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<b>let </b><font color="Maroon" title="c1">W</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>;<br/><b>let </b><font color="Maroon" title="c2">G</font> be   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Graph</a>;<br/><b>let </b><font color="Maroon" title="c3">v1</font>, <font color="Maroon" title="c4">v2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c2">G</font>;<br/><b>let </b><font color="Maroon" title="c5">D</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default">the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">G</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>;<br/>









<b>assume </b><a NAME="E1:75"/><i><font color="Green" title="E59">A1</font></i>: 
<font color="Maroon" title="c5">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_5.html#K4" title="GRAPH_5:func.4">AcyclicPaths</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">v2</font>
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="graph_5.html#M1" title="GRAPH_5:mode.1">Element</a> of <font color="Maroon" title="c5">D</font>) <b>-&gt; </b>  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> =  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> $1,<font color="Maroon" title="c1">W</font>;<br/>
<b>consider </b><font color="Maroon" title="c6">x</font> being    <a href="graph_5.html#M1" title="GRAPH_5:mode.1">Element</a> of <font color="Maroon" title="c5">D</font><b> such that </b><br/><a NAME="E3:75"/><i><font color="Green" title="E60">A2</font></i>: 
for <font color="Olive" title="b1">y</font> being   <a href="graph_5.html#M1" title="GRAPH_5:mode.1">Element</a> of <font color="Maroon" title="c5">D</font> holds  <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">x</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">y</font>)
 <b>from </b><i><a class="ref" href="graph_5.html#S2" target="_self" title="GRAPH_5:sch.2">GRAPH_5:sch 2</a>();<br/></i>
<a NAME="E4:75"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_5.html#K4" title="GRAPH_5:func.4">AcyclicPaths</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">v2</font>
 
<b>by </b><i><a class="txt" href="graph_5.html#E1:75"><i><font color="Green" title="E59">A1</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">p</font> being  <a href="graph_1.html#V8" title="GRAPH_1:attr.8">oriented</a> <a href="graph_4.html#V1" title="GRAPH_4:attr.1">Simple</a>  <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c2">G</font><b> such that </b><br/><a NAME="E6:75"/><i><font color="Green" title="E61">A3</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font> &amp; <font color="Maroon" title="c7">p</font> <a href="graph_5.html#R3" title="GRAPH_5:pred.3">is_acyclicpath_of</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">v2</font> )
 ;<br/>
<b>take </b>
<font color="Maroon" title="c7">p</font>
;<br/>

<b>thus </b><a NAME="E7:75"/>
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E6:75"><i><font color="Green" title="E61">A3</font></i></a></i>;<br/>

<b>let </b><font color="Maroon" title="c8">pe</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">G</font>;<br/>

<b>assume </b><a NAME="E8:75"/>
<font color="Maroon" title="c8">pe</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c9">y</font> = <font color="Maroon" title="c8">pe</font> as    <a href="graph_5.html#M1" title="GRAPH_5:mode.1">Element</a> of <font color="Maroon" title="c5">D</font> ;<br/>
<a NAME="E10:75"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">x</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c9">y</font>)
 
<b>by </b><i><a class="txt" href="graph_5.html#E3:75"><i><font color="Green" title="E60">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:75"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c1">W</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c8">pe</font>,<font color="Maroon" title="c1">W</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E6:75"><i><font color="Green" title="E61">A3</font></i></a></i>;<br/>


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