<?xml version="1.0"?>
<div class="add">

<b>set </b><font color="Maroon" title="c4">IT</font> =  <a href="lexbfs.html#K14" title="LEXBFS:func.14">LexBFS:LabelAdjacent</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">k</font>;<br/>
<b>set </b><font color="Maroon" title="c5">VL</font> =  <a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K14" title="LEXBFS:func.14">LexBFS:LabelAdjacent</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">k</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c6">V2</font> =  <a href="lexbfs.html#K6" title="LEXBFS:func.6">the_V2Label_of</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K14" title="LEXBFS:func.14">LexBFS:LabelAdjacent</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">k</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c7">VLG</font> =  <a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font>;<br/>
<b>set </b><font color="Maroon" title="c8">V2G</font> =  <a href="lexbfs.html#K6" title="LEXBFS:func.6">the_V2Label_of</a> <font color="Maroon" title="c1">G</font>;<br/>
<a NAME="E1:138_1_1"/><i><font color="Green" title="E65">A1</font></i>: 
 <a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K14" title="LEXBFS:func.14">LexBFS:LabelAdjacent</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="ref" href="glib_000.html#T12" title="GLIB_000:th.12">GLIB_000:12</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="138_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:138_1_1_1"/><i><font color="Green" title="E66">A2</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K6" title="LEXBFS:func.6">the_V2Label_of</a> <span class="p2">(<span class="default"><a href="lexbfs.html#K14" title="LEXBFS:func.14">LexBFS:LabelAdjacent</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span>
 ;<br/><b>consider </b><font color="Maroon" title="c10">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:138_1_1_1"/>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K6" title="LEXBFS:func.6">the_V2Label_of</a> <span class="p2">(<span class="default"><a href="lexbfs.html#K14" title="LEXBFS:func.14">LexBFS:LabelAdjacent</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E4:138_1_1_1"/><i><font color="Green" title="E67">A3</font></i>: 
<span class="p1">(<span class="default"><a href="lexbfs.html#K6" title="LEXBFS:func.6">the_V2Label_of</a> <span class="p2">(<span class="default"><a href="lexbfs.html#K14" title="LEXBFS:func.14">LexBFS:LabelAdjacent</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E1:138_1_1_1"><i><font color="Green" title="E66">A2</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="138_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="E1:138_1_1_1_1"/>
(  not <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> or ( <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; not <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="138_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:138_1_1_1_1_1"/><i><font color="Green" title="E68">A4</font></i>: 
(  not <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="138_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="138_1_1_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="E1:138_1_1_1_1_1_1_1"/>
(  not <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> )
 </span><b>by </b><i><a class="txt" href="lexbfs.html#E1:138_1_1_1_1_1"><i><font color="Green" title="E68">A4</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="138_1_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:138_1_1_1_1_1_1_1_1"/>
not <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:138_1_1_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="lexbfs.html#K6" title="LEXBFS:func.6">the_V2Label_of</a> <span class="p2">(<span class="default"><a href="lexbfs.html#K14" title="LEXBFS:func.14">LexBFS:LabelAdjacent</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K6" title="LEXBFS:func.6">the_V2Label_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="lexbfs.html#K2" title="LEXBFS:func.2">.</a> <font color="Maroon" title="c10">y</font>
 <b>by </b><i><a class="ref" href="lexbfs.html#T32" target="_self" title="LEXBFS:th.32">Th32</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="138_1_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:138_1_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:138_1_1_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="lexbfs.html#K6" title="LEXBFS:func.6">the_V2Label_of</a> <span class="p2">(<span class="default"><a href="lexbfs.html#K14" title="LEXBFS:func.14">LexBFS:LabelAdjacent</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K6" title="LEXBFS:func.6">the_V2Label_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="lexbfs.html#K2" title="LEXBFS:func.2">.</a> <font color="Maroon" title="c10">y</font>
