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<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="closure1.html#NM3">MSClosureOperator</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="E1:80_1_2"/><i><font color="Green">E32</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> the <a href="msualg_1.html#U4">Sorts</a> of <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 the <a href="msualg_1.html#U4">Sorts</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 the <a href="struct_0.html#U1">carrier</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> the <a href="closure1.html#U1" target="_self">Family</a> of <font color="Maroon">c<sub>2</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>3</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>
 <b>and </b><br/><a NAME="E2:80_1_2"/><i><font color="Green">E33</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> the <a href="msualg_1.html#U4">Sorts</a> of <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 the <a href="msualg_1.html#U4">Sorts</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 the <a href="struct_0.html#U1">carrier</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> the <a href="closure1.html#U1" target="_self">Family</a> of <font color="Maroon">c<sub>2</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>set </b><font color="Maroon">c<sub>5</sub></font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_2_1"><b>now </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="E1:80_1_2_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="pboole.html#R1">in</a>  <a href="mssubfam.html#K7">bool</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E2:80_1_2_1"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is    <a href="pboole.html#M2">Element</a> of  <a href="mssubfam.html#K7">bool</a> the <a href="msualg_1.html#U4">Sorts</a> of <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>then consider </b><font color="Maroon">c<sub>7</sub></font> being  <a href="relat_1.html#V2">V5</a> <a href="mssubfam.html#NM1">MSSubsetFamily</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="E4:80_1_2_1"/><i><font color="Green">E35</font></i>: 
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> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="pboole.html#R1">in</a> <font color="Maroon">c<sub>7</sub></font> iff ( <font color="Olive">b<sub>1</sub></font> <a href="pboole.html#R1">in</a> the <a href="closure1.html#U1" target="_self">Family</a> of <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>6</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/><b>thus </b><font color="Maroon">c<sub>3</sub></font> <a href="closure1.html#K1" target="_self">..</a> <font color="Maroon">c<sub>6</sub></font> = 
 <a href="mssubfam.html#K6">meet</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="closure1.html#E1:80_1_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="closure1.html#E2:80_1_2_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="closure1.html#E4:80_1_2_1"><i><font color="Green">E35</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="closure1.html#K1" target="_self">..</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="closure1.html#E2:80_1_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="closure1.html#E2:80_1_2_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="closure1.html#E4:80_1_2_1"><i><font color="Green">E35</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:80_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="closure1.html#T7" target="_self">Th7</a></i>;<br/>


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