<?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">D</font> be   <a href="closure2.html#NM4" title="CLOSURE2:NM.4">ClosureSystem</a> of <font color="Maroon" title="c1">S</font>;<br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="closure2.html#NM3" title="CLOSURE2:NM.3">SetOp</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">D</font>;<br/>





<b>assume </b><a NAME="E1:98"/><i><font color="Green" title="E40">A1</font></i>: 
for <font color="Olive" title="b1">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#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">D</font> holds  <font color="Maroon" title="c3">f</font> <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>  <a href="closure2.html#K11" title="CLOSURE2:func.11">Cl</a> <font color="Olive" title="b1">x</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c4">M</font> = the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">D</font>;<br/>
<a NAME="E2:98"/><i><font color="Green" title="E41">A2</font></i>: 
<font color="Maroon" title="c3">f</font> is <a href="closure2.html#V7" title="CLOSURE2:attr.7">reflexive</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be    <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#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">D</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="closure2.html#D12" target="_self" title="CLOSURE2:def.12">CLOSURE2:def 12</a><br/>

<b>consider </b><font color="Maroon" title="c6">F</font> being   <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">D</font><b> such that </b><br/><a NAME="E2:98_1"/><i><font color="Green" title="E42">A3</font></i>: 
 <a href="closure2.html#K11" title="CLOSURE2:func.11">Cl</a> <font color="Maroon" title="c5">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="c6">F</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span>
 <b>and </b><br/><a NAME="E3:98_1"/><i><font color="Green" title="E43">A4</font></i>: 
<font color="Maroon" title="c6">F</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#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">D</font> : ( <font color="Maroon" title="c5">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="E4:98_1"/><i><font color="Green" title="E44">A5</font></i>: 
for <font color="Olive" title="b1">Z1</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">Z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> holds <br/><font color="Maroon" title="c5">x</font> <a href="closure2.html#NR2" title="CLOSURE2:NR.2">c='</a> <font color="Olive" title="b1">Z1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">Z1</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="E1:98_1_1"/>
<font color="Maroon" title="c7">Z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">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#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">D</font><b> such that </b><br/><a NAME="E3:98_1_1"/><i><font color="Green" title="E45">A6</font></i>: 
( <font color="Maroon" title="c7">Z1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c8">a</font> &amp; <font color="Maroon" title="c5">x</font> <a href="closure2.html#NR2" title="CLOSURE2:NR.2">c='</a> <font color="Maroon" title="c8">a</font> )
 <b>and </b><br/><a NAME="E4:98_1_1"/>
<font color="Maroon" title="c8">a</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="closure2.html#E3:98_1"><i><font color="Green" title="E43">A4</font></i></a></i>;<br/>
<b>thus </b><a NAME="E5:98_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="closure2.html#NR2" title="CLOSURE2:NR.2">c='</a> <font color="Maroon" title="c7">Z1</font>
 <b>by </b><i><a class="txt" href="closure2.html#E3:98_1_1"><i><font color="Green" title="E45">A6</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:98_1"/>
<font color="Maroon" title="c3">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c5">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="c6">F</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span>
 
<b>by </b><i><a class="txt" href="closure2.html#E1:98"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="closure2.html#E2:98_1"><i><font color="Green" title="E42">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:98_1"/>
<font color="Maroon" title="c5">x</font> <a href="closure2.html#NR2" title="CLOSURE2:NR.2">c='</a> <font color="Maroon" title="c3">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="txt" href="closure2.html#E4:98_1"><i><font color="Green" title="E44">A5</font></i></a>, <a class="ref" href="closure2.html#T25" target="_self" title="CLOSURE2:th.25">Th25</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:98"/><i><font color="Green" title="E42">A7</font></i>: 
<font color="Maroon" title="c3">f</font> is <a href="closure2.html#V8" title="CLOSURE2:attr.8">monotonic</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> be    <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#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">D</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="closure2.html#D13" target="_self" title="CLOSURE2:def.13">CLOSURE2:def 13</a><br/>

