<?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    <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>3</sub></font> be    <a href="ordinal4.html#M2">Ordinal-Sequence</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><a NAME="E1:35"/><i><font color="Green">E20</font></i>: 
(  <a href="ordinal1.html#K5">omega</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="ordinal2.html#V2">increasing</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="ordinal2.html#V3">continuous</a> )
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="ordinal1.html#NM2">Ordinal</a>,   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font>) <b>-&gt; </b>   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> =  <a href="ordinal4.html#K6">succ</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">a<sub>2</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="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> =  <a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</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="E3:35"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>4</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> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E4:35"/><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> holds <font color="Maroon">c<sub>4</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> <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="ordinal4.html#K4">.</a> <font color="Olive">b<sub>1</sub></font>)
 <b>and </b><br/><a NAME="E5:35"/>
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>  holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="ordinal4.html#K4">.</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="zf_refle.html#S4">ZF_REFLE:sch 4</a>();<br/></i>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="ordinal1.html#K2">|</a> <a href="ordinal1.html#K5">omega</a> ;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <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> <a href="ordinal1.html#K5">omega</a> </span>)</span>;<br/>
<a NAME="E6:35"/><i><font color="Green">E23</font></i>: 
(  <a href="ordinal1.html#K5">omega</a>  <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="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> )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E1:35"><i><font color="Green">E20</font></i></a>, <a class="ref" href="ordinal1.html#D10">ORDINAL1:def 10</a>, <a class="ref" href="ordinal4.html#D3">ORDINAL4:def 3</a></i>;<br/>
<a NAME="E7:35"/><b>then </b><i><font color="Green">E24</font></i>: 
 <a href="ordinal1.html#K5">omega</a>  <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E8:35"/><b>then </b><i><font color="Green">E25</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> <a href="ordinal1.html#K5">omega</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K5">omega</a>  &amp;  <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> <a href="ordinal1.html#K5">omega</a> </span>)</span> <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> <a href="ordinal1.html#K5">omega</a> </span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E9:35"/>
(  <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> <a href="ordinal1.html#K5">omega</a> </span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <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> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T99">RELAT_1:99</a>, <a class="ref" href="ordinal4.html#D3">ORDINAL4:def 3</a></i>;<br/>
<a NAME="E10:35"/><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> <a href="ordinal1.html#K5">omega</a> </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;  <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="E11:35"/><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> <a href="ordinal1.html#K5">omega</a> </span>)</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="E12:35"/><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> <a href="ordinal1.html#K5">omega</a> </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#E1:35"><i><font color="Green">E20</font></i></a>, <a class="txt" href="zfrefle1.html#E8:35"><i><font color="Green">E25</font></i></a>, <a class="ref" href="zfrefle1.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E13:35"/><b>then </b>
 <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> <a href="ordinal1.html#K5">omega</a> </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="zf_refle.html#T28">ZF_REFLE:28</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> =  <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> <a href="ordinal1.html#K5">omega</a> </span>)</span> 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="ordinal4.html#D2">ORDINAL4:def 2</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/>

<a NAME="E15:35"/>
(  <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>  &amp;  <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="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="ordinal4.html#T35">ORDINAL4:35</a>, <a class="ref" href="ordinal4.html#T36">ORDINAL4:36</a></i>;<br/>
<a NAME="E16:35"/><b>then </b><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>2</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> <a href="ordinal1.html#K5">omega</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E3:35"><i><font color="Green">E21</font></i></a>, <a class="ref" href="funct_1.html#T73">FUNCT_1:73</a>, <a class="ref" href="ordinal2.html#T19">ORDINAL2:19</a></i>;<br/>
<b>hence </b><a NAME="E17:35"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="ordinal2.html#T27">ORDINAL2:27</a></i>;<br/>

