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

<b>let </b><font color="Maroon">G</font> be  <a href="glib_000.html#V2">finite</a> <a href="glib_003.html#V8">nonnegative-weighted</a> <a href="glib_003.html#NM1">WGraph</a>;<br/><b>let </b><font color="Maroon">source</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>, <font color="Maroon">sink</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>





<b>assume </b><a NAME="E1:80"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">source</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">sink</font>
 ;<br/>

<b>set </b><font color="Maroon">CS</font> =  <a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><span class="p1">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="glib_005.html#R1" target="_self">has_valid_flow_from</a> <font color="Maroon">source</font>,<font color="Maroon">sink</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1"><b>now </b></a><div class="add"><b>set </b><font color="Maroon">G0</font> = <span class="p1">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> 0;<br/><a NAME="E1:80_1"/><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K3">ELabelSelector</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> 0</span>)</span>
 <b>by </b><i/>;<br/><a NAME="E2:80_1"/><b>then </b><i><font color="Green">E82</font></i>: 
<span class="p1">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> 0 <a href="glib_000.html#R5">==</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="glib_003.html#T8">GLIB_003:8</a></i>;<br/><b>hence </b><a NAME="E3:80_1"/>
( <font color="Maroon">source</font> is   <a href="glib_000.html#NM10">Vertex</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> 0</span>)</span> &amp; <font color="Maroon">sink</font> is   <a href="glib_000.html#NM10">Vertex</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> 0</span>)</span> )
 <b>by </b><i><a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>;<br/><a NAME="E4:80_1"/><i><font color="Green">E89</font></i>: 
 <a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="glib_000.html#T11">GLIB_000:11</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:80_1_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K7">the_Edges_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> 0</span>)</span>
 ;<br/><a NAME="E2:80_1_1"/><b>then </b><i><font color="Green">E92</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="glib_005.html#D1" target="_self">Def1</a>, <a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>;<br/><b>hence </b><a NAME="E3:80_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> 0</span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
 <b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><a NAME="E4:80_1_1"/>
<span class="p1">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> 0 is <a href="glib_003.html#V8">nonnegative-weighted</a>
 <b>by </b><i><a class="ref" href="glib_003.html#D28">GLIB_003:def 28</a></i>;<br/><a NAME="E5:80_1_1"/><b>then </b><i><font color="Green">E93</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> 0</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="graph_5.html#K8">Real&gt;=0</a> 
 <b>by </b><i><a class="ref" href="glib_003.html#D14">GLIB_003:def 14</a></i>;<br/> <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> 0</span>)</span></span>)</span> = 
 <a href="glib_000.html#K7">the_Edges_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
 <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font>
<b>by </b><i><a class="ref" href="glib_005.html#D1" target="_self">Def1</a>, <a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>
;<br/><a NAME="E7:80_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> 0</span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> 0</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E8:80_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> 0</span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="graph_5.html#K8">Real&gt;=0</a> 
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E10:80_1_1"/><i><font color="Green">E94</font></i>: 
( <font color="Maroon">r</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"><span class="p3">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> 0</span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i><a class="ref" href="graph_5.html#D12">GRAPH_5:def 12</a></i>;<br/><b>thus </b><a NAME="E11:80_1_1"/>
<span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> 0</span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> 0</span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
 <b>by </b><i>, , <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:80_1"/>
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K7">the_Edges_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> 0</span>)</span> holds <br/>( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> 0</span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Olive">e</font> &amp; <span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> 0</span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Olive">e</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> 0</span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Olive">e</font> )
 ;<br/><b>let </b><font color="Maroon">v</font> be   <a href="glib_000.html#NM10">Vertex</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> 0</span>)</span>;<br/><b>assume </b><a NAME="E7:80_1"/>
( <font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">source</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">sink</font> )
 ;<br/><b>set </b><font color="Maroon">B1</font> =  <a href="polynom1.html#K16">EmptyBag</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span>;<br/><b>set </b><font color="Maroon">B2</font> =  <a href="polynom1.html#K16">EmptyBag</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span>;<br/><b>set </b><font color="Maroon">E1</font> = <span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> 0</span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span>;<br/><b>set </b><font color="Maroon">E2</font> = <span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> 0</span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:80_1_2"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font> <a href="glib_000.html#K27">.edgesIn()</a> 
 ;<br/><a NAME="E2:80_1_2"/><b>then </b>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K7">the_Edges_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> 0</span>)</span>
 ;<br/><a NAME="E3:80_1_2"/><b>then </b><i><font color="Green">E96</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="glib_005.html#D1" target="_self">Def1</a>, <a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> 0</span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> = 
