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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" target="_self">Walk</a> of <font color="Maroon">G</font>;<br/>



<b>assume </b><a NAME="E1:218"/>
not <font color="Maroon">W</font> is <a href="glib_001.html#V3" target="_self">trivial</a>
 ;<br/>

<a NAME="E2:218"/><b>then </b><i><font color="Green">E39</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 3
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">lenW2</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span>;<br/>
<a NAME="E3:218"/>
2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font>
 
<b>by </b><i><a class="txt" href="glib_001.html#E46"><i><font color="Green">Lemma149</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">lenW2</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> as  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/>
<b>take </b>
<font color="Maroon">lenW2</font>
;<br/>

<b>thus </b><a NAME="E5:218"/>
<font color="Maroon">lenW2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2
 ;<br/>

<b>set </b><font color="Maroon">W1</font> = <font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">lenW2</font>;<br/>
<b>set </b><font color="Maroon">e</font> = <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">lenW2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/>
<a NAME="E6:218"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">lenW2</font> &amp; <font color="Maroon">lenW2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 0 )
 
<b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a>, <a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">lenW2</font></span>)</span> <a href="glib_001.html#K10" target="_self">.addEdge</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">lenW2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
<font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<span class="p1">(<span class="default"><font color="Maroon">lenW2</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>
<b>by </b><i>, <a class="ref" href="jordan12.html#T3">JORDAN12:3</a></i>
<br/>.= 
<font color="Maroon">W</font>
<b>by </b><i/>
;<br/><br/>


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