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<b>let </b><font color="Maroon" title="c1">G1</font>, <font color="Maroon" title="c2">G2</font> be   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>;<br/><b>let </b><font color="Maroon" title="c3">W1</font> be    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G1</font>;<br/><b>let </b><font color="Maroon" title="c4">W2</font> be    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c2">G2</font>;<br/><b>let </b><font color="Maroon" title="c5">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>









<b>assume </b><a NAME="E1:209"/><i><font color="Green" title="E139">A1</font></i>: 
<font color="Maroon" title="c3">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W2</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="209_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="209_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:209_1_1"/>
( <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">W1</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  or  not <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">W1</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="209_1_1_1"><b>suppose </b></a><a NAME="E1:209_1_1_1"/><i><font color="Green" title="E140">A2</font></i>: 
<font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">W1</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 ;<br/><div class="add"><a NAME="E2:209_1_1_1"/><b>then </b><i><font color="Green" title="E141">A3</font></i>: 
<font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <b>by </b><i><a class="txt" href="glib_001.html#E1:209"><i><font color="Green" title="E139">A1</font></i></a>, <a class="ref" href="glib_001.html#T77" target="_self" title="GLIB_001:th.77">Th77</a></i>;<br/><a NAME="E3:209_1_1_1"/>
( <font color="Maroon" title="c3">W1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <font color="Maroon" title="c5">v</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="c4">W2</font> &amp; <font color="Maroon" title="c4">W2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <font color="Maroon" title="c5">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v</font> &amp; ( for <font color="Olive" title="b1">n</font> being <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">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="c4">W2</font> &amp; <font color="Maroon" title="c4">W2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v</font> holds <br/><font color="Maroon" title="c3">W1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <font color="Maroon" title="c5">v</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> ) )
 <b>by </b><i><a class="txt" href="glib_001.html#E1:209"><i><font color="Green" title="E139">A1</font></i></a>, <a class="txt" href="glib_001.html#E1:209_1_1_1"><i><font color="Green" title="E140">A2</font></i></a>, <a class="ref" href="glib_001.html#D19" target="_self" title="GLIB_001:def.19">Def19</a></i>;<br/><b>hence </b><a NAME="E4:209_1_1_1"/>
<font color="Maroon" title="c3">W1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <font color="Maroon" title="c5">v</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E2:209_1_1_1"><i><font color="Green" title="E141">A3</font></i></a>, <a class="ref" href="glib_001.html#D19" target="_self" title="GLIB_001:def.19">Def19</a></i>;<br/><a NAME="E5:209_1_1_1"/>
( <font color="Maroon" title="c3">W1</font> <a href="glib_001.html#K18" title="GLIB_001:func.18">.rfind</a> <font color="Maroon" title="c5">v</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="c4">W2</font> &amp; <font color="Maroon" title="c4">W2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="glib_001.html#K18" title="GLIB_001:func.18">.rfind</a> <font color="Maroon" title="c5">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v</font> &amp; ( for <font color="Olive" title="b1">n</font> being <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">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="c4">W2</font> &amp; <font color="Maroon" title="c4">W2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v</font> holds <br/><font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">W1</font> <a href="glib_001.html#K18" title="GLIB_001:func.18">.rfind</a> <font color="Maroon" title="c5">v</font> ) )
 <b>by </b><i><a class="txt" href="glib_001.html#E1:209"><i><font color="Green" title="E139">A1</font></i></a>, <a class="txt" href="glib_001.html#E1:209_1_1_1"><i><font color="Green" title="E140">A2</font></i></a>, <a class="ref" href="glib_001.html#D21" target="_self" title="GLIB_001:def.21">Def21</a></i>;<br/><b>hence </b><a NAME="E6:209_1_1_1"/>
<font color="Maroon" title="c3">W1</font> <a href="glib_001.html#K18" title="GLIB_001:func.18">.rfind</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K18" title="GLIB_001:func.18">.rfind</a> <font color="Maroon" title="c5">v</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E2:209_1_1_1"><i><font color="Green" title="E141">A3</font></i></a>, <a class="ref" href="glib_001.html#D21" target="_self" title="GLIB_001:def.21">Def21</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="209_1_1_2"><b>suppose </b></a><a NAME="E1:209_1_1_2"/><i><font color="Green" title="E140">A4</font></i>: 
not <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">W1</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 ;<br/><div class="add"><a NAME="E2:209_1_1_2"/><b>then </b><i><font color="Green" title="E141">A5</font></i>: 
not <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <b>by </b><i><a class="txt" href="glib_001.html#E1:209"><i><font color="Green" title="E139">A1</font></i></a>, <a class="ref" href="glib_001.html#T77" target="_self" title="GLIB_001:th.77">Th77</a></i>;<br/><b>thus </b><font color="Maroon" title="c3">W1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <font color="Maroon" title="c5">v</font> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">W2</font>
<b>by </b><i><a class="txt" href="glib_001.html#E1:209"><i><font color="Green" title="E139">A1</font></i></a>, <a class="txt" href="glib_001.html#E1:209_1_1_2"><i><font color="Green" title="E140">A4</font></i></a>, <a class="ref" href="glib_001.html#D19" target="_self" title="GLIB_001:def.19">Def19</a></i>
<br/>.= 
<font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <font color="Maroon" title="c5">v</font>
<b>by </b><i><a class="txt" href="glib_001.html#E2:209_1_1_2"><i><font color="Green" title="E141">A5</font></i></a>, <a class="ref" href="glib_001.html#D19" target="_self" title="GLIB_001:def.19">Def19</a></i>
;<br/><b>thus </b><font color="Maroon" title="c3">W1</font> <a href="glib_001.html#K18" title="GLIB_001:func.18">.rfind</a> <font color="Maroon" title="c5">v</font> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">W2</font>
<b>by </b><i><a class="txt" href="glib_001.html#E1:209"><i><font color="Green" title="E139">A1</font></i></a>, <a class="txt" href="glib_001.html#E1:209_1_1_2"><i><font color="Green" title="E140">A4</font></i></a>, <a class="ref" href="glib_001.html#D21" target="_self" title="GLIB_001:def.21">Def21</a></i>
<br/>.= 
<font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K18" title="GLIB_001:func.18">.rfind</a> <font color="Maroon" title="c5">v</font>
<b>by </b><i><a class="txt" href="glib_001.html#E2:209_1_1_2"><i><font color="Green" title="E141">A5</font></i></a>, <a class="ref" href="glib_001.html#D21" target="_self" title="GLIB_001:def.21">Def21</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:209"/>
( <font color="Maroon" title="c3">W1</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K16" title="GLIB_001:func.16">.find</a> <font color="Maroon" title="c5">v</font> &amp; <font color="Maroon" title="c3">W1</font> <a href="glib_001.html#K18" title="GLIB_001:func.18">.rfind</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W2</font> <a href="glib_001.html#K18" title="GLIB_001:func.18">.rfind</a> <font color="Maroon" title="c5">v</font> )
 ;<br/>


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