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

<b>let </b><font color="Maroon">M</font> be   <a href="card_1.html#NM1">Cardinal</a>;<br/><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:63"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/> <a href="card_1.html#K1">Card</a> <font color="Olive">Z</font> <a href="card_1.html#NR2" target="_self">&lt;`</a> <font color="Maroon">M</font>
 <b>and </b><br/><a NAME="E2:63"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">X</font> is <a href="ordinal1.html#V6">c=-linear</a>
 ;<br/>

<b>consider </b><font color="Maroon">XX</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:63"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">XX</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp;  <a href="tarski.html#K3">union</a> <font color="Maroon">XX</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">X</font> )
 <b>and </b><br/><a NAME="E5:63"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">XX</font> &amp; <font color="Olive">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/> ex <font color="Olive">Z1</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">Z1</font> <a href="hidden.html#R2">in</a> <font color="Olive">Z</font> &amp; ( for <font color="Olive">Z2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z2</font> <a href="hidden.html#R2">in</a> <font color="Olive">Z</font> holds <br/><font color="Olive">Z1</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Z2</font> ) )
 <b>by </b><i>, </i>;<br/>
<div><i><font color="Green">E58</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Z1</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">Z2</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:63_1"/>
( <font color="Maroon">Z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XX</font> &amp; <font color="Maroon">Z2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XX</font> )
 ;<br/><a NAME="E2:63_1"/><b>then </b>
<font color="Maroon">Z1</font>,<font color="Maroon">Z2</font> <a href="xboole_0.html#R3">are_c=-comparable</a> 
 <b>by </b><i>, , <a class="ref" href="ordinal1.html#D9">ORDINAL1:def 9</a></i>;<br/><b>hence </b><a NAME="E3:63_1"/>
( <font color="Maroon">Z1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z2</font> or <font color="Maroon">Z2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z1</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D9">XBOOLE_0:def 9</a></i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">R</font> being   <a href="relat_1.html#NM1">Relation</a><b> such that </b><br/><a NAME="E8:63"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">R</font> <a href="wellord1.html#R2">well_orders</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="wellord2.html#T26">WELLORD2:26</a></i>;<br/>
<a NAME="E9:63"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">R</font> <a href="relat_2.html#R1">is_reflexive_in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">X</font> &amp; <font color="Maroon">R</font> <a href="relat_2.html#R8">is_transitive_in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">X</font> &amp; <font color="Maroon">R</font> <a href="relat_2.html#R4">is_antisymmetric_in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">X</font> &amp; <font color="Maroon">R</font> <a href="relat_2.html#R6">is_connected_in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">X</font> &amp; <font color="Maroon">R</font> <a href="wellord1.html#R1">is_well_founded_in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">X</font> )
 
<b>by </b><i>, <a class="ref" href="wellord1.html#D5">WELLORD1:def 5</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>(  ex <font color="Olive">Z1</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">Z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XX</font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">Z1</font> &amp; not <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">Z1</font> ) or ( ( for <font color="Olive">Z1</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XX</font> holds <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">Z1</font> iff <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">Z1</font> ) ) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font></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="E11:63"/><i><font color="Green">E63</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>  <a href="tarski.html#K3">union</a> <font color="Maroon">X</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">X</font> &amp; <font color="Maroon">S<sub>2</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="E12:63"/><i><font color="Green">E64</font></i>: 
 <a href="relat_1.html#K3">field</a> <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:63_2"/>
 <a href="relat_1.html#K3">field</a> <font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</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="63_2_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="E1:63_2_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>
 ;<br/>

<a NAME="E2:63_2_1"/><b>then </b>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <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> or <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/>;<br/>
<b>hence </b><a NAME="E3:63_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">X</font>
 <b>by </b><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="E2:63_2"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E3:63_2"/><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">R</font> &amp; ( for <font color="Olive">Z1</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XX</font> holds <br/>( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">Z1</font> iff <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">Z1</font> ) ) )
 
<b>by </b><i>, <a class="ref" href="relat_2.html#D1">RELAT_2:def 1</a></i>;<br/>
<a NAME="E4:63_2"/><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>, </i>;<br/>
<b>hence </b><a NAME="E5:63_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>
 <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="E13:63"/>
<font color="Maroon">Q</font> is <a href="wellord1.html#V2">well-ordering</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:63_3"/>
<font color="Maroon">Q</font> is <a href="relat_2.html#V1">reflexive</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="wellord1.html#D4">WELLORD1:def 4</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_3_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="relat_2.html#D1">RELAT_2:def 1</a>,<a class="ref" href="relat_2.html#D9">RELAT_2:def 9</a><br/>

