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

<b>let </b><font color="Maroon">W</font> be   <a href="classes2.html#NM1">Universe</a>;<br/>

<b>assume </b><a NAME="E1:43"/>
 <a href="ordinal1.html#K5">omega</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 ;<br/>

<b>then consider </b><font color="Maroon">phi</font> being    <a href="ordinal4.html#M2">Ordinal-Sequence</a> of <font color="Maroon">W</font><b> such that </b><br/><a NAME="E3:43"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">phi</font> is <a href="ordinal2.html#V2">increasing</a> &amp; <font color="Maroon">phi</font> is <a href="ordinal2.html#V3">continuous</a> &amp; ( for <font color="Olive">a</font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">W</font><br/> for <font color="Olive">M</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>   st <font color="Maroon">phi</font> <a href="ordinal4.html#K4">.</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">a</font> &amp; <font color="Olive">M</font> <a href="hidden.html#R1">=</a>  <a href="classes1.html#K4">Rank</a> <font color="Olive">a</font> holds <br/><font color="Olive">M</font> <a href="zfrefle1.html#R3" target="_self">is_elementary_subsystem_of</a> <font color="Maroon">W</font> ) )
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">phi</font>
;<br/>

<b>thus </b><a NAME="E4:43"/>
( <font color="Maroon">phi</font> is <a href="ordinal2.html#V2">increasing</a> &amp; <font color="Maroon">phi</font> is <a href="ordinal2.html#V3">continuous</a> )
 <b>by </b><i/>;<br/>

<b>let </b><font color="Maroon">a</font> be    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">W</font>;<br/><b>let </b><font color="Maroon">M</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E5:43"/>
( <font color="Maroon">phi</font> <a href="ordinal4.html#K4">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> &amp; <font color="Maroon">M</font> <a href="hidden.html#R1">=</a>  <a href="classes1.html#K4">Rank</a> <font color="Maroon">a</font> )
 ;<br/>

<a NAME="E6:43"/><b>then </b>
<font color="Maroon">M</font> <a href="zfrefle1.html#R3" target="_self">is_elementary_subsystem_of</a> <font color="Maroon">W</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:43"/>
<font color="Maroon">M</font> <a href="zfrefle1.html#R2" target="_self">&lt;==&gt;</a> <font color="Maroon">W</font>
 <b>by </b><i/>;<br/>


</div>
