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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>



<b>assume </b><a NAME="E1:23"/><i><font color="Green">E18</font></i>: 
 <a href="classes1.html#K3" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="classes1.html#K3" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:23"/>
(  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>
<a NAME="E3:23"/><b>then </b><i><font color="Green">E19</font></i>: 
(  <a href="classes1.html#K3" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>1</sub></font>,<a href="xboole_0.html#K1">{}</a>  <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K3" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp;  <a href="classes1.html#K3" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>1</sub></font>,<a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="txt" href="classes1.html#E15"><i><font color="Green">Lemma10</font></i></a>, <a class="ref" href="classes1.html#T19" target="_self">Th19</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:23"/>
 <a href="classes1.html#K3" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="classes1.html#R1" target="_self">is_Tarski-Class_of</a> <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:23_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K3" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="classes1.html#E3:23"><i><font color="Green">E19</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="classes1.html#D3" target="_self">CLASSES1:def 3</a><br/>

<a NAME="E2:23_1"/><i><font color="Green">E20</font></i>: 
 <a href="classes1.html#K3" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>1</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K3" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>2</sub></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">b<sub>1</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K3" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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> <span class="p3">(<span class="default"><a href="classes1.html#K3" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></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">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="classes1.html#E15"><i><font color="Green">Lemma10</font></i></a></i>;<br/>
<a NAME="E3:23_1"/>
 <a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>1</sub></font> <a href="classes1.html#R1" target="_self">is_Tarski-Class_of</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="classes1.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E4:23_1"/><b>then </b><i><font color="Green">E21</font></i>: 
 <a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>1</sub></font> <a href="classes1.html#NR1" target="_self">is_Tarski-Class</a> 
 
<b>by </b><i><a class="ref" href="classes1.html#D3" target="_self">Def3</a></i>;<br/>
<b>thus </b><a NAME="E5:23_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K3" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K3" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="classes1.html#D1" target="_self">CLASSES1:def 1</a>,<a class="ref" href="classes1.html#D2" target="_self">CLASSES1:def 2</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:23_1_1"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K3" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<a NAME="E2:23_1_1"/>
 <a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>1</sub></font> <a href="classes1.html#R1" target="_self">is_Tarski-Class_of</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="classes1.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E3:23_1_1"/><b>then </b>
 <a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>1</sub></font> <a href="classes1.html#NR1" target="_self">is_Tarski-Class</a> 
 
<b>by </b><i><a class="ref" href="classes1.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E4:23_1_1"/><b>then </b>
 <a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>1</sub></font> is <a href="classes1.html#V1" target="_self">subset-closed</a>
 
<b>by </b><i><a class="ref" href="classes1.html#D2" target="_self">Def2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="classes1.html#E1:23_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="classes1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E6:23_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K3" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="classes1.html#E1:23_1_1"><i><font color="Green">E22</font></i></a></i>;<br/>
<a NAME="E7:23_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>1</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K3" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>2</sub></font> ) </span>}</span> 
 
;<br/>
<a NAME="E8:23_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>1</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K3" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>2</sub></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">b<sub>1</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K3" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E9:23_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K3" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="classes1.html#E1:23"><i><font color="Green">E18</font></i></a>, <a class="txt" href="classes1.html#E2:23_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E6:23_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K3" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> holds <br/> <a href="zfmisc_1.html#K1">bool</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K3" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:23_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K3" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:23_1_2"/><b>then </b>
 <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>3</sub></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">b<sub>1</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K3" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> </span>}</span> 
 
;<br/>
<a NAME="E3:23_1_2"/><b>then </b>
 <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>1</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K3" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>2</sub></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">b<sub>1</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K3" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:23_1_2"/>
 <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K3" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="classes1.html#E1:23"><i><font color="Green">E18</font></i></a>, <a class="txt" href="classes1.html#E2:23_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E7:23_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K3" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E8:23_1"/><i><font color="Green">E23</font></i>: 
not <font color="Maroon">c<sub>3</sub></font>, <a href="classes1.html#K3" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R2">are_equipotent</a> 
 ;<br/>

<a NAME="E9:23_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="classes1.html#E7:23_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E10:23_1"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font>, <a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R2">are_equipotent</a>  or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="classes1.html#E4:23_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="classes1.html#D2" target="_self">Def2</a></i>;<br/>
<b>hence </b><a NAME="E11:23_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K3" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="classes1.html#E1:23"><i><font color="Green">E18</font></i></a>, <a class="txt" href="classes1.html#E7:23_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="classes1.html#E8:23_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="classes1.html#T13" target="_self">Th13</a>, <a class="ref" href="card_1.html#T44">CARD_1:44</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:23"/>
(  <a href="classes1.html#K3" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K3" target="_self">Tarski-Class</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="classes1.html#D4" target="_self">Def4</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>


</div>
