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

<b>let </b><font color="Maroon" title="c1">R</font>, <font color="Maroon" title="c2">S</font> be   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a>;<br/>



<b>assume </b><a NAME="E1:62"/><i><font color="Green" title="E58">A1</font></i>: 
( <font color="Maroon" title="c1">R</font> is <a href="wellord1.html#V2" title="WELLORD1:attr.2">well-ordering</a> &amp; <font color="Maroon" title="c2">S</font> is <a href="wellord1.html#V2" title="WELLORD1:attr.2">well-ordering</a> )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> $1</span>)</span>,<font color="Maroon" title="c2">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a>  );<br/>
<b>consider </b><font color="Maroon" title="c3">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:62"/><i><font color="Green" title="E59">A2</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z</font> iff ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E4:62"/><i><font color="Green" title="E60">A3</font></i>: 
<font color="Maroon" title="c3">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">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" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>thus </b><a NAME="E1:62_1"/>
(  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font> )
 <b>by </b><i><a class="txt" href="wellord1.html#E3:62"><i><font color="Green" title="E59">A2</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:62"/><i><font color="Green" title="E61">A4</font></i>: 
( <font color="Maroon" title="c2">S</font> is <a href="relat_2.html#V1" title="RELAT_2:attr.1">reflexive</a> &amp; <font color="Maroon" title="c2">S</font> is <a href="relat_2.html#V4" title="RELAT_2:attr.4">antisymmetric</a> &amp; <font color="Maroon" title="c1">R</font> is <a href="relat_2.html#V6" title="RELAT_2:attr.6">connected</a> &amp; <font color="Maroon" title="c1">R</font> is <a href="relat_2.html#V1" title="RELAT_2:attr.1">reflexive</a> )
 
<b>by </b><i><a class="txt" href="wellord1.html#E1:62"><i><font color="Green" title="E58">A1</font></i></a>, <a class="ref" href="wellord1.html#D4" target="_self" title="WELLORD1:def.4">Def4</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( $2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> $1</span>)</span>,<font color="Maroon" title="c2">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> $2</span>)</span> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a>  );<br/>
<a NAME="E6:62"/><i><font color="Green" title="E62">A5</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font>] holds <br/><font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">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" title="c4">a</font>, <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>





<b>assume </b><a NAME="E1:62_2"/><i><font color="Green" title="E63">A6</font></i>: 
( <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c4">a</font></span>)</span>,<font color="Maroon" title="c2">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a>  &amp; <font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c4">a</font></span>)</span>,<font color="Maroon" title="c2">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c6">c</font></span>)</span> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a>  )
 ;<br/>

<a NAME="E2:62_2"/><b>then </b>
<font color="Maroon" title="c2">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c5">b</font></span>)</span>,<font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a> 
 
<b>by </b><i><a class="ref" href="wellord1.html#T50" target="_self" title="WELLORD1:th.50">Th50</a></i>;<br/>
<b>hence </b><a NAME="E3:62_2"/>
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">c</font>
 <b>by </b><i><a class="txt" href="wellord1.html#E1:62"><i><font color="Green" title="E58">A1</font></i></a>, <a class="txt" href="wellord1.html#E1:62_2"><i><font color="Green" title="E63">A6</font></i></a>, <a class="ref" href="wellord1.html#T52" target="_self" title="WELLORD1:th.52">Th52</a>, <a class="ref" href="wellord1.html#T58" target="_self" title="WELLORD1:th.58">Th58</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c4">F</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E8:62"/><i><font color="Green" title="E63">A7</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font> iff ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>] ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S1" title="FUNCT_1:sch.1">FUNCT_1:sch 1</a>(<a class="txt" href="wellord1.html#E6:62"><i><font color="Green" title="E62">A5</font></i></a>);<br/></i>
<a NAME="E9:62"/><i><font color="Green" title="E64">A8</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c2">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" title="c5">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:62_3"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:62_3"/><i><font color="Green" title="E65">A9</font></i>: 
( <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">F</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:62_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font>
 
<b>by </b><i><a class="txt" href="wellord1.html#E3:62_3"><i><font color="Green" title="E65">A9</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/>
<b>hence </b><a NAME="E5:62_3"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="wellord1.html#E8:62"><i><font color="Green" title="E63">A7</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:62"/><i><font color="Green" title="E65">A10</font></i>: 
<font color="Maroon" title="c3">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">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" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z</font> holds <br/><font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">F</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">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" title="c5">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:62_4_1"/><i><font color="Green" title="E66">A11</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z</font>
 ;<br/>

<a NAME="E2:62_4_1"/><b>then </b><i><font color="Green" title="E67">A12</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font>
 
<b>by </b><i><a class="txt" href="wellord1.html#E3:62"><i><font color="Green" title="E59">A2</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:62_4_1"/><i><font color="Green" title="E68">A13</font></i>: 
( <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c5">a</font></span>)</span>,<font color="Maroon" title="c2">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a>  )
 <b>by </b><i><a class="txt" href="wellord1.html#E3:62"><i><font color="Green" title="E59">A2</font></i></a>, <a class="txt" href="wellord1.html#E1:62_4_1"><i><font color="Green" title="E66">A11</font></i></a></i>;<br/>
<a NAME="E5:62_4_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font>
 
<b>by </b><i><a class="txt" href="wellord1.html#E8:62"><i><font color="Green" title="E63">A7</font></i></a>, <a class="txt" href="wellord1.html#E2:62_4_1"><i><font color="Green" title="E67">A12</font></i></a>, <a class="txt" href="wellord1.html#E4:62_4_1"><i><font color="Green" title="E68">A13</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:62_4_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">F</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D4" title="RELAT_1:def.4">RELAT_1:def 4</a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c5">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:62_4"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:62_4"/><i><font color="Green" title="E66">A14</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D4" title="RELAT_1:def.4">RELAT_1:def 4</a></i>;<br/>
<a NAME="E5:62_4"/>
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c5">a</font></span>)</span>,<font color="Maroon" title="c2">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a>  )
 
