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



<b>assume </b><a NAME="E1:62"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">R</font> is <a href="wellord1.html#V2" target="_self">well-ordering</a> &amp; <font color="Maroon">S</font> is <a href="wellord1.html#V2" target="_self">well-ordering</a> )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">S</font> &amp; <font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,<font color="Maroon">S</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Olive">b</font></span>)</span> <a href="wellord1.html#R4" target="_self">are_isomorphic</a>  );<br/>
<b>consider </b><font color="Maroon">Z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:62"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> iff ( <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> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E4:62"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_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="E1:62_1"/>
(  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> or <font color="Maroon">x</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#T32" target="_self">Th32</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:62"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">S</font> is <a href="relat_2.html#V1">reflexive</a> &amp; <font color="Maroon">S</font> is <a href="relat_2.html#V4">antisymmetric</a> &amp; <font color="Maroon">R</font> is <a href="relat_2.html#V6">connected</a> &amp; <font color="Maroon">R</font> is <a href="relat_2.html#V1">reflexive</a> )
 
<b>by </b><i><a class="ref" href="wellord1.html#T31" target="_self">Th31</a>, </i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">S</font> &amp; <font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,<font color="Maroon">S</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a<sub>2</sub></font></span>)</span> <a href="wellord1.html#R4" target="_self">are_isomorphic</a>  );<br/>
<a NAME="E6:62"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">a</font>,<font color="Olive">b</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">a</font>,<font color="Olive">c</font>] holds <br/><font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_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>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>assume </b><a NAME="E1:62_2"/><i><font color="Green">E37</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">S</font> &amp; <font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">S</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">b</font></span>)</span> <a href="wellord1.html#R4" target="_self">are_isomorphic</a>  &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">S</font> &amp; <font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">S</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c</font></span>)</span> <a href="wellord1.html#R4" target="_self">are_isomorphic</a>  )
 ;<br/>

<a NAME="E2:62_2"/><b>then </b>
<font color="Maroon">S</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">b</font></span>)</span>,<font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font></span>)</span> <a href="wellord1.html#R4" target="_self">are_isomorphic</a> 
 
<b>by </b><i><a class="ref" href="wellord1.html#T16" target="_self">Th16</a></i>;<br/>
<a NAME="E3:62_2"/><b>then </b>
<font color="Maroon">S</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">b</font></span>)</span>,<font color="Maroon">S</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c</font></span>)</span> <a href="wellord1.html#R4" target="_self">are_isomorphic</a> 
 
<b>by </b><i>, <a class="ref" href="wellord1.html#T18" target="_self">Th18</a></i>;<br/>
<b>hence </b><a NAME="E4:62_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="wellord1.html#T31" target="_self">Th31</a>, , <a class="ref" href="wellord1.html#T23" target="_self">Th23</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">F</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E8:62"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</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">a</font>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> iff ( <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> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">a</font>,<font color="Olive">b</font>] ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S1">FUNCT_1:sch 1</a>(<a class="ref" href="wellord1.html#T36" target="_self">Th36</a>);<br/></i>
<a NAME="E9:62"/><i><font color="Green">E39</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_3"><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="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:62_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:62_3"/><i><font color="Green">E40</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">F</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</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/>
<a NAME="E4:62_3"/>
<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">F</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>
<b>hence </b><a NAME="E5:62_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">S</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E10:62"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:62_4"/>
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> <font color="Maroon">Z</font> holds <br/><font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_4_1"><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="E1:62_4_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/>

<a NAME="E2:62_4_1"/><b>then </b><i><font color="Green">E65</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>
 
<b>by </b><i><a class="ref" href="wellord1.html#T32" target="_self">Th32</a></i>;<br/>
<b>consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:62_4_1"/><i><font color="Green">E66</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">S</font> &amp; <font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">S</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">b</font></span>)</span> <a href="wellord1.html#R4" target="_self">are_isomorphic</a>  )
 <b>by </b><i><a class="ref" href="wellord1.html#T32" target="_self">Th32</a>, <a class="txt" href="wellord1.html#E4:62_4"><i><font color="Green">E67</font></i></a></i>;<br/>
<a NAME="E5:62_4_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> <font color="Maroon">F</font>
 
