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

<b>let </b><font color="Maroon">S</font> be    <a href="struct_0.html#L1">1-sorted</a> ;<br/><b>let </b><font color="Maroon">D</font> be   <a href="closure1.html#NM2">MSClosureSystem</a> of <font color="Maroon">S</font>;<br/>



<b>set </b><font color="Maroon">M</font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">D</font>;<br/>
<b>set </b><font color="Maroon">F</font> = the <a href="closure1.html#U1" target="_self">Family</a> of <font color="Maroon">D</font>;<br/>
<b>set </b><font color="Maroon">I</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>;<br/>
<b>consider </b><font color="Maroon">MS</font> being   <a href="mssubfam.html#NM1">MSSubsetFamily</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">D</font><b> such that </b><br/><a NAME="E2:82"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">MS</font> <a href="pboole.html#R6">=</a>  <a href="closure1.html#K6" target="_self">MSFixPoints</a> <span class="p1">(<span class="default"><a href="closure1.html#K8" target="_self">ClSys-&gt;ClOp</a> <font color="Maroon">D</font></span>)</span>
 <b>and </b><br/><a NAME="E3:82"/><i><font color="Green">E36</font></i>: 
 <a href="closure1.html#K7" target="_self">ClOp-&gt;ClSys</a> <span class="p1">(<span class="default"><a href="closure1.html#K8" target="_self">ClSys-&gt;ClOp</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="closure1.html#G1" target="_self">MSClosureStr</a>(# the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">D</font>,<font color="Maroon">MS</font> #)
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">X1</font> being    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E5:82"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">X1</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">D</font>
 <b>by </b><i><a class="ref" href="pboole.html#T146">PBOOLE:146</a></i>;<br/>
<a NAME="E6:82"/>
the <a href="closure1.html#U1" target="_self">Family</a> of <font color="Maroon">D</font> <a href="pboole.html#R6">=</a>  <a href="closure1.html#K6" target="_self">MSFixPoints</a> <span class="p1">(<span class="default"><a href="closure1.html#K8" target="_self">ClSys-&gt;ClOp</a> <font color="Maroon">D</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:82_1"/>
the <a href="closure1.html#U1" target="_self">Family</a> of <font color="Maroon">D</font> <a href="pboole.html#R2">c=</a>  <a href="closure1.html#K6" target="_self">MSFixPoints</a> <span class="p1">(<span class="default"><a href="closure1.html#K8" target="_self">ClSys-&gt;ClOp</a> <font color="Maroon">D</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D13">PBOOLE:def 13</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</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="E1:82_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 ;<br/>

<b>let </b><font color="Maroon">x</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="E2:82_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="closure1.html#U1" target="_self">Family</a> of <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>

<a NAME="E3:82_1_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
<b>by </b><i><a class="txt" href="closure1.html#E3:82"><i><font color="Green">E36</font></i></a>, <a class="ref" href="pzfmisc1.html#T1">PZFMISC1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">X</font> = <font color="Maroon">X1</font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">x</font></span>)</span> as    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E5:82_1_1"/><i><font color="Green">E40</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/>
<a NAME="E6:82_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>then </b><i><font color="Green">E42</font></i>: <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="txt" href="closure1.html#E1:82_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
;<br/>
<a NAME="E8:82_1_1"/>
the <a href="closure1.html#U1" target="_self">Family</a> of <font color="Maroon">D</font> <a href="pboole.html#R2">c=</a>  <a href="mssubfam.html#K7">bool</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E9:82_1_1"/><b>then </b><i><font color="Green">E57</font></i>: 
the <a href="closure1.html#U1" target="_self">Family</a> of <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="mssubfam.html#K7">bool</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">D</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="txt" href="closure1.html#E3:82"><i><font color="Green">E36</font></i></a>, <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>
<a NAME="E10:82_1_1"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="mssubfam.html#K7">bool</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">D</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="txt" href="closure1.html#E5:82"><i><font color="Green">E37</font></i></a></i>;<br/>
<a NAME="E11:82_1_1"/>
<font color="Maroon">X</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">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D17">PBOOLE:def 17</a><br/>

