<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></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="E1:43"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="card_lar.html#V4" target="_self">Mahlo</a>
 ;<br/>

<a NAME="E2:43"/><b>then </b><i><font color="Green">E33</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="card_lar.html#D10" target="_self">Def10</a></i>;<br/>
<b>assume </b><a NAME="E3:43"/>
not <font color="Maroon">c<sub>1</sub></font> is <a href="card_5.html#V1">regular</a>
 ;<br/>

<a NAME="E4:43"/><b>then </b><i><font color="Green">E34</font></i>: 
 <a href="card_5.html#K1">cf</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="card_5.html#D4">CARD_5:def 4</a></i>;<br/>
<a NAME="E5:43"/>
 <a href="card_5.html#K1">cf</a> <font color="Maroon">c<sub>1</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="card_5.html#D2">CARD_5:def 2</a></i>;<br/>
<a NAME="E6:43"/><b>then </b>
 <a href="card_5.html#K1">cf</a> <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E4:43"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/>
<a NAME="E7:43"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="card_5.html#K1">cf</a> <font color="Maroon">c<sub>1</sub></font> <a href="card_1.html#NR2" target="_self">&lt;`</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T21">ORDINAL1:21</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="ordinal2.html#NM1">Ordinal-Sequence</a><b> such that </b><br/><a NAME="E9:43"/><i><font color="Green">E36</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_5.html#K1">cf</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E10:43"/><i><font color="Green">E37</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E11:43"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="ordinal2.html#V2">increasing</a>
 <b>and </b><br/><a NAME="E12:43"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K4">sup</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E13:43"/>
<font color="Maroon">c<sub>2</sub></font> is   <a href="card_3.html#NM1">Cardinal-Function</a>
 <b>and </b><br/><a NAME="E14:43"/>
not 0 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="card_5.html#T41">CARD_5:41</a></i>;<br/>
<a NAME="E15:43"/><i><font color="Green">E39</font></i>: 
 <a href="ordinal1.html#K1">succ</a> <span class="p1">(<span class="default"><a href="card_5.html#K1">cf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E7:43"><i><font color="Green">E35</font></i></a>, <a class="ref" href="ordinal1.html#T41">ORDINAL1:41</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> =  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="card_lar.html#E10:43"><i><font color="Green">E37</font></i></a></i>;<br/>
<a NAME="E17:43"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K3">sup</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E12:43"><i><font color="Green">E38</font></i></a>, <a class="ref" href="ordinal2.html#T35">ORDINAL2:35</a></i>;<br/>
<a NAME="E18:43"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="card_lar.html#V1" target="_self">unbounded</a>
 
<b>by </b><i><a class="ref" href="card_lar.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E19:43"/><b>then </b><i><font color="Green">E41</font></i>: 
 <a href="card_5.html#K1">cf</a> <font color="Maroon">c<sub>1</sub></font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="card_lar.html#T21" target="_self">Th21</a></i>;<br/>
<a NAME="E20:43"/>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>3</sub></font> <a href="ordinal1.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_5.html#K1">cf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="card_lar.html#E9:43"><i><font color="Green">E36</font></i></a>, <a class="ref" href="card_2.html#T80">CARD_2:80</a></i>;<br/>
<a NAME="E21:43"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>3</sub></font> <a href="ordinal1.html#R1">c=</a>  <a href="card_5.html#K1">cf</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="card_1.html#D5">CARD_1:def 5</a></i>;<br/>
<a NAME="E22:43"/><b>then </b><i><font color="Green">E42</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_5.html#K1">cf</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E19:43"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E23:43"/>
 <a href="card_lar.html#K2" target="_self">limpoints</a> <font color="Maroon">c<sub>3</sub></font> is <a href="card_lar.html#V1" target="_self">unbounded</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:43_1"/>
not  <a href="card_lar.html#K2" target="_self">limpoints</a> <font color="Maroon">c<sub>3</sub></font> is <a href="card_lar.html#V1" target="_self">unbounded</a>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E3:43_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E4:43_1"/><i><font color="Green">E43</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span>  <a href="card_lar.html#R3" target="_self">is_club_in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="card_lar.html#E18:43"><i><font color="Green">E40</font></i></a>, <a class="ref" href="card_lar.html#T20" target="_self">Th20</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span> ;<br/>
<a NAME="E5:43_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span>  <a href="card_lar.html#R2" target="_self">is_closed_in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E4:43_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="card_lar.html#D3" target="_self">Def3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span>  as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="card_lar.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E7:43_1"/>
( <font color="Maroon">c<sub>6</sub></font> is <a href="card_lar.html#V2" target="_self">closed</a> &amp; <font color="Maroon">c<sub>6</sub></font> is <a href="card_lar.html#V1" target="_self">unbounded</a> )
 
<b>by </b><i><a class="txt" href="card_lar.html#E4:43_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="card_lar.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E8:43_1"/><b>then </b>
not <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="card_5.html#V1">regular</a> </span>}</span>  <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>6</sub></font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="txt" href="card_lar.html#E2:43"><i><font color="Green">E33</font></i></a>, <a class="ref" href="card_lar.html#D8" target="_self">Def8</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:43_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K8">/\</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="card_5.html#V1">regular</a> </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E11:43_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="card_lar.html#E10:43_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E13:43_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E14:43_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/>
<a NAME="E15:43_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="card_5.html#V1">regular</a> </span>}</span> 
 
