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<b>let </b><font color="Maroon" title="c3">SF</font> be   <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of <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">SF</font> is <a href="closure2.html#V4" title="CLOSURE2:attr.4">absolutely-multiplicative</a> implies <font color="Maroon" title="c3">SF</font> is <a href="closure2.html#V5" title="CLOSURE2:attr.5">properly-upper-bound</a> )</span><br/>

<b>assume </b><a NAME="E1:45_1_1"/><i><font color="Green" title="E25">A1</font></i>: 
<font color="Maroon" title="c3">SF</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"> <font color="Maroon" title="c3">SF</font> is <a href="closure2.html#V5" title="CLOSURE2:attr.5">properly-upper-bound</a></span><br/>

<b>reconsider </b><font color="Maroon" title="c4">a</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as   <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of <font color="Maroon" title="c2">M</font> <b>by </b><i><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/>
<a NAME="E3:45_1_1"/>
<span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="pboole.html#K1" title="PBOOLE:func.1">[0]</a> <font color="Maroon" title="c1">I</font>
 
<b>by </b><i><a class="ref" href="closure2.html#D4" target="_self" title="CLOSURE2:def.4">Def4</a></i>;<br/>
<a NAME="E4:45_1_1"/><b>then </b><i><font color="Green" title="E26">A2</font></i>: 
 <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="c4">a</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c2">M</font>
 
<b>by </b><i><a class="ref" href="mssubfam.html#T41" title="MSSUBFAM:th.41">MSSUBFAM:41</a></i>;<br/>
<a NAME="E5:45_1_1"/>
<font color="Maroon" title="c4">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">SF</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/>
<b>hence </b><a NAME="E6:45_1_1"/>
<font color="Maroon" title="c2">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">SF</font>
 <b>by </b><i><a class="txt" href="#E1:45_1_1"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="#E4:45_1_1"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="closure2.html#D8" target="_self" title="CLOSURE2:def.8">Def8</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/>


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