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

<b>let </b><font color="Maroon" title="c1">S</font> be    <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</a> ;<br/><b>let </b><font color="Maroon" title="c2">A</font> be    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>;<br/><b>let </b><font color="Maroon" title="c3">J</font> be   <a href="closure1.html#NM3" title="CLOSURE1:NM.3">MSClosureOperator</a> of <font color="Maroon" title="c2">A</font>;<br/>





<b>set </b><font color="Maroon" title="c4">I</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>;<br/>
<b>set </b><font color="Maroon" title="c5">M</font> = the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="closure1.html#K7" title="CLOSURE1:func.7">ClOp-&gt;ClSys</a> <font color="Maroon" title="c3">J</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c6">j</font> =  <a href="closure1.html#K8" title="CLOSURE1:func.8">ClSys-&gt;ClOp</a> <span class="p1">(<span class="default"><a href="closure1.html#K7" title="CLOSURE1:func.7">ClOp-&gt;ClSys</a> <font color="Maroon" title="c3">J</font></span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c7">D</font> being   <a href="mssubfam.html#NM1" title="MSSUBFAM:NM.1">MSSubsetFamily</a> of <font color="Maroon" title="c2">A</font><b> such that </b><br/><a NAME="E2:81"/><i><font color="Green" title="E33">A1</font></i>: 
<font color="Maroon" title="c7">D</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="closure1.html#K6" title="CLOSURE1:func.6">MSFixPoints</a> <font color="Maroon" title="c3">J</font>
 <b>and </b><br/><a NAME="E3:81"/><i><font color="Green" title="E34">A2</font></i>: 
 <a href="closure1.html#K7" title="CLOSURE1:func.7">ClOp-&gt;ClSys</a> <font color="Maroon" title="c3">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="closure1.html#G1" title="CLOSURE1:aggr.1">MSClosureStr</a>(# <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c7">D</font> #)
 <b>by </b><i><a class="ref" href="closure1.html#D14" target="_self" title="CLOSURE1:def.14">Def14</a></i>;<br/>
<a NAME="E4:81"/><i><font color="Green" title="E35">A3</font></i>: 
the <a href="closure1.html#U1" title="CLOSURE1:sel.1">Family</a> of <span class="p1">(<span class="default"><a href="closure1.html#K7" title="CLOSURE1:func.7">ClOp-&gt;ClSys</a> <font color="Maroon" title="c3">J</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c7">D</font>
 
<b>by </b><i><a class="txt" href="closure1.html#E3:81"><i><font color="Green" title="E34">A2</font></i></a></i>;<br/>
<a NAME="E5:81"/>
for <font color="Olive" title="b1">X</font> being   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b1">X</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a>  <a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">A</font> holds <br/><span class="p1">(<span class="default"><a href="closure1.html#K8" title="CLOSURE1:func.8">ClSys-&gt;ClOp</a> <span class="p2">(<span class="default"><a href="closure1.html#K7" title="CLOSURE1:func.7">ClOp-&gt;ClSys</a> <font color="Maroon" title="c3">J</font></span>)</span></span>)</span> <a href="closure1.html#K1" title="CLOSURE1:func.1">..</a> <font color="Olive" title="b1">X</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">J</font> <a href="closure1.html#K1" title="CLOSURE1:func.1">..</a> <font color="Olive" title="b1">X</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" title="c8">X</font> be    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>;<br/>

<b>assume </b><a NAME="E1:81_1"/>
<font color="Maroon" title="c8">X</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a>  <a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">A</font>
 ;<br/>

<a NAME="E2:81_1"/><b>then </b><i><font color="Green" title="E36">A4</font></i>: 
<font color="Maroon" title="c8">X</font> is    <a href="pboole.html#M2" title="PBOOLE:mode.2">Element</a> of  <a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="closure1.html#K7" title="CLOSURE1:func.7">ClOp-&gt;ClSys</a> <font color="Maroon" title="c3">J</font></span>)</span>
 
