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<div class="add">

<b>let </b><font color="Maroon" title="c1">G</font> be  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="glib_003.html#V8" title="GLIB_003:attr.8">nonnegative-weighted</a> <a href="glib_003.html#NM1" title="GLIB_003:NM.1">WGraph</a>;<br/><b>let </b><font color="Maroon" title="c2">source</font>, <font color="Maroon" title="c3">sink</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>;<br/><b>let </b><font color="Maroon" title="c4">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>





<b>assume </b><a NAME="E1:80"/><i><font color="Green" title="E39">A1</font></i>: 
<font color="Maroon" title="c2">source</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">sink</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c5">CS</font> =  <a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b><span class="p1">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> $1 <a href="glib_005.html#R1" title="GLIB_005:pred.1">has_valid_flow_from</a> <font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font>;<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" title="c6">G0</font> = <span class="p1">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> 0;<br/><a NAME="E1:80_1"/><i><font color="Green" title="E40">A2</font></i>: 
<span class="p1">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K3" title="GLIB_003:func.3">ELabelSelector</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> 0</span>)</span>
 <b>by </b><i><a class="ref" href="glib_005.html#D23" target="_self" title="GLIB_005:def.23">Def23</a></i>;<br/><a NAME="E2:80_1"/><b>then </b><i><font color="Green" title="E41">A3</font></i>: 
<span class="p1">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> 0 <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="ref" href="glib_003.html#T8" title="GLIB_003:th.8">GLIB_003:8</a></i>;<br/><b>hence </b><a NAME="E3:80_1"/>
( <font color="Maroon" title="c2">source</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> 0</span>)</span> &amp; <font color="Maroon" title="c3">sink</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> 0</span>)</span> )
 <b>by </b><i><a class="ref" href="glib_000.html#D36" title="GLIB_000:def.36">GLIB_000:def 36</a></i>;<br/><a NAME="E4:80_1"/><i><font color="Green" title="E42">A4</font></i>: 
 <a href="glib_005.html#K1" title="GLIB_005:func.1">the_ELabel_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> 0</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> 0
 <b>by </b><i><a class="txt" href="glib_005.html#E1:80_1"><i><font color="Green" title="E40">A2</font></i></a>, <a class="ref" href="glib_000.html#T11" title="GLIB_000:th.11">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" title="c7">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:80_1_1"/>
<font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> 0</span>)</span>
 ;<br/><a NAME="E2:80_1_1"/><b>then </b><i><font color="Green" title="E43">A5</font></i>: 
<font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="glib_005.html#E2:80_1"><i><font color="Green" title="E41">A3</font></i></a>, <a class="ref" href="glib_000.html#D36" title="GLIB_000:def.36">GLIB_000:def 36</a></i>;<br/><b>hence </b><a NAME="E3:80_1_1"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K1" title="GLIB_005:func.1">the_ELabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> 0</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font>
 <b>by </b><i><a class="txt" href="glib_005.html#E4:80_1"><i><font color="Green" title="E42">A4</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/><a NAME="E4:80_1_1"/>
<span class="p1">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> 0 is <a href="glib_003.html#V8" title="GLIB_003:attr.8">nonnegative-weighted</a>
 <b>by </b><i><a class="ref" href="glib_003.html#D28" title="GLIB_003:def.28">GLIB_003:def 28</a></i>;<br/><a NAME="E5:80_1_1"/><b>then </b><i><font color="Green" title="E44">A6</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> 0</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="graph_5.html#K8" title="GRAPH_5:func.8">Real&gt;=0</a> 
 <b>by </b><i><a class="ref" href="glib_003.html#D14" title="GLIB_003:def.14">GLIB_003:def 14</a></i>;<br/> <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> 0</span>)</span></span>)</span> = 
 <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
<br/>.= 
 <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font>
<b>by </b><i><a class="txt" href="glib_005.html#E2:80_1"><i><font color="Green" title="E41">A3</font></i></a>, <a class="ref" href="glib_000.html#D36" title="GLIB_000:def.36">GLIB_000:def 36</a></i>
;<br/><a NAME="E7:80_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> 0</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> 0</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_005.html#E2:80_1_1"><i><font color="Green" title="E43">A5</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">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" title="GLIB_003:func.5">the_Weight_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> 0</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_5.html#K8" title="GRAPH_5:func.8">Real&gt;=0</a> 
 <b>by </b><i><a class="txt" href="glib_005.html#E5:80_1_1"><i><font color="Green" title="E44">A6</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c8">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E10:80_1_1"/><i><font color="Green" title="E45">A7</font></i>: 
