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

<b>let </b><font color="Maroon">M</font> be   <a href="card_1.html#NM1">Cardinal</a>;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>( not <font color="Maroon">a<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a> implies  ex <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K3">alef</a> <font color="Olive">A</font> );<br/>
<a NAME="E1:6"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1">{}</a> ]
 
;<br/>
<a NAME="E2:6"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">K</font> being  <a href="card_1.html#NM1">Cardinal</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">K</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="card_1.html#K2">nextcard</a> <font color="Olive">K</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">K</font> be   <a href="card_1.html#NM1">Cardinal</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:6_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">K</font>]
 <b>and </b><br/><a NAME="E2:6_1"/><i><font color="Green">E26</font></i>: 
not  <a href="card_1.html#K2">nextcard</a> <font color="Maroon">K</font> is <a href="finset_1.html#V1">finite</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:6_1_1"/>
<font color="Maroon">K</font> is <a href="finset_1.html#V1">finite</a>
 ;<br/><b>then reconsider </b><font color="Maroon">K'</font> = <font color="Maroon">K</font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  ;<br/><a NAME="E3:6_1_1"/>
(  <a href="card_1.html#K1">Card</a> <font color="Maroon">K</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">K'</font></span>)</span> &amp;  <a href="card_1.html#K1">Card</a> <font color="Maroon">K</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K</font> )
 <b>by </b><i><a class="ref" href="card_1.html#D5">CARD_1:def 5</a></i>;<br/><a NAME="E4:6_1_1"/><b>then </b>
 <a href="card_1.html#K2">nextcard</a> <font color="Maroon">K</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">K'</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="card_1.html#T76">CARD_1:76</a></i>;<br/><b>hence </b><a NAME="E5:6_1_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon">A</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E5:6_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">K</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K3">alef</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/>
<b>take </b>
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font>
;<br/>

<b>thus </b><a NAME="E6:6_1"/>
 <a href="card_1.html#K2">nextcard</a> <font color="Maroon">K</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K3">alef</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="card_1.html#T39">CARD_1:39</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:6"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">K</font> being  <a href="card_1.html#NM1">Cardinal</a>  st <font color="Olive">K</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">K</font> <a href="card_1.html#NR3" target="_self">is_limit_cardinal</a>  &amp; ( for <font color="Olive">N</font> being  <a href="card_1.html#NM1">Cardinal</a>  st <font color="Olive">N</font> <a href="card_1.html#NR2" target="_self">&lt;`</a> <font color="Olive">K</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">N</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">K</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">K</font> be   <a href="card_1.html#NM1">Cardinal</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:6_2"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">K</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">K</font> <a href="card_1.html#NR3" target="_self">is_limit_cardinal</a>  )
 <b>and </b><br/><a NAME="E2:6_2"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">N</font> being  <a href="card_1.html#NM1">Cardinal</a>  st <font color="Olive">N</font> <a href="card_1.html#NR2" target="_self">&lt;`</a> <font color="Maroon">K</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">N</font>]
 <b>and </b><br/><a NAME="E3:6_2"/><i><font color="Green">E46</font></i>: 
not <font color="Maroon">K</font> is <a href="finset_1.html#V1">finite</a>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">N</font> being  <a href="card_1.html#NM1">Cardinal</a> st <br/>( <font color="Olive">N</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; not <font color="Olive">N</font> is <a href="finset_1.html#V1">finite</a> );<br/>
<b>consider </b><font color="Maroon">X</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:6_2"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K3">alef</a> <font color="Olive">A</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">A</font> );<br/>
<a NAME="E6:6_2"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">x</font>,<font color="Olive">z</font>] holds <br/><font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">z</font>
 
<b>by </b><i><a class="ref" href="card_1.html#T42">CARD_1:42</a></i>;<br/>
<a NAME="E7:6_2"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:6_2_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>then consider </b><font color="Maroon">N</font> being   <a href="card_1.html#NM1">Cardinal</a><b> such that </b><br/><a NAME="E3:6_2_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">N</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; not <font color="Maroon">N</font> is <a href="finset_1.html#V1">finite</a> )
 <b>by </b><i/>;<br/>
<a NAME="E4:6_2_1"/>
<font color="Maroon">N</font> <a href="card_1.html#NR2" target="_self">&lt;`</a> <font color="Maroon">K</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E5:6_2_1"/><b>then </b>
 ex <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K3">alef</a> <font color="Olive">A</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:6_2_1"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">x</font>,<font color="Olive">y</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E9:6_2"/><i><font color="Green">E53</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">x</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S2">FUNCT_1:sch 2</a>(, );<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:6_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 ;<br/><b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:6_2_2"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon">A</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E5:6_2_2"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K3">alef</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> )
 <b>by </b><i>, </i>;<br/><b>thus </b><a NAME="E6:6_2_2"/>
<font color="Maroon">x</font> is   <a href="ordinal1.html#NM2">Ordinal</a>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E7:6_2_2"/>
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</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="E1:6_2_2_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">z'</font> = <font color="Maroon">z</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E3:6_2_2_1"/>
(  <a href="card_1.html#K3">alef</a> <font color="Maroon">z'</font> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K3">alef</a> <font color="Maroon">A</font> &amp;  <a href="card_1.html#K3">alef</a> <font color="Maroon">A</font> <a href="card_1.html#NR2" target="_self">&lt;`</a> <font color="Maroon">K</font> )
 
