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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">e</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>







<b>set </b><font color="Maroon">W</font> = <font color="Maroon">G</font> <a href="glib_001.html#K2" target="_self">.walkOf</a> <font color="Maroon">x</font>,<font color="Maroon">e</font>,<font color="Maroon">y</font>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:204_1"/>
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">G</font>
 ;<br/><a NAME="E2:204_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2" target="_self">.walkOf</a> <font color="Maroon">x</font>,<font color="Maroon">e</font>,<font color="Maroon">y</font></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">e</font></span><a href="finseq_1.html#K9">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="glib_001.html#D21" target="_self">Def21</a></i>;<br/><b>hence </b><a NAME="E3:204_1"/>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2" target="_self">.walkOf</a> <font color="Maroon">x</font>,<font color="Maroon">e</font>,<font color="Maroon">y</font></span>)</span> <a href="glib_001.html#K14" target="_self">.edges()</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">e</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:204"/>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2" target="_self">.walkOf</a> <font color="Maroon">x</font>,<font color="Maroon">e</font>,<font color="Maroon">y</font></span>)</span> <a href="glib_001.html#K14" target="_self">.edges()</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">e</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E3:204"/><b>then </b>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2" target="_self">.walkOf</a> <font color="Maroon">x</font>,<font color="Maroon">e</font>,<font color="Maroon">y</font></span>)</span> <a href="glib_001.html#K14" target="_self">.edges()</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">n</font> being  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:204"/><i><font color="Green">E39</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2" target="_self">.walkOf</a> <font color="Maroon">x</font>,<font color="Maroon">e</font>,<font color="Maroon">y</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2" target="_self">.walkOf</a> <font color="Maroon">x</font>,<font color="Maroon">e</font>,<font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> )
 <b>by </b><i/>;<br/>
<a NAME="E6:204"/><i><font color="Green">E40</font></i>: 
<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 <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="204_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:204_2"/>
not <font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">G</font>
 ;<br/><a NAME="E2:204_2"/><b>then </b>
<font color="Maroon">G</font> <a href="glib_001.html#K2" target="_self">.walkOf</a> <font color="Maroon">x</font>,<font color="Maroon">e</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="glib_001.html#K1" target="_self">.walkOf</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:204_2"/>
contradiction
 <b>by </b><i>, <a class="txt" href="glib_001.html#E42"><i><font color="Green">Lemma145</font></i></a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:204"/>
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">G</font>
 ;<br/>


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