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<b>let </b><font color="Maroon">X</font> be  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><a NAME="E1:2"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">X</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">X</font> implies  ex <font color="Olive">Y</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> st  <a href="card_1.html#K4">card</a> <font color="Olive">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E2:2"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:2_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">X</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">e</font> =  <a href="xboole_0.html#K1">{}</a>  as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>
<b>take </b>
<font color="Maroon">e</font>
;<br/>

<b>thus </b><a NAME="E3:2_1"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="card_1.html#T78">CARD_1:78</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E31</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:2_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/><b>thus </b><a NAME="E2:2_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:2_2_1"/>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:2_2_1"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>then consider </b><font color="Maroon">Y</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E4:2_2_1"/><i><font color="Green">E34</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/>
<a NAME="E5:2_2_1"/>
not <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>, , </i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:2_2_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E8:2_2_1"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">Z</font> = <font color="Maroon">Y</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<b>take </b>
<font color="Maroon">Z</font>
;<br/>

<b>thus </b><a NAME="E10:2_2_1"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="card_2.html#T54">CARD_2:54</a>, , </i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E4:2"/>
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(, );<br/></i>
<b>hence </b><a NAME="E5:2"/>
 ex <font color="Olive">Y</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> st  <a href="card_1.html#K4">card</a> <font color="Olive">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/>


</div>
