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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="relat_1.html#NM1">Relation</a>;<br/>



<b>assume </b><a NAME="E1:63"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="wellord1.html#V2" target="_self">well-ordering</a> )
 ;<br/>

<a NAME="E2:63"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <font color="Maroon">c<sub>1</sub></font> is <a href="wellord1.html#V2" target="_self">well-ordering</a>
 
<b>by </b><i><a class="ref" href="wellord1.html#T32" target="_self">Th32</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">c<sub>3</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E1:63_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="wellord1.html#R4" target="_self">are_isomorphic</a>  )
 ;<br/><a NAME="E2:63_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <font color="Maroon">c<sub>1</sub></font> is <a href="wellord1.html#V2" target="_self">well-ordering</a>
 <b>by </b><i><a class="txt" href="wellord1.html#E1:63"><i><font color="Green">E59</font></i></a>, <a class="ref" href="wellord1.html#T32" target="_self">Th32</a></i>;<br/><a NAME="E3:63_1"/><b>then </b><i><font color="Green">E62</font></i>: 
(  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="wellord1.html#E1:63"><i><font color="Green">E59</font></i></a>, <a class="ref" href="wellord1.html#T39" target="_self">Th39</a>, <a class="ref" href="wellord1.html#T40" target="_self">Th40</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:63_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="wellord1.html#R3" target="_self">is_isomorphism_of</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="wellord1.html#E1:63_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="wellord1.html#D8" target="_self">Def8</a></i>;<br/><a NAME="E6:63_1"/><i><font color="Green">E64</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="wellord1.html#E3:63_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="wellord1.html#E5:63_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="wellord1.html#D7" target="_self">Def7</a></i>;<br/><a NAME="E7:63_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="wellord1.html#E3:63_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="wellord1.html#T13" target="_self">Th13</a></i>;<br/><a NAME="E8:63_1"/><b>then </b><i><font color="Green">E65</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="wellord1.html#E1:63"><i><font color="Green">E59</font></i></a>, <a class="txt" href="wellord1.html#E6:63_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:63_1_1"/><i><font color="Green">E66</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><a NAME="E2:63_1_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="wellord1.html#E5:63_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="wellord1.html#T45" target="_self">Th45</a></i>;<br/><a NAME="E3:63_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="wellord1.html#T16" target="_self">Th16</a></i>;<br/><b>hence </b><a NAME="E4:63_1_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="wellord1.html#E5:63_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="wellord1.html#E1:63_1_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="wellord1.html#T16" target="_self">Th16</a>, <a class="ref" href="wellord1.html#T45" target="_self">Th45</a></i>;<br/></div><b>end;</b></div><a NAME="E10:63_1"/><b>then </b><i><font color="Green">E66</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="wellord1.html#E1:63"><i><font color="Green">E59</font></i></a>, <a class="txt" href="wellord1.html#E1:63_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="wellord1.html#E3:63_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="wellord1.html#E6:63_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="wellord1.html#E8:63_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="wellord1.html#T43" target="_self">Th43</a></i>;<br/><a NAME="E11:63_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="wellord1.html#E1:63"><i><font color="Green">E59</font></i></a>, <a class="txt" href="wellord1.html#E1:63_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="wellord1.html#E3:63_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="wellord1.html#E6:63_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E12:63_1"/><b>then </b><i><font color="Green">E67</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="wellord1.html#E6:63_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="wellord1.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E13:63_1"/><b>then </b><i><font color="Green">E68</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="wellord1.html#T16" target="_self">Th16</a></i>;<br/><a NAME="E14:63_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="relat_2.html#V4">antisymmetric</a>
 <b>by </b><i><a class="txt" href="wellord1.html#E1:63"><i><font color="Green">E59</font></i></a>, <a class="ref" href="wellord1.html#D4" target="_self">Def4</a></i>;<br/><b>hence </b><a NAME="E15:63_1"/>
contradiction
 <b>by </b><i><a class="txt" href="wellord1.html#E10:63_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="wellord1.html#E12:63_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="wellord1.html#E13:63_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="wellord1.html#E3"><i><font color="Green">Lemma3</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:63"/>
( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#R4" target="_self">are_isomorphic</a>  or ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#R4" target="_self">are_isomorphic</a>  ) )
 <b>by </b><i><a class="txt" href="wellord1.html#E1:63"><i><font color="Green">E59</font></i></a>, <a class="txt" href="wellord1.html#E2:63"><i><font color="Green">E60</font></i></a>, <a class="ref" href="wellord1.html#T63" target="_self">Th63</a></i>;<br/>


</div>
