<?xml version="1.0"?>
<div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] <b>means </b> <a href="classes1.html#K2" target="_self">Tarski-Class</a> <font color="Maroon">X</font>,<font color="Maroon">a<sub>1</sub></font> is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">X</font></span>)</span>;<br/>
<a NAME="E1:15_1_1"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">X</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_1"><b>proof </b></a><div class="add">

<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="E1:15_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">X</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:15_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:15_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:15_1_1"/><b>then </b><i><font color="Green">E25</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1">{}</a> ]
 
<b>by </b><i/>;<br/>
<a NAME="E3:15_1_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="Maroon">S<sub>1</sub></font>[<font color="Olive">B</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1">succ</a> <font color="Olive">B</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:15_1_1_2"/>
 <a href="classes1.html#K2" target="_self">Tarski-Class</a> <font color="Maroon">X</font>,<font color="Maroon">B</font> is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">X</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">S</font> =  <a href="classes1.html#K2" target="_self">Tarski-Class</a> <font color="Maroon">X</font>,<font color="Maroon">B</font> as   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">X</font></span>)</span> ;<br/>
<b>set </b><font color="Maroon">Y</font> =  <a href="classes1.html#K2" target="_self">Tarski-Class</a> <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">B</font></span>)</span>;<br/>
<a NAME="E3:15_1_1_2"/>
 <a href="classes1.html#K2" target="_self">Tarski-Class</a> <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">B</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_2_1"><b>proof </b></a><div class="add">

<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="E1:15_1_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K2" target="_self">Tarski-Class</a> <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">B</font></span>)</span>
 ;<br/>

<a NAME="E2:15_1_1_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">X</font> :  ex <font color="Olive">v</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">X</font> st <br/>( <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Olive">u</font> <a href="tarski.html#R1">c=</a> <font color="Olive">v</font> ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Olive">v</font></span>)</span> where <font color="Olive">v</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">X</font> : <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">S</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">X</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:15_1_1_2_1"/><b>then </b><i><font color="Green">E29</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">X</font> :  ex <font color="Olive">v</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">X</font> st <br/>( <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Olive">u</font> <a href="tarski.html#R1">c=</a> <font color="Olive">v</font> ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Olive">v</font></span>)</span> where <font color="Olive">v</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">X</font> : <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> </span>}</span>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">S</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">X</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<div><i><font color="Green">E31</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_1_1_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">X</font> :  ex <font color="Olive">v</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">X</font> st <br/>( <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Olive">u</font> <a href="tarski.html#R1">c=</a> <font color="Olive">v</font> ) </span>}</span> 
 ;<br/><a NAME="E2:15_1_1_2_1_1"/><b>then </b>
 ex <font color="Olive">u</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">X</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">u</font> &amp;  ex <font color="Olive">v</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">X</font> st <br/>( <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Olive">u</font> <a href="tarski.html#R1">c=</a> <font color="Olive">v</font> ) )
 ;<br/><b>hence </b><a NAME="E3:15_1_1_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">X</font>
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_1_1_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Olive">v</font></span>)</span> where <font color="Olive">v</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">X</font> : <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> </span>}</span> 
 ;<br/><a NAME="E2:15_1_1_2_1_2"/><b>then </b>
 ex <font color="Olive">v</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">X</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Olive">v</font> &amp; <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 ;<br/><b>hence </b><a NAME="E3:15_1_1_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:15_1_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">X</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:15_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1">succ</a> <font color="Maroon">B</font>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:15_1_1"/><i><font color="Green">E32</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#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">B</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  &amp; ( for <font color="Olive">C</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">C</font> <a href="hidden.html#R2">in</a> <font color="Olive">B</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">C</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_3"><b>proof </b></a><div class="add">

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

<b>assume </b><b>that </b><br/><a NAME="E1:15_1_1_3"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">B</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  )
 <b>and </b><br/><a NAME="E2:15_1_1_3"/>
for <font color="Olive">C</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> holds <br/> <a href="classes1.html#K2" target="_self">Tarski-Class</a> <font color="Maroon">X</font>,<font color="Olive">C</font> is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">X</font></span>)</span>
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="ordinal1.html#NM2">Ordinal</a>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="classes1.html#K2" target="_self">Tarski-Class</a> <font color="Maroon">X</font>,<font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">L</font> being   <a href="ordinal1.html#NM3">T-Sequence</a><b> such that </b><br/><a NAME="E4:15_1_1_3"/><i><font color="Green">E34</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> &amp; ( for <font color="Olive">C</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> holds <br/><font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Olive">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">C</font>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S2">ORDINAL2:sch 2</a>();<br/></i>
<a NAME="E5:15_1_1_3"/>
(  <a href="classes1.html#K2" target="_self">Tarski-Class</a> <font color="Maroon">X</font>,<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">X</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">X</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">X</font> )
 
<b>by </b><i>, , , <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>hence </b><a NAME="E6:15_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">B</font>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:15_1_1"/>
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="ordinal2.html#S1">ORDINAL2:sch 1</a>(, , );<br/></i>
<b>hence </b><a NAME="E6:15_1_1"/>
 <a href="classes1.html#K2" target="_self">Tarski-Class</a> <font color="Maroon">X</font>,<font color="Maroon">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">X</font></span>)</span>
 ;<br/>


</div>
