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<b>let </b><font color="Maroon" title="c1">G</font> be  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="glib_003.html#V7" title="GLIB_003:attr.7">real-weighted</a> <a href="glib_003.html#V9" title="GLIB_003:attr.9">real-elabeled</a> <a href="glib_005.html#V1" title="GLIB_005:attr.1">complete-elabeled</a> <a href="glib_003.html#NM4" title="GLIB_003:NM.4">WEGraph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">source</font>, <font color="Olive" title="b2">sink</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">source</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">sink</font> &amp; <font color="Maroon" title="c1">G</font> <a href="glib_005.html#R1" title="GLIB_005:pred.1">has_valid_flow_from</a> <font color="Olive" title="b1">source</font>,<font color="Olive" title="b2">sink</font> &amp; ( for <font color="Olive" title="b3">P</font> being <a href="glib_005.html#V7" title="GLIB_005:attr.7">augmenting</a> <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G</font>  holds not <font color="Olive" title="b3">P</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Olive" title="b1">source</font>,<font color="Olive" title="b2">sink</font> ) holds <br/><font color="Maroon" title="c1">G</font> <a href="glib_005.html#R2" title="GLIB_005:pred.2">has_maximum_flow_from</a> <font color="Olive" title="b1">source</font>,<font color="Olive" title="b2">sink</font></span><br/><b>let </b><font color="Maroon" title="c2">source</font>, <font color="Maroon" title="c3">sink</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">source</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">sink</font> &amp; <font color="Maroon" title="c1">G</font> <a href="glib_005.html#R1" title="GLIB_005:pred.1">has_valid_flow_from</a> <font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font> &amp; ( for <font color="Olive" title="b1">P</font> being <a href="glib_005.html#V7" title="GLIB_005:attr.7">augmenting</a> <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G</font>  holds not <font color="Olive" title="b1">P</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font> ) implies <font color="Maroon" title="c1">G</font> <a href="glib_005.html#R2" title="GLIB_005:pred.2">has_maximum_flow_from</a> <font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font> )</span><br/>



<b>assume </b><a NAME="E1:81"/><i><font color="Green" title="E40">A1</font></i>: 
( <font color="Maroon" title="c2">source</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">sink</font> &amp; <font color="Maroon" title="c1">G</font> <a href="glib_005.html#R1" title="GLIB_005:pred.1">has_valid_flow_from</a> <font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font> &amp; ( for <font color="Olive" title="b1">P</font> being <a href="glib_005.html#V7" title="GLIB_005:attr.7">augmenting</a> <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G</font>  holds not <font color="Olive" title="b1">P</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">G</font> <a href="glib_005.html#R2" title="GLIB_005:pred.2">has_maximum_flow_from</a> <font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c4">src</font> = <font color="Maroon" title="c2">source</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="ref" href="glib_005.html#D7" target="_self" title="GLIB_005:def.7">Def7</a></i>;<br/>
<b>set </b><font color="Maroon" title="c5">CS</font> =  <a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font>;<br/>
<b>set </b><font color="Maroon" title="c6">Gn</font> =  <a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font>;<br/>
<b>set </b><font color="Maroon" title="c7">V</font> = <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> ;<br/>
<b>set </b><font color="Maroon" title="c8">Gn1</font> = <span class="p1">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</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" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/>
<a NAME="E3:81"/><i><font color="Green" title="E41">A2</font></i>: 
(  <a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c1">G</font> &amp;  <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font> &amp;  <a href="glib_003.html#K6" title="GLIB_003:func.6">the_ELabel_of</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_005.html#K1" title="GLIB_005:func.1">the_ELabel_of</a> <font color="Maroon" title="c1">G</font> )
 
<b>by </b><i><a class="ref" href="glib_005.html#T10" target="_self" title="GLIB_005:th.10">Th10</a></i>;<br/>
<a NAME="E4:81"/><b>then </b><i><font color="Green" title="E42">A3</font></i>: 
(  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font> &amp;  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font> &amp;  <a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font> &amp;  <a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G</font> )
 
<b>by </b><i><a class="ref" href="glib_000.html#D36" title="GLIB_000:def.36">GLIB_000:def 36</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">V</font> = <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <b>by </b><i><a class="txt" href="glib_005.html#E3:81"><i><font color="Green" title="E41">A2</font></i></a>, <a class="ref" href="glib_000.html#D36" title="GLIB_000:def.36">GLIB_000:def 36</a></i>;<br/>
<a NAME="E6:81"/><i><font color="Green" title="E43">A4</font></i>: 
