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

<b>let </b><font color="Maroon">B</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">X</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">FR</font> be   <a href="setfam_1.html#NM1">Subset-Family</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>let </b><font color="Maroon">G</font> be   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">B</font>;<br/>









<b>assume </b><a NAME="E1:47"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">R</font> <a href="relset_2.html#K5" target="_self">.:^</a> <font color="Maroon">X</font></span>)</span> where <font color="Olive">R</font> is   <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> : <font color="Olive">R</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FR</font> </span>}</span> 
 ;<br/>

<a NAME="E2:47"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> holds <br/> <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><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="E1:47_1"/>
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:47_1"/><b>then </b>
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>;<br/>
<b>hence </b><a NAME="E3:47_1"/>
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#T11">SETFAM_1:11</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:47_2"/>
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">X</font> <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="47_2_1"><b>suppose </b></a><a NAME="E1:47_2_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:47_2_1"/><b>then </b><i><font color="Green">E25</font></i>: 
<span class="p1">(<span class="default"><a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">FR</font></span>)</span> <a href="relset_2.html#K5" target="_self">.:^</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 <b>by </b><i/>;<br/><a NAME="E3:47_2_1"/>
<font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">B</font></span><a href="tarski.html#K1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:47_2_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 ;<br/>

<b>then consider </b><font color="Maroon">R</font> being   <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><b> such that </b><br/><a NAME="E3:47_2_1_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> <a href="relset_2.html#K5" target="_self">.:^</a> <font color="Maroon">X</font> &amp; <font color="Maroon">R</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FR</font> )
 <b>by </b><i><a class="txt" href="relset_2.html#E1:47"><i><font color="Green">E21</font></i></a></i>;<br/>
<a NAME="E4:47_2_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="txt" href="relset_2.html#E2:47_2_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="relset_2.html#E1:47_2_1"><i><font color="Green">E23</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E5:47_2_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">B</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:47_2_1"/><b>then </b>
( <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">B</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T39">ZFMISC_1:39</a></i>;<br/><b>hence </b><a NAME="E5:47_2_1"/>
<span class="p1">(<span class="default"><a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">FR</font></span>)</span> <a href="relset_2.html#K5" target="_self">.:^</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="txt" href="relset_2.html#E2:47"><i><font color="Green">E22</font></i></a>, <a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_2_2"><b>suppose </b></a><a NAME="E1:47_2_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:47_2_2_1"/>
( <font color="Maroon">FR</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">FR</font> <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="47_2_2_1_1"><b>suppose </b></a><a NAME="E1:47_2_2_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">FR</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:47_2_2_1_1"/><b>then </b>
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">FR</font> <a href="hidden.html#R1">=</a> <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>
 <b>by </b><i><a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>;<br/><a NAME="E3:47_2_2_1_1"/><b>then </b><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default"><a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">FR</font></span>)</span> <a href="relset_2.html#K5" target="_self">.:^</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="relset_2.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E4:47_2_2_1_1"/>
<font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">B</font></span><a href="tarski.html#K1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2_2_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:47_2_2_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 ;<br/>

