<?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="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="closure1.html#NM3">MSClosureOperator</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>





<b>set </b><font color="Maroon">c<sub>4</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="closure1.html#K7" target="_self">ClOp-&gt;ClSys</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="closure1.html#K8" target="_self">ClSys-&gt;ClOp</a> <span class="p1">(<span class="default"><a href="closure1.html#K7" target="_self">ClOp-&gt;ClSys</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="mssubfam.html#NM1">MSSubsetFamily</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E2:81"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="pboole.html#R6">=</a>  <a href="closure1.html#K6" target="_self">MSFixPoints</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E3:81"/><i><font color="Green">E34</font></i>: 
 <a href="closure1.html#K7" target="_self">ClOp-&gt;ClSys</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="closure1.html#G1" target="_self">MSClosureStr</a>(# <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font> #)
 <b>by </b><i><a class="ref" href="closure1.html#D14" target="_self">Def14</a></i>;<br/>
<a NAME="E4:81"/><i><font color="Green">E35</font></i>: 
the <a href="closure1.html#U1" target="_self">Family</a> of <span class="p1">(<span class="default"><a href="closure1.html#K7" target="_self">ClOp-&gt;ClSys</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="closure1.html#E3:81"><i><font color="Green">E34</font></i></a></i>;<br/>
<a NAME="E5:81"/>
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="pboole.html#R1">in</a>  <a href="mssubfam.html#K7">bool</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><span class="p1">(<span class="default"><a href="closure1.html#K8" target="_self">ClSys-&gt;ClOp</a> <span class="p2">(<span class="default"><a href="closure1.html#K7" target="_self">ClOp-&gt;ClSys</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="closure1.html#K1" target="_self">..</a> <font color="Olive">b<sub>1</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="closure1.html#K1" target="_self">..</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1"><b>proof </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="E1:81_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="pboole.html#R1">in</a>  <a href="mssubfam.html#K7">bool</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:81_1"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>8</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 <span class="p1">(<span class="default"><a href="closure1.html#K7" target="_self">ClOp-&gt;ClSys</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="closure1.html#E3:81"><i><font color="Green">E34</font></i></a>, <a class="ref" href="mssubfam.html#T11">MSSUBFAM:11</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>9</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 <span class="p1">(<span class="default"><a href="closure1.html#K7" target="_self">ClOp-&gt;ClSys</a> <font color="Maroon">c<sub>3</sub></font></span>)</span><b> such that </b><br/><a NAME="E4:81_1"/><i><font color="Green">E37</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>9</sub></font> iff ( <font color="Olive">b<sub>1</sub></font> <a href="pboole.html#R1">in</a>  <a href="closure1.html#K6" target="_self">MSFixPoints</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="pboole.html#R2">c=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="closure1.html#E2:81"><i><font color="Green">E33</font></i></a>, <a class="txt" href="closure1.html#E4:81"><i><font color="Green">E35</font></i></a>, <a class="ref" href="closure1.html#T31" target="_self">Th31</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:81_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>10</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/><a NAME="E2:81_1_1"/><i><font color="Green">E39</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><a href="mssubfam.html#K7">bool</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>,<span class="p1">(<span class="default"><a href="mssubfam.html#K7">bool</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <b>by </b><i><a class="txt" href="closure1.html#E1:81_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/><a NAME="E4:81_1_1"/>
<span class="p1">(<span class="default"><a href="mssubfam.html#K7">bool</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="closure1.html#E1:81_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="mboolean.html#D1">MBOOLEAN:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>, <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> ;<br/><b>consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="setfam_1.html#NM1">Subset-Family</a> of <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><a href="closure1.html#K7" target="_self">ClOp-&gt;ClSys</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span><b> such that </b><br/><a NAME="E7:81_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>
 <b>and </b><br/><a NAME="E8:81_1_1"/><i><font color="Green">E41</font></i>: 
<span class="p1">(<span class="default"><a href="mssubfam.html#K6">meet</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="closure1.html#E1:81_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="mssubfam.html#D2">MSSUBFAM:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <span class="p1">(<span class="default"><a href="closure1.html#K6" target="_self">MSFixPoints</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="closure1.html#E2:81"><i><font color="Green">E33</font></i></a>, <a class="txt" href="closure1.html#E4:81"><i><font color="Green">E35</font></i></a>, <a class="txt" href="closure1.html#E1:81_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="pboole.html#D16">PBOOLE:def 16</a></i>;<br/><a NAME="E10:81_1_1"/>
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:81_1_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</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="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>14</sub></font> : <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="closure1.html#E2:81"><i><font color="Green">E33</font></i></a>, <a class="txt" href="closure1.html#E4:81"><i><font color="Green">E35</font></i></a>, <a class="txt" href="closure1.html#E2:81_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="closure1.html#E4:81_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="closure1.html#E1:81_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="closure1.html#T32" target="_self">Th32</a></i>;<br/>