<font color="Maroon" title="c2">source</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">V</font>
 
<b>by </b><i><a class="ref" href="glib_005.html#T10" target="_self" title="GLIB_005:th.10">Th10</a></i>;<br/>
<a NAME="E7:81"/><i><font color="Green" title="E44">A5</font></i>: 
( <font color="Maroon" title="c3">sink</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  iff  ex <font color="Olive" title="b1">P</font> being  <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b1">P</font> is <a href="glib_005.html#V7" title="GLIB_005:attr.7">augmenting</a> &amp; <font color="Olive" title="b1">P</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c4">src</font>,<font color="Maroon" title="c3">sink</font> ) )
 
<b>by </b><i><a class="ref" href="glib_005.html#T9" target="_self" title="GLIB_005:th.9">Th9</a></i>;<br/>
<b>set </b><font color="Maroon" title="c10">E1</font> = <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c9">V</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">V</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c11">E2</font> = <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">V</font></span>)</span>,<font color="Maroon" title="c9">V</font>;<br/>
<a NAME="E8:81"/><i><font color="Green" title="E45">A6</font></i>: 
<font color="Maroon" title="c1">G</font> <a href="glib_005.html#K2" title="GLIB_005:func.2">.flow</a> <font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K1" title="GLIB_005:func.1">the_ELabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="pre_circ.html#K3" title="PRE_CIRC:func.3">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c9">V</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">V</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K1" title="GLIB_005:func.1">the_ELabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="pre_circ.html#K3" title="PRE_CIRC:func.3">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">V</font></span>)</span>,<font color="Maroon" title="c9">V</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="glib_005.html#E1:81"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="glib_005.html#E6:81"><i><font color="Green" title="E43">A4</font></i></a>, <a class="txt" href="glib_005.html#E7:81"><i><font color="Green" title="E44">A5</font></i></a>, <a class="ref" href="glib_005.html#T11" target="_self" title="GLIB_005:th.11">Th11</a></i>;<br/>
<b>set </b><font color="Maroon" title="c12">A1</font> = <span class="p1">(<span class="default"><a href="glib_005.html#K1" title="GLIB_005:func.1">the_ELabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="pre_circ.html#K3" title="PRE_CIRC:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c9">V</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">V</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c13">B1</font> = <span class="p1">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="pre_circ.html#K3" title="PRE_CIRC:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c9">V</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">V</font></span>)</span></span>)</span>;<br/>
<a NAME="E9:81"/><i><font color="Green" title="E46">A7</font></i>: 
 <a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font> is <a href="glib_000.html#V13" title="GLIB_000:attr.13">halting</a>
 
<b>by </b><i><a class="ref" href="glib_005.html#T6" target="_self" title="GLIB_005:th.6">Th6</a></i>;<br/>
<a NAME="E10:81"/><i><font color="Green" title="E47">A8</font></i>: 
<span class="p1">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</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" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_005.html#K4" title="GLIB_005:func.4">AP:Step</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span>
 
<b>by </b><i><a class="ref" href="glib_005.html#D15" target="_self" title="GLIB_005:def.15">Def15</a></i>;<br/>
<a NAME="E11:81"/><i><font color="Green" title="E48">A9</font></i>: 
<span class="p1">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</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" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font>
 
<b>by </b><i><a class="txt" href="glib_005.html#E9:81"><i><font color="Green" title="E46">A7</font></i></a>, <a class="ref" href="glib_000.html#D57" title="GLIB_000:def.57">GLIB_000:def 57</a></i>;<br/>
<b>set </b><font color="Maroon" title="c14">e</font> =  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" title="GLIB_005:func.3">AP:NextBestEdges</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span></span>)</span>;<br/>
<div><i><font color="Green" title="E49">A10</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:81_1"/><i><font color="Green" title="E50">A11</font></i>: 
 <a href="glib_005.html#K3" title="GLIB_005:func.3">AP:NextBestEdges</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:81_1"/><b>then </b><i><font color="Green" title="E51">A12</font></i>: 
 <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" title="GLIB_005:func.3">AP:NextBestEdges</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_005.html#K3" title="GLIB_005:func.3">AP:NextBestEdges</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:81_1_1_1"/>
