtheorem6.p1.3.m3.3.3.1.1.1.3.2.1" stretchy="false" xref="S7.Thmtheorem6.p1.3.m3.3.3.1.1.1.cmml">(</mo><mi id="S7.Thmtheorem6.p1.3.m3.1.1" xref="S7.Thmtheorem6.p1.3.m3.1.1.cmml">X</mi><mo id="S7.Thmtheorem6.p1.3.m3.3.3.1.1.1.3.2.2" stretchy="false" xref="S7.Thmtheorem6.p1.3.m3.3.3.1.1.1.cmml">)</mo></mrow></mrow><mo id="S7.Thmtheorem6.p1.3.m3.4.4.2.2.3" xref="S7.Thmtheorem6.p1.3.m3.4.4.2.3.cmml">,</mo><mrow id="S7.Thmtheorem6.p1.3.m3.4.4.2.2.2" xref="S7.Thmtheorem6.p1.3.m3.4.4.2.2.2.cmml"><mi id="S7.Thmtheorem6.p1.3.m3.4.4.2.2.2.2" xref="S7.Thmtheorem6.p1.3.m3.4.4.2.2.2.2.cmml">A</mi><mo id="S7.Thmtheorem6.p1.3.m3.4.4.2.2.2.1" xref="S7.Thmtheorem6.p1.3.m3.4.4.2.2.2.1.cmml">⁢</mo><mrow id="S7.Thmtheorem6.p1.3.m3.4.4.2.2.2.3.2" xref="S7.Thmtheorem6.p1.3.m3.4.4.2.2.2.cmml"><mo id="S7.Thmtheorem6.p1.3.m3.4.4.2.2.2.3.2.1" stretchy="false" xref="S7.Thmtheorem6.p1.3.m3.4.4.2.2.2.cmml">(</mo><mi id="S7.Thmtheorem6.p1.3.m3.2.2" xref="S7.Thmtheorem6.p1.3.m3.2.2.cmml">Y</mi><mo id="S7.Thmtheorem6.p1.3.m3.4.4.2.2.2.3.2.2" stretchy="false" xref="S7.Thmtheorem6.p1.3.m3.4.4.2.2.2.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem6.p1.3.m3.4b"><apply id="S7.Thmtheorem6.p1.3.m3.4.4.cmml" xref="S7.Thmtheorem6.p1.3.m3.4.4"><csymbol cd="latexml" id="S7.Thmtheorem6.p1.3.m3.4.4.3.cmml" xref="S7.Thmtheorem6.p1.3.m3.4.4.3">precedes-or-equals</csymbol><ci id="S7.Thmtheorem6.p1.3.m3.4.4.4.cmml" xref="S7.Thmtheorem6.p1.3.m3.4.4.4">𝐵</ci><list id="S7.Thmtheorem6.p1.3.m3.4.4.2.3.cmml" xref="S7.Thmtheorem6.p1.3.m3.4.4.2.2"><apply id="S7.Thmtheorem6.p1.3.m3.3.3.1.1.1.cmml" xref="S7.Thmtheorem6.p1.3.m3.3.3.1.1.1"><times id="S7.Thmtheorem6.p1.3.m3.3.3.1.1.1.1.cmml" xref="S7.Thmtheorem6.p1.3.m3.3.3.1.1.1.1"></times><ci id="S7.Thmtheorem6.p1.3.m3.3.3.1.1.1.2.cmml" xref="S7.Thmtheorem6.p1.3.m3.3.3.1.1.1.2">𝐴</ci><ci id="S7.Thmtheorem6.p1.3.m3.1.1.cmml" xref="S7.Thmtheorem6.p1.3.m3.1.1">𝑋</ci></apply><apply id="S7.Thmtheorem6.p1.3.m3.4.4.2.2.2.cmml" xref="S7.Thmtheorem6.p1.3.m3.4.4.2.2.2"><times id="S7.Thmtheorem6.p1.3.m3.4.4.2.2.2.1.cmml" xref="S7.Thmtheorem6.p1.3.m3.4.4.2.2.2.1"></times><ci id="S7.Thmtheorem6.p1.3.m3.4.4.2.2.2.2.cmml" xref="S7.Thmtheorem6.p1.3.m3.4.4.2.2.2.2">𝐴</ci><ci id="S7.Thmtheorem6.p1.3.m3.2.2.cmml" xref="S7.Thmtheorem6.p1.3.m3.2.2">𝑌</ci></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem6.p1.3.m3.4c">B\preceq A(X),A(Y)</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem6.p1.3.m3.4d">italic_B ⪯ italic_A ( italic_X ) , italic_A ( italic_Y )</annotation></semantics></math>, then <math alttext="|B|<\mathfrak{c}" class="ltx_Math" display="inline" id="S7.Thmtheorem6.p1.4.m4.1"><semantics id="S7.Thmtheorem6.p1.4.m4.1a"><mrow id="S7.Thmtheorem6.p1.4.m4.1.2" xref="S7.Thmtheorem6.p1.4.m4.1.2.cmml"><mrow id="S7.Thmtheorem6.p1.4.m4.1.2.2.2" xref="S7.Thmtheorem6.p1.4.m4.1.2.2.1.cmml"><mo id="S7.Thmtheorem6.p1.4.m4.1.2.2.2.1" stretchy="false" xref="S7.Thmtheorem6.p1.4.m4.1.2.2.1.1.cmml">|</mo><mi id="S7.Thmtheorem6.p1.4.m4.1.1" xref="S7.Thmtheorem6.p1.4.m4.1.1.cmml">B</mi><mo id="S7.Thmtheorem6.p1.4.m4.1.2.2.2.2" stretchy="false" xref="S7.Thmtheorem6.p1.4.m4.1.2.2.1.1.cmml">|</mo></mrow><mo id="S7.Thmtheorem6.p1.4.m4.1.2.1" xref="S7.Thmtheorem6.p1.4.m4.1.2.1.cmml"><</mo><mi id="S7.Thmtheorem6.p1.4.m4.1.2.3" xref="S7.Thmtheorem6.p1.4.m4.1.2.3.cmml">𝔠</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem6.p1.4.m4.1b"><apply id="S7.Thmtheorem6.p1.4.m4.1.2.cmml" xref="S7.Thmtheorem6.p1.4.m4.1.2"><lt id="S7.Thmtheorem6.p1.4.m4.1.2.1.cmml" xref="S7.Thmtheorem6.p1.4.m4.1.2.1"></lt><apply id="S7.Thmtheorem6.p1.4.m4.1.2.2.1.cmml" xref="S7.Thmtheorem6.p1.4.m4.1.2.2.2"><abs id="S7.Thmtheorem6.p1.4.m4.1.2.2.1.1.cmml" xref="S7.Thmtheorem6.p1.4.m4.1.2.2.2.1"></abs><ci id="S7.Thmtheorem6.p1.4.m4.1.1.cmml" xref="S7.Thmtheorem6.p1.4.m4.1.1">𝐵</ci></apply><ci id="S7.Thmtheorem6.p1.4.m4.1.2.3.cmml" xref="S7.Thmtheorem6.p1.4.m4.1.2.3">𝔠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem6.p1.4.m4.1c">|B|<\mathfrak{c}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem6.p1.4.m4.1d">| italic_B | < fraktur_c</annotation></semantics></math>.</p> </div> </div> <div class="ltx_proof" id="S7.SS1.2"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S7.SS1.2.p1"> <p class="ltx_p" id="S7.SS1.2.p1.9">Suppose that <math alttext="|B|=\mathfrak{c}" class="ltx_Math" display="inline" id="S7.SS1.2.p1.1.m1.1"><semantics id="S7.SS1.2.p1.1.m1.1a"><mrow id="S7.SS1.2.p1.1.m1.1.2" xref="S7.SS1.2.p1.1.m1.1.2.cmml"><mrow id="S7.SS1.2.p1.1.m1.1.2.2.2" xref="S7.SS1.2.p1.1.m1.1.2.2.1.cmml"><mo id="S7.SS1.2.p1.1.m1.1.2.2.2.1" stretchy="false" xref="S7.SS1.2.p1.1.m1.1.2.2.1.1.cmml">|</mo><mi id="S7.SS1.2.p1.1.m1.1.1" xref="S7.SS1.2.p1.1.m1.1.1.cmml">B</mi><mo id="S7.SS1.2.p1.1.m1.1.2.2.2.2" stretchy="false" xref="S7.SS1.2.p1.1.m1.1.2.2.1.1.cmml">|</mo></mrow><mo id="S7.SS1.2.p1.1.m1.1.2.1" xref="S7.SS1.2.p1.1.m1.1.2.1.cmml">=</mo><mi id="S7.SS1.2.p1.1.m1.1.2.3" xref="S7.SS1.2.p1.1.m1.1.2.3.cmml">𝔠</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.2.p1.1.m1.1b"><apply id="S7.SS1.2.p1.1.m1.1.2.cmml" xref="S7.SS1.2.p1.1.m1.1.2"><eq id="S7.SS1.2.p1.1.m1.1.2.1.cmml" xref="S7.SS1.2.p1.1.m1.1.2.1"></eq><apply id="S7.SS1.2.p1.1.m1.1.2.2.1.cmml" xref="S7.SS1.2.p1.1.m1.1.2.2.2"><abs id="S7.SS1.2.p1.1.m1.1.2.2.1.1.cmml" xref="S7.SS1.2.p1.1.m1.1.2.2.2.1"></abs><ci id="S7.SS1.2.p1.1.m1.1.1.cmml" xref="S7.SS1.2.p1.1.m1.1.1">𝐵</ci></apply><ci id="S7.SS1.2.p1.1.m1.1.2.3.cmml" xref="S7.SS1.2.p1.1.m1.1.2.3">𝔠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.2.p1.1.m1.1c">|B|=\mathfrak{c}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.2.p1.1.m1.1d">| italic_B | = fraktur_c</annotation></semantics></math> and that <math alttext="B\preceq A(X)" class="ltx_Math" display="inline" id="S7.SS1.2.p1.2.m2.1"><semantics id="S7.SS1.2.p1.2.m2.1a"><mrow id="S7.SS1.2.p1.2.m2.1.2" xref="S7.SS1.2.p1.2.m2.1.2.cmml"><mi id="S7.SS1.2.p1.2.m2.1.2.2" xref="S7.SS1.2.p1.2.m2.1.2.2.cmml">B</mi><mo id="S7.SS1.2.p1.2.m2.1.2.1" xref="S7.SS1.2.p1.2.m2.1.2.1.cmml">⪯</mo><mrow id="S7.SS1.2.p1.2.m2.1.2.3" xref="S7.SS1.2.p1.2.m2.1.2.3.cmml"><mi id="S7.SS1.2.p1.2.m2.1.2.3.2" xref="S7.SS1.2.p1.2.m2.1.2.3.2.cmml">A</mi><mo id="S7.SS1.2.p1.2.m2.1.2.3.1" xref="S7.SS1.2.p1.2.m2.1.2.3.1.cmml">⁢</mo><mrow id="S7.SS1.2.p1.2.m2.1.2.3.3.2" xref="S7.SS1.2.p1.2.m2.1.2.3.cmml"><mo id="S7.SS1.2.p1.2.m2.1.2.3.3.2.1" stretchy="false" xref="S7.SS1.2.p1.2.m2.1.2.3.cmml">(</mo><mi id="S7.SS1.2.p1.2.m2.1.1" xref="S7.SS1.2.p1.2.m2.1.1.cmml">X</mi><mo id="S7.SS1.2.p1.2.m2.1.2.3.3.2.2" stretchy="false" xref="S7.SS1.2.p1.2.m2.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.2.p1.2.m2.1b"><apply id="S7.SS1.2.p1.2.m2.1.2.cmml" xref="S7.SS1.2.p1.2.m2.1.2"><csymbol cd="latexml" id="S7.SS1.2.p1.2.m2.1.2.1.cmml" xref="S7.SS1.2.p1.2.m2.1.2.1">precedes-or-equals</csymbol><ci id="S7.SS1.2.p1.2.m2.1.2.2.cmml" xref="S7.SS1.2.p1.2.m2.1.2.2">𝐵</ci><apply id="S7.SS1.2.p1.2.m2.1.2.3.cmml" xref="S7.SS1.2.p1.2.m2.1.2.3"><times id="S7.SS1.2.p1.2.m2.1.2.3.1.cmml" xref="S7.SS1.2.p1.2.m2.1.2.3.1"></times><ci id="S7.SS1.2.p1.2.m2.1.2.3.2.cmml" xref="S7.SS1.2.p1.2.m2.1.2.3.2">𝐴</ci><ci id="S7.SS1.2.p1.2.m2.1.1.cmml" xref="S7.SS1.2.p1.2.m2.1.1">𝑋</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.2.p1.2.m2.1c">B\preceq A(X)</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.2.p1.2.m2.1d">italic_B ⪯ italic_A ( italic_X )</annotation></semantics></math>, we may assume in fact that <math alttext="B\subseteq A(X)" class="ltx_Math" display="inline" id="S7.SS1.2.p1.3.m3.1"><semantics id="S7.SS1.2.p1.3.m3.1a"><mrow id="S7.SS1.2.p1.3.m3.1.2" xref="S7.SS1.2.p1.3.m3.1.2.cmml"><mi id="S7.SS1.2.p1.3.m3.1.2.2" xref="S7.SS1.2.p1.3.m3.1.2.2.cmml">B</mi><mo id="S7.SS1.2.p1.3.m3.1.2.1" xref="S7.SS1.2.p1.3.m3.1.2.1.cmml">⊆</mo><mrow id="S7.SS1.2.p1.3.m3.1.2.3" xref="S7.SS1.2.p1.3.m3.1.2.3.cmml"><mi id="S7.SS1.2.p1.3.m3.1.2.3.2" xref="S7.SS1.2.p1.3.m3.1.2.3.2.cmml">A</mi><mo id="S7.SS1.2.p1.3.m3.1.2.3.1" xref="S7.SS1.2.p1.3.m3.1.2.3.1.cmml">⁢</mo><mrow id="S7.SS1.2.p1.3.m3.1.2.3.3.2" xref="S7.SS1.2.p1.3.m3.1.2.3.cmml"><mo id="S7.SS1.2.p1.3.m3.1.2.3.3.2.1" stretchy="false" xref="S7.SS1.2.p1.3.m3.1.2.3.cmml">(</mo><mi id="S7.SS1.2.p1.3.m3.1.1" xref="S7.SS1.2.p1.3.m3.1.1.cmml">X</mi><mo id="S7.SS1.2.p1.3.m3.1.2.3.3.2.2" stretchy="false" xref="S7.SS1.2.p1.3.m3.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.2.p1.3.m3.1b"><apply id="S7.SS1.2.p1.3.m3.1.2.cmml" xref="S7.SS1.2.p1.3.m3.1.2"><subset id="S7.SS1.2.p1.3.m3.1.2.1.cmml" xref="S7.SS1.2.p1.3.m3.1.2.1"></subset><ci id="S7.SS1.2.p1.3.m3.1.2.2.cmml" xref="S7.SS1.2.p1.3.m3.1.2.2">𝐵</ci><apply id="S7.SS1.2.p1.3.m3.1.2.3.cmml" xref="S7.SS1.2.p1.3.m3.1.2.3"><times id="S7.SS1.2.p1.3.m3.1.2.3.1.cmml" xref="S7.SS1.2.p1.3.m3.1.2.3.1"></times><ci id="S7.SS1.2.p1.3.m3.1.2.3.2.cmml" xref="S7.SS1.2.p1.3.m3.1.2.3.2">𝐴</ci><ci id="S7.SS1.2.p1.3.m3.1.1.cmml" xref="S7.SS1.2.p1.3.m3.1.1">𝑋</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.2.p1.3.m3.1c">B\subseteq A(X)</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.2.p1.3.m3.1d">italic_B ⊆ italic_A ( italic_X )</annotation></semantics></math>. Since for each <math alttext="\alpha" class="ltx_Math" display="inline" id="S7.SS1.2.p1.4.m4.1"><semantics id="S7.SS1.2.p1.4.m4.1a"><mi id="S7.SS1.2.p1.4.m4.1.1" xref="S7.SS1.2.p1.4.m4.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.2.p1.4.m4.1b"><ci id="S7.SS1.2.p1.4.m4.1.1.cmml" xref="S7.SS1.2.p1.4.m4.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.2.p1.4.m4.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.2.p1.4.m4.1d">italic_α</annotation></semantics></math>, <math alttext="|A_{\alpha}|<\max(\aleph_{0},|\alpha|)" class="ltx_Math" display="inline" id="S7.SS1.2.p1.5.m5.5"><semantics id="S7.SS1.2.p1.5.m5.5a"><mrow id="S7.SS1.2.p1.5.m5.5.5" xref="S7.SS1.2.p1.5.m5.5.5.cmml"><mrow id="S7.SS1.2.p1.5.m5.3.3.1.1" xref="S7.SS1.2.p1.5.m5.3.3.1.2.cmml"><mo id="S7.SS1.2.p1.5.m5.3.3.1.1.2" stretchy="false" xref="S7.SS1.2.p1.5.m5.3.3.1.2.1.cmml">|</mo><msub id="S7.SS1.2.p1.5.m5.3.3.1.1.1" xref="S7.SS1.2.p1.5.m5.3.3.1.1.1.cmml"><mi id="S7.SS1.2.p1.5.m5.3.3.1.1.1.2" xref="S7.SS1.2.p1.5.m5.3.3.1.1.1.2.cmml">A</mi><mi id="S7.SS1.2.p1.5.m5.3.3.1.1.1.3" xref="S7.SS1.2.p1.5.m5.3.3.1.1.1.3.cmml">α</mi></msub><mo id="S7.SS1.2.p1.5.m5.3.3.1.1.3" stretchy="false" xref="S7.SS1.2.p1.5.m5.3.3.1.2.1.cmml">|</mo></mrow><mo id="S7.SS1.2.p1.5.m5.5.5.4" xref="S7.SS1.2.p1.5.m5.5.5.4.cmml"><</mo><mrow id="S7.SS1.2.p1.5.m5.5.5.3.2" xref="S7.SS1.2.p1.5.m5.5.5.3.3.cmml"><mi id="S7.SS1.2.p1.5.m5.2.2" xref="S7.SS1.2.p1.5.m5.2.2.cmml">max</mi><mo id="S7.SS1.2.p1.5.m5.5.5.3.2a" xref="S7.SS1.2.p1.5.m5.5.5.3.3.cmml">⁡</mo><mrow id="S7.SS1.2.p1.5.m5.5.5.3.2.2" xref="S7.SS1.2.p1.5.m5.5.5.3.3.cmml"><mo id="S7.SS1.2.p1.5.m5.5.5.3.2.2.3" stretchy="false" xref="S7.SS1.2.p1.5.m5.5.5.3.3.cmml">(</mo><msub id="S7.SS1.2.p1.5.m5.4.4.2.1.1.1" xref="S7.SS1.2.p1.5.m5.4.4.2.1.1.1.cmml"><mi id="S7.SS1.2.p1.5.m5.4.4.2.1.1.1.2" mathvariant="normal" xref="S7.SS1.2.p1.5.m5.4.4.2.1.1.1.2.cmml">ℵ</mi><mn id="S7.SS1.2.p1.5.m5.4.4.2.1.1.1.3" xref="S7.SS1.2.p1.5.m5.4.4.2.1.1.1.3.cmml">0</mn></msub><mo id="S7.SS1.2.p1.5.m5.5.5.3.2.2.4" xref="S7.SS1.2.p1.5.m5.5.5.3.3.cmml">,</mo><mrow id="S7.SS1.2.p1.5.m5.5.5.3.2.2.2.2" xref="S7.SS1.2.p1.5.m5.5.5.3.2.2.2.1.cmml"><mo id="S7.SS1.2.p1.5.m5.5.5.3.2.2.2.2.1" stretchy="false" xref="S7.SS1.2.p1.5.m5.5.5.3.2.2.2.1.1.cmml">|</mo><mi id="S7.SS1.2.p1.5.m5.1.1" xref="S7.SS1.2.p1.5.m5.1.1.cmml">α</mi><mo id="S7.SS1.2.p1.5.m5.5.5.3.2.2.2.2.2" stretchy="false" xref="S7.SS1.2.p1.5.m5.5.5.3.2.2.2.1.1.cmml">|</mo></mrow><mo id="S7.SS1.2.p1.5.m5.5.5.3.2.2.5" stretchy="false" xref="S7.SS1.2.p1.5.m5.5.5.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.2.p1.5.m5.5b"><apply id="S7.SS1.2.p1.5.m5.5.5.cmml" xref="S7.SS1.2.p1.5.m5.5.5"><lt id="S7.SS1.2.p1.5.m5.5.5.4.cmml" xref="S7.SS1.2.p1.5.m5.5.5.4"></lt><apply id="S7.SS1.2.p1.5.m5.3.3.1.2.cmml" xref="S7.SS1.2.p1.5.m5.3.3.1.1"><abs id="S7.SS1.2.p1.5.m5.3.3.1.2.1.cmml" xref="S7.SS1.2.p1.5.m5.3.3.1.1.2"></abs><apply id="S7.SS1.2.p1.5.m5.3.3.1.1.1.cmml" xref="S7.SS1.2.p1.5.m5.3.3.1.1.1"><csymbol cd="ambiguous" id="S7.SS1.2.p1.5.m5.3.3.1.1.1.1.cmml" xref="S7.SS1.2.p1.5.m5.3.3.1.1.1">subscript</csymbol><ci id="S7.SS1.2.p1.5.m5.3.3.1.1.1.2.cmml" xref="S7.SS1.2.p1.5.m5.3.3.1.1.1.2">𝐴</ci><ci id="S7.SS1.2.p1.5.m5.3.3.1.1.1.3.cmml" xref="S7.SS1.2.p1.5.m5.3.3.1.1.1.3">𝛼</ci></apply></apply><apply id="S7.SS1.2.p1.5.m5.5.5.3.3.cmml" xref="S7.SS1.2.p1.5.m5.5.5.3.2"><max id="S7.SS1.2.p1.5.m5.2.2.cmml" xref="S7.SS1.2.p1.5.m5.2.2"></max><apply id="S7.SS1.2.p1.5.m5.4.4.2.1.1.1.cmml" xref="S7.SS1.2.p1.5.m5.4.4.2.1.1.1"><csymbol cd="ambiguous" id="S7.SS1.2.p1.5.m5.4.4.2.1.1.1.1.cmml" xref="S7.SS1.2.p1.5.m5.4.4.2.1.1.1">subscript</csymbol><ci id="S7.SS1.2.p1.5.m5.4.4.2.1.1.1.2.cmml" xref="S7.SS1.2.p1.5.m5.4.4.2.1.1.1.2">ℵ</ci><cn id="S7.SS1.2.p1.5.m5.4.4.2.1.1.1.3.cmml" type="integer" xref="S7.SS1.2.p1.5.m5.4.4.2.1.1.1.3">0</cn></apply><apply id="S7.SS1.2.p1.5.m5.5.5.3.2.2.2.1.cmml" xref="S7.SS1.2.p1.5.m5.5.5.3.2.2.2.2"><abs id="S7.SS1.2.p1.5.m5.5.5.3.2.2.2.1.1.cmml" xref="S7.SS1.2.p1.5.m5.5.5.3.2.2.2.2.1"></abs><ci id="S7.SS1.2.p1.5.m5.1.1.cmml" xref="S7.SS1.2.p1.5.m5.1.1">𝛼</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.2.p1.5.m5.5c">|A_{\alpha}|<\max(\aleph_{0},|\alpha|)</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.2.p1.5.m5.5d">| italic_A start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT | < roman_max ( roman_ℵ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , | italic_α | )</annotation></semantics></math>, we conclude that <math alttext="X^{\prime}:=\{\alpha<\mathfrak{c}:B\cap A_{\alpha}\neq\varnothing\}" class="ltx_Math" display="inline" id="S7.SS1.2.p1.6.m6.2"><semantics id="S7.SS1.2.p1.6.m6.2a"><mrow id="S7.SS1.2.p1.6.m6.2.2" xref="S7.SS1.2.p1.6.m6.2.2.cmml"><msup id="S7.SS1.2.p1.6.m6.2.2.4" xref="S7.SS1.2.p1.6.m6.2.2.4.cmml"><mi id="S7.SS1.2.p1.6.m6.2.2.4.2" xref="S7.SS1.2.p1.6.m6.2.2.4.2.cmml">X</mi><mo id="S7.SS1.2.p1.6.m6.2.2.4.3" xref="S7.SS1.2.p1.6.m6.2.2.4.3.cmml">′</mo></msup><mo id="S7.SS1.2.p1.6.m6.2.2.3" lspace="0.278em" rspace="0.278em" xref="S7.SS1.2.p1.6.m6.2.2.3.cmml">:=</mo><mrow id="S7.SS1.2.p1.6.m6.2.2.2.2" xref="S7.SS1.2.p1.6.m6.2.2.2.3.cmml"><mo id="S7.SS1.2.p1.6.m6.2.2.2.2.3" stretchy="false" xref="S7.SS1.2.p1.6.m6.2.2.2.3.1.cmml">{</mo><mrow id="S7.SS1.2.p1.6.m6.1.1.1.1.1" xref="S7.SS1.2.p1.6.m6.1.1.1.1.1.cmml"><mi id="S7.SS1.2.p1.6.m6.1.1.1.1.1.2" xref="S7.SS1.2.p1.6.m6.1.1.1.1.1.2.cmml">α</mi><mo id="S7.SS1.2.p1.6.m6.1.1.1.1.1.1" xref="S7.SS1.2.p1.6.m6.1.1.1.1.1.1.cmml"><</mo><mi id="S7.SS1.2.p1.6.m6.1.1.1.1.1.3" xref="S7.SS1.2.p1.6.m6.1.1.1.1.1.3.cmml">𝔠</mi></mrow><mo id="S7.SS1.2.p1.6.m6.2.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S7.SS1.2.p1.6.m6.2.2.2.3.1.cmml">:</mo><mrow id="S7.SS1.2.p1.6.m6.2.2.2.2.2" xref="S7.SS1.2.p1.6.m6.2.2.2.2.2.cmml"><mrow id="S7.SS1.2.p1.6.m6.2.2.2.2.2.2" xref="S7.SS1.2.p1.6.m6.2.2.2.2.2.2.cmml"><mi id="S7.SS1.2.p1.6.m6.2.2.2.2.2.2.2" xref="S7.SS1.2.p1.6.m6.2.2.2.2.2.2.2.cmml">B</mi><mo id="S7.SS1.2.p1.6.m6.2.2.2.2.2.2.1" xref="S7.SS1.2.p1.6.m6.2.2.2.2.2.2.1.cmml">∩</mo><msub id="S7.SS1.2.p1.6.m6.2.2.2.2.2.2.3" xref="S7.SS1.2.p1.6.m6.2.2.2.2.2.2.3.cmml"><mi id="S7.SS1.2.p1.6.m6.2.2.2.2.2.2.3.2" xref="S7.SS1.2.p1.6.m6.2.2.2.2.2.2.3.2.cmml">A</mi><mi id="S7.SS1.2.p1.6.m6.2.2.2.2.2.2.3.3" xref="S7.SS1.2.p1.6.m6.2.2.2.2.2.2.3.3.cmml">α</mi></msub></mrow><mo id="S7.SS1.2.p1.6.m6.2.2.2.2.2.1" xref="S7.SS1.2.p1.6.m6.2.2.2.2.2.1.cmml">≠</mo><mi id="S7.SS1.2.p1.6.m6.2.2.2.2.2.3" mathvariant="normal" xref="S7.SS1.2.p1.6.m6.2.2.2.2.2.3.cmml">∅</mi></mrow><mo id="S7.SS1.2.p1.6.m6.2.2.2.2.5" stretchy="false" xref="S7.SS1.2.p1.6.m6.2.2.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.2.p1.6.m6.2b"><apply id="S7.SS1.2.p1.6.m6.2.2.cmml" xref="S7.SS1.2.p1.6.m6.2.2"><csymbol cd="latexml" id="S7.SS1.2.p1.6.m6.2.2.3.cmml" xref="S7.SS1.2.p1.6.m6.2.2.3">assign</csymbol><apply id="S7.SS1.2.p1.6.m6.2.2.4.cmml" xref="S7.SS1.2.p1.6.m6.2.2.4"><csymbol cd="ambiguous" id="S7.SS1.2.p1.6.m6.2.2.4.1.cmml" xref="S7.SS1.2.p1.6.m6.2.2.4">superscript</csymbol><ci id="S7.SS1.2.p1.6.m6.2.2.4.2.cmml" xref="S7.SS1.2.p1.6.m6.2.2.4.2">𝑋</ci><ci id="S7.SS1.2.p1.6.m6.2.2.4.3.cmml" xref="S7.SS1.2.p1.6.m6.2.2.4.3">′</ci></apply><apply id="S7.SS1.2.p1.6.m6.2.2.2.3.cmml" xref="S7.SS1.2.p1.6.m6.2.2.2.2"><csymbol cd="latexml" id="S7.SS1.2.p1.6.m6.2.2.2.3.1.cmml" xref="S7.SS1.2.p1.6.m6.2.2.2.2.3">conditional-set</csymbol><apply id="S7.SS1.2.p1.6.m6.1.1.1.1.1.cmml" xref="S7.SS1.2.p1.6.m6.1.1.1.1.1"><lt id="S7.SS1.2.p1.6.m6.1.1.1.1.1.1.cmml" xref="S7.SS1.2.p1.6.m6.1.1.1.1.1.1"></lt><ci id="S7.SS1.2.p1.6.m6.1.1.1.1.1.2.cmml" xref="S7.SS1.2.p1.6.m6.1.1.1.1.1.2">𝛼</ci><ci id="S7.SS1.2.p1.6.m6.1.1.1.1.1.3.cmml" xref="S7.SS1.2.p1.6.m6.1.1.1.1.1.3">𝔠</ci></apply><apply id="S7.SS1.2.p1.6.m6.2.2.2.2.2.cmml" xref="S7.SS1.2.p1.6.m6.2.2.2.2.2"><neq id="S7.SS1.2.p1.6.m6.2.2.2.2.2.1.cmml" xref="S7.SS1.2.p1.6.m6.2.2.2.2.2.1"></neq><apply id="S7.SS1.2.p1.6.m6.2.2.2.2.2.2.cmml" xref="S7.SS1.2.p1.6.m6.2.2.2.2.2.2"><intersect id="S7.SS1.2.p1.6.m6.2.2.2.2.2.2.1.cmml" xref="S7.SS1.2.p1.6.m6.2.2.2.2.2.2.1"></intersect><ci id="S7.SS1.2.p1.6.m6.2.2.2.2.2.2.2.cmml" xref="S7.SS1.2.p1.6.m6.2.2.2.2.2.2.2">𝐵</ci><apply id="S7.SS1.2.p1.6.m6.2.2.2.2.2.2.3.cmml" xref="S7.SS1.2.p1.6.m6.2.2.2.2.2.2.3"><csymbol cd="ambiguous" id="S7.SS1.2.p1.6.m6.2.2.2.2.2.2.3.1.cmml" xref="S7.SS1.2.p1.6.m6.2.2.2.2.2.2.3">subscript</csymbol><ci id="S7.SS1.2.p1.6.m6.2.2.2.2.2.2.3.2.cmml" xref="S7.SS1.2.p1.6.m6.2.2.2.2.2.2.3.2">𝐴</ci><ci id="S7.SS1.2.p1.6.m6.2.2.2.2.2.2.3.3.cmml" xref="S7.SS1.2.p1.6.m6.2.2.2.2.2.2.3.3">𝛼</ci></apply></apply><emptyset id="S7.SS1.2.p1.6.m6.2.2.2.2.2.3.cmml" xref="S7.SS1.2.p1.6.m6.2.2.2.2.2.3"></emptyset></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.2.p1.6.m6.2c">X^{\prime}:=\{\alpha<\mathfrak{c}:B\cap A_{\alpha}\neq\varnothing\}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.2.p1.6.m6.2d">italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT := { italic_α < fraktur_c : italic_B ∩ italic_A start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ≠ ∅ }</annotation></semantics></math> is cofinal. Since <math alttext="\sup(X\cap Y)<\mathfrak{c}" class="ltx_Math" display="inline" id="S7.SS1.2.p1.7.m7.1"><semantics id="S7.SS1.2.p1.7.m7.1a"><mrow id="S7.SS1.2.p1.7.m7.1.1" xref="S7.SS1.2.p1.7.m7.1.1.cmml"><mrow id="S7.SS1.2.p1.7.m7.1.1.1" xref="S7.SS1.2.p1.7.m7.1.1.1.cmml"><mo id="S7.SS1.2.p1.7.m7.1.1.1.2" rspace="0em" xref="S7.SS1.2.p1.7.m7.1.1.1.2.cmml">sup</mo><mrow id="S7.SS1.2.p1.7.m7.1.1.1.1.1" xref="S7.SS1.2.p1.7.m7.1.1.1.1.1.1.cmml"><mo id="S7.SS1.2.p1.7.m7.1.1.1.1.1.2" stretchy="false" xref="S7.SS1.2.p1.7.m7.1.1.1.1.1.1.cmml">(</mo><mrow id="S7.SS1.2.p1.7.m7.1.1.1.1.1.1" xref="S7.SS1.2.p1.7.m7.1.1.1.1.1.1.cmml"><mi id="S7.SS1.2.p1.7.m7.1.1.1.1.1.1.2" xref="S7.SS1.2.p1.7.m7.1.1.1.1.1.1.2.cmml">X</mi><mo id="S7.SS1.2.p1.7.m7.1.1.1.1.1.1.1" xref="S7.SS1.2.p1.7.m7.1.1.1.1.1.1.1.cmml">∩</mo><mi id="S7.SS1.2.p1.7.m7.1.1.1.1.1.1.3" xref="S7.SS1.2.p1.7.m7.1.1.1.1.1.1.3.cmml">Y</mi></mrow><mo id="S7.SS1.2.p1.7.m7.1.1.1.1.1.3" stretchy="false" xref="S7.SS1.2.p1.7.m7.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S7.SS1.2.p1.7.m7.1.1.2" xref="S7.SS1.2.p1.7.m7.1.1.2.cmml"><</mo><mi id="S7.SS1.2.p1.7.m7.1.1.3" xref="S7.SS1.2.p1.7.m7.1.1.3.cmml">𝔠</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.2.p1.7.m7.1b"><apply id="S7.SS1.2.p1.7.m7.1.1.cmml" xref="S7.SS1.2.p1.7.m7.1.1"><lt id="S7.SS1.2.p1.7.m7.1.1.2.cmml" xref="S7.SS1.2.p1.7.m7.1.1.2"></lt><apply id="S7.SS1.2.p1.7.m7.1.1.1.cmml" xref="S7.SS1.2.p1.7.m7.1.1.1"><csymbol cd="latexml" id="S7.SS1.2.p1.7.m7.1.1.1.2.cmml" xref="S7.SS1.2.p1.7.m7.1.1.1.2">supremum</csymbol><apply id="S7.SS1.2.p1.7.m7.1.1.1.1.1.1.cmml" xref="S7.SS1.2.p1.7.m7.1.1.1.1.1"><intersect id="S7.SS1.2.p1.7.m7.1.1.1.1.1.1.1.cmml" xref="S7.SS1.2.p1.7.m7.1.1.1.1.1.1.1"></intersect><ci id="S7.SS1.2.p1.7.m7.1.1.1.1.1.1.2.cmml" xref="S7.SS1.2.p1.7.m7.1.1.1.1.1.1.2">𝑋</ci><ci id="S7.SS1.2.p1.7.m7.1.1.1.1.1.1.3.cmml" xref="S7.SS1.2.p1.7.m7.1.1.1.1.1.1.3">𝑌</ci></apply></apply><ci id="S7.SS1.2.p1.7.m7.1.1.3.cmml" xref="S7.SS1.2.p1.7.m7.1.1.3">𝔠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.2.p1.7.m7.1c">\sup(X\cap Y)<\mathfrak{c}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.2.p1.7.m7.1d">roman_sup ( italic_X ∩ italic_Y ) < fraktur_c</annotation></semantics></math>, <math alttext="X^{\prime}\setminus Y" class="ltx_Math" display="inline" id="S7.SS1.2.p1.8.m8.1"><semantics id="S7.SS1.2.p1.8.m8.1a"><mrow id="S7.SS1.2.p1.8.m8.1.1" xref="S7.SS1.2.p1.8.m8.1.1.cmml"><msup id="S7.SS1.2.p1.8.m8.1.1.2" xref="S7.SS1.2.p1.8.m8.1.1.2.cmml"><mi id="S7.SS1.2.p1.8.m8.1.1.2.2" xref="S7.SS1.2.p1.8.m8.1.1.2.2.cmml">X</mi><mo id="S7.SS1.2.p1.8.m8.1.1.2.3" xref="S7.SS1.2.p1.8.m8.1.1.2.3.cmml">′</mo></msup><mo id="S7.SS1.2.p1.8.m8.1.1.1" xref="S7.SS1.2.p1.8.m8.1.1.1.cmml">∖</mo><mi id="S7.SS1.2.p1.8.m8.1.1.3" xref="S7.SS1.2.p1.8.m8.1.1.3.cmml">Y</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.2.p1.8.m8.1b"><apply id="S7.SS1.2.p1.8.m8.1.1.cmml" xref="S7.SS1.2.p1.8.m8.1.1"><setdiff id="S7.SS1.2.p1.8.m8.1.1.1.cmml" xref="S7.SS1.2.p1.8.m8.1.1.1"></setdiff><apply id="S7.SS1.2.p1.8.m8.1.1.2.cmml" xref="S7.SS1.2.p1.8.m8.1.1.2"><csymbol cd="ambiguous" id="S7.SS1.2.p1.8.m8.1.1.2.1.cmml" xref="S7.SS1.2.p1.8.m8.1.1.2">superscript</csymbol><ci id="S7.SS1.2.p1.8.m8.1.1.2.2.cmml" xref="S7.SS1.2.p1.8.m8.1.1.2.2">𝑋</ci><ci id="S7.SS1.2.p1.8.m8.1.1.2.3.cmml" xref="S7.SS1.2.p1.8.m8.1.1.2.3">′</ci></apply><ci id="S7.SS1.2.p1.8.m8.1.1.3.cmml" xref="S7.SS1.2.p1.8.m8.1.1.3">𝑌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.2.p1.8.m8.1c">X^{\prime}\setminus Y</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.2.p1.8.m8.1d">italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∖ italic_Y</annotation></semantics></math> is also cofinal, so by the previous lemma, <math alttext="B\npreceq A(Y)" class="ltx_Math" display="inline" id="S7.SS1.2.p1.9.m9.1"><semantics id="S7.SS1.2.p1.9.m9.1a"><mrow id="S7.SS1.2.p1.9.m9.1.2" xref="S7.SS1.2.p1.9.m9.1.2.cmml"><mi id="S7.SS1.2.p1.9.m9.1.2.2" xref="S7.SS1.2.p1.9.m9.1.2.2.cmml">B</mi><mo id="S7.SS1.2.p1.9.m9.1.2.1" xref="S7.SS1.2.p1.9.m9.1.2.1.cmml">⋠</mo><mrow id="S7.SS1.2.p1.9.m9.1.2.3" xref="S7.SS1.2.p1.9.m9.1.2.3.cmml"><mi id="S7.SS1.2.p1.9.m9.1.2.3.2" xref="S7.SS1.2.p1.9.m9.1.2.3.2.cmml">A</mi><mo id="S7.SS1.2.p1.9.m9.1.2.3.1" xref="S7.SS1.2.p1.9.m9.1.2.3.1.cmml">⁢</mo><mrow id="S7.SS1.2.p1.9.m9.1.2.3.3.2" xref="S7.SS1.2.p1.9.m9.1.2.3.cmml"><mo id="S7.SS1.2.p1.9.m9.1.2.3.3.2.1" stretchy="false" xref="S7.SS1.2.p1.9.m9.1.2.3.cmml">(</mo><mi id="S7.SS1.2.p1.9.m9.1.1" xref="S7.SS1.2.p1.9.m9.1.1.cmml">Y</mi><mo id="S7.SS1.2.p1.9.m9.1.2.3.3.2.2" stretchy="false" xref="S7.SS1.2.p1.9.m9.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.2.p1.9.m9.1b"><apply id="S7.SS1.2.p1.9.m9.1.2.cmml" xref="S7.SS1.2.p1.9.m9.1.2"><csymbol cd="latexml" id="S7.SS1.2.p1.9.m9.1.2.1.cmml" xref="S7.SS1.2.p1.9.m9.1.2.1">not-precedes-nor-equals</csymbol><ci id="S7.SS1.2.p1.9.m9.1.2.2.cmml" xref="S7.SS1.2.p1.9.m9.1.2.2">𝐵</ci><apply id="S7.SS1.2.p1.9.m9.1.2.3.cmml" xref="S7.SS1.2.p1.9.m9.1.2.3"><times id="S7.SS1.2.p1.9.m9.1.2.3.1.cmml" xref="S7.SS1.2.p1.9.m9.1.2.3.1"></times><ci id="S7.SS1.2.p1.9.m9.1.2.3.2.cmml" xref="S7.SS1.2.p1.9.m9.1.2.3.2">𝐴</ci><ci id="S7.SS1.2.p1.9.m9.1.1.cmml" xref="S7.SS1.2.p1.9.m9.1.1">𝑌</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.2.p1.9.m9.1c">B\npreceq A(Y)</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.2.p1.9.m9.1d">italic_B ⋠ italic_A ( italic_Y )</annotation></semantics></math>. ∎</p> </div> </div> <div class="ltx_para" id="S7.SS1.p7"> <p class="ltx_p" id="S7.SS1.p7.8">Note that this already implies <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S7.Thmtheorem3" title="Theorem 7.3. ‣ 7.1. On a basis for all uncountable linear orders ‣ 7. A two element basis for the Aronszajn lines ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">7.3</span></a> under <math alttext="\mathsf{CH}" class="ltx_Math" display="inline" id="S7.SS1.p7.1.m1.1"><semantics id="S7.SS1.p7.1.m1.1a"><mi id="S7.SS1.p7.1.m1.1.1" xref="S7.SS1.p7.1.m1.1.1.cmml">𝖢𝖧</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.p7.1.m1.1b"><ci id="S7.SS1.p7.1.m1.1.1.cmml" xref="S7.SS1.p7.1.m1.1.1">𝖢𝖧</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p7.1.m1.1c">\mathsf{CH}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p7.1.m1.1d">sansserif_CH</annotation></semantics></math>, since any <math alttext="\trianglelefteq" class="ltx_Math" display="inline" id="S7.SS1.p7.2.m2.1"><semantics id="S7.SS1.p7.2.m2.1a"><mi id="S7.SS1.p7.2.m2.1.1" mathvariant="normal" xref="S7.SS1.p7.2.m2.1.1.cmml">⊴</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.p7.2.m2.1b"><ci id="S7.SS1.p7.2.m2.1.1.cmml" xref="S7.SS1.p7.2.m2.1.1">⊴</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p7.2.m2.1c">\trianglelefteq</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p7.2.m2.1d">⊴</annotation></semantics></math>-basis is an <math alttext="\preceq" class="ltx_Math" display="inline" id="S7.SS1.p7.3.m3.1"><semantics id="S7.SS1.p7.3.m3.1a"><mo id="S7.SS1.p7.3.m3.1.1" xref="S7.SS1.p7.3.m3.1.1.cmml">⪯</mo><annotation-xml encoding="MathML-Content" id="S7.SS1.p7.3.m3.1b"><csymbol cd="latexml" id="S7.SS1.p7.3.m3.1.1.cmml" xref="S7.SS1.p7.3.m3.1.1">precedes-or-equals</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p7.3.m3.1c">\preceq</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p7.3.m3.1d">⪯</annotation></semantics></math>-basis. What we show now, is that for a particular way of choosing the <math alttext="A_{\alpha}" class="ltx_Math" display="inline" id="S7.SS1.p7.4.m4.1"><semantics id="S7.SS1.p7.4.m4.1a"><msub id="S7.SS1.p7.4.m4.1.1" xref="S7.SS1.p7.4.m4.1.1.cmml"><mi id="S7.SS1.p7.4.m4.1.1.2" xref="S7.SS1.p7.4.m4.1.1.2.cmml">A</mi><mi id="S7.SS1.p7.4.m4.1.1.3" xref="S7.SS1.p7.4.m4.1.1.3.cmml">α</mi></msub><annotation-xml encoding="MathML-Content" id="S7.SS1.p7.4.m4.1b"><apply id="S7.SS1.p7.4.m4.1.1.cmml" xref="S7.SS1.p7.4.m4.1.1"><csymbol cd="ambiguous" id="S7.SS1.p7.4.m4.1.1.1.cmml" xref="S7.SS1.p7.4.m4.1.1">subscript</csymbol><ci id="S7.SS1.p7.4.m4.1.1.2.cmml" xref="S7.SS1.p7.4.m4.1.1.2">𝐴</ci><ci id="S7.SS1.p7.4.m4.1.1.3.cmml" xref="S7.SS1.p7.4.m4.1.1.3">𝛼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p7.4.m4.1c">A_{\alpha}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p7.4.m4.1d">italic_A start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT</annotation></semantics></math> we can achieve the following property: for <math alttext="X\subseteq\mathfrak{c}" class="ltx_Math" display="inline" id="S7.SS1.p7.5.m5.1"><semantics id="S7.SS1.p7.5.m5.1a"><mrow id="S7.SS1.p7.5.m5.1.1" xref="S7.SS1.p7.5.m5.1.1.cmml"><mi id="S7.SS1.p7.5.m5.1.1.2" xref="S7.SS1.p7.5.m5.1.1.2.cmml">X</mi><mo id="S7.SS1.p7.5.m5.1.1.1" xref="S7.SS1.p7.5.m5.1.1.1.cmml">⊆</mo><mi id="S7.SS1.p7.5.m5.1.1.3" xref="S7.SS1.p7.5.m5.1.1.3.cmml">𝔠</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.p7.5.m5.1b"><apply id="S7.SS1.p7.5.m5.1.1.cmml" xref="S7.SS1.p7.5.m5.1.1"><subset id="S7.SS1.p7.5.m5.1.1.1.cmml" xref="S7.SS1.p7.5.m5.1.1.1"></subset><ci id="S7.SS1.p7.5.m5.1.1.2.cmml" xref="S7.SS1.p7.5.m5.1.1.2">𝑋</ci><ci id="S7.SS1.p7.5.m5.1.1.3.cmml" xref="S7.SS1.p7.5.m5.1.1.3">𝔠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p7.5.m5.1c">X\subseteq\mathfrak{c}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p7.5.m5.1d">italic_X ⊆ fraktur_c</annotation></semantics></math> and <math alttext="B\subseteq\mathbb{R}" class="ltx_Math" display="inline" id="S7.SS1.p7.6.m6.1"><semantics id="S7.SS1.p7.6.m6.1a"><mrow id="S7.SS1.p7.6.m6.1.1" xref="S7.SS1.p7.6.m6.1.1.cmml"><mi id="S7.SS1.p7.6.m6.1.1.2" xref="S7.SS1.p7.6.m6.1.1.2.cmml">B</mi><mo id="S7.SS1.p7.6.m6.1.1.1" xref="S7.SS1.p7.6.m6.1.1.1.cmml">⊆</mo><mi id="S7.SS1.p7.6.m6.1.1.3" xref="S7.SS1.p7.6.m6.1.1.3.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.p7.6.m6.1b"><apply id="S7.SS1.p7.6.m6.1.1.cmml" xref="S7.SS1.p7.6.m6.1.1"><subset id="S7.SS1.p7.6.m6.1.1.1.cmml" xref="S7.SS1.p7.6.m6.1.1.1"></subset><ci id="S7.SS1.p7.6.m6.1.1.2.cmml" xref="S7.SS1.p7.6.m6.1.1.2">𝐵</ci><ci id="S7.SS1.p7.6.m6.1.1.3.cmml" xref="S7.SS1.p7.6.m6.1.1.3">ℝ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p7.6.m6.1c">B\subseteq\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p7.6.m6.1d">italic_B ⊆ blackboard_R</annotation></semantics></math>, if <math alttext="\aleph_{0}<|B|<\mathfrak{c}=|X|" class="ltx_Math" display="inline" id="S7.SS1.p7.7.m7.2"><semantics id="S7.SS1.p7.7.m7.2a"><mrow id="S7.SS1.p7.7.m7.2.3" xref="S7.SS1.p7.7.m7.2.3.cmml"><msub id="S7.SS1.p7.7.m7.2.3.2" xref="S7.SS1.p7.7.m7.2.3.2.cmml"><mi id="S7.SS1.p7.7.m7.2.3.2.2" mathvariant="normal" xref="S7.SS1.p7.7.m7.2.3.2.2.cmml">ℵ</mi><mn id="S7.SS1.p7.7.m7.2.3.2.3" xref="S7.SS1.p7.7.m7.2.3.2.3.cmml">0</mn></msub><mo id="S7.SS1.p7.7.m7.2.3.3" xref="S7.SS1.p7.7.m7.2.3.3.cmml"><</mo><mrow id="S7.SS1.p7.7.m7.2.3.4.2" xref="S7.SS1.p7.7.m7.2.3.4.1.cmml"><mo id="S7.SS1.p7.7.m7.2.3.4.2.1" stretchy="false" xref="S7.SS1.p7.7.m7.2.3.4.1.1.cmml">|</mo><mi id="S7.SS1.p7.7.m7.1.1" xref="S7.SS1.p7.7.m7.1.1.cmml">B</mi><mo id="S7.SS1.p7.7.m7.2.3.4.2.2" stretchy="false" xref="S7.SS1.p7.7.m7.2.3.4.1.1.cmml">|</mo></mrow><mo id="S7.SS1.p7.7.m7.2.3.5" xref="S7.SS1.p7.7.m7.2.3.5.cmml"><</mo><mi id="S7.SS1.p7.7.m7.2.3.6" xref="S7.SS1.p7.7.m7.2.3.6.cmml">𝔠</mi><mo id="S7.SS1.p7.7.m7.2.3.7" xref="S7.SS1.p7.7.m7.2.3.7.cmml">=</mo><mrow id="S7.SS1.p7.7.m7.2.3.8.2" xref="S7.SS1.p7.7.m7.2.3.8.1.cmml"><mo id="S7.SS1.p7.7.m7.2.3.8.2.1" stretchy="false" xref="S7.SS1.p7.7.m7.2.3.8.1.1.cmml">|</mo><mi id="S7.SS1.p7.7.m7.2.2" xref="S7.SS1.p7.7.m7.2.2.cmml">X</mi><mo id="S7.SS1.p7.7.m7.2.3.8.2.2" stretchy="false" xref="S7.SS1.p7.7.m7.2.3.8.1.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.p7.7.m7.2b"><apply id="S7.SS1.p7.7.m7.2.3.cmml" xref="S7.SS1.p7.7.m7.2.3"><and id="S7.SS1.p7.7.m7.2.3a.cmml" xref="S7.SS1.p7.7.m7.2.3"></and><apply id="S7.SS1.p7.7.m7.2.3b.cmml" xref="S7.SS1.p7.7.m7.2.3"><lt id="S7.SS1.p7.7.m7.2.3.3.cmml" xref="S7.SS1.p7.7.m7.2.3.3"></lt><apply id="S7.SS1.p7.7.m7.2.3.2.cmml" xref="S7.SS1.p7.7.m7.2.3.2"><csymbol cd="ambiguous" id="S7.SS1.p7.7.m7.2.3.2.1.cmml" xref="S7.SS1.p7.7.m7.2.3.2">subscript</csymbol><ci id="S7.SS1.p7.7.m7.2.3.2.2.cmml" xref="S7.SS1.p7.7.m7.2.3.2.2">ℵ</ci><cn id="S7.SS1.p7.7.m7.2.3.2.3.cmml" type="integer" xref="S7.SS1.p7.7.m7.2.3.2.3">0</cn></apply><apply id="S7.SS1.p7.7.m7.2.3.4.1.cmml" xref="S7.SS1.p7.7.m7.2.3.4.2"><abs id="S7.SS1.p7.7.m7.2.3.4.1.1.cmml" xref="S7.SS1.p7.7.m7.2.3.4.2.1"></abs><ci id="S7.SS1.p7.7.m7.1.1.cmml" xref="S7.SS1.p7.7.m7.1.1">𝐵</ci></apply></apply><apply id="S7.SS1.p7.7.m7.2.3c.cmml" xref="S7.SS1.p7.7.m7.2.3"><lt id="S7.SS1.p7.7.m7.2.3.5.cmml" xref="S7.SS1.p7.7.m7.2.3.5"></lt><share href="https://arxiv.org/html/2503.13728v1#S7.SS1.p7.7.m7.2.3.4.cmml" id="S7.SS1.p7.7.m7.2.3d.cmml" xref="S7.SS1.p7.7.m7.2.3"></share><ci id="S7.SS1.p7.7.m7.2.3.6.cmml" xref="S7.SS1.p7.7.m7.2.3.6">𝔠</ci></apply><apply id="S7.SS1.p7.7.m7.2.3e.cmml" xref="S7.SS1.p7.7.m7.2.3"><eq id="S7.SS1.p7.7.m7.2.3.7.cmml" xref="S7.SS1.p7.7.m7.2.3.7"></eq><share href="https://arxiv.org/html/2503.13728v1#S7.SS1.p7.7.m7.2.3.6.cmml" id="S7.SS1.p7.7.m7.2.3f.cmml" xref="S7.SS1.p7.7.m7.2.3"></share><apply id="S7.SS1.p7.7.m7.2.3.8.1.cmml" xref="S7.SS1.p7.7.m7.2.3.8.2"><abs id="S7.SS1.p7.7.m7.2.3.8.1.1.cmml" xref="S7.SS1.p7.7.m7.2.3.8.2.1"></abs><ci id="S7.SS1.p7.7.m7.2.2.cmml" xref="S7.SS1.p7.7.m7.2.2">𝑋</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p7.7.m7.2c">\aleph_{0}<|B|<\mathfrak{c}=|X|</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p7.7.m7.2d">roman_ℵ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT < | italic_B | < fraktur_c = | italic_X |</annotation></semantics></math>, then <math alttext="A(X)\ntrianglerighteq B" class="ltx_Math" display="inline" id="S7.SS1.p7.8.m8.1"><semantics id="S7.SS1.p7.8.m8.1a"><mrow id="S7.SS1.p7.8.m8.1.2" xref="S7.SS1.p7.8.m8.1.2.cmml"><mrow id="S7.SS1.p7.8.m8.1.2.2" xref="S7.SS1.p7.8.m8.1.2.2.cmml"><mi id="S7.SS1.p7.8.m8.1.2.2.2" xref="S7.SS1.p7.8.m8.1.2.2.2.cmml">A</mi><mo id="S7.SS1.p7.8.m8.1.2.2.1" xref="S7.SS1.p7.8.m8.1.2.2.1.cmml">⁢</mo><mrow id="S7.SS1.p7.8.m8.1.2.2.3.2" xref="S7.SS1.p7.8.m8.1.2.2.cmml"><mo id="S7.SS1.p7.8.m8.1.2.2.3.2.1" stretchy="false" xref="S7.SS1.p7.8.m8.1.2.2.cmml">(</mo><mi id="S7.SS1.p7.8.m8.1.1" xref="S7.SS1.p7.8.m8.1.1.cmml">X</mi><mo id="S7.SS1.p7.8.m8.1.2.2.3.2.2" stretchy="false" xref="S7.SS1.p7.8.m8.1.2.2.cmml">)</mo></mrow></mrow><mo id="S7.SS1.p7.8.m8.1.2.1" xref="S7.SS1.p7.8.m8.1.2.1.cmml">⋭</mo><mi id="S7.SS1.p7.8.m8.1.2.3" xref="S7.SS1.p7.8.m8.1.2.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.p7.8.m8.1b"><apply id="S7.SS1.p7.8.m8.1.2.cmml" xref="S7.SS1.p7.8.m8.1.2"><csymbol cd="latexml" id="S7.SS1.p7.8.m8.1.2.1.cmml" xref="S7.SS1.p7.8.m8.1.2.1">not-contains-nor-equals</csymbol><apply id="S7.SS1.p7.8.m8.1.2.2.cmml" xref="S7.SS1.p7.8.m8.1.2.2"><times id="S7.SS1.p7.8.m8.1.2.2.1.cmml" xref="S7.SS1.p7.8.m8.1.2.2.1"></times><ci id="S7.SS1.p7.8.m8.1.2.2.2.cmml" xref="S7.SS1.p7.8.m8.1.2.2.2">𝐴</ci><ci id="S7.SS1.p7.8.m8.1.1.cmml" xref="S7.SS1.p7.8.m8.1.1">𝑋</ci></apply><ci id="S7.SS1.p7.8.m8.1.2.3.cmml" xref="S7.SS1.p7.8.m8.1.2.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p7.8.m8.1c">A(X)\ntrianglerighteq B</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p7.8.m8.1d">italic_A ( italic_X ) ⋭ italic_B</annotation></semantics></math>. Together with the previous lemma, this immediately yields a proof of <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S7.Thmtheorem3" title="Theorem 7.3. ‣ 7.1. On a basis for all uncountable linear orders ‣ 7. A two element basis for the Aronszajn lines ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">7.3</span></a>.</p> </div> <div class="ltx_para" id="S7.SS1.p8"> <p class="ltx_p" id="S7.SS1.p8.16">Let <math alttext="L" class="ltx_Math" display="inline" id="S7.SS1.p8.1.m1.1"><semantics id="S7.SS1.p8.1.m1.1a"><mi id="S7.SS1.p8.1.m1.1.1" xref="S7.SS1.p8.1.m1.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.p8.1.m1.1b"><ci id="S7.SS1.p8.1.m1.1.1.cmml" xref="S7.SS1.p8.1.m1.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p8.1.m1.1c">L</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p8.1.m1.1d">italic_L</annotation></semantics></math> be a linear order. Consider a pair of subsets <math alttext="(X,Y)" class="ltx_Math" display="inline" id="S7.SS1.p8.2.m2.2"><semantics id="S7.SS1.p8.2.m2.2a"><mrow id="S7.SS1.p8.2.m2.2.3.2" xref="S7.SS1.p8.2.m2.2.3.1.cmml"><mo id="S7.SS1.p8.2.m2.2.3.2.1" stretchy="false" xref="S7.SS1.p8.2.m2.2.3.1.cmml">(</mo><mi id="S7.SS1.p8.2.m2.1.1" xref="S7.SS1.p8.2.m2.1.1.cmml">X</mi><mo id="S7.SS1.p8.2.m2.2.3.2.2" xref="S7.SS1.p8.2.m2.2.3.1.cmml">,</mo><mi id="S7.SS1.p8.2.m2.2.2" xref="S7.SS1.p8.2.m2.2.2.cmml">Y</mi><mo id="S7.SS1.p8.2.m2.2.3.2.3" stretchy="false" xref="S7.SS1.p8.2.m2.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.p8.2.m2.2b"><interval closure="open" id="S7.SS1.p8.2.m2.2.3.1.cmml" xref="S7.SS1.p8.2.m2.2.3.2"><ci id="S7.SS1.p8.2.m2.1.1.cmml" xref="S7.SS1.p8.2.m2.1.1">𝑋</ci><ci id="S7.SS1.p8.2.m2.2.2.cmml" xref="S7.SS1.p8.2.m2.2.2">𝑌</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p8.2.m2.2c">(X,Y)</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p8.2.m2.2d">( italic_X , italic_Y )</annotation></semantics></math> such that <math alttext="X" class="ltx_Math" display="inline" id="S7.SS1.p8.3.m3.1"><semantics id="S7.SS1.p8.3.m3.1a"><mi id="S7.SS1.p8.3.m3.1.1" xref="S7.SS1.p8.3.m3.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.p8.3.m3.1b"><ci id="S7.SS1.p8.3.m3.1.1.cmml" xref="S7.SS1.p8.3.m3.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p8.3.m3.1c">X</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p8.3.m3.1d">italic_X</annotation></semantics></math> is exactly the set of lower bounds of <math alttext="Y" class="ltx_Math" display="inline" id="S7.SS1.p8.4.m4.1"><semantics id="S7.SS1.p8.4.m4.1a"><mi id="S7.SS1.p8.4.m4.1.1" xref="S7.SS1.p8.4.m4.1.1.cmml">Y</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.p8.4.m4.1b"><ci id="S7.SS1.p8.4.m4.1.1.cmml" xref="S7.SS1.p8.4.m4.1.1">𝑌</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p8.4.m4.1c">Y</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p8.4.m4.1d">italic_Y</annotation></semantics></math>, and <math alttext="Y" class="ltx_Math" display="inline" id="S7.SS1.p8.5.m5.1"><semantics id="S7.SS1.p8.5.m5.1a"><mi id="S7.SS1.p8.5.m5.1.1" xref="S7.SS1.p8.5.m5.1.1.cmml">Y</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.p8.5.m5.1b"><ci id="S7.SS1.p8.5.m5.1.1.cmml" xref="S7.SS1.p8.5.m5.1.1">𝑌</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p8.5.m5.1c">Y</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p8.5.m5.1d">italic_Y</annotation></semantics></math> is the set of upper bounds of <math alttext="X" class="ltx_Math" display="inline" id="S7.SS1.p8.6.m6.1"><semantics id="S7.SS1.p8.6.m6.1a"><mi id="S7.SS1.p8.6.m6.1.1" xref="S7.SS1.p8.6.m6.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.p8.6.m6.1b"><ci id="S7.SS1.p8.6.m6.1.1.cmml" xref="S7.SS1.p8.6.m6.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p8.6.m6.1c">X</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p8.6.m6.1d">italic_X</annotation></semantics></math>. We furthermore ask <math alttext="X" class="ltx_Math" display="inline" id="S7.SS1.p8.7.m7.1"><semantics id="S7.SS1.p8.7.m7.1a"><mi id="S7.SS1.p8.7.m7.1.1" xref="S7.SS1.p8.7.m7.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.p8.7.m7.1b"><ci id="S7.SS1.p8.7.m7.1.1.cmml" xref="S7.SS1.p8.7.m7.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p8.7.m7.1c">X</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p8.7.m7.1d">italic_X</annotation></semantics></math> and <math alttext="Y" class="ltx_Math" display="inline" id="S7.SS1.p8.8.m8.1"><semantics id="S7.SS1.p8.8.m8.1a"><mi id="S7.SS1.p8.8.m8.1.1" xref="S7.SS1.p8.8.m8.1.1.cmml">Y</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.p8.8.m8.1b"><ci id="S7.SS1.p8.8.m8.1.1.cmml" xref="S7.SS1.p8.8.m8.1.1">𝑌</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p8.8.m8.1c">Y</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p8.8.m8.1d">italic_Y</annotation></semantics></math> to be nonempty. Such a pair is called a <em class="ltx_emph ltx_font_italic" id="S7.SS1.p8.16.1">gap</em> of <math alttext="L" class="ltx_Math" display="inline" id="S7.SS1.p8.9.m9.1"><semantics id="S7.SS1.p8.9.m9.1a"><mi id="S7.SS1.p8.9.m9.1.1" xref="S7.SS1.p8.9.m9.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.p8.9.m9.1b"><ci id="S7.SS1.p8.9.m9.1.1.cmml" xref="S7.SS1.p8.9.m9.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p8.9.m9.1c">L</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p8.9.m9.1d">italic_L</annotation></semantics></math> if <math alttext="X\cap Y=\varnothing" class="ltx_Math" display="inline" id="S7.SS1.p8.10.m10.1"><semantics id="S7.SS1.p8.10.m10.1a"><mrow id="S7.SS1.p8.10.m10.1.1" xref="S7.SS1.p8.10.m10.1.1.cmml"><mrow id="S7.SS1.p8.10.m10.1.1.2" xref="S7.SS1.p8.10.m10.1.1.2.cmml"><mi id="S7.SS1.p8.10.m10.1.1.2.2" xref="S7.SS1.p8.10.m10.1.1.2.2.cmml">X</mi><mo id="S7.SS1.p8.10.m10.1.1.2.1" xref="S7.SS1.p8.10.m10.1.1.2.1.cmml">∩</mo><mi id="S7.SS1.p8.10.m10.1.1.2.3" xref="S7.SS1.p8.10.m10.1.1.2.3.cmml">Y</mi></mrow><mo id="S7.SS1.p8.10.m10.1.1.1" xref="S7.SS1.p8.10.m10.1.1.1.cmml">=</mo><mi id="S7.SS1.p8.10.m10.1.1.3" mathvariant="normal" xref="S7.SS1.p8.10.m10.1.1.3.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.p8.10.m10.1b"><apply id="S7.SS1.p8.10.m10.1.1.cmml" xref="S7.SS1.p8.10.m10.1.1"><eq id="S7.SS1.p8.10.m10.1.1.1.cmml" xref="S7.SS1.p8.10.m10.1.1.1"></eq><apply id="S7.SS1.p8.10.m10.1.1.2.cmml" xref="S7.SS1.p8.10.m10.1.1.2"><intersect id="S7.SS1.p8.10.m10.1.1.2.1.cmml" xref="S7.SS1.p8.10.m10.1.1.2.1"></intersect><ci id="S7.SS1.p8.10.m10.1.1.2.2.cmml" xref="S7.SS1.p8.10.m10.1.1.2.2">𝑋</ci><ci id="S7.SS1.p8.10.m10.1.1.2.3.cmml" xref="S7.SS1.p8.10.m10.1.1.2.3">𝑌</ci></apply><emptyset id="S7.SS1.p8.10.m10.1.1.3.cmml" xref="S7.SS1.p8.10.m10.1.1.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p8.10.m10.1c">X\cap Y=\varnothing</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p8.10.m10.1d">italic_X ∩ italic_Y = ∅</annotation></semantics></math>. Note that in this case <math alttext="X" class="ltx_Math" display="inline" id="S7.SS1.p8.11.m11.1"><semantics id="S7.SS1.p8.11.m11.1a"><mi id="S7.SS1.p8.11.m11.1.1" xref="S7.SS1.p8.11.m11.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.p8.11.m11.1b"><ci id="S7.SS1.p8.11.m11.1.1.cmml" xref="S7.SS1.p8.11.m11.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p8.11.m11.1c">X</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p8.11.m11.1d">italic_X</annotation></semantics></math> has no right endpoint, and <math alttext="Y" class="ltx_Math" display="inline" id="S7.SS1.p8.12.m12.1"><semantics id="S7.SS1.p8.12.m12.1a"><mi id="S7.SS1.p8.12.m12.1.1" xref="S7.SS1.p8.12.m12.1.1.cmml">Y</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.p8.12.m12.1b"><ci id="S7.SS1.p8.12.m12.1.1.cmml" xref="S7.SS1.p8.12.m12.1.1">𝑌</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p8.12.m12.1c">Y</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p8.12.m12.1d">italic_Y</annotation></semantics></math> has no left endpoint. Let <math alttext="G(L)" class="ltx_Math" display="inline" id="S7.SS1.p8.13.m13.1"><semantics id="S7.SS1.p8.13.m13.1a"><mrow id="S7.SS1.p8.13.m13.1.2" xref="S7.SS1.p8.13.m13.1.2.cmml"><mi id="S7.SS1.p8.13.m13.1.2.2" xref="S7.SS1.p8.13.m13.1.2.2.cmml">G</mi><mo id="S7.SS1.p8.13.m13.1.2.1" xref="S7.SS1.p8.13.m13.1.2.1.cmml">⁢</mo><mrow id="S7.SS1.p8.13.m13.1.2.3.2" xref="S7.SS1.p8.13.m13.1.2.cmml"><mo id="S7.SS1.p8.13.m13.1.2.3.2.1" stretchy="false" xref="S7.SS1.p8.13.m13.1.2.cmml">(</mo><mi id="S7.SS1.p8.13.m13.1.1" xref="S7.SS1.p8.13.m13.1.1.cmml">L</mi><mo id="S7.SS1.p8.13.m13.1.2.3.2.2" stretchy="false" xref="S7.SS1.p8.13.m13.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.p8.13.m13.1b"><apply id="S7.SS1.p8.13.m13.1.2.cmml" xref="S7.SS1.p8.13.m13.1.2"><times id="S7.SS1.p8.13.m13.1.2.1.cmml" xref="S7.SS1.p8.13.m13.1.2.1"></times><ci id="S7.SS1.p8.13.m13.1.2.2.cmml" xref="S7.SS1.p8.13.m13.1.2.2">𝐺</ci><ci id="S7.SS1.p8.13.m13.1.1.cmml" xref="S7.SS1.p8.13.m13.1.1">𝐿</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p8.13.m13.1c">G(L)</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p8.13.m13.1d">italic_G ( italic_L )</annotation></semantics></math> be the set of gaps of <math alttext="L" class="ltx_Math" display="inline" id="S7.SS1.p8.14.m14.1"><semantics id="S7.SS1.p8.14.m14.1a"><mi id="S7.SS1.p8.14.m14.1.1" xref="S7.SS1.p8.14.m14.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.p8.14.m14.1b"><ci id="S7.SS1.p8.14.m14.1.1.cmml" xref="S7.SS1.p8.14.m14.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p8.14.m14.1c">L</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p8.14.m14.1d">italic_L</annotation></semantics></math>. This is naturally linearly ordered by letting <math alttext="(X,Y)\leq(X^{\prime},Y^{\prime})" class="ltx_Math" display="inline" id="S7.SS1.p8.15.m15.4"><semantics id="S7.SS1.p8.15.m15.4a"><mrow id="S7.SS1.p8.15.m15.4.4" xref="S7.SS1.p8.15.m15.4.4.cmml"><mrow id="S7.SS1.p8.15.m15.4.4.4.2" xref="S7.SS1.p8.15.m15.4.4.4.1.cmml"><mo id="S7.SS1.p8.15.m15.4.4.4.2.1" stretchy="false" xref="S7.SS1.p8.15.m15.4.4.4.1.cmml">(</mo><mi id="S7.SS1.p8.15.m15.1.1" xref="S7.SS1.p8.15.m15.1.1.cmml">X</mi><mo id="S7.SS1.p8.15.m15.4.4.4.2.2" xref="S7.SS1.p8.15.m15.4.4.4.1.cmml">,</mo><mi id="S7.SS1.p8.15.m15.2.2" xref="S7.SS1.p8.15.m15.2.2.cmml">Y</mi><mo id="S7.SS1.p8.15.m15.4.4.4.2.3" stretchy="false" xref="S7.SS1.p8.15.m15.4.4.4.1.cmml">)</mo></mrow><mo id="S7.SS1.p8.15.m15.4.4.3" xref="S7.SS1.p8.15.m15.4.4.3.cmml">≤</mo><mrow id="S7.SS1.p8.15.m15.4.4.2.2" xref="S7.SS1.p8.15.m15.4.4.2.3.cmml"><mo id="S7.SS1.p8.15.m15.4.4.2.2.3" stretchy="false" xref="S7.SS1.p8.15.m15.4.4.2.3.cmml">(</mo><msup id="S7.SS1.p8.15.m15.3.3.1.1.1" xref="S7.SS1.p8.15.m15.3.3.1.1.1.cmml"><mi id="S7.SS1.p8.15.m15.3.3.1.1.1.2" xref="S7.SS1.p8.15.m15.3.3.1.1.1.2.cmml">X</mi><mo id="S7.SS1.p8.15.m15.3.3.1.1.1.3" xref="S7.SS1.p8.15.m15.3.3.1.1.1.3.cmml">′</mo></msup><mo id="S7.SS1.p8.15.m15.4.4.2.2.4" xref="S7.SS1.p8.15.m15.4.4.2.3.cmml">,</mo><msup id="S7.SS1.p8.15.m15.4.4.2.2.2" xref="S7.SS1.p8.15.m15.4.4.2.2.2.cmml"><mi id="S7.SS1.p8.15.m15.4.4.2.2.2.2" xref="S7.SS1.p8.15.m15.4.4.2.2.2.2.cmml">Y</mi><mo id="S7.SS1.p8.15.m15.4.4.2.2.2.3" xref="S7.SS1.p8.15.m15.4.4.2.2.2.3.cmml">′</mo></msup><mo id="S7.SS1.p8.15.m15.4.4.2.2.5" stretchy="false" xref="S7.SS1.p8.15.m15.4.4.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.p8.15.m15.4b"><apply id="S7.SS1.p8.15.m15.4.4.cmml" xref="S7.SS1.p8.15.m15.4.4"><leq id="S7.SS1.p8.15.m15.4.4.3.cmml" xref="S7.SS1.p8.15.m15.4.4.3"></leq><interval closure="open" id="S7.SS1.p8.15.m15.4.4.4.1.cmml" xref="S7.SS1.p8.15.m15.4.4.4.2"><ci id="S7.SS1.p8.15.m15.1.1.cmml" xref="S7.SS1.p8.15.m15.1.1">𝑋</ci><ci id="S7.SS1.p8.15.m15.2.2.cmml" xref="S7.SS1.p8.15.m15.2.2">𝑌</ci></interval><interval closure="open" id="S7.SS1.p8.15.m15.4.4.2.3.cmml" xref="S7.SS1.p8.15.m15.4.4.2.2"><apply id="S7.SS1.p8.15.m15.3.3.1.1.1.cmml" xref="S7.SS1.p8.15.m15.3.3.1.1.1"><csymbol cd="ambiguous" id="S7.SS1.p8.15.m15.3.3.1.1.1.1.cmml" xref="S7.SS1.p8.15.m15.3.3.1.1.1">superscript</csymbol><ci id="S7.SS1.p8.15.m15.3.3.1.1.1.2.cmml" xref="S7.SS1.p8.15.m15.3.3.1.1.1.2">𝑋</ci><ci id="S7.SS1.p8.15.m15.3.3.1.1.1.3.cmml" xref="S7.SS1.p8.15.m15.3.3.1.1.1.3">′</ci></apply><apply id="S7.SS1.p8.15.m15.4.4.2.2.2.cmml" xref="S7.SS1.p8.15.m15.4.4.2.2.2"><csymbol cd="ambiguous" id="S7.SS1.p8.15.m15.4.4.2.2.2.1.cmml" xref="S7.SS1.p8.15.m15.4.4.2.2.2">superscript</csymbol><ci id="S7.SS1.p8.15.m15.4.4.2.2.2.2.cmml" xref="S7.SS1.p8.15.m15.4.4.2.2.2.2">𝑌</ci><ci id="S7.SS1.p8.15.m15.4.4.2.2.2.3.cmml" xref="S7.SS1.p8.15.m15.4.4.2.2.2.3">′</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p8.15.m15.4c">(X,Y)\leq(X^{\prime},Y^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p8.15.m15.4d">( italic_X , italic_Y ) ≤ ( italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_Y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> iff <math alttext="X\subseteq X^{\prime}" class="ltx_Math" display="inline" id="S7.SS1.p8.16.m16.1"><semantics id="S7.SS1.p8.16.m16.1a"><mrow id="S7.SS1.p8.16.m16.1.1" xref="S7.SS1.p8.16.m16.1.1.cmml"><mi id="S7.SS1.p8.16.m16.1.1.2" xref="S7.SS1.p8.16.m16.1.1.2.cmml">X</mi><mo id="S7.SS1.p8.16.m16.1.1.1" xref="S7.SS1.p8.16.m16.1.1.1.cmml">⊆</mo><msup id="S7.SS1.p8.16.m16.1.1.3" xref="S7.SS1.p8.16.m16.1.1.3.cmml"><mi id="S7.SS1.p8.16.m16.1.1.3.2" xref="S7.SS1.p8.16.m16.1.1.3.2.cmml">X</mi><mo id="S7.SS1.p8.16.m16.1.1.3.3" xref="S7.SS1.p8.16.m16.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.p8.16.m16.1b"><apply id="S7.SS1.p8.16.m16.1.1.cmml" xref="S7.SS1.p8.16.m16.1.1"><subset id="S7.SS1.p8.16.m16.1.1.1.cmml" xref="S7.SS1.p8.16.m16.1.1.1"></subset><ci id="S7.SS1.p8.16.m16.1.1.2.cmml" xref="S7.SS1.p8.16.m16.1.1.2">𝑋</ci><apply id="S7.SS1.p8.16.m16.1.1.3.cmml" xref="S7.SS1.p8.16.m16.1.1.3"><csymbol cd="ambiguous" id="S7.SS1.p8.16.m16.1.1.3.1.cmml" xref="S7.SS1.p8.16.m16.1.1.3">superscript</csymbol><ci id="S7.SS1.p8.16.m16.1.1.3.2.cmml" xref="S7.SS1.p8.16.m16.1.1.3.2">𝑋</ci><ci id="S7.SS1.p8.16.m16.1.1.3.3.cmml" xref="S7.SS1.p8.16.m16.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p8.16.m16.1c">X\subseteq X^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p8.16.m16.1d">italic_X ⊆ italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>. By taking preimages the following is easily proved (or see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib5" title="">5</a>, Proposition 5.3]</cite>).</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="S7.Thmtheorem7"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S7.Thmtheorem7.1.1.1">Proposition 7.7</span></span><span class="ltx_text ltx_font_bold" id="S7.Thmtheorem7.2.2">.</span> </h6> <div class="ltx_para" id="S7.Thmtheorem7.p1"> <p class="ltx_p" id="S7.Thmtheorem7.p1.2">If <math alttext="B\trianglelefteq A" class="ltx_Math" display="inline" id="S7.Thmtheorem7.p1.1.m1.1"><semantics id="S7.Thmtheorem7.p1.1.m1.1a"><mrow id="S7.Thmtheorem7.p1.1.m1.1.1" xref="S7.Thmtheorem7.p1.1.m1.1.1.cmml"><mi id="S7.Thmtheorem7.p1.1.m1.1.1.2" xref="S7.Thmtheorem7.p1.1.m1.1.1.2.cmml">B</mi><mo id="S7.Thmtheorem7.p1.1.m1.1.1.1" xref="S7.Thmtheorem7.p1.1.m1.1.1.1.cmml">⁢</mo><mi id="S7.Thmtheorem7.p1.1.m1.1.1.3" mathvariant="normal" xref="S7.Thmtheorem7.p1.1.m1.1.1.3.cmml">⊴</mi><mo id="S7.Thmtheorem7.p1.1.m1.1.1.1a" xref="S7.Thmtheorem7.p1.1.m1.1.1.1.cmml">⁢</mo><mi id="S7.Thmtheorem7.p1.1.m1.1.1.4" xref="S7.Thmtheorem7.p1.1.m1.1.1.4.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem7.p1.1.m1.1b"><apply id="S7.Thmtheorem7.p1.1.m1.1.1.cmml" xref="S7.Thmtheorem7.p1.1.m1.1.1"><times id="S7.Thmtheorem7.p1.1.m1.1.1.1.cmml" xref="S7.Thmtheorem7.p1.1.m1.1.1.1"></times><ci id="S7.Thmtheorem7.p1.1.m1.1.1.2.cmml" xref="S7.Thmtheorem7.p1.1.m1.1.1.2">𝐵</ci><ci id="S7.Thmtheorem7.p1.1.m1.1.1.3.cmml" xref="S7.Thmtheorem7.p1.1.m1.1.1.3">⊴</ci><ci id="S7.Thmtheorem7.p1.1.m1.1.1.4.cmml" xref="S7.Thmtheorem7.p1.1.m1.1.1.4">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem7.p1.1.m1.1c">B\trianglelefteq A</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem7.p1.1.m1.1d">italic_B ⊴ italic_A</annotation></semantics></math>, then <math alttext="G(B)\preceq G(A)" class="ltx_Math" display="inline" id="S7.Thmtheorem7.p1.2.m2.2"><semantics id="S7.Thmtheorem7.p1.2.m2.2a"><mrow id="S7.Thmtheorem7.p1.2.m2.2.3" xref="S7.Thmtheorem7.p1.2.m2.2.3.cmml"><mrow id="S7.Thmtheorem7.p1.2.m2.2.3.2" xref="S7.Thmtheorem7.p1.2.m2.2.3.2.cmml"><mi id="S7.Thmtheorem7.p1.2.m2.2.3.2.2" xref="S7.Thmtheorem7.p1.2.m2.2.3.2.2.cmml">G</mi><mo id="S7.Thmtheorem7.p1.2.m2.2.3.2.1" xref="S7.Thmtheorem7.p1.2.m2.2.3.2.1.cmml">⁢</mo><mrow id="S7.Thmtheorem7.p1.2.m2.2.3.2.3.2" xref="S7.Thmtheorem7.p1.2.m2.2.3.2.cmml"><mo id="S7.Thmtheorem7.p1.2.m2.2.3.2.3.2.1" stretchy="false" xref="S7.Thmtheorem7.p1.2.m2.2.3.2.cmml">(</mo><mi id="S7.Thmtheorem7.p1.2.m2.1.1" xref="S7.Thmtheorem7.p1.2.m2.1.1.cmml">B</mi><mo id="S7.Thmtheorem7.p1.2.m2.2.3.2.3.2.2" stretchy="false" xref="S7.Thmtheorem7.p1.2.m2.2.3.2.cmml">)</mo></mrow></mrow><mo id="S7.Thmtheorem7.p1.2.m2.2.3.1" xref="S7.Thmtheorem7.p1.2.m2.2.3.1.cmml">⪯</mo><mrow id="S7.Thmtheorem7.p1.2.m2.2.3.3" xref="S7.Thmtheorem7.p1.2.m2.2.3.3.cmml"><mi id="S7.Thmtheorem7.p1.2.m2.2.3.3.2" xref="S7.Thmtheorem7.p1.2.m2.2.3.3.2.cmml">G</mi><mo id="S7.Thmtheorem7.p1.2.m2.2.3.3.1" xref="S7.Thmtheorem7.p1.2.m2.2.3.3.1.cmml">⁢</mo><mrow id="S7.Thmtheorem7.p1.2.m2.2.3.3.3.2" xref="S7.Thmtheorem7.p1.2.m2.2.3.3.cmml"><mo id="S7.Thmtheorem7.p1.2.m2.2.3.3.3.2.1" stretchy="false" xref="S7.Thmtheorem7.p1.2.m2.2.3.3.cmml">(</mo><mi id="S7.Thmtheorem7.p1.2.m2.2.2" xref="S7.Thmtheorem7.p1.2.m2.2.2.cmml">A</mi><mo id="S7.Thmtheorem7.p1.2.m2.2.3.3.3.2.2" stretchy="false" xref="S7.Thmtheorem7.p1.2.m2.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem7.p1.2.m2.2b"><apply id="S7.Thmtheorem7.p1.2.m2.2.3.cmml" xref="S7.Thmtheorem7.p1.2.m2.2.3"><csymbol cd="latexml" id="S7.Thmtheorem7.p1.2.m2.2.3.1.cmml" xref="S7.Thmtheorem7.p1.2.m2.2.3.1">precedes-or-equals</csymbol><apply id="S7.Thmtheorem7.p1.2.m2.2.3.2.cmml" xref="S7.Thmtheorem7.p1.2.m2.2.3.2"><times id="S7.Thmtheorem7.p1.2.m2.2.3.2.1.cmml" xref="S7.Thmtheorem7.p1.2.m2.2.3.2.1"></times><ci id="S7.Thmtheorem7.p1.2.m2.2.3.2.2.cmml" xref="S7.Thmtheorem7.p1.2.m2.2.3.2.2">𝐺</ci><ci id="S7.Thmtheorem7.p1.2.m2.1.1.cmml" xref="S7.Thmtheorem7.p1.2.m2.1.1">𝐵</ci></apply><apply id="S7.Thmtheorem7.p1.2.m2.2.3.3.cmml" xref="S7.Thmtheorem7.p1.2.m2.2.3.3"><times id="S7.Thmtheorem7.p1.2.m2.2.3.3.1.cmml" xref="S7.Thmtheorem7.p1.2.m2.2.3.3.1"></times><ci id="S7.Thmtheorem7.p1.2.m2.2.3.3.2.cmml" xref="S7.Thmtheorem7.p1.2.m2.2.3.3.2">𝐺</ci><ci id="S7.Thmtheorem7.p1.2.m2.2.2.cmml" xref="S7.Thmtheorem7.p1.2.m2.2.2">𝐴</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem7.p1.2.m2.2c">G(B)\preceq G(A)</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem7.p1.2.m2.2d">italic_G ( italic_B ) ⪯ italic_G ( italic_A )</annotation></semantics></math>.</p> </div> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S7.Thmtheorem8"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S7.Thmtheorem8.1.1.1">Lemma 7.8</span></span><span class="ltx_text ltx_font_bold" id="S7.Thmtheorem8.2.2">.</span> </h6> <div class="ltx_para" id="S7.Thmtheorem8.p1"> <p class="ltx_p" id="S7.Thmtheorem8.p1.5">Let <math alttext="A,B\subseteq\mathbb{R}" class="ltx_Math" display="inline" id="S7.Thmtheorem8.p1.1.m1.2"><semantics id="S7.Thmtheorem8.p1.1.m1.2a"><mrow id="S7.Thmtheorem8.p1.1.m1.2.3" xref="S7.Thmtheorem8.p1.1.m1.2.3.cmml"><mrow id="S7.Thmtheorem8.p1.1.m1.2.3.2.2" xref="S7.Thmtheorem8.p1.1.m1.2.3.2.1.cmml"><mi id="S7.Thmtheorem8.p1.1.m1.1.1" xref="S7.Thmtheorem8.p1.1.m1.1.1.cmml">A</mi><mo id="S7.Thmtheorem8.p1.1.m1.2.3.2.2.1" xref="S7.Thmtheorem8.p1.1.m1.2.3.2.1.cmml">,</mo><mi id="S7.Thmtheorem8.p1.1.m1.2.2" xref="S7.Thmtheorem8.p1.1.m1.2.2.cmml">B</mi></mrow><mo id="S7.Thmtheorem8.p1.1.m1.2.3.1" xref="S7.Thmtheorem8.p1.1.m1.2.3.1.cmml">⊆</mo><mi id="S7.Thmtheorem8.p1.1.m1.2.3.3" xref="S7.Thmtheorem8.p1.1.m1.2.3.3.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem8.p1.1.m1.2b"><apply id="S7.Thmtheorem8.p1.1.m1.2.3.cmml" xref="S7.Thmtheorem8.p1.1.m1.2.3"><subset id="S7.Thmtheorem8.p1.1.m1.2.3.1.cmml" xref="S7.Thmtheorem8.p1.1.m1.2.3.1"></subset><list id="S7.Thmtheorem8.p1.1.m1.2.3.2.1.cmml" xref="S7.Thmtheorem8.p1.1.m1.2.3.2.2"><ci id="S7.Thmtheorem8.p1.1.m1.1.1.cmml" xref="S7.Thmtheorem8.p1.1.m1.1.1">𝐴</ci><ci id="S7.Thmtheorem8.p1.1.m1.2.2.cmml" xref="S7.Thmtheorem8.p1.1.m1.2.2">𝐵</ci></list><ci id="S7.Thmtheorem8.p1.1.m1.2.3.3.cmml" xref="S7.Thmtheorem8.p1.1.m1.2.3.3">ℝ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem8.p1.1.m1.2c">A,B\subseteq\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem8.p1.1.m1.2d">italic_A , italic_B ⊆ blackboard_R</annotation></semantics></math>. If <math alttext="A\trianglerighteq B" class="ltx_Math" display="inline" id="S7.Thmtheorem8.p1.2.m2.1"><semantics id="S7.Thmtheorem8.p1.2.m2.1a"><mrow id="S7.Thmtheorem8.p1.2.m2.1.1" xref="S7.Thmtheorem8.p1.2.m2.1.1.cmml"><mi id="S7.Thmtheorem8.p1.2.m2.1.1.2" xref="S7.Thmtheorem8.p1.2.m2.1.1.2.cmml">A</mi><mo id="S7.Thmtheorem8.p1.2.m2.1.1.1" xref="S7.Thmtheorem8.p1.2.m2.1.1.1.cmml">⁢</mo><mi id="S7.Thmtheorem8.p1.2.m2.1.1.3" mathvariant="normal" xref="S7.Thmtheorem8.p1.2.m2.1.1.3.cmml">⊵</mi><mo id="S7.Thmtheorem8.p1.2.m2.1.1.1a" xref="S7.Thmtheorem8.p1.2.m2.1.1.1.cmml">⁢</mo><mi id="S7.Thmtheorem8.p1.2.m2.1.1.4" xref="S7.Thmtheorem8.p1.2.m2.1.1.4.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem8.p1.2.m2.1b"><apply id="S7.Thmtheorem8.p1.2.m2.1.1.cmml" xref="S7.Thmtheorem8.p1.2.m2.1.1"><times id="S7.Thmtheorem8.p1.2.m2.1.1.1.cmml" xref="S7.Thmtheorem8.p1.2.m2.1.1.1"></times><ci id="S7.Thmtheorem8.p1.2.m2.1.1.2.cmml" xref="S7.Thmtheorem8.p1.2.m2.1.1.2">𝐴</ci><ci id="S7.Thmtheorem8.p1.2.m2.1.1.3.cmml" xref="S7.Thmtheorem8.p1.2.m2.1.1.3">⊵</ci><ci id="S7.Thmtheorem8.p1.2.m2.1.1.4.cmml" xref="S7.Thmtheorem8.p1.2.m2.1.1.4">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem8.p1.2.m2.1c">A\trianglerighteq B</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem8.p1.2.m2.1d">italic_A ⊵ italic_B</annotation></semantics></math> and <math alttext="|\mathbb{R}\setminus A|<\mathfrak{c}" class="ltx_Math" display="inline" id="S7.Thmtheorem8.p1.3.m3.1"><semantics id="S7.Thmtheorem8.p1.3.m3.1a"><mrow id="S7.Thmtheorem8.p1.3.m3.1.1" xref="S7.Thmtheorem8.p1.3.m3.1.1.cmml"><mrow id="S7.Thmtheorem8.p1.3.m3.1.1.1.1" xref="S7.Thmtheorem8.p1.3.m3.1.1.1.2.cmml"><mo id="S7.Thmtheorem8.p1.3.m3.1.1.1.1.2" stretchy="false" xref="S7.Thmtheorem8.p1.3.m3.1.1.1.2.1.cmml">|</mo><mrow id="S7.Thmtheorem8.p1.3.m3.1.1.1.1.1" xref="S7.Thmtheorem8.p1.3.m3.1.1.1.1.1.cmml"><mi id="S7.Thmtheorem8.p1.3.m3.1.1.1.1.1.2" xref="S7.Thmtheorem8.p1.3.m3.1.1.1.1.1.2.cmml">ℝ</mi><mo id="S7.Thmtheorem8.p1.3.m3.1.1.1.1.1.1" xref="S7.Thmtheorem8.p1.3.m3.1.1.1.1.1.1.cmml">∖</mo><mi id="S7.Thmtheorem8.p1.3.m3.1.1.1.1.1.3" xref="S7.Thmtheorem8.p1.3.m3.1.1.1.1.1.3.cmml">A</mi></mrow><mo id="S7.Thmtheorem8.p1.3.m3.1.1.1.1.3" stretchy="false" xref="S7.Thmtheorem8.p1.3.m3.1.1.1.2.1.cmml">|</mo></mrow><mo id="S7.Thmtheorem8.p1.3.m3.1.1.2" xref="S7.Thmtheorem8.p1.3.m3.1.1.2.cmml"><</mo><mi id="S7.Thmtheorem8.p1.3.m3.1.1.3" xref="S7.Thmtheorem8.p1.3.m3.1.1.3.cmml">𝔠</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem8.p1.3.m3.1b"><apply id="S7.Thmtheorem8.p1.3.m3.1.1.cmml" xref="S7.Thmtheorem8.p1.3.m3.1.1"><lt id="S7.Thmtheorem8.p1.3.m3.1.1.2.cmml" xref="S7.Thmtheorem8.p1.3.m3.1.1.2"></lt><apply id="S7.Thmtheorem8.p1.3.m3.1.1.1.2.cmml" xref="S7.Thmtheorem8.p1.3.m3.1.1.1.1"><abs id="S7.Thmtheorem8.p1.3.m3.1.1.1.2.1.cmml" xref="S7.Thmtheorem8.p1.3.m3.1.1.1.1.2"></abs><apply id="S7.Thmtheorem8.p1.3.m3.1.1.1.1.1.cmml" xref="S7.Thmtheorem8.p1.3.m3.1.1.1.1.1"><setdiff id="S7.Thmtheorem8.p1.3.m3.1.1.1.1.1.1.cmml" xref="S7.Thmtheorem8.p1.3.m3.1.1.1.1.1.1"></setdiff><ci id="S7.Thmtheorem8.p1.3.m3.1.1.1.1.1.2.cmml" xref="S7.Thmtheorem8.p1.3.m3.1.1.1.1.1.2">ℝ</ci><ci id="S7.Thmtheorem8.p1.3.m3.1.1.1.1.1.3.cmml" xref="S7.Thmtheorem8.p1.3.m3.1.1.1.1.1.3">𝐴</ci></apply></apply><ci id="S7.Thmtheorem8.p1.3.m3.1.1.3.cmml" xref="S7.Thmtheorem8.p1.3.m3.1.1.3">𝔠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem8.p1.3.m3.1c">|\mathbb{R}\setminus A|<\mathfrak{c}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem8.p1.3.m3.1d">| blackboard_R ∖ italic_A | < fraktur_c</annotation></semantics></math>, then <math alttext="B" class="ltx_Math" display="inline" id="S7.Thmtheorem8.p1.4.m4.1"><semantics id="S7.Thmtheorem8.p1.4.m4.1a"><mi id="S7.Thmtheorem8.p1.4.m4.1.1" xref="S7.Thmtheorem8.p1.4.m4.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem8.p1.4.m4.1b"><ci id="S7.Thmtheorem8.p1.4.m4.1.1.cmml" xref="S7.Thmtheorem8.p1.4.m4.