<b>by </b><i><a class="txt" href="wellord1.html#E1:62_4_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="wellord1.html#E11:62"><i><font color="Green">E68</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E6:62_4_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a></i>;<br/>


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

<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="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:62_4"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</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="E4:62_4"/><i><font color="Green">E67</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">F</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a></i>;<br/>
<a NAME="E5:62_4"/>
( <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">S</font> &amp; <font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">S</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">b</font></span>)</span> <a href="wellord1.html#R4" target="_self">are_isomorphic</a>  )
 
<b>by </b><i>, <a class="txt" href="wellord1.html#E4:62_4"><i><font color="Green">E67</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:62_4"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="ref" href="wellord1.html#T32" target="_self">Th32</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E68</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="62_5"><b>now </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="E1:62_5"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/><b>let </b><font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E2:62_5"/><i><font color="Green">E70</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>
 ;<br/><a NAME="E3:62_5"/><i><font color="Green">E71</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>
 <b>by </b><i><a class="ref" href="wellord1.html#T32" target="_self">Th32</a>, <a class="txt" href="wellord1.html#E11:62"><i><font color="Green">E68</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:62_5"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">S</font> &amp; <font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">S</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c</font></span>)</span> <a href="wellord1.html#R4" target="_self">are_isomorphic</a>  )
 <b>by </b><i><a class="ref" href="wellord1.html#T32" target="_self">Th32</a>, <a class="txt" href="wellord1.html#E11:62"><i><font color="Green">E68</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_5_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:62_5_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 ;<br/><a NAME="E2:62_5_1"/><i><font color="Green">E90</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>by </b><i><a class="txt" href="wellord1.html#E1:62_5"><i><font color="Green">E69</font></i></a>, <a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/><b>set </b><font color="Maroon">P</font> = <font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font></span>)</span>;<br/><b>set </b><font color="Maroon">Q</font> = <font color="Maroon">S</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c</font></span>)</span>;<br/><a NAME="E3:62_5_1"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font></span>)</span> is <a href="wellord1.html#V2" target="_self">well-ordering</a>
 <b>by </b><i><a class="ref" href="wellord1.html#T31" target="_self">Th31</a>, <a class="ref" href="wellord1.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E4:62_5_1"/><b>then </b><i><font color="Green">E92</font></i>: 
 <a href="wellord1.html#K3" target="_self">canonical_isomorphism_of</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="wellord1.html#R3" target="_self">is_isomorphism_of</a> <font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">S</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c</font></span>)</span>
 <b>by </b><i><a class="txt" href="wellord1.html#E3:62_5"><i><font color="Green">E71</font></i></a>, <a class="ref" href="wellord1.html#T21" target="_self">Th21</a></i>;<br/><a NAME="E5:62_5_1"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font> &amp;  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="wellord1.html#T31" target="_self">Th31</a>, <a class="txt" href="wellord1.html#E1:62_5"><i><font color="Green">E69</font></i></a>, <a class="txt" href="wellord1.html#E5:62_5"><i><font color="Green">E72</font></i></a>, , <a class="txt" href="wellord1.html#E6:62"><i><font color="Green">E36</font></i></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="E7:62_5_1"/><i><font color="Green">E106</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">S</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p3">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">d</font></span>)</span> <a href="wellord1.html#R4" target="_self">are_isomorphic</a>  )
 <b>by </b><i><a class="ref" href="wellord1.html#T37" target="_self">Th37</a>, <a class="ref" href="wellord1.html#T38" target="_self">Th38</a>, <a class="ref" href="wellord1.html#T26" target="_self">Th26</a></i>;<br/><a NAME="E8:62_5_1"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">S</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <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"><font color="Maroon">S</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="wellord1.html#T31" target="_self">Th31</a>, <a class="txt" href="wellord1.html#E6:62"><i><font color="Green">E36</font></i></a></i>;<br/><a NAME="E9:62_5_1"/><i><font color="Green">E108</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="wellord1.html#T31" target="_self">Th31</a>, <a class="txt" href="wellord1.html#E2:62_5"><i><font color="Green">E70</font></i></a>, <a class="ref" href="wellord1.html#T39" target="_self">Th39</a>, <a class="ref" href="wellord1.