( <font color="Maroon" title="c8">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> 0</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font> &amp; <font color="Maroon" title="c8">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 )
 <b>by </b><i><a class="ref" href="graph_5.html#D12" title="GRAPH_5:def.12">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" title="GLIB_005:func.1">the_ELabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> 0</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> 0</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font>
 <b>by </b><i><a class="txt" href="glib_005.html#E2:80_1_1"><i><font color="Green" title="E43">A5</font></i></a>, <a class="txt" href="glib_005.html#E4:80_1"><i><font color="Green" title="E42">A4</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a>, <a class="txt" href="glib_005.html#E10:80_1_1"><i><font color="Green" title="E45">A7</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:80_1"/>
for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> 0</span>)</span> holds <br/>( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K1" title="GLIB_005:func.1">the_ELabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> 0</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">e</font> &amp; <span class="p1">(<span class="default"><a href="glib_005.html#K1" title="GLIB_005:func.1">the_ELabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> 0</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> 0</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">e</font> )
 ;<br/><b>let </b><font color="Maroon" title="c7">v</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> 0</span>)</span>;<br/><b>assume </b><a NAME="E7:80_1"/>
( <font color="Maroon" title="c7">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">source</font> &amp; <font color="Maroon" title="c7">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">sink</font> )
 ;<br/><b>set </b><font color="Maroon" title="c8">B1</font> =  <a href="polynom1.html#K16" title="POLYNOM1:func.16">EmptyBag</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span>;<br/><b>set </b><font color="Maroon" title="c9">B2</font> =  <a href="polynom1.html#K16" title="POLYNOM1:func.16">EmptyBag</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span>;<br/><b>set </b><font color="Maroon" title="c10">E1</font> = <span class="p1">(<span class="default"><a href="glib_005.html#K1" title="GLIB_005:func.1">the_ELabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> 0</span>)</span></span>)</span> <a href="pre_circ.html#K3" title="PRE_CIRC:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span>;<br/><b>set </b><font color="Maroon" title="c11">E2</font> = <span class="p1">(<span class="default"><a href="glib_005.html#K1" title="GLIB_005:func.1">the_ELabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> 0</span>)</span></span>)</span> <a href="pre_circ.html#K3" title="PRE_CIRC:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.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" title="c12">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:80_1_2"/><i><font color="Green" title="E43">A8</font></i>: 
<font color="Maroon" title="c12">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> 
 ;<br/><a NAME="E2:80_1_2"/><b>then </b>
<font color="Maroon" title="c12">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> 0</span>)</span>
 ;<br/><a NAME="E3:80_1_2"/><b>then </b><i><font color="Green" title="E44">A9</font></i>: 
<font color="Maroon" title="c12">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="glib_005.html#E2:80_1"><i><font color="Green" title="E41">A3</font></i></a>, <a class="ref" href="glib_000.html#D36" title="GLIB_000:def.36">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" title="GLIB_005:func.1">the_ELabel_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> 0</span>)</span></span>)</span> <a href="pre_circ.html#K3" title="PRE_CIRC:func.3">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">e</font> = 
<span class="p1">(<span class="default"><a href="glib_005.html#K1" title="GLIB_005:func.1">the_ELabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> 0</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">e</font>
<b>by </b><i><a class="txt" href="glib_005.html#E1:80_1_2"><i><font color="Green" title="E43">A8</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="glib_005.html#E4:80_1"><i><font color="Green" title="E42">A4</font></i></a>, <a class="txt" href="glib_005.html#E3:80_1_2"><i><font color="Green" title="E44">A9</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">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" title="POLYNOM1:func.16">EmptyBag</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c12">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" title="GLIB_005:func.1">the_ELabel_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> 0</span>)</span></span>)</span> <a href="pre_circ.html#K3" title="PRE_CIRC:func.3">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">e</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T56" title="POLYNOM1:th.56">POLYNOM1:56</a></i>;<br/></div><b>end;</b></div><b>then </b><i><font color="Green" title="E43">A10</font></i>:  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" title="GLIB_005:func.1">the_ELabel_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> 0</span>)</span></span>)</span> <a href="pre_circ.html#K3" title="PRE_CIRC:func.3">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> = 
