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

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

<b>set </b><font color="Maroon">M</font> =  <a href="card_1.html#K1">Card</a> <font color="Maroon">A</font>;<br/>
<a NAME="E1:12"/>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">A</font>,<font color="Maroon">A</font> <a href="wellord2.html#R2">are_equipotent</a> 
 
<b>by </b><i><a class="ref" href="card_1.html#D5">CARD_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:12"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">A</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="wellord2.html#D4">WELLORD2:def 4</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>not <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K3">Union</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">X</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:12"/><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">X</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">A</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>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="ordinal1.html#NM3">T-Sequence</a> <b>by </b><i>, <a class="ref" href="ordinal1.html#D7">ORDINAL1:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="ordinal2.html#NM1">Ordinal-Sequence</a> <b>by </b><i>, <a class="ref" href="ordinal2.html#D8">ORDINAL2:def 8</a></i>;<br/>
<b>set </b><font color="Maroon">R</font> =  <a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">X</font>;<br/>
<b>set </b><font color="Maroon">B</font> =  <a href="wellord2.html#K2">order_type_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">X</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">phi</font> =  <a href="wellord1.html#K3">canonical_isomorphism_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <span class="p2">(<span class="default"><a href="wellord2.html#K2">order_type_of</a> <span class="p3">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">X</font></span>)</span>;<br/>
<b>take </b>
 <a href="wellord2.html#K2">order_type_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">X</font></span>)</span>
;<br/>

<a NAME="E8:12"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_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>thus </b><a NAME="E1:12_1"/>
(  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">A</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E9:12"/><b>then </b><i><font color="Green">E26</font></i>: 
(  <a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">X</font> is <a href="wellord1.html#V2">well-ordering</a> &amp;  <a href="wellord2.html#K1">RelIncl</a> <span class="p1">(<span class="default"><a href="wellord2.html#K2">order_type_of</a> <span class="p2">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">X</font></span>)</span></span>)</span> is <a href="wellord1.html#V2">well-ordering</a> )
 
<b>by </b><i><a class="ref" href="wellord2.html#T9">WELLORD2:9</a></i>;<br/>
<a NAME="E10:12"/><b>then </b>
 <a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">X</font>, <a href="wellord2.html#K1">RelIncl</a> <span class="p1">(<span class="default"><a href="wellord2.html#K2">order_type_of</a> <span class="p2">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="wellord1.html#R4">are_isomorphic</a> 
 
<b>by </b><i><a class="ref" href="wellord2.html#D2">WELLORD2:def 2</a></i>;<br/>
<a NAME="E11:12"/><b>then </b>
 <a href="wellord2.html#K1">RelIncl</a> <span class="p1">(<span class="default"><a href="wellord2.html#K2">order_type_of</a> <span class="p2">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">X</font></span>)</span></span>)</span>, <a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">X</font> <a href="wellord1.html#R4">are_isomorphic</a> 
 
<b>by </b><i><a class="ref" href="wellord1.html#T50">WELLORD1:50</a></i>;<br/>
<a NAME="E12:12"/><b>then </b><i><font color="Green">E42</font></i>: 
(  <a href="wellord1.html#K3">canonical_isomorphism_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <span class="p2">(<span class="default"><a href="wellord2.html#K2">order_type_of</a> <span class="p3">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">X</font></span>)</span> <a href="wellord1.html#R3">is_isomorphism_of</a>  <a href="wellord2.html#K1">RelIncl</a> <span class="p1">(<span class="default"><a href="wellord2.html#K2">order_type_of</a> <span class="p2">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">X</font></span>)</span></span>)</span>, <a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">X</font> &amp;  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <span class="p2">(<span class="default"><a href="wellord2.html#K2">order_type_of</a> <span class="p3">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="wellord2.html#K2">order_type_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">X</font></span>)</span> &amp;  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> )
 
