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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>(  ex <font color="Olive">Gn</font> being <a href="glib_000.html#V2">finite</a> <a href="glib_003.html#V7">real-weighted</a> <a href="glib_003.html#V9">real-elabeled</a> <a href="glib_005.html#V1" target="_self">complete-elabeled</a> <a href="glib_003.html#NM4">WEGraph</a> ex <font color="Olive">source'</font>, <font color="Olive">sink'</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Olive">Gn</font> st <br/>( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">Gn</font> &amp; <font color="Olive">source'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">source</font> &amp; <font color="Olive">sink'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">sink</font> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="glib_005.html#K13" target="_self">FF:Step</a> <font color="Olive">Gn</font>,<font color="Olive">source'</font>,<font color="Olive">sink'</font> ) or ( ( for <font color="Olive">Gn</font> being <a href="glib_000.html#V2">finite</a> <a href="glib_003.html#V7">real-weighted</a> <a href="glib_003.html#V9">real-elabeled</a> <a href="glib_005.html#V1" target="_self">complete-elabeled</a> <a href="glib_003.html#NM4">WEGraph</a><br/> for <font color="Olive">source'</font>, <font color="Olive">sink'</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Olive">Gn</font>  holds <br/>(  not <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">Gn</font> or  not <font color="Olive">source'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">source</font> or  not <font color="Olive">sink'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">sink</font> ) ) &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>3</sub></font> ) );<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">n</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x</font> be    <a href="hidden.html#M1">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">Gn</font> being <a href="glib_000.html#V2">finite</a> <a href="glib_003.html#V7">real-weighted</a> <a href="glib_003.html#V9">real-elabeled</a> <a href="glib_005.html#V1" target="_self">complete-elabeled</a> <a href="glib_003.html#NM4">WEGraph</a> ex <font color="Olive">source'</font>, <font color="Olive">sink'</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Olive">Gn</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">Gn</font> &amp; <font color="Olive">source'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">source</font> &amp; <font color="Olive">sink'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">sink</font> ) or for <font color="Olive">Gn</font> being <a href="glib_000.html#V2">finite</a> <a href="glib_003.html#V7">real-weighted</a> <a href="glib_003.html#V9">real-elabeled</a> <a href="glib_005.html#V1" target="_self">complete-elabeled</a> <a href="glib_003.html#NM4">WEGraph</a><br/> for <font color="Olive">source'</font>, <font color="Olive">sink'</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Olive">Gn</font>  holds <br/>(  not <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">Gn</font> or  not <font color="Olive">source'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">source</font> or  not <font color="Olive">sink'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">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">Gn</font> being <a href="glib_000.html#V2">finite</a> <a href="glib_003.html#V7">real-weighted</a> <a href="glib_003.html#V9">real-elabeled</a> <a href="glib_005.html#V1" target="_self">complete-elabeled</a> <a href="glib_003.html#NM4">WEGraph</a> ex <font color="Olive">source'</font>, <font color="Olive">sink'</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Olive">Gn</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">Gn</font> &amp; <font color="Olive">source'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">source</font> &amp; <font color="Olive">sink'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">sink</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">Gn</font> being  <a href="glib_000.html#V2">finite</a> <a href="glib_003.html#V7">real-weighted</a> <a href="glib_003.html#V9">real-elabeled</a> <a href="glib_005.html#V1" target="_self">complete-elabeled</a> <a href="glib_003.html#NM4">WEGraph</a>, <font color="Maroon">source'</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Gn</font>, <font color="Maroon">sink'</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Gn</font><b> such that </b><br/><a NAME="E3:70_1_1_1_1_1_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Gn</font> &amp; <font color="Maroon">source'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">source</font> &amp; <font color="Maroon">sink'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">sink</font> )
 ;<br/><b>set </b><font color="Maroon">y</font> =  <a href="glib_005.html#K13" target="_self">FF:Step</a> <font color="Maroon">Gn</font>,<font color="Maroon">source'</font>,<font color="Maroon">sink'</font>;<br/><a NAME="E4:70_1_1_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">x</font>, <a href="glib_005.html#K13" target="_self">FF:Step</a> <font color="Maroon">Gn</font>,<font color="Maroon">source'</font>,<font color="Maroon">sink'</font>]
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:70_1_1_1_1_1_1"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">x</font>,<font color="Olive">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">Gn</font> being <a href="glib_000.html#V2">finite</a> <a href="glib_003.html#V7">real-weighted</a> <a href="glib_003.html#V9">real-elabeled</a> <a href="glib_005.html#V1" target="_self">complete-elabeled</a> <a href="glib_003.html#NM4">WEGraph</a><br/> for <font color="Olive">source'</font>, <font color="Olive">sink'</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Olive">Gn</font>  holds <br/>(  not <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">Gn</font> or  not <font color="Olive">source'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">source</font> or  not <font color="Olive">sink'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">sink</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:70_1_1_1_1_1_2"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">x</font>,<font color="Olive">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">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">x</font>,<font color="Olive">y</font>]
 ;<br/></div><b>end;</b></div>
<a NAME="E2:70_1_1"/><b>then </b><i><font color="Green">E51</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">x</font>,<font color="Olive">y</font>]
 
