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

<b>let </b><font color="Maroon">G</font> be  <a href="glib_000.html#V2">finite</a> <a href="glib_003.html#V7">real-weighted</a> <a href="glib_003.html#V9">real-elabeled</a> <a href="glib_005.html#V1" target="_self">complete-elabeled</a> <a href="glib_003.html#NM4">WEGraph</a>;<br/><b>let </b><font color="Maroon">source</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">sink</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:84"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">source</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">sink</font> &amp; <font color="Maroon">G</font> <a href="glib_005.html#R1" target="_self">has_valid_flow_from</a> <font color="Maroon">source</font>,<font color="Maroon">sink</font> &amp; ( for <font color="Olive">P</font> being <a href="glib_005.html#V7" target="_self">augmenting</a> <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font>  holds not <font color="Olive">P</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">source</font>,<font color="Maroon">sink</font> ) )
 ;<br/>

<b>reconsider </b><font color="Maroon">src</font> = <font color="Maroon">source</font> as   <a href="glib_000.html#NM10">Vertex</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>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">src</font>;<br/>
<b>set </b><font color="Maroon">Gn</font> =  <a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">src</font>;<br/>
<b>set </b><font color="Maroon">V</font> = <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">src</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> ;<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">src</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</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">src</font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/>
<a NAME="E3:84"/><i><font color="Green">E51</font></i>: 
(  <a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">src</font> <a href="glib_000.html#R5">==</a> <font color="Maroon">G</font> &amp;  <a href="glib_003.html#K5">the_Weight_of</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">src</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">G</font> &amp;  <a href="glib_003.html#K6">the_ELabel_of</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">src</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E4:84"/><b>then </b><i><font color="Green">E82</font></i>: 
(  <a href="glib_000.html#K6">the_Vertices_of</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">src</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font> &amp;  <a href="glib_000.html#K7">the_Edges_of</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">src</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font> &amp;  <a href="glib_000.html#K10">the_Source_of</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">src</font></span>)</span> <a href="funct_2.html#R1">=</a>  <a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">G</font> &amp;  <a href="glib_000.html#K11">the_Target_of</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">src</font></span>)</span> <a href="funct_2.html#R1">=</a>  <a href="glib_000.html#K11">the_Target_of</a> <font color="Maroon">G</font> )
 
<b>by </b><i><a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>;<br/>
<b>reconsider </b><font color="Maroon">V</font> = <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">src</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  as   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <b>by </b><i>, <a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>;<br/>
<a NAME="E6:84"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">source</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 
<b>by </b><i/>;<br/>
<a NAME="E7:84"/><i><font color="Green">E92</font></i>: 
( <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">src</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  iff  ex <font color="Olive">P</font> being  <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">P</font> is <a href="glib_005.html#V7" target="_self">augmenting</a> &amp; <font color="Olive">P</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">src</font>,<font color="Maroon">sink</font> ) )
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">E1</font> = <font color="Maroon">G</font> <a href="glib_000.html#K24">.edgesDBetween</a> <font color="Maroon">V</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">V</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">E2</font> = <font color="Maroon">G</font> <a href="glib_000.html#K24">.edgesDBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">V</font></span>)</span>,<font color="Maroon">V</font>;<br/>
<a NAME="E8:84"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">G</font> <a href="glib_005.html#K2" target="_self">.flow</a> <font color="Maroon">source</font>,<font color="Maroon">sink</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="uproots.html#K3">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K24">.edgesDBetween</a> <font color="Maroon">V</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">V</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="uproots.html#K3">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K24">.edgesDBetween</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">V</font></span>)</span>,<font color="Maroon">V</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="glib_005.html#D1" target="_self">Def1</a>, , , </i>;<br/>
<b>set </b><font color="Maroon">A1</font> = <span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K24">.edgesDBetween</a> <font color="Maroon">V</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">V</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">B1</font> = <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="pre_circ.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K24">.edgesDBetween</a> <font color="Maroon">V</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">V</font></span>)</span></span>)</span>;<br/>
<a NAME="E9:84"/><i><font color="Green">E94</font></i>: 
 <a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font> is <a href="glib_000.html#V13">halting</a>
 
<b>by </b><i/>;<br/>