<a NAME="E18:35"/><i><font color="Green">E27</font></i>: 
 <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> <a href="ordinal1.html#K5">omega</a> </span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</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="E1:35_1"/><i><font color="Green">E28</font></i>: 
<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>4</sub></font> <a href="ordinal1.html#K2">|</a> <a href="ordinal1.html#K5">omega</a> </span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:35_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>9</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> <a href="ordinal1.html#K5">omega</a> </span>)</span> &amp; <font color="Maroon">c<sub>8</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> <a href="ordinal1.html#K5">omega</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</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>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="txt" href="zfrefle1.html#E3:35_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E5:35_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="ordinal1.html#K2">|</a> <a href="ordinal1.html#K5">omega</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> is   <a href="ordinal1.html#NM2">Ordinal</a>
 
;<br/>
<b>hence </b><a NAME="E6:35_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="zfrefle1.html#E1:35_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="zfrefle1.html#E3:35_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="ordinal2.html#T27">ORDINAL2:27</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E19:35"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="ordinal1.html#K2">|</a> <a href="ordinal1.html#K5">omega</a>  is <a href="ordinal2.html#V2">increasing</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal2.html#D16">ORDINAL2:def 16</a><br/><b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>



<b>assume </b><a NAME="E1:35_2"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</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>  <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> <a href="ordinal1.html#K5">omega</a> </span>)</span> )
 ;<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>8</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">a<sub>1</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> <a href="ordinal1.html#K5">omega</a> </span>)</span> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  implies <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="ordinal1.html#K2">|</a> <a href="ordinal1.html#K5">omega</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="ordinal1.html#K2">|</a> <a href="ordinal1.html#K5">omega</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> );<br/>
<a NAME="E2:35_2"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="xboole_0.html#K1">{}</a> ]
 
;<br/>
<a NAME="E3:35_2"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[ <a href="ordinal1.html#K1">succ</a> <font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:35_2_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>10</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> <a href="ordinal1.html#K5">omega</a> </span>)</span> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  implies <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="ordinal1.html#K2">|</a> <a href="ordinal1.html#K5">omega</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="ordinal1.html#K2">|</a> <a href="ordinal1.html#K5">omega</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> )
 <b>and </b><br/><a NAME="E2:35_2_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <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> <a href="ordinal1.html#K5">omega</a> </span>)</span> &amp;  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>

