<?xml version="1.0"?>
<div class="add">

<b>set </b><font color="Maroon">VLG</font> =  <a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font>;<br/>
<b>set </b><font color="Maroon">V2G</font> =  <a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <font color="Maroon">G</font>;<br/>
<b>set </b><font color="Maroon">VG</font> =  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font>;<br/>
<b>set </b><font color="Maroon">F</font> = <span class="p1">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <font color="Maroon">G</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p3">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">S</font> =  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <font color="Maroon">G</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p4">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span></span>)</span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="132_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:132_1_1_1"/>
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font> or  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="132_1_1_1_1"><b>suppose </b></a><a NAME="E1:132_1_1_1_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:132_1_1_1_1"/>
( (  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font> implies  ex <font color="Olive">b<sub>1</sub></font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> ) &amp; ( not  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font> implies  ex <font color="Olive">b<sub>1</sub></font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> ex <font color="Olive">S</font> being non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <a href="numbers.html#K5">NAT</a> </span>)</span> ex <font color="Olive">B</font> being non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <a href="numbers.html#K5">NAT</a> </span>)</span> ex <font color="Olive">F</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Olive">S</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Olive">F</font> &amp; <font color="Olive">F</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <font color="Maroon">G</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p3">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span> &amp; ( for <font color="Olive">x</font> being <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">S</font> holds <br/><font color="Olive">x</font>,1 <a href="uproots.html#K2">-bag</a>  <a href="hidden.html#R2">in</a> <font color="Olive">B</font> ) &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">B</font> holds <br/> ex <font color="Olive">y</font> being <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">S</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font>,1 <a href="uproots.html#K2">-bag</a>  ) ) &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><font color="Olive">F</font> <a href="relat_1.html#K10">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K11">support</a> <span class="p4">(<span class="default"><a href="lexbfs.html#K4" target="_self">max</a> <font color="Olive">B</font>,<span class="p5">(<span class="default"><a href="bagorder.html#K4">InvLexOrder</a> <a href="numbers.html#K5">NAT</a> </span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> ) ) )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="132_1_1_1_2"><b>suppose </b></a><a NAME="E1:132_1_1_1_2"/><i><font color="Green">E36</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font>
 ;<br/><div class="add"><a NAME="E2:132_1_1_1_2"/><i><font color="Green">E38</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <font color="Maroon">G</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p4">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <font color="Maroon">G</font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <font color="Maroon">G</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p4">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <font color="Maroon">G</font></span>)</span> )
 <b>by </b><i><a class="ref" href="relat_1.html#T89">RELAT_1:89</a>, <a class="ref" href="relat_1.html#T99">RELAT_1:99</a></i>;<br/><a NAME="E3:132_1_1_1_2"/><i><font color="Green">E39</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <font color="Maroon">G</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p4">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="lexbfs.html#T4" target="_self">Th4</a>, <a class="ref" href="relat_1.html#T89">RELAT_1:89</a></i>;<br/><a NAME="E4:132_1_1_1_2"/><i><font color="Green">E40</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="glib_003.html#T7">GLIB_003:7</a></i>;<br/><a NAME="E5:132_1_1_1_2"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <font color="Maroon">G</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p4">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p3">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/><a NAME="E6:132_1_1_1_2"/><b>then </b><i><font color="Green">E42</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <font color="Maroon">G</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p4">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="lexbfs.html#T4" target="_self">Th4</a>, <a class="ref" href="xboole_1.html#T49">XBOOLE_1:49</a></i>;<br/><div><i><font color="Green">E45</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="132_1_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:132_1_1_1_2_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <font color="Maroon">G</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p4">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:132_1_1_1_2_1"/><b>then </b>
