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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">Y2</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">Y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b2">Y1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">Z</font></span>)</span> where <font color="Olive" title="b2">Y1</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : <font color="Olive" title="b2">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> </span>}</span>  &amp; <font color="Olive" title="b1">Y2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> $1 );<br/>
<b>set </b><font color="Maroon" title="c4">IT</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">Y</font> where <font color="Olive" title="b1">Y</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">Y</font>] </span>}</span> ;<br/>
<a NAME="E1:22_1_1"/><i><font color="Green" title="E18">A1</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">Y</font> where <font color="Olive" title="b1">Y</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">Y</font>] </span>}</span>  is   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font>
 
<b>from </b><i><a class="ref" href="domain_1.html#S7" title="DOMAIN_1:sch.7">DOMAIN_1:sch 7</a>();<br/></i>
<a NAME="E2:22_1_1"/>
<font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 
<b>by </b><i><a class="ref" href="card_fil.html#T5" target="_self" title="CARD_FIL:th.5">Th5</a></i>;<br/>
<a NAME="E3:22_1_1"/><b>then </b>
<font color="Maroon" title="c1">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">Y1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">Z</font></span>)</span> where <font color="Olive" title="b1">Y1</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : <font color="Olive" title="b1">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> </span>}</span> 
 
;<br/>
<a NAME="E4:22_1_1"/><b>then </b>
<font color="Maroon" title="c3">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">Y1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">Z</font></span>)</span> where <font color="Olive" title="b1">Y1</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : <font color="Olive" title="b1">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E5:22_1_1"/><b>then </b>
<font color="Maroon" title="c3">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">Y</font> where <font color="Olive" title="b1">Y</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> :  ex <font color="Olive" title="b2">Y2</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b2">Y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b3">Y1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">Z</font></span>)</span> where <font color="Olive" title="b3">Y1</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : <font color="Olive" title="b3">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> </span>}</span>  &amp; <font color="Olive" title="b2">Y2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Y</font> ) </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E6:22_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">Y</font> where <font color="Olive" title="b1">Y</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">Y</font>] </span>}</span>  is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="#E1:22_1_1"><i><font color="Green" title="E18">A1</font></i></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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