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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>; <i><font color="Red">:: according to </font></i><a class="ref" href="card_fil.html#D14" target="_self" title="CARD_FIL:def.14">CARD_FIL:def 14</a><br/>

<b>assume </b><a NAME="E1:49"/>
<font color="Maroon" title="c1">N</font> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a>  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ;<br/>

<a NAME="E2:49"/><b>then </b>
<font color="Maroon" title="c1">N</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="ref" href="card_4.html#T9" title="CARD_4:th.9">CARD_4:9</a></i>;<br/>
<a NAME="E3:49"/><b>then </b><i><font color="Green" title="E33">A1</font></i>: 
 <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">N</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="ref" href="finset_1.html#T24" title="FINSET_1:th.24">FINSET_1:24</a></i>;<br/>
<a NAME="E4:49"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">N</font></span>)</span>, <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">N</font> <a href="tarski.html#R2" title="TARSKI:pred.2">are_equipotent</a> 
 
<b>by </b><i><a class="ref" href="card_1.html#D5" title="CARD_1:def.5">CARD_1:def 5</a></i>;<br/>
<a NAME="E5:49"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">N</font></span>)</span> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="txt" href="card_fil.html#E3:49"><i><font color="Green" title="E33">A1</font></i></a>, <a class="ref" href="card_1.html#T68" title="CARD_1:th.68">CARD_1:68</a></i>;<br/>
<a NAME="E6:49"/><b>then </b>
 <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> 2,<span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">N</font></span>)</span> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="ref" href="card_2.html#T44" title="CARD_2:th.44">CARD_2:44</a></i>;<br/>
<a NAME="E7:49"/><b>then </b>
 <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> 2,<font color="Maroon" title="c1">N</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="ref" href="card_1.html#D5" title="CARD_1:def.5">CARD_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E8:49"/>
 <a href="card_2.html#K3" title="CARD_2:func.3">exp</a> 2,<font color="Maroon" title="c1">N</font> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a>  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 <b>by </b><i><a class="ref" href="card_4.html#T9" title="CARD_4:th.9">CARD_4:9</a></i>;<br/>


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