<b>then consider </b><font color="Maroon">R</font> being   <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><b> such that </b><br/><a NAME="E3:47_2_2_1_1_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> <a href="relset_2.html#K5" target="_self">.:^</a> <font color="Maroon">X</font> &amp; <font color="Maroon">R</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FR</font> )
 <b>by </b><i><a class="txt" href="relset_2.html#E1:47"><i><font color="Green">E21</font></i></a></i>;<br/>
<b>thus </b><a NAME="E4:47_2_2_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">B</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><a NAME="E5:47_2_2_1_1"/><b>then </b>
( <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">B</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T39">ZFMISC_1:39</a></i>;<br/><b>hence </b><a NAME="E6:47_2_2_1_1"/>
<span class="p1">(<span class="default"><a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">FR</font></span>)</span> <a href="relset_2.html#K5" target="_self">.:^</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="txt" href="relset_2.html#E2:47"><i><font color="Green">E22</font></i></a>, , <a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_2_2_1_2"><b>suppose </b></a><a NAME="E1:47_2_2_1_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">FR</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_2_2_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:47_2_2_1_2_1"/>
( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">B</font> <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="47_2_2_1_2_1_1"><b>suppose </b></a><a NAME="E1:47_2_2_1_2_1_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:47_2_2_1_2_1_1"/><b>then </b>
<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> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T113">ZFMISC_1:113</a></i>;<br/><a NAME="E3:47_2_2_1_2_1_1"/><b>then </b><i><font color="Green">E70</font></i>: 
( <font color="Maroon">FR</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">FR</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T39">ZFMISC_1:39</a>, <a class="ref" href="zfmisc_1.html#T1">ZFMISC_1:1</a></i>;<br/><a NAME="E4:47_2_2_1_2_1_1"/><b>then </b>
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">FR</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>;<br/><a NAME="E5:47_2_2_1_2_1_1"/><b>then </b>
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">FR</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="setfam_1.html#T11">SETFAM_1:11</a></i>;<br/><a NAME="E6:47_2_2_1_2_1_1"/><b>then </b><i><font color="Green">E71</font></i>: 
<span class="p1">(<span class="default"><a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">FR</font></span>)</span> <a href="relset_2.html#K5" target="_self">.:^</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E7:47_2_2_1_2_1_1"/>
<font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2_2_1_2_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:47_2_2_1_2_1_1_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 ;<br/>

<b>consider </b><font color="Maroon">R</font> being   <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><b> such that </b><br/><a NAME="E3:47_2_2_1_2_1_1_1"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> <a href="relset_2.html#K5" target="_self">.:^</a> <font color="Maroon">X</font> &amp; <font color="Maroon">R</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FR</font> )
 <b>by </b><i><a class="ref" href="relset_2.html#T2" target="_self">Th2</a>, <a class="txt" href="relset_2.html#E1:47"><i><font color="Green">E21</font></i></a></i>;<br/>
<a NAME="E4:47_2_2_1_2_1_1_1"/>
<font color="Maroon">R</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="relset_2.html#T3" target="_self">Th3</a>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:47_2_2_1_2_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, , <a class="ref" href="relset_2.html#T3" target="_self">Th3</a></i>;<br/>
<b>hence </b><a NAME="E6:47_2_2_1_2_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E8:47_2_2_1_2_1_1"/><b>then </b>
( <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T39">ZFMISC_1:39</a></i>;<br/><b>hence </b><a NAME="E9:47_2_2_1_2_1_1"/>
<span class="p1">(<span class="default"><a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">FR</font></span>)</span> <a href="relset_2.html#K5" target="_self">.:^</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="txt" href="relset_2.html#E2:47"><i><font color="Green">E22</font></i></a>, , , <a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_2_2_1_2_1_2"><b>suppose </b></a><a NAME="E1:47_2_2_1_2_1_2"/>
<font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">B</font> = <font color="Maroon">B</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/><b>thus </b><a NAME="E3:47_2_2_1_2_1_2"/>
<span class="p1">(<span class="default"><a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">FR</font></span>)</span> <a href="relset_2.html#K5" target="_self">.:^</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">G</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2_2_1_2_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:47_2_2_1_2_1_2_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">FR</font></span>)</span> <a href="relset_2.html#K5" target="_self">.:^</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:47_2_2_1_2_1_2_1"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font> <b>by </b><i/>;<br/>
<a NAME="E4:47_2_2_1_2_1_2_1"/>
( <font color="Maroon">G</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  implies <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">G</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2_2_1_2_1_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:47_2_2_1_2_1_2_1_1"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E2:47_2_2_1_2_1_2_1_1"/><b>then </b><i><font color="Green">E81</font></i>: 
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>;<br/>
<a NAME="E3:47_2_2_1_2_1_2_1_1"/>
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> <font color="Maroon">G</font> holds <br/><font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2_2_1_2_1_2_1_1_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="E1:47_2_2_1_2_1_2_1_1_1"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 ;<br/>

