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<b>let </b><font color="Maroon" title="c1">I</font> be    <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">D</font> being <a href="mssubfam.html#V4" title="MSSUBFAM:attr.4">absolutely-multiplicative</a> <a href="mssubfam.html#NM1" title="MSSUBFAM:NM.1">MSSubsetFamily</a> of <font color="Olive" title="b1">M</font><br/> for <font color="Olive" title="b3">J</font> being  <a href="closure1.html#NM1" title="CLOSURE1:NM.1">MSSetOp</a> of <font color="Olive" title="b1">M</font>  st ( for <font color="Olive" title="b4">X</font> being   <a href="pboole.html#M2" title="PBOOLE:mode.2">Element</a> of  <a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Olive" title="b1">M</font><br/> for <font color="Olive" title="b5">SF</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">V5</a> <a href="mssubfam.html#NM1" title="MSSUBFAM:NM.1">MSSubsetFamily</a> of <font color="Olive" title="b1">M</font>  st ( for <font color="Olive" title="b6">Y</font> being   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font> holds <br/> ( <font color="Olive" title="b6">Y</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Olive" title="b5">SF</font> iff ( <font color="Olive" title="b6">Y</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Olive" title="b2">D</font> &amp; <font color="Olive" title="b4">X</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b6">Y</font> ) ) ) holds <br/><font color="Olive" title="b3">J</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Olive" title="b4">X</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <font color="Olive" title="b5">SF</font> ) holds <br/><font color="Olive" title="b3">J</font> is <a href="closure1.html#V3" title="CLOSURE1:attr.3">idempotent</a></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">D</font> being <a href="mssubfam.html#V4" title="MSSUBFAM:attr.4">absolutely-multiplicative</a> <a href="mssubfam.html#NM1" title="MSSUBFAM:NM.1">MSSubsetFamily</a> of <font color="Maroon" title="c2">M</font><br/> for <font color="Olive" title="b2">J</font> being  <a href="closure1.html#NM1" title="CLOSURE1:NM.1">MSSetOp</a> of <font color="Maroon" title="c2">M</font>  st ( for <font color="Olive" title="b3">X</font> being   <a href="pboole.html#M2" title="PBOOLE:mode.2">Element</a> of  <a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font><br/> for <font color="Olive" title="b4">SF</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">V5</a> <a href="mssubfam.html#NM1" title="MSSUBFAM:NM.1">MSSubsetFamily</a> of <font color="Maroon" title="c2">M</font>  st ( for <font color="Olive" title="b5">Y</font> being   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font> holds <br/> ( <font color="Olive" title="b5">Y</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Olive" title="b4">SF</font> iff ( <font color="Olive" title="b5">Y</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Olive" title="b1">D</font> &amp; <font color="Olive" title="b3">X</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b5">Y</font> ) ) ) holds <br/><font color="Olive" title="b2">J</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Olive" title="b3">X</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <font color="Olive" title="b4">SF</font> ) holds <br/><font color="Olive" title="b2">J</font> is <a href="closure1.html#V3" title="CLOSURE1:attr.3">idempotent</a></span><br/><b>let </b><font color="Maroon" title="c3">D</font> be  <a href="mssubfam.html#V4" title="MSSUBFAM:attr.4">absolutely-multiplicative</a> <a href="mssubfam.html#NM1" title="MSSUBFAM:NM.1">MSSubsetFamily</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"> for <font color="Olive" title="b1">J</font> being  <a href="closure1.html#NM1" title="CLOSURE1:NM.1">MSSetOp</a> of <font color="Maroon" title="c2">M</font>  st ( for <font color="Olive" title="b2">X</font> being   <a href="pboole.html#M2" title="PBOOLE:mode.2">Element</a> of  <a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font><br/> for <font color="Olive" title="b3">SF</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">V5</a> <a href="mssubfam.html#NM1" title="MSSUBFAM:NM.1">MSSubsetFamily</a> of <font color="Maroon" title="c2">M</font>  st ( for <font color="Olive" title="b4">Y</font> being   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font> holds <br/> ( <font color="Olive" title="b4">Y</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Olive" title="b3">SF</font> iff ( <font color="Olive" title="b4">Y</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c3">D</font> &amp; <font color="Olive" title="b2">X</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b4">Y</font> ) ) ) holds <br/><font color="Olive" title="b1">J</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Olive" title="b2">X</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <font color="Olive" title="b3">SF</font> ) holds <br/><font color="Olive" title="b1">J</font> is <a href="closure1.html#V3" title="CLOSURE1:attr.3">idempotent</a></span><br/><b>let </b><font color="Maroon" title="c4">J</font> be   <a href="closure1.html#NM1" title="CLOSURE1:NM.1">MSSetOp</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"> ( ( for <font color="Olive" title="b1">X</font> being   <a href="pboole.html#M2" title="PBOOLE:mode.2">Element</a> of  <a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font><br/> for <font color="Olive" title="b2">SF</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">V5</a> <a href="mssubfam.html#NM1" title="MSSUBFAM:NM.1">MSSubsetFamily</a> of <font color="Maroon" title="c2">M</font>  st ( for <font color="Olive" title="b3">Y</font> being   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font> holds <br/> ( <font color="Olive" title="b3">Y</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Olive" title="b2">SF</font> iff ( <font color="Olive" title="b3">Y</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c3">D</font> &amp; <font color="Olive" title="b1">X</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b3">Y</font> ) ) ) holds <br/><font color="Maroon" title="c4">J</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Olive" title="b1">X</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <font color="Olive" title="b2">SF</font> ) implies <font color="Maroon" title="c4">J</font> is <a href="closure1.html#V3" title="CLOSURE1:attr.3">idempotent</a> )</span><br/>







