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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="card_1.html#NM1">Cardinal</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>





<b>assume </b><a NAME="E1:22"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="card_fil.html#R1" target="_self">is_multiplicative_with</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="card_fil.html#D4" target="_self">CARD_FIL:def 4</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E2:22"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="card_fil.html#NK1" target="_self">dual</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E3:22"/><i><font color="Green">E17</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>4</sub></font> <a href="card_1.html#NR2" target="_self">&lt;`</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E4:22"/>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E2:22"><i><font color="Green">E16</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="card_fil.html#NK1" target="_self">dual</a> <font color="Maroon">c<sub>5</sub></font>;<br/>
<a NAME="E6:22"/><i><font color="Green">E18</font></i>: 
 <a href="card_fil.html#NK1" target="_self">dual</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E2:22"><i><font color="Green">E16</font></i></a>, <a class="ref" href="setfam_1.html#T52">SETFAM_1:52</a></i>;<br/>
<a NAME="E7:22"/><i><font color="Green">E19</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="card_fil.html#NK1" target="_self">dual</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="card_fil.html#T9" target="_self">Th9</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">a<sub>1</sub></font> <a href="subset_1.html#K3">`</a> ;<br/>
<a NAME="E8:22"/>
 <a href="card_1.html#K1">Card</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> </span>}</span>  <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>from </b><i><a class="ref" href="trees_2.html#S2">TREES_2:sch 2</a>();<br/></i>
<a NAME="E9:22"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_fil.html#NK1" target="_self">dual</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="card_1.html#NR2" target="_self">&lt;`</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E3:22"><i><font color="Green">E17</font></i></a>, <a class="txt" href="card_fil.html#E7:22"><i><font color="Green">E19</font></i></a>, <a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a></i>;<br/>
<a NAME="E10:22"/><b>then </b><i><font color="Green">E20</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="setfam_1.html#K6">meet</a> <span class="p3">(<span class="default"><a href="card_fil.html#NK1" target="_self">dual</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E1:22"><i><font color="Green">E15</font></i></a>, <a class="txt" href="card_fil.html#E6:22"><i><font color="Green">E18</font></i></a>, <a class="ref" href="card_fil.html#D3" target="_self">Def3</a></i>;<br/>
<span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K3">`</a>  = 
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="subset_1.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>

<br/>.= 
 <a href="setfam_1.html#K6">meet</a> <span class="p1">(<span class="default"><a href="card_fil.html#NK1" target="_self">dual</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="setfam_1.html#T47">SETFAM_1:47</a></i>
;<br/>
<b>hence </b><a NAME="E12:22"/>
 <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#NK1" target="_self">dual</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="card_fil.html#E10:22"><i><font color="Green">E20</font></i></a>, <a class="ref" href="setfam_1.html#D8">SETFAM_1:def 8</a></i>;<br/>


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