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<b>let </b><font color="Maroon">S</font> be    <a href="struct_0.html#L1">1-sorted</a> ;<br/><b>let </b><font color="Maroon">D</font> be   <a href="closure2.html#NM4">ClosureSystem</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">a</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">D</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="closure2.html#NM3">SetOp</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">D</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:97"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">f</font> <a href="closure2.html#K7" 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="E2:97"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">x</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">D</font> holds  <font color="Maroon">f</font> <a href="closure2.html#K7" target="_self">.</a> <font color="Olive">x</font> <a href="pboole.html#R6">=</a>  <a href="closure2.html#K11" target="_self">Cl</a> <font color="Olive">x</font>
 ;<br/>

<b>set </b><font color="Maroon">F</font> = the <a href="closure2.html#U1" target="_self">Family</a> of <font color="Maroon">D</font>;<br/>
<b>set </b><font color="Maroon">M</font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">D</font>;<br/>
<b>consider </b><font color="Maroon">SF</font> being   <a href="closure2.html#NM1">SubsetFamily</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">D</font><b> such that </b><br/><a NAME="E4:97"/><i><font color="Green">E40</font></i>: 
 <a href="closure2.html#K11" target="_self">Cl</a> <font color="Maroon">a</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">SF</font></span><a href="closure2.html#K5" target="_self">:|</a></span>
 <b>and </b><br/><a NAME="E5:97"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">SF</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> 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">D</font> : ( <font color="Maroon">a</font> <a href="pboole.html#R2">c=</a> <font color="Olive">X</font> &amp; <font color="Olive">X</font> <a href="hidden.html#R2">in</a> the <a href="closure2.html#U1" target="_self">Family</a> of <font color="Maroon">D</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="closure2.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E6:97"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">a</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">SF</font></span><a href="closure2.html#K5" target="_self">:|</a></span>
 
<b>by </b><i><a class="txt" href="closure2.html#E2:97"><i><font color="Green">E39</font></i></a>, <a class="txt" href="closure2.html#E4:97"><i><font color="Green">E40</font></i></a>, </i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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">D</font>] <b>means </b><font color="Maroon">a</font> <a href="closure2.html#NR2" target="_self">c='</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <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">D</font>] <b>means </b>( <font color="Maroon">a</font> <a href="closure2.html#NR2" target="_self">c='</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<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">D</font> );<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <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">D</font>] <b>means </b>( <font color="Maroon">a<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">D</font> &amp; <font color="Maroon">a</font> <a href="closure2.html#NR2" target="_self">c='</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E7:97"/><i><font color="Green">E47</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">w</font>) where <font color="Olive">w</font> 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">D</font> : ( <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">w</font>) <a href="hidden.html#R2">in</a> the <a href="closure2.html#U1" target="_self">Family</a> of <font color="Maroon">D</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">w</font>] ) </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="closure2.html#U1" target="_self">Family</a> of <font color="Maroon">D</font>
 
<b>from </b><i><a class="ref" href="fraenkel.html#S17">FRAENKEL:sch 17</a>();<br/></i>
<a NAME="E8:97"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">q</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">D</font> holds <br/> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">q</font>] iff <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">q</font>] )
 
;<br/>
<a NAME="E9:97"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">s</font>) where <font color="Olive">s</font> 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">D</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">s</font>] </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b</font>) where <font color="Olive">b</font> 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">D</font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b</font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="fraenkel.html#S3">FRAENKEL:sch 3</a>(<a class="txt" href="closure2.html#E5:97"><i><font color="Green">E45</font></i></a>);<br/></i>
<b>hence </b><a NAME="E10:97"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="closure2.html#U1" target="_self">Family</a> of <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="closure2.html#D1" target="_self">Def1</a>, , , </i>;<br/>


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