 <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16" title="POLYNOM1:func.16">EmptyBag</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="glib_004.html#T6" title="GLIB_004:th.6">GLIB_004:6</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="uproots.html#T13" title="UPROOTS:th.13">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" title="c12">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:80_1_4"/><i><font color="Green" title="E44">A11</font></i>: 
<font color="Maroon" title="c12">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> 
 ;<br/><a NAME="E2:80_1_4"/><b>then </b>
<font color="Maroon" title="c12">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> 0</span>)</span>
 ;<br/><a NAME="E3:80_1_4"/><b>then </b><i><font color="Green" title="E45">A12</font></i>: 
<font color="Maroon" title="c12">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="glib_005.html#E2:80_1"><i><font color="Green" title="E41">A3</font></i></a>, <a class="ref" href="glib_000.html#D36" title="GLIB_000:def.36">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" title="GLIB_005:func.1">the_ELabel_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> 0</span>)</span></span>)</span> <a href="pre_circ.html#K3" title="PRE_CIRC:func.3">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">e</font> = 
<span class="p1">(<span class="default"><a href="glib_005.html#K1" title="GLIB_005:func.1">the_ELabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> 0</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">e</font>
<b>by </b><i><a class="txt" href="glib_005.html#E1:80_1_4"><i><font color="Green" title="E44">A11</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="glib_005.html#E4:80_1"><i><font color="Green" title="E42">A4</font></i></a>, <a class="txt" href="glib_005.html#E3:80_1_4"><i><font color="Green" title="E45">A12</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">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" title="POLYNOM1:func.16">EmptyBag</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c12">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" title="GLIB_005:func.1">the_ELabel_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> 0</span>)</span></span>)</span> <a href="pre_circ.html#K3" title="PRE_CIRC:func.3">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">e</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T56" title="POLYNOM1:th.56">POLYNOM1:56</a></i>;<br/></div><b>end;</b></div><b>then </b> <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" title="GLIB_005:func.1">the_ELabel_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> 0</span>)</span></span>)</span> <a href="pre_circ.html#K3" title="PRE_CIRC:func.3">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> = 
 <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16" title="POLYNOM1:func.16">EmptyBag</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="glib_004.html#T6" title="GLIB_004:th.6">GLIB_004:6</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="uproots.html#T13" title="UPROOTS:th.13">UPROOTS:13</a></i>
;<br/><b>hence </b><a NAME="E12:80_1"/>
 <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" title="GLIB_005:func.1">the_ELabel_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> 0</span>)</span></span>)</span> <a href="pre_circ.html#K3" title="PRE_CIRC:func.3">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" title="GLIB_005:func.1">the_ELabel_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> 0</span>)</span></span>)</span> <a href="pre_circ.html#K3" title="PRE_CIRC:func.3">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_005.html#E9:80_1"><i><font color="Green" title="E43">A10</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:80"/><b>then </b><i><font color="Green" title="E40">A13</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="ref" href="glib_005.html#D7" target="_self" title="GLIB_005:def.7">Def7</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>set </b><font color="Maroon" title="c7">Gn</font> = <span class="p1">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> <font color="Maroon" title="c6">n</font>;<br/><b>set </b><font color="Maroon" title="c8">Gn1</font> = <span class="p1">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/><b>assume </b><a NAME="E1:80_2"/><i><font color="Green" title="E41">A14</font></i>: 
<span class="p1">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> <font color="Maroon" title="c6">n</font> <a href="glib_005.html#R1" title="GLIB_005:pred.1">has_valid_flow_from</a> <font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font>
 ;<br/><b>consider </b><font color="Maroon" title="c9">source'</font>, <font color="Maroon" title="c10">sink'</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> <font color="Maroon" title="c6">n</font></span>)</span><b> such that </b><br/><a NAME="E3:80_2"/><i><font color="Green" title="E42">A15</font></i>: 