<b>by </b><i>, <a class="ref" href="wellord1.html#D9">WELLORD1:def 9</a>, <a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/>
<a NAME="E13:12"/><b>then </b><i><font color="Green">E43</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="wellord1.html#K3">canonical_isomorphism_of</a> <span class="p2">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <span class="p3">(<span class="default"><a href="wellord2.html#K2">order_type_of</a> <span class="p4">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="wellord2.html#K2">order_type_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">X</font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="wellord1.html#K3">canonical_isomorphism_of</a> <span class="p2">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <span class="p3">(<span class="default"><a href="wellord2.html#K2">order_type_of</a> <span class="p4">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp;  <a href="wellord1.html#K3">canonical_isomorphism_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <span class="p2">(<span class="default"><a href="wellord2.html#K2">order_type_of</a> <span class="p3">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">X</font></span>)</span> is <a href="funct_1.html#V2">one-to-one</a> &amp; ( for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="wellord2.html#K1">RelIncl</a> <span class="p1">(<span class="default"><a href="wellord2.html#K2">order_type_of</a> <span class="p2">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">X</font></span>)</span></span>)</span> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="wellord2.html#K2">order_type_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">X</font></span>)</span> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="wellord2.html#K2">order_type_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">X</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wellord1.html#K3">canonical_isomorphism_of</a> <span class="p4">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <span class="p5">(<span class="default"><a href="wellord2.html#K2">order_type_of</a> <span class="p0">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wellord1.html#K3">canonical_isomorphism_of</a> <span class="p4">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <span class="p5">(<span class="default"><a href="wellord2.html#K2">order_type_of</a> <span class="p0">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">X</font> ) ) ) )
 
<b>by </b><i><a class="ref" href="wellord1.html#D7">WELLORD1:def 7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">phi</font> =  <a href="wellord1.html#K3">canonical_isomorphism_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <span class="p2">(<span class="default"><a href="wellord2.html#K2">order_type_of</a> <span class="p3">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">X</font></span>)</span> as   <a href="ordinal1.html#NM3">T-Sequence</a> <b>by </b><i><a class="ref" href="ordinal1.html#D7">ORDINAL1:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">phi</font> = <font color="Maroon">phi</font> as   <a href="ordinal2.html#NM1">Ordinal-Sequence</a> <b>by </b><i>, , <a class="ref" href="ordinal2.html#D8">ORDINAL2:def 8</a></i>;<br/>
<b>thus </b><a NAME="E16:12"/>
 <a href="wellord2.html#K2">order_type_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">X</font></span>)</span> <a href="ordinal1.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="wellord2.html#T17">WELLORD2:17</a></i>;<br/>

<a NAME="E17:12"/><i><font color="Green">E44</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">phi</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="wellord2.html#K2">order_type_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">X</font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">phi</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, , , <a class="ref" href="relat_1.html#T45">RELAT_1:45</a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">xi</font> = <font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">phi</font> as   <a href="ordinal1.html#NM3">T-Sequence</a> <b>by </b><i><a class="ref" href="ordinal1.html#D7">ORDINAL1:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">xi</font> = <font color="Maroon">xi</font> as   <a href="ordinal2.html#NM1">Ordinal-Sequence</a> <b>by </b><i>, <a class="ref" href="ordinal2.html#D8">ORDINAL2:def 8</a></i>;<br/>
<b>take </b>
<font color="Maroon">xi</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="zfrefle1.html#D5">ZFREFLE1:def 5</a><br/>

<b>thus </b><a NAME="E20:12"/>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">xi</font> <a href="hidden.html#R1">=</a>  <a href="wellord2.html#K2">order_type_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">X</font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">xi</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 <b>by </b><i>, , , <a class="ref" href="relat_1.html#T45">RELAT_1:45</a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>

<b>thus </b><a NAME="E21:12"/>
<font color="Maroon">xi</font> is <a href="ordinal2.html#V2">increasing</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="ordinal1.html#NM2">Ordinal</a>, <font color="Maroon">b</font> be   <a href="ordinal1.html#NM2">Ordinal</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal2.html#D16">ORDINAL2:def 16</a><br/>

<b>assume </b><a NAME="E1:12_2"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">xi</font> )
 ;<br/>

