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

<b>thus </b><a NAME="E1:63"/>
for <font color="Olive" title="b1">X</font>, <font color="Olive" title="b2">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; <font color="Olive" title="b2">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">X</font> holds <br/><font color="Olive" title="b2">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 <b>by </b><i><a class="ref" href="classes1.html#T47" title="CLASSES1:th.47">CLASSES1:47</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="classes1.html#D1" title="CLASSES1:def.1">CLASSES1:def 1</a>,<a class="ref" href="classes1.html#D2" title="CLASSES1:def.2">CLASSES1:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  or  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  ) ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  or <font color="Olive">b<sub>1</sub></font>, <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">are_equipotent</a>  or <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  ) ) )</span><br/>

<b>thus </b><a NAME="E2:63"/>
for <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  holds <br/> <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  or <font color="Olive">b<sub>1</sub></font>, <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">are_equipotent</a>  or <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  )</span><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" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  implies  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  )</span><br/>

<b>assume </b><a NAME="E1:63_1"/>
<font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span><br/>

<b>then consider </b><font color="Maroon" title="c2">A</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E3:63_1"/><i><font color="Green" title="E48">A1</font></i>: 
( <font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c2">A</font> )
 <b>by </b><i><a class="txt" href="classes2.html#E75"><i><font color="Green" title="E45">Lm5</font></i></a>, <a class="ref" href="classes1.html#T35" title="CLASSES1:th.35">CLASSES1:35</a></i>;<br/>
<a NAME="E4:63_1"/>
(  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp;  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">A</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="classes2.html#E3:63_1"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="classes2.html#E75"><i><font color="Green" title="E45">Lm5</font></i></a>, <a class="ref" href="classes1.html#T48" title="CLASSES1:th.48">CLASSES1:48</a>, <a class="ref" href="ordinal1.html#T41" title="ORDINAL1:th.41">ORDINAL1:41</a></i>;<br/>
<b>hence </b><a NAME="E5:63_1"/>
 <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 <b>by </b><i><a class="txt" href="classes2.html#E75"><i><font color="Green" title="E45">Lm5</font></i></a>, <a class="ref" href="classes1.html#T35" title="CLASSES1:th.35">CLASSES1:35</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>thus </b><a NAME="E3:63"/>
for <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  or <font color="Olive" title="b1">X</font>, <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a>  or <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  or <font color="Maroon" title="c1">X</font>, <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a>  or <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  )</span><br/>

<b>assume </b><a NAME="E1:63_2"/><i><font color="Green" title="E48">A2</font></i>: 
( <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; not <font color="Maroon" title="c1">X</font>, <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a>  &amp; not <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:63_2"/><b>then </b>
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">X</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  )
 
<b>by </b><i><a class="ref" href="classes2.html#T69" target="_self" title="CLASSES2:th.69">Th69</a>, <a class="ref" href="card_1.html#T21" title="CARD_1:th.21">CARD_1:21</a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a></i>;<br/>
<a NAME="E3:63_2"/><b>then </b>
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  )
 
<b>by </b><i><a class="ref" href="card_1.html#T13" title="CARD_1:th.13">CARD_1:13</a>, <a class="ref" href="card_1.html#T84" title="CARD_1:th.84">CARD_1:84</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c2">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E5:63_2"/><i><font color="Green" title="E49">A3</font></i>: 
<font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">X</font>
 ;<br/>
<a NAME="E6:63_2"/><i><font color="Green" title="E50">A4</font></i>: 
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">n</font> &amp;  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  )
 