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E10:62_5_1"/><i><font color="Green">E109</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="wellord1.html#T31" target="_self">Th31</a>, <a class="txt" href="wellord1.html#E3:62_5"><i><font color="Green">E71</font></i></a>, <a class="ref" href="wellord1.html#T40" target="_self">Th40</a>, <a class="ref" href="wellord1.html#T41" target="_self">Th41</a>, <a class="ref" href="wellord1.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E11:62_5_1"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">b</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="wellord1.html#T31" target="_self">Th31</a>, <a class="txt" href="wellord1.html#E2:62_5"><i><font color="Green">E70</font></i></a>, <a class="txt" href="wellord1.html#E1:62_5_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="wellord1.html#T39" target="_self">Th39</a>, <a class="ref" href="wellord1.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E12:62_5_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">d</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="wellord1.html#T40" target="_self">Th40</a>, <a class="ref" href="wellord1.html#T41" target="_self">Th41</a>, </i>;<br/><a NAME="E13:62_5_1"/><b>then </b><i><font color="Green">E111</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/><a NAME="E14:62_5_1"/><b>then </b><i><font color="Green">E112</font></i>: 
<font color="Maroon">S</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">d</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="wellord1.html#T31" target="_self">Th31</a>, <a class="txt" href="wellord1.html#E3:62_5"><i><font color="Green">E71</font></i></a>, <a class="ref" href="wellord1.html#T40" target="_self">Th40</a>, <a class="ref" href="wellord1.html#T41" target="_self">Th41</a>, <a class="ref" href="wellord1.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E15:62_5_1"/>
<span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="wellord1.html#D7" target="_self">Def7</a>, </i>;<br/><a NAME="E16:62_5_1"/><b>then </b>
<font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">b</font></span>)</span>,<font color="Maroon">S</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">d</font></span>)</span> <a href="wellord1.html#R4" target="_self">are_isomorphic</a> 
 <b>by </b><i><a class="ref" href="wellord1.html#T40" target="_self">Th40</a>, <a class="ref" href="wellord1.html#T42" target="_self">Th42</a>, <a class="ref" href="wellord1.html#T43" target="_self">Th43</a>, <a class="ref" href="wellord1.html#D8" target="_self">Def8</a>, </i>;<br/><b>hence </b><a NAME="E17:62_5_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="ref" href="wellord1.html#T32" target="_self">Th32</a>, <a class="txt" href="wellord1.html#E1:62_5_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="wellord1.html#T45" target="_self">Th45</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:62_5"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="txt" href="wellord1.html#E11:62"><i><font color="Green">E68</font></i></a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E113</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="62_6"><b>now </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="E1:62_6"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 ;<br/><b>let </b><font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E2:62_6"/><i><font color="Green">E115</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">S</font>
 ;<br/><b>consider </b><font color="Maroon">c</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:62_6"/><i><font color="Green">E116</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">F</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</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/><a NAME="E5:62_6"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><a NAME="E6:62_6"/><b>then </b><i><font color="Green">E117</font></i>: 
( <font color="Maroon">c</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">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">S</font> &amp; <font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c</font></span>)</span>,<font color="Maroon">S</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font></span>)</span> <a href="wellord1.html#R4" target="_self">are_isomorphic</a>  )
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_6_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:62_6_1"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 ;<br/><a NAME="E2:62_6_1"/><i><font color="Green">E119</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">S</font>
 <b>by </b><i>, <a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/><b>set </b><font color="Maroon">P</font> = <font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c</font></span>)</span>;<br/><b>set </b><font color="Maroon">Q</font> = <font color="Maroon">S</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font></span>)</span>;<br/><a NAME="E3:62_6_1"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">S</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c</font></span>)</span> <a href="wellord1.html#R4" target="_self">are_isomorphic</a> 
 <b>by </b><i>, <a class="ref" href="wellord1.html#T16" target="_self">Th16</a></i>;<br/><a NAME="E4:62_6_1"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">S</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font></span>)</span> is <a href="wellord1.html#V2" target="_self">well-ordering</a>
 <b>by </b><i><a class="ref" href="wellord1.html#T31" target="_self">Th31</a>, <a class="ref" href="wellord1.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E5:62_6_1"/><b>then </b><i><font color="Green">E122</font></i>: 
 <a href="wellord1.html#K3" target="_self">canonical_isomorphism_of</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="wellord1.html#R3" target="_self">is_isomorphism_of</a> <font color="Maroon">S</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c</font></span>)</span>
