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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>thus </b><a NAME="E1:47"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  implies 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> <font color="Maroon">c<sub>1</sub></font> holds <br/> <a href="ordinal1.html#K1" target="_self">succ</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:47_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a> 
 ;<br/>

<a NAME="E2:47_1"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#D6" target="_self">Def6</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E3:47_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/>

<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:47_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>2</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> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="ordinal1.html#E2:47_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E6:47_1"/>
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> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="ordinal1.html#E5:47_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E7:47_1"/><b>then </b><i><font color="Green">E30</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E8:47_1"/><i><font color="Green">E31</font></i>: 
<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#E5:47_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="ordinal1.html#T23" target="_self">Th23</a></i>;<br/>
<a NAME="E9:47_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="ordinal1.html#E5:47_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="ordinal1.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E10:47_1"/><b>then </b>
 <a href="ordinal1.html#K1" target="_self">succ</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="ordinal1.html#E7:47_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E11:47_1"/><b>then </b>
(  <a href="ordinal1.html#K1" target="_self">succ</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> or  <a href="ordinal1.html#K1" target="_self">succ</a> <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/>
<a NAME="E12:47_1"/><b>then </b><i><font color="Green">E32</font></i>: 
(  <a href="ordinal1.html#K1" target="_self">succ</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> or  <a href="ordinal1.html#K1" target="_self">succ</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="ordinal1.html#E8:47_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="ordinal1.html#T21" target="_self">Th21</a></i>;<br/>
<a NAME="E13:47_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="ordinal1.html#E5:47_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="ordinal1.html#D2" target="_self">Def2</a></i>;<br/>
<b>hence </b><a NAME="E14:47_1"/>
 <a href="ordinal1.html#K1" target="_self">succ</a> <font color="Maroon">c<sub>2</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#E5:47_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="ordinal1.html#E12:47_1"><i><font color="Green">E32</font></i></a></i>;<br/>


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

<b>assume </b><a NAME="E2:47"/><i><font color="Green">E28</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> <font color="Maroon">c<sub>1</sub></font> holds <br/> <a href="ordinal1.html#K1" target="_self">succ</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:47_2"/><i><font color="Green">E29</font></i>: 
<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="E2:47_2"/><b>then </b>
<font color="Maroon">c<sub>2</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/><a NAME="E3:47_2"/><b>then </b><i><font color="Green">E30</font></i>: 
 <a href="ordinal1.html#K1" target="_self">succ</a> <font color="Maroon">c<sub>2</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#E2:47"><i><font color="Green">E28</font></i></a>, <a class="txt" href="ordinal1.html#E1:47_2"><i><font color="Green">E29</font></i></a></i>;<br/><a NAME="E4:47_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1" target="_self">succ</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T10" target="_self">Th10</a></i>;<br/><b>hence </b><a NAME="E5:47_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</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#E3:47_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:47"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="ordinal1.html#R1" target="_self">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:47_3"/>
<font color="Maroon">c<sub>2</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>3</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:47_3"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>2</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> <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="E4:47_3"/>
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="ordinal1.html#E3:47_3"><i><font color="Green">E30</font></i></a>, <a class="ref" href="ordinal1.html#D2" target="_self">Def2</a></i>;<br/><b>hence </b><a NAME="E5:47_3"/>
<font color="Maroon">c<sub>2</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#E3:47_3"><i><font color="Green">E30</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:47"/><b>then </b>
 <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>1</sub></font> <a href="ordinal1.html#R1" target="_self">c=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E7:47"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</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#E4:47"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E8:47"/>
<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="ordinal1.html#D6" target="_self">Def6</a></i>;<br/>


</div>
