<?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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">D</font> being  <a href="closure2.html#NM4" title="CLOSURE2:NM.4">ClosureSystem</a> of <font color="Maroon" title="c1">S</font> holds   <a href="closure2.html#K10" title="CLOSURE2:func.10">ClOp-&gt;ClSys</a> <span class="p1">(<span class="default"><a href="closure2.html#K12" title="CLOSURE2:func.12">ClSys-&gt;ClOp</a> <font color="Olive" title="b1">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="closure2.html#G1" title="CLOSURE2:aggr.1">ClosureStr</a>(# the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Olive" title="b1">D</font>,the <a href="closure2.html#U1" title="CLOSURE2:sel.1">Family</a> of <font color="Olive" title="b1">D</font> #)</span><br/><b>let </b><font color="Maroon" title="c2">D</font> be   <a href="closure2.html#NM4" title="CLOSURE2:NM.4">ClosureSystem</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="closure2.html#K10" title="CLOSURE2:func.10">ClOp-&gt;ClSys</a> <span class="p1">(<span class="default"><a href="closure2.html#K12" title="CLOSURE2:func.12">ClSys-&gt;ClOp</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="closure2.html#G1" title="CLOSURE2:aggr.1">ClosureStr</a>(# the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font>,the <a href="closure2.html#U1" title="CLOSURE2:sel.1">Family</a> of <font color="Maroon" title="c2">D</font> #)</span><br/>



<b>set </b><font color="Maroon" title="c3">M</font> = the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font>;<br/>
<b>set </b><font color="Maroon" title="c4">F</font> = the <a href="closure2.html#U1" title="CLOSURE2:sel.1">Family</a> of <font color="Maroon" title="c2">D</font>;<br/>
<b>set </b><font color="Maroon" title="c5">d</font> = the <a href="closure2.html#U1" title="CLOSURE2:sel.1">Family</a> of <span class="p1">(<span class="default"><a href="closure2.html#K10" title="CLOSURE2:func.10">ClOp-&gt;ClSys</a> <span class="p2">(<span class="default"><a href="closure2.html#K12" title="CLOSURE2:func.12">ClSys-&gt;ClOp</a> <font color="Maroon" title="c2">D</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c6">f</font> =  <a href="closure2.html#K12" title="CLOSURE2:func.12">ClSys-&gt;ClOp</a> <font color="Maroon" title="c2">D</font>;<br/>
<a NAME="E1:100"/><i><font color="Green" title="E41">A1</font></i>: 
the <a href="closure2.html#U1" title="CLOSURE2:sel.1">Family</a> of <span class="p1">(<span class="default"><a href="closure2.html#K10" title="CLOSURE2:func.10">ClOp-&gt;ClSys</a> <span class="p2">(<span class="default"><a href="closure2.html#K12" title="CLOSURE2:func.12">ClSys-&gt;ClOp</a> <font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font> : <span class="p1">(<span class="default"><a href="closure2.html#K12" title="CLOSURE2:func.12">ClSys-&gt;ClOp</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Olive" title="b1">x</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b1">x</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="closure2.html#D23" target="_self" title="CLOSURE2:def.23">Def23</a></i>;<br/>
<a NAME="E2:100"/>
the <a href="closure2.html#U1" title="CLOSURE2:sel.1">Family</a> of <font color="Maroon" title="c2">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="closure2.html#U1" title="CLOSURE2:sel.1">Family</a> of <span class="p1">(<span class="default"><a href="closure2.html#K10" title="CLOSURE2:func.10">ClOp-&gt;ClSys</a> <span class="p2">(<span class="default"><a href="closure2.html#K12" title="CLOSURE2:func.12">ClSys-&gt;ClOp</a> <font color="Maroon" title="c2">D</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:100_1"/>
the <a href="closure2.html#U1" title="CLOSURE2:sel.1">Family</a> of <font color="Maroon" title="c2">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="closure2.html#U1" title="CLOSURE2:sel.1">Family</a> of <span class="p1">(<span class="default"><a href="closure2.html#K10" title="CLOSURE2:func.10">ClOp-&gt;ClSys</a> <span class="p2">(<span class="default"><a href="closure2.html#K12" title="CLOSURE2:func.12">ClSys-&gt;ClOp</a> <font color="Maroon" title="c2">D</font></span>)</span></span>)</span>
  <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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> the <a href="closure2.html#U1" title="CLOSURE2:sel.1">Family</a> of <span class="p1">(<span class="default"><a href="closure2.html#K10" title="CLOSURE2:func.10">ClOp-&gt;ClSys</a> <span class="p2">(<span class="default"><a href="closure2.html#K12" title="CLOSURE2:func.12">ClSys-&gt;ClOp</a> <font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="closure2.html#U1" title="CLOSURE2:sel.1">Family</a> of <font color="Maroon" title="c2">D</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c7">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="closure2.html#U1" title="CLOSURE2:sel.1">Family</a> of <font color="Maroon" title="c2">D</font> or <font color="Maroon" title="c7">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="closure2.html#U1" title="CLOSURE2:sel.1">Family</a> of <span class="p1">(<span class="default"><a href="closure2.html#K10" title="CLOSURE2:func.10">ClOp-&gt;ClSys</a> <span class="p2">(<span class="default"><a href="closure2.html#K12" title="CLOSURE2:func.12">ClSys-&gt;ClOp</a> <font color="Maroon" title="c2">D</font></span>)</span></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:100_1_1"/><i><font color="Green" title="E42">A2</font></i>: 
<font color="Maroon" title="c7">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="closure2.html#U1" title="CLOSURE2:sel.1">Family</a> of <font color="Maroon" title="c2">D</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="closure2.html#U1" title="CLOSURE2:sel.1">Family</a> of <span class="p1">(<span class="default"><a href="closure2.html#K10" title="CLOSURE2:func.10">ClOp-&gt;ClSys</a> <span class="p2">(<span class="default"><a href="closure2.html#K12" title="CLOSURE2:func.12">ClSys-&gt;ClOp</a> <font color="Maroon" title="c2">D</font></span>)</span></span>)</span></span><br/>

