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

<b>let </b><font color="Maroon" title="c1">M</font> be  non <a href="card_4.html#V1" title="CARD_4:attr.1">countable</a> <a href="card_5.html#NM1" title="CARD_5:NM.1">Aleph</a>;<br/>

<b>assume </b><a NAME="E1:43"/>
<font color="Maroon" title="c1">M</font> is <a href="card_lar.html#V4" title="CARD_LAR:attr.4">Mahlo</a>
 ;<br/>

<a NAME="E2:43"/><b>then </b><i><font color="Green" title="E33">A1</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">N</font> where <font color="Olive" title="b1">N</font> is  <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a> <a href="card_1.html#V1" title="CARD_1:attr.1">cardinal</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font> : <font color="Olive" title="b1">N</font> is <a href="card_5.html#V1" title="CARD_5:attr.1">regular</a> </span>}</span>  <a href="card_lar.html#R4" title="CARD_LAR:pred.4">is_stationary_in</a> <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="ref" href="card_lar.html#D10" target="_self" title="CARD_LAR:def.10">Def10</a></i>;<br/>
<b>assume </b><a NAME="E3:43"/>
not <font color="Maroon" title="c1">M</font> is <a href="card_5.html#V1" title="CARD_5:attr.1">regular</a>
 ;<br/>

<a NAME="E4:43"/><b>then </b><i><font color="Green" title="E34">A2</font></i>: 
 <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="ref" href="card_5.html#D4" title="CARD_5:def.4">CARD_5:def 4</a></i>;<br/>
<a NAME="E5:43"/>
 <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">M</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="ref" href="card_5.html#D2" title="CARD_5:def.2">CARD_5:def 2</a></i>;<br/>
<a NAME="E6:43"/><b>then </b>
 <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">M</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E4:43"><i><font color="Green" title="E34">A2</font></i></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/>
<a NAME="E7:43"/><b>then </b><i><font color="Green" title="E35">A3</font></i>: 
 <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">M</font> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a> <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c2">xi</font> being   <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a><b> such that </b><br/><a NAME="E9:43"/><i><font color="Green" title="E36">A4</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">M</font>
 <b>and </b><br/><a NAME="E10:43"/><i><font color="Green" title="E37">A5</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">xi</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">M</font>
 <b>and </b><br/><a NAME="E11:43"/>
<font color="Maroon" title="c2">xi</font> is <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a>
 <b>and </b><br/><a NAME="E12:43"/><i><font color="Green" title="E38">A6</font></i>: 
<font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c2">xi</font>
 <b>and </b><br/><a NAME="E13:43"/>
<font color="Maroon" title="c2">xi</font> is   <a href="card_3.html#NM1" title="CARD_3:NM.1">Cardinal-Function</a>
 <b>and </b><br/><a NAME="E14:43"/>
not  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">xi</font>
 <b>by </b><i><a class="ref" href="card_5.html#T41" title="CARD_5:th.41">CARD_5:41</a></i>;<br/>
<a NAME="E15:43"/><i><font color="Green" title="E39">A7</font></i>: 
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E7:43"><i><font color="Green" title="E35">A3</font></i></a>, <a class="ref" href="ordinal1.html#T41" title="ORDINAL1:th.41">ORDINAL1:41</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c3">RNG</font> =  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">xi</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">M</font> <b>by </b><i><a class="txt" href="card_lar.html#E10:43"><i><font color="Green" title="E37">A5</font></i></a></i>;<br/>
<a NAME="E17:43"/>
<font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <font color="Maroon" title="c3">RNG</font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E12:43"><i><font color="Green" title="E38">A6</font></i></a>, <a class="ref" href="ordinal2.html#T35" title="ORDINAL2:th.35">ORDINAL2:35</a></i>;<br/>
<a NAME="E18:43"/><b>then </b><i><font color="Green" title="E40">A8</font></i>: 
<font color="Maroon" title="c3">RNG</font> is <a href="card_lar.html#V1" title="CARD_LAR:attr.1">unbounded</a>
 
<b>by </b><i><a class="ref" href="card_lar.html#D4" target="_self" title="CARD_LAR:def.4">Def4</a></i>;<br/>
<a NAME="E19:43"/><b>then </b><i><font color="Green" title="E41">A9</font></i>: 
 <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">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="c3">RNG</font>
 
<b>by </b><i><a class="ref" href="card_lar.html#T21" target="_self" title="CARD_LAR:th.21">Th21</a></i>;<br/>
<a NAME="E20:43"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c3">RNG</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> <span class="p1">(<span class="default"><a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">M</font></span>)</span>
 
