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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:75"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="glib_005.html#R1" target="_self">has_valid_flow_from</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being <a href="glib_005.html#V7" target="_self">augmenting</a> <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">c<sub>1</sub></font> holds not <font color="Olive">b<sub>1</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> ) )
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="glib_005.html#E1:75"><i><font color="Green">E41</font></i></a>, <a class="ref" href="glib_005.html#D7" target="_self">Def7</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> ;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></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="E3:75"/><i><font color="Green">E42</font></i>: 
(  <a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="glib_000.html#R5">==</a> <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="glib_005.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E4:75"/><b>then </b><i><font color="Green">E43</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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#R1">=</a>  <a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#R1">=</a>  <a href="glib_000.html#K11">the_Target_of</a> <font color="Maroon">c<sub>1</sub></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">c<sub>9</sub></font> = <span class="p1">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="glib_005.html#E3:75"><i><font color="Green">E42</font></i></a>, <a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>;<br/>
<a NAME="E6:75"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="glib_005.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E7:75"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  iff ex <font color="Olive">b<sub>1</sub></font> being  <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="glib_005.html#V7" target="_self">augmenting</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font> ) )
 
<b>by </b><i><a class="ref" href="glib_005.html#T9" target="_self">Th9</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K24">.edgesDBetween</a> <font color="Maroon">c<sub>9</sub></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">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></font>;<br/>
<a NAME="E8:75"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="glib_005.html#K2" target="_self">.flow</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K24">.edgesDBetween</a> <font color="Maroon">c<sub>9</sub></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">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>9</sub></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">c<sub>1</sub></font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="glib_005.html#E1:75"><i><font color="Green">E41</font></i></a>, <a class="txt" href="glib_005.html#E6:75"><i><font color="Green">E44</font></i></a>, <a class="txt" href="glib_005.html#E7:75"><i><font color="Green">E45</font></i></a>, <a class="ref" href="glib_005.html#T11" target="_self">Th11</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K24">.edgesDBetween</a> <font color="Maroon">c<sub>9</sub></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">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>13</sub></font> = <span class="p1">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K24">.edgesDBetween</a> <font color="Maroon">c<sub>9</sub></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">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>;<br/>
<a NAME="E9:75"/><i><font color="Green">E47</font></i>: 
 <a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font> is <a href="glib_000.html#V13">halting</a>
 
<b>by </b><i><a class="ref" href="glib_005.html#T6" target="_self">Th6</a></i>;<br/>
 <a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="glib_000.html#K41">.Result()</a> 

<br/>.= 
<span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span>

;<br/>
<a NAME="E11:75"/><b>then </b><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="glib_005.html#D15" target="_self">Def15</a></i>;<br/>
<i><font color="Green">E49</font></i>: <span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span>
<b>by </b><i><a class="txt" href="glib_005.html#E9:75"><i><font color="Green">E47</font></i></a>, <a class="ref" href="glib_000.html#D57">GLIB_000:def 57</a></i>
<br/>.= 
 <a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font>