 <b>by </b><i><a class="ref" href="wellord1.html#T38" target="_self">Th38</a>, <a class="ref" href="wellord1.html#T21" target="_self">Th21</a></i>;<br/><a NAME="E6:62_6_1"/><i><font color="Green">E123</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font> &amp;  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="wellord1.html#T31" target="_self">Th31</a>, , , , <a class="txt" href="wellord1.html#E6:62"><i><font color="Green">E36</font></i></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="E8:62_6_1"/><i><font color="Green">E124</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" target="_self">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="wellord1.html#K1" target="_self">-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" target="_self">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p3">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">d</font></span>)</span> <a href="wellord1.html#R4" target="_self">are_isomorphic</a>  )
 <b>by </b><i><a class="ref" href="wellord1.html#T39" target="_self">Th39</a>, <a class="ref" href="wellord1.html#T40" target="_self">Th40</a>, <a class="ref" href="wellord1.html#T26" target="_self">Th26</a></i>;<br/><a NAME="E9:62_6_1"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <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"><font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="wellord1.html#T31" target="_self">Th31</a>, <a class="txt" href="wellord1.html#E6:62"><i><font color="Green">E36</font></i></a></i>;<br/><a NAME="E10:62_6_1"/><i><font color="Green">E126</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="wellord1.html#T31" target="_self">Th31</a>, , <a class="ref" href="wellord1.html#T41" target="_self">Th41</a>, <a class="ref" href="wellord1.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E11:62_6_1"/><i><font color="Green">E127</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="wellord1.html#T31" target="_self">Th31</a>, , <a class="ref" href="wellord1.html#T42" target="_self">Th42</a>, <a class="ref" href="wellord1.html#T43" target="_self">Th43</a>, <a class="ref" href="wellord1.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E12:62_6_1"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">S</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">b</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="wellord1.html#T31" target="_self">Th31</a>, , <a class="ref" href="wellord1.html#T37" target="_self">Th37</a>, <a class="ref" href="wellord1.html#T41" target="_self">Th41</a>, <a class="ref" href="wellord1.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E13:62_6_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">d</font>,<font color="Maroon">c</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#T42" target="_self">Th42</a>, <a class="ref" href="wellord1.html#T43" target="_self">Th43</a>, </i>;<br/><a NAME="E14:62_6_1"/><b>then </b><i><font color="Green">E129</font></i>: 
<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="relat_1.html#T30">RELAT_1:30</a></i>;<br/><a NAME="E15:62_6_1"/><b>then </b><i><font color="Green">E130</font></i>: 
<font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">d</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="wellord1.html#T31" target="_self">Th31</a>, , <a class="ref" href="wellord1.html#T42" target="_self">Th42</a>, <a class="ref" href="wellord1.html#T43" target="_self">Th43</a>, <a class="ref" href="wellord1.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E16:62_6_1"/>
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="wellord1.html#D8" target="_self">Def8</a>, </i>;<br/><a NAME="E17:62_6_1"/><b>then </b>
<font color="Maroon">S</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">b</font></span>)</span>,<font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">d</font></span>)</span> <a href="wellord1.html#R4" target="_self">are_isomorphic</a> 
 <b>by </b><i><a class="ref" href="wellord1.html#T42" target="_self">Th42</a>, <a class="ref" href="wellord1.html#D7" target="_self">Def7</a>, <a class="ref" href="wellord1.html#T45" target="_self">Th45</a>, <a class="ref" href="wellord1.html#T49" target="_self">Th49</a>, </i>;<br/><a NAME="E18:62_6_1"/><b>then </b>
<font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">d</font></span>)</span>,<font color="Maroon">S</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">b</font></span>)</span> <a href="wellord1.html#R4" target="_self">are_isomorphic</a> 
 <b>by </b><i><a class="ref" href="wellord1.html#T16" target="_self">Th16</a></i>;<br/><a NAME="E19:62_6_1"/><b>then </b>
<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">F</font>
 <b>by </b><i>, <a class="ref" href="wellord1.html#T37" target="_self">Th37</a>, <a class="ref" href="wellord1.html#T47" target="_self">Th47</a></i>;<br/><a NAME="E20:62_6_1"/><b>then </b>
( <font color="Maroon">d</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">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><b>hence </b><a NAME="E21:62_6_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 <b>by </b><i><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="E8:62_6"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E13:62"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">F</font> <a href="wellord1.html#R3" target="_self">is_isomorphism_of</a> <font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">S</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_7"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:62_7"/>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</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" target="_self">|_2</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font></span>)</span></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="wellord1.html#T31" target="_self">Th31</a>, <a class="ref" href="wellord1.html#T33" target="_self">Th33</a>, , , <a class="ref" href="wellord1.html#T10" target="_self">Th10</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="wellord1.html#D7" target="_self">WELLORD1:def 7</a><br/>