<b>by </b><i><a class="txt" href="wellord1.html#E8:62"><i><font color="Green" title="E63">A7</font></i></a>, <a class="txt" href="wellord1.html#E4:62_4"><i><font color="Green" title="E66">A14</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:62_4"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z</font>
 <b>by </b><i><a class="txt" href="wellord1.html#E3:62"><i><font color="Green" title="E59">A2</font></i></a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green" title="E66">A15</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" title="c5">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:62_5"/><i><font color="Green" title="E67">A16</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z</font>
 ;<br/><b>let </b><font color="Maroon" title="c6">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E2:62_5"/><i><font color="Green" title="E68">A17</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font>
 ;<br/><a NAME="E3:62_5"/><i><font color="Green" title="E69">A18</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="wellord1.html#E3:62"><i><font color="Green" title="E59">A2</font></i></a>, <a class="txt" href="wellord1.html#E1:62_5"><i><font color="Green" title="E67">A16</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c7">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:62_5"/><i><font color="Green" title="E70">A19</font></i>: 
( <font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c5">a</font></span>)</span>,<font color="Maroon" title="c2">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c7">c</font></span>)</span> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a>  )
 <b>by </b><i><a class="txt" href="wellord1.html#E3:62"><i><font color="Green" title="E59">A2</font></i></a>, <a class="txt" href="wellord1.html#E1:62_5"><i><font color="Green" title="E67">A16</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" title="E71">A20</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font>
 ;<br/><a NAME="E2:62_5_1"/><i><font color="Green" title="E72">A21</font></i>: 
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="wellord1.html#E2:62_5"><i><font color="Green" title="E68">A17</font></i></a>, <a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a></i>;<br/><b>set </b><font color="Maroon" title="c8">P</font> = <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c5">a</font></span>)</span>;<br/><b>set </b><font color="Maroon" title="c9">Q</font> = <font color="Maroon" title="c2">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c7">c</font></span>)</span>;<br/><a NAME="E3:62_5_1"/><i><font color="Green" title="E73">A22</font></i>: 
<font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c5">a</font></span>)</span> is <a href="wellord1.html#V2" title="WELLORD1:attr.2">well-ordering</a>
 <b>by </b><i><a class="txt" href="wellord1.html#E1:62"><i><font color="Green" title="E58">A1</font></i></a>, <a class="ref" href="wellord1.html#T32" target="_self" title="WELLORD1:th.32">Th32</a></i>;<br/><a NAME="E4:62_5_1"/><b>then </b><i><font color="Green" title="E74">A23</font></i>: 
 <a href="wellord1.html#K3" title="WELLORD1:func.3">canonical_isomorphism_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c5">a</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c7">c</font></span>)</span></span>)</span> <a href="wellord1.html#R3" title="WELLORD1:pred.3">is_isomorphism_of</a> <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c5">a</font></span>)</span>,<font color="Maroon" title="c2">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c7">c</font></span>)</span>
 <b>by </b><i><a class="txt" href="wellord1.html#E5:62_5"><i><font color="Green" title="E70">A19</font></i></a>, <a class="ref" href="wellord1.html#D9" target="_self" title="WELLORD1:def.9">Def9</a></i>;<br/><a NAME="E5:62_5_1"/><i><font color="Green" title="E75">A24</font></i>: 
( <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c5">a</font> &amp;  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c5">a</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c5">a</font> )
 <b>by </b><i><a class="txt" href="wellord1.html#E1:62"><i><font color="Green" title="E58">A1</font></i></a>, <a class="txt" href="wellord1.html#E2:62_5"><i><font color="Green" title="E68">A17</font></i></a>, <a class="txt" href="wellord1.html#E1:62_5_1"><i><font color="Green" title="E71">A20</font></i></a>, <a class="ref" href="wellord1.html#D1" target="_self" title="WELLORD1:def.1">Def1</a>, <a class="ref" href="wellord1.html#T40" target="_self" title="WELLORD1:th.40">Th40</a></i>;<br/><b>then consider </b><font color="Maroon" title="c10">d</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:62_5_1"/><i><font color="Green" title="E76">A25</font></i>: 
( <font color="Maroon" title="c10">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c7">c</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c5">a</font></span>)</span></span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c5">a</font></span>)</span></span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c6">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c7">c</font></span>)</span></span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c7">c</font></span>)</span></span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c10">d</font></span>)</span> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a>  )
 <b>by </b><i><a class="txt" href="wellord1.html#E3:62_5_1"><i><font color="Green" title="E73">A22</font></i></a>, <a class="txt" href="wellord1.html#E4:62_5_1"><i><font color="Green" title="E74">A23</font></i></a>, <a class="ref" href="wellord1.html#T61" target="_self" title="WELLORD1:th.61">Th61</a></i>;<br/><a NAME="E8:62_5_1"/><i><font color="Green" title="E77">A26</font></i>: 
<font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c7">c</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="wellord1.html#E1:62"><i><font color="Green" title="E58">A1</font></i></a>, <a class="ref" href="wellord1.html#T40" target="_self" title="WELLORD1:th.40">Th40</a></i>;<br/><a NAME="E9:62_5_1"/><i><font color="Green" title="E78">A27</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c5">a</font></span>)</span></span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="txt" href="wellord1.html#E1:62"><i><font color="Green" title="E58">A1</font></i></a>, <a class="txt" href="wellord1.html#E3:62_5"><i><font color="Green" title="E69">A18</font></i></a>, <a class="txt" href="wellord1.html#E5:62_5_1"><i><font color="Green" title="E75">A24</font></i></a>, <a class="ref" href="wellord1.html#T35" target="_self" title="WELLORD1:th.35">Th35</a></i>;<br/><a NAME="E10:62_5_1"/><i><font color="Green" title="E79">A28</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c7">c</font></span>)</span></span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c10">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c10">d</font>
 <b>by </b><i><a class="txt" href="wellord1.html#E1:62"><i><font color="Green" title="E58">A1</font></i></a>, <a class="txt" href="wellord1.html#E5:62_5"><i><font color="Green" title="E70">A19</font></i></a>, <a class="txt" href="wellord1.html#E7:62_5_1"><i><font color="Green" title="E76">A25</font></i></a>, <a class="txt" href="wellord1.html#E8:62_5_1"><i><font color="Green" title="E77">A26</font></i></a>, <a class="ref" href="wellord1.html#T35" target="_self" title="WELLORD1:th.35">Th35</a></i>;<br/><a NAME="E11:62_5_1"/><i><font color="Green" title="E80">A29</font></i>: 
<font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c6">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c5">a</font>
 <b>by </b><i><a class="txt" href="wellord1.html#E1:62"><i><font color="Green" title="E58">A1</font></i></a>, <a class="txt" href="wellord1.html#E3:62_5"><i><font color="Green" title="E69">A18</font></i></a>, <a class="txt" href="wellord1.html#E2:62_5_1"><i><font color="Green" title="E72">A21</font></i></a>, <a class="txt" href="wellord1.html#E5:62_5_1"><i><font color="Green" title="E75">A24</font></i></a>, <a class="ref" href="wellord1.html#T38" target="_self" title="WELLORD1:th.38">Th38</a></i>;<br/><a NAME="E12:62_5_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">d</font>,<font color="Maroon" title="c7">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="wellord1.html#E7:62_5_1"><i><font color="Green" title="E76">A25</font></i></a>, <a class="txt" href="wellord1.html#E8:62_5_1"><i><font color="Green" title="E77">A26</font></i></a>, <a class="ref" href="wellord1.html#D1" target="_self" title="WELLORD1:def.1">Def1</a></i>;<br/><a NAME="E13:62_5_1"/><b>then </b><i><font color="Green" title="E81">A30</font></i>: 
