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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K1" title="RELSET_1:func.1">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#K1" title="RELSET_1:func.1">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/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K1" title="RELSET_1:func.1">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#K1" title="RELSET_1:func.1">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/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K1" title="RELSET_1:func.1">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#K1" title="RELSET_1:func.1">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/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:75"/><span class="lab"><font color="Green" title="E52">A1</font></span>: 
 not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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>
 <span class="kw">and </span><br/><a NAME="E2:75"/><span class="lab"><font color="Green" title="E53">A2</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">set </span><font color="Maroon" title="c4">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/>
<span class="kw">set </span><font color="Maroon" title="c5">V2G</font> = <font color="Maroon" title="c2">L</font> <a href="lexbfs.html#K8" title="LEXBFS:func.8">`2</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">VLG</font> = <font color="Maroon" title="c2">L</font> <a href="lexbfs.html#K7" title="LEXBFS:func.7">`1</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">w</font> =  <a href="lexbfs.html#K10" title="LEXBFS:func.10">LexBFS:PickUnnumbered</a> <font color="Maroon" title="c2">L</font>;<br/>
<span class="kw">consider </span><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="pre_poly.html#K15" title="PRE_POLY:func.15">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><span class="kw"> such that </span><br/><a NAME="E4:75"/><span class="lab"><font color="Green" title="E54">A3</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E5:75"/><span class="lab"><font color="Green" title="E55">A4</font></span>: 
<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#K5" title="RELSET_1:func.5">|</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#K1" title="RELSET_1:func.1">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>
 <span class="kw">and </span><br/><a NAME="E6:75"/><span class="lab"><font color="Green" title="E56">A5</font></span>: 
 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>
 <span class="kw">and </span><br/><a NAME="E7:75"/><span class="lab"><font color="Green" title="E57">A6</font></span>: 
 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>  )
 <span class="kw">and </span><br/><a NAME="E8:75"/><span class="lab"><font color="Green" title="E58">A7</font></span>: 
 <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="pre_poly.html#K13" title="PRE_POLY:func.13">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:75"><span class="lab"><font color="Green" title="E53">A2</font></span></a>, <a class="ref" href="lexbfs.html#D12" target="_self" title="LEXBFS:def.12">Def12</a></span>;<br/>
<a NAME="E9:75"/><span class="lab"><font color="Green" title="E59">A8</font></span>: 
 <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#K1" title="RELSET_1:func.1">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#K1" title="RELSET_1:func.1">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:75"><span class="lab"><font color="Green" title="E55">A4</font></span></a>, <a class="ref" href="relat_1.html#T61" title="RELAT_1:th.61">RELAT_1:61</a></span>;<br/>
<span class="kw">set </span><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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lexbfs.html#D5" target="_self" title="LEXBFS:def.5">Def5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E12:75"/><span class="lab"><font color="Green" title="E60">A9</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E13:75"/><span class="lab"><font color="Green" title="E61">A10</font></span>: 
 <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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:75"><span class="lab"><font color="Green" title="E57">A6</font></span></a></span>;<br/>
<a NAME="E14:75"/><span class="lab"><font color="Green" title="E62">A11</font></span>: 
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K13" title="PRE_POLY:func.13">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:75"><span class="lab"><font color="Green" title="E61">A10</font></span></a>, <a class="ref" href="uproots.html#T8" title="UPROOTS:th.8">UPROOTS:8</a></span>;<br/>
<a NAME="E15:75"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A12</font></span>: 
 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="pre_poly.html#K13" title="PRE_POLY:func.13">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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:75"><span class="lab"><font color="Green" title="E54">A3</font></span></a>, <a class="txt" href="#E12:75"><span class="lab"><font color="Green" title="E60">A9</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>;<br/>
<a NAME="E16:75"/><span class="kw">then </span>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:75"><span class="lab"><font color="Green" title="E58">A7</font></span></a>, <a class="ref" href="funct_1.html#D7" title="FUNCT_1:def.7">FUNCT_1:def 7</a></span>;<br/>
<a NAME="E17:75"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A13</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:75"><span class="lab"><font color="Green" title="E55">A4</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>;<br/>
<a NAME="E18:75"/>
<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="pre_poly.html#K13" title="PRE_POLY:func.13">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:75"><span class="lab"><font color="Green" title="E58">A7</font></span></a>, <a class="txt" href="#E15:75"><span class="lab"><font color="Green" title="E63">A12</font></span></a>, <a class="ref" href="funct_1.html#D7" title="FUNCT_1:def.7">FUNCT_1:def 7</a></span>;<br/>
<a NAME="E19:75"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A14</font></span>: 
(<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>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:75"><span class="lab"><font color="Green" title="E61">A10</font></span></a>, <a class="txt" href="#E14:75"><span class="lab"><font color="Green" title="E62">A11</font></span></a>, <a class="txt" href="#E17:75"><span class="lab"><font color="Green" title="E64">A13</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E20:75"/><span class="lab"><font color="Green" title="E66">A15</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></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"><span class="kw">suppose </span></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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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"/><span class="kw">then </span>
<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#K1" title="RELSET_1:func.1">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E3:75_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A16</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:75"><span class="lab"><font color="Green" title="E66">A15</font></span></a>, <a class="txt" href="#E9:75"><span class="lab"><font color="Green" title="E59">A8</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E4:75_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A17</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:75"><span class="lab"><font color="Green" title="E54">A3</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<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#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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:75"><span class="lab"><font color="Green" title="E55">A4</font></span></a>, <a class="txt" href="#E3:75_1_1"><span class="lab"><font color="Green" title="E67">A16</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>;<br/><a NAME="E6:75_1_1"/><span class="kw">then </span>
(<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:75"><span class="lab"><font color="Green" title="E56">A5</font></span></a>, <a class="txt" href="#E4:75_1_1"><span class="lab"><font color="Green" title="E68">A17</font></span></a></span>;<br/><span class="kw">hence </span><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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:75"><span class="lab"><font color="Green" title="E65">A14</font></span></a>, <a class="ref" href="lexbfs.html#D5" target="_self" title="LEXBFS:def.5">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_2"><span class="kw">suppose </span></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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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"/><span class="kw">then </span>
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:75"><span class="lab"><font color="Green" title="E66">A15</font></span></a>, <a class="ref" href="funct_1.html#D2" title="FUNCT_1:def.2">FUNCT_1:def 2</a></span>;<br/><a NAME="E3:75_1_2"/><span class="kw">then </span>
(<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="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="uproots.html#T9" title="UPROOTS:th.9">UPROOTS:9</a></span>;<br/><span class="kw">hence </span><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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="termord.html#T9" title="TERMORD:th.9">TERMORD:9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
