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

<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#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><a NAME="E1:98"/><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>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>  <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>4</sub></font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E2:98"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="closure2.html#V7" target="_self">reflexive</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</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>; <i><font color="Red">:: according to </font></i><a class="ref" href="closure2.html#D12" target="_self">CLOSURE2:def 12</a><br/>

<b>consider </b><font color="Maroon">c<sub>6</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="E2:98_1"/><i><font color="Green">E42</font></i>: 
 <a href="closure2.html#K11" target="_self">Cl</a> <font color="Maroon">c<sub>5</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>6</sub></font></span><a href="closure2.html#K5" target="_self">:|</a></span>
 <b>and </b><br/><a NAME="E3:98_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>6</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>5</sub></font> <a href="closure2.html#NR2" target="_self">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="E4:98_1"/><i><font color="Green">E44</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> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="closure2.html#NR2" target="_self">c='</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</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="E1:98_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>8</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><b> such that </b><br/><a NAME="E3:98_1_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="closure2.html#NR2" target="_self">c='</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>and </b><br/><a NAME="E4:98_1_1"/>
<font color="Maroon">c<sub>8</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#E3:98_1"><i><font color="Green">E43</font></i></a></i>;<br/>
<b>thus </b><a NAME="E5:98_1_1"/>
<font color="Maroon">c<sub>5</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="txt" href="closure2.html#E3:98_1_1"><i><font color="Green">E45</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:98_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="closure2.html#K7" target="_self">.</a> <font color="Maroon">c<sub>5</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>6</sub></font></span><a href="closure2.html#K5" target="_self">:|</a></span>
 
<b>by </b><i><a class="txt" href="closure2.html#E1:98"><i><font color="Green">E40</font></i></a>, <a class="txt" href="closure2.html#E2:98_1"><i><font color="Green">E42</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:98_1"/>
<font color="Maroon">c<sub>5</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>5</sub></font>
 <b>by </b><i><a class="txt" href="closure2.html#E4:98_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="closure2.html#T25" target="_self">Th25</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:98"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="closure2.html#V8" target="_self">monotonic</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</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>; <i><font color="Red">:: according to </font></i><a class="ref" href="closure2.html#D13" target="_self">CLOSURE2:def 13</a><br/>

<b>assume </b><a NAME="E1:98_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="closure2.html#NR2" target="_self">c='</a> <font color="Maroon">c<sub>6</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="E3:98_2"/><i><font color="Green">E44</font></i>: 
 <a href="closure2.html#K11" target="_self">Cl</a> <font color="Maroon">c<sub>5</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="E4:98_2"/><i><font color="Green">E45</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>5</sub></font> <a href="closure2.html#NR2" target="_self">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/>
<b>consider </b><font color="Maroon">c<sub>8</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="E6:98_2"/><i><font color="Green">E46</font></i>: 
 <a href="closure2.html#K11" target="_self">Cl</a> <font color="Maroon">c<sub>6</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>8</sub></font></span><a href="closure2.html#K5" target="_self">:|</a></span>
 <b>and </b><br/><a NAME="E7:98_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>8</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>6</sub></font> <a href="closure2.html#NR2" target="_self">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="E8:98_2"/>
<span class="p1"><a href="closure2.html#K5" target="_self">|:</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="closure2.html#K5" target="_self">:|</a></span> <a href="closure2.html#NR2" target="_self">c='</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a><br/>

<b>assume </b><a NAME="E1:98_2_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>thus </b><a NAME="E2:98_2_1"/>
<span class="p1"><a href="closure2.html#K5" target="_self">|:</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="closure2.html#K5" target="_self">:|</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</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> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:98_2_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="closure2.html#K5" target="_self">|:</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="closure2.html#K5" target="_self">:|</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:98_2_1_1_1"/>
( <font color="Maroon">c<sub>8</sub></font> is <a href="xboole_0.html#V1">empty</a> or  not <font color="Maroon">c<sub>8</sub></font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:98_2_1_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>8</sub></font> as  <a href="xboole_0.html#V1">empty</a> <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> ;<br/><a NAME="E3:98_2_1_1_1_1"/>
<span class="p1"><a href="closure2.html#K5" target="_self">|:</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="closure2.html#K5" target="_self">:|</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="txt" href="closure2.html#E1:98_2_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="pboole.html#D15">PBOOLE:def 15</a></i>;<br/><b>hence </b><a NAME="E4:98_2_1_1_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</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> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="closure2.html#E1:98_2_1_1"><i><font color="Green">E49</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:98_2_1_1_1_2"/>
not <font color="Maroon">c<sub>8</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:98_2_1_1_1_2"/><b>then </b>
<span class="p1"><a href="closure2.html#K5" target="_self">|:</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="closure2.html#K5" target="_self">:|</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> 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="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="closure2.html#E1:98_2_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="closure2.html#T15" target="_self">Th15</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>11</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><b> such that </b><br/><a NAME="E4:98_2_1_1_1_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E5:98_2_1_1_1_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="closure2.html#E1:98_2_1_1"><i><font color="Green">E49</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>12</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><b> such that </b><br/><a NAME="E7:98_2_1_1_1_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>and </b><br/><a NAME="E8:98_2_1_1_1_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="closure2.html#NR2" target="_self">c='</a> <font color="Maroon">c<sub>12</sub></font>
 <b>and </b><br/><a NAME="E9:98_2_1_1_1_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>12</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#E7:98_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="closure2.html#E5:98_2_1_1_1_2"><i><font color="Green">E51</font></i></a></i>;<br/><a NAME="E10:98_2_1_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="closure2.html#NR2" target="_self">c='</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="closure2.html#E1:98_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="closure2.html#E7:98_2_1_1_1_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="closure2.html#E8:98_2_1_1_1_2"><i><font color="Green">E53</font></i></a>, <a class="ref" href="pboole.html#T15">PBOOLE:15</a></i>;<br/><a NAME="E11:98_2_1_1_1_2"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="closure2.html#E4:98_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="closure2.html#E7:98_2_1_1_1_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="closure2.html#E9:98_2_1_1_1_2"><i><font color="Green">E54</font></i></a></i>;<br/><a NAME="E12:98_2_1_1_1_2"/><b>then </b>
<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> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> 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="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="closure2.html#E1:98_2_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="closure2.html#T15" target="_self">Th15</a></i>;<br/><b>hence </b><a NAME="E13:98_2_1_1_1_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</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> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="closure2.html#E4:98_2_1_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="closure2.html#E11:98_2_1_1_1_2"><i><font color="Green">E55</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E9:98_2"/><b>then </b>
 <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> <a href="closure2.html#NR2" target="_self">c='</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>8</sub></font></span><a href="closure2.html#K5" target="_self">:|</a></span>
 
