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<b>let </b><font color="Maroon" title="c1">S</font> be    <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</a> ;<br/><b>let </b><font color="Maroon" title="c2">D</font> be   <a href="closure2.html#NM4" title="CLOSURE2:NM.4">ClosureSystem</a> of <font color="Maroon" title="c1">S</font>;<br/><b>let </b><font color="Maroon" title="c3">a</font> be    <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">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/><b>let </b><font color="Maroon" title="c4">f</font> be   <a href="closure2.html#NM3" title="CLOSURE2:NM.3">SetOp</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="E1:96"/><i><font color="Green" title="E39">A1</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="closure2.html#U1" title="CLOSURE2:sel.1">Family</a> of <font color="Maroon" title="c2">D</font>
 <b>and </b><br/><a NAME="E2:96"/><i><font color="Green" title="E40">A2</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">D</font> holds  <font color="Maroon" title="c4">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Olive" title="b1">x</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="closure2.html#K11" title="CLOSURE2:func.11">Cl</a> <font color="Olive" title="b1">x</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c5">F</font> being   <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</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="E4:96"/><i><font color="Green" title="E41">A3</font></i>: 
 <a href="closure2.html#K11" title="CLOSURE2:func.11">Cl</a> <font color="Maroon" title="c3">a</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c5">F</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span>
 <b>and </b><br/><a NAME="E5:96"/><i><font color="Green" title="E42">A4</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is    <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">D</font> : ( <font color="Maroon" title="c3">a</font> <a href="closure2.html#NR2" title="CLOSURE2:NR.2">c='</a> <font color="Olive" title="b1">X</font> &amp; <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="closure2.html#U1" title="CLOSURE2:sel.1">Family</a> of <font color="Maroon" title="c2">D</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="closure2.html#D24" target="_self" title="CLOSURE2:def.24">Def24</a></i>;<br/>
<a NAME="E6:96"/><i><font color="Green" title="E43">A5</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c3">a</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c5">F</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span>
 
<b>by </b><i><a class="txt" href="closure2.html#E2:96"><i><font color="Green" title="E40">A2</font></i></a>, <a class="txt" href="closure2.html#E4:96"><i><font color="Green" title="E41">A3</font></i></a></i>;<br/>
<a NAME="E7:96"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font>
 
<b>by </b><i><a class="txt" href="closure2.html#E1:96"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="closure2.html#E5:96"><i><font color="Green" title="E42">A4</font></i></a></i>;<br/>
<a NAME="E8:96"/><b>then </b>
<font color="Maroon" title="c3">a</font> <a href="closure2.html#NR1" title="CLOSURE2:NR.1">in'</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c5">F</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span>
 
<b>by </b><i><a class="ref" href="closure2.html#T22" target="_self" title="CLOSURE2:th.22">Th22</a></i>;<br/>
<b>hence </b><a NAME="E9:96"/>
<font color="Maroon" title="c4">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c3">a</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">a</font>
 <b>by </b><i><a class="txt" href="closure2.html#E6:96"><i><font color="Green" title="E43">A5</font></i></a>, <a class="ref" href="mssubfam.html#T43" title="MSSUBFAM:th.43">MSSUBFAM:43</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D13" title="PBOOLE:def.13">PBOOLE:def 13</a><br/>

<a NAME="E10:96"/>
for <font color="Olive" title="b1">Z1</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>  st <font color="Olive" title="b1">Z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font> holds <br/><font color="Maroon" title="c3">a</font> <a href="closure2.html#NR2" title="CLOSURE2:NR.2">c='</a> <font color="Olive" title="b1">Z1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">Z1</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="E1:96_1"/>
<font color="Maroon" title="c6">Z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">b</font> being    <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">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> such that </b><br/><a NAME="E3:96_1"/><i><font color="Green" title="E44">A6</font></i>: 
( <font color="Maroon" title="c6">Z1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c7">b</font> &amp; <font color="Maroon" title="c3">a</font> <a href="closure2.html#NR2" title="CLOSURE2:NR.2">c='</a> <font color="Maroon" title="c7">b</font> )
 <b>and </b><br/><a NAME="E4:96_1"/>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="closure2.html#U1" title="CLOSURE2:sel.1">Family</a> of <font color="Maroon" title="c2">D</font>
 <b>by </b><i><a class="txt" href="closure2.html#E5:96"><i><font color="Green" title="E42">A4</font></i></a></i>;<br/>
<b>thus </b><a NAME="E5:96_1"/>
<font color="Maroon" title="c3">a</font> <a href="closure2.html#NR2" title="CLOSURE2:NR.2">c='</a> <font color="Maroon" title="c6">Z1</font>
 <b>by </b><i><a class="txt" href="closure2.html#E3:96_1"><i><font color="Green" title="E44">A6</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:96"/>
<font color="Maroon" title="c3">a</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c4">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c3">a</font>
 <b>by </b><i><a class="txt" href="closure2.html#E6:96"><i><font color="Green" title="E43">A5</font></i></a>, <a class="ref" href="closure2.html#T25" target="_self" title="CLOSURE2:th.25">Th25</a></i>;<br/>


</div>
