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<div class="add">

<span class="kw">let </span><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#NM2" title="GLIB_000:NM.2">_Graph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3 implies <font color="Maroon" title="c1">G</font> is  <a href="chord.html#V6" title="CHORD:attr.6">chordal</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:113"/><span class="lab"><font color="Green" title="E90">A1</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">G</font> is  <a href="chord.html#V6" title="CHORD:attr.6">chordal</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_1"><span class="kw">now </span></a><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c2">n4</font> = <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 3</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/><span class="kw">reconsider </span><font color="Maroon" title="c3">n3</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/><span class="kw">reconsider </span><font color="Maroon" title="c4">n2</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/><span class="kw">reconsider </span><font color="Maroon" title="c5">n1</font> = <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></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/><span class="kw">let </span><font color="Maroon" title="c6">W</font> be    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">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 implies  not <font color="Maroon" title="c6">W</font> is  <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a>  )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E5:113_1"/><span class="lab"><font color="Green" title="E91">A2</font></span>: 
<font color="Maroon" title="c6">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
 <span class="kw">and </span><br/><a NAME="E6:113_1"/><span class="lab"><font color="Green" title="E92">A3</font></span>: 
<font color="Maroon" title="c6">W</font> is  <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">set </span><font color="Maroon" title="c7">x3</font> = <font color="Maroon" title="c6">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n3</font>;<br/><span class="kw">set </span><font color="Maroon" title="c8">x2</font> = <font color="Maroon" title="c6">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n2</font>;<br/><span class="kw">set </span><font color="Maroon" title="c9">x4</font> = <font color="Maroon" title="c6">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">n4</font>;<br/><span class="kw">set </span><font color="Maroon" title="c10">x1</font> = <font color="Maroon" title="c6">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n1</font>;<br/><a NAME="E7:113_1"/>
<font color="Maroon" title="c6">W</font> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:113_1"><span class="lab"><font color="Green" title="E91">A2</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E8:113_1"/><span class="kw">then </span>
2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">W</font> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 4
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>;<br/><a NAME="E9:113_1"/><span class="kw">then </span>
<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="c6">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#NR2" title="XXREAL_0:NR.2">&gt;=</a> 8 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/><a NAME="E10:113_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A4</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">W</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 9
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T112" title="GLIB_001:th.112">GLIB_001:112</a></span>;<br/><a NAME="E11:113_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A5</font></span>: 
<font color="Maroon" title="c2">n4</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">W</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E12:113_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">A6</font></span>: 
<font color="Maroon" title="c6">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">n4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:113_1"><span class="lab"><font color="Green" title="E92">A3</font></span></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></span>;<br/><a NAME="E13:113_1"/>
<font color="Maroon" title="c4">n2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:113_1"><span class="lab"><font color="Green" title="E93">A4</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E14:113_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">A7</font></span>: 
<font color="Maroon" title="c6">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:113_1"><span class="lab"><font color="Green" title="E92">A3</font></span></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></span>;<br/><a NAME="E15:113_1"/><span class="lab"><font color="Green" title="E97">A8</font></span>: 
<font color="Maroon" title="c3">n3</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:113_1"><span class="lab"><font color="Green" title="E93">A4</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E16:113_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A9</font></span>: 
<font color="Maroon" title="c6">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:113_1"><span class="lab"><font color="Green" title="E92">A3</font></span></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></span>;<br/><a NAME="E17:113_1"/><span class="lab"><font color="Green" title="E99">A10</font></span>: 
<font color="Maroon" title="c6">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">n4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:113_1"><span class="lab"><font color="Green" title="E92">A3</font></span></a>, <a class="txt" href="#E11:113_1"><span class="lab"><font color="Green" title="E94">A5</font></span></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></span>;<br/><a NAME="E18:113_1"/><span class="lab"><font color="Green" title="E100">A11</font></span>: 
<font color="Maroon" title="c6">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">n4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:113_1"><span class="lab"><font color="Green" title="E92">A3</font></span></a>, <a class="txt" href="#E11:113_1"><span class="lab"><font color="Green" title="E94">A5</font></span></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">n4</font></span>)</span></span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> implies <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font> )</span><br/><span class="kw">assume </span><a NAME="E1:113_1_1"/><span class="lab"><font color="Green" title="E101">A12</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">n4</font></span>)</span></span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="113_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:113_1_1_1_1"/>
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n1</font> or <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n2</font> or <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n3</font> or <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">n4</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:113_1_1"><span class="lab"><font color="Green" title="E101">A12</font></span></a>, <a class="ref" href="enumset1.html#D2" title="ENUMSET1:def.2">ENUMSET1:def 2</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="113_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:113_1_1_1_1_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:113_1_1_1_1_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:113_1"><span class="lab"><font color="Green" title="E93">A4</font></span></a>, <a class="ref" href="glib_001.html#T7" title="GLIB_001:th.7">GLIB_001:7</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="113_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:113_1_1_1_1_2"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:113_1_1_1_1_2"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:113_1"><span class="lab"><font color="Green" title="E93">A4</font></span></a>, <a class="ref" href="glib_001.html#T7" title="GLIB_001:th.7">GLIB_001:7</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="113_1_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:113_1_1_1_1_3"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n3</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:113_1_1_1_1_3"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:113_1"><span class="lab"><font color="Green" title="E93">A4</font></span></a>, <a class="ref" href="glib_001.html#T7" title="GLIB_001:th.7">GLIB_001:7</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="113_1_1_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:113_1_1_1_1_4"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">n4</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:113_1_1_1_1_4"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:113_1"><span class="lab"><font color="Green" title="E93">A4</font></span></a>, <a class="ref" href="glib_001.html#T7" title="GLIB_001:th.7">GLIB_001:7</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:113_1_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E20:113_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A13</font></span>: 
<span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">n4</font></span>)</span></span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><a NAME="E21:113_1"/>
<font color="Maroon" title="c6">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:113_1"><span class="lab"><font color="Green" title="E92">A3</font></span></a>, <a class="txt" href="#E15:113_1"><span class="lab"><font color="Green" title="E97">A8</font></span></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></span>;<br/><a NAME="E22:113_1"/><span class="kw">then </span>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">n4</font></span>)</span></span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:113_1"><span class="lab"><font color="Green" title="E96">A7</font></span></a>, <a class="txt" href="#E12:113_1"><span class="lab"><font color="Green" title="E95">A6</font></span></a>, <a class="txt" href="#E16:113_1"><span class="lab"><font color="Green" title="E98">A9</font></span></a>, <a class="txt" href="#E18:113_1"><span class="lab"><font color="Green" title="E100">A11</font></span></a>, <a class="txt" href="#E17:113_1"><span class="lab"><font color="Green" title="E99">A10</font></span></a>, <a class="ref" href="card_2.html#T59" title="CARD_2:th.59">CARD_2:59</a></span>;<br/><a NAME="E23:113_1"/><span class="kw">then </span>
4 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:113_1"><span class="lab"><font color="Green" title="E101">A13</font></span></a>, <a class="ref" href="nat_1.html#T43" title="NAT_1:th.43">NAT_1:43</a></span>;<br/><span class="kw">hence </span><a NAME="E24:113_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:113"><span class="lab"><font color="Green" title="E90">A1</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E3:113"/><span class="kw">then </span>
 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/>
<span class="kw">hence </span><a NAME="E4:113"/>
<font color="Maroon" title="c1">G</font> is  <a href="chord.html#V6" title="CHORD:attr.6">chordal</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#D11" target="_self" title="CHORD:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
