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<b>let </b><font color="Maroon">V</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">U</font> be    <a href="hidden.html#M1">set</a> ;<br/><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#V7">finite</a> <a href="graph_1.html#NM1">Graph</a>;<br/><b>let </b><font color="Maroon">P</font> be  <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">v1</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">v2</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font>;<br/>













<b>assume </b><a NAME="E1:96"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">W</font> <a href="graph_5.html#R5" target="_self">is_weight&gt;=0of</a> <font color="Maroon">G</font> &amp; <font color="Maroon">P</font> <a href="graph_5.html#R8" target="_self">is_shortestpath_of</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">V</font>,<font color="Maroon">W</font> &amp; <font color="Maroon">v1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v2</font> &amp; <font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U</font> &amp; ( for <font color="Olive">Q</font> being <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">G</font><br/> for <font color="Olive">v3</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font>  st not <font color="Olive">v3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; <font color="Olive">Q</font> <a href="graph_5.html#R8" target="_self">is_shortestpath_of</a> <font color="Maroon">v1</font>,<font color="Olive">v3</font>,<font color="Maroon">V</font>,<font color="Maroon">W</font> holds <br/> <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">P</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">Q</font>,<font color="Maroon">W</font> ) )
 ;<br/>

<a NAME="E2:96"/><b>then </b>
<font color="Maroon">P</font> <a href="graph_5.html#R2" target="_self">is_orientedpath_of</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">V</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:96"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">P</font> <a href="graph_5.html#R2" target="_self">is_orientedpath_of</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">U</font>
 
<b>by </b><i>, <a class="txt" href="graph_5.html#E1:96"><i><font color="Green">E39</font></i></a></i>;<br/>
<a NAME="E4:96"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">P</font> <a href="graph_5.html#R7" target="_self">is_shortestpath_of</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">W</font>
 
<b>by </b><i>, <a class="ref" href="graph_5.html#T9" target="_self">Th9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">q</font> be  <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">G</font>;<br/><b>assume </b><a NAME="E1:96_1"/>
<font color="Maroon">q</font> <a href="graph_5.html#R2" target="_self">is_orientedpath_of</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">U</font>
 ;<br/><a NAME="E2:96_1"/><b>then </b>
<font color="Maroon">q</font> <a href="graph_5.html#R1" target="_self">is_orientedpath_of</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:96_1"/>
 <a href="graph_5.html#K10" target="_self">cost</a> <font color="Maroon">P</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">q</font>,<font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="graph_5.html#T10" target="_self">Th10</a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:96"/>
<font color="Maroon">P</font> <a href="graph_5.html#R8" target="_self">is_shortestpath_of</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">U</font>,<font color="Maroon">W</font>
 <b>by </b><i>, </i>;<br/>


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