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





<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:77_1"/>
( ( not <font color="Maroon">n</font> is <a href="abian.html#V1">even</a> &amp; 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> <font color="Maroon">W</font> ) or <font color="Maroon">n</font> is <a href="abian.html#V1">even</a> or  not 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> or  not <font color="Maroon">n</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="77_1_1"><b>suppose </b></a><a NAME="E1:77_1_1"/><i><font color="Green">E39</font></i>: 
( not <font color="Maroon">n</font> is <a href="abian.html#V1">even</a> &amp; 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> <font color="Maroon">W</font> )
 ;<br/><div class="add"><b>set </b><font color="Maroon">f</font> = <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">n</font></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:77_1_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <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">n</font></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> 
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:77_1_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/><a NAME="E4:77_1_1_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">n</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">x</font> )
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E6:77_1_1_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</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">W</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span> )
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:77_1_1_1"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">y</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,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E8:77_1_1_1"/><i><font color="Green">E45</font></i>: 
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">x</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,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E9:77_1_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="glib_001.html#D8" target="_self">Def8</a>, , </i>;<br/><a NAME="E10:77_1_1_1"/><i><font color="Green">E47</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">x</font> &amp; 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">x</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,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E11:77_1_1_1"/><b>then </b>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D8" target="_self">Def8</a>, </i>;<br/><a NAME="E12:77_1_1_1"/><b>then </b>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">x</font> <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="glib_001.html#D8" target="_self">Def8</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E13:77_1_1_1"/><b>then </b>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D10" target="_self">Def10</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E14:77_1_1_1"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">x</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#K12" target="_self">.edgeSeq()</a> </span>)</span>
 <b>by </b><i/>;<br/><a NAME="E15:77_1_1_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</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#K12" target="_self">.edgeSeq()</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E16:77_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E17:77_1_1_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> 
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:77_1_1"/>
<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">n</font></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">W</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2"><b>suppose </b></a><a NAME="E1:77_1_2"/>
( <font color="Maroon">n</font> is <a href="abian.html#V1">even</a> or  not 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> or  not <font color="Maroon">n</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"><b>hence </b><a NAME="E2:77_1_2"/>
<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">n</font></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">W</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