<b>assume </b><a NAME="E1:63_3_1"/><i><font color="Green">E90</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>
 ;<br/>

<a NAME="E2:63_3_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">R</font> &amp; ( for <font color="Olive">Z1</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XX</font> holds <br/>( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">Z1</font> iff <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">Z1</font> ) ) )
 
<b>by </b><i>, , <a class="ref" href="relat_2.html#D1">RELAT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:63_3_1"/>
<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>, , </i>;<br/>


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

<b>thus </b><a NAME="E2:63_3"/>
<font color="Maroon">Q</font> is <a href="relat_2.html#V8">transitive</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_3_2"><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="relat_2.html#D8">RELAT_2:def 8</a>,<a class="ref" href="relat_2.html#D16">RELAT_2:def 16</a><br/><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/>





<b>assume </b><a NAME="E1:63_3_2"/><i><font color="Green">E91</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; <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; <font color="Maroon">z</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"><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> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> )
 ;<br/>

<div><i><font color="Green">E98</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="63_3_2_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">Z1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E1:63_3_2_1"/><i><font color="Green">E99</font></i>: 
( <font color="Maroon">Z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XX</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z1</font> &amp; not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z1</font> )
 ;<br/><b>given </b><font color="Maroon">Z2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E2:63_3_2_1"/><i><font color="Green">E100</font></i>: 
( <font color="Maroon">Z2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XX</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z2</font> &amp; not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z2</font> )
 ;<br/><a NAME="E3:63_3_2_1"/>
( <font color="Maroon">Z1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z2</font> or <font color="Maroon">Z2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z1</font> )
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E4:63_3_2_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</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>, , , , </i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E101</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="63_3_2_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">Z1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E1:63_3_2_2"/><i><font color="Green">E102</font></i>: 
( <font color="Maroon">Z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XX</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z1</font> &amp; not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z1</font> )
 ;<br/><b>assume </b><a NAME="E2:63_3_2_2"/>
( ( for <font color="Olive">Z1</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XX</font> holds <br/>( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">Z1</font> iff <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Olive">Z1</font> ) ) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 ;<br/><a NAME="E3:63_3_2_2"/><b>then </b>
not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z1</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:63_3_2_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</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>, , , </i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E103</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="63_3_2_3"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">Z1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E1:63_3_2_3"/><i><font color="Green">E105</font></i>: 
( <font color="Maroon">Z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XX</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z1</font> &amp; not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z1</font> )
 ;<br/><b>assume </b><a NAME="E2:63_3_2_3"/>
( ( for <font color="Olive">Z1</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XX</font> holds <br/>( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">Z1</font> iff <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">Z1</font> ) ) &amp; <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">R</font> )
 ;<br/><a NAME="E3:63_3_2_3"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z1</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:63_3_2_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</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>, , , </i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_3_2_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:63_3_2_4"/><i><font color="Green">E106</font></i>: 
( ( for <font color="Olive">Z1</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XX</font> holds <br/>( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">Z1</font> iff <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">Z1</font> ) ) &amp; <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">R</font> &amp; ( for <font color="Olive">Z1</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XX</font> holds <br/>( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">Z1</font> iff <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Olive">Z1</font> ) ) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 ;<br/><a NAME="E2:63_3_2_4"/><b>then </b><i><font color="Green">E107</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</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>, , , <a class="ref" href="relat_2.html#D8">RELAT_2:def 8</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_3_2_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:63_3_2_4_1"/>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XX</font>
 ;<br/><a NAME="E2:63_3_2_4_1"/><b>then </b>
( ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> implies <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> ) &amp; ( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> ) &amp; ( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> implies <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> ) &amp; ( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> implies <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> ) )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:63_3_2_4_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> iff <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:63_3_2_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</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="ref" href="card_3.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:63_3_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</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>, , , , </i>;<br/>