<b>consider </b><font color="Maroon">R</font> being   <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><b> such that </b><br/><a NAME="E3:47_2_2_1_2_1_2_1_1_1"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> <a href="relset_2.html#K5" target="_self">.:^</a> <font color="Maroon">X</font> &amp; <font color="Maroon">R</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FR</font> )
 <b>by </b><i><a class="ref" href="relset_2.html#T3" target="_self">Th3</a>, <a class="txt" href="relset_2.html#E1:47"><i><font color="Green">E21</font></i></a></i>;<br/>
<a NAME="E4:47_2_2_1_2_1_2_1_1_1"/>
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">X</font> holds <br/><font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="relat_1.html#K9">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2_2_1_2_1_2_1_1_1_1"><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="E1:47_2_2_1_2_1_2_1_1_1_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:47_2_2_1_2_1_2_1_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">FR</font></span>)</span> <a href="relat_1.html#K9">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E3:47_2_2_1_2_1_2_1_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">FR</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:47_2_2_1_2_1_2_1_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">FR</font>
 
<b>by </b><i><a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a>, </i>;<br/>
<a NAME="E5:47_2_2_1_2_1_2_1_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="relset_2.html#T4" target="_self">Th4</a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:47_2_2_1_2_1_2_1_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="relat_1.html#K9">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:47_2_2_1_2_1_2_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="relset_2.html#T4" target="_self">Th4</a>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:47_2_2_1_2_1_2_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a>, , <a class="ref" href="relset_2.html#T2" target="_self">Th2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:47_2_2_1_2_1_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>;<br/>


</div><b>end;</b></div><b>let </b><font color="Maroon">a</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="E4:47_2_2_1_2_1_2"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">G</font>
 ;<br/><b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font> ;<br/><b>consider </b><font color="Maroon">R</font> being   <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><b> such that </b><br/><a NAME="E7:47_2_2_1_2_1_2"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">R</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FR</font>
 <b>by </b><i>, <a class="ref" href="subset_1.html#T10">SUBSET_1:10</a></i>;<br/><a NAME="E8:47_2_2_1_2_1_2"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">R</font> <a href="relset_2.html#K5" target="_self">.:^</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="txt" href="relset_2.html#E1:47"><i><font color="Green">E21</font></i></a></i>;<br/><a NAME="E9:47_2_2_1_2_1_2"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">G</font>
 <b>by </b><i>, , <a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>;<br/><a NAME="E10:47_2_2_1_2_1_2"/>
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">X</font> holds <br/><font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">FR</font></span>)</span> <a href="relat_1.html#K9">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2_2_1_2_1_2_2"><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="E1:47_2_2_1_2_1_2_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:47_2_2_1_2_1_2_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> <font color="Maroon">FR</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2_2_1_2_1_2_2_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:47_2_2_1_2_1_2_2_1"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FR</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">P</font> = <font color="Maroon">P</font> as   <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>set </b><font color="Maroon">S</font> = <font color="Maroon">P</font> <a href="relset_2.html#K5" target="_self">.:^</a> <font color="Maroon">X</font>;<br/>
<a NAME="E3:47_2_2_1_2_1_2_2_1"/>
<font color="Maroon">P</font> <a href="relset_2.html#K5" target="_self">.:^</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="txt" href="relset_2.html#E1:47"><i><font color="Green">E21</font></i></a>, <a class="ref" href="relset_2.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E4:47_2_2_1_2_1_2_2_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="relset_2.html#K5" target="_self">.:^</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="relset_2.html#T2" target="_self">Th2</a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<a NAME="E5:47_2_2_1_2_1_2_2_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="relat_1.html#K9">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, <a class="ref" href="relset_2.html#T3" target="_self">Th3</a></i>;<br/>
<b>hence </b><a NAME="E6:47_2_2_1_2_1_2_2_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E3:47_2_2_1_2_1_2_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">FR</font>
 
<b>by </b><i><a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a>, </i>;<br/>
<a NAME="E4:47_2_2_1_2_1_2_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">FR</font>
 
<b>by </b><i><a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a>, </i>;<br/>
<b>hence </b><a NAME="E5:47_2_2_1_2_1_2_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">FR</font></span>)</span> <a href="relat_1.html#K9">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E11:47_2_2_1_2_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">FR</font></span>)</span> <a href="relset_2.html#K5" target="_self">.:^</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div>
