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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>(  ex <font color="Olive" title="b1">Gn</font> being <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="glib_003.html#V7" title="GLIB_003:attr.7">real-weighted</a> <a href="glib_003.html#V9" title="GLIB_003:attr.9">real-elabeled</a> <a href="glib_005.html#V1" title="GLIB_005:attr.1">complete-elabeled</a> <a href="glib_003.html#NM4" title="GLIB_003:NM.4">WEGraph</a> ex <font color="Olive" title="b2">source'</font>, <font color="Olive" title="b3">sink'</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Olive" title="b1">Gn</font> st <br/>( $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Gn</font> &amp; <font color="Olive" title="b2">source'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">source</font> &amp; <font color="Olive" title="b3">sink'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">sink</font> &amp; $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_005.html#K13" title="GLIB_005:func.13">FF:Step</a> <font color="Olive" title="b1">Gn</font>,<font color="Olive" title="b2">source'</font>,<font color="Olive" title="b3">sink'</font> ) or ( ( for <font color="Olive" title="b1">Gn</font> being <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="glib_003.html#V7" title="GLIB_003:attr.7">real-weighted</a> <a href="glib_003.html#V9" title="GLIB_003:attr.9">real-elabeled</a> <a href="glib_005.html#V1" title="GLIB_005:attr.1">complete-elabeled</a> <a href="glib_003.html#NM4" title="GLIB_003:NM.4">WEGraph</a><br/> for <font color="Olive" title="b2">source'</font>, <font color="Olive" title="b3">sink'</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Olive" title="b1">Gn</font>  holds <br/>(  not $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Gn</font> or  not <font color="Olive" title="b2">source'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">source</font> or  not <font color="Olive" title="b3">sink'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">sink</font> ) ) &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $3 ) );<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">n</font>, <font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:70_1_1_1_1_1"/>
(  ex <font color="Olive" title="b1">Gn</font> being <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="glib_003.html#V7" title="GLIB_003:attr.7">real-weighted</a> <a href="glib_003.html#V9" title="GLIB_003:attr.9">real-elabeled</a> <a href="glib_005.html#V1" title="GLIB_005:attr.1">complete-elabeled</a> <a href="glib_003.html#NM4" title="GLIB_003:NM.4">WEGraph</a> ex <font color="Olive" title="b2">source'</font>, <font color="Olive" title="b3">sink'</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Olive" title="b1">Gn</font> st <br/>( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Gn</font> &amp; <font color="Olive" title="b2">source'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">source</font> &amp; <font color="Olive" title="b3">sink'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">sink</font> ) or for <font color="Olive" title="b1">Gn</font> being <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="glib_003.html#V7" title="GLIB_003:attr.7">real-weighted</a> <a href="glib_003.html#V9" title="GLIB_003:attr.9">real-elabeled</a> <a href="glib_005.html#V1" title="GLIB_005:attr.1">complete-elabeled</a> <a href="glib_003.html#NM4" title="GLIB_003:NM.4">WEGraph</a><br/> for <font color="Olive" title="b2">source'</font>, <font color="Olive" title="b3">sink'</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Olive" title="b1">Gn</font>  holds <br/>(  not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Gn</font> or  not <font color="Olive" title="b2">source'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">source</font> or  not <font color="Olive" title="b3">sink'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">sink</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:70_1_1_1_1_1_1"/>
