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<b>let </b><font color="Maroon">M</font> be  non <a href="card_4.html#V1">countable</a> <a href="card_5.html#NM1">Aleph</a>;<br/>

<b>assume </b><a NAME="E1:44"/>
<font color="Maroon">M</font> is <a href="card_lar.html#V4" target="_self">Mahlo</a>
 ;<br/>

<a NAME="E2:44"/><b>then </b><i><font color="Green">E22</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">N</font> where <font color="Olive">N</font> is  <a href="finset_1.html#V1">infinite</a> <a href="card_1.html#V1">cardinal</a>  <a href="subset_1.html#M1">Element</a> of <font color="Maroon">M</font> : <font color="Olive">N</font> is <a href="card_5.html#V1">regular</a> </span>}</span>  <a href="card_lar.html#R4" target="_self">is_stationary_in</a> <font color="Maroon">M</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">REG</font> = <span class="p1">{<span class="default"> <font color="Olive">N</font> where <font color="Olive">N</font> is  <a href="finset_1.html#V1">infinite</a> <a href="card_1.html#V1">cardinal</a>  <a href="subset_1.html#M1">Element</a> of <font color="Maroon">M</font> : <font color="Olive">N</font> is <a href="card_5.html#V1">regular</a> </span>}</span>  as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">M</font> <b>by </b><i/>;<br/>
<b>assume </b><a NAME="E4:44"/>
not <font color="Maroon">M</font> is <a href="card_1.html#V2">limit</a>
 ;<br/>

<b>then consider </b><font color="Maroon">M1</font> being   <a href="card_1.html#NM1">Cardinal</a><b> such that </b><br/><a NAME="E6:44"/><i><font color="Green">E23</font></i>: 
 <a href="card_1.html#K2">nextcard</a> <font color="Maroon">M1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="card_1.html#D7">CARD_1:def 7</a></i>;<br/>
<a NAME="E7:44"/>
<font color="Maroon">M1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="card_lar.html#D2" target="_self">Def2</a>, <a class="ref" href="card_1.html#T32">CARD_1:32</a></i>;<br/>
<a NAME="E8:44"/><b>then </b>
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">M1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T41">ORDINAL1:41</a></i>;<br/>
<a NAME="E9:44"/><b>then </b>
( <font color="Maroon">M</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">M1</font></span>)</span> is <a href="card_lar.html#V2" target="_self">closed</a> &amp; <font color="Maroon">M</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">M1</font></span>)</span> is <a href="card_lar.html#V1" target="_self">unbounded</a> )
 
<b>by </b><i/>;<br/>
<a NAME="E10:44"/><b>then </b>
<font color="Maroon">REG</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="subset_1.html#K6">\</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">M1</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="card_lar.html#D1" target="_self">Def1</a>, </i>;<br/>
<b>then consider </b><font color="Maroon">M2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:44"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">M2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">REG</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="subset_1.html#K6">\</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">M1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E13:44"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">M2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">REG</font> &amp; <font color="Maroon">M2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">M1</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="card_lar.html#D3" target="_self">Def3</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">N</font> being  <a href="finset_1.html#V1">infinite</a> <a href="card_1.html#V1">cardinal</a>  <a href="subset_1.html#M1">Element</a> of <font color="Maroon">M</font><b> such that </b><br/><a NAME="E15:44"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">N</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M2</font>
 <b>and </b><br/><a NAME="E16:44"/>
<font color="Maroon">N</font> is <a href="card_5.html#V1">regular</a>
 ;<br/>
<a NAME="E17:44"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">N</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; not <font color="Maroon">N</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">M1</font> )
 
<b>by </b><i><a class="ref" href="card_lar.html#D4" target="_self">Def4</a>, <a class="ref" href="card_lar.html#D5" target="_self">Def5</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E18:44"/><b>then </b>
not <font color="Maroon">N</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">M1</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T34">ORDINAL1:34</a></i>;<br/>
<a NAME="E19:44"/><b>then </b>
<font color="Maroon">M1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T26">ORDINAL1:26</a></i>;<br/>
<a NAME="E20:44"/><b>then </b>
 <a href="card_1.html#K2">nextcard</a> <font color="Maroon">M1</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a> <font color="Maroon">N</font>
 
<b>by </b><i><a class="ref" href="card_4.html#T22">CARD_4:22</a></i>;<br/>
<a NAME="E21:44"/><b>then </b>
<font color="Maroon">N</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 
<b>by </b><i><a class="ref" href="card_lar.html#D2" target="_self">Def2</a>, <a class="ref" href="card_lar.html#T2" target="_self">Th2</a></i>;<br/>
<b>hence </b><a NAME="E22:44"/>
contradiction
 ;<br/>


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