<b>by </b><i><a class="txt" href="card_lar.html#E10:43_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>9</sub></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">c<sub>1</sub></font><b> such that </b><br/><a NAME="E17:43_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E18:43_1"/>
<font color="Maroon">c<sub>9</sub></font> is <a href="card_5.html#V1">regular</a>
 ;<br/>
<b>thus </b><a NAME="E19:43_1"/>
contradiction
 <b>by </b><i><a class="txt" href="card_lar.html#E13:43_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="card_lar.html#E17:43_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="ordinal1.html#T42">ORDINAL1:42</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E24:43"/><b>then </b>
(  <a href="card_lar.html#K2" target="_self">limpoints</a> <font color="Maroon">c<sub>3</sub></font> is <a href="card_lar.html#V2" target="_self">closed</a> &amp;  <a href="card_lar.html#K2" target="_self">limpoints</a> <font color="Maroon">c<sub>3</sub></font> is <a href="card_lar.html#V1" target="_self">unbounded</a> )
 
<b>by </b><i><a class="ref" href="card_lar.html#T19" target="_self">Th19</a></i>;<br/>
<a NAME="E25:43"/><b>then </b>
( <span class="p1">(<span class="default"><a href="card_lar.html#K2" target="_self">limpoints</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <span class="p2">(<span class="default"><a href="card_5.html#K1">cf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> is <a href="card_lar.html#V2" target="_self">closed</a> &amp; <span class="p1">(<span class="default"><a href="card_lar.html#K2" target="_self">limpoints</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <span class="p2">(<span class="default"><a href="card_5.html#K1">cf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> is <a href="card_lar.html#V1" target="_self">unbounded</a> )
 
<b>by </b><i><a class="txt" href="card_lar.html#E15:43"><i><font color="Green">E39</font></i></a>, <a class="ref" href="card_lar.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E26:43"/><b>then </b>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="card_5.html#V1">regular</a> </span>}</span>  <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_lar.html#K2" target="_self">limpoints</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <span class="p3">(<span class="default"><a href="card_5.html#K1">cf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></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="txt" href="card_lar.html#E2:43"><i><font color="Green">E33</font></i></a>, <a class="ref" href="card_lar.html#D8" target="_self">Def8</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E28:43"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_lar.html#K2" target="_self">limpoints</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <span class="p3">(<span class="default"><a href="card_5.html#K1">cf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="card_5.html#V1">regular</a> </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E29:43"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="card_lar.html#K2" target="_self">limpoints</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <span class="p2">(<span class="default"><a href="card_5.html#K1">cf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="card_lar.html#E28:43"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E30:43"/><b>then </b><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_lar.html#K2" target="_self">limpoints</a> <font color="Maroon">c<sub>3</sub></font> &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <span class="p1">(<span class="default"><a href="card_5.html#K1">cf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E31:43"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="card_5.html#V1">regular</a> </span>}</span> 
 
<b>by </b><i><a class="txt" href="card_lar.html#E28:43"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></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">c<sub>1</sub></font><b> such that </b><br/><a NAME="E33:43"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E34:43"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="card_5.html#V1">regular</a>
 ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means ( not <font color="Maroon">a<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Maroon">a<sub>1</sub></font> is <a href="ordinal1.html#V4">being_limit_ordinal</a> &amp;  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E35:43"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 
<b>by </b><i><a class="txt" href="card_lar.html#E30:43"><i><font color="Green">E44</font></i></a>, <a class="txt" href="card_lar.html#E33:43"><i><font color="Green">E45</font></i></a></i>;<br/>
<a NAME="E36:43"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font>]
 
<b>from </b><i><a class="ref" href="card_fil.html#S1">CARD_FIL:sch 1</a>(<a class="txt" href="card_lar.html#E35:43"><i><font color="Green">E47</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>5</sub></font> <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E38:43"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="card_lar.html#V1" target="_self">unbounded</a>
 
<b>by </b><i><a class="txt" href="card_lar.html#E36:43"><i><font color="Green">E48</font></i></a>, <a class="ref" href="card_lar.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E39:43"/><b>then </b><i><font color="Green">E49</font></i>: 
 <a href="card_5.html#K1">cf</a> <font color="Maroon">c<sub>5</sub></font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="card_lar.html#T21" target="_self">Th21</a></i>;<br/>
<a NAME="E40:43"/>
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E41:43"/><b>then </b><i><font color="Green">E50</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>6</sub></font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="card_1.html#T27">CARD_1:27</a></i>;<br/>
<a NAME="E42:43"/><i><font color="Green">E51</font></i>: 
 <a href="card_5.html#K1">cf</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E34:43"><i><font color="Green">E46</font></i></a>, <a class="ref" href="card_5.html#D4">CARD_5:def 4</a></i>;<br/>
<a NAME="E43:43"/>
not <font color="Maroon">c<sub>5</sub></font> <a href="ordinal1.html#R1">c=</a>  <a href="card_5.html#K1">cf</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E30:43"><i><font color="Green">E44</font></i></a>, <a class="txt" href="card_lar.html#E33:43"><i><font color="Green">E45</font></i></a>, <a class="ref" href="ordinal1.html#T34">ORDINAL1:34</a></i>;<br/>
<a NAME="E44:43"/><b>then </b>
 <a href="card_5.html#K1">cf</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T26">ORDINAL1:26</a></i>;<br/>
<a NAME="E45:43"/><b>then </b>
 <a href="card_5.html#K1">cf</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E39:43"><i><font color="Green">E49</font></i></a>, <a class="txt" href="card_lar.html#E42:43"><i><font color="Green">E51</font></i></a></i>;<br/>
<a NAME="E46:43"/><b>then </b>
 <a href="card_5.html#K1">cf</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E41:43"><i><font color="Green">E50</font></i></a></i>;<br/>
<b>hence </b><a NAME="E47:43"/>
contradiction
 <b>by </b><i><a class="txt" href="card_lar.html#E22:43"><i><font color="Green">E42</font></i></a></i>;<br/>


</div>
