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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="ordinal1.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="zfrefle1.html#R4">is_cofinal_with</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E1:13"/><i><font color="Green">E23</font></i>: 
 ex <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B</font>]
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">C</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E3:13"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">C</font>]
 <b>and </b><br/><a NAME="E4:13"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B</font>] holds <br/><font color="Maroon">C</font> <a href="ordinal1.html#R1">c=</a> <font color="Olive">B</font>
 <b>from </b><i><a class="ref" href="ordinal1.html#S1">ORDINAL1:sch 1</a>();<br/></i>
<b>take </b><font color="Maroon">M</font> =  <a href="card_1.html#K1">Card</a> <font color="Maroon">C</font>;<br/>

<b>consider </b><font color="Maroon">B</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E6:13"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">B</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">M</font> &amp; <font color="Maroon">C</font> <a href="zfrefle1.html#R4">is_cofinal_with</a> <font color="Maroon">B</font> )
 <b>by </b><i/>;<br/>
<a NAME="E7:13"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">M</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="card_1.html#T24">CARD_1:24</a></i>;<br/>
<a NAME="E8:13"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">B</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">C</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E9:13"/><b>then </b>
( <font color="Maroon">B</font> <a href="ordinal1.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="zfrefle1.html#R4">is_cofinal_with</a> <font color="Maroon">B</font> )
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a>, <a class="ref" href="zfrefle1.html#T21">ZFREFLE1:21</a></i>;<br/>
<a NAME="E10:13"/><b>then </b>
<font color="Maroon">C</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">B</font>
 
<b>by </b><i/>;<br/>
<a NAME="E11:13"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E12:13"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>thus </b><a NAME="E13:13"/>
( <font color="Maroon">M</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="zfrefle1.html#R4">is_cofinal_with</a> <font color="Maroon">M</font> )
 <b>by </b><i>, , , , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>

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

<b>assume </b><a NAME="E14:13"/>
( <font color="Maroon">A</font> <a href="zfrefle1.html#R4">is_cofinal_with</a> <font color="Maroon">B</font> &amp; not <font color="Maroon">M</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">B</font> )
 ;<br/>

<a NAME="E15:13"/><b>then </b>
( not <font color="Maroon">B</font> <a href="ordinal1.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">A</font> &amp; <font color="Maroon">B</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">M</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E16:13"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


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