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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="classes2.html#NM1">Universe</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="zf_refle.html#NM2">DOMAIN-Sequence</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:36"/><i><font color="Green">E21</font></i>: 
 <a href="ordinal1.html#K5">omega</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E2:36"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Olive">b<sub>2</sub></font>
 <b>and </b><br/><a NAME="E3:36"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K3">Union</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM2">ZF-formula</a>ex <font color="Olive">b<sub>2</sub></font> being   <a href="ordinal4.html#M2">Ordinal-Sequence</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="ordinal2.html#V2">increasing</a> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="ordinal2.html#V3">continuous</a> &amp; ( for <font color="Olive">b<sub>3</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>3</sub></font> holds <br/>for <font color="Olive">b<sub>4</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Olive">b<sub>3</sub></font> holds <br/>(  <a href="zf_refle.html#K4">Union</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Olive">b<sub>4</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Olive">b<sub>1</sub></font> iff <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Olive">b<sub>1</sub></font> ) ) );<br/>
<a NAME="E4:36"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_lang.html#K9">WFF</a>  holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:36_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_lang.html#K9">WFF</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="zf_lang.html#NM2">ZF-formula</a> <b>by </b><i><a class="ref" href="zf_lang.html#T14">ZF_LANG:14</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="ordinal4.html#M2">Ordinal-Sequence</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:36_1"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> is <a href="ordinal2.html#V2">increasing</a> &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="ordinal2.html#V3">continuous</a> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>1</sub></font> holds <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/>(  <a href="zf_refle.html#K4">Union</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Olive">b<sub>2</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>4</sub></font> iff <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>4</sub></font> ) ) )
 <b>by </b><i><a class="txt" href="zfrefle1.html#E1:36"><i><font color="Green">E21</font></i></a>, <a class="txt" href="zfrefle1.html#E2:36"><i><font color="Green">E22</font></i></a>, <a class="txt" href="zfrefle1.html#E3:36"><i><font color="Green">E23</font></i></a>, <a class="ref" href="zf_refle.html#T29">ZF_REFLE:29</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="hidden.html#M1">set</a>  ;<br/>
<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/>

<a NAME="E6:36_1"/>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="ordinal4.html#D3">ORDINAL4:def 3</a></i>;<br/>
<b>hence </b><a NAME="E7:36_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>

<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>thus </b><a NAME="E8:36_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="ordinal2.html#V2">increasing</a> &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="ordinal2.html#V3">continuous</a> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>1</sub></font> holds <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/>(  <a href="zf_refle.html#K4">Union</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Olive">b<sub>2</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>4</sub></font> iff <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>4</sub></font> ) ) )
 <b>by </b><i><a class="txt" href="zfrefle1.html#E4:36_1"><i><font color="Green">E25</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:36"/><i><font color="Green">E25</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="zf_lang.html#K9">WFF</a>  &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 <b>and </b><br/><a NAME="E7:36"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_lang.html#K9">WFF</a>  holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="wellord2.html#S1">WELLORD2:sch 1</a>(<a class="txt" href="zfrefle1.html#E4:36"><i><font color="Green">E24</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K9">WFF</a> , <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="zfrefle1.html#E6:36"><i><font color="Green">E25</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<a NAME="E9:36"/>
( <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K5">NAT</a> ,<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="ordinal1.html#K5">omega</a> ,<a href="ordinal1.html#K5">omega</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="wellord2.html#R2">are_equipotent</a>  &amp; <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="ordinal1.html#K5">omega</a> ,<a href="ordinal1.html#K5">omega</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E1:36"><i><font color="Green">E21</font></i></a>, <a class="ref" href="classes2.html#T67">CLASSES2:67</a></i>;<br/>
<a NAME="E10:36"/><b>then </b>
(  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K5">NAT</a> ,<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="ordinal1.html#K5">omega</a> ,<a href="ordinal1.html#K5">omega</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> &amp;  <a href="zf_lang.html#K9">WFF</a>  <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K5">NAT</a> ,<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="zfmisc_1.html#K1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="ordinal1.html#K5">omega</a> ,<a href="ordinal1.html#K5">omega</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="classes2.html#T65">CLASSES2:65</a>, <a class="ref" href="zf_lang1.html#T146">ZF_LANG1:146</a></i>;<br/>
<a NAME="E11:36"/><b>then </b>
(  <a href="card_1.html#K1">Card</a> <a href="zf_lang.html#K9">WFF</a>  <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="ordinal1.html#K5">omega</a> ,<a href="ordinal1.html#K5">omega</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="ordinal1.html#K5">omega</a> ,<a href="ordinal1.html#K5">omega</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="card_1.html#T27">CARD_1:27</a>, <a class="ref" href="classes2.html#T1">CLASSES2:1</a></i>;<br/>
<a NAME="E12:36"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <a href="zf_lang.html#K9">WFF</a>  <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="ordinal4.html#M2">Ordinal-Sequence</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E14:36"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> is <a href="ordinal2.html#V2">increasing</a> &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="ordinal2.html#V3">continuous</a> )
 <b>and </b><br/><a NAME="E15:36"/>
<font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E16:36"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="relat_1.html#K9">.:</a> <span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="zf_lang.html#K9">WFF</a> ,<span class="p4"><a href="tarski.html#K1">{</a><span class="default"><span class="p5">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E17:36"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K4">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="zfrefle1.html#T14" target="_self">Th14</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>thus </b><a NAME="E18:36"/>
( <font color="Maroon">c<sub>5</sub></font> is <a href="ordinal2.html#V2">increasing</a> &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="ordinal2.html#V3">continuous</a> )
 <b>by </b><i><a class="txt" href="zfrefle1.html#E14:36"><i><font color="Green">E27</font></i></a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E19:36"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/>

