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

<b>set </b><font color="Maroon" title="c5">PT</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</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="c1">G</font> : <font color="Olive" title="b1">p</font> <a href="graph_5.html#R4" title="GRAPH_5:pred.4">is_acyclicpath_of</a> <font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font>,<font color="Maroon" title="c4">V</font> </span>}</span> ;<br/>
<a NAME="E1:47_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</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="c1">G</font> : <font color="Olive" title="b1">p</font> <a href="graph_5.html#R4" title="GRAPH_5:pred.4">is_acyclicpath_of</a> <font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font>,<font color="Maroon" title="c4">V</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:47_1_1_1"/>
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</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="c1">G</font> : <font color="Olive" title="b1">p</font> <a href="graph_5.html#R4" title="GRAPH_5:pred.4">is_acyclicpath_of</a> <font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font>,<font color="Maroon" title="c4">V</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">p</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="c1">G</font><b> such that </b><br/><a NAME="E3:47_1_1_1"/><i><font color="Green" title="E40">A1</font></i>: 
( <font color="Maroon" title="c6">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font> &amp; <font color="Maroon" title="c7">p</font> <a href="graph_5.html#R4" title="GRAPH_5:pred.4">is_acyclicpath_of</a> <font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font>,<font color="Maroon" title="c4">V</font> )
 ;<br/>
<b>thus </b><a NAME="E4:47_1_1_1"/>
<font color="Maroon" title="c6">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="c1">G</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> 
 <b>by </b><i><a class="txt" href="graph_5.html#E3:47_1_1_1"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:47_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</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="c1">G</font> : <font color="Olive" title="b1">p</font> <a href="graph_5.html#R4" title="GRAPH_5:pred.4">is_acyclicpath_of</a> <font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font>,<font color="Maroon" title="c4">V</font> </span>}</span>  is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default">the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>
 ;<br/>


</div>
