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

<b>set </b><font color="Maroon">CS</font> =  <a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>;<br/>
<b>set </b><font color="Maroon">FAP</font> =  <a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">v</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  holds <br/> ex <font color="Olive">P</font> being <a href="glib_001.html#V6">vertex-distinct</a> <a href="glib_005.html#V7" target="_self">augmenting</a> <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">P</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">source</font>,<font color="Olive">v</font> &amp; <font color="Olive">P</font> <a href="glib_001.html#K13">.vertices()</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  &amp; ( for <font color="Olive">n</font> being <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Olive">P</font> holds <br/><font color="Olive">P</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">P</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ) );<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">v</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:48_1_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> 0</span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 ;<br/><a NAME="E2:48_1_1_1"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">source</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i/>;<br/><a NAME="E3:48_1_1_1"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">source</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>set </b><font color="Maroon">P</font> = <font color="Maroon">G</font> <a href="glib_001.html#K1">.walkOf</a> <font color="Maroon">source</font>;<br/><b>take </b><font color="Maroon">P</font> = <font color="Maroon">G</font> <a href="glib_001.html#K1">.walkOf</a> <font color="Maroon">source</font>;<br/><b>thus </b><a NAME="E4:48_1_1_1"/>
<font color="Maroon">P</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">source</font>,<font color="Maroon">v</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T14">GLIB_001:14</a></i>;<br/><a NAME="E5:48_1_1_1"/>
<font color="Maroon">P</font> <a href="glib_001.html#K13">.vertices()</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">source</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="glib_001.html#T91">GLIB_001:91</a></i>;<br/><b>hence </b><a NAME="E6:48_1_1_1"/>
<font color="Maroon">P</font> <a href="glib_001.html#K13">.vertices()</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> 0</span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i/>;<br/><b>let </b><font color="Maroon">n</font> be  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E7:48_1_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P</font>
 ;<br/><a NAME="E8:48_1_1_1"/><b>then </b><i><font color="Green">E51</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> <font color="Maroon">P</font> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E9:48_1_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E10:48_1_1_1"/>
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T14">GLIB_001:14</a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:48_1_1"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_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="E1:48_1_1_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/><b>set </b><font color="Maroon">Gn</font> = <span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font>;<br/><b>set </b><font color="Maroon">Gn1</font> = <span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>set </b><font color="Maroon">Next</font> =  <a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span>;<br/><b>set </b><font color="Maroon">e</font> =  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>;<br/><a NAME="E2:48_1_1_2"/><i><font color="Green">E92</font></i>: 
<span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_005.html#K4" target="_self">AP:Step</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:48_1_1_2"/><i><font color="Green">E93</font></i>: 
( <span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font> <a href="glib_000.html#R5">==</a> <font color="Maroon">G</font> &amp;  <a href="glib_003.html#K6">the_ELabel_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_003.html#K6">the_ELabel_of</a> <font color="Maroon">G</font> &amp;  <a href="glib_003.html#K5">the_Weight_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">G</font> )
 <b>by </b><i/>;<br/><a NAME="E4:48_1_1_2"/>
<font color="Maroon">source</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">source</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E5:48_1_1_2"/><b>then </b><i><font color="Green">E94</font></i>: 
<font color="Maroon">source</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> 0</span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i/>;<br/><a NAME="E6:48_1_1_2"/><i><font color="Green">E95</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> 0</span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_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="E1:48_1_1_2_1_1"/>
(  <a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or (  <a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; not <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  ) or (  <a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:48_1_1_2_1_1_1"/>
 <a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:48_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:48_1_1_2_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:48_1_1_2_1_1_2"/><i><font color="Green">E96</font></i>: 
(  <a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; not <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  )
 ;<br/><div class="add"><a NAME="E2:48_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E109</font></i>: 
<span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K13">.labelVertex</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E3:48_1_1_2_1_1_2"/><i><font color="Green">E110</font></i>: 
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E4:48_1_1_2_1_1_2"/>
(  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="glib_005.html#R3" target="_self">is_forward_labeling_in</a> <span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font> or  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="glib_005.html#R4" target="_self">is_backward_labeling_in</a> <span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E5:48_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E111</font></i>: 
( <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K6">the_ELabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i>, , </i>;<br/><a NAME="E6:48_1_1_2_1_1_2"/><i><font color="Green">E112</font></i>: 
<span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><a NAME="E7:48_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E128</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="glib_003.html#T59">GLIB_003:59</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">v</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:48_1_1_2_1_1_2_1"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_2_1_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_1_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:48_1_1_2_1_1_2_1_1_1"/>
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  or <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 </span><b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_1_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:48_1_1_2_1_1_2_1_1_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">P</font> being  <a href="glib_001.html#V6">vertex-distinct</a> <a href="glib_005.html#V7" target="_self">augmenting</a> <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E3:48_1_1_2_1_1_2_1_1_1_1"/><i><font color="Green">E130</font></i>: 
( <font color="Maroon">P</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">source</font>,<font color="Maroon">v</font> &amp; <font color="Maroon">P</font> <a href="glib_001.html#K13">.vertices()</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  &amp; ( for <font color="Olive">n</font> being <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P</font> holds <br/><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ) )