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem8.p1.4.m4.1c">B</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem8.p1.4.m4.1d">italic_B</annotation></semantics></math> is countable or <math alttext="|B|=\mathfrak{c}" class="ltx_Math" display="inline" id="S7.Thmtheorem8.p1.5.m5.1"><semantics id="S7.Thmtheorem8.p1.5.m5.1a"><mrow id="S7.Thmtheorem8.p1.5.m5.1.2" xref="S7.Thmtheorem8.p1.5.m5.1.2.cmml"><mrow id="S7.Thmtheorem8.p1.5.m5.1.2.2.2" xref="S7.Thmtheorem8.p1.5.m5.1.2.2.1.cmml"><mo id="S7.Thmtheorem8.p1.5.m5.1.2.2.2.1" stretchy="false" xref="S7.Thmtheorem8.p1.5.m5.1.2.2.1.1.cmml">|</mo><mi id="S7.Thmtheorem8.p1.5.m5.1.1" xref="S7.Thmtheorem8.p1.5.m5.1.1.cmml">B</mi><mo id="S7.Thmtheorem8.p1.5.m5.1.2.2.2.2" stretchy="false" xref="S7.Thmtheorem8.p1.5.m5.1.2.2.1.1.cmml">|</mo></mrow><mo id="S7.Thmtheorem8.p1.5.m5.1.2.1" xref="S7.Thmtheorem8.p1.5.m5.1.2.1.cmml">=</mo><mi id="S7.Thmtheorem8.p1.5.m5.1.2.3" xref="S7.Thmtheorem8.p1.5.m5.1.2.3.cmml">𝔠</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem8.p1.5.m5.1b"><apply id="S7.Thmtheorem8.p1.5.m5.1.2.cmml" xref="S7.Thmtheorem8.p1.5.m5.1.2"><eq id="S7.Thmtheorem8.p1.5.m5.1.2.1.cmml" xref="S7.Thmtheorem8.p1.5.m5.1.2.1"></eq><apply id="S7.Thmtheorem8.p1.5.m5.1.2.2.1.cmml" xref="S7.Thmtheorem8.p1.5.m5.1.2.2.2"><abs id="S7.Thmtheorem8.p1.5.m5.1.2.2.1.1.cmml" xref="S7.Thmtheorem8.p1.5.m5.1.2.2.2.1"></abs><ci id="S7.Thmtheorem8.p1.5.m5.1.1.cmml" xref="S7.Thmtheorem8.p1.5.m5.1.1">𝐵</ci></apply><ci id="S7.Thmtheorem8.p1.5.m5.1.2.3.cmml" xref="S7.Thmtheorem8.p1.5.m5.1.2.3">𝔠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem8.p1.5.m5.1c">|B|=\mathfrak{c}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem8.p1.5.m5.1d">| italic_B | = fraktur_c</annotation></semantics></math>.</p> </div> </div> <div class="ltx_proof" id="S7.SS1.4"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S7.SS1.3.p1"> <p class="ltx_p" id="S7.SS1.3.p1.10">Let <math alttext="f:A\twoheadrightarrow B" class="ltx_Math" display="inline" id="S7.SS1.3.p1.1.m1.1"><semantics id="S7.SS1.3.p1.1.m1.1a"><mrow id="S7.SS1.3.p1.1.m1.1.1" xref="S7.SS1.3.p1.1.m1.1.1.cmml"><mi id="S7.SS1.3.p1.1.m1.1.1.2" xref="S7.SS1.3.p1.1.m1.1.1.2.cmml">f</mi><mo id="S7.SS1.3.p1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S7.SS1.3.p1.1.m1.1.1.1.cmml">:</mo><mrow id="S7.SS1.3.p1.1.m1.1.1.3" xref="S7.SS1.3.p1.1.m1.1.1.3.cmml"><mi id="S7.SS1.3.p1.1.m1.1.1.3.2" xref="S7.SS1.3.p1.1.m1.1.1.3.2.cmml">A</mi><mo id="S7.SS1.3.p1.1.m1.1.1.3.1" stretchy="false" xref="S7.SS1.3.p1.1.m1.1.1.3.1.cmml">↠</mo><mi id="S7.SS1.3.p1.1.m1.1.1.3.3" xref="S7.SS1.3.p1.1.m1.1.1.3.3.cmml">B</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.3.p1.1.m1.1b"><apply id="S7.SS1.3.p1.1.m1.1.1.cmml" xref="S7.SS1.3.p1.1.m1.1.1"><ci id="S7.SS1.3.p1.1.m1.1.1.1.cmml" xref="S7.SS1.3.p1.1.m1.1.1.1">:</ci><ci id="S7.SS1.3.p1.1.m1.1.1.2.cmml" xref="S7.SS1.3.p1.1.m1.1.1.2">𝑓</ci><apply id="S7.SS1.3.p1.1.m1.1.1.3.cmml" xref="S7.SS1.3.p1.1.m1.1.1.3"><ci id="S7.SS1.3.p1.1.m1.1.1.3.1.cmml" xref="S7.SS1.3.p1.1.m1.1.1.3.1">↠</ci><ci id="S7.SS1.3.p1.1.m1.1.1.3.2.cmml" xref="S7.SS1.3.p1.1.m1.1.1.3.2">𝐴</ci><ci id="S7.SS1.3.p1.1.m1.1.1.3.3.cmml" xref="S7.SS1.3.p1.1.m1.1.1.3.3">𝐵</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.3.p1.1.m1.1c">f:A\twoheadrightarrow B</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.3.p1.1.m1.1d">italic_f : italic_A ↠ italic_B</annotation></semantics></math> witness that <math alttext="A\trianglerighteq B" class="ltx_Math" display="inline" id="S7.SS1.3.p1.2.m2.1"><semantics id="S7.SS1.3.p1.2.m2.1a"><mrow id="S7.SS1.3.p1.2.m2.1.1" xref="S7.SS1.3.p1.2.m2.1.1.cmml"><mi id="S7.SS1.3.p1.2.m2.1.1.2" xref="S7.SS1.3.p1.2.m2.1.1.2.cmml">A</mi><mo id="S7.SS1.3.p1.2.m2.1.1.1" xref="S7.SS1.3.p1.2.m2.1.1.1.cmml">⁢</mo><mi id="S7.SS1.3.p1.2.m2.1.1.3" mathvariant="normal" xref="S7.SS1.3.p1.2.m2.1.1.3.cmml">⊵</mi><mo id="S7.SS1.3.p1.2.m2.1.1.1a" xref="S7.SS1.3.p1.2.m2.1.1.1.cmml">⁢</mo><mi id="S7.SS1.3.p1.2.m2.1.1.4" xref="S7.SS1.3.p1.2.m2.1.1.4.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.3.p1.2.m2.1b"><apply id="S7.SS1.3.p1.2.m2.1.1.cmml" xref="S7.SS1.3.p1.2.m2.1.1"><times id="S7.SS1.3.p1.2.m2.1.1.1.cmml" xref="S7.SS1.3.p1.2.m2.1.1.1"></times><ci id="S7.SS1.3.p1.2.m2.1.1.2.cmml" xref="S7.SS1.3.p1.2.m2.1.1.2">𝐴</ci><ci id="S7.SS1.3.p1.2.m2.1.1.3.cmml" xref="S7.SS1.3.p1.2.m2.1.1.3">⊵</ci><ci id="S7.SS1.3.p1.2.m2.1.1.4.cmml" xref="S7.SS1.3.p1.2.m2.1.1.4">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.3.p1.2.m2.1c">A\trianglerighteq B</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.3.p1.2.m2.1d">italic_A ⊵ italic_B</annotation></semantics></math>, so <math alttext="f\in\mathscr{F}" class="ltx_Math" display="inline" id="S7.SS1.3.p1.3.m3.1"><semantics id="S7.SS1.3.p1.3.m3.1a"><mrow id="S7.SS1.3.p1.3.m3.1.1" xref="S7.SS1.3.p1.3.m3.1.1.cmml"><mi id="S7.SS1.3.p1.3.m3.1.1.2" xref="S7.SS1.3.p1.3.m3.1.1.2.cmml">f</mi><mo id="S7.SS1.3.p1.3.m3.1.1.1" xref="S7.SS1.3.p1.3.m3.1.1.1.cmml">∈</mo><mi class="ltx_font_mathscript" id="S7.SS1.3.p1.3.m3.1.1.3" xref="S7.SS1.3.p1.3.m3.1.1.3.cmml">ℱ</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.3.p1.3.m3.1b"><apply id="S7.SS1.3.p1.3.m3.1.1.cmml" xref="S7.SS1.3.p1.3.m3.1.1"><in id="S7.SS1.3.p1.3.m3.1.1.1.cmml" xref="S7.SS1.3.p1.3.m3.1.1.1"></in><ci id="S7.SS1.3.p1.3.m3.1.1.2.cmml" xref="S7.SS1.3.p1.3.m3.1.1.2">𝑓</ci><ci id="S7.SS1.3.p1.3.m3.1.1.3.cmml" xref="S7.SS1.3.p1.3.m3.1.1.3">ℱ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.3.p1.3.m3.1c">f\in\mathscr{F}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.3.p1.3.m3.1d">italic_f ∈ script_F</annotation></semantics></math>. Because <math alttext="|\mathbb{R}\setminus A|<\mathfrak{c}" class="ltx_Math" display="inline" id="S7.SS1.3.p1.4.m4.1"><semantics id="S7.SS1.3.p1.4.m4.1a"><mrow id="S7.SS1.3.p1.4.m4.1.1" xref="S7.SS1.3.p1.4.m4.1.1.cmml"><mrow id="S7.SS1.3.p1.4.m4.1.1.1.1" xref="S7.SS1.3.p1.4.m4.1.1.1.2.cmml"><mo id="S7.SS1.3.p1.4.m4.1.1.1.1.2" stretchy="false" xref="S7.SS1.3.p1.4.m4.1.1.1.2.1.cmml">|</mo><mrow id="S7.SS1.3.p1.4.m4.1.1.1.1.1" xref="S7.SS1.3.p1.4.m4.1.1.1.1.1.cmml"><mi id="S7.SS1.3.p1.4.m4.1.1.1.1.1.2" xref="S7.SS1.3.p1.4.m4.1.1.1.1.1.2.cmml">ℝ</mi><mo id="S7.SS1.3.p1.4.m4.1.1.1.1.1.1" xref="S7.SS1.3.p1.4.m4.1.1.1.1.1.1.cmml">∖</mo><mi id="S7.SS1.3.p1.4.m4.1.1.1.1.1.3" xref="S7.SS1.3.p1.4.m4.1.1.1.1.1.3.cmml">A</mi></mrow><mo id="S7.SS1.3.p1.4.m4.1.1.1.1.3" stretchy="false" xref="S7.SS1.3.p1.4.m4.1.1.1.2.1.cmml">|</mo></mrow><mo id="S7.SS1.3.p1.4.m4.1.1.2" xref="S7.SS1.3.p1.4.m4.1.1.2.cmml"><</mo><mi id="S7.SS1.3.p1.4.m4.1.1.3" xref="S7.SS1.3.p1.4.m4.1.1.3.cmml">𝔠</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.3.p1.4.m4.1b"><apply id="S7.SS1.3.p1.4.m4.1.1.cmml" xref="S7.SS1.3.p1.4.m4.1.1"><lt id="S7.SS1.3.p1.4.m4.1.1.2.cmml" xref="S7.SS1.3.p1.4.m4.1.1.2"></lt><apply id="S7.SS1.3.p1.4.m4.1.1.1.2.cmml" xref="S7.SS1.3.p1.4.m4.1.1.1.1"><abs id="S7.SS1.3.p1.4.m4.1.1.1.2.1.cmml" xref="S7.SS1.3.p1.4.m4.1.1.1.1.2"></abs><apply id="S7.SS1.3.p1.4.m4.1.1.1.1.1.cmml" xref="S7.SS1.3.p1.4.m4.1.1.1.1.1"><setdiff id="S7.SS1.3.p1.4.m4.1.1.1.1.1.1.cmml" xref="S7.SS1.3.p1.4.m4.1.1.1.1.1.1"></setdiff><ci id="S7.SS1.3.p1.4.m4.1.1.1.1.1.2.cmml" xref="S7.SS1.3.p1.4.m4.1.1.1.1.1.2">ℝ</ci><ci id="S7.SS1.3.p1.4.m4.1.1.1.1.1.3.cmml" xref="S7.SS1.3.p1.4.m4.1.1.1.1.1.3">𝐴</ci></apply></apply><ci id="S7.SS1.3.p1.4.m4.1.1.3.cmml" xref="S7.SS1.3.p1.4.m4.1.1.3">𝔠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.3.p1.4.m4.1c">|\mathbb{R}\setminus A|<\mathfrak{c}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.3.p1.4.m4.1d">| blackboard_R ∖ italic_A | < fraktur_c</annotation></semantics></math>, <math alttext="A=\mathbb{R}\cup E" class="ltx_Math" display="inline" id="S7.SS1.3.p1.5.m5.1"><semantics id="S7.SS1.3.p1.5.m5.1a"><mrow id="S7.SS1.3.p1.5.m5.1.1" xref="S7.SS1.3.p1.5.m5.1.1.cmml"><mi id="S7.SS1.3.p1.5.m5.1.1.2" xref="S7.SS1.3.p1.5.m5.1.1.2.cmml">A</mi><mo id="S7.SS1.3.p1.5.m5.1.1.1" xref="S7.SS1.3.p1.5.m5.1.1.1.cmml">=</mo><mrow id="S7.SS1.3.p1.5.m5.1.1.3" xref="S7.SS1.3.p1.5.m5.1.1.3.cmml"><mi id="S7.SS1.3.p1.5.m5.1.1.3.2" xref="S7.SS1.3.p1.5.m5.1.1.3.2.cmml">ℝ</mi><mo id="S7.SS1.3.p1.5.m5.1.1.3.1" xref="S7.SS1.3.p1.5.m5.1.1.3.1.cmml">∪</mo><mi id="S7.SS1.3.p1.5.m5.1.1.3.3" xref="S7.SS1.3.p1.5.m5.1.1.3.3.cmml">E</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.3.p1.5.m5.1b"><apply id="S7.SS1.3.p1.5.m5.1.1.cmml" xref="S7.SS1.3.p1.5.m5.1.1"><eq id="S7.SS1.3.p1.5.m5.1.1.1.cmml" xref="S7.SS1.3.p1.5.m5.1.1.1"></eq><ci id="S7.SS1.3.p1.5.m5.1.1.2.cmml" xref="S7.SS1.3.p1.5.m5.1.1.2">𝐴</ci><apply id="S7.SS1.3.p1.5.m5.1.1.3.cmml" xref="S7.SS1.3.p1.5.m5.1.1.3"><union id="S7.SS1.3.p1.5.m5.1.1.3.1.cmml" xref="S7.SS1.3.p1.5.m5.1.1.3.1"></union><ci id="S7.SS1.3.p1.5.m5.1.1.3.2.cmml" xref="S7.SS1.3.p1.5.m5.1.1.3.2">ℝ</ci><ci id="S7.SS1.3.p1.5.m5.1.1.3.3.cmml" xref="S7.SS1.3.p1.5.m5.1.1.3.3">𝐸</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.3.p1.5.m5.1c">A=\mathbb{R}\cup E</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.3.p1.5.m5.1d">italic_A = blackboard_R ∪ italic_E</annotation></semantics></math> where <math alttext="|E|<\mathfrak{c}" class="ltx_Math" display="inline" id="S7.SS1.3.p1.6.m6.1"><semantics id="S7.SS1.3.p1.6.m6.1a"><mrow id="S7.SS1.3.p1.6.m6.1.2" xref="S7.SS1.3.p1.6.m6.1.2.cmml"><mrow id="S7.SS1.3.p1.6.m6.1.2.2.2" xref="S7.SS1.3.p1.6.m6.1.2.2.1.cmml"><mo id="S7.SS1.3.p1.6.m6.1.2.2.2.1" stretchy="false" xref="S7.SS1.3.p1.6.m6.1.2.2.1.1.cmml">|</mo><mi id="S7.SS1.3.p1.6.m6.1.1" xref="S7.SS1.3.p1.6.m6.1.1.cmml">E</mi><mo id="S7.SS1.3.p1.6.m6.1.2.2.2.2" stretchy="false" xref="S7.SS1.3.p1.6.m6.1.2.2.1.1.cmml">|</mo></mrow><mo id="S7.SS1.3.p1.6.m6.1.2.1" xref="S7.SS1.3.p1.6.m6.1.2.1.cmml"><</mo><mi id="S7.SS1.3.p1.6.m6.1.2.3" xref="S7.SS1.3.p1.6.m6.1.2.3.cmml">𝔠</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.3.p1.6.m6.1b"><apply id="S7.SS1.3.p1.6.m6.1.2.cmml" xref="S7.SS1.3.p1.6.m6.1.2"><lt id="S7.SS1.3.p1.6.m6.1.2.1.cmml" xref="S7.SS1.3.p1.6.m6.1.2.1"></lt><apply id="S7.SS1.3.p1.6.m6.1.2.2.1.cmml" xref="S7.SS1.3.p1.6.m6.1.2.2.2"><abs id="S7.SS1.3.p1.6.m6.1.2.2.1.1.cmml" xref="S7.SS1.3.p1.6.m6.1.2.2.2.1"></abs><ci id="S7.SS1.3.p1.6.m6.1.1.cmml" xref="S7.SS1.3.p1.6.m6.1.1">𝐸</ci></apply><ci id="S7.SS1.3.p1.6.m6.1.2.3.cmml" xref="S7.SS1.3.p1.6.m6.1.2.3">𝔠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.3.p1.6.m6.1c">|E|<\mathfrak{c}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.3.p1.6.m6.1d">| italic_E | < fraktur_c</annotation></semantics></math>. Since <math alttext="\mathbb{R}" class="ltx_Math" display="inline" id="S7.SS1.3.p1.7.m7.1"><semantics id="S7.SS1.3.p1.7.m7.1a"><mi id="S7.SS1.3.p1.7.m7.1.1" xref="S7.SS1.3.p1.7.m7.1.1.cmml">ℝ</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.3.p1.7.m7.1b"><ci id="S7.SS1.3.p1.7.m7.1.1.cmml" xref="S7.SS1.3.p1.7.m7.1.1">ℝ</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.3.p1.7.m7.1c">\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.3.p1.7.m7.1d">blackboard_R</annotation></semantics></math> is complete (i.e., has no gaps), <math alttext="|G(A)|\leq|E|<\mathfrak{c}" class="ltx_Math" display="inline" id="S7.SS1.3.p1.8.m8.3"><semantics id="S7.SS1.3.p1.8.m8.3a"><mrow id="S7.SS1.3.p1.8.m8.3.3" xref="S7.SS1.3.p1.8.m8.3.3.cmml"><mrow id="S7.SS1.3.p1.8.m8.3.3.1.1" xref="S7.SS1.3.p1.8.m8.3.3.1.2.cmml"><mo id="S7.SS1.3.p1.8.m8.3.3.1.1.2" stretchy="false" xref="S7.SS1.3.p1.8.m8.3.3.1.2.1.cmml">|</mo><mrow id="S7.SS1.3.p1.8.m8.3.3.1.1.1" xref="S7.SS1.3.p1.8.m8.3.3.1.1.1.cmml"><mi id="S7.SS1.3.p1.8.m8.3.3.1.1.1.2" xref="S7.SS1.3.p1.8.m8.3.3.1.1.1.2.cmml">G</mi><mo id="S7.SS1.3.p1.8.m8.3.3.1.1.1.1" xref="S7.SS1.3.p1.8.m8.3.3.1.1.1.1.cmml">⁢</mo><mrow id="S7.SS1.3.p1.8.m8.3.3.1.1.1.3.2" xref="S7.SS1.3.p1.8.m8.3.3.1.1.1.cmml"><mo id="S7.SS1.3.p1.8.m8.3.3.1.1.1.3.2.1" stretchy="false" xref="S7.SS1.3.p1.8.m8.3.3.1.1.1.cmml">(</mo><mi id="S7.SS1.3.p1.8.m8.1.1" xref="S7.SS1.3.p1.8.m8.1.1.cmml">A</mi><mo id="S7.SS1.3.p1.8.m8.3.3.1.1.1.3.2.2" stretchy="false" xref="S7.SS1.3.p1.8.m8.3.3.1.1.1.cmml">)</mo></mrow></mrow><mo id="S7.SS1.3.p1.8.m8.3.3.1.1.3" stretchy="false" xref="S7.SS1.3.p1.8.m8.3.3.1.2.1.cmml">|</mo></mrow><mo id="S7.SS1.3.p1.8.m8.3.3.3" xref="S7.SS1.3.p1.8.m8.3.3.3.cmml">≤</mo><mrow id="S7.SS1.3.p1.8.m8.3.3.4.2" xref="S7.SS1.3.p1.8.m8.3.3.4.1.cmml"><mo id="S7.SS1.3.p1.8.m8.3.3.4.2.1" stretchy="false" xref="S7.SS1.3.p1.8.m8.3.3.4.1.1.cmml">|</mo><mi id="S7.SS1.3.p1.8.m8.2.2" xref="S7.SS1.3.p1.8.m8.2.2.cmml">E</mi><mo id="S7.SS1.3.p1.8.m8.3.3.4.2.2" stretchy="false" xref="S7.SS1.3.p1.8.m8.3.3.4.1.1.cmml">|</mo></mrow><mo id="S7.SS1.3.p1.8.m8.3.3.5" xref="S7.SS1.3.p1.8.m8.3.3.5.cmml"><</mo><mi id="S7.SS1.3.p1.8.m8.3.3.6" xref="S7.SS1.3.p1.8.m8.3.3.6.cmml">𝔠</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.3.p1.8.m8.3b"><apply id="S7.SS1.3.p1.8.m8.3.3.cmml" xref="S7.SS1.3.p1.8.m8.3.3"><and id="S7.SS1.3.p1.8.m8.3.3a.cmml" xref="S7.SS1.3.p1.8.m8.3.3"></and><apply id="S7.SS1.3.p1.8.m8.3.3b.cmml" xref="S7.SS1.3.p1.8.m8.3.3"><leq id="S7.SS1.3.p1.8.m8.3.3.3.cmml" xref="S7.SS1.3.p1.8.m8.3.3.3"></leq><apply id="S7.SS1.3.p1.8.m8.3.3.1.2.cmml" xref="S7.SS1.3.p1.8.m8.3.3.1.1"><abs id="S7.SS1.3.p1.8.m8.3.3.1.2.1.cmml" xref="S7.SS1.3.p1.8.m8.3.3.1.1.2"></abs><apply id="S7.SS1.3.p1.8.m8.3.3.1.1.1.cmml" xref="S7.SS1.3.p1.8.m8.3.3.1.1.1"><times id="S7.SS1.3.p1.8.m8.3.3.1.1.1.1.cmml" xref="S7.SS1.3.p1.8.m8.3.3.1.1.1.1"></times><ci id="S7.SS1.3.p1.8.m8.3.3.1.1.1.2.cmml" xref="S7.SS1.3.p1.8.m8.3.3.1.1.1.2">𝐺</ci><ci id="S7.SS1.3.p1.8.m8.1.1.cmml" xref="S7.SS1.3.p1.8.m8.1.1">𝐴</ci></apply></apply><apply id="S7.SS1.3.p1.8.m8.3.3.4.1.cmml" xref="S7.SS1.3.p1.8.m8.3.3.4.2"><abs id="S7.SS1.3.p1.8.m8.3.3.4.1.1.cmml" xref="S7.SS1.3.p1.8.m8.3.3.4.2.1"></abs><ci id="S7.SS1.3.p1.8.m8.2.2.cmml" xref="S7.SS1.3.p1.8.m8.2.2">𝐸</ci></apply></apply><apply id="S7.SS1.3.p1.8.m8.3.3c.cmml" xref="S7.SS1.3.p1.8.m8.3.3"><lt id="S7.SS1.3.p1.8.m8.3.3.5.cmml" xref="S7.SS1.3.p1.8.m8.3.3.5"></lt><share href="https://arxiv.org/html/2503.13728v1#S7.SS1.3.p1.8.m8.3.3.4.cmml" id="S7.SS1.3.p1.8.m8.3.3d.cmml" xref="S7.SS1.3.p1.8.m8.3.3"></share><ci id="S7.SS1.3.p1.8.m8.3.3.6.cmml" xref="S7.SS1.3.p1.8.m8.3.3.6">𝔠</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.3.p1.8.m8.3c">|G(A)|\leq|E|<\mathfrak{c}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.3.p1.8.m8.3d">| italic_G ( italic_A ) | ≤ | italic_E | < fraktur_c</annotation></semantics></math>. Then the previous proposition yields <math alttext="|G(B)|<\mathfrak{c}" class="ltx_Math" display="inline" id="S7.SS1.3.p1.9.m9.2"><semantics id="S7.SS1.3.p1.9.m9.2a"><mrow id="S7.SS1.3.p1.9.m9.2.2" xref="S7.SS1.3.p1.9.m9.2.2.cmml"><mrow id="S7.SS1.3.p1.9.m9.2.2.1.1" xref="S7.SS1.3.p1.9.m9.2.2.1.2.cmml"><mo id="S7.SS1.3.p1.9.m9.2.2.1.1.2" stretchy="false" xref="S7.SS1.3.p1.9.m9.2.2.1.2.1.cmml">|</mo><mrow id="S7.SS1.3.p1.9.m9.2.2.1.1.1" xref="S7.SS1.3.p1.9.m9.2.2.1.1.1.cmml"><mi id="S7.SS1.3.p1.9.m9.2.2.1.1.1.2" xref="S7.SS1.3.p1.9.m9.2.2.1.1.1.2.cmml">G</mi><mo id="S7.SS1.3.p1.9.m9.2.2.1.1.1.1" xref="S7.SS1.3.p1.9.m9.2.2.1.1.1.1.cmml">⁢</mo><mrow id="S7.SS1.3.p1.9.m9.2.2.1.1.1.3.2" xref="S7.SS1.3.p1.9.m9.2.2.1.1.1.cmml"><mo id="S7.SS1.3.p1.9.m9.2.2.1.1.1.3.2.1" stretchy="false" xref="S7.SS1.3.p1.9.m9.2.2.1.1.1.cmml">(</mo><mi id="S7.SS1.3.p1.9.m9.1.1" xref="S7.SS1.3.p1.9.m9.1.1.cmml">B</mi><mo id="S7.SS1.3.p1.9.m9.2.2.1.1.1.3.2.2" stretchy="false" xref="S7.SS1.3.p1.9.m9.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S7.SS1.3.p1.9.m9.2.2.1.1.3" stretchy="false" xref="S7.SS1.3.p1.9.m9.2.2.1.2.1.cmml">|</mo></mrow><mo id="S7.SS1.3.p1.9.m9.2.2.2" xref="S7.SS1.3.p1.9.m9.2.2.2.cmml"><</mo><mi id="S7.SS1.3.p1.9.m9.2.2.3" xref="S7.SS1.3.p1.9.m9.2.2.3.cmml">𝔠</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.3.p1.9.m9.2b"><apply id="S7.SS1.3.p1.9.m9.2.2.cmml" xref="S7.SS1.3.p1.9.m9.2.2"><lt id="S7.SS1.3.p1.9.m9.2.2.2.cmml" xref="S7.SS1.3.p1.9.m9.2.2.2"></lt><apply id="S7.SS1.3.p1.9.m9.2.2.1.2.cmml" xref="S7.SS1.3.p1.9.m9.2.2.1.1"><abs id="S7.SS1.3.p1.9.m9.2.2.1.2.1.cmml" xref="S7.SS1.3.p1.9.m9.2.2.1.1.2"></abs><apply id="S7.SS1.3.p1.9.m9.2.2.1.1.1.cmml" xref="S7.SS1.3.p1.9.m9.2.2.1.1.1"><times id="S7.SS1.3.p1.9.m9.2.2.1.1.1.1.cmml" xref="S7.SS1.3.p1.9.m9.2.2.1.1.1.1"></times><ci id="S7.SS1.3.p1.9.m9.2.2.1.1.1.2.cmml" xref="S7.SS1.3.p1.9.m9.2.2.1.1.1.2">𝐺</ci><ci id="S7.SS1.3.p1.9.m9.1.1.cmml" xref="S7.SS1.3.p1.9.m9.1.1">𝐵</ci></apply></apply><ci id="S7.SS1.3.p1.9.m9.2.2.3.cmml" xref="S7.SS1.3.p1.9.m9.2.2.3">𝔠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.3.p1.9.m9.2c">|G(B)|<\mathfrak{c}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.3.p1.9.m9.2d">| italic_G ( italic_B ) | < fraktur_c</annotation></semantics></math> also. Now suppose towards a contradiction that <math alttext="\aleph_{0}<|B|<\mathfrak{c}" class="ltx_Math" display="inline" id="S7.SS1.3.p1.10.m10.1"><semantics id="S7.SS1.3.p1.10.m10.1a"><mrow id="S7.SS1.3.p1.10.m10.1.2" xref="S7.SS1.3.p1.10.m10.1.2.cmml"><msub id="S7.SS1.3.p1.10.m10.1.2.2" xref="S7.SS1.3.p1.10.m10.1.2.2.cmml"><mi id="S7.SS1.3.p1.10.m10.1.2.2.2" mathvariant="normal" xref="S7.SS1.3.p1.10.m10.1.2.2.2.cmml">ℵ</mi><mn id="S7.SS1.3.p1.10.m10.1.2.2.3" xref="S7.SS1.3.p1.10.m10.1.2.2.3.cmml">0</mn></msub><mo id="S7.SS1.3.p1.10.m10.1.2.3" xref="S7.SS1.3.p1.10.m10.1.2.3.cmml"><</mo><mrow id="S7.SS1.3.p1.10.m10.1.2.4.2" xref="S7.SS1.3.p1.10.m10.1.2.4.1.cmml"><mo id="S7.SS1.3.p1.10.m10.1.2.4.2.1" stretchy="false" xref="S7.SS1.3.p1.10.m10.1.2.4.1.1.cmml">|</mo><mi id="S7.SS1.3.p1.10.m10.1.1" xref="S7.SS1.3.p1.10.m10.1.1.cmml">B</mi><mo id="S7.SS1.3.p1.10.m10.1.2.4.2.2" stretchy="false" xref="S7.SS1.3.p1.10.m10.1.2.4.1.1.cmml">|</mo></mrow><mo id="S7.SS1.3.p1.10.m10.1.2.5" xref="S7.SS1.3.p1.10.m10.1.2.5.cmml"><</mo><mi id="S7.SS1.3.p1.10.m10.1.2.6" xref="S7.SS1.3.p1.10.m10.1.2.6.cmml">𝔠</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.3.p1.10.m10.1b"><apply id="S7.SS1.3.p1.10.m10.1.2.cmml" xref="S7.SS1.3.p1.10.m10.1.2"><and id="S7.SS1.3.p1.10.m10.1.2a.cmml" xref="S7.SS1.3.p1.10.m10.1.2"></and><apply id="S7.SS1.3.p1.10.m10.1.2b.cmml" xref="S7.SS1.3.p1.10.m10.1.2"><lt id="S7.SS1.3.p1.10.m10.1.2.3.cmml" xref="S7.SS1.3.p1.10.m10.1.2.3"></lt><apply id="S7.SS1.3.p1.10.m10.1.2.2.cmml" xref="S7.SS1.3.p1.10.m10.1.2.2"><csymbol cd="ambiguous" id="S7.SS1.3.p1.10.m10.1.2.2.1.cmml" xref="S7.SS1.3.p1.10.m10.1.2.2">subscript</csymbol><ci id="S7.SS1.3.p1.10.m10.1.2.2.2.cmml" xref="S7.SS1.3.p1.10.m10.1.2.2.2">ℵ</ci><cn id="S7.SS1.3.p1.10.m10.1.2.2.3.cmml" type="integer" xref="S7.SS1.3.p1.10.m10.1.2.2.3">0</cn></apply><apply id="S7.SS1.3.p1.10.m10.1.2.4.1.cmml" xref="S7.SS1.3.p1.10.m10.1.2.4.2"><abs id="S7.SS1.3.p1.10.m10.1.2.4.1.1.cmml" xref="S7.SS1.3.p1.10.m10.1.2.4.2.1"></abs><ci id="S7.SS1.3.p1.10.m10.1.1.cmml" xref="S7.SS1.3.p1.10.m10.1.1">𝐵</ci></apply></apply><apply id="S7.SS1.3.p1.10.m10.1.2c.cmml" xref="S7.SS1.3.p1.10.m10.1.2"><lt id="S7.SS1.3.p1.10.m10.1.2.5.cmml" xref="S7.SS1.3.p1.10.m10.1.2.5"></lt><share href="https://arxiv.org/html/2503.13728v1#S7.SS1.3.p1.10.m10.1.2.4.cmml" id="S7.SS1.3.p1.10.m10.1.2d.cmml" xref="S7.SS1.3.p1.10.m10.1.2"></share><ci id="S7.SS1.3.p1.10.m10.1.2.6.cmml" xref="S7.SS1.3.p1.10.m10.1.2.6">𝔠</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.3.p1.10.m10.1c">\aleph_{0}<|B|<\mathfrak{c}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.3.p1.10.m10.1d">roman_ℵ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT < | italic_B | < fraktur_c</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S7.SS1.4.p2"> <p class="ltx_p" id="S7.SS1.4.p2.16">Since <math alttext="B" class="ltx_Math" display="inline" id="S7.SS1.4.p2.1.m1.1"><semantics id="S7.SS1.4.p2.1.m1.1a"><mi id="S7.SS1.4.p2.1.m1.1.1" xref="S7.SS1.4.p2.1.m1.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.4.p2.1.m1.1b"><ci id="S7.SS1.4.p2.1.m1.1.1.cmml" xref="S7.SS1.4.p2.1.m1.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.4.p2.1.m1.1c">B</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.4.p2.1.m1.1d">italic_B</annotation></semantics></math> is uncountable, there is <math alttext="Q\subseteq B" class="ltx_Math" display="inline" id="S7.SS1.4.p2.2.m2.1"><semantics id="S7.SS1.4.p2.2.m2.1a"><mrow id="S7.SS1.4.p2.2.m2.1.1" xref="S7.SS1.4.p2.2.m2.1.1.cmml"><mi id="S7.SS1.4.p2.2.m2.1.1.2" xref="S7.SS1.4.p2.2.m2.1.1.2.cmml">Q</mi><mo id="S7.SS1.4.p2.2.m2.1.1.1" xref="S7.SS1.4.p2.2.m2.1.1.1.cmml">⊆</mo><mi id="S7.SS1.4.p2.2.m2.1.1.3" xref="S7.SS1.4.p2.2.m2.1.1.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.4.p2.2.m2.1b"><apply id="S7.SS1.4.p2.2.m2.1.1.cmml" xref="S7.SS1.4.p2.2.m2.1.1"><subset id="S7.SS1.4.p2.2.m2.1.1.1.cmml" xref="S7.SS1.4.p2.2.m2.1.1.1"></subset><ci id="S7.SS1.4.p2.2.m2.1.1.2.cmml" xref="S7.SS1.4.p2.2.m2.1.1.2">𝑄</ci><ci id="S7.SS1.4.p2.2.m2.1.1.3.cmml" xref="S7.SS1.4.p2.2.m2.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.4.p2.2.m2.1c">Q\subseteq B</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.4.p2.2.m2.1d">italic_Q ⊆ italic_B</annotation></semantics></math> such that <math alttext="Q\cong\mathbb{Q}" class="ltx_Math" display="inline" id="S7.SS1.4.p2.3.m3.1"><semantics id="S7.SS1.4.p2.3.m3.1a"><mrow id="S7.SS1.4.p2.3.m3.1.1" xref="S7.SS1.4.p2.3.m3.1.1.cmml"><mi id="S7.SS1.4.p2.3.m3.1.1.2" xref="S7.SS1.4.p2.3.m3.1.1.2.cmml">Q</mi><mo id="S7.SS1.4.p2.3.m3.1.1.1" xref="S7.SS1.4.p2.3.m3.1.1.1.cmml">≅</mo><mi id="S7.SS1.4.p2.3.m3.1.1.3" xref="S7.SS1.4.p2.3.m3.1.1.3.cmml">ℚ</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.4.p2.3.m3.1b"><apply id="S7.SS1.4.p2.3.m3.1.1.cmml" xref="S7.SS1.4.p2.3.m3.1.1"><approx id="S7.SS1.4.p2.3.m3.1.1.1.cmml" xref="S7.SS1.4.p2.3.m3.1.1.1"></approx><ci id="S7.SS1.4.p2.3.m3.1.1.2.cmml" xref="S7.SS1.4.p2.3.m3.1.1.2">𝑄</ci><ci id="S7.SS1.4.p2.3.m3.1.1.3.cmml" xref="S7.SS1.4.p2.3.m3.1.1.3">ℚ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.4.p2.3.m3.1c">Q\cong\mathbb{Q}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.4.p2.3.m3.1d">italic_Q ≅ blackboard_Q</annotation></semantics></math>. It is well known that <math alttext="|G(\mathbb{Q})|=\mathfrak{c}" class="ltx_Math" display="inline" id="S7.SS1.4.p2.4.m4.2"><semantics id="S7.SS1.4.p2.4.m4.2a"><mrow id="S7.SS1.4.p2.4.m4.2.2" xref="S7.SS1.4.p2.4.m4.2.2.cmml"><mrow id="S7.SS1.4.p2.4.m4.2.2.1.1" xref="S7.SS1.4.p2.4.m4.2.2.1.2.cmml"><mo id="S7.SS1.4.p2.4.m4.2.2.1.1.2" stretchy="false" xref="S7.SS1.4.p2.4.m4.2.2.1.2.1.cmml">|</mo><mrow id="S7.SS1.4.p2.4.m4.2.2.1.1.1" xref="S7.SS1.4.p2.4.m4.2.2.1.1.1.cmml"><mi id="S7.SS1.4.p2.4.m4.2.2.1.1.1.2" xref="S7.SS1.4.p2.4.m4.2.2.1.1.1.2.cmml">G</mi><mo id="S7.SS1.4.p2.4.m4.2.2.1.1.1.1" xref="S7.SS1.4.p2.4.m4.2.2.1.1.1.1.cmml">⁢</mo><mrow id="S7.SS1.4.p2.4.m4.2.2.1.1.1.3.2" xref="S7.SS1.4.p2.4.m4.2.2.1.1.1.cmml"><mo id="S7.SS1.4.p2.4.m4.2.2.1.1.1.3.2.1" stretchy="false" xref="S7.SS1.4.p2.4.m4.2.2.1.1.1.cmml">(</mo><mi id="S7.SS1.4.p2.4.m4.1.1" xref="S7.SS1.4.p2.4.m4.1.1.cmml">ℚ</mi><mo id="S7.SS1.4.p2.4.m4.2.2.1.1.1.3.2.2" stretchy="false" xref="S7.SS1.4.p2.4.m4.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S7.SS1.4.p2.4.m4.2.2.1.1.3" stretchy="false" xref="S7.SS1.4.p2.4.m4.2.2.1.2.1.cmml">|</mo></mrow><mo id="S7.SS1.4.p2.4.m4.2.2.2" xref="S7.SS1.4.p2.4.m4.2.2.2.cmml">=</mo><mi id="S7.SS1.4.p2.4.m4.2.2.3" xref="S7.SS1.4.p2.4.m4.2.2.3.cmml">𝔠</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.4.p2.4.m4.2b"><apply id="S7.SS1.4.p2.4.m4.2.2.cmml" xref="S7.SS1.4.p2.4.m4.2.2"><eq id="S7.SS1.4.p2.4.m4.2.2.2.cmml" xref="S7.SS1.4.p2.4.m4.2.2.2"></eq><apply id="S7.SS1.4.p2.4.m4.2.2.1.2.cmml" xref="S7.SS1.4.p2.4.m4.2.2.1.1"><abs id="S7.SS1.4.p2.4.m4.2.2.1.2.1.cmml" xref="S7.SS1.4.p2.4.m4.2.2.1.1.2"></abs><apply id="S7.SS1.4.p2.4.m4.2.2.1.1.1.cmml" xref="S7.SS1.4.p2.4.m4.2.2.1.1.1"><times id="S7.SS1.4.p2.4.m4.2.2.1.1.1.1.cmml" xref="S7.SS1.4.p2.4.m4.2.2.1.1.1.1"></times><ci id="S7.SS1.4.p2.4.m4.2.2.1.1.1.2.cmml" xref="S7.SS1.4.p2.4.m4.2.2.1.1.1.2">𝐺</ci><ci id="S7.SS1.4.p2.4.m4.1.1.cmml" xref="S7.SS1.4.p2.4.m4.1.1">ℚ</ci></apply></apply><ci id="S7.SS1.4.p2.4.m4.2.2.3.cmml" xref="S7.SS1.4.p2.4.m4.2.2.3">𝔠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.4.p2.4.m4.2c">|G(\mathbb{Q})|=\mathfrak{c}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.4.p2.4.m4.2d">| italic_G ( blackboard_Q ) | = fraktur_c</annotation></semantics></math>, thus <math alttext="|G(Q)|=\mathfrak{c}" class="ltx_Math" display="inline" id="S7.SS1.4.p2.5.m5.2"><semantics id="S7.SS1.4.p2.5.m5.2a"><mrow id="S7.SS1.4.p2.5.m5.2.2" xref="S7.SS1.4.p2.5.m5.2.2.cmml"><mrow id="S7.SS1.4.p2.5.m5.2.2.1.1" xref="S7.SS1.4.p2.5.m5.2.2.1.2.cmml"><mo id="S7.SS1.4.p2.5.m5.2.2.1.1.2" stretchy="false" xref="S7.SS1.4.p2.5.m5.2.2.1.2.1.cmml">|</mo><mrow id="S7.SS1.4.p2.5.m5.2.2.1.1.1" xref="S7.SS1.4.p2.5.m5.2.2.1.1.1.cmml"><mi id="S7.SS1.4.p2.5.m5.2.2.1.1.1.2" xref="S7.SS1.4.p2.5.m5.2.2.1.1.1.2.cmml">G</mi><mo id="S7.SS1.4.p2.5.m5.2.2.1.1.1.1" xref="S7.SS1.4.p2.5.m5.2.2.1.1.1.1.cmml">⁢</mo><mrow id="S7.SS1.4.p2.5.m5.2.2.1.1.1.3.2" xref="S7.SS1.4.p2.5.m5.2.2.1.1.1.cmml"><mo id="S7.SS1.4.p2.5.m5.2.2.1.1.1.3.2.1" stretchy="false" xref="S7.SS1.4.p2.5.m5.2.2.1.1.1.cmml">(</mo><mi id="S7.SS1.4.p2.5.m5.1.1" xref="S7.SS1.4.p2.5.m5.1.1.cmml">Q</mi><mo id="S7.SS1.4.p2.5.m5.2.2.1.1.1.3.2.2" stretchy="false" xref="S7.SS1.4.p2.5.m5.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S7.SS1.4.p2.5.m5.2.2.1.1.3" stretchy="false" xref="S7.SS1.4.p2.5.m5.2.2.1.2.1.cmml">|</mo></mrow><mo id="S7.SS1.4.p2.5.m5.2.2.2" xref="S7.SS1.4.p2.5.m5.2.2.2.cmml">=</mo><mi id="S7.SS1.4.p2.5.m5.2.2.3" xref="S7.SS1.4.p2.5.m5.2.2.3.cmml">𝔠</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.4.p2.5.m5.2b"><apply id="S7.SS1.4.p2.5.m5.2.2.cmml" xref="S7.SS1.4.p2.5.m5.2.2"><eq id="S7.SS1.4.p2.5.m5.2.2.2.cmml" xref="S7.SS1.4.p2.5.m5.2.2.2"></eq><apply id="S7.SS1.4.p2.5.m5.2.2.1.2.cmml" xref="S7.SS1.4.p2.5.m5.2.2.1.1"><abs id="S7.SS1.4.p2.5.m5.2.2.1.2.1.cmml" xref="S7.SS1.4.p2.5.m5.2.2.1.1.2"></abs><apply id="S7.SS1.4.p2.5.m5.2.2.1.1.1.cmml" xref="S7.SS1.4.p2.5.m5.2.2.1.1.1"><times id="S7.SS1.4.p2.5.m5.2.2.1.1.1.1.cmml" xref="S7.SS1.4.p2.5.m5.2.2.1.1.1.1"></times><ci id="S7.SS1.4.p2.5.m5.2.2.1.1.1.2.cmml" xref="S7.SS1.4.p2.5.m5.2.2.1.1.1.2">𝐺</ci><ci id="S7.SS1.4.p2.5.m5.1.1.cmml" xref="S7.SS1.4.p2.5.m5.1.1">𝑄</ci></apply></apply><ci id="S7.SS1.4.p2.5.m5.2.2.3.cmml" xref="S7.SS1.4.p2.5.m5.2.2.3">𝔠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.4.p2.5.m5.2c">|G(Q)|=\mathfrak{c}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.4.p2.5.m5.2d">| italic_G ( italic_Q ) | = fraktur_c</annotation></semantics></math> also. For <math alttext="(X,Y)\in G(Q)" class="ltx_Math" display="inline" id="S7.SS1.4.p2.6.m6.3"><semantics id="S7.SS1.4.p2.6.m6.3a"><mrow id="S7.SS1.4.p2.6.m6.3.4" xref="S7.SS1.4.p2.6.m6.3.4.cmml"><mrow id="S7.SS1.4.p2.6.m6.3.4.2.2" xref="S7.SS1.4.p2.6.m6.3.4.2.1.cmml"><mo id="S7.SS1.4.p2.6.m6.3.4.2.2.1" stretchy="false" xref="S7.SS1.4.p2.6.m6.3.4.2.1.cmml">(</mo><mi id="S7.SS1.4.p2.6.m6.1.1" xref="S7.SS1.4.p2.6.m6.1.1.cmml">X</mi><mo id="S7.SS1.4.p2.6.m6.3.4.2.2.2" xref="S7.SS1.4.p2.6.m6.3.4.2.1.cmml">,</mo><mi id="S7.SS1.4.p2.6.m6.2.2" xref="S7.SS1.4.p2.6.m6.2.2.cmml">Y</mi><mo id="S7.SS1.4.p2.6.m6.3.4.2.2.3" stretchy="false" xref="S7.SS1.4.p2.6.m6.3.4.2.1.cmml">)</mo></mrow><mo id="S7.SS1.4.p2.6.m6.3.4.1" xref="S7.SS1.4.p2.6.m6.3.4.1.cmml">∈</mo><mrow id="S7.SS1.4.p2.6.m6.3.4.3" xref="S7.SS1.4.p2.6.m6.3.4.3.cmml"><mi id="S7.SS1.4.p2.6.m6.3.4.3.2" xref="S7.SS1.4.p2.6.m6.3.4.3.2.cmml">G</mi><mo id="S7.SS1.4.p2.6.m6.3.4.3.1" xref="S7.SS1.4.p2.6.m6.3.4.3.1.cmml">⁢</mo><mrow id="S7.SS1.4.p2.6.m6.3.4.3.3.2" xref="S7.SS1.4.p2.6.m6.3.4.3.cmml"><mo id="S7.SS1.4.p2.6.m6.3.4.3.3.2.1" stretchy="false" xref="S7.SS1.4.p2.6.m6.3.4.3.cmml">(</mo><mi id="S7.SS1.4.p2.6.m6.3.3" xref="S7.SS1.4.p2.6.m6.3.3.cmml">Q</mi><mo id="S7.SS1.4.p2.6.m6.3.4.3.3.2.2" stretchy="false" xref="S7.SS1.4.p2.6.m6.3.4.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.4.p2.6.m6.3b"><apply id="S7.SS1.4.p2.6.m6.3.4.cmml" xref="S7.SS1.4.p2.6.m6.3.4"><in id="S7.SS1.4.p2.6.m6.3.4.1.cmml" xref="S7.SS1.4.p2.6.m6.3.4.1"></in><interval closure="open" id="S7.SS1.4.p2.6.m6.3.4.2.1.cmml" xref="S7.SS1.4.p2.6.m6.3.4.2.2"><ci id="S7.SS1.4.p2.6.m6.1.1.cmml" xref="S7.SS1.4.p2.6.m6.1.1">𝑋</ci><ci id="S7.SS1.4.p2.6.m6.2.2.cmml" xref="S7.SS1.4.p2.6.m6.2.2">𝑌</ci></interval><apply id="S7.SS1.4.p2.6.m6.3.4.3.cmml" xref="S7.SS1.4.p2.6.m6.3.4.3"><times id="S7.SS1.4.p2.6.m6.3.4.3.1.cmml" xref="S7.SS1.4.p2.6.m6.3.4.3.1"></times><ci id="S7.SS1.4.p2.6.m6.3.4.3.2.cmml" xref="S7.SS1.4.p2.6.m6.3.4.3.2">𝐺</ci><ci id="S7.SS1.4.p2.6.m6.3.3.cmml" xref="S7.SS1.4.p2.6.m6.3.3">𝑄</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.4.p2.6.m6.3c">(X,Y)\in G(Q)</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.4.p2.6.m6.3d">( italic_X , italic_Y ) ∈ italic_G ( italic_Q )</annotation></semantics></math> let <math alttext="b(X,Y)" class="ltx_Math" display="inline" id="S7.SS1.4.p2.7.m7.2"><semantics id="S7.SS1.4.p2.7.m7.2a"><mrow id="S7.SS1.4.p2.7.m7.2.3" xref="S7.SS1.4.p2.7.m7.2.3.cmml"><mi id="S7.SS1.4.p2.7.m7.2.3.2" xref="S7.SS1.4.p2.7.m7.2.3.2.cmml">b</mi><mo id="S7.SS1.4.p2.7.m7.2.3.1" xref="S7.SS1.4.p2.7.m7.2.3.1.cmml">⁢</mo><mrow id="S7.SS1.4.p2.7.m7.2.3.3.2" xref="S7.SS1.4.p2.7.m7.2.3.3.1.cmml"><mo id="S7.SS1.4.p2.7.m7.2.3.3.2.1" stretchy="false" xref="S7.SS1.4.p2.7.m7.2.3.3.1.cmml">(</mo><mi id="S7.SS1.4.p2.7.m7.1.1" xref="S7.SS1.4.p2.7.m7.1.1.cmml">X</mi><mo id="S7.SS1.4.p2.7.m7.2.3.3.2.2" xref="S7.SS1.4.p2.7.m7.2.3.3.1.cmml">,</mo><mi id="S7.SS1.4.p2.7.m7.2.2" xref="S7.SS1.4.p2.7.m7.2.2.cmml">Y</mi><mo id="S7.SS1.4.p2.7.m7.2.3.3.2.3" stretchy="false" xref="S7.SS1.4.p2.7.m7.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.4.p2.7.m7.2b"><apply id="S7.SS1.4.p2.7.m7.2.3.cmml" xref="S7.SS1.4.p2.7.m7.2.3"><times id="S7.SS1.4.p2.7.m7.2.3.1.cmml" xref="S7.SS1.4.p2.7.m7.2.3.1"></times><ci id="S7.SS1.4.p2.7.m7.2.3.2.cmml" xref="S7.SS1.4.p2.7.m7.2.3.2">𝑏</ci><interval closure="open" id="S7.SS1.4.p2.7.m7.2.3.3.1.cmml" xref="S7.SS1.4.p2.7.m7.2.3.3.2"><ci id="S7.SS1.4.p2.7.m7.1.1.cmml" xref="S7.SS1.4.p2.7.m7.1.1">𝑋</ci><ci id="S7.SS1.4.p2.7.m7.2.2.cmml" xref="S7.SS1.4.p2.7.m7.2.2">𝑌</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.4.p2.7.m7.2c">b(X,Y)</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.4.p2.7.m7.2d">italic_b ( italic_X , italic_Y )</annotation></semantics></math> be any <math alttext="b\in B" class="ltx_Math" display="inline" id="S7.SS1.4.p2.8.m8.1"><semantics id="S7.SS1.4.p2.8.m8.1a"><mrow id="S7.SS1.4.p2.8.m8.1.1" xref="S7.SS1.4.p2.8.m8.1.1.cmml"><mi id="S7.SS1.4.p2.8.m8.1.1.2" xref="S7.SS1.4.p2.8.m8.1.1.2.cmml">b</mi><mo id="S7.SS1.4.p2.8.m8.1.1.1" xref="S7.SS1.4.p2.8.m8.1.1.1.cmml">∈</mo><mi id="S7.SS1.4.p2.8.m8.1.1.3" xref="S7.SS1.4.p2.8.m8.1.1.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.4.p2.8.m8.1b"><apply id="S7.SS1.4.p2.8.m8.1.1.cmml" xref="S7.SS1.4.p2.8.m8.1.1"><in id="S7.SS1.4.p2.8.m8.1.1.1.cmml" xref="S7.SS1.4.p2.8.m8.1.1.1"></in><ci id="S7.SS1.4.p2.8.m8.1.1.2.cmml" xref="S7.SS1.4.p2.8.m8.1.1.2">𝑏</ci><ci id="S7.SS1.4.p2.8.m8.1.1.3.cmml" xref="S7.SS1.4.p2.8.m8.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.4.p2.8.m8.1c">b\in B</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.4.p2.8.m8.1d">italic_b ∈ italic_B</annotation></semantics></math> such that for all <math alttext="x\in X" class="ltx_Math" display="inline" id="S7.SS1.4.p2.9.m9.1"><semantics id="S7.SS1.4.p2.9.m9.1a"><mrow id="S7.SS1.4.p2.9.m9.1.1" xref="S7.SS1.4.p2.9.m9.1.1.cmml"><mi id="S7.SS1.4.p2.9.m9.1.1.2" xref="S7.SS1.4.p2.9.m9.1.1.2.cmml">x</mi><mo id="S7.SS1.4.p2.9.m9.1.1.1" xref="S7.SS1.4.p2.9.m9.1.1.1.cmml">∈</mo><mi id="S7.SS1.4.p2.9.m9.1.1.3" xref="S7.SS1.4.p2.9.m9.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.4.p2.9.m9.1b"><apply id="S7.SS1.4.p2.9.m9.1.1.cmml" xref="S7.SS1.4.p2.9.m9.1.1"><in id="S7.SS1.4.p2.9.m9.1.1.1.cmml" xref="S7.SS1.4.p2.9.m9.1.1.1"></in><ci id="S7.SS1.4.p2.9.m9.1.1.2.cmml" xref="S7.SS1.4.p2.9.m9.1.1.2">𝑥</ci><ci id="S7.SS1.4.p2.9.m9.1.1.3.cmml" xref="S7.SS1.4.p2.9.m9.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.4.p2.9.m9.1c">x\in X</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.4.p2.9.m9.1d">italic_x ∈ italic_X</annotation></semantics></math> and <math alttext="y\in Y" class="ltx_Math" display="inline" id="S7.SS1.4.p2.10.m10.1"><semantics id="S7.SS1.4.p2.10.m10.1a"><mrow id="S7.SS1.4.p2.10.m10.1.1" xref="S7.SS1.4.p2.10.m10.1.1.cmml"><mi id="S7.SS1.4.p2.10.m10.1.1.2" xref="S7.SS1.4.p2.10.m10.1.1.2.cmml">y</mi><mo id="S7.SS1.4.p2.10.m10.1.1.1" xref="S7.SS1.4.p2.10.m10.1.1.1.cmml">∈</mo><mi id="S7.SS1.4.p2.10.m10.1.1.3" xref="S7.SS1.4.p2.10.m10.1.1.3.cmml">Y</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.4.p2.10.m10.1b"><apply id="S7.SS1.4.p2.10.m10.1.1.cmml" xref="S7.SS1.4.p2.10.m10.1.1"><in id="S7.SS1.4.p2.10.m10.1.1.1.cmml" xref="S7.SS1.4.p2.10.m10.1.1.1"></in><ci id="S7.SS1.4.p2.10.m10.1.1.2.cmml" xref="S7.SS1.4.p2.10.m10.1.1.2">𝑦</ci><ci id="S7.SS1.4.p2.10.m10.1.1.3.cmml" xref="S7.SS1.4.p2.10.m10.1.1.3">𝑌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.4.p2.10.m10.1c">y\in Y</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.4.p2.10.m10.1d">italic_y ∈ italic_Y</annotation></semantics></math>, <math alttext="X\leq b\leq Y" class="ltx_Math" display="inline" id="S7.SS1.4.p2.11.m11.1"><semantics id="S7.SS1.4.p2.11.m11.1a"><mrow id="S7.SS1.4.p2.11.m11.1.1" xref="S7.SS1.4.p2.11.m11.1.1.cmml"><mi id="S7.SS1.4.p2.11.m11.1.1.2" xref="S7.SS1.4.p2.11.m11.1.1.2.cmml">X</mi><mo id="S7.SS1.4.p2.11.m11.1.1.3" xref="S7.SS1.4.p2.11.m11.1.1.3.cmml">≤</mo><mi id="S7.SS1.4.p2.11.m11.1.1.4" xref="S7.SS1.4.p2.11.m11.1.1.4.cmml">b</mi><mo id="S7.SS1.4.p2.11.m11.1.1.5" xref="S7.SS1.4.p2.11.m11.1.1.5.cmml">≤</mo><mi id="S7.SS1.4.p2.11.m11.1.1.6" xref="S7.SS1.4.p2.11.m11.1.1.6.cmml">Y</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.4.p2.11.m11.1b"><apply id="S7.SS1.4.p2.11.m11.1.1.cmml" xref="S7.SS1.4.p2.11.m11.1.1"><and id="S7.SS1.4.p2.11.m11.1.1a.cmml" xref="S7.SS1.4.p2.11.m11.1.1"></and><apply id="S7.SS1.4.p2.11.m11.1.1b.cmml" xref="S7.SS1.4.p2.11.m11.1.1"><leq id="S7.SS1.4.p2.11.m11.1.1.3.cmml" xref="S7.SS1.4.p2.11.m11.1.1.3"></leq><ci id="S7.SS1.4.p2.11.m11.1.1.2.cmml" xref="S7.SS1.4.p2.11.m11.1.1.2">𝑋</ci><ci id="S7.SS1.4.p2.11.m11.1.1.4.cmml" xref="S7.SS1.4.p2.11.m11.1.1.4">𝑏</ci></apply><apply id="S7.SS1.4.p2.11.m11.1.1c.cmml" xref="S7.SS1.4.p2.11.m11.1.1"><leq id="S7.SS1.4.p2.11.m11.1.1.5.cmml" xref="S7.SS1.4.p2.11.m11.1.1.5"></leq><share href="https://arxiv.org/html/2503.13728v1#S7.SS1.4.p2.11.m11.1.1.4.cmml" id="S7.SS1.4.p2.11.m11.1.1d.cmml" xref="S7.SS1.4.p2.11.m11.1.1"></share><ci id="S7.SS1.4.p2.11.m11.1.1.6.cmml" xref="S7.SS1.4.p2.11.m11.1.1.6">𝑌</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.4.p2.11.m11.1c">X\leq b\leq Y</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.4.p2.11.m11.1d">italic_X ≤ italic_b ≤ italic_Y</annotation></semantics></math>, if it exists. Then <math alttext="b(X,Y)" class="ltx_Math" display="inline" id="S7.SS1.4.p2.12.m12.2"><semantics id="S7.SS1.4.p2.12.m12.2a"><mrow id="S7.SS1.4.p2.12.m12.2.3" xref="S7.SS1.4.p2.12.m12.2.3.cmml"><mi id="S7.SS1.4.p2.12.m12.2.3.2" xref="S7.SS1.4.p2.12.m12.2.3.2.cmml">b</mi><mo id="S7.SS1.4.p2.12.m12.2.3.1" xref="S7.SS1.4.p2.12.m12.2.3.1.cmml">⁢</mo><mrow id="S7.SS1.4.p2.12.m12.2.3.3.2" xref="S7.SS1.4.p2.12.m12.2.3.3.1.cmml"><mo id="S7.SS1.4.p2.12.m12.2.3.3.2.1" stretchy="false" xref="S7.SS1.4.p2.12.m12.2.3.3.1.cmml">(</mo><mi id="S7.SS1.4.p2.12.m12.1.1" xref="S7.SS1.4.p2.12.m12.1.1.cmml">X</mi><mo id="S7.SS1.4.p2.12.m12.2.3.3.2.2" xref="S7.SS1.4.p2.12.m12.2.3.3.1.cmml">,</mo><mi id="S7.SS1.4.p2.12.m12.2.2" xref="S7.SS1.4.p2.12.m12.2.2.cmml">Y</mi><mo id="S7.SS1.4.p2.12.m12.2.3.3.2.3" stretchy="false" xref="S7.SS1.4.p2.12.m12.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.4.p2.12.m12.2b"><apply id="S7.SS1.4.p2.12.m12.2.3.cmml" xref="S7.SS1.4.p2.12.m12.2.3"><times id="S7.SS1.4.p2.12.m12.2.3.1.cmml" xref="S7.SS1.4.p2.12.m12.2.3.1"></times><ci id="S7.SS1.4.p2.12.m12.2.3.2.cmml" xref="S7.SS1.4.p2.12.m12.2.3.2">𝑏</ci><interval closure="open" id="S7.SS1.4.p2.12.m12.2.3.3.1.cmml" xref="S7.SS1.4.p2.12.m12.2.3.3.2"><ci id="S7.SS1.4.p2.12.m12.1.1.cmml" xref="S7.SS1.4.p2.12.m12.1.1">𝑋</ci><ci id="S7.SS1.4.p2.12.m12.2.2.cmml" xref="S7.SS1.4.p2.12.m12.2.2">𝑌</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.4.p2.12.m12.2c">b(X,Y)</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.4.p2.12.m12.2d">italic_b ( italic_X , italic_Y )</annotation></semantics></math> is injective when restricted to defined values. Since <math alttext="|B|<\mathfrak{c}" class="ltx_Math" display="inline" id="S7.SS1.4.p2.13.m13.1"><semantics id="S7.SS1.4.p2.13.m13.1a"><mrow id="S7.SS1.4.p2.13.m13.1.2" xref="S7.SS1.4.p2.13.m13.1.2.cmml"><mrow id="S7.SS1.4.p2.13.m13.1.2.2.2" xref="S7.SS1.4.p2.13.m13.1.2.2.1.cmml"><mo id="S7.SS1.4.p2.13.m13.1.2.2.2.1" stretchy="false" xref="S7.SS1.4.p2.13.m13.1.2.2.1.1.cmml">|</mo><mi id="S7.SS1.4.p2.13.m13.1.1" xref="S7.SS1.4.p2.13.m13.1.1.cmml">B</mi><mo id="S7.SS1.4.p2.13.m13.1.2.2.2.2" stretchy="false" xref="S7.SS1.4.p2.13.m13.1.2.2.1.1.cmml">|</mo></mrow><mo id="S7.SS1.4.p2.13.m13.1.2.1" xref="S7.SS1.4.p2.13.m13.1.2.1.cmml"><</mo><mi id="S7.SS1.4.p2.13.m13.1.2.3" xref="S7.SS1.4.p2.13.m13.1.2.3.cmml">𝔠</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.4.p2.13.m13.1b"><apply id="S7.SS1.4.p2.13.m13.1.2.cmml" xref="S7.SS1.4.p2.13.m13.1.2"><lt id="S7.SS1.4.p2.13.m13.1.2.1.cmml" xref="S7.SS1.4.p2.13.m13.1.2.1"></lt><apply id="S7.SS1.4.p2.13.m13.1.2.2.1.cmml" xref="S7.SS1.4.p2.13.m13.1.2.2.2"><abs id="S7.SS1.4.p2.13.m13.1.2.2.1.1.cmml" xref="S7.SS1.4.p2.13.m13.1.2.2.2.1"></abs><ci id="S7.SS1.4.p2.13.m13.1.1.cmml" xref="S7.SS1.4.p2.13.m13.1.1">𝐵</ci></apply><ci id="S7.SS1.4.p2.13.m13.1.2.3.cmml" xref="S7.SS1.4.p2.13.m13.1.2.3">𝔠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.4.p2.13.m13.1c">|B|<\mathfrak{c}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.4.p2.13.m13.1d">| italic_B | < fraktur_c</annotation></semantics></math>, we conclude that it is undefined for most elements of <math alttext="G(Q)" class="ltx_Math" display="inline" id="S7.SS1.4.p2.14.m14.1"><semantics id="S7.SS1.4.p2.14.m14.1a"><mrow id="S7.SS1.4.p2.14.m14.1.2" xref="S7.SS1.4.p2.14.m14.1.2.cmml"><mi id="S7.SS1.4.p2.14.m14.1.2.2" xref="S7.SS1.4.p2.14.m14.1.2.2.cmml">G</mi><mo id="S7.SS1.4.p2.14.m14.1.2.1" xref="S7.SS1.4.p2.14.m14.1.2.1.cmml">⁢</mo><mrow id="S7.SS1.4.p2.14.m14.1.2.3.2" xref="S7.SS1.4.p2.14.m14.1.2.cmml"><mo id="S7.SS1.4.p2.14.m14.1.2.3.2.1" stretchy="false" xref="S7.SS1.4.p2.14.m14.1.2.cmml">(</mo><mi id="S7.SS1.4.p2.14.m14.1.1" xref="S7.SS1.4.p2.14.m14.1.1.cmml">Q</mi><mo id="S7.SS1.4.p2.14.m14.1.2.3.2.2" stretchy="false" xref="S7.SS1.4.p2.14.m14.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.4.p2.14.m14.1b"><apply id="S7.SS1.4.p2.14.m14.1.2.cmml" xref="S7.SS1.4.p2.14.m14.1.2"><times id="S7.SS1.4.p2.14.m14.1.2.1.cmml" xref="S7.SS1.4.p2.14.m14.1.2.1"></times><ci id="S7.SS1.4.p2.14.m14.1.2.2.cmml" xref="S7.SS1.4.p2.14.m14.1.2.2">𝐺</ci><ci id="S7.SS1.4.p2.14.m14.1.1.cmml" xref="S7.SS1.4.p2.14.m14.1.1">𝑄</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.4.p2.14.m14.1c">G(Q)</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.4.p2.14.m14.1d">italic_G ( italic_Q )</annotation></semantics></math>. It should be clear that this induces at least <math alttext="\mathfrak{c}" class="ltx_Math" display="inline" id="S7.SS1.4.p2.15.m15.1"><semantics id="S7.SS1.4.p2.15.m15.1a"><mi id="S7.SS1.4.p2.15.m15.1.1" xref="S7.SS1.4.p2.15.m15.1.1.cmml">𝔠</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.4.p2.15.m15.1b"><ci id="S7.SS1.4.p2.15.m15.1.1.cmml" xref="S7.SS1.4.p2.15.m15.1.1">𝔠</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.4.p2.15.m15.1c">\mathfrak{c}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.4.p2.15.m15.1d">fraktur_c</annotation></semantics></math> gaps in <math alttext="B" class="ltx_Math" display="inline" id="S7.SS1.4.p2.16.m16.1"><semantics id="S7.SS1.4.p2.16.m16.1a"><mi id="S7.SS1.4.p2.16.m16.1.1" xref="S7.SS1.4.p2.16.m16.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.4.p2.16.m16.1b"><ci id="S7.SS1.4.p2.16.m16.1.1.cmml" xref="S7.SS1.4.p2.16.m16.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.4.p2.16.m16.1c">B</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.4.p2.16.m16.1d">italic_B</annotation></semantics></math>. Which is a contradiction. ∎</p> </div> </div> <div class="ltx_para" id="S7.SS1.p9"> <p class="ltx_p" id="S7.SS1.p9.3">We now describe the way to choose the <math alttext="A_{\alpha}" class="ltx_Math" display="inline" id="S7.SS1.p9.1.m1.1"><semantics id="S7.SS1.p9.1.m1.1a"><msub id="S7.SS1.p9.1.m1.1.1" xref="S7.SS1.p9.1.m1.1.1.cmml"><mi id="S7.SS1.p9.1.m1.1.1.2" xref="S7.SS1.p9.1.m1.1.1.2.cmml">A</mi><mi id="S7.SS1.p9.1.m1.1.1.3" xref="S7.SS1.p9.1.m1.1.1.3.cmml">α</mi></msub><annotation-xml encoding="MathML-Content" id="S7.SS1.p9.1.m1.1b"><apply id="S7.SS1.p9.1.m1.1.1.cmml" xref="S7.SS1.p9.1.m1.1.1"><csymbol cd="ambiguous" id="S7.SS1.p9.1.m1.1.1.1.cmml" xref="S7.SS1.p9.1.m1.1.1">subscript</csymbol><ci id="S7.SS1.p9.1.m1.1.1.2.cmml" xref="S7.SS1.p9.1.m1.1.1.2">𝐴</ci><ci id="S7.SS1.p9.1.m1.1.1.3.cmml" xref="S7.SS1.p9.1.m1.1.1.3">𝛼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p9.1.m1.1c">A_{\alpha}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p9.1.m1.1d">italic_A start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT</annotation></semantics></math>. At step <math alttext="\alpha" class="ltx_Math" display="inline" id="S7.SS1.p9.2.m2.1"><semantics id="S7.SS1.p9.2.m2.1a"><mi id="S7.SS1.p9.2.m2.1.1" xref="S7.SS1.p9.2.m2.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.p9.2.m2.1b"><ci id="S7.SS1.p9.2.m2.1.1.cmml" xref="S7.SS1.p9.2.m2.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p9.2.m2.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p9.2.m2.1d">italic_α</annotation></semantics></math> we add elements to <math alttext="A_{\alpha}" class="ltx_Math" display="inline" id="S7.SS1.p9.3.m3.1"><semantics id="S7.SS1.p9.3.m3.1a"><msub id="S7.SS1.p9.3.m3.1.1" xref="S7.SS1.p9.3.m3.1.1.cmml"><mi id="S7.SS1.p9.3.m3.1.1.2" xref="S7.SS1.p9.3.m3.1.1.2.cmml">A</mi><mi id="S7.SS1.p9.3.m3.1.1.3" xref="S7.SS1.p9.3.m3.1.1.3.cmml">α</mi></msub><annotation-xml encoding="MathML-Content" id="S7.SS1.p9.3.m3.1b"><apply id="S7.SS1.p9.3.m3.1.1.cmml" xref="S7.SS1.p9.3.m3.1.1"><csymbol cd="ambiguous" id="S7.SS1.p9.3.m3.1.1.1.cmml" xref="S7.SS1.p9.3.m3.1.1">subscript</csymbol><ci id="S7.SS1.p9.3.m3.1.1.2.cmml" xref="S7.SS1.p9.3.m3.1.1.2">𝐴</ci><ci id="S7.SS1.p9.3.m3.1.1.3.cmml" xref="S7.SS1.p9.3.m3.1.1.3">𝛼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p9.3.m3.1c">A_{\alpha}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p9.3.m3.1d">italic_A start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT</annotation></semantics></math> according to the following:</p> <ol class="ltx_enumerate" id="S7.I1"> <li class="ltx_item" id="S7.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(1)</span> <div class="ltx_para" id="S7.I1.i1.p1"> <p class="ltx_p" id="S7.I1.i1.p1.4">For each <math alttext="\xi<\alpha" class="ltx_Math" display="inline" id="S7.I1.i1.p1.1.m1.1"><semantics id="S7.I1.i1.p1.1.m1.1a"><mrow id="S7.I1.i1.p1.1.m1.1.1" xref="S7.I1.i1.p1.1.m1.1.1.cmml"><mi id="S7.I1.i1.p1.1.m1.1.1.2" xref="S7.I1.i1.p1.1.m1.1.1.2.cmml">ξ</mi><mo id="S7.I1.i1.p1.1.m1.1.1.1" xref="S7.I1.i1.p1.1.m1.1.1.1.cmml"><</mo><mi id="S7.I1.i1.p1.1.m1.1.1.3" xref="S7.I1.i1.p1.1.m1.1.1.3.cmml">α</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.I1.i1.p1.1.m1.1b"><apply id="S7.I1.i1.p1.1.m1.1.1.cmml" xref="S7.I1.i1.p1.1.m1.1.1"><lt id="S7.I1.i1.p1.1.m1.1.1.1.cmml" xref="S7.I1.i1.p1.1.m1.1.1.1"></lt><ci id="S7.I1.i1.p1.1.m1.1.1.2.cmml" xref="S7.I1.i1.p1.1.m1.1.1.2">𝜉</ci><ci id="S7.I1.i1.p1.1.m1.1.1.3.cmml" xref="S7.I1.i1.p1.1.m1.1.1.3">𝛼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I1.i1.p1.1.m1.1c">\xi<\alpha</annotation><annotation encoding="application/x-llamapun" id="S7.I1.i1.p1.1.m1.1d">italic_ξ < italic_α</annotation></semantics></math> we ask if <math alttext="|\mathbb{R}\setminus\operatorname{dom}(f_{\xi})|=\mathfrak{c}" class="ltx_Math" display="inline" id="S7.I1.i1.p1.2.m2.2"><semantics id="S7.I1.i1.p1.2.m2.2a"><mrow id="S7.I1.i1.p1.2.m2.2.2" xref="S7.I1.i1.p1.2.m2.2.2.cmml"><mrow id="S7.I1.i1.p1.2.m2.2.2.1.1" xref="S7.I1.i1.p1.2.m2.2.2.1.2.cmml"><mo id="S7.I1.i1.p1.2.m2.2.2.1.1.2" stretchy="false" xref="S7.I1.i1.p1.2.m2.2.2.1.2.1.cmml">|</mo><mrow id="S7.I1.i1.p1.2.m2.2.2.1.1.1" xref="S7.I1.i1.p1.2.m2.2.2.1.1.1.cmml"><mi id="S7.I1.i1.p1.2.m2.2.2.1.1.1.3" xref="S7.I1.i1.p1.2.m2.2.2.1.1.1.3.cmml">ℝ</mi><mo id="S7.I1.i1.p1.2.m2.2.2.1.1.1.2" xref="S7.I1.i1.p1.2.m2.2.2.1.1.1.2.cmml">∖</mo><mrow id="S7.I1.i1.p1.2.m2.2.2.1.1.1.1.1" xref="S7.I1.i1.p1.2.m2.2.2.1.1.1.1.2.cmml"><mi id="S7.I1.i1.p1.2.m2.1.1" xref="S7.I1.i1.p1.2.m2.1.1.cmml">dom</mi><mo id="S7.I1.i1.p1.2.m2.2.2.1.1.1.1.1a" xref="S7.I1.i1.p1.2.m2.2.2.1.1.1.1.2.cmml">⁡</mo><mrow id="S7.I1.i1.p1.2.m2.2.2.1.1.1.1.1.1" xref="S7.I1.i1.p1.2.m2.2.2.1.1.1.1.2.cmml"><mo id="S7.I1.i1.p1.2.m2.2.2.1.1.1.1.1.1.2" stretchy="false" xref="S7.I1.i1.p1.2.m2.2.2.1.1.1.1.2.cmml">(</mo><msub id="S7.I1.i1.p1.2.m2.2.2.1.1.1.1.1.1.1" xref="S7.I1.i1.p1.2.m2.2.2.1.1.1.1.1.1.1.cmml"><mi id="S7.I1.i1.p1.2.m2.2.2.1.1.1.1.1.1.1.2" xref="S7.I1.i1.p1.2.m2.2.2.1.1.1.1.1.1.1.2.cmml">f</mi><mi id="S7.I1.i1.p1.2.m2.2.2.1.1.1.1.1.1.1.3" xref="S7.I1.i1.p1.2.m2.2.2.1.1.1.1.1.1.1.3.cmml">ξ</mi></msub><mo id="S7.I1.i1.p1.2.m2.2.2.1.1.1.1.1.1.3" stretchy="false" xref="S7.I1.i1.p1.2.m2.2.2.1.1.1.1.2.cmml">)</mo></mrow></mrow></mrow><mo id="S7.I1.i1.p1.2.m2.2.2.1.1.3" stretchy="false" xref="S7.I1.i1.p1.2.m2.2.2.1.2.1.cmml">|</mo></mrow><mo id="S7.I1.i1.p1.2.m2.2.2.2" xref="S7.I1.i1.p1.2.m2.2.2.2.cmml">=</mo><mi id="S7.I1.i1.p1.2.m2.2.2.3" xref="S7.I1.i1.p1.2.m2.2.2.3.cmml">𝔠</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.I1.i1.p1.2.m2.2b"><apply id="S7.I1.i1.p1.2.m2.2.2.cmml" xref="S7.I1.i1.p1.2.m2.2.2"><eq id="S7.I1.i1.p1.2.m2.2.2.2.cmml" xref="S7.I1.i1.p1.2.m2.2.2.2"></eq><apply id="S7.I1.i1.p1.2.m2.2.2.1.2.cmml" xref="S7.I1.i1.p1.2.m2.2.2.1.1"><abs id="S7.I1.i1.p1.2.m2.2.2.1.2.1.cmml" xref="S7.I1.i1.p1.2.m2.2.2.1.1.2"></abs><apply id="S7.I1.i1.p1.2.m2.2.2.1.1.1.cmml" xref="S7.I1.i1.p1.2.m2.2.2.1.1.1"><setdiff id="S7.I1.i1.p1.2.m2.2.2.1.1.1.2.cmml" xref="S7.I1.i1.p1.2.m2.2.2.1.1.1.2"></setdiff><ci id="S7.I1.i1.p1.2.m2.2.2.1.1.1.3.cmml" xref="S7.I1.i1.p1.2.m2.2.2.1.1.1.3">ℝ</ci><apply id="S7.I1.i1.p1.2.m2.2.2.1.1.1.1.2.cmml" xref="S7.I1.i1.p1.2.m2.2.2.1.1.1.1.1"><ci id="S7.I1.i1.p1.2.m2.1.1.cmml" xref="S7.I1.i1.p1.2.m2.1.1">dom</ci><apply id="S7.I1.i1.p1.2.m2.2.2.1.1.1.1.1.1.1.cmml" xref="S7.I1.i1.p1.2.m2.2.2.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S7.I1.i1.p1.2.m2.2.2.1.1.1.1.1.1.1.1.cmml" xref="S7.I1.i1.p1.2.m2.2.2.1.1.1.1.1.1.1">subscript</csymbol><ci id="S7.I1.i1.p1.2.m2.2.2.1.1.1.1.1.1.1.2.cmml" xref="S7.I1.i1.p1.2.m2.2.2.1.1.1.1.1.1.1.2">𝑓</ci><ci id="S7.I1.i1.p1.2.m2.2.2.1.1.1.1.1.1.1.3.cmml" xref="S7.I1.i1.p1.2.m2.2.2.1.1.1.1.1.1.1.3">𝜉</ci></apply></apply></apply></apply><ci id="S7.I1.i1.p1.2.m2.2.2.3.cmml" xref="S7.I1.i1.p1.2.m2.2.2.3">𝔠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I1.i1.p1.2.m2.2c">|\mathbb{R}\setminus\operatorname{dom}(f_{\xi})|=\mathfrak{c}</annotation><annotation encoding="application/x-llamapun" id="S7.I1.i1.p1.2.m2.2d">| blackboard_R ∖ roman_dom ( italic_f start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT ) | = fraktur_c</annotation></semantics></math>. If the answer is yes, we add any element of <math alttext="(\mathbb{R}\setminus\operatorname{dom}(f_{\xi}))\setminus Z_{\alpha}" class="ltx_Math" display="inline" id="S7.I1.i1.p1.3.m3.2"><semantics id="S7.I1.i1.p1.3.m3.2a"><mrow id="S7.I1.i1.p1.3.m3.2.2" xref="S7.I1.i1.p1.3.m3.2.2.cmml"><mrow id="S7.I1.i1.p1.3.m3.2.2.1.1" xref="S7.I1.i1.p1.3.m3.2.2.1.1.1.cmml"><mo id="S7.I1.i1.p1.3.m3.2.2.1.1.2" stretchy="false" xref="S7.I1.i1.p1.3.m3.2.2.1.1.1.cmml">(</mo><mrow id="S7.I1.i1.p1.3.m3.2.2.1.1.1" xref="S7.I1.i1.p1.3.m3.2.2.1.1.1.cmml"><mi id="S7.I1.i1.p1.3.m3.2.2.1.1.1.3" xref="S7.I1.i1.p1.3.m3.2.2.1.1.1.3.cmml">ℝ</mi><mo id="S7.I1.i1.p1.3.m3.2.2.1.1.1.2" xref="S7.I1.i1.p1.3.m3.2.2.1.1.1.2.cmml">∖</mo><mrow id="S7.I1.i1.p1.3.m3.2.2.1.1.1.1.1" xref="S7.I1.i1.p1.3.m3.2.2.1.1.1.1.2.cmml"><mi id="S7.I1.i1.p1.3.m3.1.1" xref="S7.I1.i1.p1.3.m3.1.1.cmml">dom</mi><mo id="S7.I1.i1.p1.3.m3.2.2.1.1.1.1.1a" xref="S7.I1.i1.p1.3.m3.2.2.1.1.1.1.2.cmml">⁡</mo><mrow id="S7.I1.i1.p1.3.m3.2.2.1.1.1.1.1.1" xref="S7.I1.i1.p1.3.m3.2.2.1.1.1.1.2.cmml"><mo id="S7.I1.i1.p1.3.m3.2.2.1.1.1.1.1.1.2" stretchy="false" xref="S7.I1.i1.p1.3.m3.2.2.1.1.1.1.2.cmml">(</mo><msub id="S7.I1.i1.p1.3.m3.2.2.1.1.1.1.1.1.1" xref="S7.I1.i1.p1.3.m3.2.2.1.1.1.1.1.1.1.cmml"><mi id="S7.I1.i1.p1.3.m3.2.2.1.1.1.1.1.1.1.2" xref="S7.I1.i1.p1.3.m3.2.2.1.1.1.1.1.1.1.2.cmml">f</mi><mi id="S7.I1.i1.p1.3.m3.2.2.1.1.1.1.1.1.1.3" xref="S7.I1.i1.p1.3.m3.2.2.1.1.1.1.1.1.1.3.cmml">ξ</mi></msub><mo id="S7.I1.i1.p1.3.m3.2.2.1.1.1.1.1.1.3" stretchy="false" xref="S7.I1.i1.p1.3.m3.2.2.1.1.1.1.2.cmml">)</mo></mrow></mrow></mrow><mo id="S7.I1.i1.p1.3.m3.2.2.1.1.3" stretchy="false" xref="S7.I1.i1.p1.3.m3.2.2.1.1.1.cmml">)</mo></mrow><mo id="S7.I1.i1.p1.3.m3.2.2.2" xref="S7.I1.i1.p1.3.m3.2.2.2.cmml">∖</mo><msub id="S7.I1.i1.p1.3.m3.2.2.3" xref="S7.I1.i1.p1.3.m3.2.2.3.cmml"><mi id="S7.I1.i1.p1.3.m3.2.2.3.2" xref="S7.I1.i1.p1.3.m3.2.2.3.2.cmml">Z</mi><mi id="S7.I1.i1.p1.3.m3.2.2.3.3" xref="S7.I1.i1.p1.3.m3.2.2.3.3.cmml">α</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.I1.i1.p1.3.m3.2b"><apply id="S7.I1.i1.p1.3.m3.2.2.cmml" xref="S7.I1.i1.p1.3.m3.2.2"><setdiff id="S7.I1.i1.p1.3.m3.2.2.2.cmml" xref="S7.I1.i1.p1.3.m3.2.2.2"></setdiff><apply id="S7.I1.i1.p1.3.m3.2.2.1.1.1.cmml" xref="S7.I1.i1.p1.3.m3.2.2.1.1"><setdiff id="S7.I1.i1.p1.3.m3.2.2.1.1.1.2.cmml" xref="S7.I1.i1.p1.3.m3.2.2.1.1.1.2"></setdiff><ci id="S7.I1.i1.p1.3.m3.2.2.1.1.1.3.cmml" xref="S7.I1.i1.p1.3.m3.2.2.1.1.1.3">ℝ</ci><apply id="S7.I1.i1.p1.3.m3.2.2.1.1.1.1.2.cmml" xref="S7.I1.i1.p1.3.m3.2.2.1.1.1.1.1"><ci id="S7.I1.i1.p1.3.m3.1.1.cmml" xref="S7.I1.i1.p1.3.m3.1.1">dom</ci><apply id="S7.I1.i1.p1.3.m3.2.2.1.1.1.1.1.1.1.cmml" xref="S7.I1.i1.p1.3.m3.2.2.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S7.I1.i1.p1.3.m3.2.2.1.1.1.1.1.1.1.1.cmml" xref="S7.I1.i1.p1.3.m3.2.2.1.1.1.1.1.1.1">subscript</csymbol><ci id="S7.I1.i1.p1.3.m3.2.2.1.1.1.1.1.1.1.2.cmml" xref="S7.I1.i1.p1.3.m3.2.2.1.1.1.1.1.1.1.2">𝑓</ci><ci id="S7.I1.i1.p1.3.m3.2.2.1.1.1.1.1.1.1.3.cmml" xref="S7.I1.i1.p1.3.m3.2.2.1.1.1.1.1.1.1.3">𝜉</ci></apply></apply></apply><apply id="S7.I1.i1.p1.3.m3.2.2.3.cmml" xref="S7.I1.i1.p1.3.m3.2.2.3"><csymbol cd="ambiguous" id="S7.I1.i1.p1.3.m3.2.2.3.1.cmml" xref="S7.I1.i1.p1.3.m3.2.2.3">subscript</csymbol><ci id="S7.I1.i1.p1.3.m3.2.2.3.2.cmml" xref="S7.I1.i1.p1.3.m3.2.2.3.2">𝑍</ci><ci id="S7.I1.i1.p1.3.m3.2.2.3.3.cmml" xref="S7.I1.i1.p1.3.m3.2.2.3.3">𝛼</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I1.i1.p1.3.m3.2c">(\mathbb{R}\setminus\operatorname{dom}(f_{\xi}))\setminus Z_{\alpha}</annotation><annotation encoding="application/x-llamapun" id="S7.I1.i1.p1.3.m3.2d">( blackboard_R ∖ roman_dom ( italic_f start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT ) ) ∖ italic_Z start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT</annotation></semantics></math> to <math alttext="A_{\alpha}" class="ltx_Math" display="inline" id="S7.I1.i1.p1.4.m4.1"><semantics id="S7.I1.i1.p1.4.m4.1a"><msub id="S7.I1.i1.p1.4.m4.1.1" xref="S7.I1.i1.p1.4.m4.1.1.cmml"><mi id="S7.I1.i1.p1.4.m4.1.1.2" xref="S7.I1.i1.p1.4.m4.1.1.2.cmml">A</mi><mi id="S7.I1.i1.p1.4.m4.1.1.3" xref="S7.I1.i1.p1.4.m4.1.1.3.cmml">α</mi></msub><annotation-xml encoding="MathML-Content" id="S7.I1.i1.p1.4.m4.1b"><apply id="S7.I1.i1.p1.4.m4.1.1.cmml" xref="S7.I1.i1.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S7.I1.i1.p1.4.m4.1.1.1.cmml" xref="S7.I1.i1.p1.4.m4.1.1">subscript</csymbol><ci id="S7.I1.i1.p1.4.m4.1.1.2.cmml" xref="S7.I1.i1.p1.4.m4.1.1.2">𝐴</ci><ci id="S7.I1.i1.p1.4.m4.1.1.3.cmml" xref="S7.I1.i1.p1.4.m4.1.1.3">𝛼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I1.i1.p1.4.m4.1c">A_{\alpha}</annotation><annotation encoding="application/x-llamapun" id="S7.I1.i1.p1.4.m4.1d">italic_A start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S7.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(2)</span> <div class="ltx_para" id="S7.I1.i2.p1"> <p class="ltx_p" id="S7.I1.i2.p1.5">For each <math alttext="\xi<\alpha" class="ltx_Math" display="inline" id="S7.I1.i2.p1.1.m1.1"><semantics id="S7.I1.i2.p1.1.m1.1a"><mrow id="S7.I1.i2.p1.1.m1.1.1" xref="S7.I1.i2.p1.1.m1.1.1.cmml"><mi id="S7.I1.i2.p1.1.m1.1.1.2" xref="S7.I1.i2.p1.1.m1.1.1.2.cmml">ξ</mi><mo id="S7.I1.i2.p1.1.m1.1.1.1" xref="S7.I1.i2.p1.1.m1.1.1.1.cmml"><</mo><mi id="S7.I1.i2.p1.1.m1.1.1.3" xref="S7.I1.i2.p1.1.m1.1.1.3.cmml">α</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.I1.i2.p1.1.m1.1b"><apply id="S7.I1.i2.p1.1.m1.1.1.cmml" xref="S7.I1.i2.p1.1.m1.1.1"><lt id="S7.I1.i2.p1.1.m1.1.1.1.cmml" xref="S7.I1.i2.p1.1.m1.1.1.1"></lt><ci id="S7.I1.i2.p1.1.m1.1.1.2.cmml" xref="S7.I1.i2.p1.1.m1.1.1.2">𝜉</ci><ci id="S7.I1.i2.p1.1.m1.1.1.3.cmml" xref="S7.I1.i2.p1.1.m1.1.1.3">𝛼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I1.i2.p1.1.m1.1c">\xi<\alpha</annotation><annotation encoding="application/x-llamapun" id="S7.I1.i2.p1.1.m1.1d">italic_ξ < italic_α</annotation></semantics></math> we ask if <math alttext="|\operatorname{ran}(f_{\xi})|=\mathfrak{c}" class="ltx_Math" display="inline" id="S7.I1.i2.p1.2.m2.2"><semantics id="S7.I1.i2.p1.2.m2.2a"><mrow id="S7.I1.i2.p1.2.m2.2.2" xref="S7.I1.i2.p1.2.m2.2.2.cmml"><mrow id="S7.I1.i2.p1.2.m2.2.2.1.1" xref="S7.I1.i2.p1.2.m2.2.2.1.2.cmml"><mo id="S7.I1.i2.p1.2.m2.2.2.1.1.2" stretchy="false" xref="S7.I1.i2.p1.2.m2.2.2.1.2.1.cmml">|</mo><mrow id="S7.I1.i2.p1.2.m2.2.2.1.1.1.1" xref="S7.I1.i2.p1.2.m2.2.2.1.1.1.2.cmml"><mi id="S7.I1.i2.p1.2.m2.1.1" xref="S7.I1.i2.p1.2.m2.1.1.cmml">ran</mi><mo id="S7.I1.i2.p1.2.m2.2.2.1.1.1.1a" xref="S7.I1.i2.p1.2.m2.2.2.1.1.1.2.cmml">⁡</mo><mrow id="S7.I1.i2.p1.2.m2.2.2.1.1.1.1.1" xref="S7.I1.i2.p1.2.m2.2.2.1.1.1.2.cmml"><mo id="S7.I1.i2.p1.2.m2.2.2.1.1.1.1.1.2" stretchy="false" xref="S7.I1.i2.p1.2.m2.2.2.1.1.1.2.cmml">(</mo><msub id="S7.I1.i2.p1.2.m2.2.2.1.1.1.1.1.1" xref="S7.I1.i2.p1.2.m2.2.2.1.1.1.1.1.1.cmml"><mi id="S7.I1.i2.p1.2.m2.2.2.1.1.1.1.1.1.2" xref="S7.I1.i2.p1.2.m2.2.2.1.1.1.1.1.1.2.cmml">f</mi><mi id="S7.I1.i2.p1.2.m2.2.2.1.1.1.1.1.1.3" xref="S7.I1.i2.p1.2.m2.2.2.1.1.1.1.1.1.3.cmml">ξ</mi></msub><mo id="S7.I1.i2.p1.2.m2.2.2.1.1.1.1.1.3" stretchy="false" xref="S7.I1.i2.p1.2.m2.2.2.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S7.I1.i2.p1.2.m2.2.2.1.1.3" stretchy="false" xref="S7.I1.i2.p1.2.m2.2.2.1.2.1.cmml">|</mo></mrow><mo id="S7.I1.i2.p1.2.m2.2.2.2" xref="S7.I1.i2.p1.2.m2.2.2.2.cmml">=</mo><mi id="S7.I1.i2.p1.2.m2.2.2.3" xref="S7.I1.i2.p1.2.m2.2.2.3.cmml">𝔠</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.I1.i2.p1.2.m2.2b"><apply id="S7.I1.i2.p1.2.m2.2.2.cmml" xref="S7.I1.i2.p1.2.m2.2.2"><eq id="S7.I1.i2.p1.2.m2.2.2.2.cmml" xref="S7.I1.i2.p1.2.m2.2.2.2"></eq><apply id="S7.I1.i2.p1.2.m2.2.2.1.2.cmml" xref="S7.I1.i2.p1.2.m2.2.2.1.1"><abs id="S7.I1.i2.p1.2.m2.2.2.1.2.1.cmml" xref="S7.I1.i2.p1.2.m2.2.2.1.1.2"></abs><apply id="S7.I1.i2.p1.2.m2.2.2.1.1.1.2.cmml" xref="S7.I1.i2.p1.2.m2.2.2.1.1.1.1"><ci id="S7.I1.i2.p1.2.m2.1.1.cmml" xref="S7.I1.i2.p1.2.m2.1.1">ran</ci><apply id="S7.I1.i2.p1.2.m2.2.2.1.1.1.1.1.1.cmml" xref="S7.I1.i2.p1.2.m2.2.2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S7.I1.i2.p1.2.m2.2.2.1.1.1.1.1.1.1.cmml" xref="S7.I1.i2.p1.2.m2.2.2.1.1.1.1.1.1">subscript</csymbol><ci id="S7.I1.i2.p1.2.m2.2.2.1.1.1.1.1.1.2.cmml" xref="S7.I1.i2.p1.2.m2.2.2.1.1.1.1.1.1.2">𝑓</ci><ci id="S7.I1.i2.p1.2.m2.2.2.1.1.1.1.1.1.3.cmml" xref="S7.I1.i2.p1.2.m2.2.2.1.1.1.1.1.1.3">𝜉</ci></apply></apply></apply><ci id="S7.I1.i2.p1.2.m2.2.2.3.cmml" xref="S7.I1.i2.p1.2.m2.2.2.3">𝔠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I1.i2.p1.2.m2.2c">|\operatorname{ran}(f_{\xi})|=\mathfrak{c}</annotation><annotation encoding="application/x-llamapun" id="S7.I1.i2.p1.2.m2.2d">| roman_ran ( italic_f start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT ) | = fraktur_c</annotation></semantics></math>. If the answer is yes, we add to <math alttext="A_{\alpha}" class="ltx_Math" display="inline" id="S7.I1.i2.p1.3.m3.1"><semantics id="S7.I1.i2.p1.3.m3.1a"><msub id="S7.I1.i2.p1.3.m3.1.1" xref="S7.I1.i2.p1.3.m3.1.1.cmml"><mi id="S7.I1.i2.p1.3.m3.1.1.2" xref="S7.I1.i2.p1.3.m3.1.1.2.cmml">A</mi><mi id="S7.I1.i2.p1.3.m3.1.1.3" xref="S7.I1.i2.p1.3.m3.1.1.3.cmml">α</mi></msub><annotation-xml encoding="MathML-Content" id="S7.I1.i2.p1.3.m3.1b"><apply id="S7.I1.i2.p1.3.m3.1.1.cmml" xref="S7.I1.i2.