 <b>by </b><i><a class="ref" href="lexbfs.html#T33" target="_self" title="LEXBFS:th.33">Th33</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:138_1_1_1_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="txt" href="lexbfs.html#E4:138_1_1_1"><i><font color="Green" title="E67">A3</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="138_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:138_1_1_1_1_2"/>
( <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; not <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> )
 ;<br/><div class="add"><a NAME="E2:138_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E68">A5</font></i>: 
<span class="p1">(<span class="default"><a href="lexbfs.html#K6" title="LEXBFS:func.6">the_V2Label_of</a> <span class="p2">(<span class="default"><a href="lexbfs.html#K14" title="LEXBFS:func.14">LexBFS:LabelAdjacent</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K6" title="LEXBFS:func.6">the_V2Label_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="lexbfs.html#K2" title="LEXBFS:func.2">.</a> <font color="Maroon" title="c10">y</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">k</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="lexbfs.html#T34" target="_self" title="LEXBFS:th.34">Th34</a></i>;<br/><a NAME="E3:138_1_1_1_1_2"/>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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/><a NAME="E4:138_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E69">A6</font></i>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">k</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/><a NAME="E5:138_1_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="lexbfs.html#K6" title="LEXBFS:func.6">the_V2Label_of</a> <span class="p2">(<span class="default"><a href="lexbfs.html#K14" title="LEXBFS:func.14">LexBFS:LabelAdjacent</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="txt" href="lexbfs.html#E2:138_1_1_1_1_2"><i><font color="Green" title="E68">A5</font></i></a>, <a class="txt" href="lexbfs.html#E4:138_1_1_1_1_2"><i><font color="Green" title="E69">A6</font></i></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/><b>hence </b><a NAME="E6:138_1_1_1_1_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="txt" href="lexbfs.html#E4:138_1_1_1"><i><font color="Green" title="E67">A3</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E3:138_1_1"/><b>then </b><i><font color="Green" title="E66">A7</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K6" title="LEXBFS:func.6">the_V2Label_of</a> <span class="p2">(<span class="default"><a href="lexbfs.html#K14" title="LEXBFS:func.14">LexBFS:LabelAdjacent</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="138_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:138_1_1_2"/><i><font color="Green" title="E67">A8</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K6" title="LEXBFS:func.6">the_V2Label_of</a> <span class="p2">(<span class="default"><a href="lexbfs.html#K14" title="LEXBFS:func.14">LexBFS:LabelAdjacent</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span>
 ;<br/><b>consider </b><font color="Maroon" title="c10">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:138_1_1_2"/>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K6" title="LEXBFS:func.6">the_V2Label_of</a> <span class="p2">(<span class="default"><a href="lexbfs.html#K14" title="LEXBFS:func.14">LexBFS:LabelAdjacent</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E4:138_1_1_2"/><i><font color="Green" title="E68">A9</font></i>: 
<span class="p1">(<span class="default"><a href="lexbfs.html#K6" title="LEXBFS:func.6">the_V2Label_of</a> <span class="p2">(<span class="default"><a href="lexbfs.html#K14" title="LEXBFS:func.14">LexBFS:LabelAdjacent</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E1:138_1_1_2"><i><font color="Green" title="E67">A8</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="138_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:138_1_1_2_1"/>
(  not <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> or ( <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; not <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="138_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:138_1_1_2_1_1"/><i><font color="Green" title="E69">A10</font></i>: 
(  not <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="138_1_1_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="138_1_1_2_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="E1:138_1_1_2_1_1_1_1"/>
(  not <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> )
 </span><b>by </b><i><a class="txt" href="lexbfs.html#E1:138_1_1_2_1_1"><i><font color="Green" title="E69">A10</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="138_1_1_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:138_1_1_2_1_1_1_1_1"/>
