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

<span class="kw">let </span><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#V1" title="GLIB_005:attr.1">natural-weighted</a>  <a href="glib_003.html#NM1" title="GLIB_003:NM.1">WGraph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">source</font>, <font color="Olive" title="b2">sink</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font><br/> for <font color="Olive" title="b3">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">source</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">sink</font> holds <br/><span class="p1">(<span class="default"><a href="glib_005.html#K13" title="GLIB_005:func.13">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Olive" title="b1">source</font>,<font color="Olive" title="b2">sink</font></span>)</span> <a href="glib_005.html#K12" title="GLIB_005:func.12">.</a> <font color="Olive" title="b3">n</font> <a href="glib_005.html#R1" title="GLIB_005:pred.1">has_valid_flow_from</a> <font color="Olive" title="b1">source</font>,<font color="Olive" title="b2">sink</font></span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <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> holds <br/><span class="p1">(<span class="default"><a href="glib_005.html#K13" title="GLIB_005:func.13">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#K12" title="GLIB_005:func.12">.</a> <font color="Olive" title="b1">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></span><br/><span class="kw">let </span><font color="Maroon" title="c4">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">source</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">sink</font> implies <span class="p1">(<span class="default"><a href="glib_005.html#K13" title="GLIB_005:func.13">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#K12" title="GLIB_005:func.12">.</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> )</span><br/>





<span class="kw">set </span><font color="Maroon" title="c5">CS</font> =  <a href="glib_005.html#K13" title="GLIB_005:func.13">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/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span><span class="p1">(<span class="default"><a href="glib_005.html#K13" title="GLIB_005:func.13">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#K12" title="GLIB_005:func.12">.</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="47_1"><span class="kw">now </span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c6">G0</font> = <span class="p1">(<span class="default"><a href="glib_005.html#K13" title="GLIB_005:func.13">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#K12" title="GLIB_005:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/><a NAME="E1:47_1"/><span class="lab"><font color="Green" title="E39">A1</font></span>: 
<span class="p1">(<span class="default"><a href="glib_005.html#K13" title="GLIB_005:func.13">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#K12" title="GLIB_005:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="pboole.html#R6" title="PBOOLE:pred.6">=</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="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_005.html#D20" target="_self" title="GLIB_005:def.20">Def20</a></span>;<br/><span class="kw">hence </span><a NAME="E2:47_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> <font color="Maroon" title="c1">G</font> holds <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K13" title="GLIB_005:func.13">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#K12" title="GLIB_005:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K13" title="GLIB_005:func.13">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#K12" title="GLIB_005:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_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> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">v</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">source</font> &amp; <font color="Olive" title="b1">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">sink</font> holds <br/> <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K13" title="GLIB_005:func.13">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#K12" title="GLIB_005:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">v</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.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"><span class="p3">(<span class="default"><a href="glib_005.html#K13" title="GLIB_005:func.13">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#K12" title="GLIB_005:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a> </span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c7">v</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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> implies  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K13" title="GLIB_005:func.13">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#K12" title="GLIB_005:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.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"><span class="p3">(<span class="default"><a href="glib_005.html#K13" title="GLIB_005:func.13">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#K12" title="GLIB_005:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</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">.edgesOut()</a> </span>)</span></span>)</span> )</span><br/><span class="kw">set </span><font color="Maroon" title="c8">B1</font> =  <a href="pre_poly.html#K17" title="PRE_POLY:func.17">EmptyBag</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a> </span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c9">B2</font> =  <a href="pre_poly.html#K17" title="PRE_POLY:func.17">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">.edgesOut()</a> </span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c10">E1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K13" title="GLIB_005:func.13">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#K12" title="GLIB_005:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a> </span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c11">E2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K13" title="GLIB_005:func.13">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#K12" title="GLIB_005:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</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">.edgesOut()</a> </span>)</span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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">.edgesOut()</a>  implies <span class="p1">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">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">.edgesOut()</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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"><span class="p3">(<span class="default"><a href="glib_005.html#K13" title="GLIB_005:func.13">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#K12" title="GLIB_005:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</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">.edgesOut()</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e</font> )</span><br/><span class="kw">assume </span><a NAME="E1:47_1_1"/><span class="lab"><font color="Green" title="E40">A2</font></span>: 
