<?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>, <font color="Maroon">c<sub>3</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>





<b>assume </b><a NAME="E1:31"/><i><font color="Green">E25</font></i>: 
1 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] means 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">a<sub>1</sub></font> holds <br/> <a href="ordinal2.html#K12">exp</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K12">exp</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E2:31"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1">{}</a> ]
 
;<br/>
<a NAME="E3:31"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1">succ</a> <font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E1:31_1"/><i><font color="Green">E28</font></i>: 
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>4</sub></font> holds <br/> <a href="ordinal2.html#K12">exp</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K12">exp</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E2:31_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E3:31_1"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T34">ORDINAL1:34</a></i>;<br/>
<a NAME="E4:31_1"/><i><font color="Green">E30</font></i>: 
 <a href="ordinal2.html#K12">exp</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K12">exp</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="ordinal4.html#E1:31"><i><font color="Green">E25</font></i></a>, <a class="ref" href="ordinal4.html#T23" target="_self">Th23</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:31_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E2:31_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="ordinal4.html#E3:31_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><a NAME="E3:31_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T21">ORDINAL1:21</a></i>;<br/><b>hence </b><a NAME="E4:31_1_1"/>
 <a href="ordinal2.html#K12">exp</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K12">exp</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="ordinal4.html#E1:31_1"><i><font color="Green">E28</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:31_1"/>
 <a href="ordinal2.html#K12">exp</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K12">exp</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="ordinal4.html#E4:31_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:31"/><i><font color="Green">E28</font></i>: 
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#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:31_2"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>4</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>4</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  )
 <b>and </b><br/><a NAME="E2:31_2"/><i><font color="Green">E30</font></i>: 
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>4</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 ;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E3:31_2"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<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<sub>1</sub></font>,<font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="ordinal2.html#NM1">Ordinal-Sequence</a><b> such that </b><br/><a NAME="E5:31_2"/><i><font color="Green">E32</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</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>4</sub></font> holds <br/><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S3">ORDINAL2:sch 3</a>();<br/></i>
<a NAME="E6:31_2"/><i><font color="Green">E33</font></i>: 
 <a href="ordinal2.html#K12">exp</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K8">lim</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="ordinal4.html#E1:31_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="ordinal4.html#E5:31_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="ordinal2.html#T62">ORDINAL2:62</a></i>;<br/>
<a NAME="E7:31_2"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="ordinal2.html#V2">increasing</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></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="E1:31_2_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/>

<a NAME="E2:31_2_1"/><b>then </b>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>8</sub></font>] )
 
<b>by </b><i><a class="txt" href="ordinal4.html#E2:31_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="ordinal4.html#E5:31_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/>
<a NAME="E3:31_2_1"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K12">exp</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K12">exp</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font> &amp;  <a href="ordinal2.html#K12">exp</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K12">exp</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font> )
 
<b>by </b><i><a class="txt" href="ordinal4.html#E5:31_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="ordinal4.html#E1:31_2_1"><i><font color="Green">E34</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:31_2_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E8:31_2"/><b>then </b><i><font color="Green">E34</font></i>: 
 <a href="ordinal2.html#K4">sup</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K8">lim</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="ordinal4.html#E1:31_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="ordinal4.html#E5:31_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="ordinal4.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E9:31_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K12">exp</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="ordinal4.html#E3:31_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="ordinal4.html#E5:31_2"><i><font color="Green">E32</font></i></a></i>;<br/>
<a NAME="E10:31_2"/><b>then </b>
(  <a href="ordinal2.html#K12">exp</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> &amp;  <a href="ordinal2.html#K4">sup</a> <font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="ordinal4.html#E3:31_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="ordinal4.html#E5:31_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E11:31_2"/>
 <a href="ordinal2.html#K12">exp</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K12">exp</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="ordinal4.html#E6:31_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="ordinal4.html#E8:31_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="ordinal2.html#T27">ORDINAL2:27</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:31"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="ordinal2.html#S1">ORDINAL2:sch 1</a>(<a class="txt" href="ordinal4.html#E2:31"><i><font color="Green">E26</font></i></a>, <a class="txt" href="ordinal4.html#E3:31"><i><font color="Green">E27</font></i></a>, <a class="txt" href="ordinal4.html#E4:31"><i><font color="Green">E28</font></i></a>);<br/></i>
<b>hence </b><a NAME="E6:31"/>
(  not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> or  <a href="ordinal2.html#K12">exp</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K12">exp</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 ;<br/>


</div>
