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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#NM3" title="LEXBFS:NM.3">LexBFS:Labeling</a> of <font color="Maroon" title="c1">G</font><br/> for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st 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="Olive" title="b1">L</font> <a href="lexbfs.html#K7" title="LEXBFS:func.7">`1</a> </span>)</span> &amp;  <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> <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> <font color="Maroon" title="c1">G</font> holds <br/><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="lexbfs.html#K17" title="LEXBFS:func.17">.</a> <font color="Olive" title="b2">x</font></span>)</span>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a>  <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</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="lexbfs.html#K17" title="LEXBFS:func.17">.</a> <span class="p2">(<span class="default"><a href="lexbfs.html#K10" title="LEXBFS:func.10">LexBFS:PickUnnumbered</a> <font color="Olive" title="b1">L</font></span>)</span></span>)</span>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> , <a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><br/><b>let </b><font color="Maroon" title="c2">L</font> be   <a href="lexbfs.html#NM3" title="LEXBFS:NM.3">LexBFS: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">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st not <font color="Olive" title="b1">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> &amp;  <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 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> <font color="Maroon" title="c1">G</font> holds <br/><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="lexbfs.html#K17" title="LEXBFS:func.17">.</a> <font color="Olive" title="b1">x</font></span>)</span>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a>  <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</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="lexbfs.html#K17" title="LEXBFS:func.17">.</a> <span class="p2">(<span class="default"><a href="lexbfs.html#K10" title="LEXBFS:func.10">LexBFS:PickUnnumbered</a> <font color="Maroon" title="c2">L</font></span>)</span></span>)</span>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> , <a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><br/><b>let </b><font color="Maroon" title="c3">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"> ( not <font color="Maroon" title="c3">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> &amp;  <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 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> <font color="Maroon" title="c1">G</font> implies <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="lexbfs.html#K17" title="LEXBFS:func.17">.</a> <font color="Maroon" title="c3">x</font></span>)</span>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a>  <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</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="lexbfs.html#K17" title="LEXBFS:func.17">.</a> <span class="p2">(<span class="default"><a href="lexbfs.html#K10" title="LEXBFS:func.10">LexBFS:PickUnnumbered</a> <font color="Maroon" title="c2">L</font></span>)</span></span>)</span>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> , <a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:75"/><i><font color="Green" title="E52">A1</font></i>: 
not <font color="Maroon" title="c3">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>
 <b>and </b><br/><a NAME="E2:75"/><i><font color="Green" title="E53">A3</font></i>: 
 <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 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> <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"> <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="lexbfs.html#K17" title="LEXBFS:func.17">.</a> <font color="Maroon" title="c3">x</font></span>)</span>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a>  <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</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="lexbfs.html#K17" title="LEXBFS:func.17">.</a> <span class="p2">(<span class="default"><a href="lexbfs.html#K10" title="LEXBFS:func.10">LexBFS:PickUnnumbered</a> <font color="Maroon" title="c2">L</font></span>)</span></span>)</span>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> , <a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><br/>

<b>set </b><font color="Maroon" title="c4">w</font> =  <a href="lexbfs.html#K10" title="LEXBFS:func.10">LexBFS:PickUnnumbered</a> <font color="Maroon" title="c2">L</font>;<br/>
<b>set </b><font color="Maroon" title="c5">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="c6">V2G</font> = <font color="Maroon" title="c2">L</font> <a href="lexbfs.html#K8" title="LEXBFS:func.8">`2</a> ;<br/>
<b>set </b><font color="Maroon" title="c7">VG</font> =  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font>;<br/>
<a NAME="E3:75"/><i><font color="Green" title="E54">A2</font></i>: 
 <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> <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> <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">S</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span>, <font color="Maroon" title="c9">B</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span>, <font color="Maroon" title="c10">F</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E5:75"/><i><font color="Green" title="E55">A4</font></i>: 
<font color="Maroon" title="c8">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">F</font>
 <b>and </b><br/><a NAME="E6:75"/><i><font color="Green" title="E56">A5</font></i>: 
<font color="Maroon" title="c10">F</font> <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="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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>
 <b>and </b><br/><a NAME="E7:75"/><i><font color="Green" title="E57">A6</font></i>: 
