<?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>set </b><font color="Maroon">GS</font> =  <a href="lexbfs.html#K26" target="_self">MCS:CSeq</a> <font color="Maroon">G</font>;<br/>
<a NAME="E1:212"/><i><font color="Green">E34</font></i>: 
 <a href="lexbfs.html#K26" target="_self">MCS:CSeq</a> <font color="Maroon">G</font> is <a href="lexbfs.html#V13" target="_self">fixed-vertices</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="212_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="lexbfs.html#D19" target="_self">LEXBFS:def 19</a><br/>

<a NAME="E1:212_1"/>
<span class="p1">(<span class="default"><a href="lexbfs.html#K26" target="_self">MCS: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"><a href="lexbfs.html#K26" target="_self">MCS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">m</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:212_1"/>
 <a href="glib_000.html#K6">the_Vertices_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K26" target="_self">MCS: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="glib_000.html#K6">the_Vertices_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K26" target="_self">MCS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i><a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:212"/>
<span class="p1">(<span class="default"><a href="lexbfs.html#K26" target="_self">MCS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="lexbfs.html#K20" target="_self">MCS:Init</a> <font color="Maroon">G</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:212"/><b>then </b>
 <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#K26" target="_self">MCS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> 0</span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E4:212"/><b>then </b><i><font color="Green">E36</font></i>: 
 <a href="lexbfs.html#K26" target="_self">MCS:CSeq</a> <font color="Maroon">G</font> is <a href="lexbfs.html#V18" target="_self">vlabel-initially-empty</a>
 
<b>by </b><i/>;<br/>
<a NAME="E5:212"/><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><a href="lexbfs.html#K26" target="_self">MCS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> 
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="212_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:212_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K26" target="_self">MCS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a> 
 ;<br/><b>reconsider </b><font color="Maroon">w</font> =  <a href="lexbfs.html#K21" target="_self">MCS:PickUnnumbered</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K26" target="_self">MCS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">n</font></span>)</span> as    <a href="hidden.html#M1">set</a>  ;<br/><b>take </b><font color="Maroon">w</font> = <font color="Maroon">w</font>;<br/><a NAME="E3:212_2"/><i><font color="Green">E40</font></i>: 
 <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"><span class="p3">(<span class="default"><a href="lexbfs.html#K26" target="_self">MCS: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="hidden.html#R1">=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K26" target="_self">MCS: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="E4:212_2"/>
 <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#K26" target="_self">MCS: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="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K26" target="_self">MCS: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="txt" href="lexbfs.html#E18"><i><font color="Green">Lemma126</font></i></a>, , </i>;<br/><b>hence </b><a NAME="E5:212_2"/>
not <font color="Maroon">w</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#K26" target="_self">MCS: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>
 <b>by </b><i><a class="txt" href="lexbfs.html#E19"><i><font color="Green">Lemma127</font></i></a>, </i>;<br/><a NAME="E6:212_2"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <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#K26" target="_self">MCS: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>
 <b>by </b><i><a class="txt" href="lexbfs.html#E18"><i><font color="Green">Lemma126</font></i></a>, , </i>;<br/><b>hence </b><a NAME="E7:212_2"/>
 <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#K26" target="_self">MCS: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> <a href="hidden.html#R1">=</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#K26" target="_self">MCS: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="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lexbfs.html#K26" target="_self">MCS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="lexbfs.html#E18"><i><font color="Green">Lemma126</font></i></a>, , </i>;<br/></div><b>end;</b></div>
<a NAME="E7:212"/><b>then </b>
 <a href="lexbfs.html#K26" target="_self">MCS:CSeq</a> <font color="Maroon">G</font> is <a href="lexbfs.html#V19" target="_self">adds-one-at-a-step</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:212"/>
 <a href="lexbfs.html#K26" target="_self">MCS:CSeq</a> <font color="Maroon">G</font> is <a href="lexbfs.html#V20" target="_self">vlabel-numbering</a>
 <b>by </b><i>, , </i>;<br/>


</div>
