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

<b>set </b><font color="Maroon" title="c2">VLG</font> =  <a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font>;<br/>
<b>set </b><font color="Maroon" title="c3">V2G</font> =  <a href="lexbfs.html#K6" title="LEXBFS:func.6">the_V2Label_of</a> <font color="Maroon" title="c1">G</font>;<br/>
<b>set </b><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/>
<b>let </b><font color="Maroon" title="c5">IT1</font>, <font color="Maroon" title="c6">IT2</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>;<br/>

<b>thus </b><a NAME="E1:132_1_2"/>
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></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> &amp; <font color="Maroon" title="c5">IT1</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"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> &amp; <font color="Maroon" title="c6">IT2</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"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> implies <font color="Maroon" title="c5">IT1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">IT2</font> )
 ;<br/>

<b>assume </b><a NAME="E2:132_1_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></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>
 ;<br/>

<b>assume </b><a NAME="E3:132_1_2"/>
 ex <font color="Olive" title="b1">S1</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> ex <font color="Olive" title="b2">B1</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> ex <font color="Olive" title="b3">F1</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b1">S1</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="Olive" title="b3">F1</font> &amp; <font color="Olive" title="b3">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K6" title="LEXBFS:func.6">the_V2Label_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</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="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span></span>)</span> &amp; ( for <font color="Olive" title="b4">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="b4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">S1</font> holds <br/><font color="Olive" title="b4">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="Olive" title="b2">B1</font> ) &amp; ( for <font color="Olive" title="b4">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">B1</font> holds <br/> ex <font color="Olive" title="b5">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="b5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">S1</font> &amp; <font color="Olive" title="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">y</font>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a>  ) ) &amp; <font color="Maroon" title="c5">IT1</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="Olive" title="b3">F1</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">support</a> <span class="p4">(<span class="default"><a href="lexbfs.html#K4" title="LEXBFS:func.4">max</a> <font color="Olive" title="b2">B1</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> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">S1</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="c8">B1</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span>, <font color="Maroon" title="c9">F1</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E5:132_1_2"/><i><font color="Green" title="E61">A25</font></i>: 
( <font color="Maroon" title="c7">S1</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="c9">F1</font> &amp; <font color="Maroon" title="c9">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K6" title="LEXBFS:func.6">the_V2Label_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</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="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span></span>)</span> &amp; ( 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="c7">S1</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="c8">B1</font> ) &amp; ( 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="c8">B1</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="c7">S1</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>  ) ) &amp; <font color="Maroon" title="c5">IT1</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="c9">F1</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">support</a> <span class="p4">(<span class="default"><a href="lexbfs.html#K4" title="LEXBFS:func.4">max</a> <font color="Maroon" title="c8">B1</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> )
 ;<br/>
<b>assume </b><a NAME="E6:132_1_2"/>
 ex <font color="Olive" title="b1">S2</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> ex <font color="Olive" title="b2">B2</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> ex <font color="Olive" title="b3">F2</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b1">S2</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="Olive" title="b3">F2</font> &amp; <font color="Olive" title="b3">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K6" title="LEXBFS:func.6">the_V2Label_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</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="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span></span>)</span> &amp; ( for <font color="Olive" title="b4">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="b4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">S2</font> holds <br/><font color="Olive" title="b4">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="Olive" title="b2">B2</font> ) &amp; ( for <font color="Olive" title="b4">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">B2</font> holds <br/> ex <font color="Olive" title="b5">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="b5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">S2</font> &amp; <font color="Olive" title="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">y</font>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a>  ) ) &amp; <font color="Maroon" title="c6">IT2</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="Olive" title="b3">F2</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">support</a> <span class="p4">(<span class="default"><a href="lexbfs.html#K4" title="LEXBFS:func.4">max</a> <font color="Olive" title="b2">B2</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> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">S2</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="c11">B2</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span>, <font color="Maroon" title="c12">F2</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E8:132_1_2"/><i><font color="Green" title="E62">A26</font></i>: 
( <font color="Maroon" title="c10">S2</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="c12">F2</font> &amp; <font color="Maroon" title="c12">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K6" title="LEXBFS:func.6">the_V2Label_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</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="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span></span>)</span> &amp; ( 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="c10">S2</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="c11">B2</font> ) &amp; ( 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="c11">B2</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="c10">S2</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>  ) ) &amp; <font color="Maroon" title="c6">IT2</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="c12">F2</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K10" title="POLYNOM1:func.10">support</a> <span class="p4">(<span class="default"><a href="lexbfs.html#K4" title="LEXBFS:func.4">max</a> <font color="Maroon" title="c11">B2</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> )
 ;<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" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:132_1_2_1"/><i><font color="Green" title="E63">A27</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B1</font>
 ;<br/><a NAME="E2:132_1_2_1"/>
 ex <font color="Olive" title="b1">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="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S1</font> &amp; <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</font>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a>  )
 <b>by </b><i><a class="txt" href="lexbfs.html#E5:132_1_2"><i><font color="Green" title="E61">A25</font></i></a>, <a class="txt" href="lexbfs.html#E1:132_1_2_1"><i><font color="Green" title="E63">A27</font></i></a></i>;<br/><b>hence </b><a NAME="E3:132_1_2_1"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">B2</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E5:132_1_2"><i><font color="Green" title="E61">A25</font></i></a>, <a class="txt" href="lexbfs.html#E8:132_1_2"><i><font color="Green" title="E62">A26</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E10:132_1_2"/><b>then </b><i><font color="Green" title="E63">A28</font></i>: 
<font color="Maroon" title="c8">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">B2</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">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" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:132_1_2_2"/><i><font color="Green" title="E64">A29</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">B2</font>
 ;<br/><a NAME="E2:132_1_2_2"/>
 ex <font color="Olive" title="b1">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="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">S2</font> &amp; <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</font>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a>  )
 <b>by </b><i><a class="txt" href="lexbfs.html#E8:132_1_2"><i><font color="Green" title="E62">A26</font></i></a>, <a class="txt" href="lexbfs.html#E1:132_1_2_2"><i><font color="Green" title="E64">A29</font></i></a></i>;<br/><b>hence </b><a NAME="E3:132_1_2_2"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B1</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E5:132_1_2"><i><font color="Green" title="E61">A25</font></i></a>, <a class="txt" href="lexbfs.html#E8:132_1_2"><i><font color="Green" title="E62">A26</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E12:132_1_2"/><b>then </b>
<font color="Maroon" title="c11">B2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">B1</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E13:132_1_2"/>
<font color="Maroon" title="c5">IT1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">IT2</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E5:132_1_2"><i><font color="Green" title="E61">A25</font></i></a>, <a class="txt" href="lexbfs.html#E8:132_1_2"><i><font color="Green" title="E62">A26</font></i></a>, <a class="txt" href="lexbfs.html#E10:132_1_2"><i><font color="Green" title="E63">A28</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div>