<b>assume </b><a NAME="E1:76"/><i><font color="Green" title="E31">A1</font></i>: 
for <font color="Olive" title="b1">X</font> being   <a href="pboole.html#M2" title="PBOOLE:mode.2">Element</a> of  <a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font><br/> for <font color="Olive" title="b2">SF</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">V5</a> <a href="mssubfam.html#NM1" title="MSSUBFAM:NM.1">MSSubsetFamily</a> of <font color="Maroon" title="c2">M</font>  st ( for <font color="Olive" title="b3">Y</font> being   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font> holds <br/> ( <font color="Olive" title="b3">Y</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Olive" title="b2">SF</font> iff ( <font color="Olive" title="b3">Y</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c3">D</font> &amp; <font color="Olive" title="b1">X</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b3">Y</font> ) ) ) holds <br/><font color="Maroon" title="c4">J</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Olive" title="b1">X</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <font color="Olive" title="b2">SF</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">J</font> is <a href="closure1.html#V3" title="CLOSURE1:attr.3">idempotent</a></span><br/>

<b>let </b><font color="Maroon" title="c5">X</font> be    <a href="pboole.html#M2" title="PBOOLE:mode.2">Element</a> of  <a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="closure1.html#D4" target="_self" title="CLOSURE1:def.4">CLOSURE1:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">J</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Maroon" title="c5">X</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c4">J</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Maroon" title="c5">X</font></span>)</span></span><br/>

<b>consider </b><font color="Maroon" title="c6">SF</font> being  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V5</a> <a href="mssubfam.html#NM1" title="MSSUBFAM:NM.1">MSSubsetFamily</a> of <font color="Maroon" title="c2">M</font><b> such that </b><br/><a NAME="E3:76"/><i><font color="Green" title="E32">A2</font></i>: 
for <font color="Olive" title="b1">Y</font> being   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font> holds <br/> ( <font color="Olive" title="b1">Y</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c6">SF</font> iff ( <font color="Olive" title="b1">Y</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c3">D</font> &amp; <font color="Maroon" title="c5">X</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b1">Y</font> ) )
 <b>by </b><i><a class="ref" href="closure1.html#T31" target="_self" title="CLOSURE1:th.31">Th31</a></i>;<br/>
<a NAME="E4:76"/><i><font color="Green" title="E33">A3</font></i>: 
<font color="Maroon" title="c3">D</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a>  <a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>] <b>means </b><font color="Maroon" title="c5">X</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> $1;<br/>
<a NAME="E5:76"/>
( ( for <font color="Olive" title="b1">Y</font> being   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font> holds <br/> ( <font color="Olive" title="b1">Y</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c6">SF</font> iff ( <font color="Olive" title="b1">Y</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c3">D</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">Y</font>] ) ) ) implies <font color="Maroon" title="c6">SF</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">D</font> )
 
<b>from </b><i><a class="ref" href="closure1.html#S1" target="_self" title="CLOSURE1:sch.1">CLOSURE1:sch 1</a>();<br/></i>
<a NAME="E6:76"/><b>then </b><i><font color="Green" title="E34">A4</font></i>: 
 <a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <font color="Maroon" title="c6">SF</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c3">D</font>
 
<b>by </b><i><a class="txt" href="closure1.html#E3:76"><i><font color="Green" title="E32">A2</font></i></a>, <a class="ref" href="mssubfam.html#D6" title="MSSUBFAM:def.6">MSSUBFAM:def 6</a></i>;<br/>
<a NAME="E7:76"/><b>then </b><i><font color="Green" title="E35">A5</font></i>: 
 <a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <font color="Maroon" title="c6">SF</font> is    <a href="pboole.html#M2" title="PBOOLE:mode.2">Element</a> of  <a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">M</font>
 
<b>by </b><i><a class="txt" href="closure1.html#E4:76"><i><font color="Green" title="E33">A3</font></i></a>, <a class="ref" href="mssubfam.html#T11" title="MSSUBFAM:th.11">MSSUBFAM:11</a>, <a class="ref" href="pboole.html#T9" title="PBOOLE:th.9">PBOOLE:9</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c4">J</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Maroon" title="c5">X</font> = 
 <a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <font color="Maroon" title="c6">SF</font>
<b>by </b><i><a class="txt" href="closure1.html#E1:76"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="closure1.html#E3:76"><i><font color="Green" title="E32">A2</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">J</font> <a href="closure1.html#K1" title="CLOSURE1:func.1">..</a> <span class="p1">(<span class="default"><a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <font color="Maroon" title="c6">SF</font></span>)</span>
<b>by </b><i><a class="txt" href="closure1.html#E1:76"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="closure1.html#E6:76"><i><font color="Green" title="E34">A4</font></i></a>, <a class="txt" href="closure1.html#E7:76"><i><font color="Green" title="E35">A5</font></i></a>, <a class="ref" href="closure1.html#T34" target="_self" title="CLOSURE1:th.34">Th34</a></i>
<br/>.= 
<font color="Maroon" title="c4">J</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="closure1.html#K3" title="CLOSURE1:func.3">..</a> <font color="Maroon" title="c5">X</font></span>)</span>
<b>by </b><i><a class="txt" href="closure1.html#E1:76"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="closure1.html#E3:76"><i><font color="Green" title="E32">A2</font></i></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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