 ex <font color="Olive" title="b1">Gn</font> being <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="glib_003.html#V7" title="GLIB_003:attr.7">real-weighted</a> <a href="glib_003.html#V9" title="GLIB_003:attr.9">real-elabeled</a> <a href="glib_005.html#V1" title="GLIB_005:attr.1">complete-elabeled</a> <a href="glib_003.html#NM4" title="GLIB_003:NM.4">WEGraph</a> ex <font color="Olive" title="b2">source'</font>, <font color="Olive" title="b3">sink'</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Olive" title="b1">Gn</font> st <br/>( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Gn</font> &amp; <font color="Olive" title="b2">source'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">source</font> &amp; <font color="Olive" title="b3">sink'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">sink</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c6">Gn</font> being  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="glib_003.html#V7" title="GLIB_003:attr.7">real-weighted</a> <a href="glib_003.html#V9" title="GLIB_003:attr.9">real-elabeled</a> <a href="glib_005.html#V1" title="GLIB_005:attr.1">complete-elabeled</a> <a href="glib_003.html#NM4" title="GLIB_003:NM.4">WEGraph</a>, <font color="Maroon" title="c7">source'</font>, <font color="Maroon" title="c8">sink'</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c6">Gn</font><b> such that </b><br/><a NAME="E3:70_1_1_1_1_1_1"/><i><font color="Green" title="E30">A1</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Gn</font> &amp; <font color="Maroon" title="c7">source'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">source</font> &amp; <font color="Maroon" title="c8">sink'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">sink</font> )
 ;<br/><b>set </b><font color="Maroon" title="c9">y</font> =  <a href="glib_005.html#K13" title="GLIB_005:func.13">FF:Step</a> <font color="Maroon" title="c6">Gn</font>,<font color="Maroon" title="c7">source'</font>,<font color="Maroon" title="c8">sink'</font>;<br/><a NAME="E4:70_1_1_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c5">x</font>, <a href="glib_005.html#K13" title="GLIB_005:func.13">FF:Step</a> <font color="Maroon" title="c6">Gn</font>,<font color="Maroon" title="c7">source'</font>,<font color="Maroon" title="c8">sink'</font>]
 <b>by </b><i><a class="txt" href="glib_005.html#E3:70_1_1_1_1_1_1"><i><font color="Green" title="E30">A1</font></i></a></i>;<br/><b>hence </b><a NAME="E5:70_1_1_1_1_1_1"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c5">x</font>,<font color="Olive" title="b1">y</font>]
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:70_1_1_1_1_1_2"/>
for <font color="Olive" title="b1">Gn</font> being <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="glib_003.html#V7" title="GLIB_003:attr.7">real-weighted</a> <a href="glib_003.html#V9" title="GLIB_003:attr.9">real-elabeled</a> <a href="glib_005.html#V1" title="GLIB_005:attr.1">complete-elabeled</a> <a href="glib_003.html#NM4" title="GLIB_003:NM.4">WEGraph</a><br/> for <font color="Olive" title="b2">source'</font>, <font color="Olive" title="b3">sink'</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Olive" title="b1">Gn</font>  holds <br/>(  not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Gn</font> or  not <font color="Olive" title="b2">source'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">source</font> or  not <font color="Olive" title="b3">sink'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">sink</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:70_1_1_1_1_1_2"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c5">x</font>,<font color="Olive" title="b1">y</font>]
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E2:70_1_1_1"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c5">x</font>,<font color="Olive" title="b1">y</font>]
 ;<br/></div><b>end;</b></div>
<a NAME="E2:70_1_1"/><b>then </b><i><font color="Green" title="E30">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> <br/> for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   ex <font color="Olive" title="b3">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font>]
 
;<br/>
<a NAME="E3:70_1_1"/><i><font color="Green" title="E31">A3</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> <br/> for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y1</font>, <font color="Olive" title="b4">y2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b4">y2</font>] holds <br/><font color="Olive" title="b3">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">y2</font>
 
;<br/>
