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<a NAME="E1:85_1_1"/>
 <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>  <a href="mssubfam.html#K5" title="MSSUBFAM:func.5">bool</a> <font color="Maroon" title="c2">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T49" title="PBOOLE:th.49">PBOOLE:49</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">a</font> =  <a href="pboole.html#K1" title="PBOOLE:func.1">[[0]]</a> <font color="Maroon" title="c1">I</font> as   <a href="mssubfam.html#NM1" title="MSSUBFAM:NM.1">MSSubsetFamily</a> of <font color="Maroon" title="c2">M</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c4">SF</font> be   <a href="mssubfam.html#NM1" title="MSSUBFAM:NM.1">MSSubsetFamily</a> of <font color="Maroon" title="c2">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">SF</font> is  <a href="mssubfam.html#V4" title="MSSUBFAM:attr.4">absolutely-multiplicative</a>  implies <font color="Maroon" title="c4">SF</font> is  <a href="mssubfam.html#V5" title="MSSUBFAM:attr.5">properly-upper-bound</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E3:85_1_1"/><span class="lab"><font color="Green" title="E21">A1</font></span>: 
<font color="Maroon" title="c4">SF</font> is  <a href="mssubfam.html#V4" title="MSSUBFAM:attr.4">absolutely-multiplicative</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">SF</font> is  <a href="mssubfam.html#V5" title="MSSUBFAM:attr.5">properly-upper-bound</a> </span><br/>

<a NAME="E4:85_1_1"/>
(  <a href="mssubfam.html#K4" title="MSSUBFAM:func.4">meet</a> <font color="Maroon" title="c3">a</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c2">M</font> &amp;  <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="c4">SF</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mssubfam.html#T41" target="_self" title="MSSUBFAM:th.41">Th41</a>, <a class="ref" href="pboole.html#T49" title="PBOOLE:th.49">PBOOLE:49</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:85_1_1"/>
<font color="Maroon" title="c2">M</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c4">SF</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:85_1_1"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="ref" href="mssubfam.html#D6" target="_self" title="MSSUBFAM:def.6">Def6</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="mssubfam.html#D7" target="_self" title="MSSUBFAM:def.7">MSSUBFAM:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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