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<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Graph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">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="c1">G</font><br/> for <font color="Olive" title="b3">W</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><br/> for <font color="Olive" title="b4">V</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b5">v1</font>, <font color="Olive" title="b6">v2</font> being  <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">p</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Olive" title="b5">v1</font>,<font color="Olive" title="b6">v2</font>,<font color="Olive" title="b4">V</font>,<font color="Olive" title="b3">W</font> &amp; <font color="Olive" title="b2">q</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Olive" title="b5">v1</font>,<font color="Olive" title="b6">v2</font>,<font color="Olive" title="b4">V</font>,<font color="Olive" title="b3">W</font> holds <br/> <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">W</font></span><br/><b>let </b><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">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="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">W</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><br/> for <font color="Olive" title="b2">V</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b3">v1</font>, <font color="Olive" title="b4">v2</font> being  <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c1">G</font>  st <font color="Maroon" title="c2">p</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Olive" title="b3">v1</font>,<font color="Olive" title="b4">v2</font>,<font color="Olive" title="b2">V</font>,<font color="Olive" title="b1">W</font> &amp; <font color="Maroon" title="c3">q</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Olive" title="b3">v1</font>,<font color="Olive" title="b4">v2</font>,<font color="Olive" title="b2">V</font>,<font color="Olive" title="b1">W</font> holds <br/> <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c2">p</font>,<font color="Olive" title="b1">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c3">q</font>,<font color="Olive" title="b1">W</font></span><br/><b>let </b><font color="Maroon" title="c4">W</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">V</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">v1</font>, <font color="Olive" title="b3">v2</font> being  <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c1">G</font>  st <font color="Maroon" title="c2">p</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Olive" title="b2">v1</font>,<font color="Olive" title="b3">v2</font>,<font color="Olive" title="b1">V</font>,<font color="Maroon" title="c4">W</font> &amp; <font color="Maroon" title="c3">q</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Olive" title="b2">v1</font>,<font color="Olive" title="b3">v2</font>,<font color="Olive" title="b1">V</font>,<font color="Maroon" title="c4">W</font> holds <br/> <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">W</font></span><br/><b>let </b><font color="Maroon" title="c5">V</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v2</font> being  <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c1">G</font>  st <font color="Maroon" title="c2">p</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b2">v2</font>,<font color="Maroon" title="c5">V</font>,<font color="Maroon" title="c4">W</font> &amp; <font color="Maroon" title="c3">q</font> <a href="graph_5.html#R8" title="GRAPH_5:pred.8">is_shortestpath_of</a> <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b2">v2</font>,<font color="Maroon" title="c5">V</font>,<font color="Maroon" title="c4">W</font> holds <br/> <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">W</font></span><br/><b>let </b><font color="Maroon" title="c6">v1</font>, <font color="Maroon" title="c7">v2</font> be   <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">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="c5">V</font>,<font color="Maroon" title="c4">W</font> &amp; <font color="Maroon" title="c3">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="Maroon" title="c7">v2</font>,<font color="Maroon" title="c5">V</font>,<font color="Maroon" title="c4">W</font> implies  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">W</font> )</span><br/>













<b>assume </b><a NAME="E1:11"/><i><font color="Green" title="E12">A1</font></i>: 
( <font color="Maroon" title="c2">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="c5">V</font>,<font color="Maroon" title="c4">W</font> &amp; <font color="Maroon" title="c3">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="Maroon" title="c7">v2</font>,<font color="Maroon" title="c5">V</font>,<font color="Maroon" title="c4">W</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">W</font></span><br/>

<a NAME="E2:11"/><b>then </b><i><font color="Green" title="E13">A2</font></i>: 
( <font color="Maroon" title="c2">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="c5">V</font> &amp; ( for <font color="Olive" title="b1">r</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="c1">G</font>  st <font color="Olive" title="b1">r</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="c5">V</font> holds <br/> <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">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">r</font>,<font color="Maroon" title="c4">W</font> ) )
 
<b>by </b><i><a class="ref" href="graph_5.html#D18" title="GRAPH_5:def.18">GRAPH_5:def 18</a></i>;<br/>
<a NAME="E3:11"/>
( <font color="Maroon" title="c3">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="c5">V</font> &amp; ( for <font color="Olive" title="b1">r</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="c1">G</font>  st <font color="Olive" title="b1">r</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="c5">V</font> holds <br/> <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">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">r</font>,<font color="Maroon" title="c4">W</font> ) )
 
<b>by </b><i><a class="txt" href="graphsp.html#E1:11"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="graph_5.html#D18" title="GRAPH_5:def.18">GRAPH_5:def 18</a></i>;<br/>
<a NAME="E4:11"/><b>then </b><i><font color="Green" title="E14">A3</font></i>: 
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">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="c3">q</font>,<font color="Maroon" title="c4">W</font>
 
<b>by </b><i><a class="txt" href="graphsp.html#E1:11"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="graph_5.html#D18" title="GRAPH_5:def.18">GRAPH_5:def 18</a></i>;<br/>
<a NAME="E5:11"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">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="c2">p</font>,<font color="Maroon" title="c4">W</font>
 
<b>by </b><i><a class="txt" href="graphsp.html#E1:11"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="graphsp.html#E2:11"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="graph_5.html#D18" title="GRAPH_5:def.18">GRAPH_5:def 18</a></i>;<br/>
<b>hence </b><a NAME="E6:11"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">W</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E4:11"><i><font color="Green" title="E14">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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