 <b>by </b><i/>;<br/><b>take </b><font color="Maroon">P</font> = <font color="Maroon">P</font>;<br/><a NAME="E4:48_1_1_2_1_1_2_1_1_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E5:48_1_1_2_1_1_2_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:48_1_1_2_1_1_2_1_1_1_1"/>
( <font color="Maroon">P</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">source</font>,<font color="Maroon">v</font> &amp; <font color="Maroon">P</font> <a href="glib_001.html#K13">.vertices()</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  &amp; ( for <font color="Olive">n</font> being <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P</font> holds <br/><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ) )
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_1_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:48_1_1_2_1_1_2_1_1_1_2"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:48_1_1_2_1_1_2_1_1_1_2"/><b>then </b><i><font color="Green">E132</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_2_1_1_2_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_1_1_2_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="E1:48_1_1_2_1_1_2_1_1_1_2_1_1"/>
( <font color="Maroon">source</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> or <font color="Maroon">source</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_1_1_2_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:48_1_1_2_1_1_2_1_1_1_2_1_1_1"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">source</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 ;<br/><div class="add"><b>set </b><font color="Maroon">W2</font> = <font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">source</font>,<span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span>;<br/><a NAME="E2:48_1_1_2_1_1_2_1_1_1_2_1_1_1"/>
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">source</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, , <a class="ref" href="glib_000.html#D15">GLIB_000:def 15</a></i>;<br/><a NAME="E3:48_1_1_2_1_1_2_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E134</font></i>: 
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">source</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#T91">GLIB_000:91</a></i>;<br/><a NAME="E4:48_1_1_2_1_1_2_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E135</font></i>: 
<font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">source</font>,<span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">source</font>,<span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K11">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="glib_001.html#D5">GLIB_001:def 5</a></i>;<br/><a NAME="E5:48_1_1_2_1_1_2_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E137</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">.walkOf</a> <font color="Maroon">source</font>,<span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></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">.walkOf</a> <font color="Maroon">source</font>,<span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">source</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">source</font>,<span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">source</font>,<span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_2_1_1_2_1_1_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:48_1_1_2_1_1_2_1_1_1_2_1_1_1_1"/>
<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">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">source</font>,<span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 ;<br/><a NAME="E2:48_1_1_2_1_1_2_1_1_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/><a NAME="E3:48_1_1_2_1_1_2_1_1_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:48_1_1_2_1_1_2_1_1_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E5:48_1_1_2_1_1_2_1_1_1_2_1_1_1_1"/><b>then </b>
( 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> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/><a NAME="E6:48_1_1_2_1_1_2_1_1_1_2_1_1_1_1"/><b>then </b><i><font color="Green">E138</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/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:48_1_1_2_1_1_2_1_1_1_2_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">source</font>,<span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R2">DJoins</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">source</font>,<span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">source</font>,<span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">G</font>
 ;<br/><a NAME="E2:48_1_1_2_1_1_2_1_1_1_2_1_1_1_1_1"/><b>then </b>
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="glib_000.html#R2">DJoins</a> <font color="Maroon">source</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="glib_005.html#D1" target="_self">Def1</a>, , <a class="ref" href="glib_000.html#T91">GLIB_000:91</a></i>;<br/><b>hence </b><a NAME="E3:48_1_1_2_1_1_2_1_1_1_2_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="glib_003.html#K6">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">source</font>,<span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p5">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">source</font>,<span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p5">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="glib_000.html#D16">GLIB_000:def 16</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E8:48_1_1_2_1_1_2_1_1_1_2_1_1_1_1"/>
not <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">source</font>,<span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R2">DJoins</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">source</font>,<span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">source</font>,<span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">G</font>
 ;<br/><b>thus </b><a NAME="E9:48_1_1_2_1_1_2_1_1_1_2_1_1_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K6">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">source</font>,<span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p5">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a>, <a class="ref" href="glib_005.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">W2</font> = <font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">source</font>,<span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> as  <a href="glib_005.html#V7" target="_self">augmenting</a> <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font> <b>by </b><i/>;<br/><div><i><font color="Green">E139</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_2_1_1_2_1_1_1_2_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:48_1_1_2_1_1_2_1_1_1_2_1_1_1_2"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W2</font>
 ;<br/><a NAME="E2:48_1_1_2_1_1_2_1_1_1_2_1_1_1_2"/><i><font color="Green">E141</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_2_1_1_2_1_1_1_2_1_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_1_1_2_1_1_2_1_1_1_2_1_1_1_2_1"/>
<font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><a NAME="E2:48_1_1_2_1_1_2_1_1_1_2_1_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:48_1_1_2_1_1_2_1_1_1_2_1_1_1_2_1"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="glib_001.html#T1">GLIB_001:1</a></i>;<br/><b>hence </b><a NAME="E4:48_1_1_2_1_1_2_1_1_1_2_1_1_1_2_1"/>
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 3
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:48_1_1_2_1_1_2_1_1_1_2_1_1_1_2"/>
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 3 )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_2_1_1_2_1_1_1_2_1_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">n</font> be  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:48_1_1_2_1_1_2_1_1_1_2_1_1_1_3"/><i><font color="Green">E142</font></i>: 
( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W2</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">W2</font> &amp; <font color="Maroon">W2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 ;<br/><a NAME="E2:48_1_1_2_1_1_2_1_1_1_2_1_1_1_3"/><b>then </b>
( ( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 3 ) &amp; ( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 3 ) )
 <b>by </b><i><a class="ref" href="glib_005.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E3:48_1_1_2_1_1_2_1_1_1_2_1_1_1_3"/>
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i>, , , , </i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">W2</font> = <font color="Maroon">W2</font> as  <a href="glib_001.html#V6">vertex-distinct</a> <a href="glib_005.html#V7" target="_self">augmenting</a> <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="glib_001.html#D29">GLIB_001:def 29</a></i>;<br/><b>take </b><font color="Maroon">W2</font> = <font color="Maroon">W2</font>;<br/><b>thus </b><a NAME="E11:48_1_1_2_1_1_2_1_1_1_2_1_1_1"/>
<font color="Maroon">W2</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">source</font>,<font color="Maroon">v</font>
 <b>by </b><i>, , <a class="ref" href="glib_001.html#T16">GLIB_001:16</a></i>;<br/><a NAME="E12:48_1_1_2_1_1_2_1_1_1_2_1_1_1"/><i><font color="Green">E143</font></i>: 
<font color="Maroon">W2</font> <a href="glib_001.html#K13">.vertices()</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">source</font>,<font color="Maroon">v</font></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i>, , <a class="ref" href="glib_001.html#T92">GLIB_001:92</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_2_1_1_2_1_1_1_2_1_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:48_1_1_2_1_1_2_1_1_1_2_1_1_1_4"/><i><font color="Green">E144</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font> <a href="glib_001.html#K13">.vertices()</a> 
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_2_1_1_2_1_1_1_2_1_1_1_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_1_1_2_1_1_1_2_1_1_1_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:48_1_1_2_1_1_2_1_1_1_2_1_1_1_4_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">source</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 </span><b>by </b><i>, , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_1_1_2_1_1_1_2_1_1_1_4_1_1_1"><b>suppose </b></a><a NAME="E1:48_1_1_2_1_1_2_1_1_1_2_1_1_1_4_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">source</font>
 ;<br/><div class="add"><a NAME="E2:48_1_1_2_1_1_2_1_1_1_2_1_1_1_4_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">source</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:48_1_1_2_1_1_2_1_1_1_2_1_1_1_4_1_1_1"/><b>then </b><i><font color="Green">E145</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> 0</span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i/>;<br/><a NAME="E4:48_1_1_2_1_1_2_1_1_1_2_1_1_1_4_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> 0</span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:48_1_1_2_1_1_2_1_1_1_2_1_1_1_4_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_1_1_2_1_1_1_2_1_1_1_4_1_1_2"><b>suppose </b></a><a NAME="E1:48_1_1_2_1_1_2_1_1_1_2_1_1_1_4_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:48_1_1_2_1_1_2_1_1_1_2_1_1_1_4_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:48_1_1_2_1_1_2_1_1_1_2_1_1_1_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E14:48_1_1_2_1_1_2_1_1_1_2_1_1_1"/>
<font color="Maroon">W2</font> <a href="glib_001.html#K13">.vertices()</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>let </b><font color="Maroon">n</font> be  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E15:48_1_1_2_1_1_2_1_1_1_2_1_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">W2</font>
 ;<br/><a NAME="E16:48_1_1_2_1_1_2_1_1_1_2_1_1_1"/><b>then </b>
( 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> 3 )
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E17:48_1_1_2_1_1_2_1_1_1_2_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 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> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E18:48_1_1_2_1_1_2_1_1_1_2_1_1_1"/><b>then </b>
( 1 <a href="nat_1.html#K1">+</a> 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> 2 <a href="nat_1.html#K2">*</a> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E19:48_1_1_2_1_1_2_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 2
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E20:48_1_1_2_1_1_2_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E146</font></i>: 
( <font color="Maroon">W2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> &amp; <font color="Maroon">W2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/><a NAME="E21:48_1_1_2_1_1_2_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">W2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E22:48_1_1_2_1_1_2_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E147</font></i>: 
<font color="Maroon">W2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E23:48_1_1_2_1_1_2_1_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">W2</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="glib_003.html#T46">GLIB_003:46</a></i>;<br/><b>hence </b><a NAME="E24:48_1_1_2_1_1_2_1_1_1_2_1_1_1"/>
<font color="Maroon">W2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">W2</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_1_1_2_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:48_1_1_2_1_1_2_1_1_1_2_1_1_2"/>
<font color="Maroon">source</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">W</font> being  <a href="glib_001.html#V6">vertex-distinct</a> <a href="glib_005.html#V7" target="_self">augmenting</a> <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E3:48_1_1_2_1_1_2_1_1_1_2_1_1_2"/><i><font color="Green">E148</font></i>: 
( <font color="Maroon">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">source</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">W</font> <a href="glib_001.html#K13">.vertices()</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  &amp; ( for <font color="Olive">n</font> being <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">W</font> holds <br/><font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ) )
 <b>by </b><i>, </i>;<br/><b>set </b><font color="Maroon">W2</font> = <font color="Maroon">W</font> <a href="glib_001.html#K10">.addEdge</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>;<br/><a NAME="E4:48_1_1_2_1_1_2_1_1_1_2_1_1_2"/><i><font color="Green">E149</font></i>: 
( <font color="Maroon">W</font> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">source</font> &amp; <font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="glib_001.html#D23">GLIB_001:def 23</a></i>;<br/><a NAME="E5:48_1_1_2_1_1_2_1_1_1_2_1_1_2"/><b>then </b>
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a> ,<span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#D15">GLIB_000:def 15</a></i>;<br/><a NAME="E6:48_1_1_2_1_1_2_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E150</font></i>: 
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a> ,<span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#T91">GLIB_000:91</a></i>;<br/><a NAME="E7:48_1_1_2_1_1_2_1_1_1_2_1_1_2"/><i><font color="Green">E151</font></i>: 
not <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="glib_001.html#K13">.vertices()</a> 
 <b>by </b><i>, </i>;<br/><b>then reconsider </b><font color="Maroon">W2</font> = <font color="Maroon">W</font> <a href="glib_001.html#K10">.addEdge</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> as  <a href="glib_001.html#V6">vertex-distinct</a>  <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G</font> <b>by </b><i>, <a class="ref" href="glib_001.html#T156">GLIB_001:156</a></i>;<br/><a NAME="E9:48_1_1_2_1_1_2_1_1_1_2_1_1_2"/>
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="glib_000.html#R2">DJoins</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a> ,<span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, , <a class="ref" href="glib_000.html#D16">GLIB_000:def 16</a></i>;<br/><a NAME="E10:48_1_1_2_1_1_2_1_1_1_2_1_1_2"/><b>then </b>
