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

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

<b>consider </b><font color="Maroon">Field</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:66"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Field</font> &amp; ( for <font color="Olive">X</font>, <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Field</font> &amp; <font color="Olive">Y</font> <a href="tarski.html#R1">c=</a> <font color="Olive">X</font> holds <br/><font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Field</font> ) &amp; ( for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Field</font> holds <br/> <a href="zfmisc_1.html#K1">bool</a> <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Field</font> ) &amp; ( for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Field</font> or <font color="Olive">X</font>,<font color="Maroon">Field</font> <a href="tarski.html#R2">are_equipotent</a>  or <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Field</font> ) ) )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T136">ZFMISC_1:136</a></i>;<br/>
<a NAME="E3:66"/>
for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K3" target="_self">On</a> <font color="Maroon">Field</font> holds <br/>( <font color="Olive">X</font> is   <a href="ordinal1.html#NM2">Ordinal</a> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a>  <a href="ordinal1.html#K3" target="_self">On</a> <font color="Maroon">Field</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:66_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K3" target="_self">On</a> <font color="Maroon">Field</font>
 ;<br/>

<b>hence </b><a NAME="E2:66_1"/>
<font color="Maroon">X</font> is   <a href="ordinal1.html#NM2">Ordinal</a>
 <b>by </b><i><a class="ref" href="ordinal1.html#T14" target="_self">Th14</a></i>;<br/>

<b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">X</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i>, <a class="ref" href="ordinal1.html#T14" target="_self">Th14</a></i>;<br/>
<b>let </b><font color="Maroon">y</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="E4:66_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E5:66_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">B</font> = <font color="Maroon">y</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i/>;<br/>
<a NAME="E7:66_1"/>
( <font color="Maroon">B</font> <a href="ordinal1.html#R1" target="_self">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Field</font> )
 
<b>by </b><i>, , , <a class="ref" href="ordinal1.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E8:66_1"/><b>then </b>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Field</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:66_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K3" target="_self">On</a> <font color="Maroon">Field</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T14" target="_self">Th14</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">ON</font> =  <a href="ordinal1.html#K3" target="_self">On</a> <font color="Maroon">Field</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="ref" href="ordinal1.html#T3" target="_self">Th3</a></i>;<br/>
<b>take </b>
<font color="Maroon">ON</font>
;<br/>

<b>thus </b><a NAME="E5:66"/>
<font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ON</font>
 <b>by </b><i>, <a class="ref" href="ordinal1.html#T14" target="_self">Th14</a></i>;<br/>

<a NAME="E6:66"/>
for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ON</font> holds <br/> <a href="ordinal1.html#K1" target="_self">succ</a> <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ON</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:66_2"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ON</font>
 ;<br/>

<a NAME="E2:66_2"/><b>then </b>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Field</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E3:66_2"/><b>then </b><i><font color="Green">E23</font></i>: 
 <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Field</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:66_2"/>
 <a href="ordinal1.html#K1" target="_self">succ</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:66_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1" target="_self">succ</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E2:66_2_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:66_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:66_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">A</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:66_2"/><b>then </b>
 <a href="ordinal1.html#K1" target="_self">succ</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Field</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:66_2"/>
 <a href="ordinal1.html#K1" target="_self">succ</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ON</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T14" target="_self">Th14</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:66"/>
<font color="Maroon">ON</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a> 
 <b>by </b><i/>;<br/>


</div>
