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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#NM2">_Graph</a>;<br/>

<b>given </b><font color="Maroon">S</font> being    <a href="chord.html#M3" target="_self">VertexScheme</a> of <font color="Maroon">G</font><b> such that </b><a NAME="E1:137"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">S</font> is <a href="chord.html#V7" target="_self">perfect</a>
 ;<br/>

<b>let </b><font color="Maroon">P</font> be    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="chord.html#D11" target="_self">CHORD:def 11</a><br/>

<b>assume </b><a NAME="E2:137"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">P</font> <a href="glib_001.html#K15">.length()</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 3 &amp; <font color="Maroon">P</font> is <a href="glib_001.html#V8">Cycle-like</a> )
 ;<br/>

<a NAME="E3:137"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">P</font> is <a href="glib_001.html#V5">Path-like</a>
 
<b>by </b><i><a class="ref" href="chord.html#T22" target="_self">Th22</a>, <a class="ref" href="glib_001.html#D31">GLIB_001:def 31</a></i>;<br/>
<a NAME="E4:137"/><i><font color="Green">E56</font></i>: 
not <font color="Maroon">P</font> <a href="glib_001.html#K13">.vertices()</a>  is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="glib_001.html#T89">GLIB_001:89</a></i>;<br/>
<a NAME="E5:137"/><i><font color="Green">E59</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">S</font> <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/>;<br/>
<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:137"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="glib_001.html#K13">.vertices()</a> 
 <b>and </b><br/><a NAME="E8:137"/><i><font color="Green">E67</font></i>: 
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="glib_001.html#K13">.vertices()</a>  holds <br/><font color="Maroon">x</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">S</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">y</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">S</font>
 <b>by </b><i>, <a class="ref" href="chord.html#T24" target="_self">Th24</a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">n</font> = <font color="Maroon">x</font> <a href="chord.html#K3" target="_self">..</a> <font color="Maroon">S</font>;<br/>
<a NAME="E10:137"/><i><font color="Green">E68</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">S</font> <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/>;<br/>
<a NAME="E11:137"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">x</font> <a href="chord.html#K3" target="_self">..</a> <font color="Maroon">S</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="finseq_4.html#T31">FINSEQ_4:31</a></i>;<br/>
<a NAME="E12:137"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">x</font> <a href="chord.html#K3" target="_self">..</a> <font color="Maroon">S</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">S</font>
 
<b>by </b><i>, <a class="ref" href="finseq_4.html#T30">FINSEQ_4:30</a></i>;<br/>
<a NAME="E13:137"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">S</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="137_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:137_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="glib_001.html#K13">.vertices()</a> 
 ;<br/><b>reconsider </b><font color="Maroon">z</font> = <font color="Maroon">y</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> <b>by </b><i/>;<br/><a NAME="E3:137_1"/>
<font color="Maroon">x</font> <a href="chord.html#K3" target="_self">..</a> <font color="Maroon">S</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">z</font> <a href="chord.html#K3" target="_self">..</a> <font color="Maroon">S</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:137_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> <a href="chord.html#K4" target="_self">.followSet</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="chord.html#K3" target="_self">..</a> <font color="Maroon">S</font></span>)</span>
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div>
<a NAME="E15:137"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">P</font> <a href="glib_001.html#K13">.vertices()</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> <a href="chord.html#K4" target="_self">.followSet</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="chord.html#K3" target="_self">..</a> <font color="Maroon">S</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">H</font> being   <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> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="chord.html#K3" target="_self">..</a> <font color="Maroon">S</font></span>)</span></span>)</span>;<br/>
<a NAME="E17:137"/><i><font color="Green">E74</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">S</font> <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/>;<br/>
<a NAME="E18:137"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">x</font> <a href="chord.html#K3" target="_self">..</a> <font color="Maroon">S</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="finseq_4.html#T31">FINSEQ_4:31</a></i>;<br/>
<a NAME="E19:137"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> <a href="chord.html#K4" target="_self">.followSet</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="chord.html#K3" target="_self">..</a> <font color="Maroon">S</font></span>)</span>
 
