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<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="E1:37"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">A</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  )
 ;<br/>

<b>consider </b><font color="Maroon">L</font> being   <a href="ordinal1.html#NM3">T-Sequence</a><b> such that </b><br/><a NAME="E3:37"/><i><font color="Green">E26</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp; ( for <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">B</font>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S2">ORDINAL2:sch 2</a>();<br/></i>
<a NAME="E4:37"/><i><font color="Green">E29</font></i>: 
 <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">L</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="classes1.html#E4:37"><i><font color="Green">E29</font></i></a>, <a class="txt" href="classes1.html#E3:37"><i><font color="Green">E26</font></i></a></i>;<br/>
<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>thus </b><a NAME="E5:37"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">A</font> implies  ex <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4" target="_self">Rank</a> <font color="Olive">B</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:37_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">A</font>
 ;<br/>

<b>then consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:37_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:37_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="txt" href="classes1.html#E3:37_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="txt" href="classes1.html#E5:37_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<b>take </b>
<font color="Maroon">y</font>
;<br/>

<b>thus </b><a NAME="E7:37_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="txt" href="classes1.html#E4:37"><i><font color="Green">E29</font></i></a>, <a class="txt" href="classes1.html#E3:37_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="classes1.html#E5:37_1"><i><font color="Green">E32</font></i></a></i>;<br/>


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

<b>given </b><font color="Maroon">B</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><a NAME="E6:37"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">B</font> )
 ;<br/>

<a NAME="E7:37"/>
<font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="txt" href="classes1.html#E4:37"><i><font color="Green">E29</font></i></a>, </i>;<br/>
<a NAME="E8:37"/><b>then </b>
 <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="txt" href="classes1.html#E4:37"><i><font color="Green">E29</font></i></a>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E9:37"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">A</font>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div>
