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<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>;<br/><b>let </b><font color="Maroon" title="c2">W</font> be    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G</font>;<br/><b>let </b><font color="Maroon" title="c3">m</font>, <font color="Maroon" title="c4">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>







<b>set </b><font color="Maroon" title="c7">W2</font> = <font color="Maroon" title="c2">W</font> <a href="glib_001.html#K9" title="GLIB_001:func.9">.remove</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font>;<br/>
<b>set </b><font color="Maroon" title="c8">WA</font> = <font color="Maroon" title="c2">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c3">m</font>;<br/>
<b>set </b><font color="Maroon" title="c9">WB</font> = <font color="Maroon" title="c2">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c4">n</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W</font></span>)</span>;<br/>
<b>assume </b><a NAME="E1:59"/><i><font color="Green" title="E59">A1</font></i>: 
<font color="Maroon" title="c2">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_1_1"/>
( ( not <font color="Maroon" title="c3">m</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> &amp; not <font color="Maroon" title="c4">n</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> &amp; <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> &amp; <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W</font> &amp; <font color="Maroon" title="c2">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> ) or <font color="Maroon" title="c3">m</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> or <font color="Maroon" title="c4">n</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> or  not <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> or  not <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W</font> or  not <font color="Maroon" title="c2">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_1"><b>suppose </b></a><a NAME="E1:59_1_1_1"/><i><font color="Green" title="E60">A2</font></i>: 
( not <font color="Maroon" title="c3">m</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> &amp; not <font color="Maroon" title="c4">n</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> &amp; <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> &amp; <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W</font> &amp; <font color="Maroon" title="c2">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> )
 ;<br/><div class="add"><a NAME="E2:59_1_1_1"/><b>then </b><i><font color="Green" title="E61">A3</font></i>: 
( not 1 is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> &amp; <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W</font> )
 <b>by </b><i><a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a>, <a class="ref" href="jordan12.html#T3" title="JORDAN12:th.3">JORDAN12:3</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:59_1_1_1"/><i><font color="Green" title="E62">A4</font></i>: 
<font color="Maroon" title="c2">W</font> <a href="glib_001.html#K9" title="GLIB_001:func.9">.remove</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c3">m</font></span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c4">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E1:59_1_1_1"><i><font color="Green" title="E60">A2</font></i></a>, <a class="ref" href="glib_001.html#D12" target="_self" title="GLIB_001:def.12">Def12</a></i>;<br/><i><font color="Green" title="E63">A5</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c3">m</font></span>)</span> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  = 
<font color="Maroon" title="c2">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font>
<b>by </b><i><a class="txt" href="glib_001.html#E1:59_1_1_1"><i><font color="Green" title="E60">A2</font></i></a>, <a class="txt" href="glib_001.html#E2:59_1_1_1"><i><font color="Green" title="E61">A3</font></i></a>, <a class="txt" href="glib_001.html#E57"><i><font color="Green" title="E49">Lm20</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c4">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W</font></span>)</span></span>)</span> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> 
<b>by </b><i><a class="txt" href="glib_001.html#E1:59_1_1_1"><i><font color="Green" title="E60">A2</font></i></a>, <a class="txt" href="glib_001.html#E57"><i><font color="Green" title="E49">Lm20</font></i></a></i>
;<br/><i><font color="Green" title="E64">A6</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c3">m</font></span>)</span> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  = 
<font color="Maroon" title="c2">W</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> 
<b>by </b><i><a class="txt" href="glib_001.html#E1:59_1_1_1"><i><font color="Green" title="E60">A2</font></i></a>, <a class="txt" href="glib_001.html#E2:59_1_1_1"><i><font color="Green" title="E61">A3</font></i></a>, <a class="txt" href="glib_001.html#E57"><i><font color="Green" title="E49">Lm20</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">x</font>
<b>by </b><i><a class="txt" href="glib_001.html#E1:59"><i><font color="Green" title="E59">A1</font></i></a>, <a class="ref" href="glib_001.html#D23" target="_self" title="GLIB_001:def.23">Def23</a></i>
;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c4">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W</font></span>)</span></span>)</span> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  = 
<font color="Maroon" title="c2">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
<b>by </b><i><a class="txt" href="glib_001.html#E1:59_1_1_1"><i><font color="Green" title="E60">A2</font></i></a>, <a class="txt" href="glib_001.html#E57"><i><font color="Green" title="E49">Lm20</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c6">y</font>
<b>by </b><i><a class="txt" href="glib_001.html#E1:59"><i><font color="Green" title="E59">A1</font></i></a>, <a class="ref" href="glib_001.html#D23" target="_self" title="GLIB_001:def.23">Def23</a></i>
;<br/><b>hence </b><a NAME="E7:59_1_1_1"/>
<font color="Maroon" title="c2">W</font> <a href="glib_001.html#K9" title="GLIB_001:func.9">.remove</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E3:59_1_1_1"><i><font color="Green" title="E62">A4</font></i></a>, <a class="txt" href="glib_001.html#E4:59_1_1_1"><i><font color="Green" title="E63">A5</font></i></a>, <a class="txt" href="glib_001.html#E5:59_1_1_1"><i><font color="Green" title="E64">A6</font></i></a>, <a class="txt" href="glib_001.html#E52"><i><font color="Green" title="E44">Lm15</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_2"><b>suppose </b></a><a NAME="E1:59_1_1_2"/>
( <font color="Maroon" title="c3">m</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> or <font color="Maroon" title="c4">n</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> or  not <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> or  not <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W</font> or  not <font color="Maroon" title="c2">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:59_1_1_2"/>
<font color="Maroon" title="c2">W</font> <a href="glib_001.html#K9" title="GLIB_001:func.9">.remove</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E1:59"><i><font color="Green" title="E59">A1</font></i></a>, <a class="ref" href="glib_001.html#D12" target="_self" title="GLIB_001:def.12">Def12</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:59"/>
<font color="Maroon" title="c2">W</font> <a href="glib_001.html#K9" title="GLIB_001:func.9">.remove</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>
 ;<br/>


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