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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>] <b>means </b>( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font> implies $1 <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c3">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font> );<br/>
<a NAME="E1:47_1_1"/><i><font color="Green" title="E32">A1</font></i>: 
for <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st ( for <font color="Olive" title="b2">C</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b2">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">C</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</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" title="c4">A</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">C</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">C</font>] ) implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">A</font>] )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:47_1_1_1"/><i><font color="Green" title="E33">A2</font></i>: 
for <font color="Olive" title="b1">C</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Olive" title="b1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font> holds <br/><font color="Olive" title="b1">C</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c3">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font>
 <b>and </b><br/><a NAME="E2:47_1_1_1"/><i><font color="Green" title="E34">A3</font></i>: 
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">A</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c3">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font></span><br/>

<a NAME="E3:47_1_1_1"/><i><font color="Green" title="E35">A4</font></i>: 
( <font color="Maroon" title="c3">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font> &amp; <font color="Maroon" title="c1">W</font> <a href="classes1.html#NR1" title="CLASSES1:NR.1">is_Tarski-Class</a>  )
 
<b>by </b><i><a class="ref" href="ordinal4.html#D2" target="_self" title="ORDINAL4:def.2">Def2</a></i>;<br/>
<a NAME="E4:47_1_1_1"/><b>then </b>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c3">B1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font>
 
<b>by </b><i><a class="txt" href="ordinal4.html#E2:47_1_1_1"><i><font color="Green" title="E34">A3</font></i></a>, <a class="ref" href="classes2.html#T67" title="CLASSES2:th.67">CLASSES2:67</a></i>;<br/>
<a NAME="E5:47_1_1_1"/><b>then </b><i><font color="Green" title="E36">A5</font></i>: 
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c3">B1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c3">B1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c3">B1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">W</font> )
 
<b>by </b><i><a class="ref" href="card_2.html#T18" title="CARD_2:th.18">CARD_2:18</a>, <a class="ref" href="classes2.html#T1" title="CLASSES2:th.1">CLASSES2:1</a></i>;<br/>
<a NAME="E6:47_1_1_1"/>
<font color="Maroon" title="c4">A</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c3">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">W</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" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c3">B1</font> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font> )</span><br/>

<b>assume </b><a NAME="E1:47_1_1_1_1"/><i><font color="Green" title="E37">A6</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c3">B1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c6">B</font> = <font color="Maroon" title="c5">x</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/>
<a NAME="E3:47_1_1_1_1"/>
<font color="Maroon" title="c3">B1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="ordinal4.html#E1:47_1_1_1_1"><i><font color="Green" title="E37">A6</font></i></a>, <a class="ref" href="ordinal2.html#T55" title="ORDINAL2:th.55">ORDINAL2:55</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">C</font>, <font color="Maroon" title="c8">D</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E5:47_1_1_1_1"/><i><font color="Green" title="E38">A7</font></i>: 
( <font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c3">B1</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c8">D</font> &amp; <font color="Maroon" title="c8">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font> )
 <b>by </b><i><a class="ref" href="ordinal3.html#T55" title="ORDINAL3:th.55">ORDINAL3:55</a></i>;<br/>
<a NAME="E6:47_1_1_1_1"/><i><font color="Green" title="E39">A8</font></i>: 
( <font color="Maroon" title="c3">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">W</font> &amp; <font color="Maroon" title="c7">C</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c3">B1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c6">B</font> )
 
<b>by </b><i><a class="txt" href="ordinal4.html#E3:47_1_1_1"><i><font color="Green" title="E35">A4</font></i></a>, <a class="txt" href="ordinal4.html#E5:47_1_1_1_1"><i><font color="Green" title="E38">A7</font></i></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a>, <a class="ref" href="ordinal3.html#T27" title="ORDINAL3:th.27">ORDINAL3:27</a></i>;<br/>
<a NAME="E7:47_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c7">C</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c3">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c3">B1</font>
 
<b>by </b><i><a class="txt" href="ordinal4.html#E1:47_1_1_1_1"><i><font color="Green" title="E37">A6</font></i></a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></i>;<br/>
<a NAME="E8:47_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c7">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">W</font> )
 
<b>by </b><i><a class="txt" href="ordinal4.html#E2:47_1_1_1"><i><font color="Green" title="E34">A3</font></i></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a>, <a class="ref" href="ordinal3.html#T37" title="ORDINAL3:th.37">ORDINAL3:37</a></i>;<br/>
<a NAME="E9:47_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c8">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font> &amp; <font color="Maroon" title="c7">C</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c3">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font> )
 
<b>by </b><i><a class="txt" href="ordinal4.html#E1:47_1_1_1"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="ordinal4.html#E5:47_1_1_1_1"><i><font color="Green" title="E38">A7</font></i></a>, <a class="txt" href="ordinal4.html#E6:47_1_1_1_1"><i><font color="Green" title="E39">A8</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">CB</font> = <font color="Maroon" title="c7">C</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c3">B1</font>, <font color="Maroon" title="c10">D</font> = <font color="Maroon" title="c8">D</font> as    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> <b>by </b><i><a class="ref" href="ordinal4.html#D2" target="_self" title="ORDINAL4:def.2">Def2</a></i>;<br/>
<a NAME="E11:47_1_1_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">CB</font> <a href="ordinal4.html#K7" title="ORDINAL4:func.7">+^</a> <font color="Maroon" title="c10">D</font>
 
<b>by </b><i><a class="txt" href="ordinal4.html#E5:47_1_1_1_1"><i><font color="Green" title="E38">A7</font></i></a></i>;<br/>
<b>hence </b><a NAME="E12:47_1_1_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font>
 <b>by </b><i><a class="ref" href="ordinal4.html#D2" target="_self" title="ORDINAL4:def.2">Def2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:47_1_1_1"/>
<font color="Maroon" title="c4">A</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c3">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font>
 <b>by </b><i><a class="txt" href="ordinal4.html#E5:47_1_1_1"><i><font color="Green" title="E36">A5</font></i></a>, <a class="ref" href="classes1.html#T2" title="CLASSES1:th.2">CLASSES1:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E2:47_1_1"/><i><font color="Green" title="E33">A9</font></i>: 
for <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>]
 
<b>from </b><i><a class="ref" href="ordinal1.html#S2" title="ORDINAL1:sch.2">ORDINAL1:sch 2</a>(<a class="txt" href="ordinal4.html#E1:47_1_1"><i><font color="Green" title="E32">A1</font></i></a>);<br/></i>
<a NAME="E3:47_1_1"/>
<font color="Maroon" title="c2">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font>
 
<b>by </b><i><a class="ref" href="ordinal4.html#D2" target="_self" title="ORDINAL4:def.2">Def2</a></i>;<br/>
<a NAME="E4:47_1_1"/><b>then </b>
<font color="Maroon" title="c2">A1</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c3">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font>
 
<b>by </b><i><a class="txt" href="ordinal4.html#E2:47_1_1"><i><font color="Green" title="E33">A9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:47_1_1"/>
<font color="Maroon" title="c2">A1</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c3">B1</font> is    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>
 <b>by </b><i><a class="ref" href="ordinal4.html#D2" target="_self" title="ORDINAL4:def.2">Def2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
