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<b>let </b><font color="Maroon" title="c1">I</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">M</font> be    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>;<br/>



<b>thus </b><a NAME="E1:78"/>
 <a href="mssubfam.html#K3" title="MSSUBFAM:func.3">bool</a> <font color="Maroon" title="c2">M</font> is <a href="mssubfam.html#V1" title="MSSUBFAM:attr.1">additive</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">A</font> be    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="mssubfam.html#D3" target="_self" title="MSSUBFAM:def.3">MSSUBFAM:def 3</a><br/><b>let </b><font color="Maroon" title="c4">B</font> be    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>;<br/>



<b>assume </b><a NAME="E1:78_1"/>
( <font color="Maroon" title="c3">A</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a>  <a href="mssubfam.html#K3" title="MSSUBFAM:func.3">bool</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c4">B</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a>  <a href="mssubfam.html#K3" title="MSSUBFAM:func.3">bool</a> <font color="Maroon" title="c2">M</font> )
 ;<br/>

<a NAME="E2:78_1"/><b>then </b>
( <font color="Maroon" title="c3">A</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c4">B</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c2">M</font> )
 
<b>by </b><i><a class="ref" href="mboolean.html#T1" title="MBOOLEAN:th.1">MBOOLEAN:1</a></i>;<br/>
<a NAME="E3:78_1"/><b>then </b>
<font color="Maroon" title="c3">A</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c4">B</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c2">M</font>
 
<b>by </b><i><a class="ref" href="pboole.html#T18" title="PBOOLE:th.18">PBOOLE:18</a></i>;<br/>
<b>hence </b><a NAME="E4:78_1"/>
<font color="Maroon" title="c3">A</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c4">B</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a>  <a href="mssubfam.html#K3" title="MSSUBFAM:func.3">bool</a> <font color="Maroon" title="c2">M</font>
 <b>by </b><i><a class="ref" href="mboolean.html#T1" title="MBOOLEAN:th.1">MBOOLEAN:1</a></i>;<br/>


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

<b>thus </b><a NAME="E2:78"/>
 <a href="mssubfam.html#K3" title="MSSUBFAM:func.3">bool</a> <font color="Maroon" title="c2">M</font> is <a href="mssubfam.html#V2" title="MSSUBFAM:attr.2">absolutely-additive</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">F</font> be   <a href="mssubfam.html#NM1" title="MSSUBFAM:NM.1">MSSubsetFamily</a> of <font color="Maroon" title="c2">M</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="mssubfam.html#D4" target="_self" title="MSSUBFAM:def.4">MSSUBFAM:def 4</a><br/>

<b>assume </b><a NAME="E1:78_2"/>
<font color="Maroon" title="c3">F</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a>  <a href="mssubfam.html#K3" title="MSSUBFAM:func.3">bool</a> <font color="Maroon" title="c2">M</font>
 ;<br/>

<a NAME="E2:78_2"/>
 <a href="mboolean.html#K2" title="MBOOLEAN:func.2">union</a> <font color="Maroon" title="c3">F</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c2">M</font>
 
<b>by </b><i><a class="ref" href="mssubfam.html#T40" target="_self" title="MSSUBFAM:th.40">Th40</a></i>;<br/>
<b>hence </b><a NAME="E3:78_2"/>
 <a href="mboolean.html#K2" title="MBOOLEAN:func.2">union</a> <font color="Maroon" title="c3">F</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a>  <a href="mssubfam.html#K3" title="MSSUBFAM:func.3">bool</a> <font color="Maroon" title="c2">M</font>
 <b>by </b><i><a class="ref" href="mboolean.html#T19" title="MBOOLEAN:th.19">MBOOLEAN:19</a></i>;<br/>


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

<b>thus </b><a NAME="E3:78"/>
 <a href="mssubfam.html#K3" title="MSSUBFAM:func.3">bool</a> <font color="Maroon" title="c2">M</font> is <a href="mssubfam.html#V3" title="MSSUBFAM:attr.3">multiplicative</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">A</font> be    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="mssubfam.html#D5" target="_self" title="MSSUBFAM:def.5">MSSUBFAM:def 5</a><br/><b>let </b><font color="Maroon" title="c4">B</font> be    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>;<br/>



<b>assume </b><a NAME="E1:78_3"/>
( <font color="Maroon" title="c3">A</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a>  <a href="mssubfam.html#K3" title="MSSUBFAM:func.3">bool</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c4">B</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a>  <a href="mssubfam.html#K3" title="MSSUBFAM:func.3">bool</a> <font color="Maroon" title="c2">M</font> )
 ;<br/>

