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<b>set </b><font color="Maroon">CS</font> =  <a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>( <span class="p1">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font> is <a href="glib_005.html#V4" target="_self">natural-elabeled</a> &amp; <span class="p1">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font> is <a href="glib_005.html#V3" target="_self">natural-weighted</a> );<br/>
<a NAME="E1:72_1_1"/>
<span class="p1">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <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/>
<a NAME="E2:72_1_1"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<div><i><font color="Green">E51</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1_1_1"><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="E1:72_1_1_1"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/><b>set </b><font color="Maroon">Gn</font> = <span class="p1">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> <font color="Maroon">n</font>;<br/><b>set </b><font color="Maroon">Gn1</font> = <span class="p1">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <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>;<br/><b>reconsider </b><font color="Maroon">Gn</font> = <span class="p1">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <a href="glib_005.html#K14" target="_self">.</a> <font color="Maroon">n</font> as  <a href="glib_000.html#V2">finite</a> <a href="glib_005.html#V1" target="_self">complete-elabeled</a> <a href="glib_005.html#V3" target="_self">natural-weighted</a> <a href="glib_005.html#V4" target="_self">natural-elabeled</a> <a href="glib_003.html#NM4">WEGraph</a> <b>by </b><i/>;<br/><b>consider </b><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="E4:72_1_1_1"/><i><font color="Green">E89</font></i>: 
( <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> &amp; <span class="p1">(<span class="default"><a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font></span>)</span> <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">Gn</font>,<font color="Maroon">source'</font>,<font color="Maroon">sink'</font> )
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E5:72_1_1_1"/>
<font color="Maroon">S<sub>1</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="E4:72_1_1"/><i><font color="Green">E92</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>1</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>
<a NAME="E5:72_1_1"/>
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1_1_2"><b>proof </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="E1:72_1_1_2"/>
<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>hence </b><a NAME="E2:72_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:72_1_1"/>
(  <a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font> is <a href="glib_005.html#V5" target="_self">natural-weighted</a> &amp;  <a href="glib_005.html#K15" target="_self">FF:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>,<font color="Maroon">sink</font> is <a href="glib_005.html#V6" target="_self">natural-elabeled</a> )
 <b>by </b><i>, </i>;<br/>


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