(  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" title="GLIB_005:func.3">AP:NextBestEdges</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span></span>)</span> <a href="glib_005.html#R3" title="GLIB_005:pred.3">is_forward_labeling_in</a>  <a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font> or  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" title="GLIB_005:func.3">AP:NextBestEdges</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span></span>)</span> <a href="glib_005.html#R4" title="GLIB_005:pred.4">is_backward_labeling_in</a>  <a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font> )
 </span><b>by </b><i><a class="txt" href="glib_005.html#E1:81_1"><i><font color="Green" title="E50">A11</font></i></a>, <a class="ref" href="glib_005.html#D13" target="_self" title="GLIB_005:def.13">Def13</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_1_1_1"><b>suppose </b></a><a NAME="E1:81_1_1_1_1"/>
 <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" title="GLIB_005:func.3">AP:NextBestEdges</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span></span>)</span> <a href="glib_005.html#R3" title="GLIB_005:pred.3">is_forward_labeling_in</a>  <a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:81_1_1_1_1"/><b>then </b><i><font color="Green" title="E52">A13</font></i>: 
( <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" title="GLIB_005:func.3">AP:NextBestEdges</a> <span class="p3">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">V</font> &amp; not <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" title="GLIB_005:func.3">AP:NextBestEdges</a> <span class="p3">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">V</font> )
 <b>by </b><i><a class="ref" href="glib_005.html#D10" target="_self" title="GLIB_005:def.10">Def10</a></i>;<br/><a NAME="E3:81_1_1_1_1"/><b>then </b><i><font color="Green" title="E53">A14</font></i>: 
 <a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span> <a href="glib_003.html#K13" title="GLIB_003:func.13">.labelVertex</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_005.html#K3" title="GLIB_005:func.3">AP:NextBestEdges</a> <span class="p4">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" title="GLIB_005:func.3">AP:NextBestEdges</a> <span class="p3">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_005.html#E10:81"><i><font color="Green" title="E47">A8</font></i></a>, <a class="txt" href="glib_005.html#E11:81"><i><font color="Green" title="E48">A9</font></i></a>, <a class="txt" href="glib_005.html#E1:81_1"><i><font color="Green" title="E50">A11</font></i></a>, <a class="ref" href="glib_005.html#D14" target="_self" title="GLIB_005:def.14">Def14</a></i>;<br/><a NAME="E4:81_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" title="GLIB_005:func.3">AP:NextBestEdges</a> <span class="p3">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_005.html#E2:81_1"><i><font color="Green" title="E51">A12</font></i></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/><b>hence </b><a NAME="E5:81_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_005.html#E2:81_1_1_1_1"><i><font color="Green" title="E52">A13</font></i></a>, <a class="txt" href="glib_005.html#E3:81_1_1_1_1"><i><font color="Green" title="E53">A14</font></i></a>, <a class="ref" href="glib_003.html#T64" title="GLIB_003:th.64">GLIB_003:64</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_1_1_2"><b>suppose </b></a><a NAME="E1:81_1_1_1_2"/>
 <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" title="GLIB_005:func.3">AP:NextBestEdges</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span></span>)</span> <a href="glib_005.html#R4" title="GLIB_005:pred.4">is_backward_labeling_in</a>  <a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:81_1_1_1_2"/><b>then </b><i><font color="Green" title="E52">A15</font></i>: 
( <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" title="GLIB_005:func.3">AP:NextBestEdges</a> <span class="p3">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">V</font> &amp; not <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" title="GLIB_005:func.3">AP:NextBestEdges</a> <span class="p3">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">V</font> )
 <b>by </b><i><a class="ref" href="glib_005.html#D11" target="_self" title="GLIB_005:def.11">Def11</a></i>;<br/><a NAME="E3:81_1_1_1_2"/><b>then </b><i><font color="Green" title="E53">A16</font></i>: 
 <a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span> <a href="glib_003.html#K13" title="GLIB_003:func.13">.labelVertex</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_005.html#K3" title="GLIB_005:func.3">AP:NextBestEdges</a> <span class="p4">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" title="GLIB_005:func.3">AP:NextBestEdges</a> <span class="p3">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_005.html#E10:81"><i><font color="Green" title="E47">A8</font></i></a>, <a class="txt" href="glib_005.html#E11:81"><i><font color="Green" title="E48">A9</font></i></a>, <a class="txt" href="glib_005.html#E1:81_1"><i><font color="Green" title="E50">A11</font></i></a>, <a class="ref" href="glib_005.html#D14" target="_self" title="GLIB_005:def.14">Def14</a></i>;<br/><a NAME="E4:81_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" title="GLIB_005:func.3">AP:NextBestEdges</a> <span class="p3">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_005.html#E2:81_1"><i><font