<?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">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>





<b>assume </b><a NAME="E1:66"/>
<font color="Maroon">W</font> <a href="glib_001.html#K3" target="_self">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 ;<br/>

<a NAME="E2:66"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_1_1"/>
( ( not <font color="Maroon">m</font> is <a href="abian.html#V1">even</a> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> ) or <font color="Maroon">m</font> is <a href="abian.html#V1">even</a> or  not <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_1_1"><b>suppose </b></a><a NAME="E1:66_1_1_1"/><i><font color="Green">E40</font></i>: 
( not <font color="Maroon">m</font> is <a href="abian.html#V1">even</a> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> )
 ;<br/><div class="add"><a NAME="E2:66_1_1_1"/><b>then </b>
( not 1 is <a href="abian.html#V1">even</a> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 <b>by </b><i><a class="ref" href="jordan12.html#T3">JORDAN12:3</a>, <a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/><a NAME="E3:66_1_1_1"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> 1,<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,1</span>)</span> <a href="glib_001.html#K7" target="_self">.append</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> <font color="Maroon">m</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span></span>)</span>
 <b>by </b><i>, , </i>;<br/><a NAME="E4:66_1_1_1"/><i><font color="Green">E43</font></i>: 
1 <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="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/><b>then </b><i><font color="Green">E44</font></i>: <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,1</span>)</span> <a href="glib_001.html#K4" target="_self">.last()</a>  = 
<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
<b>by </b><i>, <a class="ref" href="jordan12.html#T3">JORDAN12:3</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> <font color="Maroon">m</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span></span>)</span> <a href="glib_001.html#K3" target="_self">.first()</a> 
<b>by </b><i>, </i>
;<br/><a NAME="E6:66_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> 1,<font color="Maroon">m</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,1</span>)</span></span>)</span> <a href="nat_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">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> <font color="Maroon">m</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="glib_001.html#E3:66_1_1_1"><i><font color="Green">E42</font></i></a></i>;<br/><a NAME="E7:66_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> 1,<font color="Maroon">m</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> <font color="Maroon">m</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="jordan12.html#T3">JORDAN12:3</a>, <a class="ref" href="glib_001.html#D2" target="_self">Def2</a>, </i>;<br/><a NAME="E8:66_1_1_1"/><b>then </b><i><font color="Green">E45</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> 1,<font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> <font color="Maroon">m</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span></span>)</span>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:66_1_1_1_2"/>
( 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">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> 1,<font color="Maroon">m</font></span>)</span> )
 ;<br/><a NAME="E2:66_1_1_1_2"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> 1,<font color="Maroon">m</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_1_1_1_2_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,1</span>)</span> or  ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> <font color="Maroon">m</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span></span>)</span> &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">k</font> ) )
 </span><b>by </b><i>, , </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:66_1_1_1_2_1_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,1</span>)</span>
 ;<br/><div class="add"><a NAME="E2:66_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E48</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">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,1</span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:66_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="jordan12.html#T3">JORDAN12:3</a>, <a class="ref" href="glib_001.html#D2" target="_self">Def2</a>, </i>;<br/><a NAME="E4:66_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> 1,<font color="Maroon">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,1</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
<b>by </b><i>, , <a class="txt" href="glib_001.html#E8:66_1_1_1"><i><font color="Green">E45</font></i></a></i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K5" target="_self">.vertexAt</a> 1</span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="ref" href="jordan12.html#T3">JORDAN12:3</a>, <a class="ref" href="glib_001.html#D2" target="_self">Def2</a>, , </i>
<br/>.= 
<font color="Maroon">W</font> <a href="glib_001.html#K5" target="_self">.vertexAt</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>
<br/>.= 
<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
<b>by </b><i>, <a class="ref" href="jordan12.html#T3">JORDAN12:3</a>, <a class="ref" href="glib_001.html#D2" target="_self">Def2</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> <font color="Maroon">m</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span></span>)</span> <a href="glib_001.html#K3" target="_self">.first()</a> 
<b>by </b><i>, <a class="txt" href="glib_001.html#E3:66_1_1_1_2_1_1_1"><i><font color="Green">E49</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> <font color="Maroon">m</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 1

;<br/><b>hence </b><a NAME="E6:66_1_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> 1,<font color="Maroon">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> <font color="Maroon">m</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D5" target="_self">Def5</a>, , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:66_1_1_1_2_1_1_2"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> <font color="Maroon">m</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span></span>)</span> &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">k</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:66_1_1_1_2_1_1_2"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> <font color="Maroon">m</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span></span>)</span> &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> )
 ;<br/><a NAME="E4:66_1_1_1_2_1_1_2"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="jordan12.html#T3">JORDAN12:3</a>, <a class="ref" href="glib_001.html#D2" target="_self">Def2</a>, </i>;<br/><b>hence </b><a NAME="E5:66_1_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> 1,<font color="Maroon">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> <font color="Maroon">m</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, , , <a class="txt" href="glib_001.html#E2:66_1_1_1_2"><i><font color="Green">E46</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:66_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> 1,<font color="Maroon">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> <font color="Maroon">m</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E10:66_1_1_1"/>
<font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> 1,<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> <font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></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#T18">FINSEQ_1:18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_1_2"><b>suppose </b></a><a NAME="E1:66_1_1_2"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">m</font> is <a href="abian.html#V1">even</a> or  not <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> )
 ;<br/><div class="add"><a NAME="E2:66_1_1_2"/><b>then </b>
<font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> <font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:66_1_1_2"/>
<font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> 1,<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> <font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:66"/>
<font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> 1,<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> <font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span>
 ;<br/>


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