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<b>let </b><font color="Maroon" title="c1">W</font> be   <a href="classes2.html#NM1" title="CLASSES2:NM.1">Universe</a>;<br/>

<b>assume </b><a NAME="E1:43"/>
 <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c2">phi</font> being    <a href="ordinal4.html#M2" title="ORDINAL4:mode.2">Ordinal-Sequence</a> of <font color="Maroon" title="c1">W</font><b> such that </b><br/><a NAME="E3:43"/><i><font color="Green" title="E28">A1</font></i>: 
( <font color="Maroon" title="c2">phi</font> is <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a> &amp; <font color="Maroon" title="c2">phi</font> is <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a> &amp; ( for <font color="Olive" title="b1">a</font> being   <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font><br/> for <font color="Olive" title="b2">M</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon" title="c2">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Olive" title="b1">a</font> holds <br/><font color="Olive" title="b2">M</font> <a href="zfrefle1.html#R3" title="ZFREFLE1:pred.3">is_elementary_subsystem_of</a> <font color="Maroon" title="c1">W</font> ) )
 <b>by </b><i><a class="ref" href="zfrefle1.html#T35" target="_self" title="ZFREFLE1:th.35">Th35</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c2">phi</font>
;<br/>

<b>thus </b><a NAME="E4:43"/>
( <font color="Maroon" title="c2">phi</font> is <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a> &amp; <font color="Maroon" title="c2">phi</font> is <a href="ordinal2.html#V3" title="ORDINAL2:attr.3">continuous</a> )
 <b>by </b><i><a class="txt" href="zfrefle1.html#E3:43"><i><font color="Green" title="E28">A1</font></i></a></i>;<br/>

<b>let </b><font color="Maroon" title="c3">a</font> be    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>;<br/><b>let </b><font color="Maroon" title="c4">M</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><a NAME="E5:43"/>
( <font color="Maroon" title="c2">phi</font> <a href="ordinal4.html#K4" title="ORDINAL4:func.4">.</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c4">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c3">a</font> )
 ;<br/>

<a NAME="E6:43"/><b>then </b>
<font color="Maroon" title="c4">M</font> <a href="zfrefle1.html#R3" title="ZFREFLE1:pred.3">is_elementary_subsystem_of</a> <font color="Maroon" title="c1">W</font>
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E3:43"><i><font color="Green" title="E28">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:43"/>
<font color="Maroon" title="c4">M</font> <a href="zfrefle1.html#R2" title="ZFREFLE1:pred.2">&lt;==&gt;</a> <font color="Maroon" title="c1">W</font>
 <b>by </b><i><a class="ref" href="zfrefle1.html#T9" target="_self" title="ZFREFLE1:th.9">Th9</a></i>;<br/>


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