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<div class="add">

<b>let </b><font color="Maroon" title="c1">I</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"> for <font color="Olive" title="b1">M</font> being   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font> holds <br/> (  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Olive" title="b1">M</font> is <a href="closure2.html#V1" title="CLOSURE2:attr.1">additive</a> &amp;  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Olive" title="b1">M</font> is <a href="closure2.html#V2" title="CLOSURE2:attr.2">absolutely-additive</a> &amp;  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Olive" title="b1">M</font> is <a href="closure2.html#V3" title="CLOSURE2:attr.3">multiplicative</a> &amp;  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Olive" title="b1">M</font> is <a href="closure2.html#V4" title="CLOSURE2:attr.4">absolutely-multiplicative</a> &amp;  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Olive" title="b1">M</font> is <a href="closure2.html#V5" title="CLOSURE2:attr.5">properly-upper-bound</a> &amp;  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Olive" title="b1">M</font> is <a href="closure2.html#V6" title="CLOSURE2:attr.6">properly-lower-bound</a> )</span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font> is <a href="closure2.html#V1" title="CLOSURE2:attr.1">additive</a> &amp;  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font> is <a href="closure2.html#V2" title="CLOSURE2:attr.2">absolutely-additive</a> &amp;  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font> is <a href="closure2.html#V3" title="CLOSURE2:attr.3">multiplicative</a> &amp;  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font> is <a href="closure2.html#V4" title="CLOSURE2:attr.4">absolutely-multiplicative</a> &amp;  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font> is <a href="closure2.html#V5" title="CLOSURE2:attr.5">properly-upper-bound</a> &amp;  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font> is <a href="closure2.html#V6" title="CLOSURE2:attr.6">properly-lower-bound</a> )</span><br/>



<b>thus </b><a NAME="E1:40"/>
 <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font> is <a href="closure2.html#V1" title="CLOSURE2:attr.1">additive</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font> is <a href="closure2.html#V2" title="CLOSURE2:attr.2">absolutely-additive</a> &amp;  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font> is <a href="closure2.html#V3" title="CLOSURE2:attr.3">multiplicative</a> &amp;  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font> is <a href="closure2.html#V4" title="CLOSURE2:attr.4">absolutely-multiplicative</a> &amp;  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font> is <a href="closure2.html#V5" title="CLOSURE2:attr.5">properly-upper-bound</a> &amp;  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font> is <a href="closure2.html#V6" title="CLOSURE2:attr.6">properly-lower-bound</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_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="closure2.html#D5" target="_self" title="CLOSURE2:def.5">CLOSURE2:def 5</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">B</font> being   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>  st <font color="Maroon" title="c3">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font> holds <br/><font color="Maroon" title="c3">A</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font></span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c4">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font> implies <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="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font> )</span><br/>



<b>assume </b><a NAME="E1:40_1"/>
( <font color="Maroon" title="c3">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c4">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font></span><br/>

<b>hence </b><a NAME="E2:40_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="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font>
 <b>by </b><i><a class="ref" href="closure2.html#T11" target="_self" title="CLOSURE2:th.11">Th11</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>thus </b><a NAME="E2:40"/>
 <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font> is <a href="closure2.html#V2" title="CLOSURE2:attr.2">absolutely-additive</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font> is <a href="closure2.html#V3" title="CLOSURE2:attr.3">multiplicative</a> &amp;  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font> is <a href="closure2.html#V4" title="CLOSURE2:attr.4">absolutely-multiplicative</a> &amp;  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font> is <a href="closure2.html#V5" title="CLOSURE2:attr.5">properly-upper-bound</a> &amp;  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font> is <a href="closure2.html#V6" title="CLOSURE2:attr.6">properly-lower-bound</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">F</font> be   <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of <font color="Maroon" title="c2">M</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="closure2.html#D6" target="_self" title="CLOSURE2:def.6">CLOSURE2:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font> implies  <a href="mboolean.html#K2" title="MBOOLEAN:func.2">union</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c3">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>  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font> )</span><br/>

<b>assume </b><a NAME="E1:40_2"/>
<font color="Maroon" title="c3">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="mboolean.html#K2" title="MBOOLEAN:func.2">union</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c3">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>  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font></span><br/>