<b>thus </b><a NAME="E20:36"/>
<font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a>  <a href="zf_refle.html#K4">Union</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="zf_refle.html#T24">ZF_REFLE:24</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="zfrefle1.html#D3" target="_self">ZFREFLE1:def 3</a><br/>

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">c<sub>6</sub></font>;<br/>



<b>set </b><font color="Maroon">c<sub>9</sub></font> =  <a href="zf_refle.html#K4">Union</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E21:36"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_lang.html#K9">WFF</a> 
 
<b>by </b><i><a class="ref" href="zf_lang.html#T14">ZF_LANG:14</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="zf_lang.html#NM2">ZF-formula</a>, <font color="Maroon">c<sub>11</sub></font> being    <a href="ordinal4.html#M2">Ordinal-Sequence</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E23:36"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> is <a href="ordinal2.html#V2">increasing</a> &amp; <font color="Maroon">c<sub>11</sub></font> is <a href="ordinal2.html#V3">continuous</a> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>11</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>1</sub></font> holds <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/>(  <a href="zf_refle.html#K4">Union</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Olive">b<sub>2</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>10</sub></font> iff <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>10</sub></font> ) ) )
 <b>by </b><i><a class="txt" href="zfrefle1.html#E7:36"><i><font color="Green">E26</font></i></a></i>;<br/>
<a NAME="E24:36"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="ref" href="ordinal4.html#T36">ORDINAL4:36</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] means ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  implies <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E25:36"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[ <a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font>]
 
<b>by </b><i><a class="ref" href="ordinal4.html#T35">ORDINAL4:35</a></i>;<br/>
<a NAME="E26:36"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[ <a href="ordinal4.html#K6">succ</a> <font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E1:36_2"/>
(  <a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>11</sub></font> &amp;  <a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="ordinal4.html#T36">ORDINAL4:36</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E2:36_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<span class="p2">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<span class="p2">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="zf_lang.html#K9">WFF</a> ,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E6:36"><i><font color="Green">E25</font></i></a>, <a class="txt" href="zfrefle1.html#E21:36"><i><font color="Green">E31</font></i></a>, <a class="txt" href="zfrefle1.html#E23:36"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_5.html#T45">FUNCT_5:45</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E3:36_2"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="relat_1.html#K9">.:</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="zf_lang.html#K9">WFF</a> ,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E4:36_2"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="relat_1.html#K9">.:</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="zf_lang.html#K9">WFF</a> ,<span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E5:36_2"/><b>then </b>
( <font color="Maroon">c<sub>11</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="relat_1.html#K9">.:</a> <span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="zf_lang.html#K9">WFF</a> ,<span class="p4"><a href="tarski.html#K1">{</a><span class="default"><span class="p5">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="relat_1.html#K9">.:</a> <span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="zf_lang.html#K9">WFF</a> ,<span class="p4"><a href="tarski.html#K1">{</a><span class="default"><span class="p5">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E16:36"><i><font color="Green">E28</font></i></a>, <a class="ref" href="ordinal2.html#T27">ORDINAL2:27</a></i>;<br/>
<b>hence </b><a NAME="E6:36_2"/>
( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>12</sub></font>] implies <font color="Maroon">S<sub>3</sub></font>[ <a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>12</sub></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="E27:36"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>2</sub></font>] ) holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:36_3"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  )
 <b>and </b><br/><a NAME="E2:36_3"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/><font color="Maroon">c<sub>11</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>and </b><br/><a NAME="E3:36_3"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<i><font color="Green">E40</font></i>: <font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font> = 
 <a href="ordinal2.html#K4">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="zfrefle1.html#E17:36"><i><font color="Green">E29</font></i></a>, <a class="txt" href="zfrefle1.html#E1:36_3"><i><font color="Green">E37</font></i></a></i>
<br/>.= 
 <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span>

;<br/>
<b>let </b><font color="Maroon">c<sub>13</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="zf_refle.html#D1">ZF_REFLE:def 1</a><br/>

<b>assume </b><a NAME="E5:36_3"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/>

