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<a NAME="E1:15_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#NK1" target="_self">dual</a> <font color="Maroon">c<sub>2</sub></font> iff <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="setfam_1.html#D8">SETFAM_1:def 8</a></i>;<br/>
<a NAME="E2:15_1_1"/><b>then </b>
( not <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#NK1" target="_self">dual</a> <font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#NK1" target="_self">dual</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#NK1" target="_self">dual</a> <font color="Maroon">c<sub>2</sub></font> implies <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#NK1" target="_self">dual</a> <font color="Maroon">c<sub>2</sub></font> ) &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#NK1" target="_self">dual</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> implies <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#NK1" target="_self">dual</a> <font color="Maroon">c<sub>2</sub></font> ) ) ) )
 
<b>by </b><i><a class="ref" href="card_fil.html#T7" target="_self">Th7</a></i>;<br/>
<b>hence </b><a NAME="E3:15_1_1"/>
 <a href="card_fil.html#NK1" target="_self">dual</a> <font color="Maroon">c<sub>2</sub></font> is    <a href="card_fil.html#M2" target="_self">Ideal</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="card_fil.html#D2" target="_self">Def2</a></i>;<br/>


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