<b>by </b><i><a class="txt" href="card_lar.html#E9:43"><i><font color="Green" title="E36">A4</font></i></a>, <a class="ref" href="card_2.html#T80" title="CARD_2:th.80">CARD_2:80</a></i>;<br/>
<a NAME="E21:43"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c3">RNG</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="ref" href="card_1.html#D5" title="CARD_1:def.5">CARD_1:def 5</a></i>;<br/>
<a NAME="E22:43"/><b>then </b><i><font color="Green" title="E42">A10</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c3">RNG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E19:43"><i><font color="Green" title="E41">A9</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="E23:43"/>
 <a href="card_lar.html#K2" title="CARD_LAR:func.2">limpoints</a> <font color="Maroon" title="c3">RNG</font> is <a href="card_lar.html#V1" title="CARD_LAR:attr.1">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" title="CARD_LAR:func.2">limpoints</a> <font color="Maroon" title="c3">RNG</font> is <a href="card_lar.html#V1" title="CARD_LAR:attr.1">unbounded</a>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">B1</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E3:43_1"/>
<font color="Maroon" title="c4">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font>
 <b>and </b><br/><a NAME="E4:43_1"/><i><font color="Green" title="E43">A11</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">B2</font></span>)</span> where <font color="Olive" title="b1">B2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font> : ( <font color="Olive" title="b1">B2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">RNG</font> &amp; <font color="Maroon" title="c4">B1</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="Olive" title="b1">B2</font> ) </span>}</span>  <a href="card_lar.html#R3" title="CARD_LAR:pred.3">is_club_in</a> <font color="Maroon" title="c1">M</font>
 <b>by </b><i><a class="txt" href="card_lar.html#E18:43"><i><font color="Green" title="E40">A8</font></i></a>, <a class="ref" href="card_lar.html#T20" target="_self" title="CARD_LAR:th.20">Th20</a></i>;<br/>
<b>set </b><font color="Maroon" title="c5">SUCC</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">B2</font></span>)</span> where <font color="Olive" title="b1">B2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font> : ( <font color="Olive" title="b1">B2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">RNG</font> &amp; <font color="Maroon" title="c4">B1</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="Olive" title="b1">B2</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" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">B2</font></span>)</span> where <font color="Olive" title="b1">B2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font> : ( <font color="Olive" title="b1">B2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">RNG</font> &amp; <font color="Maroon" title="c4">B1</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="Olive" title="b1">B2</font> ) </span>}</span>  <a href="card_lar.html#R2" title="CARD_LAR:pred.2">is_closed_in</a> <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E4:43_1"><i><font color="Green" title="E43">A11</font></i></a>, <a class="ref" href="card_lar.html#D3" target="_self" title="CARD_LAR:def.3">Def3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">SUCC</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">B2</font></span>)</span> where <font color="Olive" title="b1">B2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font> : ( <font color="Olive" title="b1">B2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">RNG</font> &amp; <font color="Maroon" title="c4">B1</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="Olive" title="b1">B2</font> ) </span>}</span>  as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">M</font> <b>by </b><i><a class="ref" href="card_lar.html#D2" target="_self" title="CARD_LAR:def.2">Def2</a></i>;<br/>
<a NAME="E7:43_1"/>
( <font color="Maroon" title="c6">SUCC</font> is <a href="card_lar.html#V2" title="CARD_LAR:attr.2">closed</a> &amp; <font color="Maroon" title="c6">SUCC</font> is <a href="card_lar.html#V1" title="CARD_LAR:attr.1">unbounded</a> )
 
<b>by </b><i><a class="txt" href="card_lar.html#E4:43_1"><i><font color="Green" title="E43">A11</font></i></a>, <a class="ref" href="card_lar.html#T2" target="_self" title="CARD_LAR:th.2">Th2</a></i>;<br/>
<a NAME="E8:43_1"/><b>then </b>
not <span class="p1">{<span class="default"> <font color="Olive" title="b1">N</font> where <font color="Olive" title="b1">N</font> is  <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a> <a href="card_1.html#V1" title="CARD_1:attr.1">cardinal</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font> : <font color="Olive" title="b1">N</font> is <a href="card_5.html#V1" title="CARD_5:attr.1">regular</a> </span>}</span>  <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">SUCC</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="txt" href="card_lar.html#E2:43"><i><font color="Green" title="E33">A1</font></i></a>, <a class="ref" href="card_lar.html#D8" target="_self" title="CARD_LAR:def.8">Def8</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E10:43_1"/><i><font color="Green" title="E44">A12</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">SUCC</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">N</font> where <font color="Olive" title="b1">N</font> is  <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a> <a href="card_1.html#V1" title="CARD_1:attr.1">cardinal</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font> : <font color="Olive" title="b1">N</font> is <a href="card_5.html#V1" title="CARD_5:attr.1">regular</a> </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E11:43_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">B2</font></span>)</span> where <font color="Olive" title="b1">B2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font> : ( <font color="Olive" title="b1">B2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">RNG</font> &amp; <font color="Maroon" title="c4">B1</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="Olive" title="b1">B2</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="card_lar.html#E10:43_1"><i><font color="Green" title="E44">A12</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">B2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font><b> such that </b><br/><a NAME="E13:43_1"/><i><font color="Green" title="E45">A13</font></i>: 
<font color="Maroon" title="c7">x</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="c8">B2</font>
 <b>and </b><br/><a NAME="E14:43_1"/>
( <font color="Maroon" title="c8">B2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">RNG</font> &amp; <font color="Maroon" title="c4">B1</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="c8">B2</font> )
 ;<br/>
<a NAME="E15:43_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">N</font> where <font color="Olive" title="b1">N</font> is  <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a> <a href="card_1.html#V1" title="CARD_1:attr.1">cardinal</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font> : <font color="Olive" title="b1">N</font> is <a href="card_5.html#V1" title="CARD_5:attr.1">regular</a> </span>}</span> 
 
