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

<b>let </b><font color="Maroon">R</font> be   <a href="relat_1.html#NM1">Relation</a>;<br/>

<b>assume </b><a NAME="E1:13"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">R</font> is <a href="wellord1.html#V2">well-ordering</a>
 ;<br/>

<b>assume </b><a NAME="E2:13"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> holds <br/> ex <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Olive">a</font></span>)</span>, <a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Olive">A</font> <a href="wellord1.html#R4">are_isomorphic</a> 
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a> holds <br/> ( <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> iff <font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>, <a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Olive">A</font> <a href="wellord1.html#R4">are_isomorphic</a>  );<br/>
<a NAME="E3:13"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">b</font>, <font color="Olive">c</font>, <font color="Olive">d</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b</font>,<font color="Olive">c</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b</font>,<font color="Olive">d</font>] holds <br/><font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>proof </b></a><div class="add">

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





<b>assume </b><b>that </b><br/><a NAME="E1:13_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 <b>and </b><br/><a NAME="E2:13_1"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a> holds <br/> ( <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> iff <font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">b</font></span>)</span>, <a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Olive">A</font> <a href="wellord1.html#R4">are_isomorphic</a>  )
 <b>and </b><br/><a NAME="E3:13_1"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a> holds <br/> ( <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> iff <font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">b</font></span>)</span>, <a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Olive">B</font> <a href="wellord1.html#R4">are_isomorphic</a>  )
 ;<br/>

<b>consider </b><font color="Maroon">A</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E5:13_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">b</font></span>)</span>, <a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Maroon">A</font> <a href="wellord1.html#R4">are_isomorphic</a> 
 <b>by </b><i>, </i>;<br/>
<a NAME="E6:13_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> )
 
<b>by </b><i>, <a class="ref" href="wellord2.html#D1" target="_self">Def1</a>, </i>;<br/>
<b>hence </b><a NAME="E7:13_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:13"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> holds <br/> ex <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>,<font color="Olive">b</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:13_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 ;<br/>

<b>then consider </b><font color="Maroon">A</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E3:13_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">a</font></span>)</span>, <a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Maroon">A</font> <a href="wellord1.html#R4">are_isomorphic</a> 
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">A</font> as    <a href="hidden.html#M1">set</a>  ;<br/>
<b>take </b>
<font color="Maroon">b</font>
;<br/>

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

<b>thus </b><a NAME="E5:13_2"/>
( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> implies <font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">a</font></span>)</span>, <a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Maroon">B</font> <a href="wellord1.html#R4">are_isomorphic</a>  )
 <b>by </b><i/>;<br/>

<b>assume </b><a NAME="E6:13_2"/>
<font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">a</font></span>)</span>, <a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Maroon">B</font> <a href="wellord1.html#R4">are_isomorphic</a> 
 ;<br/>

<b>hence </b><a NAME="E7:13_2"/>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">H</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:13"/><i><font color="Green">E34</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> &amp; ( for <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b</font>,<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">b</font>] ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S2">FUNCT_1:sch 2</a>(, );<br/></i>
<a NAME="E7:13"/>
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">H</font> holds <br/><font color="Olive">a</font> is   <a href="ordinal1.html#NM2">Ordinal</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:13_3"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">H</font>
 ;<br/>

<b>then consider </b><font color="Maroon">c</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:13_3"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">A</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E5:13_3"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c</font></span>)</span>, <a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Maroon">A</font> <a href="wellord1.html#R4">are_isomorphic</a> 
 <b>by </b><i>, , <a class="ref" href="wellord2.html#D1" target="_self">Def1</a></i>;<br/>
<b>thus </b><a NAME="E6:13_3"/>
<font color="Maroon">b</font> is   <a href="ordinal1.html#NM2">Ordinal</a>
 <b>by </b><i>, <a class="ref" href="wellord2.html#D1" target="_self">Def1</a>, </i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">C</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E9:13"/><i><font color="Green">E50</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T36">ORDINAL1:36</a></i>;<br/>
<a NAME="E10:13"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <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">C</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E11:13"/><i><font color="Green">E52</font></i>: 
 <a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Maroon">C</font> is <a href="wellord1.html#V2">well-ordering</a>
 
