<?xml version="1.0"?>
<div class="add">

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





<b>set </b><font color="Maroon">W1</font> = <font color="Maroon">G</font> <a href="glib_001.html#K2" target="_self">.walkOf</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> </span>)</span>,<font color="Maroon">e</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> </span>)</span> <a href="glib_000.html#K30">.adj</a> <font color="Maroon">e</font></span>)</span>;<br/>
<b>assume </b><a NAME="E1:69"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> ,<font color="Maroon">x</font>,<font color="Maroon">G</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="glib_000.html#T16">GLIB_000:16</a></i>;<br/>
<a NAME="E3:69"/><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> </span>)</span> <a href="glib_000.html#K30">.adj</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font>
 
<b>by </b><i><a class="ref" href="glib_001.html#D2" target="_self">Def2</a>, <a class="ref" href="glib_000.html#T69">GLIB_000:69</a></i>;<br/>
<a NAME="E4:69"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">G</font> <a href="glib_001.html#K2" target="_self">.walkOf</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> </span>)</span>,<font color="Maroon">e</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> </span>)</span> <a href="glib_000.html#K30">.adj</a> <font color="Maroon">e</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> </span>)</span>,<font color="Maroon">e</font>,<font color="Maroon">x</font></span><a href="finseq_1.html#K11">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="glib_001.html#D2" target="_self">Def2</a>, </i>;<br/>
<a NAME="E5:69"/>
<font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2" target="_self">.walkOf</a> <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> </span>)</span>,<font color="Maroon">e</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> </span>)</span> <a href="glib_000.html#K30">.adj</a> <font color="Maroon">e</font></span>)</span></span>)</span> <a href="glib_001.html#K3" target="_self">.first()</a> 
 
<b>by </b><i><a class="ref" href="glib_001.html#D2" target="_self">Def2</a>, , </i>;<br/>
<b>then </b><i><font color="Green">E43</font></i>: <font color="Maroon">W</font> <a href="glib_001.html#K7" target="_self">.append</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2" target="_self">.walkOf</a> <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> </span>)</span>,<font color="Maroon">e</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> </span>)</span> <a href="glib_000.html#K30">.adj</a> <font color="Maroon">e</font></span>)</span></span>)</span> = 
<font color="Maroon">W</font> <a href="graph_2.html#K4">^'</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2" target="_self">.walkOf</a> <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> </span>)</span>,<font color="Maroon">e</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> </span>)</span> <a href="glib_000.html#K30">.adj</a> <font color="Maroon">e</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">W</font> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default">2,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2" target="_self">.walkOf</a> <span class="p4">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> </span>)</span>,<font color="Maroon">e</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> </span>)</span> <a href="glib_000.html#K30">.adj</a> <font color="Maroon">e</font></span>)</span></span>)</span></span>)</span> <a href="graph_2.html#K2">-cut</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2" target="_self">.walkOf</a> <span class="p3">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> </span>)</span>,<font color="Maroon">e</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> </span>)</span> <a href="glib_000.html#K30">.adj</a> <font color="Maroon">e</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="graph_2.html#D2">GRAPH_2:def 2</a></i>
<br/>.= 
<font color="Maroon">W</font> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default">2,3 <a href="graph_2.html#K2">-cut</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2" target="_self">.walkOf</a> <span class="p3">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> </span>)</span>,<font color="Maroon">e</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> </span>)</span> <a href="glib_000.html#K30">.adj</a> <font color="Maroon">e</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="glib_001.html#D3" target="_self">Def3</a>, <a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>
;<br/>
<a NAME="E7:69"/><i><font color="Green">E44</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> <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> </span>)</span>,<font color="Maroon">e</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> </span>)</span> <a href="glib_000.html#K30">.adj</a> <font color="Maroon">e</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 3 &amp; <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2" target="_self">.walkOf</a> <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> </span>)</span>,<font color="Maroon">e</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> </span>)</span> <a href="glib_000.html#K30">.adj</a> <font color="Maroon">e</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2" target="_self">.walkOf</a> <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> </span>)</span>,<font color="Maroon">e</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> </span>)</span> <a href="glib_000.html#K30">.adj</a> <font color="Maroon">e</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 
<b>by </b><i><a class="ref" href="glib_001.html#D3" target="_self">Def3</a>, <a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/>
<a NAME="E8:69"/><b>then </b>
( 2,2 <a href="graph_2.html#K2">-cut</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2" target="_self">.walkOf</a> <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> </span>)</span>,<font color="Maroon">e</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> </span>)</span> <a href="glib_000.html#K30">.adj</a> <font color="Maroon">e</font></span>)</span></span>)</span> <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> &amp; 3,3 <a href="graph_2.html#K2">-cut</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2" target="_self">.walkOf</a> <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> </span>)</span>,<font color="Maroon">e</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> </span>)</span> <a href="glib_000.html#K30">.adj</a> <font color="Maroon">e</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9">*&gt;</a></span> )
 
<b>by </b><i><a class="ref" href="graph_2.html#T6">GRAPH_2:6</a></i>;<br/>
<a NAME="E9:69"/><b>then </b>
<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="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K1">+</a> 1</span>)</span>,3 <a href="graph_2.html#K2">-cut</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2" target="_self">.walkOf</a> <span class="p3">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> </span>)</span>,<font color="Maroon">e</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> </span>)</span> <a href="glib_000.html#K30">.adj</a> <font color="Maroon">e</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>,3 <a href="graph_2.html#K2">-cut</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2" target="_self">.walkOf</a> <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> </span>)</span>,<font color="Maroon">e</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> </span>)</span> <a href="glib_000.html#K30">.adj</a> <font color="Maroon">e</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="graph_2.html#T8">GRAPH_2:8</a></i>;<br/>
<a NAME="E10:69"/><b>then </b>
<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="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R1">=</a> 2,3 <a href="graph_2.html#K2">-cut</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2" target="_self">.walkOf</a> <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> </span>)</span>,<font color="Maroon">e</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> </span>)</span> <a href="glib_000.html#K30">.adj</a> <font color="Maroon">e</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="graph_2.html#T6">GRAPH_2:6</a></i>;<br/>
<a NAME="E11:69"/><b>then </b>
<font color="Maroon">W</font> <a href="glib_001.html#K7" target="_self">.append</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2" target="_self">.walkOf</a> <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> </span>)</span>,<font color="Maroon">e</font>,<font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">e</font>,<font color="Maroon">x</font></span><a href="finseq_1.html#K10">*&gt;</a></span>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#D9">FINSEQ_1:def 9</a></i>;<br/>
<b>hence </b><a NAME="E12:69"/>
<font color="Maroon">W</font> <a href="glib_001.html#K10" target="_self">.addEdge</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">e</font>,<font color="Maroon">x</font></span><a href="finseq_1.html#K10">*&gt;</a></span>
 <b>by </b><i/>;<br/>


</div>
