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

<b>let </b><font color="Maroon" title="c1">F</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>;<br/><b>let </b><font color="Maroon" title="c2">D</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><a NAME="E1:6"/><i><font color="Green" title="E6">A1</font></i>: 
for <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font> holds <br/>( not <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> &amp; <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">D</font> )
 ;<br/>

<b>set </b><font color="Maroon" title="c3">W0</font> =  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</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"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">D</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a>] <b>means </b>( $1 is <a href="wellord1.html#V2" title="WELLORD1:attr.2">well-ordering</a> &amp; ( for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</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> $1 holds <br/>( $1 <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font> &amp; <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">$1 <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b1">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</font> ) ) );<br/>
<b>consider </b><font color="Maroon" title="c4">G</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:6"/><i><font color="Green" title="E7">A2</font></i>: 
for <font color="Olive" title="b1">W</font> being  <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a> holds <br/> ( <font color="Olive" title="b1">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G</font> iff ( <font color="Olive" title="b1">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</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"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">D</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">W</font>] ) )
 <b>from </b><i><a class="ref" href="wellset1.html#S1" target="_self" title="WELLSET1:sch.1">WELLSET1:sch 1</a>();<br/></i>
<a NAME="E4:6"/><i><font color="Green" title="E8">A3</font></i>: 
for <font color="Olive" title="b1">W1</font>, <font color="Olive" title="b2">W2</font> being  <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a>  st <font color="Olive" title="b1">W1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G</font> &amp; <font color="Olive" title="b2">W2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G</font> &amp; not <!--HUMANRECFAILED-->( <font color="Olive" title="b1">W1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">W2</font> &amp; ( for <font color="Olive" title="b3">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">x</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="Olive" title="b1">W1</font> holds <br/><font color="Olive" title="b1">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b3">x</font> ) ) holds <br/>( <font color="Olive" title="b2">W2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">W1</font> &amp; ( for <font color="Olive" title="b3">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">x</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="Olive" title="b2">W2</font> holds <br/><font color="Olive" title="b2">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b3">x</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">W1</font>, <font color="Maroon" title="c6">W2</font> be   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a>;<br/>



<b>assume </b><a NAME="E1:6_1"/><i><font color="Green" title="E9">A4</font></i>: 
( <font color="Maroon" title="c5">W1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G</font> &amp; <font color="Maroon" title="c6">W2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G</font> )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( $1 <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="c6">W2</font> &amp; <font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> $1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> $1</span>)</span> );<br/>
<b>consider </b><font color="Maroon" title="c7">C</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:6_1"/><i><font color="Green" title="E10">A5</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font> iff ( <font color="Olive" title="b1">x</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="c5">W1</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E4:6_1"/><i><font color="Green" title="E11">A6</font></i>: 
( <font color="Maroon" title="c5">W1</font> is <a href="wellord1.html#V2" title="WELLORD1:attr.2">well-ordering</a> &amp; <font color="Maroon" title="c6">W2</font> is <a href="wellord1.html#V2" title="WELLORD1:attr.2">well-ordering</a> )
 
<b>by </b><i><a class="txt" href="wellset1.html#E3:6"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="wellset1.html#E1:6_1"><i><font color="Green" title="E9">A4</font></i></a></i>;<br/>
<a NAME="E5:6_1"/><i><font color="Green" title="E12">A7</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font> holds <br/><font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:6_1_1"/><i><font color="Green" title="E13">A8</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font>
 ;<br/>

<a NAME="E2:6_1_1"/>
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">x</font> iff <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">x</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<a NAME="E1:6_1_1_1"/><i><font color="Green" title="E14">A9</font></i>: 
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">x</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E4:6_1"><i><font color="Green" title="E11">A6</font></i></a>, <a class="ref" href="wellord1.html#T40" title="WELLORD1:th.40">WELLORD1:40</a></i>;<br/>
<a NAME="E2:6_1_1_1"/>
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">x</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E4:6_1"><i><font color="Green" title="E11">A6</font></i></a>, <a class="ref" href="wellord1.html#T40" title="WELLORD1:th.40">WELLORD1:40</a></i>;<br/>
<b>hence </b><a NAME="E3:6_1_1_1"/>
( <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">x</font> iff <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">x</font> )
 <b>by </b><i><a class="txt" href="wellset1.html#E3:6_1"><i><font color="Green" title="E10">A5</font></i></a>, <a class="txt" href="wellset1.html#E1:6_1_1"><i><font color="Green" title="E13">A8</font></i></a>, <a class="txt" href="wellset1.html#E1:6_1_1_1"><i><font color="Green" title="E14">A9</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:6_1_1"/>
<font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">x</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:6_1"/><i><font color="Green" title="E13">A10</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font> holds <br/><font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b1">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:6_1_2"/><i><font color="Green" title="E14">A11</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font>
 ;<br/>

<a NAME="E2:6_1_2"/>
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">x</font> holds <br/><font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:6_1_2_1"/><i><font color="Green" title="E15">A12</font></i>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">x</font>
 ;<br/>

<a NAME="E2:6_1_2_1"/><b>then </b><i><font color="Green" title="E16">A13</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c8">x</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="c5">W1</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#D1" title="WELLORD1:def.1">WELLORD1:def 1</a></i>;<br/>
<a NAME="E3:6_1_2_1"/><b>then </b><i><font color="Green" title="E17">A14</font></i>: 
<font color="Maroon" title="c9">y</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="c5">W1</font>
 
