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<b>let </b><font color="Maroon">Z</font> be  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/>

<a NAME="E1:69"/><i><font color="Green">E35</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">Z</font>
 
;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">Z</font> being <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>   st  <a href="card_1.html#K4">card</a> <font color="Olive">Z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; ( for <font color="Olive">X</font>, <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Olive">Z</font> &amp; <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Olive">Z</font> &amp; not <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> holds <br/><font color="Olive">Y</font> <a href="tarski.html#R1">c=</a> <font color="Olive">X</font> ) holds <br/> <a href="tarski.html#K3">union</a> <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <font color="Olive">Z</font>;<br/>
<a NAME="E2:69"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="ref" href="card_2.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E3:69"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<b>by </b><i><a class="txt" href="card_2.html#E8"><i><font color="Green">Lemma48</font></i></a></i>;<br/>
<a NAME="E4:69"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>hence </b><a NAME="E5:69"/>
( <font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; ( for <font color="Olive">X</font>, <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; not <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> holds <br/><font color="Olive">Y</font> <a href="tarski.html#R1">c=</a> <font color="Olive">X</font> ) implies  <a href="tarski.html#K3">union</a> <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 <b>by </b><i/>;<br/>


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