<?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>let </b><font color="Maroon">IT1</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>, <font color="Maroon">IT2</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>;<br/>

<b>thus </b><a NAME="E1:132_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> &amp; <font color="Maroon">IT1</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; <font color="Maroon">IT2</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> implies <font color="Maroon">IT1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">IT2</font> )
 ;<br/>

<b>assume </b><a NAME="E2:132_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#NR2" target="_self">&lt;&gt;</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font>
 ;<br/>

<b>assume </b><a NAME="E3:132_1_2"/>
 ex <font color="Olive">S1</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">B1</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">F1</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Olive">S1</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Olive">F1</font> &amp; <font color="Olive">F1</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">S1</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">B1</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">B1</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">S1</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">IT1</font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><font color="Olive">F1</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">B1</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/>

<b>then consider </b><font color="Maroon">S1</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>, <font color="Maroon">B1</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>, <font color="Maroon">F1</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:132_1_2"/><i><font color="Green">E103</font></i>: 
( <font color="Maroon">S1</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font> &amp; <font color="Maroon">F1</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="Maroon">S1</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">B1</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="Maroon">B1</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">S1</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">IT1</font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><font color="Maroon">F1</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="Maroon">B1</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/>
<b>assume </b><a NAME="E6:132_1_2"/>
 ex <font color="Olive">S2</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">B2</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">F2</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Olive">S2</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Olive">F2</font> &amp; <font color="Olive">F2</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">S2</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">B2</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">B2</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">S2</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">IT2</font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><font color="Olive">F2</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">B2</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/>

<b>then consider </b><font color="Maroon">S2</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>, <font color="Maroon">B2</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>, <font color="Maroon">F2</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E8:132_1_2"/><i><font color="Green">E104</font></i>: 
( <font color="Maroon">S2</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F2</font> &amp; <font color="Maroon">F2</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="Maroon">S2</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">B2</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="Maroon">B2</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">S2</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">IT2</font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><font color="Maroon">F2</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="Maroon">B2</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><a class="txt" onclick="hs2(this)" href="javascript:()" title="132_1_2_1"><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_2_1"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font>
 ;<br/><a NAME="E2:132_1_2_1"/>
 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">S1</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>, </i>;<br/><b>hence </b><a NAME="E3:132_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B2</font>
 <b>by </b><i>, <a class="ref" href="lexbfs.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E10:132_1_2"/><b>then </b><i><font color="Green">E117</font></i>: 
<font color="Maroon">B1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B2</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="132_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_2_2"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B2</font>
 ;<br/><a NAME="E2:132_1_2_2"/>
 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">S2</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="ref" href="lexbfs.html#D1" target="_self">Def1</a>, </i>;<br/><b>hence </b><a NAME="E3:132_1_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font>
 <b>by </b><i>, <a class="ref" href="lexbfs.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E12:132_1_2"/><b>then </b>
<font color="Maroon">B2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B1</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E13:132_1_2"/>
<font color="Maroon">IT1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">IT2</font>
 <b>by </b><i>, <a class="ref" href="lexbfs.html#D1" target="_self">Def1</a>, , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
