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<b>let </b><font color="Maroon" title="c1">N</font> be   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a>;<br/><b>let </b><font color="Maroon" title="c2">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">S</font> be  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="c2">X</font>;<br/>





<b>assume </b><a NAME="E1:22"/><i><font color="Green" title="E15">A1</font></i>: 
<font color="Maroon" title="c3">S</font> <a href="card_fil.html#R1" title="CARD_FIL:pred.1">is_multiplicative_with</a> <font color="Maroon" title="c1">N</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c4">S1</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="card_fil.html#D4" target="_self" title="CARD_FIL:def.4">CARD_FIL:def 4</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E2:22"/><i><font color="Green" title="E16">A2</font></i>: 
<font color="Maroon" title="c4">S1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_fil.html#NK1" title="CARD_FIL:NK.1">dual</a> <font color="Maroon" title="c3">S</font>
 <b>and </b><br/><a NAME="E3:22"/><i><font color="Green" title="E17">A3</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c4">S1</font> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a> <font color="Maroon" title="c1">N</font>
 ;<br/>

<a NAME="E4:22"/>
<font color="Maroon" title="c4">S1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E2:22"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">S2</font> = <font color="Maroon" title="c4">S1</font> as  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="c2">X</font> ;<br/>
<b>set </b><font color="Maroon" title="c6">S3</font> =  <a href="card_fil.html#NK1" title="CARD_FIL:NK.1">dual</a> <font color="Maroon" title="c5">S2</font>;<br/>
<a NAME="E6:22"/><i><font color="Green" title="E18">A4</font></i>: 
 <a href="card_fil.html#NK1" title="CARD_FIL:NK.1">dual</a> <font color="Maroon" title="c5">S2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">S</font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E2:22"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="setfam_1.html#T52" title="SETFAM_1:th.52">SETFAM_1:52</a></i>;<br/>
<a NAME="E7:22"/><i><font color="Green" title="E19">A5</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">Y</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> 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="c2">X</font> : <font color="Olive" title="b1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">S2</font> </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_fil.html#NK1" title="CARD_FIL:NK.1">dual</a> <font color="Maroon" title="c5">S2</font>
 
<b>by </b><i><a class="ref" href="card_fil.html#T9" target="_self" title="CARD_FIL:th.9">Th9</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c2">X</font> = $1 <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> ;<br/>
<a NAME="E8:22"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<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="c2">X</font> : <font color="Olive" title="b1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">S2</font> </span>}</span>  <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c5">S2</font>
 
<b>from </b><i><a class="ref" href="trees_2.html#S2" title="TREES_2:sch.2">TREES_2:sch 2</a>();<br/></i>
<a NAME="E9:22"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="card_fil.html#NK1" title="CARD_FIL:NK.1">dual</a> <font color="Maroon" title="c5">S2</font></span>)</span> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a> <font color="Maroon" title="c1">N</font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E3:22"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="card_fil.html#E7:22"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></i>;<br/>
<a NAME="E10:22"/><b>then </b><i><font color="Green" title="E20">A6</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <span class="p3">(<span class="default"><a href="card_fil.html#NK1" title="CARD_FIL:NK.1">dual</a> <font color="Maroon" title="c5">S2</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <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="c3">S</font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E1:22"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="card_fil.html#E6:22"><i><font color="Green" title="E18">A4</font></i></a>, <a class="ref" href="card_fil.html#D3" target="_self" title="CARD_FIL:def.3">Def3</a></i>;<br/>
<span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c5">S2</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  = 
<span class="p1">(<span class="default"><a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c5">S2</font></span>)</span>

<br/>.= 
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <span class="p1">(<span class="default"><a href="card_fil.html#NK1" title="CARD_FIL:NK.1">dual</a> <font color="Maroon" title="c5">S2</font></span>)</span>
<b>by </b><i><a class="ref" href="setfam_1.html#T47" title="SETFAM_1:th.47">SETFAM_1:47</a></i>
;<br/>
<b>hence </b><a NAME="E12:22"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c4">S1</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="c3">S</font>
 <b>by </b><i><a class="txt" href="card_fil.html#E10:22"><i><font color="Green" title="E20">A6</font></i></a>, <a class="ref" href="setfam_1.html#D8" title="SETFAM_1:def.8">SETFAM_1:def 8</a></i>;<br/>


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