<b>thus </b><a NAME="E2:62_7"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">F</font> is <a href="funct_1.html#V2">one-to-one</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_7_1"><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="E1:62_7_1"/><i><font color="Green">E133</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">F</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">F</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">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> )
 ;<br/>

<a NAME="E2:62_7_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<span class="p2">(<span class="default"><font color="Maroon">F</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> <font color="Maroon">F</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>
<a NAME="E3:62_7_1"/><b>then </b><i><font color="Green">E134</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">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">S</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="wellord1.html#R4" target="_self">are_isomorphic</a>  &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; <font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">b</font></span>)</span>,<font color="Maroon">S</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="wellord1.html#R4" target="_self">are_isomorphic</a>  )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:62_7_1"/><b>then </b>
<font color="Maroon">S</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span></span>)</span>,<font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">b</font></span>)</span> <a href="wellord1.html#R4" target="_self">are_isomorphic</a> 
 
<b>by </b><i><a class="ref" href="wellord1.html#T16" target="_self">Th16</a></i>;<br/>
<a NAME="E5:62_7_1"/><b>then </b>
<font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">b</font></span>)</span> <a href="wellord1.html#R4" target="_self">are_isomorphic</a> 
 
<b>by </b><i>, <a class="ref" href="wellord1.html#T18" target="_self">Th18</a></i>;<br/>
<b>hence </b><a NAME="E6:62_7_1"/>
<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#T31" target="_self">Th31</a>, , <a class="ref" href="wellord1.html#T23" target="_self">Th23</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="E3:62_7"/>
( <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> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font></span>)</span> implies ( <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" target="_self">|_2</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font></span>)</span></span>)</span> &amp; <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" target="_self">|_2</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font></span>)</span></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</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> <font color="Maroon">S</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_7_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:62_7_2"/><i><font color="Green">E135</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> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font></span>)</span>
 ;<br/>

<b>hence </b><a NAME="E2:62_7_2"/>
( <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" target="_self">|_2</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font></span>)</span></span>)</span> &amp; <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" target="_self">|_2</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>

<a NAME="E3:62_7_2"/><b>then </b><i><font color="Green">E136</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">F</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">F</font> )
 
<b>by </b><i><a class="txt" href="wellord1.html#E2"><i><font color="Green">Lemma18</font></i></a></i>;<br/>
<a NAME="E4:62_7_2"/><b>then </b><i><font color="Green">E137</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<span class="p2">(<span class="default"><font color="Maroon">F</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> <font color="Maroon">F</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>
<a NAME="E5:62_7_2"/><b>then </b><i><font color="Green">E138</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">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">S</font> &amp; <font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">S</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="wellord1.html#R4" target="_self">are_isomorphic</a>  )
 