for <font color="Olive" title="b1">x</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">S</font> holds <br/><font color="Olive" title="b1">x</font>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B</font>
 <b>and </b><br/><a NAME="E8:75"/><i><font color="Green" title="E58">A7</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B</font> holds <br/> ex <font color="Olive" title="b2">y</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">S</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a>  )
 <b>and </b><br/><a NAME="E9:75"/><i><font color="Green" title="E59">A8</font></i>: 
 <a href="lexbfs.html#K10" title="LEXBFS:func.10">LexBFS:PickUnnumbered</a> <font color="Maroon" title="c2">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <span class="p4">(<span class="default"><a href="lexbfs.html#K4" title="LEXBFS:func.4">max</a> <font color="Maroon" title="c9">B</font>,<span class="p5">(<span class="default"><a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:75"><i><font color="Green" title="E53">A3</font></i></a>, <a class="ref" href="lexbfs.html#D12" target="_self" title="LEXBFS:def.12">Def29</a></i>;<br/>
<b>set </b><font color="Maroon" title="c11">mw</font> =  <a href="lexbfs.html#K4" title="LEXBFS:func.4">max</a> <font color="Maroon" title="c9">B</font>,<span class="p1">(<span class="default"><a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span>;<br/>
<a NAME="E10:75"/>
 <a href="lexbfs.html#K4" title="LEXBFS:func.4">max</a> <font color="Maroon" title="c9">B</font>,<span class="p1">(<span class="default"><a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B</font>
 
<b>by </b><i><a class="ref" href="lexbfs.html#D5" target="_self" title="LEXBFS:def.5">Def5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">y</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E12:75"/><i><font color="Green" title="E60">A9</font></i>: 
( <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">S</font> &amp;  <a href="lexbfs.html#K4" title="LEXBFS:func.4">max</a> <font color="Maroon" title="c9">B</font>,<span class="p1">(<span class="default"><a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c12">y</font>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a>  )
 <b>by </b><i><a class="txt" href="#E8:75"><i><font color="Green" title="E58">A7</font></i></a></i>;<br/>
<a NAME="E13:75"/><i><font color="Green" title="E61">A10</font></i>: 
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K4" title="LEXBFS:func.4">max</a> <font color="Maroon" title="c9">B</font>,<span class="p2">(<span class="default"><a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E12:75"><i><font color="Green" title="E60">A9</font></i></a>, <a class="ref" href="uproots.html#T10" title="UPROOTS:th.10">UPROOTS:10</a></i>;<br/>
<a NAME="E14:75"/><b>then </b>
not <font color="Maroon" title="c10">F</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <span class="p3">(<span class="default"><a href="lexbfs.html#K4" title="LEXBFS:func.4">max</a> <font color="Maroon" title="c9">B</font>,<span class="p4">(<span class="default"><a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="txt" href="#E5:75"><i><font color="Green" title="E55">A4</font></i></a>, <a class="txt" href="#E12:75"><i><font color="Green" title="E60">A9</font></i></a>, <a class="ref" href="funct_1.html#T142" title="FUNCT_1:th.142">FUNCT_1:142</a></i>;<br/>
<a NAME="E15:75"/><b>then </b><i><font color="Green" title="E62">A11</font></i>: 
(  <a href="lexbfs.html#K10" title="LEXBFS:func.10">LexBFS:PickUnnumbered</a> <font color="Maroon" title="c2">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">F</font> &amp; <font color="Maroon" title="c10">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K10" title="LEXBFS:func.10">LexBFS:PickUnnumbered</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K9" title="POLYNOM1:func.9">support</a> <span class="p3">(<span class="default"><a href="lexbfs.html#K4" title="LEXBFS:func.4">max</a> <font color="Maroon" title="c9">B</font>,<span class="p4">(<span class="default"><a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="txt" href="#E9:75"><i><font color="Green" title="E59">A8</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E16:75"/><b>then </b>
<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#K17" title="LEXBFS:func.17">.</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K10" title="LEXBFS:func.10">LexBFS:PickUnnumbered</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K10" title="LEXBFS:func.10">LexBFS:PickUnnumbered</a> <font color="Maroon" title="c2">L</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E6:75"><i><font color="Green" title="E56">A5</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>
<a NAME="E17:75"/><b>then </b><i><font color="Green" title="E63">A12</font></i>: 
<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="lexbfs.html#K17" title="LEXBFS:func.17">.</a> <span class="p2">(<span class="default"><a href="lexbfs.html#K10" title="LEXBFS:func.10">LexBFS:PickUnnumbered</a> <font color="Maroon" title="c2">L</font></span>)</span></span>)</span>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a>  <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="lexbfs.html#K4" title="LEXBFS:func.4">max</a> <font color="Maroon" title="c9">B</font>,<span class="p1">(<span class="default"><a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="#E12:75"><i><font color="Green" title="E60">A9</font></i></a>, <a class="txt" href="#E13:75"><i><font color="Green" title="E61">A10</font></i></a>, <a class="txt" href="#E15:75"><i><font color="Green" title="E62">A11</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E18:75"/><i><font color="Green" title="E64">A13</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#K8" title="LEXBFS:func.8">`2</a> </span>)</span></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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>
 