<b>by </b><i><a class="txt" href="closure1.html#E3:81"><i><font color="Green" title="E34">A2</font></i></a>, <a class="ref" href="mssubfam.html#T11" title="MSSUBFAM:th.11">MSSUBFAM:11</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">SF</font> being  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V5</a> <a href="mssubfam.html#NM1" title="MSSUBFAM:NM.1">MSSubsetFamily</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="closure1.html#K7" title="CLOSURE1:func.7">ClOp-&gt;ClSys</a> <font color="Maroon" title="c3">J</font></span>)</span><b> such that </b><br/><a NAME="E4:81_1"/><i><font color="Green" title="E37">A5</font></i>: 
for <font color="Olive" title="b1">Y</font> being   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> holds <br/> ( <font color="Olive" title="b1">Y</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c9">SF</font> iff ( <font color="Olive" title="b1">Y</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a>  <a href="closure1.html#K6" title="CLOSURE1:func.6">MSFixPoints</a> <font color="Maroon" title="c3">J</font> &amp; <font color="Maroon" title="c8">X</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b1">Y</font> ) )
 <b>by </b><i><a class="txt" href="closure1.html#E2:81"><i><font color="Green" title="E33">A1</font></i></a>, <a class="txt" href="closure1.html#E4:81"><i><font color="Green" title="E35">A3</font></i></a>, <a class="ref" href="closure1.html#T31" target="_self" title="CLOSURE1:th.31">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" title="c10">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:81_1_1"/><i><font color="Green" title="E38">A6</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>
 ;<br/><a NAME="E2:81_1_1"/><i><font color="Green" title="E39">A7</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">f</font> = <font color="Maroon" title="c3">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font>,<span class="p1">(<span class="default"><a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <b>by </b><i><a class="txt" href="closure1.html#E1:81_1_1"><i><font color="Green" title="E38">A6</font></i></a>, <a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></i>;<br/><a NAME="E4:81_1_1"/>
<span class="p1">(<span class="default"><a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="closure1.html#E1:81_1_1"><i><font color="Green" title="E38">A6</font></i></a>, <a class="ref" href="mboolean.html#D1" title="MBOOLEAN:def.1">MBOOLEAN:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c12">f</font> = <font color="Maroon" title="c11">f</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span>, <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span> ;<br/><b>consider </b><font color="Maroon" title="c13">Q</font> being   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p2">(<span class="default"><a href="closure1.html#K7" title="CLOSURE1:func.7">ClOp-&gt;ClSys</a> <font color="Maroon" title="c3">J</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span><b> such that </b><br/><a NAME="E7:81_1_1"/><i><font color="Green" title="E40">A8</font></i>: 
<font color="Maroon" title="c13">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">SF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font>
 <b>and </b><br/><a NAME="E8:81_1_1"/><i><font color="Green" title="E41">A9</font></i>: 
<span class="p1">(<span class="default"><a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <font color="Maroon" title="c9">SF</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c13">Q</font>
 <b>by </b><i><a class="txt" href="closure1.html#E1:81_1_1"><i><font color="Green" title="E38">A6</font></i></a>, <a class="ref" href="mssubfam.html#D2" title="MSSUBFAM:def.2">MSSUBFAM:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c14">Di</font> = <span class="p1">(<span class="default"><a href="closure1.html#K6" title="CLOSURE1:func.6">MSFixPoints</a> <font color="Maroon" title="c3">J</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="closure1.html#E2:81"><i><font color="Green" title="E33">A1</font></i></a>, <a class="txt" href="closure1.html#E4:81"><i><font color="Green" title="E35">A3</font></i></a>, <a class="txt" href="closure1.html#E1:81_1_1"><i><font color="Green" title="E38">A6</font></i></a>, <a class="ref" href="pboole.html#D16" title="PBOOLE:def.16">PBOOLE:def 16</a></i>;<br/><a NAME="E10:81_1_1"/>
 <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c13">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</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" title="E42">A10</font></i>: 
<font color="Maroon" title="c9">SF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c14">Di</font> : <font color="Maroon" title="c8">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">z</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="closure1.html#E2:81"><i><font color="Green" title="E33">A1</font></i></a>, <a class="txt" href="closure1.html#E4:81"><i><font color="Green" title="E35">A3</font></i></a>, <a class="txt" href="closure1.html#E2:81_1"><i><font color="Green" title="E36">A4</font></i></a>, <a class="txt" href="closure1.html#E4:81_1"><i><font color="Green" title="E37">A5</font></i></a>, <a class="txt" href="closure1.html#E1:81_1_1"><i><font color="Green" title="E38">A6</font></i></a>, <a class="ref" href="closure1.html#T32" target="_self" title="CLOSURE1:th.32">Th32</a></i>;<br/>
<a NAME="E2:81_1_1_1"/>
<font color="Maroon" title="c8">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font>
 
