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<b>set </b><font color="Maroon" title="c4">CS</font> =  <a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c2">sink</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b><span class="p1">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c2">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> <font color="Maroon" title="c1">G</font> is <a href="glib_003.html#V8" title="GLIB_003:attr.8">nonnegative-weighted</a>;<br/>
<a NAME="E1:68_1_1"/>
<span class="p1">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c2">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K3" title="GLIB_003:func.3">ELabelSelector</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="glib_005.html#D23" target="_self" title="GLIB_005:def.23">Def23</a></i>;<br/>
<a NAME="E2:68_1_1"/><b>then </b><i><font color="Green" title="E31">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<div><i><font color="Green" title="E32">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/><b>assume </b><a NAME="E1:68_1_1_1"/><i><font color="Green" title="E33">A3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><b>set </b><font color="Maroon" title="c6">Gn</font> = <span class="p1">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c2">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> <font color="Maroon" title="c5">n</font>;<br/><b>set </b><font color="Maroon" title="c7">Gn1</font> = <span class="p1">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c2">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/><b>reconsider </b><font color="Maroon" title="c8">Gn</font> = <span class="p1">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c2">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> <font color="Maroon" title="c5">n</font> as  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="glib_003.html#V8" title="GLIB_003:attr.8">nonnegative-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#E1:68_1_1_1"><i><font color="Green" title="E33">A3</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c9">source'</font>, <font color="Maroon" title="c10">sink'</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c8">Gn</font><b> such that </b><br/><a NAME="E4:68_1_1_1"/><i><font color="Green" title="E34">A4</font></i>: 
( <font color="Maroon" title="c9">source'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">source</font> &amp; <font color="Maroon" title="c10">sink'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">sink</font> &amp; <span class="p1">(<span class="default"><a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c2">sink</font></span>)</span> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_005.html#K13" title="GLIB_005:func.13">FF:Step</a> <font color="Maroon" title="c8">Gn</font>,<font color="Maroon" title="c9">source'</font>,<font color="Maroon" title="c10">sink'</font> )
 <b>by </b><i><a class="ref" href="glib_005.html#D23" target="_self" title="GLIB_005:def.23">Def23</a></i>;<br/><b>thus </b><a NAME="E5:68_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <b>by </b><i><a class="txt" href="glib_005.html#E4:68_1_1_1"><i><font color="Green" title="E34">A4</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E4:68_1_1"/><i><font color="Green" title="E33">A5</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="glib_005.html#E2:68_1_1"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="glib_005.html#E3:68_1_1"><i><font color="Green" title="E32">A2</font></i></a>);<br/></i>
<a NAME="E5:68_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  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font>]</span><br/>

<a NAME="E1:68_1_1_2"/>
<font color="Maroon" title="c5">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>hence </b><a NAME="E2:68_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font>]
 <b>by </b><i><a class="txt" href="glib_005.html#E4:68_1_1"><i><font color="Green" title="E33">A5</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:68_1_1"/>
 <a href="glib_005.html#K15" title="GLIB_005:func.15">FF:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c2">sink</font> is <a href="glib_003.html#V16" title="GLIB_003:attr.16">nonnegative-weighted</a>
 <b>by </b><i><a class="ref" href="glib_003.html#D28" title="GLIB_003:def.28">GLIB_003:def 28</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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