<?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>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] <b>means </b>( <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>  &amp; <font color="Maroon">a<sub>1</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  &amp; ( for <font color="Olive">fi</font> being  <a href="ordinal2.html#NM1">Ordinal-Sequence</a>  st  <a href="relat_1.html#K1">dom</a> <font color="Olive">fi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></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<sub>1</sub></font> holds <br/><font color="Olive">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> ) holds <br/>for <font color="Olive">D</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  holds not <font color="Olive">D</font> <a href="ordinal2.html#R1">is_limes_of</a> <font color="Olive">fi</font> ) );<br/>
<b>let </b><font color="Maroon">A</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E1:27"/><i><font color="Green">E13</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>  &amp; ( for <font color="Olive">fi</font> being  <a href="ordinal2.html#NM1">Ordinal-Sequence</a>  st  <a href="relat_1.html#K1">dom</a> <font color="Olive">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="Olive">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> ) holds <br/>for <font color="Olive">D</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  holds not <font color="Olive">D</font> <a href="ordinal2.html#R1">is_limes_of</a> <font color="Olive">fi</font> ) )
 ;<br/>

<a NAME="E2:27"/><i><font color="Green">E25</font></i>: 
 ex <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>]
 
<b>by </b><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="E4:27"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">A</font>]
 <b>and </b><br/><a NAME="E5:27"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">A1</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">A1</font>] holds <br/><font color="Maroon">A</font> <a href="ordinal1.html#R1">c=</a> <font color="Olive">A1</font>
 <b>from </b><i><a class="ref" href="ordinal1.html#S1">ORDINAL1:sch 1</a>();<br/></i>
<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="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<font color="Maroon">a<sub>1</sub></font>;<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="E7:27"/><i><font color="Green">E28</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> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">B</font>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S3">ORDINAL2:sch 3</a>();<br/></i>
<div><i><font color="Green">E29</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_1"/>
( <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> 1 )
 ;<br/><a NAME="E2:27_1"/><b>then </b>
(  <a href="xboole_0.html#K1">{}</a>  <a href="ordinal2.html#R1">is_limes_of</a> <font color="Maroon">fi</font> or 1 <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"><i><font color="Green">Lemma23</font></i></a>, <a class="txt" href="ordinal4.html#E2:27"><i><font color="Green">E25</font></i></a>, , </i>;<br/><b>hence </b><a NAME="E3:27_1"/>
contradiction
 <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#E2:27"><i><font color="Green">E25</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:27"/><b>then </b>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="ordinal3.html#T10">ORDINAL3:10</a></i>;<br/>
<a NAME="E10:27"/><b>then </b>
1 <a href="ordinal1.html#R1">c=</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T33">ORDINAL1:33</a>, </i>;<br/>
<a NAME="E11:27"/><b>then </b>
1 <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="txt" href="ordinal4.html#E4:27"><i><font color="Green">E26</font></i></a>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/>
<a NAME="E12:27"/><b>then </b><i><font color="Green">E36</font></i>: 
1 <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T21">ORDINAL1:21</a></i>;<br/>
<a NAME="E13:27"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">B2</font>, <font color="Olive">B1</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">B1</font> <a href="hidden.html#R2">in</a> <font color="Olive">B2</font> &amp; <font color="Olive">B2</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">B1</font> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<font color="Olive">B2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] <b>means </b>for <font color="Olive">B1</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">B1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>1</sub></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">B1</font> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E1:27_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="xboole_0.html#K1">{}</a> ]
 
;<br/>
<a NAME="E2:27_2"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">B</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[ <a href="ordinal1.html#K1">succ</a> <font color="Olive">B</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_1"><b>proof </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="E1:27_2_1"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">B1</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">B1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">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">B1</font> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<font color="Maroon">B</font>
 ;<br/>

<b>let </b><font color="Maroon">B1</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E2:27_2_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">B1</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">B</font> &amp;  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/>

