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

<b>assume </b><b>that </b><br/><a NAME="E1:70_1_2"/><i><font color="Green">E130</font></i>: 
( <font color="Maroon">IT1</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> &amp; ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  ex <font color="Olive">source'</font>, <font color="Olive">sink'</font> being  <a href="glib_000.html#NM10">Vertex</a> of <span class="p1">(<span class="default"><font color="Maroon">IT1</font> <a href="glib_005.html#K14" target="_self">.</a> <font color="Olive">n</font></span>)</span> st <br/>( <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">IT1</font> <a href="glib_005.html#K14" target="_self">.</a> <span class="p1">(<span class="default"><font color="Olive">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">IT1</font> <a href="glib_005.html#K14" target="_self">.</a> <font color="Olive">n</font></span>)</span>,<font color="Olive">source'</font>,<font color="Olive">sink'</font> ) ) )
 <b>and </b><br/><a NAME="E2:70_1_2"/><i><font color="Green">E131</font></i>: 
( <font color="Maroon">IT2</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> &amp; ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  ex <font color="Olive">source'</font>, <font color="Olive">sink'</font> being  <a href="glib_000.html#NM10">Vertex</a> of <span class="p1">(<span class="default"><font color="Maroon">IT2</font> <a href="glib_005.html#K14" target="_self">.</a> <font color="Olive">n</font></span>)</span> st <br/>( <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">IT2</font> <a href="glib_005.html#K14" target="_self">.</a> <span class="p1">(<span class="default"><font color="Olive">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">IT2</font> <a href="glib_005.html#K14" target="_self">.</a> <font color="Olive">n</font></span>)</span>,<font color="Olive">source'</font>,<font color="Olive">sink'</font> ) ) )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><font color="Maroon">IT1</font> <a href="glib_005.html#K14" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">IT2</font> <a href="glib_005.html#K14" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E3:70_1_2"/>
<font color="Maroon">IT1</font> <a href="glib_005.html#K14" target="_self">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">IT2</font> <a href="glib_005.html#K14" target="_self">.</a> 0
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:70_1_2"/><b>then </b><i><font color="Green">E132</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<div><i><font color="Green">E133</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">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_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/><a NAME="E2:70_1_2_1"/><b>then </b>
<font color="Maroon">IT1</font> <a href="glib_005.html#K14" target="_self">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">IT2</font> <a href="glib_005.html#K14" target="_self">.</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E3:70_1_2_1"/><b>then </b><i><font color="Green">E134</font></i>: 
<font color="Maroon">IT1</font> <a href="glib_005.html#K14" target="_self">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">IT2</font> <a href="glib_005.html#K14" target="_self">.</a> <font color="Maroon">n</font>
 ;<br/><b>consider </b><font color="Maroon">source1</font> being   <a href="glib_000.html#NM10">Vertex</a> of <span class="p1">(<span class="default"><font color="Maroon">IT1</font> <a href="glib_005.html#K14" target="_self">.</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">sink1</font> being   <a href="glib_000.html#NM10">Vertex</a> of <span class="p1">(<span class="default"><font color="Maroon">IT1</font> <a href="glib_005.html#K14" target="_self">.</a> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E5:70_1_2_1"/><i><font color="Green">E135</font></i>: 
( <font color="Maroon">source1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">source</font> &amp; <font color="Maroon">sink1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">sink</font> &amp; <font color="Maroon">IT1</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> <span class="p1">(<span class="default"><font color="Maroon">IT1</font> <a href="glib_005.html#K14" target="_self">.</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">source1</font>,<font color="Maroon">sink1</font> )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">source2</font> being   <a href="glib_000.html#NM10">Vertex</a> of <span class="p1">(<span class="default"><font color="Maroon">IT2</font> <a href="glib_005.html#K14" target="_self">.</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">sink2</font> being   <a href="glib_000.html#NM10">Vertex</a> of <span class="p1">(<span class="default"><font color="Maroon">IT2</font> <a href="glib_005.html#K14" target="_self">.</a> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E7:70_1_2_1"/><i><font color="Green">E137</font></i>: 
( <font color="Maroon">source2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">source</font> &amp; <font color="Maroon">sink2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">sink</font> &amp; <font color="Maroon">IT2</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> <span class="p1">(<span class="default"><font color="Maroon">IT2</font> <a href="glib_005.html#K14" target="_self">.</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">source2</font>,<font color="Maroon">sink2</font> )
 <b>by </b><i/>;<br/><a NAME="E8:70_1_2_1"/>
<font color="Maroon">IT1</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> <font color="Maroon">IT2</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>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E9:70_1_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E6:70_1_2"/>
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="E7:70_1_2"/><b>then </b>
for <font color="Olive">n</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/><font color="Maroon">IT1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">IT2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font>
 
;<br/>
<b>hence </b><a NAME="E8:70_1_2"/>
<font color="Maroon">IT1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">IT2</font>
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


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