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<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<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> implies <font color="Maroon">A1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> );<br/>
<a NAME="E1:46_2_1"/><i><font color="Green">E13</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="46_2_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="E1:46_2_1_1"/><i><font color="Green">E25</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> &amp; <font color="Olive">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> holds <br/><font color="Maroon">A1</font> <a href="ordinal2.html#K10">+^</a> <font color="Olive">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>and </b><br/><a NAME="E2:46_2_1_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 ;<br/>

<a NAME="E3:46_2_1_1"/>
(  <a href="ordinal4.html#K2" target="_self">0-element_of</a> <font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> &amp;  <a href="ordinal4.html#K3" target="_self">1-element_of</a> <font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> )
 
<b>by </b><i><a class="txt" href="ordinal4.html#E4:46_2_1_1"><i><font color="Green">E27</font></i></a></i>;<br/>
<a NAME="E4:46_2_1_1"/><b>then </b><i><font color="Green">E27</font></i>: 
( <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="ordinal4.html#K2" target="_self">0-element_of</a> <font color="Maroon">W</font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> &amp; <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="ordinal4.html#K3" target="_self">1-element_of</a> <font color="Maroon">W</font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> &amp; <font color="Maroon">A1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> )
 
<b>by </b><i><a class="txt" href="ordinal4.html#E4:46_2_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="classes2.html#T63">CLASSES2:63</a></i>;<br/>
<a NAME="E5:46_2_1_1"/><b>then </b>
( <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A1</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K2" target="_self">0-element_of</a> <font color="Maroon">W</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> &amp; <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">B</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K3" target="_self">1-element_of</a> <font color="Maroon">W</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> )
 
<b>by </b><i>, <a class="ref" href="classes2.html#T67">CLASSES2:67</a></i>;<br/>
<a NAME="E6:46_2_1_1"/><b>then </b>
( <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A1</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K2" target="_self">0-element_of</a> <font color="Maroon">W</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">B</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K3" target="_self">1-element_of</a> <font color="Maroon">W</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ordinal1.html#K1">succ</a> <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">W</font> <a href="classes1.html#NR1" target="_self">is_Tarski-Class</a>  &amp;  <a href="ordinal4.html#K2" target="_self">0-element_of</a> <font color="Maroon">W</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="ordinal4.html#K3" target="_self">1-element_of</a> <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="classes2.html#T66">CLASSES2:66</a></i>;<br/>
<a NAME="E7:46_2_1_1"/><b>then </b><i><font color="Green">E28</font></i>: 
(  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A1</font>,<span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K2" target="_self">0-element_of</a> <font color="Maroon">W</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">B</font>,<span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K3" target="_self">1-element_of</a> <font color="Maroon">W</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A1</font>,<span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K2" target="_self">0-element_of</a> <font color="Maroon">W</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">B</font>,<span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K3" target="_self">1-element_of</a> <font color="Maroon">W</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">W</font> )
 
<b>by </b><i><a class="ref" href="card_2.html#T16">CARD_2:16</a>, <a class="ref" href="classes2.html#T1">CLASSES2:1</a></i>;<br/>
<a NAME="E8:46_2_1_1"/>
<font color="Maroon">A1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_2_1_1_1"><b>proof </b></a><div class="add">

<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="E1:46_2_1_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">B</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">A</font> = <font color="Maroon">x</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="E3:46_2_1_1_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W</font> &amp; ( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> or <font color="Maroon">A1</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A</font> ) )
 
<b>by </b><i>, <a class="ref" href="ordinal1.html#T26">ORDINAL1:26</a>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_2_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:46_2_1_1_1_1"/>
<font color="Maroon">A1</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A</font>
 ;<br/><b>then consider </b><font color="Maroon">C</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E3:46_2_1_1_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="ordinal3.html#T30">ORDINAL3:30</a></i>;<br/><a NAME="E4:46_2_1_1_1_1"/>
( <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W</font> )
 <b>by </b><i>, , <a class="txt" href="ordinal4.html#E3:46_2_1_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a>, <a class="ref" href="ordinal3.html#T25">ORDINAL3:25</a></i>;<br/><b>hence </b><a NAME="E5:46_2_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="ordinal4.html#D1" target="_self">Def1</a>, <a class="txt" href="ordinal4.html#E3:46_2_1_1_1"><i><font color="Green">E36</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:46_2_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:46_2_1_1"/>
<font color="Maroon">A1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>by </b><i>, <a class="ref" href="classes1.html#T2">CLASSES1:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:46_2_1"/><i><font color="Green">E42</font></i>: 
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>
<a NAME="E3:46_2_1"/>
<font color="Maroon">B1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:46_2_1"/><b>then </b>
<font color="Maroon">A1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">B1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 
<b>by </b><i><a class="ref" href="ordinal4.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E5:46_2_1"/>
<font color="Maroon">A1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">B1</font> is    <a href="ordinal4.html#M1" target="_self">Ordinal</a> of <font color="Maroon">W</font>
 <b>by </b><i/>;<br/>


</div>