;<br/>
<b>set </b><font color="Maroon">c<sub>14</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>;<br/>
<div><i><font color="Green">E50</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="75_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:75_3"/><i><font color="Green">E51</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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:75_3"/><b>then </b><i><font color="Green">E52</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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:75_3_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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 </span><b>by </b><i><a class="txt" href="glib_005.html#E1:75_3"><i><font color="Green">E51</font></i></a>, <a class="ref" href="glib_005.html#D13" target="_self">Def13</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_3_1_1_1"><b>suppose </b></a><a NAME="E1:75_3_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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><a NAME="E2:75_3_1_1_1"/><b>then </b><i><font color="Green">E53</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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="ref" href="glib_005.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E3:75_3_1_1_1"/><b>then </b><i><font color="Green">E54</font></i>: 
 <a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_005.html#E11:75"><i><font color="Green">E48</font></i></a>, <a class="txt" href="glib_005.html#E12:75"><i><font color="Green">E49</font></i></a>, <a class="txt" href="glib_005.html#E1:75_3"><i><font color="Green">E51</font></i></a>, <a class="ref" href="glib_005.html#D14" target="_self">Def14</a></i>;<br/><a NAME="E4:75_3_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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_005.html#E2:75_3"><i><font color="Green">E52</font></i></a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><b>hence </b><a NAME="E5:75_3_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_005.html#E2:75_3_1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="glib_005.html#E3:75_3_1_1_1"><i><font color="Green">E54</font></i></a>, <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="75_3_1_1_2"><b>suppose </b></a><a NAME="E1:75_3_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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><a NAME="E2:75_3_1_1_2"/><b>then </b><i><font color="Green">E53</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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="ref" href="glib_005.html#D11" target="_self">Def11</a></i>;<br/><a NAME="E3:75_3_1_1_2"/><b>then </b><i><font color="Green">E54</font></i>: 
 <a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_005.html#E11:75"><i><font color="Green">E48</font></i></a>, <a class="txt" href="glib_005.html#E12:75"><i><font color="Green">E49</font></i></a>, <a class="txt" href="glib_005.html#E1:75_3"><i><font color="Green">E51</font></i></a>, <a class="ref" href="glib_005.html#D14" target="_self">Def14</a></i>;<br/><a NAME="E4:75_3_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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_005.html#E2:75_3"><i><font color="Green">E52</font></i></a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><b>hence </b><a NAME="E5:75_3_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_005.html#E2:75_3_1_1_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="glib_005.html#E3:75_3_1_1_2"><i><font color="Green">E54</font></i></a>, <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="E4:75_3"/>
contradiction
 ;<br/></div><b>end;</b></div>
<div><i><font color="Green">E51</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="75_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>15</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:75_4"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K24">.edgesDBetween</a> <font color="Maroon">c<sub>9</sub></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">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 ;<br/><a NAME="E2:75_4"/><b>then </b><i><font color="Green">E53</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">c<sub>1</sub></font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K24">.edgesDBetween</a> <font color="Maroon">c<sub>9</sub></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">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></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">c<sub>1</sub></font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K24">.edgesDBetween</a> <font color="Maroon">c<sub>9</sub></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">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="txt" href="glib_005.html#E3:75"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><a NAME="E3:75_4"/>
<font color="Maroon">c<sub>15</sub></font> <a href="glib_000.html#R4">DSJoins</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="glib_005.html#E1:75_4"><i><font color="Green">E52</font></i></a>, <a class="ref" href="glib_000.html#D33">GLIB_000:def 33</a></i>;<br/><a NAME="E4:75_4"/><b>then </b><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>15</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="glib_005.html#E4:75"><i><font color="Green">E43</font></i></a>, <a class="ref" href="glib_000.html#D18">GLIB_000:def 18</a></i>;<br/><a NAME="E5:75_4"/><b>then </b><i><font color="Green">E55</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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E6:75_4"/><i><font color="Green">E56</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">c<sub>1</sub></font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K24">.edgesDBetween</a> <font color="Maroon">c<sub>9</sub></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">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></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">c<sub>1</sub></font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K24">.edgesDBetween</a> <font color="Maroon">c<sub>9</sub></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">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="glib_005.html#E1:75"><i><font color="Green">E41</font></i></a>, <a class="txt" href="glib_005.html#E2:75_4"><i><font color="Green">E53</font></i></a>, <a class="txt" href="glib_005.html#E4:75_4"><i><font color="Green">E54</font></i></a>, <a class="txt" href="glib_005.html#E3:75"><i><font color="Green">E42</font></i></a>, <a class="txt" href="glib_005.html#E4:75"><i><font color="Green">E43</font></i></a>, <a class="ref" href="glib_005.html#D7" target="_self">Def7</a></i>;<br/><b>set </b><font color="Maroon">c<sub>16</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:75_4_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">c<sub>1</sub></font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K24">.edgesDBetween</a> <font color="Maroon">c<sub>9</sub></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">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></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">c<sub>1</sub></font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K24">.edgesDBetween</a> <font color="Maroon">c<sub>9</sub></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">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/><a NAME="E2:75_4_1"/><b>then </b>
<font color="Maroon">c<sub>15</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="glib_005.html#E2:75_4"><i><font color="Green">E53</font></i></a>, <a class="txt" href="glib_005.html#E4:75_4"><i><font color="Green">E54</font></i></a>, <a class="txt" href="glib_005.html#E5:75_4"><i><font color="Green">E55</font></i></a>, <a class="ref" href="glib_005.html#D10" target="_self">Def10</a></i>;<br/><b>hence </b><a NAME="E3:75_4_1"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_005.html#E13:75"><i><font color="Green">E50</font></i></a>, <a class="ref" href="glib_005.html#D13" target="_self">Def13</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E8:75_4"/>