<a NAME="E10:84"/><i><font color="Green">E95</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">src</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</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">src</font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span> <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"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E11:84"/><i><font color="Green">E96</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">src</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</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">src</font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">src</font>
 
<b>by </b><i><a class="ref" href="glib_005.html#D3" target="_self">Def3</a>, <a class="ref" href="glib_000.html#D57">GLIB_000:def 57</a></i>;<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"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span></span>)</span>;<br/>
<div><i><font color="Green">E109</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:84_1"/><i><font color="Green">E110</font></i>: 
 <a href="glib_005.html#K3" target="_self">AP:NextBestEdges</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">src</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:84_1"/><b>then </b><i><font color="Green">E111</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"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">src</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"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:84_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"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span></span>)</span> <a href="glib_005.html#R3" target="_self">is_forward_labeling_in</a>  <a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">src</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"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span></span>)</span> <a href="glib_005.html#R4" target="_self">is_backward_labeling_in</a>  <a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">src</font> )
 </span><b>by </b><i><a class="txt" href="glib_005.html#E3:84"><i><font color="Green">E51</font></i></a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_1_1_1_1"><b>suppose </b></a><a NAME="E1:84_1_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"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span></span>)</span> <a href="glib_005.html#R3" target="_self">is_forward_labeling_in</a>  <a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">src</font>
 ;<br/><div class="add"><a NAME="E2:84_1_1_1_1"/><b>then </b><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"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">src</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"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; not <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_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">src</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"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> )
 <b>by </b><i/>;<br/><a NAME="E3:84_1_1_1_1"/><b>then </b><i><font color="Green">E128</font></i>: 
 <a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">src</font> <a href="hidden.html#R1">=</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">src</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"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">src</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"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">src</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"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="glib_005.html#D4" target="_self">Def4</a>, <a class="ref" href="glib_005.html#D5" target="_self">Def5</a>, <a class="txt" href="glib_005.html#E3:84"><i><font color="Green">E51</font></i></a>, </i>;<br/><a NAME="E4:84_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_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">src</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"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">src</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"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><b>hence </b><a NAME="E5:84_1_1_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="glib_003.html#T64">GLIB_003:64</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_1_1_1_2"><b>suppose </b></a><a NAME="E1:84_1_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"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span></span>)</span> <a href="glib_005.html#R4" target="_self">is_backward_labeling_in</a>  <a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">src</font>
 ;<br/><div class="add"><a NAME="E2:84_1_1_1_2"/><b>then </b><i><font color="Green">E129</font></i>: 
( <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_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">src</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"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; not <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_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">src</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"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> )
 <b>by </b><i/>;<br/><a NAME="E3:84_1_1_1_2"/><b>then </b><i><font color="Green">E130</font></i>: 
 <a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">src</font> <a href="hidden.html#R1">=</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">src</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"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">src</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"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">src</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"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="glib_005.html#D4" target="_self">Def4</a>, <a class="ref" href="glib_005.html#D5" target="_self">Def5</a>, <a class="txt" href="glib_005.html#E3:84"><i><font color="Green">E51</font></i></a>, </i>;<br/><a NAME="E4:84_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_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">src</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"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">src</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"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><b>hence </b><a NAME="E5:84_1_1_1_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="glib_003.html#T64">GLIB_003:64</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:84_1"/>
