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<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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">A</font>, <font color="Olive" title="b2">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b2">B</font> holds <br/> <a href="classes1.html#K3" title="CLASSES1:func.3">Tarski-Class</a> <font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K3" title="CLASSES1:func.3">Tarski-Class</a> <font color="Maroon" title="c1">X</font>,<font color="Olive" title="b2">B</font></span><br/><b>let </b><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</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"> ( <font color="Maroon" title="c2">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">B</font> implies  <a href="classes1.html#K3" title="CLASSES1:func.3">Tarski-Class</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K3" title="CLASSES1:func.3">Tarski-Class</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">B</font> )</span><br/>





<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>( <font color="Maroon" title="c2">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> $1 implies  <a href="classes1.html#K3" title="CLASSES1:func.3">Tarski-Class</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K3" title="CLASSES1:func.3">Tarski-Class</a> <font color="Maroon" title="c1">X</font>,$1 );<br/>
<a NAME="E1:21"/><i><font color="Green" title="E16">A1</font></i>: 
for <font color="Olive" title="b1">B</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">B</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">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" title="c4">B</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">B</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">B</font>] )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:21_1"/><i><font color="Green" title="E17">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">B</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">C</font>]
 <b>and </b><br/><a NAME="E2:21_1"/><i><font color="Green" title="E18">A3</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c4">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="classes1.html#K3" title="CLASSES1:func.3">Tarski-Class</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K3" title="CLASSES1:func.3">Tarski-Class</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c4">B</font></span><br/>

<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>  <a href="classes1.html#K3" title="CLASSES1:func.3">Tarski-Class</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">A</font> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K3" title="CLASSES1:func.3">Tarski-Class</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c4">B</font> )</span><br/>

<b>assume </b><a NAME="E3:21_1"/><i><font color="Green" title="E19">A4</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K3" title="CLASSES1:func.3">Tarski-Class</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">A</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>  <a href="classes1.html#K3" title="CLASSES1:func.3">Tarski-Class</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c4">B</font></span><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" title="E20">A5</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">B</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>  <a href="classes1.html#K3" title="CLASSES1:func.3">Tarski-Class</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c4">B</font></span><br/><a NAME="E2:21_1_1"/><b>then </b>
<font color="Maroon" title="c2">A</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c4">B</font>
 <b>by </b><i><a class="txt" href="classes1.html#E2:21_1"><i><font color="Green" title="E18">A3</font></i></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/><a NAME="E3:21_1_1"/><b>then </b><i><font color="Green" title="E21">A6</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a></i>;<br/><a NAME="E4:21_1_1"/>
( ( <font color="Maroon" title="c2">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c4">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">A</font> ) iff <font color="Maroon" title="c2">A</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c4">B</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/><a NAME="E5:21_1_1"/><b>then </b><i><font color="Green" title="E22">A7</font></i>: 
<font color="Maroon" title="c4">B</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="classes1.html#E2:21_1"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="classes1.html#E1:21_1_1"><i><font color="Green" title="E20">A5</font></i></a>, <a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a></i>;<br/><div><i><font color="Green" title="E23">A8</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" title="c6">C</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><a NAME="E2:21_1_1_1"/><i><font color="Green" title="E24">A9</font></i>: 
<font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c6">C</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>  <a href="classes1.html#K3" title="CLASSES1:func.3">Tarski-Class</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c4">B</font></span><br/><a NAME="E3:21_1_1_1"/>
( <font color="Maroon" title="c2">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c6">C</font> &amp; <font color="Maroon" title="c6">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> )
 <b>by </b><i><a class="txt" href="classes1.html#E3:21_1_1"><i><font color="Green" title="E21">A6</font></i></a>, <a class="txt" href="classes1.html#E2:21_1_1_1"><i><font color="Green" title="E24">A9</font></i></a>, <a class="ref" href="ordinal1.html#T34" title="ORDINAL1:th.34">ORDINAL1:34</a></i>;<br/><a NAME="E4:21_1_1_1"/><b>then </b>
(  <a href="classes1.html#K3" title="CLASSES1:func.3">Tarski-Class</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K3" title="CLASSES1:func.3">Tarski-Class</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c6">C</font> &amp;  <a href="classes1.html#K3" title="CLASSES1:func.3">Tarski-Class</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c6">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K3" title="CLASSES1:func.3">Tarski-Class</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c4">B</font> )
 <b>by </b><i><a class="txt" href="classes1.html#E1:21_1"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="classes1.html#E2:21_1_1_1"><i><font color="Green" title="E24">A9</font></i></a>, <a class="ref" href="classes1.html#T18" target="_self" title="CLASSES1:th.18">Th18</a></i>;<br/><a NAME="E5:21_1_1_1"/><b>then </b>
 <a href="classes1.html#K3" title="CLASSES1:func.3">Tarski-Class</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K3" title="CLASSES1:func.3">Tarski-Class</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c4">B</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E6:21_1_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K3" title="CLASSES1:func.3">Tarski-Class</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c4">B</font>
 <b>by </b><i><a class="txt" href="classes1.html#E3:21_1"><i><font color="Green" title="E19">A4</font></i></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><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" title="b1">C</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  holds <font color="Maroon" title="c4">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">C</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>  <a href="classes1.html#K3" title="CLASSES1:func.3">Tarski-Class</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c4">B</font></span><br/><a NAME="E2:21_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">B</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#T42" title="ORDINAL1:th.42">ORDINAL1:42</a></i>;<br/><a NAME="E3:21_1_1_2"/><b>then </b>
 <a href="classes1.html#K3" title="CLASSES1:func.3">Tarski-Class</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <font color="Maroon" title="c1">X</font> :  ex <font color="Olive" title="b2">C</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Olive" title="b2">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K3" title="CLASSES1:func.3">Tarski-Class</a> <font color="Maroon" title="c1">X</font>,<font color="Olive" title="b2">C</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="classes1.html#E5:21_1_1"><i><font color="Green" title="E22">A7</font></i></a>, <a class="ref" href="classes1.html#T12" target="_self" title="CLASSES1:th.12">Th12</a></i>;<br/><b>hence </b><a NAME="E4:21_1_1_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K3" title="CLASSES1:func.3">Tarski-Class</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c4">B</font>
 <b>by </b><i><a class="txt" href="classes1.html#E3:21_1"><i><font color="Green" title="E19">A4</font></i></a>, <a class="txt" href="classes1.html#E3:21_1_1"><i><font color="Green" title="E21">A6</font></i></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="E8:21_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K3" title="CLASSES1:func.3">Tarski-Class</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c4">B</font>
 <b>by </b><i><a class="txt" href="classes1.html#E6:21_1_1"><i><font color="Green" title="E23">A8</font></i></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="E5:21_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K3" title="CLASSES1:func.3">Tarski-Class</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c4">B</font>
 <b>by </b><i><a class="txt" href="classes1.html#E3:21_1"><i><font color="Green" title="E19">A4</font></i></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="E2:21"/>
for <font color="Olive" title="b1">B</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">B</font>]
 
<b>from </b><i><a class="ref" href="ordinal1.html#S2" title="ORDINAL1:sch.2">ORDINAL1:sch 2</a>(<a class="txt" href="classes1.html#E1:21"><i><font color="Green" title="E16">A1</font></i></a>);<br/></i>
<b>hence </b><a NAME="E3:21"/>
( <font color="Maroon" title="c2">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">B</font> implies  <a href="classes1.html#K3" title="CLASSES1:func.3">Tarski-Class</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K3" title="CLASSES1:func.3">Tarski-Class</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">B</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
