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

<b>given </b><font color="Maroon">c<sub>1</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E1:42"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> iff <font color="Olive">b<sub>1</sub></font> is   <a href="ordinal1.html#NM2">Ordinal</a> )
 ;<br/>

<a NAME="E2:42"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="ordinal1.html#V1" target="_self">epsilon-transitive</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal1.html#D2" target="_self">ORDINAL1:def 2</a><br/>

<b>assume </b><a NAME="E1:42_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:42_1"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is   <a href="ordinal1.html#NM2">Ordinal</a>
 
<b>by </b><i><a class="txt" href="ordinal1.html#E1:42"><i><font color="Green">E25</font></i></a></i>;<br/>
<b>thus </b><a NAME="E3:42_1"/>
<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="42_1_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> ; <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="E1:42_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:42_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> is   <a href="ordinal1.html#NM2">Ordinal</a>
 
<b>by </b><i><a class="txt" href="ordinal1.html#E2:42_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="ordinal1.html#T23" target="_self">Th23</a></i>;<br/>
<b>hence </b><a NAME="E3:42_1_1"/>
<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="ordinal1.html#E1:42"><i><font color="Green">E25</font></i></a></i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E3:42"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="ordinal1.html#V2" target="_self">epsilon-connected</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</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>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:42_2"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>

<a NAME="E2:42_2"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> is   <a href="ordinal1.html#NM2">Ordinal</a> &amp; <font color="Maroon">c<sub>3</sub></font> is   <a href="ordinal1.html#NM2">Ordinal</a> )
 
<b>by </b><i><a class="txt" href="ordinal1.html#E1:42"><i><font color="Green">E25</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:42_2"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="ordinal1.html#T24" target="_self">Th24</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:42"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> is   <a href="ordinal1.html#NM2">Ordinal</a>
 
<b>by </b><i><a class="txt" href="ordinal1.html#E2:42"><i><font color="Green">E26</font></i></a>, <a class="ref" href="ordinal1.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E5:42"/><b>then </b>
<font color="Maroon">c<sub>1</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#E1:42"><i><font color="Green">E25</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:42"/>
contradiction
 ;<br/>


</div>
