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<b>let </b><font color="Maroon" title="c1">G</font> be  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="glib_005.html#V3" title="GLIB_005:attr.3">natural-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>assume </b><a NAME="E1:83"/><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="c4">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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:83_1"/><i><font color="Green" title="E40">A2</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>   holds <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="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <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="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ;<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>( $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#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</span>)</span> <a href="glib_005.html#K2" title="GLIB_005:func.2">.flow</a> <font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font> &amp; <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> $1</span>)</span> <a href="glib_005.html#K2" title="GLIB_005:func.2">.flow</a> <font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font> is    <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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1"><b>now </b></a><div class="add"><b>set </b><font color="Maroon" title="c5">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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/><b>set </b><font color="Maroon" title="c6">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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="pre_circ.html#K3" title="PRE_CIRC:func.3">|</a> <span class="p1">(<span class="default"><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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">sink</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>;<br/><b>set </b><font color="Maroon" title="c7">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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="pre_circ.html#K3" title="PRE_CIRC:func.3">|</a> <span class="p1">(<span class="default"><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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">sink</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>;<br/><a NAME="E1:83_1_1"/><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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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>
 <b>by </b><i><a class="txt" href="glib_005.html#E1:83"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="glib_005.html#T18" target="_self" title="GLIB_005:th.18">Th18</a></i>;<br/><a NAME="E2:83_1_1"/><b>then </b><i><font color="Green" title="E42">A4</font></i>: 
<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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="glib_005.html#K2" title="GLIB_005:func.2">.flow</a> <font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K1" title="GLIB_005:func.1">the_ELabel_of</a> <span class="p4">(<span class="default"><span class="p5">(<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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="pre_circ.html#K3" title="PRE_CIRC:func.3">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">sink</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K1" title="GLIB_005:func.1">the_ELabel_of</a> <span class="p4">(<span class="default"><span class="p5">(<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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="pre_circ.html#K3" title="PRE_CIRC:func.3">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">sink</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="glib_005.html#D8" target="_self" title="GLIB_005:def.8">Def8</a></i>;<br/><a NAME="E3:83_1_1"/><i><font color="Green" title="E43">A5</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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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="E4:83_1_1"/><b>then </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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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/><a NAME="E5:83_1_1"/><b>then </b><i><font color="Green" title="E44">A6</font></i>: 
 <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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font>
 <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="E6:83_1_1"/><i><font color="Green" title="E45">A7</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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="glib_005.html#E3:83_1_1"><i><font color="Green" title="E43">A5</font></i></a>, <a class="ref" href="glib_000.html#T11" title="GLIB_000:th.11">GLIB_000:11</a></i>;<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"><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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">sink</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></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"><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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">sink</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:83_1_1_1"/><i><font color="Green" title="E46">A8</font></i>: 
<font color="Maroon" title="c10">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">sink</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/><a NAME="E2:83_1_1_1"/><b>then </b>
<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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="pre_circ.html#K3" title="PRE_CIRC:func.3">|</a> <span class="p2">(<span class="default"><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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">sink</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c10">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c10">e</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/><b>hence </b><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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="pre_circ.html#K3" title="PRE_CIRC:func.3">|</a> <span class="p2">(<span class="default"><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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">sink</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c10">e</font> = 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="glib_005.html#E5:83_1_1"><i><font color="Green" title="E44">A6</font></i></a>, <a class="txt" href="glib_005.html#E6:83_1_1"><i><font color="Green" title="E45">A7</font></i></a>, <a class="txt" href="glib_005.html#E1:83_1_1_1"><i><font color="Green" title="E46">A8</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom1.html#K16" title="POLYNOM1:func.16">EmptyBag</a> <span class="p2">(<span class="default"><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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">sink</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c10">e</font>
