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<b>let </b><font color="Maroon" title="c1">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>set </b><font color="Maroon" title="c2">n</font> =  <a href="graph_1.html#K3" title="GRAPH_1:func.3">EdgesCount</a> <font color="Maroon" title="c1">G</font>;<br/>
<b>set </b><font color="Maroon" title="c3">D</font> = the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font>;<br/>
<b>set </b><font color="Maroon" title="c4">A</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <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="c1">G</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  :  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_1.html#K3" title="GRAPH_1:func.3">EdgesCount</a> <font color="Maroon" title="c1">G</font> </span>}</span> ;<br/>
<a NAME="E1:84"/>
the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">G</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="ref" href="graph_1.html#D8" title="GRAPH_1:def.8">GRAPH_1:def 8</a></i>;<br/>
<a NAME="E2:84"/><b>then </b><i><font color="Green" title="E67">A1</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <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="c1">G</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  :  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_1.html#K3" title="GRAPH_1:func.3">EdgesCount</a> <font color="Maroon" title="c1">G</font> </span>}</span>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="ref" href="graph_5.html#T4" target="_self" title="GRAPH_5:th.4">Th4</a></i>;<br/>
<a NAME="E3:84"/>
 <a href="graph_5.html#K7" title="GRAPH_5:func.7">AcyclicPaths</a> <font color="Maroon" title="c1">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <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="c1">G</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  :  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_1.html#K3" title="GRAPH_1:func.3">EdgesCount</a> <font color="Maroon" title="c1">G</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</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="E1:84_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_5.html#K7" title="GRAPH_5:func.7">AcyclicPaths</a> <font color="Maroon" title="c1">G</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">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="E3:84_1"/><i><font color="Green" title="E68">A2</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font>
 ;<br/>
<a NAME="E4:84_1"/><i><font color="Green" title="E69">A3</font></i>: 
<font color="Maroon" title="c6">p</font> is    <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="c1">G</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E5:84_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_1.html#K3" title="GRAPH_1:func.3">EdgesCount</a> <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="ref" href="graph_5.html#T62" target="_self" title="GRAPH_5:th.62">Th62</a></i>;<br/>
<b>hence </b><a NAME="E6:84_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <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="c1">G</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  :  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_1.html#K3" title="GRAPH_1:func.3">EdgesCount</a> <font color="Maroon" title="c1">G</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="graph_5.html#E3:84_1"><i><font color="Green" title="E68">A2</font></i></a>, <a class="txt" href="graph_5.html#E4:84_1"><i><font color="Green" title="E69">A3</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:84"/>
 <a href="graph_5.html#K7" title="GRAPH_5:func.7">AcyclicPaths</a> <font color="Maroon" title="c1">G</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="graph_5.html#E2:84"><i><font color="Green" title="E67">A1</font></i></a>, <a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a></i>;<br/>


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