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

<b>consider </b><font color="Maroon">c<sub>1</sub></font> being   <a href="ordinal2.html#NM1">Ordinal-Sequence</a><b> such that </b><br/><a NAME="E2:48_1"/><i><font color="Green">E14</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">F<sub>1</sub></font>() &amp; (  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">F<sub>1</sub></font>() implies <font color="Maroon">c<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>2</sub></font>() ) &amp; ( 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">F<sub>1</sub></font>() holds <br/><font color="Maroon">c<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="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>) ) &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>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">F<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/><font color="Maroon">c<sub>1</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">F<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Olive">b<sub>1</sub></font></span>)</span>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S11" target="_self">ORDINAL2:sch 11</a>();<br/></i>
<b>reconsider </b><font color="Maroon">c<sub>2</sub></font> =  <a href="ordinal2.html#K1" target="_self">last</a> <font color="Maroon">c<sub>1</sub></font> as   <a href="ordinal1.html#NM2">Ordinal</a> ;<br/>
<b>thus </b><a NAME="E4:48_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a>ex <font color="Olive">b<sub>2</sub></font> being  <a href="ordinal2.html#NM1">Ordinal-Sequence</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K1" target="_self">last</a> <font color="Olive">b<sub>2</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Olive">b<sub>2</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>2</sub></font>() &amp; ( for <font color="Olive">b<sub>3</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st  <a href="ordinal1.html#K1">succ</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Olive">b<sub>2</sub></font> <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>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span>) ) &amp; ( for <font color="Olive">b<sub>3</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Olive">b<sub>3</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>3</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  holds <br/><font color="Olive">b<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">b<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Olive">b<sub>3</sub></font></span>)</span>) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">c<sub>2</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>1</sub></font>
;<br/>

<b>thus </b><a NAME="E1:48_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K1" target="_self">last</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">F<sub>1</sub></font>() )
 <b>by </b><i><a class="txt" href="ordinal2.html#E2:48_1"><i><font color="Green">E14</font></i></a></i>;<br/>

<a NAME="E2:48_1_1"/>
(  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">F<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">F<sub>1</sub></font>() )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>
<a NAME="E3:48_1_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">F<sub>1</sub></font>()
 
<b>by </b><i><a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/>
<b>hence </b><a NAME="E4:48_1_1"/>
( <font color="Maroon">c<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>2</sub></font>() &amp; ( 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">F<sub>1</sub></font>() holds <br/><font color="Maroon">c<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="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>) ) &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>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">F<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/><font color="Maroon">c<sub>1</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">F<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Olive">b<sub>1</sub></font></span>)</span>) ) )
 <b>by </b><i><a class="txt" href="ordinal2.html#E2:48_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="ordinal1.html#T21">ORDINAL1:21</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>given </b><font color="Maroon">c<sub>5</sub></font> being   <a href="ordinal2.html#NM1">Ordinal-Sequence</a><b> such that </b><a NAME="E5:48_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K1" target="_self">last</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E6:48_1"/><i><font color="Green">E16</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>and </b><br/><a NAME="E7:48_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>5</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>2</sub></font>()
 <b>and </b><br/><a NAME="E8:48_1"/><i><font color="Green">E18</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">F<sub>1</sub></font>() holds <br/><font color="Maroon">c<sub>5</sub></font> <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>3</sub></font>(<font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>)
 <b>and </b><br/><a NAME="E9:48_1"/><i><font color="Green">E19</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>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">F<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/><font color="Maroon">c<sub>5</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">F<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Olive">b<sub>1</sub></font></span>)</span>)
 ;<br/>

<b>given </b><font color="Maroon">c<sub>6</sub></font> being   <a href="ordinal2.html#NM1">Ordinal-Sequence</a><b> such that </b><a NAME="E10:48_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K1" target="_self">last</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E11:48_1"/><i><font color="Green">E21</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>and </b><br/><a NAME="E12:48_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>6</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>2</sub></font>()
 <b>and </b><br/><a NAME="E13:48_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">F<sub>1</sub></font>() holds <br/><font color="Maroon">c<sub>6</sub></font> <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>3</sub></font>(<font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>)
 <b>and </b><br/><a NAME="E14:48_1"/><i><font color="Green">E24</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>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">F<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/><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">F<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Olive">b<sub>1</sub></font></span>)</span>)
 ;<br/>

