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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">a</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>, <font color="Maroon">b</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>;<br/>



<b>assume </b><a NAME="E1:85"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; not <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="chord.html#R1" target="_self">are_adjacent</a>  )
 ;<br/>

<b>let </b><font color="Maroon">S</font> be    <a href="chord.html#M2" target="_self">VertexSeparator</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font>;<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><a NAME="E2:85"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 ;<br/>

<a NAME="E3:85"/><i><font color="Green">E53</font></i>: 
( not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; not <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 
<b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">x</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E5:85"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="glib_001.html#K13">.vertices()</a>  )
 <b>by </b><i>, , </i>;<br/>
<b>consider </b><font color="Maroon">n</font> being  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:85"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> &amp; <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i>, <a class="ref" href="glib_001.html#T88">GLIB_001:88</a></i>;<br/>
<a NAME="E8:85"/><i><font color="Green">E66</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:85_1"/>
1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E2:85_1"/><b>then </b>
<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="glib_001.html#K3">.first()</a> 
 ;<br/><b>hence </b><a NAME="E3:85_1"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="glib_001.html#D23">GLIB_001:def 23</a></i>;<br/></div><b>end;</b></div>
<a NAME="E10:85"/><b>then </b><i><font color="Green">E67</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:85_2"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font>
 ;<br/><a NAME="E2:85_2"/><b>then </b>
<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a> 
 ;<br/><b>hence </b><a NAME="E3:85_2"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="glib_001.html#D23">GLIB_001:def 23</a></i>;<br/></div><b>end;</b></div>
<a NAME="E12:85"/><b>then </b>
<font color="Maroon">n</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="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E13:85"/>
 ex <font color="Olive">k</font> being <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a> st <br/>( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">k</font> &amp; <font color="Olive">k</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">W</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 <b>by </b><i>, , </i>;<br/>


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