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<b>let </b><font color="Maroon">U</font> be    <a href="hidden.html#M1">set</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">G</font> be  <a href="graph_1.html#V3">oriented</a> <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">W</font> be   <a href="funct_2.html#NM1">Function</a> of the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font>, <a href="graph_5.html#K8">Real&gt;=0</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="E1:20"/><i><font color="Green">E34</font></i>: 
( the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">U</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">V</font> &amp; <font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U</font> &amp; ( for <font color="Olive">v3</font>, <font color="Olive">v4</font> being  <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">G</font>  st <font color="Olive">v3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U</font> &amp; <font color="Olive">v4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> holds <br/>for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Olive">e</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> or  not <font color="Olive">e</font> <a href="graph_4.html#R1">orientedly_joins</a> <font color="Olive">v3</font>,<font color="Olive">v4</font> ) ) )
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:20_1"/><i><font color="Green">E52</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">U</font>,<font color="Maroon">W</font>
 ;<br/><a NAME="E2:20_1"/><b>then </b>
( <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">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>  st <font color="Olive">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">U</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">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="E3:20_1"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">P</font> <a href="graph_5.html#R1">is_orientedpath_of</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>
 <b>by </b><i><a class="ref" href="graph_5.html#D4">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">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="E1:20_1_1"/>
<font color="Maroon">Q</font> <a href="graph_5.html#R1">is_orientedpath_of</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>
 ;<br/><a NAME="E2:20_1_1"/><b>then </b>
<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">U</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:20_1_1"/>
 <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/></div><b>end;</b></div><b>hence </b><a NAME="E5:20_1"/>
<font color="Maroon">P</font> <a href="graph_5.html#R7">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#D17">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">E54</font></i>: 
<font color="Maroon">P</font> <a href="graph_5.html#R7">is_shortestpath_of</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">W</font>
 ;<br/><a NAME="E2:20_2"/><b>then </b>
( <font color="Maroon">P</font> <a href="graph_5.html#R1">is_orientedpath_of</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</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>  st <font color="Olive">Q</font> <a href="graph_5.html#R1">is_orientedpath_of</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</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">Q</font>,<font color="Maroon">W</font> ) )
 <b>by </b><i><a class="ref" href="graph_5.html#D17">GRAPH_5:def 17</a></i>;<br/><a NAME="E3:20_2"/><b>then </b><i><font color="Green">E62</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">U</font>
 <b>by </b><i>, </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">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="E1:20_2_1"/>
<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">U</font>
 ;<br/><a NAME="E2:20_2_1"/><b>then </b>
<font color="Maroon">Q</font> <a href="graph_5.html#R1">is_orientedpath_of</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>
 <b>by </b><i><a class="ref" href="graph_5.html#D4">GRAPH_5:def 4</a></i>;<br/><b>hence </b><a NAME="E3:20_2_1"/>
 <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#D17">GRAPH_5:def 17</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:20_2"/>
<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">U</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/></div>
<b>end;</b></div>


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