;<br/>
<a NAME="E3:70_1_1"/><i><font color="Green">E82</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">x</font>, <font color="Olive">y1</font>, <font color="Olive">y2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">x</font>,<font color="Olive">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">x</font>,<font color="Olive">y2</font>] holds <br/><font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">y2</font>
 
;<br/>
<b>consider </b><font color="Maroon">IT</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:70_1_1"/><i><font color="Green">E89</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">IT</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K3">ELabelSelector</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> 0</span>)</span> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font>,<font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>] ) )
 <b>from </b><i><a class="ref" href="recdef_1.html#S1">RECDEF_1:sch 1</a>(, );<br/></i>
<b>reconsider </b><font color="Maroon">IT</font> = <font color="Maroon">IT</font> as    <a href="pboole.html#M1">ManySortedSet</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b> ex <font color="Olive">Gn</font> being <a href="glib_000.html#V2">finite</a> <a href="glib_003.html#V7">real-weighted</a> <a href="glib_003.html#V9">real-elabeled</a> <a href="glib_005.html#V1" target="_self">complete-elabeled</a> <a href="glib_003.html#NM4">WEGraph</a> st <br/>( <font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">Gn</font> &amp; <font color="Olive">Gn</font> <a href="glib_000.html#R5">==</a> <font color="Maroon">G</font> );<br/>
<b>reconsider </b><font color="Maroon">E0</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> 0 as   <a href="funct_2.html#NM1">Function</a> of  <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font>, <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="funcop_1.html#T58">FUNCOP_1:58</a></i>;<br/>
<b>set </b><font color="Maroon">G0</font> = <font color="Maroon">G</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K3">ELabelSelector</a> ,<font color="Maroon">E0</font>;<br/>
<a NAME="E8:70_1_1"/>
<font color="Maroon">G</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K3">ELabelSelector</a> ,<font color="Maroon">E0</font> <a href="glib_000.html#R5">==</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="glib_003.html#T8">GLIB_003:8</a></i>;<br/>
<a NAME="E9:70_1_1"/><b>then </b><i><font color="Green">E92</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<b>by </b><i/>;<br/>
<div><i><font color="Green">E93</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">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:70_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>]
 ;<br/><b>then consider </b><font color="Maroon">Gn</font> being  <a href="glib_000.html#V2">finite</a> <a href="glib_003.html#V7">real-weighted</a> <a href="glib_003.html#V9">real-elabeled</a> <a href="glib_005.html#V1" target="_self">complete-elabeled</a> <a href="glib_003.html#NM4">WEGraph</a><b> such that </b><br/><a NAME="E3:70_1_1_2"/><i><font color="Green">E94</font></i>: 
( <font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Gn</font> &amp; <font color="Maroon">Gn</font> <a href="glib_000.html#R5">==</a> <font color="Maroon">G</font> )
 ;<br/><b>reconsider </b><font color="Maroon">source'</font> = <font color="Maroon">source</font>, <font color="Maroon">sink'</font> = <font color="Maroon">sink</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Gn</font> <b>by </b><i>, <a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>;<br/><a NAME="E5:70_1_1_2"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">Gn</font>,<font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>]
 <b>by </b><i>, </i>;<br/><b>set </b><font color="Maroon">P</font> =  <a href="glib_005.html#K7" target="_self">AP:GetAugPath</a> <font color="Maroon">Gn</font>,<font color="Maroon">source'</font>,<font color="Maroon">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">sink</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">Gn</font>,<font color="Maroon">source'</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  or  not <font color="Maroon">sink</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">Gn</font>,<font color="Maroon">source'</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:70_1_1_2_1_1_1"/>
<font color="Maroon">sink</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">Gn</font>,<font color="Maroon">source'</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 ;<br/><div class="add"><b>then </b><font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