 <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/><b>hence </b><a NAME="E3:132_1_1_1_2_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div><a NAME="E8:132_1_1_1_2"/><b>then </b><i><font color="Green">E46</font></i>: 
not  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <font color="Maroon">G</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p4">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="ref" href="relat_1.html#T65">RELAT_1:65</a></i>;<br/><div><i><font color="Green">E47</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="132_1_1_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:132_1_1_1_2_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <font color="Maroon">G</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p4">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span></span>)</span>
 ;<br/><a NAME="E2:132_1_1_1_2_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i><a class="ref" href="lexbfs.html#D1" target="_self">Def1</a>, </i>;<br/><a NAME="E3:132_1_1_1_2_2"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <font color="Maroon">G</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:132_1_1_1_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E5:132_1_1_1_2_2"/>
<font color="Maroon">x</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><a NAME="E10:132_1_1_1_2"/><b>then </b><i><font color="Green">E50</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <font color="Maroon">G</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p4">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span></span>)</span> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <a href="numbers.html#K5">NAT</a> 
 ;<br/><a NAME="E11:132_1_1_1_2"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <font color="Maroon">G</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p4">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span></span>)</span> holds  <font color="Olive">x</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i>, </i>;<br/><a NAME="E12:132_1_1_1_2"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <font color="Maroon">G</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p4">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span></span>)</span> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/><b>then reconsider </b><font color="Maroon">S</font> =  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <font color="Maroon">G</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p4">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span></span>)</span> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="lexbfs.html#V1" target="_self">with_finite-elements</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <a href="numbers.html#K5">NAT</a> </span>)</span> <b>by </b><i><a class="ref" href="lexbfs.html#D2" target="_self">Def2</a>, <a class="ref" href="relat_1.html#T65">RELAT_1:65</a>, , <a class="ref" href="tarski.html#D3">TARSKI:def 3</a>, , , <a class="ref" href="finset_1.html#T26">FINSET_1:26</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>() <b>-&gt; </b> non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="lexbfs.html#V1" target="_self">with_finite-elements</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <a href="numbers.html#K5">NAT</a> </span>)</span> = <font color="Maroon">S</font>;<br/><b>deffunc </b><font color="Maroon">H<sub>2</sub></font>( <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <a href="numbers.html#K5">NAT</a>  = <font color="Maroon">a<sub>1</sub></font>,1 <a href="uproots.html#K2">-bag</a> ;<br/><b>deffunc </b><font color="Maroon">H<sub>3</sub></font>() <b>-&gt; </b> non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="lexbfs.html#V1" target="_self">with_finite-elements</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <a href="numbers.html#K5">NAT</a> </span>)</span> = <font color="Maroon">S</font>;<br/><a NAME="E14:132_1_1_1_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">H<sub>3</sub></font>() is <a href="finset_1.html#V1">finite</a>
 ;<br/><b>set </b><font color="Maroon">B</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">x</font>) where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">H<sub>1</sub></font>() : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>3</sub></font>() </span>}</span> ;<br/><a NAME="E15:132_1_1_1_2"/><i><font color="Green">E90</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">x</font>) where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">H<sub>1</sub></font>() : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>3</sub></font>() </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 <b>from </b><i><a class="ref" href="fraenkel.html#S21">FRAENKEL:sch 21</a>();<br/></i><b>consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E17:132_1_1_1_2"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">z</font> = <font color="Maroon">z</font> as  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="lexbfs.html#T6" target="_self">Th6</a>, </i>;<br/><a NAME="E19:132_1_1_1_2"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">z</font>,1 <a href="uproots.html#K2">-bag</a>  <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">x</font>) where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">H<sub>1</sub></font>() : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>3</sub></font>() </span>}</span> 
