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

<b>thus </b><a NAME="E1:19_1_2"/>
 ex <font color="Olive">G</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> st <br/>for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> holds   <a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Olive">x</font>,<font color="Olive">G</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Olive">x</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">G</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K1">meet</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Olive">x</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> </span>)</span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> : verum </span>}</span> ;<br/>
<a NAME="E1:19_1_2_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K1">meet</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Olive">x</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> </span>)</span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> : verum </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_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> ; <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="E1:19_1_2_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K1">meet</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Olive">x</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> </span>)</span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:19_1_2_1_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> 
 ;<br/>
<a NAME="E4:19_1_2_1_1"/>
<font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e</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="E1:19_1_2_1_1_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font>
 ;<br/>

<b>consider </b><font color="Maroon">T</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:19_1_2_1_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">T</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">T</font> = <font color="Maroon">T</font> as    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="t_1topsp.html#T3" target="_self">Th3</a>, </i>;<br/>
<a NAME="E5:19_1_2_1_1_1"/>
 <a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Maroon">T</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="t_1topsp.html#T3" target="_self">Th3</a></i>;<br/>
<b>then consider </b><font color="Maroon">f</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:19_1_2_1_1_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> 
 ;<br/>
<b>consider </b><font color="Maroon">S</font> being    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E9:19_1_2_1_1_1"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Maroon">S</font> &amp; <font color="Maroon">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 <b>by </b><i/>;<br/>
<a NAME="E10:19_1_2_1_1_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="t_1topsp.html#T2" target="_self">Th2</a>, , , , <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E11:19_1_2_1_1_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:19_1_2_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">G</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K1">meet</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Olive">x</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> </span>)</span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> : verum </span>}</span>  as   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font> ;<br/>
<a NAME="E3:19_1_2_1"/>
<font color="Maroon">G</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_1_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_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="E1:19_1_2_1_2_1"/>
( <font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2_1_2_1_1"><b>case </b></a><a NAME="E1:19_1_2_1_2_1_1"/>
<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"><b>thus </b><a NAME="E2:19_1_2_1_2_1_1"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_1_2_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:19_1_2_1_2_1_1_1"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>thus </b><a NAME="E2:19_1_2_1_2_1_1_1"/>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">G</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_1_2_1_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="E1:19_1_2_1_2_1_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> ;<br/>
<b>consider </b><font color="Maroon">T</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:19_1_2_1_2_1_1_1_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">T</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">T</font> = <font color="Maroon">T</font> as    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="t_1topsp.html#T2" target="_self">Th2</a>, </i>;<br/>
<a NAME="E6:19_1_2_1_2_1_1_1_1"/><i><font color="Green">E58</font></i>: 
 <a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">a1</font>,<font color="Maroon">T</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">a1</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="t_1topsp.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E7:19_1_2_1_2_1_1_1_1"/>
for <font color="Olive">T</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">T</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">a1</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span>  holds <br/><font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> <font color="Olive">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_1_2_1_1_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:19_1_2_1_2_1_1_1_1_1"/>
<font color="Maroon">T</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">a1</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">A</font> being    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:19_1_2_1_2_1_1_1_1_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">a1</font>,<font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E4:19_1_2_1_2_1_1_1_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>
<b>thus </b><a NAME="E5:19_1_2_1_2_1_1_1_1_1"/>
<font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="t_1topsp.html#T3" target="_self">Th3</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:19_1_2_1_2_1_1_1_1"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">a1</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> 
 
<b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<a NAME="E9:19_1_2_1_2_1_1_1_1"/>
 <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">a1</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span>  <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 
;<br/>
<b>hence </b><a NAME="E10:19_1_2_1_2_1_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="t_1topsp.html#T3" target="_self">Th3</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div><b>let </b><font color="Maroon">A</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E3:19_1_2_1_2_1_1"/><i><font color="Green">E61</font></i>: 
<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">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E5:19_1_2_1_2_1_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> 
 ;<br/><b>consider </b><font color="Maroon">T</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:19_1_2_1_2_1_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">T</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">T</font> = <font color="Maroon">T</font> as    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="t_1topsp.html#T3" target="_self">Th3</a>, </i>;<br/><a NAME="E9:19_1_2_1_2_1_1"/><i><font color="Green">E64</font></i>: 
 <a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Maroon">T</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="t_1topsp.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E10:19_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> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span>  holds <br/><font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_1_2_1_1_2"><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="E1:19_1_2_1_2_1_1_2"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> 
 ;<br/>