<a NAME="E3:27_2_1"/><i><font color="Green">E47</font></i>: 
 <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> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_2_1_1_1_1"/>
( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">B</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="27_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:27_2_1_1_1_1_1"/>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:27_2_1_1_1_1_1"/>
 <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> 
 <b>by </b><i><a class="ref" href="ordinal2.html#T60">ORDINAL2:60</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:27_2_1_1_1_1_2"/>
<font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:27_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E50</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="ordinal3.html#T10">ORDINAL3:10</a></i>;<br/><a NAME="E3:27_2_1_1_1_1_2"/>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a></i>;<br/><a NAME="E4:27_2_1_1_1_1_2"/><b>then </b>
<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="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/><b>hence </b><a NAME="E5:27_2_1_1_1_1_2"/>
 <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> 
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:27_2_1_1"/>
 <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> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:27_2_1"/><i><font color="Green">E51</font></i>: 
(  <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">B</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">B</font></span>)</span> &amp; 1 <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">B</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<font color="Maroon">B</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">B</font> )
 
<b>by </b><i><a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a>, <a class="ref" href="ordinal2.html#T56">ORDINAL2:56</a>, <a class="ref" href="ordinal2.html#T61">ORDINAL2:61</a></i>;<br/>
<a NAME="E5:27_2_1"/><b>then </b><i><font color="Green">E73</font></i>: 
(  <a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <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">B</font></span>)</span> &amp; <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">B1</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">B</font> )
 
<b>by </b><i><a class="txt" href="ordinal4.html#E5:27"><i><font color="Green">E27</font></i></a>, , , <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a>, <a class="ref" href="ordinal1.html#T34">ORDINAL1:34</a>, <a class="ref" href="ordinal2.html#T57">ORDINAL2:57</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_2_1_2"/>
<font color="Maroon">B1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">B</font>
 ;<br/><a NAME="E2:27_2_1_2"/><b>then </b>
<font color="Maroon">B1</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><a NAME="E3:27_2_1_2"/><b>then </b>
<font color="Maroon">B1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T21">ORDINAL1:21</a></i>;<br/><a NAME="E4:27_2_1_2"/><b>then </b>
 <a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<font color="Maroon">B1</font> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<font color="Maroon">B</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E5:27_2_1_2"/>
 <a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<font color="Maroon">B1</font> <a href="hidden.html#R2">in</a>  <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">B</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:27_2_1"/>
 <a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<font color="Maroon">B1</font> <a href="hidden.html#R2">in</a>  <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">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="ordinal4.html#E5:27"><i><font color="Green">E27</font></i></a>, , , <a class="ref" href="ordinal2.html#T57">ORDINAL2:57</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:27_2"/><i><font color="Green">E74</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#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">B</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  &amp; ( for <font color="Olive">D</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">D</font> <a href="hidden.html#R2">in</a> <font color="Olive">B</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">D</font>] ) holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">B</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_2"><b>proof </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><b>that </b><br/><a NAME="E1:27_2_2"/><i><font color="Green">E75</font></i>: 
( <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">B</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  )
 <b>and </b><br/><a NAME="E2:27_2_2"/><i><font color="Green">E76</font></i>: 
for <font color="Olive">D</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">D</font>]
 ;<br/>

<b>let </b><font color="Maroon">B1</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E3:27_2_2"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">B1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/>

<a NAME="E4:27_2_2"/><b>then </b>
not <font color="Maroon">A</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T7">ORDINAL1:7</a></i>;<br/>
<b>then consider </b><font color="Maroon">psi</font> being   <a href="ordinal2.html#NM1">Ordinal-Sequence</a><b> such that </b><br/><a NAME="E6:27_2_2"/><i><font color="Green">E78</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">psi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> &amp; ( for <font color="Olive">D</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> holds <br/><font color="Maroon">psi</font> <a href="funct_1.html#K1">.</a> <font color="Olive">D</font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<font color="Olive">D</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">psi</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E7:27_2_2"/><i><font color="Green">E79</font></i>: 
 <a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K8">lim</a> <font color="Maroon">psi</font>
 
<b>by </b><i>, , <a class="ref" href="ordinal2.html#T62">ORDINAL2:62</a></i>;<br/>
<a NAME="E8:27_2_2"/>
<font color="Maroon">psi</font> is <a href="ordinal2.html#V2">increasing</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">B1</font> be   <a href="ordinal1.html#NM2">Ordinal</a>, <font color="Maroon">B2</font> be   <a href="ordinal1.html#NM2">Ordinal</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal2.html#D16">ORDINAL2:def 16</a><br/>