not <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:138_1_1_2_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="lexbfs.html#K6" title="LEXBFS:func.6">the_V2Label_of</a> <span class="p2">(<span class="default"><a href="lexbfs.html#K14" title="LEXBFS:func.14">LexBFS:LabelAdjacent</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K6" title="LEXBFS:func.6">the_V2Label_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="lexbfs.html#K2" title="LEXBFS:func.2">.</a> <font color="Maroon" title="c10">y</font>
 <b>by </b><i><a class="ref" href="lexbfs.html#T32" target="_self" title="LEXBFS:th.32">Th32</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="138_1_1_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:138_1_1_2_1_1_1_1_2"/>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:138_1_1_2_1_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="lexbfs.html#K6" title="LEXBFS:func.6">the_V2Label_of</a> <span class="p2">(<span class="default"><a href="lexbfs.html#K14" title="LEXBFS:func.14">LexBFS:LabelAdjacent</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K6" title="LEXBFS:func.6">the_V2Label_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="lexbfs.html#K2" title="LEXBFS:func.2">.</a> <font color="Maroon" title="c10">y</font>
 <b>by </b><i><a class="ref" href="lexbfs.html#T33" target="_self" title="LEXBFS:th.33">Th33</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:138_1_1_2_1_1"/>
<font color="Maroon" title="c9">x</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="lexbfs.html#E4:138_1_1_2"><i><font color="Green" title="E68">A9</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="138_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:138_1_1_2_1_2"/>
( <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; not <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> )
 ;<br/><div class="add"><a NAME="E2:138_1_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="lexbfs.html#K6" title="LEXBFS:func.6">the_V2Label_of</a> <span class="p2">(<span class="default"><a href="lexbfs.html#K14" title="LEXBFS:func.14">LexBFS:LabelAdjacent</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K6" title="LEXBFS:func.6">the_V2Label_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="lexbfs.html#K2" title="LEXBFS:func.2">.</a> <font color="Maroon" title="c10">y</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">k</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="lexbfs.html#T34" target="_self" title="LEXBFS:th.34">Th34</a></i>;<br/><b>hence </b><a NAME="E3:138_1_1_2_1_2"/>
<font color="Maroon" title="c9">x</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="lexbfs.html#E4:138_1_1_2"><i><font color="Green" title="E68">A9</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E5:138_1_1"/><b>then </b>
(  <a href="lexbfs.html#K6" title="LEXBFS:func.6">the_V2Label_of</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K14" title="LEXBFS:func.14">LexBFS:LabelAdjacent</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">k</font></span>)</span> is <a href="finset_1.html#V2" title="FINSET_1:attr.2">finite-yielding</a> &amp;  <a href="lexbfs.html#K6" title="LEXBFS:func.6">the_V2Label_of</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K14" title="LEXBFS:func.14">LexBFS:LabelAdjacent</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">k</font></span>)</span> is <a href="lexbfs.html#V2" title="LEXBFS:attr.2">natsubset-yielding</a> )
 
<b>by </b><i><a class="txt" href="lexbfs.html#E3:138_1_1"><i><font color="Green" title="E66">A7</font></i></a>, <a class="ref" href="lexbfs.html#D3" target="_self" title="LEXBFS:def.3">Def3</a>, <a class="txt" href="lexbfs.html#E12"><i><font color="Green" title="E11">Lm2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:138_1_1"/>
 <a href="lexbfs.html#K14" title="LEXBFS:func.14">LexBFS:LabelAdjacent</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">k</font> is  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="lexbfs.html#V3" title="LEXBFS:attr.3">natural-vlabeled</a> <a href="lexbfs.html#V6" title="LEXBFS:attr.6">finite-v2labeled</a> <a href="lexbfs.html#V7" title="LEXBFS:attr.7">natsubset-v2labeled</a> <a href="lexbfs.html#NM3" title="LEXBFS:NM.3">VVGraph</a>
 <b>by </b><i><a class="txt" href="lexbfs.html#E1:138_1_1"><i><font color="Green" title="E65">A1</font></i></a>, <a class="ref" href="lexbfs.html#D7" target="_self" title="LEXBFS:def.7">Def7</a>, <a class="ref" href="lexbfs.html#D12" target="_self" title="LEXBFS:def.12">Def12</a>, <a class="ref" href="lexbfs.html#D13" target="_self" title="LEXBFS:def.13">Def13</a></i>;<br/>


</div>
