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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="E1:97"/><i><font color="Green">E39</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> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E2:97"/><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>set </b><font color="Maroon">c<sub>5</sub></font> = the <a href="closure2.html#U1" target="_self">Family</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</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="E4:97"/><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>7</sub></font></span><a href="closure2.html#K5" target="_self">:|</a></span>
 <b>and </b><br/><a NAME="E5:97"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>7</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="pboole.html#R2">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="E6:97"/><i><font color="Green">E43</font></i>: 
<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>7</sub></font></span><a href="closure2.html#K5" target="_self">:|</a></span>
 
<b>by </b><i><a class="txt" href="closure2.html#E1:97"><i><font color="Green">E39</font></i></a>, <a class="txt" href="closure2.html#E2:97"><i><font color="Green">E40</font></i></a>, <a class="txt" href="closure2.html#E4:97"><i><font color="Green">E41</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">c<sub>2</sub></font>] means <font color="Maroon">c<sub>3</sub></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">c<sub>2</sub></font>] means ( <font color="Maroon">c<sub>3</sub></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">c<sub>2</sub></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">c<sub>2</sub></font>] means ( <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">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></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">E44</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<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">H<sub>1</sub></font>(<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> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="closure2.html#U1" target="_self">Family</a> of <font color="Maroon">c<sub>2</sub></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">E45</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 <br/>( <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] iff <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] )
 
;<br/>
<a NAME="E9:97"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<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">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></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<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">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="fraenkel.html#S3">FRAENKEL:sch 3</a>(<a class="txt" href="closure2.html#E8:97"><i><font color="Green">E45</font></i></a>);<br/></i>
<b>hence </b><a NAME="E10:97"/>
<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>by </b><i><a class="txt" href="closure2.html#E5:97"><i><font color="Green">E42</font></i></a>, <a class="txt" href="closure2.html#E6:97"><i><font color="Green">E43</font></i></a>, <a class="txt" href="closure2.html#E7:97"><i><font color="Green">E44</font></i></a>, <a class="ref" href="closure2.html#D8" target="_self">Def8</a></i>;<br/>


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