<b>by </b><i><a class="ref" href="mssubfam.html#T46">MSSUBFAM:46</a></i>;<br/>
<a NAME="E10:98_2"/><b>then </b>
 <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> <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>6</sub></font>
 
<b>by </b><i><a class="txt" href="closure2.html#E1:98"><i><font color="Green">E40</font></i></a>, <a class="txt" href="closure2.html#E6:98_2"><i><font color="Green">E46</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:98_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="closure2.html#K7" target="_self">.</a> <font color="Maroon">c<sub>5</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>6</sub></font>
 <b>by </b><i><a class="txt" href="closure2.html#E1:98"><i><font color="Green">E40</font></i></a>, <a class="txt" href="closure2.html#E3:98_2"><i><font color="Green">E44</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:98"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="closure2.html#V9" target="_self">idempotent</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</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>; <i><font color="Red">:: according to </font></i><a class="ref" href="closure2.html#D14" target="_self">CLOSURE2:def 14</a><br/>

<b>consider </b><font color="Maroon">c<sub>6</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="E2:98_3"/><i><font color="Green">E43</font></i>: 
 <a href="closure2.html#K11" target="_self">Cl</a> <font color="Maroon">c<sub>5</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>6</sub></font></span><a href="closure2.html#K5" target="_self">:|</a></span>
 <b>and </b><br/><a NAME="E3:98_3"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>6</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>5</sub></font> <a href="closure2.html#NR2" target="_self">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="E4:98_3"/>
<font color="Maroon">c<sub>6</sub></font> <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:98_3_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>8</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><b> such that </b><br/><a NAME="E3:98_3_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="closure2.html#NR2" target="_self">c='</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</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#E3:98_3"><i><font color="Green">E44</font></i></a></i>;<br/>
<b>thus </b><a NAME="E4:98_3_1"/>
<font color="Maroon">c<sub>7</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#E3:98_3_1"><i><font color="Green">E45</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:98_3"/><b>then </b><i><font color="Green">E45</font></i>: 
 <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> 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="ref" href="closure2.html#D8" target="_self">Def8</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>3</sub></font> <a href="closure2.html#K7" target="_self">.</a> <font color="Maroon">c<sub>5</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>
<b>by </b><i><a class="txt" href="closure2.html#E1:98"><i><font color="Green">E40</font></i></a>, <a class="txt" href="closure2.html#E2:98_3"><i><font color="Green">E43</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="mssubfam.html#K6">meet</a> <span class="p2"><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></span>)</span>
<b>by </b><i><a class="txt" href="closure2.html#E1:98"><i><font color="Green">E40</font></i></a>, <a class="txt" href="closure2.html#E5:98_3"><i><font color="Green">E45</font></i></a>, <a class="ref" href="closure2.html#T40" target="_self">Th40</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="closure2.html#K7" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="closure2.html#K7" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="closure2.html#E1:98"><i><font color="Green">E40</font></i></a>, <a class="txt" href="closure2.html#E2:98_3"><i><font color="Green">E43</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:98"/>
( <font color="Maroon">c<sub>3</sub></font> is <a href="closure2.html#V7" target="_self">reflexive</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="closure2.html#V8" target="_self">monotonic</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="closure2.html#V9" target="_self">idempotent</a> )
 <b>by </b><i><a class="txt" href="closure2.html#E2:98"><i><font color="Green">E41</font></i></a>, <a class="txt" href="closure2.html#E3:98"><i><font color="Green">E42</font></i></a></i>;<br/>


</div>