<b>assume </b><a NAME="E1:98_2"/><i><font color="Green" title="E43">A8</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="closure2.html#NR2" title="CLOSURE2:NR.2">c='</a> <font color="Maroon" title="c6">y</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c7">Fx</font> being   <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">D</font><b> such that </b><br/><a NAME="E3:98_2"/><i><font color="Green" title="E44">A9</font></i>: 
 <a href="closure2.html#K11" title="CLOSURE2:func.11">Cl</a> <font color="Maroon" title="c5">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="c7">Fx</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span>
 <b>and </b><br/><a NAME="E4:98_2"/><i><font color="Green" title="E45">A10</font></i>: 
<font color="Maroon" title="c7">Fx</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#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">D</font> : ( <font color="Maroon" title="c5">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/>
<b>consider </b><font color="Maroon" title="c8">Fy</font> being   <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">D</font><b> such that </b><br/><a NAME="E6:98_2"/><i><font color="Green" title="E46">A11</font></i>: 
 <a href="closure2.html#K11" title="CLOSURE2:func.11">Cl</a> <font color="Maroon" title="c6">y</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="c8">Fy</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span>
 <b>and </b><br/><a NAME="E7:98_2"/><i><font color="Green" title="E47">A12</font></i>: 
<font color="Maroon" title="c8">Fy</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#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">D</font> : ( <font color="Maroon" title="c6">y</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="E8:98_2"/>
<span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c8">Fy</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span> <a href="closure2.html#NR2" title="CLOSURE2:NR.2">c='</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c7">Fx</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">i</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="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a><br/>

<b>assume </b><a NAME="E1:98_2_1"/><i><font color="Green" title="E48">A13</font></i>: 
<font color="Maroon" title="c9">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/>

<b>thus </b><a NAME="E2:98_2_1"/>
<span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c8">Fy</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="c9">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c7">Fx</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="c9">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">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><br/>

