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<b>let </b><font color="Maroon" title="c1">G</font> be  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="glib_000.html#V4" title="GLIB_000:attr.4">trivial</a> <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>;<br/><b>let </b><font color="Maroon" title="c2">v</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>;<br/>



<b>consider </b><font color="Maroon" title="c3">v1</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E2:133"/><i><font color="Green" title="E100">A1</font></i>: 
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">v1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="glib_000.html#T25" title="GLIB_000:th.25">GLIB_000:25</a></i>;<br/>
<a NAME="E3:133"/><i><font color="Green" title="E101">A2</font></i>: 
<font color="Maroon" title="c3">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font>
 
<b>by </b><i><a class="txt" href="chord.html#E2:133"><i><font color="Green" title="E100">A1</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>set </b><font color="Maroon" title="c4">S</font> = <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>;<br/>
<a NAME="E4:133"/><i><font color="Green" title="E102">A3</font></i>: 
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T102" title="FINSEQ_3:th.102">FINSEQ_3:102</a></i>;<br/>
<a NAME="E5:133"/><i><font color="Green" title="E103">A4</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="txt" href="chord.html#E2:133"><i><font color="Green" title="E100">A1</font></i></a>, <a class="txt" href="chord.html#E3:133"><i><font color="Green" title="E101">A2</font></i></a>, <a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">S</font> = <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> as    <a href="chord.html#M3" title="CHORD:mode.3">VertexScheme</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="chord.html#E4:133"><i><font color="Green" title="E102">A3</font></i></a>, <a class="txt" href="chord.html#E5:133"><i><font color="Green" title="E103">A4</font></i></a>, <a class="ref" href="chord.html#D12" target="_self" title="CHORD:def.12">Def12</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">S</font>
;<br/>

<b>thus </b><a NAME="E7:133"/>
<font color="Maroon" title="c5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 ;<br/>

<b>let </b><font color="Maroon" title="c6">n</font> be  non <a href="xxreal_0.html#NV4" title="XXREAL_0:NV.4">zero</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="chord.html#D13" target="_self" title="CHORD:def.13">CHORD:def 13</a><br/>

<b>assume </b><a NAME="E8:133"/>
<font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">S</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c7">Gf</font> be   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="chord.html#K4" title="CHORD:func.4">.followSet</a> <font color="Maroon" title="c6">n</font></span>)</span>;<br/>

<b>thus </b><a NAME="E9:133"/>
for <font color="Olive" title="b1">v</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c7">Gf</font>  st <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> holds <br/><font color="Olive" title="b1">v</font> is <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a>
 <b>by </b><i><a class="ref" href="chord.html#T65" target="_self" title="CHORD:th.65">Th65</a></i>;<br/>


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