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<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><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>] <b>means </b>( <font color="Maroon">A</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">a<sub>1</sub></font> implies  <a href="classes1.html#K3" target="_self">Tarski-Class</a> <font color="Maroon">X</font>,<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K3" target="_self">Tarski-Class</a> <font color="Maroon">X</font>,<font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E1:21"/><i><font color="Green">E25</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>1</sub></font>[<font color="Olive">C</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_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="E1:21_1"/><i><font color="Green">E26</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="Olive">C</font>]
 <b>and </b><br/><a NAME="E2:21_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">A</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">B</font>
 ;<br/>

<b>let </b><font color="Maroon">x</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="E3:21_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K3" target="_self">Tarski-Class</a> <font color="Maroon">X</font>,<font color="Maroon">A</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">B</font>
 ;<br/><a NAME="E2:21_1_1"/><b>then </b>
<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="classes1.html#D1" target="_self">Def1</a>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><a NAME="E3:21_1_1"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T21">ORDINAL1:21</a></i>;<br/><a NAME="E4:21_1_1"/>
( ( <font color="Maroon">A</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">B</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">A</font> ) iff <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="E5:21_1_1"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">B</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="classes1.html#D1" target="_self">Def1</a>, , <a class="ref" href="ordinal1.html#T21">ORDINAL1:21</a></i>;<br/><div><i><font color="Green">E35</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_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="E1:21_1_1_1"/><i><font color="Green">E36</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="E2:21_1_1_1"/>
( <font color="Maroon">A</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">C</font> &amp; <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="classes1.html#D3" target="_self">Def3</a>, , <a class="ref" href="ordinal1.html#T34">ORDINAL1:34</a></i>;<br/><a NAME="E3:21_1_1_1"/><b>then </b>
(  <a href="classes1.html#K3" target="_self">Tarski-Class</a> <font color="Maroon">X</font>,<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K3" target="_self">Tarski-Class</a> <font color="Maroon">X</font>,<font color="Maroon">C</font> &amp;  <a href="classes1.html#K3" target="_self">Tarski-Class</a> <font color="Maroon">X</font>,<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K3" target="_self">Tarski-Class</a> <font color="Maroon">X</font>,<font color="Maroon">B</font> )
 <b>by </b><i>, , </i>;<br/><a NAME="E4:21_1_1_1"/><b>then </b>
 <a href="classes1.html#K3" target="_self">Tarski-Class</a> <font color="Maroon">X</font>,<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K3" target="_self">Tarski-Class</a> <font color="Maroon">X</font>,<font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E5:21_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K3" target="_self">Tarski-Class</a> <font color="Maroon">X</font>,<font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="classes1.html#D2" target="_self">Def2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_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="E2:21_1_1_2"/><b>then </b>
<font color="Maroon">B</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/><a NAME="E3:21_1_1_2"/><b>then </b>
 <a href="classes1.html#K3" target="_self">Tarski-Class</a> <font color="Maroon">X</font>,<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">v</font> where <font color="Olive">v</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">X</font> :  ex <font color="Olive">C</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Olive">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">v</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K3" target="_self">Tarski-Class</a> <font color="Maroon">X</font>,<font color="Olive">C</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="classes1.html#D4" target="_self">Def4</a>, </i>;<br/><b>hence </b><a NAME="E4:21_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K3" target="_self">Tarski-Class</a> <font color="Maroon">X</font>,<font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="classes1.html#D2" target="_self">Def2</a>, <a class="ref" href="classes1.html#D3" target="_self">Def3</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E8:21_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K3" target="_self">Tarski-Class</a> <font color="Maroon">X</font>,<font color="Maroon">B</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:21_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K3" target="_self">Tarski-Class</a> <font color="Maroon">X</font>,<font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="classes1.html#D2" target="_self">Def2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:21"/>
for <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B</font>]
 
<b>from </b><i><a class="ref" href="ordinal1.html#S2">ORDINAL1:sch 2</a>();<br/></i>
<b>hence </b><a NAME="E3:21"/>
( <font color="Maroon">A</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">B</font> implies  <a href="classes1.html#K3" target="_self">Tarski-Class</a> <font color="Maroon">X</font>,<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K3" target="_self">Tarski-Class</a> <font color="Maroon">X</font>,<font color="Maroon">B</font> )
 ;<br/>


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