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

<b>thus </b><a NAME="E3:63_3"/>
<font color="Maroon">Q</font> is <a href="relat_2.html#V4">antisymmetric</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_3_3"><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="relat_2.html#D4">RELAT_2:def 4</a>,<a class="ref" href="relat_2.html#D12">RELAT_2:def 12</a><br/><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:63_3_3"/><i><font color="Green">E108</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; <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"><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> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">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> )
 ;<br/>

<a NAME="E2:63_3_3"/><b>then </b><i><font color="Green">E109</font></i>: 
( (  ex <font color="Olive">Z1</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">Z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XX</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">Z1</font> &amp; not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">Z1</font> ) or ( ( for <font color="Olive">Z1</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XX</font> holds <br/>( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">Z1</font> iff <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">Z1</font> ) ) &amp; <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">R</font> ) ) &amp; (  ex <font color="Olive">Z1</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">Z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XX</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">Z1</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">Z1</font> ) or ( ( for <font color="Olive">Z1</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XX</font> holds <br/>( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">Z1</font> iff <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">Z1</font> ) ) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">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">R</font> ) ) )
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_3_3_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">Z1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E1:63_3_3_1"/><i><font color="Green">E110</font></i>: 
( <font color="Maroon">Z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XX</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z1</font> &amp; not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z1</font> )
 ;<br/><b>given </b><font color="Maroon">Z2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E2:63_3_3_1"/><i><font color="Green">E111</font></i>: 
( <font color="Maroon">Z2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XX</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z2</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z2</font> )
 ;<br/><a NAME="E3:63_3_3_1"/>
( <font color="Maroon">Z1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z2</font> or <font color="Maroon">Z2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z1</font> )
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E4:63_3_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:63_3_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i>, , , , <a class="ref" href="relat_2.html#D4">RELAT_2:def 4</a></i>;<br/>


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

<b>thus </b><a NAME="E4:63_3"/>
<font color="Maroon">Q</font> is <a href="relat_2.html#V6">connected</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_3_4"><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="relat_2.html#D6">RELAT_2:def 6</a>,<a class="ref" href="relat_2.html#D14">RELAT_2:def 14</a><br/><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:63_3_4"/><i><font color="Green">E117</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; <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; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_3_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:63_3_4_1"/><i><font color="Green">E118</font></i>: 
for <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XX</font> holds <br/>( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">Z</font> iff <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">Z</font> )
 ;<br/><a NAME="E2:63_3_4_1"/><b>then </b>
( ( <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">R</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">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">R</font> ) &amp; ( for <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XX</font> holds <br/>( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">Z</font> iff <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">Z</font> ) ) )
 <b>by </b><i>, , , <a class="ref" href="relat_2.html#D6">RELAT_2:def 6</a></i>;<br/><b>hence </b><a NAME="E3:63_3_4_1"/>
( <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> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">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>, , , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:63_3_4"/>
( <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> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">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>, , </i>;<br/>


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

<b>thus </b><a NAME="E5:63_3"/>
<font color="Maroon">Q</font> is <a href="wellord1.html#V1">well_founded</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_3_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="wellord1.html#D2">WELLORD1:def 2</a><br/>

<b>assume </b><a NAME="E1:63_3_5"/><i><font color="Green">E119</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">Q</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> ;<br/>
<b>consider </b><font color="Maroon">Z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:63_3_5"/><i><font color="Green">E120</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XX</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E4:63_3_5"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">XX</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_3_5_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>thus </b><a NAME="E1:63_3_5_1"/>
(  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XX</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>;<br/>
<a NAME="E6:63_3_5"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">XX</font>
 
<b>by </b><i>, , , <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">Z1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:63_3_5"/><i><font color="Green">E122</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z1</font> &amp; <font color="Maroon">Z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XX</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E9:63_3_5"/>
<font color="Maroon">Z1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E10:63_3_5"/><b>then </b>
<font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">Z1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:63_3_5"/><i><font color="Green">E123</font></i>: 
( <font color="Maroon">Z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; ( for <font color="Olive">Z2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> holds <br/><font color="Maroon">Z1</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Z2</font> ) )
 <b>by </b><i>, </i>;<br/>
<a NAME="E13:63_3_5"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">Z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XX</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E14:63_3_5"/><b>then </b>
( <font color="Maroon">Z1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z1</font> &amp; <font color="Maroon">Z1</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">X</font> &amp; <font color="Maroon">Z1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i>, , , <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E15:63_3_5"/><b>then </b>
( <font color="Maroon">Z1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">X</font> &amp; <font color="Maroon">Z1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E17:63_3_5"/><i><font color="Green">E135</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font> &amp; ( for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font> holds <br/><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">R</font> ) )
 <b>by </b><i>, <a class="ref" href="wellord1.html#T9">WELLORD1:9</a></i>;<br/>
<b>take </b>
<font color="Maroon">x</font>
;<br/>

