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

<b>let </b><font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">B</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">X1</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">X2</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">R</font> be   <a href="subset_1.html#NM2">Subset</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>









<b>assume </b><a NAME="E1:40"/>
<font color="Maroon">X1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X2</font>
 ;<br/>

<a NAME="E2:40"/><b>then </b><i><font color="Green">E21</font></i>: 
<span class="p1"><a href="relset_2.html#NK1" target="_self">{_{</a><span class="default"><font color="Maroon">X1</font></span><a href="relset_2.html#NK1" target="_self">}_}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="relset_2.html#NK1" target="_self">{_{</a><span class="default"><font color="Maroon">X2</font></span><a href="relset_2.html#NK1" target="_self">}_}</a></span>
 
<b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E3:40"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="relset_2.html#K5" target="_self">.:^</a> <font color="Maroon">X2</font>
 ;<br/>

<a NAME="E4:40"/><b>then </b><i><font color="Green">E23</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_2.html#K3" target="_self">.:</a> <font color="Maroon">R</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p2"><a href="relset_2.html#NK1" target="_self">{_{</a><span class="default"><font color="Maroon">X2</font></span><a href="relset_2.html#NK1" target="_self">}_}</a></span></span>)</span>
 
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_1"/>
( <span class="p1">(<span class="default"><a href="relset_2.html#K3" target="_self">.:</a> <font color="Maroon">R</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="relset_2.html#NK1" target="_self">{_{</a><span class="default"><font color="Maroon">X2</font></span><a href="relset_2.html#NK1" target="_self">}_}</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <span class="p1">(<span class="default"><a href="relset_2.html#K3" target="_self">.:</a> <font color="Maroon">R</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="relset_2.html#NK1" target="_self">{_{</a><span class="default"><font color="Maroon">X2</font></span><a href="relset_2.html#NK1" target="_self">}_}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1"><b>suppose </b></a><a NAME="E1:40_1_1"/><i><font color="Green">E25</font></i>: 
<span class="p1">(<span class="default"><a href="relset_2.html#K3" target="_self">.:</a> <font color="Maroon">R</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="relset_2.html#NK1" target="_self">{_{</a><span class="default"><font color="Maroon">X2</font></span><a href="relset_2.html#NK1" target="_self">}_}</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_1_1_1"/>
( <span class="p1">(<span class="default"><a href="relset_2.html#K3" target="_self">.:</a> <font color="Maroon">R</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="relset_2.html#NK1" target="_self">{_{</a><span class="default"><font color="Maroon">X1</font></span><a href="relset_2.html#NK1" target="_self">}_}</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <span class="p1">(<span class="default"><a href="relset_2.html#K3" target="_self">.:</a> <font color="Maroon">R</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="relset_2.html#NK1" target="_self">{_{</a><span class="default"><font color="Maroon">X1</font></span><a href="relset_2.html#NK1" target="_self">}_}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_1_1"><b>suppose </b></a><a NAME="E1:40_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="relset_2.html#K3" target="_self">.:</a> <font color="Maroon">R</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="relset_2.html#NK1" target="_self">{_{</a><span class="default"><font color="Maroon">X1</font></span><a href="relset_2.html#NK1" target="_self">}_}</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:40_1_1_1_1"/><b>then </b>
 <a href="setfam_1.html#K8">Intersect</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_2.html#K3" target="_self">.:</a> <font color="Maroon">R</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p2"><a href="relset_2.html#NK1" target="_self">{_{</a><span class="default"><font color="Maroon">X1</font></span><a href="relset_2.html#NK1" target="_self">}_}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>;<br/><a NAME="E3:40_1_1_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_2.html#K3" target="_self">.:</a> <font color="Maroon">R</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p2"><a href="relset_2.html#NK1" target="_self">{_{</a><span class="default"><font color="Maroon">X1</font></span><a href="relset_2.html#NK1" target="_self">}_}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="relset_2.html#T1" target="_self">Th1</a></i>;<br/><b>hence </b><a NAME="E4:40_1_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="relset_2.html#K5" target="_self">.:^</a> <font color="Maroon">X1</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_1_2"><b>suppose </b></a><a NAME="E1:40_1_1_1_2"/><i><font color="Green">E26</font></i>: 