<a NAME="E2:81_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="mssubfam.html#K7">bool</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="closure1.html#E3:81"><i><font color="Green">E34</font></i></a>, <a class="txt" href="closure1.html#E2:81_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="closure1.html#E1:81_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="pboole.html#D17">PBOOLE:def 17</a></i>;<br/>
<a NAME="E3:81_1_1_1"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> is    <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="closure1.html#E1:81_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="mboolean.html#D1">MBOOLEAN:def 1</a></i>;<br/>
<a NAME="E4:81_1_1_1"/><i><font color="Green">E44</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:81_1_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="closure1.html#E3:81_1_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<a NAME="E6:81_1_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="closure1.html#E1:81_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="closure1.html#E3:81_1_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="closure1.html#T26" target="_self">Th26</a></i>;<br/>
<a NAME="E7:81_1_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="closure1.html#E1:81_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="closure1.html#E4:81_1_1_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="closure1.html#E5:81_1_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="closure1.html#D13" target="_self">Def13</a></i>;<br/>
<a NAME="E8:81_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="closure1.html#E1:81_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="closure1.html#E3:81_1_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="closure1.html#T24" target="_self">Th24</a></i>;<br/>
<a NAME="E9:81_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>14</sub></font> : <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="closure1.html#E7:81_1_1_1"><i><font color="Green">E46</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:81_1_1_1"/>
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="closure1.html#E7:81_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="closure1.html#E1:81_1_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="mssubfam.html#T2">MSSUBFAM:2</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<a NAME="E11:81_1_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="closure1.html#E1:81_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="closure1.html#E7:81_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="pboole.html#D16">PBOOLE:def 16</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>15</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:81_1_1_1_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>16</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>14</sub></font><b> such that </b><br/><a NAME="E3:81_1_1_1_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="closure1.html#E7:81_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="closure1.html#E1:81_1_1_1"><i><font color="Green">E42</font></i></a></i>;<br/><a NAME="E4:81_1_1_1_1"/>
 <a href="closure1.html#K6" target="_self">MSFixPoints</a> <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/><a NAME="E5:81_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="mssubfam.html#K7">bool</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="closure1.html#E1:81_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/><a NAME="E6:81_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="closure1.html#E1:81_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="mboolean.html#D1">MBOOLEAN:def 1</a></i>;<br/><a NAME="E7:81_1_1_1_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>16</sub></font> is    <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>17</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E9:81_1_1_1_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="closure1.html#E1:81_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="closure1.html#D13" target="_self">Def13</a></i>;<br/><b>thus </b><a NAME="E10:81_1_1_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="closure1.html#E1:81_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="closure1.html#E3:81_1_1_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="closure1.html#E3:81_1_1_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="closure1.html#E7:81_1_1_1_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="closure1.html#E9:81_1_1_1_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="closure1.html#T25" target="_self">Th25</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E13:81_1_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="closure1.html#E11:81_1_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="mssubfam.html#T5">MSSUBFAM:5</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="closure1.html#K8" target="_self">ClSys-&gt;ClOp</a> <span class="p3">(<span class="default"><a href="closure1.html#K7" target="_self">ClOp-&gt;ClSys</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="closure1.html#K1" target="_self">..</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> = 
<font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="closure1.html#E2:81"><i><font color="Green">E33</font></i></a>, <a class="txt" href="closure1.html#E3:81"><i><font color="Green">E34</font></i></a>, <a class="txt" href="closure1.html#E2:81_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="closure1.html#E4:81_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="closure1.html#E8:81_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="closure1.html#D15" target="_self">Def15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="closure1.html#K1" target="_self">..</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="closure1.html#E1:81_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="closure1.html#E2:81_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="pralg_1.html#D17">PRALG_1:def 17</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:81_1"/>
<span class="p1">(<span class="default"><a href="closure1.html#K8" target="_self">ClSys-&gt;ClOp</a> <span class="p2">(<span class="default"><a href="closure1.html#K7" target="_self">ClOp-&gt;ClSys</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="closure1.html#K1" target="_self">..</a> <font color="Maroon">c<sub>8</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="closure1.html#K1" target="_self">..</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:81"/>
 <a href="closure1.html#K8" target="_self">ClSys-&gt;ClOp</a> <span class="p1">(<span class="default"><a href="closure1.html#K7" target="_self">ClOp-&gt;ClSys</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="closure1.html#E3:81"><i><font color="Green">E34</font></i></a>, <a class="ref" href="closure1.html#T7" target="_self">Th7</a></i>;<br/>


</div>
