<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">W</font> be   <a href="funct_1.html#NM1">Function</a>;<br/><b>let </b><font color="Maroon">G</font> be   <a href="graph_1.html#NM1">Graph</a>;<br/><b>let </b><font color="Maroon">pe</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">qe</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font>;<br/>







<b>assume </b><a NAME="E1:78"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">W</font> <a href="graph_5.html#R6" target="_self">is_weight_of</a> <font color="Maroon">G</font>
 ;<br/>

<b>set </b><font color="Maroon">r</font> = <font color="Maroon">pe</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">qe</font>;<br/>
<b>set </b><font color="Maroon">f</font> =  <a href="graph_5.html#K9" target="_self">RealSequence</a> <span class="p1">(<span class="default"><font color="Maroon">pe</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">qe</font></span>)</span>,<font color="Maroon">W</font>;<br/>
<b>set </b><font color="Maroon">g</font> =  <a href="graph_5.html#K9" target="_self">RealSequence</a> <font color="Maroon">pe</font>,<font color="Maroon">W</font>;<br/>
<b>set </b><font color="Maroon">h</font> =  <a href="graph_5.html#K9" target="_self">RealSequence</a> <font color="Maroon">qe</font>,<font color="Maroon">W</font>;<br/>
<a NAME="E2:78"/><i><font color="Green">E45</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">pe</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">qe</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="graph_5.html#K9" target="_self">RealSequence</a> <span class="p2">(<span class="default"><font color="Maroon">pe</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">qe</font></span>)</span>,<font color="Maroon">W</font></span>)</span> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">pe</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">qe</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="graph_5.html#K9" target="_self">RealSequence</a> <span class="p2">(<span class="default"><font color="Maroon">pe</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">qe</font></span>)</span>,<font color="Maroon">W</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">pe</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">qe</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> ) )
 
<b>by </b><i>, <a class="txt" href="graph_5.html#E7:78"><i><font color="Green">E53</font></i></a></i>;<br/>
<a NAME="E3:78"/><i><font color="Green">E47</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">pe</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="graph_5.html#K9" target="_self">RealSequence</a> <font color="Maroon">pe</font>,<font color="Maroon">W</font></span>)</span> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">pe</font> holds <br/><span class="p1">(<span class="default"><a href="graph_5.html#K9" target="_self">RealSequence</a> <font color="Maroon">pe</font>,<font color="Maroon">W</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">pe</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> ) )
 
<b>by </b><i>, <a class="txt" href="graph_5.html#E7:78"><i><font color="Green">E53</font></i></a></i>;<br/>
<a NAME="E4:78"/><i><font color="Green">E48</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">qe</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="graph_5.html#K9" target="_self">RealSequence</a> <font color="Maroon">qe</font>,<font color="Maroon">W</font></span>)</span> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">qe</font> holds <br/><span class="p1">(<span class="default"><a href="graph_5.html#K9" target="_self">RealSequence</a> <font color="Maroon">qe</font>,<font color="Maroon">W</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">qe</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> ) )
 
