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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="E1:184_1"/><i><font color="Green">E39</font></i>: 
<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="E2:184_1"/><b>then </b>
 <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> <a href="hidden.html#R1">=</a> 3
 <b>by </b><i/>;<br/><a NAME="E3:184_1"/><b>then </b>
2 <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<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> </span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i/>;<br/><a NAME="E4:184_1"/><b>then </b>
2 <a href="nat_1.html#K2">*</a> 1 <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<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> </span>)</span></span>)</span>
 ;<br/><a NAME="E5:184_1"/><b>then </b><i><font color="Green">E40</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<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> </span>)</span> <a href="hidden.html#R1">=</a> 1
 ;<br/><a NAME="E6:184_1"/><b>then </b><i><font color="Green">E42</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<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> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</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="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/><a NAME="E7:184_1"/><i><font color="Green">E43</font></i>: 
<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> <span class="p1"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">e</font>,<font color="Maroon">y</font></span><a href="finseq_1.html#K11">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="glib_001.html#D22" target="_self">Def22</a>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="184_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:184_1_1"/><i><font color="Green">E44</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<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> </span>)</span> )
 ;<br/><a NAME="E2:184_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="glib_001.html#D25" target="_self">Def25</a>, , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2">(<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> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> = 
<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> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="glib_001.html#D25" target="_self">Def25</a>, </i>
<br/>.= 
<font color="Maroon">e</font>
<b>by </b><i><a class="ref" href="glib_001.html#D24" target="_self">Def24</a>, <a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>
;<br/><b>hence </b><a NAME="E4:184_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<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> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <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> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D26" target="_self">Def26</a>, <a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E9:184_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#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#D23" target="_self">Def23</a>, <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:184"/>
<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>
 ;<br/>

<a NAME="E3:184"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<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> </span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<a NAME="E4:184"/><b>then </b><i><font color="Green">E46</font></i>: 
 <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> <a href="hidden.html#R1">=</a> <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/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="184_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:184_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="E2:184_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="E3:184_2"/>
contradiction
 <b>by </b><i><a class="ref" href="glib_001.html#D22" target="_self">Def22</a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:184"/>
<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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