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<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">A</font>, <font color="Olive" title="b2">M</font>, <font color="Olive" title="b3">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="Olive" title="b1">A</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b2">M</font> &amp; <font color="Olive" title="b3">B</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b2">M</font> holds <br/><span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b3">B</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> is   <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of <font color="Olive" title="b2">M</font></span><br/><b>let </b><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">M</font>, <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="c2">A</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c4">B</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">M</font> implies <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">B</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> is   <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of <font color="Maroon" title="c3">M</font> )</span><br/>







<b>assume </b><a NAME="E1:15"/>
( <font color="Maroon" title="c2">A</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c4">B</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">M</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">B</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> is   <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of <font color="Maroon" title="c3">M</font></span><br/>

<a NAME="E2:15"/><b>then </b>
( <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> is   <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of <font color="Maroon" title="c3">M</font> &amp; <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">B</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> is   <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of <font color="Maroon" title="c3">M</font> )
 
<b>by </b><i><a class="ref" href="closure2.html#T8" target="_self" title="CLOSURE2:th.8">Th8</a></i>;<br/>
<a NAME="E3:15"/><b>then </b>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">B</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> is   <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of <font color="Maroon" title="c3">M</font>
 
<b>by </b><i><a class="ref" href="closure2.html#T4" target="_self" title="CLOSURE2:th.4">Th4</a></i>;<br/>
<b>hence </b><a NAME="E4:15"/>
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">B</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> is   <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of <font color="Maroon" title="c3">M</font>
 <b>by </b><i><a class="ref" href="enumset1.html#T41" title="ENUMSET1:th.41">ENUMSET1:41</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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