<?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">A</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><br/> for <font color="Olive" title="b2">J</font> being <a href="closure1.html#V1" title="CLOSURE1:attr.1">reflexive</a> <a href="closure1.html#V2" title="CLOSURE1:attr.2">monotonic</a> <a href="closure1.html#NM1" title="CLOSURE1:NM.1">MSSetOp</a> of <font color="Olive" title="b1">A</font><br/> for <font color="Olive" title="b3">D</font> being  <a href="mssubfam.html#NM1" title="MSSUBFAM:NM.1">MSSubsetFamily</a> of <font color="Olive" title="b1">A</font>  st <font color="Olive" title="b3">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="Olive" title="b2">J</font> holds <br/> <a href="closure1.html#G1" title="CLOSURE1:aggr.1">MSClosureStr</a>(# <font color="Olive" title="b1">A</font>,<font color="Olive" title="b3">D</font> #) is   <a href="closure1.html#NM2" title="CLOSURE1:NM.2">MSClosureSystem</a> of <font color="Maroon" title="c1">S</font></span><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>; <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">J</font> being <a href="closure1.html#V1" title="CLOSURE1:attr.1">reflexive</a> <a href="closure1.html#V2" title="CLOSURE1:attr.2">monotonic</a> <a href="closure1.html#NM1" title="CLOSURE1:NM.1">MSSetOp</a> of <font color="Maroon" title="c2">A</font><br/> for <font color="Olive" title="b2">D</font> being  <a href="mssubfam.html#NM1" title="MSSUBFAM:NM.1">MSSubsetFamily</a> of <font color="Maroon" title="c2">A</font>  st <font color="Olive" title="b2">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="Olive" title="b1">J</font> holds <br/> <a href="closure1.html#G1" title="CLOSURE1:aggr.1">MSClosureStr</a>(# <font color="Maroon" title="c2">A</font>,<font color="Olive" title="b2">D</font> #) is   <a href="closure1.html#NM2" title="CLOSURE1:NM.2">MSClosureSystem</a> of <font color="Maroon" title="c1">S</font></span><br/><b>let </b><font color="Maroon" title="c3">J</font> be  <a href="closure1.html#V1" title="CLOSURE1:attr.1">reflexive</a> <a href="closure1.html#V2" title="CLOSURE1:attr.2">monotonic</a> <a href="closure1.html#NM1" title="CLOSURE1:NM.1">MSSetOp</a> of <font color="Maroon" title="c2">A</font>; <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="mssubfam.html#NM1" title="MSSUBFAM:NM.1">MSSubsetFamily</a> of <font color="Maroon" title="c2">A</font>  st <font color="Olive" title="b1">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> holds <br/> <a href="closure1.html#G1" title="CLOSURE1:aggr.1">MSClosureStr</a>(# <font color="Maroon" title="c2">A</font>,<font color="Olive" title="b1">D</font> #) is   <a href="closure1.html#NM2" title="CLOSURE1:NM.2">MSClosureSystem</a> of <font color="Maroon" title="c1">S</font></span><br/><b>let </b><font color="Maroon" title="c4">D</font> be   <a href="mssubfam.html#NM1" title="MSSUBFAM:NM.1">MSSubsetFamily</a> of <font color="Maroon" title="c2">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">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> implies  <a href="closure1.html#G1" title="CLOSURE1:aggr.1">MSClosureStr</a>(# <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">D</font> #) is   <a href="closure1.html#NM2" title="CLOSURE1:NM.2">MSClosureSystem</a> of <font color="Maroon" title="c1">S</font> )</span><br/>







<b>assume </b><a NAME="E1:69"/><i><font color="Green" title="E25">A1</font></i>: 
<font color="Maroon" title="c4">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="closure1.html#G1" title="CLOSURE1:aggr.1">MSClosureStr</a>(# <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">D</font> #) is   <a href="closure1.html#NM2" title="CLOSURE1:NM.2">MSClosureSystem</a> of <font color="Maroon" title="c1">S</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c5">CS</font> =  <a href="closure1.html#G1" title="CLOSURE1:aggr.1">MSClosureStr</a>(# <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">D</font> #) as    <a href="closure1.html#L1" title="CLOSURE1:struct.1">MSClosureStr</a> of <font color="Maroon" title="c1">S</font> ;<br/>
<a NAME="E3:69"/>
<font color="Maroon" title="c5">CS</font> is <a href="closure1.html#V9" title="CLOSURE1:attr.9">absolutely-multiplicative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><b>proof </b></a><div class="add">

