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

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



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] <b>means </b> <a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<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> ;<br/>
<b>assume </b><a NAME="E1:29"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E2:29"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1">{}</a> ]
 
<b>by </b><i><a class="ref" href="ordinal2.html#T60">ORDINAL2:60</a></i>;<br/>
<a NAME="E3:29"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">A</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1">succ</a> <font color="Olive">A</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:29_1"/><i><font color="Green">E27</font></i>: 
 <a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E2:29_1"/>
 <a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="ordinal2.html#K11">*^</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="ordinal2.html#T61">ORDINAL2:61</a></i>;<br/>
<b>hence </b><a NAME="E3:29_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font>]
 <b>by </b><i><a class="txt" href="ordinal4.html#E1"><i><font color="Green">Lemma23</font></i></a>, <a class="txt" href="ordinal4.html#E3:29"><i><font color="Green">E26</font></i></a>, <a class="ref" href="ordinal3.html#T34">ORDINAL3:34</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:29"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">A</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  &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="Olive">A</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">A</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2"><b>proof </b></a><div class="add">

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

<b>assume </b><b>that </b><br/><a NAME="E1:29_2"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">A</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  )
 <b>and </b><br/><a NAME="E2:29_2"/><i><font color="Green">E36</font></i>: 
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/> <a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<font color="Olive">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon">fi</font> being   <a href="ordinal2.html#NM1">Ordinal-Sequence</a><b> such that </b><br/><a NAME="E4:29_2"/><i><font color="Green">E37</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fi</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">fi</font> <a href="funct_1.html#K1">.</a> <font color="Olive">B</font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<font color="Olive">B</font> ) &amp;  ex <font color="Olive">D</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">D</font> <a href="ordinal2.html#R1">is_limes_of</a> <font color="Maroon">fi</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E5:29_2"/><i><font color="Green">E42</font></i>: 
 <a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K8">lim</a> <font color="Maroon">fi</font>
 
<b>by </b><i>, <a class="txt" href="ordinal4.html#E1:29_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="ordinal2.html#T62">ORDINAL2:62</a></i>;<br/>
<b>consider </b><font color="Maroon">D</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E7:29_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">D</font> <a href="ordinal2.html#R1">is_limes_of</a> <font color="Maroon">fi</font>
 <b>by </b><i><a class="txt" href="ordinal4.html#E1:29_2"><i><font color="Green">E29</font></i></a></i>;<br/>
<a NAME="E8:29_2"/><i><font color="Green">E45</font></i>: 
 <a href="ordinal2.html#K8">lim</a> <font color="Maroon">fi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font>
 
<b>by </b><i>, <a class="ref" href="ordinal2.html#D14">ORDINAL2:def 14</a></i>;<br/>
<b>assume </b><a NAME="E9:29_2"/>
 <a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<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="E11:29_2"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fi</font> &amp; ( for <font color="Olive">D</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Maroon">B</font> <a href="ordinal1.html#R1">c=</a> <font color="Olive">D</font> &amp; <font color="Olive">D</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fi</font> holds <br/><font color="Maroon">fi</font> <a href="funct_1.html#K1">.</a> <font color="Olive">D</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 <b>by </b><i><a class="ref" href="ordinal4.html#D1" target="_self">Def1</a>, , , <a class="ref" href="ordinal2.html#D13">ORDINAL2:def 13</a></i>;<br/>
<a NAME="E12:29_2"/>
( <font color="Maroon">fi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<font color="Maroon">B</font> &amp;  <a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">fi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="txt" href="ordinal4.html#E4:29"><i><font color="Green">E28</font></i></a>, <a class="txt" href="ordinal4.html#E1:29_2"><i><font color="Green">E29</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E13:29_2"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:29"/>
for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">A</font>]
 
<b>from </b><i><a class="ref" href="ordinal2.html#S1">ORDINAL2:sch 1</a>(, <a class="txt" href="ordinal4.html#E2:29"><i><font color="Green">E25</font></i></a>, <a class="txt" href="ordinal4.html#E3:29"><i><font color="Green">E26</font></i></a>);<br/></i>
<b>hence </b><a NAME="E6:29"/>
 <a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>


</div>