<a NAME="E3:35_2_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <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#E6:35"><i><font color="Green">E23</font></i></a>, <a class="txt" href="zfrefle1.html#E8:35"><i><font color="Green">E25</font></i></a>, <a class="txt" href="zfrefle1.html#E2:35_2_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>8</sub></font> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> 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="E5:35_2_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a></i>;<br/>
<a NAME="E6:35_2_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="ref" href="ordinal2.html#T49">ORDINAL2:49</a></i>;<br/>
<a NAME="E7:35_2_1"/><b>then </b><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="ordinal2.html#K10">+^</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>8</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>10</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> <a href="ordinal1.html#K5">omega</a> </span>)</span> )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E2:35_2_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="zfrefle1.html#E3:35_2_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>8</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>10</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="E9:35_2_1"/>
(  <a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> &amp; ( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">c<sub>10</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="ordinal2.html#T45">ORDINAL2:45</a>, <a class="ref" href="ordinal4.html#T36">ORDINAL4:36</a></i>;<br/>
<a NAME="E10:35_2_1"/><b>then </b>
( ( <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="ordinal1.html#K2">|</a> <a href="ordinal1.html#K5">omega</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="ordinal1.html#K2">|</a> <a href="ordinal1.html#K5">omega</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal4.html#K6">succ</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>12</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> <a href="ordinal1.html#K5">omega</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="ordinal4.html#K6">succ</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> ) or <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E1:35"><i><font color="Green">E20</font></i></a>, <a class="txt" href="zfrefle1.html#E4:35"><i><font color="Green">E22</font></i></a>, <a class="txt" href="zfrefle1.html#E1:35_2_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="zfrefle1.html#E7:35_2_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a>, <a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a>, <a class="ref" href="ordinal4.html#T10">ORDINAL4:10</a></i>;<br/>
<a NAME="E11:35_2_1"/><b>then </b>
( ( <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="ordinal1.html#K2">|</a> <a href="ordinal1.html#K5">omega</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>11</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> <a href="ordinal1.html#K5">omega</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> ) or <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E2:35_2_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a>, <a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a></i>;<br/>
<a NAME="E12:35_2_1"/><b>then </b><i><font color="Green">E36</font></i>: 
( ( <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="ordinal1.html#K2">|</a> <a href="ordinal1.html#K5">omega</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="ordinal1.html#K2">|</a> <a href="ordinal1.html#K5">omega</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> ) or <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:35_2_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:35_2_1_1"/><b>then </b><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a>, <a class="ref" href="ordinal2.html#T44">ORDINAL2:44</a>, <a class="ref" href="ordinal2.html#T45">ORDINAL2:45</a>, <a class="ref" href="ordinal4.html#T36">ORDINAL4:36</a></i>;<br/><a NAME="E3:35_2_1_1"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal4.html#K6">succ</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="ordinal4.html#K6">succ</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="ordinal1.html#K2">|</a> <a href="ordinal1.html#K5">omega</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="ordinal1.html#K2">|</a> <a href="ordinal1.html#K5">omega</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="txt" href="zfrefle1.html#E1:35"><i><font color="Green">E20</font></i></a>, <a class="txt" href="zfrefle1.html#E4:35"><i><font color="Green">E22</font></i></a>, <a class="txt" href="zfrefle1.html#E2:35_2_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="zfrefle1.html#E7:35_2_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a>, <a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a>, <a class="ref" href="ordinal4.html#T10">ORDINAL4:10</a></i>;<br/><b>hence </b><a NAME="E4:35_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="ordinal1.html#K2">|</a> <a href="ordinal1.html#K5">omega</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="ordinal1.html#K2">|</a> <a href="ordinal1.html#K5">omega</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="ordinal2.html#K10">+^</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="zfrefle1.html#E2:35_2_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:35_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="ordinal1.html#K2">|</a> <a href="ordinal1.html#K5">omega</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="ordinal1.html#K2">|</a> <a href="ordinal1.html#K5">omega</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="ordinal2.html#K10">+^</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="zfrefle1.html#E12:35_2_1"><i><font color="Green">E36</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:35_2"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">S<sub>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="35_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E1:35_2_2"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>10</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>10</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  )
 ;<br/>

<a NAME="E2:35_2_2"/><b>then </b>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="ref" href="ordinal3.html#T10">ORDINAL3:10</a></i>;<br/>
<a NAME="E3:35_2_2"/><b>then </b>
(  <a href="ordinal1.html#K5">omega</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>10</sub></font> )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E1:35_2_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="ordinal1.html#D12">ORDINAL1:def 12</a>, <a class="ref" href="ordinal3.html#T27">ORDINAL3:27</a></i>;<br/>
<b>hence </b><a NAME="E4:35_2_2"/>
( ex <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp;  not <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) or <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>10</sub></font>] )
 <b>by </b><i><a class="txt" href="zfrefle1.html#E8:35"><i><font color="Green">E25</font></i></a>, <a class="ref" href="ordinal1.html#T7">ORDINAL1:7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:35_2"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> 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="ordinal2.html#S1">ORDINAL2:sch 1</a>(<a class="txt" href="zfrefle1.html#E2:35_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="zfrefle1.html#E3:35_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="zfrefle1.html#E4:35_2"><i><font color="Green">E32</font></i></a>);<br/></i>
<a NAME="E6:35_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Olive">b<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>  )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E1:35_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="ordinal3.html#T31">ORDINAL3:31</a></i>;<br/>
<b>hence </b><a NAME="E7:35_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="ordinal1.html#K2">|</a> <a href="ordinal1.html#K5">omega</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="ordinal1.html#K2">|</a> <a href="ordinal1.html#K5">omega</a> </span>)</span> <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#E1:35_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="zfrefle1.html#E5:35_2"><i><font color="Green">E33</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:35"/><i><font color="Green">E29</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> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="ordinal4.html#T36">ORDINAL4:36</a></i>;<br/>
<a NAME="E21:35"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E22:35"/><b>then </b><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>7</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#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#K2">|</a> <font color="Maroon">c<sub>7</sub></font> is <a href="ordinal2.html#V2">increasing</a> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</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>3</sub></font> )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E1:35"><i><font color="Green">E20</font></i></a>, <a class="txt" href="zfrefle1.html#E8:35"><i><font color="Green">E25</font></i></a>, <a class="txt" href="zfrefle1.html#E16:35"><i><font color="Green">E26</font></i></a>, <a class="txt" href="zfrefle1.html#E19:35"><i><font color="Green">E28</font></i></a>, <a class="ref" href="ordinal2.html#T19">ORDINAL2:19</a>, <a class="ref" href="ordinal2.html#T27">ORDINAL2:27</a>, <a class="ref" href="ordinal4.html#T15">ORDINAL4:15</a>, <a class="ref" href="ordinal4.html#T16">ORDINAL4:16</a>, <a class="ref" href="ordinal4.html#T36">ORDINAL4:36</a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E23:35"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="ordinal2.html#R1">is_limes_of</a> <font color="Maroon">c<sub>3</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>7</sub></font> &amp;  <a href="ordinal2.html#K4">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K8">lim</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> &amp;  <a href="ordinal2.html#K4">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <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>3</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E1:35"><i><font color="Green">E20</font></i></a>, <a class="ref" href="ordinal2.html#D17">ORDINAL2:def 17</a>, <a class="ref" href="ordinal4.html#T8">ORDINAL4:8</a></i>;<br/>
<a NAME="E24:35"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>7</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>3</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="ordinal2.html#D14">ORDINAL2:def 14</a></i>;<br/>
<b>thus </b><a NAME="E25:35"/>
<font color="Maroon">c<sub>3</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font>
  <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="35_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="zf_refle.html#D1">ZF_REFLE:def 1</a><br/>