<font color="Maroon" title="c10">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a></i>;<br/><a NAME="E14:62_5_1"/><b>then </b><i><font color="Green" title="E82">A31</font></i>: 
<font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c10">d</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c7">c</font>
 <b>by </b><i><a class="txt" href="wellord1.html#E1:62"><i><font color="Green" title="E58">A1</font></i></a>, <a class="txt" href="wellord1.html#E5:62_5"><i><font color="Green" title="E70">A19</font></i></a>, <a class="txt" href="wellord1.html#E7:62_5_1"><i><font color="Green" title="E76">A25</font></i></a>, <a class="txt" href="wellord1.html#E8:62_5_1"><i><font color="Green" title="E77">A26</font></i></a>, <a class="ref" href="wellord1.html#T38" target="_self" title="WELLORD1:th.38">Th38</a></i>;<br/><a NAME="E15:62_5_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c5">a</font></span>)</span></span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c6">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="wellord1.html#E11:62_5_1"><i><font color="Green" title="E80">A29</font></i></a>, <a class="ref" href="wellord1.html#T29" target="_self" title="WELLORD1:th.29">Th29</a></i>;<br/><a NAME="E16:62_5_1"/><b>then </b>
<font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c6">b</font></span>)</span>,<font color="Maroon" title="c2">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c10">d</font></span>)</span> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a> 
 <b>by </b><i><a class="txt" href="wellord1.html#E7:62_5_1"><i><font color="Green" title="E76">A25</font></i></a>, <a class="txt" href="wellord1.html#E9:62_5_1"><i><font color="Green" title="E78">A27</font></i></a>, <a class="txt" href="wellord1.html#E10:62_5_1"><i><font color="Green" title="E79">A28</font></i></a>, <a class="txt" href="wellord1.html#E14:62_5_1"><i><font color="Green" title="E82">A31</font></i></a>, <a class="ref" href="wellord1.html#T29" target="_self" title="WELLORD1:th.29">Th29</a></i>;<br/><b>hence </b><a NAME="E17:62_5_1"/>
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z</font>
 <b>by </b><i><a class="txt" href="wellord1.html#E3:62"><i><font color="Green" title="E59">A2</font></i></a>, <a class="txt" href="wellord1.html#E2:62_5_1"><i><font color="Green" title="E72">A21</font></i></a>, <a class="txt" href="wellord1.html#E13:62_5_1"><i><font color="Green" title="E81">A30</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:62_5"/>
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z</font>
 <b>by </b><i><a class="txt" href="wellord1.html#E1:62_5"><i><font color="Green" title="E67">A16</font></i></a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E67">A32</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" title="c5">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:62_6"/><i><font color="Green" title="E68">A33</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">F</font>
 ;<br/><b>let </b><font color="Maroon" title="c6">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E2:62_6"/><i><font color="Green" title="E69">A34</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 ;<br/><b>consider </b><font color="Maroon" title="c7">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:62_6"/><i><font color="Green" title="E70">A35</font></i>: 
( <font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">F</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">c</font> )
 <b>by </b><i><a class="txt" href="wellord1.html#E1:62_6"><i><font color="Green" title="E68">A33</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:62_6"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c5">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font>
 <b>by </b><i><a class="txt" href="wellord1.html#E4:62_6"><i><font color="Green" title="E70">A35</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/><a NAME="E6:62_6"/><b>then </b><i><font color="Green" title="E71">A36</font></i>: 
( <font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c7">c</font></span>)</span>,<font color="Maroon" title="c2">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a>  )
 <b>by </b><i><a class="txt" href="wellord1.html#E8:62"><i><font color="Green" title="E63">A7</font></i></a></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" title="E72">A37</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font>
 ;<br/><a NAME="E2:62_6_1"/><i><font color="Green" title="E73">A38</font></i>: 
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="wellord1.html#E2:62_6"><i><font color="Green" title="E69">A34</font></i></a>, <a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a></i>;<br/><b>set </b><font color="Maroon" title="c8">P</font> = <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c7">c</font></span>)</span>;<br/><b>set </b><font color="Maroon" title="c9">Q</font> = <font color="Maroon" title="c2">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c5">a</font></span>)</span>;<br/><a NAME="E3:62_6_1"/><i><font color="Green" title="E74">A39</font></i>: 
<font color="Maroon" title="c2">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c5">a</font></span>)</span>,<font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c7">c</font></span>)</span> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a> 
 <b>by </b><i><a class="txt" href="wellord1.html#E6:62_6"><i><font color="Green" title="E71">A36</font></i></a>, <a class="ref" href="wellord1.html#T50" target="_self" title="WELLORD1:th.50">Th50</a></i>;<br/><a NAME="E4:62_6_1"/><i><font color="Green" title="E75">A40</font></i>: 
<font color="Maroon" title="c2">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c5">a</font></span>)</span> is <a href="wellord1.html#V2" title="WELLORD1:attr.2">well-ordering</a>
 <b>by </b><i><a class="txt" href="wellord1.html#E1:62"><i><font color="Green" title="E58">A1</font></i></a>, <a class="ref" href="wellord1.html#T32" target="_self" title="WELLORD1:th.32">Th32</a></i>;<br/><a NAME="E5:62_6_1"/><b>then </b><i><font color="Green" title="E76">A41</font></i>: 
 <a href="wellord1.html#K3" title="WELLORD1:func.3">canonical_isomorphism_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c5">a</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c7">c</font></span>)</span></span>)</span> <a href="wellord1.html#R3" title="WELLORD1:pred.3">is_isomorphism_of</a> <font color="Maroon" title="c2">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c5">a</font></span>)</span>,<font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c7">c</font></span>)</span>
 <b>by </b><i><a class="txt" href="wellord1.html#E3:62_6_1"><i><font color="Green" title="E74">A39</font></i></a>, <a class="ref" href="wellord1.html#D9" target="_self" title="WELLORD1:def.9">Def9</a></i>;<br/><a NAME="E6:62_6_1"/><i><font color="Green" title="E77">A42</font></i>: 
( <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c5">a</font> &amp;  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c5">a</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c5">a</font> )
 <b>by </b><i><a class="txt" href="wellord1.html#E1:62"><i><font color="Green" title="E58">A1</font></i></a>, <a class="txt" href="wellord1.html#E2:62_6"><i><font color="Green" title="E69">A34</font></i></a>, <a class="txt" href="wellord1.html#E1:62_6_1"><i><font color="Green" title="E72">A37</font></i></a>, <a class="ref" href="wellord1.html#D1" target="_self" title="WELLORD1:def.1">Def1</a>, <a class="ref" href="wellord1.html#T40" target="_self" title="WELLORD1:th.40">Th40</a></i>;<br/><b>then consider </b><font color="Maroon" title="c10">d</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:62_6_1"/><i><font color="Green" title="E78">A43</font></i>: 