<b>by </b><i/>;<br/>
<a NAME="E6:62_7_2"/><i><font color="Green">E139</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> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">S</font> &amp; <font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">b</font></span>)</span>,<font color="Maroon">S</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="wellord1.html#R4" target="_self">are_isomorphic</a>  )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:62_7_2"/><i><font color="Green">E140</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>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:62_7_2"/><b>then </b><i><font color="Green">E141</font></i>: 
<font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#T31" target="_self">Th31</a>, <a class="ref" href="wellord1.html#T37" target="_self">Th37</a>, <a class="ref" href="wellord1.html#T38" target="_self">Th38</a>, <a class="ref" href="wellord1.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E9:62_7_2"/>
( <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E10:62_7_2"/><b>then </b><i><font color="Green">E142</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</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> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E11:62_7_2"/><i><font color="Green">E143</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</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> <font color="Maroon">S</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_7_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:62_7_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/>

<a NAME="E2:62_7_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</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> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#T35" target="_self">Th35</a>, <a class="ref" href="wellord1.html#T37" target="_self">Th37</a>, </i>;<br/>
<b>hence </b><a NAME="E3:62_7_2_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</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> <font color="Maroon">S</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="wellord1.html#T41" target="_self">Th41</a>, </i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_7_2_2"><b>now </b></a><div class="add"><b>set </b><font color="Maroon">P</font> = <font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">b</font></span>)</span>;<br/><b>assume </b><a NAME="E1:62_7_2_2"/>
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 ;<br/><a NAME="E2:62_7_2_2"/><b>then </b><i><font color="Green">E144</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">b</font> &amp;  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="wellord1.html#T31" target="_self">Th31</a>, <a class="ref" href="wellord1.html#T39" target="_self">Th39</a>, , <a class="txt" href="wellord1.html#E6:62"><i><font color="Green">E36</font></i></a></i>;<br/><a NAME="E3:62_7_2_2"/><b>then </b><i><font color="Green">E145</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="wellord1.html#T31" target="_self">Th31</a>, <a class="ref" href="wellord1.html#T38" target="_self">Th38</a>, <a class="ref" href="wellord1.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E4:62_7_2_2"/><i><font color="Green">E146</font></i>: 
<font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">b</font></span>)</span> is <a href="wellord1.html#V2" target="_self">well-ordering</a>
 <b>by </b><i><a class="ref" href="wellord1.html#T31" target="_self">Th31</a>, <a class="ref" href="wellord1.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E5:62_7_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p3">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">S</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="wellord1.html#R4" target="_self">are_isomorphic</a> 
 <b>by </b><i><a class="ref" href="wellord1.html#T37" target="_self">Th37</a>, <a class="ref" href="wellord1.html#T40" target="_self">Th40</a>, <a class="ref" href="wellord1.html#D7" target="_self">Def7</a>, </i>;<br/><a NAME="E6:62_7_2_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</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> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="wellord1.html#T31" target="_self">Th31</a>, <a class="ref" href="wellord1.html#T37" target="_self">Th37</a>, <a class="ref" href="wellord1.html#T38" target="_self">Th38</a>, <a class="ref" href="wellord1.html#T43" target="_self">Th43</a>, <a class="ref" href="wellord1.html#T45" target="_self">Th45</a>, <a class="ref" href="wellord1.html#T29" target="_self">Th29</a></i>;<br/><b>hence </b><a NAME="E7:62_7_2_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</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> <font color="Maroon">S</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="wellord1.html#T41" target="_self">Th41</a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E13:62_7_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</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> <font color="Maroon">S</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="wellord1.html#T42" target="_self">Th42</a></i>;<br/>


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

<b>assume </b><a NAME="E4:62_7"/><i><font color="Green">E147</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" target="_self">|_2</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font></span>)</span></span>)</span> &amp; <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" target="_self">|_2</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font></span>)</span></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</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> <font color="Maroon">S</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> )
 ;<br/>

<a NAME="E5:62_7"/><b>then </b><i><font color="Green">E148</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">F</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">F</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</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> <font color="Maroon">S</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="wellord1.html#E2"><i><font color="Green">Lemma18</font></i></a></i>;<br/>
<a NAME="E6:62_7"/><i><font color="Green">E149</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</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> <font color="Maroon">S</font>
 
<b>by </b><i>, </i>;<br/>
<b>assume </b><a NAME="E7:62_7"/>
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> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font></span>)</span>
 ;<br/>