<b>assume </b><a NAME="E1:35_3"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E3:35_3"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>9</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>3</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="zfrefle1.html#E24:35"><i><font color="Green">E32</font></i></a>, <a class="ref" href="ordinal2.html#T29">ORDINAL2:29</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:35_3"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="zfrefle1.html#E22:35"><i><font color="Green">E31</font></i></a>, <a class="txt" href="zfrefle1.html#E3:35_3"><i><font color="Green">E33</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>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="txt" href="zfrefle1.html#E5:35_3"><i><font color="Green">E34</font></i></a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E8:35_3"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>12</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> <a href="ordinal1.html#K5">omega</a> </span>)</span> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="zfrefle1.html#E5:35_3"><i><font color="Green">E34</font></i></a>, <a class="ref" href="ordinal2.html#T29">ORDINAL2:29</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:35_3"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a>  &amp; <font color="Maroon">c<sub>12</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> <a href="ordinal1.html#K5">omega</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="txt" href="zfrefle1.html#E8:35"><i><font color="Green">E25</font></i></a>, <a class="txt" href="zfrefle1.html#E8:35_3"><i><font color="Green">E35</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>14</sub></font> = <font color="Maroon">c<sub>13</sub></font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="txt" href="zfrefle1.html#E10:35_3"><i><font color="Green">E36</font></i></a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E12:35_3"/>
<font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a>  <a href="ordinal1.html#K5">omega</a> 
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E10:35_3"><i><font color="Green">E36</font></i></a>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E13:35_3"/><b>then </b>
<font color="Maroon">c<sub>14</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#E1:35"><i><font color="Green">E20</font></i></a>, <a class="ref" href="classes1.html#D1">CLASSES1:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</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="ordinal4.html#D2">ORDINAL4:def 2</a></i>;<br/>
<a NAME="E15:35_3"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E8:35_3"><i><font color="Green">E35</font></i></a>, <a class="ref" href="ordinal2.html#T27">ORDINAL2:27</a></i>;<br/>
<a NAME="E16:35_3"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">c<sub>12</sub></font> iff ( <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>12</sub></font> ) )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/>
<a NAME="E17:35_3"/><b>then </b>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> or ( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> ) )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E21:35"><i><font color="Green">E30</font></i></a>, <a class="txt" href="zfrefle1.html#E8:35_3"><i><font color="Green">E35</font></i></a>, <a class="txt" href="zfrefle1.html#E15:35_3"><i><font color="Green">E37</font></i></a>, <a class="ref" href="ordinal1.html#T21">ORDINAL1:21</a></i>;<br/>
<a NAME="E18:35_3"/><b>then </b><i><font color="Green">E38</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <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>12</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E1:35"><i><font color="Green">E20</font></i></a>, <a class="ref" href="ordinal2.html#D16">ORDINAL2:def 16</a></i>;<br/>
<a NAME="E19:35_3"/>
(  <a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a>  &amp;  <a href="ordinal4.html#K6">succ</a> <font color="Maroon">c<sub>15</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> )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E10:35_3"><i><font color="Green">E36</font></i></a>, <a class="ref" href="ordinal1.html#T41">ORDINAL1:41</a>, <a class="ref" href="ordinal2.html#T19">ORDINAL2:19</a>, <a class="ref" href="ordinal4.html#T36">ORDINAL4:36</a></i>;<br/>
<a NAME="E20:35_3"/><b>then </b><i><font color="Green">E39</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="ordinal1.html#K2">|</a> <a href="ordinal1.html#K5">omega</a> </span>)</span> <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>15</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</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>15</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</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>15</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> <a href="ordinal1.html#K5">omega</a> </span>)</span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E8:35"><i><font color="Green">E25</font></i></a>, <a class="txt" href="zfrefle1.html#E22:35"><i><font color="Green">E31</font></i></a>, <a class="txt" href="zfrefle1.html#E5:35_3"><i><font color="Green">E34</font></i></a>, <a class="txt" href="zfrefle1.html#E10:35_3"><i><font color="Green">E36</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a>, <a class="ref" href="funct_1.html#T73">FUNCT_1:73</a>, <a class="ref" href="ordinal4.html#T36">ORDINAL4:36</a></i>;<br/>
<a NAME="E21:35_3"/><b>then </b>
( <font color="Maroon">c<sub>4</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>15</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ordinal4.html#K6">succ</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal4.html#K6">succ</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="ordinal4.html#K4">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E4:35"><i><font color="Green">E22</font></i></a>, <a class="txt" href="zfrefle1.html#E18:35_3"><i><font color="Green">E38</font></i></a>, <a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E22:35_3"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</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>15</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>4</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>15</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E20:35_3"><i><font color="Green">E39</font></i></a>, <a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a>, <a class="ref" href="ordinal2.html#T27">ORDINAL2:27</a></i>;<br/>
<a NAME="E23:35_3"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/>
<b>hence </b><a NAME="E24:35_3"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="zfrefle1.html#E3:35_3"><i><font color="Green">E33</font></i></a>, <a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a></i>;<br/>