color="Green" title="E51">A12</font></i></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/><b>hence </b><a NAME="E5:81_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_005.html#E2:81_1_1_1_2"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="glib_005.html#E3:81_1_1_1_2"><i><font color="Green" title="E53">A16</font></i></a>, <a class="ref" href="glib_003.html#T64" title="GLIB_003:th.64">GLIB_003:64</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:81_1"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><i><font color="Green" title="E50">A17</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="81_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c15">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c9">V</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">V</font></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" title="GLIB_005:func.1">the_ELabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="pre_circ.html#K3" title="PRE_CIRC:func.3">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c9">V</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">V</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="pre_circ.html#K3" title="PRE_CIRC:func.3">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c9">V</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">V</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</font> )</span><br/><b>assume </b><a NAME="E1:81_2"/><i><font color="Green" title="E51">A18</font></i>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c9">V</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">V</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" title="GLIB_005:func.1">the_ELabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="pre_circ.html#K3" title="PRE_CIRC:func.3">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c9">V</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">V</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="pre_circ.html#K3" title="PRE_CIRC:func.3">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c9">V</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">V</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</font></span><br/><a NAME="E2:81_2"/><b>then </b><i><font color="Green" title="E52">A19</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" title="GLIB_005:func.1">the_ELabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="pre_circ.html#K3" title="PRE_CIRC:func.3">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c9">V</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">V</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K6" title="GLIB_003:func.6">the_ELabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="pre_circ.html#K3" title="PRE_CIRC:func.3">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c9">V</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">V</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</font> )
 <b>by </b><i><a class="txt" href="glib_005.html#E3:81"><i><font color="Green" title="E41">A2</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/><a NAME="E3:81_2"/>
<font color="Maroon" title="c15">x</font> <a href="glib_000.html#R4" title="GLIB_000:pred.4">DSJoins</a> <font color="Maroon" title="c9">V</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">V</font>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="glib_005.html#E1:81_2"><i><font color="Green" title="E51">A18</font></i></a>, <a class="ref" href="glib_000.html#D33" title="GLIB_000:def.33">GLIB_000:def 33</a></i>;<br/><a NAME="E4:81_2"/><b>then </b><i><font color="Green" title="E53">A20</font></i>: 
( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">V</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">V</font> )
 <b>by </b><i><a class="txt" href="glib_005.html#E4:81"><i><font color="Green" title="E42">A3</font></i></a>, <a class="ref" href="glib_000.html#D18" title="GLIB_000:def.18">GLIB_000:def 18</a></i>;<br/><a NAME="E5:81_2"/><b>then </b><i><font color="Green" title="E54">A21</font></i>: 
not <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">V</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E6:81_2"/><i><font color="Green" title="E55">A22</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" title="GLIB_005:func.1">the_ELabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="pre_circ.html#K3" title="PRE_CIRC:func.3">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c9">V</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">V</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="pre_circ.html#K3" title="PRE_CIRC:func.3">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c9">V</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">V</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</font>
 <b>by </b><i><a class="txt" href="glib_005.html#E1:81"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="glib_005.html#E3:81"><i><font color="Green" title="E41">A2</font></i></a>, <a class="txt" href="glib_005.html#E4:81"><i><font color="Green" title="E42">A3</font></i></a>, <a class="txt" href="glib_005.html#E2:81_2"><i><font color="Green" title="E52">A19</font></i></a>, <a class="txt" href="glib_005.html#E4:81_2"><i><font color="Green" title="E53">A20</font></i></a>, <a class="ref" href="glib_005.html#D7" target="_self" title="GLIB_005:def.7">Def7</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:81_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" title="GLIB_005:func.1">the_ELabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="pre_circ.html#K3" title="PRE_CIRC:func.3">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c9">V</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">V</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="pre_circ.html#K3" title="PRE_CIRC:func.3">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c9">V</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">V</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:81_2_1"/><b>then </b>
