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<b>let </b><font color="Maroon">G</font> be  <a href="glib_000.html#V2">finite</a> <a href="glib_000.html#V4">trivial</a> <a href="glib_000.html#NM2">_Graph</a>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>;<br/>



<b>consider </b><font color="Maroon">v1</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E2:133"/><i><font color="Green">E30</font></i>: 
 <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v1</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="glib_000.html#T25">GLIB_000:25</a></i>;<br/>
<a NAME="E3:133"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 
<b>by </b><i><a class="ref" href="chord.html#T21" target="_self">Th21</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>set </b><font color="Maroon">S</font> = <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">v</font></span><a href="finseq_1.html#K12">*&gt;</a></span>;<br/>
<a NAME="E4:133"/><i><font color="Green">E53</font></i>: 
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">v</font></span><a href="finseq_1.html#K12">*&gt;</a></span> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T102">FINSEQ_3:102</a></i>;<br/>
<a NAME="E5:133"/><i><font color="Green">E56</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">v</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="chord.html#T21" target="_self">Th21</a>, <a class="ref" href="chord.html#T22" target="_self">Th22</a>, <a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<b>reconsider </b><font color="Maroon">S</font> = <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">v</font></span><a href="finseq_1.html#K12">*&gt;</a></span> as    <a href="chord.html#M3" target="_self">VertexScheme</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="chord.html#T23" target="_self">Th23</a>, <a class="ref" href="chord.html#T24" target="_self">Th24</a>, </i>;<br/>
<b>take </b>
<font color="Maroon">S</font>
;<br/>

<b>thus </b><a NAME="E7:133"/>
<font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">v</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 ;<br/>

<b>let </b><font color="Maroon">n</font> be  non <a href="xboole_0.html#V1">zero</a> <a href="nat_1.html#NM1">Nat</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="chord.html#D13" target="_self">CHORD:def 13</a><br/>

<b>assume </b><a NAME="E8:133"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font>
 ;<br/>

<b>let </b><font color="Maroon">Gf</font> be   <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">G</font>,<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="chord.html#K4" target="_self">.followSet</a> <font color="Maroon">n</font></span>)</span>;<br/>

<b>thus </b><a NAME="E9:133"/>
for <font color="Olive">v</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Gf</font>  st <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> holds <br/><font color="Olive">v</font> is <a href="chord.html#V3" target="_self">simplicial</a>
 <b>by </b><i/>;<br/>


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