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

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

<b>assume </b><a NAME="E1:56"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">R</font> is <a href="wellord1.html#V2" target="_self">well-ordering</a>
 ;<br/>

<a NAME="E2:56"/><b>then </b><i><font color="Green">E20</font></i>: 
<font color="Maroon">R</font> is <a href="relat_2.html#V4">antisymmetric</a>
 
<b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E3:56"/><i><font color="Green">E27</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/>

<b>assume </b><a NAME="E4:56"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">R</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> <a href="wellord1.html#R4" target="_self">are_isomorphic</a> 
 ;<br/>

<b>set </b><font color="Maroon">S</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">F</font> =  <a href="wellord1.html#K3" target="_self">canonical_isomorphism_of</a> <font color="Maroon">R</font>,<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>;<br/>
<a NAME="E5:56"/><i><font color="Green">E36</font></i>: 
 <a href="wellord1.html#K3" target="_self">canonical_isomorphism_of</a> <font color="Maroon">R</font>,<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#R3" target="_self">is_isomorphism_of</a> <font color="Maroon">R</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>
 
<b>by </b><i><a class="ref" href="wellord1.html#T23" target="_self">Th23</a>, <a class="ref" href="wellord1.html#T26" target="_self">Th26</a>, <a class="ref" href="wellord1.html#T21" target="_self">Th21</a></i>;<br/>
<a NAME="E6:56"/><b>then </b><i><font color="Green">E37</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="wellord1.html#K3" target="_self">canonical_isomorphism_of</a> <font color="Maroon">R</font>,<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></span>)</span> <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> <span class="p1">(<span class="default"><a href="wellord1.html#K3" target="_self">canonical_isomorphism_of</a> <font color="Maroon">R</font>,<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></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> <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> &amp;  <a href="wellord1.html#K3" target="_self">canonical_isomorphism_of</a> <font color="Maroon">R</font>,<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> is <a href="funct_1.html#V2">one-to-one</a> &amp; ( for <font color="Olive">c</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">c</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">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="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"><span class="p3">(<span class="default"><a href="wellord1.html#K3" target="_self">canonical_isomorphism_of</a> <font color="Maroon">R</font>,<span class="p4">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p5">(<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>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">c</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wellord1.html#K3" target="_self">canonical_isomorphism_of</a> <font color="Maroon">R</font>,<span class="p4">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p5">(<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>)</span> <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">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> ) ) ) )
 
<b>by </b><i><a class="txt" href="wellord1.html#E9:56"><i><font color="Green">E40</font></i></a></i>;<br/>
<a NAME="E7:56"/><b>then </b><i><font color="Green">E38</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="wellord1.html#K3" target="_self">canonical_isomorphism_of</a> <font color="Maroon">R</font>,<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></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="txt" href="wellord1.html#E1:56"><i><font color="Green">E19</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:56_1"/><i><font color="Green">E39</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</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> &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> )
 ;<br/><a NAME="E2:56_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"><a href="wellord1.html#K3" target="_self">canonical_isomorphism_of</a> <font color="Maroon">R</font>,<span class="p4">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p5">(<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>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wellord1.html#K3" target="_self">canonical_isomorphism_of</a> <font color="Maroon">R</font>,<span class="p4">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p5">(<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>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font></span>)</span></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"><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#T29" target="_self">Th29</a>, <a class="txt" href="wellord1.html#E9:56"><i><font color="Green">E40</font></i></a></i>;<br/><b>hence </b><a NAME="E3:56_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"><a href="wellord1.html#K3" target="_self">canonical_isomorphism_of</a> <font color="Maroon">R</font>,<span class="p4">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p5">(<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>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wellord1.html#K3" target="_self">canonical_isomorphism_of</a> <font color="Maroon">R</font>,<span class="p4">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p5">(<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>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i/>;<br/><a NAME="E4:56_1"/>
( <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">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#T32" target="_self">Th32</a>, <a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/><b>hence </b><a NAME="E5:56_1"/>
<span class="p1">(<span class="default"><a href="wellord1.html#K3" target="_self">canonical_isomorphism_of</a> <font color="Maroon">R</font>,<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></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="wellord1.html#K3" target="_self">canonical_isomorphism_of</a> <font color="Maroon">R</font>,<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></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font>
 <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="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:56"/><b>then </b><i><font color="Green">E40</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"><span class="p3">(<span class="default"><a href="wellord1.html#K3" target="_self">canonical_isomorphism_of</a> <font color="Maroon">R</font>,<span class="p4">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p5">(<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>)</span> <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">R</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#T23" target="_self">Th23</a>, <a class="ref" href="wellord1.html#T25" target="_self">Th25</a>, <a class="ref" href="wellord1.html#T30" target="_self">Th30</a>, <a class="ref" href="wellord1.html#T31" target="_self">Th31</a>, </i>;<br/>
<a NAME="E10:56"/>
 <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#T23" target="_self">Th23</a>, <a class="txt" href="wellord1.html#E5:56"><i><font color="Green">E36</font></i></a></i>;<br/>
<a NAME="E11:56"/><b>then </b>
<span class="p1">(<span class="default"><a href="wellord1.html#K3" target="_self">canonical_isomorphism_of</a> <font color="Maroon">R</font>,<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></span>)</span> <a href="funct_1.html#K1">.</a> <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">a</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#T25" target="_self">Th25</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="E12:56"/><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"><a href="wellord1.html#K3" target="_self">canonical_isomorphism_of</a> <font color="Maroon">R</font>,<span class="p4">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p5">(<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>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</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> &amp; <span class="p1">(<span class="default"><a href="wellord1.html#K3" target="_self">canonical_isomorphism_of</a> <font color="Maroon">R</font>,<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></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E13:56"/>
contradiction
 <b>by </b><i><a class="ref" href="wellord1.html#T24" target="_self">Th24</a>, <a class="ref" href="wellord1.html#T32" target="_self">Th32</a>, </i>;<br/>


</div>