<a NAME="E2:19_1_2_1_2_1_1_2"/><b>then </b>
 ex <font color="Olive">T</font> being   <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> st <br/>( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Olive">T</font> &amp; <font color="Olive">T</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 
;<br/>
<b>hence </b><a NAME="E3:19_1_2_1_2_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E11:19_1_2_1_2_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, , <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/><b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E13:19_1_2_1_2_1_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="t_1topsp.html#T2" target="_self">Th2</a></i>;<br/><b>let </b><font color="Maroon">B</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E14:19_1_2_1_2_1_1"/>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 ;<br/><b>then consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E16:19_1_2_1_2_1_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">y</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> 
 ;<br/><b>thus </b><a NAME="E17:19_1_2_1_2_1_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> or <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_1_2_1_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:19_1_2_1_2_1_1_3"/><i><font color="Green">E68</font></i>: 
not <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span>  is <a href="xboole_0.html#V1">empty</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_1_2_1_1_3_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">T</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:19_1_2_1_2_1_1_3_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">T</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">T</font> = <font color="Maroon">T</font> as    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> <b>by </b><i>, </i>;<br/>
<a NAME="E4:19_1_2_1_2_1_1_3_1"/>
 <a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Maroon">T</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:19_1_2_1_2_1_1_3_1"/>
not <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span>  is <a href="xboole_0.html#V1">empty</a>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:19_1_2_1_2_1_1_3"/><i><font color="Green">E70</font></i>: 
not <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">y</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span>  is <a href="xboole_0.html#V1">empty</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_1_2_1_1_3_2"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">T</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:19_1_2_1_2_1_1_3_2"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">T</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">T</font> = <font color="Maroon">T</font> as    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> <b>by </b><i>, </i>;<br/>
<a NAME="E4:19_1_2_1_2_1_1_3_2"/>
 <a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">y</font>,<font color="Maroon">T</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">y</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:19_1_2_1_2_1_1_3_2"/>
not <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">y</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span>  is <a href="xboole_0.html#V1">empty</a>
 ;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_1_2_1_1_3_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:19_1_2_1_2_1_1_3_3"/>
<font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">B</font>
 ;<br/><b>then consider </b><font color="Maroon">c</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:19_1_2_1_2_1_1_3_3"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><a NAME="E4:19_1_2_1_2_1_1_3_3"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1">meet</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> </span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1">meet</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">y</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> </span>)</span>
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E5:19_1_2_1_2_1_1_3_3"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">(<span class="default"><span class="p2">{<span class="default"> <span class="p3">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">y</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> </span>)</span>
 <b>by </b><i>, , <a class="ref" href="setfam_1.html#T10">SETFAM_1:10</a></i>;<br/><div><i><font color="Green">E75</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_1_2_1_1_3_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">T</font> be    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E1:19_1_2_1_2_1_1_3_3_1"/>
<font color="Maroon">T</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/><a NAME="E2:19_1_2_1_2_1_1_3_3_1"/><b>then </b>
 <a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Maroon">T</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> 
 ;<br/><a NAME="E3:19_1_2_1_2_1_1_3_3_1"/><b>then </b>
 <a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Maroon">T</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">y</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E4:19_1_2_1_2_1_1_3_3_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E76</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_1_2_1_1_3_3_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">T</font> be    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E1:19_1_2_1_2_1_1_3_3_2"/>
<font color="Maroon">T</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/><a NAME="E2:19_1_2_1_2_1_1_3_3_2"/><b>then </b>
 <a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">y</font>,<font color="Maroon">T</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">y</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> 
 ;<br/><a NAME="E3:19_1_2_1_2_1_1_3_3_2"/><b>then </b>
 <a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">y</font>,<font color="Maroon">T</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">y</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E4:19_1_2_1_2_1_1_3_3_2"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">y</font>,<font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/></div><b>end;</b></div><a NAME="E8:19_1_2_1_2_1_1_3_3"/><i><font color="Green">E77</font></i>: 
for <font color="Olive">T</font> being   <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font>  st <font color="Olive">T</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> holds <br/> ex <font color="Olive">A</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">A</font> <a href="hidden.html#R2">in</a>  <a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Olive">T</font> &amp; <font color="Olive">A</font> <a href="hidden.html#R2">in</a>  <a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">y</font>,<font color="Olive">T</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_1_2_1_1_3_3_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">T</font> be    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E1:19_1_2_1_2_1_1_3_3_3"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">T</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<b>take </b>
<font color="Maroon">c</font>
;<br/>