<a NAME="E6:36_3"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="ref" href="ordinal4.html#T36">ORDINAL4:36</a></i>;<br/>
<a NAME="E7:36_3"/><b>then </b><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="ordinal2.html#R1">is_limes_of</a> <font color="Maroon">c<sub>11</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>12</sub></font> is <a href="ordinal2.html#V2">increasing</a> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>11</sub></font> )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E23:36"><i><font color="Green">E32</font></i></a>, <a class="txt" href="zfrefle1.html#E1:36_3"><i><font color="Green">E37</font></i></a>, <a class="txt" href="zfrefle1.html#E3:36_3"><i><font color="Green">E39</font></i></a>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a>, <a class="ref" href="ordinal2.html#D17">ORDINAL2:def 17</a>, <a class="ref" href="ordinal4.html#T15">ORDINAL4:15</a></i>;<br/>
<a NAME="E8:36_3"/><b>then </b><i><font color="Green">E43</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K8">lim</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="ordinal2.html#D14">ORDINAL2:def 14</a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<b>then </b><font color="Maroon">c<sub>11</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font> = 
 <a href="ordinal2.html#K4">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="zfrefle1.html#E1:36_3"><i><font color="Green">E37</font></i></a>, <a class="txt" href="zfrefle1.html#E3:36_3"><i><font color="Green">E39</font></i></a>, <a class="txt" href="zfrefle1.html#E7:36_3"><i><font color="Green">E42</font></i></a>, <a class="ref" href="ordinal4.html#T8">ORDINAL4:8</a></i>
<br/>.= 
 <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span>

;<br/>
<b>then consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E11:36_3"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="zfrefle1.html#E5:36_3"><i><font color="Green">E41</font></i></a>, <a class="ref" href="ordinal2.html#T29">ORDINAL2:29</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>15</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E13:36_3"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="txt" href="zfrefle1.html#E11:36_3"><i><font color="Green">E44</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="txt" href="zfrefle1.html#E13:36_3"><i><font color="Green">E45</font></i></a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E15:36_3"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="zf_refle.html#T8">ZF_REFLE:8</a></i>;<br/>
<a NAME="E16:36_3"/><b>then </b>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E8:36_3"><i><font color="Green">E43</font></i></a>, <a class="txt" href="zfrefle1.html#E13:36_3"><i><font color="Green">E45</font></i></a>, <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>16</sub></font> as    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="zf_refle.html#T8">ZF_REFLE:8</a></i>;<br/>
<a NAME="E18:36_3"/>
(  <a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> &amp;  <a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E1:36_3"><i><font color="Green">E37</font></i></a>, <a class="txt" href="zfrefle1.html#E8:36_3"><i><font color="Green">E43</font></i></a>, <a class="txt" href="zfrefle1.html#E13:36_3"><i><font color="Green">E45</font></i></a>, <a class="ref" href="ordinal1.html#T41">ORDINAL1:41</a>, <a class="ref" href="ordinal4.html#T36">ORDINAL4:36</a></i>;<br/>
<a NAME="E19:36_3"/><b>then </b>
(  <a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> &amp;  <a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>11</sub></font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T73">FUNCT_1:73</a>, <a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a>, <a class="ref" href="ordinal4.html#T36">ORDINAL4:36</a></i>;<br/>
<a NAME="E20:36_3"/><b>then </b>
( <font color="Maroon">c<sub>11</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>17</sub></font> )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E23:36"><i><font color="Green">E32</font></i></a>, <a class="txt" href="zfrefle1.html#E2:36_3"><i><font color="Green">E38</font></i></a>, <a class="txt" href="zfrefle1.html#E4:36_3"><i><font color="Green">E40</font></i></a>, <a class="txt" href="zfrefle1.html#E13:36_3"><i><font color="Green">E45</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a>, <a class="ref" href="ordinal2.html#T27">ORDINAL2:27</a>, <a class="ref" href="ordinal2.html#D16">ORDINAL2:def 16</a></i>;<br/>
<a NAME="E21:36_3"/><b>then </b>
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font> )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E11:36_3"><i><font color="Green">E44</font></i></a>, <a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a></i>;<br/>
<b>hence </b><a NAME="E22:36_3"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E28:36"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="zf_refle.html#S5">ZF_REFLE:sch 5</a>(<a class="txt" href="zfrefle1.html#E25:36"><i><font color="Green">E34</font></i></a>, <a class="txt" href="zfrefle1.html#E26:36"><i><font color="Green">E35</font></i></a>, <a class="txt" href="zfrefle1.html#E27:36"><i><font color="Green">E36</font></i></a>);<br/></i>
<a NAME="E29:36"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>11</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E19:36"><i><font color="Green">E30</font></i></a>, <a class="txt" href="zfrefle1.html#E23:36"><i><font color="Green">E32</font></i></a>, <a class="txt" href="zfrefle1.html#E24:36"><i><font color="Green">E33</font></i></a>, <a class="ref" href="ordinal4.html#T10">ORDINAL4:10</a></i>;<br/>
<a NAME="E30:36"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E31:36"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>7</sub></font> iff  <a href="zf_refle.html#K4">Union</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="zf_refle.html#K4">Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>8</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="zfrefle1.html#E19:36"><i><font color="Green">E30</font></i></a>, <a class="txt" href="zfrefle1.html#E23:36"><i><font color="Green">E32</font></i></a></i>;<br/>


</div>