<span class="p1">(<span class="default"><a href="relset_2.html#K3" target="_self">.:</a> <font color="Maroon">R</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="relset_2.html#NK1" target="_self">{_{</a><span class="default"><font color="Maroon">X1</font></span><a href="relset_2.html#NK1" target="_self">}_}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:40_1_1_1_2"/>
for <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_2.html#K3" target="_self">.:</a> <font color="Maroon">R</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="relset_2.html#NK1" target="_self">{_{</a><span class="default"><font color="Maroon">X1</font></span><a href="relset_2.html#NK1" target="_self">}_}</a></span> holds <br/><font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_1_2_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:40_1_1_1_2_1"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_2.html#K3" target="_self">.:</a> <font color="Maroon">R</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="relset_2.html#NK1" target="_self">{_{</a><span class="default"><font color="Maroon">X1</font></span><a href="relset_2.html#NK1" target="_self">}_}</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:40_1_1_1_2_1"/><i><font color="Green">E32</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">Y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_2.html#K3" target="_self">.:</a> <font color="Maroon">R</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="relset_2.html#NK1" target="_self">{_{</a><span class="default"><font color="Maroon">X1</font></span><a href="relset_2.html#NK1" target="_self">}_}</a></span> )
 <b>by </b><i><a class="ref" href="relat_1.html#D13">RELAT_1:def 13</a></i>;<br/>
<b>thus </b><a NAME="E4:40_1_1_1_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="txt" href="relset_2.html#E3:40"><i><font color="Green">E22</font></i></a>, , , <a class="ref" href="relat_1.html#D13">RELAT_1:def 13</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:40_1_1_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_2.html#K3" target="_self">.:</a> <font color="Maroon">R</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p2"><a href="relset_2.html#NK1" target="_self">{_{</a><span class="default"><font color="Maroon">X1</font></span><a href="relset_2.html#NK1" target="_self">}_}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a>, <a class="txt" href="relset_2.html#E4:40"><i><font color="Green">E23</font></i></a></i>;<br/><a NAME="E4:40_1_1_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_2.html#K3" target="_self">.:</a> <font color="Maroon">R</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p2"><a href="relset_2.html#NK1" target="_self">{_{</a><span class="default"><font color="Maroon">X1</font></span><a href="relset_2.html#NK1" target="_self">}_}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="relset_2.html#E4:40"><i><font color="Green">E23</font></i></a>, <a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>;<br/><b>hence </b><a NAME="E5:40_1_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="relset_2.html#K5" target="_self">.:^</a> <font color="Maroon">X1</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_2"><b>suppose </b></a><a NAME="E1:40_1_2"/>
<span class="p1">(<span class="default"><a href="relset_2.html#K3" target="_self">.:</a> <font color="Maroon">R</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="relset_2.html#NK1" target="_self">{_{</a><span class="default"><font color="Maroon">X2</font></span><a href="relset_2.html#NK1" target="_self">}_}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:40_1_2"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_2.html#K3" target="_self">.:</a> <font color="Maroon">R</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p2"><a href="relset_2.html#NK1" target="_self">{_{</a><span class="default"><font color="Maroon">X2</font></span><a href="relset_2.html#NK1" target="_self">}_}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="relset_2.html#E2:40"><i><font color="Green">E21</font></i></a>, <a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_1_2_1"/>
( <span class="p1">(<span class="default"><a href="relset_2.html#K3" target="_self">.:</a> <font color="Maroon">R</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="relset_2.html#NK1" target="_self">{_{</a><span class="default"><font color="Maroon">X1</font></span><a href="relset_2.html#NK1" target="_self">}_}</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <span class="p1">(<span class="default"><a href="relset_2.html#K3" target="_self">.:</a> <font color="Maroon">R</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="relset_2.html#NK1" target="_self">{_{</a><span class="default"><font color="Maroon">X1</font></span><a href="relset_2.html#NK1" target="_self">}_}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_2_1_1"><b>suppose </b></a><a NAME="E1:40_1_2_1_1"/>