<font color="Maroon" title="c15">x</font> <a href="glib_005.html#R3" title="GLIB_005:pred.3">is_forward_labeling_in</a>  <a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font>
 <b>by </b><i><a class="txt" href="glib_005.html#E2:81_2"><i><font color="Green" title="E52">A19</font></i></a>, <a class="txt" href="glib_005.html#E4:81_2"><i><font color="Green" title="E53">A20</font></i></a>, <a class="txt" href="glib_005.html#E5:81_2"><i><font color="Green" title="E54">A21</font></i></a>, <a class="ref" href="glib_005.html#D10" target="_self" title="GLIB_005:def.10">Def10</a></i>;<br/><b>hence </b><a NAME="E3:81_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_005.html#E12:81"><i><font color="Green" title="E49">A10</font></i></a>, <a class="ref" href="glib_005.html#D13" target="_self" title="GLIB_005:def.13">Def13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E8:81_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" title="GLIB_005:func.1">the_ELabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="pre_circ.html#K3" title="PRE_CIRC:func.3">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c9">V</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">V</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="pre_circ.html#K3" title="PRE_CIRC:func.3">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c9">V</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">V</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</font>
 <b>by </b><i><a class="txt" href="glib_005.html#E6:81_2"><i><font color="Green" title="E55">A22</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>set </b><font color="Maroon" title="c15">A2</font> = <span class="p1">(<span class="default"><a href="glib_005.html#K1" title="GLIB_005:func.1">the_ELabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="pre_circ.html#K3" title="PRE_CIRC:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">V</font></span>)</span>,<font color="Maroon" title="c9">V</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c16">B2</font> =  <a href="polynom1.html#K15" title="POLYNOM1:func.15">EmptyBag</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">V</font></span>)</span>,<font color="Maroon" title="c9">V</font></span>)</span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c17">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">V</font></span>)</span>,<font color="Maroon" title="c9">V</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" title="GLIB_005:func.1">the_ELabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="pre_circ.html#K3" title="PRE_CIRC:func.3">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">V</font></span>)</span>,<font color="Maroon" title="c9">V</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">EmptyBag</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">V</font></span>)</span>,<font color="Maroon" title="c9">V</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">x</font> )</span><br/><b>assume </b><a NAME="E1:81_3"/><i><font color="Green" title="E51">A23</font></i>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">V</font></span>)</span>,<font color="Maroon" title="c9">V</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" title="GLIB_005:func.1">the_ELabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="pre_circ.html#K3" title="PRE_CIRC:func.3">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">V</font></span>)</span>,<font color="Maroon" title="c9">V</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">EmptyBag</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">V</font></span>)</span>,<font color="Maroon" title="c9">V</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">x</font></span><br/><a NAME="E2:81_3"/><b>then </b><i><font color="Green" title="E52">A24</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" title="GLIB_005:func.1">the_ELabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="pre_circ.html#K3" title="PRE_CIRC:func.3">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">V</font></span>)</span>,<font color="Maroon" title="c9">V</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K6" title="GLIB_003:func.6">the_ELabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">x</font>
 <b>by </b><i><a class="txt" href="glib_005.html#E3:81"><i><font color="Green" title="E41">A2</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/><a NAME="E3:81_3"/>