<b>assume </b><a NAME="E1:98_2_1_1"/><i><font color="Green" title="E49">A14</font></i>: 
<font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c8">Fy</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="c9">i</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:98_2_1_1_1"/>
( <font color="Maroon" title="c8">Fy</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or  not <font color="Maroon" title="c8">Fy</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:98_2_1_1_1_1"/>
<font color="Maroon" title="c8">Fy</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c11">Fy'</font> = <font color="Maroon" title="c8">Fy</font> as  <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#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">D</font> ;<br/><a NAME="E3:98_2_1_1_1_1"/>
<span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c11">Fy'</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="c9">i</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="txt" href="closure2.html#E1:98_2_1"><i><font color="Green" title="E48">A13</font></i></a>, <a class="ref" href="pboole.html#D15" title="PBOOLE:def.15">PBOOLE:def 15</a></i>;<br/><b>hence </b><a NAME="E4:98_2_1_1_1_1"/>
<font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c7">Fx</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="c9">i</font>
 <b>by </b><i><a class="txt" href="closure2.html#E1:98_2_1_1"><i><font color="Green" title="E49">A14</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:98_2_1_1_1_2"/>
not <font color="Maroon" title="c8">Fy</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><a NAME="E2:98_2_1_1_1_2"/><b>then </b>
<span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c8">Fy</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="c9">i</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">e</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span> where <font color="Olive" title="b1">e</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#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">D</font> : <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Fy</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="closure2.html#E1:98_2_1"><i><font color="Green" title="E48">A13</font></i></a>, <a class="ref" href="closure2.html#T15" target="_self" title="CLOSURE2:th.15">Th15</a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">w</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#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">D</font><b> such that </b><br/><a NAME="E4:98_2_1_1_1_2"/><i><font color="Green" title="E50">A15</font></i>: 
<font color="Maroon" title="c10">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font>
 <b>and </b><br/><a NAME="E5:98_2_1_1_1_2"/><i><font color="Green" title="E51">A16</font></i>: 
<font color="Maroon" title="c11">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Fy</font>
 <b>by </b><i><a class="txt" href="closure2.html#E1:98_2_1_1"><i><font color="Green" title="E49">A14</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c12">r</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#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">D</font><b> such that </b><br/><a NAME="E7:98_2_1_1_1_2"/><i><font color="Green" title="E52">A17</font></i>: 
<font color="Maroon" title="c12">r</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c11">w</font>
 <b>and </b><br/><a NAME="E8:98_2_1_1_1_2"/><i><font color="Green" title="E53">A18</font></i>: 
<font color="Maroon" title="c6">y</font> <a href="closure2.html#NR2" title="CLOSURE2:NR.2">c='</a> <font color="Maroon" title="c12">r</font>
 <b>and </b><br/><a NAME="E9:98_2_1_1_1_2"/><i><font color="Green" title="E54">A19</font></i>: 
<font color="Maroon" title="c12">r</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="closure2.html#E7:98_2"><i><font color="Green" title="E47">A12</font></i></a>, <a class="txt" href="closure2.html#E5:98_2_1_1_1_2"><i><font color="Green" title="E51">A16</font></i></a></i>;<br/><a NAME="E10:98_2_1_1_1_2"/>
<font color="Maroon" title="c5">x</font> <a href="closure2.html#NR2" title="CLOSURE2:NR.2">c='</a> <font color="Maroon" title="c11">w</font>
 <b>by </b><i><a class="txt" href="closure2.html#E1:98_2"><i><font color="Green" title="E43">A8</font></i></a>, <a class="txt" href="closure2.html#E7:98_2_1_1_1_2"><i><font color="Green" title="E52">A17</font></i></a>, <a class="txt" href="closure2.html#E8:98_2_1_1_1_2"><i><font color="Green" title="E53">A18</font></i></a>, <a class="ref" href="pboole.html#T15" title="PBOOLE:th.15">PBOOLE:15</a></i>;<br/><a NAME="E11:98_2_1_1_1_2"/><b>then </b><i><font color="Green" title="E55">A20</font></i>: 
<font color="Maroon" title="c11">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Fx</font>
 <b>by </b><i><a class="txt" href="closure2.html#E4:98_2"><i><font color="Green" title="E45">A10</font></i></a>, <a class="txt" href="closure2.html#E7:98_2_1_1_1_2"><i><font color="Green" title="E52">A17</font></i></a>, <a class="txt" href="closure2.html#E9:98_2_1_1_1_2"><i><font color="Green" title="E54">A19</font></i></a></i>;<br/><a NAME="E12:98_2_1_1_1_2"/><b>then </b>
<span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c7">Fx</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="c9">i</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">e</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span> where <font color="Olive" title="b1">e</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#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">D</font> : <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Fx</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="closure2.html#E1:98_2_1"><i><font color="Green" title="E48">A13</font></i></a>, <a class="ref" href="closure2.html#T15" target="_self" title="CLOSURE2:th.15">Th15</a></i>;<br/><b>hence </b><a NAME="E13:98_2_1_1_1_2"/>
<font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c7">Fx</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="c9">i</font>
 <b>by </b><i><a class="txt" href="closure2.html#E4:98_2_1_1_1_2"><i><font color="Green" title="E50">A15</font></i></a>, <a class="txt" href="closure2.html#E11:98_2_1_1_1_2"><i><font color="Green" title="E55">A20</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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


</div><b>end;</b></div>
<a NAME="E9:98_2"/><b>then </b>
 <a href="mssubfam.html#K6" 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="c7">Fx</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span> <a href="closure2.html#NR2" title="CLOSURE2:NR.2">c='</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="c8">Fy</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span>
 
<b>by </b><i><a class="ref" href="mssubfam.html#T46" title="MSSUBFAM:th.46">MSSUBFAM:46</a></i>;<br/>
<a NAME="E10:98_2"/><b>then </b>
 <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="c7">Fx</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span> <a href="closure2.html#NR2" title="CLOSURE2:NR.2">c='</a> <font color="Maroon" title="c3">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c6">y</font>
 