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S7.I1.i2.p1.3.m3.1.1.1.cmml" xref="S7.I1.i2.p1.3.m3.1.1">subscript</csymbol><ci id="S7.I1.i2.p1.3.m3.1.1.2.cmml" xref="S7.I1.i2.p1.3.m3.1.1.2">𝐴</ci><ci id="S7.I1.i2.p1.3.m3.1.1.3.cmml" xref="S7.I1.i2.p1.3.m3.1.1.3">𝛼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I1.i2.p1.3.m3.1c">A_{\alpha}</annotation><annotation encoding="application/x-llamapun" id="S7.I1.i2.p1.3.m3.1d">italic_A start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT</annotation></semantics></math> any <math alttext="a\in\operatorname{dom}(f_{\xi})\setminus Z_{\alpha}" class="ltx_Math" display="inline" id="S7.I1.i2.p1.4.m4.2"><semantics id="S7.I1.i2.p1.4.m4.2a"><mrow id="S7.I1.i2.p1.4.m4.2.2" xref="S7.I1.i2.p1.4.m4.2.2.cmml"><mi id="S7.I1.i2.p1.4.m4.2.2.3" xref="S7.I1.i2.p1.4.m4.2.2.3.cmml">a</mi><mo id="S7.I1.i2.p1.4.m4.2.2.2" xref="S7.I1.i2.p1.4.m4.2.2.2.cmml">∈</mo><mrow id="S7.I1.i2.p1.4.m4.2.2.1" xref="S7.I1.i2.p1.4.m4.2.2.1.cmml"><mrow id="S7.I1.i2.p1.4.m4.2.2.1.1.1" xref="S7.I1.i2.p1.4.m4.2.2.1.1.2.cmml"><mi id="S7.I1.i2.p1.4.m4.1.1" xref="S7.I1.i2.p1.4.m4.1.1.cmml">dom</mi><mo id="S7.I1.i2.p1.4.m4.2.2.1.1.1a" xref="S7.I1.i2.p1.4.m4.2.2.1.1.2.cmml">⁡</mo><mrow id="S7.I1.i2.p1.4.m4.2.2.1.1.1.1" xref="S7.I1.i2.p1.4.m4.2.2.1.1.2.cmml"><mo id="S7.I1.i2.p1.4.m4.2.2.1.1.1.1.2" stretchy="false" xref="S7.I1.i2.p1.4.m4.2.2.1.1.2.cmml">(</mo><msub id="S7.I1.i2.p1.4.m4.2.2.1.1.1.1.1" xref="S7.I1.i2.p1.4.m4.2.2.1.1.1.1.1.cmml"><mi id="S7.I1.i2.p1.4.m4.2.2.1.1.1.1.1.2" xref="S7.I1.i2.p1.4.m4.2.2.1.1.1.1.1.2.cmml">f</mi><mi id="S7.I1.i2.p1.4.m4.2.2.1.1.1.1.1.3" xref="S7.I1.i2.p1.4.m4.2.2.1.1.1.1.1.3.cmml">ξ</mi></msub><mo id="S7.I1.i2.p1.4.m4.2.2.1.1.1.1.3" stretchy="false" xref="S7.I1.i2.p1.4.m4.2.2.1.1.2.cmml">)</mo></mrow></mrow><mo id="S7.I1.i2.p1.4.m4.2.2.1.2" xref="S7.I1.i2.p1.4.m4.2.2.1.2.cmml">∖</mo><msub id="S7.I1.i2.p1.4.m4.2.2.1.3" xref="S7.I1.i2.p1.4.m4.2.2.1.3.cmml"><mi id="S7.I1.i2.p1.4.m4.2.2.1.3.2" xref="S7.I1.i2.p1.4.m4.2.2.1.3.2.cmml">Z</mi><mi id="S7.I1.i2.p1.4.m4.2.2.1.3.3" xref="S7.I1.i2.p1.4.m4.2.2.1.3.3.cmml">α</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.I1.i2.p1.4.m4.2b"><apply id="S7.I1.i2.p1.4.m4.2.2.cmml" xref="S7.I1.i2.p1.4.m4.2.2"><in id="S7.I1.i2.p1.4.m4.2.2.2.cmml" xref="S7.I1.i2.p1.4.m4.2.2.2"></in><ci id="S7.I1.i2.p1.4.m4.2.2.3.cmml" xref="S7.I1.i2.p1.4.m4.2.2.3">𝑎</ci><apply id="S7.I1.i2.p1.4.m4.2.2.1.cmml" xref="S7.I1.i2.p1.4.m4.2.2.1"><setdiff id="S7.I1.i2.p1.4.m4.2.2.1.2.cmml" xref="S7.I1.i2.p1.4.m4.2.2.1.2"></setdiff><apply id="S7.I1.i2.p1.4.m4.2.2.1.1.2.cmml" xref="S7.I1.i2.p1.4.m4.2.2.1.1.1"><ci id="S7.I1.i2.p1.4.m4.1.1.cmml" xref="S7.I1.i2.p1.4.m4.1.1">dom</ci><apply id="S7.I1.i2.p1.4.m4.2.2.1.1.1.1.1.cmml" xref="S7.I1.i2.p1.4.m4.2.2.1.1.1.1.1"><csymbol cd="ambiguous" id="S7.I1.i2.p1.4.m4.2.2.1.1.1.1.1.1.cmml" xref="S7.I1.i2.p1.4.m4.2.2.1.1.1.1.1">subscript</csymbol><ci id="S7.I1.i2.p1.4.m4.2.2.1.1.1.1.1.2.cmml" xref="S7.I1.i2.p1.4.m4.2.2.1.1.1.1.1.2">𝑓</ci><ci id="S7.I1.i2.p1.4.m4.2.2.1.1.1.1.1.3.cmml" xref="S7.I1.i2.p1.4.m4.2.2.1.1.1.1.1.3">𝜉</ci></apply></apply><apply id="S7.I1.i2.p1.4.m4.2.2.1.3.cmml" xref="S7.I1.i2.p1.4.m4.2.2.1.3"><csymbol cd="ambiguous" id="S7.I1.i2.p1.4.m4.2.2.1.3.1.cmml" xref="S7.I1.i2.p1.4.m4.2.2.1.3">subscript</csymbol><ci id="S7.I1.i2.p1.4.m4.2.2.1.3.2.cmml" xref="S7.I1.i2.p1.4.m4.2.2.1.3.2">𝑍</ci><ci id="S7.I1.i2.p1.4.m4.2.2.1.3.3.cmml" xref="S7.I1.i2.p1.4.m4.2.2.1.3.3">𝛼</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I1.i2.p1.4.m4.2c">a\in\operatorname{dom}(f_{\xi})\setminus Z_{\alpha}</annotation><annotation encoding="application/x-llamapun" id="S7.I1.i2.p1.4.m4.2d">italic_a ∈ roman_dom ( italic_f start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT ) ∖ italic_Z start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="f_{\xi}(a)\notin Z_{\alpha}" class="ltx_Math" display="inline" id="S7.I1.i2.p1.5.m5.1"><semantics id="S7.I1.i2.p1.5.m5.1a"><mrow id="S7.I1.i2.p1.5.m5.1.2" xref="S7.I1.i2.p1.5.m5.1.2.cmml"><mrow id="S7.I1.i2.p1.5.m5.1.2.2" xref="S7.I1.i2.p1.5.m5.1.2.2.cmml"><msub id="S7.I1.i2.p1.5.m5.1.2.2.2" xref="S7.I1.i2.p1.5.m5.1.2.2.2.cmml"><mi id="S7.I1.i2.p1.5.m5.1.2.2.2.2" xref="S7.I1.i2.p1.5.m5.1.2.2.2.2.cmml">f</mi><mi id="S7.I1.i2.p1.5.m5.1.2.2.2.3" xref="S7.I1.i2.p1.5.m5.1.2.2.2.3.cmml">ξ</mi></msub><mo id="S7.I1.i2.p1.5.m5.1.2.2.1" xref="S7.I1.i2.p1.5.m5.1.2.2.1.cmml">⁢</mo><mrow id="S7.I1.i2.p1.5.m5.1.2.2.3.2" xref="S7.I1.i2.p1.5.m5.1.2.2.cmml"><mo id="S7.I1.i2.p1.5.m5.1.2.2.3.2.1" stretchy="false" xref="S7.I1.i2.p1.5.m5.1.2.2.cmml">(</mo><mi id="S7.I1.i2.p1.5.m5.1.1" xref="S7.I1.i2.p1.5.m5.1.1.cmml">a</mi><mo id="S7.I1.i2.p1.5.m5.1.2.2.3.2.2" stretchy="false" xref="S7.I1.i2.p1.5.m5.1.2.2.cmml">)</mo></mrow></mrow><mo id="S7.I1.i2.p1.5.m5.1.2.1" xref="S7.I1.i2.p1.5.m5.1.2.1.cmml">∉</mo><msub id="S7.I1.i2.p1.5.m5.1.2.3" xref="S7.I1.i2.p1.5.m5.1.2.3.cmml"><mi id="S7.I1.i2.p1.5.m5.1.2.3.2" xref="S7.I1.i2.p1.5.m5.1.2.3.2.cmml">Z</mi><mi id="S7.I1.i2.p1.5.m5.1.2.3.3" xref="S7.I1.i2.p1.5.m5.1.2.3.3.cmml">α</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.I1.i2.p1.5.m5.1b"><apply id="S7.I1.i2.p1.5.m5.1.2.cmml" xref="S7.I1.i2.p1.5.m5.1.2"><notin id="S7.I1.i2.p1.5.m5.1.2.1.cmml" xref="S7.I1.i2.p1.5.m5.1.2.1"></notin><apply id="S7.I1.i2.p1.5.m5.1.2.2.cmml" xref="S7.I1.i2.p1.5.m5.1.2.2"><times id="S7.I1.i2.p1.5.m5.1.2.2.1.cmml" xref="S7.I1.i2.p1.5.m5.1.2.2.1"></times><apply id="S7.I1.i2.p1.5.m5.1.2.2.2.cmml" xref="S7.I1.i2.p1.5.m5.1.2.2.2"><csymbol cd="ambiguous" id="S7.I1.i2.p1.5.m5.1.2.2.2.1.cmml" xref="S7.I1.i2.p1.5.m5.1.2.2.2">subscript</csymbol><ci id="S7.I1.i2.p1.5.m5.1.2.2.2.2.cmml" xref="S7.I1.i2.p1.5.m5.1.2.2.2.2">𝑓</ci><ci id="S7.I1.i2.p1.5.m5.1.2.2.2.3.cmml" xref="S7.I1.i2.p1.5.m5.1.2.2.2.3">𝜉</ci></apply><ci id="S7.I1.i2.p1.5.m5.1.1.cmml" xref="S7.I1.i2.p1.5.m5.1.1">𝑎</ci></apply><apply id="S7.I1.i2.p1.5.m5.1.2.3.cmml" xref="S7.I1.i2.p1.5.m5.1.2.3"><csymbol cd="ambiguous" id="S7.I1.i2.p1.5.m5.1.2.3.1.cmml" xref="S7.I1.i2.p1.5.m5.1.2.3">subscript</csymbol><ci id="S7.I1.i2.p1.5.m5.1.2.3.2.cmml" xref="S7.I1.i2.p1.5.m5.1.2.3.2">𝑍</ci><ci id="S7.I1.i2.p1.5.m5.1.2.3.3.cmml" xref="S7.I1.i2.p1.5.m5.1.2.3.3">𝛼</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I1.i2.p1.5.m5.1c">f_{\xi}(a)\notin Z_{\alpha}</annotation><annotation encoding="application/x-llamapun" id="S7.I1.i2.p1.5.m5.1d">italic_f start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT ( italic_a ) ∉ italic_Z start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> </li> </ol> <p class="ltx_p" id="S7.SS1.p9.7">If neither step (1) or (2) adds elements to <math alttext="A_{\alpha}" class="ltx_Math" display="inline" id="S7.SS1.p9.4.m1.1"><semantics id="S7.SS1.p9.4.m1.1a"><msub id="S7.SS1.p9.4.m1.1.1" xref="S7.SS1.p9.4.m1.1.1.cmml"><mi id="S7.SS1.p9.4.m1.1.1.2" xref="S7.SS1.p9.4.m1.1.1.2.cmml">A</mi><mi id="S7.SS1.p9.4.m1.1.1.3" xref="S7.SS1.p9.4.m1.1.1.3.cmml">α</mi></msub><annotation-xml encoding="MathML-Content" id="S7.SS1.p9.4.m1.1b"><apply id="S7.SS1.p9.4.m1.1.1.cmml" xref="S7.SS1.p9.4.m1.1.1"><csymbol cd="ambiguous" id="S7.SS1.p9.4.m1.1.1.1.cmml" xref="S7.SS1.p9.4.m1.1.1">subscript</csymbol><ci id="S7.SS1.p9.4.m1.1.1.2.cmml" xref="S7.SS1.p9.4.m1.1.1.2">𝐴</ci><ci id="S7.SS1.p9.4.m1.1.1.3.cmml" xref="S7.SS1.p9.4.m1.1.1.3">𝛼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p9.4.m1.1c">A_{\alpha}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p9.4.m1.1d">italic_A start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT</annotation></semantics></math>, simply let <math alttext="A_{\alpha}" class="ltx_Math" display="inline" id="S7.SS1.p9.5.m2.1"><semantics id="S7.SS1.p9.5.m2.1a"><msub id="S7.SS1.p9.5.m2.1.1" xref="S7.SS1.p9.5.m2.1.1.cmml"><mi id="S7.SS1.p9.5.m2.1.1.2" xref="S7.SS1.p9.5.m2.1.1.2.cmml">A</mi><mi id="S7.SS1.p9.5.m2.1.1.3" xref="S7.SS1.p9.5.m2.1.1.3.cmml">α</mi></msub><annotation-xml encoding="MathML-Content" id="S7.SS1.p9.5.m2.1b"><apply id="S7.SS1.p9.5.m2.1.1.cmml" xref="S7.SS1.p9.5.m2.1.1"><csymbol cd="ambiguous" id="S7.SS1.p9.5.m2.1.1.1.cmml" xref="S7.SS1.p9.5.m2.1.1">subscript</csymbol><ci id="S7.SS1.p9.5.m2.1.1.2.cmml" xref="S7.SS1.p9.5.m2.1.1.2">𝐴</ci><ci id="S7.SS1.p9.5.m2.1.1.3.cmml" xref="S7.SS1.p9.5.m2.1.1.3">𝛼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p9.5.m2.1c">A_{\alpha}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p9.5.m2.1d">italic_A start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT</annotation></semantics></math> be singleton and disjoint from <math alttext="Z_{\alpha}" class="ltx_Math" display="inline" id="S7.SS1.p9.6.m3.1"><semantics id="S7.SS1.p9.6.m3.1a"><msub id="S7.SS1.p9.6.m3.1.1" xref="S7.SS1.p9.6.m3.1.1.cmml"><mi id="S7.SS1.p9.6.m3.1.1.2" xref="S7.SS1.p9.6.m3.1.1.2.cmml">Z</mi><mi id="S7.SS1.p9.6.m3.1.1.3" xref="S7.SS1.p9.6.m3.1.1.3.cmml">α</mi></msub><annotation-xml encoding="MathML-Content" id="S7.SS1.p9.6.m3.1b"><apply id="S7.SS1.p9.6.m3.1.1.cmml" xref="S7.SS1.p9.6.m3.1.1"><csymbol cd="ambiguous" id="S7.SS1.p9.6.m3.1.1.1.cmml" xref="S7.SS1.p9.6.m3.1.1">subscript</csymbol><ci id="S7.SS1.p9.6.m3.1.1.2.cmml" xref="S7.SS1.p9.6.m3.1.1.2">𝑍</ci><ci id="S7.SS1.p9.6.m3.1.1.3.cmml" xref="S7.SS1.p9.6.m3.1.1.3">𝛼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p9.6.m3.1c">Z_{\alpha}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p9.6.m3.1d">italic_Z start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT</annotation></semantics></math>. Note that clearly <math alttext="|A_{\alpha}|\leq\max\{\aleph_{0},|\alpha|\}" class="ltx_Math" display="inline" id="S7.SS1.p9.7.m4.5"><semantics id="S7.SS1.p9.7.m4.5a"><mrow id="S7.SS1.p9.7.m4.5.5" xref="S7.SS1.p9.7.m4.5.5.cmml"><mrow id="S7.SS1.p9.7.m4.3.3.1.1" xref="S7.SS1.p9.7.m4.3.3.1.2.cmml"><mo id="S7.SS1.p9.7.m4.3.3.1.1.2" stretchy="false" xref="S7.SS1.p9.7.m4.3.3.1.2.1.cmml">|</mo><msub id="S7.SS1.p9.7.m4.3.3.1.1.1" xref="S7.SS1.p9.7.m4.3.3.1.1.1.cmml"><mi id="S7.SS1.p9.7.m4.3.3.1.1.1.2" xref="S7.SS1.p9.7.m4.3.3.1.1.1.2.cmml">A</mi><mi id="S7.SS1.p9.7.m4.3.3.1.1.1.3" xref="S7.SS1.p9.7.m4.3.3.1.1.1.3.cmml">α</mi></msub><mo id="S7.SS1.p9.7.m4.3.3.1.1.3" stretchy="false" xref="S7.SS1.p9.7.m4.3.3.1.2.1.cmml">|</mo></mrow><mo id="S7.SS1.p9.7.m4.5.5.4" xref="S7.SS1.p9.7.m4.5.5.4.cmml">≤</mo><mrow id="S7.SS1.p9.7.m4.5.5.3.2" xref="S7.SS1.p9.7.m4.5.5.3.3.cmml"><mi id="S7.SS1.p9.7.m4.2.2" xref="S7.SS1.p9.7.m4.2.2.cmml">max</mi><mo id="S7.SS1.p9.7.m4.5.5.3.2a" xref="S7.SS1.p9.7.m4.5.5.3.3.cmml">⁡</mo><mrow id="S7.SS1.p9.7.m4.5.5.3.2.2" xref="S7.SS1.p9.7.m4.5.5.3.3.cmml"><mo id="S7.SS1.p9.7.m4.5.5.3.2.2.3" stretchy="false" xref="S7.SS1.p9.7.m4.5.5.3.3.cmml">{</mo><msub id="S7.SS1.p9.7.m4.4.4.2.1.1.1" xref="S7.SS1.p9.7.m4.4.4.2.1.1.1.cmml"><mi id="S7.SS1.p9.7.m4.4.4.2.1.1.1.2" mathvariant="normal" xref="S7.SS1.p9.7.m4.4.4.2.1.1.1.2.cmml">ℵ</mi><mn id="S7.SS1.p9.7.m4.4.4.2.1.1.1.3" xref="S7.SS1.p9.7.m4.4.4.2.1.1.1.3.cmml">0</mn></msub><mo id="S7.SS1.p9.7.m4.5.5.3.2.2.4" xref="S7.SS1.p9.7.m4.5.5.3.3.cmml">,</mo><mrow id="S7.SS1.p9.7.m4.5.5.3.2.2.2.2" xref="S7.SS1.p9.7.m4.5.5.3.2.2.2.1.cmml"><mo id="S7.SS1.p9.7.m4.5.5.3.2.2.2.2.1" stretchy="false" xref="S7.SS1.p9.7.m4.5.5.3.2.2.2.1.1.cmml">|</mo><mi id="S7.SS1.p9.7.m4.1.1" xref="S7.SS1.p9.7.m4.1.1.cmml">α</mi><mo id="S7.SS1.p9.7.m4.5.5.3.2.2.2.2.2" stretchy="false" xref="S7.SS1.p9.7.m4.5.5.3.2.2.2.1.1.cmml">|</mo></mrow><mo id="S7.SS1.p9.7.m4.5.5.3.2.2.5" stretchy="false" xref="S7.SS1.p9.7.m4.5.5.3.3.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.p9.7.m4.5b"><apply id="S7.SS1.p9.7.m4.5.5.cmml" xref="S7.SS1.p9.7.m4.5.5"><leq id="S7.SS1.p9.7.m4.5.5.4.cmml" xref="S7.SS1.p9.7.m4.5.5.4"></leq><apply id="S7.SS1.p9.7.m4.3.3.1.2.cmml" xref="S7.SS1.p9.7.m4.3.3.1.1"><abs id="S7.SS1.p9.7.m4.3.3.1.2.1.cmml" xref="S7.SS1.p9.7.m4.3.3.1.1.2"></abs><apply id="S7.SS1.p9.7.m4.3.3.1.1.1.cmml" xref="S7.SS1.p9.7.m4.3.3.1.1.1"><csymbol cd="ambiguous" id="S7.SS1.p9.7.m4.3.3.1.1.1.1.cmml" xref="S7.SS1.p9.7.m4.3.3.1.1.1">subscript</csymbol><ci id="S7.SS1.p9.7.m4.3.3.1.1.1.2.cmml" xref="S7.SS1.p9.7.m4.3.3.1.1.1.2">𝐴</ci><ci id="S7.SS1.p9.7.m4.3.3.1.1.1.3.cmml" xref="S7.SS1.p9.7.m4.3.3.1.1.1.3">𝛼</ci></apply></apply><apply id="S7.SS1.p9.7.m4.5.5.3.3.cmml" xref="S7.SS1.p9.7.m4.5.5.3.2"><max id="S7.SS1.p9.7.m4.2.2.cmml" xref="S7.SS1.p9.7.m4.2.2"></max><apply id="S7.SS1.p9.7.m4.4.4.2.1.1.1.cmml" xref="S7.SS1.p9.7.m4.4.4.2.1.1.1"><csymbol cd="ambiguous" id="S7.SS1.p9.7.m4.4.4.2.1.1.1.1.cmml" xref="S7.SS1.p9.7.m4.4.4.2.1.1.1">subscript</csymbol><ci id="S7.SS1.p9.7.m4.4.4.2.1.1.1.2.cmml" xref="S7.SS1.p9.7.m4.4.4.2.1.1.1.2">ℵ</ci><cn id="S7.SS1.p9.7.m4.4.4.2.1.1.1.3.cmml" type="integer" xref="S7.SS1.p9.7.m4.4.4.2.1.1.1.3">0</cn></apply><apply id="S7.SS1.p9.7.m4.5.5.3.2.2.2.1.cmml" xref="S7.SS1.p9.7.m4.5.5.3.2.2.2.2"><abs id="S7.SS1.p9.7.m4.5.5.3.2.2.2.1.1.cmml" xref="S7.SS1.p9.7.m4.5.5.3.2.2.2.2.1"></abs><ci id="S7.SS1.p9.7.m4.1.1.cmml" xref="S7.SS1.p9.7.m4.1.1">𝛼</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p9.7.m4.5c">|A_{\alpha}|\leq\max\{\aleph_{0},|\alpha|\}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p9.7.m4.5d">| italic_A start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT | ≤ roman_max { roman_ℵ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , | italic_α | }</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S7.Thmtheorem9"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S7.Thmtheorem9.1.1.1">Lemma 7.9</span></span><span class="ltx_text ltx_font_bold" id="S7.Thmtheorem9.2.2">.</span> </h6> <div class="ltx_para" id="S7.Thmtheorem9.p1"> <p class="ltx_p" id="S7.Thmtheorem9.p1.5">Let <math alttext="X\subseteq\mathfrak{c}" class="ltx_Math" display="inline" id="S7.Thmtheorem9.p1.1.m1.1"><semantics id="S7.Thmtheorem9.p1.1.m1.1a"><mrow id="S7.Thmtheorem9.p1.1.m1.1.1" xref="S7.Thmtheorem9.p1.1.m1.1.1.cmml"><mi id="S7.Thmtheorem9.p1.1.m1.1.1.2" xref="S7.Thmtheorem9.p1.1.m1.1.1.2.cmml">X</mi><mo id="S7.Thmtheorem9.p1.1.m1.1.1.1" xref="S7.Thmtheorem9.p1.1.m1.1.1.1.cmml">⊆</mo><mi id="S7.Thmtheorem9.p1.1.m1.1.1.3" xref="S7.Thmtheorem9.p1.1.m1.1.1.3.cmml">𝔠</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem9.p1.1.m1.1b"><apply id="S7.Thmtheorem9.p1.1.m1.1.1.cmml" xref="S7.Thmtheorem9.p1.1.m1.1.1"><subset id="S7.Thmtheorem9.p1.1.m1.1.1.1.cmml" xref="S7.Thmtheorem9.p1.1.m1.1.1.1"></subset><ci id="S7.Thmtheorem9.p1.1.m1.1.1.2.cmml" xref="S7.Thmtheorem9.p1.1.m1.1.1.2">𝑋</ci><ci id="S7.Thmtheorem9.p1.1.m1.1.1.3.cmml" xref="S7.Thmtheorem9.p1.1.m1.1.1.3">𝔠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem9.p1.1.m1.1c">X\subseteq\mathfrak{c}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem9.p1.1.m1.1d">italic_X ⊆ fraktur_c</annotation></semantics></math> be such that <math alttext="|X|=\mathfrak{c}" class="ltx_Math" display="inline" id="S7.Thmtheorem9.p1.2.m2.1"><semantics id="S7.Thmtheorem9.p1.2.m2.1a"><mrow id="S7.Thmtheorem9.p1.2.m2.1.2" xref="S7.Thmtheorem9.p1.2.m2.1.2.cmml"><mrow id="S7.Thmtheorem9.p1.2.m2.1.2.2.2" xref="S7.Thmtheorem9.p1.2.m2.1.2.2.1.cmml"><mo id="S7.Thmtheorem9.p1.2.m2.1.2.2.2.1" stretchy="false" xref="S7.Thmtheorem9.p1.2.m2.1.2.2.1.1.cmml">|</mo><mi id="S7.Thmtheorem9.p1.2.m2.1.1" xref="S7.Thmtheorem9.p1.2.m2.1.1.cmml">X</mi><mo id="S7.Thmtheorem9.p1.2.m2.1.2.2.2.2" stretchy="false" xref="S7.Thmtheorem9.p1.2.m2.1.2.2.1.1.cmml">|</mo></mrow><mo id="S7.Thmtheorem9.p1.2.m2.1.2.1" xref="S7.Thmtheorem9.p1.2.m2.1.2.1.cmml">=</mo><mi id="S7.Thmtheorem9.p1.2.m2.1.2.3" xref="S7.Thmtheorem9.p1.2.m2.1.2.3.cmml">𝔠</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem9.p1.2.m2.1b"><apply id="S7.Thmtheorem9.p1.2.m2.1.2.cmml" xref="S7.Thmtheorem9.p1.2.m2.1.2"><eq id="S7.Thmtheorem9.p1.2.m2.1.2.1.cmml" xref="S7.Thmtheorem9.p1.2.m2.1.2.1"></eq><apply id="S7.Thmtheorem9.p1.2.m2.1.2.2.1.cmml" xref="S7.Thmtheorem9.p1.2.m2.1.2.2.2"><abs id="S7.Thmtheorem9.p1.2.m2.1.2.2.1.1.cmml" xref="S7.Thmtheorem9.p1.2.m2.1.2.2.2.1"></abs><ci id="S7.Thmtheorem9.p1.2.m2.1.1.cmml" xref="S7.Thmtheorem9.p1.2.m2.1.1">𝑋</ci></apply><ci id="S7.Thmtheorem9.p1.2.m2.1.2.3.cmml" xref="S7.Thmtheorem9.p1.2.m2.1.2.3">𝔠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem9.p1.2.m2.1c">|X|=\mathfrak{c}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem9.p1.2.m2.1d">| italic_X | = fraktur_c</annotation></semantics></math>, and <math alttext="B\subseteq\mathbb{R}" class="ltx_Math" display="inline" id="S7.Thmtheorem9.p1.3.m3.1"><semantics id="S7.Thmtheorem9.p1.3.m3.1a"><mrow id="S7.Thmtheorem9.p1.3.m3.1.1" xref="S7.Thmtheorem9.p1.3.m3.1.1.cmml"><mi id="S7.Thmtheorem9.p1.3.m3.1.1.2" xref="S7.Thmtheorem9.p1.3.m3.1.1.2.cmml">B</mi><mo id="S7.Thmtheorem9.p1.3.m3.1.1.1" xref="S7.Thmtheorem9.p1.3.m3.1.1.1.cmml">⊆</mo><mi id="S7.Thmtheorem9.p1.3.m3.1.1.3" xref="S7.Thmtheorem9.p1.3.m3.1.1.3.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem9.p1.3.m3.1b"><apply id="S7.Thmtheorem9.p1.3.m3.1.1.cmml" xref="S7.Thmtheorem9.p1.3.m3.1.1"><subset id="S7.Thmtheorem9.p1.3.m3.1.1.1.cmml" xref="S7.Thmtheorem9.p1.3.m3.1.1.1"></subset><ci id="S7.Thmtheorem9.p1.3.m3.1.1.2.cmml" xref="S7.Thmtheorem9.p1.3.m3.1.1.2">𝐵</ci><ci id="S7.Thmtheorem9.p1.3.m3.1.1.3.cmml" xref="S7.Thmtheorem9.p1.3.m3.1.1.3">ℝ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem9.p1.3.m3.1c">B\subseteq\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem9.p1.3.m3.1d">italic_B ⊆ blackboard_R</annotation></semantics></math> be such that <math alttext="\aleph_{0}<|B|<\mathfrak{c}" class="ltx_Math" display="inline" id="S7.Thmtheorem9.p1.4.m4.1"><semantics id="S7.Thmtheorem9.p1.4.m4.1a"><mrow id="S7.Thmtheorem9.p1.4.m4.1.2" xref="S7.Thmtheorem9.p1.4.m4.1.2.cmml"><msub id="S7.Thmtheorem9.p1.4.m4.1.2.2" xref="S7.Thmtheorem9.p1.4.m4.1.2.2.cmml"><mi id="S7.Thmtheorem9.p1.4.m4.1.2.2.2" mathvariant="normal" xref="S7.Thmtheorem9.p1.4.m4.1.2.2.2.cmml">ℵ</mi><mn id="S7.Thmtheorem9.p1.4.m4.1.2.2.3" xref="S7.Thmtheorem9.p1.4.m4.1.2.2.3.cmml">0</mn></msub><mo id="S7.Thmtheorem9.p1.4.m4.1.2.3" xref="S7.Thmtheorem9.p1.4.m4.1.2.3.cmml"><</mo><mrow id="S7.Thmtheorem9.p1.4.m4.1.2.4.2" xref="S7.Thmtheorem9.p1.4.m4.1.2.4.1.cmml"><mo id="S7.Thmtheorem9.p1.4.m4.1.2.4.2.1" stretchy="false" xref="S7.Thmtheorem9.p1.4.m4.1.2.4.1.1.cmml">|</mo><mi id="S7.Thmtheorem9.p1.4.m4.1.1" xref="S7.Thmtheorem9.p1.4.m4.1.1.cmml">B</mi><mo id="S7.Thmtheorem9.p1.4.m4.1.2.4.2.2" stretchy="false" xref="S7.Thmtheorem9.p1.4.m4.1.2.4.1.1.cmml">|</mo></mrow><mo id="S7.Thmtheorem9.p1.4.m4.1.2.5" xref="S7.Thmtheorem9.p1.4.m4.1.2.5.cmml"><</mo><mi id="S7.Thmtheorem9.p1.4.m4.1.2.6" xref="S7.Thmtheorem9.p1.4.m4.1.2.6.cmml">𝔠</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem9.p1.4.m4.1b"><apply id="S7.Thmtheorem9.p1.4.m4.1.2.cmml" xref="S7.Thmtheorem9.p1.4.m4.1.2"><and id="S7.Thmtheorem9.p1.4.m4.1.2a.cmml" xref="S7.Thmtheorem9.p1.4.m4.1.2"></and><apply id="S7.Thmtheorem9.p1.4.m4.1.2b.cmml" xref="S7.Thmtheorem9.p1.4.m4.1.2"><lt id="S7.Thmtheorem9.p1.4.m4.1.2.3.cmml" xref="S7.Thmtheorem9.p1.4.m4.1.2.3"></lt><apply id="S7.Thmtheorem9.p1.4.m4.1.2.2.cmml" xref="S7.Thmtheorem9.p1.4.m4.1.2.2"><csymbol cd="ambiguous" id="S7.Thmtheorem9.p1.4.m4.1.2.2.1.cmml" xref="S7.Thmtheorem9.p1.4.m4.1.2.2">subscript</csymbol><ci id="S7.Thmtheorem9.p1.4.m4.1.2.2.2.cmml" xref="S7.Thmtheorem9.p1.4.m4.1.2.2.2">ℵ</ci><cn id="S7.Thmtheorem9.p1.4.m4.1.2.2.3.cmml" type="integer" xref="S7.Thmtheorem9.p1.4.m4.1.2.2.3">0</cn></apply><apply id="S7.Thmtheorem9.p1.4.m4.1.2.4.1.cmml" xref="S7.Thmtheorem9.p1.4.m4.1.2.4.2"><abs id="S7.Thmtheorem9.p1.4.m4.1.2.4.1.1.cmml" xref="S7.Thmtheorem9.p1.4.m4.1.2.4.2.1"></abs><ci id="S7.Thmtheorem9.p1.4.m4.1.1.cmml" xref="S7.Thmtheorem9.p1.4.m4.1.1">𝐵</ci></apply></apply><apply id="S7.Thmtheorem9.p1.4.m4.1.2c.cmml" xref="S7.Thmtheorem9.p1.4.m4.1.2"><lt id="S7.Thmtheorem9.p1.4.m4.1.2.5.cmml" xref="S7.Thmtheorem9.p1.4.m4.1.2.5"></lt><share href="https://arxiv.org/html/2503.13728v1#S7.Thmtheorem9.p1.4.m4.1.2.4.cmml" id="S7.Thmtheorem9.p1.4.m4.1.2d.cmml" xref="S7.Thmtheorem9.p1.4.m4.1.2"></share><ci id="S7.Thmtheorem9.p1.4.m4.1.2.6.cmml" xref="S7.Thmtheorem9.p1.4.m4.1.2.6">𝔠</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem9.p1.4.m4.1c">\aleph_{0}<|B|<\mathfrak{c}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem9.p1.4.m4.1d">roman_ℵ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT < | italic_B | < fraktur_c</annotation></semantics></math>. Then <math alttext="A(X)\ntrianglerighteq B" class="ltx_Math" display="inline" id="S7.Thmtheorem9.p1.5.m5.1"><semantics id="S7.Thmtheorem9.p1.5.m5.1a"><mrow id="S7.Thmtheorem9.p1.5.m5.1.2" xref="S7.Thmtheorem9.p1.5.m5.1.2.cmml"><mrow id="S7.Thmtheorem9.p1.5.m5.1.2.2" xref="S7.Thmtheorem9.p1.5.m5.1.2.2.cmml"><mi id="S7.Thmtheorem9.p1.5.m5.1.2.2.2" xref="S7.Thmtheorem9.p1.5.m5.1.2.2.2.cmml">A</mi><mo id="S7.Thmtheorem9.p1.5.m5.1.2.2.1" xref="S7.Thmtheorem9.p1.5.m5.1.2.2.1.cmml">⁢</mo><mrow id="S7.Thmtheorem9.p1.5.m5.1.2.2.3.2" xref="S7.Thmtheorem9.p1.5.m5.1.2.2.cmml"><mo id="S7.Thmtheorem9.p1.5.m5.1.2.2.3.2.1" stretchy="false" xref="S7.Thmtheorem9.p1.5.m5.1.2.2.cmml">(</mo><mi id="S7.Thmtheorem9.p1.5.m5.1.1" xref="S7.Thmtheorem9.p1.5.m5.1.1.cmml">X</mi><mo id="S7.Thmtheorem9.p1.5.m5.1.2.2.3.2.2" stretchy="false" xref="S7.Thmtheorem9.p1.5.m5.1.2.2.cmml">)</mo></mrow></mrow><mo id="S7.Thmtheorem9.p1.5.m5.1.2.1" xref="S7.Thmtheorem9.p1.5.m5.1.2.1.cmml">⋭</mo><mi id="S7.Thmtheorem9.p1.5.m5.1.2.3" xref="S7.Thmtheorem9.p1.5.m5.1.2.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem9.p1.5.m5.1b"><apply id="S7.Thmtheorem9.p1.5.m5.1.2.cmml" xref="S7.Thmtheorem9.p1.5.m5.1.2"><csymbol cd="latexml" id="S7.Thmtheorem9.p1.5.m5.1.2.1.cmml" xref="S7.Thmtheorem9.p1.5.m5.1.2.1">not-contains-nor-equals</csymbol><apply id="S7.Thmtheorem9.p1.5.m5.1.2.2.cmml" xref="S7.Thmtheorem9.p1.5.m5.1.2.2"><times id="S7.Thmtheorem9.p1.5.m5.1.2.2.1.cmml" xref="S7.Thmtheorem9.p1.5.m5.1.2.2.1"></times><ci id="S7.Thmtheorem9.p1.5.m5.1.2.2.2.cmml" xref="S7.Thmtheorem9.p1.5.m5.1.2.2.2">𝐴</ci><ci id="S7.Thmtheorem9.p1.5.m5.1.1.cmml" xref="S7.Thmtheorem9.p1.5.m5.1.1">𝑋</ci></apply><ci id="S7.Thmtheorem9.p1.5.m5.1.2.3.cmml" xref="S7.Thmtheorem9.p1.5.m5.1.2.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem9.p1.5.m5.1c">A(X)\ntrianglerighteq B</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem9.p1.5.m5.1d">italic_A ( italic_X ) ⋭ italic_B</annotation></semantics></math>.</p> </div> </div> <div class="ltx_proof" id="S7.SS1.8"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S7.SS1.5.p1"> <p class="ltx_p" id="S7.SS1.5.p1.5">Suppose <math alttext="A(X)\trianglerighteq B" class="ltx_Math" display="inline" id="S7.SS1.5.p1.1.m1.1"><semantics id="S7.SS1.5.p1.1.m1.1a"><mrow id="S7.SS1.5.p1.1.m1.1.2" xref="S7.SS1.5.p1.1.m1.1.2.cmml"><mi id="S7.SS1.5.p1.1.m1.1.2.2" xref="S7.SS1.5.p1.1.m1.1.2.2.cmml">A</mi><mo id="S7.SS1.5.p1.1.m1.1.2.1" xref="S7.SS1.5.p1.1.m1.1.2.1.cmml">⁢</mo><mrow id="S7.SS1.5.p1.1.m1.1.2.3.2" xref="S7.SS1.5.p1.1.m1.1.2.cmml"><mo id="S7.SS1.5.p1.1.m1.1.2.3.2.1" stretchy="false" xref="S7.SS1.5.p1.1.m1.1.2.cmml">(</mo><mi id="S7.SS1.5.p1.1.m1.1.1" xref="S7.SS1.5.p1.1.m1.1.1.cmml">X</mi><mo id="S7.SS1.5.p1.1.m1.1.2.3.2.2" stretchy="false" xref="S7.SS1.5.p1.1.m1.1.2.cmml">)</mo></mrow><mo id="S7.SS1.5.p1.1.m1.1.2.1a" xref="S7.SS1.5.p1.1.m1.1.2.1.cmml">⁢</mo><mi id="S7.SS1.5.p1.1.m1.1.2.4" mathvariant="normal" xref="S7.SS1.5.p1.1.m1.1.2.4.cmml">⊵</mi><mo id="S7.SS1.5.p1.1.m1.1.2.1b" xref="S7.SS1.5.p1.1.m1.1.2.1.cmml">⁢</mo><mi id="S7.SS1.5.p1.1.m1.1.2.5" xref="S7.SS1.5.p1.1.m1.1.2.5.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.5.p1.1.m1.1b"><apply id="S7.SS1.5.p1.1.m1.1.2.cmml" xref="S7.SS1.5.p1.1.m1.1.2"><times id="S7.SS1.5.p1.1.m1.1.2.1.cmml" xref="S7.SS1.5.p1.1.m1.1.2.1"></times><ci id="S7.SS1.5.p1.1.m1.1.2.2.cmml" xref="S7.SS1.5.p1.1.m1.1.2.2">𝐴</ci><ci id="S7.SS1.5.p1.1.m1.1.1.cmml" xref="S7.SS1.5.p1.1.m1.1.1">𝑋</ci><ci id="S7.SS1.5.p1.1.m1.1.2.4.cmml" xref="S7.SS1.5.p1.1.m1.1.2.4">⊵</ci><ci id="S7.SS1.5.p1.1.m1.1.2.5.cmml" xref="S7.SS1.5.p1.1.m1.1.2.5">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.5.p1.1.m1.1c">A(X)\trianglerighteq B</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.5.p1.1.m1.1d">italic_A ( italic_X ) ⊵ italic_B</annotation></semantics></math> and that <math alttext="\aleph_{0}<|B|<\mathfrak{c}" class="ltx_Math" display="inline" id="S7.SS1.5.p1.2.m2.1"><semantics id="S7.SS1.5.p1.2.m2.1a"><mrow id="S7.SS1.5.p1.2.m2.1.2" xref="S7.SS1.5.p1.2.m2.1.2.cmml"><msub id="S7.SS1.5.p1.2.m2.1.2.2" xref="S7.SS1.5.p1.2.m2.1.2.2.cmml"><mi id="S7.SS1.5.p1.2.m2.1.2.2.2" mathvariant="normal" xref="S7.SS1.5.p1.2.m2.1.2.2.2.cmml">ℵ</mi><mn id="S7.SS1.5.p1.2.m2.1.2.2.3" xref="S7.SS1.5.p1.2.m2.1.2.2.3.cmml">0</mn></msub><mo id="S7.SS1.5.p1.2.m2.1.2.3" xref="S7.SS1.5.p1.2.m2.1.2.3.cmml"><</mo><mrow id="S7.SS1.5.p1.2.m2.1.2.4.2" xref="S7.SS1.5.p1.2.m2.1.2.4.1.cmml"><mo id="S7.SS1.5.p1.2.m2.1.2.4.2.1" stretchy="false" xref="S7.SS1.5.p1.2.m2.1.2.4.1.1.cmml">|</mo><mi id="S7.SS1.5.p1.2.m2.1.1" xref="S7.SS1.5.p1.2.m2.1.1.cmml">B</mi><mo id="S7.SS1.5.p1.2.m2.1.2.4.2.2" stretchy="false" xref="S7.SS1.5.p1.2.m2.1.2.4.1.1.cmml">|</mo></mrow><mo id="S7.SS1.5.p1.2.m2.1.2.5" xref="S7.SS1.5.p1.2.m2.1.2.5.cmml"><</mo><mi id="S7.SS1.5.p1.2.m2.1.2.6" xref="S7.SS1.5.p1.2.m2.1.2.6.cmml">𝔠</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.5.p1.2.m2.1b"><apply id="S7.SS1.5.p1.2.m2.1.2.cmml" xref="S7.SS1.5.p1.2.m2.1.2"><and id="S7.SS1.5.p1.2.m2.1.2a.cmml" xref="S7.SS1.5.p1.2.m2.1.2"></and><apply id="S7.SS1.5.p1.2.m2.1.2b.cmml" xref="S7.SS1.5.p1.2.m2.1.2"><lt id="S7.SS1.5.p1.2.m2.1.2.3.cmml" xref="S7.SS1.5.p1.2.m2.1.2.3"></lt><apply id="S7.SS1.5.p1.2.m2.1.2.2.cmml" xref="S7.SS1.5.p1.2.m2.1.2.2"><csymbol cd="ambiguous" id="S7.SS1.5.p1.2.m2.1.2.2.1.cmml" xref="S7.SS1.5.p1.2.m2.1.2.2">subscript</csymbol><ci id="S7.SS1.5.p1.2.m2.1.2.2.2.cmml" xref="S7.SS1.5.p1.2.m2.1.2.2.2">ℵ</ci><cn id="S7.SS1.5.p1.2.m2.1.2.2.3.cmml" type="integer" xref="S7.SS1.5.p1.2.m2.1.2.2.3">0</cn></apply><apply id="S7.SS1.5.p1.2.m2.1.2.4.1.cmml" xref="S7.SS1.5.p1.2.m2.1.2.4.2"><abs id="S7.SS1.5.p1.2.m2.1.2.4.1.1.cmml" xref="S7.SS1.5.p1.2.m2.1.2.4.2.1"></abs><ci id="S7.SS1.5.p1.2.m2.1.1.cmml" xref="S7.SS1.5.p1.2.m2.1.1">𝐵</ci></apply></apply><apply id="S7.SS1.5.p1.2.m2.1.2c.cmml" xref="S7.SS1.5.p1.2.m2.1.2"><lt id="S7.SS1.5.p1.2.m2.1.2.5.cmml" xref="S7.SS1.5.p1.2.m2.1.2.5"></lt><share href="https://arxiv.org/html/2503.13728v1#S7.SS1.5.p1.2.m2.1.2.4.cmml" id="S7.SS1.5.p1.2.m2.1.2d.cmml" xref="S7.SS1.5.p1.2.m2.1.2"></share><ci id="S7.SS1.5.p1.2.m2.1.2.6.cmml" xref="S7.SS1.5.p1.2.m2.1.2.6">𝔠</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.5.p1.2.m2.1c">\aleph_{0}<|B|<\mathfrak{c}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.5.p1.2.m2.1d">roman_ℵ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT < | italic_B | < fraktur_c</annotation></semantics></math>. Let <math alttext="f:A(X)\twoheadrightarrow B" class="ltx_Math" display="inline" id="S7.SS1.5.p1.3.m3.1"><semantics id="S7.SS1.5.p1.3.m3.1a"><mrow id="S7.SS1.5.p1.3.m3.1.2" xref="S7.SS1.5.p1.3.m3.1.2.cmml"><mi id="S7.SS1.5.p1.3.m3.1.2.2" xref="S7.SS1.5.p1.3.m3.1.2.2.cmml">f</mi><mo id="S7.SS1.5.p1.3.m3.1.2.1" lspace="0.278em" rspace="0.278em" xref="S7.SS1.5.p1.3.m3.1.2.1.cmml">:</mo><mrow id="S7.SS1.5.p1.3.m3.1.2.3" xref="S7.SS1.5.p1.3.m3.1.2.3.cmml"><mrow id="S7.SS1.5.p1.3.m3.1.2.3.2" xref="S7.SS1.5.p1.3.m3.1.2.3.2.cmml"><mi id="S7.SS1.5.p1.3.m3.1.2.3.2.2" xref="S7.SS1.5.p1.3.m3.1.2.3.2.2.cmml">A</mi><mo id="S7.SS1.5.p1.3.m3.1.2.3.2.1" xref="S7.SS1.5.p1.3.m3.1.2.3.2.1.cmml">⁢</mo><mrow id="S7.SS1.5.p1.3.m3.1.2.3.2.3.2" xref="S7.SS1.5.p1.3.m3.1.2.3.2.cmml"><mo id="S7.SS1.5.p1.3.m3.1.2.3.2.3.2.1" stretchy="false" xref="S7.SS1.5.p1.3.m3.1.2.3.2.cmml">(</mo><mi id="S7.SS1.5.p1.3.m3.1.1" xref="S7.SS1.5.p1.3.m3.1.1.cmml">X</mi><mo id="S7.SS1.5.p1.3.m3.1.2.3.2.3.2.2" stretchy="false" xref="S7.SS1.5.p1.3.m3.1.2.3.2.cmml">)</mo></mrow></mrow><mo id="S7.SS1.5.p1.3.m3.1.2.3.1" stretchy="false" xref="S7.SS1.5.p1.3.m3.1.2.3.1.cmml">↠</mo><mi id="S7.SS1.5.p1.3.m3.1.2.3.3" xref="S7.SS1.5.p1.3.m3.1.2.3.3.cmml">B</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.5.p1.3.m3.1b"><apply id="S7.SS1.5.p1.3.m3.1.2.cmml" xref="S7.SS1.5.p1.3.m3.1.2"><ci id="S7.SS1.5.p1.3.m3.1.2.1.cmml" xref="S7.SS1.5.p1.3.m3.1.2.1">:</ci><ci id="S7.SS1.5.p1.3.m3.1.2.2.cmml" xref="S7.SS1.5.p1.3.m3.1.2.2">𝑓</ci><apply id="S7.SS1.5.p1.3.m3.1.2.3.cmml" xref="S7.SS1.5.p1.3.m3.1.2.3"><ci id="S7.SS1.5.p1.3.m3.1.2.3.1.cmml" xref="S7.SS1.5.p1.3.m3.1.2.3.1">↠</ci><apply id="S7.SS1.5.p1.3.m3.1.2.3.2.cmml" xref="S7.SS1.5.p1.3.m3.1.2.3.2"><times id="S7.SS1.5.p1.3.m3.1.2.3.2.1.cmml" xref="S7.SS1.5.p1.3.m3.1.2.3.2.1"></times><ci id="S7.SS1.5.p1.3.m3.1.2.3.2.2.cmml" xref="S7.SS1.5.p1.3.m3.1.2.3.2.2">𝐴</ci><ci id="S7.SS1.5.p1.3.m3.1.1.cmml" xref="S7.SS1.5.p1.3.m3.1.1">𝑋</ci></apply><ci id="S7.SS1.5.p1.3.m3.1.2.3.3.cmml" xref="S7.SS1.5.p1.3.m3.1.2.3.3">𝐵</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.5.p1.3.m3.1c">f:A(X)\twoheadrightarrow B</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.5.p1.3.m3.1d">italic_f : italic_A ( italic_X ) ↠ italic_B</annotation></semantics></math> be an epimorphism, and find <math alttext="\alpha" class="ltx_Math" display="inline" id="S7.SS1.5.p1.4.m4.1"><semantics id="S7.SS1.5.p1.4.m4.1a"><mi id="S7.SS1.5.p1.4.m4.1.1" xref="S7.SS1.5.p1.4.m4.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.5.p1.4.m4.1b"><ci id="S7.SS1.5.p1.4.m4.1.1.cmml" xref="S7.SS1.5.p1.4.m4.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.5.p1.4.m4.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.5.p1.4.m4.1d">italic_α</annotation></semantics></math> such that <math alttext="f\subseteq f_{\alpha}" class="ltx_Math" display="inline" id="S7.SS1.5.p1.5.m5.1"><semantics id="S7.SS1.5.p1.5.m5.1a"><mrow id="S7.SS1.5.p1.5.m5.1.1" xref="S7.SS1.5.p1.5.m5.1.1.cmml"><mi id="S7.SS1.5.p1.5.m5.1.1.2" xref="S7.SS1.5.p1.5.m5.1.1.2.cmml">f</mi><mo id="S7.SS1.5.p1.5.m5.1.1.1" xref="S7.SS1.5.p1.5.m5.1.1.1.cmml">⊆</mo><msub id="S7.SS1.5.p1.5.m5.1.1.3" xref="S7.SS1.5.p1.5.m5.1.1.3.cmml"><mi id="S7.SS1.5.p1.5.m5.1.1.3.2" xref="S7.SS1.5.p1.5.m5.1.1.3.2.cmml">f</mi><mi id="S7.SS1.5.p1.5.m5.1.1.3.3" xref="S7.SS1.5.p1.5.m5.1.1.3.3.cmml">α</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.5.p1.5.m5.1b"><apply id="S7.SS1.5.p1.5.m5.1.1.cmml" xref="S7.SS1.5.p1.5.m5.1.1"><subset id="S7.SS1.5.p1.5.m5.1.1.1.cmml" xref="S7.SS1.5.p1.5.m5.1.1.1"></subset><ci id="S7.SS1.5.p1.5.m5.1.1.2.cmml" xref="S7.SS1.5.p1.5.m5.1.1.2">𝑓</ci><apply id="S7.SS1.5.p1.5.m5.1.1.3.cmml" xref="S7.SS1.5.p1.5.m5.1.1.3"><csymbol cd="ambiguous" id="S7.SS1.5.p1.5.m5.1.1.3.1.cmml" xref="S7.SS1.5.p1.5.m5.1.1.3">subscript</csymbol><ci id="S7.SS1.5.p1.5.m5.1.1.3.2.cmml" xref="S7.SS1.5.p1.5.m5.1.1.3.2">𝑓</ci><ci id="S7.SS1.5.p1.5.m5.1.1.3.3.cmml" xref="S7.SS1.5.p1.5.m5.1.1.3.3">𝛼</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.5.p1.5.m5.1c">f\subseteq f_{\alpha}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.5.p1.5.m5.1d">italic_f ⊆ italic_f start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S7.SS1.6.p2"> <p class="ltx_p" id="S7.SS1.6.p2.9">First note that <math alttext="|\mathbb{R}\setminus\operatorname{dom}(f_{\alpha})|=\mathfrak{c}" class="ltx_Math" display="inline" id="S7.SS1.6.p2.1.m1.2"><semantics id="S7.SS1.6.p2.1.m1.2a"><mrow id="S7.SS1.6.p2.1.m1.2.2" xref="S7.SS1.6.p2.1.m1.2.2.cmml"><mrow id="S7.SS1.6.p2.1.m1.2.2.1.1" xref="S7.SS1.6.p2.1.m1.2.2.1.2.cmml"><mo id="S7.SS1.6.p2.1.m1.2.2.1.1.2" stretchy="false" xref="S7.SS1.6.p2.1.m1.2.2.1.2.1.cmml">|</mo><mrow id="S7.SS1.6.p2.1.m1.2.2.1.1.1" xref="S7.SS1.6.p2.1.m1.2.2.1.1.1.cmml"><mi id="S7.SS1.6.p2.1.m1.2.2.1.1.1.3" xref="S7.SS1.6.p2.1.m1.2.2.1.1.1.3.cmml">ℝ</mi><mo id="S7.SS1.6.p2.1.m1.2.2.1.1.1.2" xref="S7.SS1.6.p2.1.m1.2.2.1.1.1.2.cmml">∖</mo><mrow id="S7.SS1.6.p2.1.m1.2.2.1.1.1.1.1" xref="S7.SS1.6.p2.1.m1.2.2.1.1.1.1.2.cmml"><mi id="S7.SS1.6.p2.1.m1.1.1" xref="S7.SS1.6.p2.1.m1.1.1.cmml">dom</mi><mo id="S7.SS1.6.p2.1.m1.2.2.1.1.1.1.1a" xref="S7.SS1.6.p2.1.m1.2.2.1.1.1.1.2.cmml">⁡</mo><mrow id="S7.SS1.6.p2.1.m1.2.2.1.1.1.1.1.1" xref="S7.SS1.6.p2.1.m1.2.2.1.1.1.1.2.cmml"><mo id="S7.SS1.6.p2.1.m1.2.2.1.1.1.1.1.1.2" stretchy="false" xref="S7.SS1.6.p2.1.m1.2.2.1.1.1.1.2.cmml">(</mo><msub id="S7.SS1.6.p2.1.m1.2.2.1.1.1.1.1.1.1" xref="S7.SS1.6.p2.1.m1.2.2.1.1.1.1.1.1.1.cmml"><mi id="S7.SS1.6.p2.1.m1.2.2.1.1.1.1.1.1.1.2" xref="S7.SS1.6.p2.1.m1.2.2.1.1.1.1.1.1.1.2.cmml">f</mi><mi id="S7.SS1.6.p2.1.m1.2.2.1.1.1.1.1.1.1.3" xref="S7.SS1.6.p2.1.m1.2.2.1.1.1.1.1.1.1.3.cmml">α</mi></msub><mo id="S7.SS1.6.p2.1.m1.2.2.1.1.1.1.1.1.3" stretchy="false" xref="S7.SS1.6.p2.1.m1.2.2.1.1.1.1.2.cmml">)</mo></mrow></mrow></mrow><mo id="S7.SS1.6.p2.1.m1.2.2.1.1.3" stretchy="false" xref="S7.SS1.6.p2.1.m1.2.2.1.2.1.cmml">|</mo></mrow><mo id="S7.SS1.6.p2.1.m1.2.2.2" xref="S7.SS1.6.p2.1.m1.2.2.2.cmml">=</mo><mi id="S7.SS1.6.p2.1.m1.2.2.3" xref="S7.SS1.6.p2.1.m1.2.2.3.cmml">𝔠</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.6.p2.1.m1.2b"><apply id="S7.SS1.6.p2.1.m1.2.2.cmml" xref="S7.SS1.6.p2.1.m1.2.2"><eq id="S7.SS1.6.p2.1.m1.2.2.2.cmml" xref="S7.SS1.6.p2.1.m1.2.2.2"></eq><apply id="S7.SS1.6.p2.1.m1.2.2.1.2.cmml" xref="S7.SS1.6.p2.1.m1.2.2.1.1"><abs id="S7.SS1.6.p2.1.m1.2.2.1.2.1.cmml" xref="S7.SS1.6.p2.1.m1.2.2.1.1.2"></abs><apply id="S7.SS1.6.p2.1.m1.2.2.1.1.1.cmml" xref="S7.SS1.6.p2.1.m1.2.2.1.1.1"><setdiff id="S7.SS1.6.p2.1.m1.2.2.1.1.1.2.cmml" xref="S7.SS1.6.p2.1.m1.2.2.1.1.1.2"></setdiff><ci id="S7.SS1.6.p2.1.m1.2.2.1.1.1.3.cmml" xref="S7.SS1.6.p2.1.m1.2.2.1.1.1.3">ℝ</ci><apply id="S7.SS1.6.p2.1.m1.2.2.1.1.1.1.2.cmml" xref="S7.SS1.6.p2.1.m1.2.2.1.1.1.1.1"><ci id="S7.SS1.6.p2.1.m1.1.1.cmml" xref="S7.SS1.6.p2.1.m1.1.1">dom</ci><apply id="S7.SS1.6.p2.1.m1.2.2.1.1.1.1.1.1.1.cmml" xref="S7.SS1.6.p2.1.m1.2.2.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S7.SS1.6.p2.1.m1.2.2.1.1.1.1.1.1.1.1.cmml" xref="S7.SS1.6.p2.1.m1.2.2.1.1.1.1.1.1.1">subscript</csymbol><ci id="S7.SS1.6.p2.1.m1.2.2.1.1.1.1.1.1.1.2.cmml" xref="S7.SS1.6.p2.1.m1.2.2.1.1.1.1.1.1.1.2">𝑓</ci><ci id="S7.SS1.6.p2.1.m1.2.2.1.1.1.1.1.1.1.3.cmml" xref="S7.SS1.6.p2.1.m1.2.2.1.1.1.1.1.1.1.3">𝛼</ci></apply></apply></apply></apply><ci id="S7.SS1.6.p2.1.m1.2.2.3.cmml" xref="S7.SS1.6.p2.1.m1.2.2.3">𝔠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.6.p2.1.m1.2c">|\mathbb{R}\setminus\operatorname{dom}(f_{\alpha})|=\mathfrak{c}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.6.p2.1.m1.2d">| blackboard_R ∖ roman_dom ( italic_f start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ) | = fraktur_c</annotation></semantics></math> is impossible: suppose this is the case, then taking <math alttext="\beta>\alpha" class="ltx_Math" display="inline" id="S7.SS1.6.p2.2.m2.1"><semantics id="S7.SS1.6.p2.2.m2.1a"><mrow id="S7.SS1.6.p2.2.m2.1.1" xref="S7.SS1.6.p2.2.m2.1.1.cmml"><mi id="S7.SS1.6.p2.2.m2.1.1.2" xref="S7.SS1.6.p2.2.m2.1.1.2.cmml">β</mi><mo id="S7.SS1.6.p2.2.m2.1.1.1" xref="S7.SS1.6.p2.2.m2.1.1.1.cmml">></mo><mi id="S7.SS1.6.p2.2.m2.1.1.3" xref="S7.SS1.6.p2.2.m2.1.1.3.cmml">α</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.6.p2.2.m2.1b"><apply id="S7.SS1.6.p2.2.m2.1.1.cmml" xref="S7.SS1.6.p2.2.m2.1.1"><gt id="S7.SS1.6.p2.2.m2.1.1.1.cmml" xref="S7.SS1.6.p2.2.m2.1.1.1"></gt><ci id="S7.SS1.6.p2.2.m2.1.1.2.cmml" xref="S7.SS1.6.p2.2.m2.1.1.2">𝛽</ci><ci id="S7.SS1.6.p2.2.m2.1.1.3.cmml" xref="S7.SS1.6.p2.2.m2.1.1.3">𝛼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.6.p2.2.m2.1c">\beta>\alpha</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.6.p2.2.m2.1d">italic_β > italic_α</annotation></semantics></math> in <math alttext="X" class="ltx_Math" display="inline" id="S7.SS1.6.p2.3.m3.1"><semantics id="S7.SS1.6.p2.3.m3.1a"><mi id="S7.SS1.6.p2.3.m3.1.1" xref="S7.SS1.6.p2.3.m3.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.6.p2.3.m3.1b"><ci id="S7.SS1.6.p2.3.m3.1.1.cmml" xref="S7.SS1.6.p2.3.m3.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.6.p2.3.m3.1c">X</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.6.p2.3.m3.1d">italic_X</annotation></semantics></math>, we see that the step (1) of the construction implies that <math alttext="A(X)" class="ltx_Math" display="inline" id="S7.SS1.6.p2.4.m4.1"><semantics id="S7.SS1.6.p2.4.m4.1a"><mrow id="S7.SS1.6.p2.4.m4.1.2" xref="S7.SS1.6.p2.4.m4.1.2.cmml"><mi id="S7.SS1.6.p2.4.m4.1.2.2" xref="S7.SS1.6.p2.4.m4.1.2.2.cmml">A</mi><mo id="S7.SS1.6.p2.4.m4.1.2.1" xref="S7.SS1.6.p2.4.m4.1.2.1.cmml">⁢</mo><mrow id="S7.SS1.6.p2.4.m4.1.2.3.2" xref="S7.SS1.6.p2.4.m4.1.2.cmml"><mo id="S7.SS1.6.p2.4.m4.1.2.3.2.1" stretchy="false" xref="S7.SS1.6.p2.4.m4.1.2.cmml">(</mo><mi id="S7.SS1.6.p2.4.m4.1.1" xref="S7.SS1.6.p2.4.m4.1.1.cmml">X</mi><mo id="S7.SS1.6.p2.4.m4.1.2.3.2.2" stretchy="false" xref="S7.SS1.6.p2.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.6.p2.4.m4.1b"><apply id="S7.SS1.6.p2.4.m4.1.2.cmml" xref="S7.SS1.6.p2.4.m4.1.2"><times id="S7.SS1.6.p2.4.m4.1.2.1.cmml" xref="S7.SS1.6.p2.4.m4.1.2.1"></times><ci id="S7.SS1.6.p2.4.m4.1.2.2.cmml" xref="S7.SS1.6.p2.4.m4.1.2.2">𝐴</ci><ci id="S7.SS1.6.p2.4.m4.1.1.cmml" xref="S7.SS1.6.p2.4.m4.1.1">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.6.p2.4.m4.1c">A(X)</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.6.p2.4.m4.1d">italic_A ( italic_X )</annotation></semantics></math> contains some <math alttext="a\notin\operatorname{dom}(f_{\alpha})" class="ltx_Math" display="inline" id="S7.SS1.6.p2.5.m5.2"><semantics id="S7.SS1.6.p2.5.m5.2a"><mrow id="S7.SS1.6.p2.5.m5.2.2" xref="S7.SS1.6.p2.5.m5.2.2.cmml"><mi id="S7.SS1.6.p2.5.m5.2.2.3" xref="S7.SS1.6.p2.5.m5.2.2.3.cmml">a</mi><mo id="S7.SS1.6.p2.5.m5.2.2.2" xref="S7.SS1.6.p2.5.m5.2.2.2.cmml">∉</mo><mrow id="S7.SS1.6.p2.5.m5.2.2.1.1" xref="S7.SS1.6.p2.5.m5.2.2.1.2.cmml"><mi id="S7.SS1.6.p2.5.m5.1.1" xref="S7.SS1.6.p2.5.m5.1.1.cmml">dom</mi><mo id="S7.SS1.6.p2.5.m5.2.2.1.1a" xref="S7.SS1.6.p2.5.m5.2.2.1.2.cmml">⁡</mo><mrow id="S7.SS1.6.p2.5.m5.2.2.1.1.1" xref="S7.SS1.6.p2.5.m5.2.2.1.2.cmml"><mo id="S7.SS1.6.p2.5.m5.2.2.1.1.1.2" stretchy="false" xref="S7.SS1.6.p2.5.m5.2.2.1.2.cmml">(</mo><msub id="S7.SS1.6.p2.5.m5.2.2.1.1.1.1" xref="S7.SS1.6.p2.5.m5.2.2.1.1.1.1.cmml"><mi id="S7.SS1.6.p2.5.m5.2.2.1.1.1.1.2" xref="S7.SS1.6.p2.5.m5.2.2.1.1.1.1.2.cmml">f</mi><mi id="S7.SS1.6.p2.5.m5.2.2.1.1.1.1.3" xref="S7.SS1.6.p2.5.m5.2.2.1.1.1.1.3.cmml">α</mi></msub><mo id="S7.SS1.6.p2.5.m5.2.2.1.1.1.3" stretchy="false" xref="S7.SS1.6.p2.5.m5.2.2.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.6.p2.5.m5.2b"><apply id="S7.SS1.6.p2.5.m5.2.2.cmml" xref="S7.SS1.6.p2.5.m5.2.2"><notin id="S7.SS1.6.p2.5.m5.2.2.2.cmml" xref="S7.SS1.6.p2.5.m5.2.2.2"></notin><ci id="S7.SS1.6.p2.5.m5.2.2.3.cmml" xref="S7.SS1.6.p2.5.m5.2.2.3">𝑎</ci><apply id="S7.SS1.6.p2.5.m5.2.2.1.2.cmml" xref="S7.SS1.6.p2.5.m5.2.2.1.1"><ci id="S7.SS1.6.p2.5.m5.1.1.cmml" xref="S7.SS1.6.p2.5.m5.1.1">dom</ci><apply id="S7.SS1.6.p2.5.m5.2.2.1.1.1.1.cmml" xref="S7.SS1.6.p2.5.m5.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S7.SS1.6.p2.5.m5.2.2.1.1.1.1.1.cmml" xref="S7.SS1.6.p2.5.m5.2.2.1.1.1.1">subscript</csymbol><ci id="S7.SS1.6.p2.5.m5.2.2.1.1.1.1.2.cmml" xref="S7.SS1.6.p2.5.m5.2.2.1.1.1.1.2">𝑓</ci><ci id="S7.SS1.6.p2.5.m5.2.2.1.1.1.1.3.cmml" xref="S7.SS1.6.p2.5.m5.2.2.1.1.1.1.3">𝛼</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.6.p2.5.m5.2c">a\notin\operatorname{dom}(f_{\alpha})</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.6.p2.5.m5.2d">italic_a ∉ roman_dom ( italic_f start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT )</annotation></semantics></math>, but then also <math alttext="a\notin\operatorname{dom}(f)" class="ltx_Math" display="inline" id="S7.SS1.6.p2.6.m6.2"><semantics id="S7.SS1.6.p2.6.m6.2a"><mrow id="S7.SS1.6.p2.6.m6.2.3" xref="S7.SS1.6.p2.6.m6.2.3.cmml"><mi id="S7.SS1.6.p2.6.m6.2.3.2" xref="S7.SS1.6.p2.6.m6.2.3.2.cmml">a</mi><mo id="S7.SS1.6.p2.6.m6.2.3.1" xref="S7.SS1.6.p2.6.m6.2.3.1.cmml">∉</mo><mrow id="S7.SS1.6.p2.6.m6.2.3.3.2" xref="S7.SS1.6.p2.6.m6.2.3.3.1.cmml"><mi id="S7.SS1.6.p2.6.m6.1.1" xref="S7.SS1.6.p2.6.m6.1.1.cmml">dom</mi><mo id="S7.SS1.6.p2.6.m6.2.3.3.2a" xref="S7.SS1.6.p2.6.m6.2.3.3.1.cmml">⁡</mo><mrow id="S7.SS1.6.p2.6.m6.2.3.3.2.1" xref="S7.SS1.6.p2.6.m6.2.3.3.1.cmml"><mo id="S7.SS1.6.p2.6.m6.2.3.3.2.1.1" stretchy="false" xref="S7.SS1.6.p2.6.m6.2.3.3.1.cmml">(</mo><mi id="S7.SS1.6.p2.6.m6.2.2" xref="S7.SS1.6.p2.6.m6.2.2.cmml">f</mi><mo id="S7.SS1.6.p2.6.m6.2.3.3.2.1.2" stretchy="false" xref="S7.SS1.6.p2.6.m6.2.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.6.p2.6.m6.2b"><apply id="S7.SS1.6.p2.6.m6.2.3.cmml" xref="S7.SS1.6.p2.6.m6.2.3"><notin id="S7.SS1.6.p2.6.m6.2.3.1.cmml" xref="S7.SS1.6.p2.6.m6.2.3.1"></notin><ci id="S7.SS1.6.p2.6.m6.2.3.2.cmml" xref="S7.SS1.6.p2.6.m6.2.3.2">𝑎</ci><apply id="S7.SS1.6.p2.6.m6.2.3.3.1.cmml" xref="S7.SS1.6.p2.6.m6.2.3.3.2"><ci id="S7.SS1.6.p2.6.m6.1.1.cmml" xref="S7.SS1.6.p2.6.m6.1.1">dom</ci><ci id="S7.SS1.6.p2.6.m6.2.2.cmml" xref="S7.SS1.6.p2.6.m6.2.2">𝑓</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.6.p2.6.m6.2c">a\notin\operatorname{dom}(f)</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.6.p2.6.m6.2d">italic_a ∉ roman_dom ( italic_f )</annotation></semantics></math> which is a contradiction. So, <math alttext="|\mathbb{R}\setminus\operatorname{dom}(f_{\alpha})|<\mathfrak{c}" class="ltx_Math" display="inline" id="S7.SS1.6.p2.7.m7.2"><semantics id="S7.SS1.6.p2.7.m7.2a"><mrow id="S7.SS1.6.p2.7.m7.2.2" xref="S7.SS1.6.p2.7.m7.2.2.cmml"><mrow id="S7.SS1.6.p2.7.m7.2.2.1.1" xref="S7.SS1.6.p2.7.m7.2.2.1.2.cmml"><mo id="S7.SS1.6.p2.7.m7.2.2.1.1.2" stretchy="false" xref="S7.SS1.6.p2.7.m7.2.2.1.2.1.cmml">|</mo><mrow id="S7.SS1.6.p2.7.m7.2.2.1.1.1" xref="S7.SS1.6.p2.7.m7.2.2.1.1.1.cmml"><mi id="S7.SS1.6.p2.7.m7.2.2.1.1.1.3" xref="S7.SS1.6.p2.7.m7.2.2.1.1.1.3.cmml">ℝ</mi><mo id="S7.SS1.6.p2.7.m7.2.2.1.1.1.2" xref="S7.SS1.6.p2.7.m7.2.2.1.1.1.2.cmml">∖</mo><mrow id="S7.SS1.6.p2.7.m7.2.2.1.1.1.1.1" xref="S7.SS1.6.p2.7.m7.2.2.1.1.1.1.2.cmml"><mi id="S7.SS1.6.p2.7.m7.1.1" xref="S7.SS1.6.p2.7.m7.1.1.cmml">dom</mi><mo id="S7.SS1.6.p2.7.m7.2.2.1.1.1.1.1a" xref="S7.SS1.6.p2.7.m7.2.2.1.1.1.1.2.cmml">⁡</mo><mrow id="S7.SS1.6.p2.7.m7.2.2.1.1.1.1.1.1" xref="S7.SS1.6.p2.7.m7.2.2.1.1.1.1.2.cmml"><mo id="S7.SS1.6.p2.7.m7.2.2.1.1.1.1.1.1.2" stretchy="false" xref="S7.SS1.6.p2.7.m7.2.2.1.1.1.1.2.cmml">(</mo><msub id="S7.SS1.6.p2.7.m7.2.2.1.1.1.1.1.1.1" xref="S7.SS1.6.p2.7.m7.2.2.1.1.1.1.1.1.1.cmml"><mi id="S7.SS1.6.p2.7.m7.2.2.1.1.1.1.1.1.1.2" xref="S7.SS1.6.p2.7.m7.2.2.1.1.1.1.1.1.1.2.cmml">f</mi><mi id="S7.SS1.6.p2.7.m7.2.2.1.1.1.1.1.1.1.3" xref="S7.SS1.6.p2.7.m7.2.2.1.1.1.1.1.1.1.3.cmml">α</mi></msub><mo id="S7.SS1.6.p2.7.m7.2.2.1.1.1.1.1.1.3" stretchy="false" xref="S7.SS1.6.p2.7.m7.2.2.1.1.1.1.2.cmml">)</mo></mrow></mrow></mrow><mo id="S7.SS1.6.p2.7.m7.2.2.1.1.3" stretchy="false" xref="S7.SS1.6.p2.7.m7.2.2.1.2.1.cmml">|</mo></mrow><mo id="S7.SS1.6.p2.7.m7.2.2.2" xref="S7.SS1.6.p2.7.m7.2.2.2.cmml"><</mo><mi id="S7.SS1.6.p2.7.m7.2.2.3" xref="S7.SS1.6.p2.7.m7.2.2.3.cmml">𝔠</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.6.p2.7.m7.2b"><apply id="S7.SS1.6.p2.7.m7.2.2.cmml" xref="S7.SS1.6.p2.7.m7.2.2"><lt id="S7.SS1.6.p2.7.m7.2.2.2.cmml" xref="S7.SS1.6.p2.7.m7.2.2.2"></lt><apply id="S7.SS1.6.p2.7.m7.2.2.1.2.cmml" xref="S7.SS1.6.p2.7.m7.2.2.1.1"><abs id="S7.SS1.6.p2.7.m7.2.2.1.2.1.cmml" xref="S7.SS1.6.p2.7.m7.2.2.1.1.2"></abs><apply id="S7.SS1.6.p2.7.m7.2.2.1.1.1.cmml" xref="S7.SS1.6.p2.7.m7.2.2.1.1.1"><setdiff id="S7.SS1.6.p2.7.m7.2.2.1.1.1.2.cmml" xref="S7.SS1.6.p2.7.m7.2.2.1.1.1.2"></setdiff><ci id="S7.SS1.6.p2.7.m7.2.2.1.1.1.3.cmml" xref="S7.SS1.6.p2.7.m7.2.2.1.1.1.3">ℝ</ci><apply id="S7.SS1.6.p2.7.m7.2.2.1.1.1.1.2.cmml" xref="S7.SS1.6.p2.7.m7.2.2.1.1.1.1.1"><ci id="S7.SS1.6.p2.7.m7.1.1.cmml" xref="S7.SS1.6.p2.7.m7.1.1">dom</ci><apply id="S7.SS1.6.p2.7.m7.2.2.1.1.1.1.1.1.1.cmml" xref="S7.SS1.6.p2.7.m7.2.2.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S7.SS1.6.p2.7.m7.2.2.1.1.1.1.1.1.1.1.cmml" xref="S7.SS1.6.p2.7.m7.2.2.1.1.1.1.1.1.1">subscript</csymbol><ci id="S7.SS1.6.p2.7.m7.2.2.1.1.1.1.1.1.1.2.cmml" xref="S7.SS1.6.p2.7.m7.2.2.1.1.1.1.1.1.1.2">𝑓</ci><ci id="S7.SS1.6.p2.7.m7.2.2.1.1.1.1.1.1.1.3.cmml" xref="S7.SS1.6.p2.7.m7.2.2.1.1.1.1.1.1.1.3">𝛼</ci></apply></apply></apply></apply><ci id="S7.SS1.6.p2.7.m7.2.2.3.cmml" xref="S7.SS1.6.p2.7.m7.2.2.3">𝔠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.6.p2.7.m7.2c">|\mathbb{R}\setminus\operatorname{dom}(f_{\alpha})|<\mathfrak{c}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.6.p2.7.m7.2d">| blackboard_R ∖ roman_dom ( italic_f start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ) | < fraktur_c</annotation></semantics></math>, and thus by <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S7.Thmtheorem8" title="Lemma 7.8. ‣ 7.1. On a basis for all uncountable linear orders ‣ 7. A two element basis for the Aronszajn lines ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">7.8</span></a>, and using the fact that <math alttext="B" class="ltx_Math" display="inline" id="S7.SS1.6.p2.8.m8.1"><semantics id="S7.SS1.6.p2.8.m8.1a"><mi id="S7.SS1.6.p2.8.m8.1.1" xref="S7.SS1.6.p2.8.m8.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.6.p2.8.m8.1b"><ci id="S7.SS1.6.p2.8.m8.1.1.cmml" xref="S7.SS1.6.p2.8.m8.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.6.p2.8.m8.1c">B</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.6.p2.8.m8.1d">italic_B</annotation></semantics></math> is uncountable, we conclude that <math alttext="|\operatorname{ran}(f_{\alpha})|=\mathfrak{c}" class="ltx_Math" display="inline" id="S7.SS1.6.p2.9.m9.2"><semantics id="S7.SS1.6.p2.9.m9.2a"><mrow id="S7.SS1.6.p2.9.m9.2.2" xref="S7.SS1.6.p2.9.m9.2.2.cmml"><mrow id="S7.SS1.6.p2.9.m9.2.2.1.1" xref="S7.SS1.6.p2.9.m9.2.2.1.2.cmml"><mo id="S7.SS1.6.p2.9.m9.2.2.1.1.2" stretchy="false" xref="S7.SS1.6.p2.9.m9.2.2.1.2.1.cmml">|</mo><mrow id="S7.SS1.6.p2.9.m9.2.2.1.1.1.1" xref="S7.SS1.6.p2.9.m9.2.2.1.1.1.2.cmml"><mi id="S7.SS1.6.p2.9.m9.1.1" xref="S7.SS1.6.p2.9.m9.1.1.cmml">ran</mi><mo id="S7.SS1.6.p2.9.m9.2.2.1.1.1.1a" xref="S7.SS1.6.p2.9.m9.2.2.1.1.1.2.cmml">⁡</mo><mrow id="S7.SS1.6.p2.9.m9.2.2.1.1.1.1.1" xref="S7.SS1.6.p2.9.m9.2.2.1.1.1.2.cmml"><mo id="S7.SS1.6.p2.9.m9.2.2.1.1.1.1.1.2" stretchy="false" xref="S7.SS1.6.p2.9.m9.2.2.1.1.1.2.cmml">(</mo><msub id="S7.SS1.6.p2.9.m9.2.2.1.1.1.1.1.1" xref="S7.SS1.6.p2.9.m9.2.2.1.1.1.1.1.1.cmml"><mi id="S7.SS1.6.p2.9.m9.2.2.1.1.1.1.1.1.2" xref="S7.SS1.6.p2.9.m9.2.2.1.1.1.1.1.1.2.cmml">f</mi><mi id="S7.SS1.6.p2.9.m9.2.2.1.1.1.1.1.1.3" xref="S7.SS1.6.p2.9.m9.2.2.1.1.1.1.1.1.3.cmml">α</mi></msub><mo id="S7.SS1.6.p2.9.m9.2.2.1.1.1.1.1.3" stretchy="false" xref="S7.SS1.6.p2.9.m9.2.2.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S7.SS1.6.p2.9.m9.2.2.1.1.3" stretchy="false" xref="S7.SS1.6.p2.9.m9.2.2.1.2.1.cmml">|</mo></mrow><mo id="S7.SS1.6.p2.9.m9.2.2.2" xref="S7.SS1.6.p2.9.m9.2.2.2.cmml">=</mo><mi id="S7.SS1.6.p2.9.m9.2.2.3" xref="S7.SS1.6.p2.9.m9.2.2.3.cmml">𝔠</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.6.p2.9.m9.2b"><apply id="S7.SS1.6.p2.9.m9.2.2.cmml" xref="S7.SS1.6.p2.9.m9.2.2"><eq id="S7.SS1.6.p2.9.m9.2.2.2.cmml" xref="S7.SS1.6.p2.9.m9.2.2.2"></eq><apply id="S7.SS1.6.p2.9.m9.2.2.1.2.cmml" xref="S7.SS1.6.p2.9.m9.2.2.1.1"><abs id="S7.SS1.6.p2.9.m9.2.2.1.2.1.cmml" xref="S7.SS1.6.p2.9.m9.2.2.1.1.2"></abs><apply id="S7.SS1.6.p2.9.m9.2.2.1.1.1.2.cmml" xref="S7.SS1.6.p2.9.m9.2.2.1.1.1.1"><ci id="S7.SS1.6.p2.9.m9.1.1.cmml" xref="S7.SS1.6.p2.9.m9.1.1">ran</ci><apply id="S7.SS1.6.p2.9.m9.2.2.1.1.1.1.1.1.cmml" xref="S7.SS1.6.p2.9.m9.2.2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S7.SS1.6.p2.9.m9.2.2.1.1.1.1.1.1.1.cmml" xref="S7.SS1.6.p2.9.m9.2.2.1.1.1.1.1.1">subscript</csymbol><ci id="S7.SS1.6.p2.9.m9.2.2.1.1.1.1.1.1.2.cmml" xref="S7.SS1.6.p2.9.m9.2.2.1.1.1.1.1.1.2">𝑓</ci><ci id="S7.SS1.6.p2.9.m9.2.2.1.1.1.1.1.1.3.cmml" xref="S7.SS1.6.p2.9.m9.2.2.1.1.1.1.1.1.3">𝛼</ci></apply></apply></apply><ci id="S7.SS1.6.p2.9.m9.2.2.3.cmml" xref="S7.SS1.6.p2.9.m9.2.2.3">𝔠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.6.p2.9.m9.2c">|\operatorname{ran}(f_{\alpha})|=\mathfrak{c}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.6.p2.9.m9.2d">| roman_ran ( italic_f start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ) | = fraktur_c</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S7.SS1.7.p3"> <p class="ltx_p" id="S7.SS1.7.p3.9">Let <math alttext="X^{\prime}:=X\setminus(\alpha+1)" class="ltx_Math" display="inline" id="S7.SS1.7.p3.1.m1.1"><semantics id="S7.SS1.7.p3.1.m1.1a"><mrow id="S7.SS1.7.p3.1.m1.1.1" xref="S7.SS1.7.p3.1.m1.1.1.cmml"><msup id="S7.SS1.7.p3.1.m1.1.1.3" xref="S7.SS1.7.p3.1.m1.1.1.3.cmml"><mi id="S7.SS1.7.p3.1.m1.1.1.3.2" xref="S7.SS1.7.p3.1.m1.1.1.3.2.cmml">X</mi><mo id="S7.SS1.7.p3.1.m1.1.1.3.3" xref="S7.SS1.7.p3.1.m1.1.1.3.3.cmml">′</mo></msup><mo id="S7.SS1.7.p3.1.m1.1.1.2" lspace="0.278em" rspace="0.278em" xref="S7.SS1.7.p3.1.m1.1.1.2.cmml">:=</mo><mrow id="S7.SS1.7.p3.1.m1.1.1.1" xref="S7.SS1.7.p3.1.m1.1.1.1.cmml"><mi id="S7.SS1.7.p3.1.m1.1.1.1.3" xref="S7.SS1.7.p3.1.m1.1.1.1.3.cmml">X</mi><mo id="S7.SS1.7.p3.1.m1.1.1.1.2" xref="S7.SS1.7.p3.1.m1.1.1.1.2.cmml">∖</mo><mrow id="S7.SS1.7.p3.1.m1.1.1.1.1.1" xref="S7.SS1.7.p3.1.m1.1.1.1.1.1.1.cmml"><mo id="S7.SS1.7.p3.1.m1.1.1.1.1.1.2" stretchy="false" xref="S7.SS1.7.p3.1.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="S7.SS1.7.p3.1.m1.1.1.1.1.1.1" xref="S7.SS1.7.p3.1.m1.1.1.1.1.1.1.cmml"><mi id="S7.SS1.7.p3.1.m1.1.1.1.1.1.1.2" xref="S7.SS1.7.p3.1.m1.1.1.1.1.1.1.2.cmml">α</mi><mo id="S7.SS1.7.p3.1.m1.1.1.1.1.1.1.1" xref="S7.SS1.7.p3.1.m1.1.1.1.1.1.1.1.cmml">+</mo><mn id="S7.SS1.7.p3.1.m1.1.1.1.1.1.1.3" xref="S7.SS1.7.p3.1.m1.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S7.SS1.7.p3.1.m1.1.1.1.1.1.3" stretchy="false" xref="S7.SS1.7.p3.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.7.p3.1.m1.1b"><apply id="S7.SS1.7.p3.1.m1.1.1.cmml" xref="S7.SS1.7.p3.1.m1.1.1"><csymbol cd="latexml" id="S7.SS1.7.p3.1.m1.1.1.2.cmml" xref="S7.SS1.7.p3.1.m1.1.1.2">assign</csymbol><apply id="S7.SS1.7.p3.1.m1.1.1.3.cmml" xref="S7.SS1.7.p3.1.m1.1.1.3"><csymbol cd="ambiguous" id="S7.SS1.7.p3.1.m1.1.1.3.1.cmml" xref="S7.SS1.7.p3.1.m1.1.1.3">superscript</csymbol><ci id="S7.SS1.7.p3.1.m1.1.1.3.2.cmml" xref="S7.SS1.7.p3.1.m1.1.1.3.2">𝑋</ci><ci id="S7.SS1.7.p3.1.m1.1.1.3.3.cmml" xref="S7.SS1.7.p3.1.m1.1.1.3.3">′</ci></apply><apply id="S7.SS1.7.p3.1.m1.1.1.1.cmml" xref="S7.SS1.7.p3.1.m1.1.1.1"><setdiff id="S7.SS1.7.p3.1.m1.1.1.1.2.cmml" xref="S7.SS1.7.p3.1.m1.1.1.1.2"></setdiff><ci id="S7.SS1.7.p3.1.m1.1.1.1.3.cmml" xref="S7.SS1.7.p3.1.m1.1.1.1.3">𝑋</ci><apply id="S7.SS1.7.p3.1.m1.1.1.1.1.1.1.cmml" xref="S7.SS1.7.p3.1.m1.1.1.1.1.1"><plus id="S7.SS1.7.p3.1.m1.1.1.1.1.1.1.1.cmml" xref="S7.SS1.7.p3.1.m1.1.1.1.1.1.1.1"></plus><ci id="S7.SS1.7.p3.1.m1.1.1.1.1.1.1.2.cmml" xref="S7.SS1.7.p3.1.m1.1.1.1.1.1.1.2">𝛼</ci><cn id="S7.SS1.7.p3.1.m1.1.1.1.1.1.1.3.cmml" type="integer" xref="S7.SS1.7.p3.1.m1.1.1.1.1.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.7.p3.1.m1.1c">X^{\prime}:=X\setminus(\alpha+1)</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.7.p3.1.m1.1d">italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT := italic_X ∖ ( italic_α + 1 )</annotation></semantics></math>, and note that by hypothesis, <math alttext="|X^{\prime}|=\mathfrak{c}" class="ltx_Math" display="inline" id="S7.SS1.7.p3.2.m2.1"><semantics id="S7.SS1.7.p3.2.m2.1a"><mrow id="S7.SS1.7.p3.2.m2.1.1" xref="S7.SS1.7.p3.2.m2.1.1.cmml"><mrow id="S7.SS1.7.p3.2.m2.1.1.1.1" xref="S7.SS1.7.p3.2.m2.1.1.1.2.cmml"><mo id="S7.SS1.7.p3.2.m2.1.1.1.1.2" stretchy="false" xref="S7.SS1.7.p3.2.m2.1.1.1.2.1.cmml">|</mo><msup id="S7.SS1.7.p3.2.m2.1.1.1.1.1" xref="S7.SS1.7.p3.2.m2.1.1.1.1.1.cmml"><mi id="S7.SS1.7.p3.2.m2.1.1.1.1.1.2" xref="S7.SS1.7.p3.2.m2.1.1.1.1.1.2.cmml">X</mi><mo id="S7.SS1.7.p3.2.m2.1.1.1.1.1.3" xref="S7.SS1.7.p3.2.m2.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S7.SS1.7.p3.2.m2.1.1.1.1.3" stretchy="false" xref="S7.SS1.7.p3.2.m2.1.1.1.2.1.cmml">|</mo></mrow><mo id="S7.SS1.7.p3.2.m2.1.1.2" xref="S7.SS1.7.p3.2.m2.1.1.2.cmml">=</mo><mi id="S7.SS1.7.p3.2.m2.1.1.3" xref="S7.SS1.7.p3.2.m2.1.1.3.cmml">𝔠</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.7.p3.2.m2.1b"><apply id="S7.SS1.7.p3.2.m2.1.1.cmml" xref="S7.SS1.7.p3.2.m2.1.1"><eq id="S7.SS1.7.p3.2.m2.1.1.2.cmml" xref="S7.SS1.7.p3.2.m2.1.1.2"></eq><apply id="S7.SS1.7.p3.2.m2.1.1.1.2.cmml" xref="S7.SS1.7.p3.2.m2.1.1.1.1"><abs id="S7.SS1.7.p3.2.m2.1.1.1.2.1.cmml" xref="S7.SS1.7.p3.2.m2.1.1.1.1.2"></abs><apply id="S7.SS1.7.p3.2.m2.1.1.1.1.1.cmml" xref="S7.SS1.7.p3.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S7.SS1.7.p3.2.m2.1.1.1.1.1.1.cmml" xref="S7.SS1.7.p3.2.m2.1.1.1.1.1">superscript</csymbol><ci id="S7.SS1.7.p3.2.m2.1.1.1.1.1.2.cmml" xref="S7.SS1.7.p3.2.m2.1.1.1.1.1.2">𝑋</ci><ci id="S7.SS1.7.p3.2.m2.1.1.1.1.1.3.cmml" xref="S7.SS1.7.p3.2.m2.1.1.1.1.1.3">′</ci></apply></apply><ci id="S7.SS1.7.p3.2.m2.1.1.3.cmml" xref="S7.SS1.7.p3.2.m2.1.1.3">𝔠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.7.p3.2.m2.1c">|X^{\prime}|=\mathfrak{c}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.7.p3.2.m2.1d">| italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT | = fraktur_c</annotation></semantics></math>. Now, for each <math alttext="\beta\in X^{\prime}" class="ltx_Math" display="inline" id="S7.SS1.7.p3.3.m3.1"><semantics id="S7.SS1.7.p3.3.m3.1a"><mrow id="S7.SS1.7.p3.3.m3.1.1" xref="S7.SS1.7.p3.3.m3.1.1.cmml"><mi id="S7.SS1.7.p3.3.m3.1.1.2" xref="S7.SS1.7.p3.3.m3.1.1.2.cmml">β</mi><mo id="S7.SS1.7.p3.3.m3.1.1.1" xref="S7.SS1.7.p3.3.m3.1.1.1.cmml">∈</mo><msup id="S7.SS1.7.p3.3.m3.1.1.3" xref="S7.SS1.7.p3.3.m3.1.1.3.cmml"><mi id="S7.SS1.7.p3.3.m3.1.1.3.2" xref="S7.SS1.7.p3.3.m3.1.1.3.2.cmml">X</mi><mo id="S7.SS1.7.p3.3.m3.1.1.3.3" xref="S7.SS1.7.p3.3.m3.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.7.p3.3.m3.1b"><apply id="S7.SS1.7.p3.3.m3.1.1.cmml" xref="S7.SS1.7.p3.3.m3.1.1"><in id="S7.SS1.7.p3.3.m3.1.1.1.cmml" xref="S7.SS1.7.p3.3.m3.1.1.1"></in><ci id="S7.SS1.7.p3.3.m3.1.1.2.cmml" xref="S7.SS1.7.p3.3.m3.1.1.2">𝛽</ci><apply id="S7.SS1.7.p3.3.m3.1.1.3.cmml" xref="S7.SS1.7.p3.3.m3.1.1.3"><csymbol cd="ambiguous" id="S7.SS1.7.p3.3.m3.1.1.3.1.cmml" xref="S7.SS1.7.p3.3.m3.1.1.3">superscript</csymbol><ci id="S7.SS1.7.p3.3.m3.1.1.3.2.cmml" xref="S7.SS1.7.p3.3.m3.1.1.3.2">𝑋</ci><ci id="S7.SS1.7.p3.3.m3.1.1.3.3.cmml" xref="S7.SS1.7.p3.3.m3.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.7.p3.3.m3.1c">\beta\in X^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.7.p3.3.m3.1d">italic_β ∈ italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> pick <math alttext="a_{\beta}\in A_{\beta}\cap\operatorname{dom}(f_{\alpha})" class="ltx_Math" display="inline" id="S7.SS1.7.p3.4.m4.2"><semantics id="S7.SS1.7.p3.4.m4.2a"><mrow id="S7.SS1.7.p3.4.m4.2.2" xref="S7.SS1.7.p3.4.m4.2.2.cmml"><msub id="S7.SS1.7.p3.4.m4.2.2.3" xref="S7.SS1.7.p3.4.m4.2.2.3.cmml"><mi id="S7.SS1.7.p3.4.m4.2.2.3.2" xref="S7.SS1.7.p3.4.m4.2.2.3.2.cmml">a</mi><mi id="S7.SS1.7.p3.4.m4.2.2.3.3" xref="S7.SS1.7.p3.4.m4.2.2.3.3.cmml">β</mi></msub><mo id="S7.SS1.7.p3.4.m4.2.2.2" xref="S7.SS1.7.p3.4.m4.2.2.2.cmml">∈</mo><mrow id="S7.SS1.7.p3.4.m4.2.2.1" xref="S7.SS1.7.p3.4.m4.2.2.1.cmml"><msub id="S7.SS1.7.p3.4.m4.2.2.1.3" xref="S7.SS1.7.p3.4.m4.2.2.1.3.cmml"><mi id="S7.SS1.7.p3.4.m4.2.2.1.3.2" xref="S7.SS1.7.p3.4.m4.2.2.1.3.2.cmml">A</mi><mi id="S7.SS1.7.p3.4.m4.2.2.1.3.3" xref="S7.SS1.7.p3.4.m4.2.2.1.3.3.cmml">β</mi></msub><mo id="S7.SS1.7.p3.4.m4.2.2.1.2" xref="S7.SS1.7.p3.4.m4.2.2.1.2.cmml">∩</mo><mrow id="S7.SS1.7.p3.4.m4.2.2.1.1.1" xref="S7.SS1.7.p3.4.m4.2.2.1.1.2.cmml"><mi id="S7.SS1.7.p3.4.m4.1.1" xref="S7.SS1.7.p3.4.m4.1.1.cmml">dom</mi><mo id="S7.SS1.7.p3.4.m4.2.2.1.1.1a" xref="S7.SS1.7.p3.4.m4.2.2.1.1.2.cmml">⁡</mo><mrow id="S7.SS1.7.p3.4.m4.2.2.1.1.1.1" xref="S7.SS1.7.p3.4.m4.2.2.1.1.2.cmml"><mo id="S7.SS1.7.p3.4.m4.2.2.1.1.1.1.2" stretchy="false" xref="S7.SS1.7.p3.4.m4.2.2.1.1.2.cmml">(</mo><msub id="S7.SS1.7.p3.4.m4.2.2.1.1.1.1.1" xref="S7.SS1.7.p3.4.m4.2.2.1.1.1.1.1.cmml"><mi id="S7.SS1.7.p3.4.m4.2.2.1.1.1.1.1.2" xref="S7.SS1.7.p3.4.m4.2.2.1.1.1.1.1.2.cmml">f</mi><mi id="S7.SS1.7.p3.4.m4.2.2.1.1.1.1.1.3" xref="S7.SS1.7.p3.4.m4.2.2.1.1.1.1.1.3.cmml">α</mi></msub><mo id="S7.SS1.7.p3.4.m4.2.2.1.1.1.1.3" stretchy="false" xref="S7.SS1.7.p3.4.m4.2.2.1.1.2.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.7.p3.4.m4.2b"><apply id="S7.SS1.7.p3.4.m4.2.2.cmml" xref="S7.SS1.7.p3.4.m4.2.2"><in id="S7.SS1.7.p3.4.m4.2.2.2.cmml" xref="S7.SS1.7.p3.4.m4.2.2.2"></in><apply id="S7.SS1.7.p3.4.m4.2.2.3.cmml" xref="S7.SS1.7.p3.4.m4.2.2.3"><csymbol cd="ambiguous" id="S7.SS1.7.p3.4.m4.2.2.3.1.cmml" xref="S7.SS1.7.p3.4.m4.2.2.3">subscript</csymbol><ci id="S7.SS1.7.p3.4.m4.2.2.3.2.cmml" xref="S7.SS1.7.p3.4.m4.2.2.3.2">𝑎</ci><ci id="S7.SS1.7.p3.4.m4.2.2.3.3.cmml" xref="S7.SS1.7.p3.4.m4.2.2.3.3">𝛽</ci></apply><apply id="S7.SS1.7.p3.4.m4.2.2.1.cmml" xref="S7.SS1.7.p3.4.m4.2.2.1"><intersect id="S7.SS1.7.p3.4.m4.2.2.1.2.cmml" xref="S7.SS1.7.p3.4.m4.2.2.1.2"></intersect><apply id="S7.SS1.7.p3.4.m4.2.2.1.3.cmml" xref="S7.SS1.7.p3.4.m4.2.2.1.3"><csymbol cd="ambiguous" id="S7.SS1.7.p3.4.m4.2.2.1.3.1.cmml" xref="S7.SS1.7.p3.4.m4.2.2.1.3">subscript</csymbol><ci id="S7.SS1.7.p3.4.m4.2.2.1.3.2.cmml" xref="S7.SS1.7.p3.4.m4.2.2.1.3.2">𝐴</ci><ci id="S7.SS1.7.p3.4.m4.2.2.1.3.3.cmml" xref="S7.SS1.7.p3.4.m4.2.2.1.3.3">𝛽</ci></apply><apply id="S7.SS1.7.p3.4.m4.2.2.1.1.2.cmml" xref="S7.SS1.7.p3.4.m4.2.2.1.1.1"><ci id="S7.SS1.7.p3.4.m4.1.1.cmml" xref="S7.SS1.7.p3.4.m4.1.1">dom</ci><apply id="S7.SS1.7.p3.4.m4.2.2.1.1.1.1.1.cmml" xref="S7.SS1.7.p3.4.m4.2.2.1.1.1.1.1"><csymbol cd="ambiguous" id="S7.SS1.7.p3.4.m4.2.2.1.1.1.1.1.1.cmml" xref="S7.SS1.7.p3.4.m4.2.2.1.1.1.1.1">subscript</csymbol><ci id="S7.SS1.7.p3.4.m4.2.2.1.1.1.1.1.2.cmml" xref="S7.SS1.7.p3.4.m4.2.2.1.1.1.1.1.2">𝑓</ci><ci id="S7.SS1.7.p3.4.m4.2.2.1.1.1.1.1.3.cmml" xref="S7.SS1.7.p3.4.m4.2.2.1.1.1.1.1.3">𝛼</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.7.p3.4.m4.2c">a_{\beta}\in A_{\beta}\cap\operatorname{dom}(f_{\alpha})</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.7.p3.4.m4.2d">italic_a start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT ∈ italic_A start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT ∩ roman_dom ( italic_f start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT )</annotation></semantics></math> such that <math alttext="f_{\alpha}(a_{\beta})\notin Z_{\beta}" class="ltx_Math" display="inline" id="S7.SS1.7.p3.5.m5.1"><semantics id="S7.SS1.7.p3.5.m5.1a"><mrow id="S7.SS1.7.p3.5.m5.1.1" xref="S7.SS1.7.p3.5.m5.1.1.cmml"><mrow id="S7.SS1.7.p3.5.m5.1.1.1" xref="S7.SS1.7.p3.5.m5.1.1.1.cmml"><msub id="S7.SS1.7.p3.5.m5.1.1.1.3" xref="S7.SS1.7.p3.5.m5.1.1.1.3.cmml"><mi id="S7.SS1.7.p3.5.m5.1.1.1.3.2" xref="S7.SS1.7.p3.5.m5.1.1.1.3.2.cmml">f</mi><mi id="S7.SS1.7.p3.5.m5.1.1.1.3.3" xref="S7.SS1.7.p3.5.m5.1.1.1.3.3.cmml">α</mi></msub><mo id="S7.SS1.7.p3.5.m5.1.1.1.2" xref="S7.SS1.7.p3.5.m5.1.1.1.2.cmml">⁢</mo><mrow id="S7.SS1.7.p3.5.m5.1.1.1.1.1" xref="S7.SS1.7.p3.5.m5.1.1.1.1.1.1.cmml"><mo id="S7.SS1.7.p3.5.m5.1.1.1.1.1.2" stretchy="false" xref="S7.SS1.7.p3.5.m5.1.1.1.1.1.1.cmml">(</mo><msub id="S7.SS1.7.p3.5.m5.1.1.1.1.1.1" xref="S7.SS1.7.p3.5.m5.1.1.1.1.1.1.cmml"><mi id="S7.SS1.7.p3.5.m5.1.1.1.1.1.1.2" xref="S7.SS1.7.p3.5.m5.1.1.1.1.1.1.2.cmml">a</mi><mi id="S7.SS1.7.p3.5.m5.1.1.1.1.1.1.3" xref="S7.SS1.7.p3.5.m5.1.1.1.1.1.1.3.cmml">β</mi></msub><mo id="S7.SS1.7.p3.5.m5.1.1.1.1.1.3" stretchy="false" xref="S7.SS1.7.p3.5.m5.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S7.SS1.7.p3.5.m5.1.1.2" xref="S7.SS1.7.p3.5.m5.1.1.2.cmml">∉</mo><msub id="S7.SS1.7.p3.5.m5.1.1.3" xref="S7.SS1.7.p3.5.m5.1.1.3.cmml"><mi id="S7.SS1.7.p3.5.m5.1.1.3.2" xref="S7.SS1.7.p3.5.m5.1.1.3.2.cmml">Z</mi><mi id="S7.SS1.7.p3.5.m5.1.1.3.3" xref="S7.SS1.7.p3.5.m5.1.1.3.3.cmml">β</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.7.p3.5.m5.1b"><apply id="S7.SS1.7.p3.5.m5.1.1.cmml" xref="S7.SS1.7.p3.5.m5.1.1"><notin id="S7.SS1.7.p3.5.m5.1.1.2.cmml" xref="S7.SS1.7.p3.5.m5.1.1.2"></notin><apply id="S7.SS1.7.p3.5.m5.1.1.1.cmml" xref="S7.SS1.7.p3.5.m5.1.1.1"><times id="S7.SS1.7.p3.5.m5.1.1.1.2.cmml" xref="S7.SS1.7.p3.5.m5.1.1.1.2"></times><apply id="S7.SS1.7.p3.5.m5.1.1.1.3.cmml" xref="S7.SS1.7.p3.5.m5.1.1.1.3"><csymbol cd="ambiguous" id="S7.SS1.7.p3.5.m5.1.1.1.3.1.cmml" xref="S7.SS1.7.p3.5.m5.1.1.1.3">subscript</csymbol><ci id="S7.SS1.7.p3.5.m5.1.1.1.3.2.cmml" xref="S7.SS1.7.p3.5.m5.1.1.1.3.2">𝑓</ci><ci id="S7.SS1.7.p3.5.m5.1.1.1.3.3.cmml" xref="S7.SS1.7.p3.5.m5.1.1.1.3.3">𝛼</ci></apply><apply id="S7.SS1.7.p3.5.m5.1.1.1.1.1.1.cmml" xref="S7.SS1.7.p3.5.m5.1.1.1.1.1"><csymbol cd="ambiguous" id="S7.SS1.7.p3.5.m5.1.1.1.1.1.1.1.cmml" xref="S7.SS1.7.p3.5.m5.1.1.1.1.1">subscript</csymbol><ci id="S7.SS1.7.p3.5.m5.1.1.1.1.1.1.2.cmml" xref="S7.SS1.7.p3.5.m5.1.1.1.1.1.1.2">𝑎</ci><ci id="S7.SS1.7.p3.5.m5.1.1.1.1.1.1.3.cmml" xref="S7.SS1.7.p3.5.m5.1.1.1.1.1.1.3">𝛽</ci></apply></apply><apply id="S7.SS1.7.p3.5.m5.1.1.3.cmml" xref="S7.SS1.7.p3.5.m5.1.1.3"><csymbol cd="ambiguous" id="S7.SS1.7.p3.5.m5.1.1.3.1.cmml" xref="S7.SS1.7.p3.5.m5.1.1.3">subscript</csymbol><ci id="S7.SS1.7.p3.5.m5.1.1.3.2.cmml" xref="S7.SS1.7.p3.5.m5.1.1.3.2">𝑍</ci><ci id="S7.SS1.7.p3.5.m5.1.1.3.3.cmml" xref="S7.SS1.7.p3.5.m5.1.1.3.3">𝛽</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.7.p3.5.m5.1c">f_{\alpha}(a_{\beta})\notin Z_{\beta}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.7.p3.5.m5.1d">italic_f start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT ) ∉ italic_Z start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT</annotation></semantics></math>. This exists by step (2) of the construction. We claim that if <math alttext="\beta<\gamma" class="ltx_Math" display="inline" id="S7.SS1.7.p3.6.m6.1"><semantics id="S7.SS1.7.p3.6.m6.1a"><mrow id="S7.SS1.7.p3.6.m6.1.1" xref="S7.SS1.7.p3.6.m6.1.1.cmml"><mi id="S7.SS1.7.p3.6.m6.1.1.2" xref="S7.SS1.7.p3.6.m6.1.1.2.cmml">β</mi><mo id="S7.SS1.7.p3.6.m6.1.1.1" xref="S7.SS1.7.p3.6.m6.1.1.1.cmml"><</mo><mi id="S7.SS1.7.p3.6.m6.1.1.3" xref="S7.SS1.7.p3.6.m6.1.1.3.cmml">γ</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.7.p3.6.m6.1b"><apply id="S7.SS1.7.p3.6.m6.1.1.cmml" xref="S7.SS1.7.p3.6.m6.1.1"><lt id="S7.SS1.7.p3.6.m6.1.1.1.cmml" xref="S7.SS1.7.p3.6.m6.1.1.1"></lt><ci id="S7.SS1.7.p3.6.m6.1.1.2.cmml" xref="S7.SS1.7.p3.6.m6.1.1.2">𝛽</ci><ci id="S7.SS1.7.p3.6.m6.1.1.3.cmml" xref="S7.SS1.7.p3.6.m6.1.1.3">𝛾</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.7.p3.6.m6.1c">\beta<\gamma</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.7.p3.6.m6.1d">italic_β < italic_γ</annotation></semantics></math> are in <math alttext="X^{\prime}" class="ltx_Math" display="inline" id="S7.SS1.7.p3.7.m7.1"><semantics id="S7.SS1.7.p3.7.m7.1a"><msup id="S7.SS1.7.p3.7.m7.1.1" xref="S7.SS1.7.p3.7.m7.1.1.cmml"><mi id="S7.SS1.7.p3.7.m7.1.1.2" xref="S7.SS1.7.p3.7.m7.1.1.2.cmml">X</mi><mo id="S7.SS1.7.p3.7.m7.1.1.3" xref="S7.SS1.7.p3.7.m7.