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<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">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>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>set </b><font color="Maroon">D</font> = the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font>;<br/>
<b>assume </b><a NAME="E1:79"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">qe</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">qe</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">pe</font> &amp; <font color="Maroon">W</font> <a href="graph_5.html#R5" target="_self">is_weight&gt;=0of</a> <font color="Maroon">G</font> )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font>  st <font color="Olive">q</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">q</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Olive">p</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/> <a href="graph_5.html#K10" target="_self">cost</a> <font color="Olive">q</font>,<font color="Maroon">W</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10" target="_self">cost</a> <font color="Olive">p</font>,<font color="Maroon">W</font>;<br/>
<a NAME="E2:79"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">W</font> <a href="graph_5.html#R6" target="_self">is_weight_of</a> <font color="Maroon">G</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:79"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</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>, <font color="Maroon">q</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="E1:79_1"/>
( <font color="Maroon">q</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/>

<a NAME="E2:79_1"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E3:79_1"/><b>then </b>
 <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">q</font>,<font color="Maroon">W</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, <a class="ref" href="graph_5.html#T7" target="_self">Th7</a></i>;<br/>
<b>hence </b><a NAME="E4:79_1"/>
 <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">q</font>,<font color="Maroon">W</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">p</font>,<font color="Maroon">W</font>
 <b>by </b><i>, <a class="ref" href="graph_5.html#T6" target="_self">Th6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:79"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:79_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</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>, <font color="Maroon">q</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="E1:79_2_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">q</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><b>then consider </b><font color="Maroon">q1</font> being   <a href="finseq_1.html#NM1">FinSequence</a>, <font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:79_2_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T21">FINSEQ_2:21</a></i>;<br/><b>consider </b><font color="Maroon">p1</font> being   <a href="finseq_1.html#NM1">FinSequence</a>, <font color="Maroon">p2</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E5:79_2_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">p2</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">q1</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">p2</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="graph_5.html#T8" target="_self">Th8</a>, </i>;<br/><a NAME="E6:79_2_1"/>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="graph_5.html#T8" target="_self">Th8</a>, <a class="ref" href="finseq_2.html#T19">FINSEQ_2:19</a></i>;<br/><a NAME="E7:79_2_1"/><b>then </b><i><font color="Green">E62</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E8:79_2_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">p1</font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9">*&gt;</a></span> is    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</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="finseq_1.html#T50">FINSEQ_1:50</a></i>;<br/><b>then reconsider </b><font color="Maroon">p1</font> = <font color="Maroon">p1</font> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="finseq_1.html#T50">FINSEQ_1:50</a></i>;<br/><b>reconsider </b><font color="Maroon">y</font> = <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9">*&gt;</a></span> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="graph_5.html#T9" target="_self">Th9</a>, <a class="ref" href="finseq_1.html#T50">FINSEQ_1:50</a></i>;<br/><b>reconsider </b><font color="Maroon">p2</font> = <font color="Maroon">p2</font> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</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="finseq_1.html#T50">FINSEQ_1:50</a></i>;<br/><b>reconsider </b><font color="Maroon">q1</font> = <font color="Maroon">q1</font> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="graph_5.html#T8" target="_self">Th8</a>, <a class="ref" href="finseq_1.html#T50">FINSEQ_1:50</a></i>;<br/><a NAME="E13:79_2_1"/><i><font color="Green">E64</font></i>: 
 <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">q</font>,<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">q1</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">y</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="txt" href="graph_5.html#E8:79_2_1"><i><font color="Green">E63</font></i></a></i>;<br/><i><font color="Green">E65</font></i>:  <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">p</font>,<font color="Maroon">W</font> = 
<span class="p1">(<span class="default"><a href="graph_5.html#K10" target="_self">cost</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">y</font></span>)</span>,<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">p2</font>,<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="txt" href="graph_5.html#E8:79_2_1"><i><font color="Green">E63</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">p1</font>,<font color="Maroon">W</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">y</font>,<font color="Maroon">W</font></span>)</span></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">p2</font>,<font color="Maroon">W</font></span>)</span>
<b>by </b><i>, <a class="txt" href="graph_5.html#E8:79_2_1"><i><font color="Green">E63</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">p1</font>,<font color="Maroon">W</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">p2</font>,<font color="Maroon">W</font></span>)</span></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">y</font>,<font color="Maroon">W</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="graph_5.html#K10" target="_self">cost</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p2</font></span>)</span>,<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">y</font>,<font color="Maroon">W</font></span>)</span>
<b>by </b><i>, <a class="txt" href="graph_5.html#E8:79_2_1"><i><font color="Green">E63</font></i></a></i>
;<br/><a NAME="E15:79_2_1"/>
<font color="Maroon">q1</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i>, <a class="ref" href="graph_5.html#T8" target="_self">Th8</a>, <a class="ref" href="finseq_3.html#T98">FINSEQ_3:98</a></i>;<br/><a NAME="E16:79_2_1"/><b>then </b>
 <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">q1</font>,<font color="Maroon">W</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10" target="_self">cost</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p2</font></span>)</span>,<font color="Maroon">W</font>
 <b>by </b><i>, <a class="txt" href="graph_5.html#E9"><i><font color="Green">Lemma72</font></i></a>, <a class="txt" href="graph_5.html#E10"><i><font color="Green">Lemma73</font></i></a></i>;<br/><b>hence </b><a NAME="E17:79_2_1"/>
 <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">q</font>,<font color="Maroon">W</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">p</font>,<font color="Maroon">W</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:79_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:79"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<a NAME="E6:79"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[ <a href="finseq_1.html#K3">len</a> <font color="Maroon">qe</font>]
 
;<br/>
<b>hence </b><a NAME="E7:79"/>
 <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">qe</font>,<font color="Maroon">W</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">pe</font>,<font color="Maroon">W</font>
 <b>by </b><i/>;<br/>


</div>