<b>by </b><i>, , , , <a class="ref" href="card_1.html#T41">CARD_1:41</a></i>;<br/>
<a NAME="E4:6_2_2_1"/><b>then </b>
(  <a href="card_1.html#K3">alef</a> <font color="Maroon">z'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; not  <a href="card_1.html#K3">alef</a> <font color="Maroon">z'</font> is <a href="finset_1.html#V1">finite</a> )
 
<b>by </b><i>, <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/>
<a NAME="E5:6_2_2_1"/><b>then </b><i><font color="Green">E58</font></i>: 
 <a href="card_1.html#K3">alef</a> <font color="Maroon">z'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:6_2_2_1"/><b>then </b>
 ex <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>(  <a href="card_1.html#K3">alef</a> <font color="Maroon">z'</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K3">alef</a> <font color="Olive">A</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="card_1.html#K3">alef</a> <font color="Maroon">z'</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">A</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E7:6_2_2_1"/><b>then </b>
<font color="Maroon">z'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="card_1.html#K3">alef</a> <font color="Maroon">z'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_1.html#T42">CARD_1:42</a></i>;<br/>
<b>hence </b><a NAME="E8:6_2_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">A</font> =  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="ref" href="ordinal1.html#T31">ORDINAL1:31</a></i>;<br/>
<b>take </b>
<font color="Maroon">A</font>
;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="ordinal1.html#NM2">Ordinal</a>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="card_1.html#K3">alef</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">L</font> being   <a href="ordinal1.html#NM3">T-Sequence</a><b> such that </b><br/><a NAME="E13:6_2"/><i><font color="Green">E61</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp; ( for <font color="Olive">B</font> being  <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> holds <br/><font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">B</font>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S2">ORDINAL2:sch 2</a>();<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">B</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>assume </b><a NAME="E1:6_2_3"/>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:6_2_3"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon">C</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E5:6_2_3"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K3">alef</a> <font color="Maroon">C</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E6:6_2_3"/>
(  <a href="card_1.html#K3">alef</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K2">nextcard</a> <span class="p1">(<span class="default"><a href="card_1.html#K3">alef</a> <font color="Maroon">C</font></span>)</span> &amp;  <a href="card_1.html#K3">alef</a> <font color="Maroon">C</font> <a href="card_1.html#NR2" target="_self">&lt;`</a> <font color="Maroon">K</font> )
 <b>by </b><i>, , , <a class="ref" href="card_1.html#T39">CARD_1:39</a></i>;<br/><a NAME="E7:6_2_3"/><b>then </b>
(  <a href="card_1.html#K3">alef</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">K</font> &amp;  <a href="card_1.html#K3">alef</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">C</font></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a> <font color="Maroon">K</font> )
 <b>by </b><i>, <a class="ref" href="card_1.html#D7">CARD_1:def 7</a>, <a class="ref" href="card_4.html#T22">CARD_4:22</a></i>;<br/><a NAME="E8:6_2_3"/><b>then </b>
(  <a href="card_1.html#K3">alef</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; not  <a href="card_1.html#K3">alef</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">C</font></span>)</span> is <a href="finset_1.html#V1">finite</a> )
 <b>by </b><i>, <a class="ref" href="card_1.html#T13">CARD_1:13</a></i>;<br/><a NAME="E9:6_2_3"/><b>then </b><i><font color="Green">E64</font></i>: 
 <a href="card_1.html#K3">alef</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><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="E11:6_2_3"/><i><font color="Green">E66</font></i>: 
(  <a href="card_1.html#K3">alef</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K3">alef</a> <font color="Maroon">D</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="card_1.html#K3">alef</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> )
 <b>by </b><i/>;<br/><a NAME="E12:6_2_3"/>
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font>
 <b>by </b><i>, <a class="ref" href="card_1.html#T42">CARD_1:42</a></i>;<br/><b>hence </b><a NAME="E13:6_2_3"/>
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, , , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div>
<a NAME="E15:6_2"/><b>then </b>
<font color="Maroon">A</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#T41">ORDINAL1:41</a></i>;<br/>
<a NAME="E16:6_2"/><b>then </b><i><font color="Green">E67</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or  <a href="card_1.html#K3">alef</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4">sup</a> <font color="Maroon">L</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="card_1.html#T40">CARD_1:40</a></i>;<br/>
<a NAME="E17:6_2"/>
 <a href="ordinal2.html#K4">sup</a> <font color="Maroon">L</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">K</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2_4"><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="E1:6_2_4"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K4">sup</a> <font color="Maroon">L</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E3:6_2_4"/>
<font color="Maroon">x</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/>;<br/>
<b>then consider </b><font color="Maroon">C</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E5:6_2_4"/><i><font color="Green">E70</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> &amp; <font color="Maroon">x'</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">C</font> )
 <b>by </b><i><a class="ref" href="ordinal2.html#T29">ORDINAL2:29</a></i>;<br/>
<b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:6_2_4"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L</font> &amp; <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">y</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E9:6_2_4"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K3">alef</a> <font color="Maroon">y</font> &amp; not  <a href="card_1.html#K3">alef</a> <font color="Maroon">y</font> is <a href="finset_1.html#V1">finite</a> )
 
