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<b>let </b><font color="Maroon" title="c1">P</font> be    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ;<br/>

<b>assume </b><a NAME="E1:41"/>
not <font color="Maroon" title="c1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c2">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:41"/><i><font color="Green" title="E23">A1</font></i>: 
<font color="Maroon" title="c2">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E4:41"/>
<font color="Maroon" title="c2">Z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="eqrel_1.html#E3:41"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="eqrel_1.html#D6" target="_self" title="EQREL_1:def.6">Def6</a></i>;<br/>
<a NAME="E5:41"/><b>then </b>
 ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Z</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:41"/>
contradiction
 <b>by </b><i><a class="txt" href="eqrel_1.html#E3:41"><i><font color="Green" title="E23">A1</font></i></a></i>;<br/>


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