<b>by </b><i><a class="txt" href="closure2.html#E1:98"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="closure2.html#E6:98_2"><i><font color="Green" title="E46">A11</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:98_2"/>
<font color="Maroon" title="c3">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c5">x</font> <a href="closure2.html#NR2" title="CLOSURE2:NR.2">c='</a> <font color="Maroon" title="c3">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c6">y</font>
 <b>by </b><i><a class="txt" href="closure2.html#E1:98"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="closure2.html#E3:98_2"><i><font color="Green" title="E44">A9</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:98"/>
<font color="Maroon" title="c3">f</font> is <a href="closure2.html#V9" title="CLOSURE2:attr.9">idempotent</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be    <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#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">D</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="closure2.html#D14" target="_self" title="CLOSURE2:def.14">CLOSURE2:def 14</a><br/>

<b>consider </b><font color="Maroon" title="c6">F</font> being   <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">D</font><b> such that </b><br/><a NAME="E2:98_3"/><i><font color="Green" title="E43">A21</font></i>: 
 <a href="closure2.html#K11" title="CLOSURE2:func.11">Cl</a> <font color="Maroon" title="c5">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="c6">F</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span>
 <b>and </b><br/><a NAME="E3:98_3"/><i><font color="Green" title="E44">A22</font></i>: 
<font color="Maroon" title="c6">F</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#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">D</font> : ( <font color="Maroon" title="c5">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="E4:98_3"/>
<font color="Maroon" title="c6">F</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 <font color="Maroon" title="c2">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">k</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><br/>

<b>assume </b><a NAME="E1:98_3_1"/>
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">q</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#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">D</font><b> such that </b><br/><a NAME="E3:98_3_1"/><i><font color="Green" title="E45">A23</font></i>: 
( <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q</font> &amp; <font color="Maroon" title="c5">x</font> <a href="closure2.html#NR2" title="CLOSURE2:NR.2">c='</a> <font color="Maroon" title="c8">q</font> &amp; <font color="Maroon" title="c8">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="closure2.html#E3:98_3"><i><font color="Green" title="E44">A22</font></i></a></i>;<br/>
<b>thus </b><a NAME="E4:98_3_1"/>
<font color="Maroon" title="c7">k</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="closure2.html#E3:98_3_1"><i><font color="Green" title="E45">A23</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:98_3"/><b>then </b><i><font color="Green" title="E45">A24</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="c6">F</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span> <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#D8" target="_self" title="CLOSURE2:def.8">Def8</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c3">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c5">x</font> = 
 <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="c6">F</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span>
<b>by </b><i><a class="txt" href="closure2.html#E1:98"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="closure2.html#E2:98_3"><i><font color="Green" title="E43">A21</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <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="c6">F</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span></span>)</span>
<b>by </b><i><a class="txt" href="closure2.html#E1:98"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="closure2.html#E5:98_3"><i><font color="Green" title="E45">A24</font></i></a>, <a class="ref" href="closure2.html#T40" target="_self" title="CLOSURE2:th.40">Th40</a></i>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c5">x</font></span>)</span>
<b>by </b><i><a class="txt" href="closure2.html#E1:98"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="closure2.html#E2:98_3"><i><font color="Green" title="E43">A21</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:98"/>
( <font color="Maroon" title="c3">f</font> is <a href="closure2.html#V7" title="CLOSURE2:attr.7">reflexive</a> &amp; <font color="Maroon" title="c3">f</font> is <a href="closure2.html#V8" title="CLOSURE2:attr.8">monotonic</a> &amp; <font color="Maroon" title="c3">f</font> is <a href="closure2.html#V9" title="CLOSURE2:attr.9">idempotent</a> )
 <b>by </b><i><a class="txt" href="closure2.html#E2:98"><i><font color="Green" title="E41">A2</font></i></a>, <a class="txt" href="closure2.html#E3:98"><i><font color="Green" title="E42">A7</font></i></a></i>;<br/>


</div>