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="glib_000.html#R2">DJoins</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a> ,<font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#T91">GLIB_000:91</a></i>;<br/><b>then reconsider </b><font color="Maroon">W2</font> = <font color="Maroon">W2</font> as  <a href="glib_001.html#V6">vertex-distinct</a> <a href="glib_005.html#V7" target="_self">augmenting</a> <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font> <b>by </b><i>, , , </i>;<br/><b>take </b><font color="Maroon">W2</font> = <font color="Maroon">W2</font>;<br/><b>thus </b><a NAME="E12:48_1_1_2_1_1_2_1_1_1_2_1_1_2"/>
<font color="Maroon">W2</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">source</font>,<font color="Maroon">v</font>
 <b>by </b><i>, , , <a class="ref" href="glib_001.html#T64">GLIB_001:64</a></i>;<br/><a NAME="E13:48_1_1_2_1_1_2_1_1_1_2_1_1_2"/><i><font color="Green">E152</font></i>: 
<font color="Maroon">W2</font> <a href="glib_001.html#K13">.vertices()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K13">.vertices()</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, , <a class="ref" href="glib_001.html#T96">GLIB_001:96</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_2_1_1_2_1_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:48_1_1_2_1_1_2_1_1_1_2_1_1_2_1"/><i><font color="Green">E153</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font> <a href="glib_001.html#K13">.vertices()</a> 
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_2_1_1_2_1_1_1_2_1_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_1_1_2_1_1_1_2_1_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:48_1_1_2_1_1_2_1_1_1_2_1_1_2_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="glib_001.html#K13">.vertices()</a>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span> )
 </span><b>by </b><i><a class="ref" href="glib_005.html#D1" target="_self">Def1</a>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_1_1_2_1_1_1_2_1_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:48_1_1_2_1_1_2_1_1_1_2_1_1_2_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="glib_001.html#K13">.vertices()</a> 
 ;<br/><div class="add"><a NAME="E2:48_1_1_2_1_1_2_1_1_1_2_1_1_2_1_1_1_1"/><b>then </b><i><font color="Green">E154</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i/>;<br/><a NAME="E3:48_1_1_2_1_1_2_1_1_1_2_1_1_2_1_1_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E4:48_1_1_2_1_1_2_1_1_1_2_1_1_2_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:48_1_1_2_1_1_2_1_1_1_2_1_1_2_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_1_1_2_1_1_1_2_1_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:48_1_1_2_1_1_2_1_1_1_2_1_1_2_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:48_1_1_2_1_1_2_1_1_1_2_1_1_2_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:48_1_1_2_1_1_2_1_1_1_2_1_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E15:48_1_1_2_1_1_2_1_1_1_2_1_1_2"/>
<font color="Maroon">W2</font> <a href="glib_001.html#K13">.vertices()</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>let </b><font color="Maroon">n</font> be  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E16:48_1_1_2_1_1_2_1_1_1_2_1_1_2"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">W2</font>
 ;<br/><a NAME="E17:48_1_1_2_1_1_2_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E155</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> <font color="Maroon">W2</font> )
 <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="48_1_1_2_1_1_2_1_1_1_2_1_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_1_1_2_1_1_1_2_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:48_1_1_2_1_1_2_1_1_1_2_1_1_2_2_1"/>
( <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> <a href="xxreal_0.html#NR4" target="_self">&gt;</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="48_1_1_2_1_1_2_1_1_1_2_1_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:48_1_1_2_1_1_2_1_1_1_2_1_1_2_2_1_1"/><i><font color="Green">E156</font></i>: 
<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"><a NAME="E2:48_1_1_2_1_1_2_1_1_1_2_1_1_2_2_1_1"/><b>then </b><i><font color="Green">E157</font></i>: 
<font color="Maroon">n</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="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:48_1_1_2_1_1_2_1_1_1_2_1_1_2_2_1_1"/><b>then </b><i><font color="Green">E158</font></i>: 
<font color="Maroon">W2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T66">GLIB_001:66</a></i>;<br/><a NAME="E4:48_1_1_2_1_1_2_1_1_1_2_1_1_2_2_1_1"/>
<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>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E5:48_1_1_2_1_1_2_1_1_1_2_1_1_2_2_1_1"/><b>then </b><i><font color="Green">E159</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 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="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E6:48_1_1_2_1_1_2_1_1_1_2_1_1_2_2_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E7:48_1_1_2_1_1_2_1_1_1_2_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <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_005.html#D2" target="_self">Def2</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E8:48_1_1_2_1_1_2_1_1_1_2_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon">W2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T66">GLIB_001:66</a></i>;<br/><b>hence </b><a NAME="E9:48_1_1_2_1_1_2_1_1_1_2_1_1_2_2_1_1"/>
<font color="Maroon">W2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">W2</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_1_1_2_1_1_1_2_1_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:48_1_1_2_1_1_2_1_1_1_2_1_1_2_2_1_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font>
 ;<br/><div class="add"><a NAME="E2:48_1_1_2_1_1_2_1_1_1_2_1_1_2_2_1_2"/><b>then </b><i><font color="Green">E160</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:48_1_1_2_1_1_2_1_1_1_2_1_1_2_2_1_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="glib_001.html#T65">GLIB_001:65</a></i>;<br/><a NAME="E4:48_1_1_2_1_1_2_1_1_1_2_1_1_2_2_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E5:48_1_1_2_1_1_2_1_1_1_2_1_1_2_2_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E6:48_1_1_2_1_1_2_1_1_1_2_1_1_2_2_1_2"/><b>then </b><i><font color="Green">E161</font></i>: 
<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> <font color="Maroon">W</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E7:48_1_1_2_1_1_2_1_1_1_2_1_1_2_2_1_2"/><b>then </b>
<font color="Maroon">n</font> <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> <font color="Maroon">W</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><a NAME="E8:48_1_1_2_1_1_2_1_1_1_2_1_1_2_2_1_2"/><b>then </b><i><font color="Green">E162</font></i>: 
( <font color="Maroon">W2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> &amp; <font color="Maroon">W2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 <b>by </b><i>, , , <a class="ref" href="glib_001.html#T66">GLIB_001:66</a></i>;<br/><a NAME="E9:48_1_1_2_1_1_2_1_1_1_2_1_1_2_2_1_2"/><i><font color="Green">E163</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E10:48_1_1_2_1_1_2_1_1_1_2_1_1_2_2_1_2"/>
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="glib_003.html#T46">GLIB_003:46</a></i>;<br/><b>hence </b><a NAME="E11:48_1_1_2_1_1_2_1_1_1_2_1_1_2_2_1_2"/>
<font color="Maroon">W2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">W2</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="glib_005.html#D2" target="_self">Def2</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E19:48_1_1_2_1_1_2_1_1_1_2_1_1_2"/>
<font color="Maroon">W2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">W2</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:48_1_1_2_1_1_2_1_1_1_2"/>
 ex <font color="Olive">P</font> being <a href="glib_001.html#V6">vertex-distinct</a> <a href="glib_005.html#V7" target="_self">augmenting</a> <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">P</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">source</font>,<font color="Maroon">v</font> &amp; <font color="Olive">P</font> <a href="glib_001.html#K13">.vertices()</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  &amp; ( for <font color="Olive">n</font> being <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Olive">P</font> holds <br/><font color="Olive">P</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">P</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ) )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:48_1_1_2_1_1_2_1"/>