<a NAME="E2:12_2"/><b>then </b><i><font color="Green">E46</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">xi</font> &amp; <font color="Maroon">a</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">xi</font> is <a href="funct_1.html#V2">one-to-one</a> )
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T46">FUNCT_1:46</a>, <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E3:12_2"/><b>then </b><i><font color="Green">E47</font></i>: 
( <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="wellord2.html#K1">RelIncl</a> <span class="p1">(<span class="default"><a href="wellord2.html#K2">order_type_of</a> <span class="p2">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">X</font></span>)</span></span>)</span> &amp; <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 
<b>by </b><i>, , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a>, <a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>, <font color="Maroon">b'</font> = <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> as   <a href="ordinal1.html#NM2">Ordinal</a> ;<br/>
<b>reconsider </b><font color="Maroon">a''</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a'</font>, <font color="Maroon">b''</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b'</font> as   <a href="ordinal1.html#NM2">Ordinal</a> ;<br/>
<a NAME="E6:12_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">X</font>
 
<b>by </b><i>, , <a class="ref" href="wellord1.html#D7">WELLORD1:def 7</a></i>;<br/>
<a NAME="E7:12_2"/><b>then </b>
<font color="Maroon">a'</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">b'</font>
 
<b>by </b><i>, <a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/>
<a NAME="E8:12_2"/><b>then </b>
<font color="Maroon">a'</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">b'</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/>
<a NAME="E9:12_2"/><b>then </b><i><font color="Green">E49</font></i>: 
( <font color="Maroon">a'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b'</font> &amp; not <font color="Maroon">b''</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K3">Union</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">b'</font></span>)</span> &amp; <font color="Maroon">a''</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b''</font> )
 
<b>by </b><i>, , , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a>, <a class="ref" href="ordinal1.html#T21">ORDINAL1:21</a></i>;<br/>
<a NAME="E10:12_2"/><b>then </b>
( <font color="Maroon">a''</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">b'</font></span>)</span> &amp;  <a href="card_3.html#K3">Union</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">b'</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">b'</font></span>)</span></span>)</span> )
 
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T73">FUNCT_1:73</a></i>;<br/>
<a NAME="E11:12_2"/><b>then </b>
( <font color="Maroon">a''</font> <a href="ordinal1.html#R1">c=</a>  <a href="card_3.html#K3">Union</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">b'</font></span>)</span> &amp;  <a href="card_3.html#K3">Union</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">b'</font></span>)</span> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">b''</font> )
 
<b>by </b><i>, <a class="ref" href="ordinal1.html#T26">ORDINAL1:26</a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E12:12_2"/><b>then </b><i><font color="Green">E50</font></i>: 
( <font color="Maroon">a''</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">b''</font> &amp; <font color="Maroon">a''</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> &amp; <font color="Maroon">b''</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E13:12_2"/><b>then </b>
<font color="Maroon">a''</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">b''</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/>
<b>hence </b><a NAME="E14:12_2"/>
<font color="Maroon">xi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">xi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="ordinal1.html#T21">ORDINAL1:21</a></i>;<br/>