 <b>by </b><i><a class="ref" href="lexbfs.html#T6" target="_self">Th6</a></i>;<br/><div><i><font color="Green">E93</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="132_1_1_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:132_1_1_1_2_3"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">x</font>) where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">H<sub>1</sub></font>() : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>3</sub></font>() </span>}</span> 
 ;<br/><b>consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E3:132_1_1_1_2_3"/><i><font color="Green">E95</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>,1 <a href="uproots.html#K2">-bag</a>  &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E4:132_1_1_1_2_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14">Bags</a> <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="lexbfs.html#D3" target="_self">Def3</a></i>;<br/></div><b>end;</b></div><b>reconsider </b><font color="Maroon">B</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">x</font>) where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">H<sub>1</sub></font>() : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>3</sub></font>() </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <a href="numbers.html#K5">NAT</a> </span>)</span> <b>by </b><i><a class="ref" href="lexbfs.html#T7" target="_self">Th7</a>, , <a class="ref" href="lexbfs.html#T5" target="_self">Th5</a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E22:132_1_1_1_2"/><i><font color="Green">E96</font></i>: 
for <font color="Olive">x</font> being <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> holds <br/><font color="Olive">x</font>,1 <a href="uproots.html#K2">-bag</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/><a NAME="E23:132_1_1_1_2"/><i><font color="Green">E97</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> holds <br/> ex <font color="Olive">y</font> being <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font>,1 <a href="uproots.html#K2">-bag</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="132_1_1_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:132_1_1_1_2_4"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>

<b>consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E3:132_1_1_1_2_4"/><i><font color="Green">E99</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>,1 <a href="uproots.html#K2">-bag</a>  &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 <b>by </b><i><a class="txt" href="lexbfs.html#E12"><i><font color="Green">Lemma78</font></i></a></i>;<br/>
<b>thus </b><a NAME="E4:132_1_1_1_2_4"/>
 ex <font color="Olive">y</font> being <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font>,1 <a href="uproots.html#K2">-bag</a>  )
 <b>by </b><i><a class="txt" href="lexbfs.html#E13"><i><font color="Green">Lemma79</font></i></a></i>;<br/>


</div><b>end;</b></div><b>set </b><font color="Maroon">mw</font> =  <a href="lexbfs.html#K4" target="_self">max</a> <font color="Maroon">B</font>,<span class="p1">(<span class="default"><a href="bagorder.html#K4">InvLexOrder</a> <a href="numbers.html#K5">NAT</a> </span>)</span>;<br/><b>set </b><font color="Maroon">IT</font> =  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <font color="Maroon">G</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p5">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="relat_1.html#K10">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K11">support</a> <span class="p4">(<span class="default"><a href="lexbfs.html#K4" target="_self">max</a> <font color="Maroon">B</font>,<span class="p5">(<span class="default"><a href="bagorder.html#K4">InvLexOrder</a> <a href="numbers.html#K5">NAT</a> </span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span>;<br/><a NAME="E24:132_1_1_1_2"/>
 <a href="lexbfs.html#K4" target="_self">max</a> <font color="Maroon">B</font>,<span class="p1">(<span class="default"><a href="bagorder.html#K4">InvLexOrder</a> <a href="numbers.html#K5">NAT</a> </span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">y</font> being  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E26:132_1_1_1_2"/><i><font color="Green">E100</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp;  <a href="lexbfs.html#K4" target="_self">max</a> <font color="Maroon">B</font>,<span class="p1">(<span class="default"><a href="bagorder.html#K4">InvLexOrder</a> <a href="numbers.html#K5">NAT</a> </span>)</span> <a href="pboole.html#R6">=</a> <font color="Maroon">y</font>,1 <a href="uproots.html#K2">-bag</a>  )
 <b>by </b><i><a class="ref" href="lexbfs.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E27:132_1_1_1_2"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K11">support</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K4" target="_self">max</a> <font color="Maroon">B</font>,<span class="p2">(<span class="default"><a href="bagorder.html#K4">InvLexOrder</a> <a href="numbers.html#K5">NAT</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="lexbfs.html#E12"><i><font color="Green">Lemma78</font></i></a>, <a class="ref" href="uproots.html#T10">UPROOTS:10</a></i>;<br/><a NAME="E28:132_1_1_1_2"/><i><font color="Green">E102</font></i>: 
not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <font color="Maroon">G</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p4">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <span class="p3">(<span class="default"><a href="lexbfs.html#K4" target="_self">max</a> <font color="Maroon">B</font>,<span class="p4">(<span class="default"><a href="bagorder.html#K4">InvLexOrder</a> <a href="numbers.html#K5">NAT</a> </span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="txt" href="lexbfs.html#E12"><i><font color="Green">Lemma78</font></i></a>, <a class="txt" href="lexbfs.html#E13"><i><font color="Green">Lemma79</font></i></a>, <a class="ref" href="funct_1.html#T142">FUNCT_1:142</a></i>;<br/><a NAME="E29:132_1_1_1_2"/>