<b>by </b><i/>;<br/>
<div><i><font color="Green">E53</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:13_4"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">H</font>
 ;<br/><b>then consider </b><font color="Maroon">b'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:13_4"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">b'</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b'</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>set </b><font color="Maroon">V</font> = <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">b'</font>;<br/><b>set </b><font color="Maroon">P</font> = <font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">b'</font></span>)</span>;<br/><b>consider </b><font color="Maroon">A</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E5:13_4"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">b'</font></span>)</span>, <a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Maroon">A</font> <a href="wellord1.html#R4">are_isomorphic</a> 
 <b>by </b><i>, , </i>;<br/><a NAME="E6:13_4"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i>, , </i>;<br/><b>let </b><font color="Maroon">c</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E7:13_4"/><i><font color="Green">E60</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c</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" target="_self">RelIncl</a> <font color="Maroon">C</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_4_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 ;<br/><a NAME="E2:13_4_1"/>
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <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">C</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:13_4_1"/><b>then </b><i><font color="Green">E62</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 <b>by </b><i>, <a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/><b>then reconsider </b><font color="Maroon">B</font> = <font color="Maroon">c</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="E5:13_4_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">B</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A</font>
 <b>by </b><i>, , , </i>;<br/><a NAME="E6:13_4_1"/><b>then </b>
<font color="Maroon">B</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">A</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><a NAME="E7:13_4_1"/><b>then </b><i><font color="Green">E64</font></i>: 
( <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <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">A</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="ordinal1.html#T21">ORDINAL1:21</a></i>;<br/><a NAME="E8:13_4_1"/><i><font color="Green">E65</font></i>: 
 <a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Maroon">A</font> is <a href="wellord1.html#V2">well-ordering</a>
 <b>by </b><i/>;<br/><a NAME="E9:13_4_1"/>
 <a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Maroon">A</font>,<font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">b'</font></span>)</span> <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="E10:13_4_1"/><b>then </b>
 <a href="wellord1.html#K3">canonical_isomorphism_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">b'</font></span>)</span></span>)</span> <a href="wellord1.html#R3">is_isomorphism_of</a>  <a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Maroon">A</font>,<font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">b'</font></span>)</span>
 <b>by </b><i><a class="txt" href="wellord2.html#E4:13"><i><font color="Green">E32</font></i></a>, <a class="ref" href="wellord1.html#D9">WELLORD1:def 9</a></i>;<br/><b>then consider </b><font color="Maroon">d</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:13_4_1"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <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> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">b'</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Maroon">A</font></span>)</span> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Maroon">A</font></span>)</span> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">b'</font></span>)</span></span>)</span> <a href="wellord1.html#K2">|_2</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> <span class="p3">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">b'</font></span>)</span></span>)</span> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">d</font></span>)</span> <a href="wellord1.html#R4">are_isomorphic</a>  )
 <b>by </b><i>, <a class="txt" href="wellord2.html#E4:13"><i><font color="Green">E32</font></i></a>, <a class="ref" href="wellord1.html#T61">WELLORD1:61</a></i>;<br/><a NAME="E13:13_4_1"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">b'</font> &amp; <font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> )
 <b>by </b><i>, <a class="ref" href="wellord1.html#T19">WELLORD1:19</a></i>;<br/><a NAME="E14:13_4_1"/><b>then </b><i><font color="Green">E68</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">d</font>,<font color="Maroon">b'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/><a NAME="E15:13_4_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Maroon">A</font></span>)</span> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">B</font>
 <b>by </b><i>, </i>;<br/><a NAME="E16:13_4_1"/><i><font color="Green">E70</font></i>: 
<span class="p1">(<span class="default"><a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Maroon">A</font></span>)</span> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Maroon">B</font>
 <b>by </b><i>, </i>;<br/><a NAME="E17:13_4_1"/><i><font color="Green">E71</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">b'</font></span>)</span></span>)</span> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">d</font>
 <b>by </b><i>, , , <a class="txt" href="wellord2.html#E6:13"><i><font color="Green">E34</font></i></a>, <a class="ref" href="wellord1.html#T35">WELLORD1:35</a></i>;<br/><a NAME="E18:13_4_1"/>
<font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">d</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">b'</font>
 <b>by </b><i>, , , <a class="txt" href="wellord2.html#E6:13"><i><font color="Green">E34</font></i></a>, <a class="ref" href="wellord2.html#T2" target="_self">Th2</a>, <a class="ref" href="wellord1.html#T37">WELLORD1:37</a></i>;<br/><a NAME="E19:13_4_1"/><b>then </b>
 <a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Maroon">B</font>,<font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">d</font></span>)</span> <a href="wellord1.html#R4">are_isomorphic</a> 
 <b>by </b><i>, <a class="ref" href="wellord2.html#T3" target="_self">Th3</a>, <a class="ref" href="wellord2.html#T4" target="_self">Th4</a>, <a class="ref" href="wellord2.html#T5" target="_self">Th5</a>, <a class="ref" href="wellord1.html#T29">WELLORD1:29</a></i>;<br/><a NAME="E20:13_4_1"/><b>then </b>
<font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">d</font></span>)</span>, <a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Maroon">B</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="E21:13_4_1"/><b>then </b>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font>
 <b>by </b><i>, <a class="txt" href="wellord2.html#E6:13"><i><font color="Green">E34</font></i></a></i>;<br/><b>hence </b><a NAME="E22:13_4_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">H</font>
 <b>by </b><i>, <a class="txt" href="wellord2.html#E6:13"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E9:13_4"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="txt" href="wellord2.html#E3:13"><i><font color="Green">E23</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E13:13"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">H</font> is   <a href="ordinal1.html#NM2">Ordinal</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_5"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_5_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E1:13_5_1"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Maroon">C</font></span>)</span> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">a</font> )
 ;<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="wellord2.html#E3:13"><i><font color="Green">E23</font></i></a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/><a NAME="E3:13_5_1"/>
<span class="p1">(<span class="default"><a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Maroon">C</font></span>)</span> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="txt" href="wellord2.html#E3:13"><i><font color="Green">E23</font></i></a>, </i>;<br/><b>hence </b><a NAME="E4:13_5_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">H</font> is   <a href="ordinal1.html#NM2">Ordinal</a>
 <b>by </b><i><a class="txt" href="wellord2.html#E3:13"><i><font color="Green">E23</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:13_5"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">H</font> is   <a href="ordinal1.html#NM2">Ordinal</a>
 <b>by </b><i><a class="ref" href="wellord2.html#D1" target="_self">Def1</a>, , <a class="txt" href="wellord2.html#E1:13"><i><font color="Green">E21</font></i></a>, <a class="txt" href="wellord2.html#E2:13"><i><font color="Green">E22</font></i></a>, <a class="ref" href="wellord1.html#T36">WELLORD1:36</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">A</font> =  <a href="relat_1.html#K2">rng</a> <font color="Maroon">H</font> as   <a href="ordinal1.html#NM2">Ordinal</a> ;<br/>
<a NAME="E15:13"/><i><font color="Green">E73</font></i>: 
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H</font> holds <br/><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font> is   <a href="ordinal1.html#NM2">Ordinal</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_6"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:13_6"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H</font>
 ;<br/>