<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="E4:6_1_2_1"/><i><font color="Green" title="E18">A15</font></i>: 
<font color="Maroon" title="c8">x</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="c5">W1</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E2:6_1_2_1"><i><font color="Green" title="E16">A13</font></i></a>, <a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a></i>;<br/>
<a NAME="E5:6_1_2_1"/><b>then </b><i><font color="Green" title="E19">A16</font></i>: 
<font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c9">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">x</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E4:6_1"><i><font color="Green" title="E11">A6</font></i></a>, <a class="txt" href="wellset1.html#E1:6_1_2_1"><i><font color="Green" title="E15">A12</font></i></a>, <a class="txt" href="wellset1.html#E3:6_1_2_1"><i><font color="Green" title="E17">A14</font></i></a>, <a class="ref" href="wellord1.html#T38" title="WELLORD1:th.38">WELLORD1:38</a></i>;<br/>
<a NAME="E6:6_1_2_1"/><i><font color="Green" title="E20">A17</font></i>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">x</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E5:6_1"><i><font color="Green" title="E12">A7</font></i></a>, <a class="txt" href="wellset1.html#E1:6_1_2"><i><font color="Green" title="E14">A11</font></i></a>, <a class="txt" href="wellset1.html#E1:6_1_2_1"><i><font color="Green" title="E15">A12</font></i></a></i>;<br/>
<a NAME="E7:6_1_2_1"/><b>then </b><i><font color="Green" title="E21">A18</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c8">x</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="c6">W2</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#D1" title="WELLORD1:def.1">WELLORD1:def 1</a></i>;<br/>
<a NAME="E8:6_1_2_1"/><b>then </b><i><font color="Green" title="E22">A19</font></i>: 
<font color="Maroon" title="c9">y</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="c6">W2</font>
 
<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="E9:6_1_2_1"/><i><font color="Green" title="E23">A20</font></i>: 
<font color="Maroon" title="c8">x</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="c6">W2</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E7:6_1_2_1"><i><font color="Green" title="E21">A18</font></i></a>, <a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a></i>;<br/>
<a NAME="E10:6_1_2_1"/><b>then </b><i><font color="Green" title="E24">A21</font></i>: 
<font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c9">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">x</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E4:6_1"><i><font color="Green" title="E11">A6</font></i></a>, <a class="txt" href="wellset1.html#E6:6_1_2_1"><i><font color="Green" title="E20">A17</font></i></a>, <a class="txt" href="wellset1.html#E8:6_1_2_1"><i><font color="Green" title="E22">A19</font></i></a>, <a class="ref" href="wellord1.html#T38" title="WELLORD1:th.38">WELLORD1:38</a></i>;<br/>
<a NAME="E11:6_1_2_1"/>
<font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c9">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c9">y</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:6_1_2_1_1"/><i><font color="Green" title="E25">A22</font></i>: 
<font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c9">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_2_1_1_1"><b>proof </b></a><div class="add">

<font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c9">y</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c9">y</font>
<b>by </b><i><a class="txt" href="wellset1.html#E4:6_1"><i><font color="Green" title="E11">A6</font></i></a>, <a class="txt" href="wellset1.html#E1:6_1_2_1"><i><font color="Green" title="E15">A12</font></i></a>, <a class="txt" href="wellset1.html#E4:6_1_2_1"><i><font color="Green" title="E18">A15</font></i></a>, <a class="ref" href="wellord1.html#T35" title="WELLORD1:th.35">WELLORD1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c9">y</font>
<b>by </b><i><a class="txt" href="wellset1.html#E3:6_1"><i><font color="Green" title="E10">A5</font></i></a>, <a class="txt" href="wellset1.html#E1:6_1_2"><i><font color="Green" title="E14">A11</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c9">y</font>
<b>by </b><i><a class="txt" href="wellset1.html#E4:6_1"><i><font color="Green" title="E11">A6</font></i></a>, <a class="txt" href="wellset1.html#E6:6_1_2_1"><i><font color="Green" title="E20">A17</font></i></a>, <a class="txt" href="wellset1.html#E9:6_1_2_1"><i><font color="Green" title="E23">A20</font></i></a>, <a class="ref" href="wellord1.html#T35" title="WELLORD1:th.35">WELLORD1:35</a></i>
;<br/>
<b>hence </b><a NAME="E2:6_1_2_1_1_1"/>
<font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c9">y</font>
 ;<br/>