<b>thus </b><a NAME="E2:19_1_2_1_2_1_1_3_3_3"/>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Maroon">T</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">y</font>,<font color="Maroon">T</font> )
 <b>by </b><i>, , <a class="ref" href="t_1topsp.html#T4" target="_self">Th4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E9:19_1_2_1_2_1_1_3_3"/><i><font color="Green">E79</font></i>: 
for <font color="Olive">T</font> being   <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font>  st <font color="Olive">T</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> holds <br/> <a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Olive">T</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">y</font>,<font color="Olive">T</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_1_2_1_1_3_3_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">T</font> be    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E1:19_1_2_1_2_1_1_3_3_4"/>
<font color="Maroon">T</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon">A</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:19_1_2_1_2_1_1_3_3_4"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Maroon">T</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">y</font>,<font color="Maroon">T</font> )
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E4:19_1_2_1_2_1_1_3_3_4"/>
 <a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Maroon">T</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">y</font>,<font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E10:19_1_2_1_2_1_1_3_3"/><i><font color="Green">E81</font></i>: 
for <font color="Olive">T</font> being   <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font>  st <font color="Olive">T</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> holds <br/> <a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Olive">T</font> <a href="hidden.html#R1">=</a>  <a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">y</font>,<font color="Olive">T</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_1_2_1_1_3_3_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">T</font> be    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E1:19_1_2_1_2_1_1_3_3_5"/>
<font color="Maroon">T</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<a NAME="E2:19_1_2_1_2_1_1_3_3_5"/><b>then </b>
 <a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Maroon">T</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">y</font>,<font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="t_1topsp.html#T4" target="_self">Th4</a></i>;<br/>
<b>hence </b><a NAME="E3:19_1_2_1_2_1_1_3_3_5"/>
 <a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">y</font>,<font color="Maroon">T</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E11:19_1_2_1_2_1_1_3_3"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">y</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_1_2_1_1_3_3_6"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:19_1_2_1_2_1_1_3_3_6"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">y</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> 
  <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="19_1_2_1_2_1_1_3_3_6_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="E1:19_1_2_1_2_1_1_3_3_6_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">T</font> being    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:19_1_2_1_2_1_1_3_3_6_1"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">T</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Maroon">T</font> )
 <b>by </b><i/>;<br/>
<a NAME="E4:19_1_2_1_2_1_1_3_3_6_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">y</font>,<font color="Maroon">T</font>
 
<b>by </b><i>, <a class="ref" href="t_1topsp.html#T5" target="_self">Th5</a></i>;<br/>
<b>hence </b><a NAME="E5:19_1_2_1_2_1_1_3_3_6_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">y</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="t_1topsp.html#T5" target="_self">Th5</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:19_1_2_1_2_1_1_3_3_6"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">y</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_1_2_1_1_3_3_6_2"><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="E1:19_1_2_1_2_1_1_3_3_6_2"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">y</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">T</font> being    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:19_1_2_1_2_1_1_3_3_6_2"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">T</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">y</font>,<font color="Maroon">T</font> )
 <b>by </b><i/>;<br/>
<a NAME="E4:19_1_2_1_2_1_1_3_3_6_2"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Maroon">T</font>
 
<b>by </b><i>, <a class="ref" href="t_1topsp.html#T5" target="_self">Th5</a></i>;<br/>
<b>hence </b><a NAME="E5:19_1_2_1_2_1_1_3_3_6_2"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="t_1topsp.html#T5" target="_self">Th5</a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div><b>hence </b><a NAME="E12:19_1_2_1_2_1_1_3_3"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> or <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:19_1_2_1_2_1_1_3"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> or <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font> )
 ;<br/>


</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2_1_2_1_2"><b>case </b></a><a NAME="E1:19_1_2_1_2_1_2"/>
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:19_1_2_1_2_1_2"/>
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">G</font> = <font color="Maroon">G</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> ;<br/>
<b>take </b>
<font color="Maroon">G</font>
;<br/>

<a NAME="E5:19_1_2_1"/>
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> holds   <a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Olive">x</font>,<font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Olive">x</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/>

<a NAME="E1:19_1_2_1_3"/><i><font color="Green">E84</font></i>: 
 <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span>  <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 
;<br/>
<a NAME="E2:19_1_2_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_1_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:19_1_2_1_3_1"/><i><font color="Green">E85</font></i>: 
not <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span>  is <a href="xboole_0.html#V1">empty</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_1_3_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">T</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:19_1_2_1_3_1_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">T</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">T</font> = <font color="Maroon">T</font> as    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="t_1topsp.html#T3" target="_self">Th3</a>, </i>;<br/>
<a NAME="E4:19_1_2_1_3_1_1"/>
 <a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Maroon">T</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="t_1topsp.html#T3" target="_self">Th3</a></i>;<br/>
<b>hence </b><a NAME="E5:19_1_2_1_3_1_1"/>
not <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span>  is <a href="xboole_0.html#V1">empty</a>
 ;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_1_3_1_2"><b>now </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="E1:19_1_2_1_3_1_2"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> 
 ;<br/><a NAME="E2:19_1_2_1_3_1_2"/><b>then </b>
 ex <font color="Olive">T</font> being   <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> st <br/>( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Olive">T</font> &amp; <font color="Olive">T</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 ;<br/><b>hence </b><a NAME="E3:19_1_2_1_3_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:19_1_2_1_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> 
 <b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:19_1_2_1_3"/>
 <a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="t_1topsp.html#T2" target="_self">Th2</a>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:19_1_2_1"/>
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> holds   <a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Olive">x</font>,<font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Olive">x</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> 
 ;<br/>


</div><b>end;</b></div>


</div>