<b>by </b><i><a class="txt" href="card_lar.html#E10:43_1"><i><font color="Green" title="E44">A12</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">N1</font> being  <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a> <a href="card_1.html#V1" title="CARD_1:attr.1">cardinal</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font><b> such that </b><br/><a NAME="E17:43_1"/><i><font color="Green" title="E46">A14</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">N1</font>
 <b>and </b><br/><a NAME="E18:43_1"/>
<font color="Maroon" title="c9">N1</font> is <a href="card_5.html#V1" title="CARD_5:attr.1">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" title="E45">A13</font></i></a>, <a class="txt" href="card_lar.html#E17:43_1"><i><font color="Green" title="E46">A14</font></i></a>, <a class="ref" href="ordinal1.html#T42" title="ORDINAL1:th.42">ORDINAL1:42</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E24:43"/><b>then </b>
(  <a href="card_lar.html#K2" title="CARD_LAR:func.2">limpoints</a> <font color="Maroon" title="c3">RNG</font> is <a href="card_lar.html#V2" title="CARD_LAR:attr.2">closed</a> &amp;  <a href="card_lar.html#K2" title="CARD_LAR:func.2">limpoints</a> <font color="Maroon" title="c3">RNG</font> is <a href="card_lar.html#V1" title="CARD_LAR:attr.1">unbounded</a> )
 
<b>by </b><i><a class="ref" href="card_lar.html#T19" target="_self" title="CARD_LAR:th.19">Th19</a></i>;<br/>
<a NAME="E25:43"/><b>then </b>
( <span class="p1">(<span class="default"><a href="card_lar.html#K2" title="CARD_LAR:func.2">limpoints</a> <font color="Maroon" title="c3">RNG</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p2">(<span class="default"><a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span> is <a href="card_lar.html#V2" title="CARD_LAR:attr.2">closed</a> &amp; <span class="p1">(<span class="default"><a href="card_lar.html#K2" title="CARD_LAR:func.2">limpoints</a> <font color="Maroon" title="c3">RNG</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p2">(<span class="default"><a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span> is <a href="card_lar.html#V1" title="CARD_LAR:attr.1">unbounded</a> )
 
<b>by </b><i><a class="txt" href="card_lar.html#E15:43"><i><font color="Green" title="E39">A7</font></i></a>, <a class="ref" href="card_lar.html#T12" target="_self" title="CARD_LAR:th.12">Th12</a></i>;<br/>
<a NAME="E26:43"/><b>then </b>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">N</font> where <font color="Olive" title="b1">N</font> is  <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a> <a href="card_1.html#V1" title="CARD_1:attr.1">cardinal</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font> : <font color="Olive" title="b1">N</font> is <a href="card_5.html#V1" title="CARD_5:attr.1">regular</a> </span>}</span>  <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_lar.html#K2" title="CARD_LAR:func.2">limpoints</a> <font color="Maroon" title="c3">RNG</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p3">(<span class="default"><a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="card_lar.html#E2:43"><i><font color="Green" title="E33">A1</font></i></a>, <a class="ref" href="card_lar.html#D8" target="_self" title="CARD_LAR:def.8">Def8</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E28:43"/><i><font color="Green" title="E43">A15</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_lar.html#K2" title="CARD_LAR:func.2">limpoints</a> <font color="Maroon" title="c3">RNG</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p3">(<span class="default"><a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">N</font> where <font color="Olive" title="b1">N</font> is  <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a> <a href="card_1.html#V1" title="CARD_1:attr.1">cardinal</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font> : <font color="Olive" title="b1">N</font> is <a href="card_5.html#V1" title="CARD_5:attr.1">regular</a> </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E29:43"/><i><font color="Green" title="E44">A16</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="card_lar.html#K2" title="CARD_LAR:func.2">limpoints</a> <font color="Maroon" title="c3">RNG</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p2">(<span class="default"><a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="card_lar.html#E28:43"><i><font color="Green" title="E43">A15</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E30:43"/><b>then </b><i><font color="Green" title="E45">A17</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_lar.html#K2" title="CARD_LAR:func.2">limpoints</a> <font color="Maroon" title="c3">RNG</font> &amp; not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">M</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E31:43"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">N</font> where <font color="Olive" title="b1">N</font> is  <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a> <a href="card_1.html#V1" title="CARD_1:attr.1">cardinal</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font> : <font color="Olive" title="b1">N</font> is <a href="card_5.html#V1" title="CARD_5:attr.1">regular</a> </span>}</span> 
 
