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<b>let </b><font color="Maroon" title="c1">Z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">R</font>, <font color="Maroon" title="c3">S</font> be   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a>;<br/><b>let </b><font color="Maroon" title="c4">F</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>;<br/>







<b>assume </b><a NAME="E1:58"/><i><font color="Green" title="E54">A1</font></i>: 
( <font color="Maroon" title="c2">R</font> is <a href="wellord1.html#V2" title="WELLORD1:attr.2">well-ordering</a> &amp; <font color="Maroon" title="c1">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="c2">R</font> &amp; <font color="Maroon" title="c4">F</font> <a href="wellord1.html#R3" title="WELLORD1:pred.3">is_isomorphism_of</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">S</font> )
 ;<br/>

<a NAME="E2:58"/><b>then </b><i><font color="Green" title="E55">A2</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">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> <font color="Maroon" title="c2">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="c3">S</font> &amp; <font color="Maroon" title="c4">F</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; ( 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="c2">R</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="c2">R</font> &amp; <font color="Olive" title="b2">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">R</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="Olive" title="b1">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="Olive" title="b2">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="c3">S</font> ) ) ) )
 
<b>by </b><i><a class="ref" href="wellord1.html#D7" target="_self" title="WELLORD1:def.7">Def7</a></i>;<br/>
<a NAME="E3:58"/><i><font color="Green" title="E56">A3</font></i>: 
<font color="Maroon" title="c1">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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c1">Z</font></span>)</span>
 
<b>by </b><i><a class="txt" href="wellord1.html#E1:58"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="wellord1.html#T39" target="_self" title="WELLORD1:th.39">Th39</a></i>;<br/>
<a NAME="E4:58"/>
( <font color="Maroon" title="c4">F</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c1">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c3">S</font> is <a href="wellord1.html#V2" title="WELLORD1:attr.2">well-ordering</a> )
 
<b>by </b><i><a class="txt" href="wellord1.html#E1:58"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="wellord1.html#T54" target="_self" title="WELLORD1:th.54">Th54</a>, <a class="ref" href="relat_1.html#T144" title="RELAT_1:th.144">RELAT_1:144</a></i>;<br/>
<a NAME="E5:58"/><b>then </b><i><font color="Green" title="E57">A4</font></i>: 
<font color="Maroon" title="c4">F</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c1">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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c1">Z</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="wellord1.html#E2:58"><i><font color="Green" title="E55">A2</font></i></a>, <a class="ref" href="wellord1.html#T39" target="_self" title="WELLORD1:th.39">Th39</a></i>;<br/>
<b>thus </b><a NAME="E6:58"/>
<font color="Maroon" title="c4">F</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">Z</font> <a href="wellord1.html#R3" title="WELLORD1:pred.3">is_isomorphism_of</a> <font color="Maroon" title="c2">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c1">Z</font>,<font color="Maroon" title="c3">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c1">Z</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:58_1"/><i><font color="Green" title="E58">A5</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">Z</font></span>)</span> <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">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c1">Z</font></span>)</span>
 <b>by </b><i><a class="txt" href="wellord1.html#E1:58"><i><font color="Green" title="E54">A1</font></i></a>, <a class="txt" href="wellord1.html#E2:58"><i><font color="Green" title="E55">A2</font></i></a>, <a class="txt" href="wellord1.html#E3:58"><i><font color="Green" title="E56">A3</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</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="E2:58_1"/><i><font color="Green" title="E59">A6</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">Z</font></span>)</span> <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="c3">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c1">Z</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="wellord1.html#E5:58"><i><font color="Green" title="E57">A4</font></i></a>, <a class="ref" href="relat_1.html#T148" title="RELAT_1:th.148">RELAT_1:148</a></i>;<br/>

<b>thus </b><a NAME="E3:58_1"/>
<font color="Maroon" title="c4">F</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">Z</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="txt" href="wellord1.html#E2:58"><i><font color="Green" title="E55">A2</font></i></a>, <a class="ref" href="funct_1.html#T84" title="FUNCT_1:th.84">FUNCT_1:84</a></i>;<br/>

<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="E4:58_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="c2">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c1">Z</font> 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="c2">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c1">Z</font></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="c2">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c1">Z</font></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"><span class="p3">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">Z</font></span>)</span> <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"><span class="p3">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">Z</font></span>)</span> <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="c3">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c1">Z</font></span>)</span> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:58_1_1"/><i><font color="Green" title="E60">A7</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="c2">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c1">Z</font>
 ;<br/>

