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<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>assume </b><a NAME="E1:21"/>
not <font color="Maroon" title="c2">M</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ;<br/>

<a NAME="E2:21"/><b>then </b><i><font color="Green" title="E18">A1</font></i>: 
not <font color="Maroon" title="c2">M</font> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a>  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="card_1.html#T83" title="CARD_1:th.83">CARD_1:83</a>, <a class="ref" href="card_1.html#T103" title="CARD_1:th.103">CARD_1:103</a></i>;<br/>
<b>assume </b><a NAME="E3:21"/>
not <font color="Maroon" title="c2">M</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c3">A</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E5:21"/><i><font color="Green" title="E19">A2</font></i>: 
<font color="Maroon" title="c2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T42" title="ORDINAL1:th.42">ORDINAL1:42</a></i>;<br/>
<a NAME="E6:21"/>
( (  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">M</font> &amp;  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">M</font> ) iff  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c2">M</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/>
<a NAME="E7:21"/><b>then </b>
(  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c3">A</font> or  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c3">A</font> )
 
<b>by </b><i><a class="txt" href="card_4.html#E2:21"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="card_4.html#E5:21"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="card_1.html#T14" title="CARD_1:th.14">CARD_1:14</a>, <a class="ref" href="card_1.html#T83" title="CARD_1:th.83">CARD_1:83</a>, <a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a></i>;<br/>
<a NAME="E8:21"/><b>then </b><i><font color="Green" title="E20">A3</font></i>: 
 <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">A</font>
 
<b>by </b><i><a class="txt" href="card_4.html#E32"><i><font color="Green" title="E15">Lm2</font></i></a>, <a class="ref" href="ordinal1.html#T34" title="ORDINAL1:th.34">ORDINAL1:34</a>, <a class="ref" href="ordinal1.html#T42" title="ORDINAL1:th.42">ORDINAL1:42</a></i>;<br/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">Card</a> 1</span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> <span class="p1">(<span class="default"><a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c3">A</font></span>)</span>
<b>by </b><i><a class="ref" href="card_2.html#T24" title="CARD_2:th.24">CARD_2:24</a></i>
<br/>.= 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default">1 <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c3">A</font></span>)</span>
<b>by </b><i><a class="ref" href="card_2.html#T24" title="CARD_2:th.24">CARD_2:24</a></i>
<br/>.= 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c3">A</font>
<b>by </b><i><a class="txt" href="card_4.html#E8:21"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="card_4.html#T31" target="_self" title="CARD_4:th.31">Th31</a></i>
;<br/>
<a NAME="E10:21"/><b>then </b>
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c3">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c3">A</font> )
 
<b>by </b><i><a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a>, <a class="ref" href="ordinal2.html#T48" title="ORDINAL2:th.48">ORDINAL2:48</a></i>;<br/>
<a NAME="E11:21"/><b>then </b>
( <font color="Maroon" title="c3">A</font>, <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c3">A</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a>  &amp; not  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c3">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">A</font> &amp;  ex <font color="Olive" title="b1">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>(  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B</font> &amp; ( for <font color="Olive" title="b2">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b2">A</font>,<font color="Olive" title="b1">B</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a>  holds <br/><font color="Olive" title="b1">B</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b2">A</font> ) ) )
 
<b>by </b><i><a class="txt" href="card_4.html#E5:21"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="card_1.html#T21" title="CARD_1:th.21">CARD_1:21</a>, <a class="ref" href="card_1.html#D1" title="CARD_1:def.1">CARD_1:def 1</a>, <a class="ref" href="ordinal1.html#T7" title="ORDINAL1:th.7">ORDINAL1:7</a></i>;<br/>
<b>hence </b><a NAME="E12:21"/>
contradiction
 ;<br/>


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