</div><b>end;</b></div>
<font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c9">y</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c9">y</font></span>)</span>
<b>by </b><i><a class="txt" href="wellset1.html#E5:6_1_2_1"><i><font color="Green" title="E19">A16</font></i></a>, <a class="ref" href="wellord1.html#T29" title="WELLORD1:th.29">WELLORD1:29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c9">y</font></span>)</span>
<b>by </b><i><a class="txt" href="wellset1.html#E3:6_1"><i><font color="Green" title="E10">A5</font></i></a>, <a class="txt" href="wellset1.html#E1:6_1_2"><i><font color="Green" title="E14">A11</font></i></a>, <a class="txt" href="wellset1.html#E1:6_1_2_1_1"><i><font color="Green" title="E25">A22</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c9">y</font></span>)</span>
<b>by </b><i><a class="txt" href="wellset1.html#E10:6_1_2_1"><i><font color="Green" title="E24">A21</font></i></a>, <a class="ref" href="wellord1.html#T29" title="WELLORD1:th.29">WELLORD1:29</a></i>
;<br/>
<b>hence </b><a NAME="E3:6_1_2_1_1"/>
<font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c9">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c9">y</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:6_1_2_1"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font>
 <b>by </b><i><a class="txt" href="wellset1.html#E3:6_1"><i><font color="Green" title="E10">A5</font></i></a>, <a class="txt" href="wellset1.html#E3:6_1_2_1"><i><font color="Green" title="E17">A14</font></i></a>, <a class="txt" href="wellset1.html#E8:6_1_2_1"><i><font color="Green" title="E22">A19</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:6_1_2"/>
<font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">C</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:6_1"/><i><font color="Green" title="E14">A23</font></i>: 
for <font color="Olive" title="b1">y1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y1</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="c5">W1</font> &amp; not <font color="Olive" title="b1">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font> holds <br/> ex <font color="Olive" title="b2">y3</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y3</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="c5">W1</font> &amp; <font color="Maroon" title="c7">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b2">y3</font> &amp; not <font color="Olive" title="b2">y3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">y1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:6_1_3"/><i><font color="Green" title="E15">A24</font></i>: 
( <font color="Maroon" title="c8">y1</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="c5">W1</font> &amp; not <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font> )
 ;<br/>

<b>set </b><font color="Maroon" title="c9">Y</font> = <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c5">W1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c7">C</font>;<br/>
<a NAME="E2:6_1_3"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c5">W1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c7">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="wellset1.html#E1:6_1_3"><i><font color="Green" title="E15">A24</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:6_1_3"/><i><font color="Green" title="E16">A25</font></i>: 
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c5">W1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c7">C</font>
 <b>and </b><br/><a NAME="E5:6_1_3"/><i><font color="Green" title="E17">A26</font></i>: 
for <font color="Olive" title="b1">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c5">W1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c7">C</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">a</font>,<font color="Olive" title="b1">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="c5">W1</font>
 <b>by </b><i><a class="txt" href="wellset1.html#E4:6_1"><i><font color="Green" title="E11">A6</font></i></a>, <a class="ref" href="wellord1.html#T10" title="WELLORD1:th.10">WELLORD1:10</a></i>;<br/>
<b>take </b><font color="Maroon" title="c11">y3</font> = <font color="Maroon" title="c10">a</font>;<br/>

<a NAME="E6:6_1_3"/>
<font color="Maroon" title="c7">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c11">y3</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:6_1_3_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font> iff <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c11">y3</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_3_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>thus </b><a NAME="E1:6_1_3_1_1"/>
( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font> implies <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c11">y3</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_3_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:6_1_3_1_1_1"/><i><font color="Green" title="E18">A27</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font>
 <b>and </b><br/><a NAME="E2:6_1_3_1_1_1"/><i><font color="Green" title="E19">A28</font></i>: 
not <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c11">y3</font>
 ;<br/>

<a NAME="E3:6_1_3_1_1_1"/>
<font color="Maroon" title="c12">x</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="c5">W1</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E3:6_1"><i><font color="Green" title="E10">A5</font></i></a>, <a class="txt" href="wellset1.html#E1:6_1_3_1_1_1"><i><font color="Green" title="E18">A27</font></i></a></i>;<br/>
<a NAME="E4:6_1_3_1_1_1"/><b>then </b><i><font color="Green" title="E20">A29</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">y3</font>,<font color="Maroon" title="c12">x</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="c5">W1</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E4:6_1"><i><font color="Green" title="E11">A6</font></i></a>, <a class="txt" href="wellset1.html#E4:6_1_3"><i><font color="Green" title="E16">A25</font></i></a>, <a class="txt" href="wellset1.html#E2:6_1_3_1_1_1"><i><font color="Green" title="E19">A28</font></i></a>, <a class="ref" href="wellset1.html#T6" target="_self" title="WELLSET1:th.6">Th6</a></i>;<br/>
<a NAME="E5:6_1_3_1_1_1"/><i><font color="Green" title="E21">A30</font></i>: 
<font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c12">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">C</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E6:6_1"><i><font color="Green" title="E13">A10</font></i></a>, <a class="txt" href="wellset1.html#E1:6_1_3_1_1_1"><i><font color="Green" title="E18">A27</font></i></a></i>;<br/>
<a NAME="E6:6_1_3_1_1_1"/>
<font color="Maroon" title="c11">y3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_3_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:6_1_3_1_1_1_1"/>
( <font color="Maroon" title="c11">y3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">x</font> implies <font color="Maroon" title="c11">y3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_3_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_1_3_1_1_1_1_1"/>
<font color="Maroon" title="c11">y3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">x</font>
 ;<br/>

<a NAME="E2:6_1_3_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c11">y3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c12">x</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E4:6_1_3_1_1_1"><i><font color="Green" title="E20">A29</font></i></a>, <a class="ref" href="wellord1.html#D1" title="WELLORD1:def.1">WELLORD1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:6_1_3_1_1_1_1_1"/>
<font color="Maroon" title="c11">y3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font>
 <b>by </b><i><a class="txt" href="wellset1.html#E5:6_1_3_1_1_1"><i><font color="Green" title="E21">A30</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:6_1_3_1_1_1_1"/>
<font color="Maroon" title="c11">y3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font>
 <b>by </b><i><a class="txt" href="wellset1.html#E1:6_1_3_1_1_1"><i><font color="Green" title="E18">A27</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:6_1_3_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="wellset1.html#E4:6_1_3"><i><font color="Green" title="E16">A25</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>