<b>by </b><i><a class="ref" href="polynom1.html#T56" title="POLYNOM1:th.56">POLYNOM1:56</a></i>
;<br/><br/></div><b>end;</b></div><b>then </b><i><font color="Green" title="E46">A9</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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="pre_circ.html#K3" title="PRE_CIRC:func.3">|</a> <span class="p2">(<span class="default"><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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">sink</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></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"><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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">sink</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="glib_004.html#T6" title="GLIB_004:th.6">GLIB_004:6</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="uproots.html#T13" title="UPROOTS:th.13">UPROOTS:13</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:83_1_1_3"/><i><font color="Green" title="E47">A10</font></i>: 
<font color="Maroon" title="c10">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">sink</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/><a NAME="E2:83_1_1_3"/><b>then </b>
<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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="pre_circ.html#K3" title="PRE_CIRC:func.3">|</a> <span class="p2">(<span class="default"><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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">sink</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c10">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c10">e</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/><b>hence </b><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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="pre_circ.html#K3" title="PRE_CIRC:func.3">|</a> <span class="p2">(<span class="default"><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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">sink</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c10">e</font> = 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="glib_005.html#E5:83_1_1"><i><font color="Green" title="E44">A6</font></i></a>, <a class="txt" href="glib_005.html#E6:83_1_1"><i><font color="Green" title="E45">A7</font></i></a>, <a class="txt" href="glib_005.html#E1:83_1_1_3"><i><font color="Green" title="E47">A10</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom1.html#K16" title="POLYNOM1:func.16">EmptyBag</a> <span class="p2">(<span class="default"><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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">sink</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c10">e</font>
<b>by </b><i><a class="ref" href="polynom1.html#T56" title="POLYNOM1:th.56">POLYNOM1:56</a></i>
;<br/><br/></div><b>end;</b></div><b>then </b><i><font color="Green" title="E47">A11</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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="pre_circ.html#K3" title="PRE_CIRC:func.3">|</a> <span class="p2">(<span class="default"><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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">sink</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></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"><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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">sink</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="glib_004.html#T6" title="GLIB_004:th.6">GLIB_004:6</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="uproots.html#T13" title="UPROOTS:th.13">UPROOTS:13</a></i>
;<br/><b>hence </b><a NAME="E11:83_1_1"/>
<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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="glib_005.html#K2" title="GLIB_005:func.2">.flow</a> <font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="glib_005.html#E1:83_1_1"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="glib_005.html#E8:83_1_1"><i><font color="Green" title="E46">A9</font></i></a>, <a class="ref" href="glib_005.html#D8" target="_self" title="GLIB_005:def.8">Def8</a></i>;<br/><b>thus </b><a NAME="E12:83_1_1"/>
<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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="glib_005.html#K2" title="GLIB_005:func.2">.flow</a> <font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font> is    <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>by </b><i><a class="txt" href="glib_005.html#E2:83_1_1"><i><font color="Green" title="E42">A4</font></i></a>, <a class="txt" href="glib_005.html#E8:83_1_1"><i><font color="Green" title="E46">A9</font></i></a>, <a class="txt" href="glib_005.html#E10:83_1_1"><i><font color="Green" title="E47">A11</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E3:83_1"/><b>then </b><i><font color="Green" title="E41">A12</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">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="c6">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="c5">n</font>;<br/><b>set </b><font color="Maroon" title="c7">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="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/><b>assume </b><a NAME="E1:83_1_2"/><i><font color="Green" title="E42">A13</font></i>: 
( <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_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="c5">n</font></span>)</span> <a href="glib_005.html#K2" title="GLIB_005:func.2">.flow</a> <font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font> &amp; <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="c5">n</font></span>)</span> <a href="glib_005.html#K2" title="GLIB_005:func.2">.flow</a> <font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font> is    <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/><a NAME="E2:83_1_2"/><i><font color="Green" title="E43">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="c5">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> &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="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <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> )
