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

<b>let </b><font color="Maroon">G</font> be  <a href="glib_000.html#V2">finite</a> <a href="glib_000.html#NM2">_Graph</a>;<br/><b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><a NAME="E1:165"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> 
 ;<br/>

<b>set </b><font color="Maroon">CSN</font> = <span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font>;<br/>
<b>set </b><font color="Maroon">VLN</font> =  <a href="glib_003.html#K7">the_VLabel_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">V2N</font> =  <a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">CN1</font> = <span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>;<br/>
<b>set </b><font color="Maroon">VN1</font> =  <a href="glib_003.html#K7">the_VLabel_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">V21</font> =  <a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">w</font> =  <a href="lexbfs.html#K13" target="_self">LexBFS:PickUnnumbered</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">k</font> = <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">n</font>;<br/>
<a NAME="E2:165"/><i><font color="Green">E36</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="lexbfs.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E3:165"/>
<font color="Maroon">G</font> <a href="glib_000.html#R5">==</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="lexbfs.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E4:165"/><b>then </b><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> 
 
<b>by </b><i><a class="ref" href="glib_000.html#T93">GLIB_000:93</a></i>;<br/>
<a NAME="E5:165"/>
<span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="lexbfs.html#K17" target="_self">LexBFS:Step</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="lexbfs.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E6:165"/><b>then </b><i><font color="Green">E39</font></i>: 
<span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="lexbfs.html#K16" target="_self">LexBFS:Update</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="lexbfs.html#K13" target="_self">LexBFS:PickUnnumbered</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<font color="Maroon">n</font>
 
<b>by </b><i>, , , </i>;<br/>
<b>set </b><font color="Maroon">CSlv</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K13">.labelVertex</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K13" target="_self">LexBFS:PickUnnumbered</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">n</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">VLv</font> =  <a href="glib_003.html#K7">the_VLabel_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K13">.labelVertex</a> <span class="p2">(<span class="default"><a href="lexbfs.html#K13" target="_self">LexBFS:PickUnnumbered</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">n</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">V2v</font> =  <a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K13">.labelVertex</a> <span class="p2">(<span class="default"><a href="lexbfs.html#K13" target="_self">LexBFS:PickUnnumbered</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">n</font></span>)</span></span>)</span>;<br/>
<a NAME="E7:165"/><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font> <a href="glib_000.html#R5">==</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K13">.labelVertex</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K13" target="_self">LexBFS:PickUnnumbered</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="glib_003.html#T47">GLIB_003:47</a></i>;<br/>
<a NAME="E8:165"/>
<span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font> <a href="glib_000.html#R5">==</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="lexbfs.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E9:165"/><b>then </b><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K13">.labelVertex</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K13" target="_self">LexBFS:PickUnnumbered</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">n</font></span>)</span> <a href="glib_000.html#R5">==</a> <font color="Maroon">G</font>
 