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

<b>thus </b><a NAME="E2:6_1_3_1_1"/>
( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c11">y3</font> implies <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_3_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:6_1_3_1_1_2"/><i><font color="Green" title="E18">A31</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c11">y3</font>
 <b>and </b><br/><a NAME="E2:6_1_3_1_1_2"/><i><font color="Green" title="E19">A32</font></i>: 
not <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font>
 ;<br/>

<a NAME="E3:6_1_3_1_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c11">y3</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="c5">W1</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E1:6_1_3_1_1_2"><i><font color="Green" title="E18">A31</font></i></a>, <a class="ref" href="wellord1.html#D1" title="WELLORD1:def.1">WELLORD1:def 1</a></i>;<br/>
<a NAME="E4:6_1_3_1_1_2"/><b>then </b><i><font color="Green" title="E20">A33</font></i>: 
<font color="Maroon" title="c12">x</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="c5">W1</font>
 
<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="E5:6_1_3_1_1_2"/><b>then </b>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c5">W1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c7">C</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E2:6_1_3_1_1_2"><i><font color="Green" title="E19">A32</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E6:6_1_3_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">y3</font>,<font color="Maroon" title="c12">x</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="c5">W1</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E5:6_1_3"><i><font color="Green" title="E17">A26</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:6_1_3_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="wellset1.html#E4:6_1"><i><font color="Green" title="E11">A6</font></i></a>, <a class="txt" href="wellset1.html#E4:6_1_3"><i><font color="Green" title="E16">A25</font></i></a>, <a class="txt" href="wellset1.html#E1:6_1_3_1_1_2"><i><font color="Green" title="E18">A31</font></i></a>, <a class="txt" href="wellset1.html#E4:6_1_3_1_1_2"><i><font color="Green" title="E20">A33</font></i></a>, <a class="ref" href="wellset1.html#T7" target="_self" title="WELLSET1:th.7">Th7</a></i>;<br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:6_1_3_1"/>
<font color="Maroon" title="c7">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c11">y3</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:6_1_3"/>
( <font color="Maroon" title="c11">y3</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="c5">W1</font> &amp; <font color="Maroon" title="c7">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c11">y3</font> &amp; not <font color="Maroon" title="c11">y3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font> )
 <b>by </b><i><a class="txt" href="wellset1.html#E4:6_1_3"><i><font color="Green" title="E16">A25</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:6_1"/><i><font color="Green" title="E15">A34</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font> holds <br/><font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b1">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:6_1_4"/><i><font color="Green" title="E16">A35</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c9">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:6_1_4"/><i><font color="Green" title="E17">A36</font></i>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">x</font>
 ;<br/>

<a NAME="E3:6_1_4"/><b>then </b><i><font color="Green" title="E18">A37</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c8">x</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="c6">W2</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#D1" title="WELLORD1:def.1">WELLORD1:def 1</a></i>;<br/>
<a NAME="E4:6_1_4"/><b>then </b><i><font color="Green" title="E19">A38</font></i>: 
<font color="Maroon" title="c9">y</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="c6">W2</font>
 
<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="E5:6_1_4"/><i><font color="Green" title="E20">A39</font></i>: 
<font color="Maroon" title="c8">x</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="c6">W2</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E3:6_1_4"><i><font color="Green" title="E18">A37</font></i></a>, <a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a></i>;<br/>
<a NAME="E6:6_1_4"/><b>then </b><i><font color="Green" title="E21">A40</font></i>: 
<font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c9">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">x</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E4:6_1"><i><font color="Green" title="E11">A6</font></i></a>, <a class="txt" href="wellset1.html#E2:6_1_4"><i><font color="Green" title="E17">A36</font></i></a>, <a class="txt" href="wellset1.html#E4:6_1_4"><i><font color="Green" title="E19">A38</font></i></a>, <a class="ref" href="wellord1.html#T38" title="WELLORD1:th.38">WELLORD1:38</a></i>;<br/>
<a NAME="E7:6_1_4"/><i><font color="Green" title="E22">A41</font></i>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">x</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E5:6_1"><i><font color="Green" title="E12">A7</font></i></a>, <a class="txt" href="wellset1.html#E1:6_1_4"><i><font color="Green" title="E16">A35</font></i></a>, <a class="txt" href="wellset1.html#E2:6_1_4"><i><font color="Green" title="E17">A36</font></i></a></i>;<br/>
<a NAME="E8:6_1_4"/><b>then </b><i><font color="Green" title="E23">A42</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c8">x</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="c5">W1</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#D1" title="WELLORD1:def.1">WELLORD1:def 1</a></i>;<br/>
<a NAME="E9:6_1_4"/><b>then </b><i><font color="Green" title="E24">A43</font></i>: 
<font color="Maroon" title="c9">y</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="c5">W1</font>
 
