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





<b>assume </b><a NAME="E1:59"/><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">F</font> <a href="wellord1.html#R3" target="_self">is_isomorphism_of</a> <font color="Maroon">R</font>,<font color="Maroon">S</font> )
 ;<br/>

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

<b>assume </b><a NAME="E2:59"/><i><font color="Green">E20</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>
 ;<br/>

<a NAME="E3:59"/><i><font color="Green">E27</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><a class="ref" href="wellord1.html#T26" target="_self">Th26</a>, </i>;<br/>
<b>take </b><font color="Maroon">b</font> = <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>;<br/>

<b>thus </b><a NAME="E4:59"/>
<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="wellord1.html#T29" target="_self">Th29</a>, <a class="ref" href="wellord1.html#T30" target="_self">Th30</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>

<a NAME="E5:59"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="relat_1.html#K9">.:</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> holds <br/><font color="Olive">c</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">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:59_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="relat_1.html#K9">.:</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>then consider </b><font color="Maroon">d</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:59_1"/><i><font color="Green">E36</font></i>: 
( <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">d</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; <font color="Maroon">c</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#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E4:59_1"/><i><font color="Green">E37</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">d</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> &amp; <font color="Maroon">d</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> )
 
<b>by </b><i><a class="ref" href="wellord1.html#T32" target="_self">Th32</a>, </i>;<br/>
<a NAME="E5:59_1"/><b>then </b><i><font color="Green">E38</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#T26" target="_self">Th26</a>, <a class="ref" href="wellord1.html#T32" target="_self">Th32</a>, </i>;<br/>
<a NAME="E6:59_1"/>
<font color="Maroon">c</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#T29" target="_self">Th29</a>, <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="wellord1.html#T33" target="_self">Th33</a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E7:59_1"/>
<font color="Maroon">c</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">b</font>
 <b>by </b><i><a class="ref" href="wellord1.html#T35" target="_self">Th35</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:59"/>
for <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">c</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">b</font> holds <br/><font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="relat_1.html#K9">.:</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:59_2"/>
<font color="Maroon">c</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">b</font>
 ;<br/>

<a NAME="E2:59_2"/><b>then </b><i><font color="Green">E39</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:59_2"/><b>then </b><i><font color="Green">E40</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>
 
<b>by </b><i><a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>
<a NAME="E4:59_2"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font></span>)</span>
 
<b>by </b><i><a class="ref" href="wellord1.html#T30" target="_self">Th30</a>, <a class="ref" href="funct_1.html#T57">FUNCT_1:57</a></i>;<br/>
<a NAME="E5:59_2"/>
(  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> )
 
<b>by </b><i><a class="ref" href="wellord1.html#T30" target="_self">Th30</a>, <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>;<br/>
<a NAME="E6:59_2"/><b>then </b><i><font color="Green">E64</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <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>
 
<b>by </b><i><a class="ref" href="wellord1.html#T30" target="_self">Th30</a>, <a class="ref" href="wellord1.html#T33" target="_self">Th33</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E7:59_2"/><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="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font></span>)</span>,<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#T26" target="_self">Th26</a>, <a class="ref" href="wellord1.html#T29" target="_self">Th29</a>, <a class="ref" href="wellord1.html#T32" target="_self">Th32</a>, <a class="ref" href="wellord1.html#T35" target="_self">Th35</a>, <a class="txt" href="wellord1.html#E3:59_2"><i><font color="Green">E40</font></i></a></i>;<br/>
<a NAME="E8:59_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</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>
 
<b>by </b><i><a class="ref" href="wellord1.html#T32" target="_self">Th32</a>, <a class="ref" href="wellord1.html#T35" target="_self">Th35</a>, </i>;<br/>
<b>hence </b><a NAME="E9:59_2"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="relat_1.html#K9">.:</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>
 <b>by </b><i><a class="ref" href="wellord1.html#T30" target="_self">Th30</a>, <a class="ref" href="wellord1.html#T35" target="_self">Th35</a>, <a class="ref" href="wellord1.html#T36" target="_self">Th36</a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:59"/>
<font color="Maroon">F</font> <a href="relat_1.html#K9">.:</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="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="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div>
