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

<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">S1</font> be    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>, <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/>





<b>assume </b><a NAME="E1:16"/><i><font color="Green">E19</font></i>: 
 <a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Maroon">S1</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">S1</font>
 ;<br/>

<b>consider </b><font color="Maroon">EQR</font> being   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:16"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">S1</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">EQR</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T43">EQREL_1:43</a></i>;<br/>
<a NAME="E4:16"/><i><font color="Green">E22</font></i>: 
 <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">EQR</font>,<font color="Maroon">x</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">S1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

<a NAME="E1:16_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">EQR</font>,<font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T28">EQREL_1:28</a></i>;<br/>
<a NAME="E2:16_1"/>
 <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">EQR</font>,<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S1</font>
 
<b>by </b><i>, <a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E3:16_1"/>
 <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">EQR</font>,<font color="Maroon">x</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">S1</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:16"/>
 <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">EQR</font>,<font color="Maroon">y</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">S1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2"><b>proof </b></a><div class="add">

<a NAME="E1:16_2"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">EQR</font>,<font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T28">EQREL_1:28</a></i>;<br/>
<a NAME="E2:16_2"/>
 <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">EQR</font>,<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S1</font>
 
<b>by </b><i>, <a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E3:16_2"/>
 <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">EQR</font>,<font color="Maroon">y</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">S1</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:16"/>
 <a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Maroon">S1</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">S1</font>
 <b>by </b><i>, <a class="ref" href="t_1topsp.html#T2" target="_self">Th2</a>, <a class="ref" href="eqrel_1.html#T32">EQREL_1:32</a></i>;<br/>


</div>
