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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:26"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K1" target="_self">Tarski-Class</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#K3" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E2:26"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st  <a href="classes1.html#K3" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="classes1.html#T22" target="_self">Th22</a></i>;<br/>
<a NAME="E3:26"/><b>then </b><i><font color="Green">E20</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#E1:26"><i><font color="Green">E19</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="E5:26"/><i><font color="Green">E21</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#E3:26"><i><font color="Green">E20</font></i></a>);<br/></i>
<a NAME="E6:26"/><i><font color="Green">E22</font></i>: 
( not <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span> &amp;  <a href="classes1.html#K3" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>2</sub></font>,<a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="txt" href="classes1.html#E1:26"><i><font color="Green">E19</font></i></a>, <a class="txt" href="classes1.html#E15"><i><font color="Green">Lemma10</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a> 
 ;<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="E3:26_1"/><i><font color="Green">E23</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#K3" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="classes1.html#E5:26"><i><font color="Green">E21</font></i></a>, <a class="txt" href="classes1.html#E6:26"><i><font color="Green">E22</font></i></a>, <a class="ref" href="classes1.html#T16" target="_self">Th16</a></i>;<br/><a NAME="E4:26_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#E5:26"><i><font color="Green">E21</font></i></a>, <a class="txt" href="classes1.html#E3:26_1"><i><font color="Green">E23</font></i></a></i>;<br/><b>hence </b><a NAME="E5:26_1"/>
contradiction
 <b>by </b><i><a class="txt" href="classes1.html#E3:26_1"><i><font color="Green">E23</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="E9:26"/><i><font color="Green">E23</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>
 <b>by </b><i><a class="ref" href="ordinal1.html#T42">ORDINAL1:42</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<a NAME="E10:26"/>
<font color="Maroon">c<sub>4</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#E9:26"><i><font color="Green">E23</font></i></a>, <a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a></i>;<br/>
<a NAME="E11:26"/><b>then </b>
not <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="ref" href="ordinal1.html#T7">ORDINAL1:7</a></i>;<br/>
<b>hence </b><a NAME="E12:26"/>
( not <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K3" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K3" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="classes1.html#E5:26"><i><font color="Green">E21</font></i></a>, <a class="txt" href="classes1.html#E9:26"><i><font color="Green">E23</font></i></a></i>;<br/>


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