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S7.SS1.7.p3.7.m7.1b"><apply id="S7.SS1.7.p3.7.m7.1.1.cmml" xref="S7.SS1.7.p3.7.m7.1.1"><csymbol cd="ambiguous" id="S7.SS1.7.p3.7.m7.1.1.1.cmml" xref="S7.SS1.7.p3.7.m7.1.1">superscript</csymbol><ci id="S7.SS1.7.p3.7.m7.1.1.2.cmml" xref="S7.SS1.7.p3.7.m7.1.1.2">𝑋</ci><ci id="S7.SS1.7.p3.7.m7.1.1.3.cmml" xref="S7.SS1.7.p3.7.m7.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.7.p3.7.m7.1c">X^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.7.p3.7.m7.1d">italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, then <math alttext="f_{\alpha}(a_{\beta})\neq f_{\beta}(a_{\gamma})" class="ltx_Math" display="inline" id="S7.SS1.7.p3.8.m8.2"><semantics id="S7.SS1.7.p3.8.m8.2a"><mrow id="S7.SS1.7.p3.8.m8.2.2" xref="S7.SS1.7.p3.8.m8.2.2.cmml"><mrow id="S7.SS1.7.p3.8.m8.1.1.1" xref="S7.SS1.7.p3.8.m8.1.1.1.cmml"><msub id="S7.SS1.7.p3.8.m8.1.1.1.3" xref="S7.SS1.7.p3.8.m8.1.1.1.3.cmml"><mi id="S7.SS1.7.p3.8.m8.1.1.1.3.2" xref="S7.SS1.7.p3.8.m8.1.1.1.3.2.cmml">f</mi><mi id="S7.SS1.7.p3.8.m8.1.1.1.3.3" xref="S7.SS1.7.p3.8.m8.1.1.1.3.3.cmml">α</mi></msub><mo id="S7.SS1.7.p3.8.m8.1.1.1.2" xref="S7.SS1.7.p3.8.m8.1.1.1.2.cmml">⁢</mo><mrow id="S7.SS1.7.p3.8.m8.1.1.1.1.1" xref="S7.SS1.7.p3.8.m8.1.1.1.1.1.1.cmml"><mo id="S7.SS1.7.p3.8.m8.1.1.1.1.1.2" stretchy="false" xref="S7.SS1.7.p3.8.m8.1.1.1.1.1.1.cmml">(</mo><msub id="S7.SS1.7.p3.8.m8.1.1.1.1.1.1" xref="S7.SS1.7.p3.8.m8.1.1.1.1.1.1.cmml"><mi id="S7.SS1.7.p3.8.m8.1.1.1.1.1.1.2" xref="S7.SS1.7.p3.8.m8.1.1.1.1.1.1.2.cmml">a</mi><mi id="S7.SS1.7.p3.8.m8.1.1.1.1.1.1.3" xref="S7.SS1.7.p3.8.m8.1.1.1.1.1.1.3.cmml">β</mi></msub><mo id="S7.SS1.7.p3.8.m8.1.1.1.1.1.3" stretchy="false" xref="S7.SS1.7.p3.8.m8.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S7.SS1.7.p3.8.m8.2.2.3" xref="S7.SS1.7.p3.8.m8.2.2.3.cmml">≠</mo><mrow id="S7.SS1.7.p3.8.m8.2.2.2" xref="S7.SS1.7.p3.8.m8.2.2.2.cmml"><msub id="S7.SS1.7.p3.8.m8.2.2.2.3" xref="S7.SS1.7.p3.8.m8.2.2.2.3.cmml"><mi id="S7.SS1.7.p3.8.m8.2.2.2.3.2" xref="S7.SS1.7.p3.8.m8.2.2.2.3.2.cmml">f</mi><mi id="S7.SS1.7.p3.8.m8.2.2.2.3.3" xref="S7.SS1.7.p3.8.m8.2.2.2.3.3.cmml">β</mi></msub><mo id="S7.SS1.7.p3.8.m8.2.2.2.2" xref="S7.SS1.7.p3.8.m8.2.2.2.2.cmml">⁢</mo><mrow id="S7.SS1.7.p3.8.m8.2.2.2.1.1" xref="S7.SS1.7.p3.8.m8.2.2.2.1.1.1.cmml"><mo id="S7.SS1.7.p3.8.m8.2.2.2.1.1.2" stretchy="false" xref="S7.SS1.7.p3.8.m8.2.2.2.1.1.1.cmml">(</mo><msub id="S7.SS1.7.p3.8.m8.2.2.2.1.1.1" xref="S7.SS1.7.p3.8.m8.2.2.2.1.1.1.cmml"><mi id="S7.SS1.7.p3.8.m8.2.2.2.1.1.1.2" xref="S7.SS1.7.p3.8.m8.2.2.2.1.1.1.2.cmml">a</mi><mi id="S7.SS1.7.p3.8.m8.2.2.2.1.1.1.3" xref="S7.SS1.7.p3.8.m8.2.2.2.1.1.1.3.cmml">γ</mi></msub><mo id="S7.SS1.7.p3.8.m8.2.2.2.1.1.3" stretchy="false" xref="S7.SS1.7.p3.8.m8.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.7.p3.8.m8.2b"><apply id="S7.SS1.7.p3.8.m8.2.2.cmml" xref="S7.SS1.7.p3.8.m8.2.2"><neq id="S7.SS1.7.p3.8.m8.2.2.3.cmml" xref="S7.SS1.7.p3.8.m8.2.2.3"></neq><apply id="S7.SS1.7.p3.8.m8.1.1.1.cmml" xref="S7.SS1.7.p3.8.m8.1.1.1"><times id="S7.SS1.7.p3.8.m8.1.1.1.2.cmml" xref="S7.SS1.7.p3.8.m8.1.1.1.2"></times><apply id="S7.SS1.7.p3.8.m8.1.1.1.3.cmml" xref="S7.SS1.7.p3.8.m8.1.1.1.3"><csymbol cd="ambiguous" id="S7.SS1.7.p3.8.m8.1.1.1.3.1.cmml" xref="S7.SS1.7.p3.8.m8.1.1.1.3">subscript</csymbol><ci id="S7.SS1.7.p3.8.m8.1.1.1.3.2.cmml" xref="S7.SS1.7.p3.8.m8.1.1.1.3.2">𝑓</ci><ci id="S7.SS1.7.p3.8.m8.1.1.1.3.3.cmml" xref="S7.SS1.7.p3.8.m8.1.1.1.3.3">𝛼</ci></apply><apply id="S7.SS1.7.p3.8.m8.1.1.1.1.1.1.cmml" xref="S7.SS1.7.p3.8.m8.1.1.1.1.1"><csymbol cd="ambiguous" id="S7.SS1.7.p3.8.m8.1.1.1.1.1.1.1.cmml" xref="S7.SS1.7.p3.8.m8.1.1.1.1.1">subscript</csymbol><ci id="S7.SS1.7.p3.8.m8.1.1.1.1.1.1.2.cmml" xref="S7.SS1.7.p3.8.m8.1.1.1.1.1.1.2">𝑎</ci><ci id="S7.SS1.7.p3.8.m8.1.1.1.1.1.1.3.cmml" xref="S7.SS1.7.p3.8.m8.1.1.1.1.1.1.3">𝛽</ci></apply></apply><apply id="S7.SS1.7.p3.8.m8.2.2.2.cmml" xref="S7.SS1.7.p3.8.m8.2.2.2"><times id="S7.SS1.7.p3.8.m8.2.2.2.2.cmml" xref="S7.SS1.7.p3.8.m8.2.2.2.2"></times><apply id="S7.SS1.7.p3.8.m8.2.2.2.3.cmml" xref="S7.SS1.7.p3.8.m8.2.2.2.3"><csymbol cd="ambiguous" id="S7.SS1.7.p3.8.m8.2.2.2.3.1.cmml" xref="S7.SS1.7.p3.8.m8.2.2.2.3">subscript</csymbol><ci id="S7.SS1.7.p3.8.m8.2.2.2.3.2.cmml" xref="S7.SS1.7.p3.8.m8.2.2.2.3.2">𝑓</ci><ci id="S7.SS1.7.p3.8.m8.2.2.2.3.3.cmml" xref="S7.SS1.7.p3.8.m8.2.2.2.3.3">𝛽</ci></apply><apply id="S7.SS1.7.p3.8.m8.2.2.2.1.1.1.cmml" xref="S7.SS1.7.p3.8.m8.2.2.2.1.1"><csymbol cd="ambiguous" id="S7.SS1.7.p3.8.m8.2.2.2.1.1.1.1.cmml" xref="S7.SS1.7.p3.8.m8.2.2.2.1.1">subscript</csymbol><ci id="S7.SS1.7.p3.8.m8.2.2.2.1.1.1.2.cmml" xref="S7.SS1.7.p3.8.m8.2.2.2.1.1.1.2">𝑎</ci><ci id="S7.SS1.7.p3.8.m8.2.2.2.1.1.1.3.cmml" xref="S7.SS1.7.p3.8.m8.2.2.2.1.1.1.3">𝛾</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.7.p3.8.m8.2c">f_{\alpha}(a_{\beta})\neq f_{\beta}(a_{\gamma})</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.7.p3.8.m8.2d">italic_f start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT ) ≠ italic_f start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT )</annotation></semantics></math>. This finishes the proof because then <math alttext="|f(A(X))|=|f_{\alpha}(A(X))|\geq|X^{\prime}|=\mathfrak{c}>|B|" class="ltx_Math" display="inline" id="S7.SS1.7.p3.9.m9.6"><semantics id="S7.SS1.7.p3.9.m9.6a"><mrow id="S7.SS1.7.p3.9.m9.6.6" xref="S7.SS1.7.p3.9.m9.6.6.cmml"><mrow id="S7.SS1.7.p3.9.m9.4.4.1.1" xref="S7.SS1.7.p3.9.m9.4.4.1.2.cmml"><mo id="S7.SS1.7.p3.9.m9.4.4.1.1.2" stretchy="false" xref="S7.SS1.7.p3.9.m9.4.4.1.2.1.cmml">|</mo><mrow id="S7.SS1.7.p3.9.m9.4.4.1.1.1" xref="S7.SS1.7.p3.9.m9.4.4.1.1.1.cmml"><mi id="S7.SS1.7.p3.9.m9.4.4.1.1.1.3" xref="S7.SS1.7.p3.9.m9.4.4.1.1.1.3.cmml">f</mi><mo id="S7.SS1.7.p3.9.m9.4.4.1.1.1.2" xref="S7.SS1.7.p3.9.m9.4.4.1.1.1.2.cmml">⁢</mo><mrow id="S7.SS1.7.p3.9.m9.4.4.1.1.1.1.1" xref="S7.SS1.7.p3.9.m9.4.4.1.1.1.1.1.1.cmml"><mo id="S7.SS1.7.p3.9.m9.4.4.1.1.1.1.1.2" stretchy="false" xref="S7.SS1.7.p3.9.m9.4.4.1.1.1.1.1.1.cmml">(</mo><mrow id="S7.SS1.7.p3.9.m9.4.4.1.1.1.1.1.1" xref="S7.SS1.7.p3.9.m9.4.4.1.1.1.1.1.1.cmml"><mi id="S7.SS1.7.p3.9.m9.4.4.1.1.1.1.1.1.2" xref="S7.SS1.7.p3.9.m9.4.4.1.1.1.1.1.1.2.cmml">A</mi><mo id="S7.SS1.7.p3.9.m9.4.4.1.1.1.1.1.1.1" xref="S7.SS1.7.p3.9.m9.4.4.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S7.SS1.7.p3.9.m9.4.4.1.1.1.1.1.1.3.2" xref="S7.SS1.7.p3.9.m9.4.4.1.1.1.1.1.1.cmml"><mo id="S7.SS1.7.p3.9.m9.4.4.1.1.1.1.1.1.3.2.1" stretchy="false" xref="S7.SS1.7.p3.9.m9.4.4.1.1.1.1.1.1.cmml">(</mo><mi id="S7.SS1.7.p3.9.m9.1.1" xref="S7.SS1.7.p3.9.m9.1.1.cmml">X</mi><mo id="S7.SS1.7.p3.9.m9.4.4.1.1.1.1.1.1.3.2.2" stretchy="false" xref="S7.SS1.7.p3.9.m9.4.4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S7.SS1.7.p3.9.m9.4.4.1.1.1.1.1.3" stretchy="false" xref="S7.SS1.7.p3.9.m9.4.4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S7.SS1.7.p3.9.m9.4.4.1.1.3" stretchy="false" xref="S7.SS1.7.p3.9.m9.4.4.1.2.1.cmml">|</mo></mrow><mo id="S7.SS1.7.p3.9.m9.6.6.5" xref="S7.SS1.7.p3.9.m9.6.6.5.cmml">=</mo><mrow id="S7.SS1.7.p3.9.m9.5.5.2.1" xref="S7.SS1.7.p3.9.m9.5.5.2.2.cmml"><mo id="S7.SS1.7.p3.9.m9.5.5.2.1.2" stretchy="false" xref="S7.SS1.7.p3.9.m9.5.5.2.2.1.cmml">|</mo><mrow id="S7.SS1.7.p3.9.m9.5.5.2.1.1" xref="S7.SS1.7.p3.9.m9.5.5.2.1.1.cmml"><msub id="S7.SS1.7.p3.9.m9.5.5.2.1.1.3" xref="S7.SS1.7.p3.9.m9.5.5.2.1.1.3.cmml"><mi id="S7.SS1.7.p3.9.m9.5.5.2.1.1.3.2" xref="S7.SS1.7.p3.9.m9.5.5.2.1.1.3.2.cmml">f</mi><mi id="S7.SS1.7.p3.9.m9.5.5.2.1.1.3.3" xref="S7.SS1.7.p3.9.m9.5.5.2.1.1.3.3.cmml">α</mi></msub><mo id="S7.SS1.7.p3.9.m9.5.5.2.1.1.2" xref="S7.SS1.7.p3.9.m9.5.5.2.1.1.2.cmml">⁢</mo><mrow id="S7.SS1.7.p3.9.m9.5.5.2.1.1.1.1" xref="S7.SS1.7.p3.9.m9.5.5.2.1.1.1.1.1.cmml"><mo id="S7.SS1.7.p3.9.m9.5.5.2.1.1.1.1.2" stretchy="false" xref="S7.SS1.7.p3.9.m9.5.5.2.1.1.1.1.1.cmml">(</mo><mrow id="S7.SS1.7.p3.9.m9.5.5.2.1.1.1.1.1" xref="S7.SS1.7.p3.9.m9.5.5.2.1.1.1.1.1.cmml"><mi id="S7.SS1.7.p3.9.m9.5.5.2.1.1.1.1.1.2" xref="S7.SS1.7.p3.9.m9.5.5.2.1.1.1.1.1.2.cmml">A</mi><mo id="S7.SS1.7.p3.9.m9.5.5.2.1.1.1.1.1.1" xref="S7.SS1.7.p3.9.m9.5.5.2.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S7.SS1.7.p3.9.m9.5.5.2.1.1.1.1.1.3.2" xref="S7.SS1.7.p3.9.m9.5.5.2.1.1.1.1.1.cmml"><mo id="S7.SS1.7.p3.9.m9.5.5.2.1.1.1.1.1.3.2.1" stretchy="false" xref="S7.SS1.7.p3.9.m9.5.5.2.1.1.1.1.1.cmml">(</mo><mi id="S7.SS1.7.p3.9.m9.2.2" xref="S7.SS1.7.p3.9.m9.2.2.cmml">X</mi><mo id="S7.SS1.7.p3.9.m9.5.5.2.1.1.1.1.1.3.2.2" stretchy="false" xref="S7.SS1.7.p3.9.m9.5.5.2.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S7.SS1.7.p3.9.m9.5.5.2.1.1.1.1.3" stretchy="false" xref="S7.SS1.7.p3.9.m9.5.5.2.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S7.SS1.7.p3.9.m9.5.5.2.1.3" stretchy="false" xref="S7.SS1.7.p3.9.m9.5.5.2.2.1.cmml">|</mo></mrow><mo id="S7.SS1.7.p3.9.m9.6.6.6" xref="S7.SS1.7.p3.9.m9.6.6.6.cmml">≥</mo><mrow id="S7.SS1.7.p3.9.m9.6.6.3.1" xref="S7.SS1.7.p3.9.m9.6.6.3.2.cmml"><mo id="S7.SS1.7.p3.9.m9.6.6.3.1.2" stretchy="false" xref="S7.SS1.7.p3.9.m9.6.6.3.2.1.cmml">|</mo><msup id="S7.SS1.7.p3.9.m9.6.6.3.1.1" xref="S7.SS1.7.p3.9.m9.6.6.3.1.1.cmml"><mi id="S7.SS1.7.p3.9.m9.6.6.3.1.1.2" xref="S7.SS1.7.p3.9.m9.6.6.3.1.1.2.cmml">X</mi><mo id="S7.SS1.7.p3.9.m9.6.6.3.1.1.3" xref="S7.SS1.7.p3.9.m9.6.6.3.1.1.3.cmml">′</mo></msup><mo id="S7.SS1.7.p3.9.m9.6.6.3.1.3" stretchy="false" xref="S7.SS1.7.p3.9.m9.6.6.3.2.1.cmml">|</mo></mrow><mo id="S7.SS1.7.p3.9.m9.6.6.7" xref="S7.SS1.7.p3.9.m9.6.6.7.cmml">=</mo><mi id="S7.SS1.7.p3.9.m9.6.6.8" xref="S7.SS1.7.p3.9.m9.6.6.8.cmml">𝔠</mi><mo id="S7.SS1.7.p3.9.m9.6.6.9" xref="S7.SS1.7.p3.9.m9.6.6.9.cmml">></mo><mrow id="S7.SS1.7.p3.9.m9.6.6.10.2" xref="S7.SS1.7.p3.9.m9.6.6.10.1.cmml"><mo id="S7.SS1.7.p3.9.m9.6.6.10.2.1" stretchy="false" xref="S7.SS1.7.p3.9.m9.6.6.10.1.1.cmml">|</mo><mi id="S7.SS1.7.p3.9.m9.3.3" xref="S7.SS1.7.p3.9.m9.3.3.cmml">B</mi><mo id="S7.SS1.7.p3.9.m9.6.6.10.2.2" stretchy="false" xref="S7.SS1.7.p3.9.m9.6.6.10.1.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.7.p3.9.m9.6b"><apply id="S7.SS1.7.p3.9.m9.6.6.cmml" xref="S7.SS1.7.p3.9.m9.6.6"><and id="S7.SS1.7.p3.9.m9.6.6a.cmml" xref="S7.SS1.7.p3.9.m9.6.6"></and><apply id="S7.SS1.7.p3.9.m9.6.6b.cmml" xref="S7.SS1.7.p3.9.m9.6.6"><eq id="S7.SS1.7.p3.9.m9.6.6.5.cmml" xref="S7.SS1.7.p3.9.m9.6.6.5"></eq><apply id="S7.SS1.7.p3.9.m9.4.4.1.2.cmml" xref="S7.SS1.7.p3.9.m9.4.4.1.1"><abs id="S7.SS1.7.p3.9.m9.4.4.1.2.1.cmml" xref="S7.SS1.7.p3.9.m9.4.4.1.1.2"></abs><apply id="S7.SS1.7.p3.9.m9.4.4.1.1.1.cmml" xref="S7.SS1.7.p3.9.m9.4.4.1.1.1"><times id="S7.SS1.7.p3.9.m9.4.4.1.1.1.2.cmml" xref="S7.SS1.7.p3.9.m9.4.4.1.1.1.2"></times><ci id="S7.SS1.7.p3.9.m9.4.4.1.1.1.3.cmml" xref="S7.SS1.7.p3.9.m9.4.4.1.1.1.3">𝑓</ci><apply id="S7.SS1.7.p3.9.m9.4.4.1.1.1.1.1.1.cmml" xref="S7.SS1.7.p3.9.m9.4.4.1.1.1.1.1"><times id="S7.SS1.7.p3.9.m9.4.4.1.1.1.1.1.1.1.cmml" xref="S7.SS1.7.p3.9.m9.4.4.1.1.1.1.1.1.1"></times><ci id="S7.SS1.7.p3.9.m9.4.4.1.1.1.1.1.1.2.cmml" xref="S7.SS1.7.p3.9.m9.4.4.1.1.1.1.1.1.2">𝐴</ci><ci id="S7.SS1.7.p3.9.m9.1.1.cmml" xref="S7.SS1.7.p3.9.m9.1.1">𝑋</ci></apply></apply></apply><apply id="S7.SS1.7.p3.9.m9.5.5.2.2.cmml" xref="S7.SS1.7.p3.9.m9.5.5.2.1"><abs id="S7.SS1.7.p3.9.m9.5.5.2.2.1.cmml" xref="S7.SS1.7.p3.9.m9.5.5.2.1.2"></abs><apply id="S7.SS1.7.p3.9.m9.5.5.2.1.1.cmml" xref="S7.SS1.7.p3.9.m9.5.5.2.1.1"><times id="S7.SS1.7.p3.9.m9.5.5.2.1.1.2.cmml" xref="S7.SS1.7.p3.9.m9.5.5.2.1.1.2"></times><apply id="S7.SS1.7.p3.9.m9.5.5.2.1.1.3.cmml" xref="S7.SS1.7.p3.9.m9.5.5.2.1.1.3"><csymbol cd="ambiguous" id="S7.SS1.7.p3.9.m9.5.5.2.1.1.3.1.cmml" xref="S7.SS1.7.p3.9.m9.5.5.2.1.1.3">subscript</csymbol><ci id="S7.SS1.7.p3.9.m9.5.5.2.1.1.3.2.cmml" xref="S7.SS1.7.p3.9.m9.5.5.2.1.1.3.2">𝑓</ci><ci id="S7.SS1.7.p3.9.m9.5.5.2.1.1.3.3.cmml" xref="S7.SS1.7.p3.9.m9.5.5.2.1.1.3.3">𝛼</ci></apply><apply id="S7.SS1.7.p3.9.m9.5.5.2.1.1.1.1.1.cmml" xref="S7.SS1.7.p3.9.m9.5.5.2.1.1.1.1"><times id="S7.SS1.7.p3.9.m9.5.5.2.1.1.1.1.1.1.cmml" xref="S7.SS1.7.p3.9.m9.5.5.2.1.1.1.1.1.1"></times><ci id="S7.SS1.7.p3.9.m9.5.5.2.1.1.1.1.1.2.cmml" xref="S7.SS1.7.p3.9.m9.5.5.2.1.1.1.1.1.2">𝐴</ci><ci id="S7.SS1.7.p3.9.m9.2.2.cmml" xref="S7.SS1.7.p3.9.m9.2.2">𝑋</ci></apply></apply></apply></apply><apply id="S7.SS1.7.p3.9.m9.6.6c.cmml" xref="S7.SS1.7.p3.9.m9.6.6"><geq id="S7.SS1.7.p3.9.m9.6.6.6.cmml" xref="S7.SS1.7.p3.9.m9.6.6.6"></geq><share href="https://arxiv.org/html/2503.13728v1#S7.SS1.7.p3.9.m9.5.5.2.cmml" id="S7.SS1.7.p3.9.m9.6.6d.cmml" xref="S7.SS1.7.p3.9.m9.6.6"></share><apply id="S7.SS1.7.p3.9.m9.6.6.3.2.cmml" xref="S7.SS1.7.p3.9.m9.6.6.3.1"><abs id="S7.SS1.7.p3.9.m9.6.6.3.2.1.cmml" xref="S7.SS1.7.p3.9.m9.6.6.3.1.2"></abs><apply id="S7.SS1.7.p3.9.m9.6.6.3.1.1.cmml" xref="S7.SS1.7.p3.9.m9.6.6.3.1.1"><csymbol cd="ambiguous" id="S7.SS1.7.p3.9.m9.6.6.3.1.1.1.cmml" xref="S7.SS1.7.p3.9.m9.6.6.3.1.1">superscript</csymbol><ci id="S7.SS1.7.p3.9.m9.6.6.3.1.1.2.cmml" xref="S7.SS1.7.p3.9.m9.6.6.3.1.1.2">𝑋</ci><ci id="S7.SS1.7.p3.9.m9.6.6.3.1.1.3.cmml" xref="S7.SS1.7.p3.9.m9.6.6.3.1.1.3">′</ci></apply></apply></apply><apply id="S7.SS1.7.p3.9.m9.6.6e.cmml" xref="S7.SS1.7.p3.9.m9.6.6"><eq id="S7.SS1.7.p3.9.m9.6.6.7.cmml" xref="S7.SS1.7.p3.9.m9.6.6.7"></eq><share href="https://arxiv.org/html/2503.13728v1#S7.SS1.7.p3.9.m9.6.6.3.cmml" id="S7.SS1.7.p3.9.m9.6.6f.cmml" xref="S7.SS1.7.p3.9.m9.6.6"></share><ci id="S7.SS1.7.p3.9.m9.6.6.8.cmml" xref="S7.SS1.7.p3.9.m9.6.6.8">𝔠</ci></apply><apply id="S7.SS1.7.p3.9.m9.6.6g.cmml" xref="S7.SS1.7.p3.9.m9.6.6"><gt id="S7.SS1.7.p3.9.m9.6.6.9.cmml" xref="S7.SS1.7.p3.9.m9.6.6.9"></gt><share href="https://arxiv.org/html/2503.13728v1#S7.SS1.7.p3.9.m9.6.6.8.cmml" id="S7.SS1.7.p3.9.m9.6.6h.cmml" xref="S7.SS1.7.p3.9.m9.6.6"></share><apply id="S7.SS1.7.p3.9.m9.6.6.10.1.cmml" xref="S7.SS1.7.p3.9.m9.6.6.10.2"><abs id="S7.SS1.7.p3.9.m9.6.6.10.1.1.cmml" xref="S7.SS1.7.p3.9.m9.6.6.10.2.1"></abs><ci id="S7.SS1.7.p3.9.m9.3.3.cmml" xref="S7.SS1.7.p3.9.m9.3.3">𝐵</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.7.p3.9.m9.6c">|f(A(X))|=|f_{\alpha}(A(X))|\geq|X^{\prime}|=\mathfrak{c}>|B|</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.7.p3.9.m9.6d">| italic_f ( italic_A ( italic_X ) ) | = | italic_f start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ( italic_A ( italic_X ) ) | ≥ | italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT | = fraktur_c > | italic_B |</annotation></semantics></math>, which is a contradiction.</p> </div> <div class="ltx_para" id="S7.SS1.8.p4"> <p class="ltx_p" id="S7.SS1.8.p4.4">To see this, note that since <math alttext="\alpha<\beta<\gamma" class="ltx_Math" display="inline" id="S7.SS1.8.p4.1.m1.1"><semantics id="S7.SS1.8.p4.1.m1.1a"><mrow id="S7.SS1.8.p4.1.m1.1.1" xref="S7.SS1.8.p4.1.m1.1.1.cmml"><mi id="S7.SS1.8.p4.1.m1.1.1.2" xref="S7.SS1.8.p4.1.m1.1.1.2.cmml">α</mi><mo id="S7.SS1.8.p4.1.m1.1.1.3" xref="S7.SS1.8.p4.1.m1.1.1.3.cmml"><</mo><mi id="S7.SS1.8.p4.1.m1.1.1.4" xref="S7.SS1.8.p4.1.m1.1.1.4.cmml">β</mi><mo id="S7.SS1.8.p4.1.m1.1.1.5" xref="S7.SS1.8.p4.1.m1.1.1.5.cmml"><</mo><mi id="S7.SS1.8.p4.1.m1.1.1.6" xref="S7.SS1.8.p4.1.m1.1.1.6.cmml">γ</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.8.p4.1.m1.1b"><apply id="S7.SS1.8.p4.1.m1.1.1.cmml" xref="S7.SS1.8.p4.1.m1.1.1"><and id="S7.SS1.8.p4.1.m1.1.1a.cmml" xref="S7.SS1.8.p4.1.m1.1.1"></and><apply id="S7.SS1.8.p4.1.m1.1.1b.cmml" xref="S7.SS1.8.p4.1.m1.1.1"><lt id="S7.SS1.8.p4.1.m1.1.1.3.cmml" xref="S7.SS1.8.p4.1.m1.1.1.3"></lt><ci id="S7.SS1.8.p4.1.m1.1.1.2.cmml" xref="S7.SS1.8.p4.1.m1.1.1.2">𝛼</ci><ci id="S7.SS1.8.p4.1.m1.1.1.4.cmml" xref="S7.SS1.8.p4.1.m1.1.1.4">𝛽</ci></apply><apply id="S7.SS1.8.p4.1.m1.1.1c.cmml" xref="S7.SS1.8.p4.1.m1.1.1"><lt id="S7.SS1.8.p4.1.m1.1.1.5.cmml" xref="S7.SS1.8.p4.1.m1.1.1.5"></lt><share href="https://arxiv.org/html/2503.13728v1#S7.SS1.8.p4.1.m1.1.1.4.cmml" id="S7.SS1.8.p4.1.m1.1.1d.cmml" xref="S7.SS1.8.p4.1.m1.1.1"></share><ci id="S7.SS1.8.p4.1.m1.1.1.6.cmml" xref="S7.SS1.8.p4.1.m1.1.1.6">𝛾</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.8.p4.1.m1.1c">\alpha<\beta<\gamma</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.8.p4.1.m1.1d">italic_α < italic_β < italic_γ</annotation></semantics></math>, <math alttext="a_{\beta}\in A_{\beta}\subseteq Z_{\gamma}" class="ltx_Math" display="inline" id="S7.SS1.8.p4.2.m2.1"><semantics id="S7.SS1.8.p4.2.m2.1a"><mrow id="S7.SS1.8.p4.2.m2.1.1" xref="S7.SS1.8.p4.2.m2.1.1.cmml"><msub id="S7.SS1.8.p4.2.m2.1.1.2" xref="S7.SS1.8.p4.2.m2.1.1.2.cmml"><mi id="S7.SS1.8.p4.2.m2.1.1.2.2" xref="S7.SS1.8.p4.2.m2.1.1.2.2.cmml">a</mi><mi id="S7.SS1.8.p4.2.m2.1.1.2.3" xref="S7.SS1.8.p4.2.m2.1.1.2.3.cmml">β</mi></msub><mo id="S7.SS1.8.p4.2.m2.1.1.3" xref="S7.SS1.8.p4.2.m2.1.1.3.cmml">∈</mo><msub id="S7.SS1.8.p4.2.m2.1.1.4" xref="S7.SS1.8.p4.2.m2.1.1.4.cmml"><mi id="S7.SS1.8.p4.2.m2.1.1.4.2" xref="S7.SS1.8.p4.2.m2.1.1.4.2.cmml">A</mi><mi id="S7.SS1.8.p4.2.m2.1.1.4.3" xref="S7.SS1.8.p4.2.m2.1.1.4.3.cmml">β</mi></msub><mo id="S7.SS1.8.p4.2.m2.1.1.5" xref="S7.SS1.8.p4.2.m2.1.1.5.cmml">⊆</mo><msub id="S7.SS1.8.p4.2.m2.1.1.6" xref="S7.SS1.8.p4.2.m2.1.1.6.cmml"><mi id="S7.SS1.8.p4.2.m2.1.1.6.2" xref="S7.SS1.8.p4.2.m2.1.1.6.2.cmml">Z</mi><mi id="S7.SS1.8.p4.2.m2.1.1.6.3" xref="S7.SS1.8.p4.2.m2.1.1.6.3.cmml">γ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.8.p4.2.m2.1b"><apply id="S7.SS1.8.p4.2.m2.1.1.cmml" xref="S7.SS1.8.p4.2.m2.1.1"><and id="S7.SS1.8.p4.2.m2.1.1a.cmml" xref="S7.SS1.8.p4.2.m2.1.1"></and><apply id="S7.SS1.8.p4.2.m2.1.1b.cmml" xref="S7.SS1.8.p4.2.m2.1.1"><in id="S7.SS1.8.p4.2.m2.1.1.3.cmml" xref="S7.SS1.8.p4.2.m2.1.1.3"></in><apply id="S7.SS1.8.p4.2.m2.1.1.2.cmml" xref="S7.SS1.8.p4.2.m2.1.1.2"><csymbol cd="ambiguous" id="S7.SS1.8.p4.2.m2.1.1.2.1.cmml" xref="S7.SS1.8.p4.2.m2.1.1.2">subscript</csymbol><ci id="S7.SS1.8.p4.2.m2.1.1.2.2.cmml" xref="S7.SS1.8.p4.2.m2.1.1.2.2">𝑎</ci><ci id="S7.SS1.8.p4.2.m2.1.1.2.3.cmml" xref="S7.SS1.8.p4.2.m2.1.1.2.3">𝛽</ci></apply><apply id="S7.SS1.8.p4.2.m2.1.1.4.cmml" xref="S7.SS1.8.p4.2.m2.1.1.4"><csymbol cd="ambiguous" id="S7.SS1.8.p4.2.m2.1.1.4.1.cmml" xref="S7.SS1.8.p4.2.m2.1.1.4">subscript</csymbol><ci id="S7.SS1.8.p4.2.m2.1.1.4.2.cmml" xref="S7.SS1.8.p4.2.m2.1.1.4.2">𝐴</ci><ci id="S7.SS1.8.p4.2.m2.1.1.4.3.cmml" xref="S7.SS1.8.p4.2.m2.1.1.4.3">𝛽</ci></apply></apply><apply id="S7.SS1.8.p4.2.m2.1.1c.cmml" xref="S7.SS1.8.p4.2.m2.1.1"><subset id="S7.SS1.8.p4.2.m2.1.1.5.cmml" xref="S7.SS1.8.p4.2.m2.1.1.5"></subset><share href="https://arxiv.org/html/2503.13728v1#S7.SS1.8.p4.2.m2.1.1.4.cmml" id="S7.SS1.8.p4.2.m2.1.1d.cmml" xref="S7.SS1.8.p4.2.m2.1.1"></share><apply id="S7.SS1.8.p4.2.m2.1.1.6.cmml" xref="S7.SS1.8.p4.2.m2.1.1.6"><csymbol cd="ambiguous" id="S7.SS1.8.p4.2.m2.1.1.6.1.cmml" xref="S7.SS1.8.p4.2.m2.1.1.6">subscript</csymbol><ci id="S7.SS1.8.p4.2.m2.1.1.6.2.cmml" xref="S7.SS1.8.p4.2.m2.1.1.6.2">𝑍</ci><ci id="S7.SS1.8.p4.2.m2.1.1.6.3.cmml" xref="S7.SS1.8.p4.2.m2.1.1.6.3">𝛾</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.8.p4.2.m2.1c">a_{\beta}\in A_{\beta}\subseteq Z_{\gamma}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.8.p4.2.m2.1d">italic_a start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT ∈ italic_A start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT ⊆ italic_Z start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT</annotation></semantics></math>, thus <math alttext="f_{\alpha}(a_{\beta})\in Z_{\gamma}" class="ltx_Math" display="inline" id="S7.SS1.8.p4.3.m3.1"><semantics id="S7.SS1.8.p4.3.m3.1a"><mrow id="S7.SS1.8.p4.3.m3.1.1" xref="S7.SS1.8.p4.3.m3.1.1.cmml"><mrow id="S7.SS1.8.p4.3.m3.1.1.1" xref="S7.SS1.8.p4.3.m3.1.1.1.cmml"><msub id="S7.SS1.8.p4.3.m3.1.1.1.3" xref="S7.SS1.8.p4.3.m3.1.1.1.3.cmml"><mi id="S7.SS1.8.p4.3.m3.1.1.1.3.2" xref="S7.SS1.8.p4.3.m3.1.1.1.3.2.cmml">f</mi><mi id="S7.SS1.8.p4.3.m3.1.1.1.3.3" xref="S7.SS1.8.p4.3.m3.1.1.1.3.3.cmml">α</mi></msub><mo id="S7.SS1.8.p4.3.m3.1.1.1.2" xref="S7.SS1.8.p4.3.m3.1.1.1.2.cmml">⁢</mo><mrow id="S7.SS1.8.p4.3.m3.1.1.1.1.1" xref="S7.SS1.8.p4.3.m3.1.1.1.1.1.1.cmml"><mo id="S7.SS1.8.p4.3.m3.1.1.1.1.1.2" stretchy="false" xref="S7.SS1.8.p4.3.m3.1.1.1.1.1.1.cmml">(</mo><msub id="S7.SS1.8.p4.3.m3.1.1.1.1.1.1" xref="S7.SS1.8.p4.3.m3.1.1.1.1.1.1.cmml"><mi id="S7.SS1.8.p4.3.m3.1.1.1.1.1.1.2" xref="S7.SS1.8.p4.3.m3.1.1.1.1.1.1.2.cmml">a</mi><mi id="S7.SS1.8.p4.3.m3.1.1.1.1.1.1.3" xref="S7.SS1.8.p4.3.m3.1.1.1.1.1.1.3.cmml">β</mi></msub><mo id="S7.SS1.8.p4.3.m3.1.1.1.1.1.3" stretchy="false" xref="S7.SS1.8.p4.3.m3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S7.SS1.8.p4.3.m3.1.1.2" xref="S7.SS1.8.p4.3.m3.1.1.2.cmml">∈</mo><msub id="S7.SS1.8.p4.3.m3.1.1.3" xref="S7.SS1.8.p4.3.m3.1.1.3.cmml"><mi id="S7.SS1.8.p4.3.m3.1.1.3.2" xref="S7.SS1.8.p4.3.m3.1.1.3.2.cmml">Z</mi><mi id="S7.SS1.8.p4.3.m3.1.1.3.3" xref="S7.SS1.8.p4.3.m3.1.1.3.3.cmml">γ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.8.p4.3.m3.1b"><apply id="S7.SS1.8.p4.3.m3.1.1.cmml" xref="S7.SS1.8.p4.3.m3.1.1"><in id="S7.SS1.8.p4.3.m3.1.1.2.cmml" xref="S7.SS1.8.p4.3.m3.1.1.2"></in><apply id="S7.SS1.8.p4.3.m3.1.1.1.cmml" xref="S7.SS1.8.p4.3.m3.1.1.1"><times id="S7.SS1.8.p4.3.m3.1.1.1.2.cmml" xref="S7.SS1.8.p4.3.m3.1.1.1.2"></times><apply id="S7.SS1.8.p4.3.m3.1.1.1.3.cmml" xref="S7.SS1.8.p4.3.m3.1.1.1.3"><csymbol cd="ambiguous" id="S7.SS1.8.p4.3.m3.1.1.1.3.1.cmml" xref="S7.SS1.8.p4.3.m3.1.1.1.3">subscript</csymbol><ci id="S7.SS1.8.p4.3.m3.1.1.1.3.2.cmml" xref="S7.SS1.8.p4.3.m3.1.1.1.3.2">𝑓</ci><ci id="S7.SS1.8.p4.3.m3.1.1.1.3.3.cmml" xref="S7.SS1.8.p4.3.m3.1.1.1.3.3">𝛼</ci></apply><apply id="S7.SS1.8.p4.3.m3.1.1.1.1.1.1.cmml" xref="S7.SS1.8.p4.3.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="S7.SS1.8.p4.3.m3.1.1.1.1.1.1.1.cmml" xref="S7.SS1.8.p4.3.m3.1.1.1.1.1">subscript</csymbol><ci id="S7.SS1.8.p4.3.m3.1.1.1.1.1.1.2.cmml" xref="S7.SS1.8.p4.3.m3.1.1.1.1.1.1.2">𝑎</ci><ci id="S7.SS1.8.p4.3.m3.1.1.1.1.1.1.3.cmml" xref="S7.SS1.8.p4.3.m3.1.1.1.1.1.1.3">𝛽</ci></apply></apply><apply id="S7.SS1.8.p4.3.m3.1.1.3.cmml" xref="S7.SS1.8.p4.3.m3.1.1.3"><csymbol cd="ambiguous" id="S7.SS1.8.p4.3.m3.1.1.3.1.cmml" xref="S7.SS1.8.p4.3.m3.1.1.3">subscript</csymbol><ci id="S7.SS1.8.p4.3.m3.1.1.3.2.cmml" xref="S7.SS1.8.p4.3.m3.1.1.3.2">𝑍</ci><ci id="S7.SS1.8.p4.3.m3.1.1.3.3.cmml" xref="S7.SS1.8.p4.3.m3.1.1.3.3">𝛾</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.8.p4.3.m3.1c">f_{\alpha}(a_{\beta})\in Z_{\gamma}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.8.p4.3.m3.1d">italic_f start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT ) ∈ italic_Z start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT</annotation></semantics></math>, and by construction <math alttext="f_{\alpha}(a_{\gamma})\notin Z_{\gamma}" class="ltx_Math" display="inline" id="S7.SS1.8.p4.4.m4.1"><semantics id="S7.SS1.8.p4.4.m4.1a"><mrow id="S7.SS1.8.p4.4.m4.1.1" xref="S7.SS1.8.p4.4.m4.1.1.cmml"><mrow id="S7.SS1.8.p4.4.m4.1.1.1" xref="S7.SS1.8.p4.4.m4.1.1.1.cmml"><msub id="S7.SS1.8.p4.4.m4.1.1.1.3" xref="S7.SS1.8.p4.4.m4.1.1.1.3.cmml"><mi id="S7.SS1.8.p4.4.m4.1.1.1.3.2" xref="S7.SS1.8.p4.4.m4.1.1.1.3.2.cmml">f</mi><mi id="S7.SS1.8.p4.4.m4.1.1.1.3.3" xref="S7.SS1.8.p4.4.m4.1.1.1.3.3.cmml">α</mi></msub><mo id="S7.SS1.8.p4.4.m4.1.1.1.2" xref="S7.SS1.8.p4.4.m4.1.1.1.2.cmml">⁢</mo><mrow id="S7.SS1.8.p4.4.m4.1.1.1.1.1" xref="S7.SS1.8.p4.4.m4.1.1.1.1.1.1.cmml"><mo id="S7.SS1.8.p4.4.m4.1.1.1.1.1.2" stretchy="false" xref="S7.SS1.8.p4.4.m4.1.1.1.1.1.1.cmml">(</mo><msub id="S7.SS1.8.p4.4.m4.1.1.1.1.1.1" xref="S7.SS1.8.p4.4.m4.1.1.1.1.1.1.cmml"><mi id="S7.SS1.8.p4.4.m4.1.1.1.1.1.1.2" xref="S7.SS1.8.p4.4.m4.1.1.1.1.1.1.2.cmml">a</mi><mi id="S7.SS1.8.p4.4.m4.1.1.1.1.1.1.3" xref="S7.SS1.8.p4.4.m4.1.1.1.1.1.1.3.cmml">γ</mi></msub><mo id="S7.SS1.8.p4.4.m4.1.1.1.1.1.3" stretchy="false" xref="S7.SS1.8.p4.4.m4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S7.SS1.8.p4.4.m4.1.1.2" xref="S7.SS1.8.p4.4.m4.1.1.2.cmml">∉</mo><msub id="S7.SS1.8.p4.4.m4.1.1.3" xref="S7.SS1.8.p4.4.m4.1.1.3.cmml"><mi id="S7.SS1.8.p4.4.m4.1.1.3.2" xref="S7.SS1.8.p4.4.m4.1.1.3.2.cmml">Z</mi><mi id="S7.SS1.8.p4.4.m4.1.1.3.3" xref="S7.SS1.8.p4.4.m4.1.1.3.3.cmml">γ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.8.p4.4.m4.1b"><apply id="S7.SS1.8.p4.4.m4.1.1.cmml" xref="S7.SS1.8.p4.4.m4.1.1"><notin id="S7.SS1.8.p4.4.m4.1.1.2.cmml" xref="S7.SS1.8.p4.4.m4.1.1.2"></notin><apply id="S7.SS1.8.p4.4.m4.1.1.1.cmml" xref="S7.SS1.8.p4.4.m4.1.1.1"><times id="S7.SS1.8.p4.4.m4.1.1.1.2.cmml" xref="S7.SS1.8.p4.4.m4.1.1.1.2"></times><apply id="S7.SS1.8.p4.4.m4.1.1.1.3.cmml" xref="S7.SS1.8.p4.4.m4.1.1.1.3"><csymbol cd="ambiguous" id="S7.SS1.8.p4.4.m4.1.1.1.3.1.cmml" xref="S7.SS1.8.p4.4.m4.1.1.1.3">subscript</csymbol><ci id="S7.SS1.8.p4.4.m4.1.1.1.3.2.cmml" xref="S7.SS1.8.p4.4.m4.1.1.1.3.2">𝑓</ci><ci id="S7.SS1.8.p4.4.m4.1.1.1.3.3.cmml" xref="S7.SS1.8.p4.4.m4.1.1.1.3.3">𝛼</ci></apply><apply id="S7.SS1.8.p4.4.m4.1.1.1.1.1.1.cmml" xref="S7.SS1.8.p4.4.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="S7.SS1.8.p4.4.m4.1.1.1.1.1.1.1.cmml" xref="S7.SS1.8.p4.4.m4.1.1.1.1.1">subscript</csymbol><ci id="S7.SS1.8.p4.4.m4.1.1.1.1.1.1.2.cmml" xref="S7.SS1.8.p4.4.m4.1.1.1.1.1.1.2">𝑎</ci><ci id="S7.SS1.8.p4.4.m4.1.1.1.1.1.1.3.cmml" xref="S7.SS1.8.p4.4.m4.1.1.1.1.1.1.3">𝛾</ci></apply></apply><apply id="S7.SS1.8.p4.4.m4.1.1.3.cmml" xref="S7.SS1.8.p4.4.m4.1.1.3"><csymbol cd="ambiguous" id="S7.SS1.8.p4.4.m4.1.1.3.1.cmml" xref="S7.SS1.8.p4.4.m4.1.1.3">subscript</csymbol><ci id="S7.SS1.8.p4.4.m4.1.1.3.2.cmml" xref="S7.SS1.8.p4.4.m4.1.1.3.2">𝑍</ci><ci id="S7.SS1.8.p4.4.m4.1.1.3.3.cmml" xref="S7.SS1.8.p4.4.m4.1.1.3.3">𝛾</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.8.p4.4.m4.1c">f_{\alpha}(a_{\gamma})\notin Z_{\gamma}</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.8.p4.4.m4.1d">italic_f start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ) ∉ italic_Z start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT</annotation></semantics></math>. ∎</p> </div> </div> </section> </section> <section class="ltx_section" id="S8"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">8. </span>On a question on Countryman lines</h2> <div class="ltx_para" id="S8.p1"> <p class="ltx_p" id="S8.p1.1">In this section we take a detour from epimorphisms, and study the following question on Countryman lines, which in our opinion is quite natural, and have not seen it asked in the literature. Recall that if <math alttext="C" class="ltx_Math" display="inline" id="S8.p1.1.m1.1"><semantics id="S8.p1.1.m1.1a"><mi id="S8.p1.1.m1.1.1" xref="S8.p1.1.m1.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S8.p1.1.m1.1b"><ci id="S8.p1.1.m1.1.1.cmml" xref="S8.p1.1.m1.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.p1.1.m1.1c">C</annotation><annotation encoding="application/x-llamapun" id="S8.p1.1.m1.1d">italic_C</annotation></semantics></math> is a Countryman line then it enjoys the property of having no two uncountable reverse isomorphic suborders. We ask if this characterizes the Countryman lines in the class of Aronszajn lines.</p> </div> <div class="ltx_para" id="S8.p2"> <p class="ltx_p" id="S8.p2.2">Usually a linear order <math alttext="L" class="ltx_Math" display="inline" id="S8.p2.1.m1.1"><semantics id="S8.p2.1.m1.1a"><mi id="S8.p2.1.m1.1.1" xref="S8.p2.1.m1.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S8.p2.1.m1.1b"><ci id="S8.p2.1.m1.1.1.cmml" xref="S8.p2.1.m1.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.p2.1.m1.1c">L</annotation><annotation encoding="application/x-llamapun" id="S8.p2.1.m1.1d">italic_L</annotation></semantics></math> is called <em class="ltx_emph ltx_font_italic" id="S8.p2.2.1">reversible</em> if <math alttext="L\cong L^{\star}" class="ltx_Math" display="inline" id="S8.p2.2.m2.1"><semantics id="S8.p2.2.m2.1a"><mrow id="S8.p2.2.m2.1.1" xref="S8.p2.2.m2.1.1.cmml"><mi id="S8.p2.2.m2.1.1.2" xref="S8.p2.2.m2.1.1.2.cmml">L</mi><mo id="S8.p2.2.m2.1.1.1" xref="S8.p2.2.m2.1.1.1.cmml">≅</mo><msup id="S8.p2.2.m2.1.1.3" xref="S8.p2.2.m2.1.1.3.cmml"><mi id="S8.p2.2.m2.1.1.3.2" xref="S8.p2.2.m2.1.1.3.2.cmml">L</mi><mo id="S8.p2.2.m2.1.1.3.3" xref="S8.p2.2.m2.1.1.3.3.cmml">⋆</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S8.p2.2.m2.1b"><apply id="S8.p2.2.m2.1.1.cmml" xref="S8.p2.2.m2.1.1"><approx id="S8.p2.2.m2.1.1.1.cmml" xref="S8.p2.2.m2.1.1.1"></approx><ci id="S8.p2.2.m2.1.1.2.cmml" xref="S8.p2.2.m2.1.1.2">𝐿</ci><apply id="S8.p2.2.m2.1.1.3.cmml" xref="S8.p2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S8.p2.2.m2.1.1.3.1.cmml" xref="S8.p2.2.m2.1.1.3">superscript</csymbol><ci id="S8.p2.2.m2.1.1.3.2.cmml" xref="S8.p2.2.m2.1.1.3.2">𝐿</ci><ci id="S8.p2.2.m2.1.1.3.3.cmml" xref="S8.p2.2.m2.1.1.3.3">⋆</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.p2.2.m2.1c">L\cong L^{\star}</annotation><annotation encoding="application/x-llamapun" id="S8.p2.2.m2.1d">italic_L ≅ italic_L start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math>, otherwise it is called irreversible. In this spirit we make the following definition.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="S8.Thmtheorem1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S8.Thmtheorem1.1.1.1">Definition 8.1</span></span><span class="ltx_text ltx_font_bold" id="S8.Thmtheorem1.2.2">.</span> </h6> <div class="ltx_para" id="S8.Thmtheorem1.p1"> <p class="ltx_p" id="S8.Thmtheorem1.p1.6">Let <math alttext="L" class="ltx_Math" display="inline" id="S8.Thmtheorem1.p1.1.m1.1"><semantics id="S8.Thmtheorem1.p1.1.m1.1a"><mi id="S8.Thmtheorem1.p1.1.m1.1.1" xref="S8.Thmtheorem1.p1.1.m1.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S8.Thmtheorem1.p1.1.m1.1b"><ci id="S8.Thmtheorem1.p1.1.m1.1.1.cmml" xref="S8.Thmtheorem1.p1.1.m1.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.Thmtheorem1.p1.1.m1.1c">L</annotation><annotation encoding="application/x-llamapun" id="S8.Thmtheorem1.p1.1.m1.1d">italic_L</annotation></semantics></math> be a linear order. For a cardinal <math alttext="\kappa" class="ltx_Math" display="inline" id="S8.Thmtheorem1.p1.2.m2.1"><semantics id="S8.Thmtheorem1.p1.2.m2.1a"><mi id="S8.Thmtheorem1.p1.2.m2.1.1" xref="S8.Thmtheorem1.p1.2.m2.1.1.cmml">κ</mi><annotation-xml encoding="MathML-Content" id="S8.Thmtheorem1.p1.2.m2.1b"><ci id="S8.Thmtheorem1.p1.2.m2.1.1.cmml" xref="S8.Thmtheorem1.p1.2.m2.1.1">𝜅</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.Thmtheorem1.p1.2.m2.1c">\kappa</annotation><annotation encoding="application/x-llamapun" id="S8.Thmtheorem1.p1.2.m2.1d">italic_κ</annotation></semantics></math> we say that <math alttext="L" class="ltx_Math" display="inline" id="S8.Thmtheorem1.p1.3.m3.1"><semantics id="S8.Thmtheorem1.p1.3.m3.1a"><mi id="S8.Thmtheorem1.p1.3.m3.1.1" xref="S8.Thmtheorem1.p1.3.m3.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S8.Thmtheorem1.p1.3.m3.1b"><ci id="S8.Thmtheorem1.p1.3.m3.1.1.cmml" xref="S8.Thmtheorem1.p1.3.m3.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.Thmtheorem1.p1.3.m3.1c">L</annotation><annotation encoding="application/x-llamapun" id="S8.Thmtheorem1.p1.3.m3.1d">italic_L</annotation></semantics></math> is <em class="ltx_emph ltx_font_italic" id="S8.Thmtheorem1.p1.4.1"><math alttext="\kappa" class="ltx_Math" display="inline" id="S8.Thmtheorem1.p1.4.1.m1.1"><semantics id="S8.Thmtheorem1.p1.4.1.m1.1a"><mi id="S8.Thmtheorem1.p1.4.1.m1.1.1" xref="S8.Thmtheorem1.p1.4.1.m1.1.1.cmml">κ</mi><annotation-xml encoding="MathML-Content" id="S8.Thmtheorem1.p1.4.1.m1.1b"><ci id="S8.Thmtheorem1.p1.4.1.m1.1.1.cmml" xref="S8.Thmtheorem1.p1.4.1.m1.1.1">𝜅</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.Thmtheorem1.p1.4.1.m1.1c">\kappa</annotation><annotation encoding="application/x-llamapun" id="S8.Thmtheorem1.p1.4.1.m1.1d">italic_κ</annotation></semantics></math>-irreversible</em> if <math alttext="L" class="ltx_Math" display="inline" id="S8.Thmtheorem1.p1.5.m4.1"><semantics id="S8.Thmtheorem1.p1.5.m4.1a"><mi id="S8.Thmtheorem1.p1.5.m4.1.1" xref="S8.Thmtheorem1.p1.5.m4.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S8.Thmtheorem1.p1.5.m4.1b"><ci id="S8.Thmtheorem1.p1.5.m4.1.1.cmml" xref="S8.Thmtheorem1.p1.5.m4.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.Thmtheorem1.p1.5.m4.1c">L</annotation><annotation encoding="application/x-llamapun" id="S8.Thmtheorem1.p1.5.m4.1d">italic_L</annotation></semantics></math> contains no two uncountable reverse isomorphic suborders of cardinality <math alttext="\kappa" class="ltx_Math" display="inline" id="S8.Thmtheorem1.p1.6.m5.1"><semantics id="S8.Thmtheorem1.p1.6.m5.1a"><mi id="S8.Thmtheorem1.p1.6.m5.1.1" xref="S8.Thmtheorem1.p1.6.m5.1.1.cmml">κ</mi><annotation-xml encoding="MathML-Content" id="S8.Thmtheorem1.p1.6.m5.1b"><ci id="S8.Thmtheorem1.p1.6.m5.1.1.cmml" xref="S8.Thmtheorem1.p1.6.m5.1.1">𝜅</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.Thmtheorem1.p1.6.m5.1c">\kappa</annotation><annotation encoding="application/x-llamapun" id="S8.Thmtheorem1.p1.6.m5.1d">italic_κ</annotation></semantics></math>.</p> </div> </div> <div class="ltx_para" id="S8.p3"> <p class="ltx_p" id="S8.p3.5">Observe that the <math alttext="\omega" class="ltx_Math" display="inline" id="S8.p3.1.m1.1"><semantics id="S8.p3.1.m1.1a"><mi id="S8.p3.1.m1.1.1" xref="S8.p3.1.m1.1.1.cmml">ω</mi><annotation-xml encoding="MathML-Content" id="S8.p3.1.m1.1b"><ci id="S8.p3.1.m1.1.1.cmml" xref="S8.p3.1.m1.1.1">𝜔</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.p3.1.m1.1c">\omega</annotation><annotation encoding="application/x-llamapun" id="S8.p3.1.m1.1d">italic_ω</annotation></semantics></math>-irreversible linear orders are exactly the infinite ordinals and their reverses. Easy examples of <math alttext="\omega_{1}" class="ltx_Math" display="inline" id="S8.p3.2.m2.1"><semantics id="S8.p3.2.m2.1a"><msub id="S8.p3.2.m2.1.1" xref="S8.p3.2.m2.1.1.cmml"><mi id="S8.p3.2.m2.1.1.2" xref="S8.p3.2.m2.1.1.2.cmml">ω</mi><mn id="S8.p3.2.m2.1.1.3" xref="S8.p3.2.m2.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S8.p3.2.m2.1b"><apply id="S8.p3.2.m2.1.1.cmml" xref="S8.p3.2.m2.1.1"><csymbol cd="ambiguous" id="S8.p3.2.m2.1.1.1.cmml" xref="S8.p3.2.m2.1.1">subscript</csymbol><ci id="S8.p3.2.m2.1.1.2.cmml" xref="S8.p3.2.m2.1.1.2">𝜔</ci><cn id="S8.p3.2.m2.1.1.3.cmml" type="integer" xref="S8.p3.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.p3.2.m2.1c">\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S8.p3.2.m2.1d">italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-irreversible uncountable orders are <math alttext="\omega_{1}" class="ltx_Math" display="inline" id="S8.p3.3.m3.1"><semantics id="S8.p3.3.m3.1a"><msub id="S8.p3.3.m3.1.1" xref="S8.p3.3.m3.1.1.cmml"><mi id="S8.p3.3.m3.1.1.2" xref="S8.p3.3.m3.1.1.2.cmml">ω</mi><mn id="S8.p3.3.m3.1.1.3" xref="S8.p3.3.m3.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S8.p3.3.m3.1b"><apply id="S8.p3.3.m3.1.1.cmml" xref="S8.p3.3.m3.1.1"><csymbol cd="ambiguous" id="S8.p3.3.m3.1.1.1.cmml" xref="S8.p3.3.m3.1.1">subscript</csymbol><ci id="S8.p3.3.m3.1.1.2.cmml" xref="S8.p3.3.m3.1.1.2">𝜔</ci><cn id="S8.p3.3.m3.1.1.3.cmml" type="integer" xref="S8.p3.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.p3.3.m3.1c">\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S8.p3.3.m3.1d">italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\omega_{1}^{\star}" class="ltx_Math" display="inline" id="S8.p3.4.m4.1"><semantics id="S8.p3.4.m4.1a"><msubsup id="S8.p3.4.m4.1.1" xref="S8.p3.4.m4.1.1.cmml"><mi id="S8.p3.4.m4.1.1.2.2" xref="S8.p3.4.m4.1.1.2.2.cmml">ω</mi><mn id="S8.p3.4.m4.1.1.2.3" xref="S8.p3.4.m4.1.1.2.3.cmml">1</mn><mo id="S8.p3.4.m4.1.1.3" xref="S8.p3.4.m4.1.1.3.cmml">⋆</mo></msubsup><annotation-xml encoding="MathML-Content" id="S8.p3.4.m4.1b"><apply id="S8.p3.4.m4.1.1.cmml" xref="S8.p3.4.m4.1.1"><csymbol cd="ambiguous" id="S8.p3.4.m4.1.1.1.cmml" xref="S8.p3.4.m4.1.1">superscript</csymbol><apply id="S8.p3.4.m4.1.1.2.cmml" xref="S8.p3.4.m4.1.1"><csymbol cd="ambiguous" id="S8.p3.4.m4.1.1.2.1.cmml" xref="S8.p3.4.m4.1.1">subscript</csymbol><ci id="S8.p3.4.m4.1.1.2.2.cmml" xref="S8.p3.4.m4.1.1.2.2">𝜔</ci><cn id="S8.p3.4.m4.1.1.2.3.cmml" type="integer" xref="S8.p3.4.m4.1.1.2.3">1</cn></apply><ci id="S8.p3.4.m4.1.1.3.cmml" xref="S8.p3.4.m4.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.p3.4.m4.1c">\omega_{1}^{\star}</annotation><annotation encoding="application/x-llamapun" id="S8.p3.4.m4.1d">italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math> and Countryman lines. As mentioned, we are interested in the following question. Is there a non Countryman <math alttext="\omega_{1}" class="ltx_Math" display="inline" id="S8.p3.5.m5.1"><semantics id="S8.p3.5.m5.1a"><msub id="S8.p3.5.m5.1.1" xref="S8.p3.5.m5.1.1.cmml"><mi id="S8.p3.5.m5.1.1.2" xref="S8.p3.5.m5.1.1.2.cmml">ω</mi><mn id="S8.p3.5.m5.1.1.3" xref="S8.p3.5.m5.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S8.p3.5.m5.1b"><apply id="S8.p3.5.m5.1.1.cmml" xref="S8.p3.5.m5.1.1"><csymbol cd="ambiguous" id="S8.p3.5.m5.1.1.1.cmml" xref="S8.p3.5.m5.1.1">subscript</csymbol><ci id="S8.p3.5.m5.1.1.2.cmml" xref="S8.p3.5.m5.1.1.2">𝜔</ci><cn id="S8.p3.5.m5.1.1.3.cmml" type="integer" xref="S8.p3.5.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.p3.5.m5.1c">\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S8.p3.5.m5.1d">italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-irreversible Aronszajn line? We have not seen an example of such a line in the literature.</p> </div> <div class="ltx_para" id="S8.p4"> <p class="ltx_p" id="S8.p4.3">Moore <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib16" title="">16</a>]</cite> proved that under <math alttext="\mathsf{PFA}" class="ltx_Math" display="inline" id="S8.p4.1.m1.1"><semantics id="S8.p4.1.m1.1a"><mi id="S8.p4.1.m1.1.1" xref="S8.p4.1.m1.1.1.cmml">𝖯𝖥𝖠</mi><annotation-xml encoding="MathML-Content" id="S8.p4.1.m1.1b"><ci id="S8.p4.1.m1.1.1.cmml" xref="S8.p4.1.m1.1.1">𝖯𝖥𝖠</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.p4.1.m1.1c">\mathsf{PFA}</annotation><annotation encoding="application/x-llamapun" id="S8.p4.1.m1.1d">sansserif_PFA</annotation></semantics></math>, every non Countryman Aronszajn line contains both orientations of any given Countryman line. Thus under <math alttext="\mathsf{PFA}" class="ltx_Math" display="inline" id="S8.p4.2.m2.1"><semantics id="S8.p4.2.m2.1a"><mi id="S8.p4.2.m2.1.1" xref="S8.p4.2.m2.1.1.cmml">𝖯𝖥𝖠</mi><annotation-xml encoding="MathML-Content" id="S8.p4.2.m2.1b"><ci id="S8.p4.2.m2.1.1.cmml" xref="S8.p4.2.m2.1.1">𝖯𝖥𝖠</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.p4.2.m2.1c">\mathsf{PFA}</annotation><annotation encoding="application/x-llamapun" id="S8.p4.2.m2.1d">sansserif_PFA</annotation></semantics></math> there is no such example. It is well known that under <math alttext="\mathsf{PFA}" class="ltx_Math" display="inline" id="S8.p4.3.m3.1"><semantics id="S8.p4.3.m3.1a"><mi id="S8.p4.3.m3.1.1" xref="S8.p4.3.m3.1.1.cmml">𝖯𝖥𝖠</mi><annotation-xml encoding="MathML-Content" id="S8.p4.3.m3.1b"><ci id="S8.p4.3.m3.1.1.cmml" xref="S8.p4.3.m3.1.1">𝖯𝖥𝖠</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.p4.3.m3.1c">\mathsf{PFA}</annotation><annotation encoding="application/x-llamapun" id="S8.p4.3.m3.1d">sansserif_PFA</annotation></semantics></math> there are no Suslin trees, and that any lexicographic ordering of a Suslin tree is a non Countryman Aronszajn line. Thus it is natural to look for an example that is the lexicographic ordering of a Suslin tree.</p> </div> <div class="ltx_para" id="S8.p5"> <p class="ltx_p" id="S8.p5.5">Let <math alttext="\mathbb{P}" class="ltx_Math" display="inline" id="S8.p5.1.m1.1"><semantics id="S8.p5.1.m1.1a"><mi id="S8.p5.1.m1.1.1" xref="S8.p5.1.m1.1.1.cmml">ℙ</mi><annotation-xml encoding="MathML-Content" id="S8.p5.1.m1.1b"><ci id="S8.p5.1.m1.1.1.cmml" xref="S8.p5.1.m1.1.1">ℙ</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.p5.1.m1.1c">\mathbb{P}</annotation><annotation encoding="application/x-llamapun" id="S8.p5.1.m1.1d">blackboard_P</annotation></semantics></math> be the forcing for adding a single Cohen real. Here we take it as the finite partial functions from <math alttext="\omega" class="ltx_Math" display="inline" id="S8.p5.2.m2.1"><semantics id="S8.p5.2.m2.1a"><mi id="S8.p5.2.m2.1.1" xref="S8.p5.2.m2.1.1.cmml">ω</mi><annotation-xml encoding="MathML-Content" id="S8.p5.2.m2.1b"><ci id="S8.p5.2.m2.1.1.cmml" xref="S8.p5.2.m2.1.1">𝜔</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.p5.2.m2.1c">\omega</annotation><annotation encoding="application/x-llamapun" id="S8.p5.2.m2.1d">italic_ω</annotation></semantics></math> to itself, ordered by reverse inclusion. It is well known that if <math alttext="G" class="ltx_Math" display="inline" id="S8.p5.3.m3.1"><semantics id="S8.p5.3.m3.1a"><mi id="S8.p5.3.m3.1.1" xref="S8.p5.3.m3.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S8.p5.3.m3.1b"><ci id="S8.p5.3.m3.1.1.cmml" xref="S8.p5.3.m3.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.p5.3.m3.1c">G</annotation><annotation encoding="application/x-llamapun" id="S8.p5.3.m3.1d">italic_G</annotation></semantics></math> is a generic filter for <math alttext="\mathbb{P}" class="ltx_Math" display="inline" id="S8.p5.4.m4.1"><semantics id="S8.p5.4.m4.1a"><mi id="S8.p5.4.m4.1.1" xref="S8.p5.4.m4.1.1.cmml">ℙ</mi><annotation-xml encoding="MathML-Content" id="S8.p5.4.m4.1b"><ci id="S8.p5.4.m4.1.1.cmml" xref="S8.p5.4.m4.1.1">ℙ</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.p5.4.m4.1c">\mathbb{P}</annotation><annotation encoding="application/x-llamapun" id="S8.p5.4.m4.1d">blackboard_P</annotation></semantics></math>, then in <math alttext="V[G]" class="ltx_Math" display="inline" id="S8.p5.5.m5.1"><semantics id="S8.p5.5.m5.1a"><mrow id="S8.p5.5.m5.1.2" xref="S8.p5.5.m5.1.2.cmml"><mi id="S8.p5.5.m5.1.2.2" xref="S8.p5.5.m5.1.2.2.cmml">V</mi><mo id="S8.p5.5.m5.1.2.1" xref="S8.p5.5.m5.1.2.1.cmml">⁢</mo><mrow id="S8.p5.5.m5.1.2.3.2" xref="S8.p5.5.m5.1.2.3.1.cmml"><mo id="S8.p5.5.m5.1.2.3.2.1" stretchy="false" xref="S8.p5.5.m5.1.2.3.1.1.cmml">[</mo><mi id="S8.p5.5.m5.1.1" xref="S8.p5.5.m5.1.1.cmml">G</mi><mo id="S8.p5.5.m5.1.2.3.2.2" stretchy="false" xref="S8.p5.5.m5.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S8.p5.5.m5.1b"><apply id="S8.p5.5.m5.1.2.cmml" xref="S8.p5.5.m5.1.2"><times id="S8.p5.5.m5.1.2.1.cmml" xref="S8.p5.5.m5.1.2.1"></times><ci id="S8.p5.5.m5.1.2.2.cmml" xref="S8.p5.5.m5.1.2.2">𝑉</ci><apply id="S8.p5.5.m5.1.2.3.1.cmml" xref="S8.p5.5.m5.1.2.3.2"><csymbol cd="latexml" id="S8.p5.5.m5.1.2.3.1.1.cmml" xref="S8.p5.5.m5.1.2.3.2.1">delimited-[]</csymbol><ci id="S8.p5.5.m5.1.1.cmml" xref="S8.p5.5.m5.1.1">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.p5.5.m5.1c">V[G]</annotation><annotation encoding="application/x-llamapun" id="S8.p5.5.m5.1d">italic_V [ italic_G ]</annotation></semantics></math> there is a Suslin tree. The original proof is due to Shelah (see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib18" title="">18</a>]</cite>). We will prove that in this same model there is an example of an Aronszajn line with the desired properties.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S8.Thmtheorem2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S8.Thmtheorem2.1.1.1">Theorem 8.2</span></span><span class="ltx_text ltx_font_bold" id="S8.Thmtheorem2.2.2">.</span> </h6> <div class="ltx_para" id="S8.Thmtheorem2.p1"> <p class="ltx_p" id="S8.Thmtheorem2.p1.3"><span class="ltx_text ltx_font_italic" id="S8.Thmtheorem2.p1.3.3">If <math alttext="\mathbb{P}" class="ltx_Math" display="inline" id="S8.Thmtheorem2.p1.1.1.m1.1"><semantics id="S8.Thmtheorem2.p1.1.1.m1.1a"><mi id="S8.Thmtheorem2.p1.1.1.m1.1.1" xref="S8.Thmtheorem2.p1.1.1.m1.1.1.cmml">ℙ</mi><annotation-xml encoding="MathML-Content" id="S8.Thmtheorem2.p1.1.1.m1.1b"><ci id="S8.Thmtheorem2.p1.1.1.m1.1.1.cmml" xref="S8.Thmtheorem2.p1.1.1.m1.1.1">ℙ</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.Thmtheorem2.p1.1.1.m1.1c">\mathbb{P}</annotation><annotation encoding="application/x-llamapun" id="S8.Thmtheorem2.p1.1.1.m1.1d">blackboard_P</annotation></semantics></math> is the forcing for adding a single Cohen real, then in <math alttext="V[G]" class="ltx_Math" display="inline" id="S8.Thmtheorem2.p1.2.2.m2.1"><semantics id="S8.Thmtheorem2.p1.2.2.m2.1a"><mrow id="S8.Thmtheorem2.p1.2.2.m2.1.2" xref="S8.Thmtheorem2.p1.2.2.m2.1.2.cmml"><mi id="S8.Thmtheorem2.p1.2.2.m2.1.2.2" xref="S8.Thmtheorem2.p1.2.2.m2.1.2.2.cmml">V</mi><mo id="S8.Thmtheorem2.p1.2.2.m2.1.2.1" xref="S8.Thmtheorem2.p1.2.2.m2.1.2.1.cmml">⁢</mo><mrow id="S8.Thmtheorem2.p1.2.2.m2.1.2.3.2" xref="S8.Thmtheorem2.p1.2.2.m2.1.2.3.1.cmml"><mo id="S8.Thmtheorem2.p1.2.2.m2.1.2.3.2.1" stretchy="false" xref="S8.Thmtheorem2.p1.2.2.m2.1.2.3.1.1.cmml">[</mo><mi id="S8.Thmtheorem2.p1.2.2.m2.1.1" xref="S8.Thmtheorem2.p1.2.2.m2.1.1.cmml">G</mi><mo id="S8.Thmtheorem2.p1.2.2.m2.1.2.3.2.2" stretchy="false" xref="S8.Thmtheorem2.p1.2.2.m2.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S8.Thmtheorem2.p1.2.2.m2.1b"><apply id="S8.Thmtheorem2.p1.2.2.m2.1.2.cmml" xref="S8.Thmtheorem2.p1.2.2.m2.1.2"><times id="S8.Thmtheorem2.p1.2.2.m2.1.2.1.cmml" xref="S8.Thmtheorem2.p1.2.2.m2.1.2.1"></times><ci id="S8.Thmtheorem2.p1.2.2.m2.1.2.2.cmml" xref="S8.Thmtheorem2.p1.2.2.m2.1.2.2">𝑉</ci><apply id="S8.Thmtheorem2.p1.2.2.m2.1.2.3.1.cmml" xref="S8.Thmtheorem2.p1.2.2.m2.1.2.3.2"><csymbol cd="latexml" id="S8.Thmtheorem2.p1.2.2.m2.1.2.3.1.1.cmml" xref="S8.Thmtheorem2.p1.2.2.m2.1.2.3.2.1">delimited-[]</csymbol><ci id="S8.Thmtheorem2.p1.2.2.m2.1.1.cmml" xref="S8.Thmtheorem2.p1.2.2.m2.1.1">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.Thmtheorem2.p1.2.2.m2.1c">V[G]</annotation><annotation encoding="application/x-llamapun" id="S8.Thmtheorem2.p1.2.2.m2.1d">italic_V [ italic_G ]</annotation></semantics></math> there is a <math alttext="\omega_{1}" class="ltx_Math" display="inline" id="S8.Thmtheorem2.p1.3.3.m3.1"><semantics id="S8.Thmtheorem2.p1.3.3.m3.1a"><msub id="S8.Thmtheorem2.p1.3.3.m3.1.1" xref="S8.Thmtheorem2.p1.3.3.m3.1.1.cmml"><mi id="S8.Thmtheorem2.p1.3.3.m3.1.1.2" xref="S8.Thmtheorem2.p1.3.3.m3.1.1.2.cmml">ω</mi><mn id="S8.Thmtheorem2.p1.3.3.m3.1.1.3" xref="S8.Thmtheorem2.p1.3.3.m3.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S8.Thmtheorem2.p1.3.3.m3.1b"><apply id="S8.Thmtheorem2.p1.3.3.m3.1.1.cmml" xref="S8.Thmtheorem2.p1.3.3.m3.1.1"><csymbol cd="ambiguous" id="S8.Thmtheorem2.p1.3.3.m3.1.1.1.cmml" xref="S8.Thmtheorem2.p1.3.3.m3.1.1">subscript</csymbol><ci id="S8.Thmtheorem2.p1.3.3.m3.1.1.2.cmml" xref="S8.Thmtheorem2.p1.3.3.m3.1.1.2">𝜔</ci><cn id="S8.Thmtheorem2.p1.3.3.m3.1.1.3.cmml" type="integer" xref="S8.Thmtheorem2.p1.3.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.Thmtheorem2.p1.3.3.m3.1c">\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S8.Thmtheorem2.p1.3.3.m3.1d">italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-irreversible lexicographically ordered Suslin tree.</span></p> </div> </div> <div class="ltx_para" id="S8.p6"> <p class="ltx_p" id="S8.p6.1">Our proof will be a modest modification of Todorcevic’s proof of Shelah’s theorem (see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib22" title="">22</a>]</cite>). For details see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib7" title="">7</a>, Proposition 20.7]</cite>.</p> </div> <div class="ltx_para" id="S8.p7"> <p class="ltx_p" id="S8.p7.2">Fix any uncountable and coherent <math alttext="T\subseteq\omega^{<\omega_{1}}" class="ltx_Math" display="inline" id="S8.p7.1.m1.1"><semantics id="S8.p7.1.m1.1a"><mrow id="S8.p7.1.m1.1.1" xref="S8.p7.1.m1.1.1.cmml"><mi id="S8.p7.1.m1.1.1.2" xref="S8.p7.1.m1.1.1.2.cmml">T</mi><mo id="S8.p7.1.m1.1.1.1" xref="S8.p7.1.m1.1.1.1.cmml">⊆</mo><msup id="S8.p7.1.m1.1.1.3" xref="S8.p7.1.m1.1.1.3.cmml"><mi id="S8.p7.1.m1.1.1.3.2" xref="S8.p7.1.m1.1.1.3.2.cmml">ω</mi><mrow id="S8.p7.1.m1.1.1.3.3" xref="S8.p7.1.m1.1.1.3.3.cmml"><mi id="S8.p7.1.m1.1.1.3.3.2" xref="S8.p7.1.m1.1.1.3.3.2.cmml"></mi><mo id="S8.p7.1.m1.1.1.3.3.1" xref="S8.p7.1.m1.1.1.3.3.1.cmml"><</mo><msub id="S8.p7.1.m1.1.1.3.3.3" xref="S8.p7.1.m1.1.1.3.3.3.cmml"><mi id="S8.p7.1.m1.1.1.3.3.3.2" xref="S8.p7.1.m1.1.1.3.3.3.2.cmml">ω</mi><mn id="S8.p7.1.m1.1.1.3.3.3.3" xref="S8.p7.1.m1.1.1.3.3.3.3.cmml">1</mn></msub></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S8.p7.1.m1.1b"><apply id="S8.p7.1.m1.1.1.cmml" xref="S8.p7.1.m1.1.1"><subset id="S8.p7.1.m1.1.1.1.cmml" xref="S8.p7.1.m1.1.1.1"></subset><ci id="S8.p7.1.m1.1.1.2.cmml" xref="S8.p7.1.m1.1.1.2">𝑇</ci><apply id="S8.p7.1.m1.1.1.3.cmml" xref="S8.p7.1.m1.1.1.3"><csymbol cd="ambiguous" id="S8.p7.1.m1.1.1.3.1.cmml" xref="S8.p7.1.m1.1.1.3">superscript</csymbol><ci id="S8.p7.1.m1.1.1.3.2.cmml" xref="S8.p7.1.m1.1.1.3.2">𝜔</ci><apply id="S8.p7.1.m1.1.1.3.3.cmml" xref="S8.p7.1.m1.1.1.3.3"><lt id="S8.p7.1.m1.1.1.3.3.1.cmml" xref="S8.p7.1.m1.1.1.3.3.1"></lt><csymbol cd="latexml" id="S8.p7.1.m1.1.1.3.3.2.cmml" xref="S8.p7.1.m1.1.1.3.3.2">absent</csymbol><apply id="S8.p7.1.m1.1.1.3.3.3.cmml" xref="S8.p7.1.m1.1.1.3.3.3"><csymbol cd="ambiguous" id="S8.p7.1.m1.1.1.3.3.3.1.cmml" xref="S8.p7.1.m1.1.1.3.3.3">subscript</csymbol><ci id="S8.p7.1.m1.1.1.3.3.3.2.cmml" xref="S8.p7.1.m1.1.1.3.3.3.2">𝜔</ci><cn id="S8.p7.1.m1.1.1.3.3.3.3.cmml" type="integer" xref="S8.p7.1.m1.1.1.3.3.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.p7.1.m1.1c">T\subseteq\omega^{<\omega_{1}}</annotation><annotation encoding="application/x-llamapun" id="S8.p7.1.m1.1d">italic_T ⊆ italic_ω start_POSTSUPERSCRIPT < italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> of finite-to-one functions such as the one used in the proof of <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S4.Thmtheorem5" title="Theorem 4.5. ‣ 4. Aronszajn line decompositions ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">4.5</span></a>. As mentioned in <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S4.Thmtheorem3" title="Lemma 4.3. ‣ 4. Aronszajn line decompositions ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">4.3</span></a>, the lexicographic ordering of <math alttext="T" class="ltx_Math" display="inline" id="S8.p7.2.m2.1"><semantics id="S8.p7.2.m2.1a"><mi id="S8.p7.2.m2.1.1" xref="S8.p7.2.m2.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S8.p7.2.m2.1b"><ci id="S8.p7.2.m2.1.1.cmml" xref="S8.p7.2.m2.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.p7.2.m2.1c">T</annotation><annotation encoding="application/x-llamapun" id="S8.p7.2.m2.1d">italic_T</annotation></semantics></math> is a Countryman line. We also need the following lemma, for a proof see the claim of the mentioned proposition in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib7" title="">7</a>]</cite>.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S8.Thmtheorem3"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S8.Thmtheorem3.1.1.1">Lemma 8.3</span></span><span class="ltx_text ltx_font_bold" id="S8.Thmtheorem3.2.2">.</span> </h6> <div class="ltx_para" id="S8.Thmtheorem3.p1"> <p class="ltx_p" id="S8.Thmtheorem3.p1.8">Let <math alttext="K\subseteq T" class="ltx_Math" display="inline" id="S8.Thmtheorem3.p1.1.m1.1"><semantics id="S8.Thmtheorem3.p1.1.m1.1a"><mrow id="S8.Thmtheorem3.p1.1.m1.1.1" xref="S8.Thmtheorem3.p1.1.m1.1.1.cmml"><mi id="S8.Thmtheorem3.p1.1.m1.1.1.2" xref="S8.Thmtheorem3.p1.1.m1.1.1.2.cmml">K</mi><mo id="S8.Thmtheorem3.p1.1.m1.1.1.1" xref="S8.Thmtheorem3.p1.1.m1.1.1.1.cmml">⊆</mo><mi id="S8.Thmtheorem3.p1.1.m1.1.1.3" xref="S8.Thmtheorem3.p1.1.m1.1.1.3.cmml">T</mi></mrow><annotation-xml encoding="MathML-Content" id="S8.Thmtheorem3.p1.1.m1.1b"><apply id="S8.Thmtheorem3.p1.1.m1.1.1.cmml" xref="S8.Thmtheorem3.p1.1.m1.1.1"><subset id="S8.Thmtheorem3.p1.1.m1.1.1.1.cmml" xref="S8.Thmtheorem3.p1.1.m1.1.1.1"></subset><ci id="S8.Thmtheorem3.p1.1.m1.1.1.2.cmml" xref="S8.Thmtheorem3.p1.1.m1.1.1.2">𝐾</ci><ci id="S8.Thmtheorem3.p1.1.m1.1.1.3.cmml" xref="S8.Thmtheorem3.p1.1.m1.1.1.3">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.Thmtheorem3.p1.1.m1.1c">K\subseteq T</annotation><annotation encoding="application/x-llamapun" id="S8.Thmtheorem3.p1.1.m1.1d">italic_K ⊆ italic_T</annotation></semantics></math> be uncountable, and let <math alttext="n\in\omega" class="ltx_Math" display="inline" id="S8.Thmtheorem3.p1.2.m2.1"><semantics id="S8.Thmtheorem3.p1.2.m2.1a"><mrow id="S8.Thmtheorem3.p1.2.m2.1.1" xref="S8.Thmtheorem3.p1.2.m2.1.1.cmml"><mi id="S8.Thmtheorem3.p1.2.m2.1.1.2" xref="S8.Thmtheorem3.p1.2.m2.1.1.2.cmml">n</mi><mo id="S8.Thmtheorem3.p1.2.m2.1.1.1" xref="S8.Thmtheorem3.p1.2.m2.1.1.1.cmml">∈</mo><mi id="S8.Thmtheorem3.p1.2.m2.1.1.3" xref="S8.Thmtheorem3.p1.2.m2.1.1.3.cmml">ω</mi></mrow><annotation-xml encoding="MathML-Content" id="S8.Thmtheorem3.p1.2.m2.1b"><apply id="S8.Thmtheorem3.p1.2.m2.1.1.cmml" xref="S8.Thmtheorem3.p1.2.m2.1.1"><in id="S8.Thmtheorem3.p1.2.m2.1.1.1.cmml" xref="S8.Thmtheorem3.p1.2.m2.1.1.1"></in><ci id="S8.Thmtheorem3.p1.2.m2.1.1.2.cmml" xref="S8.Thmtheorem3.p1.2.m2.1.1.2">𝑛</ci><ci id="S8.Thmtheorem3.p1.2.m2.1.1.3.cmml" xref="S8.Thmtheorem3.p1.2.m2.1.1.3">𝜔</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.Thmtheorem3.p1.2.m2.1c">n\in\omega</annotation><annotation encoding="application/x-llamapun" id="S8.Thmtheorem3.p1.2.m2.1d">italic_n ∈ italic_ω</annotation></semantics></math>. Then there is an uncountable <math alttext="K^{\prime}\subseteq K" class="ltx_Math" display="inline" id="S8.Thmtheorem3.p1.3.m3.1"><semantics id="S8.Thmtheorem3.p1.3.m3.1a"><mrow id="S8.Thmtheorem3.p1.3.m3.1.1" xref="S8.Thmtheorem3.p1.3.m3.1.1.cmml"><msup id="S8.Thmtheorem3.p1.3.m3.1.1.2" xref="S8.Thmtheorem3.p1.3.m3.1.1.2.cmml"><mi id="S8.Thmtheorem3.p1.3.m3.1.1.2.2" xref="S8.Thmtheorem3.p1.3.m3.1.1.2.2.cmml">K</mi><mo id="S8.Thmtheorem3.p1.3.m3.1.1.2.3" xref="S8.Thmtheorem3.p1.3.m3.1.1.2.3.cmml">′</mo></msup><mo id="S8.Thmtheorem3.p1.3.m3.1.1.1" xref="S8.Thmtheorem3.p1.3.m3.1.1.1.cmml">⊆</mo><mi id="S8.Thmtheorem3.p1.3.m3.1.1.3" xref="S8.Thmtheorem3.p1.3.m3.1.1.3.cmml">K</mi></mrow><annotation-xml encoding="MathML-Content" id="S8.Thmtheorem3.p1.3.m3.1b"><apply id="S8.Thmtheorem3.p1.3.m3.1.1.cmml" xref="S8.Thmtheorem3.p1.3.m3.1.1"><subset id="S8.Thmtheorem3.p1.3.m3.1.1.1.cmml" xref="S8.Thmtheorem3.p1.3.m3.1.1.1"></subset><apply id="S8.Thmtheorem3.p1.3.m3.1.1.2.cmml" xref="S8.Thmtheorem3.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="S8.Thmtheorem3.p1.3.m3.1.1.2.1.cmml" xref="S8.Thmtheorem3.p1.3.m3.1.1.2">superscript</csymbol><ci id="S8.Thmtheorem3.p1.3.m3.1.1.2.2.cmml" xref="S8.Thmtheorem3.p1.3.m3.1.1.2.2">𝐾</ci><ci id="S8.Thmtheorem3.p1.3.m3.1.1.2.3.cmml" xref="S8.Thmtheorem3.p1.3.m3.1.1.2.3">′</ci></apply><ci id="S8.Thmtheorem3.p1.3.m3.1.1.3.cmml" xref="S8.Thmtheorem3.p1.3.m3.1.1.3">𝐾</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.Thmtheorem3.p1.3.m3.1c">K^{\prime}\subseteq K</annotation><annotation encoding="application/x-llamapun" id="S8.Thmtheorem3.p1.3.m3.1d">italic_K start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⊆ italic_K</annotation></semantics></math> such that for all <math alttext="s,t\in K^{\prime}" class="ltx_Math" display="inline" id="S8.Thmtheorem3.p1.4.m4.2"><semantics id="S8.Thmtheorem3.p1.4.m4.2a"><mrow id="S8.Thmtheorem3.p1.4.m4.2.3" xref="S8.Thmtheorem3.p1.4.m4.2.3.cmml"><mrow id="S8.Thmtheorem3.p1.4.m4.2.3.2.2" xref="S8.Thmtheorem3.p1.4.m4.2.3.2.1.cmml"><mi id="S8.Thmtheorem3.p1.4.m4.1.1" xref="S8.Thmtheorem3.p1.4.m4.1.1.cmml">s</mi><mo id="S8.Thmtheorem3.p1.4.m4.2.3.2.2.1" xref="S8.Thmtheorem3.p1.4.m4.2.3.2.1.cmml">,</mo><mi id="S8.Thmtheorem3.p1.4.m4.2.2" xref="S8.Thmtheorem3.p1.4.m4.2.2.cmml">t</mi></mrow><mo id="S8.Thmtheorem3.p1.4.m4.2.3.1" xref="S8.Thmtheorem3.p1.4.m4.2.3.1.cmml">∈</mo><msup id="S8.Thmtheorem3.p1.4.m4.2.3.3" xref="S8.Thmtheorem3.p1.4.m4.2.3.3.cmml"><mi id="S8.Thmtheorem3.p1.4.m4.2.3.3.2" xref="S8.Thmtheorem3.p1.4.m4.2.3.3.2.cmml">K</mi><mo id="S8.Thmtheorem3.p1.4.m4.2.3.3.3" xref="S8.Thmtheorem3.p1.4.m4.2.3.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S8.Thmtheorem3.p1.4.m4.2b"><apply id="S8.Thmtheorem3.p1.4.m4.2.3.cmml" xref="S8.Thmtheorem3.p1.4.m4.2.3"><in id="S8.Thmtheorem3.p1.4.m4.2.3.1.cmml" xref="S8.Thmtheorem3.p1.4.m4.2.3.1"></in><list id="S8.Thmtheorem3.p1.4.m4.2.3.2.1.cmml" xref="S8.Thmtheorem3.p1.4.m4.2.3.2.2"><ci id="S8.Thmtheorem3.p1.4.m4.1.1.cmml" xref="S8.Thmtheorem3.p1.4.m4.1.1">𝑠</ci><ci id="S8.Thmtheorem3.p1.4.m4.2.2.cmml" xref="S8.Thmtheorem3.p1.4.m4.2.2">𝑡</ci></list><apply id="S8.Thmtheorem3.p1.4.m4.2.3.3.cmml" xref="S8.Thmtheorem3.p1.4.m4.2.3.3"><csymbol cd="ambiguous" id="S8.Thmtheorem3.p1.4.m4.2.3.3.1.cmml" xref="S8.Thmtheorem3.p1.4.m4.2.3.3">superscript</csymbol><ci id="S8.Thmtheorem3.p1.4.m4.2.3.3.2.cmml" xref="S8.Thmtheorem3.p1.4.m4.2.3.3.2">𝐾</ci><ci id="S8.Thmtheorem3.p1.4.m4.2.3.3.3.cmml" xref="S8.Thmtheorem3.p1.4.m4.2.3.