<b>by </b><i><a class="txt" href="#E6:75"><i><font color="Green" title="E56">A5</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:75_1"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font> or  not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_1"><b>suppose </b></a><a NAME="E1:75_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <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"> <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="lexbfs.html#K17" title="LEXBFS:func.17">.</a> <font color="Maroon" title="c3">x</font></span>)</span>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a>  <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</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="lexbfs.html#K17" title="LEXBFS:func.17">.</a> <span class="p2">(<span class="default"><a href="lexbfs.html#K10" title="LEXBFS:func.10">LexBFS:PickUnnumbered</a> <font color="Maroon" title="c2">L</font></span>)</span></span>)</span>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> , <a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><br/><div class="add"><a NAME="E2:75_1_1"/><b>then </b>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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="#E1:75"><i><font color="Green" title="E52">A1</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><a NAME="E3:75_1_1"/><b>then </b><i><font color="Green" title="E65">A14</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">F</font>
 <b>by </b><i><a class="txt" href="#E3:75"><i><font color="Green" title="E54">A2</font></i></a>, <a class="txt" href="#E18:75"><i><font color="Green" title="E64">A13</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E4:75_1_1"/><b>then </b><i><font color="Green" title="E66">A15</font></i>: 
<font color="Maroon" title="c10">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <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#K17" title="LEXBFS:func.17">.</a> <font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="txt" href="#E6:75"><i><font color="Green" title="E56">A5</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/><a NAME="E5:75_1_1"/>
<font color="Maroon" title="c10">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">S</font>
 <b>by </b><i><a class="txt" href="#E5:75"><i><font color="Green" title="E55">A4</font></i></a>, <a class="txt" href="#E3:75_1_1"><i><font color="Green" title="E65">A14</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E6:75_1_1"/><b>then </b>
<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="lexbfs.html#K17" title="LEXBFS:func.17">.</a> <font color="Maroon" title="c3">x</font></span>)</span>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B</font>
 <b>by </b><i><a class="txt" href="#E7:75"><i><font color="Green" title="E57">A6</font></i></a>, <a class="txt" href="#E4:75_1_1"><i><font color="Green" title="E66">A15</font></i></a></i>;<br/><b>hence </b><a NAME="E7:75_1_1"/>
<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="lexbfs.html#K17" title="LEXBFS:func.17">.</a> <font color="Maroon" title="c3">x</font></span>)</span>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a>  <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</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="lexbfs.html#K17" title="LEXBFS:func.17">.</a> <span class="p2">(<span class="default"><a href="lexbfs.html#K10" title="LEXBFS:func.10">LexBFS:PickUnnumbered</a> <font color="Maroon" title="c2">L</font></span>)</span></span>)</span>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> , <a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="txt" href="#E17:75"><i><font color="Green" title="E63">A12</font></i></a>, <a class="ref" href="lexbfs.html#D5" target="_self" title="LEXBFS:def.5">Def5</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_2"><b>suppose </b></a><a NAME="E1:75_1_2"/>
not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <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"> <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="lexbfs.html#K17" title="LEXBFS:func.17">.</a> <font color="Maroon" title="c3">x</font></span>)</span>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a>  <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</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="lexbfs.html#K17" title="LEXBFS:func.17">.</a> <span class="p2">(<span class="default"><a href="lexbfs.html#K10" title="LEXBFS:func.10">LexBFS:PickUnnumbered</a> <font color="Maroon" title="c2">L</font></span>)</span></span>)</span>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> , <a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><br/><div class="add"><a NAME="E2:75_1_2"/><b>then </b>
<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#K17" title="LEXBFS:func.17">.</a> <font color="Maroon" title="c3">x</font> <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="txt" href="#E3:75"><i><font color="Green" title="E54">A2</font></i></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></i>;<br/><a NAME="E3:75_1_2"/><b>then </b>
<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="lexbfs.html#K17" title="LEXBFS:func.17">.</a> <font color="Maroon" title="c3">x</font></span>)</span>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a>  <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">EmptyBag</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="uproots.html#T11" title="UPROOTS:th.11">UPROOTS:11</a></i>;<br/><b>hence </b><a NAME="E4:75_1_2"/>
<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="lexbfs.html#K17" title="LEXBFS:func.17">.</a> <font color="Maroon" title="c3">x</font></span>)</span>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a>  <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</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="lexbfs.html#K17" title="LEXBFS:func.17">.</a> <span class="p2">(<span class="default"><a href="lexbfs.html#K10" title="LEXBFS:func.10">LexBFS:PickUnnumbered</a> <font color="Maroon" title="c2">L</font></span>)</span></span>)</span>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> , <a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="termord.html#T9" title="TERMORD:th.9">TERMORD:9</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></div><b>end;</b></div>

</div>
