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

<b>consider </b><font color="Maroon">L</font> being   <a href="ordinal1.html#NM3">T-Sequence</a><b> such that </b><br/><a NAME="E2:42_1"/><i><font color="Green">E6</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">F<sub>1</sub></font>() &amp; (  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">F<sub>1</sub></font>() implies <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() ) &amp; ( for <font color="Olive">C</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st  <a href="ordinal1.html#K1">succ</a> <font color="Olive">C</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Olive">C</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">C</font>,<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Olive">C</font></span>)</span>) ) &amp; ( 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="ordinal1.html#K1">succ</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Olive">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">C</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  holds <br/><font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Olive">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">C</font>,<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="ordinal1.html#K2">|</a> <font color="Olive">C</font></span>)</span>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S5" target="_self">ORDINAL2:sch 5</a>();<br/></i>
<b>thus </b><a NAME="E3:42_1"/>
 ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  ex <font color="Olive">L</font> being  <a href="ordinal1.html#NM3">T-Sequence</a> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K1" target="_self">last</a> <font color="Olive">L</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">L</font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Olive">L</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; ( for <font color="Olive">C</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st  <a href="ordinal1.html#K1">succ</a> <font color="Olive">C</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Olive">L</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Olive">C</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">C</font>,<span class="p1">(<span class="default"><font color="Olive">L</font> <a href="funct_1.html#K1">.</a> <font color="Olive">C</font></span>)</span>) ) &amp; ( 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="ordinal1.html#K1">succ</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Olive">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">C</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  holds <br/><font color="Olive">L</font> <a href="funct_1.html#K1">.</a> <font color="Olive">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">C</font>,<span class="p1">(<span class="default"><font color="Olive">L</font> <a href="ordinal1.html#K2">|</a> <font color="Olive">C</font></span>)</span>) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon">x</font> =  <a href="ordinal2.html#K1" target="_self">last</a> <font color="Maroon">L</font>;<br/>

<b>take </b>
<font color="Maroon">L</font>
;<br/>

<b>thus </b><a NAME="E1:42_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K1" target="_self">last</a> <font color="Maroon">L</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">F<sub>1</sub></font>() )
 <b>by </b><i/>;<br/>

<a NAME="E2:42_1_1"/>
(  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="ordinal1.html#R1">c=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">F<sub>1</sub></font>() )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>
<a NAME="E3:42_1_1"/><b>then </b>
 <a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#R2">c&lt;</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/>
<b>hence </b><a NAME="E4:42_1_1"/>
( <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; ( for <font color="Olive">C</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st  <a href="ordinal1.html#K1">succ</a> <font color="Olive">C</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Olive">C</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">C</font>,<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Olive">C</font></span>)</span>) ) &amp; ( 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="ordinal1.html#K1">succ</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Olive">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">C</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  holds <br/><font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Olive">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">C</font>,<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="ordinal1.html#K2">|</a> <font color="Olive">C</font></span>)</span>) ) )
 <b>by </b><i>, <a class="ref" href="ordinal1.html#T21">ORDINAL1:21</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>given </b><font color="Maroon">L1</font> being   <a href="ordinal1.html#NM3">T-Sequence</a><b> such that </b><a NAME="E4:42_1"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K1" target="_self">last</a> <font color="Maroon">L1</font>
 <b>and </b><br/><a NAME="E5:42_1"/><i><font color="Green">E25</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">L1</font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>and </b><br/><a NAME="E6:42_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>()
 <b>and </b><br/><a NAME="E7:42_1"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">C</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st  <a href="ordinal1.html#K1">succ</a> <font color="Olive">C</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Olive">C</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">C</font>,<span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">C</font></span>)</span>)
 <b>and </b><br/><a NAME="E8:42_1"/><i><font color="Green">E28</font></i>: 
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="ordinal1.html#K1">succ</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Olive">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">C</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  holds <br/><font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">C</font>,<span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="ordinal1.html#K2">|</a> <font color="Olive">C</font></span>)</span>)
 ;<br/>

<b>given </b><font color="Maroon">L2</font> being   <a href="ordinal1.html#NM3">T-Sequence</a><b> such that </b><a NAME="E9:42_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K1" target="_self">last</a> <font color="Maroon">L2</font>
 <b>and </b><br/><a NAME="E10:42_1"/><i><font color="Green">E30</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">L2</font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>and </b><br/><a NAME="E11:42_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>()
 <b>and </b><br/><a NAME="E12:42_1"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">C</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st  <a href="ordinal1.html#K1">succ</a> <font color="Olive">C</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Olive">C</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">C</font>,<span class="p1">(<span class="default"><font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">C</font></span>)</span>)
 <b>and </b><br/><a NAME="E13:42_1"/><i><font color="Green">E54</font></i>: 
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="ordinal1.html#K1">succ</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Olive">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">C</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  holds <br/><font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">C</font>,<span class="p1">(<span class="default"><font color="Maroon">L2</font> <a href="ordinal1.html#K2">|</a> <font color="Olive">C</font></span>)</span>)
 ;<br/>

<a NAME="E14:42_1"/><i><font color="Green">E55</font></i>: 
(  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">F<sub>1</sub></font>() implies <font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() )
 
<b>by </b><i/>;<br/>
<a NAME="E15:42_1"/><i><font color="Green">E56</font></i>: 
(  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">F<sub>1</sub></font>() implies <font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() )
 
<b>by </b><i/>;<br/>
<a NAME="E16:42_1"/>
<font color="Maroon">L1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L2</font>
 
<b>from </b><i><a class="ref" href="ordinal2.html#S4" target="_self">ORDINAL2:sch 4</a>(, <a class="txt" href="ordinal2.html#E5:42_1"><i><font color="Green">E25</font></i></a>, , , , <a class="txt" href="ordinal2.html#E6:42_1"><i><font color="Green">E26</font></i></a>, , );<br/></i>
<b>hence </b><a NAME="E17:42_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i>, </i>;<br/>


</div>
