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

<b>let </b><font color="Maroon">Z</font> be    <a href="hidden.html#M1">set</a> ;<br/><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>let </b><font color="Maroon">F</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>







<b>assume </b><a NAME="E1:58"/><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">Z</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> &amp; <font color="Maroon">F</font> <a href="wellord1.html#R3" target="_self">is_isomorphism_of</a> <font color="Maroon">R</font>,<font color="Maroon">S</font> )
 ;<br/>

<a NAME="E2:58"/><b>then </b><i><font color="Green">E20</font></i>: 
(  <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> <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> &amp; <font color="Maroon">F</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; ( 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">R</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="Olive">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">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/>;<br/>
<a NAME="E3:58"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">Z</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> <font color="Maroon">Z</font></span>)</span>
 
<b>by </b><i><a class="ref" href="wellord1.html#T25" target="_self">Th25</a>, <a class="ref" href="wellord1.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E4:58"/>
( <font color="Maroon">F</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> &amp; <font color="Maroon">S</font> is <a href="wellord1.html#V2" target="_self">well-ordering</a> )
 
<b>by </b><i><a class="ref" href="wellord1.html#T25" target="_self">Th25</a>, <a class="ref" href="wellord1.html#T19" target="_self">Th19</a>, <a class="ref" href="relat_1.html#T144">RELAT_1:144</a></i>;<br/>
<a NAME="E5:58"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">F</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">Z</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">F</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">Z</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="wellord1.html#T26" target="_self">Th26</a>, <a class="ref" href="wellord1.html#T10" target="_self">Th10</a></i>;<br/>
<b>thus </b><a NAME="E6:58"/>
<font color="Maroon">F</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">Z</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> <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"><font color="Maroon">F</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">Z</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:58_1"/><i><font color="Green">E36</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">Z</font></span>)</span> <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> <font color="Maroon">Z</font></span>)</span>
 <b>by </b><i><a class="ref" href="wellord1.html#T25" target="_self">Th25</a>, <a class="ref" href="wellord1.html#T26" target="_self">Th26</a>, <a class="ref" href="wellord1.html#T29" target="_self">Th29</a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</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="E2:58_1"/><i><font color="Green">E37</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">Z</font></span>)</span> <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">F</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">Z</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="wellord1.html#T30" target="_self">Th30</a>, <a class="ref" href="relat_1.html#T148">RELAT_1:148</a></i>;<br/>

<b>thus </b><a NAME="E3:58_1"/>
<font color="Maroon">F</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">Z</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="wellord1.html#T26" target="_self">Th26</a>, <a class="ref" href="funct_1.html#T84">FUNCT_1:84</a></i>;<br/>

<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="E4:58_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">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <font color="Maroon">Z</font> 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> <font color="Maroon">Z</font></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> <font color="Maroon">Z</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">Z</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">Z</font></span>)</span> <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"><font color="Maroon">F</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">Z</font></span>)</span> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:58_1_1"/><i><font color="Green">E38</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> <font color="Maroon">Z</font>
 ;<br/>

<b>hence </b><a NAME="E2:58_1_1"/><i><font color="Green">E39</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> <font color="Maroon">Z</font></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> <font color="Maroon">Z</font></span>)</span> )
 <b>by </b><i><a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>

<a NAME="E3:58_1_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">R</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#T33" target="_self">Th33</a>, </i>;<br/>
<a NAME="E4:58_1_1"/><b>then </b><i><font color="Green">E40</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><a class="ref" href="wellord1.html#T25" target="_self">Th25</a>, <a class="txt" href="wellord1.html#E4:58_1_1"><i><font color="Green">E40</font></i></a></i>;<br/>
<a NAME="E5:58_1_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">Z</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">Z</font></span>)</span> <a href="funct_1.html#K1">.</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#T35" target="_self">Th35</a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<a NAME="E6:58_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">Z</font></span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">Z</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">Z</font></span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">Z</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="wellord1.html#T31" target="_self">Th31</a>, <a class="ref" href="wellord1.html#T35" target="_self">Th35</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E7:58_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">Z</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">Z</font></span>)</span> <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"><font color="Maroon">F</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">Z</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">Z</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="wellord1.html#T30" target="_self">Th30</a>, <a class="ref" href="wellord1.html#T32" target="_self">Th32</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><a NAME="E8:58_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">Z</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">Z</font></span>)</span> <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"><font color="Maroon">F</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">Z</font></span>)</span>
 <b>by </b><i><a class="ref" href="wellord1.html#T36" target="_self">Th36</a>, <a class="txt" href="wellord1.html#E5:58_1"><i><font color="Green">E64</font></i></a>, </i>;<br/>


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

<b>assume </b><a NAME="E5:58_1"/><i><font color="Green">E64</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> <font color="Maroon">Z</font></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> <font color="Maroon">Z</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">Z</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">Z</font></span>)</span> <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"><font color="Maroon">F</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">Z</font></span>)</span> )
 ;<br/>

<a NAME="E6:58_1"/><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> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">Z</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">Z</font></span>)</span> <a href="funct_1.html#K1">.</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="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<a NAME="E7:58_1"/><b>then </b><i><font color="Green">E65</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><a class="ref" href="wellord1.html#T33" target="_self">Th33</a>, </i>;<br/>
<a NAME="E8:58_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> )
 
<b>by </b><i><a class="ref" href="wellord1.html#T33" target="_self">Th33</a>, <a class="txt" href="wellord1.html#E2"><i><font color="Green">Lemma18</font></i></a></i>;<br/>
<a NAME="E9:58_1"/><b>then </b><i><font color="Green">E66</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><a class="ref" href="wellord1.html#T25" target="_self">Th25</a>, <a class="ref" href="wellord1.html#T35" target="_self">Th35</a>, </i>;<br/>
<a NAME="E10:58_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> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">Z</font>,<font color="Maroon">Z</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="wellord1.html#T29" target="_self">Th29</a>, <a class="ref" href="wellord1.html#T33" target="_self">Th33</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><a NAME="E11:58_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">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="ref" href="wellord1.html#T36" target="_self">Th36</a>, </i>;<br/>


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

<b>hence </b><a NAME="E7:58"/>
<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"><font color="Maroon">F</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">Z</font></span>)</span> <a href="wellord1.html#R4" target="_self">are_isomorphic</a> 
 <b>by </b><i/>;<br/>


</div>