contradiction
 ;<br/></div><b>end;</b></div>
<div><i><font color="Green">E131</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2"><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="E1:84_2"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="glib_000.html#K24">.edgesDBetween</a> <font color="Maroon">V</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">V</font></span>)</span>
 ;<br/><a NAME="E2:84_2"/><b>then </b><i><font color="Green">E133</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K24">.edgesDBetween</a> <font color="Maroon">V</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">V</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K6">the_ELabel_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">src</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K24">.edgesDBetween</a> <font color="Maroon">V</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">V</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5">the_Weight_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">src</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><a NAME="E3:84_2"/>
<font color="Maroon">x</font> <a href="glib_000.html#R4">DSJoins</a> <font color="Maroon">V</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">V</font>,<font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#D33">GLIB_000:def 33</a></i>;<br/><a NAME="E4:84_2"/><b>then </b><i><font color="Green">E134</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K7">the_Edges_of</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">src</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_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">src</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_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">src</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_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">src</font></span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">V</font> )
 <b>by </b><i>, <a class="ref" href="glib_000.html#D18">GLIB_000:def 18</a></i>;<br/><a NAME="E5:84_2"/><b>then </b><i><font color="Green">E135</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"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E6:84_2"/><i><font color="Green">E137</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K24">.edgesDBetween</a> <font color="Maroon">V</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">V</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K24">.edgesDBetween</a> <font color="Maroon">V</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">V</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="glib_005.html#D1" target="_self">Def1</a>, , , , , </i>;<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"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span></span>)</span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:84_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K24">.edgesDBetween</a> <font color="Maroon">V</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">V</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K24">.edgesDBetween</a> <font color="Maroon">V</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">V</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font>
 ;<br/><a NAME="E2:84_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="glib_005.html#R3" target="_self">is_forward_labeling_in</a>  <a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">src</font>
 <b>by </b><i>, , , </i>;<br/><b>hence </b><a NAME="E3:84_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="glib_005.html#D6" target="_self">Def6</a>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E8:84_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K24">.edgesDBetween</a> <font color="Maroon">V</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">V</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K24">.edgesDBetween</a> <font color="Maroon">V</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">V</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div>
<b>set </b><font color="Maroon">A2</font> = <span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K24">.edgesDBetween</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">V</font></span>)</span>,<font color="Maroon">V</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">B2</font> =  <a href="polynom1.html#K16">EmptyBag</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K24">.edgesDBetween</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">V</font></span>)</span>,<font color="Maroon">V</font></span>)</span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_3"><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="E1:84_3"/><i><font color="Green">E138</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="glib_000.html#K24">.edgesDBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">V</font></span>)</span>,<font color="Maroon">V</font>
 ;<br/><a NAME="E2:84_3"/><b>then </b><i><font color="Green">E139</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K24">.edgesDBetween</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">V</font></span>)</span>,<font color="Maroon">V</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K6">the_ELabel_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">src</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><a NAME="E3:84_3"/>
 <a href="polynom1.html#K16">EmptyBag</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K24">.edgesDBetween</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">V</font></span>)</span>,<font color="Maroon">V</font></span>)</span> <a href="pboole.html#R6">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K24">.edgesDBetween</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">V</font></span>)</span>,<font color="Maroon">V</font></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> 0
 <b>by </b><i><a class="ref" href="polynom1.html#D15">POLYNOM1:def 15</a></i>;<br/><a NAME="E4:84_3"/><b>then </b><i><font color="Green">E140</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K24">.edgesDBetween</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">V</font></span>)</span>,<font color="Maroon">V</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><a NAME="E5:84_3"/>