( <font color="Maroon" title="c10">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c7">c</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c5">a</font></span>)</span></span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c5">a</font></span>)</span></span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c6">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c7">c</font></span>)</span></span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c7">c</font></span>)</span></span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c10">d</font></span>)</span> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a>  )
 <b>by </b><i><a class="txt" href="wellord1.html#E4:62_6_1"><i><font color="Green" title="E75">A40</font></i></a>, <a class="txt" href="wellord1.html#E5:62_6_1"><i><font color="Green" title="E76">A41</font></i></a>, <a class="ref" href="wellord1.html#T61" target="_self" title="WELLORD1:th.61">Th61</a></i>;<br/><a NAME="E9:62_6_1"/><i><font color="Green" title="E79">A44</font></i>: 
<font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c7">c</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="wellord1.html#E1:62"><i><font color="Green" title="E58">A1</font></i></a>, <a class="ref" href="wellord1.html#T40" target="_self" title="WELLORD1:th.40">Th40</a></i>;<br/><a NAME="E10:62_6_1"/><i><font color="Green" title="E80">A45</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c5">a</font></span>)</span></span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="txt" href="wellord1.html#E1:62"><i><font color="Green" title="E58">A1</font></i></a>, <a class="txt" href="wellord1.html#E6:62_6"><i><font color="Green" title="E71">A36</font></i></a>, <a class="txt" href="wellord1.html#E6:62_6_1"><i><font color="Green" title="E77">A42</font></i></a>, <a class="ref" href="wellord1.html#T35" target="_self" title="WELLORD1:th.35">Th35</a></i>;<br/><a NAME="E11:62_6_1"/><i><font color="Green" title="E81">A46</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c7">c</font></span>)</span></span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c10">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c10">d</font>
 <b>by </b><i><a class="txt" href="wellord1.html#E1:62"><i><font color="Green" title="E58">A1</font></i></a>, <a class="txt" href="wellord1.html#E6:62_6"><i><font color="Green" title="E71">A36</font></i></a>, <a class="txt" href="wellord1.html#E8:62_6_1"><i><font color="Green" title="E78">A43</font></i></a>, <a class="txt" href="wellord1.html#E9:62_6_1"><i><font color="Green" title="E79">A44</font></i></a>, <a class="ref" href="wellord1.html#T35" target="_self" title="WELLORD1:th.35">Th35</a></i>;<br/><a NAME="E12:62_6_1"/><i><font color="Green" title="E82">A47</font></i>: 
<font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c6">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c5">a</font>
 <b>by </b><i><a class="txt" href="wellord1.html#E1:62"><i><font color="Green" title="E58">A1</font></i></a>, <a class="txt" href="wellord1.html#E6:62_6"><i><font color="Green" title="E71">A36</font></i></a>, <a class="txt" href="wellord1.html#E2:62_6_1"><i><font color="Green" title="E73">A38</font></i></a>, <a class="txt" href="wellord1.html#E6:62_6_1"><i><font color="Green" title="E77">A42</font></i></a>, <a class="ref" href="wellord1.html#T38" target="_self" title="WELLORD1:th.38">Th38</a></i>;<br/><a NAME="E13:62_6_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">d</font>,<font color="Maroon" title="c7">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="wellord1.html#E8:62_6_1"><i><font color="Green" title="E78">A43</font></i></a>, <a class="txt" href="wellord1.html#E9:62_6_1"><i><font color="Green" title="E79">A44</font></i></a>, <a class="ref" href="wellord1.html#D1" target="_self" title="WELLORD1:def.1">Def1</a></i>;<br/><a NAME="E14:62_6_1"/><b>then </b><i><font color="Green" title="E83">A48</font></i>: 
<font color="Maroon" title="c10">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a></i>;<br/><a NAME="E15:62_6_1"/><b>then </b><i><font color="Green" title="E84">A49</font></i>: 
<font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c10">d</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c7">c</font>
 <b>by </b><i><a class="txt" href="wellord1.html#E1:62"><i><font color="Green" title="E58">A1</font></i></a>, <a class="txt" href="wellord1.html#E6:62_6"><i><font color="Green" title="E71">A36</font></i></a>, <a class="txt" href="wellord1.html#E8:62_6_1"><i><font color="Green" title="E78">A43</font></i></a>, <a class="txt" href="wellord1.html#E9:62_6_1"><i><font color="Green" title="E79">A44</font></i></a>, <a class="ref" href="wellord1.html#T38" target="_self" title="WELLORD1:th.38">Th38</a></i>;<br/><a NAME="E16:62_6_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c5">a</font></span>)</span></span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c6">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="wellord1.html#E12:62_6_1"><i><font color="Green" title="E82">A47</font></i></a>, <a class="ref" href="wellord1.html#T29" target="_self" title="WELLORD1:th.29">Th29</a></i>;<br/><a NAME="E17:62_6_1"/><b>then </b>
<font color="Maroon" title="c2">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c6">b</font></span>)</span>,<font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c10">d</font></span>)</span> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a> 
 <b>by </b><i><a class="txt" href="wellord1.html#E8:62_6_1"><i><font color="Green" title="E78">A43</font></i></a>, <a class="txt" href="wellord1.html#E10:62_6_1"><i><font color="Green" title="E80">A45</font></i></a>, <a class="txt" href="wellord1.html#E11:62_6_1"><i><font color="Green" title="E81">A46</font></i></a>, <a class="txt" href="wellord1.html#E15:62_6_1"><i><font color="Green" title="E84">A49</font></i></a>, <a class="ref" href="wellord1.html#T29" target="_self" title="WELLORD1:th.29">Th29</a></i>;<br/><a NAME="E18:62_6_1"/><b>then </b>
<font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c10">d</font></span>)</span>,<font color="Maroon" title="c2">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a> 
 <b>by </b><i><a class="ref" href="wellord1.html#T50" target="_self" title="WELLORD1:th.50">Th50</a></i>;<br/><a NAME="E19:62_6_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">d</font>,<font color="Maroon" title="c6">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font>
 <b>by </b><i><a class="txt" href="wellord1.html#E8:62"><i><font color="Green" title="E63">A7</font></i></a>, <a class="txt" href="wellord1.html#E2:62_6_1"><i><font color="Green" title="E73">A38</font></i></a>, <a class="txt" href="wellord1.html#E14:62_6_1"><i><font color="Green" title="E83">A48</font></i></a></i>;<br/><a NAME="E20:62_6_1"/><b>then </b>
( <font color="Maroon" title="c10">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">F</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">d</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/><b>hence </b><a NAME="E21:62_6_1"/>
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">F</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E8:62_6"/>
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">F</font>
 <b>by </b><i><a class="txt" href="wellord1.html#E1:62_6"><i><font color="Green" title="E68">A33</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E13:62"/><i><font color="Green" title="E68">A50</font></i>: 
<font color="Maroon" title="c4">F</font> <a href="wellord1.html#R3" title="WELLORD1:pred.3">is_isomorphism_of</a> <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">F</font></span>)</span>,<font color="Maroon" title="c2">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">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#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="wellord1.html#E1:62"><i><font color="Green" title="E58">A1</font></i></a>, <a class="txt" href="wellord1.html#E4:62"><i><font color="Green" title="E60">A3</font></i></a>, <a class="txt" href="wellord1.html#E9:62"><i><font color="Green" title="E64">A8</font></i></a>, <a class="txt" href="wellord1.html#E10:62"><i><font color="Green" title="E65">A10</font></i></a>, <a class="ref" href="wellord1.html#T39" target="_self" title="WELLORD1:th.39">Th39</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="wellord1.html#D7" target="_self" title="WELLORD1:def.7">WELLORD1:def 7</a><br/>

