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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"> for <font color="Olive" title="b1">L</font> being  <a href="lexbfs.html#NM5" title="LEXBFS:NM.5">MCS:Labeling</a> of <font color="Maroon" title="c1">G</font><br/> for <font color="Olive" title="b2">v</font>, <font color="Olive" title="b3">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">L</font> <a href="lexbfs.html#K8" title="LEXBFS:func.8">`2</a> </span>)</span> &amp; <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; not <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">L</font> <a href="lexbfs.html#K7" title="LEXBFS:func.7">`1</a> </span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K20" title="LEXBFS:func.20">MCS:LabelAdjacent</a> <font color="Olive" title="b1">L</font>,<font color="Olive" title="b2">v</font></span>)</span> <a href="lexbfs.html#K8" title="LEXBFS:func.8">`2</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">L</font> <a href="lexbfs.html#K8" title="LEXBFS:func.8">`2</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b3">x</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span><br/><b>let </b><font color="Maroon" title="c2">L</font> be   <a href="lexbfs.html#NM5" title="LEXBFS:NM.5">MCS:Labeling</a> of <font color="Maroon" title="c1">G</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">v</font>, <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="lexbfs.html#K8" title="LEXBFS:func.8">`2</a> </span>)</span> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; not <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="lexbfs.html#K7" title="LEXBFS:func.7">`1</a> </span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K20" title="LEXBFS:func.20">MCS:LabelAdjacent</a> <font color="Maroon" title="c2">L</font>,<font color="Olive" title="b1">v</font></span>)</span> <a href="lexbfs.html#K8" title="LEXBFS:func.8">`2</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="lexbfs.html#K8" title="LEXBFS:func.8">`2</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span><br/><b>let </b><font color="Maroon" title="c3">v</font>, <font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="lexbfs.html#K8" title="LEXBFS:func.8">`2</a> </span>)</span> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="lexbfs.html#K7" title="LEXBFS:func.7">`1</a> </span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K20" title="LEXBFS:func.20">MCS:LabelAdjacent</a> <font color="Maroon" title="c2">L</font>,<font color="Maroon" title="c3">v</font></span>)</span> <a href="lexbfs.html#K8" title="LEXBFS:func.8">`2</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="lexbfs.html#K8" title="LEXBFS:func.8">`2</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:123"/><i><font color="Green" title="E91">A1</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="lexbfs.html#K8" title="LEXBFS:func.8">`2</a> </span>)</span> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>and </b><br/><a NAME="E2:123"/><i><font color="Green" title="E92">A2</font></i>: 
not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="lexbfs.html#K7" title="LEXBFS:func.7">`1</a> </span>)</span>
 ; <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#K20" title="LEXBFS:func.20">MCS:LabelAdjacent</a> <font color="Maroon" title="c2">L</font>,<font color="Maroon" title="c3">v</font></span>)</span> <a href="lexbfs.html#K8" title="LEXBFS:func.8">`2</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="lexbfs.html#K8" title="LEXBFS:func.8">`2</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span><br/>

<b>set </b><font color="Maroon" title="c5">GL</font> =  <a href="lexbfs.html#K20" title="LEXBFS:func.20">MCS:LabelAdjacent</a> <font color="Maroon" title="c2">L</font>,<font color="Maroon" title="c3">v</font>;<br/>
<b>set </b><font color="Maroon" title="c6">V2</font> = <span class="p1">(<span class="default"><a href="lexbfs.html#K20" title="LEXBFS:func.20">MCS:LabelAdjacent</a> <font color="Maroon" title="c2">L</font>,<font color="Maroon" title="c3">v</font></span>)</span> <a href="lexbfs.html#K8" title="LEXBFS:func.8">`2</a> ;<br/>
<b>set </b><font color="Maroon" title="c7">VLG</font> = <font color="Maroon" title="c2">L</font> <a href="lexbfs.html#K7" title="LEXBFS:func.7">`1</a> ;<br/>
<b>set </b><font color="Maroon" title="c8">V2G</font> = <font color="Maroon" title="c2">L</font> <a href="lexbfs.html#K8" title="LEXBFS:func.8">`2</a> ;<br/>
<a NAME="E3:123"/><i><font color="Green" title="E93">A3</font></i>: 
<span class="p1">(<span class="default"><a href="lexbfs.html#K20" title="LEXBFS:func.20">MCS:LabelAdjacent</a> <font color="Maroon" title="c2">L</font>,<font color="Maroon" title="c3">v</font></span>)</span> <a href="lexbfs.html#K8" title="LEXBFS:func.8">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="lexbfs.html#K8" title="LEXBFS:func.8">`2</a> </span>)</span> <a href="lexbfs.html#K3" title="LEXBFS:func.3">.incSubset</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="lexbfs.html#K7" title="LEXBFS:func.7">`1</a> </span>)</span></span>)</span></span>)</span>,1
 
<b>by </b><i><a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>;<br/>
<a NAME="E4:123"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="lexbfs.html#K7" title="LEXBFS:func.7">`1</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:123"><i><font color="Green" title="E91">A1</font></i></a>, <a class="txt" href="#E2:123"><i><font color="Green" title="E92">A2</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/>
<b>hence </b><a NAME="E5:123"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K20" title="LEXBFS:func.20">MCS:LabelAdjacent</a> <font color="Maroon" title="c2">L</font>,<font color="Maroon" title="c3">v</font></span>)</span> <a href="lexbfs.html#K8" title="LEXBFS:func.8">`2</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="lexbfs.html#K8" title="LEXBFS:func.8">`2</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="#E1:123"><i><font color="Green" title="E91">A1</font></i></a>, <a class="txt" href="#E3:123"><i><font color="Green" title="E93">A3</font></i></a>, <a class="ref" href="lexbfs.html#D4" target="_self" title="LEXBFS:def.4">Def4</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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