<?xml version="1.0"?>
<div class="add">

<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="pboole.html#M2">Element</a> of  <a href="mssubfam.html#K7">bool</a> <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="closure1.html#NM1">MSSetOp</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:74"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="closure1.html#K3" target="_self">..</a> <font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E2:74"/><i><font color="Green">E30</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>5</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>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="E4:74"/><i><font color="Green">E31</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>4</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="E5:74"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="closure1.html#K3" target="_self">..</a> <font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#R6">=</a>  <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#E2:74"><i><font color="Green">E30</font></i></a>, <a class="txt" href="closure1.html#E4:74"><i><font color="Green">E31</font></i></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>4</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E6:74"/>
( ( 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>
<b>hence </b><a NAME="E7:74"/>
<font color="Maroon">c<sub>4</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#E1:74"><i><font color="Green">E29</font></i></a>, <a class="txt" href="closure1.html#E4:74"><i><font color="Green">E31</font></i></a>, <a class="txt" href="closure1.html#E5:74"><i><font color="Green">E32</font></i></a>, <a class="ref" href="mssubfam.html#D6">MSSUBFAM:def 6</a></i>;<br/>


</div>