 ex <font color="Olive">P</font> being <a href="glib_001.html#V6">vertex-distinct</a> <a href="glib_005.html#V7" target="_self">augmenting</a> <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">P</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">source</font>,<font color="Maroon">v</font> &amp; <font color="Olive">P</font> <a href="glib_001.html#K13">.vertices()</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  &amp; ( for <font color="Olive">n</font> being <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Olive">P</font> holds <br/><font color="Olive">P</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">P</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ) )
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E9:48_1_1_2_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:48_1_1_2_1_1_3"/><i><font color="Green">E164</font></i>: 
(  <a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  )
 ;<br/><div class="add"><a NAME="E2:48_1_1_2_1_1_3"/><b>then </b><i><font color="Green">E165</font></i>: 
<span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K13">.labelVertex</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E3:48_1_1_2_1_1_3"/><i><font color="Green">E166</font></i>: 
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E4:48_1_1_2_1_1_3"/><i><font color="Green">E167</font></i>: 
(  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="glib_005.html#R3" target="_self">is_forward_labeling_in</a> <span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font> or  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="glib_005.html#R4" target="_self">is_backward_labeling_in</a> <span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E5:48_1_1_2_1_1_3"/><b>then </b><i><font color="Green">E168</font></i>: 
( not <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  &amp; <span class="p1">(<span class="default"><a href="glib_003.html#K6">the_ELabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i>, , </i>;<br/><a NAME="E6:48_1_1_2_1_1_3"/><i><font color="Green">E169</font></i>: 
<span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><a NAME="E7:48_1_1_2_1_1_3"/><b>then </b><i><font color="Green">E170</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="glib_003.html#T59">GLIB_003:59</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_2_1_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">v</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:48_1_1_2_1_1_3_1"/><i><font color="Green">E171</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_2_1_1_3_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_1_1_3_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:48_1_1_2_1_1_3_1_1_1"/>
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  or <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 </span><b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_1_1_3_1_1_1_1"><b>suppose </b></a><a NAME="E1:48_1_1_2_1_1_3_1_1_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">P</font> being  <a href="glib_001.html#V6">vertex-distinct</a> <a href="glib_005.html#V7" target="_self">augmenting</a> <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E3:48_1_1_2_1_1_3_1_1_1_1"/><i><font color="Green">E172</font></i>: 
( <font color="Maroon">P</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">source</font>,<font color="Maroon">v</font> &amp; <font color="Maroon">P</font> <a href="glib_001.html#K13">.vertices()</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  &amp; ( for <font color="Olive">n</font> being <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P</font> holds <br/><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ) )
 <b>by </b><i/>;<br/><b>take </b><font color="Maroon">P</font> = <font color="Maroon">P</font>;<br/><a NAME="E4:48_1_1_2_1_1_3_1_1_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E5:48_1_1_2_1_1_3_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:48_1_1_2_1_1_3_1_1_1_1"/>
( <font color="Maroon">P</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">source</font>,<font color="Maroon">v</font> &amp; <font color="Maroon">P</font> <a href="glib_001.html#K13">.vertices()</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  &amp; ( for <font color="Olive">n</font> being <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P</font> holds <br/><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ) )
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_1_1_3_1_1_1_2"><b>suppose </b></a><a NAME="E1:48_1_1_2_1_1_3_1_1_1_2"/><i><font color="Green">E173</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:48_1_1_2_1_1_3_1_1_1_2"/><b>then </b><i><font color="Green">E174</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_2_1_1_3_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_1_1_3_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="E1:48_1_1_2_1_1_3_1_1_1_2_1_1"/>
( <font color="Maroon">source</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> or <font color="Maroon">source</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_1_1_3_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:48_1_1_2_1_1_3_1_1_1_2_1_1_1"/><i><font color="Green">E175</font></i>: 
<font color="Maroon">source</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 ;<br/><div class="add"><b>set </b><font color="Maroon">W2</font> = <font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">source</font>,<span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span>;<br/><a NAME="E2:48_1_1_2_1_1_3_1_1_1_2_1_1_1"/>
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">source</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, , <a class="ref" href="glib_000.html#D15">GLIB_000:def 15</a></i>;<br/><a NAME="E3:48_1_1_2_1_1_3_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E176</font></i>: 
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">source</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#T91">GLIB_000:91</a></i>;<br/><a NAME="E4:48_1_1_2_1_1_3_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E177</font></i>: 
<font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">source</font>,<span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">source</font>,<span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K11">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="glib_001.html#D5">GLIB_001:def 5</a></i>;<br/><a NAME="E5:48_1_1_2_1_1_3_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E178</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">.walkOf</a> <font color="Maroon">source</font>,<span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></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">.walkOf</a> <font color="Maroon">source</font>,<span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">source</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">source</font>,<span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">source</font>,<span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/><a NAME="E6:48_1_1_2_1_1_3_1_1_1_2_1_1_1"/><b>then </b>
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="glib_000.html#R2">DJoins</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">source</font>,<span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 1,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">source</font>,<span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, , <a class="ref" href="glib_000.html#D16">GLIB_000:def 16</a></i>;<br/><a NAME="E7:48_1_1_2_1_1_3_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E179</font></i>: 
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="glib_000.html#R2">DJoins</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">source</font>,<span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 1,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">source</font>,<span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#T91">GLIB_000:91</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_2_1_1_3_1_1_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:48_1_1_2_1_1_3_1_1_1_2_1_1_1_1"/>
