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<b>let </b><font color="Maroon" title="c3">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">SF</font> is <a href="mssubfam.html#V4" title="MSSUBFAM:attr.4">absolutely-multiplicative</a> implies <font color="Maroon" title="c3">SF</font> is <a href="mssubfam.html#V3" title="MSSUBFAM:attr.3">multiplicative</a> )</span><br/>

<b>assume </b><a NAME="E1:82_1_1"/><i><font color="Green" title="E20">A1</font></i>: 
<font color="Maroon" title="c3">SF</font> is <a href="mssubfam.html#V4" title="MSSUBFAM: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="mssubfam.html#V3" title="MSSUBFAM:attr.3">multiplicative</a></span><br/>

<b>let </b><font color="Maroon" title="c4">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> <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="c4">A</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c3">SF</font> &amp; <font color="Olive" title="b1">B</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c3">SF</font> holds <br/><font color="Maroon" title="c4">A</font> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <font color="Olive" title="b1">B</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c3">SF</font></span><br/><b>let </b><font color="Maroon" title="c5">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="c4">A</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c3">SF</font> &amp; <font color="Maroon" title="c5">B</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c3">SF</font> implies <font color="Maroon" title="c4">A</font> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <font color="Maroon" title="c5">B</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c3">SF</font> )</span><br/>



<b>assume </b><a NAME="E2:82_1_1"/><i><font color="Green" title="E21">A2</font></i>: 
( <font color="Maroon" title="c4">A</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c3">SF</font> &amp; <font color="Maroon" title="c5">B</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c3">SF</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">A</font> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <font color="Maroon" title="c5">B</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c3">SF</font></span><br/>

<a NAME="E3:82_1_1"/><b>then </b><i><font color="Green" title="E22">A3</font></i>: 
( <font color="Maroon" title="c4">A</font> is    <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c5">B</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="mssubfam.html#T33" target="_self" title="MSSUBFAM:th.33">Th33</a></i>;<br/>
<a NAME="E4:82_1_1"/><b>then </b>
( <font color="Maroon" title="c4">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="c5">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#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">ab</font> = <span class="p1"><a href="pzfmisc1.html#K2" title="PZFMISC1:func.2">{</a><span class="default"><font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">B</font></span><a href="pzfmisc1.html#K2" title="PZFMISC1:func.2">}</a></span> as   <a href="mssubfam.html#NM1" title="MSSUBFAM:NM.1">MSSubsetFamily</a> of <font color="Maroon" title="c2">M</font> <b>by </b><i><a class="ref" href="mssubfam.html#T39" target="_self" title="MSSUBFAM:th.39">Th39</a></i>;<br/>
<a NAME="E6:82_1_1"/>
( <span class="p1"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c4">A</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">SF</font> &amp; <span class="p1"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c5">B</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">SF</font> )
 
<b>by </b><i><a class="txt" href="#E2:82_1_1"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="pzfmisc1.html#T36" title="PZFMISC1:th.36">PZFMISC1:36</a></i>;<br/>
<a NAME="E7:82_1_1"/><b>then </b>
<span class="p1"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c4">A</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p1"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c5">B</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">SF</font>
 
<b>by </b><i><a class="ref" href="pboole.html#T18" title="PBOOLE:th.18">PBOOLE:18</a></i>;<br/>
<a NAME="E8:82_1_1"/><b>then </b>
<span class="p1"><a href="pzfmisc1.html#K2" title="PZFMISC1:func.2">{</a><span class="default"><font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">B</font></span><a href="pzfmisc1.html#K2" title="PZFMISC1:func.2">}</a></span> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">SF</font>
 
<b>by </b><i><a class="ref" href="pzfmisc1.html#T11" title="PZFMISC1:th.11">PZFMISC1:11</a></i>;<br/>
<a NAME="E9:82_1_1"/><b>then </b>
 <a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <font color="Maroon" title="c6">ab</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c3">SF</font>
 
<b>by </b><i><a class="txt" href="#E1:82_1_1"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="mssubfam.html#D6" target="_self" title="MSSUBFAM:def.6">Def6</a></i>;<br/>
<b>hence </b><a NAME="E10:82_1_1"/>
<font color="Maroon" title="c4">A</font> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <font color="Maroon" title="c5">B</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c3">SF</font>
 <b>by </b><i><a class="txt" href="#E3:82_1_1"><i><font color="Green" title="E22">A3</font></i></a>, <a class="ref" href="mssubfam.html#T51" target="_self" title="MSSUBFAM:th.51">Th51</a></i>; <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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