<b>by </b><i><a class="txt" href="closure1.html#E3:81"><i><font color="Green" title="E34">A2</font></i></a>, <a class="txt" href="closure1.html#E2:81_1"><i><font color="Green" title="E36">A4</font></i></a>, <a class="txt" href="closure1.html#E1:81_1_1"><i><font color="Green" title="E38">A6</font></i></a>, <a class="ref" href="pboole.html#D17" title="PBOOLE:def.17">PBOOLE:def 17</a></i>;<br/>
<a NAME="E3:81_1_1_1"/><b>then </b><i><font color="Green" title="E43">A11</font></i>: 
<font color="Maroon" title="c8">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="closure1.html#E1:81_1_1"><i><font color="Green" title="E38">A6</font></i></a>, <a class="ref" href="mboolean.html#D1" title="MBOOLEAN:def.1">MBOOLEAN:def 1</a></i>;<br/>
<a NAME="E4:81_1_1_1"/><i><font color="Green" title="E44">A12</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c12">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:81_1_1_1"/><i><font color="Green" title="E45">A13</font></i>: 
<font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="closure1.html#E3:81_1_1_1"><i><font color="Green" title="E43">A11</font></i></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/>
<a NAME="E6:81_1_1_1"/>
<font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="closure1.html#E1:81_1_1"><i><font color="Green" title="E38">A6</font></i></a>, <a class="txt" href="closure1.html#E3:81_1_1_1"><i><font color="Green" title="E43">A11</font></i></a>, <a class="ref" href="closure1.html#T26" target="_self" title="CLOSURE1:th.26">Th26</a></i>;<br/>
<a NAME="E7:81_1_1_1"/><b>then </b><i><font color="Green" title="E46">A14</font></i>: 
<font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c14">Di</font>
 
<b>by </b><i><a class="txt" href="closure1.html#E1:81_1_1"><i><font color="Green" title="E38">A6</font></i></a>, <a class="txt" href="closure1.html#E4:81_1_1_1"><i><font color="Green" title="E44">A12</font></i></a>, <a class="txt" href="closure1.html#E5:81_1_1_1"><i><font color="Green" title="E45">A13</font></i></a>, <a class="ref" href="closure1.html#D13" target="_self" title="CLOSURE1:def.13">Def13</a></i>;<br/>
<a NAME="E8:81_1_1_1"/>
<font color="Maroon" title="c8">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="closure1.html#E1:81_1_1"><i><font color="Green" title="E38">A6</font></i></a>, <a class="txt" href="closure1.html#E3:81_1_1_1"><i><font color="Green" title="E43">A11</font></i></a>, <a class="ref" href="closure1.html#T24" target="_self" title="CLOSURE1:th.24">Th24</a></i>;<br/>
<a NAME="E9:81_1_1_1"/><b>then </b>
<font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c14">Di</font> : <font color="Maroon" title="c8">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">z</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="closure1.html#E7:81_1_1_1"><i><font color="Green" title="E46">A14</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:81_1_1_1"/>
 <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c13">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="closure1.html#E7:81_1_1"><i><font color="Green" title="E40">A8</font></i></a>, <a class="txt" href="closure1.html#E1:81_1_1_1"><i><font color="Green" title="E42">A10</font></i></a>, <a class="ref" href="mssubfam.html#T2" title="MSSUBFAM:th.2">MSSUBFAM:2</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><br/>