<b>assume </b><a NAME="E1:82_1_1_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:82_1_1_2_1"/>
( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> or <font color="Maroon">s</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:82_1_1_2_1_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:82_1_1_2_1_1"/>
<font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="mssubfam.html#K7">bool</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">D</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="txt" href="closure1.html#E5:82"><i><font color="Green">E37</font></i></a>, <a class="txt" href="closure1.html#E2:82_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="closure1.html#E5:82_1_1"><i><font color="Green">E40</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:82_1_1_2_1_2"/>
<font color="Maroon">s</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i</font>
 ;<br/><div class="add"><a NAME="E2:82_1_1_2_1_2"/><b>then </b>
not <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="closure1.html#E1:82_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:82_1_1_2_1_2"/><b>then </b>
<font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/><b>hence </b><a NAME="E4:82_1_1_2_1_2"/>
<font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="mssubfam.html#K7">bool</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">D</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 <b>by </b><i>, <a class="txt" href="closure1.html#E7:82_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="pboole.html#D4">PBOOLE:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">X</font> = <font color="Maroon">X</font> as    <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">D</font> ;<br/>
<b>reconsider </b><font color="Maroon">Fi</font> = the <a href="closure1.html#U1" target="_self">Family</a> of <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</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#E3:82"><i><font color="Green">E36</font></i></a>, <a class="ref" href="pboole.html#D16">PBOOLE:def 16</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <span class="p1">(<span class="default"><a href="closure1.html#K8" target="_self">ClSys-&gt;ClOp</a> <font color="Maroon">D</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><a href="mssubfam.html#K7">bool</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">D</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><a href="mssubfam.html#K7">bool</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">D</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <b>by </b><i><a class="txt" href="closure1.html#E3:82"><i><font color="Green">E36</font></i></a>, <a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>
<b>consider </b><font color="Maroon">SF</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">D</font><b> such that </b><br/><a NAME="E16:82_1_1"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">Y</font> being   <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> holds <br/> ( <font color="Olive">Y</font> <a href="pboole.html#R1">in</a> <font color="Maroon">SF</font> iff ( <font color="Olive">Y</font> <a href="pboole.html#R1">in</a> the <a href="closure1.html#U1" target="_self">Family</a> of <font color="Maroon">D</font> &amp; <font color="Maroon">X</font> <a href="pboole.html#R2">c=</a> <font color="Olive">Y</font> ) )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">Q</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 <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span><b> such that </b><br/><a NAME="E18:82_1_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">SF</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E19:82_1_1"/><i><font color="Green">E64</font></i>: 
<span class="p1">(<span class="default"><a href="mssubfam.html#K6">meet</a> <font color="Maroon">SF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="txt" href="closure1.html#E3:82"><i><font color="Green">E36</font></i></a>, <a class="ref" href="mssubfam.html#D2">MSSUBFAM:def 2</a></i>;<br/>
<a NAME="E20:82_1_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E21:82_1_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">SF</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Fi</font> : <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <font color="Olive">z</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="closure1.html#E3:82"><i><font color="Green">E36</font></i></a>, <a class="txt" href="closure1.html#E7:82_1_1"><i><font color="Green">E42</font></i></a>, </i>;<br/>
<a NAME="E22:82_1_1"/><i><font color="Green">E67</font></i>: 
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:82_1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">B</font> where <font color="Olive">B</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Fi</font> : <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Olive">B</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="closure1.html#E5:82"><i><font color="Green">E37</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:82_1_1_3"/>
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="closure1.html#E2:82_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="closure1.html#T1" target="_self">Th1</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="E3:82_1_1_3"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">Q</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#E3:82"><i><font color="Green">E36</font></i></a>, <a class="ref" href="closure1.html#T1" target="_self">Th1</a>, <a class="ref" href="pboole.html#D16">PBOOLE:def 16</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Z1</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:82_1_1_3_1"/>
<font color="Maroon">Z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 ;<br/><b>then consider </b><font color="Maroon">q</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Fi</font><b> such that </b><br/><a NAME="E3:82_1_1_3_1"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Z1</font> &amp; <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">q</font> )
 <b>by </b><i><a class="ref" href="closure1.html#T1" target="_self">Th1</a>, </i>;<br/><b>thus </b><a NAME="E4:82_1_1_3_1"/>
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z1</font>
 <b>by </b><i><a class="txt" href="closure1.html#E2:82_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="closure1.html#E3"><i><font color="Green">Lemma49</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:82_1_1_3"/>
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">Q</font>
 <b>by </b><i>, <a class="ref" href="mssubfam.html#T5">MSSUBFAM:5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E23:82_1_1"/><i><font color="Green">E70</font></i>: 
<span class="p1">(<span class="default"><a href="closure1.html#K8" target="_self">ClSys-&gt;ClOp</a> <font color="Maroon">D</font></span>)</span> <a href="closure1.html#K3" target="_self">..</a> <font color="Maroon">X</font> <a href="pboole.html#R6">=</a>  <a href="mssubfam.html#K6">meet</a> <font color="Maroon">SF</font>
 