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

<b>thus </b><a NAME="E22:12"/>
<font color="Maroon">A</font> <a href="ordinal1.html#R1">c=</a>  <a href="ordinal2.html#K4">sup</a> <font color="Maroon">xi</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3"><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="E1:12_3"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</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="E3:12_3"/><i><font color="Green">E52</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/>
<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y</font> = <font color="Maroon">y</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_3_1_1"/>
(  not <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K3">Union</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">y</font></span>)</span> or <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K3">Union</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">y</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_3_1_1_1"><b>suppose </b></a><a NAME="E1:12_3_1_1_1"/>
not <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K3">Union</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">y</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:12_3_1_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, , </i>;<br/><b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:12_3_1_1_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="wellord2.html#K2">order_type_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">X</font></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:12_3_1_1_1"/>
( <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> &amp; <font color="Maroon">xi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">xi</font> )
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E6:12_3_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K4">sup</a> <font color="Maroon">xi</font>
 <b>by </b><i><a class="ref" href="ordinal2.html#T27">ORDINAL2:27</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_3_1_1_2"><b>suppose </b></a><a NAME="E1:12_3_1_1_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K3">Union</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">y</font></span>)</span>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:12_3_1_1_2"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">y</font></span>)</span> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">z</font> = <font color="Maroon">z</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/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> );<br/><a NAME="E5:12_3_1_1_2"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/><a NAME="E6:12_3_1_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E7:12_3_1_1_2"/><b>then </b><i><font color="Green">E55</font></i>: 
 ex <font color="Olive">C</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">C</font>]
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">C</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E9:12_3_1_1_2"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">C</font>] &amp; ( for <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">B</font>] holds <br/><font color="Maroon">C</font> <a href="ordinal1.html#R1">c=</a> <font color="Olive">B</font> ) )
 <b>from </b><i><a class="ref" href="ordinal1.html#S1">ORDINAL1:sch 1</a>();<br/></i><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3_1_1_2_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:12_3_1_1_2_1"/>
<font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">A</font>
 <b>by </b><i>, , , <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/><b>assume </b><a NAME="E2:12_3_1_1_2_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K3">Union</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">C</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:12_3_1_1_2_1"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">C</font></span>)</span> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="txt" href="card_5.html#E1"><i><font color="Green">Lemma22</font></i></a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/><b>reconsider </b><font color="Maroon">fa</font> = <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>, <font color="Maroon">fc</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">C</font>, <font color="Maroon">fy</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> as   <a href="ordinal1.html#NM2">Ordinal</a> ;<br/><a NAME="E7:12_3_1_1_2_1"/>
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">C</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fa</font> &amp; <font color="Maroon">fc</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fa</font> )
 <b>by </b><i><a class="txt" href="card_5.html#E1"><i><font color="Green">Lemma22</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a>, <a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/><a NAME="E8:12_3_1_1_2_1"/><b>then </b><i><font color="Green">E61</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Maroon">fy</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> )
 <b>by </b><i>, <a class="txt" href="card_5.html#E1"><i><font color="Green">Lemma22</font></i></a>, <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E9:12_3_1_1_2_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> &amp; not <font color="Maroon">C</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">a</font> )
 <b>by </b><i>, <a class="ref" href="ordinal1.html#T7">ORDINAL1:7</a>, <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/><b>hence </b><a NAME="E10:12_3_1_1_2_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="card_5.html#E3:12"><i><font color="Green">E23</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E11:12_3_1_1_2"/><b>then </b>
<font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E13:12_3_1_1_2"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="wellord2.html#K2">order_type_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">X</font></span>)</span> &amp; <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">z</font> = <font color="Maroon">z</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="txt" href="card_5.html#E1"><i><font color="Green">Lemma22</font></i></a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/><b>reconsider </b><font color="Maroon">xz</font> = <font color="Maroon">xi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> as   <a href="ordinal1.html#NM2">Ordinal</a> ;<br/><a NAME="E16:12_3_1_1_2"/>
( <font color="Maroon">xz</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">C</font> &amp; <font color="Maroon">xz</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">xi</font> )
 <b>by </b><i>, <a class="txt" href="card_5.html#E1"><i><font color="Green">Lemma22</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E17:12_3_1_1_2"/><b>then </b>
( <font color="Maroon">xz</font> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K4">sup</a> <font color="Maroon">xi</font> &amp; <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">xz</font> )
 <b>by </b><i>, , <a class="ref" href="ordinal2.html#T27">ORDINAL2:27</a></i>;<br/><b>hence </b><a NAME="E18:12_3_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K4">sup</a> <font color="Maroon">xi</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:12_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K4">sup</a> <font color="Maroon">xi</font>
 ;<br/>


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

<a NAME="E23:12"/>
 <a href="ordinal2.html#K3">sup</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="ordinal2.html#T26">ORDINAL2:26</a></i>;<br/>
<b>hence </b><a NAME="E24:12"/>
 <a href="ordinal2.html#K4">sup</a> <font color="Maroon">xi</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="ordinal2.html#T30">ORDINAL2:30</a></i>;<br/>


</div>
