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

<b>let </b><font color="Maroon">X</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="E1:21"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">R</font> is <a href="wellord1.html#V2">well-ordering</a>
 ;<br/>

<a NAME="E2:21"/><b>then </b>
<font color="Maroon">R</font> is <a href="relat_2.html#V1">reflexive</a>
 
<b>by </b><i><a class="ref" href="wellord1.html#D4">WELLORD1:def 4</a></i>;<br/>
<a NAME="E3:21"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">R</font> <a href="relat_2.html#R1">is_reflexive_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="relat_2.html#D9">RELAT_2:def 9</a></i>;<br/>
<b>given </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><a NAME="E4:21"/><i><font color="Green">E23</font></i>: 
( <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>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="wellord2.html#D4" target="_self">WELLORD2:def 4</a><br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </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">a<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>2</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>;<br/>
<b>consider </b><font color="Maroon">Q</font> being   <a href="relat_1.html#NM1">Relation</a><b> such that </b><br/><a NAME="E6:21"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</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">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] ) )
 <b>from </b><i><a class="ref" href="relat_1.html#S1">RELAT_1:sch 1</a>();<br/></i>
<a NAME="E7:21"/><i><font color="Green">E29</font></i>: 
 <a href="relat_1.html#K3">field</a> <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:21_1"/>
 <a href="relat_1.html#K3">field</a> <font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:21_1_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">Q</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 ;<br/>

<a NAME="E2:21_1_1"/><b>then </b>
( ( for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   holds not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> ) &amp; ( for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   holds not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> ) )
 
<b>by </b><i><a class="txt" href="wellord2.html#E1:21"><i><font color="Green">E21</font></i></a></i>;<br/>
<a NAME="E3:21_1_1"/><b>then </b>
( not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Q</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Q</font> )
 
<b>by </b><i><a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a>, <a class="ref" href="relat_1.html#D5">RELAT_1:def 5</a></i>;<br/>
<a NAME="E4:21_1_1"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">Q</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">Q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E5:21_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="wellord2.html#E4:21"><i><font color="Green">E23</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:21_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E3:21_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</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="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:21_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">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</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="wellord2.html#D1" target="_self">Def1</a>, , <a class="ref" href="relat_2.html#D1">RELAT_2:def 1</a></i>;<br/>
<a NAME="E5:21_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 
<b>by </b><i><a class="txt" href="wellord2.html#E1:21"><i><font color="Green">E21</font></i></a>, <a class="txt" href="wellord2.html#E4:21"><i><font color="Green">E23</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:21_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:21"/>
<font color="Maroon">f</font> <a href="wellord1.html#R3">is_isomorphism_of</a> <font color="Maroon">Q</font>,<font color="Maroon">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:21_2"/>
(  <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">Q</font> &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> <font color="Maroon">R</font> &amp; <font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a> )
 <b>by </b><i>, <a class="txt" href="wellord2.html#E3:21"><i><font color="Green">E22</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="wellord1.html#D7">WELLORD1:def 7</a><br/>

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



<b>thus </b><a NAME="E2:21_2"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> implies ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">Q</font> &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="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</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="txt" href="wellord2.html#E1:21"><i><font color="Green">E21</font></i></a>, <a class="txt" href="wellord2.html#E3:21"><i><font color="Green">E22</font></i></a></i>;<br/>

<b>assume </b><a NAME="E3:21_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">Q</font> &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="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 ;<br/>

<b>hence </b><a NAME="E4:21_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="txt" href="wellord2.html#E1:21"><i><font color="Green">E21</font></i></a>, <a class="txt" href="wellord2.html#E3:21"><i><font color="Green">E22</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:21"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K2">"</a>  <a href="wellord1.html#R3">is_isomorphism_of</a> <font color="Maroon">R</font>,<font color="Maroon">Q</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#T49">WELLORD1:49</a></i>;<br/>
<a NAME="E10:21"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">Q</font> is <a href="wellord1.html#V2">well-ordering</a>
 
<b>by </b><i>, <a class="ref" href="wellord1.html#T54">WELLORD1:54</a></i>;<br/>
<b>take </b>
<font color="Maroon">Q</font>
;<br/>

<b>thus </b><a NAME="E11:21"/>
<font color="Maroon">Q</font> <a href="wellord1.html#R2">well_orders</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="wellord2.html#E3:21"><i><font color="Green">E22</font></i></a>, <a class="txt" href="wellord2.html#E4:21"><i><font color="Green">E23</font></i></a>, <a class="ref" href="wellord1.html#T8">WELLORD1:8</a></i>;<br/>


</div>
