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<b>given </b><font color="Maroon">S1</font> being    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font>, <font color="Maroon">S2</font> being    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">X</font><b> such that </b><a NAME="E1:19_1_1"/><i><font color="Green">E19</font></i>: 
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">S1</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> 
 <b>and </b><br/><a NAME="E2:19_1_1"/><i><font color="Green">E21</font></i>: 
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">S2</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> 
 <b>and </b><br/><a NAME="E3:19_1_1"/><i><font color="Green">E22</font></i>: 
not <font color="Maroon">S1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S2</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_1"><b>now </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_1_1"/>
 <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="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>hence </b><a NAME="E2:19_1_1_1"/>
 <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">x</font>,<font color="Maroon">S2</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:19_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="t_1topsp.html#T3" target="_self">Th3</a>, </i>;<br/>


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