<?xml version="1.0"?>
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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="relat_1.html#NM1">Relation</a>;<br/>









<b>assume </b><a NAME="E1:61"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> is <a href="wellord1.html#V2" target="_self">well-ordering</a> &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="wellord1.html#V2" target="_self">well-ordering</a> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="wellord1.html#R4" target="_self">are_isomorphic</a>  &amp; <font color="Maroon">c<sub>4</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <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/>

<b>set </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> =  <a href="wellord1.html#K3" target="_self">canonical_isomorphism_of</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>;<br/>
<a NAME="E2:61"/><i><font color="Green">E59</font></i>: 
 <a href="wellord1.html#K3" target="_self">canonical_isomorphism_of</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="wellord1.html#R3" target="_self">is_isomorphism_of</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="wellord1.html#E1:61"><i><font color="Green">E58</font></i></a>, <a class="ref" href="wellord1.html#D9" target="_self">Def9</a></i>;<br/>
<a NAME="E3:61"/>
<font color="Maroon">c<sub>4</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <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>4</sub></font>
 
<b>by </b><i><a class="ref" href="wellord1.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E4:61"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wellord1.html#K3" target="_self">canonical_isomorphism_of</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="relat_1.html#K9">.:</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="wellord1.html#R4" target="_self">are_isomorphic</a> 
 
<b>by </b><i><a class="txt" href="wellord1.html#E1:61"><i><font color="Green">E58</font></i></a>, <a class="txt" href="wellord1.html#E2:61"><i><font color="Green">E59</font></i></a>, <a class="ref" href="wellord1.html#T59" target="_self">Th59</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:61"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>10</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>5</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="wellord1.html#K3" target="_self">canonical_isomorphism_of</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="wellord1.html#E1:61"><i><font color="Green">E58</font></i></a>, <a class="txt" href="wellord1.html#E2:61"><i><font color="Green">E59</font></i></a>, <a class="ref" href="wellord1.html#T60" target="_self">Th60</a></i>;<br/>
<a NAME="E7:61"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="wellord1.html#E1:61"><i><font color="Green">E58</font></i></a>, <a class="ref" href="wellord1.html#T40" target="_self">Th40</a></i>;<br/>
<a NAME="E8:61"/><b>then </b><i><font color="Green">E63</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="wellord1.html#E1:61"><i><font color="Green">E58</font></i></a>, <a class="txt" href="wellord1.html#E6:61"><i><font color="Green">E61</font></i></a>, <a class="ref" href="wellord1.html#T35" target="_self">Th35</a></i>;<br/>
<a NAME="E9:61"/>
<font color="Maroon">c<sub>5</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="wellord1.html#R4" target="_self">are_isomorphic</a> 
 
<b>by </b><i><a class="txt" href="wellord1.html#E1:61"><i><font color="Green">E58</font></i></a>, <a class="ref" href="wellord1.html#T50" target="_self">Th50</a></i>;<br/>
<a NAME="E10:61"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></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>5</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="wellord1.html#R4" target="_self">are_isomorphic</a> 
 
<b>by </b><i><a class="txt" href="wellord1.html#E4:61"><i><font color="Green">E60</font></i></a>, <a class="txt" href="wellord1.html#E6:61"><i><font color="Green">E61</font></i></a>, <a class="ref" href="wellord1.html#T52" target="_self">Th52</a></i>;<br/>
<a NAME="E11:61"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="wellord1.html#K3" target="_self">canonical_isomorphism_of</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="wellord1.html#E2:61"><i><font color="Green">E59</font></i></a>, <a class="txt" href="wellord1.html#E7:61"><i><font color="Green">E62</font></i></a>, <a class="ref" href="wellord1.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E12:61"/><b>then </b><i><font color="Green">E65</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="wellord1.html#E6:61"><i><font color="Green">E61</font></i></a>, <a class="ref" href="relat_1.html#T144">RELAT_1:144</a></i>;<br/>
<a NAME="E13:61"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="wellord1.html#R4" target="_self">are_isomorphic</a> 
 
<b>by </b><i><a class="txt" href="wellord1.html#E8:61"><i><font color="Green">E63</font></i></a>, <a class="txt" href="wellord1.html#E10:61"><i><font color="Green">E64</font></i></a>, <a class="ref" href="wellord1.html#T29" target="_self">Th29</a></i>;<br/>
<a NAME="E14:61"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="wellord1.html#E6:61"><i><font color="Green">E61</font></i></a>, <a class="ref" href="wellord1.html#T19" target="_self">Th19</a></i>;<br/>
<b>hence </b><a NAME="E15:61"/>
<font color="Maroon">c<sub>5</sub></font> <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>5</sub></font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="wellord1.html#E1:61"><i><font color="Green">E58</font></i></a>, <a class="txt" href="wellord1.html#E8:61"><i><font color="Green">E63</font></i></a>, <a class="txt" href="wellord1.html#E12:61"><i><font color="Green">E65</font></i></a>, <a class="txt" href="wellord1.html#E13:61"><i><font color="Green">E66</font></i></a>, <a class="ref" href="wellord1.html#T58" target="_self">Th58</a></i>;<br/>

<b>hence </b><a NAME="E16:61"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="wellord1.html#E1:61"><i><font color="Green">E58</font></i></a>, <a class="ref" href="wellord1.html#T37" target="_self">Th37</a></i>;<br/>


</div>
