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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] <b>means </b>for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Olive">X</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E1:39"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st ( for <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">A</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:39_1"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B</font>]
 ;<br/>

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E2:39_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">A</font>
 ;<br/>

<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="E3:39_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<div><i><font color="Green">E32</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">A</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a> 
 ;<br/><b>then consider </b><font color="Maroon">B</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E3:39_1_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">B</font> )
 <b>by </b><i><a class="txt" href="classes1.html#E4:39_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="classes1.html#E1:39_1"><i><font color="Green">E26</font></i></a>, </i>;<br/><a NAME="E4:39_1_1"/>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="txt" href="classes1.html#E3:39_1"><i><font color="Green">E31</font></i></a>, </i>;<br/><b>hence </b><a NAME="E5:39_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="txt" href="classes1.html#E1:39_1_1"><i><font color="Green">E33</font></i></a>, , , </i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_1_2"/>
not <font color="Maroon">A</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a> 
 ;<br/><b>then consider </b><font color="Maroon">B</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E3:39_1_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T42">ORDINAL1:42</a></i>;<br/><a NAME="E4:39_1_2"/>
( <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">B</font> )
 <b>by </b><i><a class="txt" href="classes1.html#E4:39_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="classes1.html#E3:39_1_2"><i><font color="Green">E35</font></i></a>, , <a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a></i>;<br/><a NAME="E5:39_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="txt" href="classes1.html#E3:39_1"><i><font color="Green">E31</font></i></a>, </i>;<br/><b>hence </b><a NAME="E6:39_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="txt" href="classes1.html#E3:39_1_2"><i><font color="Green">E35</font></i></a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:39_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:39"/>
for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">A</font>]
 
<b>from </b><i><a class="ref" href="ordinal1.html#S2">ORDINAL1:sch 2</a>();<br/></i>
<b>hence </b><a NAME="E3:39"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">A</font>]
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a><br/>


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