<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="E10:6_1_4"/><i><font color="Green" title="E25">A44</font></i>: 
<font color="Maroon" title="c8">x</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="c5">W1</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E8:6_1_4"><i><font color="Green" title="E23">A42</font></i></a>, <a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a></i>;<br/>
<a NAME="E11:6_1_4"/><b>then </b><i><font color="Green" title="E26">A45</font></i>: 
<font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c9">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">x</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E4:6_1"><i><font color="Green" title="E11">A6</font></i></a>, <a class="txt" href="wellset1.html#E7:6_1_4"><i><font color="Green" title="E22">A41</font></i></a>, <a class="txt" href="wellset1.html#E9:6_1_4"><i><font color="Green" title="E24">A43</font></i></a>, <a class="ref" href="wellord1.html#T38" title="WELLORD1:th.38">WELLORD1:38</a></i>;<br/>
<a NAME="E12:6_1_4"/>
<font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c9">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c9">y</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_4_1"><b>proof </b></a><div class="add">

<a NAME="E1:6_1_4_1"/><i><font color="Green" title="E27">A46</font></i>: 
<font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c9">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_4_1_1"><b>proof </b></a><div class="add">

<font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c9">y</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c9">y</font>
<b>by </b><i><a class="txt" href="wellset1.html#E4:6_1"><i><font color="Green" title="E11">A6</font></i></a>, <a class="txt" href="wellset1.html#E2:6_1_4"><i><font color="Green" title="E17">A36</font></i></a>, <a class="txt" href="wellset1.html#E5:6_1_4"><i><font color="Green" title="E20">A39</font></i></a>, <a class="ref" href="wellord1.html#T35" title="WELLORD1:th.35">WELLORD1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c9">y</font>
<b>by </b><i><a class="txt" href="wellset1.html#E3:6_1"><i><font color="Green" title="E10">A5</font></i></a>, <a class="txt" href="wellset1.html#E1:6_1_4"><i><font color="Green" title="E16">A35</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c9">y</font>
<b>by </b><i><a class="txt" href="wellset1.html#E4:6_1"><i><font color="Green" title="E11">A6</font></i></a>, <a class="txt" href="wellset1.html#E7:6_1_4"><i><font color="Green" title="E22">A41</font></i></a>, <a class="txt" href="wellset1.html#E10:6_1_4"><i><font color="Green" title="E25">A44</font></i></a>, <a class="ref" href="wellord1.html#T35" title="WELLORD1:th.35">WELLORD1:35</a></i>
;<br/>
<b>hence </b><a NAME="E2:6_1_4_1_1"/>
<font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c9">y</font>
 ;<br/>


</div><b>end;</b></div>
<font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c9">y</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c9">y</font></span>)</span>
<b>by </b><i><a class="txt" href="wellset1.html#E6:6_1_4"><i><font color="Green" title="E21">A40</font></i></a>, <a class="ref" href="wellord1.html#T29" title="WELLORD1:th.29">WELLORD1:29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c9">y</font></span>)</span>
<b>by </b><i><a class="txt" href="wellset1.html#E3:6_1"><i><font color="Green" title="E10">A5</font></i></a>, <a class="txt" href="wellset1.html#E1:6_1_4"><i><font color="Green" title="E16">A35</font></i></a>, <a class="txt" href="wellset1.html#E1:6_1_4_1"><i><font color="Green" title="E27">A46</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c9">y</font></span>)</span>
<b>by </b><i><a class="txt" href="wellset1.html#E11:6_1_4"><i><font color="Green" title="E26">A45</font></i></a>, <a class="ref" href="wellord1.html#T29" title="WELLORD1:th.29">WELLORD1:29</a></i>
;<br/>
<b>hence </b><a NAME="E3:6_1_4_1"/>
<font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c9">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c9">y</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E13:6_1_4"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font>
 <b>by </b><i><a class="txt" href="wellset1.html#E3:6_1"><i><font color="Green" title="E10">A5</font></i></a>, <a class="txt" href="wellset1.html#E4:6_1_4"><i><font color="Green" title="E19">A38</font></i></a>, <a class="txt" href="wellset1.html#E9:6_1_4"><i><font color="Green" title="E24">A43</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:6_1"/><i><font color="Green" title="E16">A47</font></i>: 
for <font color="Olive" title="b1">y1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y1</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="c6">W2</font> &amp; not <font color="Olive" title="b1">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font> holds <br/> ex <font color="Olive" title="b2">y3</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y3</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="c6">W2</font> &amp; <font color="Maroon" title="c7">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b2">y3</font> &amp; not <font color="Olive" title="b2">y3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">y1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:6_1_5"/><i><font color="Green" title="E17">A48</font></i>: 
( <font color="Maroon" title="c8">y1</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="c6">W2</font> &amp; not <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font> )
 ;<br/>

<b>set </b><font color="Maroon" title="c9">Y</font> = <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">W2</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c7">C</font>;<br/>
<a NAME="E2:6_1_5"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">W2</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c7">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="wellset1.html#E1:6_1_5"><i><font color="Green" title="E17">A48</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:6_1_5"/><i><font color="Green" title="E18">A49</font></i>: 
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">W2</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c7">C</font>
 <b>and </b><br/><a NAME="E5:6_1_5"/><i><font color="Green" title="E19">A50</font></i>: 
for <font color="Olive" title="b1">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">W2</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c7">C</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">a</font>,<font color="Olive" title="b1">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="c6">W2</font>
 <b>by </b><i><a class="txt" href="wellset1.html#E4:6_1"><i><font color="Green" title="E11">A6</font></i></a>, <a class="ref" href="wellord1.html#T10" title="WELLORD1:th.10">WELLORD1:10</a></i>;<br/>
<b>take </b><font color="Maroon" title="c11">y3</font> = <font color="Maroon" title="c10">a</font>;<br/>