<span class="p1">(<span class="default"><a href="relset_2.html#K3" target="_self">.:</a> <font color="Maroon">R</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="relset_2.html#NK1" target="_self">{_{</a><span class="default"><font color="Maroon">X1</font></span><a href="relset_2.html#NK1" target="_self">}_}</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:40_1_2_1_1"/><b>then </b>
 <a href="setfam_1.html#K8">Intersect</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_2.html#K3" target="_self">.:</a> <font color="Maroon">R</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p2"><a href="relset_2.html#NK1" target="_self">{_{</a><span class="default"><font color="Maroon">X1</font></span><a href="relset_2.html#NK1" target="_self">}_}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>;<br/><a NAME="E3:40_1_2_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_2.html#K3" target="_self">.:</a> <font color="Maroon">R</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p2"><a href="relset_2.html#NK1" target="_self">{_{</a><span class="default"><font color="Maroon">X1</font></span><a href="relset_2.html#NK1" target="_self">}_}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="relset_2.html#T1" target="_self">Th1</a></i>;<br/><b>hence </b><a NAME="E4:40_1_2_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="relset_2.html#K5" target="_self">.:^</a> <font color="Maroon">X1</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_2_1_2"><b>suppose </b></a><a NAME="E1:40_1_2_1_2"/><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default"><a href="relset_2.html#K3" target="_self">.:</a> <font color="Maroon">R</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="relset_2.html#NK1" target="_self">{_{</a><span class="default"><font color="Maroon">X1</font></span><a href="relset_2.html#NK1" target="_self">}_}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:40_1_2_1_2"/>
<span class="p1">(<span class="default"><a href="relset_2.html#K3" target="_self">.:</a> <font color="Maroon">R</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="relset_2.html#NK1" target="_self">{_{</a><span class="default"><font color="Maroon">X1</font></span><a href="relset_2.html#NK1" target="_self">}_}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_2.html#K3" target="_self">.:</a> <font color="Maroon">R</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="relset_2.html#NK1" target="_self">{_{</a><span class="default"><font color="Maroon">X2</font></span><a href="relset_2.html#NK1" target="_self">}_}</a></span>
 <b>by </b><i>, <a class="ref" href="relat_1.html#T156">RELAT_1:156</a></i>;<br/><a NAME="E3:40_1_2_1_2"/><b>then </b>
 <a href="setfam_1.html#K6">meet</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_2.html#K3" target="_self">.:</a> <font color="Maroon">R</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p2"><a href="relset_2.html#NK1" target="_self">{_{</a><span class="default"><font color="Maroon">X2</font></span><a href="relset_2.html#NK1" target="_self">}_}</a></span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K6">meet</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_2.html#K3" target="_self">.:</a> <font color="Maroon">R</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p2"><a href="relset_2.html#NK1" target="_self">{_{</a><span class="default"><font color="Maroon">X1</font></span><a href="relset_2.html#NK1" target="_self">}_}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="setfam_1.html#T7">SETFAM_1:7</a>, <a class="txt" href="relset_2.html#E4:40"><i><font color="Green">E23</font></i></a></i>;<br/><a NAME="E4:40_1_2_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_2.html#K3" target="_self">.:</a> <font color="Maroon">R</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p2"><a href="relset_2.html#NK1" target="_self">{_{</a><span class="default"><font color="Maroon">X1</font></span><a href="relset_2.html#NK1" target="_self">}_}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="relset_2.html#E3:40"><i><font color="Green">E22</font></i></a></i>;<br/><a NAME="E5:40_1_2_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_2.html#K3" target="_self">.:</a> <font color="Maroon">R</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p2"><a href="relset_2.html#NK1" target="_self">{_{</a><span class="default"><font color="Maroon">X1</font></span><a href="relset_2.html#NK1" target="_self">}_}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="relset_2.html#E4:40"><i><font color="Green">E23</font></i></a>, <a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>;<br/><b>hence </b><a NAME="E6:40_1_2_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="relset_2.html#K5" target="_self">.:^</a> <font color="Maroon">X1</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

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