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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="graph_1.html#NM1">Graph</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="funct_1.html#NM1">Function</a>;<br/>





<b>assume </b><a NAME="E1:4"/><i><font color="Green">E5</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="graph_5.html#R6">is_weight_of</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="graph_5.html#K9">RealSequence</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>
<a NAME="E2:4"/><i><font color="Green">E6</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="graph_5.html#K9">RealSequence</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><span class="p1">(<span class="default"><a href="graph_5.html#K9">RealSequence</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> ) )
 
<b>by </b><i><a class="txt" href="graphsp.html#E1:4"><i><font color="Green">E5</font></i></a>, <a class="ref" href="graph_5.html#D15">GRAPH_5:def 15</a></i>;<br/>
<a NAME="E3:4"/><i><font color="Green">E7</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="graphsp.html#E1:4"><i><font color="Green">E5</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E4:4"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="graph_5.html#K9">RealSequence</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="graphsp.html#E1:4"><i><font color="Green">E5</font></i></a>, <a class="txt" href="graphsp.html#E2:4"><i><font color="Green">E6</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E5:4"/><b>then </b><i><font color="Green">E8</font></i>: 
 <a href="graph_5.html#K9">RealSequence</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="graph_5.html#K9">RealSequence</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> 1</span>)</span></span><a href="binarith.html#K13">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>
<b>thus </b> <a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> = 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="graph_5.html#K9">RealSequence</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="graph_5.html#D16">GRAPH_5:def 16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="graph_5.html#K9">RealSequence</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> 1
<b>by </b><i><a class="txt" href="graphsp.html#E5:4"><i><font color="Green">E8</font></i></a>, <a class="ref" href="finsop_1.html#T12">FINSOP_1:12</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="graphsp.html#E1:4"><i><font color="Green">E5</font></i></a>, <a class="txt" href="graphsp.html#E3:4"><i><font color="Green">E7</font></i></a>, <a class="ref" href="graph_5.html#D15">GRAPH_5:def 15</a></i>
;<br/>


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