<a NAME="E2:13_6"/><b>then </b>
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E3:13_6"/>
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> is   <a href="ordinal1.html#NM2">Ordinal</a>
 <b>by </b><i><a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon">A</font>
;<br/>

<b>take </b>
<font color="Maroon">H</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="wellord1.html#D8">WELLORD1:def 8</a><br/>

<b>thus </b><a NAME="E16:13"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="wellord1.html#D7">WELLORD1:def 7</a><br/>

<b>thus </b><a NAME="E17:13"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <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">A</font></span>)</span>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E18:13"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">H</font> is <a href="funct_1.html#V2">one-to-one</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_7"><b>proof </b></a><div class="add">

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



<b>assume </b><a NAME="E1:13_7"/><i><font color="Green">E75</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">H</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">H</font> &amp; <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">B</font> = <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="txt" href="wellord2.html#E3:13"><i><font color="Green">E23</font></i></a></i>;<br/>
<a NAME="E3:13_7"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">a</font></span>)</span>, <a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Maroon">B</font> <a href="wellord1.html#R4">are_isomorphic</a>  &amp; <font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">b</font></span>)</span>, <a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Maroon">B</font> <a href="wellord1.html#R4">are_isomorphic</a>  )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:13_7"/><b>then </b>
 <a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Maroon">B</font>,<font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">b</font></span>)</span> <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="E5:13_7"/><b>then </b>
<font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">b</font></span>)</span> <a href="wellord1.html#R4">are_isomorphic</a> 
 
