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

<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>let </b><font color="Maroon">X</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">M</font>;<br/>



<b>assume </b><a NAME="E1:38"/><i><font color="Green">E22</font></i>: 
 <a href="card_1.html#K3">alef</a> 0 <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_5.html#K1">cf</a> <font color="Maroon">M</font>
 ;<br/>

<b>assume </b><a NAME="E2:38"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">X</font> is <a href="card_lar.html#V1" target="_self">unbounded</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon">X1</font> = <font color="Maroon">X</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">M</font> <b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">B1</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E4:38"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">B1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">LB1</font> =  <a href="card_lar.html#K1" target="_self">LBound</a> <font color="Maroon">B1</font>,<font color="Maroon">X1</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">X1</font> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>3</sub></font>;<br/>
<a NAME="E6:38"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">x</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">X1</font>  ex <font color="Olive">y</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">X1</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">X1</font>;<br/>



<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">M</font> ;<br/>
<a NAME="E2:38_1"/>
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">x1</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/>
<b>then consider </b><font color="Maroon">y</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E4:38_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font>
 <b>and </b><br/><a NAME="E5:38_1"/><i><font color="Green">E47</font></i>: 
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">x1</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">y</font>
 <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">y1</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">X1</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">y1</font>
;<br/>

<a NAME="E7:38_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">x1</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a></i>;<br/>
<b>hence </b><a NAME="E8:38_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y1</font>
 <b>by </b><i><a class="ref" href="card_lar.html#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">L</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<font color="Maroon">X1</font><b> such that </b><br/><a NAME="E8:38"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">LB1</font>
 <b>and </b><br/><a NAME="E9:38"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font>,<font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>]
 <b>from </b><i><a class="ref" href="recdef_1.html#S2">RECDEF_1:sch 2</a>();<br/></i>
<a NAME="E10:38"/><i><font color="Green">E50</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E11:38"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E12:38"/><i><font color="Green">E66</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">L</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E13:38"/><b>then </b><i><font color="Green">E67</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">L</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>set </b><font color="Maroon">L2</font> = <font color="Maroon">L</font>;<br/>
<a NAME="E14:38"/>
for <font color="Olive">C</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">C</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L</font> holds <br/> ex <font color="Olive">D</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Olive">D</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L</font> &amp; <font color="Olive">C</font> <a href="hidden.html#R2">in</a> <font color="Olive">D</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:38_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L</font>
 ;<br/>

<b>consider </b><font color="Maroon">C1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:38_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">C1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E4:38_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">C1</font>
 <b>by </b><i><a class="ref" href="card_lar.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">C2</font> = <font color="Maroon">C1</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="card_lar.html#T3" target="_self">Th3</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:38_2"/>
<font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">C2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">C3</font> = <font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">C2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">M</font> ;<br/>
<b>take </b>
<font color="Maroon">C3</font>
;<br/>

<b>thus </b><a NAME="E8:38_2"/>
<font color="Maroon">C3</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="card_lar.html#D2" target="_self">Def2</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>

<b>thus </b><a NAME="E9:38_2"/>
<font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C3</font>
 <b>by </b><i><a class="ref" href="card_lar.html#D1" target="_self">Def1</a>, </i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">SUP</font> =  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">L</font></span>)</span> as  <a href="ordinal1.html#V4">being_limit_ordinal</a> <a href="finset_1.html#V1">infinite</a>  <a href="subset_1.html#M1">Element</a> of <font color="Maroon">M</font> <b>by </b><i>, <a class="ref" href="card_lar.html#D3" target="_self">Def3</a>, <a class="ref" href="card_lar.html#D5" target="_self">Def5</a>, , </i>;<br/>
<b>take </b>
<font color="Maroon">SUP</font>
;<br/>

<a NAME="E16:38"/>
 <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">SUP</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">SUP</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:38_3"/>
 <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">SUP</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">SUP</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="E3:38_3"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">B5</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SUP</font>
 <b>and </b><br/><a NAME="E4:38_3"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">X</font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">SUP</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B5</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">B6</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E6:38_3"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">B6</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E7:38_3"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">B5</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">B6</font>
 <b>by </b><i><a class="ref" href="card_lar.html#T2" target="_self">Th2</a>, <a class="ref" href="ordinal2.html#T29">ORDINAL2:29</a></i>;<br/>
<a NAME="E8:38_3"/>
<font color="Maroon">B6</font> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">L</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="ordinal2.html#T27">ORDINAL2:27</a></i>;<br/>
<a NAME="E9:38_3"/><b>then </b>
<font color="Maroon">B6</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">SUP</font>
 
<b>by </b><i><a class="ref" href="card_lar.html#D4" target="_self">Def4</a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E10:38_3"/><b>then </b>
<font color="Maroon">B6</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B5</font>
 
<b>by </b><i><a class="ref" href="card_lar.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E11:38_3"/><b>then </b>
<font color="Maroon">B6</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B6</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E12:38_3"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<a NAME="E17:38"/><b>then </b><i><font color="Green">E94</font></i>: 
<font color="Maroon">SUP</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">B4</font> where <font color="Olive">B4</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">M</font> : ( not <font color="Olive">B4</font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Olive">B4</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">B4</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">B4</font> ) </span>}</span> 
 
;<br/>
<b>reconsider </b><font color="Maroon">LB2</font> = <font color="Maroon">LB1</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">M</font> ;<br/>
<a NAME="E19:38"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">LB2</font> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">L</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="card_lar.html#D3" target="_self">Def3</a>, <a class="ref" href="ordinal2.html#T27">ORDINAL2:27</a></i>;<br/>
<a NAME="E20:38"/>
<font color="Maroon">B1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">LB2</font>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E21:38"/>
( <font color="Maroon">SUP</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">B1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SUP</font> &amp; <font color="Maroon">SUP</font> <a href="hidden.html#R2">in</a>  <a href="card_lar.html#K2" target="_self">limpoints</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="ref" href="card_lar.html#T2" target="_self">Th2</a>, <a class="ref" href="card_lar.html#T3" target="_self">Th3</a>, <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/>


</div>