<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">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">source</font>,<span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 ;<br/><a NAME="E2:48_1_1_2_1_1_3_1_1_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/><a NAME="E3:48_1_1_2_1_1_3_1_1_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:48_1_1_2_1_1_3_1_1_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E5:48_1_1_2_1_1_3_1_1_1_2_1_1_1_1"/><b>then </b>
( 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> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/><a NAME="E6:48_1_1_2_1_1_3_1_1_1_2_1_1_1_1"/><b>then </b><i><font color="Green">E180</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/><b>thus </b><a NAME="E7:48_1_1_2_1_1_3_1_1_1_2_1_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">source</font>,<span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R2">DJoins</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">source</font>,<span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">source</font>,<span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">G</font> implies <span class="p1">(<span class="default"><a href="glib_003.html#K6">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">source</font>,<span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p5">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">source</font>,<span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p5">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i>, , , , , , <a class="ref" href="glib_005.html#D1" target="_self">Def1</a></i>;<br/><b>assume </b><a NAME="E8:48_1_1_2_1_1_3_1_1_1_2_1_1_1_1"/>
not <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">source</font>,<span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R2">DJoins</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">source</font>,<span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">source</font>,<span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">G</font>
 ;<br/><b>hence </b><a NAME="E9:48_1_1_2_1_1_3_1_1_1_2_1_1_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K6">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">source</font>,<span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p5">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a>, , </i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">W2</font> = <font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">source</font>,<span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> as  <a href="glib_005.html#V7" target="_self">augmenting</a> <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font> <b>by </b><i/>;<br/><div><i><font color="Green">E181</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_2_1_1_3_1_1_1_2_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:48_1_1_2_1_1_3_1_1_1_2_1_1_1_2"/><i><font color="Green">E182</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W2</font>
 ;<br/><a NAME="E2:48_1_1_2_1_1_3_1_1_1_2_1_1_1_2"/><i><font color="Green">E183</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_2_1_1_3_1_1_1_2_1_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_1_1_2_1_1_3_1_1_1_2_1_1_1_2_1"/>
<font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><a NAME="E2:48_1_1_2_1_1_3_1_1_1_2_1_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:48_1_1_2_1_1_3_1_1_1_2_1_1_1_2_1"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="glib_001.html#T1">GLIB_001:1</a></i>;<br/><b>hence </b><a NAME="E4:48_1_1_2_1_1_3_1_1_1_2_1_1_1_2_1"/>
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 3
 <b>by </b><i><a class="ref" href="glib_005.html#D1" target="_self">Def1</a>, , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:48_1_1_2_1_1_3_1_1_1_2_1_1_1_2"/>
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 3 )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_2_1_1_3_1_1_1_2_1_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">n</font> be  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:48_1_1_2_1_1_3_1_1_1_2_1_1_1_3"/><i><font color="Green">E184</font></i>: 
( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W2</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">W2</font> &amp; <font color="Maroon">W2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 ;<br/><a NAME="E2:48_1_1_2_1_1_3_1_1_1_2_1_1_1_3"/><b>then </b>
( ( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 3 ) &amp; ( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 3 ) )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:48_1_1_2_1_1_3_1_1_1_2_1_1_1_3"/>
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="glib_005.html#D1" target="_self">Def1</a>, , , , , , </i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">W2</font> = <font color="Maroon">W2</font> as  <a href="glib_001.html#V6">vertex-distinct</a> <a href="glib_005.html#V7" target="_self">augmenting</a> <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="glib_001.html#D29">GLIB_001:def 29</a></i>;<br/><b>take </b><font color="Maroon">W2</font> = <font color="Maroon">W2</font>;<br/><b>thus </b><a NAME="E13:48_1_1_2_1_1_3_1_1_1_2_1_1_1"/>
<font color="Maroon">W2</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">source</font>,<font color="Maroon">v</font>
 <b>by </b><i>, , <a class="ref" href="glib_001.html#T16">GLIB_001:16</a></i>;<br/><a NAME="E14:48_1_1_2_1_1_3_1_1_1_2_1_1_1"/><i><font color="Green">E185</font></i>: 
<font color="Maroon">W2</font> <a href="glib_001.html#K13">.vertices()</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">source</font>,<font color="Maroon">v</font></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i>, , <a class="ref" href="glib_001.html#T92">GLIB_001:92</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_2_1_1_3_1_1_1_2_1_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:48_1_1_2_1_1_3_1_1_1_2_1_1_1_4"/><i><font color="Green">E186</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font> <a href="glib_001.html#K13">.vertices()</a> 
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_2_1_1_3_1_1_1_2_1_1_1_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_1_1_3_1_1_1_2_1_1_1_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:48_1_1_2_1_1_3_1_1_1_2_1_1_1_4_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">source</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 </span><b>by </b><i>, , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_1_1_3_1_1_1_2_1_1_1_4_1_1_1"><b>suppose </b></a><a NAME="E1:48_1_1_2_1_1_3_1_1_1_2_1_1_1_4_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">source</font>
 ;<br/><div class="add"><a NAME="E2:48_1_1_2_1_1_3_1_1_1_2_1_1_1_4_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">source</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:48_1_1_2_1_1_3_1_1_1_2_1_1_1_4_1_1_1"/><b>then </b><i><font color="Green">E187</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> 0</span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i/>;<br/><a NAME="E4:48_1_1_2_1_1_3_1_1_1_2_1_1_1_4_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> 0</span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:48_1_1_2_1_1_3_1_1_1_2_1_1_1_4_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i><a class="ref" href="glib_005.html#D2" target="_self">Def2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_1_1_3_1_1_1_2_1_1_1_4_1_1_2"><b>suppose </b></a><a NAME="E1:48_1_1_2_1_1_3_1_1_1_2_1_1_1_4_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:48_1_1_2_1_1_3_1_1_1_2_1_1_1_4_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:48_1_1_2_1_1_3_1_1_1_2_1_1_1_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E16:48_1_1_2_1_1_3_1_1_1_2_1_1_1"/>
<font color="Maroon">W2</font> <a href="glib_001.html#K13">.vertices()</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>let </b><font color="Maroon">n</font> be  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E17:48_1_1_2_1_1_3_1_1_1_2_1_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">W2</font>
