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<b>let </b><font color="Maroon" title="c1">X</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">F</font> being  <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">F</font> is <a href="finsub_1.html#V1" title="FINSUB_1:attr.1">cup-closed</a> &amp; <font color="Olive" title="b1">F</font> is <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> holds <br/><font color="Olive" title="b1">F</font> is <a href="finsub_1.html#V2" title="FINSUB_1:attr.2">cap-closed</a></span><br/><b>let </b><font color="Maroon" title="c2">S</font> be   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</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">S</font> is <a href="finsub_1.html#V1" title="FINSUB_1:attr.1">cup-closed</a> &amp; <font color="Maroon" title="c2">S</font> is <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> implies <font color="Maroon" title="c2">S</font> is <a href="finsub_1.html#V2" title="FINSUB_1:attr.2">cap-closed</a> )</span><br/>



<b>assume </b><a NAME="E1:12"/><i><font color="Green" title="E8">A1</font></i>: 
( <font color="Maroon" title="c2">S</font> is <a href="finsub_1.html#V1" title="FINSUB_1:attr.1">cup-closed</a> &amp; <font color="Maroon" title="c2">S</font> is <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">S</font> is <a href="finsub_1.html#V2" title="FINSUB_1:attr.2">cap-closed</a></span><br/>

<b>let </b><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="finsub_1.html#D2" title="FINSUB_1:def.2">FINSUB_1:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c3">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> or  not <font color="Maroon" title="c4">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> or <font color="Maroon" title="c3">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> )</span><br/>

<b>assume </b><a NAME="E2:12"/><i><font color="Green" title="E9">A2</font></i>: 
( <font color="Maroon" title="c3">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c4">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</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">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font></span><br/>

<a NAME="E3:12"/><b>then </b>
( <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c4">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> )
 
<b>by </b><i><a class="txt" href="measure1.html#E1:12"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="measure1.html#D3" target="_self" title="MEASURE1:def.3">Def3</a></i>;<br/>
<a NAME="E4:12"/><b>then </b><i><font color="Green" title="E10">A3</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="txt" href="measure1.html#E1:12"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="finsub_1.html#D1" title="FINSUB_1:def.1">FINSUB_1:def 1</a></i>;<br/>
<a NAME="E5:12"/>
<font color="Maroon" title="c3">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<b>then </b><font color="Maroon" title="c3">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">B</font> = 
<font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">B</font></span>)</span>
<b>by </b><i><a class="txt" href="measure1.html#E2:12"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>
<br/>.= 
<font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T48" title="XBOOLE_1:th.48">XBOOLE_1:48</a></i>
<br/>.= 
<font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T54" title="XBOOLE_1:th.54">XBOOLE_1:54</a></i>
;<br/>
<b>hence </b><a NAME="E7:12"/>
<font color="Maroon" title="c3">A</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="measure1.html#E1:12"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="measure1.html#E4:12"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="measure1.html#D3" target="_self" title="MEASURE1:def.3">Def3</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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