<a NAME="E6:6_1_5"/>
<font color="Maroon" title="c7">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c11">y3</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_5_1"><b>proof </b></a><div class="add">

<a NAME="E1:6_1_5_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font> iff <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c11">y3</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_5_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>thus </b><a NAME="E1:6_1_5_1_1"/>
( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font> implies <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c11">y3</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_5_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:6_1_5_1_1_1"/><i><font color="Green" title="E20">A51</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font>
 <b>and </b><br/><a NAME="E2:6_1_5_1_1_1"/><i><font color="Green" title="E21">A52</font></i>: 
not <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c11">y3</font>
 ;<br/>

<a NAME="E3:6_1_5_1_1_1"/>
<font color="Maroon" title="c12">x</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="c6">W2</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E3:6_1"><i><font color="Green" title="E10">A5</font></i></a>, <a class="txt" href="wellset1.html#E1:6_1_5_1_1_1"><i><font color="Green" title="E20">A51</font></i></a></i>;<br/>
<a NAME="E4:6_1_5_1_1_1"/><b>then </b><i><font color="Green" title="E22">A53</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">y3</font>,<font color="Maroon" title="c12">x</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="c6">W2</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E4:6_1"><i><font color="Green" title="E11">A6</font></i></a>, <a class="txt" href="wellset1.html#E4:6_1_5"><i><font color="Green" title="E18">A49</font></i></a>, <a class="txt" href="wellset1.html#E2:6_1_5_1_1_1"><i><font color="Green" title="E21">A52</font></i></a>, <a class="ref" href="wellset1.html#T6" target="_self" title="WELLSET1:th.6">Th6</a></i>;<br/>
<a NAME="E5:6_1_5_1_1_1"/><i><font color="Green" title="E23">A54</font></i>: 
<font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c12">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">C</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E8:6_1"><i><font color="Green" title="E15">A34</font></i></a>, <a class="txt" href="wellset1.html#E1:6_1_5_1_1_1"><i><font color="Green" title="E20">A51</font></i></a></i>;<br/>
<a NAME="E6:6_1_5_1_1_1"/>
( <font color="Maroon" title="c11">y3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">x</font> implies <font color="Maroon" title="c11">y3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_5_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_1_5_1_1_1_1"/>
<font color="Maroon" title="c11">y3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">x</font>
 ;<br/>

<a NAME="E2:6_1_5_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c11">y3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c12">x</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E4:6_1_5_1_1_1"><i><font color="Green" title="E22">A53</font></i></a>, <a class="ref" href="wellord1.html#D1" title="WELLORD1:def.1">WELLORD1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:6_1_5_1_1_1_1"/>
<font color="Maroon" title="c11">y3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font>
 <b>by </b><i><a class="txt" href="wellset1.html#E5:6_1_5_1_1_1"><i><font color="Green" title="E23">A54</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:6_1_5_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="wellset1.html#E4:6_1_5"><i><font color="Green" title="E18">A49</font></i></a>, <a class="txt" href="wellset1.html#E1:6_1_5_1_1_1"><i><font color="Green" title="E20">A51</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>


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

<b>thus </b><a NAME="E2:6_1_5_1_1"/>
( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c11">y3</font> implies <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_5_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:6_1_5_1_1_2"/><i><font color="Green" title="E20">A55</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c11">y3</font>
 <b>and </b><br/><a NAME="E2:6_1_5_1_1_2"/><i><font color="Green" title="E21">A56</font></i>: 
not <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font>
 ;<br/>

<a NAME="E3:6_1_5_1_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c11">y3</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="c6">W2</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E1:6_1_5_1_1_2"><i><font color="Green" title="E20">A55</font></i></a>, <a class="ref" href="wellord1.html#D1" title="WELLORD1:def.1">WELLORD1:def 1</a></i>;<br/>
<a NAME="E4:6_1_5_1_1_2"/><b>then </b><i><font color="Green" title="E22">A57</font></i>: 
<font color="Maroon" title="c12">x</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="c6">W2</font>
 
<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="E5:6_1_5_1_1_2"/><b>then </b>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">W2</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c7">C</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E2:6_1_5_1_1_2"><i><font color="Green" title="E21">A56</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E6:6_1_5_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">y3</font>,<font color="Maroon" title="c12">x</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="c6">W2</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E5:6_1_5"><i><font color="Green" title="E19">A50</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:6_1_5_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="wellset1.html#E4:6_1"><i><font color="Green" title="E11">A6</font></i></a>, <a class="txt" href="wellset1.html#E4:6_1_5"><i><font color="Green" title="E18">A49</font></i></a>, <a class="txt" href="wellset1.html#E1:6_1_5_1_1_2"><i><font color="Green" title="E20">A55</font></i></a>, <a class="txt" href="wellset1.html#E4:6_1_5_1_1_2"><i><font color="Green" title="E22">A57</font></i></a>, <a class="ref" href="wellset1.html#T7" target="_self" title="WELLSET1:th.7">Th7</a></i>;<br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:6_1_5_1"/>
<font color="Maroon" title="c7">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c11">y3</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:6_1_5"/>
( <font color="Maroon" title="c11">y3</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="c6">W2</font> &amp; <font color="Maroon" title="c7">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c11">y3</font> &amp; not <font color="Maroon" title="c11">y3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font> )
 <b>by </b><i><a class="txt" href="wellset1.html#E4:6_1_5"><i><font color="Green" title="E18">A49</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:6_1"/><i><font color="Green" title="E17">A58</font></i>: 
( <font color="Maroon" title="c7">C</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="c5">W1</font> or <font color="Maroon" title="c7">C</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="c6">W2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_6"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_1_6"/>
not <font color="Maroon" title="c7">C</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="c5">W1</font>
 ;<br/>