 ;<br/><a NAME="E18:48_1_1_2_1_1_3_1_1_1_2_1_1_1"/><b>then </b>
( 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> 3 )
 <b>by </b><i><a class="ref" href="glib_005.html#D1" target="_self">Def1</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E19:48_1_1_2_1_1_3_1_1_1_2_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 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> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E20:48_1_1_2_1_1_3_1_1_1_2_1_1_1"/><b>then </b>
( 1 <a href="nat_1.html#K1">+</a> 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> 2 <a href="nat_1.html#K2">*</a> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E21:48_1_1_2_1_1_3_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 2
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E22:48_1_1_2_1_1_3_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E188</font></i>: 
( <font color="Maroon">W2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> &amp; <font color="Maroon">W2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/><a NAME="E23:48_1_1_2_1_1_3_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">W2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E24:48_1_1_2_1_1_3_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E189</font></i>: 
<font color="Maroon">W2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E25:48_1_1_2_1_1_3_1_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">W2</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="glib_003.html#T46">GLIB_003:46</a></i>;<br/><b>hence </b><a NAME="E26:48_1_1_2_1_1_3_1_1_1_2_1_1_1"/>
<font color="Maroon">W2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">W2</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="glib_005.html#D2" target="_self">Def2</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_1_1_3_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:48_1_1_2_1_1_3_1_1_1_2_1_1_2"/>
<font color="Maroon">source</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">W</font> being  <a href="glib_001.html#V6">vertex-distinct</a> <a href="glib_005.html#V7" target="_self">augmenting</a> <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E3:48_1_1_2_1_1_3_1_1_1_2_1_1_2"/><i><font color="Green">E190</font></i>: 
( <font color="Maroon">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">source</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">W</font> <a href="glib_001.html#K13">.vertices()</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  &amp; ( for <font color="Olive">n</font> being <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">W</font> holds <br/><font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ) )
 <b>by </b><i>, </i>;<br/><b>set </b><font color="Maroon">W2</font> = <font color="Maroon">W</font> <a href="glib_001.html#K10">.addEdge</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>;<br/><a NAME="E4:48_1_1_2_1_1_3_1_1_1_2_1_1_2"/><i><font color="Green">E191</font></i>: 
( <font color="Maroon">W</font> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">source</font> &amp; <font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="glib_001.html#D23">GLIB_001:def 23</a></i>;<br/><a NAME="E5:48_1_1_2_1_1_3_1_1_1_2_1_1_2"/><b>then </b>
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a> ,<span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#D15">GLIB_000:def 15</a></i>;<br/><a NAME="E6:48_1_1_2_1_1_3_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E192</font></i>: 
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a> ,<span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#T91">GLIB_000:91</a></i>;<br/><a NAME="E7:48_1_1_2_1_1_3_1_1_1_2_1_1_2"/><i><font color="Green">E193</font></i>: 
not <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="glib_001.html#K13">.vertices()</a> 
 <b>by </b><i>, , , , </i>;<br/><b>then reconsider </b><font color="Maroon">W2</font> = <font color="Maroon">W</font> <a href="glib_001.html#K10">.addEdge</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> as  <a href="glib_001.html#V6">vertex-distinct</a>  <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G</font> <b>by </b><i>, <a class="ref" href="glib_001.html#T156">GLIB_001:156</a></i>;<br/><a NAME="E9:48_1_1_2_1_1_3_1_1_1_2_1_1_2"/>
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="glib_000.html#R2">DJoins</a> <font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a> ,<span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, , <a class="ref" href="glib_000.html#D16">GLIB_000:def 16</a></i>;<br/><a NAME="E10:48_1_1_2_1_1_3_1_1_1_2_1_1_2"/><b>then </b>
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="glib_000.html#R2">DJoins</a> <font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a> ,<span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#T91">GLIB_000:91</a></i>;<br/><b>then reconsider </b><font color="Maroon">W2</font> = <font color="Maroon">W2</font> as  <a href="glib_001.html#V6">vertex-distinct</a> <a href="glib_005.html#V7" target="_self">augmenting</a> <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font> <b>by </b><i>, , <a class="ref" href="glib_005.html#D1" target="_self">Def1</a>, </i>;<br/><b>take </b><font color="Maroon">W2</font> = <font color="Maroon">W2</font>;<br/><b>thus </b><a NAME="E12:48_1_1_2_1_1_3_1_1_1_2_1_1_2"/>
<font color="Maroon">W2</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">source</font>,<font color="Maroon">v</font>
 <b>by </b><i>, , , <a class="ref" href="glib_001.html#T64">GLIB_001:64</a></i>;<br/><a NAME="E13:48_1_1_2_1_1_3_1_1_1_2_1_1_2"/><i><font color="Green">E194</font></i>: 
<font color="Maroon">W2</font> <a href="glib_001.html#K13">.vertices()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K13">.vertices()</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, , <a class="ref" href="glib_001.html#T96">GLIB_001:96</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_2_1_1_3_1_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:48_1_1_2_1_1_3_1_1_1_2_1_1_2_1"/><i><font color="Green">E195</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font> <a href="glib_001.html#K13">.vertices()</a> 
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_2_1_1_3_1_1_1_2_1_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_1_1_3_1_1_1_2_1_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:48_1_1_2_1_1_3_1_1_1_2_1_1_2_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="glib_001.html#K13">.vertices()</a>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span> )
 </span><b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_1_1_3_1_1_1_2_1_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:48_1_1_2_1_1_3_1_1_1_2_1_1_2_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="glib_001.html#K13">.vertices()</a> 
 ;<br/><div class="add"><a NAME="E2:48_1_1_2_1_1_3_1_1_1_2_1_1_2_1_1_1_1"/><b>then </b><i><font color="Green">E196</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i/>;<br/><a NAME="E3:48_1_1_2_1_1_3_1_1_1_2_1_1_2_1_1_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E4:48_1_1_2_1_1_3_1_1_1_2_1_1_2_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:48_1_1_2_1_1_3_1_1_1_2_1_1_2_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_1_1_3_1_1_1_2_1_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:48_1_1_2_1_1_3_1_1_1_2_1_1_2_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:48_1_1_2_1_1_3_1_1_1_2_1_1_2_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:48_1_1_2_1_1_3_1_1_1_2_1_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E15:48_1_1_2_1_1_3_1_1_1_2_1_1_2"/>
<font color="Maroon">W2</font> <a href="glib_001.html#K13">.vertices()</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>let </b><font color="Maroon">n</font> be  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E16:48_1_1_2_1_1_3_1_1_1_2_1_1_2"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">W2</font>
 ;<br/><a NAME="E17:48_1_1_2_1_1_3_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E197</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> <font color="Maroon">W2</font> )
