<?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>let </b><font color="Maroon">B</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>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<font color="Maroon">a<sub>1</sub></font></span>)</span> <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>;<br/>
<a NAME="E1:37"/>
(  <a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> 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">A</font></span>)</span> &amp; 1 <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="ordinal2.html#T56">ORDINAL2:56</a>, <a class="ref" href="ordinal2.html#T60">ORDINAL2:60</a></i>;<br/>
<a NAME="E2:37"/><b>then </b><i><font color="Green">E13</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#T44">ORDINAL2:44</a></i>;<br/>
<a NAME="E3:37"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1">succ</a> <font color="Olive">B</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_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="E1:37_1"/><i><font color="Green">E26</font></i>: 
 <a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</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="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>
 ;<br/>

<b>thus </b> <a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K10">+^</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">B</font></span>)</span></span>)</span> = 
 <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> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">B</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="ordinal2.html#T45">ORDINAL2:45</a></i>
<br/>.= 
<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>,<span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">B</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="ordinal2.html#T61">ORDINAL2:61</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="ordinal2.html#K11">*^</a> <span class="p2">(<span class="default"><a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<font color="Maroon">B</font></span>)</span></span>)</span> <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="ordinal3.html#T58">ORDINAL3:58</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">B</font></span>)</span></span>)</span> <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/>


</div><b>end;</b></div>
<a NAME="E4:37"/><i><font color="Green">E27</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>1</sub></font>[<font color="Olive">D</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_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="E1:37_2"/><i><font color="Green">E28</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="E2:37_2"/><i><font color="Green">E29</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/> <a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K10">+^</a> <font color="Olive">D</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<font color="Olive">D</font></span>)</span> <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>
 ;<br/>

<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="E4:37_2"/><i><font color="Green">E36</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fi</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">fi</font> <a href="funct_1.html#K1">.</a> <font color="Olive">D</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">D</font>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S3">ORDINAL2:sch 3</a>();<br/></i>
<b>consider </b><font color="Maroon">psi</font> being   <a href="ordinal2.html#NM1">Ordinal-Sequence</a><b> such that </b><br/><a NAME="E6:37_2"/><i><font color="Green">E37</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">psi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="ordinal2.html#K10">+^</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">A</font> <a href="ordinal2.html#K10">+^</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> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">D</font>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S3">ORDINAL2:sch 3</a>();<br/></i>
<a NAME="E7:37_2"/>
( <font color="Maroon">A</font> <a href="ordinal2.html#K10">+^</a> <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">A</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">B</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  )
 
<b>by </b><i><a class="ref" href="ordinal4.html#D1" target="_self">Def1</a>, <a class="ref" href="ordinal3.html#T29">ORDINAL3:29</a>, <a class="ref" href="ordinal3.html#T32">ORDINAL3:32</a></i>;<br/>
<a NAME="E8:37_2"/><b>then </b><i><font color="Green">E42</font></i>: 
 <a href="ordinal2.html#K8">lim</a> <font color="Maroon">psi</font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">B</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="ordinal2.html#T62">ORDINAL2:62</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</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">f1</font> being   <a href="ordinal2.html#NM1">Ordinal-Sequence</a><b> such that </b><br/><a NAME="E10:37_2"/><i><font color="Green">E44</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</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">A</font> holds <br/><font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">D</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">D</font>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S3">ORDINAL2:sch 3</a>();<br/></i>
<a NAME="E11:37_2"/><i><font color="Green">E45</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">psi</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">fi</font> <a href="ordinal3.html#K4">*^</a> <span class="p3">(<span class="default"><a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<font color="Maroon">A</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="ordinal3.html#D5">ORDINAL3:def 5</a></i>;<br/>
<div><i><font color="Green">E46</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">D</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>assume </b><a NAME="E1:37_2_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">D</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font>
 ;<br/><a NAME="E2:37_2_1"/>
<font color="Maroon">A</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="ordinal3.html#T27">ORDINAL3:27</a></i>;<br/><b>hence </b><font color="Maroon">psi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">D</font> = 
 <a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<font color="Maroon">D</font>
<b>by </b><i>, , <a class="ref" href="ordinal4.html#T3" target="_self">Th3</a></i>
<br/>.= 
<font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">D</font>
<b>by </b><i>, <a class="ref" href="ordinal4.html#T3" target="_self">Th3</a></i>
;<br/><br/></div><b>end;</b></div>
<a NAME="E13:37_2"/><i><font color="Green">E50</font></i>: 
 <a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<font color="Maroon">B</font> <a href="ordinal2.html#R1">is_limes_of</a> <font color="Maroon">fi</font>
 
<b>by </b><i><a class="ref" href="ordinal4.html#D1" target="_self">Def1</a>, , </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">D</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>assume </b><a NAME="E1:37_2_2"/>
<font color="Maroon">D</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">fi</font> <a href="ordinal3.html#K4">*^</a> <span class="p2">(<span class="default"><a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<font color="Maroon">A</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:37_2_2"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">D</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="ordinal3.html#D5">ORDINAL3:def 5</a></i>;<br/><a NAME="E3:37_2_2"/><b>then </b>
<font color="Maroon">A</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="ordinal2.html#T49">ORDINAL2:49</a></i>;<br/><b>hence </b><font color="Maroon">psi</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">D</font></span>)</span> = 
 <a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">D</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<font color="Maroon">D</font></span>)</span> <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>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">fi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">D</font></span>)</span> <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>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">fi</font> <a href="ordinal3.html#K4">*^</a> <span class="p2">(<span class="default"><a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">D</font>
<b>by </b><i>, <a class="ref" href="ordinal3.html#D5">ORDINAL3:def 5</a></i>
;<br/><br/></div><b>end;</b></div>
<a NAME="E15:37_2"/><b>then </b>
( <font color="Maroon">psi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="ordinal4.html#K1" target="_self">^</a> <span class="p1">(<span class="default"><font color="Maroon">fi</font> <a href="ordinal3.html#K4">*^</a> <span class="p2">(<span class="default"><a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<font color="Maroon">A</font></span>)</span></span>)</span> &amp; <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="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> <a href="ordinal2.html#R1">is_limes_of</a> <font color="Maroon">fi</font> <a href="ordinal3.html#K4">*^</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="txt" href="ordinal4.html#E4:37_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="ordinal4.html#E6:37_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="ordinal4.html#T3" target="_self">Th3</a>, , </i>;<br/>
<a NAME="E16:37_2"/><b>then </b>
<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="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> <a href="ordinal2.html#R1">is_limes_of</a> <font color="Maroon">psi</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E17:37_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">B</font>]
 <b>by </b><i>, <a class="ref" href="ordinal2.html#D14">ORDINAL2:def 14</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:37"/>
for <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a> holds  <font color="Maroon">S<sub>1</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#E4:37"><i><font color="Green">E27</font></i></a>, , );<br/></i>
<b>hence </b><a NAME="E6:37"/>
 <a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</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="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>
 ;<br/>


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