<b>by </b><i>, , , </i>;<br/>
<b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">x</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">H</font> <b>by </b><i><a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/>
<a NAME="E21:137"/>
<font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="chord.html#K3" target="_self">..</a> <font color="Maroon">S</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="finseq_4.html#T29">FINSEQ_4:29</a></i>;<br/>
<a NAME="E22:137"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">y</font> is <a href="chord.html#V3" target="_self">simplicial</a>
 
<b>by </b><i><a class="ref" href="chord.html#T21" target="_self">Th21</a>, , </i>;<br/>
<a NAME="E23:137"/><i><font color="Green">E80</font></i>: 
not <font color="Maroon">S</font> <a href="chord.html#K4" target="_self">.followSet</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="chord.html#K3" target="_self">..</a> <font color="Maroon">S</font></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="chord.html#T25" target="_self">Th25</a>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">C</font> = <font color="Maroon">P</font> as    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">H</font> <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">C</font> = <font color="Maroon">C</font> as   <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">H</font> <b>by </b><i><a class="ref" href="chord.html#T23" target="_self">Th23</a>, <a class="ref" href="glib_001.html#T177">GLIB_001:177</a></i>;<br/>
<a NAME="E26:137"/>
( not <font color="Maroon">P</font> is <a href="glib_001.html#V3">trivial</a> &amp; <font color="Maroon">P</font> is <a href="glib_001.html#V1">closed</a> )
 
<b>by </b><i><a class="ref" href="chord.html#T22" target="_self">Th22</a>, <a class="ref" href="glib_001.html#D31">GLIB_001:def 31</a></i>;<br/>
<a NAME="E27:137"/><b>then </b>
( not <font color="Maroon">C</font> is <a href="glib_001.html#V3">trivial</a> &amp; <font color="Maroon">C</font> is <a href="glib_001.html#V1">closed</a> )
 
<b>by </b><i><a class="ref" href="glib_001.html#T177">GLIB_001:177</a></i>;<br/>
<a NAME="E28:137"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">C</font> is <a href="glib_001.html#V8">Cycle-like</a>
 
<b>by </b><i><a class="ref" href="glib_001.html#D31">GLIB_001:def 31</a></i>;<br/>
<a NAME="E29:137"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">C</font> <a href="glib_001.html#K15">.length()</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 3
 
<b>by </b><i><a class="ref" href="chord.html#T22" target="_self">Th22</a>, <a class="ref" href="glib_001.html#T115">GLIB_001:115</a></i>;<br/>
<a NAME="E30:137"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> <a href="glib_001.html#K13">.vertices()</a> 
 
<b>by </b><i>, <a class="ref" href="glib_001.html#T99">GLIB_001:99</a></i>;<br/>
<b>then consider </b><font color="Maroon">a</font> being  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">b</font> being  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E32:137"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font> )
 <b>and </b><br/><a NAME="E33:137"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> &amp; <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span> )
 <b>and </b><br/><a NAME="E34:137"/><i><font color="Green">E85</font></i>: 
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> <a href="glib_001.html#K14">.edges()</a>  holds <br/> not <font color="Olive">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">H</font>
 <b>by </b><i>, , <a class="txt" href="chord.html#E15"><i><font color="Green">Lemma49</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">e</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E36:137"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">H</font>
 <b>by </b><i>, , </i>;<br/>
<a NAME="E37:137"/>
<font color="Maroon">C</font> is <a href="chord.html#V5" target="_self">chordal</a>
 
<b>by </b><i>, , <a class="ref" href="chord.html#T27" target="_self">Th27</a>, , <a class="txt" href="chord.html#E23:137"><i><font color="Green">E80</font></i></a></i>;<br/>
<b>hence </b><a NAME="E38:137"/>
<font color="Maroon">P</font> is <a href="chord.html#V5" target="_self">chordal</a>
 <b>by </b><i>, <a class="txt" href="chord.html#E33:137"><i><font color="Green">E84</font></i></a></i>;<br/>


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