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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>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/> not <font color="Olive">X</font>, <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">a<sub>1</sub></font> <a href="tarski.html#R2">are_equipotent</a> ;<br/>
<a NAME="E1:47"/><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="47_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="E1:47_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="E2:47_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/>

<div><i><font color="Green">E31</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">B</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><a NAME="E1:47_1_1"/><i><font color="Green">E32</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>
 ;<br/><a NAME="E2:47_1_1"/>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a></i>;<br/><a NAME="E3:47_1_1"/><b>then </b>
( <font color="Maroon">B</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A</font> &amp;  <a href="classes1.html#K4" target="_self">Rank</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">B</font></span>)</span> )
 <b>by </b><i><a class="txt" href="classes1.html#E1:47_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/><a NAME="E4:47_1_1"/><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> &amp; not  <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">B</font>, <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">A</font> <a href="tarski.html#R2">are_equipotent</a>  &amp;  <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</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#E3:47"><i><font color="Green">E37</font></i></a>, , , <a class="ref" href="card_1.html#T29">CARD_1:29</a></i>;<br/><b>hence </b><a NAME="E5:47_1_1"/>
not <font color="Maroon">X</font>, <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">A</font> <a href="tarski.html#R2">are_equipotent</a> 
 <b>by </b><i><a class="ref" href="card_1.html#T44">CARD_1:44</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_1_2"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; ( for <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  holds <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ordinal1.html#K1">succ</a> <font color="Olive">B</font> ) )
 ;<br/><a NAME="E2:47_1_2"/><b>then </b>
<font color="Maroon">A</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#T42">ORDINAL1:42</a></i>;<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="E4:47_1_2"/><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#E3:47"><i><font color="Green">E37</font></i></a>, , </i>;<br/><a NAME="E5:47_1_2"/><i><font color="Green">E35</font></i>: 
not <font color="Maroon">X</font>, <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">B</font> <a href="tarski.html#R2">are_equipotent</a> 
 <b>by </b><i><a class="txt" href="classes1.html#E2:47"><i><font color="Green">E36</font></i></a>, </i>;<br/><a NAME="E6:47_1_2"/>
 <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">B</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>, </i>;<br/><a NAME="E7:47_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> &amp;  <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">A</font> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E8:47_1_2"/>
not <font color="Maroon">X</font>, <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">A</font> <a href="tarski.html#R2">are_equipotent</a> 
 <b>by </b><i>, <a class="ref" href="card_1.html#T44">CARD_1:44</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:47_1"/>
not <font color="Maroon">X</font>, <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">A</font> <a href="tarski.html#R2">are_equipotent</a> 
 <b>by </b><i><a class="txt" href="classes1.html#E3:47"><i><font color="Green">E37</font></i></a>, , <a class="txt" href="classes1.html#E1:47_1"><i><font color="Green">E26</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:47"/><i><font color="Green">E36</font></i>: 
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>let </b><font color="Maroon">A</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E3:47"/><i><font color="Green">E37</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/>

<a NAME="E4:47"/><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">A</font> &amp; not <font color="Maroon">X</font>, <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">A</font> <a href="tarski.html#R2">are_equipotent</a>  )
 
<b>by </b><i><a class="txt" href="classes1.html#E2:47"><i><font color="Green">E36</font></i></a>, </i>;<br/>
<a NAME="E5:47"/><b>then </b>
(  <a href="card_1.html#K1">Card</a> <font color="Maroon">X</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">A</font></span>)</span> &amp;  <a href="card_1.html#K1">Card</a> <font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">A</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="card_1.html#T21">CARD_1:21</a>, <a class="ref" href="card_1.html#T27">CARD_1:27</a></i>;<br/>
<b>hence </b><a NAME="E6:47"/>
( not <font color="Maroon">X</font>, <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">A</font> <a href="tarski.html#R2">are_equipotent</a>  &amp;  <a href="card_1.html#K1">Card</a> <font color="Maroon">X</font> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">A</font></span>)</span> )
 <b>by </b><i><a class="txt" href="classes1.html#E2:47"><i><font color="Green">E36</font></i></a>, <a class="txt" href="classes1.html#E3:47"><i><font color="Green">E37</font></i></a>, <a class="ref" href="card_1.html#T13">CARD_1:13</a></i>;<br/>


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