<b>by </b><i>, <a class="txt" href="graph_5.html#E7:78"><i><font color="Green">E53</font></i></a></i>;<br/>
<a NAME="E5:78"/><b>then </b><i><font color="Green">E51</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="graph_5.html#K9" target="_self">RealSequence</a> <span class="p2">(<span class="default"><font color="Maroon">pe</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">qe</font></span>)</span>,<font color="Maroon">W</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">pe</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">qe</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">pe</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="graph_5.html#K9" target="_self">RealSequence</a> <font color="Maroon">pe</font>,<font color="Maroon">W</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">qe</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="graph_5.html#K9" target="_self">RealSequence</a> <font color="Maroon">qe</font>,<font color="Maroon">W</font></span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E6:78"/><b>then </b><i><font color="Green">E52</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="graph_5.html#K9" target="_self">RealSequence</a> <span class="p2">(<span class="default"><font color="Maroon">pe</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">qe</font></span>)</span>,<font color="Maroon">W</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="graph_5.html#K9" target="_self">RealSequence</a> <font color="Maroon">pe</font>,<font color="Maroon">W</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="graph_5.html#K9" target="_self">RealSequence</a> <font color="Maroon">qe</font>,<font color="Maroon">W</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<div><i><font color="Green">E53</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:78_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="graph_5.html#K9" target="_self">RealSequence</a> <font color="Maroon">pe</font>,<font color="Maroon">W</font></span>)</span>
 ;<br/><a NAME="E2:78_1"/><b>then </b><i><font color="Green">E62</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="graph_5.html#K9" target="_self">RealSequence</a> <font color="Maroon">pe</font>,<font color="Maroon">W</font></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:78_1"/><i><font color="Green">E63</font></i>: 
<span class="p1">(<span class="default"><a href="graph_5.html#K9" target="_self">RealSequence</a> <font color="Maroon">pe</font>,<font color="Maroon">W</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">pe</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="graph_5.html#T8" target="_self">Th8</a></i>;<br/><a NAME="E4:78_1"/><i><font color="Green">E64</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">pe</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">qe</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pe</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="graph_5.html#T8" target="_self">Th8</a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E5:78_1"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="graph_5.html#K9" target="_self">RealSequence</a> <span class="p2">(<span class="default"><font color="Maroon">pe</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">qe</font></span>)</span>,<font color="Maroon">W</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="graph_5.html#E9"><i><font color="Green">Lemma72</font></i></a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E6:78_1"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="graph_5.html#K9" target="_self">RealSequence</a> <span class="p2">(<span class="default"><font color="Maroon">pe</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">qe</font></span>)</span>,<font color="Maroon">W</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_5.html#E9"><i><font color="Green">Lemma72</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E7:78_1"/>
<span class="p1">(<span class="default"><a href="graph_5.html#K9" target="_self">RealSequence</a> <span class="p2">(<span class="default"><font color="Maroon">pe</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">qe</font></span>)</span>,<font color="Maroon">W</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="graph_5.html#K9" target="_self">RealSequence</a> <font color="Maroon">pe</font>,<font color="Maroon">W</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="txt" href="graph_5.html#E10"><i><font color="Green">Lemma73</font></i></a>, <a class="ref" href="graph_5.html#T9" target="_self">Th9</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:78_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="graph_5.html#K9" target="_self">RealSequence</a> <font color="Maroon">qe</font>,<font color="Maroon">W</font></span>)</span>
 ;<br/><a NAME="E2:78_2"/><b>then </b><i><font color="Green">E66</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="graph_5.html#K9" target="_self">RealSequence</a> <font color="Maroon">qe</font>,<font color="Maroon">W</font></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:78_2"/><i><font color="Green">E67</font></i>: 
<span class="p1">(<span class="default"><a href="graph_5.html#K9" target="_self">RealSequence</a> <font color="Maroon">qe</font>,<font color="Maroon">W</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">qe</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_5.html#E3:78_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="graph_5.html#T8" target="_self">Th8</a></i>;<br/><a NAME="E4:78_2"/><i><font color="Green">E68</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">pe</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">qe</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="graph_5.html#K9" target="_self">RealSequence</a> <font color="Maroon">pe</font>,<font color="Maroon">W</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">qe</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E3:78_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="graph_5.html#E4:78_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="graph_5.html#T8" target="_self">Th8</a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E5:78_2"/><i><font color="Green">E87</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="graph_5.html#K9" target="_self">RealSequence</a> <font color="Maroon">pe</font>,<font color="Maroon">W</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E9"><i><font color="Green">Lemma72</font></i></a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E6:78_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="graph_5.html#K9" target="_self">RealSequence</a> <font color="Maroon">pe</font>,<font color="Maroon">W</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="graph_5.html#K9" target="_self">RealSequence</a> <span class="p2">(<span class="default"><font color="Maroon">pe</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">qe</font></span>)</span>,<font color="Maroon">W</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="graph_5.html#E9"><i><font color="Green">Lemma72</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E7:78_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="graph_5.html#K9" target="_self">RealSequence</a> <font color="Maroon">pe</font>,<font color="Maroon">W</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="graph_5.html#K9" target="_self">RealSequence</a> <span class="p2">(<span class="default"><font color="Maroon">pe</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">qe</font></span>)</span>,<font color="Maroon">W</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E8:78_2"/>
<span class="p1">(<span class="default"><a href="graph_5.html#K9" target="_self">RealSequence</a> <span class="p2">(<span class="default"><font color="Maroon">pe</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">qe</font></span>)</span>,<font color="Maroon">W</font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="graph_5.html#K9" target="_self">RealSequence</a> <font color="Maroon">pe</font>,<font color="Maroon">W</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="graph_5.html#K9" target="_self">RealSequence</a> <font color="Maroon">qe</font>,<font color="Maroon">W</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E1:78_2"><i><font color="Green">E65</font></i></a>, <a class="txt" href="graph_5.html#E10"><i><font color="Green">Lemma73</font></i></a>, <a class="ref" href="graph_5.html#T9" target="_self">Th9</a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:78"/><b>then </b><i><font color="Green">E88</font></i>: 
 <a href="graph_5.html#K9" target="_self">RealSequence</a> <span class="p1">(<span class="default"><font color="Maroon">pe</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">qe</font></span>)</span>,<font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="graph_5.html#K9" target="_self">RealSequence</a> <font color="Maroon">pe</font>,<font color="Maroon">W</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="graph_5.html#K9" target="_self">RealSequence</a> <font color="Maroon">qe</font>,<font color="Maroon">W</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="finseq_3.html#T43">FINSEQ_3:43</a></i>;<br/>
<b>thus </b><a NAME="E10:78"/>
 <a href="graph_5.html#K10" target="_self">cost</a> <span class="p1">(<span class="default"><font color="Maroon">pe</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">qe</font></span>)</span>,<font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">pe</font>,<font color="Maroon">W</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">qe</font>,<font color="Maroon">W</font></span>)</span>
 <b>by </b><i><a class="ref" href="graph_5.html#T8" target="_self">Th8</a>, <a class="ref" href="rvsum_1.html#T105">RVSUM_1:105</a></i>;<br/>


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