<b>consider </b><font color="Maroon" title="c4">IT</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E5:70_1_1"/><i><font color="Green" title="E32">A4</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c4">IT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K3" title="GLIB_003:func.3">ELabelSelector</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> 0</span>)</span> &amp; ( 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>,<font color="Maroon" title="c4">IT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c4">IT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>] ) )
 <b>from </b><i><a class="ref" href="recdef_1.html#S1" title="RECDEF_1:sch.1">RECDEF_1:sch 1</a>(<a class="txt" href="glib_005.html#E2:70_1_1"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="glib_005.html#E3:70_1_1"><i><font color="Green" title="E31">A3</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c5">IT</font> = <font color="Maroon" title="c4">IT</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="glib_005.html#E5:70_1_1"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b> ex <font color="Olive" title="b1">Gn</font> being <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="glib_003.html#V7" title="GLIB_003:attr.7">real-weighted</a> <a href="glib_003.html#V9" title="GLIB_003:attr.9">real-elabeled</a> <a href="glib_005.html#V1" title="GLIB_005:attr.1">complete-elabeled</a> <a href="glib_003.html#NM4" title="GLIB_003:NM.4">WEGraph</a> st <br/>( <font color="Maroon" title="c5">IT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Gn</font> &amp; <font color="Olive" title="b1">Gn</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c1">G</font> );<br/>
<b>reconsider </b><font color="Maroon" title="c6">E0</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> 0 as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="ref" href="funcop_1.html#T58" title="FUNCOP_1:th.58">FUNCOP_1:58</a></i>;<br/>
<b>set </b><font color="Maroon" title="c7">G0</font> = <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> ,<font color="Maroon" title="c6">E0</font>;<br/>
<a NAME="E8:70_1_1"/>
<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> ,<font color="Maroon" title="c6">E0</font> <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="E9:70_1_1"/><b>then </b><i><font color="Green" title="E33">A5</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<b>by </b><i><a class="txt" href="glib_005.html#E5:70_1_1"><i><font color="Green" title="E32">A4</font></i></a></i>;<br/>
<div><i><font color="Green" title="E34">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:70_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">n</font>]
 ;<br/><b>then consider </b><font color="Maroon" title="c9">Gn</font> being  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="glib_003.html#V7" title="GLIB_003:attr.7">real-weighted</a> <a href="glib_003.html#V9" title="GLIB_003:attr.9">real-elabeled</a> <a href="glib_005.html#V1" title="GLIB_005:attr.1">complete-elabeled</a> <a href="glib_003.html#NM4" title="GLIB_003:NM.4">WEGraph</a><b> such that </b><br/><a NAME="E3:70_1_1_2"/><i><font color="Green" title="E35">A7</font></i>: 
( <font color="Maroon" title="c5">IT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">Gn</font> &amp; <font color="Maroon" title="c9">Gn</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c1">G</font> )
 ;<br/><b>reconsider </b><font color="Maroon" title="c10">source'</font> = <font color="Maroon" title="c2">source</font>, <font color="Maroon" title="c11">sink'</font> = <font color="Maroon" title="c3">sink</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c9">Gn</font> <b>by </b><i><a class="txt" href="glib_005.html#E3:70_1_1_2"><i><font color="Green" title="E35">A7</font></i></a>, <a class="ref" href="glib_000.html#D36" title="GLIB_000:def.36">GLIB_000:def 36</a></i>;<br/><a NAME="E5:70_1_1_2"/><i><font color="Green" title="E36">A8</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">n</font>,<font color="Maroon" title="c9">Gn</font>,<font color="Maroon" title="c5">IT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>]
