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

<b>let </b><font color="Maroon">A</font> be  <a href="ordinal1.html#V4">being_limit_ordinal</a> <a href="finset_1.html#V1">infinite</a> <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">X</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">A</font>;<br/>



<b>assume </b><a NAME="E1:32"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">X</font> is <a href="card_lar.html#V1" target="_self">unbounded</a>
 ;<br/>

<b>assume </b><a NAME="E2:32"/>
not  <a href="card_lar.html#K2" target="_self">limpoints</a> <font color="Maroon">X</font> is <a href="card_lar.html#V1" target="_self">unbounded</a>
 ;<br/>

<b>then consider </b><font color="Maroon">B1</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E4:32"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">B1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E5:32"/><i><font color="Green">E44</font></i>: 
 <a href="card_lar.html#K2" target="_self">limpoints</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B1</font>
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">B1</font>
;<br/>

<b>set </b><font color="Maroon">SUCC</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Olive">B2</font></span>)</span> where <font color="Olive">B2</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> : ( <font color="Olive">B2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">B1</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Olive">B2</font> ) </span>}</span> ;<br/>
<a NAME="E6:32"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Olive">B2</font></span>)</span> where <font color="Olive">B2</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> : ( <font color="Olive">B2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">B1</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Olive">B2</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:32_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">x</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">B2</font></span>)</span> where <font color="Olive">B2</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> : ( <font color="Olive">B2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">B1</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Olive">B2</font> ) </span>}</span> 
 ;<br/>

<b>consider </b><font color="Maroon">B2</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E3:32_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">B2</font>
 <b>and </b><br/><a NAME="E4:32_1"/>
( <font color="Maroon">B2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">B1</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">B2</font> )
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E5:32_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="ordinal1.html#T41">ORDINAL1:41</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">SUCC</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Olive">B2</font></span>)</span> where <font color="Olive">B2</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> : ( <font color="Olive">B2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">B1</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Olive">B2</font> ) </span>}</span>  as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">A</font> ;<br/>
<a NAME="E8:32"/>
for <font color="Olive">B</font> being <a href="ordinal1.html#V4">being_limit_ordinal</a> <a href="finset_1.html#V1">infinite</a> <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp;  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><font color="Maroon">SUCC</font> <a href="subset_1.html#K8">/\</a> <font color="Olive">B</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> holds <br/><font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SUCC</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">B</font> be  <a href="ordinal1.html#V4">being_limit_ordinal</a> <a href="finset_1.html#V1">infinite</a> <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E1:32_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">B0</font> = <font color="Maroon">B</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> <b>by </b><i/>;<br/>
<a NAME="E3:32_2"/>
not  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><font color="Maroon">SUCC</font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_2_1"/><i><font color="Green">E48</font></i>: 
 <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><font color="Maroon">SUCC</font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 ;<br/>

<a NAME="E2:32_2_1"/>
 <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_2_1_1"/>
not  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 ;<br/>

<b>then consider </b><font color="Maroon">B5</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E3:32_2_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">B5</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E4:32_2_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">X</font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B5</font>
 <b>by </b><i/>;<br/>
<a NAME="E5:32_2_1_1"/>
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">B5</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
<b>by </b><i>, <a class="ref" href="ordinal1.html#T41">ORDINAL1:41</a></i>;<br/>
<a NAME="E6:32_2_1_1"/><b>then </b>
 <a href="ordinal1.html#K1">succ</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">B5</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T41">ORDINAL1:41</a></i>;<br/>
<b>then consider </b><font color="Maroon">B6</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E8:32_2_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">B6</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SUCC</font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E9:32_2_1_1"/><i><font color="Green">E66</font></i>: 
 <a href="ordinal1.html#K1">succ</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">B5</font></span>)</span> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">B6</font>
 <b>by </b><i>, <a class="ref" href="ordinal2.html#T29">ORDINAL2:29</a></i>;<br/>
<a NAME="E10:32_2_1_1"/>
<font color="Maroon">B6</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SUCC</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">B7</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E12:32_2_1_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">B6</font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">B7</font>
 <b>and </b><br/><a NAME="E13:32_2_1_1"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">B7</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">B1</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">B7</font> )
 ;<br/>
<a NAME="E14:32_2_1_1"/>
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">B5</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">B7</font>
 
<b>by </b><i>, <a class="ref" href="card_lar.html#D1" target="_self">Def1</a>, <a class="ref" href="ordinal1.html#T33">ORDINAL1:33</a></i>;<br/>
<a NAME="E15:32_2_1_1"/><b>then </b><i><font color="Green">E69</font></i>: 
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">B5</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">B7</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T34">ORDINAL1:34</a></i>;<br/>
<a NAME="E16:32_2_1_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">B6</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E17:32_2_1_1"/>
<font color="Maroon">B7</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">B7</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a></i>;<br/>
<a NAME="E18:32_2_1_1"/><b>then </b>
<font color="Maroon">B7</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="card_lar.html#D1" target="_self">Def1</a>, <a class="ref" href="card_lar.html#D4" target="_self">Def4</a>, <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/>
<a NAME="E19:32_2_1_1"/><b>then </b>
<font color="Maroon">B7</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="card_lar.html#D2" target="_self">Def2</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E20:32_2_1_1"/><b>then </b>
<font color="Maroon">B7</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B5</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E21:32_2_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="card_lar.html#D3" target="_self">Def3</a>, <a class="ref" href="ordinal1.html#T33">ORDINAL1:33</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:32_2_1"/><b>then </b>
<font color="Maroon">B0</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">B10</font> where <font color="Olive">B10</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> : ( not <font color="Olive">B10</font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Olive">B10</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">X</font> <a href="subset_1.html#K8">/\</a> <font color="Olive">B10</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">B10</font> ) </span>}</span> 
 