<b>thus </b><a NAME="E18:63_3_5"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="txt" href="card_3.html#E1:63"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>

<b>assume </b><a NAME="E19:63_3_5"/><i><font color="Green">E136</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a><br/>

<b>consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font>;<br/>
<a NAME="E21:63_3_5"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z1</font>
 
<b>by </b><i><a class="txt" href="card_3.html#E1:63"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E22:63_3_5"/><i><font color="Green">E138</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E23:63_3_5"/><b>then </b><i><font color="Green">E139</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">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> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 
<b>by </b><i><a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_3_5_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">Z2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E1:63_3_5_2"/><i><font color="Green">E140</font></i>: 
( <font color="Maroon">Z2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XX</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z2</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z2</font> )
 ;<br/><a NAME="E2:63_3_5_2"/>
<font color="Maroon">Z2</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="card_3.html#D2" target="_self">Def2</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:63_3_5_2"/><b>then </b>
<font color="Maroon">Z2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i>, <a class="ref" href="card_3.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E4:63_3_5_2"/><b>then </b>
<font color="Maroon">Z1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z2</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:63_3_5_2"/>
contradiction
 <b>by </b><i><a class="txt" href="card_3.html#E2:63"><i><font color="Green">E47</font></i></a>, <a class="ref" href="card_3.html#D2" target="_self">Def2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E25:63_3_5"/><b>then </b><i><font color="Green">E141</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">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">R</font> )
 
<b>by </b><i>, <a class="ref" href="card_3.html#D1" target="_self">Def1</a>, <a class="txt" href="card_3.html#E2:63"><i><font color="Green">E47</font></i></a>, </i>;<br/>
<a NAME="E26:63_3_5"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E27:63_3_5"/><b>then </b>
( <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">R</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">X</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">X</font> )
 
<b>by </b><i>, <a class="txt" href="card_3.html#E1:63"><i><font color="Green">E45</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E28:63_3_5"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="card_3.html#D2" target="_self">Def2</a>, <a class="ref" href="relat_2.html#D4">RELAT_2:def 4</a></i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E14:63"/><b>then </b>
<font color="Maroon">Q</font>, <a href="wellord2.html#K1">RelIncl</a> <span class="p1">(<span class="default"><a href="wellord2.html#K2">order_type_of</a> <font color="Maroon">Q</font></span>)</span> <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/>
<a NAME="E15:63"/><b>then </b>
 <a href="wellord2.html#K1">RelIncl</a> <span class="p1">(<span class="default"><a href="wellord2.html#K2">order_type_of</a> <font color="Maroon">Q</font></span>)</span>,<font color="Maroon">Q</font> <a href="wellord1.html#R4">are_isomorphic</a> 
 
<b>by </b><i><a class="ref" href="wellord1.html#T50">WELLORD1:50</a></i>;<br/>
<b>then consider </b><font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E17:63"/><i><font color="Green">E142</font></i>: 
<font color="Maroon">g</font> <a href="wellord1.html#R3">is_isomorphism_of</a>  <a href="wellord2.html#K1">RelIncl</a> <span class="p1">(<span class="default"><a href="wellord2.html#K2">order_type_of</a> <font color="Maroon">Q</font></span>)</span>,<font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="wellord1.html#D8">WELLORD1:def 8</a></i>;<br/>
<a NAME="E18:63"/>
 <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <span class="p2">(<span class="default"><a href="wellord2.html#K2">order_type_of</a> <font color="Maroon">Q</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="wellord2.html#K2">order_type_of</a> <font color="Maroon">Q</font>
 