<a NAME="E2:6_1_6"/><b>then </b><i><font color="Green" title="E18">A59</font></i>: 
 ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font> &amp; not <font color="Olive" title="b1">x</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="c5">W1</font> ) or ( <font color="Olive" title="b1">x</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="c5">W1</font> &amp; not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font> ) )
 
<b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>
<b>assume </b><a NAME="E3:6_1_6"/>
not <font color="Maroon" title="c7">C</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="c6">W2</font>
 ;<br/>

<a NAME="E4:6_1_6"/><b>then </b><i><font color="Green" title="E19">A60</font></i>: 
 ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font> &amp; not <font color="Olive" title="b1">x</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="c6">W2</font> ) or ( <font color="Olive" title="b1">x</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="c6">W2</font> &amp; not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font> ) )
 
<b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">y3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:6_1_6"/><i><font color="Green" title="E20">A61</font></i>: 
( <font color="Maroon" title="c8">y3</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="c5">W1</font> &amp; <font color="Maroon" title="c7">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">y3</font> &amp; not <font color="Maroon" title="c8">y3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font> )
 <b>by </b><i><a class="txt" href="wellset1.html#E3:6_1"><i><font color="Green" title="E10">A5</font></i></a>, <a class="txt" href="wellset1.html#E7:6_1"><i><font color="Green" title="E14">A23</font></i></a>, <a class="txt" href="wellset1.html#E2:6_1_6"><i><font color="Green" title="E18">A59</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">y4</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:6_1_6"/><i><font color="Green" title="E21">A62</font></i>: 
( <font color="Maroon" title="c9">y4</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="c6">W2</font> &amp; <font color="Maroon" title="c7">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c9">y4</font> &amp; not <font color="Maroon" title="c9">y4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font> )
 <b>by </b><i><a class="txt" href="wellset1.html#E3:6_1"><i><font color="Green" title="E10">A5</font></i></a>, <a class="txt" href="wellset1.html#E9:6_1"><i><font color="Green" title="E16">A47</font></i></a>, <a class="txt" href="wellset1.html#E4:6_1_6"><i><font color="Green" title="E19">A60</font></i></a></i>;<br/>
<a NAME="E9:6_1_6"/><i><font color="Green" title="E22">A63</font></i>: 
<font color="Maroon" title="c8">y3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y4</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_6_1"><b>proof </b></a><div class="add">

<font color="Maroon" title="c8">y3</font> = 
<font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c9">y4</font></span>)</span>
<b>by </b><i><a class="txt" href="wellset1.html#E3:6"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="wellset1.html#E1:6_1"><i><font color="Green" title="E9">A4</font></i></a>, <a class="txt" href="wellset1.html#E6:6_1_6"><i><font color="Green" title="E20">A61</font></i></a>, <a class="txt" href="wellset1.html#E8:6_1_6"><i><font color="Green" title="E21">A62</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c9">y4</font>
<b>by </b><i><a class="txt" href="wellset1.html#E3:6"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="wellset1.html#E1:6_1"><i><font color="Green" title="E9">A4</font></i></a>, <a class="txt" href="wellset1.html#E8:6_1_6"><i><font color="Green" title="E21">A62</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E2:6_1_6_1"/>
<font color="Maroon" title="c8">y3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y4</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E10:6_1_6"/>
<font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">y3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">y3</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_6_2"><b>proof </b></a><div class="add">

<a NAME="E1:6_1_6_2"/>
for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">y3</font></span>)</span> iff <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">y3</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_6_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<a NAME="E1:6_1_6_2_1"/>
( <font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">W1</font> &amp; <font color="Maroon" title="c10">z</font> <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="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">y3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">y3</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> iff ( <font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">W2</font> &amp; <font color="Maroon" title="c10">z</font> <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="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">y3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">y3</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_6_2_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:6_1_6_2_1_1"/>
( <font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">W1</font> &amp; <font color="Maroon" title="c10">z</font> <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="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">y3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">y3</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> implies ( <font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">W2</font> &amp; <font color="Maroon" title="c10">z</font> <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="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">y3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">y3</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_6_2_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:6_1_6_2_1_1_1"/><i><font color="Green" title="E23">A64</font></i>: 
<font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">W1</font>
 <b>and </b><br/><a NAME="E2:6_1_6_2_1_1_1"/><i><font color="Green" title="E24">A65</font></i>: 
<font color="Maroon" title="c10">z</font> <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="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">y3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">y3</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ;<br/>

