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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>



<b>thus </b><a NAME="E1:78"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="ordinal1.html#R1">c=</a>  <a href="classes1.html#K6" target="_self">the_rank_of</a> <font color="Maroon">c<sub>1</sub></font> implies for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="ordinal1.html#R1">c=</a>  <a href="classes1.html#K6" target="_self">the_rank_of</a> <font color="Olive">b<sub>2</sub></font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:78_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="ordinal1.html#R1">c=</a>  <a href="classes1.html#K6" target="_self">the_rank_of</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E2:78_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E3:78_1"/><b>then </b>
not  <a href="classes1.html#K6" target="_self">the_rank_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="classes1.html#E1:78_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="ordinal1.html#T7">ORDINAL1:7</a></i>;<br/>
<a NAME="E4:78_1"/><b>then </b>
not <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="classes1.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E5:78_1"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<a NAME="E7:78_1"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="classes1.html#E5:78_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>thus </b><a NAME="E8:78_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="classes1.html#E5:78_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>

<a NAME="E9:78_1"/>
not  <a href="classes1.html#K6" target="_self">the_rank_of</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="classes1.html#E7:78_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="classes1.html#T74" target="_self">Th74</a></i>;<br/>
<b>hence </b><a NAME="E10:78_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="ordinal1.html#R1">c=</a>  <a href="classes1.html#K6" target="_self">the_rank_of</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T26">ORDINAL1:26</a></i>;<br/>


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

<b>assume </b><a NAME="E2:78"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="ordinal1.html#R1">c=</a>  <a href="classes1.html#K6" target="_self">the_rank_of</a> <font color="Olive">b<sub>2</sub></font> )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></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="E3:78"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:78"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="ordinal1.html#R1">c=</a>  <a href="classes1.html#K6" target="_self">the_rank_of</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="classes1.html#E2:78"><i><font color="Green">E58</font></i></a>, <a class="txt" href="classes1.html#E3:78"><i><font color="Green">E59</font></i></a></i>;<br/>
<a NAME="E7:78"/>
 <a href="classes1.html#K6" target="_self">the_rank_of</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K6" target="_self">the_rank_of</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="classes1.html#E6:78"><i><font color="Green">E60</font></i></a>, <a class="ref" href="classes1.html#T76" target="_self">Th76</a></i>;<br/>
<b>hence </b><a NAME="E8:78"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K6" target="_self">the_rank_of</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="classes1.html#E6:78"><i><font color="Green">E60</font></i></a>, <a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a></i>;<br/>


</div>
