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

<b>thus </b><a NAME="E1:63"/>
for <font color="Olive">X</font>, <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4">Rank</a> <a href="ordinal1.html#K5">omega</a>  &amp; <font color="Olive">Y</font> <a href="tarski.html#R1">c=</a> <font color="Olive">X</font> holds <br/><font color="Olive">Y</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4">Rank</a> <a href="ordinal1.html#K5">omega</a> 
 <b>by </b><i><a class="ref" href="classes1.html#T47">CLASSES1:47</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="classes1.html#D1">CLASSES1:def 1</a>,<a class="ref" href="classes1.html#D2">CLASSES1:def 2</a><br/>

<b>thus </b><a NAME="E2:63"/>
for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4">Rank</a> <a href="ordinal1.html#K5">omega</a>  holds <br/> <a href="zfmisc_1.html#K1">bool</a> <font color="Olive">X</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4">Rank</a> <a href="ordinal1.html#K5">omega</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:63_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4">Rank</a> <a href="ordinal1.html#K5">omega</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">A</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E3:63_1"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a>  &amp; <font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4">Rank</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="classes1.html#T35">CLASSES1:35</a>, <a class="ref" href="ordinal2.html#T19">ORDINAL2:19</a></i>;<br/>
<a NAME="E4:63_1"/>
(  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a>  &amp;  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="classes2.html#T21" target="_self">Th21</a>, <a class="ref" href="classes1.html#T48">CLASSES1:48</a>, <a class="ref" href="ordinal1.html#T41">ORDINAL1:41</a>, <a class="ref" href="ordinal2.html#T19">ORDINAL2:19</a></i>;<br/>
<b>hence </b><a NAME="E5:63_1"/>
 <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4">Rank</a> <a href="ordinal1.html#K5">omega</a> 
 <b>by </b><i><a class="ref" href="classes1.html#T35">CLASSES1:35</a>, <a class="ref" href="ordinal2.html#T19">ORDINAL2:19</a></i>;<br/>


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

<b>thus </b><a NAME="E3:63"/>
for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Olive">X</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4">Rank</a> <a href="ordinal1.html#K5">omega</a>  or <font color="Olive">X</font>, <a href="classes1.html#K4">Rank</a> <a href="ordinal1.html#K5">omega</a>  <a href="wellord2.html#R2">are_equipotent</a>  or <font color="Olive">X</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4">Rank</a> <a href="ordinal1.html#K5">omega</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:63_2"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4">Rank</a> <a href="ordinal1.html#K5">omega</a>  &amp; not <font color="Maroon">X</font>, <a href="classes1.html#K4">Rank</a> <a href="ordinal1.html#K5">omega</a>  <a href="wellord2.html#R2">are_equipotent</a>  &amp; not <font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4">Rank</a> <a href="ordinal1.html#K5">omega</a>  )
 ;<br/>

<a NAME="E2:63_2"/><b>then </b>
(  <a href="card_1.html#K1">Card</a> <font color="Maroon">X</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <a href="ordinal1.html#K5">omega</a>  &amp;  <a href="card_1.html#K1">Card</a> <font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="card_1.html#K1">Card</a> <a href="ordinal1.html#K5">omega</a>  )
 
<b>by </b><i><a class="ref" href="classes2.html#T14" target="_self">Th14</a>, <a class="ref" href="card_1.html#T21">CARD_1:21</a>, <a class="ref" href="card_1.html#T27">CARD_1:27</a></i>;<br/>
<a NAME="E3:63_2"/><b>then </b>
(  <a href="card_1.html#K1">Card</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">X</font> &amp;  <a href="card_1.html#K1">Card</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a>  )
 
<b>by </b><i><a class="ref" href="card_1.html#T13">CARD_1:13</a>, <a class="ref" href="card_1.html#T84">CARD_1:84</a></i>;<br/>
<b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:63_2"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">X</font>
 ;<br/>
<a NAME="E6:63_2"/><i><font color="Green">E31</font></i>: 
(  <a href="card_1.html#K1">Card</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">n</font> &amp;  <a href="ordinal1.html#K5">omega</a>  <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4">Rank</a> <a href="ordinal1.html#K5">omega</a>  )
 
<b>by </b><i><a class="ref" href="classes2.html#T22" target="_self">Th22</a>, <a class="ref" href="card_1.html#D5">CARD_1:def 5</a>, <a class="ref" href="classes1.html#T44">CLASSES1:44</a></i>;<br/>
<a NAME="E7:63_2"/><b>then </b>
( <font color="Maroon">X</font>,<font color="Maroon">n</font> <a href="wellord2.html#R2">are_equipotent</a>  &amp; <font color="Maroon">n</font> is <a href="finset_1.html#V1">finite</a> )
 
<b>by </b><i><a class="ref" href="card_1.html#T21">CARD_1:21</a>, <a class="ref" href="card_1.html#T69">CARD_1:69</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">X</font> = <font color="Maroon">X</font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="card_1.html#T68">CARD_1:68</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">a<sub>2</sub></font> <a href="ordinal1.html#R1">c=</a>  <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E9:63_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">X</font> <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="classes2.html#T21" target="_self">Th21</a>, , <a class="ref" href="ordinal2.html#T19">ORDINAL2:19</a></i>;<br/>
<a NAME="E10:63_2"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   holds <br/>( <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] or <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">y</font>,<font color="Olive">x</font>] )
 
;<br/>
<a NAME="E11:63_2"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">y</font>,<font color="Olive">z</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">z</font>]
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E13:63_2"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; ( for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">x</font>,<font color="Olive">y</font>] ) )
 <b>from </b><i><a class="ref" href="card_3.html#S3">CARD_3:sch 3</a>(, <a class="ref" href="classes2.html#T23" target="_self">Th23</a>, );<br/></i>
<b>set </b><font color="Maroon">A</font> =  <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">x</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:63_2_1"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E2:63_2_1"/><b>then </b>
(  <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">Y</font> <a href="ordinal1.html#R1">c=</a>  <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">x</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:63_2_1"/>
 <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <span class="p1">(<span class="default"><a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="ordinal1.html#T34">ORDINAL1:34</a></i>;<br/></div><b>end;</b></div>
<a NAME="E15:63_2"/><b>then </b><i><font color="Green">E54</font></i>: 
 <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">X</font> <a href="ordinal1.html#R1">c=</a>  <a href="ordinal1.html#K1">succ</a> <span class="p1">(<span class="default"><a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i><a class="ref" href="classes1.html#T77">CLASSES1:77</a></i>;<br/>
<a NAME="E16:63_2"/>
 <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 
<b>by </b><i><a class="ref" href="classes2.html#T21" target="_self">Th21</a>, , <a class="ref" href="classes1.html#T74">CLASSES1:74</a></i>;<br/>
<a NAME="E17:63_2"/><b>then </b>
 <a href="ordinal1.html#K1">succ</a> <span class="p1">(<span class="default"><a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#T41">ORDINAL1:41</a>, <a class="ref" href="ordinal2.html#T19">ORDINAL2:19</a></i>;<br/>
<a NAME="E18:63_2"/><b>then </b>
 <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 
<b>by </b><i><a class="txt" href="classes2.html#E15:63_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a></i>;<br/>
<b>hence </b><a NAME="E19:63_2"/>
contradiction
 <b>by </b><i><a class="ref" href="classes2.html#T21" target="_self">Th21</a>, <a class="ref" href="classes1.html#T74">CLASSES1:74</a></i>;<br/>


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


</div>