<a NAME="E2:78_3"/><b>then </b>
( <font color="Maroon" title="c3">A</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c4">B</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c2">M</font> )
 
<b>by </b><i><a class="ref" href="mboolean.html#T1" title="MBOOLEAN:th.1">MBOOLEAN:1</a></i>;<br/>
<a NAME="E3:78_3"/><b>then </b>
<font color="Maroon" title="c3">A</font> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <font color="Maroon" title="c4">B</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c2">M</font>
 
<b>by </b><i><a class="ref" href="mboolean.html#T15" title="MBOOLEAN:th.15">MBOOLEAN:15</a></i>;<br/>
<b>hence </b><a NAME="E4:78_3"/>
<font color="Maroon" title="c3">A</font> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <font color="Maroon" title="c4">B</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a>  <a href="mssubfam.html#K3" title="MSSUBFAM:func.3">bool</a> <font color="Maroon" title="c2">M</font>
 <b>by </b><i><a class="ref" href="mboolean.html#T1" title="MBOOLEAN:th.1">MBOOLEAN:1</a></i>;<br/>


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

<b>thus </b><a NAME="E4:78"/>
 <a href="mssubfam.html#K3" title="MSSUBFAM:func.3">bool</a> <font color="Maroon" title="c2">M</font> is <a href="mssubfam.html#V4" title="MSSUBFAM:attr.4">absolutely-multiplicative</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">F</font> be   <a href="mssubfam.html#NM1" title="MSSUBFAM:NM.1">MSSubsetFamily</a> of <font color="Maroon" title="c2">M</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="mssubfam.html#D6" target="_self" title="MSSUBFAM:def.6">MSSUBFAM:def 6</a><br/>

<b>assume </b><a NAME="E1:78_4"/>
<font color="Maroon" title="c3">F</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a>  <a href="mssubfam.html#K3" title="MSSUBFAM:func.3">bool</a> <font color="Maroon" title="c2">M</font>
 ;<br/>

<a NAME="E2:78_4"/>
 <a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <font color="Maroon" title="c3">F</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c2">M</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></i>;<br/>
<b>hence </b><a NAME="E3:78_4"/>
 <a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <font color="Maroon" title="c3">F</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a>  <a href="mssubfam.html#K3" title="MSSUBFAM:func.3">bool</a> <font color="Maroon" title="c2">M</font>
 <b>by </b><i><a class="ref" href="mboolean.html#T19" title="MBOOLEAN:th.19">MBOOLEAN:19</a></i>;<br/>


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

<b>thus </b><a NAME="E5:78"/>
 <a href="mssubfam.html#K3" title="MSSUBFAM:func.3">bool</a> <font color="Maroon" title="c2">M</font> is <a href="mssubfam.html#V5" title="MSSUBFAM:attr.5">properly-upper-bound</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_5"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:78_5"/>
<font color="Maroon" title="c2">M</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a>  <a href="mssubfam.html#K3" title="MSSUBFAM:func.3">bool</a> <font color="Maroon" title="c2">M</font>
 <b>by </b><i><a class="ref" href="mboolean.html#T19" title="MBOOLEAN:th.19">MBOOLEAN:19</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="mssubfam.html#D7" target="_self" title="MSSUBFAM:def.7">MSSUBFAM:def 7</a><br/>


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

<b>thus </b><a NAME="E6:78"/>
 <a href="mssubfam.html#K3" title="MSSUBFAM:func.3">bool</a> <font color="Maroon" title="c2">M</font> is <a href="mssubfam.html#V6" title="MSSUBFAM:attr.6">properly-lower-bound</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_6"><b>proof </b></a><div class="add">

<a NAME="E1:78_6"/>
 <a href="pboole.html#K1" title="PBOOLE:func.1">[0]</a> <font color="Maroon" title="c1">I</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c2">M</font>
 
<b>by </b><i><a class="ref" href="mboolean.html#T5" title="MBOOLEAN:th.5">MBOOLEAN:5</a></i>;<br/>
<b>hence </b><a NAME="E2:78_6"/>
 <a href="pboole.html#K1" title="PBOOLE:func.1">[0]</a> <font color="Maroon" title="c1">I</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a>  <a href="mssubfam.html#K3" title="MSSUBFAM:func.3">bool</a> <font color="Maroon" title="c2">M</font>
 <b>by </b><i><a class="ref" href="mboolean.html#T1" title="MBOOLEAN:th.1">MBOOLEAN:1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="mssubfam.html#D8" target="_self" title="MSSUBFAM:def.8">MSSUBFAM:def 8</a><br/>


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


</div>
