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<a NAME="E1:15_1_1"/>
for <font color="Olive" title="b1">Y1</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> holds <br/> ( <font color="Olive" title="b1">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#NK1" title="CARD_FIL:NK.1">dual</a> <font color="Maroon" title="c2">F</font> iff <font color="Olive" title="b1">Y1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <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="setfam_1.html#D8" title="SETFAM_1:def.8">SETFAM_1:def 8</a></i>;<br/>
<a NAME="E2:15_1_1"/><b>then </b>
( not <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#NK1" title="CARD_FIL:NK.1">dual</a> <font color="Maroon" title="c2">F</font> &amp; ( for <font color="Olive" title="b1">Y1</font>, <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> holds <br/> ( ( <font color="Olive" title="b1">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#NK1" title="CARD_FIL:NK.1">dual</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Olive" title="b2">Y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#NK1" title="CARD_FIL:NK.1">dual</a> <font color="Maroon" title="c2">F</font> implies <font color="Olive" title="b1">Y1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">Y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#NK1" title="CARD_FIL:NK.1">dual</a> <font color="Maroon" title="c2">F</font> ) &amp; ( <font color="Olive" title="b1">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#NK1" title="CARD_FIL:NK.1">dual</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Olive" title="b2">Y2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Y1</font> implies <font color="Olive" title="b2">Y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#NK1" title="CARD_FIL:NK.1">dual</a> <font color="Maroon" title="c2">F</font> ) ) ) )
 
<b>by </b><i><a class="ref" href="card_fil.html#T7" target="_self" title="CARD_FIL:th.7">Th7</a></i>;<br/>
<b>hence </b><a NAME="E3:15_1_1"/>
 <a href="card_fil.html#NK1" title="CARD_FIL:NK.1">dual</a> <font color="Maroon" title="c2">F</font> is    <a href="card_fil.html#M2" title="CARD_FIL:mode.2">Ideal</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="card_fil.html#D2" target="_self" title="CARD_FIL:def.2">Def2</a></i>;<br/>


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