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

<b>let </b><font color="Maroon" title="c1">W</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>;<br/><b>let </b><font color="Maroon" title="c2">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="c3">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="c2">G</font>;<br/><b>let </b><font color="Maroon" title="c4">v1</font>, <font color="Maroon" title="c5">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="c2">G</font>;<br/>









<b>assume </b><a NAME="E1:92"/><i><font color="Green" title="E73">A1</font></i>: 
( <font color="Maroon" title="c3">P</font> <a href="graph_5.html#R1" title="GRAPH_5:pred.1">is_orientedpath_of</a> <font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c5">v2</font> &amp; <font color="Maroon" title="c1">W</font> <a href="graph_5.html#R5" title="GRAPH_5:pred.5">is_weight&gt;=0of</a> <font color="Maroon" title="c2">G</font> )
 ;<br/>

<a NAME="E2:92"/><b>then </b>
 <a href="graph_5.html#K4" title="GRAPH_5:func.4">AcyclicPaths</a> <font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c5">v2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="graph_5.html#T47" target="_self" title="GRAPH_5:th.47">Th47</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">r</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">G</font><b> such that </b><br/><a NAME="E4:92"/><i><font color="Green" title="E74">A2</font></i>: 
( <font color="Maroon" title="c6">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_5.html#K4" title="GRAPH_5:func.4">AcyclicPaths</a> <font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c5">v2</font> &amp; ( for <font color="Olive" title="b1">s</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">G</font>  st <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_5.html#K4" title="GRAPH_5:func.4">AcyclicPaths</a> <font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c5">v2</font> holds <br/> <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c6">r</font>,<font color="Maroon" title="c1">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">s</font>,<font color="Maroon" title="c1">W</font> ) )
 <b>by </b><i><a class="ref" href="graph_5.html#T56" target="_self" title="GRAPH_5:th.56">Th56</a></i>;<br/>
<a NAME="E5:92"/>
<font color="Maroon" title="c6">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">t</font> where <font color="Olive" title="b1">t</font> is  <a href="graph_1.html#V8" title="GRAPH_1:attr.8">oriented</a> <a href="graph_4.html#V1" title="GRAPH_4:attr.1">Simple</a>  <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c2">G</font> : <font color="Olive" title="b1">t</font> <a href="graph_5.html#R3" title="GRAPH_5:pred.3">is_acyclicpath_of</a> <font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c5">v2</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="graph_5.html#E4:92"><i><font color="Green" title="E74">A2</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">t</font> being  <a href="graph_1.html#V8" title="GRAPH_1:attr.8">oriented</a> <a href="graph_4.html#V1" title="GRAPH_4:attr.1">Simple</a>  <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c2">G</font><b> such that </b><br/><a NAME="E7:92"/><i><font color="Green" title="E75">A3</font></i>: 
( <font color="Maroon" title="c6">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t</font> &amp; <font color="Maroon" title="c7">t</font> <a href="graph_5.html#R3" title="GRAPH_5:pred.3">is_acyclicpath_of</a> <font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c5">v2</font> )
 ;<br/>
<b>take </b>
<font color="Maroon" title="c7">t</font>
;<br/>

<b>thus </b><a NAME="E8:92"/>
<font color="Maroon" title="c7">t</font> <a href="graph_5.html#R1" title="GRAPH_5:pred.1">is_orientedpath_of</a> <font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c5">v2</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E7:92"><i><font color="Green" title="E75">A3</font></i></a>, <a class="ref" href="graph_5.html#D6" target="_self" title="GRAPH_5:def.6">Def6</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="graph_5.html#D17" target="_self" title="GRAPH_5:def.17">GRAPH_5:def 17</a><br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon" title="c8">s</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="c2">G</font>;<br/><b>assume </b><a NAME="E1:92_1"/><i><font color="Green" title="E76">A4</font></i>: 
<font color="Maroon" title="c8">s</font> <a href="graph_5.html#R1" title="GRAPH_5:pred.1">is_orientedpath_of</a> <font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c5">v2</font>
 ;<br/><a NAME="E2:92_1"/><b>then </b>
 <a href="graph_5.html#K6" title="GRAPH_5:func.6">AcyclicPaths</a> <font color="Maroon" title="c8">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="graph_5.html#T47" target="_self" title="GRAPH_5:th.47">Th47</a></i>;<br/><b>then consider </b><font color="Maroon" title="c9">x</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">G</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> <b> such that </b><br/><a NAME="E4:92_1"/><i><font color="Green" title="E77">A5</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_5.html#K6" title="GRAPH_5:func.6">AcyclicPaths</a> <font color="Maroon" title="c8">s</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T10" title="SUBSET_1:th.10">SUBSET_1:10</a></i>;<br/><a NAME="E5:92_1"/><i><font color="Green" title="E78">A6</font></i>: 
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c1">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">s</font>,<font color="Maroon" title="c1">W</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E1:92"><i><font color="Green" title="E73">A1</font></i></a>, <a class="txt" href="graph_5.html#E4:92_1"><i><font color="Green" title="E77">A5</font></i></a>, <a class="ref" href="graph_5.html#T60" target="_self" title="GRAPH_5:th.60">Th60</a></i>;<br/><a NAME="E6:92_1"/>
 <a href="graph_5.html#K6" title="GRAPH_5:func.6">AcyclicPaths</a> <font color="Maroon" title="c8">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="graph_5.html#K4" title="GRAPH_5:func.4">AcyclicPaths</a> <font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c5">v2</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E1:92_1"><i><font color="Green" title="E76">A4</font></i></a>, <a class="ref" href="graph_5.html#T44" target="_self" title="GRAPH_5:th.44">Th44</a></i>;<br/><a NAME="E7:92_1"/><b>then </b>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c6">r</font>,<font color="Maroon" title="c1">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="c9">x</font>,<font color="Maroon" title="c1">W</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E4:92"><i><font color="Green" title="E74">A2</font></i></a>, <a class="txt" href="graph_5.html#E4:92_1"><i><font color="Green" title="E77">A5</font></i></a></i>;<br/><b>hence </b><a NAME="E8:92_1"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c7">t</font>,<font color="Maroon" title="c1">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">s</font>,<font color="Maroon" title="c1">W</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E7:92"><i><font color="Green" title="E75">A3</font></i></a>, <a class="txt" href="graph_5.html#E5:92_1"><i><font color="Green" title="E78">A6</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div>
<b>end;</b></div>


</div>
