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<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:14_1_1_1"/><i><font color="Green" title="E13">A1</font></i>: 
( not <font color="Maroon" title="c3">n</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> &amp; <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W</font> )
 ;<br/><b>reconsider </b><font color="Maroon" title="c4">n1</font> = <font color="Maroon" title="c3">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_1_1_1_1_1"/>
( <font color="Maroon" title="c3">n</font> <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="c2">W</font> or <font color="Maroon" title="c3">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:14_1_1_1_1_1_1"/>
<font color="Maroon" title="c3">n</font> <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="c2">W</font>
 ;<br/><div class="add"><a NAME="E2:14_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
 ;<br/><b>hence </b><a NAME="E3:14_1_1_1_1_1_1"/>
<font color="Maroon" title="c2">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:14_1_1_1_1_1_2"/>
<font color="Maroon" title="c3">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W</font>
 ;<br/><div class="add"><a NAME="E2:14_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E1:14_1_1_1"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:14_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c2">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E1:14_1_1_1"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="glib_001.html#D3" target="_self" title="GLIB_001:def.3">Def3</a></i>;<br/><b>hence </b><a NAME="E4:14_1_1_1_1_1_2"/>
<font color="Maroon" title="c2">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T16" title="GLIB_000:th.16">GLIB_000:16</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:14_1_1_1"/>
<font color="Maroon" title="c2">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>
 ;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E2:14_1_1"/>
( ( ( <font color="Maroon" title="c3">n</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> or  not <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W</font> ) implies <font color="Maroon" title="c2">W</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> holds  verum ) )
 ;<br/>


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