 <a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">Gn</font>,<span class="p1">(<span class="default"><a href="glib_005.html#K7" target="_self">AP:GetAugPath</a> <font color="Maroon">Gn</font>,<font color="Maroon">source'</font>,<font color="Maroon">sink'</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">Gn</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K3">ELabelSelector</a> ,<span class="p1">(<span class="default"><a href="glib_005.html#K11" target="_self">FF:PushFlow</a> <font color="Maroon">Gn</font>,<span class="p2">(<span class="default"><a href="glib_005.html#K7" target="_self">AP:GetAugPath</a> <font color="Maroon">Gn</font>,<font color="Maroon">source'</font>,<font color="Maroon">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" target="_self">FF:Step</a> <font color="Maroon">Gn</font>,<font color="Maroon">source'</font>,<font color="Maroon">sink'</font> <a href="glib_000.html#R5">==</a> <font color="Maroon">Gn</font>
 <b>by </b><i>, <a class="ref" href="glib_003.html#T8">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">sink</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">Gn</font>,<font color="Maroon">source'</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:70_1_1_2_1_1_2"/>
 <a href="glib_005.html#K13" target="_self">FF:Step</a> <font color="Maroon">Gn</font>,<font color="Maroon">source'</font>,<font color="Maroon">sink'</font> <a href="glib_000.html#R5">==</a> <font color="Maroon">Gn</font>
 <b>by </b><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" target="_self">FF:Step</a> <font color="Maroon">Gn</font>,<font color="Maroon">source'</font>,<font color="Maroon">sink'</font> <a href="glib_000.html#R5">==</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#T88">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">n</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E11:70_1_1"/><i><font color="Green">E96</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</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">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>consider </b><font color="Maroon">Gn</font> being  <a href="glib_000.html#V2">finite</a> <a href="glib_003.html#V7">real-weighted</a> <a href="glib_003.html#V9">real-elabeled</a> <a href="glib_005.html#V1" target="_self">complete-elabeled</a> <a href="glib_003.html#NM4">WEGraph</a><b> such that </b><br/><a NAME="E2:70_1_1_3"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Gn</font> &amp; <font color="Maroon">Gn</font> <a href="glib_000.html#R5">==</a> <font color="Maroon">G</font> )
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E3:70_1_1_3"/>
<font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> is   <a href="glib_000.html#NM2">_Graph</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">IT</font> = <font color="Maroon">IT</font> as   <a href="glib_000.html#NM11">GraphSeq</a> <b>by </b><i><a class="ref" href="glib_000.html#D55">GLIB_000:def 55</a></i>;<br/>
<div><i><font color="Green">E110</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">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:70_1_1_4"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>then consider </b><font color="Maroon">Gn</font> being  <a href="glib_000.html#V2">finite</a> <a href="glib_003.html#V7">real-weighted</a> <a href="glib_003.html#V9">real-elabeled</a> <a href="glib_005.html#V1" target="_self">complete-elabeled</a> <a href="glib_003.html#NM4">WEGraph</a><b> such that </b><br/><a NAME="E3:70_1_1_4"/><i><font color="Green">E111</font></i>: 
( <font color="Maroon">IT</font> <a href="glib_000.html#K42">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Gn</font> &amp; <font color="Maroon">Gn</font> <a href="glib_000.html#R5">==</a> <font color="Maroon">G</font> )
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E4:70_1_1_4"/>
<font color="Maroon">IT</font> <a href="glib_000.html#K42">.</a> <font color="Maroon">n</font> is  <a href="glib_000.html#V2">finite</a> <a href="glib_003.html#V7">real-weighted</a> <a href="glib_003.html#V9">real-elabeled</a> <a href="glib_005.html#V1" target="_self">complete-elabeled</a> <a href="glib_003.html#NM4">WEGraph</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E15:70_1_1"/><b>then </b>
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/> ( <font color="Maroon">IT</font> <a href="glib_000.html#K42">.</a> <font color="Olive">n</font> is <a href="glib_003.html#V1">[Weighted]</a> &amp; <font color="Maroon">IT</font> <a href="glib_000.html#K42">.</a> <font color="Olive">n</font> is <a href="glib_003.html#V2">[ELabeled]</a> )
 