<b>by </b><i>, <a class="ref" href="wellord1.html#T52">WELLORD1:52</a></i>;<br/>
<b>hence </b><a NAME="E6:13_7"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i>, , , <a class="ref" href="wellord1.html#T58">WELLORD1:58</a></i>;<br/>


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

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



<b>thus </b><a NAME="E19:13"/>
( <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> <font color="Maroon">R</font> implies ( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">H</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" target="_self">RelIncl</a> <font color="Maroon">A</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_8"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_8"/><i><font color="Green">E77</font></i>: 
<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> <font color="Maroon">R</font>
 ;<br/>

<b>hence </b><a NAME="E2:13_8"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>

<b>then reconsider </b><font color="Maroon">A'</font> = <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>, <font color="Maroon">B'</font> = <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i>, <a class="txt" href="wellord2.html#E3:13"><i><font color="Green">E23</font></i></a></i>;<br/>
<a NAME="E4:13_8"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">a</font></span>)</span>, <a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Maroon">A'</font> <a href="wellord1.html#R4">are_isomorphic</a>  &amp; <font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">b</font></span>)</span>, <a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Maroon">B'</font> <a href="wellord1.html#R4">are_isomorphic</a>  )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:13_8"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">A'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">B'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <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">A</font></span>)</span> )
 
<b>by </b><i>, , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>set </b><font color="Maroon">Z</font> = <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">b</font>;<br/>
<b>set </b><font color="Maroon">P</font> = <font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">b</font></span>)</span>;<br/>
<a NAME="E6:13_8"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">b</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_8_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_8_1"/>
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 ;<br/><a NAME="E2:13_8_1"/><b>then </b><i><font color="Green">E84</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">b</font> &amp; <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">b</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> )
 <b>by </b><i>, <a class="ref" href="wellord1.html#T13">WELLORD1:13</a>, <a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/><a NAME="E3:13_8_1"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <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> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="wellord1.html#T39">WELLORD1:39</a></i>;<br/><a NAME="E4:13_8_1"/><i><font color="Green">E86</font></i>: 
 <a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Maroon">A</font> is <a href="wellord1.html#V2">well-ordering</a>
 <b>by </b><i/>;<br/><a NAME="E5:13_8_1"/><i><font color="Green">E87</font></i>: 
( <font color="Maroon">A'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Maroon">A</font></span>)</span> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">A'</font> &amp; <font color="Maroon">B'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Maroon">A</font></span>)</span> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">B'</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E6:13_8_1"/><i><font color="Green">E88</font></i>: 
( <font color="Maroon">A'</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">B'</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A</font> )
 <b>by </b><i>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/><a NAME="E7:13_8_1"/><b>then </b><i><font color="Green">E89</font></i>: 
( <span class="p1">(<span class="default"><a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Maroon">A</font></span>)</span> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">A'</font> <a href="hidden.html#R1">=</a>  <a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Maroon">A'</font> &amp; <span class="p1">(<span class="default"><a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Maroon">A</font></span>)</span> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">B'</font> <a href="hidden.html#R1">=</a>  <a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Maroon">B'</font> )
 <b>by </b><i/>;<br/><a NAME="E8:13_8_1"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">b</font></span>)</span>,<span class="p1">(<span class="default"><a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Maroon">A</font></span>)</span> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Maroon">A</font></span>)</span> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">B'</font></span>)</span> <a href="wellord1.html#R4">are_isomorphic</a> 
 <b>by </b><i>, , <a class="txt" href="wellord2.html#E6:13"><i><font color="Green">E34</font></i></a>, </i>;<br/><a NAME="E9:13_8_1"/><i><font color="Green">E91</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">a</font>
 <b>by </b><i>, , , <a class="ref" href="wellord1.html#T35">WELLORD1:35</a></i>;<br/><a NAME="E10:13_8_1"/>
<font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">b</font>
 <b>by </b><i>, , , <a class="ref" href="wellord1.html#T37">WELLORD1:37</a></i>;<br/><a NAME="E11:13_8_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="wellord1.html#K2">|_2</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> <span class="p3">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Maroon">A</font></span>)</span> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Maroon">A</font></span>)</span> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">A'</font></span>)</span> <a href="wellord1.html#R4">are_isomorphic</a> 
 <b>by </b><i>, , <a class="ref" href="wellord2.html#T2" target="_self">Th2</a>, <a class="ref" href="wellord2.html#T4" target="_self">Th4</a>, <a class="ref" href="wellord1.html#T29">WELLORD1:29</a></i>;<br/><b>hence </b><a NAME="E12:13_8_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" target="_self">RelIncl</a> <font color="Maroon">A</font>
 <b>by </b><i>, , , <a class="txt" href="wellord2.html#E4:13"><i><font color="Green">E32</font></i></a>, <a class="ref" href="wellord2.html#T3" target="_self">Th3</a>, <a class="ref" href="wellord1.html#T62">WELLORD1:62</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:13_8"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">H</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" target="_self">RelIncl</a> <font color="Maroon">A</font>
 <b>by </b><i>, </i>;<br/>


