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<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>] <b>means </b> <a href="classes2.html#K16" title="CLASSES2:func.16">UNIVERSE</a> <font color="Maroon" title="c1">A</font> is   <a href="classes2.html#NM1" title="CLASSES2:NM.1">Universe</a>;<br/>
<a NAME="E1:72_1_1"/><i><font color="Green" title="E51">A1</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 
<b>by </b><i><a class="txt" href="classes2.html#E82"><i><font color="Green" title="E50">Lm6</font></i></a></i>;<br/>
<a NAME="E2:72_1_1"/><i><font color="Green" title="E52">A2</font></i>: 
for <font color="Olive" title="b1">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">B</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[ <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">B</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">B</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E1:72_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">B</font>]
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c3">UU</font> =  <a href="classes2.html#K16" title="CLASSES2:func.16">UNIVERSE</a> <font color="Maroon" title="c2">B</font> as   <a href="classes2.html#NM1" title="CLASSES2:NM.1">Universe</a> ;<br/>
<a NAME="E3:72_1_1_1"/>
 <a href="classes2.html#K16" title="CLASSES2:func.16">UNIVERSE</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <font color="Maroon" title="c3">UU</font>
 
<b>by </b><i><a class="txt" href="classes2.html#E82"><i><font color="Green" title="E50">Lm6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:72_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[ <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">B</font>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:72_1_1"/><i><font color="Green" title="E53">A3</font></i>: 
for <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">A</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>  &amp; <font color="Olive" title="b1">A</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  &amp; ( for <font color="Olive" title="b2">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b2">B</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>2</sub></font>[<font color="Olive" title="b2">B</font>] ) holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">A</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">A</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:72_1_1_2"/><i><font color="Green" title="E54">A4</font></i>: 
( <font color="Maroon" title="c2">A</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>  &amp; <font color="Maroon" title="c2">A</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  )
 <b>and </b><br/><a NAME="E2:72_1_1_2"/>
for <font color="Olive" title="b1">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">B</font>]
 ;<br/>

<b>consider </b><font color="Maroon" title="c3">L</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a><b> such that </b><br/><a NAME="E4:72_1_1_2"/><i><font color="Green" title="E55">A5</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> &amp; ( for <font color="Olive" title="b1">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> holds <br/><font color="Maroon" title="c3">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive" title="b1">B</font>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S2" title="ORDINAL2:sch.2">ORDINAL2:sch 2</a>();<br/></i>
 <a href="classes2.html#K16" title="CLASSES2:func.16">UNIVERSE</a> <font color="Maroon" title="c2">A</font> = 
 <a href="classes2.html#K15" title="CLASSES2:func.15">Universe_closure</a> <span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c3">L</font></span>)</span>
<b>by </b><i><a class="txt" href="classes2.html#E1:72_1_1_2"><i><font color="Green" title="E54">A4</font></i></a>, <a class="txt" href="classes2.html#E4:72_1_1_2"><i><font color="Green" title="E55">A5</font></i></a>, <a class="txt" href="classes2.html#E82"><i><font color="Green" title="E50">Lm6</font></i></a></i>
<br/>.= 
 <a href="classes2.html#K15" title="CLASSES2:func.15">Universe_closure</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">L</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="card_3.html#D4" title="CARD_3:def.4">CARD_3:def 4</a></i>
;<br/>
<b>hence </b><a NAME="E6:72_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">A</font>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:72_1_1"/>
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>2</sub></font>[<font color="Olive" title="b1">A</font>]
 
<b>from </b><i><a class="ref" href="ordinal2.html#S1" title="ORDINAL2:sch.1">ORDINAL2:sch 1</a>(<a class="txt" href="classes2.html#E1:72_1_1"><i><font color="Green" title="E51">A1</font></i></a>, <a class="txt" href="classes2.html#E2:72_1_1"><i><font color="Green" title="E52">A2</font></i></a>, <a class="txt" href="classes2.html#E3:72_1_1"><i><font color="Green" title="E53">A3</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:72_1_1"/>
(  <a href="classes2.html#K16" title="CLASSES2:func.16">UNIVERSE</a> <font color="Maroon" title="c1">A</font> is <a href="classes2.html#V1" title="CLASSES2:attr.1">universal</a> &amp; not  <a href="classes2.html#K16" title="CLASSES2:func.16">UNIVERSE</a> <font color="Maroon" title="c1">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 ;<br/>


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