<b>then reconsider </b><font color="Maroon" title="c8">q1</font> = <font color="Maroon" title="c7">q</font> as    <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font> ;<br/>
<b>consider </b><font color="Maroon" title="c9">SF</font> being   <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font><b> such that </b><br/><a NAME="E4:100_1_1"/><i><font color="Green" title="E43">A3</font></i>: 
 <a href="closure2.html#K11" title="CLOSURE2:func.11">Cl</a> <font color="Maroon" title="c8">q1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c9">SF</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span>
 <b>and </b><br/><a NAME="E5:100_1_1"/><i><font color="Green" title="E44">A4</font></i>: 
<font color="Maroon" title="c9">SF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is    <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font> : ( <font color="Maroon" title="c8">q1</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b1">X</font> &amp; <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="closure2.html#U1" title="CLOSURE2:sel.1">Family</a> of <font color="Maroon" title="c2">D</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="closure2.html#D24" target="_self" title="CLOSURE2:def.24">Def24</a></i>;<br/>
<a NAME="E6:100_1_1"/><i><font color="Green" title="E45">A5</font></i>: 
<font color="Maroon" title="c8">q1</font> <a href="closure2.html#NR2" title="CLOSURE2:NR.2">c='</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E7:100_1_1"/>
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="closure2.html#U1" title="CLOSURE2:sel.1">Family</a> of <font color="Maroon" title="c2">D</font>
 
<b>by </b><i><a class="ref" href="closure2.html#D9" target="_self" title="CLOSURE2:def.9">Def9</a></i>;<br/>
<a NAME="E8:100_1_1"/><b>then </b>
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">SF</font>
 
