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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>assume </b><a NAME="E1:114"/><i><font color="Green">E30</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 3
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="114_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">W</font> be    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G</font>;<br/><b>assume </b><a NAME="E1:114_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">W</font> <a href="glib_001.html#K15">.length()</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 3 &amp; <font color="Maroon">W</font> is <a href="glib_001.html#V8">Cycle-like</a> )
 ;<br/><b>reconsider </b><font color="Maroon">n1</font> = <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1 as  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a> ;<br/><b>reconsider </b><font color="Maroon">n2</font> = <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 as  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a> ;<br/><b>reconsider </b><font color="Maroon">n3</font> = <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1 as  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a> ;<br/><b>reconsider </b><font color="Maroon">n4</font> = <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 3</span>)</span> <a href="nat_1.html#K1">+</a> 1 as  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a> ;<br/><b>set </b><font color="Maroon">x1</font> = <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</font>;<br/><b>set </b><font color="Maroon">x2</font> = <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n2</font>;<br/><b>set </b><font color="Maroon">x3</font> = <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n3</font>;<br/><b>set </b><font color="Maroon">x4</font> = <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n4</font>;<br/><a NAME="E6:114_1"/>
<font color="Maroon">W</font> <a href="glib_001.html#K15">.length()</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 3 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E7:114_1"/><b>then </b>
2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K15">.length()</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="nat_1.html#K2">*</a> 4
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E8:114_1"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K15">.length()</a> </span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 8 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E9:114_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 9
 <b>by </b><i><a class="ref" href="glib_001.html#T113">GLIB_001:113</a></i>;<br/><a NAME="E10:114_1"/><b>then </b><i><font color="Green">E53</font></i>: 
( <font color="Maroon">n1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> &amp; <font color="Maroon">n2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> &amp; <font color="Maroon">n3</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> &amp; <font color="Maroon">n4</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E11:114_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">W</font> is <a href="glib_001.html#V5">Path-like</a>
 <b>by </b><i>, <a class="ref" href="glib_001.html#D31">GLIB_001:def 31</a></i>;<br/><a NAME="E12:114_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n2</font>
 <b>by </b><i>, , <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/><a NAME="E13:114_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n3</font>
 <b>by </b><i>, , <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/><a NAME="E14:114_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n4</font>
 <b>by </b><i>, , <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/><a NAME="E15:114_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n3</font>
 <b>by </b><i>, , <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/><a NAME="E16:114_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n4</font>
 <b>by </b><i>, , <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/><a NAME="E17:114_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n4</font>
 <b>by </b><i>, , <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/><a NAME="E18:114_1"/><i><font color="Green">E71</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1"><a href="enumset1.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n4</font></span>)</span></span><a href="enumset1.html#K2">}</a></span> <a href="hidden.html#R1">=</a> 4
 <b>by </b><i>, , , , , <a class="ref" href="chord.html#T21" target="_self">Th21</a>, <a class="ref" href="card_2.html#T78">CARD_2:78</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="114_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:114_1_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n4</font></span>)</span></span><a href="enumset1.html#K2">}</a></span>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="114_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:114_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n2</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n3</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n4</font> )
 </span><b>by </b><i><a class="ref" href="chord.html#T23" target="_self">Th23</a>, <a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:114_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:114_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T8">GLIB_001:8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:114_1_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:114_1_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T8">GLIB_001:8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:114_1_1_1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n3</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:114_1_1_1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T8">GLIB_001:8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_1_1_1_4"><b>suppose </b></a><a NAME="E1:114_1_1_1_1_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n4</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:114_1_1_1_1_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T8">GLIB_001:8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:114_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font>
 ;<br/></div><b>end;</b></div><a NAME="E20:114_1"/><b>then </b>
<span class="p1"><a href="enumset1.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n4</font></span>)</span></span><a href="enumset1.html#K2">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E21:114_1"/><b>then </b>
4 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i><a class="ref" href="chord.html#T22" target="_self">Th22</a>, <a class="ref" href="card_1.html#T80">CARD_1:80</a></i>;<br/><b>hence </b><a NAME="E22:114_1"/>
contradiction
 <b>by </b><i><a class="ref" href="chord.html#T19" target="_self">Th19</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:114"/><b>then </b>
for <font color="Olive">W</font> being   <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G</font>  st <font color="Olive">W</font> <a href="glib_001.html#K15">.length()</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 3 &amp; <font color="Olive">W</font> is <a href="glib_001.html#V8">Cycle-like</a> holds <br/><font color="Olive">W</font> is <a href="chord.html#V5" target="_self">chordal</a>
 
;<br/>
<b>hence </b><a NAME="E4:114"/>
<font color="Maroon">G</font> is <a href="chord.html#V6" target="_self">chordal</a>
 <b>by </b><i/>;<br/>


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