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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#V7">finite</a> <a href="graph_1.html#NM1">Graph</a>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be  <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</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>assume </b><a NAME="E1:94"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="graph_5.html#R2" target="_self">is_orientedpath_of</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="graph_5.html#R5" target="_self">is_weight&gt;=0of</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<a NAME="E2:94"/><b>then </b>
 <a href="graph_5.html#K5" target="_self">AcyclicPaths</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="graph_5.html#T48" target="_self">Th48</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <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><b> such that </b><br/><a NAME="E4:94"/><i><font color="Green">E75</font></i>: 
( <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>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <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> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="graph_5.html#K5" target="_self">AcyclicPaths</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>1</sub></font> holds <br/> <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>7</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="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> ) )
 <b>by </b><i><a class="ref" href="graph_5.html#T57" target="_self">Th57</a></i>;<br/>
<a NAME="E5:94"/>
<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>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="graph_5.html#E4:94"><i><font color="Green">E75</font></i></a></i>;<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="E7:94"/><i><font color="Green">E76</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>5</sub></font>,<font color="Maroon">c<sub>6</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="E8:94"/>
<font color="Maroon">c<sub>8</sub></font> <a href="graph_5.html#R2" target="_self">is_orientedpath_of</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E7:94"><i><font color="Green">E76</font></i></a>, <a class="ref" href="graph_5.html#D7" target="_self">Def7</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="graph_5.html#D18" target="_self">GRAPH_5:def 18</a><br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be  <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">c<sub>3</sub></font>;<br/><b>assume </b><a NAME="E1:94_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="graph_5.html#R2" target="_self">is_orientedpath_of</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E2:94_1"/><b>then </b>
 <a href="graph_5.html#K6" target="_self">AcyclicPaths</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="graph_5.html#T48" target="_self">Th48</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="finseq_2.html#M2">Element</a> of 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> <b> such that </b><br/><a NAME="E4:94_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="graph_5.html#K6" target="_self">AcyclicPaths</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="subset_1.html#T10">SUBSET_1:10</a></i>;<br/><a NAME="E5:94_1"/><i><font color="Green">E79</font></i>: 
 <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>10</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#E1:94"><i><font color="Green">E74</font></i></a>, <a class="txt" href="graph_5.html#E4:94_1"><i><font color="Green">E78</font></i></a>, <a class="ref" href="graph_5.html#T60" target="_self">Th60</a></i>;<br/><a NAME="E6:94_1"/>
 <a href="graph_5.html#K6" target="_self">AcyclicPaths</a> <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a>  <a href="graph_5.html#K5" target="_self">AcyclicPaths</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E1:94_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="graph_5.html#T45" target="_self">Th45</a></i>;<br/><a NAME="E7:94_1"/><b>then </b>
 <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">c<sub>7</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>10</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="graph_5.html#E4:94"><i><font color="Green">E75</font></i></a>, <a class="txt" href="graph_5.html#E4:94_1"><i><font color="Green">E78</font></i></a></i>;<br/><b>hence </b><a NAME="E8:94_1"/>
 <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#E7:94"><i><font color="Green">E76</font></i></a>, <a class="txt" href="graph_5.html#E5:94_1"><i><font color="Green">E79</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div>
<b>end;</b></div>


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