<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">.edgesOut()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">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">.edgesOut()</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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"><span class="p3">(<span class="default"><a href="glib_005.html#K13" title="GLIB_005:func.13">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#K12" title="GLIB_005:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</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">.edgesOut()</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e</font></span><br/><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K13" title="GLIB_005:func.13">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#K12" title="GLIB_005:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</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">.edgesOut()</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K13" title="GLIB_005:func.13">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#K12" title="GLIB_005:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="txt" href="#E1:47_1_1"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:47_1_1"/>
<span class="p1">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">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">.edgesOut()</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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"><span class="p3">(<span class="default"><a href="glib_005.html#K13" title="GLIB_005:func.13">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#K12" title="GLIB_005:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</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">.edgesOut()</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#T52" title="PRE_POLY:th.52">PRE_POLY:52</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A3</font></span>:  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K13" title="GLIB_005:func.13">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#K12" title="GLIB_005:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</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">.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="pre_poly.html#K17" title="PRE_POLY:func.17">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">.edgesOut()</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_004.html#T6" title="GLIB_004:th.6">GLIB_004:6</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="uproots.html#T13" title="UPROOTS:th.13">UPROOTS:13</a></span>
;<br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E5:47_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>
 <span class="kw">and </span><br/><a NAME="E6:47_1"/>
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K13" title="GLIB_005:func.13">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#K12" title="GLIB_005:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.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"><span class="p3">(<span class="default"><a href="glib_005.html#K13" title="GLIB_005:func.13">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#K12" title="GLIB_005:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</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">.edgesOut()</a> </span>)</span></span>)</span></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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#K26" title="GLIB_000:func.26">.edgesIn()</a>  implies <span class="p1">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">EmptyBag</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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"><span class="p3">(<span class="default"><a href="glib_005.html#K13" title="GLIB_005:func.13">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#K12" title="GLIB_005:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e</font> )</span><br/><span class="kw">assume </span><a NAME="E1:47_1_3"/><span class="lab"><font color="Green" title="E41">A4</font></span>: 
<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#K26" title="GLIB_000:func.26">.edgesIn()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">EmptyBag</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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"><span class="p3">(<span class="default"><a href="glib_005.html#K13" title="GLIB_005:func.13">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#K12" title="GLIB_005:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e</font></span><br/><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K13" title="GLIB_005:func.13">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#K12" title="GLIB_005:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K13" title="GLIB_005:func.13">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#K12" title="GLIB_005:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="txt" href="#E1:47_1_3"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:47_1_3"/>
<span class="p1">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">EmptyBag</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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"><span class="p3">(<span class="default"><a href="glib_005.html#K13" title="GLIB_005:func.13">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#K12" title="GLIB_005:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#T52" title="PRE_POLY:th.52">PRE_POLY:52</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">then </span> <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K13" title="GLIB_005:func.13">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#K12" title="GLIB_005:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.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="pre_poly.html#K17" title="PRE_POLY:func.17">EmptyBag</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_004.html#T6" title="GLIB_004:th.6">GLIB_004:6</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="uproots.html#T13" title="UPROOTS:th.13">UPROOTS:13</a></span>
;<br/><span class="kw">hence </span><a NAME="E9:47_1"/>
 <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K13" title="GLIB_005:func.13">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#K12" title="GLIB_005:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.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"><span class="p3">(<span class="default"><a href="glib_005.html#K13" title="GLIB_005:func.13">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#K12" title="GLIB_005:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</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">.edgesOut()</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:47_1"><span class="lab"><font color="Green" title="E40">A3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E2:47"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A5</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_005.html#D2" target="_self" title="GLIB_005:def.2">Def2</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:47"/><span class="lab"><font color="Green" title="E40">A6</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="glib_005.html#K13" title="GLIB_005:func.13">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#K12" title="GLIB_005:func.12">.</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></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="glib_005.html#K13" title="GLIB_005:func.13">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#K12" title="GLIB_005:func.12">.</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> implies <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] )</span><br/><span class="kw">set </span><font color="Maroon" title="c7">Gn</font> = <span class="p1">(<span class="default"><a href="glib_005.html#K13" title="GLIB_005:func.13">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#K12" title="GLIB_005:func.12">.</a> <font color="Maroon" title="c6">n</font>;<br/><span class="kw">set </span><font color="Maroon" title="c8">Gn1</font> = <span class="p1">(<span class="default"><a href="glib_005.html#K13" title="GLIB_005:func.13">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#K12" title="GLIB_005:func.12">.</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/><span class="kw">assume </span><a NAME="E1:47_2"/><span class="lab"><font color="Green" title="E41">A7</font></span>: 
