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<a NAME="E1:91_1_1"/><span class="lab"><font color="Green" title="E36">A1</font></span>: 
 the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="closure2.html#K10" title="CLOSURE2:func.10">ClOp-&gt;ClSys</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="closure2.html#D21" target="_self" title="CLOSURE2:def.21">Def23</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">SF</font> = <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">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] </span> } </span> ;<br/>
<a NAME="E2:91_1_1"/><span class="lab"><font color="Green" title="E37">A2</font></span>: 
<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">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] </span> } </span>  <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> <font color="Maroon" title="c3">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="closure2.html#D21" target="_self" title="CLOSURE2:def.21">Def23</a></span>;<br/>
<a NAME="E3:91_1_1"/>
<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">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] </span> } </span>  is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> <font color="Maroon" title="c2">A</font></span>)</span>
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="domain_1.html#S7" title="DOMAIN_1:sch.7">DOMAIN_1:sch 7</a>();<br/></span>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">D</font> = <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">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] </span> } </span>  as   <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of <font color="Maroon" title="c2">A</font> ;<br/>
<a NAME="E5:91_1_1"/><span class="lab"><font color="Green" title="E38">A3</font></span>: 
 <a href="closure2.html#K10" title="CLOSURE2:func.10">ClOp-&gt;ClSys</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="closure2.html#G1" title="CLOSURE2:aggr.1">ClosureStr</a>(# <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c5">D</font> #)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91_1_1"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E2:91_1_1"><span class="lab"><font color="Green" title="E37">A2</font></span></a></span>;<br/>
<a NAME="E6:91_1_1"/>
<font color="Maroon" title="c5">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="91_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="closure2.html#D7" target="_self" title="CLOSURE2:def.7">CLOSURE2:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">D</font> implies  <a href="mssubfam.html#K4" title="MSSUBFAM:func.4">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="c5">D</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:91_1_1_1"/><span class="lab"><font color="Green" title="E39">A4</font></span>: 
<font color="Maroon" title="c6">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">D</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="mssubfam.html#K4" title="MSSUBFAM:func.4">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="c5">D</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">mf</font> =  <a href="mssubfam.html#K4" title="MSSUBFAM:func.4">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="closure2.html#D1" target="_self" title="CLOSURE2:def.1">Def1</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">Z1</font> be   <a href="pboole.html#NM1" title="PBOOLE:NM.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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> implies <font color="Maroon" title="c3">g</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> )</span><br/><span class="kw">assume </span><a NAME="E1:91_1_1_1_1"/><span class="lab"><font color="Green" title="E40">A5</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">g</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></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">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> ;<br/><a NAME="E3:91_1_1_1_1"/>
<font color="Maroon" title="c8">Z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91_1_1_1"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="txt" href="#E1:91_1_1_1_1"><span class="lab"><font color="Green" title="E40">A5</font></span></a></span>;<br/><a NAME="E4:91_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A6</font></span>: 
 ex <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> <font color="Maroon" title="c2">A</font> st <br/>( <font color="Maroon" title="c8">Z1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Maroon" title="c3">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Olive" title="b1">a</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b1">a</font> )
 ;<br/><a NAME="E5:91_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="c9">y1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91_1_1_1_1"><span class="lab"><font color="Green" title="E40">A5</font></span></a>, <a class="ref" href="closure2.html#T21" target="_self" title="CLOSURE2:th.21">Th22</a>, <a class="ref" href="mssubfam.html#T43" title="MSSUBFAM:th.43">MSSUBFAM:43</a></span>;<br/><span class="kw">hence </span><a NAME="E6:91_1_1_1_1"/>
<font color="Maroon" title="c3">g</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:91_1_1_1_1"><span class="lab"><font color="Green" title="E41">A6</font></span></a>, <a class="ref" href="closure2.html#D11" target="_self" title="CLOSURE2:def.11">Def13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E4:91_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A7</font></span>: 
<font color="Maroon" title="c3">g</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="closure2.html#T24" target="_self" title="CLOSURE2:th.24">Th25</a></span>;<br/>
<a NAME="E5:91_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">g</font> <a href="closure2.html#K7" title="CLOSURE2:func.7">.</a> <font color="Maroon" title="c7">mf</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="closure2.html#D10" target="_self" title="CLOSURE2:def.10">Def12</a></span>;<br/>
<a NAME="E6:91_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">g</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:91_1_1_1"><span class="lab"><font color="Green" title="E40">A7</font></span></a>, <a class="ref" href="pboole.html#D10" title="PBOOLE:def.10">PBOOLE:def 10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:91_1_1_1"/>
 <a href="mssubfam.html#K4" title="MSSUBFAM:func.4">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="c5">D</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:91_1_1"/>
 <a href="closure2.html#K10" title="CLOSURE2:func.10">ClOp-&gt;ClSys</a> <font color="Maroon" title="c3">g</font> is  <a href="closure2.html#V15" title="CLOSURE2:attr.15">absolutely-multiplicative</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:91_1_1"><span class="lab"><font color="Green" title="E38">A3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
