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<b>set </b><font color="Maroon">VE</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font></span>)</span>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:12_1_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">G</font>
 ;<br/><a NAME="E2:12_1_1_1"/><b>then </b>
( <font color="Maroon">x</font> is   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> &amp; <font color="Maroon">y</font> is   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> )
 <b>by </b><i><a class="ref" href="glib_000.html#T16">GLIB_000:16</a></i>;<br/><a NAME="E3:12_1_1_1"/><b>then </b><i><font color="Green">E40</font></i>: 
( <font color="Maroon">x</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font></span>)</span> &amp; <font color="Maroon">y</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E4:12_1_1_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#D15">GLIB_000:def 15</a></i>;<br/><a NAME="E5:12_1_1_1"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">e</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>set </b><font color="Maroon">W</font> = <span class="p1"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">e</font>,<font color="Maroon">y</font></span><a href="finseq_1.html#K11">*&gt;</a></span>;<br/><b>reconsider </b><font color="Maroon">W</font> = <span class="p1"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">e</font>,<font color="Maroon">y</font></span><a href="finseq_1.html#K11">*&gt;</a></span> as    <a href="finseq_1.html#M2">FinSequence</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font></span>)</span> <b>by </b><i>, , <a class="ref" href="finseq_2.html#T16">FINSEQ_2:16</a></i>;<br/><a NAME="E7:12_1_1_1"/><i><font color="Green">E43</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> 3 &amp; <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> &amp; <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_1_1"><b>now </b></a><div class="add"><b>reconsider </b><font color="Maroon">aa1</font> = 1 as  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="jordan12.html#T3">JORDAN12:3</a></i>;<br/><a NAME="E2:12_1_1_1_1"/>
not <font color="Maroon">aa1</font> <a href="nat_1.html#K1">+</a> 2 is <a href="abian.html#V1">even</a>
 ;<br/><b>hence </b><a NAME="E3:12_1_1_1_1"/>
not  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> is <a href="abian.html#V1">even</a>
 <b>by </b><i><a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/><b>thus </b><a NAME="E4:12_1_1_1_1"/>
<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font>
 <b>by </b><i>, , <a class="ref" href="glib_000.html#T16">GLIB_000:16</a></i>;<br/><b>let </b><font color="Maroon">n</font> be  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E5:12_1_1_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font>
 ;<br/><a NAME="E6:12_1_1_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/><a NAME="E7:12_1_1_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E8:12_1_1_1_1"/><b>then </b>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> 0 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> 1 )
 <b>by </b><i><a class="ref" href="cqc_the1.html#T3">CQC_THE1:3</a></i>;<br/><b>hence </b><a NAME="E9:12_1_1_1_1"/>
<font color="Maroon">W</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> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">W</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> 2</span>)</span>,<font color="Maroon">G</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E9:12_1_1_1"/>
<span class="p1"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">e</font>,<font color="Maroon">y</font></span><a href="finseq_1.html#K11">*&gt;</a></span> is    <a href="glib_001.html#M3" target="_self">Walk</a> of <font color="Maroon">G</font>
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E2:12_1_1"/>
( not <font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">G</font> implies <font color="Maroon">G</font> <a href="glib_001.html#K1" target="_self">.walkOf</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span></span>)</span> is    <a href="glib_001.html#M3" target="_self">Walk</a> of <font color="Maroon">G</font> )
 ;<br/>


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