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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>



<b>assume </b><a NAME="E1:28"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>1</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="ordinal2.html#NM1">Ordinal-Sequence</a>;<br/>

<b>assume </b><a NAME="E2:28"/><i><font color="Green">E23</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K12">exp</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> ) )
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="ordinal2.html#NM1">Ordinal-Sequence</a><b> such that </b><br/><a NAME="E4:28"/><i><font color="Green">E24</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K12">exp</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> ) )
 <b>and </b><br/><a NAME="E5:28"/><i><font color="Green">E25</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>1</sub></font> <a href="ordinal2.html#R1">is_limes_of</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="ordinal4.html#E1:28"><i><font color="Green">E22</font></i></a>, <a class="txt" href="ordinal4.html#E27"><i><font color="Green">Lemma20</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:28_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></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/><b>thus </b><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> = 
 <a href="ordinal2.html#K12">exp</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="ordinal4.html#E2:28"><i><font color="Green">E23</font></i></a>, <a class="txt" href="ordinal4.html#E1:28_1"><i><font color="Green">E26</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="ordinal4.html#E4:28"><i><font color="Green">E24</font></i></a>, <a class="txt" href="ordinal4.html#E1:28_1"><i><font color="Green">E26</font></i></a></i>
;<br/></div><b>end;</b></div>
<a NAME="E7:28"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="ordinal4.html#E2:28"><i><font color="Green">E23</font></i></a>, <a class="txt" href="ordinal4.html#E4:28"><i><font color="Green">E24</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E9:28"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="ordinal2.html#R1">is_limes_of</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="ordinal4.html#E5:28"><i><font color="Green">E25</font></i></a></i>;<br/>
<font color="Maroon">c<sub>5</sub></font> = 
 <a href="ordinal2.html#K8">lim</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="ordinal4.html#E9:28"><i><font color="Green">E26</font></i></a>, <a class="ref" href="ordinal2.html#D14">ORDINAL2:def 14</a></i>
<br/>.= 
 <a href="ordinal2.html#K12">exp</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="ordinal4.html#E1:28"><i><font color="Green">E22</font></i></a>, <a class="txt" href="ordinal4.html#E2:28"><i><font color="Green">E23</font></i></a>, <a class="ref" href="ordinal2.html#T62">ORDINAL2:62</a></i>
;<br/>
<b>hence </b><a NAME="E11:28"/>
 <a href="ordinal2.html#K12">exp</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#R1">is_limes_of</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="ordinal4.html#E9:28"><i><font color="Green">E26</font></i></a></i>;<br/>


</div>
