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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="classes2.html#NM1">Universe</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>2</sub></font>, <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:18"/><i><font color="Green">E12</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>2</sub></font> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>1</sub></font>
 ;<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="hidden.html#M1">set</a>  =  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><span class="p2"><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> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relat_1.html#K9">.:</a> <span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<span class="p4"><a href="tarski.html#K1">{</a><span class="default"><span class="p5">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span></span>)</span>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1">set</a> ,  <a href="ordinal1.html#NM3">T-Sequence</a>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="ordinal2.html#K4">sup</a> <font color="Maroon">a<sub>2</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="ordinal2.html#NM1">Ordinal-Sequence</a><b> such that </b><br/><a NAME="E3:18"/><i><font color="Green">E13</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font> &amp; (  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font> implies <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 <b>and </b><br/><a NAME="E4:18"/><i><font color="Green">E14</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#K3">On</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>4</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>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>)
 <b>and </b><br/><a NAME="E5:18"/><i><font color="Green">E15</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#K3">On</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/><font color="Maroon">c<sub>4</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>2</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="ordinal2.html#S11">ORDINAL2:sch 11</a>();<br/></i>
<a NAME="E6:18"/><i><font color="Green">E16</font></i>: 
 <a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="ordinal4.html#T35">ORDINAL4:35</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] means <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E7:18"/><i><font color="Green">E17</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="classes2.html#T62">CLASSES2:62</a></i>;<br/>
<a NAME="E8:18"/><b>then </b><i><font color="Green">E18</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font>]
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E3:18"><i><font color="Green">E13</font></i></a>, <a class="txt" href="zfrefle1.html#E6:18"><i><font color="Green">E16</font></i></a>, <a class="ref" href="ordinal1.html#D10">ORDINAL1:def 10</a></i>;<br/>
<a NAME="E9:18"/><i><font color="Green">E19</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[ <a href="ordinal4.html#K6">succ</a> <font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:18_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> as    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="zf_refle.html#T8">ZF_REFLE:8</a></i>;<br/>
<a NAME="E3:18_1"/>
 <a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="zf_refle.html#T8">ZF_REFLE:8</a></i>;<br/>
<a NAME="E4:18_1"/><b>then </b><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relat_1.html#K9">.:</a> <span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<span class="p4"><a href="tarski.html#K1">{</a><span class="default"><span class="p5">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E4:18"><i><font color="Green">E14</font></i></a></i>;<br/>
<a NAME="E5:18_1"/>
 <a href="card_1.html#K1">Card</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="card_2.html#T13">CARD_2:13</a></i>;<br/>
<a NAME="E6:18_1"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relat_1.html#K9">.:</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="card_2.html#T2">CARD_2:2</a></i>;<br/>
<a NAME="E7:18_1"/><b>then </b><i><font color="Green">E21</font></i>: 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relat_1.html#K9">.:</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E1:18"><i><font color="Green">E12</font></i></a>, <a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a></i>;<br/>
<a NAME="E8:18_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E9:18_1"/><b>then </b>
(  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relat_1.html#K9">.:</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_5.html#T48">FUNCT_5:48</a>, <a class="ref" href="relat_1.html#T144">RELAT_1:144</a></i>;<br/>
<a NAME="E10:18_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relat_1.html#K9">.:</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="ordinal2.html#T9">ORDINAL2:9</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E11:18_1"/><b>then </b><i><font color="Green">E22</font></i>: 
<span class="p1">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relat_1.html#K9">.:</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E12:18_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="ordinal4.html#D2">ORDINAL4:def 2</a></i>;<br/>
<a NAME="E13:18_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relat_1.html#K9">.:</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E7:18_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="zfrefle1.html#E11:18_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="classes1.html#T2">CLASSES1:2</a>, <a class="ref" href="classes2.html#T63">CLASSES2:63</a></i>;<br/>