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

<b>assume </b><a NAME="E20:13"/><i><font color="Green">E92</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">H</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" target="_self">RelIncl</a> <font color="Maroon">A</font> )
 ;<br/>

<a NAME="E21:13"/>
<font color="Maroon">R</font> is <a href="relat_2.html#V6">connected</a>
 
<b>by </b><i>, <a class="ref" href="wellord1.html#D4">WELLORD1:def 4</a></i>;<br/>
<a NAME="E22:13"/><b>then </b><i><font color="Green">E93</font></i>: 
<font color="Maroon">R</font> <a href="relat_2.html#R6">is_connected_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="relat_2.html#D14">RELAT_2:def 14</a></i>;<br/>
<a NAME="E23:13"/>
<font color="Maroon">R</font> is <a href="relat_2.html#V1">reflexive</a>
 
<b>by </b><i>, <a class="ref" href="wellord1.html#D4">WELLORD1:def 4</a></i>;<br/>
<a NAME="E24:13"/><b>then </b><i><font color="Green">E94</font></i>: 
<font color="Maroon">R</font> <a href="relat_2.html#R1">is_reflexive_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="relat_2.html#D9">RELAT_2:def 9</a></i>;<br/>
<a NAME="E25:13"/>
 <a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Maroon">A</font> is <a href="relat_2.html#V4">antisymmetric</a>
 
<b>by </b><i/>;<br/>
<a NAME="E26:13"/><b>then </b><i><font color="Green">E95</font></i>: 
 <a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Maroon">A</font> <a href="relat_2.html#R4">is_antisymmetric_in</a>  <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">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_2.html#D12">RELAT_2:def 12</a></i>;<br/>
<b>assume </b><a NAME="E27:13"/><i><font color="Green">E96</font></i>: 
not <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> <font color="Maroon">R</font>
 ;<br/>

<a NAME="E28:13"/><b>then </b><i><font color="Green">E97</font></i>: 
<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="relat_2.html#D1">RELAT_2:def 1</a></i>;<br/>
<a NAME="E29:13"/><b>then </b><i><font color="Green">E98</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 
<b>by </b><i>, , , <a class="ref" href="relat_2.html#D6">RELAT_2:def 6</a></i>;<br/>
<b>reconsider </b><font color="Maroon">A'</font> = <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>, <font color="Maroon">B'</font> = <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i>, <a class="txt" href="wellord2.html#E3:13"><i><font color="Green">E23</font></i></a>, </i>;<br/>
<a NAME="E31:13"/><i><font color="Green">E99</font></i>: 
( <font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">a</font></span>)</span>, <a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Maroon">A'</font> <a href="wellord1.html#R4">are_isomorphic</a>  &amp; <font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">b</font></span>)</span>, <a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Maroon">B'</font> <a href="wellord1.html#R4">are_isomorphic</a>  )
 
<b>by </b><i>, </i>;<br/>
<b>set </b><font color="Maroon">Z</font> = <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">a</font>;<br/>
<b>set </b><font color="Maroon">P</font> = <font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">a</font></span>)</span>;<br/>
<a NAME="E32:13"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">a</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="E33:13"/><i><font color="Green">E101</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">a</font> &amp; <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> )
 
