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<div class="add">

<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st $1,<font color="Olive" title="b1">n</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a> ;<br/>
<b>assume </b><a NAME="E1:58"/><i><font color="Green" title="E41">A1</font></i>: 
<font color="Maroon" title="c1">X</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ;<br/>

<a NAME="E2:58"/><i><font color="Green" title="E42">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 
<b>by </b><i><a class="ref" href="card_1.html#T51" target="_self" title="CARD_1:th.51">Th51</a></i>;<br/>
<a NAME="E3:58"/><i><font color="Green" title="E43">A3</font></i>: 
for <font color="Olive" title="b1">Z</font>, <font color="Olive" title="b2">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">Y</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">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="Olive" title="b1">Z</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">Z</font>, <font color="Maroon" title="c3">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><a NAME="E1:58_1"/>
( <font color="Maroon" title="c2">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c3">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font> )
 ;<br/>

<b>given </b><font color="Maroon" title="c4">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><a NAME="E3:58_1"/><i><font color="Green" title="E44">A4</font></i>: 
<font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c4">n</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a> 
 ;<br/>

<a NAME="E4:58_1"/><i><font color="Green" title="E45">A5</font></i>: 
( <font color="Maroon" title="c2">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font> implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">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="c2">Z</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:58_1_1"/>
<font color="Maroon" title="c2">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font>
 ;<br/>

<a NAME="E2:58_1_1"/><b>then </b><i><font color="Green" title="E46">A6</font></i>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">Z</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="c3">Y</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">n</font>
;<br/>

<b>thus </b><a NAME="E3:58_1_1"/>
<font color="Maroon" title="c3">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="c2">Z</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c4">n</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a> 
 <b>by </b><i><a class="txt" href="card_1.html#E3:58_1"><i><font color="Green" title="E44">A4</font></i></a>, <a class="txt" href="card_1.html#E2:58_1_1"><i><font color="Green" title="E46">A6</font></i></a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:58_1"/>
( not <font color="Maroon" title="c2">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font> implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">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="c2">Z</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:58_1_2"/><i><font color="Green" title="E46">A7</font></i>: 
not <font color="Maroon" title="c2">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font>
 ;<br/>

<b>take </b><font color="Maroon" title="c5">m</font> = <font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/>

<i><font color="Green" title="E47">A8</font></i>: <font color="Maroon" title="c5">m</font> = 
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">n</font>
<b>by </b><i><a class="ref" href="card_1.html#T52" target="_self" title="CARD_1:th.52">Th52</a></i>
<br/>.= 
<font color="Maroon" title="c4">n</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="c4">n</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>

;<br/>
<a NAME="E3:58_1_2"/><i><font color="Green" title="E48">A9</font></i>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">Z</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">n</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a> 
 
<b>by </b><i><a class="ref" href="card_1.html#T48" target="_self" title="CARD_1:th.48">Th48</a></i>;<br/>
<div><i><font color="Green" title="E49">A10</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:58_1_2_2"/>
<font color="Maroon" title="c4">n</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">n</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><b>then consider </b><font color="Maroon" title="c6">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:58_1_2_2"/><i><font color="Green" title="E50">A11</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">n</font> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">n</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><a NAME="E4:58_1_2_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="txt" href="card_1.html#E3:58_1_2_2"><i><font color="Green" title="E50">A11</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E5:58_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="card_1.html#E3:58_1_2_2"><i><font color="Green" title="E50">A11</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_2_3"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon" title="c6">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">Y</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">Z</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/><b>assume </b><a NAME="E2:58_1_2_3"/>
<font color="Maroon" title="c3">Y</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">Z</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E3:58_1_2_3"/><b>then </b>
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">Z</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E4:58_1_2_3"/>
contradiction
 <b>by </b><i><a class="txt" href="card_1.html#E1:58_1_2"><i><font color="Green" title="E46">A7</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:58_1_2"/><b>then </b>
<font color="Maroon" title="c3">Y</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">Z</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<b>hence </b><a NAME="E7:58_1_2"/>
<font color="Maroon" title="c3">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="c2">Z</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c5">m</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a> 
 <b>by </b><i><a class="txt" href="card_1.html#E3:58_1"><i><font color="Green" title="E44">A4</font></i></a>, <a class="txt" href="card_1.html#E2:58_1_2"><i><font color="Green" title="E47">A8</font></i></a>, <a class="txt" href="card_1.html#E3:58_1_2"><i><font color="Green" title="E48">A9</font></i></a>, <a class="txt" href="card_1.html#E4:58_1_2"><i><font color="Green" title="E49">A10</font></i></a>, <a class="ref" href="card_1.html#T58" target="_self" title="CARD_1:th.58">Th58</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:58_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">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="c2">Z</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>]
 <b>by </b><i><a class="txt" href="card_1.html#E4:58_1"><i><font color="Green" title="E45">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E4:58"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">X</font>]
 <b>from </b><i><a class="ref" href="finset_1.html#S2" title="FINSET_1:sch.2">FINSET_1:sch 2</a>(<a class="txt" href="card_1.html#E1:58"><i><font color="Green" title="E41">A1</font></i></a>, <a class="txt" href="card_1.html#E2:58"><i><font color="Green" title="E42">A2</font></i></a>, <a class="txt" href="card_1.html#E3:58"><i><font color="Green" title="E43">A3</font></i></a>);</i><br/>


</div>