<b>by </b><i><a class="txt" href="#E5:100_1_1"><i><font color="Green" title="E44">A4</font></i></a>, <a class="txt" href="#E6:100_1_1"><i><font color="Green" title="E45">A5</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">SF'</font> = <font color="Maroon" title="c9">SF</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c11">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> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="closure2.html#K12" title="CLOSURE2:func.12">ClSys-&gt;ClOp</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c8">q1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> )</span><br/><b>assume </b><a NAME="E1:100_1_1_1"/><i><font color="Green" title="E46">A6</font></i>: 
<font color="Maroon" title="c11">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="closure2.html#K12" title="CLOSURE2:func.12">ClSys-&gt;ClOp</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c8">q1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span><br/><b>then consider </b><font color="Maroon" title="c12">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#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span>)</span><b> such that </b><br/><a NAME="E3:100_1_1_1"/><i><font color="Green" title="E47">A7</font></i>: 
<font color="Maroon" title="c12">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c10">SF'</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font>
 <b>and </b><br/><a NAME="E4:100_1_1_1"/><i><font color="Green" title="E48">A8</font></i>: 
<span class="p1">(<span class="default"><a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <span class="p2"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c10">SF'</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">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="c12">Q</font>
 <b>by </b><i><a class="ref" href="mssubfam.html#D2" title="MSSUBFAM:def.2">MSSUBFAM:def 2</a></i>;<br/><a NAME="E5:100_1_1_1"/>
 <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c12">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:100_1_1_1_1"/><i><font color="Green" title="E49">A9</font></i>: 
<font color="Maroon" title="c8">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">SF'</font>
 
<b>by </b><i><a class="txt" href="#E1:100_1_1"><i><font color="Green" title="E42">A2</font></i></a>, <a class="txt" href="#E5:100_1_1"><i><font color="Green" title="E44">A4</font></i></a></i>;<br/>
<a NAME="E2:100_1_1_1_1"/><i><font color="Green" title="E50">A10</font></i>: 
<font color="Maroon" title="c12">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span>)</span> where <font color="Olive" title="b1">x</font> is    <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">SF'</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="#E1:100_1_1_1"><i><font color="Green" title="E46">A6</font></i></a>, <a class="txt" href="#E3:100_1_1_1"><i><font color="Green" title="E47">A7</font></i></a>, <a class="ref" href="closure2.html#T15" target="_self" title="CLOSURE2:th.15">Th15</a></i>;<br/>
<a NAME="E3:100_1_1_1_1"/><b>then </b><i><font color="Green" title="E51">A11</font></i>: 
<font color="Maroon" title="c8">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Q</font>
 
<b>by </b><i><a class="txt" href="#E1:100_1_1_1_1"><i><font color="Green" title="E49">A9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:100_1_1_1_1"/>
 <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c12">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font>
 <b>by </b><i><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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <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="c12">Q</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c13">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Q</font> implies <font color="Maroon" title="c8">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">z</font> )</span><br/><b>assume </b><a NAME="E1:100_1_1_1_1_1"/>
<font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">z</font></span><br/><b>then consider </b><font color="Maroon" title="c14">x</font> being    <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font><b> such that </b><br/><a NAME="E3:100_1_1_1_1_1"/><i><font color="Green" title="E52">A12</font></i>: 
<font color="Maroon" title="c13">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font>
 <b>and </b><br/><a NAME="E4:100_1_1_1_1_1"/><i><font color="Green" title="E53">A13</font></i>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">SF'</font>
 <b>by </b><i><a class="txt" href="#E2:100_1_1_1_1"><i><font color="Green" title="E50">A10</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c15">xx</font> being    <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font><b> such that </b><br/><a NAME="E6:100_1_1_1_1_1"/><i><font color="Green" title="E54">A14</font></i>: 
( <font color="Maroon" title="c15">xx</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c14">x</font> &amp; <font color="Maroon" title="c8">q1</font> <a href="closure2.html#NR2" title="CLOSURE2:NR.2">c='</a> <font color="Maroon" title="c15">xx</font> )
 <b>and </b><br/><a NAME="E7:100_1_1_1_1_1"/>
<font color="Maroon" title="c15">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="closure2.html#U1" title="CLOSURE2:sel.1">Family</a> of <font color="Maroon" title="c2">D</font>
 <b>by </b><i><a class="txt" href="#E5:100_1_1"><i><font color="Green" title="E44">A4</font></i></a>, <a class="txt" href="#E4:100_1_1_1_1_1"><i><font color="Green" title="E53">A13</font></i></a></i>;<br/><b>thus </b><a NAME="E8:100_1_1_1_1_1"/>
<font color="Maroon" title="c8">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">z</font>
 <b>by </b><i><a class="txt" href="#E1:100_1_1_1"><i><font color="Green" title="E46">A6</font></i></a>, <a class="txt" href="#E3:100_1_1_1_1_1"><i><font color="Green" title="E52">A12</font></i></a>, <a class="txt" href="#E6:100_1_1_1_1_1"><i><font color="Green" title="E54">A14</font></i></a>, <a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:100_1_1_1_1"/>
<font color="Maroon" title="c8">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <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="c12">Q</font>
 <b>by </b><i><a class="txt" href="#E3:100_1_1_1_1"><i><font color="Green" title="E51">A11</font></i></a>, <a class="ref" href="mssubfam.html#T5" title="MSSUBFAM:th.5">MSSUBFAM:5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>hence </b><a NAME="E6:100_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="closure2.html#K12" title="CLOSURE2:func.12">ClSys-&gt;ClOp</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c8">q1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font>
 <b>by </b><i><a class="txt" href="#E4:100_1_1"><i><font color="Green" title="E43">A3</font></i></a>, <a class="txt" href="#E4:100_1_1_1"><i><font color="Green" title="E48">A8</font></i></a>, <a class="ref" href="closure2.html#D25" target="_self" title="CLOSURE2:def.25">Def25</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E11:100_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="closure2.html#K12" title="CLOSURE2:func.12">ClSys-&gt;ClOp</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c8">q1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c8">q1</font>
 