;<br/>
<b>then reconsider </b><font color="Maroon">IT</font> = <font color="Maroon">IT</font> as   <a href="glib_003.html#NM32">WEGraphSeq</a> <b>by </b><i><a class="ref" href="glib_003.html#D24">GLIB_003:def 24</a>, <a class="ref" href="glib_003.html#D25">GLIB_003:def 25</a></i>;<br/>
<a NAME="E17:70_1_1"/>
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/> ( <font color="Maroon">IT</font> <a href="glib_005.html#K14" target="_self">.</a> <font color="Olive">n</font> is <a href="glib_000.html#V2">finite</a> &amp; <font color="Maroon">IT</font> <a href="glib_005.html#K14" target="_self">.</a> <font color="Olive">n</font> is <a href="glib_003.html#V7">real-weighted</a> &amp; <font color="Maroon">IT</font> <a href="glib_005.html#K14" target="_self">.</a> <font color="Olive">n</font> is <a href="glib_003.html#V9">real-elabeled</a> &amp; <font color="Maroon">IT</font> <a href="glib_005.html#K14" target="_self">.</a> <font color="Olive">n</font> is <a href="glib_005.html#V1" target="_self">complete-elabeled</a> )
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">IT</font> = <font color="Maroon">IT</font> as  <a href="glib_000.html#V14">finite</a> <a href="glib_003.html#V15">real-weighted</a> <a href="glib_003.html#V17">real-elabeled</a> <a href="glib_005.html#V2" target="_self">complete-elabeled</a> <a href="glib_003.html#NM32">WEGraphSeq</a> <b>by </b><i><a class="ref" href="glib_000.html#D60">GLIB_000:def 60</a>, <a class="ref" href="glib_003.html#D27">GLIB_003:def 27</a>, <a class="ref" href="glib_003.html#D29">GLIB_003:def 29</a>, </i>;<br/>
<b>take </b>
<font color="Maroon">IT</font>
;<br/>

<b>thus </b><a NAME="E19:70_1_1"/>
<font color="Maroon">IT</font> <a href="glib_005.html#K14" target="_self">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K3">ELabelSelector</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> 0</span>)</span>
 <b>by </b><i/>;<br/>

