<?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>assume </b><a NAME="E1:39"/><i><font color="Green">E24</font></i>: 
 <a href="ordinal1.html#K5">omega</a>  <a href="hidden.html#R2">in</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="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="ordinal1.html#NM2">Ordinal</a>,   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="classes1.html#K4">Rank</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="ordinal1.html#NM3">T-Sequence</a><b> such that </b><br/><a NAME="E3:39"/><i><font color="Green">E25</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</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>2</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K3">1-element_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> ) )
 <b>and </b><br/><a NAME="E4:39"/><i><font color="Green">E26</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>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>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>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>)
 <b>and </b><br/><a NAME="E5:39"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="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>2</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>2</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#S5">ORDINAL2:sch 5</a>();<br/></i>
<a NAME="E6:39"/><i><font color="Green">E28</font></i>: 
(  <a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<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;  <a href="xboole_0.html#K1">{}</a>  <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="ref" href="ordinal4.html#T35">ORDINAL4:35</a>, <a class="ref" href="zf_refle.html#T8">ZF_REFLE:8</a></i>;<br/>
<a NAME="E7:39"/><i><font color="Green">E29</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> holds <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</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="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E1:39_1"/>
(  <a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>3</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>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</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="E2:39_1"/>
<font color="Maroon">c<sub>2</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>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="zfrefle1.html#E4:39"><i><font color="Green">E26</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:39"/><i><font color="Green">E30</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="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>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="classes1.html#K4">Rank</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E1:39_2"/>
( <font color="Maroon">c<sub>3</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>2</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>3</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="E2:39_2"/>
( <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>  implies <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="classes1.html#K4">Rank</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="zfrefle1.html#E5:39"><i><font color="Green">E27</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:39"/><i><font color="Green">E31</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="xboole_0.html#K1">{}</a>  holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="classes1.html#K4">Rank</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E1:39_3"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  or ex <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Olive">b<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="ordinal1.html#T42">ORDINAL1:42</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_3_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">c<sub>4</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><a NAME="E1:39_3_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E2:39_3_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</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#E1:39_3_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a>, <a class="ref" href="zf_refle.html#T8">ZF_REFLE:8</a></i>;<br/><a NAME="E3:39_3_1"/><b>then </b>
<font color="Maroon">c<sub>4</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="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/><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#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="E5:39_3_1"/>
<font color="Maroon">c<sub>2</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="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="zfrefle1.html#E7:39"><i><font color="Green">E29</font></i></a></i>;<br/><b>hence </b><a NAME="E6:39_3_1"/>
( <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>  implies <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="classes1.html#K4">Rank</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="zfrefle1.html#E1:39_3_1"><i><font color="Green">E33</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:39_3"/>
( <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>  implies <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="classes1.html#K4">Rank</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="zfrefle1.html#E8:39"><i><font color="Green">E30</font></i></a>, <a class="txt" href="zfrefle1.html#E1:39_3"><i><font color="Green">E32</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:39"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font> <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="39_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</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="E1:39_4"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:39_4"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>4</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>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="zfrefle1.html#E3:39"><i><font color="Green">E25</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</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#E3:39_4"><i><font color="Green">E32</font></i></a>, <a class="ref" href="zf_refle.html#T8">ZF_REFLE:8</a></i>;<br/>
<a NAME="E5:39_4"/>
(  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K3">1-element_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="classes1.html#K4">Rank</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="ref" href="zfrefle1.html#T33" target="_self">Th33</a></i>;<br/>
<b>hence </b><a NAME="E6:39_4"/>
<font color="Maroon">c<sub>3</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#E3:39"><i><font color="Green">E25</font></i></a>, <a class="txt" href="zfrefle1.html#E9:39"><i><font color="Green">E31</font></i></a>, <a class="txt" href="zfrefle1.html#E3:39_4"><i><font color="Green">E32</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as    <a href="ordinal1.html#M1">T-Sequence</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="ordinal1.html#D8">ORDINAL1:def 8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_5"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:39_5"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>4</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;  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="zfrefle1.html#E3:39"><i><font color="Green">E25</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</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#E3:39_5"><i><font color="Green">E32</font></i></a>, <a class="ref" href="zf_refle.html#T8">ZF_REFLE:8</a></i>;<br/><a NAME="E5:39_5"/>
( ( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="ordinal4.html#K3">1-element_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> ) or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="ordinal3.html#T18">ORDINAL3:18</a>, <a class="ref" href="ordinal4.html#T35">ORDINAL4:35</a></i>;<br/><a NAME="E6:39_5"/><b>then </b>
( ( <font color="Maroon">c<sub>3</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="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K3">1-element_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="ordinal4.html#K3">1-element_of</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>  ) or ( <font color="Maroon">c<sub>5</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="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="classes1.html#K4">Rank</a> <font color="Maroon">c<sub>5</sub></font> ) )
 <b>by </b><i><a class="txt" href="zfrefle1.html#E3:39"><i><font color="Green">E25</font></i></a>, <a class="txt" href="zfrefle1.html#E9:39"><i><font color="Green">E31</font></i></a>, <a class="ref" href="zf_refle.html#T8">ZF_REFLE:8</a></i>;<br/><b>hence </b><a NAME="E7:39_5"/>
contradiction
 <b>by </b><i><a class="txt" href="zfrefle1.html#E3:39_5"><i><font color="Green">E32</font></i></a>, <a class="ref" href="zfrefle1.html#T34" target="_self">Th34</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="zf_refle.html#NM2">DOMAIN-Sequence</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="zfrefle1.html#E3:39"><i><font color="Green">E25</font></i></a>, <a class="ref" href="relat_1.html#D9">RELAT_1:def 9</a>, <a class="ref" href="zf_refle.html#D5">ZF_REFLE:def 5</a></i>;<br/>
<a NAME="E14:39"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <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>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Olive">b<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font>, <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/>



