<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">X</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>



<b>assume </b><a NAME="E1:2"/><i><font color="Green" title="E3">A1</font></i>: 
<font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c1">X</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>( $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c1">X</font> implies  ex <font color="Olive" title="b1">Y</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 );<br/>
<a NAME="E2:2"/><i><font color="Green" title="E4">A2</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" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c1">X</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c3">e</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c3">e</font>
;<br/>

<b>thus </b><a NAME="E3:2_1"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="card_1.html#T78" title="CARD_1:th.78">CARD_1:78</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green" title="E5">A3</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" title="c3">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:2_2"/><i><font color="Green" title="E6">A4</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font>]
 ;<br/><b>thus </b><a NAME="E2:2_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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"/><i><font color="Green" title="E7">A5</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c1">X</font>
 ;<br/>

<a NAME="E2:2_2_1"/><b>then </b><i><font color="Green" title="E8">A6</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">Y</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E4:2_2_1"/><i><font color="Green" title="E9">A7</font></i>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c4">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="txt" href="pencil_4.html#E1:2_2"><i><font color="Green" title="E6">A4</font></i></a>, <a class="txt" href="pencil_4.html#E1:2_2_1"><i><font color="Green" title="E7">A5</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E5:2_2_1"/>
not <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Y</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a>, <a class="txt" href="pencil_4.html#E4:2_2_1"><i><font color="Green" title="E9">A7</font></i></a>, <a class="txt" href="pencil_4.html#E2:2_2_1"><i><font color="Green" title="E8">A6</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:2_2_1"/><i><font color="Green" title="E10">A8</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E8:2_2_1"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a>, <a class="txt" href="pencil_4.html#E7:2_2_1"><i><font color="Green" title="E10">A8</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">Z</font> = <font color="Maroon" title="c4">Y</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">Z</font>
;<br/>

<b>thus </b><a NAME="E10:2_2_1"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="card_2.html#T54" title="CARD_2:th.54">CARD_2:54</a>, <a class="txt" href="pencil_4.html#E7:2_2_1"><i><font color="Green" title="E10">A8</font></i></a>, <a class="txt" href="pencil_4.html#E4:2_2_1"><i><font color="Green" title="E9">A7</font></i></a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E4:2"/>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="pencil_4.html#E2:2"><i><font color="Green" title="E4">A2</font></i></a>, <a class="txt" href="pencil_4.html#E3:2"><i><font color="Green" title="E5">A3</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:2"/>
 ex <font color="Olive" title="b1">Y</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="pencil_4.html#E1:2"><i><font color="Green" title="E3">A1</font></i></a></i>;<br/>


</div>