<b>by </b><i><a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/>
<a NAME="E19:63"/><b>then </b><i><font color="Green">E143</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="wellord2.html#K2">order_type_of</a> <font color="Maroon">Q</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">X</font> &amp; <font color="Maroon">g</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; ( 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>  <a href="wellord2.html#K1">RelIncl</a> <span class="p1">(<span class="default"><a href="wellord2.html#K2">order_type_of</a> <font color="Maroon">Q</font></span>)</span> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="wellord2.html#K2">order_type_of</a> <font color="Maroon">Q</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="wellord2.html#K2">order_type_of</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">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font></span>)</span></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="ref" href="wellord1.html#D7">WELLORD1:def 7</a></i>;<br/>
<a NAME="E20:63"/><i><font color="Green">E144</font></i>: 
for <font color="Olive">Z</font>, <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XX</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">Z</font> holds <br/><font color="Maroon">Q</font> <a href="wellord1.html#K1">-Seg</a> <font color="Olive">x</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_4"><b>proof </b></a><div class="add">

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



<b>assume </b><a NAME="E1:63_4"/><i><font color="Green">E145</font></i>: 
( <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XX</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 ;<br/>

<b>let </b><font color="Maroon">y</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="E2:63_4"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x</font>
 ;<br/>

<a NAME="E3:63_4"/><b>then </b><i><font color="Green">E146</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">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="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_4_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">Z1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E1:63_4_1"/><i><font color="Green">E147</font></i>: 
( <font color="Maroon">Z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XX</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z1</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z1</font> )
 ;<br/><a NAME="E2:63_4_1"/>
( <font color="Maroon">Z1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font> or <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z1</font> )
 <b>by </b><i>, , <a class="ref" href="card_3.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E3:63_4_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i>, <a class="ref" href="card_3.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:63_4"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E21:63"/><i><font color="Green">E148</font></i>: 
for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a>  <a href="wellord2.html#K2">order_type_of</a> <font color="Maroon">Q</font> holds <br/> <a href="card_1.html#K1">Card</a> <font color="Olive">A</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="wellord1.html#K1">-Seg</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">A</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E1:63_5"/><i><font color="Green">E149</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="wellord2.html#K2">order_type_of</a> <font color="Maroon">Q</font>
 ;<br/>

<a NAME="E2:63_5"/>
<font color="Maroon">A</font>,<font color="Maroon">Q</font> <a href="wellord1.html#K1">-Seg</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font></span>)</span> <a href="wellord2.html#R2">are_equipotent</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_5_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon">f</font> = <font color="Maroon">g</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">A</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="wellord2.html#D4">WELLORD2:def 4</a><br/>

<a NAME="E1:63_5_1"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>
 
<b>by </b><i>, , <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E2:63_5_1"/><i><font color="Green">E150</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">A</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#T84">FUNCT_1:84</a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>

<b>thus </b><a NAME="E3:63_5_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font> <a href="wellord1.html#K1">-Seg</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font></span>)</span>
  <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="63_5_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="E1:63_5_1_1"/>
<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>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:63_5_1_1"/><i><font color="Green">E151</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">B</font> = <font color="Maroon">y</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="ref" href="card_3.html#D1" target="_self">Def1</a>, <a class="txt" href="card_3.html#E1:63"><i><font color="Green">E45</font></i></a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E5:63_5_1_1"/><i><font color="Green">E152</font></i>: 
( <font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="wellord2.html#K2">order_type_of</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">B</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, <a class="ref" href="card_3.html#D1" target="_self">Def1</a>, <a class="txt" href="card_3.html#E1:63"><i><font color="Green">E45</font></i></a>, <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E6:63_5_1_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">A</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="wellord2.html#K1">RelIncl</a> <span class="p1">(<span class="default"><a href="wellord2.html#K2">order_type_of</a> <font color="Maroon">Q</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">B</font> )
 
<b>by </b><i>, <a class="ref" href="card_3.html#D1" target="_self">Def1</a>, <a class="txt" href="card_3.html#E1:63"><i><font color="Green">E45</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a>, <a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/>
<a NAME="E7:63_5_1_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">g</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">Q</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E8:63_5_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> <a href="wellord1.html#K1">-Seg</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i><a class="ref" href="wellord1.html#D1">WELLORD1:def 1</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="E4:63_5_1"/><i><font color="Green">E153</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> <a href="wellord1.html#K1">-Seg</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font></span>)</span>
 ;<br/>

<a NAME="E5:63_5_1"/><b>then </b><i><font color="Green">E154</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">g</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">Q</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> )
 
