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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="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>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="closure2.html#V7" target="_self">reflexive</a> <a href="closure2.html#V8" target="_self">monotonic</a> <a href="closure2.html#NM3">SetOp</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#NM1">SubsetFamily</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>







<b>assume </b><a NAME="E1:93"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>4</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> <font color="Maroon">c<sub>2</sub></font> : <font color="Maroon">c<sub>3</sub></font> <a href="closure2.html#K7" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="pboole.html#R6">=</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> =  <a href="closure2.html#G1" target="_self">ClosureStr</a>(# <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> #) as    <a href="closure2.html#L1" target="_self">ClosureStr</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E3:93"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="closure2.html#V15" target="_self">absolutely-multiplicative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1"><b>proof </b></a><div class="add">

<a NAME="E1:93_1"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="closure2.html#V4" target="_self">absolutely-multiplicative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="closure2.html#NM1">SubsetFamily</a> of <font color="Maroon">c<sub>2</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="closure2.html#D8" target="_self">CLOSURE2:def 8</a><br/>

<b>assume </b><a NAME="E1:93_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> =  <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>6</sub></font></span><a href="closure2.html#K5" target="_self">:|</a></span> as    <a href="closure2.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6" target="_self">Bool</a> <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="closure2.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E3:93_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="closure2.html#K7" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>7</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</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="E1:93_1_1_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E2:93_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="closure2.html#E1:93_1_1"><i><font color="Green">E37</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="closure2.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6" target="_self">Bool</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E4:93_1_1_1_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="closure2.html#K7" target="_self">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="closure2.html#E1:93"><i><font color="Green">E36</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font> as    <a href="closure2.html#M1" target="_self">Element</a> of  <a href="closure2.html#K6" target="_self">Bool</a> <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="closure2.html#E1:93_1_1_1_1"><i><font color="Green">E38</font></i></a></i>;<br/><a NAME="E6:93_1_1_1_1"/>
<font color="Maroon">c<sub>10</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>6</sub></font></span><a href="closure2.html#K5" target="_self">:|</a></span>
 <b>by </b><i><a class="txt" href="closure2.html#E1:93_1_1_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="closure2.html#T22" target="_self">Th22</a></i>;<br/><a NAME="E7:93_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="closure2.html#NR2" target="_self">c='</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="mssubfam.html#T43">MSSUBFAM:43</a></i>;<br/><b>hence </b><a NAME="E8:93_1_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="closure2.html#K7" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="closure2.html#NR2" target="_self">c='</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="closure2.html#E4:93_1_1_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="closure2.html#D13" target="_self">Def13</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:93_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="closure2.html#K7" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="closure2.html#NR2" target="_self">c='</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="closure2.html#T25" target="_self">Th25</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D13">PBOOLE:def 13</a><br/>

<b>thus </b><a NAME="E3:93_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="closure2.html#NR2" target="_self">c='</a> <font color="Maroon">c<sub>3</sub></font> <a href="closure2.html#K7" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="closure2.html#D12" target="_self">Def12</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:93_1_1"/>
 <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>6</sub></font></span><a href="closure2.html#K5" target="_self">:|</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="closure2.html#E1:93"><i><font color="Green">E36</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:93_1"/>
the <a href="closure2.html#U1" target="_self">Family</a> of <font color="Maroon">c<sub>5</sub></font> is <a href="closure2.html#V4" target="_self">absolutely-multiplicative</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="closure2.html#D19" target="_self">CLOSURE2:def 19</a><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:93"/>
 <a href="closure2.html#G1" target="_self">ClosureStr</a>(# <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> #) is   <a href="closure2.html#NM4">ClosureSystem</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>


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