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<b>let </b><font color="Maroon" title="c3">Q1</font>, <font color="Maroon" title="c4">Q2</font> be   <a href="closure1.html#NM3" title="CLOSURE1:NM.3">MSClosureOperator</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">D</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:80_1_2"/><i><font color="Green" title="E32">A2</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> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">D</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 the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">D</font>  st ( for <font color="Olive" title="b3">Y</font> being   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</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> the <a href="closure1.html#U1" title="CLOSURE1:sel.1">Family</a> of <font color="Maroon" title="c2">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="c3">Q1</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>
 <b>and </b><br/><a NAME="E2:80_1_2"/><i><font color="Green" title="E33">A3</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> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">D</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 the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">D</font>  st ( for <font color="Olive" title="b3">Y</font> being   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</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> the <a href="closure1.html#U1" title="CLOSURE1:sel.1">Family</a> of <font color="Maroon" title="c2">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">Q2</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>
 ;<br/>

<b>set </b><font color="Maroon" title="c5">M</font> = the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">D</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">x</font> be    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>;<br/><b>assume </b><a NAME="E1:80_1_2_1"/>
<font color="Maroon" title="c6">x</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a>  <a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">D</font>
 ;<br/><a NAME="E2:80_1_2_1"/><b>then </b><i><font color="Green" title="E34">A4</font></i>: 
<font color="Maroon" title="c6">x</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> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">D</font>
 <b>by </b><i><a class="ref" href="mssubfam.html#T11" title="MSSUBFAM:th.11">MSSUBFAM:11</a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">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 the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">D</font><b> such that </b><br/><a NAME="E4:80_1_2_1"/><i><font color="Green" title="E35">A5</font></i>: 
for <font color="Olive" title="b1">Y</font> being   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</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="c7">SF</font> iff ( <font color="Olive" title="b1">Y</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> the <a href="closure1.html#U1" title="CLOSURE1:sel.1">Family</a> of <font color="Maroon" title="c2">D</font> &amp; <font color="Maroon" title="c6">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/><b>thus </b><font color="Maroon" title="c3">Q1</font> <a href="closure1.html#K1" title="CLOSURE1:func.1">..</a> <font color="Maroon" title="c6">x</font> = 
 <a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <font color="Maroon" title="c7">SF</font>
<b>by </b><i><a class="txt" href="closure1.html#E1:80_1_2"><i><font color="Green" title="E32">A2</font></i></a>, <a class="txt" href="closure1.html#E2:80_1_2_1"><i><font color="Green" title="E34">A4</font></i></a>, <a class="txt" href="closure1.html#E4:80_1_2_1"><i><font color="Green" title="E35">A5</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">Q2</font> <a href="closure1.html#K1" title="CLOSURE1:func.1">..</a> <font color="Maroon" title="c6">x</font>
<b>by </b><i><a class="txt" href="closure1.html#E2:80_1_2"><i><font color="Green" title="E33">A3</font></i></a>, <a class="txt" href="closure1.html#E2:80_1_2_1"><i><font color="Green" title="E34">A4</font></i></a>, <a class="txt" href="closure1.html#E4:80_1_2_1"><i><font color="Green" title="E35">A5</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:80_1_2"/>
<font color="Maroon" title="c3">Q1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c4">Q2</font>
 <b>by </b><i><a class="ref" href="closure1.html#T7" target="_self" title="CLOSURE1:th.7">Th7</a></i>;<br/>


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