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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] means <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="wellord2.html#R2">are_equipotent</a> ;<br/>
<a NAME="E1:10_1_1"/><i><font color="Green">E5</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>by </b><i><a class="ref" href="card_1.html#T4" target="_self">Th4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E3:10_1_1"/><i><font color="Green">E6</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>] &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>from </b><i><a class="ref" href="ordinal1.html#S1">ORDINAL1:sch 1</a>(<a class="txt" href="card_1.html#E1:10_1_1"><i><font color="Green">E5</font></i></a>);<br/></i>
<a NAME="E4:10_1_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="card_1.html#V1" target="_self">cardinal</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">c<sub>2</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="card_1.html#D1" target="_self">CARD_1:def 1</a><br/>

<b>thus </b><a NAME="E1:10_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E2:10_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="wellord2.html#R2">are_equipotent</a> 
 ;<br/>

<a NAME="E3:10_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="wellord2.html#R2">are_equipotent</a> 
 
<b>by </b><i><a class="txt" href="card_1.html#E3:10_1_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="wellord2.html#T22">WELLORD2:22</a></i>;<br/>
<b>hence </b><a NAME="E4:10_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="card_1.html#E3:10_1_1"><i><font color="Green">E6</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="card_1.html#NM1">Cardinal</a> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>3</sub></font>
;<br/>

<b>thus </b><a NAME="E6:10_1_1"/>
<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="wellord2.html#R2">are_equipotent</a> 
 <b>by </b><i><a class="txt" href="card_1.html#E3:10_1_1"><i><font color="Green">E6</font></i></a></i>;<br/>


</div>
