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<b>let </b><font color="Maroon" title="c1">I</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">M</font> being   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font><br/> for <font color="Olive" title="b2">EqR</font> being  <a href="msualg_5.html#NM1" title="MSUALG_5:NM.1">Equivalence_Relation</a> of <font color="Olive" title="b1">M</font> holds   <a href="msualg_5.html#K3" title="MSUALG_5:func.3">EqCl</a> <font color="Olive" title="b2">EqR</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b2">EqR</font></span><br/><b>let </b><font color="Maroon" title="c2">M</font> be    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">EqR</font> being  <a href="msualg_5.html#NM1" title="MSUALG_5:NM.1">Equivalence_Relation</a> of <font color="Maroon" title="c2">M</font> holds   <a href="msualg_5.html#K3" title="MSUALG_5:func.3">EqCl</a> <font color="Olive" title="b1">EqR</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b1">EqR</font></span><br/><b>let </b><font color="Maroon" title="c3">EqR</font> be   <a href="msualg_5.html#NM1" title="MSUALG_5:NM.1">Equivalence_Relation</a> of <font color="Maroon" title="c2">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="msualg_5.html#K3" title="MSUALG_5:func.3">EqCl</a> <font color="Maroon" title="c3">EqR</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">EqR</font></span><br/>





<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">EqR</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_5.html#K1" title="MSUALG_5:func.1">EqCl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">EqR</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Maroon" title="c4">i</font></span>)</span></span><br/><b>reconsider </b><font color="Maroon" title="c5">ER</font> = <font color="Maroon" title="c3">EqR</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Maroon" title="c4">i</font> as   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c2">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <b>by </b><i><a class="ref" href="msualg_4.html#D3" title="MSUALG_4:def.3">MSUALG_4:def 3</a></i>;<br/><b>thus </b><font color="Maroon" title="c3">EqR</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Maroon" title="c4">i</font> = 
 <a href="msualg_5.html#K1" title="MSUALG_5:func.1">EqCl</a> <font color="Maroon" title="c5">ER</font>
<b>by </b><i><a class="ref" href="msualg_5.html#T3" target="_self" title="MSUALG_5:th.3">Th3</a></i>
<br/>.= 
 <a href="msualg_5.html#K1" title="MSUALG_5:func.1">EqCl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">EqR</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Maroon" title="c4">i</font></span>)</span>

; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:8"/>
 <a href="msualg_5.html#K3" title="MSUALG_5:func.3">EqCl</a> <font color="Maroon" title="c3">EqR</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">EqR</font>
 <b>by </b><i><a class="ref" href="msualg_5.html#D3" target="_self" title="MSUALG_5:def.3">Def3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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