<b>by </b><i>, , <a class="ref" href="wellord1.html#T13">WELLORD1:13</a>, <a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/>
<a NAME="E34:13"/><b>then </b><i><font color="Green">E102</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <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> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">a</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="wellord1.html#T39">WELLORD1:39</a></i>;<br/>
<a NAME="E35:13"/><i><font color="Green">E103</font></i>: 
 <a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Maroon">A</font> is <a href="wellord1.html#V2">well-ordering</a>
 
<b>by </b><i/>;<br/>
<a NAME="E36:13"/><i><font color="Green">E104</font></i>: 
( <font color="Maroon">A'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">B'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <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">A</font></span>)</span> )
 
<b>by </b><i>, , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E37:13"/><b>then </b><i><font color="Green">E105</font></i>: 
( <font color="Maroon">A'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Maroon">A</font></span>)</span> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">A'</font> &amp; <font color="Maroon">B'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Maroon">A</font></span>)</span> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">B'</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E38:13"/><i><font color="Green">E106</font></i>: 
( <font color="Maroon">A'</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">B'</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A</font> )
 
<b>by </b><i><a class="ref" href="wellord2.html#T4" target="_self">Th4</a>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E39:13"/><b>then </b><i><font color="Green">E107</font></i>: 
( <span class="p1">(<span class="default"><a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Maroon">A</font></span>)</span> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">A'</font> <a href="hidden.html#R1">=</a>  <a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Maroon">A'</font> &amp; <span class="p1">(<span class="default"><a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Maroon">A</font></span>)</span> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">B'</font> <a href="hidden.html#R1">=</a>  <a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Maroon">B'</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E40:13"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Maroon">A</font></span>)</span> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Maroon">A</font></span>)</span> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">A'</font></span>)</span> <a href="wellord1.html#R4">are_isomorphic</a> 
 
<b>by </b><i><a class="txt" href="wellord2.html#E4:13"><i><font color="Green">E32</font></i></a>, <a class="ref" href="wellord2.html#T5" target="_self">Th5</a>, <a class="ref" href="wellord2.html#T6" target="_self">Th6</a>, </i>;<br/>
<a NAME="E41:13"/><i><font color="Green">E109</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">b</font>
 
<b>by </b><i>, , <a class="txt" href="wellord2.html#E6:13"><i><font color="Green">E34</font></i></a>, <a class="ref" href="wellord1.html#T35">WELLORD1:35</a></i>;<br/>
<a NAME="E42:13"/>
<font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">b</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">a</font>
 
<b>by </b><i>, , , <a class="ref" href="wellord1.html#T37">WELLORD1:37</a></i>;<br/>
<a NAME="E43:13"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="wellord1.html#K2">|_2</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> <span class="p3">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">b</font></span>)</span>,<span class="p1">(<span class="default"><a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Maroon">A</font></span>)</span> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Maroon">A</font></span>)</span> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">B'</font></span>)</span> <a href="wellord1.html#R4">are_isomorphic</a> 
 
<b>by </b><i><a class="txt" href="wellord2.html#E4:13"><i><font color="Green">E32</font></i></a>, <a class="ref" href="wellord2.html#T5" target="_self">Th5</a>, <a class="ref" href="wellord2.html#T7" target="_self">Th7</a>, <a class="ref" href="wellord2.html#T9" target="_self">Th9</a>, <a class="ref" href="wellord1.html#T29">WELLORD1:29</a></i>;<br/>
<a NAME="E44:13"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">B'</font>,<font color="Maroon">A'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="wellord2.html#K1" target="_self">RelIncl</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="wellord2.html#T2" target="_self">Th2</a>, <a class="ref" href="wellord2.html#T3" target="_self">Th3</a>, <a class="ref" href="wellord2.html#T4" target="_self">Th4</a>, <a class="ref" href="wellord2.html#T8" target="_self">Th8</a>, <a class="ref" href="wellord1.html#T62">WELLORD1:62</a></i>;<br/>
<a NAME="E45:13"/><b>then </b>
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font>
 
<b>by </b><i>, , <a class="ref" href="wellord2.html#T4" target="_self">Th4</a>, <a class="ref" href="relat_2.html#D4">RELAT_2:def 4</a></i>;<br/>
<b>hence </b><a NAME="E46:13"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>


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