<b>thus </b><a NAME="E2:62_7"/><i><font color="Green" title="E69">A51</font></i>: 
<font color="Maroon" title="c4">F</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">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" title="c5">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a><br/><b>let </b><font color="Maroon" title="c6">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><a NAME="E1:62_7_1"/><i><font color="Green" title="E70">A52</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">F</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">F</font> &amp; <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font> )
 ;<br/>

<a NAME="E2:62_7_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">b</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/>
<a NAME="E3:62_7_1"/><b>then </b><i><font color="Green" title="E71">A53</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c5">a</font></span>)</span>,<font color="Maroon" title="c2">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font></span>)</span></span>)</span> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a>  &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c6">b</font></span>)</span>,<font color="Maroon" title="c2">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font></span>)</span></span>)</span> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a>  )
 
<b>by </b><i><a class="txt" href="wellord1.html#E8:62"><i><font color="Green" title="E63">A7</font></i></a>, <a class="txt" href="wellord1.html#E1:62_7_1"><i><font color="Green" title="E70">A52</font></i></a></i>;<br/>
<a NAME="E4:62_7_1"/><b>then </b>
<font color="Maroon" title="c2">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a> 
 
<b>by </b><i><a class="ref" href="wellord1.html#T50" target="_self" title="WELLORD1:th.50">Th50</a></i>;<br/>
<b>hence </b><a NAME="E5:62_7_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="txt" href="wellord1.html#E1:62"><i><font color="Green" title="E58">A1</font></i></a>, <a class="txt" href="wellord1.html#E3:62_7_1"><i><font color="Green" title="E71">A53</font></i></a>, <a class="ref" href="wellord1.html#T52" target="_self" title="WELLORD1:th.52">Th52</a>, <a class="ref" href="wellord1.html#T58" target="_self" title="WELLORD1:th.58">Th58</a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>thus </b><a NAME="E3:62_7"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">F</font></span>)</span> implies ( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">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" title="E70">A54</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">F</font></span>)</span>
 ;<br/>

<b>hence </b><a NAME="E2:62_7_2"/>
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a></i>;<br/>

<a NAME="E3:62_7_2"/><b>then </b><i><font color="Green" title="E71">A55</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">F</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">F</font> )
 
<b>by </b><i><a class="ref" href="wellord1.html#T19" target="_self" title="WELLORD1:th.19">Th19</a></i>;<br/>
<a NAME="E4:62_7_2"/><b>then </b><i><font color="Green" title="E72">A56</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">b</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/>
<a NAME="E5:62_7_2"/><b>then </b><i><font color="Green" title="E73">A57</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c5">a</font></span>)</span>,<font color="Maroon" title="c2">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font></span>)</span></span>)</span> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a>  )
 
<b>by </b><i><a class="txt" href="wellord1.html#E8:62"><i><font color="Green" title="E63">A7</font></i></a></i>;<br/>
<a NAME="E6:62_7_2"/><i><font color="Green" title="E74">A58</font></i>: 
( <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c6">b</font></span>)</span>,<font color="Maroon" title="c2">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span>)</span> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a>  )
 
<b>by </b><i><a class="txt" href="wellord1.html#E8:62"><i><font color="Green" title="E63">A7</font></i></a>, <a class="txt" href="wellord1.html#E4:62_7_2"><i><font color="Green" title="E72">A56</font></i></a></i>;<br/>
<a NAME="E7:62_7_2"/><i><font color="Green" title="E75">A59</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font>
 
<b>by </b><i><a class="txt" href="wellord1.html#E1:62_7_2"><i><font color="Green" title="E70">A54</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E8:62_7_2"/><b>then </b><i><font color="Green" title="E76">A60</font></i>: 
<font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c5">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c6">b</font>
 
<b>by </b><i><a class="txt" href="wellord1.html#E1:62"><i><font color="Green" title="E58">A1</font></i></a>, <a class="txt" href="wellord1.html#E5:62_7_2"><i><font color="Green" title="E73">A57</font></i></a>, <a class="txt" href="wellord1.html#E6:62_7_2"><i><font color="Green" title="E74">A58</font></i></a>, <a class="ref" href="wellord1.html#T37" target="_self" title="WELLORD1:th.37">Th37</a></i>;<br/>
<a NAME="E9:62_7_2"/>
( <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">F</font> &amp; <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">F</font> )
 