;<br/>
<a NAME="E4:32_2_1"/><b>then </b><i><font color="Green">E71</font></i>: 
<font color="Maroon">B0</font> <a href="hidden.html#R2">in</a>  <a href="card_lar.html#K2" target="_self">limpoints</a> <font color="Maroon">X</font>
 
;<br/>
<b>consider </b><font color="Maroon">B2</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font>;<br/>
<b>consider </b><font color="Maroon">B3</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E7:32_2_1"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">B3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SUCC</font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E8:32_2_1"/>
<font color="Maroon">B2</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">B3</font>
 <b>by </b><i>, <a class="ref" href="ordinal2.html#T29">ORDINAL2:29</a></i>;<br/>
<a NAME="E9:32_2_1"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">B3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E10:32_2_1"/>
<font color="Maroon">B3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SUCC</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">B4</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E12:32_2_1"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">B3</font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">B4</font>
 <b>and </b><br/><a NAME="E13:32_2_1"/>
<font color="Maroon">B4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E14:32_2_1"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">B1</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">B4</font>
 ;<br/>
<b>thus </b><a NAME="E15:32_2_1"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="card_lar.html#D1" target="_self">Def1</a>, <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:32_2"/>
(  not  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><font color="Maroon">SUCC</font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> or <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SUCC</font> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E9:32"/><b>then </b><i><font color="Green">E95</font></i>: 
<font color="Maroon">SUCC</font> <a href="card_lar.html#R2" target="_self">is_closed_in</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<a NAME="E10:32"/>
for <font color="Olive">B2</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">B2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/> ex <font color="Olive">C</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Olive">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SUCC</font> &amp; <font color="Olive">B2</font> <a href="ordinal1.html#R1">c=</a> <font color="Olive">C</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">B2</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E1:32_3"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">B2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<b>set </b><font color="Maroon">B21</font> = <font color="Maroon">B2</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">B1</font>;<br/>
<a NAME="E2:32_3"/><i><font color="Green">E98</font></i>: 
( <font color="Maroon">B2</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">B2</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">B1</font> &amp; <font color="Maroon">B1</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">B2</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">B1</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E3:32_3"/>
<font color="Maroon">B2</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">B1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i>, , <a class="ref" href="ordinal3.html#T15">ORDINAL3:15</a></i>;<br/>
<b>then consider </b><font color="Maroon">D</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E5:32_3"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E6:32_3"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">B2</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">B1</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">D</font>
 <b>by </b><i>, </i>;<br/>
<b>take </b>
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">D</font>
;<br/>

<a NAME="E7:32_3"/>
<font color="Maroon">B2</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">B1</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">D</font>
 
<b>by </b><i>, <a class="ref" href="ordinal1.html#T34">ORDINAL1:34</a></i>;<br/>
<a NAME="E8:32_3"/><b>then </b>
<font color="Maroon">B1</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">D</font>
 
<b>by </b><i>, <a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a></i>;<br/>
<b>hence </b><a NAME="E9:32_3"/>
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SUCC</font>
 <b>by </b><i/>;<br/>

<a NAME="E10:32_3"/>
<font color="Maroon">B2</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">D</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E11:32_3"/><b>then </b>
<font color="Maroon">B2</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T34">ORDINAL1:34</a></i>;<br/>
<b>hence </b><a NAME="E12:32_3"/>
<font color="Maroon">B2</font> <a href="ordinal1.html#R1">c=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:32"/><b>then </b>
<font color="Maroon">SUCC</font> is <a href="card_lar.html#V1" target="_self">unbounded</a>
 
<b>by </b><i/>;<br/>
<a NAME="E12:32"/><b>then </b>
 <a href="ordinal2.html#K3">sup</a> <font color="Maroon">SUCC</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<a NAME="E13:32"/><b>then </b>
<font color="Maroon">SUCC</font> <a href="card_lar.html#R1" target="_self">is_unbounded_in</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E14:32"/>
( <font color="Maroon">B1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Olive">B2</font></span>)</span> where <font color="Olive">B2</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> : ( <font color="Olive">B2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">B1</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Olive">B2</font> ) </span>}</span>  <a href="card_lar.html#R3" target="_self">is_club_in</a> <font color="Maroon">A</font> )
 <b>by </b><i>, , </i>;<br/>


</div>