 <b>by </b><i><a class="txt" href="glib_005.html#E5:70_1_1"><i><font color="Green" title="E32">A4</font></i></a>, <a class="txt" href="glib_005.html#E3:70_1_1_2"><i><font color="Green" title="E35">A7</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c12">P</font> =  <a href="glib_005.html#K7" title="GLIB_005:func.7">AP:GetAugPath</a> <font color="Maroon" title="c9">Gn</font>,<font color="Maroon" title="c10">source'</font>,<font color="Maroon" title="c11">sink'</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:70_1_1_2_1_1"/>
( <font color="Maroon" title="c3">sink</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c9">Gn</font>,<font color="Maroon" title="c10">source'</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  or  not <font color="Maroon" title="c3">sink</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c9">Gn</font>,<font color="Maroon" title="c10">source'</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:70_1_1_2_1_1_1"/>
<font color="Maroon" title="c3">sink</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c9">Gn</font>,<font color="Maroon" title="c10">source'</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> 
 ;<br/><div class="add"><b>then </b><font color="Maroon" title="c5">IT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
 <a href="glib_005.html#K12" title="GLIB_005:func.12">FF:AugmentPath</a> <font color="Maroon" title="c9">Gn</font>,<span class="p1">(<span class="default"><a href="glib_005.html#K7" title="GLIB_005:func.7">AP:GetAugPath</a> <font color="Maroon" title="c9">Gn</font>,<font color="Maroon" title="c10">source'</font>,<font color="Maroon" title="c11">sink'</font></span>)</span>
<b>by </b><i><a class="txt" href="glib_005.html#E5:70_1_1_2"><i><font color="Green" title="E36">A8</font></i></a>, <a class="ref" href="glib_005.html#D22" target="_self" title="GLIB_005:def.22">Def22</a></i>
<br/>.= 
<font color="Maroon" title="c9">Gn</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"><a href="glib_005.html#K11" title="GLIB_005:func.11">FF:PushFlow</a> <font color="Maroon" title="c9">Gn</font>,<span class="p2">(<span class="default"><a href="glib_005.html#K7" title="GLIB_005:func.7">AP:GetAugPath</a> <font color="Maroon" title="c9">Gn</font>,<font color="Maroon" title="c10">source'</font>,<font color="Maroon" title="c11">sink'</font></span>)</span></span>)</span>

;<br/><b>hence </b><a NAME="E3:70_1_1_2_1_1_1"/>
 <a href="glib_005.html#K13" title="GLIB_005:func.13">FF:Step</a> <font color="Maroon" title="c9">Gn</font>,<font color="Maroon" title="c10">source'</font>,<font color="Maroon" title="c11">sink'</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c9">Gn</font>
 <b>by </b><i><a class="txt" href="glib_005.html#E5:70_1_1_2"><i><font color="Green" title="E36">A8</font></i></a>, <a class="ref" href="glib_003.html#T8" title="GLIB_003:th.8">GLIB_003:8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:70_1_1_2_1_1_2"/>
not <font color="Maroon" title="c3">sink</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c9">Gn</font>,<font color="Maroon" title="c10">source'</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:70_1_1_2_1_1_2"/>
 <a href="glib_005.html#K13" title="GLIB_005:func.13">FF:Step</a> <font color="Maroon" title="c9">Gn</font>,<font color="Maroon" title="c10">source'</font>,<font color="Maroon" title="c11">sink'</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c9">Gn</font>
 <b>by </b><i><a class="ref" href="glib_005.html#D22" target="_self" title="GLIB_005:def.22">Def22</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E7:70_1_1_2"/><b>then </b>
 <a href="glib_005.html#K13" title="GLIB_005:func.13">FF:Step</a> <font color="Maroon" title="c9">Gn</font>,<font color="Maroon" title="c10">source'</font>,<font color="Maroon" title="c11">sink'</font> <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="txt" href="glib_005.html#E3:70_1_1_2"><i><font color="Green" title="E35">A7</font></i></a>, <a class="ref" href="glib_000.html#T88" title="GLIB_000:th.88">GLIB_000:88</a></i>;<br/><b>hence </b><a NAME="E8:70_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <b>by </b><i><a class="txt" href="glib_005.html#E5:70_1_1_2"><i><font color="Green" title="E36">A8</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:70_1_1"/><i><font color="Green" title="E35">A9</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>2</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#E9:70_1_1"><i><font color="Green" title="E33">A5</font></i></a>, <a