<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>reconsider </b><font color="Maroon">n'</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>consider </b><font color="Maroon">Gn</font> being  <a href="glib_000.html#V2">finite</a> <a href="glib_003.html#V7">real-weighted</a> <a href="glib_003.html#V9">real-elabeled</a> <a href="glib_005.html#V1" target="_self">complete-elabeled</a> <a href="glib_003.html#NM4">WEGraph</a><b> such that </b><br/><a NAME="E22:70_1_1"/><i><font color="Green">E112</font></i>: 
( <font color="Maroon">IT</font> <a href="glib_005.html#K14" target="_self">.</a> <font color="Maroon">n'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Gn</font> &amp; <font color="Maroon">Gn</font> <a href="glib_000.html#R5">==</a> <font color="Maroon">G</font> )
 <b>by </b><i/>;<br/>
<a NAME="E23:70_1_1"/>
 <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">Gn</font> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font>
 
<b>by </b><i>, <a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">source'</font> = <font color="Maroon">source</font>, <font color="Maroon">sink'</font> = <font color="Maroon">sink</font> as   <a href="glib_000.html#NM10">Vertex</a> of <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="glib_005.html#K14" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">source'</font>
;<br/>

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

<b>thus </b><a NAME="E25:70_1_1"/>
( <font color="Maroon">source'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">source</font> &amp; <font color="Maroon">sink'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">sink</font> )
 ;<br/>

<a NAME="E26:70_1_1"/>
( <font color="Maroon">IT</font> <a href="glib_005.html#K14" target="_self">.</a> <font color="Maroon">n</font> is  <a href="glib_000.html#V2">finite</a> <a href="glib_003.html#V7">real-weighted</a> <a href="glib_003.html#V9">real-elabeled</a> <a href="glib_005.html#V1" target="_self">complete-elabeled</a> <a href="glib_003.html#NM4">WEGraph</a> &amp; <font color="Maroon">source'</font> is   <a href="glib_000.html#NM10">Vertex</a> of <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="glib_005.html#K14" target="_self">.</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">sink'</font> is   <a href="glib_000.html#NM10">Vertex</a> of <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="glib_005.html#K14" target="_self">.</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">IT</font> <a href="glib_005.html#K14" target="_self">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Gn</font> &amp; <font color="Maroon">source'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">source</font> &amp; <font color="Maroon">sink'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">sink</font> )
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">X</font> being  <a href="glib_000.html#V2">finite</a> <a href="glib_003.html#V7">real-weighted</a> <a href="glib_003.html#V9">real-elabeled</a> <a href="glib_005.html#V1" target="_self">complete-elabeled</a> <a href="glib_003.html#NM4">WEGraph</a>, <font color="Maroon">sourceX</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">X</font>, <font color="Maroon">sinkX</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E28:70_1_1"/><i><font color="Green">E128</font></i>: 
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Gn</font> &amp; <font color="Maroon">sourceX</font> <a href="hidden.html#R1">=</a> <font color="Maroon">source</font> &amp; <font color="Maroon">sinkX</font> <a href="hidden.html#R1">=</a> <font color="Maroon">sink</font> &amp; <font color="Maroon">IT</font> <a href="glib_005.html#K14" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">n'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_005.html#K13" target="_self">FF:Step</a> <font color="Maroon">X</font>,<font color="Maroon">sourceX</font>,<font color="Maroon">sinkX</font> )
 <b>by </b><i/>;<br/>
<a NAME="E29:70_1_1"/><i><font color="Green">E129</font></i>: 
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">IT</font> <a href="glib_005.html#K14" target="_self">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">sourceX</font> <a href="hidden.html#R1">=</a> <font color="Maroon">source'</font> &amp; <font color="Maroon">sink'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">sink</font> )
 
<b>by </b><i>, </i>;<br/>
<b>thus </b><a NAME="E30:70_1_1"/>
<font color="Maroon">IT</font> <a href="glib_005.html#K14" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_005.html#K13" target="_self">FF:Step</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="glib_005.html#K14" target="_self">.</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">source'</font>,<font color="Maroon">sink'</font>
 <b>by </b><i>, </i>;<br/>


</div>