<b>assume </b><a NAME="E1:39_6"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<a NAME="E2:39_6"/><b>then </b><i><font color="Green">E34</font></i>: 
(  <a href="classes1.html#K4">Rank</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4">Rank</a> <font color="Maroon">c<sub>6</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;  <a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="ref" href="classes1.html#T42">CLASSES1:42</a>, <a class="ref" href="ordinal1.html#T33">ORDINAL1:33</a></i>;<br/>
<a NAME="E3:39_6"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="classes1.html#K4">Rank</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E9:39"><i><font color="Green">E31</font></i></a>, <a class="txt" href="zfrefle1.html#E1:39_6"><i><font color="Green">E33</font></i></a></i>;<br/>
<a NAME="E4:39_6"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="classes1.html#K4">Rank</a> <font color="Maroon">c<sub>5</sub></font> or ( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K3">1-element_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="ordinal4.html#K3">1-element_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal4.html#K6">succ</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> ) )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E3:39"><i><font color="Green">E25</font></i></a>, <a class="txt" href="zfrefle1.html#E6:39"><i><font color="Green">E28</font></i></a>, <a class="txt" href="zfrefle1.html#E9:39"><i><font color="Green">E31</font></i></a>, <a class="ref" href="ordinal4.html#T35">ORDINAL4:35</a>, <a class="txt" href="zfrefle1.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:39_6"/>
<font color="Maroon">c<sub>4</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="zfrefle1.html#E2:39_6"><i><font color="Green">E34</font></i></a>, <a class="txt" href="zfrefle1.html#E3:39_6"><i><font color="Green">E35</font></i></a>, <a class="ref" href="classes1.html#T38">CLASSES1:38</a>, <a class="ref" href="classes1.html#T43">CLASSES1:43</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:39"/>
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="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="zf_refle.html#K5">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K3">Union</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <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="39_7"><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="E1:39_7"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>5</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>5</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  )
 ;<br/>

