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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">F<sub>1</sub></font>() <a href="ordinal1.html#K2">|</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<b>assume </b><a NAME="E1:41_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/>

<a NAME="E2:41_1"/><b>then </b><i><font color="Green">E19</font></i>: 
(  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>1</sub></font> <a href="ordinal1.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="ordinal1.html#T33">ORDINAL1:33</a></i>;<br/>
<a NAME="E3:41_1"/><b>then </b><i><font color="Green">E20</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="ordinal1.html#K2">|</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E4:41_1"/><b>then </b>
 <a href="ordinal2.html#K1" target="_self">last</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="ordinal1.html#K2">|</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="ordinal1.html#K2">|</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="ordinal2.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E5:41_1"/><b>then </b><i><font color="Green">E21</font></i>: 
 <a href="ordinal2.html#K1" target="_self">last</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="ordinal1.html#K2">|</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="ordinal2.html#E2:41_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<a NAME="E6:41_1"/>
(  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<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">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>
<a NAME="E7:41_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">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/>
<a NAME="E8:41_1"/><b>then </b>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T21">ORDINAL1:21</a></i>;<br/>
<a NAME="E9:41_1"/><b>then </b><i><font color="Green">E22</font></i>: 
( <font color="Maroon">F<sub>1</sub></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>4</sub></font>() &amp; <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="ordinal1.html#K2">|</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <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>1</sub></font>() <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="txt" href="ordinal2.html#E2:41"><i><font color="Green">E15</font></i></a>, <a class="txt" href="ordinal2.html#E3:41"><i><font color="Green">E16</font></i></a>, <a class="txt" href="ordinal2.html#E2:41_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<a NAME="E10:41_1"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st  <a href="ordinal1.html#K1">succ</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="ordinal1.html#K2">|</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="ordinal1.html#K2">|</a> <span class="p3">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:41_1_1"/><i><font color="Green">E24</font></i>: 
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:41_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a></i>;<br/>
<a NAME="E3:41_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="ordinal2.html#E1:41_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/>
<a NAME="E4:41_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="ordinal1.html#K2">|</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="ordinal1.html#K2">|</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="ordinal2.html#E1:41_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<b>hence </b><a NAME="E5:41_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="ordinal1.html#K2">|</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="ordinal1.html#K2">|</a> <span class="p3">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>)
 <b>by </b><i><a class="txt" href="ordinal2.html#E2:41"><i><font color="Green">E15</font></i></a>, <a class="txt" href="ordinal2.html#E4:41"><i><font color="Green">E17</font></i></a>, <a class="txt" href="ordinal2.html#E2:41_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="ordinal2.html#E1:41_1_1"><i><font color="Green">E24</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:41_1"/>
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>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>1</sub></font> &amp; <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>  holds <br/><span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="ordinal1.html#K2">|</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="ordinal1.html#K2">|</a> <span class="p3">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="ordinal1.html#K2">|</a> <font color="Olive">b<sub>1</sub></font></span>)</span>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:41_1_2"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</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>3</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  )
 ;<br/>

<a NAME="E2:41_1_2"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>3</sub></font>() &amp; <font color="Maroon">c<sub>3</sub></font> <a href="ordinal1.html#R1">c=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="ordinal2.html#E2:41"><i><font color="Green">E15</font></i></a>, <a class="txt" href="ordinal2.html#E2:41_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E3:41_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="ordinal1.html#K2">|</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="ordinal1.html#K2">|</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="ordinal2.html#E1:41_1_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a>, <a class="ref" href="funct_1.html#T82">FUNCT_1:82</a></i>;<br/>
<b>hence </b><a NAME="E4:41_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="ordinal1.html#K2">|</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="ordinal1.html#K2">|</a> <span class="p3">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>)
 <b>by </b><i><a class="txt" href="ordinal2.html#E2:41"><i><font color="Green">E15</font></i></a>, <a class="txt" href="ordinal2.html#E5:41"><i><font color="Green">E18</font></i></a>, <a class="txt" href="ordinal2.html#E2:41_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="ordinal2.html#E1:41_1_2"><i><font color="Green">E24</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:41_1"/>
<font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font>)
 <b>by </b><i><a class="txt" href="ordinal2.html#E1:41"><i><font color="Green">E14</font></i></a>, <a class="txt" href="ordinal2.html#E3:41_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="ordinal2.html#E5:41_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="ordinal2.html#E9:41_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="ordinal2.html#E10:41_1"><i><font color="Green">E23</font></i></a></i>;<br/>


</div>