<b>by </b><i><a class="ref" href="pboole.html#T3" title="PBOOLE:th.3">PBOOLE:3</a></i>;<br/>
<b>hence </b><a NAME="E12:100_1_1"/>
<font color="Maroon" title="c7">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="closure2.html#U1" title="CLOSURE2:sel.1">Family</a> of <span class="p1">(<span class="default"><a href="closure2.html#K10" title="CLOSURE2:func.10">ClOp-&gt;ClSys</a> <span class="p2">(<span class="default"><a href="closure2.html#K12" title="CLOSURE2:func.12">ClSys-&gt;ClOp</a> <font color="Maroon" title="c2">D</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:100"><i><font color="Green" title="E41">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>let </b><font color="Maroon" title="c7">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c7">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="closure2.html#U1" title="CLOSURE2:sel.1">Family</a> of <span class="p1">(<span class="default"><a href="closure2.html#K10" title="CLOSURE2:func.10">ClOp-&gt;ClSys</a> <span class="p2">(<span class="default"><a href="closure2.html#K12" title="CLOSURE2:func.12">ClSys-&gt;ClOp</a> <font color="Maroon" title="c2">D</font></span>)</span></span>)</span> or <font color="Maroon" title="c7">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="closure2.html#U1" title="CLOSURE2:sel.1">Family</a> of <font color="Maroon" title="c2">D</font> )</span><br/>

<b>assume </b><a NAME="E2:100_1"/>
<font color="Maroon" title="c7">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="closure2.html#U1" title="CLOSURE2:sel.1">Family</a> of <span class="p1">(<span class="default"><a href="closure2.html#K10" title="CLOSURE2:func.10">ClOp-&gt;ClSys</a> <span class="p2">(<span class="default"><a href="closure2.html#K12" title="CLOSURE2:func.12">ClSys-&gt;ClOp</a> <font color="Maroon" title="c2">D</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="closure2.html#U1" title="CLOSURE2:sel.1">Family</a> of <font color="Maroon" title="c2">D</font></span><br/>

<b>then consider </b><font color="Maroon" title="c8">x</font> being    <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font><b> such that </b><br/><a NAME="E4:100_1"/><i><font color="Green" title="E42">A15</font></i>: 
<font color="Maroon" title="c7">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x</font>
 <b>and </b><br/><a NAME="E5:100_1"/><i><font color="Green" title="E43">A16</font></i>: 
<span class="p1">(<span class="default"><a href="closure2.html#K12" title="CLOSURE2:func.12">ClSys-&gt;ClOp</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c8">x</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c8">x</font>
 <b>by </b><i><a class="txt" href="#E1:100"><i><font color="Green" title="E41">A1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">SF</font> being   <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font><b> such that </b><br/><a NAME="E7:100_1"/><i><font color="Green" title="E44">A17</font></i>: 
 <a href="closure2.html#K11" title="CLOSURE2:func.11">Cl</a> <font color="Maroon" title="c8">x</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c9">SF</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span>
 <b>and </b><br/><a NAME="E8:100_1"/><i><font color="Green" title="E45">A18</font></i>: 
<font color="Maroon" title="c9">SF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is    <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font> : ( <font color="Maroon" title="c8">x</font> <a href="closure2.html#NR2" title="CLOSURE2:NR.2">c='</a> <font color="Olive" title="b1">X</font> &amp; <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="closure2.html#U1" title="CLOSURE2:sel.1">Family</a> of <font color="Maroon" title="c2">D</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="closure2.html#D24" target="_self" title="CLOSURE2:def.24">Def24</a></i>;<br/>
<a NAME="E9:100_1"/><i><font color="Green" title="E46">A19</font></i>: 
 <a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c9">SF</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q</font>
 