<span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> 0</span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
0
<b>by </b><i>, , <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/><b>hence </b><a NAME="E5:80_1_2"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> 0</span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T56">POLYNOM1:56</a></i>;<br/></div><b>end;</b></div><b>then </b><i><font color="Green">E109</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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> 0</span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span> = 
 <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">v</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="glib_004.html#T6">GLIB_004:6</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="uproots.html#T13">UPROOTS:13</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:80_1_4"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font> <a href="glib_000.html#K28">.edgesOut()</a> 
 ;<br/><a NAME="E2:80_1_4"/><b>then </b>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K7">the_Edges_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> 0</span>)</span>
 ;<br/><a NAME="E3:80_1_4"/><b>then </b><i><font color="Green">E111</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="glib_005.html#D1" target="_self">Def1</a>, <a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> 0</span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> = 
<span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> 0</span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
0
<b>by </b><i>, , <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/><b>hence </b><a NAME="E5:80_1_4"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> 0</span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T56">POLYNOM1:56</a></i>;<br/></div><b>end;</b></div><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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> 0</span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span> = 
 <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">v</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="glib_004.html#T6">GLIB_004:6</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="uproots.html#T13">UPROOTS:13</a></i>
;<br/><b>hence </b><a NAME="E12:80_1"/>
 <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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> 0</span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span> <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_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> 0</span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E3:80"/><b>then </b><i><font color="Green">E112</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>set </b><font color="Maroon">Gn</font> = <span class="p1">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> <font color="Maroon">n</font>;<br/><b>set </b><font color="Maroon">Gn1</font> = <span class="p1">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>assume </b><a NAME="E1:80_2"/><i><font color="Green">E128</font></i>: 
<span class="p1">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> <font color="Maroon">n</font> <a href="glib_005.html#R1" target="_self">has_valid_flow_from</a> <font color="Maroon">source</font>,<font color="Maroon">sink</font>
 ;<br/><b>consider </b><font color="Maroon">source'</font> being   <a href="glib_000.html#NM10">Vertex</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">sink'</font> being   <a href="glib_000.html#NM10">Vertex</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E3:80_2"/><i><font color="Green">E129</font></i>: 
( <font color="Maroon">source'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">source</font> &amp; <font color="Maroon">sink'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">sink</font> &amp; <span class="p1">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_005.html#K13" target="_self">FF:Step</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">source'</font>,<font color="Maroon">sink'</font> )
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:80_2_1_1"/>
( <font color="Maroon">sink</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">source'</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  or  not <font color="Maroon">sink</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">source'</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_2_1_1_1"><b>suppose </b></a><a NAME="E1:80_2_1_1_1"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">sink</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">source'</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 ;<br/><div class="add"><b>set </b><font color="Maroon">P</font> =  <a href="glib_005.html#K7" target="_self">AP:GetAugPath</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">source'</font>,<font color="Maroon">sink'</font>;<br/><a NAME="E2:80_2_1_1_1"/><i><font color="Green">E131</font></i>: 
<span class="p1">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="glib_005.html#K7" target="_self">AP:GetAugPath</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">source'</font>,<font color="Maroon">sink'</font></span>)</span>
 <b>by </b><i>, , </i>;<br/><a NAME="E3:80_2_1_1_1"/>
 <a href="glib_005.html#K7" target="_self">AP:GetAugPath</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">source'</font>,<font color="Maroon">sink'</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">source'</font>,<font color="Maroon">sink'</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E4:80_2_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i>, , <a class="ref" href="glib_005.html#D1" target="_self">Def1</a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_2_1_1_2"><b>suppose </b></a><a NAME="E1:80_2_1_1_2"/>
not <font color="Maroon">sink</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">source'</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:80_2_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="ref" href="glib_005.html#D1" target="_self">Def1</a>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:80_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E5:80"/><b>then </b><i><font color="Green">E132</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 
;<br/>
<a NAME="E6:80"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, <a class="ref" href="glib_005.html#D1" target="_self">Def1</a>);<br/></i>
<b>hence </b><a NAME="E7:80"/>
<span class="p1">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> <font color="Maroon">n</font> <a href="glib_005.html#R1" target="_self">has_valid_flow_from</a> <font color="Maroon">source</font>,<font color="Maroon">sink</font>
 ;<br/>


</div>