<b>by </b><i><a class="txt" href="wellord1.html#E3:62_7_2"><i><font color="Green" title="E71">A55</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E10:62_7_2"/><b>then </b><i><font color="Green" title="E77">A61</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">F</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E11:62_7_2"/><i><font color="Green" title="E78">A62</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">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" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font>
 ;<br/>

<a NAME="E2:62_7_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="txt" href="wellord1.html#E5:62"><i><font color="Green" title="E61">A4</font></i></a>, <a class="txt" href="wellord1.html#E5:62_7_2"><i><font color="Green" title="E73">A57</font></i></a>, <a class="txt" href="wellord1.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:62_7_2_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">F</font></span>)</span>
 <b>by </b><i><a class="txt" href="wellord1.html#E10:62_7_2"><i><font color="Green" title="E77">A61</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</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" title="c7">P</font> = <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c6">b</font></span>)</span>;<br/><b>assume </b><a NAME="E1:62_7_2_2"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font>
 ;<br/><a NAME="E2:62_7_2_2"/><b>then </b><i><font color="Green" title="E79">A63</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c6">b</font> &amp;  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c6">b</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c6">b</font> )
 <b>by </b><i><a class="txt" href="wellord1.html#E1:62"><i><font color="Green" title="E58">A1</font></i></a>, <a class="txt" href="wellord1.html#E7:62_7_2"><i><font color="Green" title="E75">A59</font></i></a>, <a class="ref" href="wellord1.html#D1" target="_self" title="WELLORD1:def.1">Def1</a>, <a class="ref" href="wellord1.html#T40" target="_self" title="WELLORD1:th.40">Th40</a></i>;<br/><a NAME="E3:62_7_2_2"/><b>then </b><i><font color="Green" title="E80">A64</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c6">b</font></span>)</span></span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c5">a</font>
 <b>by </b><i><a class="txt" href="wellord1.html#E1:62"><i><font color="Green" title="E58">A1</font></i></a>, <a class="txt" href="wellord1.html#E6:62_7_2"><i><font color="Green" title="E74">A58</font></i></a>, <a class="ref" href="wellord1.html#T35" target="_self" title="WELLORD1:th.35">Th35</a></i>;<br/><a NAME="E4:62_7_2_2"/><i><font color="Green" title="E81">A65</font></i>: 
<font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c6">b</font></span>)</span> is <a href="wellord1.html#V2" title="WELLORD1:attr.2">well-ordering</a>
 <b>by </b><i><a class="txt" href="wellord1.html#E1:62"><i><font color="Green" title="E58">A1</font></i></a>, <a class="ref" href="wellord1.html#T32" target="_self" title="WELLORD1:th.32">Th32</a></i>;<br/><a NAME="E5:62_7_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c6">b</font></span>)</span></span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c6">b</font></span>)</span></span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c5">a</font></span>)</span>,<font color="Maroon" title="c2">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font></span>)</span></span>)</span> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a> 
 <b>by </b><i><a class="txt" href="wellord1.html#E5:62_7_2"><i><font color="Green" title="E73">A57</font></i></a>, <a class="txt" href="wellord1.html#E8:62_7_2"><i><font color="Green" title="E76">A60</font></i></a>, <a class="txt" href="wellord1.html#E3:62_7_2_2"><i><font color="Green" title="E80">A64</font></i></a>, <a class="ref" href="wellord1.html#T29" target="_self" title="WELLORD1:th.29">Th29</a></i>;<br/><a NAME="E6:62_7_2_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="wellord1.html#E1:62"><i><font color="Green" title="E58">A1</font></i></a>, <a class="txt" href="wellord1.html#E5:62_7_2"><i><font color="Green" title="E73">A57</font></i></a>, <a class="txt" href="wellord1.html#E6:62_7_2"><i><font color="Green" title="E74">A58</font></i></a>, <a class="txt" href="wellord1.html#E2:62_7_2_2"><i><font color="Green" title="E79">A63</font></i></a>, <a class="txt" href="wellord1.html#E4:62_7_2_2"><i><font color="Green" title="E81">A65</font></i></a>, <a class="ref" href="wellord1.html#T62" target="_self" title="WELLORD1:th.62">Th62</a></i>;<br/><b>hence </b><a NAME="E7:62_7_2_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">F</font></span>)</span>
 <b>by </b><i><a class="txt" href="wellord1.html#E10:62_7_2"><i><font color="Green" title="E77">A61</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</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" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">F</font></span>)</span>
 <b>by </b><i><a class="txt" href="wellord1.html#E11:62_7_2"><i><font color="Green" title="E78">A62</font></i></a></i>;<br/>


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

<b>assume </b><a NAME="E4:62_7"/><i><font color="Green" title="E70">A66</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">F</font></span>)</span> )
 ;<br/>

<a NAME="E5:62_7"/><b>then </b><i><font color="Green" title="E71">A67</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">F</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">F</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">F</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="wellord1.html#T19" target="_self" title="WELLORD1:th.19">Th19</a></i>;<br/>
<a NAME="E6:62_7"/><i><font color="Green" title="E72">A68</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="txt" href="wellord1.html#E4:62_7"><i><font color="Green" title="E70">A66</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>assume </b><a NAME="E7:62_7"/>
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">F</font></span>)</span>
 ;<br/>

<a NAME="E8:62_7"/><b>then </b><i><font color="Green" title="E73">A69</font></i>: 
(  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">F</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E9:62_7"/><i><font color="Green" title="E74">A70</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">b</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font> )
 
<b>by </b><i><a class="txt" href="wellord1.html#E5:62_7"><i><font color="Green" title="E71">A67</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/>
<a NAME="E10:62_7"/><b>then </b><i><font color="Green" title="E75">A71</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c5">a</font></span>)</span>,<font color="Maroon" title="c2">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font></span>)</span></span>)</span> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a>  )
 
<b>by </b><i><a class="txt" href="wellord1.html#E8:62"><i><font color="Green" title="E63">A7</font></i></a></i>;<br/>
<a NAME="E11:62_7"/><i><font color="Green" title="E76">A72</font></i>: 
( <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c6">b</font></span>)</span>,<font color="Maroon" title="c2">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span>)</span> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a>  )
 
<b>by </b><i><a class="txt" href="wellord1.html#E8:62"><i><font color="Green" title="E63">A7</font></i></a>, <a class="txt" href="wellord1.html#E9:62_7"><i><font color="Green" title="E74">A70</font></i></a></i>;<br/>
<a NAME="E12:62_7"/><i><font color="Green" title="E77">A73</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font>
 
