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

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

<b>consider </b><font color="Maroon">Class</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:22"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Class</font>
 <b>and </b><br/><a NAME="E3:22"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">Y</font>, <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Class</font> &amp; <font color="Olive">Z</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> holds <br/><font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Class</font>
 <b>and </b><br/><a NAME="E4:22"/>
for <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Class</font> holds <br/> <a href="zfmisc_1.html#K1">bool</a> <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Class</font>
 <b>and </b><br/><a NAME="E5:22"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Olive">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Class</font> or <font color="Olive">Y</font>,<font color="Maroon">Class</font> <a href="wellord2.html#R2" target="_self">are_equipotent</a>  or <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Class</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T136">ZFMISC_1:136</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> is   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>
<b>consider </b><font color="Maroon">ON</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:22"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ON</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Class</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E8:22"/>
for <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ON</font> holds <br/>( <font color="Olive">Y</font> is   <a href="ordinal1.html#NM2">Ordinal</a> &amp; <font color="Olive">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">ON</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:22_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ON</font>
 ;<br/>

<b>hence </b><a NAME="E2:22_1"/>
<font color="Maroon">Y</font> is   <a href="ordinal1.html#NM2">Ordinal</a>
 <b>by </b><i><a class="txt" href="wellord2.html#E3:22"><i><font color="Green">E22</font></i></a></i>;<br/>

<b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">Y</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="txt" href="wellord2.html#E3:22"><i><font color="Green">E22</font></i></a>, <a class="txt" href="wellord2.html#E5:22"><i><font color="Green">E23</font></i></a></i>;<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="E4:22_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<a NAME="E5:22_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">B</font> = <font color="Maroon">x</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E7:22_1"/>
( <font color="Maroon">B</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Class</font> )
 
<b>by </b><i><a class="txt" href="wellord2.html#E3:22"><i><font color="Green">E22</font></i></a>, <a class="txt" href="wellord2.html#E5:22"><i><font color="Green">E23</font></i></a>, <a class="txt" href="wellord2.html#E7:22"><i><font color="Green">E24</font></i></a>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E8:22_1"/><b>then </b>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Class</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:22_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ON</font>
 <b>by </b><i><a class="txt" href="wellord2.html#E3:22"><i><font color="Green">E22</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">ON</font> = <font color="Maroon">ON</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="ref" href="ordinal1.html#T31">ORDINAL1:31</a></i>;<br/>
<a NAME="E10:22"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">ON</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Class</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2"><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>thus </b><a NAME="E1:22_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ON</font> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Class</font> )
 <b>by </b><i><a class="txt" href="wellord2.html#E3:22"><i><font color="Green">E22</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:22"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">ON</font>,<font color="Maroon">Class</font> <a href="wellord2.html#R2" target="_self">are_equipotent</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:22_3"/>
not <font color="Maroon">ON</font>,<font color="Maroon">Class</font> <a href="wellord2.html#R2" target="_self">are_equipotent</a> 
 ;<br/>

<a NAME="E2:22_3"/><b>then </b>
<font color="Maroon">ON</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Class</font>
 
<b>by </b><i><a class="txt" href="wellord2.html#E2:22"><i><font color="Green">E21</font></i></a>, <a class="txt" href="wellord2.html#E5:22"><i><font color="Green">E23</font></i></a></i>;<br/>
<a NAME="E3:22_3"/><b>then </b>
<font color="Maroon">ON</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ON</font>
 
<b>by </b><i><a class="txt" href="wellord2.html#E3:22"><i><font color="Green">E22</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:22_3"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<a NAME="E12:22"/>
(  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Maroon">ON</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">ON</font> &amp;  <a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Maroon">ON</font> is <a href="wellord1.html#V2">well-ordering</a> )
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">R</font> being   <a href="relat_1.html#NM1">Relation</a><b> such that </b><br/><a NAME="E14:22"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">R</font> <a href="wellord1.html#R2">well_orders</a> <font color="Maroon">Class</font>
 <b>by </b><i><a class="txt" href="wellord2.html#E7:22"><i><font color="Green">E24</font></i></a>, </i>;<br/>
<b>set </b><font color="Maroon">Q</font> = <font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">Class</font>;<br/>
<a NAME="E15:22"/><i><font color="Green">E34</font></i>: 
(  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">Class</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Class</font> &amp; <font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">Class</font> is <a href="wellord1.html#V2">well-ordering</a> )
 
<b>by </b><i>, </i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span>;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E17:22"/><i><font color="Green">E48</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp; ( 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> <font color="Maroon">X</font> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>) ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S3">FUNCT_1:sch 3</a>();<br/></i>
<a NAME="E18:22"/><i><font color="Green">E49</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Class</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_4"><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="E1:22_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:22_4"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:22_4"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:22_4"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</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="E1:22_4_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<b>hence </b><a NAME="E2:22_4_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:22_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Class</font>
 <b>by </b><i><a class="ref" href="wellord2.html#D1" target="_self">Def1</a>, , , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E19:22"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">X</font>, <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="wellord2.html#R2" target="_self">are_equipotent</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_5"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">f</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="wellord2.html#D4" target="_self">WELLORD2:def 4</a><br/>

<b>thus </b><a NAME="E1:22_5"/>
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_5_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="funct_1.html#D8">FUNCT_1:def 8</a><br/><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:22_5_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 ;<br/>

<a NAME="E2:22_5_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</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> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:22_5_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T6">ZFMISC_1:6</a></i>;<br/>


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

<b>thus </b><a NAME="E2:22_5"/>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E20:22"/><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">Class</font></span>)</span> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> is <a href="wellord1.html#V2">well-ordering</a>
 
<b>by </b><i>, <a class="ref" href="wellord1.html#T32">WELLORD1:32</a></i>;<br/>
<a NAME="E21:22"/>
 <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">Class</font></span>)</span> <a href="wellord1.html#K2">|_2</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i>, , <a class="ref" href="wellord1.html#T39">WELLORD1:39</a></i>;<br/>
<b>hence </b><a NAME="E22:22"/>
 ex <font color="Olive">R</font> being  <a href="relat_1.html#NM1">Relation</a> st <font color="Olive">R</font> <a href="wellord1.html#R2">well_orders</a> <font color="Maroon">X</font>
 <b>by </b><i>, , </i>;<br/>


</div>