class="txt" href="glib_005.html#E10:70_1_1"><i><font color="Green" title="E34">A6</font></i></a>);<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">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>consider </b><font color="Maroon" title="c9">Gn</font> being  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="glib_003.html#V7" title="GLIB_003:attr.7">real-weighted</a> <a href="glib_003.html#V9" title="GLIB_003:attr.9">real-elabeled</a> <a href="glib_005.html#V1" title="GLIB_005:attr.1">complete-elabeled</a> <a href="glib_003.html#NM4" title="GLIB_003:NM.4">WEGraph</a><b> such that </b><br/><a NAME="E2:70_1_1_3"/><i><font color="Green" title="E36">A10</font></i>: 
( <font color="Maroon" title="c5">IT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">Gn</font> &amp; <font color="Maroon" title="c9">Gn</font> <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="txt" href="glib_005.html#E11:70_1_1"><i><font color="Green" title="E35">A9</font></i></a></i>;<br/><b>thus </b><a NAME="E3:70_1_1_3"/>
<font color="Maroon" title="c5">IT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> is   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>
 <b>by </b><i><a class="txt" href="glib_005.html#E2:70_1_1_3"><i><font color="Green" title="E36">A10</font></i></a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c8">IT</font> = <font color="Maroon" title="c5">IT</font> as   <a href="glib_000.html#NM11" title="GLIB_000:NM.11">GraphSeq</a> <b>by </b><i><a class="ref" href="glib_000.html#D55" title="GLIB_000:def.55">GLIB_000:def 55</a></i>;<br/>
<div><i><font color="Green" title="E36">A11</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><a NAME="E1:70_1_1_4"/>
<font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>then consider </b><font color="Maroon" title="c10">Gn</font> being  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="glib_003.html#V7" title="GLIB_003:attr.7">real-weighted</a> <a href="glib_003.html#V9" title="GLIB_003:attr.9">real-elabeled</a> <a href="glib_005.html#V1" title="GLIB_005:attr.1">complete-elabeled</a> <a href="glib_003.html#NM4" title="GLIB_003:NM.4">WEGraph</a><b> such that </b><br/><a NAME="E3:70_1_1_4"/><i><font color="Green" title="E37">A12</font></i>: 
( <font color="Maroon" title="c8">IT</font> <a href="glib_000.html#K42" title="GLIB_000:func.42">.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Gn</font> &amp; <font color="Maroon" title="c10">Gn</font> <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="txt" href="glib_005.html#E11:70_1_1"><i><font color="Green" title="E35">A9</font></i></a></i>;<br/><b>thus </b><a NAME="E4:70_1_1_4"/>
<font color="Maroon" title="c8">IT</font> <a href="glib_000.html#K42" title="GLIB_000:func.42">.</a> <font color="Maroon" title="c9">n</font> is  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="glib_003.html#V7" title="GLIB_003:attr.7">real-weighted</a> <a href="glib_003.html#V9" title="GLIB_003:attr.9">real-elabeled</a> <a href="glib_005.html#V1" title="GLIB_005:attr.1">complete-elabeled</a> <a href="glib_003.html#NM4" title="GLIB_003:NM.4">WEGraph</a>
 <b>by </b><i><a class="txt" href="glib_005.html#E3:70_1_1_4"><i><font color="Green" title="E37">A12</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E15:70_1_1"/><b>then </b>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds <br/> ( <font color="Maroon" title="c8">IT</font> <a href="glib_000.html#K42" title="GLIB_000:func.42">.</a> <font color="Olive" title="b1">n</font> is <a href="glib_003.html#V1" title="GLIB_003:attr.1">[Weighted]</a> &amp; <font color="Maroon" title="c8">IT</font> <a href="glib_000.html#K42" title="GLIB_000:func.42">.</a> <font color="Olive" title="b1">n</font> is <a href="glib_003.html#V2" title="GLIB_003:attr.2">[ELabeled]</a> )
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">IT</font> = <font color="Maroon" title="c8">IT</font> as   <a href="glib_003.html#NM32" title="GLIB_003:NM.32">WEGraphSeq</a> <b>by </b><i><a class="ref" href="glib_003.html#D24" title="GLIB_003:def.24">GLIB_003:def 24</a>, <a class="ref" href="glib_003.html#D25" title="GLIB_003:def.25">GLIB_003:def 25</a></i>;<br/>
