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<b>let </b><font color="Maroon" title="c4">IT1</font>, <font color="Maroon" title="c5">IT2</font> be  <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>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:70_1_2"/><i><font color="Green" title="E30">A15</font></i>: 
( <font color="Maroon" title="c4">IT1</font> <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> &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</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 <span class="p1">(<span class="default"><font color="Maroon" title="c4">IT1</font> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> <font color="Olive" title="b1">n</font></span>)</span> st <br/>( <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; <font color="Maroon" title="c4">IT1</font> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</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> <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="c4">IT1</font> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Olive" title="b2">source'</font>,<font color="Olive" title="b3">sink'</font> ) ) )
 <b>and </b><br/><a NAME="E2:70_1_2"/><i><font color="Green" title="E31">A16</font></i>: 
( <font color="Maroon" title="c5">IT2</font> <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> &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</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 <span class="p1">(<span class="default"><font color="Maroon" title="c5">IT2</font> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> <font color="Olive" title="b1">n</font></span>)</span> st <br/>( <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; <font color="Maroon" title="c5">IT2</font> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</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> <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="c5">IT2</font> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Olive" title="b2">source'</font>,<font color="Olive" title="b3">sink'</font> ) ) )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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>means </b><font color="Maroon" title="c4">IT1</font> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">IT2</font> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> $1;<br/>
<a NAME="E3:70_1_2"/><i><font color="Green" title="E32">A17</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<b>by </b><i><a class="txt" href="glib_005.html#E1:70_1_2"><i><font color="Green" title="E30">A15</font></i></a>, <a class="txt" href="glib_005.html#E2:70_1_2"><i><font color="Green" title="E31">A16</font></i></a></i>;<br/>
<div><i><font color="Green" title="E33">A18</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">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_2_1"/><i><font color="Green" title="E34">A19</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">n</font>]
 ;<br/><b>consider </b><font color="Maroon" title="c7">source1</font>, <font color="Maroon" title="c8">sink1</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">IT1</font> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> <font color="Maroon" title="c6">n</font></span>)</span><b> such that </b><br/><a NAME="E3:70_1_2_1"/><i><font color="Green" title="E35">A20</font></i>: 
( <font color="Maroon" title="c7">source1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">source</font> &amp; <font color="Maroon" title="c8">sink1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">sink</font> &amp; <font color="Maroon" title="c4">IT1</font> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">IT1</font> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> <font color="Maroon" title="c6">n</font></span>)</span>,<font color="Maroon" title="c7">source1</font>,<font color="Maroon" title="c8">sink1</font> )
 <b>by </b><i><a class="txt" href="glib_005.html#E1:70_1_2"><i><font color="Green" title="E30">A15</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c9">source2</font>, <font color="Maroon" title="c10">sink2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c5">IT2</font> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> <font color="Maroon" title="c6">n</font></span>)</span><b> such that </b><br/><a NAME="E5:70_1_2_1"/><i><font color="Green" title="E36">A21</font></i>: 
( <font color="Maroon" title="c9">source2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">source</font> &amp; <font color="Maroon" title="c10">sink2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">sink</font> &amp; <font color="Maroon" title="c5">IT2</font> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">IT2</font> <a href="glib_005.html#K14" title="GLIB_005:func.14">.</a> <font color="Maroon" title="c6">n</font></span>)</span>,<font color="Maroon" title="c9">source2</font>,<font color="Maroon" title="c10">sink2</font> )
 <b>by </b><i><a class="txt" href="glib_005.html#E2:70_1_2"><i><font color="Green" title="E31">A16</font></i></a></i>;<br/><b>thus </b><a NAME="E6:70_1_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">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#E1:70_1_2_1"><i><font color="Green" title="E34">A19</font></i></a>, <a class="txt" href="glib_005.html#E3:70_1_2_1"><i><font color="Green" title="E35">A20</font></i></a>, <a class="txt" href="glib_005.html#E5:70_1_2_1"><i><font color="Green" title="E36">A21</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:70_1_2"/>
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#E3:70_1_2"><i><font color="Green" title="E32">A17</font></i></a>, <a class="txt" href="glib_005.html#E4:70_1_2"><i><font color="Green" title="E33">A18</font></i></a>);<br/></i>
<a NAME="E6:70_1_2"/><b>then </b>
for <font color="Olive" title="b1">n</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><font color="Maroon" title="c4">IT1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">IT2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font>
 
;<br/>
<b>hence </b><a NAME="E7:70_1_2"/>
<font color="Maroon" title="c4">IT1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c5">IT2</font>
 <b>by </b><i><a class="ref" href="pboole.html#T3" title="PBOOLE:th.3">PBOOLE:3</a></i>;<br/>


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