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<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="card_1.html#NM1">Cardinal</a>;<br/>



<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> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">a<sub>1</sub></font> <a href="wellord2.html#R2">are_equipotent</a>  );<br/>
<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="ordinal1.html#NM2">Ordinal</a> ;<br/>
<b>assume </b><a NAME="E2:13"/>
<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E3:13"/><b>then </b><i><font color="Green">E9</font></i>: 
(  <a href="wellord2.html#K2">order_type_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">c<sub>1</sub></font> is <a href="wellord1.html#V2">well-ordering</a> &amp;  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="wellord2.html#T9">WELLORD2:9</a>, <a class="ref" href="wellord2.html#T17">WELLORD2:17</a>, <a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/>
<a NAME="E4:13"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font>, <a href="wellord2.html#K2">order_type_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="wellord2.html#R2">are_equipotent</a> 
 
<b>by </b><i><a class="ref" href="card_1.html#T22" target="_self">Th22</a></i>;<br/>
<a NAME="E5:13"/><b>then </b><i><font color="Green">E10</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="txt" href="card_1.html#E3:13"><i><font color="Green">E9</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E7:13"/><i><font color="Green">E11</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</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>4</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#E5:13"><i><font color="Green">E10</font></i></a>);<br/></i>
<a NAME="E8:13"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="card_1.html#V1" target="_self">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">c<sub>4</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="E1:13_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E2:13_1"/>
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="wellord2.html#R2">are_equipotent</a> 
 ;<br/>

<a NAME="E3:13_1"/><b>then </b><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="wellord2.html#R2">are_equipotent</a> 
 
<b>by </b><i><a class="txt" href="card_1.html#E7:13"><i><font color="Green">E11</font></i></a>, <a class="ref" href="wellord2.html#T22">WELLORD2:22</a></i>;<br/>
<b>assume </b><a NAME="E4:13_1"/><i><font color="Green">E13</font></i>: 
not <font color="Maroon">c<sub>4</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E5:13_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="card_1.html#E7:13"><i><font color="Green">E11</font></i></a>, <a class="txt" href="card_1.html#E3:13_1"><i><font color="Green">E12</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">E11</font></i></a>, <a class="txt" href="card_1.html#E4:13_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="card_1.html#NM1">Cardinal</a> ;<br/>
<a NAME="E10:13"/>
 <a href="card_1.html#K1" target="_self">Card</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="card_1.html#E7:13"><i><font color="Green">E11</font></i></a>, <a class="ref" href="card_1.html#D5" target="_self">Def5</a></i>;<br/>
<b>hence </b><a NAME="E11:13"/>
 <a href="card_1.html#K1" target="_self">Card</a> <font color="Maroon">c<sub>1</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="card_1.html#E7:13"><i><font color="Green">E11</font></i></a></i>;<br/>


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