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <font color="Maroon">G</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p5">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="relat_1.html#K10">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K11">support</a> <span class="p4">(<span class="default"><a href="lexbfs.html#K4" target="_self">max</a> <font color="Maroon">B</font>,<span class="p5">(<span class="default"><a href="bagorder.html#K4">InvLexOrder</a> <a href="numbers.html#K5">NAT</a> </span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <font color="Maroon">G</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p4">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="lexbfs.html#T8" target="_self">Th8</a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/><b>then reconsider </b><font color="Maroon">IT</font> =  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <font color="Maroon">G</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p5">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="relat_1.html#K10">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K11">support</a> <span class="p4">(<span class="default"><a href="lexbfs.html#K4" target="_self">max</a> <font color="Maroon">B</font>,<span class="p5">(<span class="default"><a href="bagorder.html#K4">InvLexOrder</a> <a href="numbers.html#K5">NAT</a> </span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> <b>by </b><i/>;<br/><a NAME="E31:132_1_1_1_2"/>
 ex <font color="Olive">S</font> being non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <a href="numbers.html#K5">NAT</a> </span>)</span> ex <font color="Olive">B</font> being non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <a href="numbers.html#K5">NAT</a> </span>)</span> ex <font color="Olive">F</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Olive">S</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Olive">F</font> &amp; <font color="Olive">F</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <font color="Maroon">G</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p3">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span> &amp; ( for <font color="Olive">x</font> being <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">S</font> holds <br/><font color="Olive">x</font>,1 <a href="uproots.html#K2">-bag</a>  <a href="hidden.html#R2">in</a> <font color="Olive">B</font> ) &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">B</font> holds <br/> ex <font color="Olive">y</font> being <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">S</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font>,1 <a href="uproots.html#K2">-bag</a>  ) ) &amp; <font color="Maroon">IT</font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><font color="Olive">F</font> <a href="relat_1.html#K10">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K11">support</a> <span class="p4">(<span class="default"><a href="lexbfs.html#K4" target="_self">max</a> <font color="Olive">B</font>,<span class="p5">(<span class="default"><a href="bagorder.html#K4">InvLexOrder</a> <a href="numbers.html#K5">NAT</a> </span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> &amp; <font color="Maroon">IT</font> is   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> )
 <b>by </b><i>, <a class="ref" href="lexbfs.html#D3" target="_self">Def3</a></i>;<br/><b>hence </b><a NAME="E32:132_1_1_1_2"/>
( (  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font> implies  ex <font color="Olive">b<sub>1</sub></font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> ) &amp; ( not  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font> implies  ex <font color="Olive">b<sub>1</sub></font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> ex <font color="Olive">S</font> being non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <a href="numbers.html#K5">NAT</a> </span>)</span> ex <font color="Olive">B</font> being non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <a href="numbers.html#K5">NAT</a> </span>)</span> ex <font color="Olive">F</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Olive">S</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Olive">F</font> &amp; <font color="Olive">F</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <font color="Maroon">G</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p3">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span> &amp; ( for <font color="Olive">x</font> being <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">S</font> holds <br/><font color="Olive">x</font>,1 <a href="uproots.html#K2">-bag</a>  <a href="hidden.html#R2">in</a> <font color="Olive">B</font> ) &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">B</font> holds <br/> ex <font color="Olive">y</font> being <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">S</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font>,1 <a href="uproots.html#K2">-bag</a>  ) ) &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><font color="Olive">F</font> <a href="relat_1.html#K10">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K11">support</a> <span class="p4">(<span class="default"><a href="lexbfs.html#K4" target="_self">max</a> <font color="Olive">B</font>,<span class="p5">(<span class="default"><a href="bagorder.html#K4">InvLexOrder</a> <a href="numbers.html#K5">NAT</a> </span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> ) ) )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
