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

<b>set </b><font color="Maroon">IT</font> =  <a href="lexbfs.html#K14" target="_self">LexBFS:LabelAdjacent</a> <font color="Maroon">G</font>,<font color="Maroon">v</font>,<font color="Maroon">k</font>;<br/>
<b>set </b><font color="Maroon">VL</font> =  <a href="glib_003.html#K7">the_VLabel_of</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K14" target="_self">LexBFS:LabelAdjacent</a> <font color="Maroon">G</font>,<font color="Maroon">v</font>,<font color="Maroon">k</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">V2</font> =  <a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K14" target="_self">LexBFS:LabelAdjacent</a> <font color="Maroon">G</font>,<font color="Maroon">v</font>,<font color="Maroon">k</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">VLG</font> =  <a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font>;<br/>
<b>set </b><font color="Maroon">V2G</font> =  <a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <font color="Maroon">G</font>;<br/>
<a NAME="E1:138_1_1"/><i><font color="Green">E34</font></i>: 
 <a href="glib_003.html#K7">the_VLabel_of</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K14" target="_self">LexBFS:LabelAdjacent</a> <font color="Maroon">G</font>,<font color="Maroon">v</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="glib_000.html#T12">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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:138_1_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <span class="p2">(<span class="default"><a href="lexbfs.html#K14" target="_self">LexBFS:LabelAdjacent</a> <font color="Maroon">G</font>,<font color="Maroon">v</font>,<font color="Maroon">k</font></span>)</span></span>)</span>
 ;<br/><b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:138_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <span class="p2">(<span class="default"><a href="lexbfs.html#K14" target="_self">LexBFS:LabelAdjacent</a> <font color="Maroon">G</font>,<font color="Maroon">v</font>,<font color="Maroon">k</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E4:138_1_1_1"/><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <span class="p2">(<span class="default"><a href="lexbfs.html#K14" target="_self">LexBFS:LabelAdjacent</a> <font color="Maroon">G</font>,<font color="Maroon">v</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="lexbfs.html#D2" target="_self">Def2</a>, <a class="ref" href="funct_1.html#D5">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">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chord.html#K2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span> or <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> or ( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chord.html#K2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span> &amp; not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">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">E39</font></i>: 
(  not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chord.html#K2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span> or <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">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">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chord.html#K2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span> or <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> )
 </span><b>by </b><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">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chord.html#K2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</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" target="_self">the_V2Label_of</a> <span class="p2">(<span class="default"><a href="lexbfs.html#K14" target="_self">LexBFS:LabelAdjacent</a> <font color="Maroon">G</font>,<font color="Maroon">v</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K2" target="_self">.</a> <font color="Maroon">y</font>
 <b>by </b><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">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">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" target="_self">the_V2Label_of</a> <span class="p2">(<span class="default"><a href="lexbfs.html#K14" target="_self">LexBFS:LabelAdjacent</a> <font color="Maroon">G</font>,<font color="Maroon">v</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K2" target="_self">.</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="lexbfs.html#D1" target="_self">Def1</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">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <a href="numbers.html#K5">NAT</a> 
 <b>by </b><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">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chord.html#K2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span> &amp; not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> )
 ;<br/><div class="add"><a NAME="E2:138_1_1_1_1_2"/><b>then </b><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <span class="p2">(<span class="default"><a href="lexbfs.html#K14" target="_self">LexBFS:LabelAdjacent</a> <font color="Maroon">G</font>,<font color="Maroon">v</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K2" target="_self">.</a> <font color="Maroon">y</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">k</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i/>;<br/><a NAME="E3:138_1_1_1_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E4:138_1_1_1_1_2"/><b>then </b><i><font color="Green">E42</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">k</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="138_1_1_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_1_1_2_1"/>
(  not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <font color="Maroon">G</font></span>)</span> or <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <font color="Maroon">G</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="138_1_1_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:138_1_1_1_1_2_1_1"/>
