<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="graph_1.html#NM1">Graph</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be  <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="funct_1.html#NM1">Function</a>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be   <a href="graph_3.html#NM2">Vertex</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>













<b>assume </b><a NAME="E1:11"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="graph_5.html#R8">is_shortestpath_of</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="graph_5.html#R8">is_shortestpath_of</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<a NAME="E2:11"/><b>then </b><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="graph_5.html#R2">is_orientedpath_of</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="graph_5.html#R2">is_orientedpath_of</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font> holds <br/> <a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10">cost</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></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">c<sub>3</sub></font> <a href="graph_5.html#R2">is_orientedpath_of</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="graph_5.html#R2">is_orientedpath_of</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font> holds <br/> <a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10">cost</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font> ) )
 
<b>by </b><i><a class="txt" href="graphsp.html#E1:11"><i><font color="Green">E12</font></i></a>, <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">E14</font></i>: 
 <a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="graphsp.html#E1:11"><i><font color="Green">E12</font></i></a>, <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">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="graphsp.html#E1:11"><i><font color="Green">E12</font></i></a>, <a class="txt" href="graphsp.html#E2:11"><i><font color="Green">E13</font></i></a>, <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">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="graph_5.html#K10">cost</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E4:11"><i><font color="Green">E14</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>


</div>