<a NAME="E17:70_1_1"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds <br/> ( <font color="Maroon" title="c9">IT</font> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> <font color="Olive" title="b1">n</font> is <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> &amp; <font color="Maroon" title="c9">IT</font> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> <font color="Olive" title="b1">n</font> is <a href="glib_003.html#V7" title="GLIB_003:attr.7">real-weighted</a> &amp; <font color="Maroon" title="c9">IT</font> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> <font color="Olive" title="b1">n</font> is <a href="glib_003.html#V9" title="GLIB_003:attr.9">real-elabeled</a> &amp; <font color="Maroon" title="c9">IT</font> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> <font color="Olive" title="b1">n</font> is <a href="glib_005.html#V1" title="GLIB_005:attr.1">complete-elabeled</a> )
 
<b>by </b><i><a class="txt" href="glib_005.html#E14:70_1_1"><i><font color="Green" title="E36">A11</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">IT</font> = <font color="Maroon" title="c9">IT</font> as  <a href="glib_000.html#V14" title="GLIB_000:attr.14">finite</a> <a href="glib_003.html#V15" title="GLIB_003:attr.15">real-weighted</a> <a href="glib_003.html#V17" title="GLIB_003:attr.17">real-elabeled</a> <a href="glib_005.html#V2" title="GLIB_005:attr.2">complete-elabeled</a> <a href="glib_003.html#NM32" title="GLIB_003:NM.32">WEGraphSeq</a> <b>by </b><i><a class="ref" href="glib_000.html#D60" title="GLIB_000:def.60">GLIB_000:def 60</a>, <a class="ref" href="glib_003.html#D27" title="GLIB_003:def.27">GLIB_003:def 27</a>, <a class="ref" href="glib_003.html#D29" title="GLIB_003:def.29">GLIB_003:def 29</a>, <a class="ref" href="glib_005.html#D2" target="_self" title="GLIB_005:def.2">Def2</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c10">IT</font>
;<br/>

<b>thus </b><a NAME="E19:70_1_1"/>
<font color="Maroon" title="c10">IT</font> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K3" title="GLIB_003:func.3">ELabelSelector</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> 0</span>)</span>
 <b>by </b><i><a class="txt" href="glib_005.html#E5:70_1_1"><i><font color="Green" title="E32">A4</font></i></a></i>;<br/>

<b>let </b><font color="Maroon" title="c11">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>reconsider </b><font color="Maroon" title="c12">n'</font> = <font color="Maroon" title="c11">n</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="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">Gn</font> being  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="glib_003.html#V7" title="GLIB_003:attr.7">real-weighted</a> <a href="glib_003.html#V9" title="GLIB_003:attr.9">real-elabeled</a> <a href="glib_005.html#V1" title="GLIB_005:attr.1">complete-elabeled</a> <a href="glib_003.html#NM4" title="GLIB_003:NM.4">WEGraph</a><b> such that </b><br/><a NAME="E22:70_1_1"/><i><font color="Green" title="E37">A13</font></i>: 
( <font color="Maroon" title="c10">IT</font> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> <font color="Maroon" title="c12">n'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">Gn</font> &amp; <font color="Maroon" title="c13">Gn</font> <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="txt" href="glib_005.html#E11:70_1_1"><i><font color="Green" title="E35">A9</font></i></a></i>;<br/>
<a NAME="E23:70_1_1"/>
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c13">Gn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="txt" href="glib_005.html#E22:70_1_1"><i><font color="Green" title="E37">A13</font></i></a>, <a class="ref" href="glib_000.html#D36" title="GLIB_000:def.36">GLIB_000:def 36</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c14">source'</font> = <font color="Maroon" title="c2">source</font>, <font color="Maroon" title="c15">sink'</font> = <font color="Maroon" title="c3">sink</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c10">IT</font> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <b>by </b><i><a class="txt" href="glib_005.html#E22:70_1_1"><i><font color="Green" title="E37">A13</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c14">source'</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c15">sink'</font>
;<br/>

<b>thus </b><a NAME="E25:70_1_1"/>
