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<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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">v</font> be   <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c2">G</font>;<br/><b>let </b><font color="Maroon" title="c4">p</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">X</font> <a href="graph_3.html#K12" title="GRAPH_3:func.12">-PathSet</a> <font color="Maroon" title="c3">v</font>;<br/><b>let </b><font color="Maroon" title="c5">Y</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>









<b>assume </b><a NAME="E1:77"/><i><font color="Green" title="E65">A1</font></i>: 
( <font color="Maroon" title="c5">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">G</font> &amp;  <a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c1">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/>

<a NAME="E2:77"/><b>then </b><i><font color="Green" title="E66">A2</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="graph_3.html#K12" title="GRAPH_3:func.12">-PathSet</a> <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  : ( <font color="Olive" title="b1">c</font> is   <a href="graph_1.html#NM4" title="GRAPH_1:NM.4">Path</a> of <font color="Maroon" title="c2">G</font> &amp; not <font color="Olive" title="b1">c</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp;  ex <font color="Olive" title="b2">vs</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b2">vs</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Olive" title="b1">c</font> &amp; <font color="Olive" title="b2">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> ) ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="graph_3.html#D11" target="_self" title="GRAPH_3:def.11">Def11</a></i>;<br/>
<a NAME="E3:77"/>
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="graph_3.html#K12" title="GRAPH_3:func.12">-PathSet</a> <font color="Maroon" title="c3">v</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c6">c</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">X</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> <b> such that </b><br/><a NAME="E5:77"/><i><font color="Green" title="E67">A3</font></i>: 
( <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">c</font> &amp; <font color="Maroon" title="c6">c</font> is   <a href="graph_1.html#NM4" title="GRAPH_1:NM.4">Path</a> of <font color="Maroon" title="c2">G</font> &amp; not <font color="Maroon" title="c6">c</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp;  ex <font color="Olive" title="b1">vs</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b1">vs</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c6">c</font> &amp; <font color="Olive" title="b1">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> ) )
 <b>by </b><i><a class="txt" href="graph_3.html#E2:77"><i><font color="Green" title="E66">A2</font></i></a></i>;<br/>
<a NAME="E6:77"/><i><font color="Green" title="E68">A4</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">p</font>
 
<b>by </b><i><a class="txt" href="graph_3.html#E5:77"><i><font color="Green" title="E67">A3</font></i></a>, <a class="ref" href="finseq_4.html#T77" title="FINSEQ_4:th.77">FINSEQ_4:77</a></i>;<br/>
<a NAME="E7:77"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Y</font>
 
<b>by </b><i><a class="txt" href="graph_3.html#E1:77"><i><font color="Green" title="E65">A1</font></i></a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E8:77"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">Y</font>
 <b>by </b><i><a class="txt" href="graph_3.html#E6:77"><i><font color="Green" title="E68">A4</font></i></a>, <a class="ref" href="nat_1.html#T44" title="NAT_1:th.44">NAT_1:44</a></i>;<br/>


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