<a NAME="E11:81_1_1_1"/><i><font color="Green" title="E47">A15</font></i>: 
<font color="Maroon" title="c13">Q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="closure1.html#E1:81_1_1"><i><font color="Green" title="E38">A6</font></i></a>, <a class="txt" href="closure1.html#E7:81_1_1"><i><font color="Green" title="E40">A8</font></i></a>, <a class="ref" href="pboole.html#D16" title="PBOOLE:def.16">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" title="c15">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:81_1_1_1_1"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">Q</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c16">x1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c14">Di</font><b> such that </b><br/><a NAME="E3:81_1_1_1_1"/><i><font color="Green" title="E48">A16</font></i>: 
( <font color="Maroon" title="c16">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x</font> &amp; <font color="Maroon" title="c8">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">x1</font> )
 <b>by </b><i><a class="txt" href="closure1.html#E7:81_1_1"><i><font color="Green" title="E40">A8</font></i></a>, <a class="txt" href="closure1.html#E1:81_1_1_1"><i><font color="Green" title="E42">A10</font></i></a></i>;<br/><a NAME="E4:81_1_1_1_1"/>
 <a href="closure1.html#K6" title="CLOSURE1:func.6">MSFixPoints</a> <font color="Maroon" title="c3">J</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a>  <a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></i>;<br/><a NAME="E5:81_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c14">Di</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="mssubfam.html#K7" title="MSSUBFAM:func.7">bool</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font>
 <b>by </b><i><a class="txt" href="closure1.html#E1:81_1_1"><i><font color="Green" title="E38">A6</font></i></a>, <a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a></i>;<br/><a NAME="E6:81_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c14">Di</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="closure1.html#E1:81_1_1"><i><font color="Green" title="E38">A6</font></i></a>, <a class="ref" href="mboolean.html#D1" title="MBOOLEAN:def.1">MBOOLEAN:def 1</a></i>;<br/><a NAME="E7:81_1_1_1_1"/><b>then </b><i><font color="Green" title="E49">A17</font></i>: 
<font color="Maroon" title="c16">x1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>consider </b><font color="Maroon" title="c17">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E9:81_1_1_1_1"/><i><font color="Green" title="E50">A18</font></i>: 
( <font color="Maroon" title="c17">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> &amp; <font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c17">g</font> &amp; <font color="Maroon" title="c17">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x1</font> )
 <b>by </b><i><a class="txt" href="closure1.html#E1:81_1_1"><i><font color="Green" title="E38">A6</font></i></a>, <a class="ref" href="closure1.html#D13" target="_self" title="CLOSURE1:def.13">Def13</a></i>;<br/><b>thus </b><a NAME="E10:81_1_1_1_1"/>
<font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">x</font>
 <b>by </b><i><a class="txt" href="closure1.html#E1:81_1_1"><i><font color="Green" title="E38">A6</font></i></a>, <a class="txt" href="closure1.html#E3:81_1_1_1"><i><font color="Green" title="E43">A11</font></i></a>, <a class="txt" href="closure1.html#E3:81_1_1_1_1"><i><font color="Green" title="E48">A16</font></i></a>, <a class="txt" href="closure1.html#E7:81_1_1_1_1"><i><font color="Green" title="E49">A17</font></i></a>, <a class="txt" href="closure1.html#E9:81_1_1_1_1"><i><font color="Green" title="E50">A18</font></i></a>, <a class="ref" href="closure1.html#T25" target="_self" title="CLOSURE1:th.25">Th25</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E13:81_1_1_1"/>
<font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c13">Q</font>
 <b>by </b><i><a class="txt" href="closure1.html#E11:81_1_1_1"><i><font color="Green" title="E47">A15</font></i></a>, <a class="ref" href="mssubfam.html#T5" title="MSSUBFAM:th.5">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" title="CLOSURE1:func.8">ClSys-&gt;ClOp</a> <span class="p3">(<span class="default"><a href="closure1.html#K7" title="CLOSURE1:func.7">ClOp-&gt;ClSys</a> <font color="Maroon" title="c3">J</font></span>)</span></span>)</span> <a href="closure1.html#K1" title="CLOSURE1:func.1">..</a> <font color="Maroon" title="c8">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> = 
<font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span>
<b>by </b><i><a class="txt" href="closure1.html#E2:81"><i><font color="Green" title="E33">A1</font></i></a>, <a class="txt" href="closure1.html#E3:81"><i><font color="Green" title="E34">A2</font></i></a>, <a class="txt" href="closure1.html#E2:81_1"><i><font color="Green" title="E36">A4</font></i></a>, <a class="txt" href="closure1.html#E4:81_1"><i><font color="Green" title="E37">A5</font></i></a>, <a class="txt" href="closure1.html#E8:81_1_1"><i><font color="Green" title="E41">A9</font></i></a>, <a class="ref" href="closure1.html#D15" target="_self" title="CLOSURE1:def.15">Def15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">J</font> <a href="closure1.html#K1" title="CLOSURE1:func.1">..</a> <font color="Maroon" title="c8">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font>
<b>by </b><i><a class="txt" href="closure1.html#E1:81_1_1"><i><font color="Green" title="E38">A6</font></i></a>, <a class="txt" href="closure1.html#E2:81_1_1"><i><font color="Green" title="E39">A7</font></i></a>, <a class="ref" href="pralg_1.html#D17" title="PRALG_1:def.17">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" title="CLOSURE1:func.8">ClSys-&gt;ClOp</a> <span class="p2">(<span class="default"><a href="closure1.html#K7" title="CLOSURE1:func.7">ClOp-&gt;ClSys</a> <font color="Maroon" title="c3">J</font></span>)</span></span>)</span> <a href="closure1.html#K1" title="CLOSURE1:func.1">..</a> <font color="Maroon" title="c8">X</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">J</font> <a href="closure1.html#K1" title="CLOSURE1:func.1">..</a> <font color="Maroon" title="c8">X</font>
 <b>by </b><i><a class="ref" href="pboole.html#T3" title="PBOOLE:th.3">PBOOLE:3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:81"/>
 <a href="closure1.html#K8" title="CLOSURE1:func.8">ClSys-&gt;ClOp</a> <span class="p1">(<span class="default"><a href="closure1.html#K7" title="CLOSURE1:func.7">ClOp-&gt;ClSys</a> <font color="Maroon" title="c3">J</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">J</font>
 <b>by </b><i><a class="txt" href="closure1.html#E3:81"><i><font color="Green" title="E34">A2</font></i></a>, <a class="ref" href="closure1.html#T7" target="_self" title="CLOSURE1:th.7">Th7</a></i>;<br/>


</div>