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.Thmtheorem3.p1.4.m4.2c">s,t\in K^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S8.Thmtheorem3.p1.4.m4.2d">italic_s , italic_t ∈ italic_K start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, if <math alttext="\xi\in\operatorname{dom}(s)\cap\operatorname{dom}(t)" class="ltx_Math" display="inline" id="S8.Thmtheorem3.p1.5.m5.4"><semantics id="S8.Thmtheorem3.p1.5.m5.4a"><mrow id="S8.Thmtheorem3.p1.5.m5.4.5" xref="S8.Thmtheorem3.p1.5.m5.4.5.cmml"><mi id="S8.Thmtheorem3.p1.5.m5.4.5.2" xref="S8.Thmtheorem3.p1.5.m5.4.5.2.cmml">ξ</mi><mo id="S8.Thmtheorem3.p1.5.m5.4.5.1" xref="S8.Thmtheorem3.p1.5.m5.4.5.1.cmml">∈</mo><mrow id="S8.Thmtheorem3.p1.5.m5.4.5.3" xref="S8.Thmtheorem3.p1.5.m5.4.5.3.cmml"><mrow id="S8.Thmtheorem3.p1.5.m5.4.5.3.2.2" xref="S8.Thmtheorem3.p1.5.m5.4.5.3.2.1.cmml"><mi id="S8.Thmtheorem3.p1.5.m5.1.1" xref="S8.Thmtheorem3.p1.5.m5.1.1.cmml">dom</mi><mo id="S8.Thmtheorem3.p1.5.m5.4.5.3.2.2a" xref="S8.Thmtheorem3.p1.5.m5.4.5.3.2.1.cmml">⁡</mo><mrow id="S8.Thmtheorem3.p1.5.m5.4.5.3.2.2.1" xref="S8.Thmtheorem3.p1.5.m5.4.5.3.2.1.cmml"><mo id="S8.Thmtheorem3.p1.5.m5.4.5.3.2.2.1.1" stretchy="false" xref="S8.Thmtheorem3.p1.5.m5.4.5.3.2.1.cmml">(</mo><mi id="S8.Thmtheorem3.p1.5.m5.2.2" xref="S8.Thmtheorem3.p1.5.m5.2.2.cmml">s</mi><mo id="S8.Thmtheorem3.p1.5.m5.4.5.3.2.2.1.2" stretchy="false" xref="S8.Thmtheorem3.p1.5.m5.4.5.3.2.1.cmml">)</mo></mrow></mrow><mo id="S8.Thmtheorem3.p1.5.m5.4.5.3.1" xref="S8.Thmtheorem3.p1.5.m5.4.5.3.1.cmml">∩</mo><mrow id="S8.Thmtheorem3.p1.5.m5.4.5.3.3.2" xref="S8.Thmtheorem3.p1.5.m5.4.5.3.3.1.cmml"><mi id="S8.Thmtheorem3.p1.5.m5.3.3" xref="S8.Thmtheorem3.p1.5.m5.3.3.cmml">dom</mi><mo id="S8.Thmtheorem3.p1.5.m5.4.5.3.3.2a" xref="S8.Thmtheorem3.p1.5.m5.4.5.3.3.1.cmml">⁡</mo><mrow id="S8.Thmtheorem3.p1.5.m5.4.5.3.3.2.1" xref="S8.Thmtheorem3.p1.5.m5.4.5.3.3.1.cmml"><mo id="S8.Thmtheorem3.p1.5.m5.4.5.3.3.2.1.1" stretchy="false" xref="S8.Thmtheorem3.p1.5.m5.4.5.3.3.1.cmml">(</mo><mi id="S8.Thmtheorem3.p1.5.m5.4.4" xref="S8.Thmtheorem3.p1.5.m5.4.4.cmml">t</mi><mo id="S8.Thmtheorem3.p1.5.m5.4.5.3.3.2.1.2" stretchy="false" xref="S8.Thmtheorem3.p1.5.m5.4.5.3.3.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S8.Thmtheorem3.p1.5.m5.4b"><apply id="S8.Thmtheorem3.p1.5.m5.4.5.cmml" xref="S8.Thmtheorem3.p1.5.m5.4.5"><in id="S8.Thmtheorem3.p1.5.m5.4.5.1.cmml" xref="S8.Thmtheorem3.p1.5.m5.4.5.1"></in><ci id="S8.Thmtheorem3.p1.5.m5.4.5.2.cmml" xref="S8.Thmtheorem3.p1.5.m5.4.5.2">𝜉</ci><apply id="S8.Thmtheorem3.p1.5.m5.4.5.3.cmml" xref="S8.Thmtheorem3.p1.5.m5.4.5.3"><intersect id="S8.Thmtheorem3.p1.5.m5.4.5.3.1.cmml" xref="S8.Thmtheorem3.p1.5.m5.4.5.3.1"></intersect><apply id="S8.Thmtheorem3.p1.5.m5.4.5.3.2.1.cmml" xref="S8.Thmtheorem3.p1.5.m5.4.5.3.2.2"><ci id="S8.Thmtheorem3.p1.5.m5.1.1.cmml" xref="S8.Thmtheorem3.p1.5.m5.1.1">dom</ci><ci id="S8.Thmtheorem3.p1.5.m5.2.2.cmml" xref="S8.Thmtheorem3.p1.5.m5.2.2">𝑠</ci></apply><apply id="S8.Thmtheorem3.p1.5.m5.4.5.3.3.1.cmml" xref="S8.Thmtheorem3.p1.5.m5.4.5.3.3.2"><ci id="S8.Thmtheorem3.p1.5.m5.3.3.cmml" xref="S8.Thmtheorem3.p1.5.m5.3.3">dom</ci><ci id="S8.Thmtheorem3.p1.5.m5.4.4.cmml" xref="S8.Thmtheorem3.p1.5.m5.4.4">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.Thmtheorem3.p1.5.m5.4c">\xi\in\operatorname{dom}(s)\cap\operatorname{dom}(t)</annotation><annotation encoding="application/x-llamapun" id="S8.Thmtheorem3.p1.5.m5.4d">italic_ξ ∈ roman_dom ( italic_s ) ∩ roman_dom ( italic_t )</annotation></semantics></math>, and <math alttext="t(\xi)<n" class="ltx_Math" display="inline" id="S8.Thmtheorem3.p1.6.m6.1"><semantics id="S8.Thmtheorem3.p1.6.m6.1a"><mrow id="S8.Thmtheorem3.p1.6.m6.1.2" xref="S8.Thmtheorem3.p1.6.m6.1.2.cmml"><mrow id="S8.Thmtheorem3.p1.6.m6.1.2.2" xref="S8.Thmtheorem3.p1.6.m6.1.2.2.cmml"><mi id="S8.Thmtheorem3.p1.6.m6.1.2.2.2" xref="S8.Thmtheorem3.p1.6.m6.1.2.2.2.cmml">t</mi><mo id="S8.Thmtheorem3.p1.6.m6.1.2.2.1" xref="S8.Thmtheorem3.p1.6.m6.1.2.2.1.cmml">⁢</mo><mrow id="S8.Thmtheorem3.p1.6.m6.1.2.2.3.2" xref="S8.Thmtheorem3.p1.6.m6.1.2.2.cmml"><mo id="S8.Thmtheorem3.p1.6.m6.1.2.2.3.2.1" stretchy="false" xref="S8.Thmtheorem3.p1.6.m6.1.2.2.cmml">(</mo><mi id="S8.Thmtheorem3.p1.6.m6.1.1" xref="S8.Thmtheorem3.p1.6.m6.1.1.cmml">ξ</mi><mo id="S8.Thmtheorem3.p1.6.m6.1.2.2.3.2.2" stretchy="false" xref="S8.Thmtheorem3.p1.6.m6.1.2.2.cmml">)</mo></mrow></mrow><mo id="S8.Thmtheorem3.p1.6.m6.1.2.1" xref="S8.Thmtheorem3.p1.6.m6.1.2.1.cmml"><</mo><mi id="S8.Thmtheorem3.p1.6.m6.1.2.3" xref="S8.Thmtheorem3.p1.6.m6.1.2.3.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S8.Thmtheorem3.p1.6.m6.1b"><apply id="S8.Thmtheorem3.p1.6.m6.1.2.cmml" xref="S8.Thmtheorem3.p1.6.m6.1.2"><lt id="S8.Thmtheorem3.p1.6.m6.1.2.1.cmml" xref="S8.Thmtheorem3.p1.6.m6.1.2.1"></lt><apply id="S8.Thmtheorem3.p1.6.m6.1.2.2.cmml" xref="S8.Thmtheorem3.p1.6.m6.1.2.2"><times id="S8.Thmtheorem3.p1.6.m6.1.2.2.1.cmml" xref="S8.Thmtheorem3.p1.6.m6.1.2.2.1"></times><ci id="S8.Thmtheorem3.p1.6.m6.1.2.2.2.cmml" xref="S8.Thmtheorem3.p1.6.m6.1.2.2.2">𝑡</ci><ci id="S8.Thmtheorem3.p1.6.m6.1.1.cmml" xref="S8.Thmtheorem3.p1.6.m6.1.1">𝜉</ci></apply><ci id="S8.Thmtheorem3.p1.6.m6.1.2.3.cmml" xref="S8.Thmtheorem3.p1.6.m6.1.2.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.Thmtheorem3.p1.6.m6.1c">t(\xi)<n</annotation><annotation encoding="application/x-llamapun" id="S8.Thmtheorem3.p1.6.m6.1d">italic_t ( italic_ξ ) < italic_n</annotation></semantics></math> or <math alttext="s(\xi)<n" class="ltx_Math" display="inline" id="S8.Thmtheorem3.p1.7.m7.1"><semantics id="S8.Thmtheorem3.p1.7.m7.1a"><mrow id="S8.Thmtheorem3.p1.7.m7.1.2" xref="S8.Thmtheorem3.p1.7.m7.1.2.cmml"><mrow id="S8.Thmtheorem3.p1.7.m7.1.2.2" xref="S8.Thmtheorem3.p1.7.m7.1.2.2.cmml"><mi id="S8.Thmtheorem3.p1.7.m7.1.2.2.2" xref="S8.Thmtheorem3.p1.7.m7.1.2.2.2.cmml">s</mi><mo id="S8.Thmtheorem3.p1.7.m7.1.2.2.1" xref="S8.Thmtheorem3.p1.7.m7.1.2.2.1.cmml">⁢</mo><mrow id="S8.Thmtheorem3.p1.7.m7.1.2.2.3.2" xref="S8.Thmtheorem3.p1.7.m7.1.2.2.cmml"><mo id="S8.Thmtheorem3.p1.7.m7.1.2.2.3.2.1" stretchy="false" xref="S8.Thmtheorem3.p1.7.m7.1.2.2.cmml">(</mo><mi id="S8.Thmtheorem3.p1.7.m7.1.1" xref="S8.Thmtheorem3.p1.7.m7.1.1.cmml">ξ</mi><mo id="S8.Thmtheorem3.p1.7.m7.1.2.2.3.2.2" stretchy="false" xref="S8.Thmtheorem3.p1.7.m7.1.2.2.cmml">)</mo></mrow></mrow><mo id="S8.Thmtheorem3.p1.7.m7.1.2.1" xref="S8.Thmtheorem3.p1.7.m7.1.2.1.cmml"><</mo><mi id="S8.Thmtheorem3.p1.7.m7.1.2.3" xref="S8.Thmtheorem3.p1.7.m7.1.2.3.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S8.Thmtheorem3.p1.7.m7.1b"><apply id="S8.Thmtheorem3.p1.7.m7.1.2.cmml" xref="S8.Thmtheorem3.p1.7.m7.1.2"><lt id="S8.Thmtheorem3.p1.7.m7.1.2.1.cmml" xref="S8.Thmtheorem3.p1.7.m7.1.2.1"></lt><apply id="S8.Thmtheorem3.p1.7.m7.1.2.2.cmml" xref="S8.Thmtheorem3.p1.7.m7.1.2.2"><times id="S8.Thmtheorem3.p1.7.m7.1.2.2.1.cmml" xref="S8.Thmtheorem3.p1.7.m7.1.2.2.1"></times><ci id="S8.Thmtheorem3.p1.7.m7.1.2.2.2.cmml" xref="S8.Thmtheorem3.p1.7.m7.1.2.2.2">𝑠</ci><ci id="S8.Thmtheorem3.p1.7.m7.1.1.cmml" xref="S8.Thmtheorem3.p1.7.m7.1.1">𝜉</ci></apply><ci id="S8.Thmtheorem3.p1.7.m7.1.2.3.cmml" xref="S8.Thmtheorem3.p1.7.m7.1.2.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.Thmtheorem3.p1.7.m7.1c">s(\xi)<n</annotation><annotation encoding="application/x-llamapun" id="S8.Thmtheorem3.p1.7.m7.1d">italic_s ( italic_ξ ) < italic_n</annotation></semantics></math>, then <math alttext="t(\xi)=s(\xi)" class="ltx_Math" display="inline" id="S8.Thmtheorem3.p1.8.m8.2"><semantics id="S8.Thmtheorem3.p1.8.m8.2a"><mrow id="S8.Thmtheorem3.p1.8.m8.2.3" xref="S8.Thmtheorem3.p1.8.m8.2.3.cmml"><mrow id="S8.Thmtheorem3.p1.8.m8.2.3.2" xref="S8.Thmtheorem3.p1.8.m8.2.3.2.cmml"><mi id="S8.Thmtheorem3.p1.8.m8.2.3.2.2" xref="S8.Thmtheorem3.p1.8.m8.2.3.2.2.cmml">t</mi><mo id="S8.Thmtheorem3.p1.8.m8.2.3.2.1" xref="S8.Thmtheorem3.p1.8.m8.2.3.2.1.cmml">⁢</mo><mrow id="S8.Thmtheorem3.p1.8.m8.2.3.2.3.2" xref="S8.Thmtheorem3.p1.8.m8.2.3.2.cmml"><mo id="S8.Thmtheorem3.p1.8.m8.2.3.2.3.2.1" stretchy="false" xref="S8.Thmtheorem3.p1.8.m8.2.3.2.cmml">(</mo><mi id="S8.Thmtheorem3.p1.8.m8.1.1" xref="S8.Thmtheorem3.p1.8.m8.1.1.cmml">ξ</mi><mo id="S8.Thmtheorem3.p1.8.m8.2.3.2.3.2.2" stretchy="false" xref="S8.Thmtheorem3.p1.8.m8.2.3.2.cmml">)</mo></mrow></mrow><mo id="S8.Thmtheorem3.p1.8.m8.2.3.1" xref="S8.Thmtheorem3.p1.8.m8.2.3.1.cmml">=</mo><mrow id="S8.Thmtheorem3.p1.8.m8.2.3.3" xref="S8.Thmtheorem3.p1.8.m8.2.3.3.cmml"><mi id="S8.Thmtheorem3.p1.8.m8.2.3.3.2" xref="S8.Thmtheorem3.p1.8.m8.2.3.3.2.cmml">s</mi><mo id="S8.Thmtheorem3.p1.8.m8.2.3.3.1" xref="S8.Thmtheorem3.p1.8.m8.2.3.3.1.cmml">⁢</mo><mrow id="S8.Thmtheorem3.p1.8.m8.2.3.3.3.2" xref="S8.Thmtheorem3.p1.8.m8.2.3.3.cmml"><mo id="S8.Thmtheorem3.p1.8.m8.2.3.3.3.2.1" stretchy="false" xref="S8.Thmtheorem3.p1.8.m8.2.3.3.cmml">(</mo><mi id="S8.Thmtheorem3.p1.8.m8.2.2" xref="S8.Thmtheorem3.p1.8.m8.2.2.cmml">ξ</mi><mo id="S8.Thmtheorem3.p1.8.m8.2.3.3.3.2.2" stretchy="false" xref="S8.Thmtheorem3.p1.8.m8.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S8.Thmtheorem3.p1.8.m8.2b"><apply id="S8.Thmtheorem3.p1.8.m8.2.3.cmml" xref="S8.Thmtheorem3.p1.8.m8.2.3"><eq id="S8.Thmtheorem3.p1.8.m8.2.3.1.cmml" xref="S8.Thmtheorem3.p1.8.m8.2.3.1"></eq><apply id="S8.Thmtheorem3.p1.8.m8.2.3.2.cmml" xref="S8.Thmtheorem3.p1.8.m8.2.3.2"><times id="S8.Thmtheorem3.p1.8.m8.2.3.2.1.cmml" xref="S8.Thmtheorem3.p1.8.m8.2.3.2.1"></times><ci id="S8.Thmtheorem3.p1.8.m8.2.3.2.2.cmml" xref="S8.Thmtheorem3.p1.8.m8.2.3.2.2">𝑡</ci><ci id="S8.Thmtheorem3.p1.8.m8.1.1.cmml" xref="S8.Thmtheorem3.p1.8.m8.1.1">𝜉</ci></apply><apply id="S8.Thmtheorem3.p1.8.m8.2.3.3.cmml" xref="S8.Thmtheorem3.p1.8.m8.2.3.3"><times id="S8.Thmtheorem3.p1.8.m8.2.3.3.1.cmml" xref="S8.Thmtheorem3.p1.8.m8.2.3.3.1"></times><ci id="S8.Thmtheorem3.p1.8.m8.2.3.3.2.cmml" xref="S8.Thmtheorem3.p1.8.m8.2.3.3.2">𝑠</ci><ci id="S8.Thmtheorem3.p1.8.m8.2.2.cmml" xref="S8.Thmtheorem3.p1.8.m8.2.2">𝜉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.Thmtheorem3.p1.8.m8.2c">t(\xi)=s(\xi)</annotation><annotation encoding="application/x-llamapun" id="S8.Thmtheorem3.p1.8.m8.2d">italic_t ( italic_ξ ) = italic_s ( italic_ξ )</annotation></semantics></math>.</p> </div> </div> <div class="ltx_para" id="S8.p8"> <p class="ltx_p" id="S8.p8.7">We turn now to prove <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S8.Thmtheorem2" title="Theorem 8.2. ‣ 8. On a question on Countryman lines ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">8.2</span></a>. Fix <math alttext="G" class="ltx_Math" display="inline" id="S8.p8.1.m1.1"><semantics id="S8.p8.1.m1.1a"><mi id="S8.p8.1.m1.1.1" xref="S8.p8.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S8.p8.1.m1.1b"><ci id="S8.p8.1.m1.1.1.cmml" xref="S8.p8.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.p8.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S8.p8.1.m1.1d">italic_G</annotation></semantics></math> a generic filter for <math alttext="\mathbb{P}" class="ltx_Math" display="inline" id="S8.p8.2.m2.1"><semantics id="S8.p8.2.m2.1a"><mi id="S8.p8.2.m2.1.1" xref="S8.p8.2.m2.1.1.cmml">ℙ</mi><annotation-xml encoding="MathML-Content" id="S8.p8.2.m2.1b"><ci id="S8.p8.2.m2.1.1.cmml" xref="S8.p8.2.m2.1.1">ℙ</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.p8.2.m2.1c">\mathbb{P}</annotation><annotation encoding="application/x-llamapun" id="S8.p8.2.m2.1d">blackboard_P</annotation></semantics></math>, and let <math alttext="\mathring{c}" class="ltx_Math" display="inline" id="S8.p8.3.m3.1"><semantics id="S8.p8.3.m3.1a"><mover accent="true" id="S8.p8.3.m3.1.1" xref="S8.p8.3.m3.1.1.cmml"><mi id="S8.p8.3.m3.1.1.2" xref="S8.p8.3.m3.1.1.2.cmml">c</mi><mo id="S8.p8.3.m3.1.1.1" xref="S8.p8.3.m3.1.1.1.cmml">̊</mo></mover><annotation-xml encoding="MathML-Content" id="S8.p8.3.m3.1b"><apply id="S8.p8.3.m3.1.1.cmml" xref="S8.p8.3.m3.1.1"><ci id="S8.p8.3.m3.1.1.1.cmml" xref="S8.p8.3.m3.1.1.1">̊</ci><ci id="S8.p8.3.m3.1.1.2.cmml" xref="S8.p8.3.m3.1.1.2">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.p8.3.m3.1c">\mathring{c}</annotation><annotation encoding="application/x-llamapun" id="S8.p8.3.m3.1d">over̊ start_ARG italic_c end_ARG</annotation></semantics></math> be the canonical name for <math alttext="\bigcup G" class="ltx_Math" display="inline" id="S8.p8.4.m4.1"><semantics id="S8.p8.4.m4.1a"><mrow id="S8.p8.4.m4.1.1" xref="S8.p8.4.m4.1.1.cmml"><mo id="S8.p8.4.m4.1.1.1" xref="S8.p8.4.m4.1.1.1.cmml">⋃</mo><mi id="S8.p8.4.m4.1.1.2" xref="S8.p8.4.m4.1.1.2.cmml">G</mi></mrow><annotation-xml encoding="MathML-Content" id="S8.p8.4.m4.1b"><apply id="S8.p8.4.m4.1.1.cmml" xref="S8.p8.4.m4.1.1"><union id="S8.p8.4.m4.1.1.1.cmml" xref="S8.p8.4.m4.1.1.1"></union><ci id="S8.p8.4.m4.1.1.2.cmml" xref="S8.p8.4.m4.1.1.2">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.p8.4.m4.1c">\bigcup G</annotation><annotation encoding="application/x-llamapun" id="S8.p8.4.m4.1d">⋃ italic_G</annotation></semantics></math>. Then clearly <math alttext="\mathbbm{1}\Vdash\mathring{c}:\check{\omega}\to\check{\omega}" class="ltx_Math" display="inline" id="S8.p8.5.m5.1"><semantics id="S8.p8.5.m5.1a"><mrow id="S8.p8.5.m5.1.1" xref="S8.p8.5.m5.1.1.cmml"><mrow id="S8.p8.5.m5.1.1.2" xref="S8.p8.5.m5.1.1.2.cmml"><mn id="S8.p8.5.m5.1.1.2.2" xref="S8.p8.5.m5.1.1.2.2.cmml">𝟙</mn><mo id="S8.p8.5.m5.1.1.2.1" xref="S8.p8.5.m5.1.1.2.1.cmml">⊩</mo><mover accent="true" id="S8.p8.5.m5.1.1.2.3" xref="S8.p8.5.m5.1.1.2.3.cmml"><mi id="S8.p8.5.m5.1.1.2.3.2" xref="S8.p8.5.m5.1.1.2.3.2.cmml">c</mi><mo id="S8.p8.5.m5.1.1.2.3.1" xref="S8.p8.5.m5.1.1.2.3.1.cmml">̊</mo></mover></mrow><mo id="S8.p8.5.m5.1.1.1" lspace="0.278em" rspace="0.278em" xref="S8.p8.5.m5.1.1.1.cmml">:</mo><mrow id="S8.p8.5.m5.1.1.3" xref="S8.p8.5.m5.1.1.3.cmml"><mover accent="true" id="S8.p8.5.m5.1.1.3.2" xref="S8.p8.5.m5.1.1.3.2.cmml"><mi id="S8.p8.5.m5.1.1.3.2.2" xref="S8.p8.5.m5.1.1.3.2.2.cmml">ω</mi><mo id="S8.p8.5.m5.1.1.3.2.1" xref="S8.p8.5.m5.1.1.3.2.1.cmml">ˇ</mo></mover><mo id="S8.p8.5.m5.1.1.3.1" stretchy="false" xref="S8.p8.5.m5.1.1.3.1.cmml">→</mo><mover accent="true" id="S8.p8.5.m5.1.1.3.3" xref="S8.p8.5.m5.1.1.3.3.cmml"><mi id="S8.p8.5.m5.1.1.3.3.2" xref="S8.p8.5.m5.1.1.3.3.2.cmml">ω</mi><mo id="S8.p8.5.m5.1.1.3.3.1" xref="S8.p8.5.m5.1.1.3.3.1.cmml">ˇ</mo></mover></mrow></mrow><annotation-xml encoding="MathML-Content" id="S8.p8.5.m5.1b"><apply id="S8.p8.5.m5.1.1.cmml" xref="S8.p8.5.m5.1.1"><ci id="S8.p8.5.m5.1.1.1.cmml" xref="S8.p8.5.m5.1.1.1">:</ci><apply id="S8.p8.5.m5.1.1.2.cmml" xref="S8.p8.5.m5.1.1.2"><csymbol cd="latexml" id="S8.p8.5.m5.1.1.2.1.cmml" xref="S8.p8.5.m5.1.1.2.1">forces</csymbol><cn id="S8.p8.5.m5.1.1.2.2.cmml" type="integer" xref="S8.p8.5.m5.1.1.2.2">1</cn><apply id="S8.p8.5.m5.1.1.2.3.cmml" xref="S8.p8.5.m5.1.1.2.3"><ci id="S8.p8.5.m5.1.1.2.3.1.cmml" xref="S8.p8.5.m5.1.1.2.3.1">̊</ci><ci id="S8.p8.5.m5.1.1.2.3.2.cmml" xref="S8.p8.5.m5.1.1.2.3.2">𝑐</ci></apply></apply><apply id="S8.p8.5.m5.1.1.3.cmml" xref="S8.p8.5.m5.1.1.3"><ci id="S8.p8.5.m5.1.1.3.1.cmml" xref="S8.p8.5.m5.1.1.3.1">→</ci><apply id="S8.p8.5.m5.1.1.3.2.cmml" xref="S8.p8.5.m5.1.1.3.2"><ci id="S8.p8.5.m5.1.1.3.2.1.cmml" xref="S8.p8.5.m5.1.1.3.2.1">ˇ</ci><ci id="S8.p8.5.m5.1.1.3.2.2.cmml" xref="S8.p8.5.m5.1.1.3.2.2">𝜔</ci></apply><apply id="S8.p8.5.m5.1.1.3.3.cmml" xref="S8.p8.5.m5.1.1.3.3"><ci id="S8.p8.5.m5.1.1.3.3.1.cmml" xref="S8.p8.5.m5.1.1.3.3.1">ˇ</ci><ci id="S8.p8.5.m5.1.1.3.3.2.cmml" xref="S8.p8.5.m5.1.1.3.3.2">𝜔</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.p8.5.m5.1c">\mathbbm{1}\Vdash\mathring{c}:\check{\omega}\to\check{\omega}</annotation><annotation encoding="application/x-llamapun" id="S8.p8.5.m5.1d">blackboard_1 ⊩ over̊ start_ARG italic_c end_ARG : overroman_ˇ start_ARG italic_ω end_ARG → overroman_ˇ start_ARG italic_ω end_ARG</annotation></semantics></math>. Now working in <math alttext="V[G]" class="ltx_Math" display="inline" id="S8.p8.6.m6.1"><semantics id="S8.p8.6.m6.1a"><mrow id="S8.p8.6.m6.1.2" xref="S8.p8.6.m6.1.2.cmml"><mi id="S8.p8.6.m6.1.2.2" xref="S8.p8.6.m6.1.2.2.cmml">V</mi><mo id="S8.p8.6.m6.1.2.1" xref="S8.p8.6.m6.1.2.1.cmml">⁢</mo><mrow id="S8.p8.6.m6.1.2.3.2" xref="S8.p8.6.m6.1.2.3.1.cmml"><mo id="S8.p8.6.m6.1.2.3.2.1" stretchy="false" xref="S8.p8.6.m6.1.2.3.1.1.cmml">[</mo><mi id="S8.p8.6.m6.1.1" xref="S8.p8.6.m6.1.1.cmml">G</mi><mo id="S8.p8.6.m6.1.2.3.2.2" stretchy="false" xref="S8.p8.6.m6.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S8.p8.6.m6.1b"><apply id="S8.p8.6.m6.1.2.cmml" xref="S8.p8.6.m6.1.2"><times id="S8.p8.6.m6.1.2.1.cmml" xref="S8.p8.6.m6.1.2.1"></times><ci id="S8.p8.6.m6.1.2.2.cmml" xref="S8.p8.6.m6.1.2.2">𝑉</ci><apply id="S8.p8.6.m6.1.2.3.1.cmml" xref="S8.p8.6.m6.1.2.3.2"><csymbol cd="latexml" id="S8.p8.6.m6.1.2.3.1.1.cmml" xref="S8.p8.6.m6.1.2.3.2.1">delimited-[]</csymbol><ci id="S8.p8.6.m6.1.1.cmml" xref="S8.p8.6.m6.1.1">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.p8.6.m6.1c">V[G]</annotation><annotation encoding="application/x-llamapun" id="S8.p8.6.m6.1d">italic_V [ italic_G ]</annotation></semantics></math> let <math alttext="T^{c}:=\{c\circ t:t\in T\}" class="ltx_Math" display="inline" id="S8.p8.7.m7.2"><semantics id="S8.p8.7.m7.2a"><mrow id="S8.p8.7.m7.2.2" xref="S8.p8.7.m7.2.2.cmml"><msup id="S8.p8.7.m7.2.2.4" xref="S8.p8.7.m7.2.2.4.cmml"><mi id="S8.p8.7.m7.2.2.4.2" xref="S8.p8.7.m7.2.2.4.2.cmml">T</mi><mi id="S8.p8.7.m7.2.2.4.3" xref="S8.p8.7.m7.2.2.4.3.cmml">c</mi></msup><mo id="S8.p8.7.m7.2.2.3" lspace="0.278em" rspace="0.278em" xref="S8.p8.7.m7.2.2.3.cmml">:=</mo><mrow id="S8.p8.7.m7.2.2.2.2" xref="S8.p8.7.m7.2.2.2.3.cmml"><mo id="S8.p8.7.m7.2.2.2.2.3" stretchy="false" xref="S8.p8.7.m7.2.2.2.3.1.cmml">{</mo><mrow id="S8.p8.7.m7.1.1.1.1.1" xref="S8.p8.7.m7.1.1.1.1.1.cmml"><mi id="S8.p8.7.m7.1.1.1.1.1.2" xref="S8.p8.7.m7.1.1.1.1.1.2.cmml">c</mi><mo id="S8.p8.7.m7.1.1.1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S8.p8.7.m7.1.1.1.1.1.1.cmml">∘</mo><mi id="S8.p8.7.m7.1.1.1.1.1.3" xref="S8.p8.7.m7.1.1.1.1.1.3.cmml">t</mi></mrow><mo id="S8.p8.7.m7.2.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S8.p8.7.m7.2.2.2.3.1.cmml">:</mo><mrow id="S8.p8.7.m7.2.2.2.2.2" xref="S8.p8.7.m7.2.2.2.2.2.cmml"><mi id="S8.p8.7.m7.2.2.2.2.2.2" xref="S8.p8.7.m7.2.2.2.2.2.2.cmml">t</mi><mo id="S8.p8.7.m7.2.2.2.2.2.1" xref="S8.p8.7.m7.2.2.2.2.2.1.cmml">∈</mo><mi id="S8.p8.7.m7.2.2.2.2.2.3" xref="S8.p8.7.m7.2.2.2.2.2.3.cmml">T</mi></mrow><mo id="S8.p8.7.m7.2.2.2.2.5" stretchy="false" xref="S8.p8.7.m7.2.2.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S8.p8.7.m7.2b"><apply id="S8.p8.7.m7.2.2.cmml" xref="S8.p8.7.m7.2.2"><csymbol cd="latexml" id="S8.p8.7.m7.2.2.3.cmml" xref="S8.p8.7.m7.2.2.3">assign</csymbol><apply id="S8.p8.7.m7.2.2.4.cmml" xref="S8.p8.7.m7.2.2.4"><csymbol cd="ambiguous" id="S8.p8.7.m7.2.2.4.1.cmml" xref="S8.p8.7.m7.2.2.4">superscript</csymbol><ci id="S8.p8.7.m7.2.2.4.2.cmml" xref="S8.p8.7.m7.2.2.4.2">𝑇</ci><ci id="S8.p8.7.m7.2.2.4.3.cmml" xref="S8.p8.7.m7.2.2.4.3">𝑐</ci></apply><apply id="S8.p8.7.m7.2.2.2.3.cmml" xref="S8.p8.7.m7.2.2.2.2"><csymbol cd="latexml" id="S8.p8.7.m7.2.2.2.3.1.cmml" xref="S8.p8.7.m7.2.2.2.2.3">conditional-set</csymbol><apply id="S8.p8.7.m7.1.1.1.1.1.cmml" xref="S8.p8.7.m7.1.1.1.1.1"><compose id="S8.p8.7.m7.1.1.1.1.1.1.cmml" xref="S8.p8.7.m7.1.1.1.1.1.1"></compose><ci id="S8.p8.7.m7.1.1.1.1.1.2.cmml" xref="S8.p8.7.m7.1.1.1.1.1.2">𝑐</ci><ci id="S8.p8.7.m7.1.1.1.1.1.3.cmml" xref="S8.p8.7.m7.1.1.1.1.1.3">𝑡</ci></apply><apply id="S8.p8.7.m7.2.2.2.2.2.cmml" xref="S8.p8.7.m7.2.2.2.2.2"><in id="S8.p8.7.m7.2.2.2.2.2.1.cmml" xref="S8.p8.7.m7.2.2.2.2.2.1"></in><ci id="S8.p8.7.m7.2.2.2.2.2.2.cmml" xref="S8.p8.7.m7.2.2.2.2.2.2">𝑡</ci><ci id="S8.p8.7.m7.2.2.2.2.2.3.cmml" xref="S8.p8.7.m7.2.2.2.2.2.3">𝑇</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.p8.7.m7.2c">T^{c}:=\{c\circ t:t\in T\}</annotation><annotation encoding="application/x-llamapun" id="S8.p8.7.m7.2d">italic_T start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT := { italic_c ∘ italic_t : italic_t ∈ italic_T }</annotation></semantics></math>. Todorcevic proved that this is always a Suslin tree. Also a proof of this is easily extracted from what follows, and not by chance, but because the proof here follows that one.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="S8.Thmtheorem4"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S8.Thmtheorem4.1.1.1">Proposition 8.4</span></span><span class="ltx_text ltx_font_bold" id="S8.Thmtheorem4.2.2">.</span> </h6> <div class="ltx_para" id="S8.Thmtheorem4.p1"> <p class="ltx_p" id="S8.Thmtheorem4.p1.3"><math alttext="V[G]\models" class="ltx_Math" display="inline" id="S8.Thmtheorem4.p1.1.m1.1"><semantics id="S8.Thmtheorem4.p1.1.m1.1a"><mrow id="S8.Thmtheorem4.p1.1.m1.1.2" xref="S8.Thmtheorem4.p1.1.m1.1.2.cmml"><mrow id="S8.Thmtheorem4.p1.1.m1.1.2.2" xref="S8.Thmtheorem4.p1.1.m1.1.2.2.cmml"><mi id="S8.Thmtheorem4.p1.1.m1.1.2.2.2" xref="S8.Thmtheorem4.p1.1.m1.1.2.2.2.cmml">V</mi><mo id="S8.Thmtheorem4.p1.1.m1.1.2.2.1" xref="S8.Thmtheorem4.p1.1.m1.1.2.2.1.cmml">⁢</mo><mrow id="S8.Thmtheorem4.p1.1.m1.1.2.2.3.2" xref="S8.Thmtheorem4.p1.1.m1.1.2.2.3.1.cmml"><mo id="S8.Thmtheorem4.p1.1.m1.1.2.2.3.2.1" stretchy="false" xref="S8.Thmtheorem4.p1.1.m1.1.2.2.3.1.1.cmml">[</mo><mi id="S8.Thmtheorem4.p1.1.m1.1.1" xref="S8.Thmtheorem4.p1.1.m1.1.1.cmml">G</mi><mo id="S8.Thmtheorem4.p1.1.m1.1.2.2.3.2.2" stretchy="false" xref="S8.Thmtheorem4.p1.1.m1.1.2.2.3.1.1.cmml">]</mo></mrow></mrow><mo id="S8.Thmtheorem4.p1.1.m1.1.2.1" xref="S8.Thmtheorem4.p1.1.m1.1.2.1.cmml">⊧</mo><mi id="S8.Thmtheorem4.p1.1.m1.1.2.3" xref="S8.Thmtheorem4.p1.1.m1.1.2.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="S8.Thmtheorem4.p1.1.m1.1b"><apply id="S8.Thmtheorem4.p1.1.m1.1.2.cmml" xref="S8.Thmtheorem4.p1.1.m1.1.2"><csymbol cd="latexml" id="S8.Thmtheorem4.p1.1.m1.1.2.1.cmml" xref="S8.Thmtheorem4.p1.1.m1.1.2.1">models</csymbol><apply id="S8.Thmtheorem4.p1.1.m1.1.2.2.cmml" xref="S8.Thmtheorem4.p1.1.m1.1.2.2"><times id="S8.Thmtheorem4.p1.1.m1.1.2.2.1.cmml" xref="S8.Thmtheorem4.p1.1.m1.1.2.2.1"></times><ci id="S8.Thmtheorem4.p1.1.m1.1.2.2.2.cmml" xref="S8.Thmtheorem4.p1.1.m1.1.2.2.2">𝑉</ci><apply id="S8.Thmtheorem4.p1.1.m1.1.2.2.3.1.cmml" xref="S8.Thmtheorem4.p1.1.m1.1.2.2.3.2"><csymbol cd="latexml" id="S8.Thmtheorem4.p1.1.m1.1.2.2.3.1.1.cmml" xref="S8.Thmtheorem4.p1.1.m1.1.2.2.3.2.1">delimited-[]</csymbol><ci id="S8.Thmtheorem4.p1.1.m1.1.1.cmml" xref="S8.Thmtheorem4.p1.1.m1.1.1">𝐺</ci></apply></apply><csymbol cd="latexml" id="S8.Thmtheorem4.p1.1.m1.1.2.3.cmml" xref="S8.Thmtheorem4.p1.1.m1.1.2.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.Thmtheorem4.p1.1.m1.1c">V[G]\models</annotation><annotation encoding="application/x-llamapun" id="S8.Thmtheorem4.p1.1.m1.1d">italic_V [ italic_G ] ⊧</annotation></semantics></math> <math alttext="(T^{c},<_{\mathrm{lex}})" class="ltx_Math" display="inline" id="S8.Thmtheorem4.p1.2.m2.2"><semantics id="S8.Thmtheorem4.p1.2.m2.2a"><mrow id="S8.Thmtheorem4.p1.2.m2.2.2.2" xref="S8.Thmtheorem4.p1.2.m2.2.2.3.cmml"><mo id="S8.Thmtheorem4.p1.2.m2.2.2.2.3" stretchy="false" xref="S8.Thmtheorem4.p1.2.m2.2.2.3.cmml">(</mo><msup id="S8.Thmtheorem4.p1.2.m2.1.1.1.1" xref="S8.Thmtheorem4.p1.2.m2.1.1.1.1.cmml"><mi id="S8.Thmtheorem4.p1.2.m2.1.1.1.1.2" xref="S8.Thmtheorem4.p1.2.m2.1.1.1.1.2.cmml">T</mi><mi id="S8.Thmtheorem4.p1.2.m2.1.1.1.1.3" xref="S8.Thmtheorem4.p1.2.m2.1.1.1.1.3.cmml">c</mi></msup><mo id="S8.Thmtheorem4.p1.2.m2.2.2.2.4" xref="S8.Thmtheorem4.p1.2.m2.2.2.3.cmml">,</mo><msub id="S8.Thmtheorem4.p1.2.m2.2.2.2.2" xref="S8.Thmtheorem4.p1.2.m2.2.2.2.2.cmml"><mo id="S8.Thmtheorem4.p1.2.m2.2.2.2.2.2" lspace="0em" rspace="0em" xref="S8.Thmtheorem4.p1.2.m2.2.2.2.2.2.cmml"><</mo><mi id="S8.Thmtheorem4.p1.2.m2.2.2.2.2.3" xref="S8.Thmtheorem4.p1.2.m2.2.2.2.2.3.cmml">lex</mi></msub><mo id="S8.Thmtheorem4.p1.2.m2.2.2.2.5" stretchy="false" xref="S8.Thmtheorem4.p1.2.m2.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S8.Thmtheorem4.p1.2.m2.2b"><interval closure="open" id="S8.Thmtheorem4.p1.2.m2.2.2.3.cmml" xref="S8.Thmtheorem4.p1.2.m2.2.2.2"><apply id="S8.Thmtheorem4.p1.2.m2.1.1.1.1.cmml" xref="S8.Thmtheorem4.p1.2.m2.1.1.1.1"><csymbol cd="ambiguous" id="S8.Thmtheorem4.p1.2.m2.1.1.1.1.1.cmml" xref="S8.Thmtheorem4.p1.2.m2.1.1.1.1">superscript</csymbol><ci id="S8.Thmtheorem4.p1.2.m2.1.1.1.1.2.cmml" xref="S8.Thmtheorem4.p1.2.m2.1.1.1.1.2">𝑇</ci><ci id="S8.Thmtheorem4.p1.2.m2.1.1.1.1.3.cmml" xref="S8.Thmtheorem4.p1.2.m2.1.1.1.1.3">𝑐</ci></apply><apply id="S8.Thmtheorem4.p1.2.m2.2.2.2.2.cmml" xref="S8.Thmtheorem4.p1.2.m2.2.2.2.2"><csymbol cd="ambiguous" id="S8.Thmtheorem4.p1.2.m2.2.2.2.2.1.cmml" xref="S8.Thmtheorem4.p1.2.m2.2.2.2.2">subscript</csymbol><lt id="S8.Thmtheorem4.p1.2.m2.2.2.2.2.2.cmml" xref="S8.Thmtheorem4.p1.2.m2.2.2.2.2.2"></lt><ci id="S8.Thmtheorem4.p1.2.m2.2.2.2.2.3.cmml" xref="S8.Thmtheorem4.p1.2.m2.2.2.2.2.3">lex</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S8.Thmtheorem4.p1.2.m2.2c">(T^{c},<_{\mathrm{lex}})</annotation><annotation encoding="application/x-llamapun" id="S8.Thmtheorem4.p1.2.m2.2d">( italic_T start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT , < start_POSTSUBSCRIPT roman_lex end_POSTSUBSCRIPT )</annotation></semantics></math> is <math alttext="\omega_{1}" class="ltx_Math" display="inline" id="S8.Thmtheorem4.p1.3.m3.1"><semantics id="S8.Thmtheorem4.p1.3.m3.1a"><msub id="S8.Thmtheorem4.p1.3.m3.1.1" xref="S8.Thmtheorem4.p1.3.m3.1.1.cmml"><mi id="S8.Thmtheorem4.p1.3.m3.1.1.2" xref="S8.Thmtheorem4.p1.3.m3.1.1.2.cmml">ω</mi><mn id="S8.Thmtheorem4.p1.3.m3.1.1.3" xref="S8.Thmtheorem4.p1.3.m3.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S8.Thmtheorem4.p1.3.m3.1b"><apply id="S8.Thmtheorem4.p1.3.m3.1.1.cmml" xref="S8.Thmtheorem4.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S8.Thmtheorem4.p1.3.m3.1.1.1.cmml" xref="S8.Thmtheorem4.p1.3.m3.1.1">subscript</csymbol><ci id="S8.Thmtheorem4.p1.3.m3.1.1.2.cmml" xref="S8.Thmtheorem4.p1.3.m3.1.1.2">𝜔</ci><cn id="S8.Thmtheorem4.p1.3.m3.1.1.3.cmml" type="integer" xref="S8.Thmtheorem4.p1.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.Thmtheorem4.p1.3.m3.1c">\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S8.Thmtheorem4.p1.3.m3.1d">italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-irreversible.</p> </div> </div> <div class="ltx_proof" id="S8.6"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S8.1.p1"> <p class="ltx_p" id="S8.1.p1.4">Suppose towards a contradiction that <math alttext="V[G]\models T^{c}\text{ is not $\omega_{1}$-irreversible}" class="ltx_Math" display="inline" id="S8.1.p1.1.m1.2"><semantics id="S8.1.p1.1.m1.2a"><mrow id="S8.1.p1.1.m1.2.3" xref="S8.1.p1.1.m1.2.3.cmml"><mrow id="S8.1.p1.1.m1.2.3.2" xref="S8.1.p1.1.m1.2.3.2.cmml"><mi id="S8.1.p1.1.m1.2.3.2.2" xref="S8.1.p1.1.m1.2.3.2.2.cmml">V</mi><mo id="S8.1.p1.1.m1.2.3.2.1" xref="S8.1.p1.1.m1.2.3.2.1.cmml">⁢</mo><mrow id="S8.1.p1.1.m1.2.3.2.3.2" xref="S8.1.p1.1.m1.2.3.2.3.1.cmml"><mo id="S8.1.p1.1.m1.2.3.2.3.2.1" stretchy="false" xref="S8.1.p1.1.m1.2.3.2.3.1.1.cmml">[</mo><mi id="S8.1.p1.1.m1.2.2" xref="S8.1.p1.1.m1.2.2.cmml">G</mi><mo id="S8.1.p1.1.m1.2.3.2.3.2.2" stretchy="false" xref="S8.1.p1.1.m1.2.3.2.3.1.1.cmml">]</mo></mrow></mrow><mo id="S8.1.p1.1.m1.2.3.1" xref="S8.1.p1.1.m1.2.3.1.cmml">⊧</mo><mrow id="S8.1.p1.1.m1.2.3.3" xref="S8.1.p1.1.m1.2.3.3.cmml"><msup id="S8.1.p1.1.m1.2.3.3.2" xref="S8.1.p1.1.m1.2.3.3.2.cmml"><mi id="S8.1.p1.1.m1.2.3.3.2.2" xref="S8.1.p1.1.m1.2.3.3.2.2.cmml">T</mi><mi id="S8.1.p1.1.m1.2.3.3.2.3" xref="S8.1.p1.1.m1.2.3.3.2.3.cmml">c</mi></msup><mo id="S8.1.p1.1.m1.2.3.3.1" xref="S8.1.p1.1.m1.2.3.3.1.cmml">⁢</mo><mrow id="S8.1.p1.1.m1.1.1.1" xref="S8.1.p1.1.m1.1.1.1c.cmml"><mtext id="S8.1.p1.1.m1.1.1.1a" xref="S8.1.p1.1.m1.1.1.1c.cmml"> is not </mtext><msub id="S8.1.p1.1.m1.1.1.1.m1.1.1" xref="S8.1.p1.1.m1.1.1.1.m1.1.1.cmml"><mi id="S8.1.p1.1.m1.1.1.1.m1.1.1.2" xref="S8.1.p1.1.m1.1.1.1.m1.1.1.2.cmml">ω</mi><mn id="S8.1.p1.1.m1.1.1.1.m1.1.1.3" xref="S8.1.p1.1.m1.1.1.1.m1.1.1.3.cmml">1</mn></msub><mtext id="S8.1.p1.1.m1.1.1.1b" xref="S8.1.p1.1.m1.1.1.1c.cmml">-irreversible</mtext></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S8.1.p1.1.m1.2b"><apply id="S8.1.p1.1.m1.2.3.cmml" xref="S8.1.p1.1.m1.2.3"><csymbol cd="latexml" id="S8.1.p1.1.m1.2.3.1.cmml" xref="S8.1.p1.1.m1.2.3.1">models</csymbol><apply id="S8.1.p1.1.m1.2.3.2.cmml" xref="S8.1.p1.1.m1.2.3.2"><times id="S8.1.p1.1.m1.2.3.2.1.cmml" xref="S8.1.p1.1.m1.2.3.2.1"></times><ci id="S8.1.p1.1.m1.2.3.2.2.cmml" xref="S8.1.p1.1.m1.2.3.2.2">𝑉</ci><apply id="S8.1.p1.1.m1.2.3.2.3.1.cmml" xref="S8.1.p1.1.m1.2.3.2.3.2"><csymbol cd="latexml" id="S8.1.p1.1.m1.2.3.2.3.1.1.cmml" xref="S8.1.p1.1.m1.2.3.2.3.2.1">delimited-[]</csymbol><ci id="S8.1.p1.1.m1.2.2.cmml" xref="S8.1.p1.1.m1.2.2">𝐺</ci></apply></apply><apply id="S8.1.p1.1.m1.2.3.3.cmml" xref="S8.1.p1.1.m1.2.3.3"><times id="S8.1.p1.1.m1.2.3.3.1.cmml" xref="S8.1.p1.1.m1.2.3.3.1"></times><apply id="S8.1.p1.1.m1.2.3.3.2.cmml" xref="S8.1.p1.1.m1.2.3.3.2"><csymbol cd="ambiguous" id="S8.1.p1.1.m1.2.3.3.2.1.cmml" xref="S8.1.p1.1.m1.2.3.3.2">superscript</csymbol><ci id="S8.1.p1.1.m1.2.3.3.2.2.cmml" xref="S8.1.p1.1.m1.2.3.3.2.2">𝑇</ci><ci id="S8.1.p1.1.m1.2.3.3.2.3.cmml" xref="S8.1.p1.1.m1.2.3.3.2.3">𝑐</ci></apply><ci id="S8.1.p1.1.m1.1.1.1c.cmml" xref="S8.1.p1.1.m1.1.1.1"><mrow id="S8.1.p1.1.m1.1.1.1.cmml" xref="S8.1.p1.1.m1.1.1.1"><mtext id="S8.1.p1.1.m1.1.1.1a.cmml" xref="S8.1.p1.1.m1.1.1.1"> is not </mtext><msub id="S8.1.p1.1.m1.1.1.1.m1.1.1.cmml" xref="S8.1.p1.1.m1.1.1.1.m1.1.1"><mi id="S8.1.p1.1.m1.1.1.1.m1.1.1.2.cmml" xref="S8.1.p1.1.m1.1.1.1.m1.1.1.2">ω</mi><mn id="S8.1.p1.1.m1.1.1.1.m1.1.1.3.cmml" xref="S8.1.p1.1.m1.1.1.1.m1.1.1.3">1</mn></msub><mtext id="S8.1.p1.1.m1.1.1.1b.cmml" xref="S8.1.p1.1.m1.1.1.1">-irreversible</mtext></mrow></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.1.p1.1.m1.2c">V[G]\models T^{c}\text{ is not $\omega_{1}$-irreversible}</annotation><annotation encoding="application/x-llamapun" id="S8.1.p1.1.m1.2d">italic_V [ italic_G ] ⊧ italic_T start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT is not italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT -irreversible</annotation></semantics></math>. Then there must be <math alttext="p\in G" class="ltx_Math" display="inline" id="S8.1.p1.2.m2.1"><semantics id="S8.1.p1.2.m2.1a"><mrow id="S8.1.p1.2.m2.1.1" xref="S8.1.p1.2.m2.1.1.cmml"><mi id="S8.1.p1.2.m2.1.1.2" xref="S8.1.p1.2.m2.1.1.2.cmml">p</mi><mo id="S8.1.p1.2.m2.1.1.1" xref="S8.1.p1.2.m2.1.1.1.cmml">∈</mo><mi id="S8.1.p1.2.m2.1.1.3" xref="S8.1.p1.2.m2.1.1.3.cmml">G</mi></mrow><annotation-xml encoding="MathML-Content" id="S8.1.p1.2.m2.1b"><apply id="S8.1.p1.2.m2.1.1.cmml" xref="S8.1.p1.2.m2.1.1"><in id="S8.1.p1.2.m2.1.1.1.cmml" xref="S8.1.p1.2.m2.1.1.1"></in><ci id="S8.1.p1.2.m2.1.1.2.cmml" xref="S8.1.p1.2.m2.1.1.2">𝑝</ci><ci id="S8.1.p1.2.m2.1.1.3.cmml" xref="S8.1.p1.2.m2.1.1.3">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.1.p1.2.m2.1c">p\in G</annotation><annotation encoding="application/x-llamapun" id="S8.1.p1.2.m2.1d">italic_p ∈ italic_G</annotation></semantics></math> and names <math alttext="\mathring{A}" class="ltx_Math" display="inline" id="S8.1.p1.3.m3.1"><semantics id="S8.1.p1.3.m3.1a"><mover accent="true" id="S8.1.p1.3.m3.1.1" xref="S8.1.p1.3.m3.1.1.cmml"><mi id="S8.1.p1.3.m3.1.1.2" xref="S8.1.p1.3.m3.1.1.2.cmml">A</mi><mo id="S8.1.p1.3.m3.1.1.1" xref="S8.1.p1.3.m3.1.1.1.cmml">̊</mo></mover><annotation-xml encoding="MathML-Content" id="S8.1.p1.3.m3.1b"><apply id="S8.1.p1.3.m3.1.1.cmml" xref="S8.1.p1.3.m3.1.1"><ci id="S8.1.p1.3.m3.1.1.1.cmml" xref="S8.1.p1.3.m3.1.1.1">̊</ci><ci id="S8.1.p1.3.m3.1.1.2.cmml" xref="S8.1.p1.3.m3.1.1.2">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.1.p1.3.m3.1c">\mathring{A}</annotation><annotation encoding="application/x-llamapun" id="S8.1.p1.3.m3.1d">over̊ start_ARG italic_A end_ARG</annotation></semantics></math>,<math alttext="\mathring{f}" class="ltx_Math" display="inline" id="S8.1.p1.4.m4.1"><semantics id="S8.1.p1.4.m4.1a"><mover accent="true" id="S8.1.p1.4.m4.1.1" xref="S8.1.p1.4.m4.1.1.cmml"><mi id="S8.1.p1.4.m4.1.1.2" xref="S8.1.p1.4.m4.1.1.2.cmml">f</mi><mo id="S8.1.p1.4.m4.1.1.1" xref="S8.1.p1.4.m4.1.1.1.cmml">̊</mo></mover><annotation-xml encoding="MathML-Content" id="S8.1.p1.4.m4.1b"><apply id="S8.1.p1.4.m4.1.1.cmml" xref="S8.1.p1.4.m4.1.1"><ci id="S8.1.p1.4.m4.1.1.1.cmml" xref="S8.1.p1.4.m4.1.1.1">̊</ci><ci id="S8.1.p1.4.m4.1.1.2.cmml" xref="S8.1.p1.4.m4.1.1.2">𝑓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.1.p1.4.m4.1c">\mathring{f}</annotation><annotation encoding="application/x-llamapun" id="S8.1.p1.4.m4.1d">over̊ start_ARG italic_f end_ARG</annotation></semantics></math> such that</p> <table class="ltx_equation ltx_eqn_table" id="S8.Ex14"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="p\Vdash\mathring{A}\text{ is uncountable and }f:\mathring{A}\subseteq T^{% \mathring{c}}\to T^{\mathring{c}}\text{ reverses }<_{\mathrm{lex}}." class="ltx_Math" display="block" id="S8.Ex14.m1.1"><semantics id="S8.Ex14.m1.1a"><mrow id="S8.Ex14.m1.1.1.1" xref="S8.Ex14.m1.1.1.1.1.cmml"><mrow id="S8.Ex14.m1.1.1.1.1" xref="S8.Ex14.m1.1.1.1.1.cmml"><mrow id="S8.Ex14.m1.1.1.1.1.2" xref="S8.Ex14.m1.1.1.1.1.2.cmml"><mi id="S8.Ex14.m1.1.1.1.1.2.2" xref="S8.Ex14.m1.1.1.1.1.2.2.cmml">p</mi><mo id="S8.Ex14.m1.1.1.1.1.2.1" xref="S8.Ex14.m1.1.1.1.1.2.1.cmml">⊩</mo><mrow id="S8.Ex14.m1.1.1.1.1.2.3" xref="S8.Ex14.m1.1.1.1.1.2.3.cmml"><mover accent="true" id="S8.Ex14.m1.1.1.1.1.2.3.2" xref="S8.Ex14.m1.1.1.1.1.2.3.2.cmml"><mi id="S8.Ex14.m1.1.1.1.1.2.3.2.2" xref="S8.Ex14.m1.1.1.1.1.2.3.2.2.cmml">A</mi><mo id="S8.Ex14.m1.1.1.1.1.2.3.2.1" xref="S8.Ex14.m1.1.1.1.1.2.3.2.1.cmml">̊</mo></mover><mo id="S8.Ex14.m1.1.1.1.1.2.3.1" xref="S8.Ex14.m1.1.1.1.1.2.3.1.cmml">⁢</mo><mtext id="S8.Ex14.m1.1.1.1.1.2.3.3" xref="S8.Ex14.m1.1.1.1.1.2.3.3a.cmml"> is uncountable and </mtext><mo id="S8.Ex14.m1.1.1.1.1.2.3.1a" xref="S8.Ex14.m1.1.1.1.1.2.3.1.cmml">⁢</mo><mi id="S8.Ex14.m1.1.1.1.1.2.3.4" xref="S8.Ex14.m1.1.1.1.1.2.3.4.cmml">f</mi></mrow></mrow><mo id="S8.Ex14.m1.1.1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S8.Ex14.m1.1.1.1.1.1.cmml">:</mo><mrow id="S8.Ex14.m1.1.1.1.1.3" xref="S8.Ex14.m1.1.1.1.1.3.cmml"><mover accent="true" id="S8.Ex14.m1.1.1.1.1.3.2" xref="S8.Ex14.m1.1.1.1.1.3.2.cmml"><mi id="S8.Ex14.m1.1.1.1.1.3.2.2" xref="S8.Ex14.m1.1.1.1.1.3.2.2.cmml">A</mi><mo id="S8.Ex14.m1.1.1.1.1.3.2.1" xref="S8.Ex14.m1.1.1.1.1.3.2.1.cmml">̊</mo></mover><mo id="S8.Ex14.m1.1.1.1.1.3.3" xref="S8.Ex14.m1.1.1.1.1.3.3.cmml">⊆</mo><msup id="S8.Ex14.m1.1.1.1.1.3.4" xref="S8.Ex14.m1.1.1.1.1.3.4.cmml"><mi id="S8.Ex14.m1.1.1.1.1.3.4.2" xref="S8.Ex14.m1.1.1.1.1.3.4.2.cmml">T</mi><mover accent="true" id="S8.Ex14.m1.1.1.1.1.3.4.3" xref="S8.Ex14.m1.1.1.1.1.3.4.3.cmml"><mi id="S8.Ex14.m1.1.1.1.1.3.4.3.2" xref="S8.Ex14.m1.1.1.1.1.3.4.3.2.cmml">c</mi><mo id="S8.Ex14.m1.1.1.1.1.3.4.3.1" xref="S8.Ex14.m1.1.1.1.1.3.4.3.1.cmml">̊</mo></mover></msup><mo id="S8.Ex14.m1.1.1.1.1.3.5" stretchy="false" xref="S8.Ex14.m1.1.1.1.1.3.5.cmml">→</mo><mrow id="S8.Ex14.m1.1.1.1.1.3.6" xref="S8.Ex14.m1.1.1.1.1.3.6.cmml"><msup id="S8.Ex14.m1.1.1.1.1.3.6.2" xref="S8.Ex14.m1.1.1.1.1.3.6.2.cmml"><mi id="S8.Ex14.m1.1.1.1.1.3.6.2.2" xref="S8.Ex14.m1.1.1.1.1.3.6.2.2.cmml">T</mi><mover accent="true" id="S8.Ex14.m1.1.1.1.1.3.6.2.3" xref="S8.Ex14.m1.1.1.1.1.3.6.2.3.cmml"><mi id="S8.Ex14.m1.1.1.1.1.3.6.2.3.2" xref="S8.Ex14.m1.1.1.1.1.3.6.2.3.2.cmml">c</mi><mo id="S8.Ex14.m1.1.1.1.1.3.6.2.3.1" xref="S8.Ex14.m1.1.1.1.1.3.6.2.3.1.cmml">̊</mo></mover></msup><mo id="S8.Ex14.m1.1.1.1.1.3.6.1" xref="S8.Ex14.m1.1.1.1.1.3.6.1.cmml">⁢</mo><mtext id="S8.Ex14.m1.1.1.1.1.3.6.3" xref="S8.Ex14.m1.1.1.1.1.3.6.3a.cmml"> reverses </mtext></mrow><msub id="S8.Ex14.m1.1.1.1.1.3.7" xref="S8.Ex14.m1.1.1.1.1.3.7.cmml"><mo id="S8.Ex14.m1.1.1.1.1.3.7.2" xref="S8.Ex14.m1.1.1.1.1.3.7.2.cmml"><</mo><mi id="S8.Ex14.m1.1.1.1.1.3.7.3" xref="S8.Ex14.m1.1.1.1.1.3.7.3.cmml">lex</mi></msub><mi id="S8.Ex14.m1.1.1.1.1.3.8" xref="S8.Ex14.m1.1.1.1.1.3.8.cmml"></mi></mrow></mrow><mo id="S8.Ex14.m1.1.1.1.2" lspace="0em" xref="S8.Ex14.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S8.Ex14.m1.1b"><apply id="S8.Ex14.m1.1.1.1.1.cmml" xref="S8.Ex14.m1.1.1.1"><ci id="S8.Ex14.m1.1.1.1.1.1.cmml" xref="S8.Ex14.m1.1.1.1.1.1">:</ci><apply id="S8.Ex14.m1.1.1.1.1.2.cmml" xref="S8.Ex14.m1.1.1.1.1.2"><csymbol cd="latexml" id="S8.Ex14.m1.1.1.1.1.2.1.cmml" xref="S8.Ex14.m1.1.1.1.1.2.1">forces</csymbol><ci id="S8.Ex14.m1.1.1.1.1.2.2.cmml" xref="S8.Ex14.m1.1.1.1.1.2.2">𝑝</ci><apply id="S8.Ex14.m1.1.1.1.1.2.3.cmml" xref="S8.Ex14.m1.1.1.1.1.2.3"><times id="S8.Ex14.m1.1.1.1.1.2.3.1.cmml" xref="S8.Ex14.m1.1.1.1.1.2.3.1"></times><apply id="S8.Ex14.m1.1.1.1.1.2.3.2.cmml" xref="S8.Ex14.m1.1.1.1.1.2.3.2"><ci id="S8.Ex14.m1.1.1.1.1.2.3.2.1.cmml" xref="S8.Ex14.m1.1.1.1.1.2.3.2.1">̊</ci><ci id="S8.Ex14.m1.1.1.1.1.2.3.2.2.cmml" xref="S8.Ex14.m1.1.1.1.1.2.3.2.2">𝐴</ci></apply><ci id="S8.Ex14.m1.1.1.1.1.2.3.3a.cmml" xref="S8.Ex14.m1.1.1.1.1.2.3.3"><mtext id="S8.Ex14.m1.1.1.1.1.2.3.3.cmml" xref="S8.Ex14.m1.1.1.1.1.2.3.3"> is uncountable and </mtext></ci><ci id="S8.Ex14.m1.1.1.1.1.2.3.4.cmml" xref="S8.Ex14.m1.1.1.1.1.2.3.4">𝑓</ci></apply></apply><apply id="S8.Ex14.m1.1.1.1.1.3.cmml" xref="S8.Ex14.m1.1.1.1.1.3"><and id="S8.Ex14.m1.1.1.1.1.3a.cmml" xref="S8.Ex14.m1.1.1.1.1.3"></and><apply id="S8.Ex14.m1.1.1.1.1.3b.cmml" xref="S8.Ex14.m1.1.1.1.1.3"><subset id="S8.Ex14.m1.1.1.1.1.3.3.cmml" xref="S8.Ex14.m1.1.1.1.1.3.3"></subset><apply id="S8.Ex14.m1.1.1.1.1.3.2.cmml" xref="S8.Ex14.m1.1.1.1.1.3.2"><ci id="S8.Ex14.m1.1.1.1.1.3.2.1.cmml" xref="S8.Ex14.m1.1.1.1.1.3.2.1">̊</ci><ci id="S8.Ex14.m1.1.1.1.1.3.2.2.cmml" xref="S8.Ex14.m1.1.1.1.1.3.2.2">𝐴</ci></apply><apply id="S8.Ex14.m1.1.1.1.1.3.4.cmml" xref="S8.Ex14.m1.1.1.1.1.3.4"><csymbol cd="ambiguous" id="S8.Ex14.m1.1.1.1.1.3.4.1.cmml" xref="S8.Ex14.m1.1.1.1.1.3.4">superscript</csymbol><ci id="S8.Ex14.m1.1.1.1.1.3.4.2.cmml" xref="S8.Ex14.m1.1.1.1.1.3.4.2">𝑇</ci><apply id="S8.Ex14.m1.1.1.1.1.3.4.3.cmml" xref="S8.Ex14.m1.1.1.1.1.3.4.3"><ci id="S8.Ex14.m1.1.1.1.1.3.4.3.1.cmml" xref="S8.Ex14.m1.1.1.1.1.3.4.3.1">̊</ci><ci id="S8.Ex14.m1.1.1.1.1.3.4.3.2.cmml" xref="S8.Ex14.m1.1.1.1.1.3.4.3.2">𝑐</ci></apply></apply></apply><apply id="S8.Ex14.m1.1.1.1.1.3c.cmml" xref="S8.Ex14.m1.1.1.1.1.3"><ci id="S8.Ex14.m1.1.1.1.1.3.5.cmml" xref="S8.Ex14.m1.1.1.1.1.3.5">→</ci><share href="https://arxiv.org/html/2503.13728v1#S8.Ex14.m1.1.1.1.1.3.4.cmml" id="S8.Ex14.m1.1.1.1.1.3d.cmml" xref="S8.Ex14.m1.1.1.1.1.3"></share><apply id="S8.Ex14.m1.1.1.1.1.3.6.cmml" xref="S8.Ex14.m1.1.1.1.1.3.6"><times id="S8.Ex14.m1.1.1.1.1.3.6.1.cmml" xref="S8.Ex14.m1.1.1.1.1.3.6.1"></times><apply id="S8.Ex14.m1.1.1.1.1.3.6.2.cmml" xref="S8.Ex14.m1.1.1.1.1.3.6.2"><csymbol cd="ambiguous" id="S8.Ex14.m1.1.1.1.1.3.6.2.1.cmml" xref="S8.Ex14.m1.1.1.1.1.3.6.2">superscript</csymbol><ci id="S8.Ex14.m1.1.1.1.1.3.6.2.2.cmml" xref="S8.Ex14.m1.1.1.1.1.3.6.2.2">𝑇</ci><apply id="S8.Ex14.m1.1.1.1.1.3.6.2.3.cmml" xref="S8.Ex14.m1.1.1.1.1.3.6.2.3"><ci id="S8.Ex14.m1.1.1.1.1.3.6.2.3.1.cmml" xref="S8.Ex14.m1.1.1.1.1.3.6.2.3.1">̊</ci><ci id="S8.Ex14.m1.1.1.1.1.3.6.2.3.2.cmml" xref="S8.Ex14.m1.1.1.1.1.3.6.2.3.2">𝑐</ci></apply></apply><ci id="S8.Ex14.m1.1.1.1.1.3.6.3a.cmml" xref="S8.Ex14.m1.1.1.1.1.3.6.3"><mtext id="S8.Ex14.m1.1.1.1.1.3.6.3.cmml" xref="S8.Ex14.m1.1.1.1.1.3.6.3"> reverses </mtext></ci></apply></apply><apply id="S8.Ex14.m1.1.1.1.1.3e.cmml" xref="S8.Ex14.m1.1.1.1.1.3"><apply id="S8.Ex14.m1.1.1.1.1.3.7.cmml" xref="S8.Ex14.m1.1.1.1.1.3.7"><csymbol cd="ambiguous" id="S8.Ex14.m1.1.1.1.1.3.7.1.cmml" xref="S8.Ex14.m1.1.1.1.1.3.7">subscript</csymbol><lt id="S8.Ex14.m1.1.1.1.1.3.7.2.cmml" xref="S8.Ex14.m1.1.1.1.1.3.7.2"></lt><ci id="S8.Ex14.m1.1.1.1.1.3.7.3.cmml" xref="S8.Ex14.m1.1.1.1.1.3.7.3">lex</ci></apply><share href="https://arxiv.org/html/2503.13728v1#S8.Ex14.m1.1.1.1.1.3.6.cmml" id="S8.Ex14.m1.1.1.1.1.3f.cmml" xref="S8.Ex14.m1.1.1.1.1.3"></share><csymbol cd="latexml" id="S8.Ex14.m1.1.1.1.1.3.8.cmml" xref="S8.Ex14.m1.1.1.1.1.3.8">absent</csymbol></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.Ex14.m1.1c">p\Vdash\mathring{A}\text{ is uncountable and }f:\mathring{A}\subseteq T^{% \mathring{c}}\to T^{\mathring{c}}\text{ reverses }<_{\mathrm{lex}}.</annotation><annotation encoding="application/x-llamapun" id="S8.Ex14.m1.1d">italic_p ⊩ over̊ start_ARG italic_A end_ARG is uncountable and italic_f : over̊ start_ARG italic_A end_ARG ⊆ italic_T start_POSTSUPERSCRIPT over̊ start_ARG italic_c end_ARG end_POSTSUPERSCRIPT → italic_T start_POSTSUPERSCRIPT over̊ start_ARG italic_c end_ARG end_POSTSUPERSCRIPT reverses < start_POSTSUBSCRIPT roman_lex end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S8.1.p1.6">To arrive to a contradiction it is enough to find some extension of <math alttext="p" class="ltx_Math" display="inline" id="S8.1.p1.5.m1.1"><semantics id="S8.1.p1.5.m1.1a"><mi id="S8.1.p1.5.m1.1.1" xref="S8.1.p1.5.m1.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S8.1.p1.5.m1.1b"><ci id="S8.1.p1.5.m1.1.1.cmml" xref="S8.1.p1.5.m1.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.1.p1.5.m1.1c">p</annotation><annotation encoding="application/x-llamapun" id="S8.1.p1.5.m1.1d">italic_p</annotation></semantics></math> that forces <math alttext="(\exists t,s\in\mathring{A})(t<_{\mathrm{lex}}s\land\mathring{f}(t)<_{\mathrm{% lex}}\mathring{f}(s))" class="ltx_Math" display="inline" id="S8.1.p1.6.m2.5"><semantics id="S8.1.p1.6.m2.5a"><mrow id="S8.1.p1.6.m2.5.5" xref="S8.1.p1.6.m2.5.5.cmml"><mrow id="S8.1.p1.6.m2.4.4.1.1" xref="S8.1.p1.6.m2.4.4.1.1.1.cmml"><mo id="S8.1.p1.6.m2.4.4.1.1.2" stretchy="false" xref="S8.1.p1.6.m2.4.4.1.1.1.cmml">(</mo><mrow id="S8.1.p1.6.m2.4.4.1.1.1" xref="S8.1.p1.6.m2.4.4.1.1.1.cmml"><mrow id="S8.1.p1.6.m2.4.4.1.1.1.1.1" xref="S8.1.p1.6.m2.4.4.1.1.1.1.2.cmml"><mrow id="S8.1.p1.6.m2.4.4.1.1.1.1.1.1" xref="S8.1.p1.6.m2.4.4.1.1.1.1.1.1.cmml"><mo id="S8.1.p1.6.m2.4.4.1.1.1.1.1.1.1" rspace="0.167em" xref="S8.1.p1.6.m2.4.4.1.1.1.1.1.1.1.cmml">∃</mo><mi id="S8.1.p1.6.m2.4.4.1.1.1.1.1.1.2" xref="S8.1.p1.6.m2.4.4.1.1.1.1.1.1.2.cmml">t</mi></mrow><mo id="S8.1.p1.6.m2.4.4.1.1.1.1.1.2" xref="S8.1.p1.6.m2.4.4.1.1.1.1.2.cmml">,</mo><mi id="S8.1.p1.6.m2.1.1" xref="S8.1.p1.6.m2.1.1.cmml">s</mi></mrow><mo id="S8.1.p1.6.m2.4.4.1.1.1.2" xref="S8.1.p1.6.m2.4.4.1.1.1.2.cmml">∈</mo><mover accent="true" id="S8.1.p1.6.m2.4.4.1.1.1.3" xref="S8.1.p1.6.m2.4.4.1.1.1.3.cmml"><mi id="S8.1.p1.6.m2.4.4.1.1.1.3.2" xref="S8.1.p1.6.m2.4.4.1.1.1.3.2.cmml">A</mi><mo id="S8.1.p1.6.m2.4.4.1.1.1.3.1" xref="S8.1.p1.6.m2.4.4.1.1.1.3.1.cmml">̊</mo></mover></mrow><mo id="S8.1.p1.6.m2.4.4.1.1.3" stretchy="false" xref="S8.1.p1.6.m2.4.4.1.1.1.cmml">)</mo></mrow><mo id="S8.1.p1.6.m2.5.5.3" xref="S8.1.p1.6.m2.5.5.3.cmml">⁢</mo><mrow id="S8.1.p1.6.m2.5.5.2.1" xref="S8.1.p1.6.m2.5.5.2.1.1.cmml"><mo id="S8.1.p1.6.m2.5.5.2.1.2" stretchy="false" xref="S8.1.p1.6.m2.5.5.2.1.1.cmml">(</mo><mrow id="S8.1.p1.6.m2.5.5.2.1.1" xref="S8.1.p1.6.m2.5.5.2.1.1.cmml"><mi id="S8.1.p1.6.m2.5.5.2.1.1.2" xref="S8.1.p1.6.m2.5.5.2.1.1.2.cmml">t</mi><msub id="S8.1.p1.6.m2.5.5.2.1.1.3" xref="S8.1.p1.6.m2.5.5.2.1.1.3.cmml"><mo id="S8.1.p1.6.m2.5.5.2.1.1.3.2" xref="S8.1.p1.6.m2.5.5.2.1.1.3.2.cmml"><</mo><mi id="S8.1.p1.6.m2.5.5.2.1.1.3.3" xref="S8.1.p1.6.m2.5.5.2.1.1.3.3.cmml">lex</mi></msub><mrow id="S8.1.p1.6.m2.5.5.2.1.1.4" xref="S8.1.p1.6.m2.5.5.2.1.1.4.cmml"><mi id="S8.1.p1.6.m2.5.5.2.1.1.4.2" xref="S8.1.p1.6.m2.5.5.2.1.1.4.2.cmml">s</mi><mo id="S8.1.p1.6.m2.5.5.2.1.1.4.1" xref="S8.1.p1.6.m2.5.5.2.1.1.4.1.cmml">∧</mo><mrow id="S8.1.p1.6.m2.5.5.2.1.1.4.3" xref="S8.1.p1.6.m2.5.5.2.1.1.4.3.cmml"><mover accent="true" id="S8.1.p1.6.m2.5.5.2.1.1.4.3.2" xref="S8.1.p1.6.m2.5.5.2.1.1.4.3.2.cmml"><mi id="S8.1.p1.6.m2.5.5.2.1.1.4.3.2.2" xref="S8.1.p1.6.m2.5.5.2.1.1.4.3.2.2.cmml">f</mi><mo id="S8.1.p1.6.m2.5.5.2.1.1.4.3.2.1" xref="S8.1.p1.6.m2.5.5.2.1.1.4.3.2.1.cmml">̊</mo></mover><mo id="S8.1.p1.6.m2.5.5.2.1.1.4.3.1" xref="S8.1.p1.6.m2.5.5.2.1.1.4.3.1.cmml">⁢</mo><mrow id="S8.1.p1.6.m2.5.5.2.1.1.4.3.3.2" xref="S8.1.p1.6.m2.5.5.2.1.1.4.3.cmml"><mo id="S8.1.p1.6.m2.5.5.2.1.1.4.3.3.2.1" stretchy="false" xref="S8.1.p1.6.m2.5.5.2.1.1.4.3.cmml">(</mo><mi id="S8.1.p1.6.m2.2.2" xref="S8.1.p1.6.m2.2.2.cmml">t</mi><mo id="S8.1.p1.6.m2.5.5.2.1.1.4.3.3.2.2" stretchy="false" xref="S8.1.p1.6.m2.5.5.2.1.1.4.3.cmml">)</mo></mrow></mrow></mrow><msub id="S8.1.p1.6.m2.5.5.2.1.1.5" xref="S8.1.p1.6.m2.5.5.2.1.1.5.cmml"><mo id="S8.1.p1.6.m2.5.5.2.1.1.5.2" xref="S8.1.p1.6.m2.5.5.2.1.1.5.2.cmml"><</mo><mi id="S8.1.p1.6.m2.5.5.2.1.1.5.3" xref="S8.1.p1.6.m2.5.5.2.1.1.5.3.cmml">lex</mi></msub><mrow id="S8.1.p1.6.m2.5.5.2.1.1.6" xref="S8.1.p1.6.m2.5.5.2.1.1.6.cmml"><mover accent="true" id="S8.1.p1.6.m2.5.5.2.1.1.6.2" xref="S8.1.p1.6.m2.5.5.2.1.1.6.2.cmml"><mi id="S8.1.p1.6.m2.5.5.2.1.1.6.2.2" xref="S8.1.p1.6.m2.5.5.2.1.1.6.2.2.cmml">f</mi><mo id="S8.1.p1.6.m2.5.5.2.1.1.6.2.1" xref="S8.1.p1.6.m2.5.5.2.1.1.6.2.1.cmml">̊</mo></mover><mo id="S8.1.p1.6.m2.5.5.2.1.1.6.1" xref="S8.1.p1.6.m2.5.5.2.1.1.6.1.cmml">⁢</mo><mrow id="S8.1.p1.6.m2.5.5.2.1.1.6.3.2" xref="S8.1.p1.6.m2.5.5.2.1.1.6.cmml"><mo id="S8.1.p1.6.m2.5.5.2.1.1.6.3.2.1" stretchy="false" xref="S8.1.p1.6.m2.5.5.2.1.1.6.cmml">(</mo><mi id="S8.1.p1.6.m2.3.3" xref="S8.1.p1.6.m2.3.3.cmml">s</mi><mo id="S8.1.p1.6.m2.5.5.2.1.1.6.3.2.2" stretchy="false" xref="S8.1.p1.6.m2.5.5.2.1.1.6.cmml">)</mo></mrow></mrow></mrow><mo id="S8.1.p1.6.m2.5.5.2.1.3" stretchy="false" xref="S8.1.p1.6.m2.5.5.2.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S8.1.p1.6.m2.5b"><apply id="S8.1.p1.6.m2.5.5.cmml" xref="S8.1.p1.6.m2.5.5"><times id="S8.1.p1.6.m2.5.5.3.cmml" xref="S8.1.p1.6.m2.5.5.3"></times><apply id="S8.1.p1.6.m2.4.4.1.1.1.cmml" xref="S8.1.p1.6.m2.4.4.1.1"><in id="S8.1.p1.6.m2.4.4.1.1.1.2.cmml" xref="S8.1.p1.6.m2.4.4.1.1.1.2"></in><list id="S8.1.p1.6.m2.4.4.1.1.1.1.2.cmml" xref="S8.1.p1.6.m2.4.4.1.1.1.1.1"><apply id="S8.1.p1.6.m2.4.4.1.1.1.1.1.1.cmml" xref="S8.1.p1.6.m2.4.4.1.1.1.1.1.1"><exists id="S8.1.p1.6.m2.4.4.1.1.1.1.1.1.1.cmml" xref="S8.1.p1.6.m2.4.4.1.1.1.1.1.1.1"></exists><ci id="S8.1.p1.6.m2.4.4.1.1.1.1.1.1.2.cmml" xref="S8.1.p1.6.m2.4.4.1.1.1.1.1.1.2">𝑡</ci></apply><ci id="S8.1.p1.6.m2.1.1.cmml" xref="S8.1.p1.6.m2.1.1">𝑠</ci></list><apply id="S8.1.p1.6.m2.4.4.1.1.1.3.cmml" xref="S8.1.p1.6.m2.4.4.1.1.1.3"><ci id="S8.1.p1.6.m2.4.4.1.1.1.3.1.cmml" xref="S8.1.p1.6.m2.4.4.1.1.1.3.1">̊</ci><ci id="S8.1.p1.6.m2.4.4.1.1.1.3.2.cmml" xref="S8.1.p1.6.m2.4.4.1.1.1.3.2">𝐴</ci></apply></apply><apply id="S8.1.p1.6.m2.5.5.2.1.1.cmml" xref="S8.1.p1.6.m2.5.5.2.1"><and id="S8.1.p1.6.m2.5.5.2.1.1a.cmml" xref="S8.1.p1.6.m2.5.5.2.1"></and><apply id="S8.1.p1.6.m2.5.5.2.1.1b.cmml" xref="S8.1.p1.6.m2.5.5.2.1"><apply id="S8.1.p1.6.m2.5.5.2.1.1.3.cmml" xref="S8.1.p1.6.m2.5.5.2.1.1.3"><csymbol cd="ambiguous" id="S8.1.p1.6.m2.5.5.2.1.1.3.1.cmml" xref="S8.1.p1.6.m2.5.5.2.1.1.3">subscript</csymbol><lt id="S8.1.p1.6.m2.5.5.2.1.1.3.2.cmml" xref="S8.1.p1.6.m2.5.5.2.1.1.3.2"></lt><ci id="S8.1.p1.6.m2.5.5.2.1.1.3.3.cmml" xref="S8.1.p1.6.m2.5.5.2.1.1.3.3">lex</ci></apply><ci id="S8.1.p1.6.m2.5.5.2.1.1.2.cmml" xref="S8.1.p1.6.m2.5.5.2.1.1.2">𝑡</ci><apply id="S8.1.p1.6.m2.5.5.2.1.1.4.cmml" xref="S8.1.p1.6.m2.5.5.2.1.1.4"><and id="S8.1.p1.6.m2.5.5.2.1.1.4.1.cmml" xref="S8.1.p1.6.m2.5.5.2.1.1.4.1"></and><ci id="S8.1.p1.6.m2.5.5.2.1.1.4.2.cmml" xref="S8.1.p1.6.m2.5.5.2.1.1.4.2">𝑠</ci><apply id="S8.1.p1.6.m2.5.5.2.1.1.4.3.cmml" xref="S8.1.p1.6.m2.5.5.2.1.1.4.3"><times id="S8.1.p1.6.m2.5.5.2.1.1.4.3.1.cmml" xref="S8.1.p1.6.m2.5.5.2.1.1.4.3.1"></times><apply id="S8.1.p1.6.m2.5.5.2.1.1.4.3.2.cmml" xref="S8.1.p1.6.m2.5.5.2.1.1.4.3.2"><ci id="S8.1.p1.6.m2.5.5.2.1.1.4.3.2.1.cmml" xref="S8.1.p1.6.m2.5.5.2.1.1.4.3.2.1">̊</ci><ci id="S8.1.p1.6.m2.5.5.2.1.1.4.3.2.2.cmml" xref="S8.1.p1.6.m2.5.5.2.1.1.4.3.2.2">𝑓</ci></apply><ci id="S8.1.p1.6.m2.2.2.cmml" xref="S8.1.p1.6.m2.2.2">𝑡</ci></apply></apply></apply><apply id="S8.1.p1.6.m2.5.5.2.1.1c.cmml" xref="S8.1.p1.6.m2.5.5.2.1"><apply id="S8.1.p1.6.m2.5.5.2.1.1.5.cmml" xref="S8.1.p1.6.m2.5.5.2.1.1.5"><csymbol cd="ambiguous" id="S8.1.p1.6.m2.5.5.2.1.1.5.1.cmml" xref="S8.1.p1.6.m2.5.5.2.1.1.5">subscript</csymbol><lt id="S8.1.p1.6.m2.5.5.2.1.1.5.2.cmml" xref="S8.1.p1.6.m2.5.5.2.1.1.5.2"></lt><ci id="S8.1.p1.6.m2.5.5.2.1.1.5.3.cmml" xref="S8.1.p1.6.m2.5.5.2.1.1.5.3">lex</ci></apply><share href="https://arxiv.org/html/2503.13728v1#S8.1.p1.6.m2.5.5.2.1.1.4.cmml" id="S8.1.p1.6.m2.5.5.2.1.1d.cmml" xref="S8.1.p1.6.m2.5.5.2.1"></share><apply id="S8.1.p1.6.m2.5.5.2.1.1.6.cmml" xref="S8.1.p1.6.m2.5.5.2.1.1.6"><times id="S8.1.p1.6.m2.5.5.2.1.1.6.1.cmml" xref="S8.1.p1.6.m2.5.5.2.1.1.6.1"></times><apply id="S8.1.p1.6.m2.5.5.2.1.1.6.2.cmml" xref="S8.1.p1.6.m2.5.5.2.1.1.6.2"><ci id="S8.1.p1.6.m2.5.5.2.1.1.6.2.1.cmml" xref="S8.1.p1.6.m2.5.5.2.1.1.6.2.1">̊</ci><ci id="S8.1.p1.6.m2.5.5.2.1.1.6.2.2.cmml" xref="S8.1.p1.6.m2.5.5.2.1.1.6.2.2">𝑓</ci></apply><ci id="S8.1.p1.6.m2.3.3.cmml" xref="S8.1.p1.6.m2.3.3">𝑠</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.1.p1.6.m2.5c">(\exists t,s\in\mathring{A})(t<_{\mathrm{lex}}s\land\mathring{f}(t)<_{\mathrm{% lex}}\mathring{f}(s))</annotation><annotation encoding="application/x-llamapun" id="S8.1.p1.6.m2.5d">( ∃ italic_t , italic_s ∈ over̊ start_ARG italic_A end_ARG ) ( italic_t < start_POSTSUBSCRIPT roman_lex end_POSTSUBSCRIPT italic_s ∧ over̊ start_ARG italic_f end_ARG ( italic_t ) < start_POSTSUBSCRIPT roman_lex end_POSTSUBSCRIPT over̊ start_ARG italic_f end_ARG ( italic_s ) )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S8.2.p2"> <p class="ltx_p" id="S8.2.p2.18">For each <math alttext="q\leq p" class="ltx_Math" display="inline" id="S8.2.p2.1.m1.1"><semantics id="S8.2.p2.1.m1.1a"><mrow id="S8.2.p2.1.m1.1.1" xref="S8.2.p2.1.m1.1.1.cmml"><mi id="S8.2.p2.1.m1.1.1.2" xref="S8.2.p2.1.m1.1.1.2.cmml">q</mi><mo id="S8.2.p2.1.m1.1.1.1" xref="S8.2.p2.1.m1.1.1.1.cmml">≤</mo><mi id="S8.2.p2.1.m1.1.1.3" xref="S8.2.p2.1.m1.1.1.3.cmml">p</mi></mrow><annotation-xml encoding="MathML-Content" id="S8.2.p2.1.m1.1b"><apply id="S8.2.p2.1.m1.1.1.cmml" xref="S8.2.p2.1.m1.1.1"><leq id="S8.2.p2.1.m1.1.1.1.cmml" xref="S8.2.p2.1.m1.1.1.1"></leq><ci id="S8.2.p2.1.m1.1.1.2.cmml" xref="S8.2.p2.1.m1.1.1.2">𝑞</ci><ci id="S8.2.p2.1.m1.1.1.3.cmml" xref="S8.2.p2.1.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.2.p2.1.m1.1c">q\leq p</annotation><annotation encoding="application/x-llamapun" id="S8.2.p2.1.m1.1d">italic_q ≤ italic_p</annotation></semantics></math>, consider the set <math alttext="A_{q}:=\{t\in T:q\Vdash\mathring{c}\circ\check{t}\in\mathring{A}\}" class="ltx_Math" display="inline" id="S8.2.p2.2.m2.2"><semantics id="S8.2.p2.2.m2.2a"><mrow id="S8.2.p2.2.m2.2.2" xref="S8.2.p2.2.m2.2.2.cmml"><msub id="S8.2.p2.2.m2.2.2.4" xref="S8.2.p2.2.m2.2.2.4.cmml"><mi id="S8.2.p2.2.m2.2.2.4.2" xref="S8.2.p2.2.m2.2.2.4.2.cmml">A</mi><mi id="S8.2.p2.2.m2.2.2.4.3" xref="S8.2.p2.2.m2.2.2.4.3.cmml">q</mi></msub><mo id="S8.2.p2.2.m2.2.2.3" lspace="0.278em" rspace="0.278em" xref="S8.2.p2.2.m2.2.2.3.cmml">:=</mo><mrow id="S8.2.p2.2.m2.2.2.2.2" xref="S8.2.p2.2.m2.2.2.2.3.cmml"><mo id="S8.2.p2.2.m2.2.2.2.2.3" stretchy="false" xref="S8.2.p2.2.m2.2.2.2.3.1.cmml">{</mo><mrow id="S8.2.p2.2.m2.1.1.1.1.1" xref="S8.2.p2.2.m2.1.1.1.1.1.cmml"><mi id="S8.2.p2.2.m2.1.1.1.1.1.2" xref="S8.2.p2.2.m2.1.1.1.1.1.2.cmml">t</mi><mo id="S8.2.p2.2.m2.1.1.1.1.1.1" xref="S8.2.p2.2.m2.1.1.1.1.1.1.cmml">∈</mo><mi id="S8.2.p2.2.m2.1.1.1.1.1.3" xref="S8.2.p2.2.m2.1.1.1.1.1.3.cmml">T</mi></mrow><mo id="S8.2.p2.2.m2.2.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S8.2.p2.2.m2.2.2.2.3.1.cmml">:</mo><mrow id="S8.2.p2.2.m2.2.2.2.2.2" xref="S8.2.p2.2.m2.2.2.2.2.2.cmml"><mi id="S8.2.p2.2.m2.2.2.2.2.2.2" xref="S8.2.p2.2.m2.2.2.2.2.2.2.cmml">q</mi><mo id="S8.2.p2.2.m2.2.2.2.2.2.3" xref="S8.2.p2.2.m2.2.2.2.2.2.3.cmml">⊩</mo><mrow id="S8.2.p2.2.m2.2.2.2.2.2.4" xref="S8.2.p2.2.m2.2.2.2.2.2.4.cmml"><mover accent="true" id="S8.2.p2.2.m2.2.2.2.2.2.4.2" xref="S8.2.p2.2.m2.2.2.2.2.2.4.2.cmml"><mi id="S8.2.p2.2.m2.2.2.2.2.2.4.2.2" xref="S8.2.p2.2.m2.2.2.2.2.2.4.2.2.cmml">c</mi><mo id="S8.2.p2.2.m2.2.2.2.2.2.4.2.1" xref="S8.2.p2.2.m2.2.2.2.2.2.4.2.1.cmml">̊</mo></mover><mo id="S8.2.p2.2.m2.2.2.2.2.2.4.1" lspace="0.222em" rspace="0.222em" xref="S8.2.p2.2.m2.2.2.2.2.2.4.1.cmml">∘</mo><mover accent="true" id="S8.2.p2.2.m2.2.2.2.2.2.4.3" xref="S8.2.p2.2.m2.2.2.2.2.2.4.3.cmml"><mi id="S8.2.p2.2.m2.2.2.2.2.2.4.3.2" xref="S8.2.p2.2.m2.2.2.2.2.2.4.3.2.cmml">t</mi><mo id="S8.2.p2.2.m2.2.2.2.2.2.4.3.1" xref="S8.2.p2.2.m2.2.2.2.2.2.4.3.1.cmml">ˇ</mo></mover></mrow><mo id="S8.2.p2.2.m2.2.2.2.2.2.5" xref="S8.2.p2.2.m2.2.2.2.2.2.5.cmml">∈</mo><mover accent="true" id="S8.2.p2.2.m2.2.2.2.2.2.6" xref="S8.2.p2.2.m2.2.2.2.2.2.6.cmml"><mi id="S8.2.p2.2.m2.2.2.2.2.2.6.2" xref="S8.2.p2.2.m2.2.2.2.2.2.6.2.cmml">A</mi><mo id="S8.2.p2.2.m2.2.2.2.2.2.6.1" xref="S8.2.p2.2.m2.2.2.2.2.2.6.1.cmml">̊</mo></mover></mrow><mo id="S8.2.p2.2.m2.2.2.2.2.5" stretchy="false" xref="S8.2.p2.2.m2.2.2.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S8.2.p2.2.m2.2b"><apply id="S8.2.p2.2.m2.2.2.cmml" xref="S8.2.p2.2.m2.2.2"><csymbol cd="latexml" id="S8.2.p2.2.m2.2.2.3.cmml" xref="S8.2.p2.2.m2.2.2.3">assign</csymbol><apply id="S8.2.p2.2.m2.2.2.4.cmml" xref="S8.2.p2.2.m2.2.2.4"><csymbol cd="ambiguous" id="S8.2.p2.2.m2.2.2.4.1.cmml" xref="S8.2.p2.2.m2.2.2.4">subscript</csymbol><ci id="S8.2.p2.2.m2.2.2.4.2.cmml" xref="S8.2.p2.2.m2.2.2.4.2">𝐴</ci><ci id="S8.2.p2.2.m2.2.2.4.3.cmml" xref="S8.2.p2.2.m2.2.2.4.3">𝑞</ci></apply><apply id="S8.2.p2.2.m2.2.2.2.3.cmml" xref="S8.2.p2.2.m2.2.2.2.2"><csymbol cd="latexml" id="S8.2.p2.2.m2.2.2.2.3.1.cmml" xref="S8.2.p2.2.m2.2.2.2.2.3">conditional-set</csymbol><apply id="S8.2.p2.2.m2.1.1.1.1.1.cmml" xref="S8.2.p2.2.m2.1.1.1.1.1"><in id="S8.2.p2.2.m2.1.1.1.1.1.1.cmml" xref="S8.2.p2.2.m2.1.1.1.1.1.1"></in><ci id="S8.2.p2.2.m2.1.1.1.1.1.2.cmml" xref="S8.2.p2.2.m2.1.1.1.1.1.2">𝑡</ci><ci id="S8.2.p2.2.m2.1.1.1.1.1.3.cmml" xref="S8.2.p2.2.m2.1.1.1.1.1.3">𝑇</ci></apply><apply id="S8.2.p2.2.m2.2.2.2.2.2.cmml" xref="S8.2.p2.2.m2.2.2.2.2.2"><and id="S8.2.p2.2.m2.2.2.2.2.2a.cmml" xref="S8.2.p2.2.m2.2.2.2.2.2"></and><apply id="S8.2.p2.2.m2.2.2.2.2.2b.cmml" xref="S8.2.p2.2.m2.2.2.2.2.2"><csymbol cd="latexml" id="S8.2.p2.2.m2.2.2.2.2.2.3.cmml" xref="S8.2.p2.2.m2.2.2.2.2.2.3">forces</csymbol><ci id="S8.2.p2.2.m2.2.2.2.2.2.2.cmml" xref="S8.2.p2.2.m2.2.2.2.2.2.2">𝑞</ci><apply id="S8.2.p2.2.m2.2.2.2.2.2.4.cmml" xref="S8.2.p2.2.m2.2.2.2.2.2.4"><compose id="S8.2.p2.2.m2.2.2.2.2.2.4.1.cmml" xref="S8.2.p2.2.m2.2.2.2.2.2.4.1"></compose><apply id="S8.2.p2.2.m2.2.2.2.2.2.4.2.cmml" xref="S8.2.p2.2.m2.2.2.2.2.2.4.2"><ci id="S8.2.p2.2.m2.2.2.2.2.2.4.2.1.cmml" xref="S8.2.p2.2.m2.2.2.2.2.2.4.2.1">̊</ci><ci id="S8.2.p2.2.m2.2.2.2.2.2.4.2.2.cmml" xref="S8.2.p2.2.m2.2.2.2.2.2.4.2.2">𝑐</ci></apply><apply id="S8.2.p2.2.m2.2.2.2.2.2.4.3.cmml" xref="S8.2.p2.2.m2.2.2.2.2.2.4.3"><ci id="S8.2.p2.2.m2.2.2.2.2.2.4.3.1.cmml" xref="S8.2.p2.2.m2.2.2.2.2.2.4.3.1">ˇ</ci><ci id="S8.2.p2.2.m2.2.2.2.2.2.4.3.2.cmml" xref="S8.2.p2.2.m2.2.2.2.2.2.4.3.2">𝑡</ci></apply></apply></apply><apply id="S8.2.p2.2.m2.2.2.2.2.2c.cmml" xref="S8.2.p2.2.m2.2.2.2.2.2"><in id="S8.2.p2.2.m2.2.2.2.2.2.5.cmml" xref="S8.2.p2.2.m2.2.2.2.2.2.5"></in><share href="https://arxiv.org/html/2503.13728v1#S8.2.p2.2.m2.2.2.2.2.2.4.cmml" id="S8.2.p2.2.m2.2.2.2.2.2d.cmml" xref="S8.2.p2.2.m2.2.2.2.2.2"></share><apply id="S8.2.p2.2.m2.2.2.2.2.2.6.cmml" xref="S8.2.p2.2.m2.2.2.2.2.2.6"><ci id="S8.2.p2.2.m2.2.2.2.2.2.6.1.cmml" xref="S8.2.p2.2.m2.2.2.2.2.2.6.1">̊</ci><ci id="S8.2.p2.2.m2.2.2.2.2.2.6.2.cmml" xref="S8.2.p2.2.m2.2.2.2.2.2.6.2">𝐴</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.2.p2.2.m2.2c">A_{q}:=\{t\in T:q\Vdash\mathring{c}\circ\check{t}\in\mathring{A}\}</annotation><annotation encoding="application/x-llamapun" id="S8.2.p2.2.m2.2d">italic_A start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT := { italic_t ∈ italic_T : italic_q ⊩ over̊ start_ARG italic_c end_ARG ∘ overroman_ˇ start_ARG italic_t end_ARG ∈ over̊ start_ARG italic_A end_ARG }</annotation></semantics></math>. Since <math alttext="\mathbb{P}" class="ltx_Math" display="inline" id="S8.2.p2.3.m3.1"><semantics id="S8.2.p2.3.m3.1a"><mi id="S8.2.p2.3.m3.1.1" xref="S8.2.p2.3.m3.1.1.cmml">ℙ</mi><annotation-xml encoding="MathML-Content" id="S8.2.p2.3.m3.1b"><ci id="S8.2.p2.3.m3.1.1.cmml" xref="S8.2.p2.3.m3.1.1">ℙ</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.2.p2.3.m3.1c">\mathbb{P}</annotation><annotation encoding="application/x-llamapun" id="S8.2.p2.3.m3.1d">blackboard_P</annotation></semantics></math> is countable, there is some <math alttext="q\leq p" class="ltx_Math" display="inline" id="S8.2.p2.4.m4.1"><semantics id="S8.2.p2.4.m4.1a"><mrow id="S8.2.p2.4.m4.1.1" xref="S8.2.p2.4.m4.1.1.cmml"><mi id="S8.2.p2.4.m4.1.1.2" xref="S8.2.p2.4.m4.1.1.2.cmml">q</mi><mo id="S8.2.p2.4.m4.1.1.1" xref="S8.2.p2.4.m4.1.1.1.cmml">≤</mo><mi id="S8.2.p2.4.m4.1.1.3" xref="S8.2.p2.4.m4.1.1.3.cmml">p</mi></mrow><annotation-xml encoding="MathML-Content" id="S8.2.p2.4.m4.1b"><apply id="S8.2.p2.4.m4.1.1.cmml" xref="S8.2.p2.4.m4.1.1"><leq id="S8.2.p2.4.m4.1.1.1.cmml" xref="S8.2.p2.4.m4.1.1.1"></leq><ci id="S8.2.p2.4.m4.1.1.2.cmml" xref="S8.2.p2.4.m4.1.1.2">𝑞</ci><ci id="S8.2.p2.4.m4.1.1.3.cmml" xref="S8.2.p2.4.m4.