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<b>let </b><font color="Maroon">I</font> be    <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">D</font> be  <a href="mssubfam.html#V4">absolutely-multiplicative</a> <a href="mssubfam.html#NM1">MSSubsetFamily</a> of <font color="Maroon">M</font>;<br/><b>let </b><font color="Maroon">A</font> be    <a href="pboole.html#M2">Element</a> of  <a href="mssubfam.html#K7">bool</a> <font color="Maroon">M</font>;<br/><b>let </b><font color="Maroon">J</font> be   <a href="closure1.html#NM1">MSSetOp</a> of <font color="Maroon">M</font>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:74"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">J</font> <a href="closure1.html#K3" target="_self">..</a> <font color="Maroon">A</font> <a href="pboole.html#R6">=</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E2:74"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">X</font> being   <a href="pboole.html#M2">Element</a> of  <a href="mssubfam.html#K7">bool</a> <font color="Maroon">M</font><br/> for <font color="Olive">SF</font> being <a href="relat_1.html#V2">V5</a> <a href="mssubfam.html#NM1">MSSubsetFamily</a> of <font color="Maroon">M</font>  st ( for <font color="Olive">Y</font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font> holds <br/> ( <font color="Olive">Y</font> <a href="pboole.html#R1">in</a> <font color="Olive">SF</font> iff ( <font color="Olive">Y</font> <a href="pboole.html#R1">in</a> <font color="Maroon">D</font> &amp; <font color="Olive">X</font> <a href="pboole.html#R2">c=</a> <font color="Olive">Y</font> ) ) ) holds <br/><font color="Maroon">J</font> <a href="closure1.html#K3" target="_self">..</a> <font color="Olive">X</font> <a href="pboole.html#R6">=</a>  <a href="mssubfam.html#K6">meet</a> <font color="Olive">SF</font>
 ;<br/>

<b>consider </b><font color="Maroon">SF</font> being  <a href="relat_1.html#V2">V5</a> <a href="mssubfam.html#NM1">MSSubsetFamily</a> of <font color="Maroon">M</font><b> such that </b><br/><a NAME="E4:74"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">Y</font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font> holds <br/> ( <font color="Olive">Y</font> <a href="pboole.html#R1">in</a> <font color="Maroon">SF</font> iff ( <font color="Olive">Y</font> <a href="pboole.html#R1">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">A</font> <a href="pboole.html#R2">c=</a> <font color="Olive">Y</font> ) )
 <b>by </b><i/>;<br/>
<a NAME="E5:74"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">J</font> <a href="closure1.html#K3" target="_self">..</a> <font color="Maroon">A</font> <a href="pboole.html#R6">=</a>  <a href="mssubfam.html#K6">meet</a> <font color="Maroon">SF</font>
 
<b>by </b><i>, </i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>] <b>means </b><font color="Maroon">A</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E6:74"/>
( ( for <font color="Olive">Y</font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font> holds <br/> ( <font color="Olive">Y</font> <a href="pboole.html#R1">in</a> <font color="Maroon">SF</font> iff ( <font color="Olive">Y</font> <a href="pboole.html#R1">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">Y</font>] ) ) ) implies <font color="Maroon">SF</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">D</font> )
 
<b>from </b><i><a class="ref" href="closure1.html#S1" target="_self">CLOSURE1:sch 1</a>();<br/></i>
<b>hence </b><a NAME="E7:74"/>
<font color="Maroon">A</font> <a href="pboole.html#R1">in</a> <font color="Maroon">D</font>
 <b>by </b><i>, , , <a class="ref" href="mssubfam.html#D6">MSSUBFAM:def 6</a></i>;<br/>


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