<font color="Maroon">x</font> <a href="glib_000.html#R4">DSJoins</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">V</font>,<font color="Maroon">V</font>,<font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#D33">GLIB_000:def 33</a></i>;<br/><a NAME="E6:84_3"/><b>then </b><i><font color="Green">E141</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K7">the_Edges_of</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">src</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_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">src</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_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">src</font></span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">V</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_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">src</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> )
 <b>by </b><i>, <a class="ref" href="glib_000.html#D18">GLIB_000:def 18</a></i>;<br/><a NAME="E7:84_3"/><b>then </b><i><font color="Green">E142</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"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<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"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span></span>)</span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:84_3_1"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K24">.edgesDBetween</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">V</font></span>)</span>,<font color="Maroon">V</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font>
 ;<br/><a NAME="E2:84_3_1"/><b>then </b>
<font color="Maroon">x</font> <a href="glib_005.html#R4" target="_self">is_backward_labeling_in</a>  <a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">src</font>
 <b>by </b><i>, , , </i>;<br/><b>hence </b><a NAME="E3:84_3_1"/>
contradiction
 <b>by </b><i><a class="ref" href="glib_005.html#D6" target="_self">Def6</a>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E9:84_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K24">.edgesDBetween</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">V</font></span>)</span>,<font color="Maroon">V</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K24">.edgesDBetween</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">V</font></span>)</span>,<font color="Maroon">V</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="glib_005.html#D1" target="_self">Def1</a>, , , , , </i>;<br/></div><b>end;</b></div>
<a NAME="E15:84"/><b>then </b>
 <a href="uproots.html#K3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K24">.edgesDBetween</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">V</font></span>)</span>,<font color="Maroon">V</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="uproots.html#K3">Sum</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K24">.edgesDBetween</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">V</font></span>)</span>,<font color="Maroon">V</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="glib_004.html#T6">GLIB_004:6</a></i>;<br/>
<a NAME="E16:84"/><b>then </b><i><font color="Green">E143</font></i>: 
 <a href="uproots.html#K3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K24">.edgesDBetween</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">V</font></span>)</span>,<font color="Maroon">V</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="uproots.html#T13">UPROOTS:13</a></i>;<br/>
<a NAME="E17:84"/><i><font color="Green">E144</font></i>: 
<font color="Maroon">G</font> <a href="glib_005.html#K2" target="_self">.flow</a> <font color="Maroon">source</font>,<font color="Maroon">sink</font> <a href="hidden.html#R1">=</a>  <a href="uproots.html#K3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K24">.edgesDBetween</a> <font color="Maroon">V</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">V</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="glib_005.html#D2" target="_self">Def2</a>, <a class="txt" href="glib_005.html#E3:84"><i><font color="Green">E51</font></i></a>, , <a class="ref" href="glib_004.html#T6">GLIB_004:6</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">X</font> be  <a href="glib_000.html#V2">finite</a> <a href="glib_003.html#V7">real-weighted</a> <a href="glib_003.html#V9">real-elabeled</a> <a href="glib_005.html#V1" target="_self">complete-elabeled</a> <a href="glib_003.html#NM4">WEGraph</a>;<br/><b>assume </b><a NAME="E1:84_4"/><i><font color="Green">E145</font></i>: 
( <font color="Maroon">X</font> <a href="glib_000.html#R5">==</a> <font color="Maroon">G</font> &amp;  <a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">G</font> &amp; <font color="Maroon">X</font> <a href="glib_005.html#R1" target="_self">has_valid_flow_from</a> <font color="Maroon">source</font>,<font color="Maroon">sink</font> )
 ;<br/><a NAME="E2:84_4"/><b>then </b><i><font color="Green">E146</font></i>: 
 <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>;<br/><b>reconsider </b><font color="Maroon">V'</font> = <font color="Maroon">V</font> as   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i>, <a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>;<br/><b>set </b><font color="Maroon">E1'</font> = <font color="Maroon">X</font> <a href="glib_000.html#K24">.edgesDBetween</a> <font color="Maroon">V'</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">V'</font></span>)</span>;<br/><a NAME="E4:84_4"/>
<font color="Maroon">X</font> <a href="glib_000.html#K24">.edgesDBetween</a> <font color="Maroon">V'</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">V'</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="glib_000.html#K24">.edgesDBetween</a> <font color="Maroon">V</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">V</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="glib_000.html#T93">GLIB_000:93</a></i>;<br/><b>hence </b><a NAME="E5:84_4"/>
<font color="Maroon">X</font> <a href="glib_005.html#K2" target="_self">.flow</a> <font color="Maroon">source</font>,<font color="Maroon">sink</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">G</font> <a href="glib_005.html#K2" target="_self">.flow</a> <font color="Maroon">source</font>,<font color="Maroon">sink</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>hence </b><a NAME="E19:84"/>
<font color="Maroon">G</font> <a href="glib_005.html#R2" target="_self">has_maximum_flow_from</a> <font color="Maroon">source</font>,<font color="Maroon">sink</font>
 <b>by </b><i><a class="ref" href="glib_005.html#D1" target="_self">Def1</a>, </i>;<br/>


</div>