<a NAME="E1:69_1"/>
<font color="Maroon" title="c4">D</font> is <a href="mssubfam.html#V4" title="MSSUBFAM:attr.4">absolutely-multiplicative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">F</font> be   <a href="mssubfam.html#NM1" title="MSSUBFAM:NM.1">MSSubsetFamily</a> of <font color="Maroon" title="c2">A</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="mssubfam.html#D6" title="MSSUBFAM:def.6">MSSUBFAM:def 6</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="c6">F</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c4">D</font> or  <a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <font color="Maroon" title="c6">F</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c4">D</font> )</span><br/>

<b>assume </b><a NAME="E1:69_1_1"/><i><font color="Green" title="E26">A2</font></i>: 
<font color="Maroon" title="c6">F</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c4">D</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <font color="Maroon" title="c6">F</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c4">D</font></span><br/>

<a NAME="E2:69_1_1"/>
 <a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <font color="Maroon" title="c6">F</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</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="E3:69_1_1"/><b>then </b><i><font color="Green" title="E27">A3</font></i>: 
 <a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <font color="Maroon" title="c6">F</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>
 
<b>by </b><i><a class="ref" href="mboolean.html#T19" title="MBOOLEAN:th.19">MBOOLEAN:19</a></i>;<br/>
<a NAME="E4:69_1_1"/>
<font color="Maroon" title="c3">J</font> <a href="closure1.html#K1" title="CLOSURE1:func.1">..</a> <span class="p1">(<span class="default"><a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <font color="Maroon" title="c6">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:69_1_1_1"/>
<font color="Maroon" title="c3">J</font> <a href="closure1.html#K1" title="CLOSURE1:func.1">..</a> <span class="p1">(<span class="default"><a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a>  <a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <font color="Maroon" title="c6">F</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D13" title="PBOOLE:def.13">PBOOLE:def 13</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <font color="Maroon" title="c6">F</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">J</font> <a href="closure1.html#K1" title="CLOSURE1:func.1">..</a> <span class="p1">(<span class="default"><a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <font color="Maroon" title="c6">F</font></span>)</span></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">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> <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">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> or <span class="p1">(<span class="default"><font color="Maroon" title="c3">J</font> <a href="closure1.html#K1" title="CLOSURE1:func.1">..</a> <span class="p2">(<span class="default"><a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <font color="Maroon" title="c6">F</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> )</span><br/>

<b>assume </b><a NAME="E1:69_1_1_1_1"/><i><font color="Green" title="E28">A4</font></i>: 
<font color="Maroon" title="c7">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"><font color="Maroon" title="c3">J</font> <a href="closure1.html#K1" title="CLOSURE1:func.1">..</a> <span class="p2">(<span class="default"><a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <font color="Maroon" title="c6">F</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c8">j</font> = <font color="Maroon" title="c3">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">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="c7">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="c7">i</font> <b>by </b><i><a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">Q</font> being   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <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="c7">i</font></span>)</span><b> such that </b><br/><a NAME="E4:69_1_1_1_1"/><i><font color="Green" title="E29">A5</font></i>: 
( <font color="Maroon" title="c9">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> &amp; <span class="p1">(<span class="default"><a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">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="c9">Q</font> )
 <b>by </b><i><a class="txt" href="#E1:69_1_1_1_1"><i><font color="Green" title="E28">A4</font></i></a>, <a class="ref" href="mssubfam.html#D2" title="MSSUBFAM:def.2">MSSUBFAM:def 2</a></i>;<br/>
<a NAME="E5:69_1_1_1_1"/><i><font color="Green" title="E30">A6</font></i>: 
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">J</font>
 
