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<b>let </b><font color="Maroon" title="c1">A</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</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>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c1">A</font> <a href="zfrefle1.html#R4" title="ZFREFLE1:pred.4">is_cofinal_with</a> $1 );<br/>
<a NAME="E1:13"/><i><font color="Green" title="E11">A1</font></i>: 
 ex <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>]
 
<b>by </b><i><a class="ref" href="card_5.html#T17" target="_self" title="CARD_5:th.17">Th17</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c2">C</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E3:13"/><i><font color="Green" title="E12">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">C</font>]
 <b>and </b><br/><a NAME="E4:13"/><i><font color="Green" title="E13">A3</font></i>: 
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="c2">C</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_5.html#E1:13"><i><font color="Green" title="E11">A1</font></i></a>);<br/></i>
<b>take </b><font color="Maroon" title="c3">M</font> =  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">C</font>;<br/>

<b>consider </b><font color="Maroon" title="c4">B</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E6:13"/><i><font color="Green" title="E14">A4</font></i>: 
( <font color="Maroon" title="c4">B</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c2">C</font> <a href="zfrefle1.html#R4" title="ZFREFLE1:pred.4">is_cofinal_with</a> <font color="Maroon" title="c4">B</font> )
 <b>by </b><i><a class="ref" href="card_5.html#T17" target="_self" title="CARD_5:th.17">Th17</a></i>;<br/>
<a NAME="E7:13"/><i><font color="Green" title="E15">A5</font></i>: 
<font color="Maroon" title="c3">M</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">C</font>
 
<b>by </b><i><a class="ref" href="card_1.html#T24" title="CARD_1:th.24">CARD_1:24</a></i>;<br/>
<a NAME="E8:13"/><b>then </b><i><font color="Green" title="E16">A6</font></i>: 
<font color="Maroon" title="c4">B</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">C</font>
 
<b>by </b><i><a class="txt" href="card_5.html#E6:13"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E9:13"/><b>then </b>
( <font color="Maroon" title="c4">B</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c1">A</font> <a href="zfrefle1.html#R4" title="ZFREFLE1:pred.4">is_cofinal_with</a> <font color="Maroon" title="c4">B</font> )
 
<b>by </b><i><a class="txt" href="card_5.html#E3:13"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="card_5.html#E6:13"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a>, <a class="ref" href="zfrefle1.html#T21" title="ZFREFLE1:th.21">ZFREFLE1:21</a></i>;<br/>
<a NAME="E10:13"/><b>then </b>
<font color="Maroon" title="c2">C</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c4">B</font>
 
<b>by </b><i><a class="txt" href="card_5.html#E4:13"><i><font color="Green" title="E13">A3</font></i></a></i>;<br/>
<a NAME="E11:13"/><b>then </b><i><font color="Green" title="E17">A7</font></i>: 
<font color="Maroon" title="c2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font>
 
<b>by </b><i><a class="txt" href="card_5.html#E8:13"><i><font color="Green" title="E16">A6</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E12:13"/><b>then </b><i><font color="Green" title="E18">A8</font></i>: 
<font color="Maroon" title="c2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font>
 
<b>by </b><i><a class="txt" href="card_5.html#E6:13"><i><font color="Green" title="E14">A4</font></i></a>, <a class="txt" href="card_5.html#E7:13"><i><font color="Green" title="E15">A5</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<b>thus </b><a NAME="E13:13"/>
( <font color="Maroon" title="c3">M</font> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c1">A</font> <a href="zfrefle1.html#R4" title="ZFREFLE1:pred.4">is_cofinal_with</a> <font color="Maroon" title="c3">M</font> )
 <b>by </b><i><a class="txt" href="card_5.html#E3:13"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="card_5.html#E6:13"><i><font color="Green" title="E14">A4</font></i></a>, <a class="txt" href="card_5.html#E7:13"><i><font color="Green" title="E15">A5</font></i></a>, <a class="txt" href="card_5.html#E11:13"><i><font color="Green" title="E17">A7</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<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="E14:13"/>
( <font color="Maroon" title="c1">A</font> <a href="zfrefle1.html#R4" title="ZFREFLE1:pred.4">is_cofinal_with</a> <font color="Maroon" title="c5">B</font> &amp; not <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> )
 ;<br/>

<a NAME="E15:13"/><b>then </b>
( not <font color="Maroon" title="c5">B</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c5">B</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">M</font> )
 
<b>by </b><i><a class="txt" href="card_5.html#E4:13"><i><font color="Green" title="E13">A3</font></i></a>, <a class="txt" href="card_5.html#E12:13"><i><font color="Green" title="E18">A8</font></i></a></i>;<br/>
<b>hence </b><a NAME="E16:13"/>
contradiction
 <b>by </b><i><a class="txt" href="card_5.html#E3:13"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="card_5.html#E12:13"><i><font color="Green" title="E18">A8</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>


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