 <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="48_1_1_2_1_1_3_1_1_1_2_1_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_1_1_3_1_1_1_2_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:48_1_1_2_1_1_3_1_1_1_2_1_1_2_2_1"/>
( <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> <a href="xxreal_0.html#NR4" target="_self">&gt;</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="48_1_1_2_1_1_3_1_1_1_2_1_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:48_1_1_2_1_1_3_1_1_1_2_1_1_2_2_1_1"/><i><font color="Green">E198</font></i>: 
<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"><a NAME="E2:48_1_1_2_1_1_3_1_1_1_2_1_1_2_2_1_1"/><b>then </b><i><font color="Green">E199</font></i>: 
<font color="Maroon">n</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="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:48_1_1_2_1_1_3_1_1_1_2_1_1_2_2_1_1"/><b>then </b><i><font color="Green">E200</font></i>: 
<font color="Maroon">W2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T66">GLIB_001:66</a></i>;<br/><a NAME="E4:48_1_1_2_1_1_3_1_1_1_2_1_1_2_2_1_1"/>
<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>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E5:48_1_1_2_1_1_3_1_1_1_2_1_1_2_2_1_1"/><b>then </b><i><font color="Green">E201</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 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="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E6:48_1_1_2_1_1_3_1_1_1_2_1_1_2_2_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E7:48_1_1_2_1_1_3_1_1_1_2_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <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_005.html#D3" target="_self">Def3</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E8:48_1_1_2_1_1_3_1_1_1_2_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon">W2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T66">GLIB_001:66</a></i>;<br/><b>hence </b><a NAME="E9:48_1_1_2_1_1_3_1_1_1_2_1_1_2_2_1_1"/>
<font color="Maroon">W2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">W2</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="glib_005.html#D2" target="_self">Def2</a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_1_1_3_1_1_1_2_1_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:48_1_1_2_1_1_3_1_1_1_2_1_1_2_2_1_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font>
 ;<br/><div class="add"><a NAME="E2:48_1_1_2_1_1_3_1_1_1_2_1_1_2_2_1_2"/><b>then </b><i><font color="Green">E202</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:48_1_1_2_1_1_3_1_1_1_2_1_1_2_2_1_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="glib_001.html#T65">GLIB_001:65</a></i>;<br/><a NAME="E4:48_1_1_2_1_1_3_1_1_1_2_1_1_2_2_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E5:48_1_1_2_1_1_3_1_1_1_2_1_1_2_2_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E6:48_1_1_2_1_1_3_1_1_1_2_1_1_2_2_1_2"/><b>then </b><i><font color="Green">E203</font></i>: 
<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> <font color="Maroon">W</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E7:48_1_1_2_1_1_3_1_1_1_2_1_1_2_2_1_2"/><b>then </b>
<font color="Maroon">n</font> <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> <font color="Maroon">W</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><a NAME="E8:48_1_1_2_1_1_3_1_1_1_2_1_1_2_2_1_2"/><b>then </b><i><font color="Green">E204</font></i>: 
( <font color="Maroon">W2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> &amp; <font color="Maroon">W2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 <b>by </b><i>, , , <a class="ref" href="glib_001.html#T66">GLIB_001:66</a></i>;<br/><a NAME="E9:48_1_1_2_1_1_3_1_1_1_2_1_1_2_2_1_2"/><i><font color="Green">E205</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E10:48_1_1_2_1_1_3_1_1_1_2_1_1_2_2_1_2"/>
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="glib_003.html#T46">GLIB_003:46</a></i>;<br/><b>hence </b><a NAME="E11:48_1_1_2_1_1_3_1_1_1_2_1_1_2_2_1_2"/>
<font color="Maroon">W2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">W2</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="glib_005.html#D2" target="_self">Def2</a>, <a class="ref" href="glib_005.html#D3" target="_self">Def3</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E19:48_1_1_2_1_1_3_1_1_1_2_1_1_2"/>
<font color="Maroon">W2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">W2</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:48_1_1_2_1_1_3_1_1_1_2"/>
 ex <font color="Olive">P</font> being <a href="glib_001.html#V6">vertex-distinct</a> <a href="glib_005.html#V7" target="_self">augmenting</a> <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">P</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">source</font>,<font color="Maroon">v</font> &amp; <font color="Olive">P</font> <a href="glib_001.html#K13">.vertices()</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  &amp; ( for <font color="Olive">n</font> being <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Olive">P</font> holds <br/><font color="Olive">P</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">P</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ) )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:48_1_1_2_1_1_3_1"/>
 ex <font color="Olive">P</font> being <a href="glib_001.html#V6">vertex-distinct</a> <a href="glib_005.html#V7" target="_self">augmenting</a> <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">P</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">source</font>,<font color="Maroon">v</font> &amp; <font color="Olive">P</font> <a href="glib_001.html#K13">.vertices()</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  &amp; ( for <font color="Olive">n</font> being <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Olive">P</font> holds <br/><font color="Olive">P</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">P</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ) )
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E9:48_1_1_2_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E8:48_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E4:48_1_1"/><b>then </b><i><font color="Green">E206</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 
;<br/>
<a NAME="E5:48_1_1"/><i><font color="Green">E207</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:48_1_1_3"/>
<font color="Maroon">sink</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 ;<br/><b>then consider </b><font color="Maroon">W</font> being  <a href="glib_001.html#V6">vertex-distinct</a> <a href="glib_005.html#V7" target="_self">augmenting</a> <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E3:48_1_1_3"/><i><font color="Green">E208</font></i>: 
( <font color="Maroon">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">source</font>,<font color="Maroon">sink</font> &amp; <font color="Maroon">W</font> <a href="glib_001.html#K13">.vertices()</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  &amp; ( for <font color="Olive">n</font> being <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">W</font> holds <br/><font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ) )
 <b>by </b><i/>;<br/><b>take </b><font color="Maroon">W</font> = <font color="Maroon">W</font>;<br/><b>thus </b><a NAME="E4:48_1_1_3"/>
( <font color="Maroon">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">source</font>,<font color="Maroon">sink</font> &amp; ( for <font color="Olive">n</font> being <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">W</font> holds <br/><font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ) )
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E7:48_1_1"/>
( not <font color="Maroon">sink</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  implies  ex <font color="Olive">b<sub>1</sub></font> being <a href="glib_001.html#V6">vertex-distinct</a> <a href="glib_005.html#V7" target="_self">augmenting</a> <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="glib_001.html#K1">.walkOf</a> <font color="Maroon">source</font> )
 ;<br/>


</div>
