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<b>consider </b><font color="Maroon" title="c1">G</font> being  <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>consider </b><font color="Maroon" title="c2">v</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="E3:115_1_1"/><i><font color="Green" title="E90">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="c2">v</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">W</font> be    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G</font>;<br/><b>assume </b><a NAME="E1:115_1_1_1"/><i><font color="Green" title="E91">A2</font></i>: 
( <font color="Maroon" title="c3">W</font> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 3 &amp; <font color="Maroon" title="c3">W</font> is <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a> )
 ;<br/><a NAME="E2:115_1_1_1"/>
2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a> </span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 3
 <b>by </b><i><a class="txt" href="chord.html#E1:115_1_1_1"><i><font color="Green" title="E91">A2</font></i></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/><a NAME="E3:115_1_1_1"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a> </span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 6 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E4:115_1_1_1"/><b>then </b><i><font color="Green" title="E92">A3</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">W</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 7
 <b>by </b><i><a class="ref" href="glib_001.html#T113" title="GLIB_001:th.113">GLIB_001:113</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c4">j3</font> = <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 as  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/><b>reconsider </b><font color="Maroon" title="c5">j5</font> = <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 as  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/><a NAME="E7:115_1_1_1"/><i><font color="Green" title="E93">A4</font></i>: 
( <font color="Maroon" title="c4">j3</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="c3">W</font> &amp; <font color="Maroon" title="c5">j5</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="c3">W</font> )
 <b>by </b><i><a class="txt" href="chord.html#E4:115_1_1_1"><i><font color="Green" title="E92">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E8:115_1_1_1"/><b>then </b>
( <font color="Maroon" title="c3">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">j3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">W</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  &amp; <font color="Maroon" title="c3">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">j5</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">W</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  )
 <b>by </b><i><a class="ref" href="glib_001.html#T88" title="GLIB_001:th.88">GLIB_001:88</a></i>;<br/><a NAME="E9:115_1_1_1"/><b>then </b>
( <font color="Maroon" title="c3">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">j3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font> &amp; <font color="Maroon" title="c3">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">j5</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#E3:115_1_1"><i><font color="Green" title="E90">A1</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E10:115_1_1_1"/><b>then </b>
not <font color="Maroon" title="c3">W</font> is <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>
 <b>by </b><i><a class="txt" href="chord.html#E7:115_1_1_1"><i><font color="Green" title="E93">A4</font></i></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></i>;<br/><b>hence </b><a NAME="E11:115_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E1:115_1_1_1"><i><font color="Green" title="E91">A2</font></i></a>, <a class="ref" href="glib_001.html#D31" title="GLIB_001:def.31">GLIB_001:def 31</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:115_1_1"/><b>then </b>
for <font color="Olive" title="b1">W</font> being   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">W</font> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 3 &amp; <font color="Olive" title="b1">W</font> is <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a> holds <br/><font color="Olive" title="b1">W</font> is <a href="chord.html#V5" title="CHORD:attr.5">chordal</a>
 
;<br/>
<a NAME="E6:115_1_1"/><b>then </b>
<font color="Maroon" title="c1">G</font> is <a href="chord.html#V6" title="CHORD:attr.6">chordal</a>
 
<b>by </b><i><a class="ref" href="chord.html#D11" target="_self" title="CHORD:def.11">Def11</a></i>;<br/>
<b>hence </b><a NAME="E7:115_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="glib_000.html#V4" title="GLIB_000:attr.4">trivial</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="chord.html#V6" title="CHORD:attr.6">chordal</a> )
 ;<br/>


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