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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] means ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> implies <font color="Maroon">a<sub>1</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> );<br/>
<a NAME="E1:47_1_1"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st ( for <font color="Olive">b<sub>2</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:47_1_1_1"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E2:47_1_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E3:47_1_1_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="classes1.html#NR1" target="_self">is_Tarski-Class</a>  )
 
<b>by </b><i><a class="ref" href="ordinal4.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E4:47_1_1_1"/><b>then </b>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="ordinal4.html#E2:47_1_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="classes2.html#T67">CLASSES2:67</a></i>;<br/>
<a NAME="E5:47_1_1_1"/><b>then </b><i><font color="Green">E33</font></i>: 
(  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="card_2.html#T18">CARD_2:18</a>, <a class="ref" href="classes2.html#T1">CLASSES2:1</a></i>;<br/>
<a NAME="E6:47_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</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="E1:47_1_1_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E3:47_1_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="ordinal4.html#E1:47_1_1_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="ordinal2.html#T55">ORDINAL2:55</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E5:47_1_1_1_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="ordinal3.html#T55">ORDINAL3:55</a></i>;<br/>
<a NAME="E6:47_1_1_1_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>3</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="ordinal4.html#E3:47_1_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="ordinal4.html#E5:47_1_1_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a>, <a class="ref" href="ordinal3.html#T27">ORDINAL3:27</a></i>;<br/>
<a NAME="E7:47_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="ordinal4.html#E1:47_1_1_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a></i>;<br/>
<a NAME="E8:47_1_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="ordinal4.html#E2:47_1_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a>, <a class="ref" href="ordinal3.html#T37">ORDINAL3:37</a></i>;<br/>
<a NAME="E9:47_1_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="ordinal4.html#E1:47_1_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="ordinal4.html#E5:47_1_1_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="ordinal4.html#E6:47_1_1_1_1"><i><font color="Green">E36</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font> as    <a href="ordinal4.html#M1" target="_self">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="ordinal4.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E11:47_1_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="ordinal4.html#K7" target="_self">+^</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="ordinal4.html#E5:47_1_1_1_1"><i><font color="Green">E35</font></i></a></i>;<br/>
<b>hence </b><a NAME="E12:47_1_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="ordinal4.html#D2" target="_self">Def2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:47_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="ordinal4.html#E5:47_1_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="classes1.html#T2">CLASSES1:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:47_1_1"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="ordinal1.html#S2">ORDINAL1:sch 2</a>(<a class="txt" href="ordinal4.html#E1:47_1_1"><i><font color="Green">E29</font></i></a>);<br/></i>
<a NAME="E3:47_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="ordinal4.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E4:47_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="ordinal4.html#E2:47_1_1"><i><font color="Green">E30</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:47_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>3</sub></font> is    <a href="ordinal4.html#M1" target="_self">Ordinal</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="ordinal4.html#D2" target="_self">Def2</a></i>;<br/>


</div>