 <b>by </b><i><a class="txt" href="glib_005.html#E1:83"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="glib_005.html#T18" target="_self" title="GLIB_005:th.18">Th18</a></i>;<br/><b>consider </b><font color="Maroon" title="c8">source'</font>, <font color="Maroon" title="c9">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="c5">n</font></span>)</span><b> such that </b><br/><a NAME="E4:83_1_2"/><i><font color="Green" title="E44">A15</font></i>: 
( <font color="Maroon" title="c8">source'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">source</font> &amp; <font color="Maroon" title="c9">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="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_005.html#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="c5">n</font></span>)</span>,<font color="Maroon" title="c8">source'</font>,<font color="Maroon" title="c9">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><i><font color="Green" title="E45">A16</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:83_1_2_1"/>
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="c5">n</font></span>)</span>,<font color="Maroon" title="c8">source'</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> 
 ;<br/><a NAME="E2:83_1_2_1"/><b>then </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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c5">n</font>
 <b>by </b><i><a class="txt" href="glib_005.html#E4:83_1_2"><i><font color="Green" title="E44">A15</font></i></a>, <a class="ref" href="glib_005.html#D22" target="_self" title="GLIB_005:def.22">Def22</a></i>;<br/><b>hence </b><a NAME="E3:83_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_005.html#E1:83_1"><i><font color="Green" title="E40">A2</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E6:83_1_2"/><b>then </b><i><font color="Green" title="E46">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="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_005.html#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="c5">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="c5">n</font></span>)</span>,<font color="Maroon" title="c8">source'</font>,<font color="Maroon" title="c9">sink'</font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_005.html#E4:83_1_2"><i><font color="Green" title="E44">A15</font></i></a>, <a class="ref" href="glib_005.html#D22" target="_self" title="GLIB_005:def.22">Def22</a></i>;<br/><b>set </b><font color="Maroon" title="c10">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="c5">n</font></span>)</span>,<font color="Maroon" title="c8">source'</font>,<font color="Maroon" title="c9">sink'</font>;<br/><a NAME="E7:83_1_2"/><i><font color="Green" title="E47">A18</font></i>: 
 <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="c5">n</font></span>)</span>,<font color="Maroon" title="c8">source'</font>,<font color="Maroon" title="c9">sink'</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c8">source'</font>,<font color="Maroon" title="c9">sink'</font>
 <b>by </b><i><a class="txt" href="glib_005.html#E4:83_1_2"><i><font color="Green" title="E44">A15</font></i></a>, <a class="txt" href="glib_005.html#E5:83_1_2"><i><font color="Green" title="E45">A16</font></i></a>, <a class="ref" href="glib_005.html#D17" target="_self" title="GLIB_005:def.17">Def17</a></i>;<br/><a NAME="E8:83_1_2"/><b>then </b>
( <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="c5">n</font></span>)</span>,<font color="Maroon" title="c8">source'</font>,<font color="Maroon" title="c9">sink'</font></span>)</span> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">source'</font> &amp; <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="c5">n</font></span>)</span>,<font color="Maroon" title="c8">source'</font>,<font color="Maroon" title="c9">sink'</font></span>)</span> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">sink'</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></i>;<br/><a NAME="E9:83_1_2"/><b>then </b>
not  <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="c5">n</font></span>)</span>,<font color="Maroon" title="c8">source'</font>,<font color="Maroon" title="c9">sink'</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a>
 <b>by </b><i><a class="txt" href="glib_005.html#E1:83"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="glib_005.html#E4:83_1_2"><i><font color="Green" title="E44">A15</font></i></a>, <a class="ref" href="glib_001.html#T128" title="GLIB_001:th.128">GLIB_001:128</a></i>;<br/><a NAME="E10:83_1_2"/><b>then </b><i><font color="Green" title="E48">A19</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><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="c5">n</font></span>)</span>,<font color="Maroon" title="c8">source'</font>,<font color="Maroon" title="c9">sink'</font></span>)</span> <a href="glib_005.html#K10" title="GLIB_005:func.10">.tolerance()</a> 
 <b>by </b><i><a class="ref" href="glib_005.html#T14" target="_self" title="GLIB_005:th.14">Th14</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">GnF</font> = <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="c5">n</font></span>)</span> <a href="glib_005.html#K2" title="GLIB_005:func.2">.flow</a> <font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font> as    <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>by </b><i><a class="txt" href="glib_005.html#E1:83_1_2"><i><font color="Green" title="E42">A13</font></i></a></i>;<br/><a NAME="E12:83_1_2"/><i><font color="Green" title="E49">A20</font></i>: 
<font color="Maroon" title="c11">GnF</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K7" title="GLIB_005:func.7">AP:GetAugPath</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> <font color="Maroon" title="c5">n</font></span>)</span>,<font color="Maroon" title="c8">source'</font>,<font color="Maroon" title="c9">sink'</font></span>)</span> <a href="glib_005.html#K10" title="GLIB_005:func.10">.tolerance()</a> </span>)</span> <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#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="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="glib_005.html#K2" title="GLIB_005:func.2">.flow</a> <font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font>
