<?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>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<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>assume </b><a NAME="E1:9"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E2:9"/><b>then </b><i><font color="Green">E24</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> &amp;  <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> <font color="Maroon">A</font> )
 
<b>by </b><i><a class="ref" href="wellord2.html#T9">WELLORD2:9</a>, <a class="ref" href="wellord2.html#T17">WELLORD2:17</a></i>;<br/>
<a NAME="E3:9"/><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="E4:9"/><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="E5:9"/><b>then </b><i><font color="Green">E25</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="E6:9"/><b>then </b><i><font color="Green">E26</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>take </b>
<font color="Maroon">phi</font>
;<br/>

<b>thus </b><a NAME="E9:9"/>
<font color="Maroon">phi</font> <a href="hidden.html#R1">=</a>  <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>thus </b><a NAME="E10:9"/>
<font color="Maroon">phi</font> is <a href="ordinal2.html#V2">increasing</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><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="E1:9_1"/><i><font color="Green">E42</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">phi</font> )
 ;<br/>

<a NAME="E2:9_1"/><b>then </b><i><font color="Green">E43</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">phi</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> )
 
<b>by </b><i><a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</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/>
<a NAME="E4:9_1"/>
<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>
 
<b>by </b><i>, , , <a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/>
<a NAME="E5:9_1"/><b>then </b>
( <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> <font color="Maroon">X</font> &amp; <font color="Maroon">a'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <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></i>;<br/>
<a NAME="E6:9_1"/><b>then </b>
( <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> )
 
<b>by </b><i>, , , <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/>
<a NAME="E7:9_1"/><b>then </b>
( <font color="Maroon">a'</font> <a href="xboole_0.html#R2">c&lt;</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> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/>
<b>hence </b><a NAME="E8:9_1"/>
<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">phi</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="E11:9"/>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">phi</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">phi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> )
 <b>by </b><i>, <a class="ref" href="wellord1.html#D7">WELLORD1:def 7</a></i>;<br/>


</div>
