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<b>let </b><font color="Maroon" title="c1">V</font>, <font color="Maroon" title="c2">U</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">W</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>;<br/><b>let </b><font color="Maroon" title="c4">G</font> be  <a href="graph_1.html#V7" title="GRAPH_1:attr.7">finite</a> <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Graph</a>;<br/><b>let </b><font color="Maroon" title="c5">P</font> be  <a href="graph_1.html#V8" title="GRAPH_1:attr.8">oriented</a>  <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c4">G</font>;<br/><b>let </b><font color="Maroon" title="c6">v1</font>, <font color="Maroon" title="c7">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="c4">G</font>;<br/>













<b>assume </b><a NAME="E1:95"/><i><font color="Green" title="E72">A1</font></i>: 
( <font color="Maroon" title="c3">W</font> <a href="graph_5.html#R5" title="GRAPH_5:pred.5">is_weight&gt;=0of</a> <font color="Maroon" title="c4">G</font> &amp; <font color="Maroon" title="c5">P</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c3">W</font> &amp; <font color="Maroon" title="c6">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">v2</font> &amp; <font color="Maroon" title="c1">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">U</font> &amp; ( for <font color="Olive" title="b1">Q</font> being <a href="graph_1.html#V8" title="GRAPH_1:attr.8">oriented</a>  <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c4">G</font><br/> for <font color="Olive" title="b2">v3</font> being   <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="c4">G</font>  st not <font color="Olive" title="b2">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">V</font> &amp; <font color="Olive" title="b1">Q</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c6">v1</font>,<font color="Olive" title="b2">v3</font>,<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c3">W</font> holds <br/> <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c3">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="Olive" title="b1">Q</font>,<font color="Maroon" title="c3">W</font> ) )
 ;<br/>

<a NAME="E2:95"/><b>then </b>
<font color="Maroon" title="c5">P</font> <a href="graph_5.html#R2" title="GRAPH_5:pred.2">is_orientedpath_of</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c1">V</font>
 
<b>by </b><i><a class="ref" href="graph_5.html#D18" target="_self" title="GRAPH_5:def.18">Def18</a></i>;<br/>
<a NAME="E3:95"/><b>then </b><i><font color="Green" title="E73">A2</font></i>: 
<font color="Maroon" title="c5">P</font> <a href="graph_5.html#R2" title="GRAPH_5:pred.2">is_orientedpath_of</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c2">U</font>
 
<b>by </b><i><a class="txt" href="graph_5.html#E1:95"><i><font color="Green" title="E72">A1</font></i></a>, <a class="ref" href="graph_5.html#T36" target="_self" title="GRAPH_5:th.36">Th36</a></i>;<br/>
<a NAME="E4:95"/><i><font color="Green" title="E74">A3</font></i>: 
<font color="Maroon" title="c5">P</font> <a href="graph_5.html#R7" title="GRAPH_5:pred.7">is_shortestpath_of</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c3">W</font>
 
<b>by </b><i><a class="txt" href="graph_5.html#E1:95"><i><font color="Green" title="E72">A1</font></i></a>, <a class="ref" href="graph_5.html#T67" target="_self" title="GRAPH_5:th.67">Th67</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">q</font> be  <a href="graph_1.html#V8" title="GRAPH_1:attr.8">oriented</a>  <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c4">G</font>;<br/><b>assume </b><a NAME="E1:95_1"/>
<font color="Maroon" title="c8">q</font> <a href="graph_5.html#R2" title="GRAPH_5:pred.2">is_orientedpath_of</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c2">U</font>
 ;<br/><a NAME="E2:95_1"/><b>then </b>
<font color="Maroon" title="c8">q</font> <a href="graph_5.html#R1" title="GRAPH_5:pred.1">is_orientedpath_of</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c7">v2</font>
 <b>by </b><i><a class="ref" href="graph_5.html#D4" target="_self" title="GRAPH_5:def.4">Def4</a></i>;<br/><b>hence </b><a NAME="E3:95_1"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c3">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">q</font>,<font color="Maroon" title="c3">W</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E4:95"><i><font color="Green" title="E74">A3</font></i></a>, <a class="ref" href="graph_5.html#D17" target="_self" title="GRAPH_5:def.17">Def17</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:95"/>
<font color="Maroon" title="c5">P</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c2">U</font>,<font color="Maroon" title="c3">W</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E3:95"><i><font color="Green" title="E73">A2</font></i></a>, <a class="ref" href="graph_5.html#D18" target="_self" title="GRAPH_5:def.18">Def18</a></i>;<br/>


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