<b>by </b><i><a class="txt" href="card_lar.html#E28:43"><i><font color="Green" title="E43">A15</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">N1</font> being  <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a> <a href="card_1.html#V1" title="CARD_1:attr.1">cardinal</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font><b> such that </b><br/><a NAME="E33:43"/><i><font color="Green" title="E46">A18</font></i>: 
<font color="Maroon" title="c5">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font>
 <b>and </b><br/><a NAME="E34:43"/><i><font color="Green" title="E47">A19</font></i>: 
<font color="Maroon" title="c5">N1</font> is <a href="card_5.html#V1" title="CARD_5:attr.1">regular</a>
 ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( not $1 is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; $1 is <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">being_limit_ordinal</a> &amp;  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">RNG</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> $1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 );<br/>
<a NAME="E35:43"/><i><font color="Green" title="E48">A20</font></i>: 
<font color="Maroon" title="c5">N1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">B1</font> where <font color="Olive" title="b1">B1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B1</font>] </span>}</span> 
 
<b>by </b><i><a class="txt" href="card_lar.html#E29:43"><i><font color="Green" title="E44">A16</font></i></a>, <a class="txt" href="card_lar.html#E33:43"><i><font color="Green" title="E46">A18</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E36:43"/><i><font color="Green" title="E49">A21</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">N1</font>]
 
<b>from </b><i><a class="ref" href="card_fil.html#S1" title="CARD_FIL:sch.1">CARD_FIL:sch 1</a>(<a class="txt" href="card_lar.html#E35:43"><i><font color="Green" title="E48">A20</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c6">RNG1</font> = <font color="Maroon" title="c5">N1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">RNG</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c5">N1</font> <b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E38:43"/>
<font color="Maroon" title="c6">RNG1</font> is <a href="card_lar.html#V1" title="CARD_LAR:attr.1">unbounded</a>
 
<b>by </b><i><a class="txt" href="card_lar.html#E36:43"><i><font color="Green" title="E49">A21</font></i></a>, <a class="ref" href="card_lar.html#D4" target="_self" title="CARD_LAR:def.4">Def4</a></i>;<br/>
<a NAME="E39:43"/><b>then </b><i><font color="Green" title="E50">A22</font></i>: 
 <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c5">N1</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="c6">RNG1</font>
 
<b>by </b><i><a class="ref" href="card_lar.html#T21" target="_self" title="CARD_LAR:th.21">Th21</a></i>;<br/>
<a NAME="E40:43"/><i><font color="Green" title="E51">A23</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c6">RNG1</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="c3">RNG</font>
 
<b>by </b><i><a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E41:43"/><i><font color="Green" title="E52">A24</font></i>: 
 <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c5">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">N1</font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E34:43"><i><font color="Green" title="E47">A19</font></i></a>, <a class="ref" href="card_5.html#D4" title="CARD_5:def.4">CARD_5:def 4</a></i>;<br/>
<a NAME="E42:43"/>
not <font color="Maroon" title="c5">N1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E30:43"><i><font color="Green" title="E45">A17</font></i></a>, <a class="txt" href="card_lar.html#E33:43"><i><font color="Green" title="E46">A18</font></i></a>, <a class="ref" href="ordinal1.html#T34" title="ORDINAL1:th.34">ORDINAL1:34</a></i>;<br/>
<a NAME="E43:43"/><b>then </b>
 <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N1</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T26" title="ORDINAL1:th.26">ORDINAL1:26</a></i>;<br/>
<a NAME="E44:43"/><b>then </b>
 <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c6">RNG1</font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E39:43"><i><font color="Green" title="E50">A22</font></i></a>, <a class="txt" href="card_lar.html#E41:43"><i><font color="Green" title="E52">A24</font></i></a></i>;<br/>
<a NAME="E45:43"/><b>then </b>
 <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c3">RNG</font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E40:43"><i><font color="Green" title="E51">A23</font></i></a></i>;<br/>
<b>hence </b><a NAME="E46:43"/>
contradiction
 <b>by </b><i><a class="txt" href="card_lar.html#E22:43"><i><font color="Green" title="E42">A10</font></i></a></i>;<br/>


</div>