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.2.p2.4.m4.1c">q\leq p</annotation><annotation encoding="application/x-llamapun" id="S8.2.p2.4.m4.1d">italic_q ≤ italic_p</annotation></semantics></math> such that <math alttext="A_{q}" class="ltx_Math" display="inline" id="S8.2.p2.5.m5.1"><semantics id="S8.2.p2.5.m5.1a"><msub id="S8.2.p2.5.m5.1.1" xref="S8.2.p2.5.m5.1.1.cmml"><mi id="S8.2.p2.5.m5.1.1.2" xref="S8.2.p2.5.m5.1.1.2.cmml">A</mi><mi id="S8.2.p2.5.m5.1.1.3" xref="S8.2.p2.5.m5.1.1.3.cmml">q</mi></msub><annotation-xml encoding="MathML-Content" id="S8.2.p2.5.m5.1b"><apply id="S8.2.p2.5.m5.1.1.cmml" xref="S8.2.p2.5.m5.1.1"><csymbol cd="ambiguous" id="S8.2.p2.5.m5.1.1.1.cmml" xref="S8.2.p2.5.m5.1.1">subscript</csymbol><ci id="S8.2.p2.5.m5.1.1.2.cmml" xref="S8.2.p2.5.m5.1.1.2">𝐴</ci><ci id="S8.2.p2.5.m5.1.1.3.cmml" xref="S8.2.p2.5.m5.1.1.3">𝑞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.2.p2.5.m5.1c">A_{q}</annotation><annotation encoding="application/x-llamapun" id="S8.2.p2.5.m5.1d">italic_A start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT</annotation></semantics></math> is uncountable. Fix such a <math alttext="q" class="ltx_Math" display="inline" id="S8.2.p2.6.m6.1"><semantics id="S8.2.p2.6.m6.1a"><mi id="S8.2.p2.6.m6.1.1" xref="S8.2.p2.6.m6.1.1.cmml">q</mi><annotation-xml encoding="MathML-Content" id="S8.2.p2.6.m6.1b"><ci id="S8.2.p2.6.m6.1.1.cmml" xref="S8.2.p2.6.m6.1.1">𝑞</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.2.p2.6.m6.1c">q</annotation><annotation encoding="application/x-llamapun" id="S8.2.p2.6.m6.1d">italic_q</annotation></semantics></math>. In similar fashion, there is <math alttext="q^{\prime}\leq q" class="ltx_Math" display="inline" id="S8.2.p2.7.m7.1"><semantics id="S8.2.p2.7.m7.1a"><mrow id="S8.2.p2.7.m7.1.1" xref="S8.2.p2.7.m7.1.1.cmml"><msup id="S8.2.p2.7.m7.1.1.2" xref="S8.2.p2.7.m7.1.1.2.cmml"><mi id="S8.2.p2.7.m7.1.1.2.2" xref="S8.2.p2.7.m7.1.1.2.2.cmml">q</mi><mo id="S8.2.p2.7.m7.1.1.2.3" xref="S8.2.p2.7.m7.1.1.2.3.cmml">′</mo></msup><mo id="S8.2.p2.7.m7.1.1.1" xref="S8.2.p2.7.m7.1.1.1.cmml">≤</mo><mi id="S8.2.p2.7.m7.1.1.3" xref="S8.2.p2.7.m7.1.1.3.cmml">q</mi></mrow><annotation-xml encoding="MathML-Content" id="S8.2.p2.7.m7.1b"><apply id="S8.2.p2.7.m7.1.1.cmml" xref="S8.2.p2.7.m7.1.1"><leq id="S8.2.p2.7.m7.1.1.1.cmml" xref="S8.2.p2.7.m7.1.1.1"></leq><apply id="S8.2.p2.7.m7.1.1.2.cmml" xref="S8.2.p2.7.m7.1.1.2"><csymbol cd="ambiguous" id="S8.2.p2.7.m7.1.1.2.1.cmml" xref="S8.2.p2.7.m7.1.1.2">superscript</csymbol><ci id="S8.2.p2.7.m7.1.1.2.2.cmml" xref="S8.2.p2.7.m7.1.1.2.2">𝑞</ci><ci id="S8.2.p2.7.m7.1.1.2.3.cmml" xref="S8.2.p2.7.m7.1.1.2.3">′</ci></apply><ci id="S8.2.p2.7.m7.1.1.3.cmml" xref="S8.2.p2.7.m7.1.1.3">𝑞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.2.p2.7.m7.1c">q^{\prime}\leq q</annotation><annotation encoding="application/x-llamapun" id="S8.2.p2.7.m7.1d">italic_q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_q</annotation></semantics></math>, an uncountable <math alttext="K\subseteq A_{q}" class="ltx_Math" display="inline" id="S8.2.p2.8.m8.1"><semantics id="S8.2.p2.8.m8.1a"><mrow id="S8.2.p2.8.m8.1.1" xref="S8.2.p2.8.m8.1.1.cmml"><mi id="S8.2.p2.8.m8.1.1.2" xref="S8.2.p2.8.m8.1.1.2.cmml">K</mi><mo id="S8.2.p2.8.m8.1.1.1" xref="S8.2.p2.8.m8.1.1.1.cmml">⊆</mo><msub id="S8.2.p2.8.m8.1.1.3" xref="S8.2.p2.8.m8.1.1.3.cmml"><mi id="S8.2.p2.8.m8.1.1.3.2" xref="S8.2.p2.8.m8.1.1.3.2.cmml">A</mi><mi id="S8.2.p2.8.m8.1.1.3.3" xref="S8.2.p2.8.m8.1.1.3.3.cmml">q</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S8.2.p2.8.m8.1b"><apply id="S8.2.p2.8.m8.1.1.cmml" xref="S8.2.p2.8.m8.1.1"><subset id="S8.2.p2.8.m8.1.1.1.cmml" xref="S8.2.p2.8.m8.1.1.1"></subset><ci id="S8.2.p2.8.m8.1.1.2.cmml" xref="S8.2.p2.8.m8.1.1.2">𝐾</ci><apply id="S8.2.p2.8.m8.1.1.3.cmml" xref="S8.2.p2.8.m8.1.1.3"><csymbol cd="ambiguous" id="S8.2.p2.8.m8.1.1.3.1.cmml" xref="S8.2.p2.8.m8.1.1.3">subscript</csymbol><ci id="S8.2.p2.8.m8.1.1.3.2.cmml" xref="S8.2.p2.8.m8.1.1.3.2">𝐴</ci><ci id="S8.2.p2.8.m8.1.1.3.3.cmml" xref="S8.2.p2.8.m8.1.1.3.3">𝑞</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.2.p2.8.m8.1c">K\subseteq A_{q}</annotation><annotation encoding="application/x-llamapun" id="S8.2.p2.8.m8.1d">italic_K ⊆ italic_A start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT</annotation></semantics></math>, and <math alttext="g:K\to T" class="ltx_Math" display="inline" id="S8.2.p2.9.m9.1"><semantics id="S8.2.p2.9.m9.1a"><mrow id="S8.2.p2.9.m9.1.1" xref="S8.2.p2.9.m9.1.1.cmml"><mi id="S8.2.p2.9.m9.1.1.2" xref="S8.2.p2.9.m9.1.1.2.cmml">g</mi><mo id="S8.2.p2.9.m9.1.1.1" lspace="0.278em" rspace="0.278em" xref="S8.2.p2.9.m9.1.1.1.cmml">:</mo><mrow id="S8.2.p2.9.m9.1.1.3" xref="S8.2.p2.9.m9.1.1.3.cmml"><mi id="S8.2.p2.9.m9.1.1.3.2" xref="S8.2.p2.9.m9.1.1.3.2.cmml">K</mi><mo id="S8.2.p2.9.m9.1.1.3.1" stretchy="false" xref="S8.2.p2.9.m9.1.1.3.1.cmml">→</mo><mi id="S8.2.p2.9.m9.1.1.3.3" xref="S8.2.p2.9.m9.1.1.3.3.cmml">T</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S8.2.p2.9.m9.1b"><apply id="S8.2.p2.9.m9.1.1.cmml" xref="S8.2.p2.9.m9.1.1"><ci id="S8.2.p2.9.m9.1.1.1.cmml" xref="S8.2.p2.9.m9.1.1.1">:</ci><ci id="S8.2.p2.9.m9.1.1.2.cmml" xref="S8.2.p2.9.m9.1.1.2">𝑔</ci><apply id="S8.2.p2.9.m9.1.1.3.cmml" xref="S8.2.p2.9.m9.1.1.3"><ci id="S8.2.p2.9.m9.1.1.3.1.cmml" xref="S8.2.p2.9.m9.1.1.3.1">→</ci><ci id="S8.2.p2.9.m9.1.1.3.2.cmml" xref="S8.2.p2.9.m9.1.1.3.2">𝐾</ci><ci id="S8.2.p2.9.m9.1.1.3.3.cmml" xref="S8.2.p2.9.m9.1.1.3.3">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.2.p2.9.m9.1c">g:K\to T</annotation><annotation encoding="application/x-llamapun" id="S8.2.p2.9.m9.1d">italic_g : italic_K → italic_T</annotation></semantics></math> in <math alttext="V" class="ltx_Math" display="inline" id="S8.2.p2.10.m10.1"><semantics id="S8.2.p2.10.m10.1a"><mi id="S8.2.p2.10.m10.1.1" xref="S8.2.p2.10.m10.1.1.cmml">V</mi><annotation-xml encoding="MathML-Content" id="S8.2.p2.10.m10.1b"><ci id="S8.2.p2.10.m10.1.1.cmml" xref="S8.2.p2.10.m10.1.1">𝑉</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.2.p2.10.m10.1c">V</annotation><annotation encoding="application/x-llamapun" id="S8.2.p2.10.m10.1d">italic_V</annotation></semantics></math> such that for all <math alttext="t\in K" class="ltx_Math" display="inline" id="S8.2.p2.11.m11.1"><semantics id="S8.2.p2.11.m11.1a"><mrow id="S8.2.p2.11.m11.1.1" xref="S8.2.p2.11.m11.1.1.cmml"><mi id="S8.2.p2.11.m11.1.1.2" xref="S8.2.p2.11.m11.1.1.2.cmml">t</mi><mo id="S8.2.p2.11.m11.1.1.1" xref="S8.2.p2.11.m11.1.1.1.cmml">∈</mo><mi id="S8.2.p2.11.m11.1.1.3" xref="S8.2.p2.11.m11.1.1.3.cmml">K</mi></mrow><annotation-xml encoding="MathML-Content" id="S8.2.p2.11.m11.1b"><apply id="S8.2.p2.11.m11.1.1.cmml" xref="S8.2.p2.11.m11.1.1"><in id="S8.2.p2.11.m11.1.1.1.cmml" xref="S8.2.p2.11.m11.1.1.1"></in><ci id="S8.2.p2.11.m11.1.1.2.cmml" xref="S8.2.p2.11.m11.1.1.2">𝑡</ci><ci id="S8.2.p2.11.m11.1.1.3.cmml" xref="S8.2.p2.11.m11.1.1.3">𝐾</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.2.p2.11.m11.1c">t\in K</annotation><annotation encoding="application/x-llamapun" id="S8.2.p2.11.m11.1d">italic_t ∈ italic_K</annotation></semantics></math>, <math alttext="q^{\prime}\Vdash\mathring{f}(\mathring{c}\circ\check{t})=\mathring{c}\circ% \check{g}(\check{t})" class="ltx_Math" display="inline" id="S8.2.p2.12.m12.2"><semantics id="S8.2.p2.12.m12.2a"><mrow id="S8.2.p2.12.m12.2.2" xref="S8.2.p2.12.m12.2.2.cmml"><msup id="S8.2.p2.12.m12.2.2.3" xref="S8.2.p2.12.m12.2.2.3.cmml"><mi id="S8.2.p2.12.m12.2.2.3.2" xref="S8.2.p2.12.m12.2.2.3.2.cmml">q</mi><mo id="S8.2.p2.12.m12.2.2.3.3" xref="S8.2.p2.12.m12.2.2.3.3.cmml">′</mo></msup><mo id="S8.2.p2.12.m12.2.2.4" xref="S8.2.p2.12.m12.2.2.4.cmml">⊩</mo><mrow id="S8.2.p2.12.m12.2.2.1" xref="S8.2.p2.12.m12.2.2.1.cmml"><mover accent="true" id="S8.2.p2.12.m12.2.2.1.3" xref="S8.2.p2.12.m12.2.2.1.3.cmml"><mi id="S8.2.p2.12.m12.2.2.1.3.2" xref="S8.2.p2.12.m12.2.2.1.3.2.cmml">f</mi><mo id="S8.2.p2.12.m12.2.2.1.3.1" xref="S8.2.p2.12.m12.2.2.1.3.1.cmml">̊</mo></mover><mo id="S8.2.p2.12.m12.2.2.1.2" xref="S8.2.p2.12.m12.2.2.1.2.cmml">⁢</mo><mrow id="S8.2.p2.12.m12.2.2.1.1.1" xref="S8.2.p2.12.m12.2.2.1.1.1.1.cmml"><mo id="S8.2.p2.12.m12.2.2.1.1.1.2" stretchy="false" xref="S8.2.p2.12.m12.2.2.1.1.1.1.cmml">(</mo><mrow id="S8.2.p2.12.m12.2.2.1.1.1.1" xref="S8.2.p2.12.m12.2.2.1.1.1.1.cmml"><mover accent="true" id="S8.2.p2.12.m12.2.2.1.1.1.1.2" xref="S8.2.p2.12.m12.2.2.1.1.1.1.2.cmml"><mi id="S8.2.p2.12.m12.2.2.1.1.1.1.2.2" xref="S8.2.p2.12.m12.2.2.1.1.1.1.2.2.cmml">c</mi><mo id="S8.2.p2.12.m12.2.2.1.1.1.1.2.1" xref="S8.2.p2.12.m12.2.2.1.1.1.1.2.1.cmml">̊</mo></mover><mo id="S8.2.p2.12.m12.2.2.1.1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S8.2.p2.12.m12.2.2.1.1.1.1.1.cmml">∘</mo><mover accent="true" id="S8.2.p2.12.m12.2.2.1.1.1.1.3" xref="S8.2.p2.12.m12.2.2.1.1.1.1.3.cmml"><mi id="S8.2.p2.12.m12.2.2.1.1.1.1.3.2" xref="S8.2.p2.12.m12.2.2.1.1.1.1.3.2.cmml">t</mi><mo id="S8.2.p2.12.m12.2.2.1.1.1.1.3.1" xref="S8.2.p2.12.m12.2.2.1.1.1.1.3.1.cmml">ˇ</mo></mover></mrow><mo id="S8.2.p2.12.m12.2.2.1.1.1.3" stretchy="false" xref="S8.2.p2.12.m12.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S8.2.p2.12.m12.2.2.5" xref="S8.2.p2.12.m12.2.2.5.cmml">=</mo><mrow id="S8.2.p2.12.m12.2.2.6" xref="S8.2.p2.12.m12.2.2.6.cmml"><mrow id="S8.2.p2.12.m12.2.2.6.2" xref="S8.2.p2.12.m12.2.2.6.2.cmml"><mover accent="true" id="S8.2.p2.12.m12.2.2.6.2.2" xref="S8.2.p2.12.m12.2.2.6.2.2.cmml"><mi id="S8.2.p2.12.m12.2.2.6.2.2.2" xref="S8.2.p2.12.m12.2.2.6.2.2.2.cmml">c</mi><mo id="S8.2.p2.12.m12.2.2.6.2.2.1" xref="S8.2.p2.12.m12.2.2.6.2.2.1.cmml">̊</mo></mover><mo id="S8.2.p2.12.m12.2.2.6.2.1" lspace="0.222em" rspace="0.222em" xref="S8.2.p2.12.m12.2.2.6.2.1.cmml">∘</mo><mover accent="true" id="S8.2.p2.12.m12.2.2.6.2.3" xref="S8.2.p2.12.m12.2.2.6.2.3.cmml"><mi id="S8.2.p2.12.m12.2.2.6.2.3.2" xref="S8.2.p2.12.m12.2.2.6.2.3.2.cmml">g</mi><mo id="S8.2.p2.12.m12.2.2.6.2.3.1" xref="S8.2.p2.12.m12.2.2.6.2.3.1.cmml">ˇ</mo></mover></mrow><mo id="S8.2.p2.12.m12.2.2.6.1" xref="S8.2.p2.12.m12.2.2.6.1.cmml">⁢</mo><mrow id="S8.2.p2.12.m12.2.2.6.3.2" xref="S8.2.p2.12.m12.1.1.cmml"><mo id="S8.2.p2.12.m12.2.2.6.3.2.1" stretchy="false" xref="S8.2.p2.12.m12.1.1.cmml">(</mo><mover accent="true" id="S8.2.p2.12.m12.1.1" xref="S8.2.p2.12.m12.1.1.cmml"><mi id="S8.2.p2.12.m12.1.1.2" xref="S8.2.p2.12.m12.1.1.2.cmml">t</mi><mo id="S8.2.p2.12.m12.1.1.1" xref="S8.2.p2.12.m12.1.1.1.cmml">ˇ</mo></mover><mo id="S8.2.p2.12.m12.2.2.6.3.2.2" stretchy="false" xref="S8.2.p2.12.m12.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S8.2.p2.12.m12.2b"><apply id="S8.2.p2.12.m12.2.2.cmml" xref="S8.2.p2.12.m12.2.2"><and id="S8.2.p2.12.m12.2.2a.cmml" xref="S8.2.p2.12.m12.2.2"></and><apply id="S8.2.p2.12.m12.2.2b.cmml" xref="S8.2.p2.12.m12.2.2"><csymbol cd="latexml" id="S8.2.p2.12.m12.2.2.4.cmml" xref="S8.2.p2.12.m12.2.2.4">forces</csymbol><apply id="S8.2.p2.12.m12.2.2.3.cmml" xref="S8.2.p2.12.m12.2.2.3"><csymbol cd="ambiguous" id="S8.2.p2.12.m12.2.2.3.1.cmml" xref="S8.2.p2.12.m12.2.2.3">superscript</csymbol><ci id="S8.2.p2.12.m12.2.2.3.2.cmml" xref="S8.2.p2.12.m12.2.2.3.2">𝑞</ci><ci id="S8.2.p2.12.m12.2.2.3.3.cmml" xref="S8.2.p2.12.m12.2.2.3.3">′</ci></apply><apply id="S8.2.p2.12.m12.2.2.1.cmml" xref="S8.2.p2.12.m12.2.2.1"><times id="S8.2.p2.12.m12.2.2.1.2.cmml" xref="S8.2.p2.12.m12.2.2.1.2"></times><apply id="S8.2.p2.12.m12.2.2.1.3.cmml" xref="S8.2.p2.12.m12.2.2.1.3"><ci id="S8.2.p2.12.m12.2.2.1.3.1.cmml" xref="S8.2.p2.12.m12.2.2.1.3.1">̊</ci><ci id="S8.2.p2.12.m12.2.2.1.3.2.cmml" xref="S8.2.p2.12.m12.2.2.1.3.2">𝑓</ci></apply><apply id="S8.2.p2.12.m12.2.2.1.1.1.1.cmml" xref="S8.2.p2.12.m12.2.2.1.1.1"><compose id="S8.2.p2.12.m12.2.2.1.1.1.1.1.cmml" xref="S8.2.p2.12.m12.2.2.1.1.1.1.1"></compose><apply id="S8.2.p2.12.m12.2.2.1.1.1.1.2.cmml" xref="S8.2.p2.12.m12.2.2.1.1.1.1.2"><ci id="S8.2.p2.12.m12.2.2.1.1.1.1.2.1.cmml" xref="S8.2.p2.12.m12.2.2.1.1.1.1.2.1">̊</ci><ci id="S8.2.p2.12.m12.2.2.1.1.1.1.2.2.cmml" xref="S8.2.p2.12.m12.2.2.1.1.1.1.2.2">𝑐</ci></apply><apply id="S8.2.p2.12.m12.2.2.1.1.1.1.3.cmml" xref="S8.2.p2.12.m12.2.2.1.1.1.1.3"><ci id="S8.2.p2.12.m12.2.2.1.1.1.1.3.1.cmml" xref="S8.2.p2.12.m12.2.2.1.1.1.1.3.1">ˇ</ci><ci id="S8.2.p2.12.m12.2.2.1.1.1.1.3.2.cmml" xref="S8.2.p2.12.m12.2.2.1.1.1.1.3.2">𝑡</ci></apply></apply></apply></apply><apply id="S8.2.p2.12.m12.2.2c.cmml" xref="S8.2.p2.12.m12.2.2"><eq id="S8.2.p2.12.m12.2.2.5.cmml" xref="S8.2.p2.12.m12.2.2.5"></eq><share href="https://arxiv.org/html/2503.13728v1#S8.2.p2.12.m12.2.2.1.cmml" id="S8.2.p2.12.m12.2.2d.cmml" xref="S8.2.p2.12.m12.2.2"></share><apply id="S8.2.p2.12.m12.2.2.6.cmml" xref="S8.2.p2.12.m12.2.2.6"><times id="S8.2.p2.12.m12.2.2.6.1.cmml" xref="S8.2.p2.12.m12.2.2.6.1"></times><apply id="S8.2.p2.12.m12.2.2.6.2.cmml" xref="S8.2.p2.12.m12.2.2.6.2"><compose id="S8.2.p2.12.m12.2.2.6.2.1.cmml" xref="S8.2.p2.12.m12.2.2.6.2.1"></compose><apply id="S8.2.p2.12.m12.2.2.6.2.2.cmml" xref="S8.2.p2.12.m12.2.2.6.2.2"><ci id="S8.2.p2.12.m12.2.2.6.2.2.1.cmml" xref="S8.2.p2.12.m12.2.2.6.2.2.1">̊</ci><ci id="S8.2.p2.12.m12.2.2.6.2.2.2.cmml" xref="S8.2.p2.12.m12.2.2.6.2.2.2">𝑐</ci></apply><apply id="S8.2.p2.12.m12.2.2.6.2.3.cmml" xref="S8.2.p2.12.m12.2.2.6.2.3"><ci id="S8.2.p2.12.m12.2.2.6.2.3.1.cmml" xref="S8.2.p2.12.m12.2.2.6.2.3.1">ˇ</ci><ci id="S8.2.p2.12.m12.2.2.6.2.3.2.cmml" xref="S8.2.p2.12.m12.2.2.6.2.3.2">𝑔</ci></apply></apply><apply id="S8.2.p2.12.m12.1.1.cmml" xref="S8.2.p2.12.m12.2.2.6.3.2"><ci id="S8.2.p2.12.m12.1.1.1.cmml" xref="S8.2.p2.12.m12.1.1.1">ˇ</ci><ci id="S8.2.p2.12.m12.1.1.2.cmml" xref="S8.2.p2.12.m12.1.1.2">𝑡</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.2.p2.12.m12.2c">q^{\prime}\Vdash\mathring{f}(\mathring{c}\circ\check{t})=\mathring{c}\circ% \check{g}(\check{t})</annotation><annotation encoding="application/x-llamapun" id="S8.2.p2.12.m12.2d">italic_q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⊩ over̊ start_ARG italic_f end_ARG ( over̊ start_ARG italic_c end_ARG ∘ overroman_ˇ start_ARG italic_t end_ARG ) = over̊ start_ARG italic_c end_ARG ∘ overroman_ˇ start_ARG italic_g end_ARG ( overroman_ˇ start_ARG italic_t end_ARG )</annotation></semantics></math>. We may assume that this is already forced by <math alttext="q" class="ltx_Math" display="inline" id="S8.2.p2.13.m13.1"><semantics id="S8.2.p2.13.m13.1a"><mi id="S8.2.p2.13.m13.1.1" xref="S8.2.p2.13.m13.1.1.cmml">q</mi><annotation-xml encoding="MathML-Content" id="S8.2.p2.13.m13.1b"><ci id="S8.2.p2.13.m13.1.1.cmml" xref="S8.2.p2.13.m13.1.1">𝑞</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.2.p2.13.m13.1c">q</annotation><annotation encoding="application/x-llamapun" id="S8.2.p2.13.m13.1d">italic_q</annotation></semantics></math>. Let <math alttext="n" class="ltx_Math" display="inline" id="S8.2.p2.14.m14.1"><semantics id="S8.2.p2.14.m14.1a"><mi id="S8.2.p2.14.m14.1.1" xref="S8.2.p2.14.m14.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S8.2.p2.14.m14.1b"><ci id="S8.2.p2.14.m14.1.1.cmml" xref="S8.2.p2.14.m14.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.2.p2.14.m14.1c">n</annotation><annotation encoding="application/x-llamapun" id="S8.2.p2.14.m14.1d">italic_n</annotation></semantics></math> be greater than any element in <math alttext="\operatorname{dom}(q)" class="ltx_Math" display="inline" id="S8.2.p2.15.m15.2"><semantics id="S8.2.p2.15.m15.2a"><mrow id="S8.2.p2.15.m15.2.3.2" xref="S8.2.p2.15.m15.2.3.1.cmml"><mi id="S8.2.p2.15.m15.1.1" xref="S8.2.p2.15.m15.1.1.cmml">dom</mi><mo id="S8.2.p2.15.m15.2.3.2a" xref="S8.2.p2.15.m15.2.3.1.cmml">⁡</mo><mrow id="S8.2.p2.15.m15.2.3.2.1" xref="S8.2.p2.15.m15.2.3.1.cmml"><mo id="S8.2.p2.15.m15.2.3.2.1.1" stretchy="false" xref="S8.2.p2.15.m15.2.3.1.cmml">(</mo><mi id="S8.2.p2.15.m15.2.2" xref="S8.2.p2.15.m15.2.2.cmml">q</mi><mo id="S8.2.p2.15.m15.2.3.2.1.2" stretchy="false" xref="S8.2.p2.15.m15.2.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S8.2.p2.15.m15.2b"><apply id="S8.2.p2.15.m15.2.3.1.cmml" xref="S8.2.p2.15.m15.2.3.2"><ci id="S8.2.p2.15.m15.1.1.cmml" xref="S8.2.p2.15.m15.1.1">dom</ci><ci id="S8.2.p2.15.m15.2.2.cmml" xref="S8.2.p2.15.m15.2.2">𝑞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.2.p2.15.m15.2c">\operatorname{dom}(q)</annotation><annotation encoding="application/x-llamapun" id="S8.2.p2.15.m15.2d">roman_dom ( italic_q )</annotation></semantics></math>. By going to an uncountable subset of <math alttext="K" class="ltx_Math" display="inline" id="S8.2.p2.16.m16.1"><semantics id="S8.2.p2.16.m16.1a"><mi id="S8.2.p2.16.m16.1.1" xref="S8.2.p2.16.m16.1.1.cmml">K</mi><annotation-xml encoding="MathML-Content" id="S8.2.p2.16.m16.1b"><ci id="S8.2.p2.16.m16.1.1.cmml" xref="S8.2.p2.16.m16.1.1">𝐾</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.2.p2.16.m16.1c">K</annotation><annotation encoding="application/x-llamapun" id="S8.2.p2.16.m16.1d">italic_K</annotation></semantics></math>, we may assume that <math alttext="K" class="ltx_Math" display="inline" id="S8.2.p2.17.m17.1"><semantics id="S8.2.p2.17.m17.1a"><mi id="S8.2.p2.17.m17.1.1" xref="S8.2.p2.17.m17.1.1.cmml">K</mi><annotation-xml encoding="MathML-Content" id="S8.2.p2.17.m17.1b"><ci id="S8.2.p2.17.m17.1.1.cmml" xref="S8.2.p2.17.m17.1.1">𝐾</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.2.p2.17.m17.1c">K</annotation><annotation encoding="application/x-llamapun" id="S8.2.p2.17.m17.1d">italic_K</annotation></semantics></math> and <math alttext="\{g(t):t\in K\}" class="ltx_Math" display="inline" id="S8.2.p2.18.m18.3"><semantics id="S8.2.p2.18.m18.3a"><mrow id="S8.2.p2.18.m18.3.3.2" xref="S8.2.p2.18.m18.3.3.3.cmml"><mo id="S8.2.p2.18.m18.3.3.2.3" stretchy="false" xref="S8.2.p2.18.m18.3.3.3.1.cmml">{</mo><mrow id="S8.2.p2.18.m18.2.2.1.1" xref="S8.2.p2.18.m18.2.2.1.1.cmml"><mi id="S8.2.p2.18.m18.2.2.1.1.2" xref="S8.2.p2.18.m18.2.2.1.1.2.cmml">g</mi><mo id="S8.2.p2.18.m18.2.2.1.1.1" xref="S8.2.p2.18.m18.2.2.1.1.1.cmml">⁢</mo><mrow id="S8.2.p2.18.m18.2.2.1.1.3.2" xref="S8.2.p2.18.m18.2.2.1.1.cmml"><mo id="S8.2.p2.18.m18.2.2.1.1.3.2.1" stretchy="false" xref="S8.2.p2.18.m18.2.2.1.1.cmml">(</mo><mi id="S8.2.p2.18.m18.1.1" xref="S8.2.p2.18.m18.1.1.cmml">t</mi><mo id="S8.2.p2.18.m18.2.2.1.1.3.2.2" rspace="0.278em" stretchy="false" xref="S8.2.p2.18.m18.2.2.1.1.cmml">)</mo></mrow></mrow><mo id="S8.2.p2.18.m18.3.3.2.4" rspace="0.278em" xref="S8.2.p2.18.m18.3.3.3.1.cmml">:</mo><mrow id="S8.2.p2.18.m18.3.3.2.2" xref="S8.2.p2.18.m18.3.3.2.2.cmml"><mi id="S8.2.p2.18.m18.3.3.2.2.2" xref="S8.2.p2.18.m18.3.3.2.2.2.cmml">t</mi><mo id="S8.2.p2.18.m18.3.3.2.2.1" xref="S8.2.p2.18.m18.3.3.2.2.1.cmml">∈</mo><mi id="S8.2.p2.18.m18.3.3.2.2.3" xref="S8.2.p2.18.m18.3.3.2.2.3.cmml">K</mi></mrow><mo id="S8.2.p2.18.m18.3.3.2.5" stretchy="false" xref="S8.2.p2.18.m18.3.3.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S8.2.p2.18.m18.3b"><apply id="S8.2.p2.18.m18.3.3.3.cmml" xref="S8.2.p2.18.m18.3.3.2"><csymbol cd="latexml" id="S8.2.p2.18.m18.3.3.3.1.cmml" xref="S8.2.p2.18.m18.3.3.2.3">conditional-set</csymbol><apply id="S8.2.p2.18.m18.2.2.1.1.cmml" xref="S8.2.p2.18.m18.2.2.1.1"><times id="S8.2.p2.18.m18.2.2.1.1.1.cmml" xref="S8.2.p2.18.m18.2.2.1.1.1"></times><ci id="S8.2.p2.18.m18.2.2.1.1.2.cmml" xref="S8.2.p2.18.m18.2.2.1.1.2">𝑔</ci><ci id="S8.2.p2.18.m18.1.1.cmml" xref="S8.2.p2.18.m18.1.1">𝑡</ci></apply><apply id="S8.2.p2.18.m18.3.3.2.2.cmml" xref="S8.2.p2.18.m18.3.3.2.2"><in id="S8.2.p2.18.m18.3.3.2.2.1.cmml" xref="S8.2.p2.18.m18.3.3.2.2.1"></in><ci id="S8.2.p2.18.m18.3.3.2.2.2.cmml" xref="S8.2.p2.18.m18.3.3.2.2.2">𝑡</ci><ci id="S8.2.p2.18.m18.3.3.2.2.3.cmml" xref="S8.2.p2.18.m18.3.3.2.2.3">𝐾</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.2.p2.18.m18.3c">\{g(t):t\in K\}</annotation><annotation encoding="application/x-llamapun" id="S8.2.p2.18.m18.3d">{ italic_g ( italic_t ) : italic_t ∈ italic_K }</annotation></semantics></math> satisfy the conclusions of <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S8.Thmtheorem3" title="Lemma 8.3. ‣ 8. On a question on Countryman lines ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">8.3</span></a>.</p> </div> <div class="ltx_para" id="S8.3.p3"> <p class="ltx_p" id="S8.3.p3.27">Since <math alttext="T" class="ltx_Math" display="inline" id="S8.3.p3.1.m1.1"><semantics id="S8.3.p3.1.m1.1a"><mi id="S8.3.p3.1.m1.1.1" xref="S8.3.p3.1.m1.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S8.3.p3.1.m1.1b"><ci id="S8.3.p3.1.m1.1.1.cmml" xref="S8.3.p3.1.m1.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.3.p3.1.m1.1c">T</annotation><annotation encoding="application/x-llamapun" id="S8.3.p3.1.m1.1d">italic_T</annotation></semantics></math> is Countryman we see that <math alttext="T" class="ltx_Math" display="inline" id="S8.3.p3.2.m2.1"><semantics id="S8.3.p3.2.m2.1a"><mi id="S8.3.p3.2.m2.1.1" xref="S8.3.p3.2.m2.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S8.3.p3.2.m2.1b"><ci id="S8.3.p3.2.m2.1.1.cmml" xref="S8.3.p3.2.m2.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.3.p3.2.m2.1c">T</annotation><annotation encoding="application/x-llamapun" id="S8.3.p3.2.m2.1d">italic_T</annotation></semantics></math> is <math alttext="\omega_{1}" class="ltx_Math" display="inline" id="S8.3.p3.3.m3.1"><semantics id="S8.3.p3.3.m3.1a"><msub id="S8.3.p3.3.m3.1.1" xref="S8.3.p3.3.m3.1.1.cmml"><mi id="S8.3.p3.3.m3.1.1.2" xref="S8.3.p3.3.m3.1.1.2.cmml">ω</mi><mn id="S8.3.p3.3.m3.1.1.3" xref="S8.3.p3.3.m3.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S8.3.p3.3.m3.1b"><apply id="S8.3.p3.3.m3.1.1.cmml" xref="S8.3.p3.3.m3.1.1"><csymbol cd="ambiguous" id="S8.3.p3.3.m3.1.1.1.cmml" xref="S8.3.p3.3.m3.1.1">subscript</csymbol><ci id="S8.3.p3.3.m3.1.1.2.cmml" xref="S8.3.p3.3.m3.1.1.2">𝜔</ci><cn id="S8.3.p3.3.m3.1.1.3.cmml" type="integer" xref="S8.3.p3.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.3.p3.3.m3.1c">\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S8.3.p3.3.m3.1d">italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-irreversible, and thus there are <math alttext="t,s\in K" class="ltx_Math" display="inline" id="S8.3.p3.4.m4.2"><semantics id="S8.3.p3.4.m4.2a"><mrow id="S8.3.p3.4.m4.2.3" xref="S8.3.p3.4.m4.2.3.cmml"><mrow id="S8.3.p3.4.m4.2.3.2.2" xref="S8.3.p3.4.m4.2.3.2.1.cmml"><mi id="S8.3.p3.4.m4.1.1" xref="S8.3.p3.4.m4.1.1.cmml">t</mi><mo id="S8.3.p3.4.m4.2.3.2.2.1" xref="S8.3.p3.4.m4.2.3.2.1.cmml">,</mo><mi id="S8.3.p3.4.m4.2.2" xref="S8.3.p3.4.m4.2.2.cmml">s</mi></mrow><mo id="S8.3.p3.4.m4.2.3.1" xref="S8.3.p3.4.m4.2.3.1.cmml">∈</mo><mi id="S8.3.p3.4.m4.2.3.3" xref="S8.3.p3.4.m4.2.3.3.cmml">K</mi></mrow><annotation-xml encoding="MathML-Content" id="S8.3.p3.4.m4.2b"><apply id="S8.3.p3.4.m4.2.3.cmml" xref="S8.3.p3.4.m4.2.3"><in id="S8.3.p3.4.m4.2.3.1.cmml" xref="S8.3.p3.4.m4.2.3.1"></in><list id="S8.3.p3.4.m4.2.3.2.1.cmml" xref="S8.3.p3.4.m4.2.3.2.2"><ci id="S8.3.p3.4.m4.1.1.cmml" xref="S8.3.p3.4.m4.1.1">𝑡</ci><ci id="S8.3.p3.4.m4.2.2.cmml" xref="S8.3.p3.4.m4.2.2">𝑠</ci></list><ci id="S8.3.p3.4.m4.2.3.3.cmml" xref="S8.3.p3.4.m4.2.3.3">𝐾</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.3.p3.4.m4.2c">t,s\in K</annotation><annotation encoding="application/x-llamapun" id="S8.3.p3.4.m4.2d">italic_t , italic_s ∈ italic_K</annotation></semantics></math> such that <math alttext="t<_{\mathrm{lex}}s" class="ltx_Math" display="inline" id="S8.3.p3.5.m5.1"><semantics id="S8.3.p3.5.m5.1a"><mrow id="S8.3.p3.5.m5.1.1" xref="S8.3.p3.5.m5.1.1.cmml"><mi id="S8.3.p3.5.m5.1.1.2" xref="S8.3.p3.5.m5.1.1.2.cmml">t</mi><msub id="S8.3.p3.5.m5.1.1.1" xref="S8.3.p3.5.m5.1.1.1.cmml"><mo id="S8.3.p3.5.m5.1.1.1.2" xref="S8.3.p3.5.m5.1.1.1.2.cmml"><</mo><mi id="S8.3.p3.5.m5.1.1.1.3" xref="S8.3.p3.5.m5.1.1.1.3.cmml">lex</mi></msub><mi id="S8.3.p3.5.m5.1.1.3" xref="S8.3.p3.5.m5.1.1.3.cmml">s</mi></mrow><annotation-xml encoding="MathML-Content" id="S8.3.p3.5.m5.1b"><apply id="S8.3.p3.5.m5.1.1.cmml" xref="S8.3.p3.5.m5.1.1"><apply id="S8.3.p3.5.m5.1.1.1.cmml" xref="S8.3.p3.5.m5.1.1.1"><csymbol cd="ambiguous" id="S8.3.p3.5.m5.1.1.1.1.cmml" xref="S8.3.p3.5.m5.1.1.1">subscript</csymbol><lt id="S8.3.p3.5.m5.1.1.1.2.cmml" xref="S8.3.p3.5.m5.1.1.1.2"></lt><ci id="S8.3.p3.5.m5.1.1.1.3.cmml" xref="S8.3.p3.5.m5.1.1.1.3">lex</ci></apply><ci id="S8.3.p3.5.m5.1.1.2.cmml" xref="S8.3.p3.5.m5.1.1.2">𝑡</ci><ci id="S8.3.p3.5.m5.1.1.3.cmml" xref="S8.3.p3.5.m5.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.3.p3.5.m5.1c">t<_{\mathrm{lex}}s</annotation><annotation encoding="application/x-llamapun" id="S8.3.p3.5.m5.1d">italic_t < start_POSTSUBSCRIPT roman_lex end_POSTSUBSCRIPT italic_s</annotation></semantics></math> and <math alttext="g(t)<_{\mathrm{lex}}g(s)" class="ltx_Math" display="inline" id="S8.3.p3.6.m6.2"><semantics id="S8.3.p3.6.m6.2a"><mrow id="S8.3.p3.6.m6.2.3" xref="S8.3.p3.6.m6.2.3.cmml"><mrow id="S8.3.p3.6.m6.2.3.2" xref="S8.3.p3.6.m6.2.3.2.cmml"><mi id="S8.3.p3.6.m6.2.3.2.2" xref="S8.3.p3.6.m6.2.3.2.2.cmml">g</mi><mo id="S8.3.p3.6.m6.2.3.2.1" xref="S8.3.p3.6.m6.2.3.2.1.cmml">⁢</mo><mrow id="S8.3.p3.6.m6.2.3.2.3.2" xref="S8.3.p3.6.m6.2.3.2.cmml"><mo id="S8.3.p3.6.m6.2.3.2.3.2.1" stretchy="false" xref="S8.3.p3.6.m6.2.3.2.cmml">(</mo><mi id="S8.3.p3.6.m6.1.1" xref="S8.3.p3.6.m6.1.1.cmml">t</mi><mo id="S8.3.p3.6.m6.2.3.2.3.2.2" stretchy="false" xref="S8.3.p3.6.m6.2.3.2.cmml">)</mo></mrow></mrow><msub id="S8.3.p3.6.m6.2.3.1" xref="S8.3.p3.6.m6.2.3.1.cmml"><mo id="S8.3.p3.6.m6.2.3.1.2" xref="S8.3.p3.6.m6.2.3.1.2.cmml"><</mo><mi id="S8.3.p3.6.m6.2.3.1.3" xref="S8.3.p3.6.m6.2.3.1.3.cmml">lex</mi></msub><mrow id="S8.3.p3.6.m6.2.3.3" xref="S8.3.p3.6.m6.2.3.3.cmml"><mi id="S8.3.p3.6.m6.2.3.3.2" xref="S8.3.p3.6.m6.2.3.3.2.cmml">g</mi><mo id="S8.3.p3.6.m6.2.3.3.1" xref="S8.3.p3.6.m6.2.3.3.1.cmml">⁢</mo><mrow id="S8.3.p3.6.m6.2.3.3.3.2" xref="S8.3.p3.6.m6.2.3.3.cmml"><mo id="S8.3.p3.6.m6.2.3.3.3.2.1" stretchy="false" xref="S8.3.p3.6.m6.2.3.3.cmml">(</mo><mi id="S8.3.p3.6.m6.2.2" xref="S8.3.p3.6.m6.2.2.cmml">s</mi><mo id="S8.3.p3.6.m6.2.3.3.3.2.2" stretchy="false" xref="S8.3.p3.6.m6.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S8.3.p3.6.m6.2b"><apply id="S8.3.p3.6.m6.2.3.cmml" xref="S8.3.p3.6.m6.2.3"><apply id="S8.3.p3.6.m6.2.3.1.cmml" xref="S8.3.p3.6.m6.2.3.1"><csymbol cd="ambiguous" id="S8.3.p3.6.m6.2.3.1.1.cmml" xref="S8.3.p3.6.m6.2.3.1">subscript</csymbol><lt id="S8.3.p3.6.m6.2.3.1.2.cmml" xref="S8.3.p3.6.m6.2.3.1.2"></lt><ci id="S8.3.p3.6.m6.2.3.1.3.cmml" xref="S8.3.p3.6.m6.2.3.1.3">lex</ci></apply><apply id="S8.3.p3.6.m6.2.3.2.cmml" xref="S8.3.p3.6.m6.2.3.2"><times id="S8.3.p3.6.m6.2.3.2.1.cmml" xref="S8.3.p3.6.m6.2.3.2.1"></times><ci id="S8.3.p3.6.m6.2.3.2.2.cmml" xref="S8.3.p3.6.m6.2.3.2.2">𝑔</ci><ci id="S8.3.p3.6.m6.1.1.cmml" xref="S8.3.p3.6.m6.1.1">𝑡</ci></apply><apply id="S8.3.p3.6.m6.2.3.3.cmml" xref="S8.3.p3.6.m6.2.3.3"><times id="S8.3.p3.6.m6.2.3.3.1.cmml" xref="S8.3.p3.6.m6.2.3.3.1"></times><ci id="S8.3.p3.6.m6.2.3.3.2.cmml" xref="S8.3.p3.6.m6.2.3.3.2">𝑔</ci><ci id="S8.3.p3.6.m6.2.2.cmml" xref="S8.3.p3.6.m6.2.2">𝑠</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.3.p3.6.m6.2c">g(t)<_{\mathrm{lex}}g(s)</annotation><annotation encoding="application/x-llamapun" id="S8.3.p3.6.m6.2d">italic_g ( italic_t ) < start_POSTSUBSCRIPT roman_lex end_POSTSUBSCRIPT italic_g ( italic_s )</annotation></semantics></math>. There are four cases depending on the extension relation of <math alttext="t" class="ltx_Math" display="inline" id="S8.3.p3.7.m7.1"><semantics id="S8.3.p3.7.m7.1a"><mi id="S8.3.p3.7.m7.1.1" xref="S8.3.p3.7.m7.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S8.3.p3.7.m7.1b"><ci id="S8.3.p3.7.m7.1.1.cmml" xref="S8.3.p3.7.m7.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.3.p3.7.m7.1c">t</annotation><annotation encoding="application/x-llamapun" id="S8.3.p3.7.m7.1d">italic_t</annotation></semantics></math> with <math alttext="s" class="ltx_Math" display="inline" id="S8.3.p3.8.m8.1"><semantics id="S8.3.p3.8.m8.1a"><mi id="S8.3.p3.8.m8.1.1" xref="S8.3.p3.8.m8.1.1.cmml">s</mi><annotation-xml encoding="MathML-Content" id="S8.3.p3.8.m8.1b"><ci id="S8.3.p3.8.m8.1.1.cmml" xref="S8.3.p3.8.m8.1.1">𝑠</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.3.p3.8.m8.1c">s</annotation><annotation encoding="application/x-llamapun" id="S8.3.p3.8.m8.1d">italic_s</annotation></semantics></math> and of <math alttext="g(t)" class="ltx_Math" display="inline" id="S8.3.p3.9.m9.1"><semantics id="S8.3.p3.9.m9.1a"><mrow id="S8.3.p3.9.m9.1.2" xref="S8.3.p3.9.m9.1.2.cmml"><mi id="S8.3.p3.9.m9.1.2.2" xref="S8.3.p3.9.m9.1.2.2.cmml">g</mi><mo id="S8.3.p3.9.m9.1.2.1" xref="S8.3.p3.9.m9.1.2.1.cmml">⁢</mo><mrow id="S8.3.p3.9.m9.1.2.3.2" xref="S8.3.p3.9.m9.1.2.cmml"><mo id="S8.3.p3.9.m9.1.2.3.2.1" stretchy="false" xref="S8.3.p3.9.m9.1.2.cmml">(</mo><mi id="S8.3.p3.9.m9.1.1" xref="S8.3.p3.9.m9.1.1.cmml">t</mi><mo id="S8.3.p3.9.m9.1.2.3.2.2" stretchy="false" xref="S8.3.p3.9.m9.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S8.3.p3.9.m9.1b"><apply id="S8.3.p3.9.m9.1.2.cmml" xref="S8.3.p3.9.m9.1.2"><times id="S8.3.p3.9.m9.1.2.1.cmml" xref="S8.3.p3.9.m9.1.2.1"></times><ci id="S8.3.p3.9.m9.1.2.2.cmml" xref="S8.3.p3.9.m9.1.2.2">𝑔</ci><ci id="S8.3.p3.9.m9.1.1.cmml" xref="S8.3.p3.9.m9.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.3.p3.9.m9.1c">g(t)</annotation><annotation encoding="application/x-llamapun" id="S8.3.p3.9.m9.1d">italic_g ( italic_t )</annotation></semantics></math> with <math alttext="g(s)" class="ltx_Math" display="inline" id="S8.3.p3.10.m10.1"><semantics id="S8.3.p3.10.m10.1a"><mrow id="S8.3.p3.10.m10.1.2" xref="S8.3.p3.10.m10.1.2.cmml"><mi id="S8.3.p3.10.m10.1.2.2" xref="S8.3.p3.10.m10.1.2.2.cmml">g</mi><mo id="S8.3.p3.10.m10.1.2.1" xref="S8.3.p3.10.m10.1.2.1.cmml">⁢</mo><mrow id="S8.3.p3.10.m10.1.2.3.2" xref="S8.3.p3.10.m10.1.2.cmml"><mo id="S8.3.p3.10.m10.1.2.3.2.1" stretchy="false" xref="S8.3.p3.10.m10.1.2.cmml">(</mo><mi id="S8.3.p3.10.m10.1.1" xref="S8.3.p3.10.m10.1.1.cmml">s</mi><mo id="S8.3.p3.10.m10.1.2.3.2.2" stretchy="false" xref="S8.3.p3.10.m10.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S8.3.p3.10.m10.1b"><apply id="S8.3.p3.10.m10.1.2.cmml" xref="S8.3.p3.10.m10.1.2"><times id="S8.3.p3.10.m10.1.2.1.cmml" xref="S8.3.p3.10.m10.1.2.1"></times><ci id="S8.3.p3.10.m10.1.2.2.cmml" xref="S8.3.p3.10.m10.1.2.2">𝑔</ci><ci id="S8.3.p3.10.m10.1.1.cmml" xref="S8.3.p3.10.m10.1.1">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.3.p3.10.m10.1c">g(s)</annotation><annotation encoding="application/x-llamapun" id="S8.3.p3.10.m10.1d">italic_g ( italic_s )</annotation></semantics></math>. In case that <math alttext="t\not\sqsubset s" class="ltx_Math" display="inline" id="S8.3.p3.11.m11.1"><semantics id="S8.3.p3.11.m11.1a"><mrow id="S8.3.p3.11.m11.1.1" xref="S8.3.p3.11.m11.1.1.cmml"><mi id="S8.3.p3.11.m11.1.1.2" xref="S8.3.p3.11.m11.1.1.2.cmml">t</mi><mo id="S8.3.p3.11.m11.1.1.1" xref="S8.3.p3.11.m11.1.1.1.cmml">⊏̸</mo><mi id="S8.3.p3.11.m11.1.1.3" xref="S8.3.p3.11.m11.1.1.3.cmml">s</mi></mrow><annotation-xml encoding="MathML-Content" id="S8.3.p3.11.m11.1b"><apply id="S8.3.p3.11.m11.1.1.cmml" xref="S8.3.p3.11.m11.1.1"><csymbol cd="latexml" id="S8.3.p3.11.m11.1.1.1.cmml" xref="S8.3.p3.11.m11.1.1.1">not-square-image-of</csymbol><ci id="S8.3.p3.11.m11.1.1.2.cmml" xref="S8.3.p3.11.m11.1.1.2">𝑡</ci><ci id="S8.3.p3.11.m11.1.1.3.cmml" xref="S8.3.p3.11.m11.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.3.p3.11.m11.1c">t\not\sqsubset s</annotation><annotation encoding="application/x-llamapun" id="S8.3.p3.11.m11.1d">italic_t ⊏̸ italic_s</annotation></semantics></math> we let <math alttext="x" class="ltx_Math" display="inline" id="S8.3.p3.12.m12.1"><semantics id="S8.3.p3.12.m12.1a"><mi id="S8.3.p3.12.m12.1.1" xref="S8.3.p3.12.m12.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S8.3.p3.12.m12.1b"><ci id="S8.3.p3.12.m12.1.1.cmml" xref="S8.3.p3.12.m12.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.3.p3.12.m12.1c">x</annotation><annotation encoding="application/x-llamapun" id="S8.3.p3.12.m12.1d">italic_x</annotation></semantics></math> and <math alttext="y" class="ltx_Math" display="inline" id="S8.3.p3.13.m13.1"><semantics id="S8.3.p3.13.m13.1a"><mi id="S8.3.p3.13.m13.1.1" xref="S8.3.p3.13.m13.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="S8.3.p3.13.m13.1b"><ci id="S8.3.p3.13.m13.1.1.cmml" xref="S8.3.p3.13.m13.1.1">𝑦</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.3.p3.13.m13.1c">y</annotation><annotation encoding="application/x-llamapun" id="S8.3.p3.13.m13.1d">italic_y</annotation></semantics></math> be such that <math alttext="t(\xi)=x" class="ltx_Math" display="inline" id="S8.3.p3.14.m14.1"><semantics id="S8.3.p3.14.m14.1a"><mrow id="S8.3.p3.14.m14.1.2" xref="S8.3.p3.14.m14.1.2.cmml"><mrow id="S8.3.p3.14.m14.1.2.2" xref="S8.3.p3.14.m14.1.2.2.cmml"><mi id="S8.3.p3.14.m14.1.2.2.2" xref="S8.3.p3.14.m14.1.2.2.2.cmml">t</mi><mo id="S8.3.p3.14.m14.1.2.2.1" xref="S8.3.p3.14.m14.1.2.2.1.cmml">⁢</mo><mrow id="S8.3.p3.14.m14.1.2.2.3.2" xref="S8.3.p3.14.m14.1.2.2.cmml"><mo id="S8.3.p3.14.m14.1.2.2.3.2.1" stretchy="false" xref="S8.3.p3.14.m14.1.2.2.cmml">(</mo><mi id="S8.3.p3.14.m14.1.1" xref="S8.3.p3.14.m14.1.1.cmml">ξ</mi><mo id="S8.3.p3.14.m14.1.2.2.3.2.2" stretchy="false" xref="S8.3.p3.14.m14.1.2.2.cmml">)</mo></mrow></mrow><mo id="S8.3.p3.14.m14.1.2.1" xref="S8.3.p3.14.m14.1.2.1.cmml">=</mo><mi id="S8.3.p3.14.m14.1.2.3" xref="S8.3.p3.14.m14.1.2.3.cmml">x</mi></mrow><annotation-xml encoding="MathML-Content" id="S8.3.p3.14.m14.1b"><apply id="S8.3.p3.14.m14.1.2.cmml" xref="S8.3.p3.14.m14.1.2"><eq id="S8.3.p3.14.m14.1.2.1.cmml" xref="S8.3.p3.14.m14.1.2.1"></eq><apply id="S8.3.p3.14.m14.1.2.2.cmml" xref="S8.3.p3.14.m14.1.2.2"><times id="S8.3.p3.14.m14.1.2.2.1.cmml" xref="S8.3.p3.14.m14.1.2.2.1"></times><ci id="S8.3.p3.14.m14.1.2.2.2.cmml" xref="S8.3.p3.14.m14.1.2.2.2">𝑡</ci><ci id="S8.3.p3.14.m14.1.1.cmml" xref="S8.3.p3.14.m14.1.1">𝜉</ci></apply><ci id="S8.3.p3.14.m14.1.2.3.cmml" xref="S8.3.p3.14.m14.1.2.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.3.p3.14.m14.1c">t(\xi)=x</annotation><annotation encoding="application/x-llamapun" id="S8.3.p3.14.m14.1d">italic_t ( italic_ξ ) = italic_x</annotation></semantics></math> and <math alttext="s(\xi)=y" class="ltx_Math" display="inline" id="S8.3.p3.15.m15.1"><semantics id="S8.3.p3.15.m15.1a"><mrow id="S8.3.p3.15.m15.1.2" xref="S8.3.p3.15.m15.1.2.cmml"><mrow id="S8.3.p3.15.m15.1.2.2" xref="S8.3.p3.15.m15.1.2.2.cmml"><mi id="S8.3.p3.15.m15.1.2.2.2" xref="S8.3.p3.15.m15.1.2.2.2.cmml">s</mi><mo id="S8.3.p3.15.m15.1.2.2.1" xref="S8.3.p3.15.m15.1.2.2.1.cmml">⁢</mo><mrow id="S8.3.p3.15.m15.1.2.2.3.2" xref="S8.3.p3.15.m15.1.2.2.cmml"><mo id="S8.3.p3.15.m15.1.2.2.3.2.1" stretchy="false" xref="S8.3.p3.15.m15.1.2.2.cmml">(</mo><mi id="S8.3.p3.15.m15.1.1" xref="S8.3.p3.15.m15.1.1.cmml">ξ</mi><mo id="S8.3.p3.15.m15.1.2.2.3.2.2" stretchy="false" xref="S8.3.p3.15.m15.1.2.2.cmml">)</mo></mrow></mrow><mo id="S8.3.p3.15.m15.1.2.1" xref="S8.3.p3.15.m15.1.2.1.cmml">=</mo><mi id="S8.3.p3.15.m15.1.2.3" xref="S8.3.p3.15.m15.1.2.3.cmml">y</mi></mrow><annotation-xml encoding="MathML-Content" id="S8.3.p3.15.m15.1b"><apply id="S8.3.p3.15.m15.1.2.cmml" xref="S8.3.p3.15.m15.1.2"><eq id="S8.3.p3.15.m15.1.2.1.cmml" xref="S8.3.p3.15.m15.1.2.1"></eq><apply id="S8.3.p3.15.m15.1.2.2.cmml" xref="S8.3.p3.15.m15.1.2.2"><times id="S8.3.p3.15.m15.1.2.2.1.cmml" xref="S8.3.p3.15.m15.1.2.2.1"></times><ci id="S8.3.p3.15.m15.1.2.2.2.cmml" xref="S8.3.p3.15.m15.1.2.2.2">𝑠</ci><ci id="S8.3.p3.15.m15.1.1.cmml" xref="S8.3.p3.15.m15.1.1">𝜉</ci></apply><ci id="S8.3.p3.15.m15.1.2.3.cmml" xref="S8.3.p3.15.m15.1.2.3">𝑦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.3.p3.15.m15.1c">s(\xi)=y</annotation><annotation encoding="application/x-llamapun" id="S8.3.p3.15.m15.1d">italic_s ( italic_ξ ) = italic_y</annotation></semantics></math>, where <math alttext="\xi" class="ltx_Math" display="inline" id="S8.3.p3.16.m16.1"><semantics id="S8.3.p3.16.m16.1a"><mi id="S8.3.p3.16.m16.1.1" xref="S8.3.p3.16.m16.1.1.cmml">ξ</mi><annotation-xml encoding="MathML-Content" id="S8.3.p3.16.m16.1b"><ci id="S8.3.p3.16.m16.1.1.cmml" xref="S8.3.p3.16.m16.1.1">𝜉</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.3.p3.16.m16.1c">\xi</annotation><annotation encoding="application/x-llamapun" id="S8.3.p3.16.m16.1d">italic_ξ</annotation></semantics></math> is the first such that <math alttext="t(\xi)\neq s(\xi)" class="ltx_Math" display="inline" id="S8.3.p3.17.m17.2"><semantics id="S8.3.p3.17.m17.2a"><mrow id="S8.3.p3.17.m17.2.3" xref="S8.3.p3.17.m17.2.3.cmml"><mrow id="S8.3.p3.17.m17.2.3.2" xref="S8.3.p3.17.m17.2.3.2.cmml"><mi id="S8.3.p3.17.m17.2.3.2.2" xref="S8.3.p3.17.m17.2.3.2.2.cmml">t</mi><mo id="S8.3.p3.17.m17.2.3.2.1" xref="S8.3.p3.17.m17.2.3.2.1.cmml">⁢</mo><mrow id="S8.3.p3.17.m17.2.3.2.3.2" xref="S8.3.p3.17.m17.2.3.2.cmml"><mo id="S8.3.p3.17.m17.2.3.2.3.2.1" stretchy="false" xref="S8.3.p3.17.m17.2.3.2.cmml">(</mo><mi id="S8.3.p3.17.m17.1.1" xref="S8.3.p3.17.m17.1.1.cmml">ξ</mi><mo id="S8.3.p3.17.m17.2.3.2.3.2.2" stretchy="false" xref="S8.3.p3.17.m17.2.3.2.cmml">)</mo></mrow></mrow><mo id="S8.3.p3.17.m17.2.3.1" xref="S8.3.p3.17.m17.2.3.1.cmml">≠</mo><mrow id="S8.3.p3.17.m17.2.3.3" xref="S8.3.p3.17.m17.2.3.3.cmml"><mi id="S8.3.p3.17.m17.2.3.3.2" xref="S8.3.p3.17.m17.2.3.3.2.cmml">s</mi><mo id="S8.3.p3.17.m17.2.3.3.1" xref="S8.3.p3.17.m17.2.3.3.1.cmml">⁢</mo><mrow id="S8.3.p3.17.m17.2.3.3.3.2" xref="S8.3.p3.17.m17.2.3.3.cmml"><mo id="S8.3.p3.17.m17.2.3.3.3.2.1" stretchy="false" xref="S8.3.p3.17.m17.2.3.3.cmml">(</mo><mi id="S8.3.p3.17.m17.2.2" xref="S8.3.p3.17.m17.2.2.cmml">ξ</mi><mo id="S8.3.p3.17.m17.2.3.3.3.2.2" stretchy="false" xref="S8.3.p3.17.m17.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S8.3.p3.17.m17.2b"><apply id="S8.3.p3.17.m17.2.3.cmml" xref="S8.3.p3.17.m17.2.3"><neq id="S8.3.p3.17.m17.2.3.1.cmml" xref="S8.3.p3.17.m17.2.3.1"></neq><apply id="S8.3.p3.17.m17.2.3.2.cmml" xref="S8.3.p3.17.m17.2.3.2"><times id="S8.3.p3.17.m17.2.3.2.1.cmml" xref="S8.3.p3.17.m17.2.3.2.1"></times><ci id="S8.3.p3.17.m17.2.3.2.2.cmml" xref="S8.3.p3.17.m17.2.3.2.2">𝑡</ci><ci id="S8.3.p3.17.m17.1.1.cmml" xref="S8.3.p3.17.m17.1.1">𝜉</ci></apply><apply id="S8.3.p3.17.m17.2.3.3.cmml" xref="S8.3.p3.17.m17.2.3.3"><times id="S8.3.p3.17.m17.2.3.3.1.cmml" xref="S8.3.p3.17.m17.2.3.3.1"></times><ci id="S8.3.p3.17.m17.2.3.3.2.cmml" xref="S8.3.p3.17.m17.2.3.3.2">𝑠</ci><ci id="S8.3.p3.17.m17.2.2.cmml" xref="S8.3.p3.17.m17.2.2">𝜉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.3.p3.17.m17.2c">t(\xi)\neq s(\xi)</annotation><annotation encoding="application/x-llamapun" id="S8.3.p3.17.m17.2d">italic_t ( italic_ξ ) ≠ italic_s ( italic_ξ )</annotation></semantics></math>. Otherwise we let <math alttext="x" class="ltx_Math" display="inline" id="S8.3.p3.18.m18.1"><semantics id="S8.3.p3.18.m18.1a"><mi id="S8.3.p3.18.m18.1.1" xref="S8.3.p3.18.m18.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S8.3.p3.18.m18.1b"><ci id="S8.3.p3.18.m18.1.1.cmml" xref="S8.3.p3.18.m18.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.3.p3.18.m18.1c">x</annotation><annotation encoding="application/x-llamapun" id="S8.3.p3.18.m18.1d">italic_x</annotation></semantics></math> and <math alttext="y" class="ltx_Math" display="inline" id="S8.3.p3.19.m19.1"><semantics id="S8.3.p3.19.m19.1a"><mi id="S8.3.p3.19.m19.1.1" xref="S8.3.p3.19.m19.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="S8.3.p3.19.m19.1b"><ci id="S8.3.p3.19.m19.1.1.cmml" xref="S8.3.p3.19.m19.1.1">𝑦</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.3.p3.19.m19.1c">y</annotation><annotation encoding="application/x-llamapun" id="S8.3.p3.19.m19.1d">italic_y</annotation></semantics></math> undefined. In similar fashion, when <math alttext="g(t)\not\sqsubset g(s)" class="ltx_Math" display="inline" id="S8.3.p3.20.m20.2"><semantics id="S8.3.p3.20.m20.2a"><mrow id="S8.3.p3.20.m20.2.3" xref="S8.3.p3.20.m20.2.3.cmml"><mrow id="S8.3.p3.20.m20.2.3.2" xref="S8.3.p3.20.m20.2.3.2.cmml"><mi id="S8.3.p3.20.m20.2.3.2.2" xref="S8.3.p3.20.m20.2.3.2.2.cmml">g</mi><mo id="S8.3.p3.20.m20.2.3.2.1" xref="S8.3.p3.20.m20.2.3.2.1.cmml">⁢</mo><mrow id="S8.3.p3.20.m20.2.3.2.3.2" xref="S8.3.p3.20.m20.2.3.2.cmml"><mo id="S8.3.p3.20.m20.2.3.2.3.2.1" stretchy="false" xref="S8.3.p3.20.m20.2.3.2.cmml">(</mo><mi id="S8.3.p3.20.m20.1.1" xref="S8.3.p3.20.m20.1.1.cmml">t</mi><mo id="S8.3.p3.20.m20.2.3.2.3.2.2" stretchy="false" xref="S8.3.p3.20.m20.2.3.2.cmml">)</mo></mrow></mrow><mo id="S8.3.p3.20.m20.2.3.1" xref="S8.3.p3.20.m20.2.3.1.cmml">⊏̸</mo><mrow id="S8.3.p3.20.m20.2.3.3" xref="S8.3.p3.20.m20.2.3.3.cmml"><mi id="S8.3.p3.20.m20.2.3.3.2" xref="S8.3.p3.20.m20.2.3.3.2.cmml">g</mi><mo id="S8.3.p3.20.m20.2.3.3.1" xref="S8.3.p3.20.m20.2.3.3.1.cmml">⁢</mo><mrow id="S8.3.p3.20.m20.2.3.3.3.2" xref="S8.3.p3.20.m20.2.3.3.cmml"><mo id="S8.3.p3.20.m20.2.3.3.3.2.1" stretchy="false" xref="S8.3.p3.20.m20.2.3.3.cmml">(</mo><mi id="S8.3.p3.20.m20.2.2" xref="S8.3.p3.20.m20.2.2.cmml">s</mi><mo id="S8.3.p3.20.m20.2.3.3.3.2.2" stretchy="false" xref="S8.3.p3.20.m20.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S8.3.p3.20.m20.2b"><apply id="S8.3.p3.20.m20.2.3.cmml" xref="S8.3.p3.20.m20.2.3"><csymbol cd="latexml" id="S8.3.p3.20.m20.2.3.1.cmml" xref="S8.3.p3.20.m20.2.3.1">not-square-image-of</csymbol><apply id="S8.3.p3.20.m20.2.3.2.cmml" xref="S8.3.p3.20.m20.2.3.2"><times id="S8.3.p3.20.m20.2.3.2.1.cmml" xref="S8.3.p3.20.m20.2.3.2.1"></times><ci id="S8.3.p3.20.m20.2.3.2.2.cmml" xref="S8.3.p3.20.m20.2.3.2.2">𝑔</ci><ci id="S8.3.p3.20.m20.1.1.cmml" xref="S8.3.p3.20.m20.1.1">𝑡</ci></apply><apply id="S8.3.p3.20.m20.2.3.3.cmml" xref="S8.3.p3.20.m20.2.3.3"><times id="S8.3.p3.20.m20.2.3.3.1.cmml" xref="S8.3.p3.20.m20.2.3.3.1"></times><ci id="S8.3.p3.20.m20.2.3.3.2.cmml" xref="S8.3.p3.20.m20.2.3.3.2">𝑔</ci><ci id="S8.3.p3.20.m20.2.2.cmml" xref="S8.3.p3.20.m20.2.2">𝑠</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.3.p3.20.m20.2c">g(t)\not\sqsubset g(s)</annotation><annotation encoding="application/x-llamapun" id="S8.3.p3.20.m20.2d">italic_g ( italic_t ) ⊏̸ italic_g ( italic_s )</annotation></semantics></math> we let <math alttext="a=g(t)(\xi)" class="ltx_Math" display="inline" id="S8.3.p3.21.m21.2"><semantics id="S8.3.p3.21.m21.2a"><mrow id="S8.3.p3.21.m21.2.3" xref="S8.3.p3.21.m21.2.3.cmml"><mi id="S8.3.p3.21.m21.2.3.2" xref="S8.3.p3.21.m21.2.3.2.cmml">a</mi><mo id="S8.3.p3.21.m21.2.3.1" xref="S8.3.p3.21.m21.2.3.1.cmml">=</mo><mrow id="S8.3.p3.21.m21.2.3.3" xref="S8.3.p3.21.m21.2.3.3.cmml"><mi id="S8.3.p3.21.m21.2.3.3.2" xref="S8.3.p3.21.m21.2.3.3.2.cmml">g</mi><mo id="S8.3.p3.21.m21.2.3.3.1" xref="S8.3.p3.21.m21.2.3.3.1.cmml">⁢</mo><mrow id="S8.3.p3.21.m21.2.3.3.3.2" xref="S8.3.p3.21.m21.2.3.3.cmml"><mo id="S8.3.p3.21.m21.2.3.3.3.2.1" stretchy="false" xref="S8.3.p3.21.m21.2.3.3.cmml">(</mo><mi id="S8.3.p3.21.m21.1.1" xref="S8.3.p3.21.m21.1.1.cmml">t</mi><mo id="S8.3.p3.21.m21.2.3.3.3.2.2" stretchy="false" xref="S8.3.p3.21.m21.2.3.3.cmml">)</mo></mrow><mo id="S8.3.p3.21.m21.2.3.3.1a" xref="S8.3.p3.21.m21.2.3.3.1.cmml">⁢</mo><mrow id="S8.3.p3.21.m21.2.3.3.4.2" xref="S8.3.p3.21.m21.2.3.3.cmml"><mo id="S8.3.p3.21.m21.2.3.3.4.2.1" stretchy="false" xref="S8.3.p3.21.m21.2.3.3.cmml">(</mo><mi id="S8.3.p3.21.m21.2.2" xref="S8.3.p3.21.m21.2.2.cmml">ξ</mi><mo id="S8.3.p3.21.m21.2.3.3.4.2.2" stretchy="false" xref="S8.3.p3.21.m21.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S8.3.p3.21.m21.2b"><apply id="S8.3.p3.21.m21.2.3.cmml" xref="S8.3.p3.21.m21.2.3"><eq id="S8.3.p3.21.m21.2.3.1.cmml" xref="S8.3.p3.21.m21.2.3.1"></eq><ci id="S8.3.p3.21.m21.2.3.2.cmml" xref="S8.3.p3.21.m21.2.3.2">𝑎</ci><apply id="S8.3.p3.21.m21.2.3.3.cmml" xref="S8.3.p3.21.m21.2.3.3"><times id="S8.3.p3.21.m21.2.3.3.1.cmml" xref="S8.3.p3.21.m21.2.3.3.1"></times><ci id="S8.3.p3.21.m21.2.3.3.2.cmml" xref="S8.3.p3.21.m21.2.3.3.2">𝑔</ci><ci id="S8.3.p3.21.m21.1.1.cmml" xref="S8.3.p3.21.m21.1.1">𝑡</ci><ci id="S8.3.p3.21.m21.2.2.cmml" xref="S8.3.p3.21.m21.2.2">𝜉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.3.p3.21.m21.2c">a=g(t)(\xi)</annotation><annotation encoding="application/x-llamapun" id="S8.3.p3.21.m21.2d">italic_a = italic_g ( italic_t ) ( italic_ξ )</annotation></semantics></math> and <math alttext="b=g(s)(\xi)" class="ltx_Math" display="inline" id="S8.3.p3.22.m22.2"><semantics id="S8.3.p3.22.m22.2a"><mrow id="S8.3.p3.22.m22.2.3" xref="S8.3.p3.22.m22.2.3.cmml"><mi id="S8.3.p3.22.m22.2.3.2" xref="S8.3.p3.22.m22.2.3.2.cmml">b</mi><mo id="S8.3.p3.22.m22.2.3.1" xref="S8.3.p3.22.m22.2.3.1.cmml">=</mo><mrow id="S8.3.p3.22.m22.2.3.3" xref="S8.3.p3.22.m22.2.3.3.cmml"><mi id="S8.3.p3.22.m22.2.3.3.2" xref="S8.3.p3.22.m22.2.3.3.2.cmml">g</mi><mo id="S8.3.p3.22.m22.2.3.3.1" xref="S8.3.p3.22.m22.2.3.3.1.cmml">⁢</mo><mrow id="S8.3.p3.22.m22.2.3.3.3.2" xref="S8.3.p3.22.m22.2.3.3.cmml"><mo id="S8.3.p3.22.m22.2.3.3.3.2.1" stretchy="false" xref="S8.3.p3.22.m22.2.3.3.cmml">(</mo><mi id="S8.3.p3.22.m22.1.1" xref="S8.3.p3.22.m22.1.1.cmml">s</mi><mo id="S8.3.p3.22.m22.2.3.3.3.2.2" stretchy="false" xref="S8.3.p3.22.m22.2.3.3.cmml">)</mo></mrow><mo id="S8.3.p3.22.m22.2.3.3.1a" xref="S8.3.p3.22.m22.2.3.3.1.cmml">⁢</mo><mrow id="S8.3.p3.22.m22.2.3.3.4.2" xref="S8.3.p3.22.m22.2.3.3.cmml"><mo id="S8.3.p3.22.m22.2.3.3.4.2.1" stretchy="false" xref="S8.3.p3.22.m22.2.3.3.cmml">(</mo><mi id="S8.3.p3.22.m22.2.2" xref="S8.3.p3.22.m22.2.2.cmml">ξ</mi><mo id="S8.3.p3.22.m22.2.3.3.4.2.2" stretchy="false" xref="S8.3.p3.22.m22.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S8.3.p3.22.m22.2b"><apply id="S8.3.p3.22.m22.2.3.cmml" xref="S8.3.p3.22.m22.2.3"><eq id="S8.3.p3.22.m22.2.3.1.cmml" xref="S8.3.p3.22.m22.2.3.1"></eq><ci id="S8.3.p3.22.m22.2.3.2.cmml" xref="S8.3.p3.22.m22.2.3.2">𝑏</ci><apply id="S8.3.p3.22.m22.2.3.3.cmml" xref="S8.3.p3.22.m22.2.3.3"><times id="S8.3.p3.22.m22.2.3.3.1.cmml" xref="S8.3.p3.22.m22.2.3.3.1"></times><ci id="S8.3.p3.22.m22.2.3.3.2.cmml" xref="S8.3.p3.22.m22.2.3.3.2">𝑔</ci><ci id="S8.3.p3.22.m22.1.1.cmml" xref="S8.3.p3.22.m22.1.1">𝑠</ci><ci id="S8.3.p3.22.m22.2.2.cmml" xref="S8.3.p3.22.m22.2.2">𝜉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.3.p3.22.m22.2c">b=g(s)(\xi)</annotation><annotation encoding="application/x-llamapun" id="S8.3.p3.22.m22.2d">italic_b = italic_g ( italic_s ) ( italic_ξ )</annotation></semantics></math> where <math alttext="\xi" class="ltx_Math" display="inline" id="S8.3.p3.23.m23.1"><semantics id="S8.3.p3.23.m23.1a"><mi id="S8.3.p3.23.m23.1.1" xref="S8.3.p3.23.m23.1.1.cmml">ξ</mi><annotation-xml encoding="MathML-Content" id="S8.3.p3.23.m23.1b"><ci id="S8.3.p3.23.m23.1.1.cmml" xref="S8.3.p3.23.m23.1.1">𝜉</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.3.p3.23.m23.1c">\xi</annotation><annotation encoding="application/x-llamapun" id="S8.3.p3.23.m23.1d">italic_ξ</annotation></semantics></math> is the first such that <math alttext="g(t)(\xi)\neq g(s)(\xi)" class="ltx_Math" display="inline" id="S8.3.p3.24.m24.4"><semantics id="S8.3.p3.24.m24.4a"><mrow id="S8.3.p3.24.m24.4.5" xref="S8.3.p3.24.m24.4.5.cmml"><mrow id="S8.3.p3.24.m24.4.5.2" xref="S8.3.p3.24.m24.4.5.2.cmml"><mi id="S8.3.p3.24.m24.4.5.2.2" xref="S8.3.p3.24.m24.4.5.2.2.cmml">g</mi><mo id="S8.3.p3.24.m24.4.5.2.1" xref="S8.3.p3.24.m24.4.5.2.1.cmml">⁢</mo><mrow id="S8.3.p3.24.m24.4.5.2.3.2" xref="S8.3.p3.24.m24.4.5.2.cmml"><mo id="S8.3.p3.24.m24.4.5.2.3.2.1" stretchy="false" xref="S8.3.p3.24.m24.4.5.2.cmml">(</mo><mi id="S8.3.p3.24.m24.1.1" xref="S8.3.p3.24.m24.1.1.cmml">t</mi><mo id="S8.3.p3.24.m24.4.5.2.3.2.2" stretchy="false" xref="S8.3.p3.24.m24.4.5.2.cmml">)</mo></mrow><mo id="S8.3.p3.24.m24.4.5.2.1a" xref="S8.3.p3.24.m24.4.5.2.1.cmml">⁢</mo><mrow id="S8.3.p3.24.m24.4.5.2.4.2" xref="S8.3.p3.24.m24.4.5.2.cmml"><mo id="S8.3.p3.24.m24.4.5.2.4.2.1" stretchy="false" xref="S8.3.p3.24.m24.4.5.2.cmml">(</mo><mi id="S8.3.p3.24.m24.2.2" xref="S8.3.p3.24.m24.2.2.cmml">ξ</mi><mo id="S8.3.p3.24.m24.4.5.2.4.2.2" stretchy="false" xref="S8.3.p3.24.m24.4.5.2.cmml">)</mo></mrow></mrow><mo id="S8.3.p3.24.m24.4.5.1" xref="S8.3.p3.24.m24.4.5.1.cmml">≠</mo><mrow id="S8.3.p3.24.m24.4.5.3" xref="S8.3.p3.24.m24.4.5.3.cmml"><mi id="S8.3.p3.24.m24.4.5.3.2" xref="S8.3.p3.24.m24.4.5.3.2.cmml">g</mi><mo id="S8.3.p3.24.m24.4.5.3.1" xref="S8.3.p3.24.m24.4.5.3.1.cmml">⁢</mo><mrow id="S8.3.p3.24.m24.4.5.3.3.2" xref="S8.3.p3.24.m24.4.5.3.cmml"><mo id="S8.3.p3.24.m24.4.5.3.3.2.1" stretchy="false" xref="S8.3.p3.24.m24.4.5.3.cmml">(</mo><mi id="S8.3.p3.24.m24.3.3" xref="S8.3.p3.24.m24.3.3.cmml">s</mi><mo id="S8.3.p3.24.m24.4.5.3.3.2.2" stretchy="false" xref="S8.3.p3.24.m24.4.5.3.cmml">)</mo></mrow><mo id="S8.3.p3.24.m24.4.5.3.1a" xref="S8.3.p3.24.m24.4.5.3.1.cmml">⁢</mo><mrow id="S8.3.p3.24.m24.4.5.3.4.2" xref="S8.3.p3.24.m24.4.5.3.cmml"><mo id="S8.3.p3.24.m24.4.5.3.4.2.1" stretchy="false" xref="S8.3.p3.24.m24.4.5.3.cmml">(</mo><mi id="S8.3.p3.24.m24.4.4" xref="S8.3.p3.24.m24.4.4.cmml">ξ</mi><mo id="S8.3.p3.24.m24.4.5.3.4.2.2" stretchy="false" xref="S8.3.p3.24.m24.4.5.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S8.3.p3.24.m24.4b"><apply id="S8.3.p3.24.m24.4.5.cmml" xref="S8.3.p3.24.m24.4.5"><neq id="S8.3.p3.24.m24.4.5.1.cmml" xref="S8.3.p3.24.m24.4.5.1"></neq><apply id="S8.3.p3.24.m24.4.5.2.cmml" xref="S8.3.p3.24.m24.4.5.2"><times id="S8.3.p3.24.m24.4.5.2.1.cmml" xref="S8.3.p3.24.m24.4.5.2.1"></times><ci id="S8.3.p3.24.m24.4.5.2.2.cmml" xref="S8.3.p3.24.m24.4.5.2.2">𝑔</ci><ci id="S8.3.p3.24.m24.1.1.cmml" xref="S8.3.p3.24.m24.1.1">𝑡</ci><ci id="S8.3.p3.24.m24.2.2.cmml" xref="S8.3.p3.24.m24.2.2">𝜉</ci></apply><apply id="S8.3.p3.24.m24.4.5.3.cmml" xref="S8.3.p3.24.m24.4.5.3"><times id="S8.3.p3.24.m24.4.5.3.1.cmml" xref="S8.3.p3.24.m24.4.5.3.1"></times><ci id="S8.3.p3.24.m24.4.5.3.2.cmml" xref="S8.3.p3.24.m24.4.5.3.2">𝑔</ci><ci id="S8.3.p3.24.m24.3.3.cmml" xref="S8.3.p3.24.m24.3.3">𝑠</ci><ci id="S8.3.p3.24.m24.4.4.cmml" xref="S8.3.p3.24.m24.4.4">𝜉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.3.p3.24.m24.4c">g(t)(\xi)\neq g(s)(\xi)</annotation><annotation encoding="application/x-llamapun" id="S8.3.p3.24.m24.4d">italic_g ( italic_t ) ( italic_ξ ) ≠ italic_g ( italic_s ) ( italic_ξ )</annotation></semantics></math>. Note that when they are defined, <math alttext="a,b,x,y>n" class="ltx_Math" display="inline" id="S8.3.p3.25.m25.4"><semantics id="S8.3.p3.25.m25.4a"><mrow id="S8.3.p3.25.m25.4.5" xref="S8.3.p3.25.m25.4.5.cmml"><mrow id="S8.3.p3.25.m25.4.5.2.2" xref="S8.3.p3.25.m25.4.5.2.1.cmml"><mi id="S8.3.p3.25.m25.1.1" xref="S8.3.p3.25.m25.1.1.cmml">a</mi><mo id="S8.3.p3.25.m25.4.5.2.2.1" xref="S8.3.p3.25.m25.4.5.2.1.cmml">,</mo><mi id="S8.3.p3.25.m25.2.2" xref="S8.3.p3.25.m25.2.2.cmml">b</mi><mo id="S8.3.p3.25.m25.4.5.2.2.2" xref="S8.3.p3.25.m25.4.5.2.1.cmml">,</mo><mi id="S8.3.p3.25.m25.3.3" xref="S8.3.p3.25.m25.3.3.cmml">x</mi><mo id="S8.3.p3.25.m25.4.5.2.2.3" xref="S8.3.p3.25.m25.4.5.2.1.cmml">,</mo><mi id="S8.3.p3.25.m25.4.4" xref="S8.3.p3.25.m25.4.4.cmml">y</mi></mrow><mo id="S8.3.p3.25.m25.4.5.1" xref="S8.3.p3.25.m25.4.5.1.cmml">></mo><mi id="S8.3.p3.25.m25.4.5.3" xref="S8.3.p3.25.m25.4.5.3.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S8.3.p3.25.m25.4b"><apply id="S8.3.p3.25.m25.4.5.cmml" xref="S8.3.p3.25.m25.4.5"><gt id="S8.3.p3.25.m25.4.5.1.cmml" xref="S8.3.p3.25.m25.4.5.1"></gt><list id="S8.3.p3.25.m25.4.5.2.1.cmml" xref="S8.3.p3.25.m25.4.5.2.2"><ci id="S8.3.p3.25.m25.1.1.cmml" xref="S8.3.p3.25.m25.1.1">𝑎</ci><ci id="S8.3.p3.25.m25.2.2.cmml" xref="S8.3.p3.25.m25.2.2">𝑏</ci><ci id="S8.3.p3.25.m25.3.3.cmml" xref="S8.3.p3.25.m25.3.3">𝑥</ci><ci id="S8.3.p3.25.m25.4.4.cmml" xref="S8.3.p3.25.m25.4.4">𝑦</ci></list><ci id="S8.3.p3.25.m25.4.5.3.cmml" xref="S8.3.p3.25.m25.4.5.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.3.p3.25.m25.4c">a,b,x,y>n</annotation><annotation encoding="application/x-llamapun" id="S8.3.p3.25.m25.4d">italic_a , italic_b , italic_x , italic_y > italic_n</annotation></semantics></math>, <math alttext="x<y" class="ltx_Math" display="inline" id="S8.3.p3.26.m26.1"><semantics id="S8.3.p3.26.m26.1a"><mrow id="S8.3.p3.26.m26.1.1" xref="S8.3.p3.26.m26.1.1.cmml"><mi id="S8.3.p3.26.m26.1.1.2" xref="S8.3.p3.26.m26.1.1.2.cmml">x</mi><mo id="S8.3.p3.26.m26.1.1.1" xref="S8.3.p3.26.m26.1.1.1.cmml"><</mo><mi id="S8.3.p3.26.m26.1.1.3" xref="S8.3.p3.26.m26.1.1.3.cmml">y</mi></mrow><annotation-xml encoding="MathML-Content" id="S8.3.p3.26.m26.1b"><apply id="S8.3.p3.26.m26.1.1.cmml" xref="S8.3.p3.26.m26.1.1"><lt id="S8.3.p3.26.m26.1.1.1.cmml" xref="S8.3.p3.26.m26.1.1.1"></lt><ci id="S8.3.p3.26.m26.1.1.2.cmml" xref="S8.3.p3.26.m26.1.1.2">𝑥</ci><ci id="S8.3.p3.26.m26.1.1.3.cmml" xref="S8.3.p3.26.m26.1.1.3">𝑦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.3.p3.26.m26.1c">x<y</annotation><annotation encoding="application/x-llamapun" id="S8.3.p3.26.m26.1d">italic_x < italic_y</annotation></semantics></math> and <math alttext="a<b" class="ltx_Math" display="inline" id="S8.3.p3.27.m27.1"><semantics id="S8.3.p3.27.m27.1a"><mrow id="S8.3.p3.27.m27.1.1" xref="S8.3.p3.27.m27.1.1.cmml"><mi id="S8.3.p3.27.m27.1.1.2" xref="S8.3.p3.27.m27.1.1.2.cmml">a</mi><mo id="S8.3.p3.27.m27.1.1.1" xref="S8.3.p3.27.m27.1.1.1.cmml"><</mo><mi id="S8.3.p3.27.m27.1.1.3" xref="S8.3.p3.27.m27.1.1.3.cmml">b</mi></mrow><annotation-xml encoding="MathML-Content" id="S8.3.p3.27.m27.1b"><apply id="S8.3.p3.27.m27.1.1.cmml" xref="S8.3.p3.27.m27.1.1"><lt id="S8.3.p3.27.m27.1.1.1.cmml" xref="S8.3.p3.27.m27.1.1.1"></lt><ci id="S8.3.p3.27.m27.1.1.2.cmml" xref="S8.3.p3.27.m27.1.1.2">𝑎</ci><ci id="S8.3.p3.27.m27.1.1.3.cmml" xref="S8.3.p3.27.m27.1.1.3">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.3.p3.27.m27.1c">a<b</annotation><annotation encoding="application/x-llamapun" id="S8.3.p3.27.m27.1d">italic_a < italic_b</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S8.4.p4"> <p class="ltx_p" id="S8.4.p4.6">If <math alttext="t\sqsubset s" class="ltx_Math" display="inline" id="S8.4.p4.1.m1.1"><semantics id="S8.4.p4.1.m1.1a"><mrow id="S8.4.p4.1.m1.1.1" xref="S8.4.p4.1.m1.1.1.cmml"><mi id="S8.4.p4.1.m1.1.1.2" xref="S8.4.p4.1.m1.1.1.2.cmml">t</mi><mo id="S8.4.p4.1.m1.1.1.1" xref="S8.4.p4.1.m1.1.1.1.cmml">⊏</mo><mi id="S8.4.p4.1.m1.1.1.3" xref="S8.4.p4.1.m1.1.1.3.cmml">s</mi></mrow><annotation-xml encoding="MathML-Content" id="S8.4.p4.1.m1.1b"><apply id="S8.4.p4.1.m1.1.1.cmml" xref="S8.4.p4.1.m1.1.1"><csymbol cd="latexml" id="S8.4.p4.1.m1.1.1.1.cmml" xref="S8.4.p4.1.m1.1.1.1">square-image-of</csymbol><ci id="S8.4.p4.1.m1.1.1.2.cmml" xref="S8.4.p4.1.m1.1.1.2">𝑡</ci><ci id="S8.4.p4.1.m1.1.1.3.cmml" xref="S8.4.p4.1.m1.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.4.p4.1.m1.1c">t\sqsubset s</annotation><annotation encoding="application/x-llamapun" id="S8.4.p4.1.m1.1d">italic_t ⊏ italic_s</annotation></semantics></math> and <math alttext="g(t)\sqsubset g(s)" class="ltx_Math" display="inline" id="S8.4.p4.2.m2.2"><semantics id="S8.4.p4.2.m2.2a"><mrow id="S8.4.p4.2.m2.2.3" xref="S8.4.p4.2.m2.2.3.cmml"><mrow id="S8.4.p4.2.m2.2.3.2" xref="S8.4.p4.2.m2.2.3.2.cmml"><mi id="S8.4.p4.2.m2.2.3.2.2" xref="S8.4.p4.2.m2.2.3.2.2.cmml">g</mi><mo id="S8.4.p4.2.m2.2.3.2.1" xref="S8.4.p4.2.m2.2.3.2.1.cmml">⁢</mo><mrow id="S8.4.p4.2.m2.2.3.2.3.2" xref="S8.4.p4.2.m2.2.3.2.cmml"><mo id="S8.4.p4.2.m2.2.3.2.3.2.1" stretchy="false" xref="S8.4.p4.2.m2.2.3.2.cmml">(</mo><mi id="S8.4.p4.2.m2.1.1" xref="S8.4.p4.2.m2.1.1.cmml">t</mi><mo id="S8.4.p4.2.m2.2.3.2.3.2.2" stretchy="false" xref="S8.4.p4.2.m2.2.3.2.cmml">)</mo></mrow></mrow><mo id="S8.4.p4.2.m2.2.3.1" xref="S8.4.p4.2.m2.2.3.1.cmml">⊏</mo><mrow id="S8.4.p4.2.m2.2.3.3" xref="S8.4.p4.2.m2.2.3.3.cmml"><mi id="S8.4.p4.2.m2.2.3.3.2" xref="S8.4.p4.2.m2.2.3.3.2.cmml">g</mi><mo id="S8.4.p4.2.m2.2.3.3.1" xref="S8.4.p4.2.m2.2.3.3.1.cmml">⁢</mo><mrow id="S8.4.p4.2.m2.2.3.3.3.2" xref="S8.4.p4.2.m2.2.3.3.cmml"><mo id="S8.4.p4.2.m2.2.3.3.3.2.1" stretchy="false" xref="S8.4.p4.2.m2.2.3.3.cmml">(</mo><mi id="S8.4.p4.2.m2.2.2" xref="S8.4.p4.2.m2.2.2.cmml">s</mi><mo id="S8.4.p4.2.m2.2.3.3.3.2.2" stretchy="false" xref="S8.4.p4.2.m2.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S8.4.p4.2.m2.2b"><apply id="S8.4.p4.2.m2.2.3.cmml" xref="S8.4.p4.2.m2.2.3"><csymbol cd="latexml" id="S8.4.p4.2.m2.2.3.1.cmml" xref="S8.4.p4.2.m2.2.3.1">square-image-of</csymbol><apply id="S8.4.p4.2.m2.2.3.2.cmml" xref="S8.4.p4.2.m2.2.3.2"><times id="S8.4.p4.2.m2.2.3.2.1.cmml" xref="S8.4.p4.2.m2.2.3.2.1"></times><ci id="S8.4.p4.2.m2.2.3.2.2.cmml" xref="S8.4.p4.2.m2.2.3.2.2">𝑔</ci><ci id="S8.4.p4.2.m2.1.1.cmml" xref="S8.4.p4.2.m2.1.1">𝑡</ci></apply><apply id="S8.4.p4.2.m2.2.3.3.cmml" xref="S8.4.p4.2.m2.2.3.3"><times id="S8.4.p4.2.m2.2.3.3.1.cmml" xref="S8.4.p4.2.m2.2.3.3.1"></times><ci id="S8.4.p4.2.m2.2.3.3.2.cmml" xref="S8.4.p4.2.m2.2.3.3.2">𝑔</ci><ci id="S8.4.p4.2.m2.2.2.cmml" xref="S8.4.p4.2.m2.2.2">𝑠</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.4.p4.2.m2.2c">g(t)\sqsubset g(s)</annotation><annotation encoding="application/x-llamapun" id="S8.4.p4.2.m2.2d">italic_g ( italic_t ) ⊏ italic_g ( italic_s )</annotation></semantics></math>, then clearly <math alttext="q\Vdash\mathring{c}\circ\check{t}\sqsubset\mathring{c}\circ\check{s}" class="ltx_Math" display="inline" id="S8.4.p4.3.m3.1"><semantics id="S8.4.p4.3.m3.1a"><mrow id="S8.4.p4.3.m3.1.1" xref="S8.4.p4.3.m3.1.1.cmml"><mi id="S8.4.p4.3.m3.1.1.2" xref="S8.4.p4.3.m3.1.1.2.cmml">q</mi><mo id="S8.4.p4.3.m3.1.1.3" xref="S8.4.p4.3.m3.1.1.3.cmml">⊩</mo><mrow id="S8.4.p4.3.m3.1.1.4" xref="S8.4.p4.3.m3.1.1.4.cmml"><mover accent="true" id="S8.4.p4.3.m3.1.1.4.2" xref="S8.4.p4.3.m3.1.1.4.2.cmml"><mi id="S8.4.p4.3.m3.1.1.4.2.2" xref="S8.4.p4.3.m3.1.1.4.2.2.cmml">c</mi><mo id="S8.4.p4.3.m3.1.1.4.2.1" xref="S8.4.p4.3.m3.1.1.4.2.1.cmml">̊</mo></mover><mo id="S8.4.p4.3.m3.1.1.4.1" lspace="0.222em" rspace="0.222em" xref="S8.4.p4.3.m3.1.1.4.1.cmml">∘</mo><mover accent="true" id="S8.4.p4.3.m3.1.1.4.3" xref="S8.4.p4.3.m3.1.1.4.3.cmml"><mi id="S8.4.p4.3.m3.1.1.4.3.2" xref="S8.4.p4.3.m3.1.1.4.3.2.cmml">t</mi><mo id="S8.4.p4.3.m3.1.1.4.3.1" xref="S8.4.p4.3.m3.1.1.4.3.1.cmml">ˇ</mo></mover></mrow><mo id="S8.4.p4.3.m3.1.1.5" xref="S8.4.p4.3.m3.1.1.5.cmml">⊏</mo><mrow id="S8.4.p4.3.m3.1.1.6" xref="S8.4.p4.3.m3.1.1.6.cmml"><mover accent="true" id="S8.4.p4.3.m3.1.1.6.2" xref="S8.4.p4.3.m3.1.1.6.2.cmml"><mi id="S8.4.p4.3.m3.1.1.6.2.2" xref="S8.4.p4.3.m3.1.1.6.2.2.cmml">c</mi><mo id="S8.4.p4.3.m3.1.1.6.2.1" xref="S8.4.p4.3.m3.1.1.6.2.1.cmml">̊</mo></mover><mo id="S8.4.p4.3.m3.1.1.6.1" lspace="0.222em" rspace="0.222em" xref="S8.4.p4.3.m3.1.1.6.1.cmml">∘</mo><mover accent="true" id="S8.4.p4.3.m3.1.1.6.3" xref="S8.4.p4.3.m3.1.1.6.3.cmml"><mi id="S8.4.p4.3.m3.1.1.6.3.2" xref="S8.4.p4.3.m3.1.1.6.3.2.cmml">s</mi><mo id="S8.4.p4.3.m3.1.1.6.3.1" xref="S8.4.p4.3.m3.1.1.6.3.1.cmml">ˇ</mo></mover></mrow></mrow><annotation-xml encoding="MathML-Content" id="S8.4.p4.3.m3.1b"><apply id="S8.4.p4.3.m3.1.1.cmml" xref="S8.4.p4.3.m3.1.1"><and id="S8.4.p4.3.m3.1.1a.cmml" xref="S8.4.p4.3.m3.1.1"></and><apply id="S8.4.p4.3.m3.1.1b.cmml" xref="S8.4.p4.3.m3.1.1"><csymbol cd="latexml" id="S8.4.p4.3.m3.1.1.3.cmml" xref="S8.4.p4.3.m3.1.1.3">forces</csymbol><ci id="S8.4.p4.3.m3.1.1.2.cmml" xref="S8.4.p4.3.m3.1.1.2">𝑞</ci><apply id="S8.4.p4.3.m3.1.1.4.cmml" xref="S8.4.p4.3.m3.1.1.4"><compose id="S8.4.p4.3.m3.1.1.4.1.cmml" xref="S8.4.p4.3.m3.1.1.4.1"></compose><apply id="S8.4.p4.3.m3.1.1.4.2.cmml" xref="S8.4.p4.3.m3.1.1.4.2"><ci id="S8.4.p4.3.m3.1.1.4.2.1.cmml" xref="S8.4.p4.3.m3.1.1.4.2.1">̊</ci><ci id="S8.4.p4.3.m3.1.1.4.2.2.cmml" xref="S8.4.p4.3.m3.1.1.4.2.2">𝑐</ci></apply><apply id="S8.4.p4.3.m3.1.1.4.3.cmml" xref="S8.4.p4.3.m3.1.1.4.3"><ci id="S8.4.p4.3.m3.1.1.4.3.1.cmml" xref="S8.4.p4.3.m3.1.1.4.3.1">ˇ</ci><ci id="S8.4.p4.3.m3.1.1.4.3.2.cmml" xref="S8.4.p4.3.m3.1.1.4.3.2">𝑡</ci></apply></apply></apply><apply id="S8.4.p4.3.m3.1.1c.cmml" xref="S8.4.p4.3.m3.1.1"><csymbol cd="latexml" id="S8.4.p4.3.m3.1.1.5.cmml" xref="S8.4.p4.3.m3.1.1.5">square-image-of</csymbol><share href="https://arxiv.org/html/2503.13728v1#S8.4.p4.3.m3.1.1.4.cmml" id="S8.4.p4.3.m3.1.1d.cmml" xref="S8.4.p4.3.m3.1.1"></share><apply id="S8.4.p4.3.m3.1.1.6.cmml" xref="S8.4.p4.3.m3.1.1.6"><compose id="S8.4.p4.3.m3.1.1.6.1.cmml" xref="S8.4.p4.3.m3.1.1.6.1"></compose><apply id="S8.4.p4.3.m3.1.1.6.2.cmml" xref="S8.4.p4.3.m3.1.1.6.2"><ci id="S8.4.p4.3.m3.1.1.6.2.1.cmml" xref="S8.4.p4.3.m3.1.1.6.2.1">̊</ci><ci id="S8.4.p4.3.m3.1.1.6.2.2.cmml" xref="S8.4.p4.3.m3.1.1.6.2.2">𝑐</ci></apply><apply id="S8.4.p4.3.m3.1.1.6.3.cmml" xref="S8.4.p4.3.m3.1.1.6.3"><ci id="S8.4.p4.3.m3.1.1.6.3.1.cmml" xref="S8.4.p4.3.m3.1.1.6.3.1">ˇ</ci><ci id="S8.4.p4.3.m3.1.1.6.3.2.cmml" xref="S8.4.p4.3.m3.1.1.6.3.2">𝑠</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.4.p4.3.m3.1c">q\Vdash\mathring{c}\circ\check{t}\sqsubset\mathring{c}\circ\check{s}</annotation><annotation encoding="application/x-llamapun" id="S8.4.p4.3.m3.1d">italic_q ⊩ over̊ start_ARG italic_c end_ARG ∘ overroman_ˇ start_ARG italic_t end_ARG ⊏ over̊ start_ARG italic_c end_ARG ∘ overroman_ˇ start_ARG italic_s end_ARG</annotation></semantics></math> and <math alttext="q\Vdash\mathring{c}\circ\check{g}(\check{t})\sqsubset\mathring{c}\circ\check{g% }(\check{s})" class="ltx_Math" display="inline" id="S8.4.p4.4.m4.2"><semantics id="S8.4.p4.4.m4.2a"><mrow id="S8.4.p4.4.m4.2.3" xref="S8.4.p4.4.m4.2.3.cmml"><mi id="S8.4.p4.4.m4.2.3.2" xref="S8.4.p4.4.m4.2.3.2.cmml">q</mi><mo id="S8.4.p4.4.m4.2.3.3" xref="S8.4.p4.4.m4.2.3.3.cmml">⊩</mo><mrow id="S8.4.p4.4.m4.2.3.4" xref="S8.4.p4.4.m4.2.3.4.cmml"><mrow id="S8.4.p4.4.m4.2.3.4.2" xref="S8.4.p4.4.m4.2.3.4.2.cmml"><mover accent="true" id="S8.4.p4.4.m4.2.3.4.2.2" xref="S8.4.p4.4.m4.2.3.4.2.2.cmml"><mi id="S8.4.p4.4.m4.2.3.4.2.2.2" xref="S8.4.p4.4.m4.2.3.4.2.2.2.cmml">c</mi><mo id="S8.4.p4.4.m4.2.3.4.2.2.1" xref="S8.4.p4.4.m4.2.3.4.2.2.1.cmml">̊</mo></mover><mo id="S8.4.p4.4.m4.2.3.4.2.1" lspace="0.222em" rspace="0.222em" xref="S8.4.p4.4.m4.2.3.4.2.1.cmml">∘</mo><mover accent="true" id="S8.4.p4.4.m4.2.3.4.2.3" xref="S8.4.p4.4.m4.2.3.4.2.3.cmml"><mi id="S8.4.p4.4.m4.2.3.4.2.3.2" xref="S8.4.p4.4.m4.2.3.4.2.3.2.cmml">g</mi><mo id="S8.4.p4.4.m4.2.3.4.2.3.1" xref="S8.4.p4.4.m4.2.3.4.2.3.1.cmml">ˇ</mo></mover></mrow><mo id="S8.4.p4.4.m4.2.3.4.1" xref="S8.4.p4.4.m4.2.3.4.1.cmml">⁢</mo><mrow id="S8.4.p4.4.m4.2.3.4.3.2" xref="S8.4.p4.4.m4.1.1.cmml"><mo id="S8.4.p4.4.m4.2.3.4.3.2.1" stretchy="false" xref="S8.4.p4.4.m4.1.1.cmml">(</mo><mover accent="true" id="S8.4.p4.4.m4.1.1" xref="S8.4.p4.4.m4.1.1.cmml"><mi id="S8.4.p4.4.m4.1.1.2" xref="S8.4.p4.4.m4.1.1.2.cmml">t</mi><mo id="S8.4.p4.4.m4.1.1.1" xref="S8.4.p4.4.m4.1.1.1.cmml">ˇ</mo></mover><mo id="S8.4.p4.4.m4.2.3.4.3.2.2" stretchy="false" xref="S8.4.p4.4.m4.1.1.cmml">)</mo></mrow></mrow><mo id="S8.4.p4.4.m4.2.3.5" xref="S8.4.p4.4.m4.2.3.5.cmml">⊏</mo><mrow id="S8.4.p4.4.m4.2.3.6" xref="S8.4.p4.4.m4.2.3.6.cmml"><mrow id="S8.4.p4.4.m4.2.3.6.2" xref="S8.4.p4.4.m4.2.3.6.2.cmml"><mover accent="true" id="S8.4.p4.4.m4.2.3.6.2.2" xref="S8.4.p4.4.m4.2.3.6.2.2.cmml"><mi id="S8.4.p4.4.m4.2.3.6.2.2.2" xref="S8.4.p4.4.m4.2.3.6.2.2.2.cmml">c</mi><mo id="S8.4.p4.4.m4.2.3.6.2.2.1" xref="S8.4.p4.4.m4.2.3.6.2.2.1.cmml">̊</mo></mover><mo id="S8.4.p4.4.m4.2.3.6.2.1" lspace="0.222em" rspace="0.222em" xref="S8.4.p4.4.m4.2.3.6.2.1.cmml">∘</mo><mover accent="true" id="S8.4.p4.4.m4.2.3.6.2.3" xref="S8.4.p4.4.m4.2.3.6.2.3.cmml"><mi id="S8.4.p4.4.m4.2.3.6.2.3.2" xref="S8.4.p4.4.m4.2.3.6.2.3.2.cmml">g</mi><mo id="S8.4.p4.4.m4.2.3.6.2.3.1" xref="S8.4.p4.4.m4.2.3.6.2.3.1.cmml">ˇ</mo></mover></mrow><mo id="S8.4.p4.4.m4.2.3.6.1" xref="S8.4.p4.4.m4.2.3.6.1.cmml">⁢</mo><mrow id="S8.4.p4.4.m4.2.3.6.3.2" xref="S8.4.p4.4.m4.2.2.cmml"><mo id="S8.4.p4.4.m4.2.3.6.3.2.1" stretchy="false" xref="S8.4.p4.4.m4.2.2.cmml">(</mo><mover accent="true" id="S8.4.p4.4.m4.2.2" xref="S8.4.p4.4.m4.2.2.cmml"><mi id="S8.4.p4.4.m4.2.2.2" xref="S8.4.p4.4.m4.2.2.2.cmml">s</mi><mo id="S8.4.p4.4.m4.2.2.1" xref="S8.4.p4.4.m4.2.2.1.cmml">ˇ</mo></mover><mo id="S8.4.p4.4.m4.2.3.6.3.2.2" stretchy="false" xref="S8.4.p4.4.m4.2.