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



<b>assume </b><a NAME="E1:63"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> &amp; <font color="Maroon">R</font> is <a href="wellord1.html#V2" target="_self">well-ordering</a> )
 ;<br/>

<a NAME="E2:63"/><b>then </b><i><font color="Green">E20</font></i>: 
<font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <font color="Maroon">Y</font> is <a href="wellord1.html#V2" target="_self">well-ordering</a>
 
<b>by </b><i><a class="ref" href="wellord1.html#T2" target="_self">Th2</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">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E1:63_1"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <font color="Maroon">Y</font></span>)</span> &amp; <font color="Maroon">R</font>,<span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <font color="Maroon">Y</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">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <font color="Maroon">Y</font></span>)</span> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font></span>)</span> <a href="wellord1.html#R4" target="_self">are_isomorphic</a>  )
 ;<br/><a NAME="E2:63_1"/>
<font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <font color="Maroon">Y</font> is <a href="wellord1.html#V2" target="_self">well-ordering</a>
 <b>by </b><i><a class="ref" href="wellord1.html#T31" target="_self">Th31</a>, <a class="ref" href="wellord1.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E3:63_1"/><b>then </b><i><font color="Green">E28</font></i>: 
(  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <font color="Maroon">Y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</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">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <font color="Maroon">Y</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">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <font color="Maroon">Y</font></span>)</span> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <font color="Maroon">Y</font></span>)</span> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="wellord1.html#T31" target="_self">Th31</a>, <a class="ref" href="wellord1.html#T10" target="_self">Th10</a>, <a class="txt" href="wellord1.html#E5:63_1"><i><font color="Green">E36</font></i></a></i>;<br/><b>consider </b><font color="Maroon">F</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:63_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">F</font> <a href="wellord1.html#R3" target="_self">is_isomorphism_of</a> <font color="Maroon">R</font>,<span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <font color="Maroon">Y</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">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <font color="Maroon">Y</font></span>)</span> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="wellord1.html#T33" target="_self">Th33</a>, </i>;<br/><a NAME="E6:63_1"/><i><font color="Green">E37</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <font color="Maroon">Y</font></span>)</span> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="wellord1.html#T35" target="_self">Th35</a>, <a class="ref" href="wellord1.html#T36" target="_self">Th36</a>, <a class="txt" href="wellord1.html#E10:63_1"><i><font color="Green">E40</font></i></a></i>;<br/><a NAME="E7:63_1"/>
<span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <font color="Maroon">Y</font></span>)</span> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="wellord1.html#T35" target="_self">Th35</a>, </i>;<br/><a NAME="E8:63_1"/><b>then </b><i><font color="Green">E38</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="wellord1.html#T31" target="_self">Th31</a>, <a class="txt" href="wellord1.html#E13:63_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</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:63_1_1"/><i><font color="Green">E39</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</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">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</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">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <font color="Maroon">Y</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">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <font color="Maroon">Y</font></span>)</span> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="wellord1.html#T36" target="_self">Th36</a>, <a class="txt" href="wellord1.html#E12:63_1"><i><font color="Green">E41</font></i></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">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <font color="Maroon">Y</font>
 <b>by </b><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">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="wellord1.html#T36" target="_self">Th36</a>, , , <a class="txt" href="wellord1.html#E12:63_1"><i><font color="Green">E41</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E10:63_1"/><b>then </b><i><font color="Green">E40</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="wellord1.html#T31" target="_self">Th31</a>, <a class="ref" href="wellord1.html#T33" target="_self">Th33</a>, <a class="ref" href="wellord1.html#T35" target="_self">Th35</a>, <a class="txt" href="wellord1.html#E13:63_1"><i><font color="Green">E64</font></i></a>, , </i>;<br/><a NAME="E11:63_1"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="wellord1.html#T31" target="_self">Th31</a>, <a class="ref" href="wellord1.html#T33" target="_self">Th33</a>, <a class="ref" href="wellord1.html#T35" target="_self">Th35</a>, <a class="txt" href="wellord1.html#E13:63_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">E41</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="txt" href="wellord1.html#E13:63_1"><i><font color="Green">E64</font></i></a>, </i>;<br/><a NAME="E13:63_1"/><b>then </b><i><font color="Green">E64</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i/>;<br/><a NAME="E14:63_1"/>
<font color="Maroon">R</font> is <a href="relat_2.html#V4">antisymmetric</a>
 <b>by </b><i><a class="ref" href="wellord1.html#T31" target="_self">Th31</a>, </i>;<br/><b>hence </b><a NAME="E15:63_1"/>
contradiction
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:63"/>
( <font color="Maroon">R</font>,<font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <font color="Maroon">Y</font> <a href="wellord1.html#R4" target="_self">are_isomorphic</a>  or  ex <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> &amp; <font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Olive">a</font></span>)</span>,<font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <font color="Maroon">Y</font> <a href="wellord1.html#R4" target="_self">are_isomorphic</a>  ) )
 <b>by </b><i><a class="ref" href="wellord1.html#T31" target="_self">Th31</a>, <a class="ref" href="wellord1.html#T32" target="_self">Th32</a>, <a class="ref" href="wellord1.html#T30" target="_self">Th30</a></i>;<br/>


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