<a NAME="E2:39_7"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="classes1.html#K4">Rank</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E8:39"><i><font color="Green">E30</font></i></a></i>;<br/>
<a NAME="E3:39_7"/><i><font color="Green">E35</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/>
<b>thus </b><a NAME="E4:39_7"/>
<font color="Maroon">c<sub>4</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K3">Union</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>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_7_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="E1:39_7_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E3:39_7_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4">Rank</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="zfrefle1.html#E1:39_7"><i><font color="Green">E33</font></i></a>, <a class="txt" href="zfrefle1.html#E2:39_7"><i><font color="Green">E34</font></i></a>, <a class="ref" href="classes1.html#T35">CLASSES1:35</a></i>;<br/>
<a NAME="E4:39_7_1"/>
<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>
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E3:39_7"><i><font color="Green">E35</font></i></a>, <a class="txt" href="zfrefle1.html#E3:39_7_1"><i><font color="Green">E36</font></i></a>, <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>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="ref" href="zf_refle.html#T8">ZF_REFLE:8</a></i>;<br/>
<a NAME="E6:39_7_1"/>
(  <a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="ordinal4.html#K6">succ</a> <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#E1:39_7"><i><font color="Green">E33</font></i></a>, <a class="txt" href="zfrefle1.html#E3:39_7_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="ordinal1.html#T41">ORDINAL1:41</a>, <a class="ref" href="zf_refle.html#T8">ZF_REFLE:8</a></i>;<br/>
<a NAME="E7:39_7_1"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="zf_refle.html#K5">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="zf_refle.html#K5">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <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> &amp;  <a href="card_3.html#K3">Union</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#R1">=</a>  <a href="tarski.html#K3">union</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> )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E3:39"><i><font color="Green">E25</font></i></a>, <a class="txt" href="zfrefle1.html#E7:39"><i><font color="Green">E29</font></i></a>, <a class="ref" href="funct_1.html#T73">FUNCT_1:73</a>, <a class="ref" href="card_3.html#D4">CARD_3:def 4</a></i>;<br/>
<a NAME="E8:39_7_1"/><b>then </b>
(  <a href="classes1.html#K4">Rank</a> <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> <a href="zf_refle.html#K5">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="zf_refle.html#K5">.</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K3">Union</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> )
 
<b>by </b><i><a class="ref" href="classes1.html#T39">CLASSES1:39</a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E9:39_7_1"/><b>then </b>
 <a href="classes1.html#K4">Rank</a> <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K3">Union</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>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E10:39_7_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K3">Union</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>
 <b>by </b><i><a class="txt" href="zfrefle1.html#E3:39_7_1"><i><font color="Green">E36</font></i></a></i>;<br/>


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

<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="E5:39_7"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K3">Union</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/>

<a NAME="E6:39_7"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</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>
 
<b>by </b><i><a class="ref" href="card_3.html#D4">CARD_3:def 4</a></i>;<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="E8:39_7"/><i><font color="Green">E36</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#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> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:39_7"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>8</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>7</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>8</sub></font> )
 <b>by </b><i><a class="txt" href="zfrefle1.html#E8:39_7"><i><font color="Green">E36</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="txt" href="zfrefle1.html#E10:39_7"><i><font color="Green">E37</font></i></a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E12:39_7"/><i><font color="Green">E38</font></i>: 
 <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="ref" href="relat_1.html#T87">RELAT_1:87</a></i>;<br/>
<a NAME="E13:39_7"/><b>then </b><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>7</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> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E10:39_7"><i><font color="Green">E37</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<a NAME="E14:39_7"/>
<font color="Maroon">c<sub>9</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#E3:39_7"><i><font color="Green">E35</font></i></a>, <a class="txt" href="zfrefle1.html#E10:39_7"><i><font color="Green">E37</font></i></a>, <a class="txt" href="zfrefle1.html#E12:39_7"><i><font color="Green">E38</font></i></a>, <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</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="E16:39_7"/>
<font color="Maroon">c<sub>4</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E14:39"><i><font color="Green">E32</font></i></a>, <a class="txt" href="zfrefle1.html#E10:39_7"><i><font color="Green">E37</font></i></a>, <a class="txt" href="zfrefle1.html#E12:39_7"><i><font color="Green">E38</font></i></a></i>;<br/>
<b>hence </b><a NAME="E17:39_7"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="zfrefle1.html#E8:39_7"><i><font color="Green">E36</font></i></a>, <a class="txt" href="zfrefle1.html#E13:39_7"><i><font color="Green">E39</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="ordinal4.html#M2">Ordinal-Sequence</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E17:39"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> is <a href="ordinal2.html#V2">increasing</a> &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="ordinal2.html#V3">continuous</a> &amp; ( 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">c<sub>5</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Olive">b<sub>1</sub></font> <a href="zfrefle1.html#R3" target="_self">is_elementary_subsystem_of</a>  <a href="zf_refle.html#K4">Union</a> <font color="Maroon">c<sub>4</sub></font> ) )
 <b>by </b><i><a class="txt" href="zfrefle1.html#E1:39"><i><font color="Green">E24</font></i></a>, <a class="txt" href="zfrefle1.html#E14:39"><i><font color="Green">E32</font></i></a>, <a class="ref" href="zfrefle1.html#T32" target="_self">Th32</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>thus </b><a NAME="E18:39"/>
( <font color="Maroon">c<sub>5</sub></font> is <a href="ordinal2.html#V2">increasing</a> &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="ordinal2.html#V3">continuous</a> )
 <b>by </b><i><a class="txt" href="zfrefle1.html#E17:39"><i><font color="Green">E33</font></i></a></i>;<br/>

