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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">assume </span><a NAME="E1:84_1_1"/><span class="lab"><font color="Green" title="E21">A1</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">let </span><font color="Maroon" title="c4">A</font> be   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>; <span class="comment"><font color="Red">:: according to </font></span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">B</font> being  <a href="pboole.html#NM1" title="PBOOLE:NM.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/><span class="kw">let </span><font color="Maroon" title="c5">B</font> be   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</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">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/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:84_1_1"/><span class="lab"><font color="Green" title="E22">A2</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E3:84_1_1"/><span class="lab"><font color="Green" title="E23">A3</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="E4:84_1_1"/><span class="lab"><font color="Green" title="E24">A4</font></span>: 
<font color="Maroon" title="c5">B</font> is    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedSubset</a> of <font color="Maroon" title="c2">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:84_1_1"><span class="lab"><font color="Green" title="E23">A3</font></span></a>, <a class="ref" href="mssubfam.html#T33" target="_self" title="MSSUBFAM:th.33">Th33</a></span>;<br/>
<a NAME="E5:84_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A5</font></span>: 
<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>
 
<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/>
<a NAME="E6:84_1_1"/><span class="lab"><font color="Green" title="E26">A6</font></span>: 
<font color="Maroon" title="c4">A</font> is    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedSubset</a> of <font color="Maroon" title="c2">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:84_1_1"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="ref" href="mssubfam.html#T33" target="_self" title="MSSUBFAM:th.33">Th33</a></span>;<br/>
<a NAME="E7:84_1_1"/><span class="kw">then </span>
<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>
 
<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">then </span><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:84_1_1"><span class="lab"><font color="Green" title="E25">A5</font></span></a>, <a class="ref" href="mssubfam.html#T39" target="_self" title="MSSUBFAM:th.39">Th39</a></span>;<br/>
<a NAME="E9:84_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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:84_1_1"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="txt" href="#E3:84_1_1"><span class="lab"><font color="Green" title="E23">A3</font></span></a>, <a class="ref" href="pzfmisc1.html#T36" title="PZFMISC1:th.36">PZFMISC1:36</a></span>;<br/>
<a NAME="E10:84_1_1"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T18" title="PBOOLE:th.18">PBOOLE:18</a></span>;<br/>
<a NAME="E11:84_1_1"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pzfmisc1.html#T11" title="PZFMISC1:th.11">PZFMISC1:11</a></span>;<br/>
<a NAME="E12:84_1_1"/><span class="kw">then </span>
 <a href="mssubfam.html#K4" title="MSSUBFAM:func.4">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:84_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>;<br/>
<span class="kw">hence </span><a NAME="E13:84_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:84_1_1"><span class="lab"><font color="Green" title="E26">A6</font></span></a>, <a class="txt" href="#E4:84_1_1"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="ref" href="mssubfam.html#T51" target="_self" title="MSSUBFAM:th.51">Th51</a></span>; <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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