 <b>by </b><i><a class="txt" href="glib_005.html#E1:83"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="glib_005.html#E2:83_1_2"><i><font color="Green" title="E43">A14</font></i></a>, <a class="txt" href="glib_005.html#E4:83_1_2"><i><font color="Green" title="E44">A15</font></i></a>, <a class="txt" href="glib_005.html#E6:83_1_2"><i><font color="Green" title="E46">A17</font></i></a>, <a class="txt" href="glib_005.html#E7:83_1_2"><i><font color="Green" title="E47">A18</font></i></a>, <a class="ref" href="glib_005.html#T16" target="_self" title="GLIB_005:th.16">Th16</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c12">Gn1F</font> = <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> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="glib_005.html#K2" title="GLIB_005:func.2">.flow</a> <font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font> as    <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/><a NAME="E14:83_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">GnF</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K7" title="GLIB_005:func.7">AP:GetAugPath</a> <span class="p4">(<span class="default"><span class="p5">(<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="c5">n</font></span>)</span>,<font color="Maroon" title="c8">source'</font>,<font color="Maroon" title="c9">sink'</font></span>)</span> <a href="glib_005.html#K10" title="GLIB_005:func.10">.tolerance()</a> </span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K7" title="GLIB_005:func.7">AP:GetAugPath</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> <font color="Maroon" title="c5">n</font></span>)</span>,<font color="Maroon" title="c8">source'</font>,<font color="Maroon" title="c9">sink'</font></span>)</span> <a href="glib_005.html#K10" title="GLIB_005:func.10">.tolerance()</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c12">Gn1F</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="glib_005.html#E10:83_1_2"><i><font color="Green" title="E48">A19</font></i></a>, <a class="txt" href="glib_005.html#E12:83_1_2"><i><font color="Green" title="E49">A20</font></i></a>, <a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/><a NAME="E15:83_1_2"/><b>then </b>
<font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c12">Gn1F</font>
 <b>by </b><i><a class="txt" href="glib_005.html#E1:83_1_2"><i><font color="Green" title="E42">A13</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E16:83_1_2"/>
<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#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="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="glib_005.html#K2" title="GLIB_005:func.2">.flow</a> <font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>thus </b><a NAME="E17:83_1_2"/>
<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> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="glib_005.html#K2" title="GLIB_005:func.2">.flow</a> <font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font> is    <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>by </b><i><a class="txt" href="glib_005.html#E12:83_1_2"><i><font color="Green" title="E49">A20</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E5:83_1"/><b>then </b><i><font color="Green" title="E42">A21</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#K2" title="NAT_1:func.2">+</a> 1]
 ;<br/><a NAME="E6:83_1"/><i><font color="Green" title="E43">A22</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>  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#E3:83_1"><i><font color="Green" title="E41">A12</font></i></a>, <a class="txt" href="glib_005.html#E5:83_1"><i><font color="Green" title="E42">A21</font></i></a>);<br/></i><b>set </b><font color="Maroon" title="c5">V</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">source</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/><a NAME="E7:83_1"/><i><font color="Green" title="E44">A23</font></i>: 
( <font color="Maroon" title="c2">source</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">source</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; not <font color="Maroon" title="c3">sink</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">source</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <b>by </b><i><a class="txt" href="glib_005.html#E1:83"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>set </b><font color="Maroon" title="c6">W1</font> = <span class="p1">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="pre_circ.html#K3" title="PRE_CIRC:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">source</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">source</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>;<br/><a NAME="E8:83_1"/>
 <a href="uproots.html#K4" title="UPROOTS:func.4">degree</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="pre_circ.html#K3" title="PRE_CIRC:func.3">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">source</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">source</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></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_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="pre_circ.html#K3" title="PRE_CIRC:func.3">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">source</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">source</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c7">N</font> =  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="pre_circ.html#K3" title="PRE_CIRC:func.3">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">source</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">source</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> as    <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="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="c7">N</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/><b>reconsider </b><font color="Maroon" title="c9">Gn1F</font> = <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> <span class="p2">(<span class="default"><font color="Maroon" title="c7">N</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="glib_005.html#K2" title="GLIB_005:func.2">.flow</a> <font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font> as    <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>by </b><i><a class="txt" href="glib_005.html#E6:83_1"><i><font color="Green" title="E43">A22</font></i></a></i>;<br/><a NAME="E11:83_1"/>
<span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="pre_circ.html#K3" title="PRE_CIRC:func.3">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">source</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">source</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">Gn1F</font>
