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<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">A</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>



<b>assume </b><a NAME="E1:80"/><i><font color="Green">E25</font></i>: 
 <a href="classes1.html#K6" target="_self">the_rank_of</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E2:80"/><b>then </b>
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font> &amp;  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font> <a href="ordinal1.html#R1">c=</a>  <a href="classes1.html#K6" target="_self">the_rank_of</a> <font color="Maroon">X</font> )
 
<b>by </b><i><a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a></i>;<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="E4:80"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">A</font> <a href="ordinal1.html#R1">c=</a>  <a href="classes1.html#K6" target="_self">the_rank_of</a> <font color="Maroon">Y</font> )
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">Y</font>
;<br/>

<a NAME="E5:80"/>
 <a href="classes1.html#K6" target="_self">the_rank_of</a> <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K6" target="_self">the_rank_of</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="classes1.html#T5" target="_self">Th5</a>, </i>;<br/>
<a NAME="E6:80"/><b>then </b>
 <a href="classes1.html#K6" target="_self">the_rank_of</a> <font color="Maroon">Y</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="ordinal1.html#T34">ORDINAL1:34</a></i>;<br/>
<b>hence </b><a NAME="E7:80"/>
( <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp;  <a href="classes1.html#K6" target="_self">the_rank_of</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="classes1.html#T5" target="_self">Th5</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


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