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<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">S1</font> be    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/>





<b>assume </b><a NAME="E1:64"/><i><font color="Green" title="E32">A1</font></i>: 
 <a href="eqrel_1.html#K11" title="EQREL_1:func.11">EqClass</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">S1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="eqrel_1.html#K11" title="EQREL_1:func.11">EqClass</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c2">S1</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c5">EQR</font> being   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:64"/><i><font color="Green" title="E33">A2</font></i>: 
<font color="Maroon" title="c2">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c5">EQR</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T43" target="_self" title="EQREL_1:th.43">Th43</a></i>;<br/>
<a NAME="E4:64"/><i><font color="Green" title="E34">A3</font></i>: 
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c5">EQR</font>,<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K11" title="EQREL_1:func.11">EqClass</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">S1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><b>proof </b></a><div class="add">

<a NAME="E1:64_1"/><i><font color="Green" title="E35">A4</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c5">EQR</font>,<font color="Maroon" title="c3">x</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T28" target="_self" title="EQREL_1:th.28">Th28</a></i>;<br/>
<a NAME="E2:64_1"/>
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c5">EQR</font>,<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S1</font>
 
<b>by </b><i><a class="txt" href="eqrel_1.html#E3:64"><i><font color="Green" title="E33">A2</font></i></a>, <a class="ref" href="eqrel_1.html#D5" target="_self" title="EQREL_1:def.5">Def5</a></i>;<br/>
<b>hence </b><a NAME="E3:64_1"/>
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c5">EQR</font>,<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K11" title="EQREL_1:func.11">EqClass</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">S1</font>
 <b>by </b><i><a class="txt" href="eqrel_1.html#E1:64_1"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="eqrel_1.html#D8" target="_self" title="EQREL_1:def.8">Def8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:64"/>
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c5">EQR</font>,<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K11" title="EQREL_1:func.11">EqClass</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c2">S1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2"><b>proof </b></a><div class="add">

<a NAME="E1:64_2"/><i><font color="Green" title="E35">A5</font></i>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c5">EQR</font>,<font color="Maroon" title="c4">y</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T28" target="_self" title="EQREL_1:th.28">Th28</a></i>;<br/>
<a NAME="E2:64_2"/>
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c5">EQR</font>,<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S1</font>
 
<b>by </b><i><a class="txt" href="eqrel_1.html#E3:64"><i><font color="Green" title="E33">A2</font></i></a>, <a class="ref" href="eqrel_1.html#D5" target="_self" title="EQREL_1:def.5">Def5</a></i>;<br/>
<b>hence </b><a NAME="E3:64_2"/>
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c5">EQR</font>,<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K11" title="EQREL_1:func.11">EqClass</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c2">S1</font>
 <b>by </b><i><a class="txt" href="eqrel_1.html#E1:64_2"><i><font color="Green" title="E35">A5</font></i></a>, <a class="ref" href="eqrel_1.html#D8" target="_self" title="EQREL_1:def.8">Def8</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:64"/>
 <a href="eqrel_1.html#K11" title="EQREL_1:func.11">EqClass</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K11" title="EQREL_1:func.11">EqClass</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c2">S1</font>
 <b>by </b><i><a class="txt" href="eqrel_1.html#E1:64"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="eqrel_1.html#E4:64"><i><font color="Green" title="E34">A3</font></i></a>, <a class="ref" href="eqrel_1.html#T32" target="_self" title="EQREL_1:th.32">Th32</a></i>;<br/>


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