 <b>by </b><i><a class="txt" href="glib_005.html#E6:83_1"><i><font color="Green" title="E43">A22</font></i></a></i>;<br/><a NAME="E12:83_1"/><b>then </b><i><font color="Green" title="E45">A24</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_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="pre_circ.html#K3" title="PRE_CIRC:func.3">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">source</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">source</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#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="p2">(<span class="default"><font color="Maroon" title="c7">N</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="glib_005.html#K2" title="GLIB_005:func.2">.flow</a> <font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E13:83_1"/><i><font color="Green" title="E46">A25</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="c7">N</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c1">G</font> &amp;  <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <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> <span class="p2">(<span class="default"><font color="Maroon" title="c7">N</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="glib_005.html#T17" target="_self" title="GLIB_005:th.17">Th17</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c10">V'</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">source</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <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> <span class="p3">(<span class="default"><font color="Maroon" title="c7">N</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></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/><b>set </b><font color="Maroon" title="c11">W2</font> = <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> <span class="p3">(<span class="default"><font color="Maroon" title="c7">N</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="pre_circ.html#K3" title="PRE_CIRC:func.3">|</a> <span class="p1">(<span class="default"><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> <span class="p3">(<span class="default"><font color="Maroon" title="c7">N</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c10">V'</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p4">(<span class="default"><span class="p5">(<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="p5">(<span class="default"><font color="Maroon" title="c7">N</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c10">V'</font></span>)</span></span>)</span>;<br/><span class="p1">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="pre_circ.html#K3" title="PRE_CIRC:func.3">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">source</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">source</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> = 
<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> <span class="p3">(<span class="default"><font color="Maroon" title="c7">N</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c10">V'</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p4">(<span class="default"><span class="p5">(<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="p5">(<span class="default"><font color="Maroon" title="c7">N</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c10">V'</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="glib_005.html#E13:83_1"><i><font color="Green" title="E46">A25</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"><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> <span class="p3">(<span class="default"><font color="Maroon" title="c7">N</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="pre_circ.html#K3" title="PRE_CIRC:func.3">|</a> <span class="p1">(<span class="default"><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> <span class="p3">(<span class="default"><font color="Maroon" title="c7">N</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c10">V'</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p4">(<span class="default"><span class="p5">(<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="p5">(<span class="default"><font color="Maroon" title="c7">N</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c10">V'</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="glib_005.html#E13:83_1"><i><font color="Green" title="E46">A25</font></i></a>, <a class="ref" href="glib_000.html#T93" title="GLIB_000:th.93">GLIB_000:93</a></i>
;<br/><a NAME="E16:83_1"/><b>then </b><i><font color="Green" title="E47">A26</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_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="pre_circ.html#K3" title="PRE_CIRC:func.3">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">source</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">source</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></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_003.html#K5" title="GLIB_003:func.5">the_Weight_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> <span class="p4">(<span class="default"><font color="Maroon" title="c7">N</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="pre_circ.html#K3" title="PRE_CIRC:func.3">|</a> <span class="p2">(<span class="default"><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> <span class="p4">(<span class="default"><font color="Maroon" title="c7">N</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c10">V'</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p5">(<span class="default"><span class="p0">(<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="p0">(<span class="default"><font color="Maroon" title="c7">N</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c10">V'</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="glib_004.html#T7" title="GLIB_004:th.7">GLIB_004:7</a></i>;<br/><a NAME="E17:83_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> <span class="p1">(<span class="default"><font color="Maroon" title="c7">N</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <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>
 <b>by </b><i><a class="txt" href="glib_005.html#E1:83"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="glib_005.html#T18" target="_self" title="GLIB_005:th.18">Th18</a></i>;<br/><b>hence </b><a NAME="E18:83_1"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_005.html#E7:83_1"><i><font color="Green" title="E44">A23</font></i></a>, <a class="txt" href="glib_005.html#E12:83_1"><i><font color="Green" title="E45">A24</font></i></a>, <a class="txt" href="glib_005.html#E16:83_1"><i><font color="Green" title="E47">A26</font></i></a>, <a class="ref" href="glib_005.html#T12" target="_self" title="GLIB_005:th.12">Th12</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:83"/>
 <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> is <a href="glib_000.html#V13" title="GLIB_000:attr.13">halting</a>
 <b>by </b><i><a class="ref" href="glib_000.html#D68" title="GLIB_000:def.68">GLIB_000:def 68</a></i>;<br/>


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