<a NAME="E2:40_2"/>
 <a href="mboolean.html#K2" title="MBOOLEAN:func.2">union</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c3">F</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span> <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" title="MSSUBFAM:th.40">MSSUBFAM:40</a></i>;<br/>
<a NAME="E3:40_2"/><b>then </b>
 <a href="mboolean.html#K2" title="MBOOLEAN:func.2">union</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c3">F</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span> is    <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of <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="E4:40_2"/>
 <a href="mboolean.html#K2" title="MBOOLEAN:func.2">union</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c3">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>  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font>
 <b>by </b><i><a class="ref" href="closure2.html#D1" target="_self" title="CLOSURE2:def.1">Def1</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>thus </b><a NAME="E3:40"/>
 <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font> is <a href="closure2.html#V3" title="CLOSURE2:attr.3">multiplicative</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font> is <a href="closure2.html#V4" title="CLOSURE2:attr.4">absolutely-multiplicative</a> &amp;  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font> is <a href="closure2.html#V5" title="CLOSURE2:attr.5">properly-upper-bound</a> &amp;  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font> is <a href="closure2.html#V6" title="CLOSURE2:attr.6">properly-lower-bound</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_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="closure2.html#D7" target="_self" title="CLOSURE2:def.7">CLOSURE2:def 7</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">B</font> being   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>  st <font color="Maroon" title="c3">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font> holds <br/><font color="Maroon" title="c3">A</font> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font></span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c4">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font> implies <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="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font> )</span><br/>



<b>assume </b><a NAME="E1:40_3"/>
( <font color="Maroon" title="c3">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c4">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font></span><br/>

<b>hence </b><a NAME="E2:40_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="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font>
 <b>by </b><i><a class="ref" href="closure2.html#T10" target="_self" title="CLOSURE2:th.10">Th10</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>thus </b><a NAME="E4:40"/>
 <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font> is <a href="closure2.html#V4" title="CLOSURE2:attr.4">absolutely-multiplicative</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font> is <a href="closure2.html#V5" title="CLOSURE2:attr.5">properly-upper-bound</a> &amp;  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font> is <a href="closure2.html#V6" title="CLOSURE2:attr.6">properly-lower-bound</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">F</font> be   <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of <font color="Maroon" title="c2">M</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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font> implies  <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="c3">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>  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font> )</span><br/>

<b>assume </b><a NAME="E1:40_4"/>
<font color="Maroon" title="c3">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</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> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c3">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>  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font></span><br/>

<b>thus </b><a NAME="E2:40_4"/>
 <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="c3">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>  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font>
 <b>by </b><i><a class="ref" href="closure2.html#D1" target="_self" title="CLOSURE2:def.1">Def1</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>thus </b><a NAME="E5:40"/>
 <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font> is <a href="closure2.html#V5" title="CLOSURE2:attr.5">properly-upper-bound</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font> is <a href="closure2.html#V6" title="CLOSURE2:attr.6">properly-lower-bound</a></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_5"><b>proof </b></a><div class="add">

<a NAME="E1:40_5"/>
<font color="Maroon" title="c2">M</font> is    <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of <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="E2:40_5"/>
<font color="Maroon" title="c2">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font>
 <b>by </b><i><a class="ref" href="closure2.html#D1" target="_self" title="CLOSURE2:def.1">Def1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="closure2.html#D9" target="_self" title="CLOSURE2:def.9">CLOSURE2: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>

<a NAME="E6:40"/>
 <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="pboole.html#T49" title="PBOOLE:th.49">PBOOLE:49</a></i>;<br/>
<a NAME="E7:40"/><b>then </b>
 <a href="pboole.html#K1" title="PBOOLE:func.1">[0]</a> <font color="Maroon" title="c1">I</font> is    <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of <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="E8:40"/>
 <a href="pboole.html#K1" title="PBOOLE:func.1">[0]</a> <font color="Maroon" title="c1">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="closure2.html#K2" title="CLOSURE2:func.2">Bool</a> <font color="Maroon" title="c2">M</font>
 <b>by </b><i><a class="ref" href="closure2.html#D1" target="_self" title="CLOSURE2:def.1">Def1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="closure2.html#D10" target="_self" title="CLOSURE2:def.10">CLOSURE2:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