( <font color="Maroon" title="c14">source'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">source</font> &amp; <font color="Maroon" title="c15">sink'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">sink</font> )
 ;<br/>

<a NAME="E26:70_1_1"/>
( <font color="Maroon" title="c10">IT</font> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> <font color="Maroon" title="c11">n</font> is  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="glib_003.html#V7" title="GLIB_003:attr.7">real-weighted</a> <a href="glib_003.html#V9" title="GLIB_003:attr.9">real-elabeled</a> <a href="glib_005.html#V1" title="GLIB_005:attr.1">complete-elabeled</a> <a href="glib_003.html#NM4" title="GLIB_003:NM.4">WEGraph</a> &amp; <font color="Maroon" title="c14">source'</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c10">IT</font> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> <font color="Maroon" title="c11">n</font></span>)</span> &amp; <font color="Maroon" title="c15">sink'</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c10">IT</font> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> <font color="Maroon" title="c11">n</font></span>)</span> &amp; <font color="Maroon" title="c10">IT</font> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">Gn</font> &amp; <font color="Maroon" title="c14">source'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">source</font> &amp; <font color="Maroon" title="c15">sink'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">sink</font> )
 
<b>by </b><i><a class="txt" href="glib_005.html#E22:70_1_1"><i><font color="Green" title="E37">A13</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c16">X</font> being  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="glib_003.html#V7" title="GLIB_003:attr.7">real-weighted</a> <a href="glib_003.html#V9" title="GLIB_003:attr.9">real-elabeled</a> <a href="glib_005.html#V1" title="GLIB_005:attr.1">complete-elabeled</a> <a href="glib_003.html#NM4" title="GLIB_003:NM.4">WEGraph</a>, <font color="Maroon" title="c17">sourceX</font>, <font color="Maroon" title="c18">sinkX</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c16">X</font><b> such that </b><br/><a NAME="E28:70_1_1"/><i><font color="Green" title="E38">A14</font></i>: 
( <font color="Maroon" title="c16">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">Gn</font> &amp; <font color="Maroon" title="c17">sourceX</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">source</font> &amp; <font color="Maroon" title="c18">sinkX</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">sink</font> &amp; <font color="Maroon" title="c10">IT</font> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_005.html#K13" title="GLIB_005:func.13">FF:Step</a> <font color="Maroon" title="c16">X</font>,<font color="Maroon" title="c17">sourceX</font>,<font color="Maroon" title="c18">sinkX</font> )
 <b>by </b><i><a class="txt" href="glib_005.html#E5:70_1_1"><i><font color="Green" title="E32">A4</font></i></a></i>;<br/>
<a NAME="E29:70_1_1"/><i><font color="Green" title="E39">A15</font></i>: 
( <font color="Maroon" title="c16">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">IT</font> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> <font color="Maroon" title="c11">n</font> &amp; <font color="Maroon" title="c17">sourceX</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">source'</font> &amp; <font color="Maroon" title="c15">sink'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">sink</font> )
 
<b>by </b><i><a class="txt" href="glib_005.html#E22:70_1_1"><i><font color="Green" title="E37">A13</font></i></a>, <a class="txt" href="glib_005.html#E28:70_1_1"><i><font color="Green" title="E38">A14</font></i></a></i>;<br/>
<b>thus </b><a NAME="E30:70_1_1"/>
<font color="Maroon" title="c10">IT</font> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0: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"><font color="Maroon" title="c10">IT</font> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> <font color="Maroon" title="c11">n</font></span>)</span>,<font color="Maroon" title="c14">source'</font>,<font color="Maroon" title="c15">sink'</font>
 <b>by </b><i><a class="txt" href="glib_005.html#E28:70_1_1"><i><font color="Green" title="E38">A14</font></i></a>, <a class="txt" href="glib_005.html#E29:70_1_1"><i><font color="Green" title="E39">A15</font></i></a></i>;<br/>


</div>
