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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">W1</font> be    <a href="glib_001.html#M3" target="_self">Walk</a> of <font color="Maroon">G</font>, <font color="Maroon">W2</font> be    <a href="glib_001.html#M3" target="_self">Walk</a> of <font color="Maroon">G</font>;<br/>



<b>assume </b><a NAME="E1:45"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">W1</font> <a href="glib_001.html#K4" target="_self">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">W2</font> <a href="glib_001.html#K3" target="_self">.first()</a> 
 ;<br/>

<b>set </b><font color="Maroon">W</font> = <font color="Maroon">W1</font> <a href="glib_001.html#K7" target="_self">.append</a> <font color="Maroon">W2</font>;<br/>
<a NAME="E2:45"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">W1</font> <a href="glib_001.html#K7" target="_self">.append</a> <font color="Maroon">W2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">W2</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:45"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W1</font>
 
<b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/>
<a NAME="E4:45"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K7" target="_self">.append</a> <font color="Maroon">W2</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font> <a href="glib_001.html#K3" target="_self">.first()</a> 
 
<b>by </b><i>, <a class="ref" href="graph_2.html#T14">GRAPH_2:14</a></i>;<br/>
<b>hence </b><a NAME="E5:45"/><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K7" target="_self">.append</a> <font color="Maroon">W2</font></span>)</span> <a href="glib_001.html#K3" target="_self">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font> <a href="glib_001.html#K3" target="_self">.first()</a> 
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:45_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W2</font> <a href="hidden.html#R1">=</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_1"><b>suppose </b></a><a NAME="E1:45_1_1_1"/><i><font color="Green">E43</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">W2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:45_1_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W2</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/><a NAME="E3:45_1_1_1"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W2</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E4:45_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K7" target="_self">.append</a> <font color="Maroon">W2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K7" target="_self">.append</a> <font color="Maroon">W2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">W2</font> <a href="glib_001.html#K4" target="_self">.last()</a> 
 <b>by </b><i>, <a class="ref" href="graph_2.html#T16">GRAPH_2:16</a></i>;<br/><b>hence </b><a NAME="E5:45_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K7" target="_self">.append</a> <font color="Maroon">W2</font></span>)</span> <a href="glib_001.html#K4" target="_self">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">W2</font> <a href="glib_001.html#K4" target="_self">.last()</a> 
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_2"><b>suppose </b></a><a NAME="E1:45_1_1_2"/><i><font color="Green">E44</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">W2</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:45_1_1_2"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">W1</font> <a href="glib_001.html#K7" target="_self">.append</a> <font color="Maroon">W2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default">2,1 <a href="graph_2.html#K2">-cut</a> <font color="Maroon">W2</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="graph_2.html#D2">GRAPH_2:def 2</a></i>;<br/><a NAME="E3:45_1_1_2"/>
2,1 <a href="graph_2.html#K2">-cut</a> <font color="Maroon">W2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="graph_2.html#D1">GRAPH_2:def 1</a></i>;<br/><a NAME="E4:45_1_1_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default">2,1 <a href="graph_2.html#K2">-cut</a> <font color="Maroon">W2</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="card_1.html#T47">CARD_1:47</a></i>;<br/><a NAME="E5:45_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default">2,1 <a href="graph_2.html#K2">-cut</a> <font color="Maroon">W2</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E6:45_1_1_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default">2,1 <a href="graph_2.html#K2">-cut</a> <font color="Maroon">W2</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E7:45_1_1_2"/><b>then </b>
2,1 <a href="graph_2.html#K2">-cut</a> <font color="Maroon">W2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><b>hence </b><a NAME="E8:45_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K7" target="_self">.append</a> <font color="Maroon">W2</font></span>)</span> <a href="glib_001.html#K4" target="_self">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">W2</font> <a href="glib_001.html#K4" target="_self">.last()</a> 
 <b>by </b><i><a class="txt" href="glib_001.html#E2:45_1_1_2"><i><font color="Green">E45</font></i></a>, , <a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:45"/>
( <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K7" target="_self">.append</a> <font color="Maroon">W2</font></span>)</span> <a href="glib_001.html#K4" target="_self">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">W2</font> <a href="glib_001.html#K4" target="_self">.last()</a>  &amp; <font color="Maroon">W1</font> <a href="glib_001.html#K7" target="_self">.append</a> <font color="Maroon">W2</font> <a href="glib_001.html#R1" target="_self">is_Walk_from</a> <font color="Maroon">W1</font> <a href="glib_001.html#K3" target="_self">.first()</a> ,<font color="Maroon">W2</font> <a href="glib_001.html#K4" target="_self">.last()</a>  )
 <b>by </b><i>, </i>;<br/>


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