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<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">P</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="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/>





<b>assume </b><a NAME="E1:7"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</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="E3:7"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">P</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/>
<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="E5:7"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</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#T45">EQREL_1:45</a></i>;<br/>
<b>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="E7:7"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</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#T45">EQREL_1:45</a></i>;<br/>
<b>thus </b><a NAME="E8:7"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i>, , , </i>;<br/>


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