<a NAME="E8:62_7"/><b>then </b><i><font color="Green">E150</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> or  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> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i/>;<br/>
<a NAME="E9:62_7"/><i><font color="Green">E151</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<span class="p2">(<span class="default"><font color="Maroon">F</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> <font color="Maroon">F</font> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>
<a NAME="E10:62_7"/><b>then </b><i><font color="Green">E152</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">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">S</font> &amp; <font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">S</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="wellord1.html#R4" target="_self">are_isomorphic</a>  )
 
<b>by </b><i/>;<br/>
<a NAME="E11:62_7"/><i><font color="Green">E153</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> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">S</font> &amp; <font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">b</font></span>)</span>,<font color="Maroon">S</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="wellord1.html#R4" target="_self">are_isomorphic</a>  )
 
<b>by </b><i>, <a class="ref" href="wellord1.html#T38" target="_self">Th38</a></i>;<br/>
<a NAME="E12:62_7"/><i><font color="Green">E154</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="wellord1.html#T35" target="_self">Th35</a>, , <a class="ref" href="wellord1.html#T37" target="_self">Th37</a>, <a class="ref" href="wellord1.html#T39" target="_self">Th39</a>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E13:62_7"/><b>then </b><i><font color="Green">E155</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="wellord1.html#T35" target="_self">Th35</a>, , <a class="ref" href="wellord1.html#T37" target="_self">Th37</a>, <a class="ref" href="wellord1.html#T39" target="_self">Th39</a>, <a class="ref" href="wellord1.html#T40" target="_self">Th40</a>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E14:62_7"/><b>then </b><i><font color="Green">E156</font></i>: 
<font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">b</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#T31" target="_self">Th31</a>, <a class="ref" href="wellord1.html#T39" target="_self">Th39</a>, <a class="ref" href="wellord1.html#T40" target="_self">Th40</a>, <a class="ref" href="wellord1.html#D4" target="_self">Def4</a></i>;<br/>
<b>set </b><font color="Maroon">P</font> = <font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font></span>)</span>;<br/>
<a NAME="E15:62_7"/><i><font color="Green">E157</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font> &amp;  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font> )
 
<b>by </b><i><a class="ref" href="wellord1.html#T31" target="_self">Th31</a>, <a class="ref" href="wellord1.html#T41" target="_self">Th41</a>, <a class="ref" href="wellord1.html#T42" target="_self">Th42</a>, , <a class="txt" href="wellord1.html#E6:62"><i><font color="Green">E36</font></i></a></i>;<br/>
<a NAME="E16:62_7"/><b>then </b><i><font color="Green">E158</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#T31" target="_self">Th31</a>, <a class="ref" href="wellord1.html#T39" target="_self">Th39</a>, <a class="ref" href="wellord1.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E17:62_7"/><i><font color="Green">E159</font></i>: 
<font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font></span>)</span> is <a href="wellord1.html#V2" target="_self">well-ordering</a>
 
<b>by </b><i><a class="ref" href="wellord1.html#T31" target="_self">Th31</a>, <a class="ref" href="wellord1.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E18:62_7"/>
<span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p3">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">b</font></span>)</span>,<font color="Maroon">S</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="wellord1.html#R4" target="_self">are_isomorphic</a> 
 
<b>by </b><i><a class="ref" href="wellord1.html#T40" target="_self">Th40</a>, <a class="ref" href="wellord1.html#T43" target="_self">Th43</a>, <a class="ref" href="wellord1.html#T45" target="_self">Th45</a>, </i>;<br/>
<a NAME="E19:62_7"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#T31" target="_self">Th31</a>, <a class="ref" href="wellord1.html#T39" target="_self">Th39</a>, <a class="ref" href="wellord1.html#T40" target="_self">Th40</a>, <a class="ref" href="wellord1.html#D7" target="_self">Def7</a>, <a class="ref" href="wellord1.html#D8" target="_self">Def8</a>, <a class="ref" href="wellord1.html#T29" target="_self">Th29</a></i>;<br/>
<a NAME="E20:62_7"/><b>then </b>
<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">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#T35" target="_self">Th35</a>, , </i>;<br/>
<b>hence </b><a NAME="E21:62_7"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="wellord1.html#T41" target="_self">Th41</a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E160</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="62_8"><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="E1:62_8"/><i><font color="Green">E161</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">Z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font> )
 ;<br/><b>given </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E2:62_8"/><i><font color="Green">E162</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">S</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">b</font> )
 ;<br/><a NAME="E3:62_8"/>
<font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">S</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">b</font></span>)</span> <a href="wellord1.html#R4" target="_self">are_isomorphic</a> 
 <b>by </b><i>, , , , </i>;<br/><a NAME="E4:62_8"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="ref" href="wellord1.html#T32" target="_self">Th32</a>, , </i>;<br/><b>hence </b><a NAME="E5:62_8"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<a NAME="E15:62"/><i><font color="Green">E163</font></i>: 
<font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <font color="Maroon">Z</font>,<font color="Maroon">S</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> <a href="wellord1.html#R4" target="_self">are_isomorphic</a> 
 
