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<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="pboole.html#M1">ManySortedSet</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="mssubfam.html#V4">absolutely-multiplicative</a> <a href="mssubfam.html#NM1">MSSubsetFamily</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="closure1.html#NM1">MSSetOp</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>







<b>assume </b><a NAME="E1:76"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="pboole.html#M2">Element</a> of  <a href="mssubfam.html#K7">bool</a> <font color="Maroon">c<sub>2</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being <a href="relat_1.html#V2">V5</a> <a href="mssubfam.html#NM1">MSSubsetFamily</a> of <font color="Maroon">c<sub>2</sub></font> st ( for <font color="Olive">b<sub>3</sub></font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Olive">b<sub>3</sub></font> <a href="pboole.html#R1">in</a> <font color="Olive">b<sub>2</sub></font> iff ( <font color="Olive">b<sub>3</sub></font> <a href="pboole.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="pboole.html#R2">c=</a> <font color="Olive">b<sub>3</sub></font> ) ) ) holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="closure1.html#K3" target="_self">..</a> <font color="Olive">b<sub>1</sub></font> <a href="pboole.html#R6">=</a>  <a href="mssubfam.html#K6">meet</a> <font color="Olive">b<sub>2</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="pboole.html#M2">Element</a> of  <a href="mssubfam.html#K7">bool</a> <font color="Maroon">c<sub>2</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="closure1.html#D4" target="_self">CLOSURE1:def 4</a><br/>

<b>consider </b><font color="Maroon">c<sub>6</sub></font> being  <a href="relat_1.html#V2">V5</a> <a href="mssubfam.html#NM1">MSSubsetFamily</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:76"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="pboole.html#R1">in</a> <font color="Maroon">c<sub>6</sub></font> iff ( <font color="Olive">b<sub>1</sub></font> <a href="pboole.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="pboole.html#R2">c=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="ref" href="closure1.html#T31" target="_self">Th31</a></i>;<br/>
<a NAME="E4:76"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#R2">c=</a>  <a href="mssubfam.html#K7">bool</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font>] means <font color="Maroon">c<sub>5</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E5:76"/>
( ( for <font color="Olive">b<sub>1</sub></font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="pboole.html#R1">in</a> <font color="Maroon">c<sub>6</sub></font> iff ( <font color="Olive">b<sub>1</sub></font> <a href="pboole.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) ) ) implies <font color="Maroon">c<sub>6</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>from </b><i><a class="ref" href="closure1.html#S1" target="_self">CLOSURE1:sch 1</a>();<br/></i>
<a NAME="E6:76"/><b>then </b><i><font color="Green">E34</font></i>: 
 <a href="mssubfam.html#K6">meet</a> <font color="Maroon">c<sub>6</sub></font> <a href="pboole.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="closure1.html#E3:76"><i><font color="Green">E32</font></i></a>, <a class="ref" href="mssubfam.html#D6">MSSUBFAM:def 6</a></i>;<br/>
<a NAME="E7:76"/><b>then </b>
 <a href="mssubfam.html#K6">meet</a> <font color="Maroon">c<sub>6</sub></font> <a href="pboole.html#R1">in</a>  <a href="mssubfam.html#K7">bool</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="closure1.html#E4:76"><i><font color="Green">E33</font></i></a>, <a class="ref" href="pboole.html#T9">PBOOLE:9</a></i>;<br/>
<a NAME="E8:76"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="mssubfam.html#K6">meet</a> <font color="Maroon">c<sub>6</sub></font> is    <a href="pboole.html#M2">Element</a> of  <a href="mssubfam.html#K7">bool</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="mssubfam.html#T11">MSSUBFAM:11</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>4</sub></font> <a href="closure1.html#K3" target="_self">..</a> <font color="Maroon">c<sub>5</sub></font> = 
 <a href="mssubfam.html#K6">meet</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="closure1.html#E1:76"><i><font color="Green">E31</font></i></a>, <a class="txt" href="closure1.html#E3:76"><i><font color="Green">E32</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="closure1.html#K1" target="_self">..</a> <span class="p1">(<span class="default"><a href="mssubfam.html#K6">meet</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="closure1.html#E1:76"><i><font color="Green">E31</font></i></a>, <a class="txt" href="closure1.html#E6:76"><i><font color="Green">E34</font></i></a>, <a class="txt" href="closure1.html#E8:76"><i><font color="Green">E35</font></i></a>, <a class="ref" href="closure1.html#T34" target="_self">Th34</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="closure1.html#K3" target="_self">..</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="closure1.html#K3" target="_self">..</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="closure1.html#E1:76"><i><font color="Green">E31</font></i></a>, <a class="txt" href="closure1.html#E3:76"><i><font color="Green">E32</font></i></a></i>
;<br/>


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