<?xml version="1.0"?>
<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>let </b><font color="Maroon" title="c2">M</font> be   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a>;<br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>] <b>means </b>( $1 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c1">X</font>,$1 <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a>  );<br/>
<b>reconsider </b><font color="Maroon" title="c3">m</font> = <font color="Maroon" title="c2">M</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> ;<br/>
<b>assume </b><a NAME="E2:13"/>
<font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">M</font>
 ;<br/>

<a NAME="E3:13"/><b>then </b><i><font color="Green" title="E9">A1</font></i>: 
(  <a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">m</font> &amp;  <a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c1">X</font> is <a href="wellord1.html#V2" title="WELLORD1:attr.2">well-ordering</a> &amp;  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> )
 
<b>by </b><i><a class="ref" href="wellord2.html#T9" title="WELLORD2:th.9">WELLORD2:9</a>, <a class="ref" href="wellord2.html#T17" title="WELLORD2:th.17">WELLORD2:17</a>, <a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></i>;<br/>
<a NAME="E4:13"/><b>then </b>
<font color="Maroon" title="c1">X</font>, <a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a> 
 
<b>by </b><i><a class="ref" href="card_1.html#T22" target="_self" title="CARD_1:th.22">Th22</a></i>;<br/>
<a NAME="E5:13"/><b>then </b><i><font color="Green" title="E10">A2</font></i>: 
 ex <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>]
 
<b>by </b><i><a class="txt" href="card_1.html#E3:13"><i><font color="Green" title="E9">A1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">A</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E7:13"/><i><font color="Green" title="E11">A3</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">A</font>] &amp; ( for <font color="Olive" title="b1">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B</font>] holds <br/><font color="Maroon" title="c4">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">B</font> ) )
 <b>from </b><i><a class="ref" href="ordinal1.html#S1" title="ORDINAL1:sch.1">ORDINAL1:sch 1</a>(<a class="txt" href="card_1.html#E5:13"><i><font color="Green" title="E10">A2</font></i></a>);<br/></i>
<a NAME="E8:13"/>
<font color="Maroon" title="c4">A</font> is <a href="card_1.html#V1" title="CARD_1:attr.1">cardinal</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon" title="c4">A</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="card_1.html#D1" target="_self" title="CARD_1:def.1">CARD_1:def 1</a><br/>

<b>thus </b><a NAME="E1:13_1"/>
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c5">B</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E2:13_1"/>
<font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c4">A</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a> 
 ;<br/>

<a NAME="E3:13_1"/><b>then </b><i><font color="Green" title="E12">A4</font></i>: 
<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">B</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a> 
 
<b>by </b><i><a class="txt" href="card_1.html#E7:13"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="wellord2.html#T22" title="WELLORD2:th.22">WELLORD2:22</a></i>;<br/>
<b>assume </b><a NAME="E4:13_1"/><i><font color="Green" title="E13">A5</font></i>: 
not <font color="Maroon" title="c4">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">B</font>
 ;<br/>

<a NAME="E5:13_1"/><b>then </b>
<font color="Maroon" title="c3">m</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">B</font>
 
<b>by </b><i><a class="txt" href="card_1.html#E7:13"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="card_1.html#E3:13_1"><i><font color="Green" title="E12">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:13_1"/>
contradiction
 <b>by </b><i><a class="txt" href="card_1.html#E7:13"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="card_1.html#E4:13_1"><i><font color="Green" title="E13">A5</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c5">A</font> = <font color="Maroon" title="c4">A</font> as   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a> ;<br/>
<a NAME="E10:13"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A</font>
 
<b>by </b><i><a class="txt" href="card_1.html#E7:13"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="card_1.html#D5" target="_self" title="CARD_1:def.5">Def5</a></i>;<br/>
<b>hence </b><a NAME="E11:13"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">X</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">M</font>
 <b>by </b><i><a class="txt" href="card_1.html#E7:13"><i><font color="Green" title="E11">A3</font></i></a></i>;<br/>


</div>
