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<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>consider </b><font color="Maroon">R</font> being   <a href="relat_1.html#NM1">Relation</a><b> such that </b><br/><a NAME="E2:6"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">R</font> <a href="wellord1.html#R2">well_orders</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="wellord2.html#T26">WELLORD2:26</a></i>;<br/>
<b>set </b><font color="Maroon">Q</font> = <font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">X</font>;<br/>
<a NAME="E3:6"/><i><font color="Green">E34</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">|_2</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp; <font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">X</font> is <a href="wellord1.html#V2">well-ordering</a> )
 
<b>by </b><i>, <a class="ref" href="wellord2.html#T25">WELLORD2:25</a></i>;<br/>
<b>take </b><font color="Maroon">A</font> =  <a href="wellord2.html#K2">order_type_of</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">X</font></span>)</span>;<br/>

<a NAME="E4:6"/>
<font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">X</font>, <a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">A</font> <a href="wellord1.html#R4">are_isomorphic</a> 
 
<b>by </b><i>, <a class="ref" href="wellord2.html#D2">WELLORD2:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:6"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">f</font> <a href="wellord1.html#R3">is_isomorphism_of</a> <font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">X</font>, <a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="wellord1.html#D8">WELLORD1:def 8</a></i>;<br/>
<a NAME="E7:6"/><i><font color="Green">E43</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> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">X</font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">A</font></span>)</span> &amp; <font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</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">|_2</a> <font color="Maroon">X</font> iff ( <font color="Olive">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">|_2</a> <font color="Maroon">X</font></span>)</span> &amp; <font color="Olive">b</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">|_2</a> <font color="Maroon">X</font></span>)</span> &amp; <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="Olive">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">b</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="wellord2.html#K1">RelIncl</a> <font color="Maroon">A</font> ) ) ) )
 
<b>by </b><i>, <a class="ref" href="wellord1.html#D7">WELLORD1:def 7</a></i>;<br/>
<b>take </b>
<font color="Maroon">f</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="wellord2.html#D4">WELLORD2:def 4</a><br/>

<b>thus </b><a NAME="E8:6"/>
( <font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> )
 <b>by </b><i>, , <a class="ref" href="wellord2.html#T25">WELLORD2:25</a>, <a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/>


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