<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">c<sub>1</sub></font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K24">.edgesDBetween</a> <font color="Maroon">c<sub>9</sub></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">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></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">c<sub>1</sub></font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K24">.edgesDBetween</a> <font color="Maroon">c<sub>9</sub></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">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="glib_005.html#E6:75_4"><i><font color="Green">E56</font></i></a>, <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">c<sub>15</sub></font> = <span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>16</sub></font> =  <a href="polynom1.html#K16">EmptyBag</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></font></span>)</span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>17</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:75_5"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></font>
 ;<br/><a NAME="E2:75_5"/><b>then </b><i><font color="Green">E53</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">c<sub>1</sub></font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="glib_005.html#E3:75"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><a NAME="E3:75_5"/>
 <a href="polynom1.html#K16">EmptyBag</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="pboole.html#R6">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></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="E4:75_5"/><b>then </b><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="glib_005.html#E1:75_5"><i><font color="Green">E52</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><a NAME="E5:75_5"/>
<font color="Maroon">c<sub>17</sub></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">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="glib_005.html#E1:75_5"><i><font color="Green">E52</font></i></a>, <a class="ref" href="glib_000.html#D33">GLIB_000:def 33</a></i>;<br/><a NAME="E6:75_5"/><b>then </b><i><font color="Green">E55</font></i>: 
( <font color="Maroon">c<sub>17</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>9</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="glib_005.html#E4:75"><i><font color="Green">E43</font></i></a>, <a class="ref" href="glib_000.html#D18">GLIB_000:def 18</a></i>;<br/><a NAME="E7:75_5"/><b>then </b><i><font color="Green">E56</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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></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">c<sub>18</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_5_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:75_5_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">c<sub>1</sub></font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font>
 ;<br/><a NAME="E2:75_5_1"/><b>then </b>
<font color="Maroon">c<sub>17</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="glib_005.html#E2:75_5"><i><font color="Green">E53</font></i></a>, <a class="txt" href="glib_005.html#E6:75_5"><i><font color="Green">E55</font></i></a>, <a class="txt" href="glib_005.html#E7:75_5"><i><font color="Green">E56</font></i></a>, <a class="ref" href="glib_005.html#D11" target="_self">Def11</a></i>;<br/><b>hence </b><a NAME="E3:75_5_1"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_005.html#E13:75"><i><font color="Green">E50</font></i></a>, <a class="ref" href="glib_005.html#D13" target="_self">Def13</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E9:75_5"/>
<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">c<sub>1</sub></font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></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">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="glib_005.html#E4:75_5"><i><font color="Green">E54</font></i></a>, <a class="txt" href="glib_005.html#E1:75"><i><font color="Green">E41</font></i></a>, <a class="txt" href="glib_005.html#E2:75_5"><i><font color="Green">E53</font></i></a>, <a class="txt" href="glib_005.html#E6:75_5"><i><font color="Green">E55</font></i></a>, <a class="txt" href="glib_005.html#E3:75"><i><font color="Green">E42</font></i></a>, <a class="txt" href="glib_005.html#E4:75"><i><font color="Green">E43</font></i></a>, <a class="ref" href="glib_005.html#D7" target="_self">Def7</a></i>;<br/></div><b>end;</b></div>
<a NAME="E16:75"/><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">c<sub>1</sub></font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></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">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></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="E17:75"/><b>then </b><i><font color="Green">E52</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">c<sub>1</sub></font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></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="E18:75"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="glib_005.html#K2" target="_self">.flow</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K24">.edgesDBetween</a> <font color="Maroon">c<sub>9</sub></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">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="glib_005.html#E8:75"><i><font color="Green">E46</font></i></a>, <a class="txt" href="glib_005.html#E14:75"><i><font color="Green">E51</font></i></a>, <a class="txt" href="glib_005.html#E17:75"><i><font color="Green">E52</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="75_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>17</sub></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="E1:75_6"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="glib_000.html#R5">==</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a>  <a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="glib_005.html#R1" target="_self">has_valid_flow_from</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 ;<br/><a NAME="E2:75_6"/><b>then </b><i><font color="Green">E55</font></i>: 
 <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">c<sub>17</sub></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">c<sub>18</sub></font> = <font color="Maroon">c<sub>9</sub></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">c<sub>17</sub></font></span>)</span> <b>by </b><i><a class="txt" href="glib_005.html#E1:75_6"><i><font color="Green">E54</font></i></a>, <a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>;<br/><b>set </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>17</sub></font> <a href="glib_000.html#K24">.edgesDBetween</a> <font color="Maroon">c<sub>18</sub></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">c<sub>17</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>;<br/><a NAME="E4:75_6"/>
<font color="Maroon">c<sub>17</sub></font> <a href="glib_000.html#K24">.edgesDBetween</a> <font color="Maroon">c<sub>18</sub></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">c<sub>17</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K24">.edgesDBetween</a> <font color="Maroon">c<sub>9</sub></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">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_005.html#E2:75_6"><i><font color="Green">E55</font></i></a>, <a class="txt" href="glib_005.html#E1:75_6"><i><font color="Green">E54</font></i></a>, <a class="ref" href="glib_000.html#T93">GLIB_000:93</a></i>;<br/><b>hence </b><a NAME="E5:75_6"/>
<font color="Maroon">c<sub>17</sub></font> <a href="glib_005.html#K2" target="_self">.flow</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="glib_005.html#K2" target="_self">.flow</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="glib_005.html#E1:75_6"><i><font color="Green">E54</font></i></a>, <a class="txt" href="glib_005.html#E6:75"><i><font color="Green">E44</font></i></a>, <a class="txt" href="glib_005.html#E1:75"><i><font color="Green">E41</font></i></a>, <a class="txt" href="glib_005.html#E7:75"><i><font color="Green">E45</font></i></a>, <a class="ref" href="glib_005.html#T12" target="_self">Th12</a>, <a class="txt" href="glib_005.html#E18:75"><i><font color="Green">E53</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E20:75"/>
<font color="Maroon">c<sub>1</sub></font> <a href="glib_005.html#R2" target="_self">has_maximum_flow_from</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="glib_005.html#E1:75"><i><font color="Green">E41</font></i></a>, <a class="ref" href="glib_005.html#D9" target="_self">Def9</a></i>;<br/>


</div>