 <a href="polynom1.html#K15" title="POLYNOM1:func.15">EmptyBag</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">V</font></span>)</span>,<font color="Maroon" title="c9">V</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">V</font></span>)</span>,<font color="Maroon" title="c9">V</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="polynom1.html#D15" title="POLYNOM1:def.15">POLYNOM1:def 15</a></i>;<br/><a NAME="E4:81_3"/><b>then </b><i><font color="Green" title="E53">A25</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">EmptyBag</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">V</font></span>)</span>,<font color="Maroon" title="c9">V</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="glib_005.html#E1:81_3"><i><font color="Green" title="E51">A23</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/><a NAME="E5:81_3"/>
<font color="Maroon" title="c17">x</font> <a href="glib_000.html#R4" title="GLIB_000:pred.4">DSJoins</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">V</font>,<font color="Maroon" title="c9">V</font>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="glib_005.html#E1:81_3"><i><font color="Green" title="E51">A23</font></i></a>, <a class="ref" href="glib_000.html#D33" title="GLIB_000:def.33">GLIB_000:def 33</a></i>;<br/><a NAME="E6:81_3"/><b>then </b><i><font color="Green" title="E54">A26</font></i>: 
( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">V</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">V</font> )
 <b>by </b><i><a class="txt" href="glib_005.html#E4:81"><i><font color="Green" title="E42">A3</font></i></a>, <a class="ref" href="glib_000.html#D18" title="GLIB_000:def.18">GLIB_000:def 18</a></i>;<br/><a NAME="E7:81_3"/><b>then </b><i><font color="Green" title="E55">A27</font></i>: 
not <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">V</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:81_3_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" title="GLIB_005:func.1">the_ELabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="pre_circ.html#K3" title="PRE_CIRC:func.3">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">V</font></span>)</span>,<font color="Maroon" title="c9">V</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:81_3_1"/><b>then </b>
<font color="Maroon" title="c17">x</font> <a href="glib_005.html#R4" title="GLIB_005:pred.4">is_backward_labeling_in</a>  <a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c4">src</font>
 <b>by </b><i><a class="txt" href="glib_005.html#E2:81_3"><i><font color="Green" title="E52">A24</font></i></a>, <a class="txt" href="glib_005.html#E6:81_3"><i><font color="Green" title="E54">A26</font></i></a>, <a class="txt" href="glib_005.html#E7:81_3"><i><font color="Green" title="E55">A27</font></i></a>, <a class="ref" href="glib_005.html#D11" target="_self" title="GLIB_005:def.11">Def11</a></i>;<br/><b>hence </b><a NAME="E3:81_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_005.html#E12:81"><i><font color="Green" title="E49">A10</font></i></a>, <a class="ref" href="glib_005.html#D13" target="_self" title="GLIB_005:def.13">Def13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E9:81_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" title="GLIB_005:func.1">the_ELabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="pre_circ.html#K3" title="PRE_CIRC:func.3">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">V</font></span>)</span>,<font color="Maroon" title="c9">V</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">EmptyBag</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">V</font></span>)</span>,<font color="Maroon" title="c9">V</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">x</font>
 <b>by </b><i><a class="txt" href="glib_005.html#E1:81"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="glib_005.html#E3:81"><i><font color="Green" title="E41">A2</font></i></a>, <a class="txt" href="glib_005.html#E4:81"><i><font color="Green" title="E42">A3</font></i></a>, <a class="txt" href="glib_005.html#E2:81_3"><i><font color="Green" title="E52">A24</font></i></a>, <a class="txt" href="glib_005.html#E4:81_3"><i><font color="Green" title="E53">A25</font></i></a>, <a class="txt" href="glib_005.html#E6:81_3"><i><font color="Green" title="E54">A26</font></i></a>, <a class="ref" href="glib_005.html#D7" target="_self" title="GLIB_005:def.7">Def7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E15:81"/><b>then </b>
 <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" title="GLIB_005:func.1">the_ELabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="pre_circ.html#K3" title="PRE_CIRC:func.3">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">V</font></span>)</span>,<font color="Maroon" title="c9">V</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">EmptyBag</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">V</font></span>)</span>,<font color="Maroon" title="c9">V</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="glib_004.html#T6" title="GLIB_004:th.6">GLIB_004:6</a></i>;<br/>
<a NAME="E16:81"/><b>then </b>
 <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" title="GLIB_005:func.1">the_ELabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="pre_circ.html#K3" title="PRE_CIRC:func.3">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">V</font></span>)</span>,<font color="Maroon" title="c9">V</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="uproots.html#T13" title="UPROOTS:th.13">UPROOTS:13</a></i>;<br/>
<a NAME="E17:81"/><b>then </b><i><font color="Green" title="E51">A28</font></i>: 