<b>by </b><i><a class="txt" href="#E4:100_1"><i><font color="Green" title="E42">A15</font></i></a>, <a class="txt" href="#E5:100_1"><i><font color="Green" title="E43">A16</font></i></a>, <a class="txt" href="#E7:100_1"><i><font color="Green" title="E44">A17</font></i></a>, <a class="ref" href="closure2.html#D25" target="_self" title="CLOSURE2:def.25">Def25</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font>] <b>means </b><font color="Maroon" title="c8">x</font> <a href="closure2.html#NR2" title="CLOSURE2:NR.2">c='</a> $1;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font>] <b>means </b>( <font color="Maroon" title="c8">x</font> <a href="closure2.html#NR2" title="CLOSURE2:NR.2">c='</a> $1 &amp; $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="closure2.html#U1" title="CLOSURE2:sel.1">Family</a> of <font color="Maroon" title="c2">D</font> );<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font>] <b>means </b>( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="closure2.html#U1" title="CLOSURE2:sel.1">Family</a> of <font color="Maroon" title="c2">D</font> &amp; <font color="Maroon" title="c8">x</font> <a href="closure2.html#NR2" title="CLOSURE2:NR.2">c='</a> $1 );<br/>
<a NAME="E10:100_1"/><i><font color="Green" title="E47">A20</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">w</font>) where <font color="Olive" title="b1">w</font> is    <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font> : ( <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">w</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="closure2.html#U1" title="CLOSURE2:sel.1">Family</a> of <font color="Maroon" title="c2">D</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">w</font>] ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="closure2.html#U1" title="CLOSURE2:sel.1">Family</a> of <font color="Maroon" title="c2">D</font>
 
<b>from </b><i><a class="ref" href="fraenkel.html#S17" title="FRAENKEL:sch.17">FRAENKEL:sch 17</a>();<br/></i>
<a NAME="E11:100_1"/><i><font color="Green" title="E48">A21</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font> holds <br/> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">a</font>] iff <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">a</font>] )
 
;<br/>
<a NAME="E12:100_1"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">a</font>) where <font color="Olive" title="b1">a</font> is    <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">a</font>] </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">b</font>) where <font color="Olive" title="b1">b</font> is    <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">b</font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="fraenkel.html#S3" title="FRAENKEL:sch.3">FRAENKEL:sch 3</a>(<a class="txt" href="#E11:100_1"><i><font color="Green" title="E48">A21</font></i></a>);<br/></i>
<b>hence </b><a NAME="E13:100_1"/>
<font color="Maroon" title="c7">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="closure2.html#U1" title="CLOSURE2:sel.1">Family</a> of <font color="Maroon" title="c2">D</font>
 <b>by </b><i><a class="txt" href="#E8:100_1"><i><font color="Green" title="E45">A18</font></i></a>, <a class="txt" href="#E9:100_1"><i><font color="Green" title="E46">A19</font></i></a>, <a class="txt" href="#E10:100_1"><i><font color="Green" title="E47">A20</font></i></a>, <a class="ref" href="closure2.html#D8" target="_self" title="CLOSURE2:def.8">Def8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:100"/>
 <a href="closure2.html#K10" title="CLOSURE2:func.10">ClOp-&gt;ClSys</a> <span class="p1">(<span class="default"><a href="closure2.html#K12" title="CLOSURE2:func.12">ClSys-&gt;ClOp</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="closure2.html#G1" title="CLOSURE2:aggr.1">ClosureStr</a>(# the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font>,the <a href="closure2.html#U1" title="CLOSURE2:sel.1">Family</a> of <font color="Maroon" title="c2">D</font> #)
 <b>by </b><i><a class="ref" href="closure2.html#D23" target="_self" title="CLOSURE2:def.23">Def23</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
