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<b>let </b><font color="Maroon">G</font> be   <a href="glib_000.html#NM2">_Graph</a>;<br/><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><b>that </b><br/><a NAME="E1:52"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">W</font> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a> 
 <b>and </b><br/><a NAME="E2:52"/><i><font color="Green">E33</font></i>: 
not <font color="Maroon">W</font> <a href="glib_001.html#K3">.first()</a> ,<font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a>  <a href="chord.html#R1" target="_self">are_adjacent</a> 
 ;<br/>

<b>set </b><font color="Maroon">l</font> = <font color="Maroon">W</font> <a href="glib_001.html#K15">.length()</a> ;<br/>
<b>assume </b><a NAME="E3:52"/>
<font color="Maroon">W</font> <a href="glib_001.html#K15">.length()</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2
 ;<br/>

<a NAME="E4:52"/><b>then </b>
<font color="Maroon">W</font> <a href="glib_001.html#K15">.length()</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="nat_1.html#K1">+</a> 1
 
;<br/>
<a NAME="E5:52"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">W</font> <a href="glib_001.html#K15">.length()</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_1"/>
( <font color="Maroon">W</font> <a href="glib_001.html#K15">.length()</a>  <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">W</font> <a href="glib_001.html#K15">.length()</a>  <a href="hidden.html#R1">=</a> 1 )
 </span><b>by </b><i><a class="ref" href="chord.html#T11" target="_self">Th11</a>, <a class="ref" href="cqc_the1.html#T2">CQC_THE1:2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1"><b>suppose </b></a><a NAME="E1:52_1_1"/>
<font color="Maroon">W</font> <a href="glib_001.html#K15">.length()</a>  <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:52_1_1"/><b>then </b>
<font color="Maroon">W</font> is <a href="glib_001.html#V3">trivial</a>
 <b>by </b><i><a class="ref" href="glib_001.html#D26">GLIB_001:def 26</a></i>;<br/><b>hence </b><a NAME="E3:52_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="chord.html#T8" target="_self">Th8</a>, <a class="ref" href="glib_001.html#T128">GLIB_001:128</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2"><b>suppose </b></a><a NAME="E1:52_1_2"/>
<font color="Maroon">W</font> <a href="glib_001.html#K15">.length()</a>  <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>then </b><i><font color="Green">E56</font></i>:  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</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
<b>by </b><i><a class="ref" href="glib_001.html#T113">GLIB_001:113</a></i>
<br/>.= 
3

;<br/><a NAME="E3:52_1_2"/>
1 <a href="hidden.html#R1">=</a> <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
 ;<br/><a NAME="E4:52_1_2"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> 1,<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="chord.html#T12" target="_self">Th12</a>, <a class="ref" href="glib_001.html#D3">GLIB_001:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">W1</font> = <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> 1 as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="glib_001.html#D3">GLIB_001:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">WlW</font> = <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="glib_001.html#T8">GLIB_001:8</a></i>;<br/><b>thus </b><a NAME="E7:52_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="chord.html#T10" target="_self">Th10</a>, <a class="ref" href="chord.html#T13" target="_self">Th13</a>, <a class="ref" href="chord.html#T12" target="_self">Th12</a>, <a class="ref" href="chord.html#T4" target="_self">Th4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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