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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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="lexbfs.html#K16" title="LEXBFS:func.16">LexBFS:CSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="lexbfs.html#K15" title="LEXBFS:func.15">``1</a>  is <a href="lexbfs.html#V4" title="LEXBFS:attr.4">eventually-constant</a></span><br/>

<b>set </b><font color="Maroon" title="c2">CS</font> =  <a href="lexbfs.html#K16" title="LEXBFS:func.16">LexBFS:CSeq</a> <font color="Maroon" title="c1">G</font>;<br/>
<b>set </b><font color="Maroon" title="c3">S</font> = <span class="p1">(<span class="default"><a href="lexbfs.html#K16" title="LEXBFS:func.16">LexBFS:CSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="lexbfs.html#K15" title="LEXBFS:func.15">``1</a> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_1"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon" title="c4">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E2:85_1"/><i><font color="Green" title="E61">A</font></i>: 
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><span class="p1">(<span class="default"><a href="lexbfs.html#K16" title="LEXBFS:func.16">LexBFS:CSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="lexbfs.html#K13" title="LEXBFS:func.13">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K16" title="LEXBFS:func.16">LexBFS:CSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="lexbfs.html#K13" title="LEXBFS:func.13">.</a> <font color="Olive" title="b1">m</font>
 <b>by </b><i><a class="ref" href="lexbfs.html#D7" target="_self" title="LEXBFS:def.7">Def16</a></i>;<br/><b>take </b><font color="Maroon" title="c5">n</font> = <font color="Maroon" title="c4">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K16" title="LEXBFS:func.16">LexBFS:CSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="lexbfs.html#K15" title="LEXBFS:func.15">``1</a> </span>)</span> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K16" title="LEXBFS:func.16">LexBFS:CSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="lexbfs.html#K15" title="LEXBFS:func.15">``1</a> </span>)</span> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Olive" title="b1">m</font></span><br/><b>let </b><font color="Maroon" title="c6">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K16" title="LEXBFS:func.16">LexBFS:CSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="lexbfs.html#K15" title="LEXBFS:func.15">``1</a> </span>)</span> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K16" title="LEXBFS:func.16">LexBFS:CSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="lexbfs.html#K15" title="LEXBFS:func.15">``1</a> </span>)</span> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Maroon" title="c6">m</font> )</span><br/><b>assume </b><a NAME="E3:85_1"/><i><font color="Green" title="E62">B</font></i>: 
<font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K16" title="LEXBFS:func.16">LexBFS:CSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="lexbfs.html#K15" title="LEXBFS:func.15">``1</a> </span>)</span> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K16" title="LEXBFS:func.16">LexBFS:CSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="lexbfs.html#K15" title="LEXBFS:func.15">``1</a> </span>)</span> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Maroon" title="c6">m</font></span><br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K16" title="LEXBFS:func.16">LexBFS:CSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="lexbfs.html#K15" title="LEXBFS:func.15">``1</a> </span>)</span> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Maroon" title="c5">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K16" title="LEXBFS:func.16">LexBFS:CSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="lexbfs.html#K13" title="LEXBFS:func.13">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="lexbfs.html#K7" title="LEXBFS:func.7">`1</a> 
<b>by </b><i><a class="ref" href="lexbfs.html#D16" target="_self" title="LEXBFS:def.16">d1stBFSLS</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K16" title="LEXBFS:func.16">LexBFS:CSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="lexbfs.html#K13" title="LEXBFS:func.13">.</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="lexbfs.html#K7" title="LEXBFS:func.7">`1</a> 
<b>by </b><i><a class="txt" href="#E2:85_1"><i><font color="Green" title="E61">A</font></i></a>, <a class="txt" href="#E3:85_1"><i><font color="Green" title="E62">B</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K16" title="LEXBFS:func.16">LexBFS:CSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="lexbfs.html#K15" title="LEXBFS:func.15">``1</a> </span>)</span> <a href="lexbfs.html#K5" title="LEXBFS:func.5">.</a> <font color="Maroon" title="c6">m</font>
<b>by </b><i><a class="ref" href="lexbfs.html#D16" target="_self" title="LEXBFS:def.16">d1stBFSLS</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:85"/>
<span class="p1">(<span class="default"><a href="lexbfs.html#K16" title="LEXBFS:func.16">LexBFS:CSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="lexbfs.html#K15" title="LEXBFS:func.15">``1</a>  is <a href="lexbfs.html#V4" title="LEXBFS:attr.4">eventually-constant</a>
 <b>by </b><i><a class="ref" href="lexbfs.html#D7" target="_self" title="LEXBFS:def.7">Def16</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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