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



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>,  <a href="ordinal1.html#NM2">Ordinal</a>] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>2</sub></font> implies  <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">a<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>2</sub></font> );<br/>
<div><i><font color="Green">E25</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">A</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] <b>means </b><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">A</font>,<font color="Maroon">a<sub>1</sub></font>];<br/><a NAME="E1:44_1"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st ( for <font color="Olive">C</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">C</font> <a href="hidden.html#R2">in</a> <font color="Olive">B</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">C</font>] ) holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">B</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><b>that </b><br/><a NAME="E1:44_1_1"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">C</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">A</font>,<font color="Olive">C</font>]
 <b>and </b><br/><a NAME="E2:44_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>

<div><i><font color="Green">E32</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">C</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><a NAME="E1:44_1_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">C</font>
 ;<br/><a NAME="E2:44_1_1_1"/>
<font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a></i>;<br/><a NAME="E3:44_1_1_1"/><b>then </b><i><font color="Green">E34</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> implies  <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">C</font> )
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:44_1_1_1_1"/><i><font color="Green">E35</font></i>: 
not <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/><a NAME="E2:44_1_1_1_1"/>
( ( <font color="Maroon">A</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">C</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">C</font> ) iff <font color="Maroon">A</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">C</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><b>hence </b><a NAME="E3:44_1_1_1_1"/>
 <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , <a class="ref" href="ordinal1.html#T21">ORDINAL1:21</a>, <a class="ref" href="ordinal1.html#T34">ORDINAL1:34</a></i>;<br/></div><b>end;</b></div><a NAME="E5:44_1_1_1"/><b>then </b>
 <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">C</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E6:44_1_1_1"/>
 <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">B</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:44_1_1_2"/>
for <font color="Olive">C</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  holds <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ordinal1.html#K1">succ</a> <font color="Olive">C</font>
 ;<br/><a NAME="E2:44_1_1_2"/><b>then </b><i><font color="Green">E36</font></i>: 
( <font color="Maroon">B</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  &amp; <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="ordinal1.html#T42">ORDINAL1:42</a></i>;<br/><a NAME="E3:44_1_1_2"/>
(  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">B</font> &amp;  <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="txt" href="classes1.html#E2:44_1_1_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="ordinal1.html#T33">ORDINAL1:33</a></i>;<br/><a NAME="E4:44_1_1_2"/><b>then </b>
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><a NAME="E5:44_1_1_2"/><b>then </b>
(  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp;  <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">A</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> <font color="Maroon">A</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="ordinal1.html#T21">ORDINAL1:21</a></i>;<br/><b>hence </b><a NAME="E6:44_1_1_2"/>
 <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">B</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:44_1_1"/>
 <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="txt" href="classes1.html#E3:44"><i><font color="Green">E37</font></i></a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E2:44_1"/>
for <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">B</font>]
 <b>from </b><i><a class="ref" href="ordinal1.html#S2">ORDINAL1:sch 2</a>();</i><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:44"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">A</font>,<font color="Maroon">B</font>]
 ;<br/>

<b>assume </b><a NAME="E3:44"/><i><font color="Green">E37</font></i>: 
(  <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">B</font> &amp; not <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 ;<br/>

<a NAME="E4:44"/><b>then </b>
( <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> or <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> )
 
<b>by </b><i><a class="ref" href="ordinal1.html#T24">ORDINAL1:24</a></i>;<br/>
<b>hence </b><a NAME="E5:44"/>
contradiction
 <b>by </b><i>, </i>;<br/>


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