<b>by </b><i>, , </i>;<br/>
<b>consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:6_2_4"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> )
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E12:6_2_4"/>
 ex <font color="Olive">D</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K3">alef</a> <font color="Olive">D</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">D</font> )
 
<b>by </b><i>, <a class="txt" href="card_5.html#E1"><i><font color="Green">Lemma22</font></i></a></i>;<br/>
<a NAME="E13:6_2_4"/><b>then </b>
( <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Maroon">K</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K</font> )
 
<b>by </b><i>, , , <a class="txt" href="card_5.html#E1"><i><font color="Green">Lemma22</font></i></a></i>;<br/>
<b>hence </b><a NAME="E14:6_2_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font>
 <b>by </b><i>, <a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:6_2"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4">sup</a> <font color="Maroon">L</font></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">K</font>
 
<b>by </b><i><a class="ref" href="card_1.html#T27">CARD_1:27</a></i>;<br/>
<a NAME="E19:6_2"/><b>then </b><i><font color="Green">E74</font></i>: 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4">sup</a> <font color="Maroon">L</font></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a> <font color="Maroon">K</font>
 
<b>by </b><i><a class="ref" href="card_1.html#D5">CARD_1:def 5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2_5"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_2_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:6_2_5_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_2_5_1_1"><b>case </b></a><a NAME="E1:6_2_5_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:6_2_5_1_1"/><b>then </b>
( not  <a href="card_1.html#K3">alef</a> 0 <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; not  <a href="card_1.html#K3">alef</a> 0 is <a href="finset_1.html#V1">finite</a> )
 <b>by </b><i>, , <a class="ref" href="relat_1.html#T65">RELAT_1:65</a></i>;<br/><a NAME="E3:6_2_5_1_1"/><b>then </b>
not  <a href="card_1.html#K3">alef</a> 0 <a href="card_1.html#NR2" target="_self">&lt;`</a> <font color="Maroon">K</font>
 <b>by </b><i/>;<br/><a NAME="E4:6_2_5_1_1"/><b>then </b>
( <font color="Maroon">K</font> <a href="ordinal1.html#R1">c=</a>  <a href="card_1.html#K3">alef</a> 0 &amp;  <a href="card_1.html#K3">alef</a> 0 <a href="ordinal1.html#R1">c=</a> <font color="Maroon">K</font> )
 <b>by </b><i>, <a class="ref" href="card_1.html#T14">CARD_1:14</a>, <a class="ref" href="card_4.html#T11">CARD_4:11</a></i>;<br/><b>hence </b><a NAME="E5:6_2_5_1_1"/>
<font color="Maroon">K</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K3">alef</a> 0
 <b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_2_5_1_2"><b>case </b></a><a NAME="E1:6_2_5_1_2"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>assume </b><a NAME="E2:6_2_5_1_2"/>
<font color="Maroon">K</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="card_1.html#K3">alef</a> <font color="Maroon">A</font>
 ;<br/><a NAME="E3:6_2_5_1_2"/><b>then </b>
(  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4">sup</a> <font color="Maroon">L</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; not  <a href="card_1.html#K3">alef</a> <font color="Maroon">A</font> is <a href="finset_1.html#V1">finite</a> )
 <b>by </b><i>, , , , <a class="ref" href="card_1.html#T13">CARD_1:13</a></i>;<br/><a NAME="E4:6_2_5_1_2"/><b>then </b><i><font color="Green">E76</font></i>: 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4">sup</a> <font color="Maroon">L</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, , </i>;<br/><b>then consider </b><font color="Maroon">B</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E6:6_2_5_1_2"/><i><font color="Green">E77</font></i>: 
(  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4">sup</a> <font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K3">alef</a> <font color="Maroon">B</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <span class="p2">(<span class="default"><a href="ordinal2.html#K4">sup</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> )
 <b>by </b><i/>;<br/><a NAME="E7:6_2_5_1_2"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 <b>by </b><i>, , , <a class="ref" href="card_1.html#T42">CARD_1:42</a></i>;<br/><a NAME="E8:6_2_5_1_2"/><b>then </b>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E9:6_2_5_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E21:6_2"/>
<font color="Maroon">K</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K3">alef</a> <font color="Maroon">A</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:6"/>
for <font color="Olive">M</font> being  <a href="card_1.html#NM1">Cardinal</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">M</font>]
 
<b>from </b><i><a class="ref" href="card_1.html#S1">CARD_1:sch 1</a>(, , );<br/></i>
<b>hence </b><a NAME="E5:6"/>
( not <font color="Maroon">M</font> is <a href="finset_1.html#V1">finite</a> implies  ex <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Maroon">M</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K3">alef</a> <font color="Olive">A</font> )
 ;<br/>


</div>