<b>by </b><i><a class="txt" href="classes2.html#E5:63_2"><i><font color="Green" title="E49">A3</font></i></a>, <a class="ref" href="classes1.html#T44" title="CLASSES1:th.44">CLASSES1:44</a></i>;<br/>
<a NAME="E7:63_2"/>
( <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">n</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a>  &amp; <font color="Maroon" title="c2">n</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 
<b>by </b><i><a class="txt" href="classes2.html#E5:63_2"><i><font color="Green" title="E49">A3</font></i></a>, <a class="ref" href="card_1.html#D5" title="CARD_1:def.5">CARD_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c3">X</font> = <font color="Maroon" title="c1">X</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="ref" href="card_1.html#T68" title="CARD_1:th.68">CARD_1:68</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> $2 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> $1;<br/>
<a NAME="E9:63_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 
;<br/>
<a NAME="E10:63_2"/><b>then </b><i><font color="Green" title="E51">A5</font></i>: 
<font color="Maroon" title="c3">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="classes2.html#E1:63_2"><i><font color="Green" title="E48">A2</font></i></a>, <a class="txt" href="classes2.html#E6:63_2"><i><font color="Green" title="E50">A4</font></i></a></i>;<br/>
<a NAME="E11:63_2"/><i><font color="Green" title="E52">A6</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>( <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] or <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</font>] )
 
;<br/>
<a NAME="E12:63_2"/><i><font color="Green" title="E53">A7</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">z</font>]
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E14:63_2"/><i><font color="Green" title="E54">A8</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; ( for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c4">x</font>,<font color="Olive" title="b1">y</font>] ) )
 <b>from </b><i><a class="ref" href="card_3.html#S3" title="CARD_3:sch.3">CARD_3:sch 3</a>(<a class="txt" href="classes2.html#E10:63_2"><i><font color="Green" title="E51">A5</font></i></a>, <a class="txt" href="classes2.html#E11:63_2"><i><font color="Green" title="E52">A6</font></i></a>, <a class="txt" href="classes2.html#E12:63_2"><i><font color="Green" title="E53">A7</font></i></a>);<br/></i>
<b>set </b><font color="Maroon" title="c5">A</font> =  <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c4">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" title="c6">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> implies  <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c6">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c4">x</font></span>)</span> )</span><br/><b>assume </b><a NAME="E1:63_2_1"/>
<font color="Maroon" title="c6">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c6">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c4">x</font></span>)</span></span><br/><a NAME="E2:63_2_1"/><b>then </b>
(  <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c6">Y</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c6">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Y</font> )
 <b>by </b><i><a class="txt" href="classes2.html#E14:63_2"><i><font color="Green" title="E54">A8</font></i></a></i>;<br/><b>hence </b><a NAME="E3:63_2_1"/>
 <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c6">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c4">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="ordinal1.html#T34" title="ORDINAL1:th.34">ORDINAL1:34</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E16:63_2"/><b>then </b><i><font color="Green" title="E55">A9</font></i>: 
 <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c3">X</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c4">x</font></span>)</span>
 
<b>by </b><i><a class="ref" href="classes1.html#T77" title="CLASSES1:th.77">CLASSES1:77</a></i>;<br/>
<a NAME="E17:63_2"/>
 <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 
<b>by </b><i><a class="txt" href="classes2.html#E1:63_2"><i><font color="Green" title="E48">A2</font></i></a>, <a class="txt" href="classes2.html#E14:63_2"><i><font color="Green" title="E54">A8</font></i></a>, <a class="ref" href="classes1.html#T74" title="CLASSES1:th.74">CLASSES1:74</a></i>;<br/>
<a NAME="E18:63_2"/><b>then </b>
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 
<b>by </b><i><a class="txt" href="classes2.html#E75"><i><font color="Green" title="E45">Lm5</font></i></a>, <a class="ref" href="ordinal1.html#T41" title="ORDINAL1:th.41">ORDINAL1:41</a></i>;<br/>
<a NAME="E19:63_2"/><b>then </b>
 <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c3">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 
<b>by </b><i><a class="txt" href="classes2.html#E16:63_2"><i><font color="Green" title="E55">A9</font></i></a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></i>;<br/>
<b>hence </b><a NAME="E20:63_2"/>
contradiction
 <b>by </b><i><a class="txt" href="classes2.html#E1:63_2"><i><font color="Green" title="E48">A2</font></i></a>, <a class="ref" href="classes1.html#T74" title="CLASSES1:th.74">CLASSES1:74</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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


</div>