<font color="Maroon" title="c1">G</font> <a href="glib_005.html#K2" title="GLIB_005:func.2">.flow</a> <font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="pre_circ.html#K3" title="PRE_CIRC:func.3">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c9">V</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">V</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="glib_005.html#E8:81"><i><font color="Green" title="E45">A6</font></i></a>, <a class="txt" href="glib_005.html#E13:81"><i><font color="Green" title="E50">A17</font></i></a>, <a class="ref" href="glib_004.html#T6" title="GLIB_004:th.6">GLIB_004:6</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c17">X</font> be  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="glib_003.html#V7" title="GLIB_003:attr.7">real-weighted</a> <a href="glib_003.html#V9" title="GLIB_003:attr.9">real-elabeled</a> <a href="glib_005.html#V1" title="GLIB_005:attr.1">complete-elabeled</a> <a href="glib_003.html#NM4" title="GLIB_003:NM.4">WEGraph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c17">X</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c1">G</font> &amp;  <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c17">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c17">X</font> <a href="glib_005.html#R1" title="GLIB_005:pred.1">has_valid_flow_from</a> <font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font> implies <font color="Maroon" title="c17">X</font> <a href="glib_005.html#K2" title="GLIB_005:func.2">.flow</a> <font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">G</font> <a href="glib_005.html#K2" title="GLIB_005:func.2">.flow</a> <font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font> )</span><br/><b>assume </b><a NAME="E1:81_4"/><i><font color="Green" title="E52">A29</font></i>: 
( <font color="Maroon" title="c17">X</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c1">G</font> &amp;  <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c17">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c17">X</font> <a href="glib_005.html#R1" title="GLIB_005:pred.1">has_valid_flow_from</a> <font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c17">X</font> <a href="glib_005.html#K2" title="GLIB_005:func.2">.flow</a> <font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">G</font> <a href="glib_005.html#K2" title="GLIB_005:func.2">.flow</a> <font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font></span><br/><a NAME="E2:81_4"/><b>then </b><i><font color="Green" title="E53">A30</font></i>: 
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c17">X</font>
 <b>by </b><i><a class="ref" href="glib_000.html#D36" title="GLIB_000:def.36">GLIB_000:def 36</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c18">V'</font> = <font color="Maroon" title="c9">V</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c17">X</font></span>)</span> <b>by </b><i><a class="txt" href="glib_005.html#E1:81_4"><i><font color="Green" title="E52">A29</font></i></a>, <a class="ref" href="glib_000.html#D36" title="GLIB_000:def.36">GLIB_000:def 36</a></i>;<br/><b>set </b><font color="Maroon" title="c19">E1'</font> = <font color="Maroon" title="c17">X</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c18">V'</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c17">X</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c18">V'</font></span>)</span>;<br/><a NAME="E4:81_4"/>
<font color="Maroon" title="c17">X</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c18">V'</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c17">X</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c18">V'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c9">V</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">V</font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_005.html#E1:81_4"><i><font color="Green" title="E52">A29</font></i></a>, <a class="txt" href="glib_005.html#E2:81_4"><i><font color="Green" title="E53">A30</font></i></a>, <a class="ref" href="glib_000.html#T93" title="GLIB_000:th.93">GLIB_000:93</a></i>;<br/><b>hence </b><a NAME="E5:81_4"/>
<font color="Maroon" title="c17">X</font> <a href="glib_005.html#K2" title="GLIB_005:func.2">.flow</a> <font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">G</font> <a href="glib_005.html#K2" title="GLIB_005:func.2">.flow</a> <font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font>
 <b>by </b><i><a class="txt" href="glib_005.html#E1:81"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="glib_005.html#E6:81"><i><font color="Green" title="E43">A4</font></i></a>, <a class="txt" href="glib_005.html#E7:81"><i><font color="Green" title="E44">A5</font></i></a>, <a class="txt" href="glib_005.html#E17:81"><i><font color="Green" title="E51">A28</font></i></a>, <a class="txt" href="glib_005.html#E1:81_4"><i><font color="Green" title="E52">A29</font></i></a>, <a class="ref" href="glib_005.html#T12" target="_self" title="GLIB_005:th.12">Th12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E19:81"/>
<font color="Maroon" title="c1">G</font> <a href="glib_005.html#R2" title="GLIB_005:pred.2">has_maximum_flow_from</a> <font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font>
 <b>by </b><i><a class="txt" href="glib_005.html#E1:81"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="glib_005.html#D9" target="_self" title="GLIB_005:def.9">Def9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