<a NAME="E14:18_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relat_1.html#K9">.:</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="classes2.html#T66">CLASSES2:66</a></i>;<br/>
<a NAME="E15:18_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E4:18_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="zf_refle.html#T28">ZF_REFLE:28</a></i>;<br/>
<b>hence </b><a NAME="E16:18_1"/>
<font color="Maroon">S<sub>2</sub></font>[ <a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>5</sub></font>]
 <b>by </b><i><a class="ref" href="ordinal1.html#D10">ORDINAL1:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:18"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font> &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="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> 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>2</sub></font>[<font color="Olive">b<sub>2</sub></font>] ) holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:18_2"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  )
 <b>and </b><br/><a NAME="E2:18_2"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E3:18_2"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="zf_refle.html#T8">ZF_REFLE:8</a></i>;<br/>
<a NAME="E4:18_2"/><b>then </b><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>4</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#K4">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E5:18"><i><font color="Green">E15</font></i></a>, <a class="txt" href="zfrefle1.html#E6:18"><i><font color="Green">E16</font></i></a>, <a class="txt" href="zfrefle1.html#E1:18_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="ordinal1.html#D10">ORDINAL1:def 10</a>, <a class="ref" href="relat_1.html#T87">RELAT_1:87</a></i>;<br/>
<a NAME="E5:18_2"/><b>then </b><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>4</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#K3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="classes1.html#D1">CLASSES1:def 1</a></i>;<br/>
<a NAME="E6:18_2"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:18_2_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:18_2_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>7</sub></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">c<sub>4</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="txt" href="zfrefle1.html#E3:18_2_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E5:18_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E3:18_2"><i><font color="Green">E23</font></i></a>, <a class="txt" href="zfrefle1.html#E4:18_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="zfrefle1.html#E3:18_2_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="zf_refle.html#T8">ZF_REFLE:8</a></i>;<br/>
<a NAME="E7:18_2_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E3:18_2_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<a NAME="E8:18_2_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E2:18_2"><i><font color="Green">E22</font></i></a>, <a class="txt" href="zfrefle1.html#E4:18_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="zfrefle1.html#E3:18_2_1"><i><font color="Green">E26</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:18_2_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="ordinal1.html#D10">ORDINAL1:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:18_2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E5:18_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="zfrefle1.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E8:18_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E5:18_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="zf_refle.html#T28">ZF_REFLE:28</a></i>;<br/>
<b>hence </b><a NAME="E9:18_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>5</sub></font>]
 <b>by </b><i><a class="ref" href="ordinal1.html#D10">ORDINAL1:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:18"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:18_3"/>
<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>4</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:18_3"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="zfrefle1.html#E3:18"><i><font color="Green">E13</font></i></a>, <a class="txt" href="zfrefle1.html#E3:18_3"><i><font color="Green">E21</font></i></a>, <a class="ref" href="zf_refle.html#T8">ZF_REFLE:8</a></i>;<br/>
<a NAME="E5:18_3"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="zf_refle.html#S5">ZF_REFLE:sch 5</a>(<a class="txt" href="zfrefle1.html#E8:18"><i><font color="Green">E18</font></i></a>, <a class="txt" href="zfrefle1.html#E9:18"><i><font color="Green">E19</font></i></a>, <a class="txt" href="zfrefle1.html#E10:18"><i><font color="Green">E20</font></i></a>);<br/></i>
<a NAME="E6:18_3"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<b>hence </b><a NAME="E7:18_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="zfrefle1.html#E3:18_3"><i><font color="Green">E21</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="ordinal4.html#M2">Ordinal-Sequence</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="zfrefle1.html#E3:18"><i><font color="Green">E13</font></i></a>, <a class="ref" href="ordinal4.html#D3">ORDINAL4:def 3</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>thus </b><a NAME="E13:18"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="ordinal2.html#V2">increasing</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</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>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>



