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<b>let </b><font color="Maroon" title="c1">G</font> be  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>;<br/>

<b>set </b><font color="Maroon" title="c2">GS</font> =  <a href="lexbfs.html#K26" title="LEXBFS:func.26">MCS:CSeq</a> <font color="Maroon" title="c1">G</font>;<br/>
<a NAME="E1:212"/><i><font color="Green" title="E123">A1</font></i>: 
 <a href="lexbfs.html#K26" title="LEXBFS:func.26">MCS:CSeq</a> <font color="Maroon" title="c1">G</font> is <a href="lexbfs.html#V13" title="LEXBFS:attr.13">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" title="c3">n</font>, <font color="Maroon" title="c4">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="lexbfs.html#D19" target="_self" title="LEXBFS:def.19">LEXBFS:def 19</a><br/>

<a NAME="E1:212_1"/>
<span class="p1">(<span class="default"><a href="lexbfs.html#K26" title="LEXBFS:func.26">MCS:CSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="lexbfs.html#K9" title="LEXBFS:func.9">.</a> <font color="Maroon" title="c3">n</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K26" title="LEXBFS:func.26">MCS:CSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="lexbfs.html#K9" title="LEXBFS:func.9">.</a> <font color="Maroon" title="c4">m</font>
 
<b>by </b><i><a class="ref" href="lexbfs.html#T79" target="_self" title="LEXBFS:th.79">Th79</a></i>;<br/>
<b>hence </b><a NAME="E2:212_1"/>
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K26" title="LEXBFS:func.26">MCS:CSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="lexbfs.html#K9" title="LEXBFS:func.9">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K26" title="LEXBFS:func.26">MCS:CSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="lexbfs.html#K9" title="LEXBFS:func.9">.</a> <font color="Maroon" title="c4">m</font></span>)</span>
 <b>by </b><i><a class="ref" href="glib_000.html#D36" title="GLIB_000:def.36">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" title="LEXBFS:func.26">MCS:CSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="lexbfs.html#K9" title="LEXBFS:func.9">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lexbfs.html#K20" title="LEXBFS:func.20">MCS:Init</a> <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="ref" href="lexbfs.html#D40" target="_self" title="LEXBFS:def.40">Def40</a></i>;<br/>
<a NAME="E3:212"/><b>then </b>
 <a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K26" title="LEXBFS:func.26">MCS:CSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="lexbfs.html#K9" title="LEXBFS:func.9">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="lexbfs.html#T69" target="_self" title="LEXBFS:th.69">Th69</a></i>;<br/>
<a NAME="E4:212"/><b>then </b><i><font color="Green" title="E124">A2</font></i>: 
 <a href="lexbfs.html#K26" title="LEXBFS:func.26">MCS:CSeq</a> <font color="Maroon" title="c1">G</font> is <a href="lexbfs.html#V18" title="LEXBFS:attr.18">vlabel-initially-empty</a>
 
<b>by </b><i><a class="ref" href="lexbfs.html#D24" target="_self" title="LEXBFS:def.24">Def24</a></i>;<br/>
<a NAME="E5:212"/><i><font color="Green" title="E125">A3</font></i>: 
<span class="p1">(<span class="default"><a href="lexbfs.html#K26" title="LEXBFS:func.26">MCS:CSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.order()</a> 
 