<a NAME="E15:48_1"/><i><font color="Green">E25</font></i>: 
(  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">F<sub>1</sub></font>() implies <font color="Maroon">c<sub>5</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>2</sub></font>() )
 
<b>by </b><i><a class="txt" href="ordinal2.html#E7:48_1"><i><font color="Green">E17</font></i></a></i>;<br/>
<a NAME="E16:48_1"/><i><font color="Green">E26</font></i>: 
(  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">F<sub>1</sub></font>() implies <font color="Maroon">c<sub>6</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>2</sub></font>() )
 
<b>by </b><i><a class="txt" href="ordinal2.html#E12:48_1"><i><font color="Green">E22</font></i></a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="ordinal1.html#NM2">Ordinal</a>,  <a href="ordinal1.html#NM2">Ordinal</a>) <b>-&gt; </b>  <a href="ordinal1.html#NM2">Ordinal</a> = <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">a<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="ordinal2.html#K3" target="_self">sup</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>);<br/>
<a NAME="E17:48_1"/><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  <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">F<sub>1</sub></font>() holds <br/><font color="Maroon">c<sub>5</sub></font> <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">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E1:48_1_2"/><i><font color="Green">E28</font></i>: 
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> as   <a href="ordinal1.html#NM2">Ordinal</a> ;<br/>
 <a href="ordinal2.html#K3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> = 
 <a href="ordinal2.html#K3" target="_self">sup</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="zfmisc_1.html#T31">ZFMISC_1:31</a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="ordinal2.html#T26" target="_self">Th26</a></i>
;<br/>
<b>hence </b><a NAME="E4:48_1_2"/>
<font color="Maroon">c<sub>5</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>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>)
 <b>by </b><i><a class="txt" href="ordinal2.html#E8:48_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="ordinal2.html#E1:48_1_2"><i><font color="Green">E28</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:48_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  <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">F<sub>1</sub></font>() holds <br/><font color="Maroon">c<sub>6</sub></font> <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">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E1:48_1_3"/><i><font color="Green">E29</font></i>: 
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> as   <a href="ordinal1.html#NM2">Ordinal</a> ;<br/>
 <a href="ordinal2.html#K3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> = 
 <a href="ordinal2.html#K3" target="_self">sup</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="zfmisc_1.html#T31">ZFMISC_1:31</a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="ordinal2.html#T26" target="_self">Th26</a></i>
;<br/>
<b>hence </b><a NAME="E4:48_1_3"/>
<font color="Maroon">c<sub>6</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>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>)
 <b>by </b><i><a class="txt" href="ordinal2.html#E13:48_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="ordinal2.html#E1:48_1_3"><i><font color="Green">E29</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E19:48_1"/><i><font color="Green">E29</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>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">F<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/><font color="Maroon">c<sub>5</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">F<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Olive">b<sub>1</sub></font></span>)</span>)
 
<b>by </b><i><a class="txt" href="ordinal2.html#E9:48_1"><i><font color="Green">E19</font></i></a></i>;<br/>
<a NAME="E20:48_1"/><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>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">F<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/><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">F<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Olive">b<sub>1</sub></font></span>)</span>)
 
<b>by </b><i><a class="txt" href="ordinal2.html#E14:48_1"><i><font color="Green">E24</font></i></a></i>;<br/>
<a NAME="E21:48_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>from </b><i><a class="ref" href="ordinal2.html#S4" target="_self">ORDINAL2:sch 4</a>(<a class="txt" href="ordinal2.html#E6:48_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="ordinal2.html#E15:48_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="ordinal2.html#E17:48_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="ordinal2.html#E19:48_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="ordinal2.html#E11:48_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="ordinal2.html#E16:48_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="ordinal2.html#E18:48_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="ordinal2.html#E20:48_1"><i><font color="Green">E30</font></i></a>);<br/></i>
<b>hence </b><a NAME="E22:48_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="ordinal2.html#E5:48_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="ordinal2.html#E10:48_1"><i><font color="Green">E20</font></i></a></i>;<br/>


</div>
