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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:31"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="ordinal1.html#V1" target="_self">epsilon-transitive</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="ordinal1.html#V2" target="_self">epsilon-connected</a> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal1.html#D4" target="_self">ORDINAL1:def 4</a><br/>

<b>then reconsider </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">c<sub>1</sub></font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="ref" href="ordinal1.html#D4" target="_self">Def4</a></i>;<br/>
<b>thus </b><a NAME="E3:31"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>1</sub></font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal1.html#D2" target="_self">ORDINAL1:def 2</a>,<a class="ref" href="ordinal1.html#D4" target="_self">ORDINAL1:def 4</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><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> ;<br/>

<b>assume </b><a NAME="E1:31_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>1</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="E3:31_1"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</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="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E4:31_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="ordinal1.html#E3:31_1"><i><font color="Green">E18</font></i></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="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="ref" href="ordinal1.html#T23" target="_self">Th23</a></i>;<br/>
<a NAME="E6:31_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="ordinal1.html#R1" target="_self">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="ordinal1.html#E3:31_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="ordinal1.html#D2" target="_self">Def2</a></i>;<br/>
<b>hence </b><a NAME="E7:31_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="ordinal1.html#E3:31_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>


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

<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="ordinal1.html#D3" target="_self">ORDINAL1:def 3</a><br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E4:31"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:31"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <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="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:31"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="ordinal1.html#E4:31"><i><font color="Green">E18</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E9:31"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="ordinal1.html#E6:31"><i><font color="Green">E19</font></i></a>, <a class="txt" href="ordinal1.html#E8:31"><i><font color="Green">E20</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="ref" href="ordinal1.html#T23" target="_self">Th23</a></i>;<br/>
<a NAME="E11:31"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> )
 
<b>by </b><i><a class="txt" href="ordinal1.html#E6:31"><i><font color="Green">E19</font></i></a>, <a class="txt" href="ordinal1.html#E8:31"><i><font color="Green">E20</font></i></a></i>;<br/>
<a NAME="E12:31"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> is   <a href="ordinal1.html#NM2">Ordinal</a> &amp; <font color="Maroon">c<sub>4</sub></font> is   <a href="ordinal1.html#NM2">Ordinal</a> )
 
<b>by </b><i><a class="ref" href="ordinal1.html#T23" target="_self">Th23</a></i>;<br/>
<b>hence </b><a NAME="E13:31"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="ordinal1.html#T24" target="_self">Th24</a></i>;<br/>


</div>