<b>by </b><i>, , </i>;<br/>
<div><i><font color="Green">E164</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="62_9"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:62_9"/><i><font color="Green">E165</font></i>: 
( <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">S</font> )
 ;<br/><a NAME="E2:62_9"/>
( <font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> &amp; <font color="Maroon">S</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">S</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> )
 <b>by </b><i><a class="txt" href="wellord1.html#E4:62"><i><font color="Green">E27</font></i></a></i>;<br/><b>hence </b><a NAME="E3:62_9"/>
<font color="Maroon">R</font>,<font color="Maroon">S</font> <a href="wellord1.html#R4" target="_self">are_isomorphic</a> 
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E166</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="62_10"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:62_10"/><i><font color="Green">E167</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 ;<br/><b>given </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E2:62_10"/><i><font color="Green">E168</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">S</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font> )
 ;<br/><b>take </b><font color="Maroon">a</font> = <font color="Maroon">a</font>;<br/><b>thus </b><a NAME="E3:62_10"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="wellord1.html#T38" target="_self">Th38</a></i>;<br/><b>thus </b><a NAME="E4:62_10"/>
<font color="Maroon">R</font>,<font color="Maroon">S</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font></span>)</span> <a href="wellord1.html#R4" target="_self">are_isomorphic</a> 
 <b>by </b><i>, <a class="ref" href="wellord1.html#T37" target="_self">Th37</a>, <a class="ref" href="wellord1.html#T38" target="_self">Th38</a>, <a class="txt" href="wellord1.html#E4:62"><i><font color="Green">E27</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_11"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:62_11"/><i><font color="Green">E169</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">S</font>
 ;<br/><b>given </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E2:62_11"/><i><font color="Green">E170</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">Z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font> )
 ;<br/><b>take </b><font color="Maroon">a</font> = <font color="Maroon">a</font>;<br/><b>thus </b><a NAME="E3:62_11"/>
<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>
 <b>by </b><i><a class="ref" href="wellord1.html#T38" target="_self">Th38</a></i>;<br/><b>thus </b><a NAME="E4:62_11"/>
<font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">S</font> <a href="wellord1.html#R4" target="_self">are_isomorphic</a> 
 <b>by </b><i>, <a class="ref" href="wellord1.html#T37" target="_self">Th37</a>, <a class="ref" href="wellord1.html#T38" target="_self">Th38</a>, <a class="txt" href="wellord1.html#E4:62"><i><font color="Green">E27</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E19:62"/>
( <font color="Maroon">R</font>,<font color="Maroon">S</font> <a href="wellord1.html#R4" target="_self">are_isomorphic</a>  or  ex <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <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> &amp; <font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Olive">a</font></span>)</span>,<font color="Maroon">S</font> <a href="wellord1.html#R4" target="_self">are_isomorphic</a>  ) or  ex <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">S</font> &amp; <font color="Maroon">R</font>,<font color="Maroon">S</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Olive">a</font></span>)</span> <a href="wellord1.html#R4" target="_self">are_isomorphic</a>  ) )
 <b>by </b><i><a class="ref" href="wellord1.html#T31" target="_self">Th31</a>, <a class="ref" href="wellord1.html#T33" target="_self">Th33</a>, , , <a class="txt" href="wellord1.html#E11:62"><i><font color="Green">E68</font></i></a>, , , , <a class="ref" href="wellord1.html#T5" target="_self">Th5</a></i>;<br/>


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