<?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">f</font> be  <a href="closure2.html#V7" title="CLOSURE2:attr.7">reflexive</a> <a href="closure2.html#V8" title="CLOSURE2:attr.8">monotonic</a> <a href="closure2.html#NM3" title="CLOSURE2:NM.3">SetOp</a> of <font color="Maroon" title="c2">A</font>;<br/><b>let </b><font color="Maroon" title="c4">D</font> be   <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of <font color="Maroon" title="c2">A</font>;<br/>







<b>assume </b><a NAME="E1:93"/><i><font color="Green" title="E36">A1</font></i>: 
<font color="Maroon" title="c4">D</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> <font color="Maroon" title="c2">A</font> : <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> <font color="Olive" title="b1">x</font> </span>}</span> 
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c5">cs</font> =  <a href="closure2.html#G1" title="CLOSURE2:aggr.1">ClosureStr</a>(# <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">D</font> #) as    <a href="closure2.html#L1" title="CLOSURE2:struct.1">ClosureStr</a> of <font color="Maroon" title="c1">S</font> ;<br/>
<a NAME="E3:93"/>
<font color="Maroon" title="c5">cs</font> is <a href="closure2.html#V15" title="CLOSURE2:attr.15">absolutely-multiplicative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1"><b>proof </b></a><div class="add">

<a NAME="E1:93_1"/>
<font color="Maroon" title="c4">D</font> is <a href="closure2.html#V4" title="CLOSURE2:attr.4">absolutely-multiplicative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">F</font> be   <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of <font color="Maroon" title="c2">A</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="closure2.html#D8" target="_self" title="CLOSURE2:def.8">CLOSURE2:def 8</a><br/>

<b>assume </b><a NAME="E1:93_1_1"/><i><font color="Green" title="E37">A2</font></i>: 
<font color="Maroon" title="c6">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">D</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c7">mf</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> as    <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> <font color="Maroon" title="c2">A</font> <b>by </b><i><a class="ref" href="closure2.html#D1" target="_self" title="CLOSURE2:def.1">Def1</a></i>;<br/>
<a NAME="E3:93_1_1"/>
<font color="Maroon" title="c3">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c7">mf</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c7">mf</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">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="E1:93_1_1_1_1"/><i><font color="Green" title="E38">A3</font></i>: 
<font color="Maroon" title="c8">Z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font>
 ;<br/><a NAME="E2:93_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c8">Z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D</font>
 <b>by </b><i><a class="txt" href="closure2.html#E1:93_1_1"><i><font color="Green" title="E37">A2</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c9">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> <font color="Maroon" title="c2">A</font><b> such that </b><br/><a NAME="E4:93_1_1_1_1"/><i><font color="Green" title="E39">A4</font></i>: 
( <font color="Maroon" title="c8">Z1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c9">a</font> &amp; <font color="Maroon" title="c3">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c9">a</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c9">a</font> )
 <b>by </b><i><a class="txt" href="closure2.html#E1:93"><i><font color="Green" title="E36">A1</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">y1</font> = <font color="Maroon" title="c8">Z1</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> <font color="Maroon" title="c2">A</font> <b>by </b><i><a class="txt" href="closure2.html#E1:93_1_1_1_1"><i><font color="Green" title="E38">A3</font></i></a></i>;<br/><a NAME="E6:93_1_1_1_1"/>
<font color="Maroon" title="c7">mf</font> <a href="closure2.html#NR2" title="CLOSURE2:NR.2">c='</a> <font color="Maroon" title="c10">y1</font>
 <b>by </b><i><a class="txt" href="closure2.html#E1:93_1_1_1_1"><i><font color="Green" title="E38">A3</font></i></a>, <a class="ref" href="closure2.html#T22" target="_self" title="CLOSURE2:th.22">Th22</a>, <a class="ref" href="mssubfam.html#T43" title="MSSUBFAM:th.43">MSSUBFAM:43</a></i>;<br/><b>hence </b><a NAME="E7:93_1_1_1_1"/>
<font color="Maroon" title="c3">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c7">mf</font> <a href="closure2.html#NR2" title="CLOSURE2:NR.2">c='</a> <font color="Maroon" title="c8">Z1</font>
 <b>by </b><i><a class="txt" href="closure2.html#E4:93_1_1_1_1"><i><font color="Green" title="E39">A4</font></i></a>, <a class="ref" href="closure2.html#D13" target="_self" title="CLOSURE2:def.13">Def13</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:93_1_1_1"/>
<font color="Maroon" title="c3">f</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c7">mf</font> <a href="closure2.html#NR2" title="CLOSURE2:NR.2">c='</a> <font color="Maroon" title="c7">mf</font>
 <b>by </b><i><a class="ref" href="closure2.html#T25" target="_self" title="CLOSURE2:th.25">Th25</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D13" title="PBOOLE:def.13">PBOOLE:def 13</a><br/>

<b>thus </b><a NAME="E3:93_1_1_1"/>
<font color="Maroon" title="c7">mf</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="c7">mf</font>
 <b>by </b><i><a class="ref" href="closure2.html#D12" target="_self" title="CLOSURE2:def.12">Def12</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:93_1_1"/>
 <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> <font color="Maroon" title="c4">D</font>
 <b>by </b><i><a class="txt" href="closure2.html#E1:93"><i><font color="Green" title="E36">A1</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:93_1"/>
the <a href="closure2.html#U1" title="CLOSURE2:sel.1">Family</a> of <font color="Maroon" title="c5">cs</font> is <a href="closure2.html#V4" title="CLOSURE2:attr.4">absolutely-multiplicative</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="closure2.html#D19" target="_self" title="CLOSURE2:def.19">CLOSURE2:def 19</a><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:93"/>
 <a href="closure2.html#G1" title="CLOSURE2:aggr.1">ClosureStr</a>(# <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">D</font> #) is   <a href="closure2.html#NM4" title="CLOSURE2:NM.4">ClosureSystem</a> of <font color="Maroon" title="c1">S</font>
 ;<br/>


</div>