<b>by </b><i><a class="ref" href="lexbfs.html#T87" target="_self" title="LEXBFS:th.87">Th87</a></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" title="c3">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:212_2"/><i><font color="Green" title="E126">A4</font></i>: 
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K26" title="LEXBFS:func.26">MCS:CSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> 
 ;<br/><b>reconsider </b><font color="Maroon" title="c4">w</font> =  <a href="lexbfs.html#K21" title="LEXBFS:func.21">MCS:PickUnnumbered</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K26" title="LEXBFS:func.26">MCS:CSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="lexbfs.html#K9" title="LEXBFS:func.9">.</a> <font color="Maroon" title="c3">n</font></span>)</span> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><b>take </b><font color="Maroon" title="c5">w</font> = <font color="Maroon" title="c4">w</font>;<br/><a NAME="E3:212_2"/><i><font color="Green" title="E127">A5</font></i>: 
 <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"><span class="p3">(<span class="default"><a href="lexbfs.html#K26" title="LEXBFS:func.26">MCS:CSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="lexbfs.html#K9" title="LEXBFS:func.9">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K26" title="LEXBFS:func.26">MCS:CSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="lexbfs.html#K9" title="LEXBFS:func.9">.</a> <font color="Maroon" title="c3">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="lexbfs.html#T80" target="_self" title="LEXBFS:th.80">Th80</a></i>;<br/><a NAME="E4:212_2"/>
 <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K26" title="LEXBFS:func.26">MCS:CSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="lexbfs.html#K9" title="LEXBFS:func.9">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K26" title="LEXBFS:func.26">MCS:CSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="lexbfs.html#K9" title="LEXBFS:func.9">.</a> <font color="Maroon" title="c3">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="lexbfs.html#E5:212"><i><font color="Green" title="E125">A3</font></i></a>, <a class="txt" href="lexbfs.html#E1:212_2"><i><font color="Green" title="E126">A4</font></i></a>, <a class="ref" href="lexbfs.html#T86" target="_self" title="LEXBFS:th.86">Th86</a></i>;<br/><b>hence </b><a NAME="E5:212_2"/>
not <font color="Maroon" title="c5">w</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K26" title="LEXBFS:func.26">MCS:CSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="lexbfs.html#K9" title="LEXBFS:func.9">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="lexbfs.html#E3:212_2"><i><font color="Green" title="E127">A5</font></i></a>, <a class="ref" href="lexbfs.html#T73" target="_self" title="LEXBFS:th.73">Th73</a></i>;<br/><a NAME="E6:212_2"/>
<font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lexbfs.html#K26" title="LEXBFS:func.26">MCS:CSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="lexbfs.html#K9" title="LEXBFS:func.9">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="lexbfs.html#E5:212"><i><font color="Green" title="E125">A3</font></i></a>, <a class="txt" href="lexbfs.html#E1:212_2"><i><font color="Green" title="E126">A4</font></i></a>, <a class="ref" href="lexbfs.html#T82" target="_self" title="LEXBFS:th.82">Th82</a></i>;<br/><b>hence </b><a NAME="E7:212_2"/>
 <a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K26" title="LEXBFS:func.26">MCS:CSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="lexbfs.html#K9" title="LEXBFS:func.9">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K26" title="LEXBFS:func.26">MCS:CSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="lexbfs.html#K9" title="LEXBFS:func.9">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">w</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&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" title="LEXBFS:func.26">MCS:CSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> </span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="lexbfs.html#E5:212"><i><font color="Green" title="E125">A3</font></i></a>, <a class="txt" href="lexbfs.html#E1:212_2"><i><font color="Green" title="E126">A4</font></i></a>, <a class="ref" href="lexbfs.html#T81" target="_self" title="LEXBFS:th.81">Th81</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:212"/><b>then </b>
 <a href="lexbfs.html#K26" title="LEXBFS:func.26">MCS:CSeq</a> <font color="Maroon" title="c1">G</font> is <a href="lexbfs.html#V19" title="LEXBFS:attr.19">adds-one-at-a-step</a>
 
<b>by </b><i><a class="ref" href="lexbfs.html#D25" target="_self" title="LEXBFS:def.25">Def25</a></i>;<br/>
<b>hence </b><a NAME="E8:212"/>
 <a href="lexbfs.html#K26" title="LEXBFS:func.26">MCS:CSeq</a> <font color="Maroon" title="c1">G</font> is <a href="lexbfs.html#V20" title="LEXBFS:attr.20">vlabel-numbering</a>
 <b>by </b><i><a class="txt" href="lexbfs.html#E1:212"><i><font color="Green" title="E123">A1</font></i></a>, <a class="txt" href="lexbfs.html#E4:212"><i><font color="Green" title="E124">A2</font></i></a>, <a class="ref" href="lexbfs.html#D26" target="_self" title="LEXBFS:def.26">Def26</a></i>;<br/>


</div>
