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<b>let </b><font color="Maroon">G</font> be   <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">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>let </b><font color="Maroon">W</font> be   <a href="funct_1.html#NM1">Function</a>;<br/><b>let </b><font color="Maroon">V</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">v1</font> be   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">v2</font> be   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">G</font>;<br/>













<b>assume </b><a NAME="E1:11"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">p</font> <a href="graph_5.html#R8">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">q</font> <a href="graph_5.html#R8">is_shortestpath_of</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">V</font>,<font color="Maroon">W</font> )
 ;<br/>

<a NAME="E2:11"/><b>then </b><i><font color="Green">E52</font></i>: 
( <font color="Maroon">p</font> <a href="graph_5.html#R2">is_orientedpath_of</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">V</font> &amp; ( for <font color="Olive">r</font> being <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">G</font>  st <font color="Olive">r</font> <a href="graph_5.html#R2">is_orientedpath_of</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">V</font> holds <br/> <a href="graph_5.html#K10">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">cost</a> <font color="Olive">r</font>,<font color="Maroon">W</font> ) )
 
<b>by </b><i><a class="ref" href="graph_5.html#D18">GRAPH_5:def 18</a></i>;<br/>
<a NAME="E3:11"/>
( <font color="Maroon">q</font> <a href="graph_5.html#R2">is_orientedpath_of</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">V</font> &amp; ( for <font color="Olive">r</font> being <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">G</font>  st <font color="Olive">r</font> <a href="graph_5.html#R2">is_orientedpath_of</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">V</font> holds <br/> <a href="graph_5.html#K10">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">cost</a> <font color="Olive">r</font>,<font color="Maroon">W</font> ) )
 
<b>by </b><i>, <a class="ref" href="graph_5.html#D18">GRAPH_5:def 18</a></i>;<br/>
<a NAME="E4:11"/><b>then </b><i><font color="Green">E53</font></i>: 
 <a href="graph_5.html#K10">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">cost</a> <font color="Maroon">q</font>,<font color="Maroon">W</font>
 
<b>by </b><i>, <a class="ref" href="graph_5.html#D18">GRAPH_5:def 18</a></i>;<br/>
<a NAME="E5:11"/>
 <a href="graph_5.html#K10">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">cost</a> <font color="Maroon">p</font>,<font color="Maroon">W</font>
 
<b>by </b><i>, , <a class="ref" href="graph_5.html#D18">GRAPH_5:def 18</a></i>;<br/>
<b>hence </b><a NAME="E6:11"/>
 <a href="graph_5.html#K10">cost</a> <font color="Maroon">p</font>,<font color="Maroon">W</font> <a href="hidden.html#R1">=</a>  <a href="graph_5.html#K10">cost</a> <font color="Maroon">q</font>,<font color="Maroon">W</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>


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