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

<b>thus </b><a NAME="E26:35"/>
<font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="zfrefle1.html#E1:35"><i><font color="Green">E20</font></i></a>, <a class="txt" href="zfrefle1.html#E20:35"><i><font color="Green">E29</font></i></a>, <a class="ref" href="ordinal4.html#T10">ORDINAL4:10</a></i>;<br/>

<b>take </b>
<font color="Maroon">c<sub>4</sub></font> <a href="ordinal1.html#K2">|</a> <a href="ordinal1.html#K5">omega</a> 
; <i><font color="Red">:: according to </font></i><a class="ref" href="zfrefle1.html#D5" target="_self">ZFREFLE1:def 5</a><br/>

<b>thus </b><a NAME="E27:35"/>
(  <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> <a href="ordinal1.html#K5">omega</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K5">omega</a>  &amp;  <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> <a href="ordinal1.html#K5">omega</a> </span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="ordinal1.html#K2">|</a> <a href="ordinal1.html#K5">omega</a>  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>4</sub></font> <a href="ordinal1.html#K2">|</a> <a href="ordinal1.html#K5">omega</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="zfrefle1.html#E7:35"><i><font color="Green">E24</font></i></a>, <a class="txt" href="zfrefle1.html#E18:35"><i><font color="Green">E27</font></i></a>, <a class="txt" href="zfrefle1.html#E19:35"><i><font color="Green">E28</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>


</div>