<b>by </b><i><a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/>
<a NAME="E6:63_5_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:63_5_1"/><i><font color="Green">E155</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">B</font> = <font color="Maroon">y</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i>, <a class="txt" href="card_3.html#E2:63"><i><font color="Green">E47</font></i></a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E10:63_5_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">A</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="wellord2.html#K1">RelIncl</a> <span class="p1">(<span class="default"><a href="wellord2.html#K2">order_type_of</a> <font color="Maroon">Q</font></span>)</span>
 
<b>by </b><i>, , , <a class="txt" href="card_3.html#E2:63"><i><font color="Green">E47</font></i></a></i>;<br/>
<a NAME="E11:63_5_1"/><b>then </b>
( <font color="Maroon">B</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, , <a class="txt" href="card_3.html#E1:63"><i><font color="Green">E45</font></i></a>, <a class="txt" href="card_3.html#E2:63"><i><font color="Green">E47</font></i></a>, <a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a>, <a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/>
<a NAME="E12:63_5_1"/><b>then </b>
<font color="Maroon">B</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/>
<a NAME="E13:63_5_1"/><b>then </b>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T21">ORDINAL1:21</a></i>;<br/>
<b>hence </b><a NAME="E14:63_5_1"/>
<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="txt" href="card_3.html#E2:63"><i><font color="Green">E47</font></i></a>, <a class="ref" href="funct_1.html#T73">FUNCT_1:73</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:63_5"/>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="wellord1.html#K1">-Seg</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="card_1.html#T21">CARD_1:21</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E22:63"/><i><font color="Green">E156</font></i>: 
 <a href="wellord2.html#K2">order_type_of</a> <font color="Maroon">Q</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">M</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_6"><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="E1:63_6"/><i><font color="Green">E157</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="wellord2.html#K2">order_type_of</a> <font color="Maroon">Q</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">A</font> = <font color="Maroon">x</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E3:63_6"/>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="card_3.html#D1" target="_self">Def1</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">Z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:63_6"/><i><font color="Green">E158</font></i>: 
( <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XX</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E6:63_6"/>
<font color="Maroon">Q</font> <a href="wellord1.html#K1">-Seg</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font>
 
<b>by </b><i>, <a class="txt" href="card_3.html#E1:63"><i><font color="Green">E45</font></i></a></i>;<br/>
<a NAME="E7:63_6"/><b>then </b>
(  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="wellord1.html#K1">-Seg</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">Z</font> &amp;  <a href="card_1.html#K1">Card</a> <font color="Maroon">Z</font> <a href="card_1.html#NR2" target="_self">&lt;`</a> <font color="Maroon">M</font> )
 
<b>by </b><i>, , <a class="txt" href="card_3.html#E1:63"><i><font color="Green">E45</font></i></a>, <a class="ref" href="card_1.html#T27">CARD_1:27</a></i>;<br/>
<a NAME="E8:63_6"/><b>then </b>
(  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="wellord1.html#K1">-Seg</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">A</font> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="wellord1.html#K1">-Seg</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="card_1.html#NR2" target="_self">&lt;`</a> <font color="Maroon">M</font> )
 
<b>by </b><i>, <a class="ref" href="card_3.html#D1" target="_self">Def1</a>, <a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a></i>;<br/>
<b>hence </b><a NAME="E9:63_6"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E23:63"/>
 <a href="wellord2.html#K2">order_type_of</a> <font color="Maroon">Q</font>, <a href="tarski.html#K3">union</a> <font color="Maroon">X</font> <a href="wellord2.html#R2">are_equipotent</a> 
 
<b>by </b><i>, <a class="ref" href="wellord2.html#D4">WELLORD2:def 4</a></i>;<br/>
<a NAME="E24:63"/><b>then </b>
(  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="wellord2.html#K2">order_type_of</a> <font color="Maroon">Q</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">X</font></span>)</span> &amp;  <a href="card_1.html#K1">Card</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> )
 
<b>by </b><i><a class="ref" href="card_1.html#T21">CARD_1:21</a>, <a class="ref" href="card_1.html#D5">CARD_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E25:63"/>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">X</font></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a> <font color="Maroon">M</font>
 <b>by </b><i>, <a class="ref" href="card_1.html#T27">CARD_1:27</a></i>;<br/>


</div>