not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <font color="Maroon">G</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:138_1_1_1_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K2" target="_self">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/><b>hence </b><a NAME="E3:138_1_1_1_1_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="138_1_1_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:138_1_1_1_1_2_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <font color="Maroon">G</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:138_1_1_1_1_2_1_2"/>
<span class="p1">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <span class="p2">(<span class="default"><a href="lexbfs.html#K14" target="_self">LexBFS:LabelAdjacent</a> <font color="Maroon">G</font>,<font color="Maroon">v</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><b>hence </b><a NAME="E3:138_1_1_1_1_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></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">E45</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <span class="p2">(<span class="default"><a href="lexbfs.html#K14" target="_self">LexBFS:LabelAdjacent</a> <font color="Maroon">G</font>,<font color="Maroon">v</font>,<font color="Maroon">k</font></span>)</span></span>)</span> is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <a href="numbers.html#K5">NAT</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:138_1_1_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <span class="p2">(<span class="default"><a href="lexbfs.html#K14" target="_self">LexBFS:LabelAdjacent</a> <font color="Maroon">G</font>,<font color="Maroon">v</font>,<font color="Maroon">k</font></span>)</span></span>)</span>
 ;<br/><b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:138_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <span class="p2">(<span class="default"><a href="lexbfs.html#K14" target="_self">LexBFS:LabelAdjacent</a> <font color="Maroon">G</font>,<font color="Maroon">v</font>,<font color="Maroon">k</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E4:138_1_1_2"/><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <span class="p2">(<span class="default"><a href="lexbfs.html#K14" target="_self">LexBFS:LabelAdjacent</a> <font color="Maroon">G</font>,<font color="Maroon">v</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">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">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chord.html#K2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span> or <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> or ( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chord.html#K2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span> &amp; not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">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">E48</font></i>: 
(  not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chord.html#K2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span> or <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">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">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chord.html#K2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span> or <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> )
 </span><b>by </b><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">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chord.html#K2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</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" target="_self">the_V2Label_of</a> <span class="p2">(<span class="default"><a href="lexbfs.html#K14" target="_self">LexBFS:LabelAdjacent</a> <font color="Maroon">G</font>,<font color="Maroon">v</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K2" target="_self">.</a> <font color="Maroon">y</font>
 <b>by </b><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">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">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" target="_self">the_V2Label_of</a> <span class="p2">(<span class="default"><a href="lexbfs.html#K14" target="_self">LexBFS:LabelAdjacent</a> <font color="Maroon">G</font>,<font color="Maroon">v</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K2" target="_self">.</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="lexbfs.html#D1" target="_self">Def1</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">x</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><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">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chord.html#K2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span> &amp; not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">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" target="_self">the_V2Label_of</a> <span class="p2">(<span class="default"><a href="lexbfs.html#K14" target="_self">LexBFS:LabelAdjacent</a> <font color="Maroon">G</font>,<font color="Maroon">v</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K2" target="_self">.</a> <font color="Maroon">y</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">k</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:138_1_1_2_1_2"/>
<font color="Maroon">x</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><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" target="_self">the_V2Label_of</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K14" target="_self">LexBFS:LabelAdjacent</a> <font color="Maroon">G</font>,<font color="Maroon">v</font>,<font color="Maroon">k</font></span>)</span> is <a href="finset_1.html#V2">finite-yielding</a> &amp;  <a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K14" target="_self">LexBFS:LabelAdjacent</a> <font color="Maroon">G</font>,<font color="Maroon">v</font>,<font color="Maroon">k</font></span>)</span> is <a href="lexbfs.html#V2" target="_self">natsubset-yielding</a> )
 
<b>by </b><i><a class="ref" href="lexbfs.html#D2" target="_self">Def2</a>, , </i>;<br/>
<b>hence </b><a NAME="E6:138_1_1"/>
 <a href="lexbfs.html#K14" target="_self">LexBFS:LabelAdjacent</a> <font color="Maroon">G</font>,<font color="Maroon">v</font>,<font color="Maroon">k</font> is  <a href="glib_000.html#V2">finite</a> <a href="lexbfs.html#V3" target="_self">natural-vlabeled</a> <a href="lexbfs.html#V6" target="_self">finite-v2labeled</a> <a href="lexbfs.html#V7" target="_self">natsubset-v2labeled</a> <a href="lexbfs.html#NM3">VVGraph</a>
 <b>by </b><i>, , , </i>;<br/>


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