<?xml version="1.0"?>
<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">R</font> be   <a href="relat_1.html#NM1">Relation</a>;<br/>





<b>assume </b><a NAME="E1:37"/><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">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> )
 ;<br/>

<b>thus </b><a NAME="E2:37"/>
(  not <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> or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> or <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> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:37_1"/>
<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>
 ;<br/>

<a NAME="E2:37_1"/><b>then </b>
<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#T10" target="_self">Th10</a>, <a class="ref" href="wellord1.html#D4" target="_self">Def4</a></i>;<br/>
<b>hence </b><a NAME="E3:37_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> or <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> )
 <b>by </b><i/>;<br/>


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

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:37_2"/>
<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>
 ;<br/><a NAME="E2:37_2"/><b>then </b>
<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/>;<br/><b>hence </b><a NAME="E3:37_2"/>
<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#T10" target="_self">Th10</a>, <a class="ref" href="wellord1.html#D4" target="_self">Def4</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:37"/>
( ( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> or <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> ) implies <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> )
 ;<br/>


</div>
