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<b>let </b><font color="Maroon">M</font> be   <a href="card_1.html#NM1">Cardinal</a>;<br/>

<b>assume </b><a NAME="E1:25"/>
not <font color="Maroon">M</font> is <a href="finset_1.html#V1">finite</a>
 ;<br/>

<a NAME="E2:25"/><b>then </b><i><font color="Green">E44</font></i>: 
not <font color="Maroon">M</font> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K3">alef</a> 0
 
<b>by </b><i>, <a class="ref" href="card_1.html#T83">CARD_1:83</a></i>;<br/>
<b>assume </b><a NAME="E3:25"/>
not <font color="Maroon">M</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">A</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E5:25"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">M</font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T42">ORDINAL1:42</a></i>;<br/>
<a NAME="E6:25"/>
( (  <a href="ordinal1.html#K5">omega</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">M</font> &amp;  <a href="ordinal1.html#K5">omega</a>  <a href="ordinal1.html#R1">c=</a> <font color="Maroon">M</font> ) iff  <a href="ordinal1.html#K5">omega</a>  <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">M</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/>
<a NAME="E7:25"/><b>then </b>
(  <a href="ordinal1.html#K5">omega</a>  <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font> or  <a href="ordinal1.html#K5">omega</a>  <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, , <a class="ref" href="card_1.html#T14">CARD_1:14</a>, <a class="ref" href="card_1.html#T83">CARD_1:83</a>, <a class="ref" href="ordinal1.html#T21">ORDINAL1:21</a></i>;<br/>
<a NAME="E8:25"/><b>then </b><i><font color="Green">E50</font></i>: 
 <a href="ordinal1.html#K5">omega</a>  <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T34">ORDINAL1:34</a>, <a class="ref" href="ordinal1.html#T42">ORDINAL1:42</a>, <a class="ref" href="ordinal2.html#T19">ORDINAL2:19</a></i>;<br/>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K10">+^</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> 1</span>)</span> <a href="card_2.html#K1">+`</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">A</font></span>)</span>
<b>by </b><i><a class="ref" href="card_2.html#T24">CARD_2:24</a></i>
<br/>.= 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default">1 <a href="ordinal2.html#K10">+^</a> <font color="Maroon">A</font></span>)</span>
<b>by </b><i><a class="ref" href="card_2.html#T24">CARD_2:24</a></i>
<br/>.= 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">A</font>
<b>by </b><i>, </i>
;<br/>
<a NAME="E10:25"/><b>then </b>
(  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font> )
 
<b>by </b><i><a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a>, <a class="ref" href="ordinal2.html#T48">ORDINAL2:48</a></i>;<br/>
<a NAME="E11:25"/><b>then </b>
( <font color="Maroon">A</font>, <a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font> <a href="wellord2.html#R2">are_equipotent</a>  &amp; not  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A</font> &amp;  ex <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>(  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> &amp; ( for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">A</font>,<font color="Olive">B</font> <a href="wellord2.html#R2">are_equipotent</a>  holds <br/><font color="Olive">B</font> <a href="ordinal1.html#R1">c=</a> <font color="Olive">A</font> ) ) )
 
<b>by </b><i>, <a class="ref" href="card_1.html#T21">CARD_1:21</a>, <a class="ref" href="card_1.html#D1">CARD_1:def 1</a>, <a class="ref" href="ordinal1.html#T7">ORDINAL1:7</a></i>;<br/>
<b>hence </b><a NAME="E12:25"/>
contradiction
 ;<br/>


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