<span class="p1">(<span class="default"><a href="glib_005.html#K13" title="GLIB_005:func.13">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#K12" title="GLIB_005:func.12">.</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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]</span><br/><a NAME="E2:47_2"/><span class="lab"><font color="Green" title="E42">A8</font></span>: 
<span class="p1">(<span class="default"><a href="glib_005.html#K13" title="GLIB_005:func.13">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#K12" title="GLIB_005:func.12">.</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="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="glib_005.html#K11" title="GLIB_005:func.11">FF:Step</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K13" title="GLIB_005:func.13">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#K12" title="GLIB_005:func.12">.</a> <font color="Maroon" title="c6">n</font></span>)</span>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_005.html#D20" target="_self" title="GLIB_005:def.20">Def20</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:47_2_1_1"/>
( <font color="Maroon" title="c3">sink</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</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#K13" title="GLIB_005:func.13">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#K12" title="GLIB_005:func.12">.</a> <font color="Maroon" title="c6">n</font></span>)</span>,<font color="Maroon" title="c2">source</font></span>)</span> or  not <font color="Maroon" title="c3">sink</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</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#K13" title="GLIB_005:func.13">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#K12" title="GLIB_005:func.12">.</a> <font color="Maroon" title="c6">n</font></span>)</span>,<font color="Maroon" title="c2">source</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:47_2_1_1_1"/><span class="lab"><font color="Green" title="E43">A9</font></span>: 
<font color="Maroon" title="c3">sink</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</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#K13" title="GLIB_005:func.13">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#K12" title="GLIB_005:func.12">.</a> <font color="Maroon" title="c6">n</font></span>)</span>,<font color="Maroon" title="c2">source</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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]</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c9">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#K13" title="GLIB_005:func.13">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#K12" title="GLIB_005:func.12">.</a> <font color="Maroon" title="c6">n</font></span>)</span>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font>;<br/><a NAME="E2:47_2_1_1_1"/><span class="lab"><font color="Green" title="E44">A10</font></span>: 
 <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#K13" title="GLIB_005:func.13">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#K12" title="GLIB_005:func.12">.</a> <font color="Maroon" title="c6">n</font></span>)</span>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_2_1_1_1"><span class="lab"><font color="Green" title="E43">A9</font></span></a>, <a class="ref" href="glib_005.html#D14" target="_self" title="GLIB_005:def.14">Def14</a></span>;<br/><a NAME="E3:47_2_1_1_1"/><span class="lab"><font color="Green" title="E45">A11</font></span>: 
 <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#K13" title="GLIB_005:func.13">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#K12" title="GLIB_005:func.12">.</a> <font color="Maroon" title="c6">n</font></span>)</span>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font> <a href="glib_005.html#R5" title="GLIB_005:pred.5">is_augmenting_wrt</a> <span class="p1">(<span class="default"><a href="glib_005.html#K13" title="GLIB_005:func.13">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#K12" title="GLIB_005:func.12">.</a> <font color="Maroon" title="c6">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_2_1_1_1"><span class="lab"><font color="Green" title="E43">A9</font></span></a>, <a class="ref" href="glib_005.html#D14" target="_self" title="GLIB_005:def.14">Def14</a></span>;<br/><a NAME="E4:47_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="glib_005.html#K13" title="GLIB_005:func.13">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#K12" title="GLIB_005:func.12">.</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="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K13" title="GLIB_005:func.13">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#K12" title="GLIB_005:func.12">.</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#K13" title="GLIB_005:func.13">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#K12" title="GLIB_005:func.12">.</a> <font color="Maroon" title="c6">n</font></span>)</span>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47_2"><span class="lab"><font color="Green" title="E42">A8</font></span></a>, <a class="txt" href="#E1:47_2_1_1_1"><span class="lab"><font color="Green" title="E43">A9</font></span></a>, <a class="ref" href="glib_005.html#D18" target="_self" title="GLIB_005:def.18">Def18</a></span>;<br/><span class="kw">hence </span><a NAME="E5:47_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]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47"><span class="lab"><font color="Green" title="E40">A6</font></span></a>, <a class="txt" href="#E1:47_2"><span class="lab"><font color="Green" title="E41">A7</font></span></a>, <a class="txt" href="#E2:47_2_1_1_1"><span class="lab"><font color="Green" title="E44">A10</font></span></a>, <a class="txt" href="#E3:47_2_1_1_1"><span class="lab"><font color="Green" title="E45">A11</font></span></a>, <a class="ref" href="glib_005.html#T14" target="_self" title="GLIB_005:th.14">Th14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:47_2_1_1_2"/>
not <font color="Maroon" title="c3">sink</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</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#K13" title="GLIB_005:func.13">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#K12" title="GLIB_005:func.12">.</a> <font color="Maroon" title="c6">n</font></span>)</span>,<font color="Maroon" title="c2">source</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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]</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:47_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]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_2"><span class="lab"><font color="Green" title="E41">A7</font></span></a>, <a class="txt" href="#E2:47_2"><span class="lab"><font color="Green" title="E42">A8</font></span></a>, <a class="ref" href="glib_005.html#D18" target="_self" title="GLIB_005:def.18">Def18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:47_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]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E5:47"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A12</font></span>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">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="E6:47"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E2:47"><span class="lab"><font color="Green" title="E39">A5</font></span></a>, <a class="txt" href="#E5:47"><span class="lab"><font color="Green" title="E41">A12</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E7:47"/>
<span class="p1">(<span class="default"><a href="glib_005.html#K13" title="GLIB_005:func.13">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#K12" title="GLIB_005:func.12">.</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