<b>assume </b><a NAME="E1:18_4"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <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>5</sub></font> )
 ;<br/>

<a NAME="E2:18_4"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> as    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="zfrefle1.html#E3:18"><i><font color="Green">E13</font></i></a>, <a class="txt" href="zfrefle1.html#E1:18_4"><i><font color="Green">E22</font></i></a>, <a class="ref" href="zf_refle.html#T8">ZF_REFLE:8</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] means ( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> implies <font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E4:18_4"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[ <a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font>]
 
<b>by </b><i><a class="ref" href="ordinal4.html#T35">ORDINAL4:35</a></i>;<br/>
<a NAME="E5:18_4"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[ <a href="ordinal4.html#K6">succ</a> <font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:18_4_1"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> implies <font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>and </b><br/><a NAME="E2:18_4_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<a NAME="E3:18_4_1"/>
(  <a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="zf_refle.html#T8">ZF_REFLE:8</a></i>;<br/>
<a NAME="E4:18_4_1"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><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="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relat_1.html#K9">.:</a> <span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<span class="p4"><a href="tarski.html#K1">{</a><span class="default"><span class="p5">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relat_1.html#K9">.:</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E4:18"><i><font color="Green">E14</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E5:18_4_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="ordinal2.html#T27">ORDINAL2:27</a></i>;<br/>
<b>hence </b><a NAME="E6:18_4_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="zfrefle1.html#E1:18_4_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="zfrefle1.html#E2:18_4_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="ordinal1.html#T13">ORDINAL1:13</a>, <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:18_4"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font> &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="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> 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>3</sub></font>[<font color="Olive">b<sub>2</sub></font>] ) holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:18_4_2"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  )
 <b>and </b><br/><a NAME="E2:18_4_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>and </b><br/><a NAME="E3:18_4_2"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<a NAME="E4:18_4_2"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="zf_refle.html#T8">ZF_REFLE:8</a></i>;<br/>
<a NAME="E5:18_4_2"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K4">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E3:18"><i><font color="Green">E13</font></i></a>, <a class="txt" href="zfrefle1.html#E5:18"><i><font color="Green">E15</font></i></a>, <a class="txt" href="zfrefle1.html#E1:18_4_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="zfrefle1.html#E3:18_4_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="funct_1.html#T73">FUNCT_1:73</a></i>;<br/>
<a NAME="E6:18_4_2"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>10</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> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="ordinal2.html#T27">ORDINAL2:27</a></i>;<br/>
<b>hence </b><a NAME="E7:18_4_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E7:18_4"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="zf_refle.html#S5">ZF_REFLE:sch 5</a>(<a class="txt" href="zfrefle1.html#E4:18_4"><i><font color="Green">E23</font></i></a>, <a class="txt" href="zfrefle1.html#E5:18_4"><i><font color="Green">E24</font></i></a>, <a class="txt" href="zfrefle1.html#E6:18_4"><i><font color="Green">E25</font></i></a>);<br/></i>
<a NAME="E8:18_4"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E1:18_4"><i><font color="Green">E22</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:18_4"/>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>


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

<b>thus </b><a NAME="E14:18"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="ordinal2.html#V3">continuous</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal2.html#D17">ORDINAL2:def 17</a><br/><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="E1:18_5"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>6</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>6</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/>

<a NAME="E2:18_5"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E3:18_5"/><b>then </b>
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>6</sub></font> is <a href="ordinal2.html#V2">increasing</a> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K4">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E3:18"><i><font color="Green">E13</font></i></a>, <a class="txt" href="zfrefle1.html#E5:18"><i><font color="Green">E15</font></i></a>, <a class="txt" href="zfrefle1.html#E13:18"><i><font color="Green">E21</font></i></a>, <a class="txt" href="zfrefle1.html#E1:18_5"><i><font color="Green">E22</font></i></a>, <a class="ref" href="ordinal4.html#T15">ORDINAL4:15</a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<b>hence </b><a NAME="E4:18_5"/>
<font color="Maroon">c<sub>7</sub></font> <a href="ordinal2.html#R1">is_limes_of</a> <font color="Maroon">c<sub>5</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="zfrefle1.html#E1:18_5"><i><font color="Green">E22</font></i></a>, <a class="ref" href="ordinal4.html#T8">ORDINAL4:8</a></i>;<br/>


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

<b>thus </b><a NAME="E15:18"/>
<font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="zfrefle1.html#E3:18"><i><font color="Green">E13</font></i></a>, <a class="txt" href="zfrefle1.html#E6:18"><i><font color="Green">E16</font></i></a>, <a class="txt" href="zfrefle1.html#E7:18"><i><font color="Green">E17</font></i></a>, <a class="ref" href="ordinal1.html#D10">ORDINAL1:def 10</a></i>;<br/>

<b>thus </b><a NAME="E16:18"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><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="ordinal4.html#K4">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relat_1.html#K9">.:</a> <span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<span class="p4"><a href="tarski.html#K1">{</a><span class="default"><span class="p5">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E1:18_6"/>
 <a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="zf_refle.html#T8">ZF_REFLE:8</a></i>;<br/>
<b>hence </b><a NAME="E2:18_6"/>
<font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><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="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relat_1.html#K9">.:</a> <span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<span class="p4"><a href="tarski.html#K1">{</a><span class="default"><span class="p5">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="zfrefle1.html#E4:18"><i><font color="Green">E14</font></i></a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E17:18"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="zf_refle.html#T8">ZF_REFLE:8</a></i>;<br/>
<b>hence </b><a NAME="E18:18"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  implies <font color="Maroon">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K4">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="zfrefle1.html#E5:18"><i><font color="Green">E15</font></i></a>, <a class="txt" href="zfrefle1.html#E6:18"><i><font color="Green">E16</font></i></a></i>;<br/>


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