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<b>let </b><font color="Maroon">I</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">M</font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>;<br/><b>let </b><font color="Maroon">EqR1</font> be   <a href="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">M</font>;<br/><b>let </b><font color="Maroon">EqR2</font> be   <a href="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">M</font>;<br/>







<b>consider </b><font color="Maroon">EqR3</font> being   <a href="msualg_4.html#NM3">ManySortedRelation</a> of <font color="Maroon">M</font><b> such that </b><br/><a NAME="E2:12"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">EqR3</font> <a href="pboole.html#R6">=</a> <font color="Maroon">EqR1</font> <a href="pboole.html#K2">\/</a> <font color="Maroon">EqR2</font> &amp; <font color="Maroon">EqR1</font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">EqR2</font> <a href="pboole.html#R6">=</a>  <a href="msualg_5.html#K3" target="_self">EqCl</a> <font color="Maroon">EqR3</font> )
 <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:12_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><b>then reconsider </b><font color="Maroon">i'</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> ;<br/><a NAME="E3:12_1"/>
<font color="Maroon">EqR3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a>  <a href="msualg_5.html#K1" target="_self">EqCl</a> <span class="p1">(<span class="default"><font color="Maroon">EqR3</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">i'</font></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:12_1"/>
<font color="Maroon">EqR3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">EqR1</font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">EqR2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:12"/>
<font color="Maroon">EqR1</font> <a href="pboole.html#K2">\/</a> <font color="Maroon">EqR2</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">EqR1</font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">EqR2</font>
 <b>by </b><i>, <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>


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