<b>by </b><i><a class="txt" href="closure1.html#E7:82_1_1"><i><font color="Green">E42</font></i></a>, </i>;<br/>
<a NAME="E24:82_1_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="closure1.html#K8" target="_self">ClSys-&gt;ClOp</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E25:82_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="txt" href="closure1.html#E3:82"><i><font color="Green">E36</font></i></a>, <a class="txt" href="closure1.html#E2:82_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/>
<b>hence </b><a NAME="E26:82_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="closure1.html#K6" target="_self">MSFixPoints</a> <span class="p2">(<span class="default"><a href="closure1.html#K8" target="_self">ClSys-&gt;ClOp</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="closure1.html#E3:82"><i><font color="Green">E36</font></i></a>, , </i>;<br/>


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

<b>let </b><font color="Maroon">i</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="E2:82_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 ;<br/>

<b>let </b><font color="Maroon">x</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="E3:82_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="closure1.html#K6" target="_self">MSFixPoints</a> <span class="p2">(<span class="default"><a href="closure1.html#K8" target="_self">ClSys-&gt;ClOp</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>

<b>then consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:82_1"/><i><font color="Green">E132</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="closure1.html#K8" target="_self">ClSys-&gt;ClOp</a> <font color="Maroon">D</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="txt" href="closure1.html#E3:82"><i><font color="Green">E36</font></i></a>, </i>;<br/>
<a NAME="E6:82_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
<b>by </b><i><a class="txt" href="closure1.html#E3:82"><i><font color="Green">E36</font></i></a>, <a class="ref" href="pzfmisc1.html#T1">PZFMISC1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">X</font> = <font color="Maroon">X1</font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">x</font></span>)</span> as    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E8:82_1"/><i><font color="Green">E133</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/>
<a NAME="E9:82_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>then </b><i><font color="Green">E134</font></i>: <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
;<br/>
<a NAME="E11:82_1"/>
 <a href="closure1.html#K6" target="_self">MSFixPoints</a> <span class="p1">(<span class="default"><a href="closure1.html#K8" target="_self">ClSys-&gt;ClOp</a> <font color="Maroon">D</font></span>)</span> <a href="pboole.html#R2">c=</a>  <a href="mssubfam.html#K7">bool</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E12:82_1"/><b>then </b><i><font color="Green">E135</font></i>: 
<span class="p1">(<span class="default"><a href="closure1.html#K6" target="_self">MSFixPoints</a> <span class="p2">(<span class="default"><a href="closure1.html#K8" target="_self">ClSys-&gt;ClOp</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="mssubfam.html#K7">bool</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">D</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="txt" href="closure1.html#E3:82"><i><font color="Green">E36</font></i></a>, <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>
<a NAME="E13:82_1"/>
<font color="Maroon">X</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">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D17">PBOOLE:def 17</a><br/>