<b>by </b><i><a class="txt" href="wellord1.html#E5:62"><i><font color="Green" title="E61">A4</font></i></a>, <a class="txt" href="wellord1.html#E5:62_7"><i><font color="Green" title="E71">A67</font></i></a>, <a class="txt" href="wellord1.html#E8:62_7"><i><font color="Green" title="E73">A69</font></i></a>, <a class="txt" href="wellord1.html#E10:62_7"><i><font color="Green" title="E75">A71</font></i></a>, <a class="txt" href="wellord1.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E13:62_7"/><b>then </b><i><font color="Green" title="E78">A74</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font>
 
<b>by </b><i><a class="txt" href="wellord1.html#E5:62"><i><font color="Green" title="E61">A4</font></i></a>, <a class="txt" href="wellord1.html#E5:62_7"><i><font color="Green" title="E71">A67</font></i></a>, <a class="txt" href="wellord1.html#E8:62_7"><i><font color="Green" title="E73">A69</font></i></a>, <a class="txt" href="wellord1.html#E10:62_7"><i><font color="Green" title="E75">A71</font></i></a>, <a class="txt" href="wellord1.html#E11:62_7"><i><font color="Green" title="E76">A72</font></i></a>, <a class="txt" href="wellord1.html#E4"><i><font color="Green" title="E4">Lm4</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E14:62_7"/><b>then </b><i><font color="Green" title="E79">A75</font></i>: 
<font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c6">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c5">a</font>
 
<b>by </b><i><a class="txt" href="wellord1.html#E1:62"><i><font color="Green" title="E58">A1</font></i></a>, <a class="txt" href="wellord1.html#E10:62_7"><i><font color="Green" title="E75">A71</font></i></a>, <a class="txt" href="wellord1.html#E11:62_7"><i><font color="Green" title="E76">A72</font></i></a>, <a class="ref" href="wellord1.html#T37" target="_self" title="WELLORD1:th.37">Th37</a></i>;<br/>
<b>set </b><font color="Maroon" title="c7">P</font> = <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c5">a</font></span>)</span>;<br/>
<a NAME="E15:62_7"/><i><font color="Green" title="E80">A76</font></i>: 
( <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c5">a</font> &amp;  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c5">a</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c5">a</font> )
 
<b>by </b><i><a class="txt" href="wellord1.html#E1:62"><i><font color="Green" title="E58">A1</font></i></a>, <a class="txt" href="wellord1.html#E5:62_7"><i><font color="Green" title="E71">A67</font></i></a>, <a class="txt" href="wellord1.html#E8:62_7"><i><font color="Green" title="E73">A69</font></i></a>, <a class="txt" href="wellord1.html#E13:62_7"><i><font color="Green" title="E78">A74</font></i></a>, <a class="ref" href="wellord1.html#D1" target="_self" title="WELLORD1:def.1">Def1</a>, <a class="ref" href="wellord1.html#T40" target="_self" title="WELLORD1:th.40">Th40</a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E16:62_7"/><b>then </b><i><font color="Green" title="E81">A77</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c5">a</font></span>)</span></span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c6">b</font>
 
<b>by </b><i><a class="txt" href="wellord1.html#E1:62"><i><font color="Green" title="E58">A1</font></i></a>, <a class="txt" href="wellord1.html#E10:62_7"><i><font color="Green" title="E75">A71</font></i></a>, <a class="ref" href="wellord1.html#T35" target="_self" title="WELLORD1:th.35">Th35</a></i>;<br/>
<a NAME="E17:62_7"/><i><font color="Green" title="E82">A78</font></i>: 
<font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c5">a</font></span>)</span> is <a href="wellord1.html#V2" title="WELLORD1:attr.2">well-ordering</a>
 
<b>by </b><i><a class="txt" href="wellord1.html#E1:62"><i><font color="Green" title="E58">A1</font></i></a>, <a class="ref" href="wellord1.html#T32" target="_self" title="WELLORD1:th.32">Th32</a></i>;<br/>
<a NAME="E18:62_7"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c5">a</font></span>)</span></span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c5">a</font></span>)</span></span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c6">b</font></span>)</span>,<font color="Maroon" title="c2">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span>)</span> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a> 
 
<b>by </b><i><a class="txt" href="wellord1.html#E11:62_7"><i><font color="Green" title="E76">A72</font></i></a>, <a class="txt" href="wellord1.html#E14:62_7"><i><font color="Green" title="E79">A75</font></i></a>, <a class="txt" href="wellord1.html#E16:62_7"><i><font color="Green" title="E81">A77</font></i></a>, <a class="ref" href="wellord1.html#T29" target="_self" title="WELLORD1:th.29">Th29</a></i>;<br/>
<a NAME="E19:62_7"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="txt" href="wellord1.html#E1:62"><i><font color="Green" title="E58">A1</font></i></a>, <a class="txt" href="wellord1.html#E10:62_7"><i><font color="Green" title="E75">A71</font></i></a>, <a class="txt" href="wellord1.html#E11:62_7"><i><font color="Green" title="E76">A72</font></i></a>, <a class="txt" href="wellord1.html#E15:62_7"><i><font color="Green" title="E80">A76</font></i></a>, <a class="txt" href="wellord1.html#E17:62_7"><i><font color="Green" title="E82">A78</font></i></a>, <a class="ref" href="wellord1.html#T62" target="_self" title="WELLORD1:th.62">Th62</a></i>;<br/>
<a NAME="E20:62_7"/><b>then </b>
<font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font>
 