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S8.4.p4.4.m4.2b"><apply id="S8.4.p4.4.m4.2.3.cmml" xref="S8.4.p4.4.m4.2.3"><and id="S8.4.p4.4.m4.2.3a.cmml" xref="S8.4.p4.4.m4.2.3"></and><apply id="S8.4.p4.4.m4.2.3b.cmml" xref="S8.4.p4.4.m4.2.3"><csymbol cd="latexml" id="S8.4.p4.4.m4.2.3.3.cmml" xref="S8.4.p4.4.m4.2.3.3">forces</csymbol><ci id="S8.4.p4.4.m4.2.3.2.cmml" xref="S8.4.p4.4.m4.2.3.2">𝑞</ci><apply id="S8.4.p4.4.m4.2.3.4.cmml" xref="S8.4.p4.4.m4.2.3.4"><times id="S8.4.p4.4.m4.2.3.4.1.cmml" xref="S8.4.p4.4.m4.2.3.4.1"></times><apply id="S8.4.p4.4.m4.2.3.4.2.cmml" xref="S8.4.p4.4.m4.2.3.4.2"><compose id="S8.4.p4.4.m4.2.3.4.2.1.cmml" xref="S8.4.p4.4.m4.2.3.4.2.1"></compose><apply id="S8.4.p4.4.m4.2.3.4.2.2.cmml" xref="S8.4.p4.4.m4.2.3.4.2.2"><ci id="S8.4.p4.4.m4.2.3.4.2.2.1.cmml" xref="S8.4.p4.4.m4.2.3.4.2.2.1">̊</ci><ci id="S8.4.p4.4.m4.2.3.4.2.2.2.cmml" xref="S8.4.p4.4.m4.2.3.4.2.2.2">𝑐</ci></apply><apply id="S8.4.p4.4.m4.2.3.4.2.3.cmml" xref="S8.4.p4.4.m4.2.3.4.2.3"><ci id="S8.4.p4.4.m4.2.3.4.2.3.1.cmml" xref="S8.4.p4.4.m4.2.3.4.2.3.1">ˇ</ci><ci id="S8.4.p4.4.m4.2.3.4.2.3.2.cmml" xref="S8.4.p4.4.m4.2.3.4.2.3.2">𝑔</ci></apply></apply><apply id="S8.4.p4.4.m4.1.1.cmml" xref="S8.4.p4.4.m4.2.3.4.3.2"><ci id="S8.4.p4.4.m4.1.1.1.cmml" xref="S8.4.p4.4.m4.1.1.1">ˇ</ci><ci id="S8.4.p4.4.m4.1.1.2.cmml" xref="S8.4.p4.4.m4.1.1.2">𝑡</ci></apply></apply></apply><apply id="S8.4.p4.4.m4.2.3c.cmml" xref="S8.4.p4.4.m4.2.3"><csymbol cd="latexml" id="S8.4.p4.4.m4.2.3.5.cmml" xref="S8.4.p4.4.m4.2.3.5">square-image-of</csymbol><share href="https://arxiv.org/html/2503.13728v1#S8.4.p4.4.m4.2.3.4.cmml" id="S8.4.p4.4.m4.2.3d.cmml" xref="S8.4.p4.4.m4.2.3"></share><apply id="S8.4.p4.4.m4.2.3.6.cmml" xref="S8.4.p4.4.m4.2.3.6"><times id="S8.4.p4.4.m4.2.3.6.1.cmml" xref="S8.4.p4.4.m4.2.3.6.1"></times><apply id="S8.4.p4.4.m4.2.3.6.2.cmml" xref="S8.4.p4.4.m4.2.3.6.2"><compose id="S8.4.p4.4.m4.2.3.6.2.1.cmml" xref="S8.4.p4.4.m4.2.3.6.2.1"></compose><apply id="S8.4.p4.4.m4.2.3.6.2.2.cmml" xref="S8.4.p4.4.m4.2.3.6.2.2"><ci id="S8.4.p4.4.m4.2.3.6.2.2.1.cmml" xref="S8.4.p4.4.m4.2.3.6.2.2.1">̊</ci><ci id="S8.4.p4.4.m4.2.3.6.2.2.2.cmml" xref="S8.4.p4.4.m4.2.3.6.2.2.2">𝑐</ci></apply><apply id="S8.4.p4.4.m4.2.3.6.2.3.cmml" xref="S8.4.p4.4.m4.2.3.6.2.3"><ci id="S8.4.p4.4.m4.2.3.6.2.3.1.cmml" xref="S8.4.p4.4.m4.2.3.6.2.3.1">ˇ</ci><ci id="S8.4.p4.4.m4.2.3.6.2.3.2.cmml" xref="S8.4.p4.4.m4.2.3.6.2.3.2">𝑔</ci></apply></apply><apply id="S8.4.p4.4.m4.2.2.cmml" xref="S8.4.p4.4.m4.2.3.6.3.2"><ci id="S8.4.p4.4.m4.2.2.1.cmml" xref="S8.4.p4.4.m4.2.2.1">ˇ</ci><ci id="S8.4.p4.4.m4.2.2.2.cmml" xref="S8.4.p4.4.m4.2.2.2">𝑠</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.4.p4.4.m4.2c">q\Vdash\mathring{c}\circ\check{g}(\check{t})\sqsubset\mathring{c}\circ\check{g% }(\check{s})</annotation><annotation encoding="application/x-llamapun" id="S8.4.p4.4.m4.2d">italic_q ⊩ over̊ start_ARG italic_c end_ARG ∘ overroman_ˇ start_ARG italic_g end_ARG ( overroman_ˇ start_ARG italic_t end_ARG ) ⊏ over̊ start_ARG italic_c end_ARG ∘ overroman_ˇ start_ARG italic_g end_ARG ( overroman_ˇ start_ARG italic_s end_ARG )</annotation></semantics></math>. This arrives gives us our desired contradiction, since <math alttext="q\Vdash\mathring{c}\circ\check{t},\mathring{c}\circ\check{s}\in\mathring{A}" class="ltx_Math" display="inline" id="S8.4.p4.5.m5.2"><semantics id="S8.4.p4.5.m5.2a"><mrow id="S8.4.p4.5.m5.2.2.2" xref="S8.4.p4.5.m5.2.2.3.cmml"><mrow id="S8.4.p4.5.m5.1.1.1.1" xref="S8.4.p4.5.m5.1.1.1.1.cmml"><mi id="S8.4.p4.5.m5.1.1.1.1.2" xref="S8.4.p4.5.m5.1.1.1.1.2.cmml">q</mi><mo id="S8.4.p4.5.m5.1.1.1.1.1" xref="S8.4.p4.5.m5.1.1.1.1.1.cmml">⊩</mo><mrow id="S8.4.p4.5.m5.1.1.1.1.3" xref="S8.4.p4.5.m5.1.1.1.1.3.cmml"><mover accent="true" id="S8.4.p4.5.m5.1.1.1.1.3.2" xref="S8.4.p4.5.m5.1.1.1.1.3.2.cmml"><mi id="S8.4.p4.5.m5.1.1.1.1.3.2.2" xref="S8.4.p4.5.m5.1.1.1.1.3.2.2.cmml">c</mi><mo id="S8.4.p4.5.m5.1.1.1.1.3.2.1" xref="S8.4.p4.5.m5.1.1.1.1.3.2.1.cmml">̊</mo></mover><mo id="S8.4.p4.5.m5.1.1.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S8.4.p4.5.m5.1.1.1.1.3.1.cmml">∘</mo><mover accent="true" id="S8.4.p4.5.m5.1.1.1.1.3.3" xref="S8.4.p4.5.m5.1.1.1.1.3.3.cmml"><mi id="S8.4.p4.5.m5.1.1.1.1.3.3.2" xref="S8.4.p4.5.m5.1.1.1.1.3.3.2.cmml">t</mi><mo id="S8.4.p4.5.m5.1.1.1.1.3.3.1" xref="S8.4.p4.5.m5.1.1.1.1.3.3.1.cmml">ˇ</mo></mover></mrow></mrow><mo id="S8.4.p4.5.m5.2.2.2.3" xref="S8.4.p4.5.m5.2.2.3a.cmml">,</mo><mrow id="S8.4.p4.5.m5.2.2.2.2" xref="S8.4.p4.5.m5.2.2.2.2.cmml"><mrow id="S8.4.p4.5.m5.2.2.2.2.2" xref="S8.4.p4.5.m5.2.2.2.2.2.cmml"><mover accent="true" id="S8.4.p4.5.m5.2.2.2.2.2.2" xref="S8.4.p4.5.m5.2.2.2.2.2.2.cmml"><mi id="S8.4.p4.5.m5.2.2.2.2.2.2.2" xref="S8.4.p4.5.m5.2.2.2.2.2.2.2.cmml">c</mi><mo id="S8.4.p4.5.m5.2.2.2.2.2.2.1" xref="S8.4.p4.5.m5.2.2.2.2.2.2.1.cmml">̊</mo></mover><mo id="S8.4.p4.5.m5.2.2.2.2.2.1" lspace="0.222em" rspace="0.222em" xref="S8.4.p4.5.m5.2.2.2.2.2.1.cmml">∘</mo><mover accent="true" id="S8.4.p4.5.m5.2.2.2.2.2.3" xref="S8.4.p4.5.m5.2.2.2.2.2.3.cmml"><mi id="S8.4.p4.5.m5.2.2.2.2.2.3.2" xref="S8.4.p4.5.m5.2.2.2.2.2.3.2.cmml">s</mi><mo id="S8.4.p4.5.m5.2.2.2.2.2.3.1" xref="S8.4.p4.5.m5.2.2.2.2.2.3.1.cmml">ˇ</mo></mover></mrow><mo id="S8.4.p4.5.m5.2.2.2.2.1" xref="S8.4.p4.5.m5.2.2.2.2.1.cmml">∈</mo><mover accent="true" id="S8.4.p4.5.m5.2.2.2.2.3" xref="S8.4.p4.5.m5.2.2.2.2.3.cmml"><mi id="S8.4.p4.5.m5.2.2.2.2.3.2" xref="S8.4.p4.5.m5.2.2.2.2.3.2.cmml">A</mi><mo id="S8.4.p4.5.m5.2.2.2.2.3.1" xref="S8.4.p4.5.m5.2.2.2.2.3.1.cmml">̊</mo></mover></mrow></mrow><annotation-xml encoding="MathML-Content" id="S8.4.p4.5.m5.2b"><apply id="S8.4.p4.5.m5.2.2.3.cmml" xref="S8.4.p4.5.m5.2.2.2"><csymbol cd="ambiguous" id="S8.4.p4.5.m5.2.2.3a.cmml" xref="S8.4.p4.5.m5.2.2.2.3">formulae-sequence</csymbol><apply id="S8.4.p4.5.m5.1.1.1.1.cmml" xref="S8.4.p4.5.m5.1.1.1.1"><csymbol cd="latexml" id="S8.4.p4.5.m5.1.1.1.1.1.cmml" xref="S8.4.p4.5.m5.1.1.1.1.1">forces</csymbol><ci id="S8.4.p4.5.m5.1.1.1.1.2.cmml" xref="S8.4.p4.5.m5.1.1.1.1.2">𝑞</ci><apply id="S8.4.p4.5.m5.1.1.1.1.3.cmml" xref="S8.4.p4.5.m5.1.1.1.1.3"><compose id="S8.4.p4.5.m5.1.1.1.1.3.1.cmml" xref="S8.4.p4.5.m5.1.1.1.1.3.1"></compose><apply id="S8.4.p4.5.m5.1.1.1.1.3.2.cmml" xref="S8.4.p4.5.m5.1.1.1.1.3.2"><ci id="S8.4.p4.5.m5.1.1.1.1.3.2.1.cmml" xref="S8.4.p4.5.m5.1.1.1.1.3.2.1">̊</ci><ci id="S8.4.p4.5.m5.1.1.1.1.3.2.2.cmml" xref="S8.4.p4.5.m5.1.1.1.1.3.2.2">𝑐</ci></apply><apply id="S8.4.p4.5.m5.1.1.1.1.3.3.cmml" xref="S8.4.p4.5.m5.1.1.1.1.3.3"><ci id="S8.4.p4.5.m5.1.1.1.1.3.3.1.cmml" xref="S8.4.p4.5.m5.1.1.1.1.3.3.1">ˇ</ci><ci id="S8.4.p4.5.m5.1.1.1.1.3.3.2.cmml" xref="S8.4.p4.5.m5.1.1.1.1.3.3.2">𝑡</ci></apply></apply></apply><apply id="S8.4.p4.5.m5.2.2.2.2.cmml" xref="S8.4.p4.5.m5.2.2.2.2"><in id="S8.4.p4.5.m5.2.2.2.2.1.cmml" xref="S8.4.p4.5.m5.2.2.2.2.1"></in><apply id="S8.4.p4.5.m5.2.2.2.2.2.cmml" xref="S8.4.p4.5.m5.2.2.2.2.2"><compose id="S8.4.p4.5.m5.2.2.2.2.2.1.cmml" xref="S8.4.p4.5.m5.2.2.2.2.2.1"></compose><apply id="S8.4.p4.5.m5.2.2.2.2.2.2.cmml" xref="S8.4.p4.5.m5.2.2.2.2.2.2"><ci id="S8.4.p4.5.m5.2.2.2.2.2.2.1.cmml" xref="S8.4.p4.5.m5.2.2.2.2.2.2.1">̊</ci><ci id="S8.4.p4.5.m5.2.2.2.2.2.2.2.cmml" xref="S8.4.p4.5.m5.2.2.2.2.2.2.2">𝑐</ci></apply><apply id="S8.4.p4.5.m5.2.2.2.2.2.3.cmml" xref="S8.4.p4.5.m5.2.2.2.2.2.3"><ci id="S8.4.p4.5.m5.2.2.2.2.2.3.1.cmml" xref="S8.4.p4.5.m5.2.2.2.2.2.3.1">ˇ</ci><ci id="S8.4.p4.5.m5.2.2.2.2.2.3.2.cmml" xref="S8.4.p4.5.m5.2.2.2.2.2.3.2">𝑠</ci></apply></apply><apply id="S8.4.p4.5.m5.2.2.2.2.3.cmml" xref="S8.4.p4.5.m5.2.2.2.2.3"><ci id="S8.4.p4.5.m5.2.2.2.2.3.1.cmml" xref="S8.4.p4.5.m5.2.2.2.2.3.1">̊</ci><ci id="S8.4.p4.5.m5.2.2.2.2.3.2.cmml" xref="S8.4.p4.5.m5.2.2.2.2.3.2">𝐴</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.4.p4.5.m5.2c">q\Vdash\mathring{c}\circ\check{t},\mathring{c}\circ\check{s}\in\mathring{A}</annotation><annotation encoding="application/x-llamapun" id="S8.4.p4.5.m5.2d">italic_q ⊩ over̊ start_ARG italic_c end_ARG ∘ overroman_ˇ start_ARG italic_t end_ARG , over̊ start_ARG italic_c end_ARG ∘ overroman_ˇ start_ARG italic_s end_ARG ∈ over̊ start_ARG italic_A end_ARG</annotation></semantics></math> and <math alttext="q\Vdash\mathring{c}\circ\check{g}(\check{t})=\mathring{f}(\mathring{c}\circ% \check{t})\land\mathring{c}\circ\check{g}(\check{s})=\mathring{f}(\mathring{c}% \circ\check{s})" class="ltx_Math" display="inline" id="S8.4.p4.6.m6.4"><semantics id="S8.4.p4.6.m6.4a"><mrow id="S8.4.p4.6.m6.4.4" xref="S8.4.p4.6.m6.4.4.cmml"><mi id="S8.4.p4.6.m6.4.4.4" xref="S8.4.p4.6.m6.4.4.4.cmml">q</mi><mo id="S8.4.p4.6.m6.4.4.5" xref="S8.4.p4.6.m6.4.4.5.cmml">⊩</mo><mrow id="S8.4.p4.6.m6.4.4.6" xref="S8.4.p4.6.m6.4.4.6.cmml"><mrow id="S8.4.p4.6.m6.4.4.6.2" xref="S8.4.p4.6.m6.4.4.6.2.cmml"><mover accent="true" id="S8.4.p4.6.m6.4.4.6.2.2" xref="S8.4.p4.6.m6.4.4.6.2.2.cmml"><mi id="S8.4.p4.6.m6.4.4.6.2.2.2" xref="S8.4.p4.6.m6.4.4.6.2.2.2.cmml">c</mi><mo id="S8.4.p4.6.m6.4.4.6.2.2.1" xref="S8.4.p4.6.m6.4.4.6.2.2.1.cmml">̊</mo></mover><mo id="S8.4.p4.6.m6.4.4.6.2.1" lspace="0.222em" rspace="0.222em" xref="S8.4.p4.6.m6.4.4.6.2.1.cmml">∘</mo><mover accent="true" id="S8.4.p4.6.m6.4.4.6.2.3" xref="S8.4.p4.6.m6.4.4.6.2.3.cmml"><mi id="S8.4.p4.6.m6.4.4.6.2.3.2" xref="S8.4.p4.6.m6.4.4.6.2.3.2.cmml">g</mi><mo id="S8.4.p4.6.m6.4.4.6.2.3.1" xref="S8.4.p4.6.m6.4.4.6.2.3.1.cmml">ˇ</mo></mover></mrow><mo id="S8.4.p4.6.m6.4.4.6.1" xref="S8.4.p4.6.m6.4.4.6.1.cmml">⁢</mo><mrow id="S8.4.p4.6.m6.4.4.6.3.2" xref="S8.4.p4.6.m6.1.1.cmml"><mo id="S8.4.p4.6.m6.4.4.6.3.2.1" stretchy="false" xref="S8.4.p4.6.m6.1.1.cmml">(</mo><mover accent="true" id="S8.4.p4.6.m6.1.1" xref="S8.4.p4.6.m6.1.1.cmml"><mi id="S8.4.p4.6.m6.1.1.2" xref="S8.4.p4.6.m6.1.1.2.cmml">t</mi><mo id="S8.4.p4.6.m6.1.1.1" xref="S8.4.p4.6.m6.1.1.1.cmml">ˇ</mo></mover><mo id="S8.4.p4.6.m6.4.4.6.3.2.2" stretchy="false" xref="S8.4.p4.6.m6.1.1.cmml">)</mo></mrow></mrow><mo id="S8.4.p4.6.m6.4.4.7" xref="S8.4.p4.6.m6.4.4.7.cmml">=</mo><mrow id="S8.4.p4.6.m6.3.3.1" xref="S8.4.p4.6.m6.3.3.1.cmml"><mrow id="S8.4.p4.6.m6.3.3.1.1" xref="S8.4.p4.6.m6.3.3.1.1.cmml"><mover accent="true" id="S8.4.p4.6.m6.3.3.1.1.3" xref="S8.4.p4.6.m6.3.3.1.1.3.cmml"><mi id="S8.4.p4.6.m6.3.3.1.1.3.2" xref="S8.4.p4.6.m6.3.3.1.1.3.2.cmml">f</mi><mo id="S8.4.p4.6.m6.3.3.1.1.3.1" xref="S8.4.p4.6.m6.3.3.1.1.3.1.cmml">̊</mo></mover><mo id="S8.4.p4.6.m6.3.3.1.1.2" xref="S8.4.p4.6.m6.3.3.1.1.2.cmml">⁢</mo><mrow id="S8.4.p4.6.m6.3.3.1.1.1.1" xref="S8.4.p4.6.m6.3.3.1.1.1.1.1.cmml"><mo id="S8.4.p4.6.m6.3.3.1.1.1.1.2" stretchy="false" xref="S8.4.p4.6.m6.3.3.1.1.1.1.1.cmml">(</mo><mrow id="S8.4.p4.6.m6.3.3.1.1.1.1.1" xref="S8.4.p4.6.m6.3.3.1.1.1.1.1.cmml"><mover accent="true" id="S8.4.p4.6.m6.3.3.1.1.1.1.1.2" xref="S8.4.p4.6.m6.3.3.1.1.1.1.1.2.cmml"><mi id="S8.4.p4.6.m6.3.3.1.1.1.1.1.2.2" xref="S8.4.p4.6.m6.3.3.1.1.1.1.1.2.2.cmml">c</mi><mo id="S8.4.p4.6.m6.3.3.1.1.1.1.1.2.1" xref="S8.4.p4.6.m6.3.3.1.1.1.1.1.2.1.cmml">̊</mo></mover><mo id="S8.4.p4.6.m6.3.3.1.1.1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S8.4.p4.6.m6.3.3.1.1.1.1.1.1.cmml">∘</mo><mover accent="true" id="S8.4.p4.6.m6.3.3.1.1.1.1.1.3" xref="S8.4.p4.6.m6.3.3.1.1.1.1.1.3.cmml"><mi id="S8.4.p4.6.m6.3.3.1.1.1.1.1.3.2" xref="S8.4.p4.6.m6.3.3.1.1.1.1.1.3.2.cmml">t</mi><mo id="S8.4.p4.6.m6.3.3.1.1.1.1.1.3.1" xref="S8.4.p4.6.m6.3.3.1.1.1.1.1.3.1.cmml">ˇ</mo></mover></mrow><mo id="S8.4.p4.6.m6.3.3.1.1.1.1.3" stretchy="false" xref="S8.4.p4.6.m6.3.3.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S8.4.p4.6.m6.3.3.1.2" xref="S8.4.p4.6.m6.3.3.1.2.cmml">∧</mo><mrow id="S8.4.p4.6.m6.3.3.1.3" xref="S8.4.p4.6.m6.3.3.1.3.cmml"><mrow id="S8.4.p4.6.m6.3.3.1.3.2" xref="S8.4.p4.6.m6.3.3.1.3.2.cmml"><mover accent="true" id="S8.4.p4.6.m6.3.3.1.3.2.2" xref="S8.4.p4.6.m6.3.3.1.3.2.2.cmml"><mi id="S8.4.p4.6.m6.3.3.1.3.2.2.2" xref="S8.4.p4.6.m6.3.3.1.3.2.2.2.cmml">c</mi><mo id="S8.4.p4.6.m6.3.3.1.3.2.2.1" xref="S8.4.p4.6.m6.3.3.1.3.2.2.1.cmml">̊</mo></mover><mo id="S8.4.p4.6.m6.3.3.1.3.2.1" lspace="0.222em" rspace="0.222em" xref="S8.4.p4.6.m6.3.3.1.3.2.1.cmml">∘</mo><mover accent="true" id="S8.4.p4.6.m6.3.3.1.3.2.3" xref="S8.4.p4.6.m6.3.3.1.3.2.3.cmml"><mi id="S8.4.p4.6.m6.3.3.1.3.2.3.2" xref="S8.4.p4.6.m6.3.3.1.3.2.3.2.cmml">g</mi><mo id="S8.4.p4.6.m6.3.3.1.3.2.3.1" xref="S8.4.p4.6.m6.3.3.1.3.2.3.1.cmml">ˇ</mo></mover></mrow><mo id="S8.4.p4.6.m6.3.3.1.3.1" xref="S8.4.p4.6.m6.3.3.1.3.1.cmml">⁢</mo><mrow id="S8.4.p4.6.m6.3.3.1.3.3.2" xref="S8.4.p4.6.m6.2.2.cmml"><mo id="S8.4.p4.6.m6.3.3.1.3.3.2.1" stretchy="false" xref="S8.4.p4.6.m6.2.2.cmml">(</mo><mover accent="true" id="S8.4.p4.6.m6.2.2" xref="S8.4.p4.6.m6.2.2.cmml"><mi id="S8.4.p4.6.m6.2.2.2" xref="S8.4.p4.6.m6.2.2.2.cmml">s</mi><mo id="S8.4.p4.6.m6.2.2.1" xref="S8.4.p4.6.m6.2.2.1.cmml">ˇ</mo></mover><mo id="S8.4.p4.6.m6.3.3.1.3.3.2.2" stretchy="false" xref="S8.4.p4.6.m6.2.2.cmml">)</mo></mrow></mrow></mrow><mo id="S8.4.p4.6.m6.4.4.8" xref="S8.4.p4.6.m6.4.4.8.cmml">=</mo><mrow id="S8.4.p4.6.m6.4.4.2" xref="S8.4.p4.6.m6.4.4.2.cmml"><mover accent="true" id="S8.4.p4.6.m6.4.4.2.3" xref="S8.4.p4.6.m6.4.4.2.3.cmml"><mi id="S8.4.p4.6.m6.4.4.2.3.2" xref="S8.4.p4.6.m6.4.4.2.3.2.cmml">f</mi><mo id="S8.4.p4.6.m6.4.4.2.3.1" xref="S8.4.p4.6.m6.4.4.2.3.1.cmml">̊</mo></mover><mo id="S8.4.p4.6.m6.4.4.2.2" xref="S8.4.p4.6.m6.4.4.2.2.cmml">⁢</mo><mrow id="S8.4.p4.6.m6.4.4.2.1.1" xref="S8.4.p4.6.m6.4.4.2.1.1.1.cmml"><mo id="S8.4.p4.6.m6.4.4.2.1.1.2" stretchy="false" xref="S8.4.p4.6.m6.4.4.2.1.1.1.cmml">(</mo><mrow id="S8.4.p4.6.m6.4.4.2.1.1.1" xref="S8.4.p4.6.m6.4.4.2.1.1.1.cmml"><mover accent="true" id="S8.4.p4.6.m6.4.4.2.1.1.1.2" xref="S8.4.p4.6.m6.4.4.2.1.1.1.2.cmml"><mi id="S8.4.p4.6.m6.4.4.2.1.1.1.2.2" xref="S8.4.p4.6.m6.4.4.2.1.1.1.2.2.cmml">c</mi><mo id="S8.4.p4.6.m6.4.4.2.1.1.1.2.1" xref="S8.4.p4.6.m6.4.4.2.1.1.1.2.1.cmml">̊</mo></mover><mo id="S8.4.p4.6.m6.4.4.2.1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S8.4.p4.6.m6.4.4.2.1.1.1.1.cmml">∘</mo><mover accent="true" id="S8.4.p4.6.m6.4.4.2.1.1.1.3" xref="S8.4.p4.6.m6.4.4.2.1.1.1.3.cmml"><mi id="S8.4.p4.6.m6.4.4.2.1.1.1.3.2" xref="S8.4.p4.6.m6.4.4.2.1.1.1.3.2.cmml">s</mi><mo id="S8.4.p4.6.m6.4.4.2.1.1.1.3.1" xref="S8.4.p4.6.m6.4.4.2.1.1.1.3.1.cmml">ˇ</mo></mover></mrow><mo id="S8.4.p4.6.m6.4.4.2.1.1.3" stretchy="false" xref="S8.4.p4.6.m6.4.4.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S8.4.p4.6.m6.4b"><apply id="S8.4.p4.6.m6.4.4.cmml" xref="S8.4.p4.6.m6.4.4"><and id="S8.4.p4.6.m6.4.4a.cmml" xref="S8.4.p4.6.m6.4.4"></and><apply id="S8.4.p4.6.m6.4.4b.cmml" xref="S8.4.p4.6.m6.4.4"><csymbol cd="latexml" id="S8.4.p4.6.m6.4.4.5.cmml" xref="S8.4.p4.6.m6.4.4.5">forces</csymbol><ci id="S8.4.p4.6.m6.4.4.4.cmml" xref="S8.4.p4.6.m6.4.4.4">𝑞</ci><apply id="S8.4.p4.6.m6.4.4.6.cmml" xref="S8.4.p4.6.m6.4.4.6"><times id="S8.4.p4.6.m6.4.4.6.1.cmml" xref="S8.4.p4.6.m6.4.4.6.1"></times><apply id="S8.4.p4.6.m6.4.4.6.2.cmml" xref="S8.4.p4.6.m6.4.4.6.2"><compose id="S8.4.p4.6.m6.4.4.6.2.1.cmml" xref="S8.4.p4.6.m6.4.4.6.2.1"></compose><apply id="S8.4.p4.6.m6.4.4.6.2.2.cmml" xref="S8.4.p4.6.m6.4.4.6.2.2"><ci id="S8.4.p4.6.m6.4.4.6.2.2.1.cmml" xref="S8.4.p4.6.m6.4.4.6.2.2.1">̊</ci><ci id="S8.4.p4.6.m6.4.4.6.2.2.2.cmml" xref="S8.4.p4.6.m6.4.4.6.2.2.2">𝑐</ci></apply><apply id="S8.4.p4.6.m6.4.4.6.2.3.cmml" xref="S8.4.p4.6.m6.4.4.6.2.3"><ci id="S8.4.p4.6.m6.4.4.6.2.3.1.cmml" xref="S8.4.p4.6.m6.4.4.6.2.3.1">ˇ</ci><ci id="S8.4.p4.6.m6.4.4.6.2.3.2.cmml" xref="S8.4.p4.6.m6.4.4.6.2.3.2">𝑔</ci></apply></apply><apply id="S8.4.p4.6.m6.1.1.cmml" xref="S8.4.p4.6.m6.4.4.6.3.2"><ci id="S8.4.p4.6.m6.1.1.1.cmml" xref="S8.4.p4.6.m6.1.1.1">ˇ</ci><ci id="S8.4.p4.6.m6.1.1.2.cmml" xref="S8.4.p4.6.m6.1.1.2">𝑡</ci></apply></apply></apply><apply id="S8.4.p4.6.m6.4.4c.cmml" xref="S8.4.p4.6.m6.4.4"><eq id="S8.4.p4.6.m6.4.4.7.cmml" xref="S8.4.p4.6.m6.4.4.7"></eq><share href="https://arxiv.org/html/2503.13728v1#S8.4.p4.6.m6.4.4.6.cmml" id="S8.4.p4.6.m6.4.4d.cmml" xref="S8.4.p4.6.m6.4.4"></share><apply id="S8.4.p4.6.m6.3.3.1.cmml" xref="S8.4.p4.6.m6.3.3.1"><and id="S8.4.p4.6.m6.3.3.1.2.cmml" xref="S8.4.p4.6.m6.3.3.1.2"></and><apply id="S8.4.p4.6.m6.3.3.1.1.cmml" xref="S8.4.p4.6.m6.3.3.1.1"><times id="S8.4.p4.6.m6.3.3.1.1.2.cmml" xref="S8.4.p4.6.m6.3.3.1.1.2"></times><apply id="S8.4.p4.6.m6.3.3.1.1.3.cmml" xref="S8.4.p4.6.m6.3.3.1.1.3"><ci id="S8.4.p4.6.m6.3.3.1.1.3.1.cmml" xref="S8.4.p4.6.m6.3.3.1.1.3.1">̊</ci><ci id="S8.4.p4.6.m6.3.3.1.1.3.2.cmml" xref="S8.4.p4.6.m6.3.3.1.1.3.2">𝑓</ci></apply><apply id="S8.4.p4.6.m6.3.3.1.1.1.1.1.cmml" xref="S8.4.p4.6.m6.3.3.1.1.1.1"><compose id="S8.4.p4.6.m6.3.3.1.1.1.1.1.1.cmml" xref="S8.4.p4.6.m6.3.3.1.1.1.1.1.1"></compose><apply id="S8.4.p4.6.m6.3.3.1.1.1.1.1.2.cmml" xref="S8.4.p4.6.m6.3.3.1.1.1.1.1.2"><ci id="S8.4.p4.6.m6.3.3.1.1.1.1.1.2.1.cmml" xref="S8.4.p4.6.m6.3.3.1.1.1.1.1.2.1">̊</ci><ci id="S8.4.p4.6.m6.3.3.1.1.1.1.1.2.2.cmml" xref="S8.4.p4.6.m6.3.3.1.1.1.1.1.2.2">𝑐</ci></apply><apply id="S8.4.p4.6.m6.3.3.1.1.1.1.1.3.cmml" xref="S8.4.p4.6.m6.3.3.1.1.1.1.1.3"><ci id="S8.4.p4.6.m6.3.3.1.1.1.1.1.3.1.cmml" xref="S8.4.p4.6.m6.3.3.1.1.1.1.1.3.1">ˇ</ci><ci id="S8.4.p4.6.m6.3.3.1.1.1.1.1.3.2.cmml" xref="S8.4.p4.6.m6.3.3.1.1.1.1.1.3.2">𝑡</ci></apply></apply></apply><apply id="S8.4.p4.6.m6.3.3.1.3.cmml" xref="S8.4.p4.6.m6.3.3.1.3"><times id="S8.4.p4.6.m6.3.3.1.3.1.cmml" xref="S8.4.p4.6.m6.3.3.1.3.1"></times><apply id="S8.4.p4.6.m6.3.3.1.3.2.cmml" xref="S8.4.p4.6.m6.3.3.1.3.2"><compose id="S8.4.p4.6.m6.3.3.1.3.2.1.cmml" xref="S8.4.p4.6.m6.3.3.1.3.2.1"></compose><apply id="S8.4.p4.6.m6.3.3.1.3.2.2.cmml" xref="S8.4.p4.6.m6.3.3.1.3.2.2"><ci id="S8.4.p4.6.m6.3.3.1.3.2.2.1.cmml" xref="S8.4.p4.6.m6.3.3.1.3.2.2.1">̊</ci><ci id="S8.4.p4.6.m6.3.3.1.3.2.2.2.cmml" xref="S8.4.p4.6.m6.3.3.1.3.2.2.2">𝑐</ci></apply><apply id="S8.4.p4.6.m6.3.3.1.3.2.3.cmml" xref="S8.4.p4.6.m6.3.3.1.3.2.3"><ci id="S8.4.p4.6.m6.3.3.1.3.2.3.1.cmml" xref="S8.4.p4.6.m6.3.3.1.3.2.3.1">ˇ</ci><ci id="S8.4.p4.6.m6.3.3.1.3.2.3.2.cmml" xref="S8.4.p4.6.m6.3.3.1.3.2.3.2">𝑔</ci></apply></apply><apply id="S8.4.p4.6.m6.2.2.cmml" xref="S8.4.p4.6.m6.3.3.1.3.3.2"><ci id="S8.4.p4.6.m6.2.2.1.cmml" xref="S8.4.p4.6.m6.2.2.1">ˇ</ci><ci id="S8.4.p4.6.m6.2.2.2.cmml" xref="S8.4.p4.6.m6.2.2.2">𝑠</ci></apply></apply></apply></apply><apply id="S8.4.p4.6.m6.4.4e.cmml" xref="S8.4.p4.6.m6.4.4"><eq id="S8.4.p4.6.m6.4.4.8.cmml" xref="S8.4.p4.6.m6.4.4.8"></eq><share href="https://arxiv.org/html/2503.13728v1#S8.4.p4.6.m6.3.3.1.cmml" id="S8.4.p4.6.m6.4.4f.cmml" xref="S8.4.p4.6.m6.4.4"></share><apply id="S8.4.p4.6.m6.4.4.2.cmml" xref="S8.4.p4.6.m6.4.4.2"><times id="S8.4.p4.6.m6.4.4.2.2.cmml" xref="S8.4.p4.6.m6.4.4.2.2"></times><apply id="S8.4.p4.6.m6.4.4.2.3.cmml" xref="S8.4.p4.6.m6.4.4.2.3"><ci id="S8.4.p4.6.m6.4.4.2.3.1.cmml" xref="S8.4.p4.6.m6.4.4.2.3.1">̊</ci><ci id="S8.4.p4.6.m6.4.4.2.3.2.cmml" xref="S8.4.p4.6.m6.4.4.2.3.2">𝑓</ci></apply><apply id="S8.4.p4.6.m6.4.4.2.1.1.1.cmml" xref="S8.4.p4.6.m6.4.4.2.1.1"><compose id="S8.4.p4.6.m6.4.4.2.1.1.1.1.cmml" xref="S8.4.p4.6.m6.4.4.2.1.1.1.1"></compose><apply id="S8.4.p4.6.m6.4.4.2.1.1.1.2.cmml" xref="S8.4.p4.6.m6.4.4.2.1.1.1.2"><ci id="S8.4.p4.6.m6.4.4.2.1.1.1.2.1.cmml" xref="S8.4.p4.6.m6.4.4.2.1.1.1.2.1">̊</ci><ci id="S8.4.p4.6.m6.4.4.2.1.1.1.2.2.cmml" xref="S8.4.p4.6.m6.4.4.2.1.1.1.2.2">𝑐</ci></apply><apply id="S8.4.p4.6.m6.4.4.2.1.1.1.3.cmml" xref="S8.4.p4.6.m6.4.4.2.1.1.1.3"><ci id="S8.4.p4.6.m6.4.4.2.1.1.1.3.1.cmml" xref="S8.4.p4.6.m6.4.4.2.1.1.1.3.1">ˇ</ci><ci id="S8.4.p4.6.m6.4.4.2.1.1.1.3.2.cmml" xref="S8.4.p4.6.m6.4.4.2.1.1.1.3.2">𝑠</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.4.p4.6.m6.4c">q\Vdash\mathring{c}\circ\check{g}(\check{t})=\mathring{f}(\mathring{c}\circ% \check{t})\land\mathring{c}\circ\check{g}(\check{s})=\mathring{f}(\mathring{c}% \circ\check{s})</annotation><annotation encoding="application/x-llamapun" id="S8.4.p4.6.m6.4d">italic_q ⊩ over̊ start_ARG italic_c end_ARG ∘ overroman_ˇ start_ARG italic_g end_ARG ( overroman_ˇ start_ARG italic_t end_ARG ) = over̊ start_ARG italic_f end_ARG ( over̊ start_ARG italic_c end_ARG ∘ overroman_ˇ start_ARG italic_t end_ARG ) ∧ over̊ start_ARG italic_c end_ARG ∘ overroman_ˇ start_ARG italic_g end_ARG ( overroman_ˇ start_ARG italic_s end_ARG ) = over̊ start_ARG italic_f end_ARG ( over̊ start_ARG italic_c end_ARG ∘ overroman_ˇ start_ARG italic_s end_ARG )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S8.5.p5"> <p class="ltx_p" id="S8.5.p5.8">If <math alttext="t\sqsubset s" class="ltx_Math" display="inline" id="S8.5.p5.1.m1.1"><semantics id="S8.5.p5.1.m1.1a"><mrow id="S8.5.p5.1.m1.1.1" xref="S8.5.p5.1.m1.1.1.cmml"><mi id="S8.5.p5.1.m1.1.1.2" xref="S8.5.p5.1.m1.1.1.2.cmml">t</mi><mo id="S8.5.p5.1.m1.1.1.1" xref="S8.5.p5.1.m1.1.1.1.cmml">⊏</mo><mi id="S8.5.p5.1.m1.1.1.3" xref="S8.5.p5.1.m1.1.1.3.cmml">s</mi></mrow><annotation-xml encoding="MathML-Content" id="S8.5.p5.1.m1.1b"><apply id="S8.5.p5.1.m1.1.1.cmml" xref="S8.5.p5.1.m1.1.1"><csymbol cd="latexml" id="S8.5.p5.1.m1.1.1.1.cmml" xref="S8.5.p5.1.m1.1.1.1">square-image-of</csymbol><ci id="S8.5.p5.1.m1.1.1.2.cmml" xref="S8.5.p5.1.m1.1.1.2">𝑡</ci><ci id="S8.5.p5.1.m1.1.1.3.cmml" xref="S8.5.p5.1.m1.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.5.p5.1.m1.1c">t\sqsubset s</annotation><annotation encoding="application/x-llamapun" id="S8.5.p5.1.m1.1d">italic_t ⊏ italic_s</annotation></semantics></math> and <math alttext="g(t)\not\sqsubset g(s)" class="ltx_Math" display="inline" id="S8.5.p5.2.m2.2"><semantics id="S8.5.p5.2.m2.2a"><mrow id="S8.5.p5.2.m2.2.3" xref="S8.5.p5.2.m2.2.3.cmml"><mrow id="S8.5.p5.2.m2.2.3.2" xref="S8.5.p5.2.m2.2.3.2.cmml"><mi id="S8.5.p5.2.m2.2.3.2.2" xref="S8.5.p5.2.m2.2.3.2.2.cmml">g</mi><mo id="S8.5.p5.2.m2.2.3.2.1" xref="S8.5.p5.2.m2.2.3.2.1.cmml">⁢</mo><mrow id="S8.5.p5.2.m2.2.3.2.3.2" xref="S8.5.p5.2.m2.2.3.2.cmml"><mo id="S8.5.p5.2.m2.2.3.2.3.2.1" stretchy="false" xref="S8.5.p5.2.m2.2.3.2.cmml">(</mo><mi id="S8.5.p5.2.m2.1.1" xref="S8.5.p5.2.m2.1.1.cmml">t</mi><mo id="S8.5.p5.2.m2.2.3.2.3.2.2" stretchy="false" xref="S8.5.p5.2.m2.2.3.2.cmml">)</mo></mrow></mrow><mo id="S8.5.p5.2.m2.2.3.1" xref="S8.5.p5.2.m2.2.3.1.cmml">⊏̸</mo><mrow id="S8.5.p5.2.m2.2.3.3" xref="S8.5.p5.2.m2.2.3.3.cmml"><mi id="S8.5.p5.2.m2.2.3.3.2" xref="S8.5.p5.2.m2.2.3.3.2.cmml">g</mi><mo id="S8.5.p5.2.m2.2.3.3.1" xref="S8.5.p5.2.m2.2.3.3.1.cmml">⁢</mo><mrow id="S8.5.p5.2.m2.2.3.3.3.2" xref="S8.5.p5.2.m2.2.3.3.cmml"><mo id="S8.5.p5.2.m2.2.3.3.3.2.1" stretchy="false" xref="S8.5.p5.2.m2.2.3.3.cmml">(</mo><mi id="S8.5.p5.2.m2.2.2" xref="S8.5.p5.2.m2.2.2.cmml">s</mi><mo id="S8.5.p5.2.m2.2.3.3.3.2.2" stretchy="false" xref="S8.5.p5.2.m2.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S8.5.p5.2.m2.2b"><apply id="S8.5.p5.2.m2.2.3.cmml" xref="S8.5.p5.2.m2.2.3"><csymbol cd="latexml" id="S8.5.p5.2.m2.2.3.1.cmml" xref="S8.5.p5.2.m2.2.3.1">not-square-image-of</csymbol><apply id="S8.5.p5.2.m2.2.3.2.cmml" xref="S8.5.p5.2.m2.2.3.2"><times id="S8.5.p5.2.m2.2.3.2.1.cmml" xref="S8.5.p5.2.m2.2.3.2.1"></times><ci id="S8.5.p5.2.m2.2.3.2.2.cmml" xref="S8.5.p5.2.m2.2.3.2.2">𝑔</ci><ci id="S8.5.p5.2.m2.1.1.cmml" xref="S8.5.p5.2.m2.1.1">𝑡</ci></apply><apply id="S8.5.p5.2.m2.2.3.3.cmml" xref="S8.5.p5.2.m2.2.3.3"><times id="S8.5.p5.2.m2.2.3.3.1.cmml" xref="S8.5.p5.2.m2.2.3.3.1"></times><ci id="S8.5.p5.2.m2.2.3.3.2.cmml" xref="S8.5.p5.2.m2.2.3.3.2">𝑔</ci><ci id="S8.5.p5.2.m2.2.2.cmml" xref="S8.5.p5.2.m2.2.2">𝑠</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.5.p5.2.m2.2c">g(t)\not\sqsubset g(s)</annotation><annotation encoding="application/x-llamapun" id="S8.5.p5.2.m2.2d">italic_g ( italic_t ) ⊏̸ italic_g ( italic_s )</annotation></semantics></math>, then letting <math alttext="q^{\prime}:=q\cup\{(a,0),(b,1)\}" class="ltx_Math" display="inline" id="S8.5.p5.3.m3.6"><semantics id="S8.5.p5.3.m3.6a"><mrow id="S8.5.p5.3.m3.6.6" xref="S8.5.p5.3.m3.6.6.cmml"><msup id="S8.5.p5.3.m3.6.6.4" xref="S8.5.p5.3.m3.6.6.4.cmml"><mi id="S8.5.p5.3.m3.6.6.4.2" xref="S8.5.p5.3.m3.6.6.4.2.cmml">q</mi><mo id="S8.5.p5.3.m3.6.6.4.3" xref="S8.5.p5.3.m3.6.6.4.3.cmml">′</mo></msup><mo id="S8.5.p5.3.m3.6.6.3" lspace="0.278em" rspace="0.278em" xref="S8.5.p5.3.m3.6.6.3.cmml">:=</mo><mrow id="S8.5.p5.3.m3.6.6.2" xref="S8.5.p5.3.m3.6.6.2.cmml"><mi id="S8.5.p5.3.m3.6.6.2.4" xref="S8.5.p5.3.m3.6.6.2.4.cmml">q</mi><mo id="S8.5.p5.3.m3.6.6.2.3" xref="S8.5.p5.3.m3.6.6.2.3.cmml">∪</mo><mrow id="S8.5.p5.3.m3.6.6.2.2.2" xref="S8.5.p5.3.m3.6.6.2.2.3.cmml"><mo id="S8.5.p5.3.m3.6.6.2.2.2.3" stretchy="false" xref="S8.5.p5.3.m3.6.6.2.2.3.cmml">{</mo><mrow id="S8.5.p5.3.m3.5.5.1.1.1.1.2" xref="S8.5.p5.3.m3.5.5.1.1.1.1.1.cmml"><mo id="S8.5.p5.3.m3.5.5.1.1.1.1.2.1" stretchy="false" xref="S8.5.p5.3.m3.5.5.1.1.1.1.1.cmml">(</mo><mi id="S8.5.p5.3.m3.1.1" xref="S8.5.p5.3.m3.1.1.cmml">a</mi><mo id="S8.5.p5.3.m3.5.5.1.1.1.1.2.2" xref="S8.5.p5.3.m3.5.5.1.1.1.1.1.cmml">,</mo><mn id="S8.5.p5.3.m3.2.2" xref="S8.5.p5.3.m3.2.2.cmml">0</mn><mo id="S8.5.p5.3.m3.5.5.1.1.1.1.2.3" stretchy="false" xref="S8.5.p5.3.m3.5.5.1.1.1.1.1.cmml">)</mo></mrow><mo id="S8.5.p5.3.m3.6.6.2.2.2.4" xref="S8.5.p5.3.m3.6.6.2.2.3.cmml">,</mo><mrow id="S8.5.p5.3.m3.6.6.2.2.2.2.2" xref="S8.5.p5.3.m3.6.6.2.2.2.2.1.cmml"><mo id="S8.5.p5.3.m3.6.6.2.2.2.2.2.1" stretchy="false" xref="S8.5.p5.3.m3.6.6.2.2.2.2.1.cmml">(</mo><mi id="S8.5.p5.3.m3.3.3" xref="S8.5.p5.3.m3.3.3.cmml">b</mi><mo id="S8.5.p5.3.m3.6.6.2.2.2.2.2.2" xref="S8.5.p5.3.m3.6.6.2.2.2.2.1.cmml">,</mo><mn id="S8.5.p5.3.m3.4.4" xref="S8.5.p5.3.m3.4.4.cmml">1</mn><mo id="S8.5.p5.3.m3.6.6.2.2.2.2.2.3" stretchy="false" xref="S8.5.p5.3.m3.6.6.2.2.2.2.1.cmml">)</mo></mrow><mo id="S8.5.p5.3.m3.6.6.2.2.2.5" stretchy="false" xref="S8.5.p5.3.m3.6.6.2.2.3.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S8.5.p5.3.m3.6b"><apply id="S8.5.p5.3.m3.6.6.cmml" xref="S8.5.p5.3.m3.6.6"><csymbol cd="latexml" id="S8.5.p5.3.m3.6.6.3.cmml" xref="S8.5.p5.3.m3.6.6.3">assign</csymbol><apply id="S8.5.p5.3.m3.6.6.4.cmml" xref="S8.5.p5.3.m3.6.6.4"><csymbol cd="ambiguous" id="S8.5.p5.3.m3.6.6.4.1.cmml" xref="S8.5.p5.3.m3.6.6.4">superscript</csymbol><ci id="S8.5.p5.3.m3.6.6.4.2.cmml" xref="S8.5.p5.3.m3.6.6.4.2">𝑞</ci><ci id="S8.5.p5.3.m3.6.6.4.3.cmml" xref="S8.5.p5.3.m3.6.6.4.3">′</ci></apply><apply id="S8.5.p5.3.m3.6.6.2.cmml" xref="S8.5.p5.3.m3.6.6.2"><union id="S8.5.p5.3.m3.6.6.2.3.cmml" xref="S8.5.p5.3.m3.6.6.2.3"></union><ci id="S8.5.p5.3.m3.6.6.2.4.cmml" xref="S8.5.p5.3.m3.6.6.2.4">𝑞</ci><set id="S8.5.p5.3.m3.6.6.2.2.3.cmml" xref="S8.5.p5.3.m3.6.6.2.2.2"><interval closure="open" id="S8.5.p5.3.m3.5.5.1.1.1.1.1.cmml" xref="S8.5.p5.3.m3.5.5.1.1.1.1.2"><ci id="S8.5.p5.3.m3.1.1.cmml" xref="S8.5.p5.3.m3.1.1">𝑎</ci><cn id="S8.5.p5.3.m3.2.2.cmml" type="integer" xref="S8.5.p5.3.m3.2.2">0</cn></interval><interval closure="open" id="S8.5.p5.3.m3.6.6.2.2.2.2.1.cmml" xref="S8.5.p5.3.m3.6.6.2.2.2.2.2"><ci id="S8.5.p5.3.m3.3.3.cmml" xref="S8.5.p5.3.m3.3.3">𝑏</ci><cn id="S8.5.p5.3.m3.4.4.cmml" type="integer" xref="S8.5.p5.3.m3.4.4">1</cn></interval></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.5.p5.3.m3.6c">q^{\prime}:=q\cup\{(a,0),(b,1)\}</annotation><annotation encoding="application/x-llamapun" id="S8.5.p5.3.m3.6d">italic_q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT := italic_q ∪ { ( italic_a , 0 ) , ( italic_b , 1 ) }</annotation></semantics></math> we see that <math alttext="q^{\prime}\in\mathbb{P}" class="ltx_Math" display="inline" id="S8.5.p5.4.m4.1"><semantics id="S8.5.p5.4.m4.1a"><mrow id="S8.5.p5.4.m4.1.1" xref="S8.5.p5.4.m4.1.1.cmml"><msup id="S8.5.p5.4.m4.1.1.2" xref="S8.5.p5.4.m4.1.1.2.cmml"><mi id="S8.5.p5.4.m4.1.1.2.2" xref="S8.5.p5.4.m4.1.1.2.2.cmml">q</mi><mo id="S8.5.p5.4.m4.1.1.2.3" xref="S8.5.p5.4.m4.1.1.2.3.cmml">′</mo></msup><mo id="S8.5.p5.4.m4.1.1.1" xref="S8.5.p5.4.m4.1.1.1.cmml">∈</mo><mi id="S8.5.p5.4.m4.1.1.3" xref="S8.5.p5.4.m4.1.1.3.cmml">ℙ</mi></mrow><annotation-xml encoding="MathML-Content" id="S8.5.p5.4.m4.1b"><apply id="S8.5.p5.4.m4.1.1.cmml" xref="S8.5.p5.4.m4.1.1"><in id="S8.5.p5.4.m4.1.1.1.cmml" xref="S8.5.p5.4.m4.1.1.1"></in><apply id="S8.5.p5.4.m4.1.1.2.cmml" xref="S8.5.p5.4.m4.1.1.2"><csymbol cd="ambiguous" id="S8.5.p5.4.m4.1.1.2.1.cmml" xref="S8.5.p5.4.m4.1.1.2">superscript</csymbol><ci id="S8.5.p5.4.m4.1.1.2.2.cmml" xref="S8.5.p5.4.m4.1.1.2.2">𝑞</ci><ci id="S8.5.p5.4.m4.1.1.2.3.cmml" xref="S8.5.p5.4.m4.1.1.2.3">′</ci></apply><ci id="S8.5.p5.4.m4.1.1.3.cmml" xref="S8.5.p5.4.m4.1.1.3">ℙ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.5.p5.4.m4.1c">q^{\prime}\in\mathbb{P}</annotation><annotation encoding="application/x-llamapun" id="S8.5.p5.4.m4.1d">italic_q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ blackboard_P</annotation></semantics></math>, <math alttext="q^{\prime}\leq q" class="ltx_Math" display="inline" id="S8.5.p5.5.m5.1"><semantics id="S8.5.p5.5.m5.1a"><mrow id="S8.5.p5.5.m5.1.1" xref="S8.5.p5.5.m5.1.1.cmml"><msup id="S8.5.p5.5.m5.1.1.2" xref="S8.5.p5.5.m5.1.1.2.cmml"><mi id="S8.5.p5.5.m5.1.1.2.2" xref="S8.5.p5.5.m5.1.1.2.2.cmml">q</mi><mo id="S8.5.p5.5.m5.1.1.2.3" xref="S8.5.p5.5.m5.1.1.2.3.cmml">′</mo></msup><mo id="S8.5.p5.5.m5.1.1.1" xref="S8.5.p5.5.m5.1.1.1.cmml">≤</mo><mi id="S8.5.p5.5.m5.1.1.3" xref="S8.5.p5.5.m5.1.1.3.cmml">q</mi></mrow><annotation-xml encoding="MathML-Content" id="S8.5.p5.5.m5.1b"><apply id="S8.5.p5.5.m5.1.1.cmml" xref="S8.5.p5.5.m5.1.1"><leq id="S8.5.p5.5.m5.1.1.1.cmml" xref="S8.5.p5.5.m5.1.1.1"></leq><apply id="S8.5.p5.5.m5.1.1.2.cmml" xref="S8.5.p5.5.m5.1.1.2"><csymbol cd="ambiguous" id="S8.5.p5.5.m5.1.1.2.1.cmml" xref="S8.5.p5.5.m5.1.1.2">superscript</csymbol><ci id="S8.5.p5.5.m5.1.1.2.2.cmml" xref="S8.5.p5.5.m5.1.1.2.2">𝑞</ci><ci id="S8.5.p5.5.m5.1.1.2.3.cmml" xref="S8.5.p5.5.m5.1.1.2.3">′</ci></apply><ci id="S8.5.p5.5.m5.1.1.3.cmml" xref="S8.5.p5.5.m5.1.1.3">𝑞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.5.p5.5.m5.1c">q^{\prime}\leq q</annotation><annotation encoding="application/x-llamapun" id="S8.5.p5.5.m5.1d">italic_q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_q</annotation></semantics></math> and <math alttext="q^{\prime}\Vdash\mathring{c}\circ\check{t}\sqsubset\mathring{c}\circ\check{s}% \land\mathring{c}\circ\check{g}(\check{t})<_{\mathrm{lex}}\mathring{c}\circ% \check{g}(\check{s})" class="ltx_Math" display="inline" id="S8.5.p5.6.m6.2"><semantics id="S8.5.p5.6.m6.2a"><mrow id="S8.5.p5.6.m6.2.3" xref="S8.5.p5.6.m6.2.3.cmml"><msup id="S8.5.p5.6.m6.2.3.2" xref="S8.5.p5.6.m6.2.3.2.cmml"><mi id="S8.5.p5.6.m6.2.3.2.2" xref="S8.5.p5.6.m6.2.3.2.2.cmml">q</mi><mo id="S8.5.p5.6.m6.2.3.2.3" xref="S8.5.p5.6.m6.2.3.2.3.cmml">′</mo></msup><mo id="S8.5.p5.6.m6.2.3.3" xref="S8.5.p5.6.m6.2.3.3.cmml">⊩</mo><mrow id="S8.5.p5.6.m6.2.3.4" xref="S8.5.p5.6.m6.2.3.4.cmml"><mover accent="true" id="S8.5.p5.6.m6.2.3.4.2" xref="S8.5.p5.6.m6.2.3.4.2.cmml"><mi id="S8.5.p5.6.m6.2.3.4.2.2" xref="S8.5.p5.6.m6.2.3.4.2.2.cmml">c</mi><mo id="S8.5.p5.6.m6.2.3.4.2.1" xref="S8.5.p5.6.m6.2.3.4.2.1.cmml">̊</mo></mover><mo id="S8.5.p5.6.m6.2.3.4.1" lspace="0.222em" rspace="0.222em" xref="S8.5.p5.6.m6.2.3.4.1.cmml">∘</mo><mover accent="true" id="S8.5.p5.6.m6.2.3.4.3" xref="S8.5.p5.6.m6.2.3.4.3.cmml"><mi id="S8.5.p5.6.m6.2.3.4.3.2" xref="S8.5.p5.6.m6.2.3.4.3.2.cmml">t</mi><mo id="S8.5.p5.6.m6.2.3.4.3.1" xref="S8.5.p5.6.m6.2.3.4.3.1.cmml">ˇ</mo></mover></mrow><mo id="S8.5.p5.6.m6.2.3.5" xref="S8.5.p5.6.m6.2.3.5.cmml">⊏</mo><mrow id="S8.5.p5.6.m6.2.3.6" xref="S8.5.p5.6.m6.2.3.6.cmml"><mrow id="S8.5.p5.6.m6.2.3.6.2" xref="S8.5.p5.6.m6.2.3.6.2.cmml"><mover accent="true" id="S8.5.p5.6.m6.2.3.6.2.2" xref="S8.5.p5.6.m6.2.3.6.2.2.cmml"><mi id="S8.5.p5.6.m6.2.3.6.2.2.2" xref="S8.5.p5.6.m6.2.3.6.2.2.2.cmml">c</mi><mo id="S8.5.p5.6.m6.2.3.6.2.2.1" xref="S8.5.p5.6.m6.2.3.6.2.2.1.cmml">̊</mo></mover><mo id="S8.5.p5.6.m6.2.3.6.2.1" lspace="0.222em" rspace="0.222em" xref="S8.5.p5.6.m6.2.3.6.2.1.cmml">∘</mo><mover accent="true" id="S8.5.p5.6.m6.2.3.6.2.3" xref="S8.5.p5.6.m6.2.3.6.2.3.cmml"><mi id="S8.5.p5.6.m6.2.3.6.2.3.2" xref="S8.5.p5.6.m6.2.3.6.2.3.2.cmml">s</mi><mo id="S8.5.p5.6.m6.2.3.6.2.3.1" xref="S8.5.p5.6.m6.2.3.6.2.3.1.cmml">ˇ</mo></mover></mrow><mo id="S8.5.p5.6.m6.2.3.6.1" xref="S8.5.p5.6.m6.2.3.6.1.cmml">∧</mo><mrow id="S8.5.p5.6.m6.2.3.6.3" xref="S8.5.p5.6.m6.2.3.6.3.cmml"><mrow id="S8.5.p5.6.m6.2.3.6.3.2" xref="S8.5.p5.6.m6.2.3.6.3.2.cmml"><mover accent="true" id="S8.5.p5.6.m6.2.3.6.3.2.2" xref="S8.5.p5.6.m6.2.3.6.3.2.2.cmml"><mi id="S8.5.p5.6.m6.2.3.6.3.2.2.2" xref="S8.5.p5.6.m6.2.3.6.3.2.2.2.cmml">c</mi><mo id="S8.5.p5.6.m6.2.3.6.3.2.2.1" xref="S8.5.p5.6.m6.2.3.6.3.2.2.1.cmml">̊</mo></mover><mo id="S8.5.p5.6.m6.2.3.6.3.2.1" lspace="0.222em" rspace="0.222em" xref="S8.5.p5.6.m6.2.3.6.3.2.1.cmml">∘</mo><mover accent="true" id="S8.5.p5.6.m6.2.3.6.3.2.3" xref="S8.5.p5.6.m6.2.3.6.3.2.3.cmml"><mi id="S8.5.p5.6.m6.2.3.6.3.2.3.2" xref="S8.5.p5.6.m6.2.3.6.3.2.3.2.cmml">g</mi><mo id="S8.5.p5.6.m6.2.3.6.3.2.3.1" xref="S8.5.p5.6.m6.2.3.6.3.2.3.1.cmml">ˇ</mo></mover></mrow><mo id="S8.5.p5.6.m6.2.3.6.3.1" xref="S8.5.p5.6.m6.2.3.6.3.1.cmml">⁢</mo><mrow id="S8.5.p5.6.m6.2.3.6.3.3.2" xref="S8.5.p5.6.m6.1.1.cmml"><mo id="S8.5.p5.6.m6.2.3.6.3.3.2.1" stretchy="false" xref="S8.5.p5.6.m6.1.1.cmml">(</mo><mover accent="true" id="S8.5.p5.6.m6.1.1" xref="S8.5.p5.6.m6.1.1.cmml"><mi id="S8.5.p5.6.m6.1.1.2" xref="S8.5.p5.6.m6.1.1.2.cmml">t</mi><mo id="S8.5.p5.6.m6.1.1.1" xref="S8.5.p5.6.m6.1.1.1.cmml">ˇ</mo></mover><mo id="S8.5.p5.6.m6.2.3.6.3.3.2.2" stretchy="false" xref="S8.5.p5.6.m6.1.1.cmml">)</mo></mrow></mrow></mrow><msub id="S8.5.p5.6.m6.2.3.7" xref="S8.5.p5.6.m6.2.3.7.cmml"><mo id="S8.5.p5.6.m6.2.3.7.2" xref="S8.5.p5.6.m6.2.3.7.2.cmml"><</mo><mi id="S8.5.p5.6.m6.2.3.7.3" xref="S8.5.p5.6.m6.2.3.7.3.cmml">lex</mi></msub><mrow id="S8.5.p5.6.m6.2.3.8" xref="S8.5.p5.6.m6.2.3.8.cmml"><mrow id="S8.5.p5.6.m6.2.3.8.2" xref="S8.5.p5.6.m6.2.3.8.2.cmml"><mover accent="true" id="S8.5.p5.6.m6.2.3.8.2.2" xref="S8.5.p5.6.m6.2.3.8.2.2.cmml"><mi id="S8.5.p5.6.m6.2.3.8.2.2.2" xref="S8.5.p5.6.m6.2.3.8.2.2.2.cmml">c</mi><mo id="S8.5.p5.6.m6.2.3.8.2.2.1" xref="S8.5.p5.6.m6.2.3.8.2.2.1.cmml">̊</mo></mover><mo id="S8.5.p5.6.m6.2.3.8.2.1" lspace="0.222em" rspace="0.222em" xref="S8.5.p5.6.m6.2.3.8.2.1.cmml">∘</mo><mover accent="true" id="S8.5.p5.6.m6.2.3.8.2.3" xref="S8.5.p5.6.m6.2.3.8.2.3.cmml"><mi id="S8.5.p5.6.m6.2.3.8.2.3.2" xref="S8.5.p5.6.m6.2.3.8.2.3.2.cmml">g</mi><mo id="S8.5.p5.6.m6.2.3.8.2.3.1" xref="S8.5.p5.6.m6.2.3.8.2.3.1.cmml">ˇ</mo></mover></mrow><mo id="S8.5.p5.6.m6.2.3.8.1" xref="S8.5.p5.6.m6.2.3.8.1.cmml">⁢</mo><mrow id="S8.5.p5.6.m6.2.3.8.3.2" xref="S8.5.p5.6.m6.2.2.cmml"><mo id="S8.5.p5.6.m6.2.3.8.3.2.1" stretchy="false" xref="S8.5.p5.6.m6.2.2.cmml">(</mo><mover accent="true" id="S8.5.p5.6.m6.2.2" xref="S8.5.p5.6.m6.2.2.cmml"><mi id="S8.5.p5.6.m6.2.2.2" xref="S8.5.p5.6.m6.2.2.2.cmml">s</mi><mo id="S8.5.p5.6.m6.2.2.1" xref="S8.5.p5.6.m6.2.2.1.cmml">ˇ</mo></mover><mo id="S8.5.p5.6.m6.2.3.8.3.2.2" stretchy="false" xref="S8.5.p5.6.m6.2.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S8.5.p5.6.m6.2b"><apply id="S8.5.p5.6.m6.2.3.cmml" xref="S8.5.p5.6.m6.2.3"><and id="S8.5.p5.6.m6.2.3a.cmml" xref="S8.5.p5.6.m6.2.3"></and><apply id="S8.5.p5.6.m6.2.3b.cmml" xref="S8.5.p5.6.m6.2.3"><csymbol cd="latexml" id="S8.5.p5.6.m6.2.3.3.cmml" xref="S8.5.p5.6.m6.2.3.3">forces</csymbol><apply id="S8.5.p5.6.m6.2.3.2.cmml" xref="S8.5.p5.6.m6.2.3.2"><csymbol cd="ambiguous" id="S8.5.p5.6.m6.2.3.2.1.cmml" xref="S8.5.p5.6.m6.2.3.2">superscript</csymbol><ci id="S8.5.p5.6.m6.2.3.2.2.cmml" xref="S8.5.p5.6.m6.2.3.2.2">𝑞</ci><ci id="S8.5.p5.6.m6.2.3.2.3.cmml" xref="S8.5.p5.6.m6.2.3.2.3">′</ci></apply><apply id="S8.5.p5.6.m6.2.3.4.cmml" xref="S8.5.p5.6.m6.2.3.4"><compose id="S8.5.p5.6.m6.2.3.4.1.cmml" xref="S8.5.p5.6.m6.2.3.4.1"></compose><apply id="S8.5.p5.6.m6.2.3.4.2.cmml" xref="S8.5.p5.6.m6.2.3.4.2"><ci id="S8.5.p5.6.m6.2.3.4.2.1.cmml" xref="S8.5.p5.6.m6.2.3.4.2.1">̊</ci><ci id="S8.5.p5.6.m6.2.3.4.2.2.cmml" xref="S8.5.p5.6.m6.2.3.4.2.2">𝑐</ci></apply><apply id="S8.5.p5.6.m6.2.3.4.3.cmml" xref="S8.5.p5.6.m6.2.3.4.3"><ci id="S8.5.p5.6.m6.2.3.4.3.1.cmml" xref="S8.5.p5.6.m6.2.3.4.3.1">ˇ</ci><ci id="S8.5.p5.6.m6.2.3.4.3.2.cmml" xref="S8.5.p5.6.m6.2.3.4.3.2">𝑡</ci></apply></apply></apply><apply id="S8.5.p5.6.m6.2.3c.cmml" xref="S8.5.p5.6.m6.2.3"><csymbol cd="latexml" id="S8.5.p5.6.m6.2.3.5.cmml" xref="S8.5.p5.6.m6.2.3.5">square-image-of</csymbol><share href="https://arxiv.org/html/2503.13728v1#S8.5.p5.6.m6.2.3.4.cmml" id="S8.5.p5.6.m6.2.3d.cmml" xref="S8.5.p5.6.m6.2.3"></share><apply id="S8.5.p5.6.m6.2.3.6.cmml" xref="S8.5.p5.6.m6.2.3.6"><and id="S8.5.p5.6.m6.2.3.6.1.cmml" xref="S8.5.p5.6.m6.2.3.6.1"></and><apply id="S8.5.p5.6.m6.2.3.6.2.cmml" xref="S8.5.p5.6.m6.2.3.6.2"><compose id="S8.5.p5.6.m6.2.3.6.2.1.cmml" xref="S8.5.p5.6.m6.2.3.6.2.1"></compose><apply id="S8.5.p5.6.m6.2.3.6.2.2.cmml" xref="S8.5.p5.6.m6.2.3.6.2.2"><ci id="S8.5.p5.6.m6.2.3.6.2.2.1.cmml" xref="S8.5.p5.6.m6.2.3.6.2.2.1">̊</ci><ci id="S8.5.p5.6.m6.2.3.6.2.2.2.cmml" xref="S8.5.p5.6.m6.2.3.6.2.2.2">𝑐</ci></apply><apply id="S8.5.p5.6.m6.2.3.6.2.3.cmml" xref="S8.5.p5.6.m6.2.3.6.2.3"><ci id="S8.5.p5.6.m6.2.3.6.2.3.1.cmml" xref="S8.5.p5.6.m6.2.3.6.2.3.1">ˇ</ci><ci id="S8.5.p5.6.m6.2.3.6.2.3.2.cmml" xref="S8.5.p5.6.m6.2.3.6.2.3.2">𝑠</ci></apply></apply><apply id="S8.5.p5.6.m6.2.3.6.3.cmml" xref="S8.5.p5.6.m6.2.3.6.3"><times id="S8.5.p5.6.m6.2.3.6.3.1.cmml" xref="S8.5.p5.6.m6.2.3.6.3.1"></times><apply id="S8.5.p5.6.m6.2.3.6.3.2.cmml" xref="S8.5.p5.6.m6.2.3.6.3.2"><compose id="S8.5.p5.6.m6.2.3.6.3.2.1.cmml" xref="S8.5.p5.6.m6.2.3.6.3.2.1"></compose><apply id="S8.5.p5.6.m6.2.3.6.3.2.2.cmml" xref="S8.5.p5.6.m6.2.3.6.3.2.2"><ci id="S8.5.p5.6.m6.2.3.6.3.2.2.1.cmml" xref="S8.5.p5.6.m6.2.3.6.3.2.2.1">̊</ci><ci id="S8.5.p5.6.m6.2.3.6.3.2.2.2.cmml" xref="S8.5.p5.6.m6.2.3.6.3.2.2.2">𝑐</ci></apply><apply id="S8.5.p5.6.m6.2.3.6.3.2.3.cmml" xref="S8.5.p5.6.m6.2.3.6.3.2.3"><ci id="S8.5.p5.6.m6.2.3.6.3.2.3.1.cmml" xref="S8.5.p5.6.m6.2.3.6.3.2.3.1">ˇ</ci><ci id="S8.5.p5.6.m6.2.3.6.3.2.3.2.cmml" xref="S8.5.p5.6.m6.2.3.6.3.2.3.2">𝑔</ci></apply></apply><apply id="S8.5.p5.6.m6.1.1.cmml" xref="S8.5.p5.6.m6.2.3.6.3.3.2"><ci id="S8.5.p5.6.m6.1.1.1.cmml" xref="S8.5.p5.6.m6.1.1.1">ˇ</ci><ci id="S8.5.p5.6.m6.1.1.2.cmml" xref="S8.5.p5.6.m6.1.1.2">𝑡</ci></apply></apply></apply></apply><apply id="S8.5.p5.6.m6.2.3e.cmml" xref="S8.5.p5.6.m6.2.3"><apply id="S8.5.p5.6.m6.2.3.7.cmml" xref="S8.5.p5.6.m6.2.3.7"><csymbol cd="ambiguous" id="S8.5.p5.6.m6.2.3.7.1.cmml" xref="S8.5.p5.6.m6.2.3.7">subscript</csymbol><lt id="S8.5.p5.6.m6.2.3.7.2.cmml" xref="S8.5.p5.6.m6.2.3.7.2"></lt><ci id="S8.5.p5.6.m6.2.3.7.3.cmml" xref="S8.5.p5.6.m6.2.3.7.3">lex</ci></apply><share href="https://arxiv.org/html/2503.13728v1#S8.5.p5.6.m6.2.3.6.cmml" id="S8.5.p5.6.m6.2.3f.cmml" xref="S8.5.p5.6.m6.2.3"></share><apply id="S8.5.p5.6.m6.2.3.8.cmml" xref="S8.5.p5.6.m6.2.3.8"><times id="S8.5.p5.6.m6.2.3.8.1.cmml" xref="S8.5.p5.6.m6.2.3.8.1"></times><apply id="S8.5.p5.6.m6.2.3.8.2.cmml" xref="S8.5.p5.6.m6.2.3.8.2"><compose id="S8.5.p5.6.m6.2.3.8.2.1.cmml" xref="S8.5.p5.6.m6.2.3.8.2.1"></compose><apply id="S8.5.p5.6.m6.2.3.8.2.2.cmml" xref="S8.5.p5.6.m6.2.3.8.2.2"><ci id="S8.5.p5.6.m6.2.3.8.2.2.1.cmml" xref="S8.5.p5.6.m6.2.3.8.2.2.1">̊</ci><ci id="S8.5.p5.6.m6.2.3.8.2.2.2.cmml" xref="S8.5.p5.6.m6.2.3.8.2.2.2">𝑐</ci></apply><apply id="S8.5.p5.6.m6.2.3.8.2.3.cmml" xref="S8.5.p5.6.m6.2.3.8.2.3"><ci id="S8.5.p5.6.m6.2.3.8.2.3.1.cmml" xref="S8.5.p5.6.m6.2.3.8.2.3.1">ˇ</ci><ci id="S8.5.p5.6.m6.2.3.8.2.3.2.cmml" xref="S8.5.p5.6.m6.2.3.8.2.3.2">𝑔</ci></apply></apply><apply id="S8.5.p5.6.m6.2.2.cmml" xref="S8.5.p5.6.m6.2.3.8.3.2"><ci id="S8.5.p5.6.m6.2.2.1.cmml" xref="S8.5.p5.6.m6.2.2.1">ˇ</ci><ci id="S8.5.p5.6.m6.2.2.2.cmml" xref="S8.5.p5.6.m6.2.2.2">𝑠</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.5.p5.6.m6.2c">q^{\prime}\Vdash\mathring{c}\circ\check{t}\sqsubset\mathring{c}\circ\check{s}% \land\mathring{c}\circ\check{g}(\check{t})<_{\mathrm{lex}}\mathring{c}\circ% \check{g}(\check{s})</annotation><annotation encoding="application/x-llamapun" id="S8.5.p5.6.m6.2d">italic_q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⊩ over̊ start_ARG italic_c end_ARG ∘ overroman_ˇ start_ARG italic_t end_ARG ⊏ over̊ start_ARG italic_c end_ARG ∘ overroman_ˇ start_ARG italic_s end_ARG ∧ over̊ start_ARG italic_c end_ARG ∘ overroman_ˇ start_ARG italic_g end_ARG ( overroman_ˇ start_ARG italic_t end_ARG ) < start_POSTSUBSCRIPT roman_lex end_POSTSUBSCRIPT over̊ start_ARG italic_c end_ARG ∘ overroman_ˇ start_ARG italic_g end_ARG ( overroman_ˇ start_ARG italic_s end_ARG )</annotation></semantics></math>, which is again a contradiction. If <math alttext="t\not\sqsubset s" class="ltx_Math" display="inline" id="S8.5.p5.7.m7.1"><semantics id="S8.5.p5.7.m7.1a"><mrow id="S8.5.p5.7.m7.1.1" xref="S8.5.p5.7.m7.1.1.cmml"><mi id="S8.5.p5.7.m7.1.1.2" xref="S8.5.p5.7.m7.1.1.2.cmml">t</mi><mo id="S8.5.p5.7.m7.1.1.1" xref="S8.5.p5.7.m7.1.1.1.cmml">⊏̸</mo><mi id="S8.5.p5.7.m7.1.1.3" xref="S8.5.p5.7.m7.1.1.3.cmml">s</mi></mrow><annotation-xml encoding="MathML-Content" id="S8.5.p5.7.m7.1b"><apply id="S8.5.p5.7.m7.1.1.cmml" xref="S8.5.p5.7.m7.1.1"><csymbol cd="latexml" id="S8.5.p5.7.m7.1.1.1.cmml" xref="S8.5.p5.7.m7.1.1.1">not-square-image-of</csymbol><ci id="S8.5.p5.7.m7.1.1.2.cmml" xref="S8.5.p5.7.m7.1.1.2">𝑡</ci><ci id="S8.5.p5.7.m7.1.1.3.cmml" xref="S8.5.p5.7.m7.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.5.p5.7.m7.1c">t\not\sqsubset s</annotation><annotation encoding="application/x-llamapun" id="S8.5.p5.7.m7.1d">italic_t ⊏̸ italic_s</annotation></semantics></math> and <math alttext="g(t)\sqsubset g(s)" class="ltx_Math" display="inline" id="S8.5.p5.8.m8.2"><semantics id="S8.5.p5.8.m8.2a"><mrow id="S8.5.p5.8.m8.2.3" xref="S8.5.p5.8.m8.2.3.cmml"><mrow id="S8.5.p5.8.m8.2.3.2" xref="S8.5.p5.8.m8.2.3.2.cmml"><mi id="S8.5.p5.8.m8.2.3.2.2" xref="S8.5.p5.8.m8.2.3.2.2.cmml">g</mi><mo id="S8.5.p5.8.m8.2.3.2.1" xref="S8.5.p5.8.m8.2.3.2.1.cmml">⁢</mo><mrow id="S8.5.p5.8.m8.2.3.2.3.2" xref="S8.5.p5.8.m8.2.3.2.cmml"><mo id="S8.5.p5.8.m8.2.3.2.3.2.1" stretchy="false" xref="S8.5.p5.8.m8.2.3.2.cmml">(</mo><mi id="S8.5.p5.8.m8.1.1" xref="S8.5.p5.8.m8.1.1.cmml">t</mi><mo id="S8.5.p5.8.m8.2.3.2.3.2.2" stretchy="false" xref="S8.5.p5.8.m8.2.3.2.cmml">)</mo></mrow></mrow><mo id="S8.5.p5.8.m8.2.3.1" xref="S8.5.p5.8.m8.2.3.1.cmml">⊏</mo><mrow id="S8.5.p5.8.m8.2.3.3" xref="S8.5.p5.8.m8.2.3.3.cmml"><mi id="S8.5.p5.8.m8.2.3.3.2" xref="S8.5.p5.8.m8.2.3.3.2.cmml">g</mi><mo id="S8.5.p5.8.m8.2.3.3.1" xref="S8.5.p5.8.m8.2.3.3.1.cmml">⁢</mo><mrow id="S8.5.p5.8.m8.2.3.3.3.2" xref="S8.5.p5.8.m8.2.3.3.cmml"><mo id="S8.5.p5.8.m8.2.3.3.3.2.1" stretchy="false" xref="S8.5.p5.8.m8.2.3.3.cmml">(</mo><mi id="S8.5.p5.8.m8.2.2" xref="S8.5.p5.8.m8.2.2.cmml">s</mi><mo id="S8.5.p5.8.m8.2.3.3.3.2.2" stretchy="false" xref="S8.5.p5.8.m8.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S8.5.p5.8.m8.2b"><apply id="S8.5.p5.8.m8.2.3.cmml" xref="S8.5.p5.8.m8.2.3"><csymbol cd="latexml" id="S8.5.p5.8.m8.2.3.1.cmml" xref="S8.5.p5.8.m8.2.3.1">square-image-of</csymbol><apply id="S8.5.p5.8.m8.2.3.2.cmml" xref="S8.5.p5.8.m8.2.3.2"><times id="S8.5.p5.8.m8.2.3.2.1.cmml" xref="S8.5.p5.8.m8.2.3.2.1"></times><ci id="S8.5.p5.8.m8.2.3.2.2.cmml" xref="S8.5.p5.8.m8.2.3.2.2">𝑔</ci><ci id="S8.5.p5.8.m8.1.1.cmml" xref="S8.5.p5.8.m8.1.1">𝑡</ci></apply><apply id="S8.5.p5.8.m8.2.3.3.cmml" xref="S8.5.p5.8.m8.2.3.3"><times id="S8.5.p5.8.m8.2.3.3.1.cmml" xref="S8.5.p5.8.m8.2.3.3.1"></times><ci id="S8.5.p5.8.m8.2.3.3.2.cmml" xref="S8.5.p5.8.m8.2.3.3.2">𝑔</ci><ci id="S8.5.p5.8.m8.2.2.cmml" xref="S8.5.p5.8.m8.2.2">𝑠</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.5.p5.8.m8.2c">g(t)\sqsubset g(s)</annotation><annotation encoding="application/x-llamapun" id="S8.5.p5.8.m8.2d">italic_g ( italic_t ) ⊏ italic_g ( italic_s )</annotation></semantics></math> the argument is identical.</p> </div> <div class="ltx_para" id="S8.6.p6"> <p class="ltx_p" id="S8.6.p6.13">Assume now that <math alttext="t\not\sqsubset s" class="ltx_Math" display="inline" id="S8.6.p6.1.m1.1"><semantics id="S8.6.p6.1.m1.1a"><mrow id="S8.6.p6.1.m1.1.1" xref="S8.6.p6.1.m1.1.1.cmml"><mi id="S8.6.p6.1.m1.1.1.2" xref="S8.6.p6.1.m1.1.1.2.cmml">t</mi><mo id="S8.6.p6.1.m1.1.1.1" xref="S8.6.p6.1.m1.1.1.1.cmml">⊏̸</mo><mi id="S8.6.p6.1.m1.1.1.3" xref="S8.6.p6.1.m1.1.1.3.cmml">s</mi></mrow><annotation-xml encoding="MathML-Content" id="S8.6.p6.1.m1.1b"><apply id="S8.6.p6.1.m1.1.1.cmml" xref="S8.6.p6.1.m1.1.1"><csymbol cd="latexml" id="S8.6.p6.1.m1.1.1.1.cmml" xref="S8.6.p6.1.m1.1.1.1">not-square-image-of</csymbol><ci id="S8.6.p6.1.m1.1.1.2.cmml" xref="S8.6.p6.1.m1.1.1.2">𝑡</ci><ci id="S8.6.p6.1.m1.1.1.3.cmml" xref="S8.6.p6.1.m1.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.6.p6.1.m1.1c">t\not\sqsubset s</annotation><annotation encoding="application/x-llamapun" id="S8.6.p6.1.m1.1d">italic_t ⊏̸ italic_s</annotation></semantics></math> and <math alttext="g(t)\not\sqsubset g(s)" class="ltx_Math" display="inline" id="S8.6.p6.2.m2.2"><semantics id="S8.6.p6.2.m2.2a"><mrow id="S8.6.p6.2.m2.2.3" xref="S8.6.p6.2.m2.2.3.cmml"><mrow id="S8.6.p6.2.m2.2.3.2" xref="S8.6.p6.2.m2.2.3.2.cmml"><mi id="S8.6.p6.2.m2.2.3.2.2" xref="S8.6.p6.2.m2.2.3.2.2.cmml">g</mi><mo id="S8.6.p6.2.m2.2.3.2.1" xref="S8.6.p6.2.m2.2.3.2.1.cmml">⁢</mo><mrow id="S8.6.p6.2.m2.2.3.2.3.2" xref="S8.6.p6.2.m2.2.3.2.cmml"><mo id="S8.6.p6.2.m2.2.3.2.3.2.1" stretchy="false" xref="S8.6.p6.2.m2.2.3.2.cmml">(</mo><mi id="S8.6.p6.2.m2.1.1" xref="S8.6.p6.2.m2.1.1.cmml">t</mi><mo id="S8.6.p6.2.m2.2.3.2.3.2.2" stretchy="false" xref="S8.6.p6.2.m2.2.3.2.cmml">)</mo></mrow></mrow><mo id="S8.6.p6.2.m2.2.3.1" xref="S8.6.p6.2.m2.2.3.1.cmml">⊏̸</mo><mrow id="S8.6.p6.2.m2.2.3.3" xref="S8.6.p6.2.m2.2.3.3.cmml"><mi id="S8.6.p6.2.m2.2.3.3.2" xref="S8.6.p6.2.m2.2.3.3.2.cmml">g</mi><mo id="S8.6.p6.2.m2.2.3.3.1" xref="S8.6.p6.2.m2.2.3.3.1.cmml">⁢</mo><mrow id="S8.6.p6.2.m2.2.3.3.3.2" xref="S8.6.p6.2.m2.2.3.3.cmml"><mo id="S8.6.p6.2.m2.2.3.3.3.2.1" stretchy="false" xref="S8.6.p6.2.m2.2.3.3.cmml">(</mo><mi id="S8.6.p6.2.m2.2.2" xref="S8.6.p6.2.m2.2.2.cmml">s</mi><mo id="S8.6.p6.2.m2.2.3.3.3.2.2" stretchy="false" xref="S8.6.p6.2.m2.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S8.6.p6.2.m2.2b"><apply id="S8.6.p6.2.m2.2.3.cmml" xref="S8.6.p6.2.m2.2.3"><csymbol cd="latexml" id="S8.6.p6.2.m2.2.3.1.cmml" xref="S8.6.p6.2.m2.2.3.1">not-square-image-of</csymbol><apply id="S8.6.p6.2.m2.2.3.2.cmml" xref="S8.6.p6.2.m2.2.3.2"><times id="S8.6.p6.2.m2.2.3.2.1.cmml" xref="S8.6.p6.2.m2.2.3.2.1"></times><ci id="S8.6.p6.2.m2.2.3.2.2.cmml" xref="S8.6.p6.2.m2.2.3.2.2">𝑔</ci><ci id="S8.6.p6.2.m2.1.1.cmml" xref="S8.6.p6.2.m2.1.1">𝑡</ci></apply><apply id="S8.6.p6.2.m2.2.3.3.cmml" xref="S8.6.p6.2.m2.2.3.3"><times id="S8.6.p6.2.m2.2.3.3.1.cmml" xref="S8.6.p6.2.m2.2.3.3.1"></times><ci id="S8.6.p6.2.m2.2.3.3.2.cmml" xref="S8.6.p6.2.m2.2.3.3.2">𝑔</ci><ci id="S8.6.p6.2.m2.2.2.cmml" xref="S8.6.p6.2.m2.2.2">𝑠</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.6.p6.2.m2.2c">g(t)\not\sqsubset g(s)</annotation><annotation encoding="application/x-llamapun" id="S8.6.p6.2.m2.2d">italic_g ( italic_t ) ⊏̸ italic_g ( italic_s )</annotation></semantics></math>. If <math alttext="x=b" class="ltx_Math" display="inline" id="S8.6.p6.3.m3.1"><semantics id="S8.6.p6.3.m3.1a"><mrow id="S8.6.p6.3.m3.1.1" xref="S8.6.p6.3.m3.1.1.cmml"><mi id="S8.6.p6.3.m3.1.1.2" xref="S8.6.p6.3.m3.1.1.2.cmml">x</mi><mo id="S8.6.p6.3.m3.1.1.1" xref="S8.6.p6.3.m3.1.1.1.cmml">=</mo><mi id="S8.6.p6.3.m3.1.1.3" xref="S8.6.p6.3.m3.1.1.3.cmml">b</mi></mrow><annotation-xml encoding="MathML-Content" id="S8.6.p6.3.m3.1b"><apply id="S8.6.p6.3.m3.1.1.cmml" xref="S8.6.p6.3.m3.1.1"><eq id="S8.6.p6.3.m3.1.1.1.cmml" xref="S8.6.p6.3.m3.1.1.1"></eq><ci id="S8.6.p6.3.m3.1.1.2.cmml" xref="S8.6.p6.3.m3.1.1.2">𝑥</ci><ci id="S8.6.p6.3.m3.1.1.3.cmml" xref="S8.6.p6.3.m3.1.1.3">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.6.p6.3.m3.1c">x=b</annotation><annotation encoding="application/x-llamapun" id="S8.6.p6.3.m3.1d">italic_x = italic_b</annotation></semantics></math>, then <math alttext="a<x=b<y" class="ltx_Math" display="inline" id="S8.6.p6.4.m4.1"><semantics id="S8.6.p6.4.m4.1a"><mrow id="S8.6.p6.4.m4.1.1" xref="S8.6.p6.4.m4.1.1.cmml"><mi id="S8.6.p6.4.m4.1.1.2" xref="S8.6.p6.4.m4.1.1.2.cmml">a</mi><mo id="S8.6.p6.4.m4.1.1.3" xref="S8.6.p6.4.m4.1.1.3.cmml"><</mo><mi id="S8.6.p6.4.m4.1.1.4" xref="S8.6.p6.4.m4.1.1.4.cmml">x</mi><mo id="S8.6.p6.4.m4.1.1.5" xref="S8.6.p6.4.m4.1.1.5.cmml">=</mo><mi id="S8.6.p6.4.m4.1.1.6" xref="S8.6.p6.4.m4.1.1.6.cmml">b</mi><mo id="S8.6.p6.4.m4.1.1.7" xref="S8.6.p6.4.m4.1.1.7.cmml"><</mo><mi id="S8.6.p6.4.m4.1.1.8" xref="S8.6.p6.4.m4.1.1.8.cmml">y</mi></mrow><annotation-xml encoding="MathML-Content" id="S8.6.p6.4.m4.1b"><apply id="S8.6.p6.4.m4.1.1.cmml" xref="S8.6.p6.4.m4.1.1"><and id="S8.6.p6.4.m4.1.1a.cmml" xref="S8.6.p6.4.m4.1.1"></and><apply id="S8.6.p6.4.m4.1.1b.cmml" xref="S8.6.p6.4.m4.1.1"><lt id="S8.6.p6.4.m4.1.1.3.cmml" xref="S8.6.p6.4.m4.1.1.3"></lt><ci id="S8.6.p6.4.m4.1.1.2.cmml" xref="S8.6.p6.4.m4.1.1.2">𝑎</ci><ci id="S8.6.p6.4.m4.1.1.4.cmml" xref="S8.6.p6.4.m4.1.1.4">𝑥</ci></apply><apply id="S8.6.p6.4.m4.1.1c.cmml" xref="S8.6.p6.4.m4.1.1"><eq id="S8.6.p6.4.m4.1.1.5.cmml" xref="S8.6.p6.4.m4.1.1.5"></eq><share href="https://arxiv.org/html/2503.13728v1#S8.6.p6.4.m4.1.1.4.cmml" id="S8.6.p6.4.m4.1.1d.cmml" xref="S8.6.p6.4.m4.1.1"></share><ci id="S8.6.p6.4.m4.1.1.6.cmml" xref="S8.6.p6.4.m4.1.1.6">𝑏</ci></apply><apply id="S8.6.p6.4.m4.1.1e.cmml" xref="S8.6.p6.4.m4.1.1"><lt id="S8.6.p6.4.m4.1.1.7.cmml" xref="S8.6.p6.4.m4.1.1.7"></lt><share href="https://arxiv.org/html/2503.13728v1#S8.6.p6.4.m4.1.1.6.cmml" id="S8.6.p6.4.m4.1.1f.cmml" xref="S8.6.p6.4.m4.1.1"></share><ci id="S8.6.p6.4.m4.1.1.8.cmml" xref="S8.6.p6.4.m4.1.1.8">𝑦</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.6.p6.4.m4.1c">a<x=b<y</annotation><annotation encoding="application/x-llamapun" id="S8.6.p6.4.m4.1d">italic_a < italic_x = italic_b < italic_y</annotation></semantics></math> and thus <math alttext="q^{\prime}:=q\cup\{(a,0),(x,1),(y,2)\}" class="ltx_Math" display="inline" id="S8.6.p6.5.m5.9"><semantics id="S8.6.p6.5.m5.9a"><mrow id="S8.6.p6.5.m5.9.9" xref="S8.6.p6.5.m5.9.9.cmml"><msup id="S8.6.p6.5.m5.9.9.5" xref="S8.6.p6.5.m5.9.9.5.cmml"><mi id="S8.6.p6.5.m5.9.9.5.2" xref="S8.6.p6.5.m5.9.9.5.2.cmml">q</mi><mo id="S8.6.p6.5.m5.9.9.5.3" xref="S8.6.p6.5.m5.9.9.5.3.cmml">′</mo></msup><mo id="S8.6.p6.5.m5.9.9.4" lspace="0.278em" rspace="0.278em" xref="S8.6.p6.5.m5.9.9.4.cmml">:=</mo><mrow id="S8.6.p6.5.m5.9.9.3" xref="S8.6.p6.5.m5.9.9.3.cmml"><mi id="S8.6.p6.5.m5.9.9.3.5" xref="S8.6.p6.5.m5.9.9.3.5.cmml">q</mi><mo id="S8.6.p6.5.m5.9.9.3.4" xref="S8.6.p6.5.m5.9.9.3.4.cmml">∪</mo><mrow id="S8.6.p6.5.m5.9.9.3.3.3" xref="S8.6.p6.5.m5.9.9.3.3.4.cmml"><mo id="S8.6.p6.5.m5.9.9.3.3.3.4" stretchy="false" xref="S8.6.p6.5.m5.9.9.3.3.4.cmml">{</mo><mrow id="S8.6.p6.5.m5.7.7.1.1.1.1.2" xref="S8.6.p6.5.m5.7.7.1.1.1.1.1.cmml"><mo id="S8.6.p6.5.m5.7.7.1.1.1.1.2.1" stretchy="false" xref="S8.6.p6.5.m5.7.7.1.1.1.1.1.cmml">(</mo><mi id="S8.6.p6.5.m5.1.1" xref="S8.6.p6.5.m5.1.1.cmml">a</mi><mo id="S8.6.p6.5.m5.7.7.1.1.1.1.2.2" xref="S8.6.p6.5.m5.7.7.1.1.1.1.1.cmml">,</mo><mn id="S8.6.p6.5.m5.2.2" xref="S8.6.p6.5.m5.2.2.cmml">0</mn><mo id="S8.6.p6.5.m5.7.7.1.1.1.1.2.3" stretchy="false" xref="S8.6.p6.5.m5.7.7.1.1.1.1.1.cmml">)</mo></mrow><mo id="S8.6.p6.5.m5.9.9.3.3.3.5" xref="S8.6.p6.5.m5.9.9.3.3.4.cmml">,</mo><mrow id="S8.6.p6.5.m5.8.8.2.2.2.2.2" xref="S8.6.p6.5.m5.8.8.2.2.2.2.1.cmml"><mo id="S8.6.p6.5.m5.8.8.2.2.2.2.2.1" stretchy="false" xref="S8.6.p6.5.m5.8.8.2.2.2.2.1.cmml">(</mo><mi id="S8.6.p6.5.m5.3.3" xref="S8.6.p6.5.m5.3.3.cmml">x</mi><mo id="S8.6.p6.5.m5.8.8.2.2.2.2.2.2" xref="S8.6.p6.5.m5.8.8.2.2.2.2.1.cmml">,</mo><mn id="S8.6.p6.5.m5.4.4" xref="S8.6.p6.5.m5.4.4.cmml">1</mn><mo id="S8.6.p6.5.m5.8.8.2.2.2.2.2.3" stretchy="false" xref="S8.6.p6.5.m5.8.8.2.2.2.2.1.cmml">)</mo></mrow><mo id="S8.6.p6.5.m5.9.9.3.3.3.6" xref="S8.6.p6.5.m5.9.9.3.3.4.cmml">,</mo><mrow id="S8.6.p6.5.m5.9.9.3.3.3.3.2" xref="S8.6.p6.5.m5.9.9.3.3.3.3.1.cmml"><mo id="S8.6.p6.5.m5.9.9.3.3.3.3.2.1" stretchy="false" xref="S8.6.p6.5.m5.9.9.3.3.3.3.1.cmml">(</mo><mi id="S8.6.p6.5.m5.5.5" xref="S8.6.p6.5.m5.5.5.cmml">y</mi><mo id="S8.6.p6.5.m5.9.9.3.3.3.3.2.2" xref="S8.6.p6.5.m5.9.9.3.3.3.3.1.cmml">,</mo><mn id="S8.6.p6.5.m5.6.6" xref="S8.6.p6.5.m5.6.6.cmml">2</mn><mo id="S8.6.p6.5.m5.9.9.3.3.3.3.2.3" stretchy="false" xref="S8.6.p6.5.m5.9.9.3.3.3.3.1.cmml">)</mo></mrow><mo id="S8.6.p6.5.m5.9.9.3.3.3.7" stretchy="false" xref="S8.6.p6.5.m5.9.9.3.3.4.