<b>assume </b><a NAME="E1:82_1_3"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:82_1_3_1"/>
( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> or <font color="Maroon">s</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_3_1_1"><b>suppose </b></a><a NAME="E1:82_1_3_1_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:82_1_3_1_1"/>
<font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="mssubfam.html#K7">bool</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">D</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="txt" href="closure1.html#E5:82"><i><font color="Green">E37</font></i></a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_3_1_2"><b>suppose </b></a><a NAME="E1:82_1_3_1_2"/>
<font color="Maroon">s</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i</font>
 ;<br/><div class="add"><a NAME="E2:82_1_3_1_2"/><b>then </b>
not <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:82_1_3_1_2"/><b>then </b>
<font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/><b>hence </b><a NAME="E4:82_1_3_1_2"/>
<font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="mssubfam.html#K7">bool</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">D</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 <b>by </b><i>, , <a class="ref" href="pboole.html#D4">PBOOLE:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">X</font> = <font color="Maroon">X</font> as    <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">D</font> ;<br/>
<b>consider </b><font color="Maroon">SF</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">D</font><b> such that </b><br/><a NAME="E16:82_1"/><i><font color="Green">E137</font></i>: 
for <font color="Olive">Y</font> being   <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> holds <br/> ( <font color="Olive">Y</font> <a href="pboole.html#R1">in</a> <font color="Maroon">SF</font> iff ( <font color="Olive">Y</font> <a href="pboole.html#R1">in</a> the <a href="closure1.html#U1" target="_self">Family</a> of <font color="Maroon">D</font> &amp; <font color="Maroon">X</font> <a href="pboole.html#R2">c=</a> <font color="Olive">Y</font> ) )
 <b>by </b><i/>;<br/>
<a NAME="E17:82_1"/><i><font color="Green">E138</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="closure1.html#K8" target="_self">ClSys-&gt;ClOp</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<i><font color="Green">E139</font></i>: <span class="p1">(<span class="default"><a href="mssubfam.html#K6">meet</a> <font color="Maroon">SF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="closure1.html#K8" target="_self">ClSys-&gt;ClOp</a> <font color="Maroon">D</font></span>)</span> <a href="closure1.html#K3" target="_self">..</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="closure1.html#E3:82"><i><font color="Green">E36</font></i></a>, , , <a class="ref" href="closure1.html#T1" target="_self">Th1</a>, <a class="ref" href="pralg_1.html#D17">PRALG_1:def 17</a></i>
;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>] <b>means </b><font color="Maroon">X</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E19:82_1"/>
( ( for <font color="Olive">Y</font> being   <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> holds <br/> ( <font color="Olive">Y</font> <a href="pboole.html#R1">in</a> <font color="Maroon">SF</font> iff ( <font color="Olive">Y</font> <a href="pboole.html#R1">in</a> the <a href="closure1.html#U1" target="_self">Family</a> of <font color="Maroon">D</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">Y</font>] ) ) ) implies <font color="Maroon">SF</font> <a href="pboole.html#R2">c=</a> the <a href="closure1.html#U1" target="_self">Family</a> of <font color="Maroon">D</font> )
 
<b>from </b><i><a class="ref" href="closure1.html#S1" target="_self">CLOSURE1:sch 1</a>();<br/></i>
<a NAME="E20:82_1"/><b>then </b>
 <a href="mssubfam.html#K6">meet</a> <font color="Maroon">SF</font> <a href="pboole.html#R1">in</a> the <a href="closure1.html#U1" target="_self">Family</a> of <font color="Maroon">D</font>
 
<b>by </b><i>, <a class="ref" href="mssubfam.html#D6">MSSUBFAM:def 6</a></i>;<br/>
<b>hence </b><a NAME="E21:82_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="closure1.html#U1" target="_self">Family</a> of <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="closure1.html#E3:82"><i><font color="Green">E36</font></i></a>, , <a class="ref" href="pboole.html#D4">PBOOLE:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:82"/>
 <a href="closure1.html#K7" target="_self">ClOp-&gt;ClSys</a> <span class="p1">(<span class="default"><a href="closure1.html#K8" target="_self">ClSys-&gt;ClOp</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="closure1.html#G1" target="_self">MSClosureStr</a>(# the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">D</font>,the <a href="closure1.html#U1" target="_self">Family</a> of <font color="Maroon">D</font> #)
 <b>by </b><i>, <a class="txt" href="closure1.html#E2:82"><i><font color="Green">E34</font></i></a></i>;<br/>


</div>
