<?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/>

<a NAME="E1:71"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4" target="_self">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6" target="_self">the_rank_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <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="Olive">b<sub>1</sub></font> holds <br/> <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="Olive">b<sub>1</sub></font> ) )
 
<b>by </b><i><a class="ref" href="classes1.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E2:71"/><i><font color="Green">E54</font></i>: 
 <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4" target="_self">Rank</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <span class="p2">(<span class="default"><a href="classes1.html#K6" target="_self">the_rank_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</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="E1:71_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:71_1"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4" target="_self">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6" target="_self">the_rank_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="classes1.html#K4" target="_self">Rank</a> <span class="p2">(<span class="default"><a href="classes1.html#K6" target="_self">the_rank_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="classes1.html#K4" target="_self">Rank</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <span class="p2">(<span class="default"><a href="classes1.html#K6" target="_self">the_rank_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="classes1.html#E1:71"><i><font color="Green">E53</font></i></a>, <a class="txt" href="classes1.html#E40"><i><font color="Green">Lemma26</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E3:71_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4" target="_self">Rank</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <span class="p2">(<span class="default"><a href="classes1.html#K6" target="_self">the_rank_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:71"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st  <a href="zfmisc_1.html#K1">bool</a> <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="Olive">b<sub>1</sub></font> holds <br/> <a href="ordinal1.html#K1">succ</a> <span class="p1">(<span class="default"><a href="classes1.html#K6" target="_self">the_rank_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="ordinal1.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:71_2"/><i><font color="Green">E55</font></i>: 
 <a href="zfmisc_1.html#K1">bool</a> <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>2</sub></font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] means <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E2:71_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#D1">ZFMISC_1:def 1</a></i>;<br/>
<a NAME="E3:71_2"/><b>then </b><i><font color="Green">E57</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>by </b><i><a class="txt" href="classes1.html#E1:71_2"><i><font color="Green">E55</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E5:71_2"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>] &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>from </b><i><a class="ref" href="ordinal1.html#S1">ORDINAL1:sch 1</a>(<a class="txt" href="classes1.html#E3:71_2"><i><font color="Green">E57</font></i></a>);<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:71_2_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> holds <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ordinal1.html#K1">succ</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/><a NAME="E2:71_2_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#T42">ORDINAL1:42</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E4:71_2_1"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="classes1.html#E5:71_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="classes1.html#E40"><i><font color="Green">Lemma26</font></i></a>, <a class="ref" href="classes1.html#T35" target="_self">Th35</a></i>;<br/><a NAME="E5:71_2_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="classes1.html#E5:71_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="classes1.html#E4:71_2_1"><i><font color="Green">E59</font></i></a></i>;<br/><b>hence </b><a NAME="E6:71_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="classes1.html#E4:71_2_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="ordinal1.html#T7">ORDINAL1:7</a></i>;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E8:71_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>
<a NAME="E9:71_2"/>
<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>4</sub></font>
 
<b>by </b><i><a class="txt" href="classes1.html#E5:71_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="classes1.html#E8:71_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="classes1.html#T36" target="_self">Th36</a></i>;<br/>
<a NAME="E10:71_2"/><b>then </b>
 <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>4</sub></font>
 
<b>by </b><i><a class="ref" href="classes1.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E11:71_2"/><b>then </b>
 <a href="classes1.html#K6" target="_self">the_rank_of</a> <font color="Maroon">c<sub>1</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#E8:71_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="ordinal1.html#T34">ORDINAL1:34</a></i>;<br/>
<a NAME="E12:71_2"/><b>then </b>
(  <a href="ordinal1.html#K1">succ</a> <span class="p1">(<span class="default"><a href="classes1.html#K6" target="_self">the_rank_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="classes1.html#E1:71_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="classes1.html#E2:71_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="classes1.html#E5:71_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="ordinal1.html#T33">ORDINAL1:33</a></i>;<br/>
<b>hence </b><a NAME="E13:71_2"/>
 <a href="ordinal1.html#K1">succ</a> <span class="p1">(<span class="default"><a href="classes1.html#K6" target="_self">the_rank_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:71"/>
 <a href="classes1.html#K6" target="_self">the_rank_of</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <span class="p1">(<span class="default"><a href="classes1.html#K6" target="_self">the_rank_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="classes1.html#E2:71"><i><font color="Green">E54</font></i></a>, <a class="ref" href="classes1.html#D8" target="_self">Def8</a></i>;<br/>


</div>