<b>by </b><i><a class="txt" href="wellord1.html#E5:62"><i><font color="Green" title="E61">A4</font></i></a>, <a class="txt" href="wellord1.html#E6:62_7"><i><font color="Green" title="E72">A68</font></i></a>, <a class="txt" href="wellord1.html#E3"><i><font color="Green" title="E3">Lm3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E21:62_7"/>
contradiction
 <b>by </b><i><a class="txt" href="wellord1.html#E2:62_7"><i><font color="Green" title="E69">A51</font></i></a>, <a class="txt" href="wellord1.html#E5:62_7"><i><font color="Green" title="E71">A67</font></i></a>, <a class="txt" href="wellord1.html#E12:62_7"><i><font color="Green" title="E77">A73</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green" title="E69">A79</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" title="c5">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E2:62_8"/><i><font color="Green" title="E70">A80</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c3">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c5">a</font> )
 ;<br/><b>given </b><font color="Maroon" title="c6">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E4:62_8"/><i><font color="Green" title="E71">A81</font></i>: 
( <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c2">S</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c6">b</font> )
 ;<br/><a NAME="E5:62_8"/>
<font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c5">a</font></span>)</span>,<font color="Maroon" title="c2">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a> 
 <b>by </b><i><a class="txt" href="wellord1.html#E10:62"><i><font color="Green" title="E65">A10</font></i></a>, <a class="txt" href="wellord1.html#E13:62"><i><font color="Green" title="E68">A50</font></i></a>, <a class="txt" href="wellord1.html#E2:62_8"><i><font color="Green" title="E70">A80</font></i></a>, <a class="txt" href="wellord1.html#E4:62_8"><i><font color="Green" title="E71">A81</font></i></a>, <a class="ref" href="wellord1.html#D8" target="_self" title="WELLORD1:def.8">Def8</a></i>;<br/><a NAME="E6:62_8"/><b>then </b>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z</font>
 <b>by </b><i><a class="txt" href="wellord1.html#E3:62"><i><font color="Green" title="E59">A2</font></i></a>, <a class="txt" href="wellord1.html#E2:62_8"><i><font color="Green" title="E70">A80</font></i></a>, <a class="txt" href="wellord1.html#E4:62_8"><i><font color="Green" title="E71">A81</font></i></a></i>;<br/><b>hence </b><a NAME="E7:62_8"/>
contradiction
 <b>by </b><i><a class="txt" href="wellord1.html#E2:62_8"><i><font color="Green" title="E70">A80</font></i></a>, <a class="ref" href="wellord1.html#D1" target="_self" title="WELLORD1:def.1">Def1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E15:62"/><i><font color="Green" title="E70">A82</font></i>: 
<font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c3">Z</font>,<font color="Maroon" title="c2">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a> 
 
<b>by </b><i><a class="txt" href="wellord1.html#E10:62"><i><font color="Green" title="E65">A10</font></i></a>, <a class="txt" href="wellord1.html#E13:62"><i><font color="Green" title="E68">A50</font></i></a>, <a class="ref" href="wellord1.html#D8" target="_self" title="WELLORD1:def.8">Def8</a></i>;<br/>
<div><i><font color="Green" title="E71">A83</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" title="E72">A84</font></i>: 
( <font color="Maroon" title="c3">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c2">S</font> )
 ;<br/><a NAME="E2:62_9"/>
( <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c2">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">S</font> )
 <b>by </b><i><a class="ref" href="wellord1.html#T30" target="_self" title="WELLORD1:th.30">Th30</a></i>;<br/><b>hence </b><a NAME="E3:62_9"/>
<font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">S</font> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a> 
 <b>by </b><i><a class="txt" href="wellord1.html#E10:62"><i><font color="Green" title="E65">A10</font></i></a>, <a class="txt" href="wellord1.html#E13:62"><i><font color="Green" title="E68">A50</font></i></a>, <a class="txt" href="wellord1.html#E1:62_9"><i><font color="Green" title="E72">A84</font></i></a>, <a class="ref" href="wellord1.html#D8" target="_self" title="WELLORD1:def.8">Def8</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E72">A85</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" title="E73">A86</font></i>: 
<font color="Maroon" title="c3">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font>
 ;<br/><b>given </b><font color="Maroon" title="c5">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E3:62_10"/><i><font color="Green" title="E74">A87</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c2">S</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c5">a</font> )
 ;<br/><b>take </b><font color="Maroon" title="c6">a</font> = <font color="Maroon" title="c5">a</font>;<br/><b>thus </b><a NAME="E4:62_10"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="wellord1.html#E3:62_10"><i><font color="Green" title="E74">A87</font></i></a></i>;<br/><b>thus </b><a NAME="E5:62_10"/>
<font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a> 
 <b>by </b><i><a class="txt" href="wellord1.html#E15:62"><i><font color="Green" title="E70">A82</font></i></a>, <a class="txt" href="wellord1.html#E1:62_10"><i><font color="Green" title="E73">A86</font></i></a>, <a class="txt" href="wellord1.html#E3:62_10"><i><font color="Green" title="E74">A87</font></i></a>, <a class="ref" href="wellord1.html#T30" target="_self" title="WELLORD1:th.30">Th30</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" title="E73">A88</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c2">S</font>
 ;<br/><b>given </b><font color="Maroon" title="c5">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E3:62_11"/><i><font color="Green" title="E74">A89</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c3">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c5">a</font> )
 ;<br/><b>take </b><font color="Maroon" title="c6">a</font> = <font color="Maroon" title="c5">a</font>;<br/><b>thus </b><a NAME="E4:62_11"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="wellord1.html#E3:62_11"><i><font color="Green" title="E74">A89</font></i></a></i>;<br/><b>thus </b><a NAME="E5:62_11"/>
<font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c6">a</font></span>)</span>,<font color="Maroon" title="c2">S</font> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a> 
 <b>by </b><i><a class="txt" href="wellord1.html#E15:62"><i><font color="Green" title="E70">A82</font></i></a>, <a class="txt" href="wellord1.html#E1:62_11"><i><font color="Green" title="E73">A88</font></i></a>, <a class="txt" href="wellord1.html#E3:62_11"><i><font color="Green" title="E74">A89</font></i></a>, <a class="ref" href="wellord1.html#T30" target="_self" title="WELLORD1:th.30">Th30</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E19:62"/>
( <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">S</font> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a>  or  ex <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b1">a</font></span>)</span>,<font color="Maroon" title="c2">S</font> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a>  ) or  ex <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="wellord1.html#R4" title="WELLORD1:pred.4">are_isomorphic</a>  ) )
 <b>by </b><i><a class="txt" href="wellord1.html#E1:62"><i><font color="Green" title="E58">A1</font></i></a>, <a class="txt" href="wellord1.html#E4:62"><i><font color="Green" title="E60">A3</font></i></a>, <a class="txt" href="wellord1.html#E9:62"><i><font color="Green" title="E64">A8</font></i></a>, <a class="txt" href="wellord1.html#E11:62"><i><font color="Green" title="E66">A15</font></i></a>, <a class="txt" href="wellord1.html#E12:62"><i><font color="Green" title="E67">A32</font></i></a>, <a class="txt" href="wellord1.html#E14:62"><i><font color="Green" title="E69">A79</font></i></a>, <a class="txt" href="wellord1.html#E16:62"><i><font color="Green" title="E71">A83</font></i></a>, <a class="txt" href="wellord1.html#E17:62"><i><font color="Green" title="E72">A85</font></i></a>, <a class="ref" href="wellord1.html#T36" target="_self" title="WELLORD1:th.36">Th36</a></i>;<br/>


</div>