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S8.6.p6.5.m5.9b"><apply id="S8.6.p6.5.m5.9.9.cmml" xref="S8.6.p6.5.m5.9.9"><csymbol cd="latexml" id="S8.6.p6.5.m5.9.9.4.cmml" xref="S8.6.p6.5.m5.9.9.4">assign</csymbol><apply id="S8.6.p6.5.m5.9.9.5.cmml" xref="S8.6.p6.5.m5.9.9.5"><csymbol cd="ambiguous" id="S8.6.p6.5.m5.9.9.5.1.cmml" xref="S8.6.p6.5.m5.9.9.5">superscript</csymbol><ci id="S8.6.p6.5.m5.9.9.5.2.cmml" xref="S8.6.p6.5.m5.9.9.5.2">𝑞</ci><ci id="S8.6.p6.5.m5.9.9.5.3.cmml" xref="S8.6.p6.5.m5.9.9.5.3">′</ci></apply><apply id="S8.6.p6.5.m5.9.9.3.cmml" xref="S8.6.p6.5.m5.9.9.3"><union id="S8.6.p6.5.m5.9.9.3.4.cmml" xref="S8.6.p6.5.m5.9.9.3.4"></union><ci id="S8.6.p6.5.m5.9.9.3.5.cmml" xref="S8.6.p6.5.m5.9.9.3.5">𝑞</ci><set id="S8.6.p6.5.m5.9.9.3.3.4.cmml" xref="S8.6.p6.5.m5.9.9.3.3.3"><interval closure="open" id="S8.6.p6.5.m5.7.7.1.1.1.1.1.cmml" xref="S8.6.p6.5.m5.7.7.1.1.1.1.2"><ci id="S8.6.p6.5.m5.1.1.cmml" xref="S8.6.p6.5.m5.1.1">𝑎</ci><cn id="S8.6.p6.5.m5.2.2.cmml" type="integer" xref="S8.6.p6.5.m5.2.2">0</cn></interval><interval closure="open" id="S8.6.p6.5.m5.8.8.2.2.2.2.1.cmml" xref="S8.6.p6.5.m5.8.8.2.2.2.2.2"><ci id="S8.6.p6.5.m5.3.3.cmml" xref="S8.6.p6.5.m5.3.3">𝑥</ci><cn id="S8.6.p6.5.m5.4.4.cmml" type="integer" xref="S8.6.p6.5.m5.4.4">1</cn></interval><interval closure="open" id="S8.6.p6.5.m5.9.9.3.3.3.3.1.cmml" xref="S8.6.p6.5.m5.9.9.3.3.3.3.2"><ci id="S8.6.p6.5.m5.5.5.cmml" xref="S8.6.p6.5.m5.5.5">𝑦</ci><cn id="S8.6.p6.5.m5.6.6.cmml" type="integer" xref="S8.6.p6.5.m5.6.6">2</cn></interval></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.6.p6.5.m5.9c">q^{\prime}:=q\cup\{(a,0),(x,1),(y,2)\}</annotation><annotation encoding="application/x-llamapun" id="S8.6.p6.5.m5.9d">italic_q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT := italic_q ∪ { ( italic_a , 0 ) , ( italic_x , 1 ) , ( italic_y , 2 ) }</annotation></semantics></math> is in <math alttext="\mathbb{P}" class="ltx_Math" display="inline" id="S8.6.p6.6.m6.1"><semantics id="S8.6.p6.6.m6.1a"><mi id="S8.6.p6.6.m6.1.1" xref="S8.6.p6.6.m6.1.1.cmml">ℙ</mi><annotation-xml encoding="MathML-Content" id="S8.6.p6.6.m6.1b"><ci id="S8.6.p6.6.m6.1.1.cmml" xref="S8.6.p6.6.m6.1.1">ℙ</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.6.p6.6.m6.1c">\mathbb{P}</annotation><annotation encoding="application/x-llamapun" id="S8.6.p6.6.m6.1d">blackboard_P</annotation></semantics></math>. But note then <math alttext="q^{\prime}\leq q" class="ltx_Math" display="inline" id="S8.6.p6.7.m7.1"><semantics id="S8.6.p6.7.m7.1a"><mrow id="S8.6.p6.7.m7.1.1" xref="S8.6.p6.7.m7.1.1.cmml"><msup id="S8.6.p6.7.m7.1.1.2" xref="S8.6.p6.7.m7.1.1.2.cmml"><mi id="S8.6.p6.7.m7.1.1.2.2" xref="S8.6.p6.7.m7.1.1.2.2.cmml">q</mi><mo id="S8.6.p6.7.m7.1.1.2.3" xref="S8.6.p6.7.m7.1.1.2.3.cmml">′</mo></msup><mo id="S8.6.p6.7.m7.1.1.1" xref="S8.6.p6.7.m7.1.1.1.cmml">≤</mo><mi id="S8.6.p6.7.m7.1.1.3" xref="S8.6.p6.7.m7.1.1.3.cmml">q</mi></mrow><annotation-xml encoding="MathML-Content" id="S8.6.p6.7.m7.1b"><apply id="S8.6.p6.7.m7.1.1.cmml" xref="S8.6.p6.7.m7.1.1"><leq id="S8.6.p6.7.m7.1.1.1.cmml" xref="S8.6.p6.7.m7.1.1.1"></leq><apply id="S8.6.p6.7.m7.1.1.2.cmml" xref="S8.6.p6.7.m7.1.1.2"><csymbol cd="ambiguous" id="S8.6.p6.7.m7.1.1.2.1.cmml" xref="S8.6.p6.7.m7.1.1.2">superscript</csymbol><ci id="S8.6.p6.7.m7.1.1.2.2.cmml" xref="S8.6.p6.7.m7.1.1.2.2">𝑞</ci><ci id="S8.6.p6.7.m7.1.1.2.3.cmml" xref="S8.6.p6.7.m7.1.1.2.3">′</ci></apply><ci id="S8.6.p6.7.m7.1.1.3.cmml" xref="S8.6.p6.7.m7.1.1.3">𝑞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.6.p6.7.m7.1c">q^{\prime}\leq q</annotation><annotation encoding="application/x-llamapun" id="S8.6.p6.7.m7.1d">italic_q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_q</annotation></semantics></math> and <math alttext="q\Vdash\mathring{c}\circ\check{t}<_{\mathrm{lex}}\mathring{c}\circ\check{s}% \land\mathring{c}\circ\check{g}(\check{t})<_{\mathrm{lex}}\mathring{c}\circ% \check{g}(\check{s})" class="ltx_Math" display="inline" id="S8.6.p6.8.m8.2"><semantics id="S8.6.p6.8.m8.2a"><mrow id="S8.6.p6.8.m8.2.3" xref="S8.6.p6.8.m8.2.3.cmml"><mi id="S8.6.p6.8.m8.2.3.2" xref="S8.6.p6.8.m8.2.3.2.cmml">q</mi><mo id="S8.6.p6.8.m8.2.3.3" xref="S8.6.p6.8.m8.2.3.3.cmml">⊩</mo><mrow id="S8.6.p6.8.m8.2.3.4" xref="S8.6.p6.8.m8.2.3.4.cmml"><mover accent="true" id="S8.6.p6.8.m8.2.3.4.2" xref="S8.6.p6.8.m8.2.3.4.2.cmml"><mi id="S8.6.p6.8.m8.2.3.4.2.2" xref="S8.6.p6.8.m8.2.3.4.2.2.cmml">c</mi><mo id="S8.6.p6.8.m8.2.3.4.2.1" xref="S8.6.p6.8.m8.2.3.4.2.1.cmml">̊</mo></mover><mo id="S8.6.p6.8.m8.2.3.4.1" lspace="0.222em" rspace="0.222em" xref="S8.6.p6.8.m8.2.3.4.1.cmml">∘</mo><mover accent="true" id="S8.6.p6.8.m8.2.3.4.3" xref="S8.6.p6.8.m8.2.3.4.3.cmml"><mi id="S8.6.p6.8.m8.2.3.4.3.2" xref="S8.6.p6.8.m8.2.3.4.3.2.cmml">t</mi><mo id="S8.6.p6.8.m8.2.3.4.3.1" xref="S8.6.p6.8.m8.2.3.4.3.1.cmml">ˇ</mo></mover></mrow><msub id="S8.6.p6.8.m8.2.3.5" xref="S8.6.p6.8.m8.2.3.5.cmml"><mo id="S8.6.p6.8.m8.2.3.5.2" xref="S8.6.p6.8.m8.2.3.5.2.cmml"><</mo><mi id="S8.6.p6.8.m8.2.3.5.3" xref="S8.6.p6.8.m8.2.3.5.3.cmml">lex</mi></msub><mrow id="S8.6.p6.8.m8.2.3.6" xref="S8.6.p6.8.m8.2.3.6.cmml"><mrow id="S8.6.p6.8.m8.2.3.6.2" xref="S8.6.p6.8.m8.2.3.6.2.cmml"><mover accent="true" id="S8.6.p6.8.m8.2.3.6.2.2" xref="S8.6.p6.8.m8.2.3.6.2.2.cmml"><mi id="S8.6.p6.8.m8.2.3.6.2.2.2" xref="S8.6.p6.8.m8.2.3.6.2.2.2.cmml">c</mi><mo id="S8.6.p6.8.m8.2.3.6.2.2.1" xref="S8.6.p6.8.m8.2.3.6.2.2.1.cmml">̊</mo></mover><mo id="S8.6.p6.8.m8.2.3.6.2.1" lspace="0.222em" rspace="0.222em" xref="S8.6.p6.8.m8.2.3.6.2.1.cmml">∘</mo><mover accent="true" id="S8.6.p6.8.m8.2.3.6.2.3" xref="S8.6.p6.8.m8.2.3.6.2.3.cmml"><mi id="S8.6.p6.8.m8.2.3.6.2.3.2" xref="S8.6.p6.8.m8.2.3.6.2.3.2.cmml">s</mi><mo id="S8.6.p6.8.m8.2.3.6.2.3.1" xref="S8.6.p6.8.m8.2.3.6.2.3.1.cmml">ˇ</mo></mover></mrow><mo id="S8.6.p6.8.m8.2.3.6.1" xref="S8.6.p6.8.m8.2.3.6.1.cmml">∧</mo><mrow id="S8.6.p6.8.m8.2.3.6.3" xref="S8.6.p6.8.m8.2.3.6.3.cmml"><mrow id="S8.6.p6.8.m8.2.3.6.3.2" xref="S8.6.p6.8.m8.2.3.6.3.2.cmml"><mover accent="true" id="S8.6.p6.8.m8.2.3.6.3.2.2" xref="S8.6.p6.8.m8.2.3.6.3.2.2.cmml"><mi id="S8.6.p6.8.m8.2.3.6.3.2.2.2" xref="S8.6.p6.8.m8.2.3.6.3.2.2.2.cmml">c</mi><mo id="S8.6.p6.8.m8.2.3.6.3.2.2.1" xref="S8.6.p6.8.m8.2.3.6.3.2.2.1.cmml">̊</mo></mover><mo id="S8.6.p6.8.m8.2.3.6.3.2.1" lspace="0.222em" rspace="0.222em" xref="S8.6.p6.8.m8.2.3.6.3.2.1.cmml">∘</mo><mover accent="true" id="S8.6.p6.8.m8.2.3.6.3.2.3" xref="S8.6.p6.8.m8.2.3.6.3.2.3.cmml"><mi id="S8.6.p6.8.m8.2.3.6.3.2.3.2" xref="S8.6.p6.8.m8.2.3.6.3.2.3.2.cmml">g</mi><mo id="S8.6.p6.8.m8.2.3.6.3.2.3.1" xref="S8.6.p6.8.m8.2.3.6.3.2.3.1.cmml">ˇ</mo></mover></mrow><mo id="S8.6.p6.8.m8.2.3.6.3.1" xref="S8.6.p6.8.m8.2.3.6.3.1.cmml">⁢</mo><mrow id="S8.6.p6.8.m8.2.3.6.3.3.2" xref="S8.6.p6.8.m8.1.1.cmml"><mo id="S8.6.p6.8.m8.2.3.6.3.3.2.1" stretchy="false" xref="S8.6.p6.8.m8.1.1.cmml">(</mo><mover accent="true" id="S8.6.p6.8.m8.1.1" xref="S8.6.p6.8.m8.1.1.cmml"><mi id="S8.6.p6.8.m8.1.1.2" xref="S8.6.p6.8.m8.1.1.2.cmml">t</mi><mo id="S8.6.p6.8.m8.1.1.1" xref="S8.6.p6.8.m8.1.1.1.cmml">ˇ</mo></mover><mo id="S8.6.p6.8.m8.2.3.6.3.3.2.2" stretchy="false" xref="S8.6.p6.8.m8.1.1.cmml">)</mo></mrow></mrow></mrow><msub id="S8.6.p6.8.m8.2.3.7" xref="S8.6.p6.8.m8.2.3.7.cmml"><mo id="S8.6.p6.8.m8.2.3.7.2" xref="S8.6.p6.8.m8.2.3.7.2.cmml"><</mo><mi id="S8.6.p6.8.m8.2.3.7.3" xref="S8.6.p6.8.m8.2.3.7.3.cmml">lex</mi></msub><mrow id="S8.6.p6.8.m8.2.3.8" xref="S8.6.p6.8.m8.2.3.8.cmml"><mrow id="S8.6.p6.8.m8.2.3.8.2" xref="S8.6.p6.8.m8.2.3.8.2.cmml"><mover accent="true" id="S8.6.p6.8.m8.2.3.8.2.2" xref="S8.6.p6.8.m8.2.3.8.2.2.cmml"><mi id="S8.6.p6.8.m8.2.3.8.2.2.2" xref="S8.6.p6.8.m8.2.3.8.2.2.2.cmml">c</mi><mo id="S8.6.p6.8.m8.2.3.8.2.2.1" xref="S8.6.p6.8.m8.2.3.8.2.2.1.cmml">̊</mo></mover><mo id="S8.6.p6.8.m8.2.3.8.2.1" lspace="0.222em" rspace="0.222em" xref="S8.6.p6.8.m8.2.3.8.2.1.cmml">∘</mo><mover accent="true" id="S8.6.p6.8.m8.2.3.8.2.3" xref="S8.6.p6.8.m8.2.3.8.2.3.cmml"><mi id="S8.6.p6.8.m8.2.3.8.2.3.2" xref="S8.6.p6.8.m8.2.3.8.2.3.2.cmml">g</mi><mo id="S8.6.p6.8.m8.2.3.8.2.3.1" xref="S8.6.p6.8.m8.2.3.8.2.3.1.cmml">ˇ</mo></mover></mrow><mo id="S8.6.p6.8.m8.2.3.8.1" xref="S8.6.p6.8.m8.2.3.8.1.cmml">⁢</mo><mrow id="S8.6.p6.8.m8.2.3.8.3.2" xref="S8.6.p6.8.m8.2.2.cmml"><mo id="S8.6.p6.8.m8.2.3.8.3.2.1" stretchy="false" xref="S8.6.p6.8.m8.2.2.cmml">(</mo><mover accent="true" id="S8.6.p6.8.m8.2.2" xref="S8.6.p6.8.m8.2.2.cmml"><mi id="S8.6.p6.8.m8.2.2.2" xref="S8.6.p6.8.m8.2.2.2.cmml">s</mi><mo id="S8.6.p6.8.m8.2.2.1" xref="S8.6.p6.8.m8.2.2.1.cmml">ˇ</mo></mover><mo id="S8.6.p6.8.m8.2.3.8.3.2.2" stretchy="false" xref="S8.6.p6.8.m8.2.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S8.6.p6.8.m8.2b"><apply id="S8.6.p6.8.m8.2.3.cmml" xref="S8.6.p6.8.m8.2.3"><and id="S8.6.p6.8.m8.2.3a.cmml" xref="S8.6.p6.8.m8.2.3"></and><apply id="S8.6.p6.8.m8.2.3b.cmml" xref="S8.6.p6.8.m8.2.3"><csymbol cd="latexml" id="S8.6.p6.8.m8.2.3.3.cmml" xref="S8.6.p6.8.m8.2.3.3">forces</csymbol><ci id="S8.6.p6.8.m8.2.3.2.cmml" xref="S8.6.p6.8.m8.2.3.2">𝑞</ci><apply id="S8.6.p6.8.m8.2.3.4.cmml" xref="S8.6.p6.8.m8.2.3.4"><compose id="S8.6.p6.8.m8.2.3.4.1.cmml" xref="S8.6.p6.8.m8.2.3.4.1"></compose><apply id="S8.6.p6.8.m8.2.3.4.2.cmml" xref="S8.6.p6.8.m8.2.3.4.2"><ci id="S8.6.p6.8.m8.2.3.4.2.1.cmml" xref="S8.6.p6.8.m8.2.3.4.2.1">̊</ci><ci id="S8.6.p6.8.m8.2.3.4.2.2.cmml" xref="S8.6.p6.8.m8.2.3.4.2.2">𝑐</ci></apply><apply id="S8.6.p6.8.m8.2.3.4.3.cmml" xref="S8.6.p6.8.m8.2.3.4.3"><ci id="S8.6.p6.8.m8.2.3.4.3.1.cmml" xref="S8.6.p6.8.m8.2.3.4.3.1">ˇ</ci><ci id="S8.6.p6.8.m8.2.3.4.3.2.cmml" xref="S8.6.p6.8.m8.2.3.4.3.2">𝑡</ci></apply></apply></apply><apply id="S8.6.p6.8.m8.2.3c.cmml" xref="S8.6.p6.8.m8.2.3"><apply id="S8.6.p6.8.m8.2.3.5.cmml" xref="S8.6.p6.8.m8.2.3.5"><csymbol cd="ambiguous" id="S8.6.p6.8.m8.2.3.5.1.cmml" xref="S8.6.p6.8.m8.2.3.5">subscript</csymbol><lt id="S8.6.p6.8.m8.2.3.5.2.cmml" xref="S8.6.p6.8.m8.2.3.5.2"></lt><ci id="S8.6.p6.8.m8.2.3.5.3.cmml" xref="S8.6.p6.8.m8.2.3.5.3">lex</ci></apply><share href="https://arxiv.org/html/2503.13728v1#S8.6.p6.8.m8.2.3.4.cmml" id="S8.6.p6.8.m8.2.3d.cmml" xref="S8.6.p6.8.m8.2.3"></share><apply id="S8.6.p6.8.m8.2.3.6.cmml" xref="S8.6.p6.8.m8.2.3.6"><and id="S8.6.p6.8.m8.2.3.6.1.cmml" xref="S8.6.p6.8.m8.2.3.6.1"></and><apply id="S8.6.p6.8.m8.2.3.6.2.cmml" xref="S8.6.p6.8.m8.2.3.6.2"><compose id="S8.6.p6.8.m8.2.3.6.2.1.cmml" xref="S8.6.p6.8.m8.2.3.6.2.1"></compose><apply id="S8.6.p6.8.m8.2.3.6.2.2.cmml" xref="S8.6.p6.8.m8.2.3.6.2.2"><ci id="S8.6.p6.8.m8.2.3.6.2.2.1.cmml" xref="S8.6.p6.8.m8.2.3.6.2.2.1">̊</ci><ci id="S8.6.p6.8.m8.2.3.6.2.2.2.cmml" xref="S8.6.p6.8.m8.2.3.6.2.2.2">𝑐</ci></apply><apply id="S8.6.p6.8.m8.2.3.6.2.3.cmml" xref="S8.6.p6.8.m8.2.3.6.2.3"><ci id="S8.6.p6.8.m8.2.3.6.2.3.1.cmml" xref="S8.6.p6.8.m8.2.3.6.2.3.1">ˇ</ci><ci id="S8.6.p6.8.m8.2.3.6.2.3.2.cmml" xref="S8.6.p6.8.m8.2.3.6.2.3.2">𝑠</ci></apply></apply><apply id="S8.6.p6.8.m8.2.3.6.3.cmml" xref="S8.6.p6.8.m8.2.3.6.3"><times id="S8.6.p6.8.m8.2.3.6.3.1.cmml" xref="S8.6.p6.8.m8.2.3.6.3.1"></times><apply id="S8.6.p6.8.m8.2.3.6.3.2.cmml" xref="S8.6.p6.8.m8.2.3.6.3.2"><compose id="S8.6.p6.8.m8.2.3.6.3.2.1.cmml" xref="S8.6.p6.8.m8.2.3.6.3.2.1"></compose><apply id="S8.6.p6.8.m8.2.3.6.3.2.2.cmml" xref="S8.6.p6.8.m8.2.3.6.3.2.2"><ci id="S8.6.p6.8.m8.2.3.6.3.2.2.1.cmml" xref="S8.6.p6.8.m8.2.3.6.3.2.2.1">̊</ci><ci id="S8.6.p6.8.m8.2.3.6.3.2.2.2.cmml" xref="S8.6.p6.8.m8.2.3.6.3.2.2.2">𝑐</ci></apply><apply id="S8.6.p6.8.m8.2.3.6.3.2.3.cmml" xref="S8.6.p6.8.m8.2.3.6.3.2.3"><ci id="S8.6.p6.8.m8.2.3.6.3.2.3.1.cmml" xref="S8.6.p6.8.m8.2.3.6.3.2.3.1">ˇ</ci><ci id="S8.6.p6.8.m8.2.3.6.3.2.3.2.cmml" xref="S8.6.p6.8.m8.2.3.6.3.2.3.2">𝑔</ci></apply></apply><apply id="S8.6.p6.8.m8.1.1.cmml" xref="S8.6.p6.8.m8.2.3.6.3.3.2"><ci id="S8.6.p6.8.m8.1.1.1.cmml" xref="S8.6.p6.8.m8.1.1.1">ˇ</ci><ci id="S8.6.p6.8.m8.1.1.2.cmml" xref="S8.6.p6.8.m8.1.1.2">𝑡</ci></apply></apply></apply></apply><apply id="S8.6.p6.8.m8.2.3e.cmml" xref="S8.6.p6.8.m8.2.3"><apply id="S8.6.p6.8.m8.2.3.7.cmml" xref="S8.6.p6.8.m8.2.3.7"><csymbol cd="ambiguous" id="S8.6.p6.8.m8.2.3.7.1.cmml" xref="S8.6.p6.8.m8.2.3.7">subscript</csymbol><lt id="S8.6.p6.8.m8.2.3.7.2.cmml" xref="S8.6.p6.8.m8.2.3.7.2"></lt><ci id="S8.6.p6.8.m8.2.3.7.3.cmml" xref="S8.6.p6.8.m8.2.3.7.3">lex</ci></apply><share href="https://arxiv.org/html/2503.13728v1#S8.6.p6.8.m8.2.3.6.cmml" id="S8.6.p6.8.m8.2.3f.cmml" xref="S8.6.p6.8.m8.2.3"></share><apply id="S8.6.p6.8.m8.2.3.8.cmml" xref="S8.6.p6.8.m8.2.3.8"><times id="S8.6.p6.8.m8.2.3.8.1.cmml" xref="S8.6.p6.8.m8.2.3.8.1"></times><apply id="S8.6.p6.8.m8.2.3.8.2.cmml" xref="S8.6.p6.8.m8.2.3.8.2"><compose id="S8.6.p6.8.m8.2.3.8.2.1.cmml" xref="S8.6.p6.8.m8.2.3.8.2.1"></compose><apply id="S8.6.p6.8.m8.2.3.8.2.2.cmml" xref="S8.6.p6.8.m8.2.3.8.2.2"><ci id="S8.6.p6.8.m8.2.3.8.2.2.1.cmml" xref="S8.6.p6.8.m8.2.3.8.2.2.1">̊</ci><ci id="S8.6.p6.8.m8.2.3.8.2.2.2.cmml" xref="S8.6.p6.8.m8.2.3.8.2.2.2">𝑐</ci></apply><apply id="S8.6.p6.8.m8.2.3.8.2.3.cmml" xref="S8.6.p6.8.m8.2.3.8.2.3"><ci id="S8.6.p6.8.m8.2.3.8.2.3.1.cmml" xref="S8.6.p6.8.m8.2.3.8.2.3.1">ˇ</ci><ci id="S8.6.p6.8.m8.2.3.8.2.3.2.cmml" xref="S8.6.p6.8.m8.2.3.8.2.3.2">𝑔</ci></apply></apply><apply id="S8.6.p6.8.m8.2.2.cmml" xref="S8.6.p6.8.m8.2.3.8.3.2"><ci id="S8.6.p6.8.m8.2.2.1.cmml" xref="S8.6.p6.8.m8.2.2.1">ˇ</ci><ci id="S8.6.p6.8.m8.2.2.2.cmml" xref="S8.6.p6.8.m8.2.2.2">𝑠</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.6.p6.8.m8.2c">q\Vdash\mathring{c}\circ\check{t}<_{\mathrm{lex}}\mathring{c}\circ\check{s}% \land\mathring{c}\circ\check{g}(\check{t})<_{\mathrm{lex}}\mathring{c}\circ% \check{g}(\check{s})</annotation><annotation encoding="application/x-llamapun" id="S8.6.p6.8.m8.2d">italic_q ⊩ over̊ start_ARG italic_c end_ARG ∘ overroman_ˇ start_ARG italic_t end_ARG < start_POSTSUBSCRIPT roman_lex end_POSTSUBSCRIPT over̊ start_ARG italic_c end_ARG ∘ overroman_ˇ start_ARG italic_s end_ARG ∧ over̊ start_ARG italic_c end_ARG ∘ overroman_ˇ start_ARG italic_g end_ARG ( overroman_ˇ start_ARG italic_t end_ARG ) < start_POSTSUBSCRIPT roman_lex end_POSTSUBSCRIPT over̊ start_ARG italic_c end_ARG ∘ overroman_ˇ start_ARG italic_g end_ARG ( overroman_ˇ start_ARG italic_s end_ARG )</annotation></semantics></math>, and we arrive at a contradiction as before. Assume now <math alttext="x\neq b" class="ltx_Math" display="inline" id="S8.6.p6.9.m9.1"><semantics id="S8.6.p6.9.m9.1a"><mrow id="S8.6.p6.9.m9.1.1" xref="S8.6.p6.9.m9.1.1.cmml"><mi id="S8.6.p6.9.m9.1.1.2" xref="S8.6.p6.9.m9.1.1.2.cmml">x</mi><mo id="S8.6.p6.9.m9.1.1.1" xref="S8.6.p6.9.m9.1.1.1.cmml">≠</mo><mi id="S8.6.p6.9.m9.1.1.3" xref="S8.6.p6.9.m9.1.1.3.cmml">b</mi></mrow><annotation-xml encoding="MathML-Content" id="S8.6.p6.9.m9.1b"><apply id="S8.6.p6.9.m9.1.1.cmml" xref="S8.6.p6.9.m9.1.1"><neq id="S8.6.p6.9.m9.1.1.1.cmml" xref="S8.6.p6.9.m9.1.1.1"></neq><ci id="S8.6.p6.9.m9.1.1.2.cmml" xref="S8.6.p6.9.m9.1.1.2">𝑥</ci><ci id="S8.6.p6.9.m9.1.1.3.cmml" xref="S8.6.p6.9.m9.1.1.3">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.6.p6.9.m9.1c">x\neq b</annotation><annotation encoding="application/x-llamapun" id="S8.6.p6.9.m9.1d">italic_x ≠ italic_b</annotation></semantics></math>. If <math alttext="y=a" class="ltx_Math" display="inline" id="S8.6.p6.10.m10.1"><semantics id="S8.6.p6.10.m10.1a"><mrow id="S8.6.p6.10.m10.1.1" xref="S8.6.p6.10.m10.1.1.cmml"><mi id="S8.6.p6.10.m10.1.1.2" xref="S8.6.p6.10.m10.1.1.2.cmml">y</mi><mo id="S8.6.p6.10.m10.1.1.1" xref="S8.6.p6.10.m10.1.1.1.cmml">=</mo><mi id="S8.6.p6.10.m10.1.1.3" xref="S8.6.p6.10.m10.1.1.3.cmml">a</mi></mrow><annotation-xml encoding="MathML-Content" id="S8.6.p6.10.m10.1b"><apply id="S8.6.p6.10.m10.1.1.cmml" xref="S8.6.p6.10.m10.1.1"><eq id="S8.6.p6.10.m10.1.1.1.cmml" xref="S8.6.p6.10.m10.1.1.1"></eq><ci id="S8.6.p6.10.m10.1.1.2.cmml" xref="S8.6.p6.10.m10.1.1.2">𝑦</ci><ci id="S8.6.p6.10.m10.1.1.3.cmml" xref="S8.6.p6.10.m10.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.6.p6.10.m10.1c">y=a</annotation><annotation encoding="application/x-llamapun" id="S8.6.p6.10.m10.1d">italic_y = italic_a</annotation></semantics></math> then <math alttext="q^{\prime}:=q\cup\{(x,0),(a,1),(b,2)\}" class="ltx_Math" display="inline" id="S8.6.p6.11.m11.9"><semantics id="S8.6.p6.11.m11.9a"><mrow id="S8.6.p6.11.m11.9.9" xref="S8.6.p6.11.m11.9.9.cmml"><msup id="S8.6.p6.11.m11.9.9.5" xref="S8.6.p6.11.m11.9.9.5.cmml"><mi id="S8.6.p6.11.m11.9.9.5.2" xref="S8.6.p6.11.m11.9.9.5.2.cmml">q</mi><mo id="S8.6.p6.11.m11.9.9.5.3" xref="S8.6.p6.11.m11.9.9.5.3.cmml">′</mo></msup><mo id="S8.6.p6.11.m11.9.9.4" lspace="0.278em" rspace="0.278em" xref="S8.6.p6.11.m11.9.9.4.cmml">:=</mo><mrow id="S8.6.p6.11.m11.9.9.3" xref="S8.6.p6.11.m11.9.9.3.cmml"><mi id="S8.6.p6.11.m11.9.9.3.5" xref="S8.6.p6.11.m11.9.9.3.5.cmml">q</mi><mo id="S8.6.p6.11.m11.9.9.3.4" xref="S8.6.p6.11.m11.9.9.3.4.cmml">∪</mo><mrow id="S8.6.p6.11.m11.9.9.3.3.3" xref="S8.6.p6.11.m11.9.9.3.3.4.cmml"><mo id="S8.6.p6.11.m11.9.9.3.3.3.4" stretchy="false" xref="S8.6.p6.11.m11.9.9.3.3.4.cmml">{</mo><mrow id="S8.6.p6.11.m11.7.7.1.1.1.1.2" xref="S8.6.p6.11.m11.7.7.1.1.1.1.1.cmml"><mo id="S8.6.p6.11.m11.7.7.1.1.1.1.2.1" stretchy="false" xref="S8.6.p6.11.m11.7.7.1.1.1.1.1.cmml">(</mo><mi id="S8.6.p6.11.m11.1.1" xref="S8.6.p6.11.m11.1.1.cmml">x</mi><mo id="S8.6.p6.11.m11.7.7.1.1.1.1.2.2" xref="S8.6.p6.11.m11.7.7.1.1.1.1.1.cmml">,</mo><mn id="S8.6.p6.11.m11.2.2" xref="S8.6.p6.11.m11.2.2.cmml">0</mn><mo id="S8.6.p6.11.m11.7.7.1.1.1.1.2.3" stretchy="false" xref="S8.6.p6.11.m11.7.7.1.1.1.1.1.cmml">)</mo></mrow><mo id="S8.6.p6.11.m11.9.9.3.3.3.5" xref="S8.6.p6.11.m11.9.9.3.3.4.cmml">,</mo><mrow id="S8.6.p6.11.m11.8.8.2.2.2.2.2" xref="S8.6.p6.11.m11.8.8.2.2.2.2.1.cmml"><mo id="S8.6.p6.11.m11.8.8.2.2.2.2.2.1" stretchy="false" xref="S8.6.p6.11.m11.8.8.2.2.2.2.1.cmml">(</mo><mi id="S8.6.p6.11.m11.3.3" xref="S8.6.p6.11.m11.3.3.cmml">a</mi><mo id="S8.6.p6.11.m11.8.8.2.2.2.2.2.2" xref="S8.6.p6.11.m11.8.8.2.2.2.2.1.cmml">,</mo><mn id="S8.6.p6.11.m11.4.4" xref="S8.6.p6.11.m11.4.4.cmml">1</mn><mo id="S8.6.p6.11.m11.8.8.2.2.2.2.2.3" stretchy="false" xref="S8.6.p6.11.m11.8.8.2.2.2.2.1.cmml">)</mo></mrow><mo id="S8.6.p6.11.m11.9.9.3.3.3.6" xref="S8.6.p6.11.m11.9.9.3.3.4.cmml">,</mo><mrow id="S8.6.p6.11.m11.9.9.3.3.3.3.2" xref="S8.6.p6.11.m11.9.9.3.3.3.3.1.cmml"><mo id="S8.6.p6.11.m11.9.9.3.3.3.3.2.1" stretchy="false" xref="S8.6.p6.11.m11.9.9.3.3.3.3.1.cmml">(</mo><mi id="S8.6.p6.11.m11.5.5" xref="S8.6.p6.11.m11.5.5.cmml">b</mi><mo id="S8.6.p6.11.m11.9.9.3.3.3.3.2.2" xref="S8.6.p6.11.m11.9.9.3.3.3.3.1.cmml">,</mo><mn id="S8.6.p6.11.m11.6.6" xref="S8.6.p6.11.m11.6.6.cmml">2</mn><mo id="S8.6.p6.11.m11.9.9.3.3.3.3.2.3" stretchy="false" xref="S8.6.p6.11.m11.9.9.3.3.3.3.1.cmml">)</mo></mrow><mo id="S8.6.p6.11.m11.9.9.3.3.3.7" stretchy="false" xref="S8.6.p6.11.m11.9.9.3.3.4.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S8.6.p6.11.m11.9b"><apply id="S8.6.p6.11.m11.9.9.cmml" xref="S8.6.p6.11.m11.9.9"><csymbol cd="latexml" id="S8.6.p6.11.m11.9.9.4.cmml" xref="S8.6.p6.11.m11.9.9.4">assign</csymbol><apply id="S8.6.p6.11.m11.9.9.5.cmml" xref="S8.6.p6.11.m11.9.9.5"><csymbol cd="ambiguous" id="S8.6.p6.11.m11.9.9.5.1.cmml" xref="S8.6.p6.11.m11.9.9.5">superscript</csymbol><ci id="S8.6.p6.11.m11.9.9.5.2.cmml" xref="S8.6.p6.11.m11.9.9.5.2">𝑞</ci><ci id="S8.6.p6.11.m11.9.9.5.3.cmml" xref="S8.6.p6.11.m11.9.9.5.3">′</ci></apply><apply id="S8.6.p6.11.m11.9.9.3.cmml" xref="S8.6.p6.11.m11.9.9.3"><union id="S8.6.p6.11.m11.9.9.3.4.cmml" xref="S8.6.p6.11.m11.9.9.3.4"></union><ci id="S8.6.p6.11.m11.9.9.3.5.cmml" xref="S8.6.p6.11.m11.9.9.3.5">𝑞</ci><set id="S8.6.p6.11.m11.9.9.3.3.4.cmml" xref="S8.6.p6.11.m11.9.9.3.3.3"><interval closure="open" id="S8.6.p6.11.m11.7.7.1.1.1.1.1.cmml" xref="S8.6.p6.11.m11.7.7.1.1.1.1.2"><ci id="S8.6.p6.11.m11.1.1.cmml" xref="S8.6.p6.11.m11.1.1">𝑥</ci><cn id="S8.6.p6.11.m11.2.2.cmml" type="integer" xref="S8.6.p6.11.m11.2.2">0</cn></interval><interval closure="open" id="S8.6.p6.11.m11.8.8.2.2.2.2.1.cmml" xref="S8.6.p6.11.m11.8.8.2.2.2.2.2"><ci id="S8.6.p6.11.m11.3.3.cmml" xref="S8.6.p6.11.m11.3.3">𝑎</ci><cn id="S8.6.p6.11.m11.4.4.cmml" type="integer" xref="S8.6.p6.11.m11.4.4">1</cn></interval><interval closure="open" id="S8.6.p6.11.m11.9.9.3.3.3.3.1.cmml" xref="S8.6.p6.11.m11.9.9.3.3.3.3.2"><ci id="S8.6.p6.11.m11.5.5.cmml" xref="S8.6.p6.11.m11.5.5">𝑏</ci><cn id="S8.6.p6.11.m11.6.6.cmml" type="integer" xref="S8.6.p6.11.m11.6.6">2</cn></interval></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.6.p6.11.m11.9c">q^{\prime}:=q\cup\{(x,0),(a,1),(b,2)\}</annotation><annotation encoding="application/x-llamapun" id="S8.6.p6.11.m11.9d">italic_q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT := italic_q ∪ { ( italic_x , 0 ) , ( italic_a , 1 ) , ( italic_b , 2 ) }</annotation></semantics></math> works, and if <math alttext="y\neq a" class="ltx_Math" display="inline" id="S8.6.p6.12.m12.1"><semantics id="S8.6.p6.12.m12.1a"><mrow id="S8.6.p6.12.m12.1.1" xref="S8.6.p6.12.m12.1.1.cmml"><mi id="S8.6.p6.12.m12.1.1.2" xref="S8.6.p6.12.m12.1.1.2.cmml">y</mi><mo id="S8.6.p6.12.m12.1.1.1" xref="S8.6.p6.12.m12.1.1.1.cmml">≠</mo><mi id="S8.6.p6.12.m12.1.1.3" xref="S8.6.p6.12.m12.1.1.3.cmml">a</mi></mrow><annotation-xml encoding="MathML-Content" id="S8.6.p6.12.m12.1b"><apply id="S8.6.p6.12.m12.1.1.cmml" xref="S8.6.p6.12.m12.1.1"><neq id="S8.6.p6.12.m12.1.1.1.cmml" xref="S8.6.p6.12.m12.1.1.1"></neq><ci id="S8.6.p6.12.m12.1.1.2.cmml" xref="S8.6.p6.12.m12.1.1.2">𝑦</ci><ci id="S8.6.p6.12.m12.1.1.3.cmml" xref="S8.6.p6.12.m12.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.6.p6.12.m12.1c">y\neq a</annotation><annotation encoding="application/x-llamapun" id="S8.6.p6.12.m12.1d">italic_y ≠ italic_a</annotation></semantics></math> then <math alttext="q^{\prime}:=q\cup\{(x,0),(y,1),(a,0),(b,1)\}" class="ltx_Math" display="inline" id="S8.6.p6.13.m13.12"><semantics id="S8.6.p6.13.m13.12a"><mrow id="S8.6.p6.13.m13.12.12" xref="S8.6.p6.13.m13.12.12.cmml"><msup id="S8.6.p6.13.m13.12.12.6" xref="S8.6.p6.13.m13.12.12.6.cmml"><mi id="S8.6.p6.13.m13.12.12.6.2" xref="S8.6.p6.13.m13.12.12.6.2.cmml">q</mi><mo id="S8.6.p6.13.m13.12.12.6.3" xref="S8.6.p6.13.m13.12.12.6.3.cmml">′</mo></msup><mo id="S8.6.p6.13.m13.12.12.5" lspace="0.278em" rspace="0.278em" xref="S8.6.p6.13.m13.12.12.5.cmml">:=</mo><mrow id="S8.6.p6.13.m13.12.12.4" xref="S8.6.p6.13.m13.12.12.4.cmml"><mi id="S8.6.p6.13.m13.12.12.4.6" xref="S8.6.p6.13.m13.12.12.4.6.cmml">q</mi><mo id="S8.6.p6.13.m13.12.12.4.5" xref="S8.6.p6.13.m13.12.12.4.5.cmml">∪</mo><mrow id="S8.6.p6.13.m13.12.12.4.4.4" xref="S8.6.p6.13.m13.12.12.4.4.5.cmml"><mo id="S8.6.p6.13.m13.12.12.4.4.4.5" stretchy="false" xref="S8.6.p6.13.m13.12.12.4.4.5.cmml">{</mo><mrow id="S8.6.p6.13.m13.9.9.1.1.1.1.2" xref="S8.6.p6.13.m13.9.9.1.1.1.1.1.cmml"><mo id="S8.6.p6.13.m13.9.9.1.1.1.1.2.1" stretchy="false" xref="S8.6.p6.13.m13.9.9.1.1.1.1.1.cmml">(</mo><mi id="S8.6.p6.13.m13.1.1" xref="S8.6.p6.13.m13.1.1.cmml">x</mi><mo id="S8.6.p6.13.m13.9.9.1.1.1.1.2.2" xref="S8.6.p6.13.m13.9.9.1.1.1.1.1.cmml">,</mo><mn id="S8.6.p6.13.m13.2.2" xref="S8.6.p6.13.m13.2.2.cmml">0</mn><mo id="S8.6.p6.13.m13.9.9.1.1.1.1.2.3" stretchy="false" xref="S8.6.p6.13.m13.9.9.1.1.1.1.1.cmml">)</mo></mrow><mo id="S8.6.p6.13.m13.12.12.4.4.4.6" xref="S8.6.p6.13.m13.12.12.4.4.5.cmml">,</mo><mrow id="S8.6.p6.13.m13.10.10.2.2.2.2.2" xref="S8.6.p6.13.m13.10.10.2.2.2.2.1.cmml"><mo id="S8.6.p6.13.m13.10.10.2.2.2.2.2.1" stretchy="false" xref="S8.6.p6.13.m13.10.10.2.2.2.2.1.cmml">(</mo><mi id="S8.6.p6.13.m13.3.3" xref="S8.6.p6.13.m13.3.3.cmml">y</mi><mo id="S8.6.p6.13.m13.10.10.2.2.2.2.2.2" xref="S8.6.p6.13.m13.10.10.2.2.2.2.1.cmml">,</mo><mn id="S8.6.p6.13.m13.4.4" xref="S8.6.p6.13.m13.4.4.cmml">1</mn><mo id="S8.6.p6.13.m13.10.10.2.2.2.2.2.3" stretchy="false" xref="S8.6.p6.13.m13.10.10.2.2.2.2.1.cmml">)</mo></mrow><mo id="S8.6.p6.13.m13.12.12.4.4.4.7" xref="S8.6.p6.13.m13.12.12.4.4.5.cmml">,</mo><mrow id="S8.6.p6.13.m13.11.11.3.3.3.3.2" xref="S8.6.p6.13.m13.11.11.3.3.3.3.1.cmml"><mo id="S8.6.p6.13.m13.11.11.3.3.3.3.2.1" stretchy="false" xref="S8.6.p6.13.m13.11.11.3.3.3.3.1.cmml">(</mo><mi id="S8.6.p6.13.m13.5.5" xref="S8.6.p6.13.m13.5.5.cmml">a</mi><mo id="S8.6.p6.13.m13.11.11.3.3.3.3.2.2" xref="S8.6.p6.13.m13.11.11.3.3.3.3.1.cmml">,</mo><mn id="S8.6.p6.13.m13.6.6" xref="S8.6.p6.13.m13.6.6.cmml">0</mn><mo id="S8.6.p6.13.m13.11.11.3.3.3.3.2.3" stretchy="false" xref="S8.6.p6.13.m13.11.11.3.3.3.3.1.cmml">)</mo></mrow><mo id="S8.6.p6.13.m13.12.12.4.4.4.8" xref="S8.6.p6.13.m13.12.12.4.4.5.cmml">,</mo><mrow id="S8.6.p6.13.m13.12.12.4.4.4.4.2" xref="S8.6.p6.13.m13.12.12.4.4.4.4.1.cmml"><mo id="S8.6.p6.13.m13.12.12.4.4.4.4.2.1" stretchy="false" xref="S8.6.p6.13.m13.12.12.4.4.4.4.1.cmml">(</mo><mi id="S8.6.p6.13.m13.7.7" xref="S8.6.p6.13.m13.7.7.cmml">b</mi><mo id="S8.6.p6.13.m13.12.12.4.4.4.4.2.2" xref="S8.6.p6.13.m13.12.12.4.4.4.4.1.cmml">,</mo><mn id="S8.6.p6.13.m13.8.8" xref="S8.6.p6.13.m13.8.8.cmml">1</mn><mo id="S8.6.p6.13.m13.12.12.4.4.4.4.2.3" stretchy="false" xref="S8.6.p6.13.m13.12.12.4.4.4.4.1.cmml">)</mo></mrow><mo id="S8.6.p6.13.m13.12.12.4.4.4.9" stretchy="false" xref="S8.6.p6.13.m13.12.12.4.4.5.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S8.6.p6.13.m13.12b"><apply id="S8.6.p6.13.m13.12.12.cmml" xref="S8.6.p6.13.m13.12.12"><csymbol cd="latexml" id="S8.6.p6.13.m13.12.12.5.cmml" xref="S8.6.p6.13.m13.12.12.5">assign</csymbol><apply id="S8.6.p6.13.m13.12.12.6.cmml" xref="S8.6.p6.13.m13.12.12.6"><csymbol cd="ambiguous" id="S8.6.p6.13.m13.12.12.6.1.cmml" xref="S8.6.p6.13.m13.12.12.6">superscript</csymbol><ci id="S8.6.p6.13.m13.12.12.6.2.cmml" xref="S8.6.p6.13.m13.12.12.6.2">𝑞</ci><ci id="S8.6.p6.13.m13.12.12.6.3.cmml" xref="S8.6.p6.13.m13.12.12.6.3">′</ci></apply><apply id="S8.6.p6.13.m13.12.12.4.cmml" xref="S8.6.p6.13.m13.12.12.4"><union id="S8.6.p6.13.m13.12.12.4.5.cmml" xref="S8.6.p6.13.m13.12.12.4.5"></union><ci id="S8.6.p6.13.m13.12.12.4.6.cmml" xref="S8.6.p6.13.m13.12.12.4.6">𝑞</ci><set id="S8.6.p6.13.m13.12.12.4.4.5.cmml" xref="S8.6.p6.13.m13.12.12.4.4.4"><interval closure="open" id="S8.6.p6.13.m13.9.9.1.1.1.1.1.cmml" xref="S8.6.p6.13.m13.9.9.1.1.1.1.2"><ci id="S8.6.p6.13.m13.1.1.cmml" xref="S8.6.p6.13.m13.1.1">𝑥</ci><cn id="S8.6.p6.13.m13.2.2.cmml" type="integer" xref="S8.6.p6.13.m13.2.2">0</cn></interval><interval closure="open" id="S8.6.p6.13.m13.10.10.2.2.2.2.1.cmml" xref="S8.6.p6.13.m13.10.10.2.2.2.2.2"><ci id="S8.6.p6.13.m13.3.3.cmml" xref="S8.6.p6.13.m13.3.3">𝑦</ci><cn id="S8.6.p6.13.m13.4.4.cmml" type="integer" xref="S8.6.p6.13.m13.4.4">1</cn></interval><interval closure="open" id="S8.6.p6.13.m13.11.11.3.3.3.3.1.cmml" xref="S8.6.p6.13.m13.11.11.3.3.3.3.2"><ci id="S8.6.p6.13.m13.5.5.cmml" xref="S8.6.p6.13.m13.5.5">𝑎</ci><cn id="S8.6.p6.13.m13.6.6.cmml" type="integer" xref="S8.6.p6.13.m13.6.6">0</cn></interval><interval closure="open" id="S8.6.p6.13.m13.12.12.4.4.4.4.1.cmml" xref="S8.6.p6.13.m13.12.12.4.4.4.4.2"><ci id="S8.6.p6.13.m13.7.7.cmml" xref="S8.6.p6.13.m13.7.7">𝑏</ci><cn id="S8.6.p6.13.m13.8.8.cmml" type="integer" xref="S8.6.p6.13.m13.8.8">1</cn></interval></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.6.p6.13.m13.12c">q^{\prime}:=q\cup\{(x,0),(y,1),(a,0),(b,1)\}</annotation><annotation encoding="application/x-llamapun" id="S8.6.p6.13.m13.12d">italic_q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT := italic_q ∪ { ( italic_x , 0 ) , ( italic_y , 1 ) , ( italic_a , 0 ) , ( italic_b , 1 ) }</annotation></semantics></math> works. ∎</p> </div> </div> <div class="ltx_theorem ltx_theorem_remark" id="S8.Thmtheorem5"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_italic" id="S8.Thmtheorem5.1.1.1">Remark 8.5</span></span><span class="ltx_text ltx_font_italic" id="S8.Thmtheorem5.2.2">.</span> </h6> <div class="ltx_para" id="S8.Thmtheorem5.p1"> <p class="ltx_p" id="S8.Thmtheorem5.p1.9">It was pointed out to us by Moore that that any coherent Aronszajn tree <math alttext="T\subseteq\omega^{<\omega_{1}}" class="ltx_Math" display="inline" id="S8.Thmtheorem5.p1.1.m1.1"><semantics id="S8.Thmtheorem5.p1.1.m1.1a"><mrow id="S8.Thmtheorem5.p1.1.m1.1.1" xref="S8.Thmtheorem5.p1.1.m1.1.1.cmml"><mi id="S8.Thmtheorem5.p1.1.m1.1.1.2" xref="S8.Thmtheorem5.p1.1.m1.1.1.2.cmml">T</mi><mo id="S8.Thmtheorem5.p1.1.m1.1.1.1" xref="S8.Thmtheorem5.p1.1.m1.1.1.1.cmml">⊆</mo><msup id="S8.Thmtheorem5.p1.1.m1.1.1.3" xref="S8.Thmtheorem5.p1.1.m1.1.1.3.cmml"><mi id="S8.Thmtheorem5.p1.1.m1.1.1.3.2" xref="S8.Thmtheorem5.p1.1.m1.1.1.3.2.cmml">ω</mi><mrow id="S8.Thmtheorem5.p1.1.m1.1.1.3.3" xref="S8.Thmtheorem5.p1.1.m1.1.1.3.3.cmml"><mi id="S8.Thmtheorem5.p1.1.m1.1.1.3.3.2" xref="S8.Thmtheorem5.p1.1.m1.1.1.3.3.2.cmml"></mi><mo id="S8.Thmtheorem5.p1.1.m1.1.1.3.3.1" xref="S8.Thmtheorem5.p1.1.m1.1.1.3.3.1.cmml"><</mo><msub id="S8.Thmtheorem5.p1.1.m1.1.1.3.3.3" xref="S8.Thmtheorem5.p1.1.m1.1.1.3.3.3.cmml"><mi id="S8.Thmtheorem5.p1.1.m1.1.1.3.3.3.2" xref="S8.Thmtheorem5.p1.1.m1.1.1.3.3.3.2.cmml">ω</mi><mn id="S8.Thmtheorem5.p1.1.m1.1.1.3.3.3.3" xref="S8.Thmtheorem5.p1.1.m1.1.1.3.3.3.3.cmml">1</mn></msub></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S8.Thmtheorem5.p1.1.m1.1b"><apply id="S8.Thmtheorem5.p1.1.m1.1.1.cmml" xref="S8.Thmtheorem5.p1.1.m1.1.1"><subset id="S8.Thmtheorem5.p1.1.m1.1.1.1.cmml" xref="S8.Thmtheorem5.p1.1.m1.1.1.1"></subset><ci id="S8.Thmtheorem5.p1.1.m1.1.1.2.cmml" xref="S8.Thmtheorem5.p1.1.m1.1.1.2">𝑇</ci><apply id="S8.Thmtheorem5.p1.1.m1.1.1.3.cmml" xref="S8.Thmtheorem5.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S8.Thmtheorem5.p1.1.m1.1.1.3.1.cmml" xref="S8.Thmtheorem5.p1.1.m1.1.1.3">superscript</csymbol><ci id="S8.Thmtheorem5.p1.1.m1.1.1.3.2.cmml" xref="S8.Thmtheorem5.p1.1.m1.1.1.3.2">𝜔</ci><apply id="S8.Thmtheorem5.p1.1.m1.1.1.3.3.cmml" xref="S8.Thmtheorem5.p1.1.m1.1.1.3.3"><lt id="S8.Thmtheorem5.p1.1.m1.1.1.3.3.1.cmml" xref="S8.Thmtheorem5.p1.1.m1.1.1.3.3.1"></lt><csymbol cd="latexml" id="S8.Thmtheorem5.p1.1.m1.1.1.3.3.2.cmml" xref="S8.Thmtheorem5.p1.1.m1.1.1.3.3.2">absent</csymbol><apply id="S8.Thmtheorem5.p1.1.m1.1.1.3.3.3.cmml" xref="S8.Thmtheorem5.p1.1.m1.1.1.3.3.3"><csymbol cd="ambiguous" id="S8.Thmtheorem5.p1.1.m1.1.1.3.3.3.1.cmml" xref="S8.Thmtheorem5.p1.1.m1.1.1.3.3.3">subscript</csymbol><ci id="S8.Thmtheorem5.p1.1.m1.1.1.3.3.3.2.cmml" xref="S8.Thmtheorem5.p1.1.m1.1.1.3.3.3.2">𝜔</ci><cn id="S8.Thmtheorem5.p1.1.m1.1.1.3.3.3.3.cmml" type="integer" xref="S8.Thmtheorem5.p1.1.m1.1.1.3.3.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.Thmtheorem5.p1.1.m1.1c">T\subseteq\omega^{<\omega_{1}}</annotation><annotation encoding="application/x-llamapun" id="S8.Thmtheorem5.p1.1.m1.1d">italic_T ⊆ italic_ω start_POSTSUPERSCRIPT < italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> is <math alttext="\omega_{1}" class="ltx_Math" display="inline" id="S8.Thmtheorem5.p1.2.m2.1"><semantics id="S8.Thmtheorem5.p1.2.m2.1a"><msub id="S8.Thmtheorem5.p1.2.m2.1.1" xref="S8.Thmtheorem5.p1.2.m2.1.1.cmml"><mi id="S8.Thmtheorem5.p1.2.m2.1.1.2" xref="S8.Thmtheorem5.p1.2.m2.1.1.2.cmml">ω</mi><mn id="S8.Thmtheorem5.p1.2.m2.1.1.3" xref="S8.Thmtheorem5.p1.2.m2.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S8.Thmtheorem5.p1.2.m2.1b"><apply id="S8.Thmtheorem5.p1.2.m2.1.1.cmml" xref="S8.Thmtheorem5.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S8.Thmtheorem5.p1.2.m2.1.1.1.cmml" xref="S8.Thmtheorem5.p1.2.m2.1.1">subscript</csymbol><ci id="S8.Thmtheorem5.p1.2.m2.1.1.2.cmml" xref="S8.Thmtheorem5.p1.2.m2.1.1.2">𝜔</ci><cn id="S8.Thmtheorem5.p1.2.m2.1.1.3.cmml" type="integer" xref="S8.Thmtheorem5.p1.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.Thmtheorem5.p1.2.m2.1c">\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S8.Thmtheorem5.p1.2.m2.1d">italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-irreversible with the natural lexicographic ordering. Note that this gives an alternative proof of <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S8.Thmtheorem2" title="Theorem 8.2. ‣ 8. On a question on Countryman lines ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">8.2</span></a>, and it also shows that <math alttext="\diamondsuit" class="ltx_Math" display="inline" id="S8.Thmtheorem5.p1.3.m3.1"><semantics id="S8.Thmtheorem5.p1.3.m3.1a"><mi id="S8.Thmtheorem5.p1.3.m3.1.1" mathvariant="normal" xref="S8.Thmtheorem5.p1.3.m3.1.1.cmml">♢</mi><annotation-xml encoding="MathML-Content" id="S8.Thmtheorem5.p1.3.m3.1b"><ci id="S8.Thmtheorem5.p1.3.m3.1.1.cmml" xref="S8.Thmtheorem5.p1.3.m3.1.1">♢</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.Thmtheorem5.p1.3.m3.1c">\diamondsuit</annotation><annotation encoding="application/x-llamapun" id="S8.Thmtheorem5.p1.3.m3.1d">♢</annotation></semantics></math> implies the existence of an <math alttext="\omega_{1}" class="ltx_Math" display="inline" id="S8.Thmtheorem5.p1.4.m4.1"><semantics id="S8.Thmtheorem5.p1.4.m4.1a"><msub id="S8.Thmtheorem5.p1.4.m4.1.1" xref="S8.Thmtheorem5.p1.4.m4.1.1.cmml"><mi id="S8.Thmtheorem5.p1.4.m4.1.1.2" xref="S8.Thmtheorem5.p1.4.m4.1.1.2.cmml">ω</mi><mn id="S8.Thmtheorem5.p1.4.m4.1.1.3" xref="S8.Thmtheorem5.p1.4.m4.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S8.Thmtheorem5.p1.4.m4.1b"><apply id="S8.Thmtheorem5.p1.4.m4.1.1.cmml" xref="S8.Thmtheorem5.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S8.Thmtheorem5.p1.4.m4.1.1.1.cmml" xref="S8.Thmtheorem5.p1.4.m4.1.1">subscript</csymbol><ci id="S8.Thmtheorem5.p1.4.m4.1.1.2.cmml" xref="S8.Thmtheorem5.p1.4.m4.1.1.2">𝜔</ci><cn id="S8.Thmtheorem5.p1.4.m4.1.1.3.cmml" type="integer" xref="S8.Thmtheorem5.p1.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.Thmtheorem5.p1.4.m4.1c">\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S8.Thmtheorem5.p1.4.m4.1d">italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-irreversible lexicographically ordered Suslin tree. This is because <math alttext="\diamondsuit" class="ltx_Math" display="inline" id="S8.Thmtheorem5.p1.5.m5.1"><semantics id="S8.Thmtheorem5.p1.5.m5.1a"><mi id="S8.Thmtheorem5.p1.5.m5.1.1" mathvariant="normal" xref="S8.Thmtheorem5.p1.5.m5.1.1.cmml">♢</mi><annotation-xml encoding="MathML-Content" id="S8.Thmtheorem5.p1.5.m5.1b"><ci id="S8.Thmtheorem5.p1.5.m5.1.1.cmml" xref="S8.Thmtheorem5.p1.5.m5.1.1">♢</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.Thmtheorem5.p1.5.m5.1c">\diamondsuit</annotation><annotation encoding="application/x-llamapun" id="S8.Thmtheorem5.p1.5.m5.1d">♢</annotation></semantics></math> implies the existence of a coherent Suslin tree (see for example <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib21" title="">21</a>, Theorem 6.9]</cite>). Too see that the claim is true one proves that the ccc forcing for specializing <math alttext="T" class="ltx_Math" display="inline" id="S8.Thmtheorem5.p1.6.m6.1"><semantics id="S8.Thmtheorem5.p1.6.m6.1a"><mi id="S8.Thmtheorem5.p1.6.m6.1.1" xref="S8.Thmtheorem5.p1.6.m6.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S8.Thmtheorem5.p1.6.m6.1b"><ci id="S8.Thmtheorem5.p1.6.m6.1.1.cmml" xref="S8.Thmtheorem5.p1.6.m6.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S8.Thmtheorem5.p1.6.m6.1c">T</annotation><annotation encoding="application/x-llamapun" id="S8.Thmtheorem5.p1.6.m6.1d">italic_T</annotation></semantics></math> (see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib8" title="">8</a>, Theorem 16.17]</cite>) makes <math alttext="(T,<_{\mathrm{lex}})" class="ltx_Math" display="inline" id="S8.Thmtheorem5.p1.7.m7.2"><semantics id="S8.Thmtheorem5.p1.7.m7.2a"><mrow id="S8.Thmtheorem5.p1.7.m7.2.2.1" xref="S8.Thmtheorem5.p1.7.m7.2.2.2.cmml"><mo id="S8.Thmtheorem5.p1.7.m7.2.2.1.2" stretchy="false" xref="S8.Thmtheorem5.p1.7.m7.2.2.2.cmml">(</mo><mi id="S8.Thmtheorem5.p1.7.m7.1.1" xref="S8.Thmtheorem5.p1.7.m7.1.1.cmml">T</mi><mo id="S8.Thmtheorem5.p1.7.m7.2.2.1.3" xref="S8.Thmtheorem5.p1.7.m7.2.2.2.cmml">,</mo><msub id="S8.Thmtheorem5.p1.7.m7.2.2.1.1" xref="S8.Thmtheorem5.p1.7.m7.2.2.1.1.cmml"><mo id="S8.Thmtheorem5.p1.7.m7.2.2.1.1.2" lspace="0em" rspace="0em" xref="S8.Thmtheorem5.p1.7.m7.2.2.1.1.2.cmml"><</mo><mi id="S8.Thmtheorem5.p1.7.m7.2.2.1.1.3" xref="S8.Thmtheorem5.p1.7.m7.2.2.1.1.3.cmml">lex</mi></msub><mo id="S8.Thmtheorem5.p1.7.m7.2.2.1.4" stretchy="false" xref="S8.Thmtheorem5.p1.7.m7.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S8.Thmtheorem5.p1.7.m7.2b"><interval closure="open" id="S8.Thmtheorem5.p1.7.m7.2.2.2.cmml" xref="S8.Thmtheorem5.p1.7.m7.2.2.1"><ci id="S8.Thmtheorem5.p1.7.m7.1.1.cmml" xref="S8.Thmtheorem5.p1.7.m7.1.1">𝑇</ci><apply id="S8.Thmtheorem5.p1.7.m7.2.2.1.1.cmml" xref="S8.Thmtheorem5.p1.7.m7.2.2.1.1"><csymbol cd="ambiguous" id="S8.Thmtheorem5.p1.7.m7.2.2.1.1.1.cmml" xref="S8.Thmtheorem5.p1.7.m7.2.2.1.1">subscript</csymbol><lt id="S8.Thmtheorem5.p1.7.m7.2.2.1.1.2.cmml" xref="S8.Thmtheorem5.p1.7.m7.2.2.1.1.2"></lt><ci id="S8.Thmtheorem5.p1.7.m7.2.2.1.1.3.cmml" xref="S8.Thmtheorem5.p1.7.m7.2.2.1.1.3">lex</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S8.Thmtheorem5.p1.7.m7.2c">(T,<_{\mathrm{lex}})</annotation><annotation encoding="application/x-llamapun" id="S8.Thmtheorem5.p1.7.m7.2d">( italic_T , < start_POSTSUBSCRIPT roman_lex end_POSTSUBSCRIPT )</annotation></semantics></math> Countryman in the extension (see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib23" title="">23</a>, 4.4]</cite>). Thus in the extension <math alttext="(T,<_{\mathrm{lex}})" class="ltx_Math" display="inline" id="S8.Thmtheorem5.p1.8.m8.2"><semantics id="S8.Thmtheorem5.p1.8.m8.2a"><mrow id="S8.Thmtheorem5.p1.8.m8.2.2.1" xref="S8.Thmtheorem5.p1.8.m8.2.2.2.cmml"><mo id="S8.Thmtheorem5.p1.8.m8.2.2.1.2" stretchy="false" xref="S8.Thmtheorem5.p1.8.m8.2.2.2.cmml">(</mo><mi id="S8.Thmtheorem5.p1.8.m8.1.1" xref="S8.Thmtheorem5.p1.8.m8.1.1.cmml">T</mi><mo id="S8.Thmtheorem5.p1.8.m8.2.2.1.3" xref="S8.Thmtheorem5.p1.8.m8.2.2.2.cmml">,</mo><msub id="S8.Thmtheorem5.p1.8.m8.2.2.1.1" xref="S8.Thmtheorem5.p1.8.m8.2.2.1.1.cmml"><mo id="S8.Thmtheorem5.p1.8.m8.2.2.1.1.2" lspace="0em" rspace="0em" xref="S8.Thmtheorem5.p1.8.m8.2.2.1.1.2.cmml"><</mo><mi id="S8.Thmtheorem5.p1.8.m8.2.2.1.1.3" xref="S8.Thmtheorem5.p1.8.m8.2.2.1.1.3.cmml">lex</mi></msub><mo id="S8.Thmtheorem5.p1.8.m8.2.2.1.4" stretchy="false" xref="S8.Thmtheorem5.p1.8.m8.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S8.Thmtheorem5.p1.8.m8.2b"><interval closure="open" id="S8.Thmtheorem5.p1.8.m8.2.2.2.cmml" xref="S8.Thmtheorem5.p1.8.m8.2.2.1"><ci id="S8.Thmtheorem5.p1.8.m8.1.1.cmml" xref="S8.Thmtheorem5.p1.8.m8.1.1">𝑇</ci><apply id="S8.Thmtheorem5.p1.8.m8.2.2.1.1.cmml" xref="S8.Thmtheorem5.p1.8.m8.2.2.1.1"><csymbol cd="ambiguous" id="S8.Thmtheorem5.p1.8.m8.2.2.1.1.1.cmml" xref="S8.Thmtheorem5.p1.8.m8.2.2.1.1">subscript</csymbol><lt id="S8.Thmtheorem5.p1.8.m8.2.2.1.1.2.cmml" xref="S8.Thmtheorem5.p1.8.m8.2.2.1.1.2"></lt><ci id="S8.Thmtheorem5.p1.8.m8.2.2.1.1.3.cmml" xref="S8.Thmtheorem5.p1.8.m8.2.2.1.1.3">lex</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S8.Thmtheorem5.p1.8.m8.2c">(T,<_{\mathrm{lex}})</annotation><annotation encoding="application/x-llamapun" id="S8.Thmtheorem5.p1.8.m8.2d">( italic_T , < start_POSTSUBSCRIPT roman_lex end_POSTSUBSCRIPT )</annotation></semantics></math> has no two uncountable reverse isomorphic suborders. But since <math alttext="\omega_{1}" class="ltx_Math" display="inline" id="S8.Thmtheorem5.p1.9.m9.1"><semantics id="S8.Thmtheorem5.p1.9.m9.1a"><msub id="S8.Thmtheorem5.p1.9.m9.1.1" xref="S8.Thmtheorem5.p1.9.m9.1.1.cmml"><mi id="S8.Thmtheorem5.p1.9.m9.1.1.2" xref="S8.Thmtheorem5.p1.9.m9.1.1.2.cmml">ω</mi><mn id="S8.Thmtheorem5.p1.9.m9.1.1.3" xref="S8.Thmtheorem5.p1.9.m9.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S8.Thmtheorem5.p1.9.m9.1b"><apply id="S8.Thmtheorem5.p1.9.m9.1.1.cmml" xref="S8.Thmtheorem5.p1.9.m9.1.1"><csymbol cd="ambiguous" id="S8.Thmtheorem5.p1.9.m9.1.1.1.cmml" xref="S8.Thmtheorem5.p1.9.m9.1.1">subscript</csymbol><ci id="S8.Thmtheorem5.p1.9.m9.1.1.2.cmml" xref="S8.Thmtheorem5.p1.9.m9.1.1.2">𝜔</ci><cn id="S8.Thmtheorem5.p1.9.m9.1.1.3.cmml" type="integer" xref="S8.Thmtheorem5.p1.9.m9.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S8.Thmtheorem5.p1.9.m9.1c">\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S8.Thmtheorem5.p1.9.m9.1d">italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> is preserved, this must already hold in the ground model.</p> </div> </div> </section> <section class="ltx_section" id="S9"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">9. </span>Concluding remarks and questions</h2> <div class="ltx_para" id="S9.p1"> <p class="ltx_p" id="S9.p1.6">As mentioned in <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S3" title="3. Strongly surjective Aronszajn lines ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">3</span></a>, this work leaves open <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#Thmquestion6" title="Question 6. ‣ Historical and mathematical context ‣ 1. Introduction ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">6</span></a> of whether <math alttext="\eta_{C}" class="ltx_Math" display="inline" id="S9.p1.1.m1.1"><semantics id="S9.p1.1.m1.1a"><msub id="S9.p1.1.m1.1.1" xref="S9.p1.1.m1.1.1.cmml"><mi id="S9.p1.1.m1.1.1.2" xref="S9.p1.1.m1.1.1.2.cmml">η</mi><mi id="S9.p1.1.m1.1.1.3" xref="S9.p1.1.m1.1.1.3.cmml">C</mi></msub><annotation-xml encoding="MathML-Content" id="S9.p1.1.m1.1b"><apply id="S9.p1.1.m1.1.1.cmml" xref="S9.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S9.p1.1.m1.1.1.1.cmml" xref="S9.p1.1.m1.1.1">subscript</csymbol><ci id="S9.p1.1.m1.1.1.2.cmml" xref="S9.p1.1.m1.1.1.2">𝜂</ci><ci id="S9.p1.1.m1.1.1.3.cmml" xref="S9.p1.1.m1.1.1.3">𝐶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S9.p1.1.m1.1c">\eta_{C}</annotation><annotation encoding="application/x-llamapun" id="S9.p1.1.m1.1d">italic_η start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT</annotation></semantics></math> (or other universal Aronszajn line) is strongly surjective under <math alttext="\mathsf{PFA}" class="ltx_Math" display="inline" id="S9.p1.2.m2.1"><semantics id="S9.p1.2.m2.1a"><mi id="S9.p1.2.m2.1.1" xref="S9.p1.2.m2.1.1.cmml">𝖯𝖥𝖠</mi><annotation-xml encoding="MathML-Content" id="S9.p1.2.m2.1b"><ci id="S9.p1.2.m2.1.1.cmml" xref="S9.p1.2.m2.1.1">𝖯𝖥𝖠</ci></annotation-xml><annotation encoding="application/x-tex" id="S9.p1.2.m2.1c">\mathsf{PFA}</annotation><annotation encoding="application/x-llamapun" id="S9.p1.2.m2.1d">sansserif_PFA</annotation></semantics></math>. Recently in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib6" title="">6</a>]</cite>, answering a question by Baumgartner, a <math alttext="\preceq" class="ltx_Math" display="inline" id="S9.p1.3.m3.1"><semantics id="S9.p1.3.m3.1a"><mo id="S9.p1.3.m3.1.1" xref="S9.p1.3.m3.1.1.cmml">⪯</mo><annotation-xml encoding="MathML-Content" id="S9.p1.3.m3.1b"><csymbol cd="latexml" id="S9.p1.3.m3.1.1.cmml" xref="S9.p1.3.m3.1.1">precedes-or-equals</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S9.p1.3.m3.1c">\preceq</annotation><annotation encoding="application/x-llamapun" id="S9.p1.3.m3.1d">⪯</annotation></semantics></math>-minimal Countryman line was constructed under <math alttext="\diamondsuit" class="ltx_Math" display="inline" id="S9.p1.4.m4.1"><semantics id="S9.p1.4.m4.1a"><mi id="S9.p1.4.m4.1.1" mathvariant="normal" xref="S9.p1.4.m4.1.1.cmml">♢</mi><annotation-xml encoding="MathML-Content" id="S9.p1.4.m4.1b"><ci id="S9.p1.4.m4.1.1.cmml" xref="S9.p1.4.m4.1.1">♢</ci></annotation-xml><annotation encoding="application/x-tex" id="S9.p1.4.m4.1c">\diamondsuit</annotation><annotation encoding="application/x-llamapun" id="S9.p1.4.m4.1d">♢</annotation></semantics></math>. As mentioned in the introduction, in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#bib.bib19" title="">19</a>]</cite> it is proved that <math alttext="\diamondsuit^{+}" class="ltx_Math" display="inline" id="S9.p1.5.m5.1"><semantics id="S9.p1.5.m5.1a"><msup id="S9.p1.5.m5.1.1" xref="S9.p1.5.m5.1.1.cmml"><mi id="S9.p1.5.m5.1.1.2" mathvariant="normal" xref="S9.p1.5.m5.1.1.2.cmml">♢</mi><mo id="S9.p1.5.m5.1.1.3" xref="S9.p1.5.m5.1.1.3.cmml">+</mo></msup><annotation-xml encoding="MathML-Content" id="S9.p1.5.m5.1b"><apply id="S9.p1.5.m5.1.1.cmml" xref="S9.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S9.p1.5.m5.1.1.1.cmml" xref="S9.p1.5.m5.1.1">superscript</csymbol><ci id="S9.p1.5.m5.1.1.2.cmml" xref="S9.p1.5.m5.1.1.2">♢</ci><plus id="S9.p1.5.m5.1.1.3.cmml" xref="S9.p1.5.m5.1.1.3"></plus></apply></annotation-xml><annotation encoding="application/x-tex" id="S9.p1.5.m5.1c">\diamondsuit^{+}</annotation><annotation encoding="application/x-llamapun" id="S9.p1.5.m5.1d">♢ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> implies the existence of a strongly surjective Suslin line. Does <math alttext="\diamondsuit" class="ltx_Math" display="inline" id="S9.p1.6.m6.1"><semantics id="S9.p1.6.m6.1a"><mi id="S9.p1.6.m6.1.1" mathvariant="normal" xref="S9.p1.6.m6.1.1.cmml">♢</mi><annotation-xml encoding="MathML-Content" id="S9.p1.6.m6.1b"><ci id="S9.p1.6.m6.1.1.cmml" xref="S9.p1.6.m6.1.1">♢</ci></annotation-xml><annotation encoding="application/x-tex" id="S9.p1.6.m6.1c">\diamondsuit</annotation><annotation encoding="application/x-llamapun" id="S9.p1.6.m6.1d">♢</annotation></semantics></math> imply the existence of a strongly surjective Aronszajn line?</p> </div> <div class="ltx_para" id="S9.p2"> <p class="ltx_p" id="S9.p2.3">The work in <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S5" title="5. An infinite antichain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Sections</span> <span class="ltx_text ltx_ref_tag">5</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S6" title="6. An infinite decreasing chain ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">6</span></a> shows that already the class of Countryman lines is not well-quasi-ordered by <math alttext="\trianglelefteq" class="ltx_Math" display="inline" id="S9.p2.1.m1.1"><semantics id="S9.p2.1.m1.1a"><mi id="S9.p2.1.m1.1.1" mathvariant="normal" xref="S9.p2.1.m1.1.1.cmml">⊴</mi><annotation-xml encoding="MathML-Content" id="S9.p2.1.m1.1b"><ci id="S9.p2.1.m1.1.1.cmml" xref="S9.p2.1.m1.1.1">⊴</ci></annotation-xml><annotation encoding="application/x-tex" id="S9.p2.1.m1.1c">\trianglelefteq</annotation><annotation encoding="application/x-llamapun" id="S9.p2.1.m1.1d">⊴</annotation></semantics></math>. This is done by considering Aronszajn lines that have decompoisitions with a stationary set of endpoints. It seems that a natural question is whether the class of <span class="ltx_text ltx_font_italic" id="S9.p2.3.1">normal</span> Aronszajn lines is well-quasi-ordered by <math alttext="\trianglelefteq" class="ltx_Math" display="inline" id="S9.p2.2.m2.1"><semantics id="S9.p2.2.m2.1a"><mi id="S9.p2.2.m2.1.1" mathvariant="normal" xref="S9.p2.2.m2.1.1.cmml">⊴</mi><annotation-xml encoding="MathML-Content" id="S9.p2.2.m2.1b"><ci id="S9.p2.2.m2.1.1.cmml" xref="S9.p2.2.m2.1.1">⊴</ci></annotation-xml><annotation encoding="application/x-tex" id="S9.p2.2.m2.1c">\trianglelefteq</annotation><annotation encoding="application/x-llamapun" id="S9.p2.2.m2.1d">⊴</annotation></semantics></math> under <math alttext="\mathsf{PFA}" class="ltx_Math" display="inline" id="S9.p2.3.m3.1"><semantics id="S9.p2.3.m3.1a"><mi id="S9.p2.3.m3.1.1" xref="S9.p2.3.m3.1.1.cmml">𝖯𝖥𝖠</mi><annotation-xml encoding="MathML-Content" id="S9.p2.3.m3.1b"><ci id="S9.p2.3.m3.1.1.cmml" xref="S9.p2.3.m3.1.1">𝖯𝖥𝖠</ci></annotation-xml><annotation encoding="application/x-tex" id="S9.p2.3.m3.1c">\mathsf{PFA}</annotation><annotation encoding="application/x-llamapun" id="S9.p2.3.m3.1d">sansserif_PFA</annotation></semantics></math>. By <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S2.Thmtheorem8" title="Theorem 2.8. ‣ 2. Aronszajn and Countryman lines ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">2.8</span></a> this is clearly true for the class of Countryman lines.</p> </div> <div class="ltx_para" id="S9.p3"> <p class="ltx_p" id="S9.p3.1">Regarding <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S8" title="8. On a question on Countryman lines ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">8</span></a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S8.Thmtheorem2" title="Theorem 8.2. ‣ 8. On a question on Countryman lines ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">8.2</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.13728v1#S8.Thmtheorem5" title="Remark 8.5. ‣ 8. On a question on Countryman lines ‣ The class of Aronszajn lines under epimorphisms"><span class="ltx_text ltx_ref_tag">Remark</span> <span class="ltx_text ltx_ref_tag">8.5</span></a> suggest the following question. Does the existence of a Suslin tree suffices to show the existence of a non Countryman <math alttext="\omega_{1}" class="ltx_Math" display="inline" id="S9.p3.1.m1.1"><semantics id="S9.p3.1.m1.1a"><msub id="S9.p3.1.m1.1.1" xref="S9.p3.1.m1.1.1.cmml"><mi id="S9.p3.1.m1.1.1.2" xref="S9.p3.1.m1.1.1.2.cmml">ω</mi><mn id="S9.p3.1.m1.1.1.3" xref="S9.p3.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S9.p3.1.m1.1b"><apply id="S9.p3.1.m1.1.1.cmml" xref="S9.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S9.p3.1.m1.1.1.1.cmml" xref="S9.p3.1.m1.1.1">subscript</csymbol><ci id="S9.p3.1.m1.1.1.2.cmml" xref="S9.p3.1.m1.1.1.2">𝜔</ci><cn id="S9.p3.1.m1.1.1.3.cmml" type="integer" xref="S9.p3.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S9.p3.1.m1.1c">\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S9.p3.1.m1.1d">italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-irreversible Aronszajn line?</p> </div> </section> <section class="ltx_bibliography" id="bib"> <h2 class="ltx_title ltx_title_bibliography">References</h2> <ul class="ltx_biblist"> <li class="ltx_bibitem" id="bib.bib1"> <span class="ltx_tag ltx_tag_bibitem">[1]</span> <span class="ltx_bibblock"> U. Abraham and S. Shelah. </span> <span class="ltx_bibblock">Isomorphism types of Aronszajn trees. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib1.1.1">Israel J. Math.</span>, 50(1-2):75–113, 1985. </span> </li> <li class="ltx_bibitem" id="bib.bib2"> <span class="ltx_tag ltx_tag_bibitem">[2]</span> <span class="ltx_bibblock"> James E. Baumgartner. </span> <span class="ltx_bibblock">All <math alttext="\aleph_{1}" class="ltx_Math" display="inline" id="bib.bib2.1.m1.1"><semantics id="bib.bib2.1.m1.1a"><msub id="bib.bib2.1.m1.1.1" xref="bib.bib2.1.m1.1.1.cmml"><mi id="bib.bib2.1.m1.1.1.2" mathvariant="normal" xref="bib.bib2.1.m1.1.1.2.cmml">ℵ</mi><mn id="bib.bib2.1.m1.1.1.3" xref="bib.bib2.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="bib.bib2.1.m1.1b"><apply id="bib.bib2.1.m1.1.1.cmml" xref="bib.bib2.1.m1.1.1"><csymbol cd="ambiguous" id="bib.bib2.1.m1.1.1.1.cmml" xref="bib.bib2.1.m1.1.1">subscript</csymbol><ci id="bib.bib2.1.m1.1.1.2.cmml" xref="bib.bib2.1.m1.1.1.2">ℵ</ci><cn id="bib.bib2.1.m1.1.1.3.cmml" type="integer" xref="bib.bib2.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="bib.bib2.1.m1.1c">\aleph_{1}</annotation><annotation encoding="application/x-llamapun" id="bib.bib2.1.m1.1d">roman_ℵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-dense sets of reals can be isomorphic. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib2.2.1">Fund. Math.</span>, 79(2):101–106, 1973. </span> </li> <li class="ltx_bibitem" id="bib.bib3"> <span class="ltx_tag ltx_tag_bibitem">[3]</span> <span class="ltx_bibblock"> James E. Baumgartner. </span> <span class="ltx_bibblock">Order types of real numbers and other uncountable orderings. </span> <span class="ltx_bibblock">In <span class="ltx_text ltx_font_italic" id="bib.bib3.1.1">Ordered sets (Banff, Alta., 1981)</span>, volume 83 of <span class="ltx_text ltx_font_italic" id="bib.bib3.2.2">NATO Adv. Study Inst. Ser. C: Math. Phys. Sci.</span>, pages 239–277. Reidel, Dordrecht-Boston, Mass., 1982. </span> </li> <li class="ltx_bibitem" id="bib.bib4"> <span class="ltx_tag ltx_tag_bibitem">[4]</span> <span class="ltx_bibblock"> Riccardo Camerlo, Raphaël Carroy, and Alberto Marcone. </span> <span class="ltx_bibblock">Epimorphisms between linear orders. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib4.1.1">Order</span>, 32(3):387–400, 2015. </span> </li> <li class="ltx_bibitem" id="bib.bib5"> <span class="ltx_tag ltx_tag_bibitem">[5]</span> <span class="ltx_bibblock"> Riccardo Camerlo, Raphaël Carroy, and Alberto Marcone. </span> <span class="ltx_bibblock">Linear orders: when embeddability and epimorphism agree. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib5.1.1">J. Math. Log.</span>, 19(1):1950003, 32, 2019. </span> </li> <li class="ltx_bibitem" id="bib.bib6"> <span class="ltx_tag ltx_tag_bibitem">[6]</span> <span class="ltx_bibblock"> James Cummings, Todd Eisworth, and Justin Tatch Moore. </span> <span class="ltx_bibblock">On minimal non-<math alttext="\sigma" class="ltx_Math" display="inline" id="bib.bib6.1.m1.1"><semantics id="bib.bib6.1.m1.1a"><mi id="bib.bib6.1.m1.1.1" xref="bib.bib6.1.m1.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="bib.bib6.1.m1.1b"><ci id="bib.bib6.1.m1.1.1.cmml" xref="bib.bib6.1.m1.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="bib.bib6.1.m1.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="bib.bib6.1.m1.1d">italic_σ</annotation></semantics></math>-scattered linear orders. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib6.2.1">Adv. Math.</span>, 441:Paper No. 109540, 36, 2024. </span> </li> <li class="ltx_bibitem" id="bib.bib7"> <span class="ltx_tag ltx_tag_bibitem">[7]</span> <span class="ltx_bibblock"> Lorenz J. Halbeisen. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib7.1.1">Combinatorial set theory</span>. </span> <span class="ltx_bibblock">Springer Monographs in Mathematics. Springer, Cham, second edition, 2017. </span> <span class="ltx_bibblock">With a gentle introduction to forcing. </span> </li> <li class="ltx_bibitem" id="bib.bib8"> <span class="ltx_tag ltx_tag_bibitem">[8]</span> <span class="ltx_bibblock"> Thomas Jech. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib8.1.1">Set theory</span>. </span> <span class="ltx_bibblock">Springer Monographs in Mathematics. Springer-Verlag, Berlin, millennium edition, 2003. </span> </li> <li class="ltx_bibitem" id="bib.bib9"> <span class="ltx_tag ltx_tag_bibitem">[9]</span> <span class="ltx_bibblock"> Péter Komjáth and Vilmos Totik. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib9.1.1">Problems and theorems in classical set theory</span>. </span> <span class="ltx_bibblock">Problem Books in Mathematics. Springer, New York, 2006. </span> </li> <li class="ltx_bibitem" id="bib.bib10"> <span class="ltx_tag ltx_tag_bibitem">[10]</span> <span class="ltx_bibblock"> Kenneth Kunen. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib10.1.1">Set theory</span>, volume 102 of <span class="ltx_text ltx_font_italic" id="bib.bib10.2.2">Studies in Logic and the Foundations of Mathematics</span>. </span> <span class="ltx_bibblock">North-Holland Publishing Co., Amsterdam-New York, 1980. </span> <span class="ltx_bibblock">An introduction to independence proofs. </span> </li> <li class="ltx_bibitem" id="bib.bib11"> <span class="ltx_tag ltx_tag_bibitem">[11]</span> <span class="ltx_bibblock"> Djuro Kurepa. </span> <span class="ltx_bibblock">Ensembles ordonnés et ramifiés de points. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib11.1.1">Math. Balkanica</span>, 7:201–204, 1977. </span> </li> <li class="ltx_bibitem" id="bib.bib12"> <span class="ltx_tag ltx_tag_bibitem">[12]</span> <span class="ltx_bibblock"> Charles Landraitis. </span> <span class="ltx_bibblock">A combinatorial property of the homomorphism relation between countable order types. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib12.1.1">J. Symbolic Logic</span>, 44(3):403–411, 1979. </span> </li> <li class="ltx_bibitem" id="bib.bib13"> <span class="ltx_tag ltx_tag_bibitem">[13]</span> <span class="ltx_bibblock"> Richard Laver. </span> <span class="ltx_bibblock">On Fraïssé’s order type conjecture. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib13.1.1">Ann. of Math. (2)</span>, 93:89–111, 1971. </span> </li> <li class="ltx_bibitem" id="bib.bib14"> <span class="ltx_tag ltx_tag_bibitem">[14]</span> <span class="ltx_bibblock"> Carlos Martinez-Ranero. </span> <span class="ltx_bibblock">Well-quasi-ordering Aronszajn lines. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib14.1.1">Fund. Math.</span>, 213(3):197–211, 2011. </span> </li> <li class="ltx_bibitem" id="bib.bib15"> <span class="ltx_tag ltx_tag_bibitem">[15]</span> <span class="ltx_bibblock"> Justin Tatch Moore. </span> <span class="ltx_bibblock">A five element basis for the uncountable linear orders. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib15.1.1">Ann. of Math. (2)</span>, 163(2):669–688, 2006. </span> </li> <li class="ltx_bibitem" id="bib.bib16"> <span class="ltx_tag ltx_tag_bibitem">[16]</span> <span class="ltx_bibblock"> Justin Tatch Moore. </span> <span class="ltx_bibblock">A universal Aronszajn line. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib16.1.1">Math. Res. Lett.</span>, 16(1):121–131, 2009. </span> </li> <li class="ltx_bibitem" id="bib.bib17"> <span class="ltx_tag ltx_tag_bibitem">[17]</span> <span class="ltx_bibblock"> Saharon Shelah. </span> <span class="ltx_bibblock">Decomposing uncountable squares to countably many chains. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib17.1.1">J. Combinatorial Theory Ser. A</span>, 21(1):110–114, 1976. </span> </li> <li class="ltx_bibitem" id="bib.bib18"> <span class="ltx_tag ltx_tag_bibitem">[18]</span> <span class="ltx_bibblock"> Saharon Shelah. </span> <span class="ltx_bibblock">Can you take Solovay’s inaccessible away? </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib18.1.1">Israel J. Math.</span>, 48(1):1–47, 1984. </span> </li> <li class="ltx_bibitem" id="bib.bib19"> <span class="ltx_tag ltx_tag_bibitem">[19]</span> <span class="ltx_bibblock"> Dániel T. Soukup. </span> <span class="ltx_bibblock">Uncountable strongly surjective linear orders. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib19.1.1">Order</span>, 36(1):43–64, 2019. </span> </li> <li class="ltx_bibitem" id="bib.bib20"> <span class="ltx_tag ltx_tag_bibitem">[20]</span> <span class="ltx_bibblock"> E. Specker. </span> <span class="ltx_bibblock">Sur un problème de Sikorski. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib20.1.1">Colloq. Math.</span>, 2:9–12, 1949. </span> </li> <li class="ltx_bibitem" id="bib.bib21"> <span class="ltx_tag ltx_tag_bibitem">[21]</span> <span class="ltx_bibblock"> Stevo Todorčević. </span> <span class="ltx_bibblock">Trees and linearly ordered sets. </span> <span class="ltx_bibblock">In <span class="ltx_text ltx_font_italic" id="bib.bib21.1.1">Handbook of set-theoretic topology</span>, pages 235–293. North-Holland, Amsterdam, 1984. </span> </li> <li class="ltx_bibitem" id="bib.bib22"> <span class="ltx_tag ltx_tag_bibitem">[22]</span> <span class="ltx_bibblock"> Stevo Todorčević. </span> <span class="ltx_bibblock">Partitioning pairs of countable sets. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib22.1.1">Proc. Amer. Math. Soc.</span>, 111(3):841–844, 1991. </span> </li> <li class="ltx_bibitem" id="bib.bib23"> <span class="ltx_tag ltx_tag_bibitem">[23]</span> <span class="ltx_bibblock"> Stevo Todorčević. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib23.1.1">Walks on ordinals and their characteristics</span>, volume 263 of <span class="ltx_text ltx_font_italic" id="bib.bib23.2.2">Progress in Mathematics</span>. </span> <span class="ltx_bibblock">Birkhäuser Verlag, Basel, 2007. </span> </li> </ul> </section> <div class="ltx_para ltx_noindent" id="p1"> <br class="ltx_break"/> <p class="ltx_p" id="p1.1">Carlos A. Martínez–Ranero <br class="ltx_break"/>Email: cmartinezr@udec.cl <br class="ltx_break"/>Homepage: www2.udec.cl/~cmartinezr <br class="ltx_break"/>Lucas Polymeris <br class="ltx_break"/>Email: lucaspolymeris@protonmail.com <br class="ltx_break"/></p> </div> <div class="ltx_para ltx_noindent" id="p2"> <p class="ltx_p" id="p2.1">Same address: <br class="ltx_break"/>Universidad de Concepción, Concepción, Chile <br class="ltx_break"/>Facultad de Ciencias Físicas y Matemáticas <br class="ltx_break"/>Departamento de Matemática <br class="ltx_break"/></p> </div> <div class="ltx_pagination ltx_role_newpage"></div> </article> </div> <footer class="ltx_page_footer"> <div class="ltx_page_logo">Generated on Mon Mar 17 21:13:06 2025 by <a class="ltx_LaTeXML_logo" href="http://dlmf.nist.gov/LaTeXML/"><span style="letter-spacing:-0.2em; margin-right:0.1em;">L<span class="ltx_font_smallcaps" style="position:relative; bottom:2.2pt;">a</span>T<span class="ltx_font_smallcaps" style="font-size:120%;position:relative; bottom:-0.2ex;">e</span></span><span style="font-size:90%; position:relative; bottom:-0.2ex;">XML</span><img alt="Mascot Sammy" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAsAAAAOCAYAAAD5YeaVAAAAAXNSR0IArs4c6QAAAAZiS0dEAP8A/wD/oL2nkwAAAAlwSFlzAAALEwAACxMBAJqcGAAAAAd0SU1FB9wKExQZLWTEaOUAAAAddEVYdENvbW1lbnQAQ3JlYXRlZCB3aXRoIFRoZSBHSU1Q72QlbgAAAdpJREFUKM9tkL+L2nAARz9fPZNCKFapUn8kyI0e4iRHSR1Kb8ng0lJw6FYHFwv2LwhOpcWxTjeUunYqOmqd6hEoRDhtDWdA8ApRYsSUCDHNt5ul13vz4w0vWCgUnnEc975arX6ORqN3VqtVZbfbTQC4uEHANM3jSqXymFI6yWazP2KxWAXAL9zCUa1Wy2tXVxheKA9YNoR8Pt+aTqe4FVVVvz05O6MBhqUIBGk8Hn8HAOVy+T+XLJfLS4ZhTiRJgqIoVBRFIoric47jPnmeB1mW/9rr9ZpSSn3Lsmir1fJZlqWlUonKsvwWwD8ymc/nXwVBeLjf7xEKhdBut9Hr9WgmkyGEkJwsy5eHG5vN5g0AKIoCAEgkEkin0wQAfN9/cXPdheu6P33fBwB4ngcAcByHJpPJl+fn54mD3Gg0NrquXxeLRQAAwzAYj8cwTZPwPH9/sVg8PXweDAauqqr2cDjEer1GJBLBZDJBs9mE4zjwfZ85lAGg2+06hmGgXq+j3+/DsixYlgVN03a9Xu8jgCNCyIegIAgx13Vfd7vdu+FweG8YRkjXdWy329+dTgeSJD3ieZ7RNO0VAXAPwDEAO5VKndi2fWrb9jWl9Esul6PZbDY9Go1OZ7PZ9z/lyuD3OozU2wAAAABJRU5ErkJggg=="/></a> </div></footer> </div> </body> </html>