<b>hence </b><a NAME="E2:58_1_1"/><i><font color="Green" title="E61">A8</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="c2">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c1">Z</font></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="c2">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c1">Z</font></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="E3:58_1_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="c2">R</font>
 
<b>by </b><i><a class="txt" href="wellord1.html#E1:58_1_1"><i><font color="Green" title="E60">A7</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="E4:58_1_1"/><b>then </b><i><font color="Green" title="E62">A9</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="c3">S</font>
 
<b>by </b><i><a class="txt" href="wellord1.html#E1:58"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="wellord1.html#D7" target="_self" title="WELLORD1:def.7">Def7</a></i>;<br/>
<a NAME="E5:58_1_1"/><i><font color="Green" title="E63">A10</font></i>: 
( <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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">Z</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</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#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">Z</font></span>)</span> <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#E1:58_1"><i><font color="Green" title="E58">A5</font></i></a>, <a class="txt" href="wellord1.html#E2:58_1_1"><i><font color="Green" title="E61">A8</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>
<a NAME="E6:58_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">Z</font></span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">Z</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">Z</font></span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">Z</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="wellord1.html#E1:58_1"><i><font color="Green" title="E58">A5</font></i></a>, <a class="txt" href="wellord1.html#E2:58_1_1"><i><font color="Green" title="E61">A8</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="E7:58_1_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"><span class="p3">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">Z</font></span>)</span> <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"><span class="p3">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">Z</font></span>)</span> <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"><font color="Maroon" title="c4">F</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c1">Z</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c1">Z</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="wellord1.html#E5:58"><i><font color="Green" title="E57">A4</font></i></a>, <a class="txt" href="wellord1.html#E2:58_1"><i><font color="Green" title="E59">A6</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><a NAME="E8:58_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">Z</font></span>)</span> <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"><span class="p3">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">Z</font></span>)</span> <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="c3">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c1">Z</font></span>)</span>
 <b>by </b><i><a class="txt" href="wellord1.html#E4:58_1_1"><i><font color="Green" title="E62">A9</font></i></a>, <a class="txt" href="wellord1.html#E5:58_1_1"><i><font color="Green" title="E63">A10</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>assume </b><a NAME="E5:58_1"/><i><font color="Green" title="E60">A11</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="c2">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c1">Z</font></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="c2">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c1">Z</font></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"><span class="p3">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">Z</font></span>)</span> <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"><span class="p3">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">Z</font></span>)</span> <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="c3">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c1">Z</font></span>)</span> )
 ;<br/>

<a NAME="E6:58_1"/><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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">Z</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</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#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">Z</font></span>)</span> <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#E1:58_1"><i><font color="Green" title="E58">A5</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>
<a NAME="E7:58_1"/><b>then </b><i><font color="Green" title="E61">A12</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="c3">S</font>
 
<b>by </b><i><a class="txt" href="wellord1.html#E5:58_1"><i><font color="Green" title="E60">A11</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="E8:58_1"/>
( <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">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">R</font> )
 
<b>by </b><i><a class="txt" href="wellord1.html#E5:58_1"><i><font color="Green" title="E60">A11</font></i></a>, <a class="ref" href="wellord1.html#T19" target="_self" title="WELLORD1:th.19">Th19</a></i>;<br/>
<a NAME="E9:58_1"/><b>then </b><i><font color="Green" title="E62">A13</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="c2">R</font>
 
<b>by </b><i><a class="txt" href="wellord1.html#E1:58"><i><font color="Green" title="E54">A1</font></i></a>, <a class="txt" href="wellord1.html#E7:58_1"><i><font color="Green" title="E61">A12</font></i></a>, <a class="ref" href="wellord1.html#D7" target="_self" title="WELLORD1:def.7">Def7</a></i>;<br/>
<a NAME="E10:58_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> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">Z</font>,<font color="Maroon" title="c1">Z</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="wellord1.html#E3:58"><i><font color="Green" title="E56">A3</font></i></a>, <a class="txt" href="wellord1.html#E5:58_1"><i><font color="Green" title="E60">A11</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><a NAME="E11:58_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="c2">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c1">Z</font>
 <b>by </b><i><a class="txt" href="wellord1.html#E9:58_1"><i><font color="Green" title="E62">A13</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="E7:58"/>
<font color="Maroon" title="c2">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c1">Z</font>,<font color="Maroon" title="c3">S</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c1">Z</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#D8" target="_self" title="WELLORD1:def.8">Def8</a></i>;<br/>


</div>