( <font color="Maroon" title="c9">source'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">source</font> &amp; <font color="Maroon" title="c10">sink'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">sink</font> &amp; <span class="p1">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_005.html#K13" title="GLIB_005:func.13">FF:Step</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> <font color="Maroon" title="c6">n</font></span>)</span>,<font color="Maroon" title="c9">source'</font>,<font color="Maroon" title="c10">sink'</font> )
 <b>by </b><i><a class="ref" href="glib_005.html#D23" target="_self" title="GLIB_005:def.23">Def23</a></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" title="c3">sink</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> <font color="Maroon" title="c6">n</font></span>)</span>,<font color="Maroon" title="c9">source'</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  or  not <font color="Maroon" title="c3">sink</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> <font color="Maroon" title="c6">n</font></span>)</span>,<font color="Maroon" title="c9">source'</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.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" title="E43">A16</font></i>: 
<font color="Maroon" title="c3">sink</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> <font color="Maroon" title="c6">n</font></span>)</span>,<font color="Maroon" title="c9">source'</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> 
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c11">P</font> =  <a href="glib_005.html#K7" title="GLIB_005:func.7">AP:GetAugPath</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> <font color="Maroon" title="c6">n</font></span>)</span>,<font color="Maroon" title="c9">source'</font>,<font color="Maroon" title="c10">sink'</font>;<br/><a NAME="E2:80_2_1_1_1"/><i><font color="Green" title="E44">A17</font></i>: 
<span class="p1">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_005.html#K12" title="GLIB_005:func.12">FF:AugmentPath</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> <font color="Maroon" title="c6">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="glib_005.html#K7" title="GLIB_005:func.7">AP:GetAugPath</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> <font color="Maroon" title="c6">n</font></span>)</span>,<font color="Maroon" title="c9">source'</font>,<font color="Maroon" title="c10">sink'</font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_005.html#E3:80_2"><i><font color="Green" title="E42">A15</font></i></a>, <a class="txt" href="glib_005.html#E1:80_2_1_1_1"><i><font color="Green" title="E43">A16</font></i></a>, <a class="ref" href="glib_005.html#D22" target="_self" title="GLIB_005:def.22">Def22</a></i>;<br/><a NAME="E3:80_2_1_1_1"/>
 <a href="glib_005.html#K7" title="GLIB_005:func.7">AP:GetAugPath</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> <font color="Maroon" title="c6">n</font></span>)</span>,<font color="Maroon" title="c9">source'</font>,<font color="Maroon" title="c10">sink'</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c9">source'</font>,<font color="Maroon" title="c10">sink'</font>
 <b>by </b><i><a class="txt" href="glib_005.html#E3:80_2"><i><font color="Green" title="E42">A15</font></i></a>, <a class="txt" href="glib_005.html#E1:80_2_1_1_1"><i><font color="Green" title="E43">A16</font></i></a>, <a class="ref" href="glib_005.html#D17" target="_self" title="GLIB_005:def.17">Def17</a></i>;<br/><b>hence </b><a NAME="E4:80_2_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <b>by </b><i><a class="txt" href="glib_005.html#E2:80_2_1_1_1"><i><font color="Green" title="E44">A17</font></i></a>, <a class="txt" href="glib_005.html#E1:80"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="glib_005.html#E1:80_2"><i><font color="Green" title="E41">A14</font></i></a>, <a class="txt" href="glib_005.html#E3:80_2"><i><font color="Green" title="E42">A15</font></i></a>, <a class="ref" href="glib_005.html#T15" target="_self" title="GLIB_005:th.15">Th15</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" title="c3">sink</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> <font color="Maroon" title="c6">n</font></span>)</span>,<font color="Maroon" title="c9">source'</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.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" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <b>by </b><i><a class="txt" href="glib_005.html#E1:80_2"><i><font color="Green" title="E41">A14</font></i></a>, <a class="txt" href="glib_005.html#E3:80_2"><i><font color="Green" title="E42">A15</font></i></a>, <a class="ref" href="glib_005.html#D22" target="_self" title="GLIB_005:def.22">Def22</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" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E5:80"/><b>then </b><i><font color="Green" title="E41">A18</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
;<br/>
<a NAME="E6:80"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="glib_005.html#E3:80"><i><font color="Green" title="E40">A13</font></i></a>, <a class="txt" href="glib_005.html#E5:80"><i><font color="Green" title="E41">A18</font></i></a>);<br/></i>
<b>hence </b><a NAME="E7:80"/>
<span class="p1">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> <font color="Maroon" title="c4">n</font> <a href="glib_005.html#R1" title="GLIB_005:pred.1">has_valid_flow_from</a> <font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font>
 ;<br/>


</div>
