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<b>let </b><font color="Maroon">G1</font> be   <a href="glib_000.html#NM2">_Graph</a>;<br/><b>let </b><font color="Maroon">G2</font> be   <a href="glib_000.html#NM2">_Graph</a>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>





<b>assume </b><a NAME="E1:249"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">G1</font> <a href="glib_000.html#R5">==</a> <font color="Maroon">G2</font> &amp; <font color="Maroon">x</font> is    <a href="glib_001.html#M2" target="_self">EdgeSeq</a> of <font color="Maroon">G1</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">es</font> = <font color="Maroon">x</font> as    <a href="glib_001.html#M2" target="_self">EdgeSeq</a> of <font color="Maroon">G1</font> ;<br/>
<b>consider </b><font color="Maroon">vs</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G1</font><b> such that </b><br/><a NAME="E4:249"/><i><font color="Green">E40</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">es</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">es</font> holds <br/><font color="Maroon">es</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font>,<font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">G1</font> ) )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">es2</font> = <font color="Maroon">es</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G2</font> <b>by </b><i><a class="txt" href="glib_001.html#E57"><i><font color="Green">Lemma161</font></i></a>, <a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="249_1"><b>now </b></a><div class="add"><b>reconsider </b><font color="Maroon">vs</font> = <font color="Maroon">vs</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G2</font> <b>by </b><i><a class="txt" href="glib_001.html#E57"><i><font color="Green">Lemma161</font></i></a>, <a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>;<br/><b>take </b><font color="Maroon">vs</font> = <font color="Maroon">vs</font>;<br/><b>thus </b><a NAME="E2:249_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">es</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i/>;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E3:249_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">es2</font> )
 ;<br/><a NAME="E4:249_1"/><b>then </b>
<font color="Maroon">es2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">G1</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:249_1"/>
<font color="Maroon">es2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">G2</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E57"><i><font color="Green">Lemma161</font></i></a>, <a class="ref" href="glib_000.html#T91">GLIB_000:91</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:249"/>
<font color="Maroon">x</font> is    <a href="glib_001.html#M2" target="_self">EdgeSeq</a> of <font color="Maroon">G2</font>
 <b>by </b><i/>;<br/>


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