<?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>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b><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">a<sub>1</sub></font> <a href="glib_000.html#R5">==</a> <font color="Maroon">G</font>;<br/>
<a NAME="E1:199"/>
<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="E2:199"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i/>;<br/>
<a NAME="E3:199"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="xcmplx_0.html#K2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="199_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:199_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<b>set </b><font color="Maroon">CSK</font> = <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">k</font>;<br/>
<b>set </b><font color="Maroon">CK1</font> = <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> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>;<br/>
<b>set </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">k</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">kk</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> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p3">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p4">(<span class="default"><span class="p5">(<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">k</font></span>)</span></span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E2:199_1"/><i><font color="Green">E39</font></i>: 
<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">k</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="E3:199_1"/><i><font color="Green">E40</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="lexbfs.html#K9" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="lexbfs.html#K25" target="_self">MCS:Step</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">k</font></span>)</span>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="199_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:199_1_1"/>
(  <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">k</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a>  or <font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a>  <a href="xxreal_0.html#R1">&lt;=</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">k</font></span>)</span></span>)</span></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="199_1_1_1"><b>suppose </b></a><a NAME="E1:199_1_1_1"/>
 <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">k</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> 
 ;<br/><div class="add"><a NAME="E2:199_1_1_1"/><b>then </b>
<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> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="lexbfs.html#K24" target="_self">MCS:Update</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">k</font></span>)</span>,<span class="p1">(<span class="default"><a href="lexbfs.html#K21" target="_self">MCS:PickUnnumbered</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">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p3">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p4">(<span class="default"><span class="p5">(<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">k</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , </i>;<br/><a NAME="E3:199_1_1_1"/><b>then </b><i><font color="Green">E42</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="lexbfs.html#K9" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="lexbfs.html#K22" target="_self">MCS:LabelAdjacent</a> <span class="p1">(<span class="default"><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">k</font></span>)</span> <a href="glib_003.html#K13">.labelVertex</a> <span class="p2">(<span class="default"><a href="lexbfs.html#K21" target="_self">MCS:PickUnnumbered</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">k</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> <span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p5">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p0">(<span class="default"><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">k</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="lexbfs.html#K21" target="_self">MCS:PickUnnumbered</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">k</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="glib_000.html#T93">GLIB_000:93</a></i>;<br/><b>set </b><font color="Maroon">CC</font> = <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">k</font></span>)</span> <a href="glib_003.html#K13">.labelVertex</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K21" target="_self">MCS:PickUnnumbered</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">k</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> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p4">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p5">(<span class="default"><span class="p0">(<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">k</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>;<br/><a NAME="E4:199_1_1_1"/><i><font color="Green">E45</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="lexbfs.html#K9" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="glib_000.html#R5">==</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">k</font></span>)</span> <a href="glib_003.html#K13">.labelVertex</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K21" target="_self">MCS:PickUnnumbered</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">k</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> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p4">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p5">(<span class="default"><span class="p0">(<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">k</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E5:199_1_1_1"/>
<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">k</font></span>)</span> <a href="glib_003.html#K13">.labelVertex</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K21" target="_self">MCS:PickUnnumbered</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">k</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> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p4">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p5">(<span class="default"><span class="p0">(<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">k</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <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">k</font>
 <b>by </b><i><a class="ref" href="glib_003.html#T47">GLIB_003:47</a></i>;<br/><a NAME="E6:199_1_1_1"/><b>then </b>
<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> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <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">k</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#T88">GLIB_000:88</a></i>;<br/><b>hence </b><a NAME="E7:199_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1]
 <b>by </b><i>, <a class="ref" href="glib_000.html#T88">GLIB_000:88</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="199_1_1_2"><b>suppose </b></a><a NAME="E1:199_1_1_2"/>
<font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a>  <a href="xxreal_0.html#R1">&lt;=</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">k</font></span>)</span></span>)</span></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:199_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1]
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E4:199"/>
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(, );<br/></i>
<b>hence </b><a NAME="E5:199"/>
<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> <font color="Maroon">G</font>
 ;<br/>


</div>
