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<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="struct_0.html#L1">1-sorted</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="closure2.html#NM4">ClosureSystem</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="closure2.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6" target="_self">Bool</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="closure2.html#NM3">SetOp</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:96"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> the <a href="closure2.html#U1" target="_self">Family</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E2:96"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="closure2.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6" target="_self">Bool</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>4</sub></font> <a href="closure2.html#K7" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="pboole.html#R6">=</a>  <a href="closure2.html#K11" target="_self">Cl</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="closure2.html#NM1">SubsetFamily</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E4:96"/><i><font color="Green">E41</font></i>: 
 <a href="closure2.html#K11" target="_self">Cl</a> <font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#R6">=</a>  <a href="mssubfam.html#K6">meet</a> <span class="p1"><a href="closure2.html#K5" target="_self">|:</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="closure2.html#K5" target="_self">:|</a></span>
 <b>and </b><br/><a NAME="E5:96"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="closure2.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6" target="_self">Bool</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> : ( <font color="Maroon">c<sub>3</sub></font> <a href="closure2.html#NR2" target="_self">c='</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="closure2.html#U1" target="_self">Family</a> of <font color="Maroon">c<sub>2</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="closure2.html#D24" target="_self">Def24</a></i>;<br/>
<a NAME="E6:96"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="closure2.html#K7" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#R6">=</a>  <a href="mssubfam.html#K6">meet</a> <span class="p1"><a href="closure2.html#K5" target="_self">|:</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="closure2.html#K5" target="_self">:|</a></span>
 
<b>by </b><i><a class="txt" href="closure2.html#E2:96"><i><font color="Green">E40</font></i></a>, <a class="txt" href="closure2.html#E4:96"><i><font color="Green">E41</font></i></a></i>;<br/>
<a NAME="E7:96"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="closure2.html#E1:96"><i><font color="Green">E39</font></i></a>, <a class="txt" href="closure2.html#E5:96"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E8:96"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="closure2.html#NR1" target="_self">in'</a> <span class="p1"><a href="closure2.html#K5" target="_self">|:</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="closure2.html#K5" target="_self">:|</a></span>
 
<b>by </b><i><a class="ref" href="closure2.html#T22" target="_self">Th22</a></i>;<br/>
<b>hence </b><a NAME="E9:96"/>
<font color="Maroon">c<sub>4</sub></font> <a href="closure2.html#K7" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="closure2.html#E6:96"><i><font color="Green">E43</font></i></a>, <a class="ref" href="mssubfam.html#T43">MSSUBFAM:43</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D13">PBOOLE:def 13</a><br/>

<a NAME="E10:96"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="closure2.html#NR2" target="_self">c='</a> <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font> be    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:96_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="closure2.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6" target="_self">Bool</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:96_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="closure2.html#NR2" target="_self">c='</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>and </b><br/><a NAME="E4:96_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> the <a href="closure2.html#U1" target="_self">Family</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="closure2.html#E5:96"><i><font color="Green">E42</font></i></a></i>;<br/>
<b>thus </b><a NAME="E5:96_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="closure2.html#NR2" target="_self">c='</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="closure2.html#E3:96_1"><i><font color="Green">E44</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:96"/>
<font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>4</sub></font> <a href="closure2.html#K7" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="closure2.html#E6:96"><i><font color="Green">E43</font></i></a>, <a class="ref" href="closure2.html#T25" target="_self">Th25</a></i>;<br/>


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