<b>by </b><i><a class="txt" href="#E1:69_1_1_1_1"><i><font color="Green" title="E28">A4</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E6:69_1_1_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="c7">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="c7">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:69_1_1_1_1"><i><font color="Green" title="E28">A4</font></i></a>, <a class="ref" href="mboolean.html#D1" title="MBOOLEAN:def.1">MBOOLEAN:def 1</a></i>;<br/>
<a NAME="E7:69_1_1_1_1"/><b>then </b><i><font color="Green" title="E31">A7</font></i>: 
<font color="Maroon" title="c8">j</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c9">Q</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="c7">i</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">x</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="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Q</font> implies <font color="Maroon" title="c8">j</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c9">Q</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">x</font> )</span><br/><b>assume </b><a NAME="E1:69_1_1_1_1_1"/><i><font color="Green" title="E32">A8</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">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">j</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c9">Q</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">x</font></span><br/><a NAME="E2:69_1_1_1_1_1"/>
<font color="Maroon" title="c9">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
 <b>by </b><i><a class="txt" href="#E1:69_1_1"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="#E1:69_1_1_1_1"><i><font color="Green" title="E28">A4</font></i></a>, <a class="txt" href="#E4:69_1_1_1_1"><i><font color="Green" title="E29">A5</font></i></a>, <a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">jj</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E4:69_1_1_1_1_1"/><i><font color="Green" title="E33">A9</font></i>: 
( <font color="Maroon" title="c11">jj</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="c7">i</font> &amp; <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">jj</font> &amp; <font color="Maroon" title="c11">jj</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font> )
 <b>by </b><i><a class="txt" href="#E1:69"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="#E1:69_1_1_1_1"><i><font color="Green" title="E28">A4</font></i></a>, <a class="txt" href="#E1:69_1_1_1_1_1"><i><font color="Green" title="E32">A8</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="E5:69_1_1_1_1_1"/>
<font color="Maroon" title="c8">j</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c9">Q</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">x</font>
 <b>by </b><i><a class="txt" href="#E1:69_1_1_1_1"><i><font color="Green" title="E28">A4</font></i></a>, <a class="txt" href="#E1:69_1_1_1_1_1"><i><font color="Green" title="E32">A8</font></i></a>, <a class="txt" href="#E4:69_1_1_1_1_1"><i><font color="Green" title="E33">A9</font></i></a>, <a class="ref" href="closure1.html#T25" target="_self" title="CLOSURE1:th.25">Th25</a>, <a class="ref" href="mssubfam.html#T2" title="MSSUBFAM:th.2">MSSUBFAM:2</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="E9:69_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c8">j</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c9">Q</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="c9">Q</font>
 
<b>by </b><i><a class="txt" href="#E7:69_1_1_1_1"><i><font color="Green" title="E31">A7</font></i></a>, <a class="ref" href="mssubfam.html#T4" title="MSSUBFAM:th.4">MSSUBFAM:4</a></i>;<br/>
<b>hence </b><a NAME="E10:69_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">J</font> <a href="closure1.html#K1" title="CLOSURE1:func.1">..</a> <span class="p2">(<span class="default"><a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <font color="Maroon" title="c6">F</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
 <b>by </b><i><a class="txt" href="#E4:69_1_1_1_1"><i><font color="Green" title="E29">A5</font></i></a>, <a class="txt" href="#E5:69_1_1_1_1"><i><font color="Green" title="E30">A6</font></i></a>, <a class="ref" href="pralg_1.html#D17" title="PRALG_1:def.17">PRALG_1:def 17</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="E2:69_1_1_1"/>
 <a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <font color="Maroon" title="c6">F</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> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="#E3:69_1_1"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="mssubfam.html#T11" title="MSSUBFAM:th.11">MSSUBFAM:11</a></i>;<br/>
<b>hence </b><a NAME="E3:69_1_1_1"/>
 <a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <font color="Maroon" title="c6">F</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">J</font> <a href="closure1.html#K1" title="CLOSURE1:func.1">..</a> <span class="p1">(<span class="default"><a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <font color="Maroon" title="c6">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="closure1.html#D2" target="_self" title="CLOSURE1:def.2">Def2</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="E5:69_1_1"/>
 <a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <font color="Maroon" title="c6">F</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c4">D</font>
 <b>by </b><i><a class="txt" href="#E1:69"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="#E3:69_1_1"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="closure1.html#T28" target="_self" title="CLOSURE1:th.28">Th28</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="E2:69_1"/>
the <a href="closure1.html#U1" title="CLOSURE1:sel.1">Family</a> of <font color="Maroon" title="c5">CS</font> is <a href="mssubfam.html#V4" title="MSSUBFAM:attr.4">absolutely-multiplicative</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="closure1.html#D9" target="_self" title="CLOSURE1:def.9">CLOSURE1:def 9</a> <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="E4:69"/>
 <a href="closure1.html#G1" title="CLOSURE1:aggr.1">MSClosureStr</a>(# <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">D</font> #) is   <a href="closure1.html#NM2" title="CLOSURE1:NM.2">MSClosureSystem</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"> verum</span><br/>


</div>
