<?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="E1:48"/>
( not <font color="Maroon">c<sub>1</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  implies ex <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1" target="_self">succ</a> <font color="Olive">b<sub>1</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:48_1"/>
not <font color="Maroon">c<sub>1</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E3:48_1"/><i><font color="Green">E28</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> &amp; not  <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="ref" href="ordinal1.html#T41" target="_self">Th41</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>2</sub></font>
;<br/>

<b>assume </b><a NAME="E4:48_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ordinal1.html#K1" target="_self">succ</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E5:48_1"/>
 <a href="ordinal1.html#K1" target="_self">succ</a> <font color="Maroon">c<sub>2</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="txt" href="ordinal1.html#E3:48_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="ordinal1.html#T33" target="_self">Th33</a></i>;<br/>
<a NAME="E6:48_1"/><b>then </b>
 <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>1</sub></font>
 
<b>by </b><i><a class="txt" href="ordinal1.html#E4:48_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/>
<b>hence </b><a NAME="E7:48_1"/>
contradiction
 <b>by </b><i><a class="txt" href="ordinal1.html#E3:48_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="ordinal1.html#T21" target="_self">Th21</a></i>;<br/>


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

<b>given </b><font color="Maroon">c<sub>2</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><a NAME="E2:48"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1" target="_self">succ</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E3:48"/><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>
 
<b>by </b><i><a class="txt" href="ordinal1.html#E2:48"><i><font color="Green">E28</font></i></a>, <a class="ref" href="ordinal1.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E4:48"/>
not  <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#E2:48"><i><font color="Green">E28</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:48"/>
not <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="txt" href="ordinal1.html#E3:48"><i><font color="Green">E29</font></i></a>, <a class="ref" href="ordinal1.html#T41" target="_self">Th41</a></i>;<br/>


</div>