<b>assume </b><a NAME="E1:27_2_2_1"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">B1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B2</font> &amp; <font color="Maroon">B2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">psi</font> )
 ;<br/>

<a NAME="E2:27_2_2_1"/><b>then </b>
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">B2</font>] &amp; <font color="Maroon">B2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">B1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 
<b>by </b><i>, , , <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/>
<a NAME="E3:27_2_2_1"/><b>then </b>
(  <a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<font color="Maroon">B1</font> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<font color="Maroon">B2</font> &amp; <font color="Maroon">psi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">B1</font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<font color="Maroon">B1</font> &amp; <font color="Maroon">psi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">B2</font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<font color="Maroon">B2</font> )
 
<b>by </b><i>, <a class="txt" href="ordinal4.html#E13:27"><i><font color="Green">E37</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:27_2_2_1"/>
<font color="Maroon">psi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">B1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">psi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">B2</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E9:27_2_2"/><b>then </b><i><font color="Green">E81</font></i>: 
 <a href="ordinal2.html#K8">lim</a> <font color="Maroon">psi</font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K4">sup</a> <font color="Maroon">psi</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E10:27_2_2"/>
<font color="Maroon">psi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">B1</font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<font color="Maroon">B1</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E11:27_2_2"/><b>then </b>
(  <a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<font color="Maroon">B1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">psi</font> &amp;  <a href="ordinal2.html#K4">sup</a> <font color="Maroon">psi</font> <a href="hidden.html#R1">=</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">psi</font></span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E12:27_2_2"/>
 <a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<font color="Maroon">B1</font> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<font color="Maroon">B</font>
 <b>by </b><i><a class="txt" href="ordinal4.html#E12:27"><i><font color="Green">E36</font></i></a>, <a class="txt" href="ordinal4.html#E13:27"><i><font color="Green">E37</font></i></a>, <a class="ref" href="ordinal2.html#T27">ORDINAL2:27</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E4:27_2"/>
for <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">B</font>]
 <b>from </b><i><a class="ref" href="ordinal2.html#S1">ORDINAL2:sch 1</a>(<a class="txt" href="ordinal4.html#E8:27"><i><font color="Green">E29</font></i></a>, <a class="ref" href="ordinal4.html#D1" target="_self">Def1</a>, );</i><br/>


</div><b>end;</b></div>
<a NAME="E14:27"/>
<font color="Maroon">fi</font> is <a href="ordinal2.html#V2">increasing</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">B1</font> be   <a href="ordinal1.html#NM2">Ordinal</a>, <font color="Maroon">B2</font> be   <a href="ordinal1.html#NM2">Ordinal</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal2.html#D16">ORDINAL2:def 16</a><br/>

<b>assume </b><a NAME="E1:27_3"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">B1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B2</font> &amp; <font color="Maroon">B2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fi</font> )
 ;<br/>

<a NAME="E2:27_3"/><b>then </b>
<font color="Maroon">B1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fi</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/>
<a NAME="E3:27_3"/><b>then </b>
(  <a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<font color="Maroon">B1</font> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<font color="Maroon">B2</font> &amp; <font color="Maroon">fi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">B1</font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<font color="Maroon">B1</font> &amp; <font color="Maroon">fi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">B2</font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<font color="Maroon">B2</font> )
 
<b>by </b><i><a class="txt" href="ordinal4.html#E2:27"><i><font color="Green">E25</font></i></a>, <a class="txt" href="ordinal4.html#E7:27"><i><font color="Green">E28</font></i></a>, <a class="txt" href="ordinal4.html#E8:27"><i><font color="Green">E29</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:27_3"/>
<font color="Maroon">fi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">B1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">B2</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E15:27"/><b>then </b>
 <a href="ordinal2.html#K4">sup</a> <font color="Maroon">fi</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"><i><font color="Green">Lemma23</font></i></a>, <a class="txt" href="ordinal4.html#E2:27"><i><font color="Green">E25</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E16:27"/>
contradiction
 <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#E2:27"><i><font color="Green">E25</font></i></a></i>;<br/>


</div>
