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<b>let </b><font color="Maroon" title="c1">U</font>, <font color="Maroon" title="c2">V</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">G</font> be  <a href="graph_1.html#V3" title="GRAPH_1:attr.3">oriented</a> <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="c4">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="c3">G</font>;<br/><b>let </b><font color="Maroon" title="c5">W</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c3">G</font>, <a href="graph_5.html#K8" title="GRAPH_5:func.8">Real&gt;=0</a> ;<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="c3">G</font>;<br/>













<b>assume </b><a NAME="E1:20"/><i><font color="Green" title="E21">A1</font></i>: 
( the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c3">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">U</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Maroon" title="c6">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">U</font> &amp; ( for <font color="Olive" title="b1">v3</font>, <font color="Olive" title="b2">v4</font> being  <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c3">G</font>  st <font color="Olive" title="b1">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">U</font> &amp; <font color="Olive" title="b2">v4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font> holds <br/>for <font color="Olive" title="b3">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c3">G</font> or  not <font color="Olive" title="b3">e</font> <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</a> <font color="Olive" title="b1">v3</font>,<font color="Olive" title="b2">v4</font> ) ) )
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:20_1"/><i><font color="Green" title="E22">A2</font></i>: 
<font color="Maroon" title="c4">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">U</font>,<font color="Maroon" title="c5">W</font>
 ;<br/><a NAME="E2:20_1"/><b>then </b>
( <font color="Maroon" title="c4">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">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="c3">G</font>  st <font color="Olive" title="b1">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="c1">U</font> holds <br/> <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">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="c5">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="E3:20_1"/><b>then </b><i><font color="Green" title="E23">A3</font></i>: 
<font color="Maroon" title="c4">P</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" title="GRAPH_5:def.4">GRAPH_5:def 4</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_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="c3">G</font>;<br/><b>assume </b><a NAME="E1:20_1_1"/>
<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>
 ;<br/><a NAME="E2:20_1_1"/><b>then </b>
<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="c1">U</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:20"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="graphsp.html#T12" target="_self" title="GRAPHSP:th.12">Th12</a></i>;<br/><b>hence </b><a NAME="E3:20_1_1"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">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="c5">W</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:20_1"><i><font color="Green" title="E22">A2</font></i></a>, <a class="ref" href="graph_5.html#D18" title="GRAPH_5:def.18">GRAPH_5:def 18</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:20_1"/>
<font color="Maroon" title="c4">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="c5">W</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E3:20_1"><i><font color="Green" title="E23">A3</font></i></a>, <a class="ref" href="graph_5.html#D17" title="GRAPH_5:def.17">GRAPH_5:def 17</a></i>;<br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:20_2"/><i><font color="Green" title="E22">A4</font></i>: 
<font color="Maroon" title="c4">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="c5">W</font>
 ;<br/><a NAME="E2:20_2"/><b>then </b>
( <font color="Maroon" title="c4">P</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> &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="c3">G</font>  st <font color="Olive" title="b1">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> holds <br/> <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">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="c5">W</font> ) )
 <b>by </b><i><a class="ref" href="graph_5.html#D17" title="GRAPH_5:def.17">GRAPH_5:def 17</a></i>;<br/><a NAME="E3:20_2"/><b>then </b><i><font color="Green" title="E23">A5</font></i>: 
<font color="Maroon" title="c4">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">U</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:20"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="graphsp.html#T12" target="_self" title="GRAPHSP:th.12">Th12</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_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="c3">G</font>;<br/><b>assume </b><a NAME="E1:20_2_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="c1">U</font>
 ;<br/><a NAME="E2:20_2_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" title="GRAPH_5:def.4">GRAPH_5:def 4</a></i>;<br/><b>hence </b><a NAME="E3:20_2_1"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">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="c5">W</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:20_2"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="graph_5.html#D17" title="GRAPH_5:def.17">GRAPH_5:def 17</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:20_2"/>
<font color="Maroon" title="c4">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">U</font>,<font color="Maroon" title="c5">W</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E3:20_2"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="graph_5.html#D18" title="GRAPH_5:def.18">GRAPH_5:def 18</a></i>;<br/></div>
<b>end;</b></div>


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