<a NAME="E19:39"/><i><font color="Green">E34</font></i>: 
 <a href="zf_refle.html#K4">Union</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_8"><b>proof </b></a><div class="add">

<a NAME="E1:39_8"/><i><font color="Green">E35</font></i>: 
 <a href="zf_refle.html#K4">Union</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_3.html#D4">CARD_3:def 4</a></i>;<br/>
<b>thus </b><a NAME="E2:39_8"/>
 <a href="zf_refle.html#K4">Union</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<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="E3:39_8"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E4:39_8"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="ordinal1.html#K3">On</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="ordinal1.html#K3">On</a> <font color="Maroon">c<sub>1</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  )
 
<b>by </b><i><a class="ref" href="classes2.html#T54">CLASSES2:54</a>, <a class="ref" href="classes2.html#T55">CLASSES2:55</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E6:39_8"/><i><font color="Green">E36</font></i>: 
( <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> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4">Rank</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="ref" href="classes1.html#T35">CLASSES1:35</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="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="zfrefle1.html#E6:39_8"><i><font color="Green">E36</font></i></a>, <a class="ref" href="zf_refle.html#T8">ZF_REFLE:8</a></i>;<br/>
<a NAME="E8:39_8"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="classes1.html#K4">Rank</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">c<sub>8</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> )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E3:39"><i><font color="Green">E25</font></i></a>, <a class="txt" href="zfrefle1.html#E9:39"><i><font color="Green">E31</font></i></a>, <a class="txt" href="zfrefle1.html#E6:39_8"><i><font color="Green">E36</font></i></a>, <a class="ref" href="classes1.html#T33">CLASSES1:33</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E9:39_8"/><b>then </b>
 <a href="classes1.html#K4">Rank</a> <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a>  <a href="zf_refle.html#K4">Union</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E1:39_8"><i><font color="Green">E35</font></i></a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<b>hence </b><a NAME="E10:39_8"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_refle.html#K4">Union</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="zfrefle1.html#E6:39_8"><i><font color="Green">E36</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/><b>let </b><font color="Maroon">c<sub>7</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E20:39"/><i><font color="Green">E35</font></i>: 
( <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> <font color="Maroon">c<sub>6</sub></font> &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="classes1.html#K4">Rank</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/>

<a NAME="E21:39"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E9:39"><i><font color="Green">E31</font></i></a></i>;<br/>
<b>hence </b><a NAME="E22:39"/>
<font color="Maroon">c<sub>7</sub></font> <a href="zfrefle1.html#R3" target="_self">is_elementary_subsystem_of</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="zfrefle1.html#E17:39"><i><font color="Green">E33</font></i></a>, <a class="txt" href="zfrefle1.html#E19:39"><i><font color="Green">E34</font></i></a>, <a class="txt" href="zfrefle1.html#E20:39"><i><font color="Green">E35</font></i></a></i>;<br/>


</div>
