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<b>let </b><font color="Maroon">I</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">M</font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>;<br/>



<b>thus </b><a NAME="E1:41"/>
 <a href="closure2.html#K2" target="_self">Bool</a> <font color="Maroon">M</font> is <a href="closure2.html#V1" target="_self">additive</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="closure2.html#D5" target="_self">CLOSURE2:def 5</a><br/><b>let </b><font color="Maroon">B</font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>;<br/>



<b>assume </b><a NAME="E1:41_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="closure2.html#K2" target="_self">Bool</a> <font color="Maroon">M</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="closure2.html#K2" target="_self">Bool</a> <font color="Maroon">M</font> )
 ;<br/>

<b>hence </b><a NAME="E2:41_1"/>
<font color="Maroon">A</font> <a href="pboole.html#K2">\/</a> <font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="closure2.html#K2" target="_self">Bool</a> <font color="Maroon">M</font>
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E2:41"/>
 <a href="closure2.html#K2" target="_self">Bool</a> <font color="Maroon">M</font> is <a href="closure2.html#V2" target="_self">absolutely-additive</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">F</font> be   <a href="closure2.html#NM1">SubsetFamily</a> of <font color="Maroon">M</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="closure2.html#D6" target="_self">CLOSURE2:def 6</a><br/>

<b>assume </b><a NAME="E1:41_2"/>
<font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="closure2.html#K2" target="_self">Bool</a> <font color="Maroon">M</font>
 ;<br/>

<a NAME="E2:41_2"/>
 <a href="mboolean.html#K2">union</a> <span class="p1"><a href="closure2.html#K5" target="_self">|:</a><span class="default"><font color="Maroon">F</font></span><a href="closure2.html#K5" target="_self">:|</a></span> <a href="pboole.html#R2">c=</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="mssubfam.html#T40">MSSUBFAM:40</a></i>;<br/>
<a NAME="E3:41_2"/><b>then </b>
 <a href="mboolean.html#K2">union</a> <span class="p1"><a href="closure2.html#K5" target="_self">|:</a><span class="default"><font color="Maroon">F</font></span><a href="closure2.html#K5" target="_self">:|</a></span> is    <a href="pboole.html#M4">ManySortedSubset</a> of <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<b>hence </b><a NAME="E4:41_2"/>
 <a href="mboolean.html#K2">union</a> <span class="p1"><a href="closure2.html#K5" target="_self">|:</a><span class="default"><font color="Maroon">F</font></span><a href="closure2.html#K5" target="_self">:|</a></span> <a href="hidden.html#R2">in</a>  <a href="closure2.html#K2" target="_self">Bool</a> <font color="Maroon">M</font>
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E3:41"/>
 <a href="closure2.html#K2" target="_self">Bool</a> <font color="Maroon">M</font> is <a href="closure2.html#V3" target="_self">multiplicative</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="closure2.html#D7" target="_self">CLOSURE2:def 7</a><br/><b>let </b><font color="Maroon">B</font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>;<br/>



<b>assume </b><a NAME="E1:41_3"/>
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="closure2.html#K2" target="_self">Bool</a> <font color="Maroon">M</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="closure2.html#K2" target="_self">Bool</a> <font color="Maroon">M</font> )
 ;<br/>

<b>hence </b><a NAME="E2:41_3"/>
<font color="Maroon">A</font> <a href="pboole.html#K3">/\</a> <font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="closure2.html#K2" target="_self">Bool</a> <font color="Maroon">M</font>
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E4:41"/>
 <a href="closure2.html#K2" target="_self">Bool</a> <font color="Maroon">M</font> is <a href="closure2.html#V4" target="_self">absolutely-multiplicative</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_4"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:41_4"/>
<font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="closure2.html#K2" target="_self">Bool</a> <font color="Maroon">M</font>
 ;<br/>

<b>thus </b><a NAME="E2:41_4"/>
 <a href="mssubfam.html#K6">meet</a> <span class="p1"><a href="closure2.html#K5" target="_self">|:</a><span class="default"><font color="Maroon">F</font></span><a href="closure2.html#K5" target="_self">:|</a></span> <a href="hidden.html#R2">in</a>  <a href="closure2.html#K2" target="_self">Bool</a> <font color="Maroon">M</font>
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E5:41"/>
 <a href="closure2.html#K2" target="_self">Bool</a> <font color="Maroon">M</font> is <a href="closure2.html#V5" target="_self">properly-upper-bound</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_5"><b>proof </b></a><div class="add">

<a NAME="E1:41_5"/>
<font color="Maroon">M</font> is    <a href="pboole.html#M4">ManySortedSubset</a> of <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<b>hence </b><a NAME="E2:41_5"/>
<font color="Maroon">M</font> <a href="hidden.html#R2">in</a>  <a href="closure2.html#K2" target="_self">Bool</a> <font color="Maroon">M</font>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="closure2.html#D9" target="_self">CLOSURE2:def 9</a><br/>


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

<a NAME="E6:41"/>
 <a href="pboole.html#K1">[0]</a> <font color="Maroon">I</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="pboole.html#T49">PBOOLE:49</a></i>;<br/>
<a NAME="E7:41"/><b>then </b>
 <a href="pboole.html#K1">[0]</a> <font color="Maroon">I</font> is    <a href="pboole.html#M4">ManySortedSubset</a> of <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<b>hence </b><a NAME="E8:41"/>
 <a href="pboole.html#K1">[0]</a> <font color="Maroon">I</font> <a href="hidden.html#R2">in</a>  <a href="closure2.html#K2" target="_self">Bool</a> <font color="Maroon">M</font>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="closure2.html#D10" target="_self">CLOSURE2:def 10</a><br/>


</div>