<b>by </b><i>, <a class="ref" href="glib_000.html#T88">GLIB_000:88</a></i>;<br/>
<a NAME="E10:165"/><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K13">.labelVertex</a> <span class="p2">(<span class="default"><a href="lexbfs.html#K13" target="_self">LexBFS:PickUnnumbered</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K13" target="_self">LexBFS:PickUnnumbered</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="glib_003.html#T59">GLIB_003:59</a></i>;<br/>
<a NAME="E11:165"/><i><font color="Green">E46</font></i>: 
 <a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K13">.labelVertex</a> <span class="p2">(<span class="default"><a href="lexbfs.html#K13" target="_self">LexBFS:PickUnnumbered</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E12:165"/><i><font color="Green">E47</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K13">.labelVertex</a> <span class="p3">(<span class="default"><a href="lexbfs.html#K13" target="_self">LexBFS:PickUnnumbered</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<div><i><font color="Green">E48</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="165_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">v</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:165_1"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">v</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"><span class="p2">(<span class="default"><a href="lexbfs.html#K13" target="_self">LexBFS:PickUnnumbered</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span> &amp; not <font color="Maroon">v</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> )
 ;<br/><a NAME="E2:165_1"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K13">.labelVertex</a> <span class="p2">(<span class="default"><a href="lexbfs.html#K13" target="_self">LexBFS:PickUnnumbered</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="chord.html#K2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K13" target="_self">LexBFS:PickUnnumbered</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, <a class="ref" href="chord.html#T51">CHORD:51</a></i>;<br/><a NAME="E3:165_1"/>
not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K13" target="_self">LexBFS:PickUnnumbered</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, <a class="ref" href="chord.html#T49">CHORD:49</a></i>;<br/><a NAME="E4:165_1"/><b>then </b>
not <font color="Maroon">v</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K13">.labelVertex</a> <span class="p3">(<span class="default"><a href="lexbfs.html#K13" target="_self">LexBFS:PickUnnumbered</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E5:165_1"/>
<span class="p1">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="lexbfs.html#K2" target="_self">.</a> <font color="Maroon">v</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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="lexbfs.html#K2" target="_self">.</a> <font color="Maroon">v</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">n</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, , , , </i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E51</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="165_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">v</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:165_2"/><i><font color="Green">E89</font></i>: 
(  not <font color="Maroon">v</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"><span class="p2">(<span class="default"><a href="lexbfs.html#K13" target="_self">LexBFS:PickUnnumbered</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span> or <font color="Maroon">v</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="165_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:165_2_1"/>
(  not <font color="Maroon">v</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"><span class="p2">(<span class="default"><a href="lexbfs.html#K13" target="_self">LexBFS:PickUnnumbered</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span> or <font color="Maroon">v</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="165_2_1_1"><b>suppose </b></a><a NAME="E1:165_2_1_1"/>
not <font color="Maroon">v</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"><span class="p2">(<span class="default"><a href="lexbfs.html#K13" target="_self">LexBFS:PickUnnumbered</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:165_2_1_1"/><b>then </b>
not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K13">.labelVertex</a> <span class="p2">(<span class="default"><a href="lexbfs.html#K13" target="_self">LexBFS:PickUnnumbered</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="chord.html#K2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K13" target="_self">LexBFS:PickUnnumbered</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, <a class="ref" href="chord.html#T51">CHORD:51</a></i>;<br/><b>hence </b><a NAME="E3:165_2_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"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="lexbfs.html#K2" target="_self">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="lexbfs.html#K2" target="_self">.</a> <font color="Maroon">v</font>
 <b>by </b><i>, <a class="txt" href="lexbfs.html#E1:165_2"><i><font color="Green">E89</font></i></a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="165_2_1_2"><b>suppose </b></a><a NAME="E1:165_2_1_2"/>
<font color="Maroon">v</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:165_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"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="lexbfs.html#K2" target="_self">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="lexbfs.html#K2" target="_self">.</a> <font color="Maroon">v</font>
 <b>by </b><i>, <a class="txt" href="lexbfs.html#E1:165_2"><i><font color="Green">E89</font></i></a>, , , <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>take </b>
 <a href="lexbfs.html#K13" target="_self">LexBFS:PickUnnumbered</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span>
;<br/>

<b>thus </b><a NAME="E15:165"/>
(  <a href="lexbfs.html#K13" target="_self">LexBFS:PickUnnumbered</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="lexbfs.html#K13" target="_self">LexBFS:PickUnnumbered</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span> &amp; ( for <font color="Olive">v</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( ( <font color="Olive">v</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"><span class="p2">(<span class="default"><a href="lexbfs.html#K13" target="_self">LexBFS:PickUnnumbered</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span> &amp; not <font color="Olive">v</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="lexbfs.html#K2" target="_self">.</a> <font color="Olive">v</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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="lexbfs.html#K2" target="_self">.</a> <font color="Olive">v</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">n</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> ) &amp; ( (  not <font color="Olive">v</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"><span class="p2">(<span class="default"><a href="lexbfs.html#K13" target="_self">LexBFS:PickUnnumbered</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span> or <font color="Olive">v</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> ) implies <span class="p1">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="lexbfs.html#K2" target="_self">.</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="lexbfs.html#K2" target="_self">.</a> <font color="Olive">v</font> ) ) ) )
 <b>by </b><i>, </i>;<br/>


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