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<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="funct_1.html#NM1">Function</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="graph_1.html#NM1">Graph</a>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be    <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default">the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">c<sub>3</sub></font> <a href="finseq_2.html#K3">*</a> </span>)</span>;<br/>











<b>assume </b><a NAME="E1:77"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="graph_5.html#K5" target="_self">AcyclicPaths</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="graph_5.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>6</sub></font>) <b>-&gt; </b>  <a href="real_1.html#NM1">Real</a> =  <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="graph_5.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E3:77"/><i><font color="Green">E64</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="graph_5.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>6</sub></font> holds <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>7</sub></font>) <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="graph_5.html#S2" target="_self">GRAPH_5:sch 2</a>();<br/></i>
<a NAME="E4:77"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="graph_5.html#K5" target="_self">AcyclicPaths</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="graph_5.html#E1:77"><i><font color="Green">E63</font></i></a></i>;<br/>
<a NAME="E5:77"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is  <a href="graph_1.html#V8">oriented</a> <a href="graph_4.html#V1">Simple</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">c<sub>3</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="graph_5.html#R4" target="_self">is_acyclicpath_of</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being  <a href="graph_1.html#V8">oriented</a> <a href="graph_4.html#V1">Simple</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E7:77"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="graph_5.html#R4" target="_self">is_acyclicpath_of</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font> )
 ;<br/>
<b>take </b>
<font color="Maroon">c<sub>8</sub></font>
;<br/>

<b>thus </b><a NAME="E8:77"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E7:77"><i><font color="Green">E65</font></i></a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>9</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>3</sub></font>;<br/>

<b>assume </b><a NAME="E9:77"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as    <a href="graph_5.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>6</sub></font> ;<br/>
<a NAME="E11:77"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>7</sub></font>) <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>10</sub></font>)
 
<b>by </b><i><a class="txt" href="graph_5.html#E3:77"><i><font color="Green">E64</font></i></a></i>;<br/>
<b>hence </b><a NAME="E12:77"/>
 <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E7:77"><i><font color="Green">E65</font></i></a></i>;<br/>


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