<b>consider </b><font color="Maroon" title="c11">z1</font>, <font color="Maroon" title="c12">z2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:6_1_6_2_1_1_1"/><i><font color="Green" title="E25">A66</font></i>: 
<font color="Maroon" title="c11">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">y3</font>
 <b>and </b><br/><a NAME="E5:6_1_6_2_1_1_1"/><i><font color="Green" title="E26">A67</font></i>: 
<font color="Maroon" title="c12">z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">y3</font>
 <b>and </b><br/><a NAME="E6:6_1_6_2_1_1_1"/><i><font color="Green" title="E27">A68</font></i>: 
<font color="Maroon" title="c10">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">z1</font>,<font color="Maroon" title="c12">z2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="wellset1.html#E2:6_1_6_2_1_1_1"><i><font color="Green" title="E24">A65</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E7:6_1_6_2_1_1_1"/><i><font color="Green" title="E28">A69</font></i>: 
<font color="Maroon" title="c11">z1</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="c6">W2</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E3:6_1"><i><font color="Green" title="E10">A5</font></i></a>, <a class="txt" href="wellset1.html#E6:6_1_6"><i><font color="Green" title="E20">A61</font></i></a>, <a class="txt" href="wellset1.html#E4:6_1_6_2_1_1_1"><i><font color="Green" title="E25">A66</font></i></a></i>;<br/>
<a NAME="E8:6_1_6_2_1_1_1"/>
( <font color="Maroon" title="c11">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c12">z2</font> or ( <font color="Maroon" title="c11">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">z2</font> &amp; not <font color="Maroon" title="c11">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c12">z2</font> ) )
 
<b>by </b><i><a class="txt" href="wellset1.html#E1:6_1_6_2_1_1_1"><i><font color="Green" title="E23">A64</font></i></a>, <a class="txt" href="wellset1.html#E6:6_1_6_2_1_1_1"><i><font color="Green" title="E27">A68</font></i></a>, <a class="ref" href="wellord1.html#D1" title="WELLORD1:def.1">WELLORD1:def 1</a></i>;<br/>
<a NAME="E9:6_1_6_2_1_1_1"/><b>then </b>
( <font color="Maroon" title="c11">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c12">z2</font> or ( <font color="Maroon" title="c11">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">z2</font> &amp; not <font color="Maroon" title="c11">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c12">z2</font> ) )
 
<b>by </b><i><a class="txt" href="wellset1.html#E5:6_1"><i><font color="Green" title="E12">A7</font></i></a>, <a class="txt" href="wellset1.html#E6:6_1_6"><i><font color="Green" title="E20">A61</font></i></a>, <a class="txt" href="wellset1.html#E5:6_1_6_2_1_1_1"><i><font color="Green" title="E26">A67</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:6_1_6_2_1_1_1"/>
( <font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">W2</font> &amp; <font color="Maroon" title="c10">z</font> <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="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">y3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">y3</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="wellset1.html#E4:6_1"><i><font color="Green" title="E11">A6</font></i></a>, <a class="txt" href="wellset1.html#E6:6_1_6"><i><font color="Green" title="E20">A61</font></i></a>, <a class="txt" href="wellset1.html#E8:6_1_6"><i><font color="Green" title="E21">A62</font></i></a>, <a class="txt" href="wellset1.html#E9:6_1_6"><i><font color="Green" title="E22">A63</font></i></a>, <a class="txt" href="wellset1.html#E2:6_1_6_2_1_1_1"><i><font color="Green" title="E24">A65</font></i></a>, <a class="txt" href="wellset1.html#E6:6_1_6_2_1_1_1"><i><font color="Green" title="E27">A68</font></i></a>, <a class="txt" href="wellset1.html#E7:6_1_6_2_1_1_1"><i><font color="Green" title="E28">A69</font></i></a>, <a class="ref" href="wellset1.html#T6" target="_self" title="WELLSET1:th.6">Th6</a>, <a class="ref" href="wellord1.html#D1" title="WELLORD1:def.1">WELLORD1:def 1</a></i>;<br/>


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

<b>thus </b><a NAME="E2:6_1_6_2_1_1"/>
( <font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">W2</font> &amp; <font color="Maroon" title="c10">z</font> <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="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">y3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">y3</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> implies ( <font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">W1</font> &amp; <font color="Maroon" title="c10">z</font> <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="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">y3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">y3</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_6_2_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:6_1_6_2_1_1_2"/><i><font color="Green" title="E23">A70</font></i>: 
<font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">W2</font>
 <b>and </b><br/><a NAME="E2:6_1_6_2_1_1_2"/><i><font color="Green" title="E24">A71</font></i>: 
<font color="Maroon" title="c10">z</font> <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="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">y3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">y3</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ;<br/>

<b>consider </b><font color="Maroon" title="c11">z1</font>, <font color="Maroon" title="c12">z2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:6_1_6_2_1_1_2"/><i><font color="Green" title="E25">A72</font></i>: 
<font color="Maroon" title="c11">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">y3</font>
 <b>and </b><br/><a NAME="E5:6_1_6_2_1_1_2"/><i><font color="Green" title="E26">A73</font></i>: 
<font color="Maroon" title="c12">z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">y3</font>
 <b>and </b><br/><a NAME="E6:6_1_6_2_1_1_2"/><i><font color="Green" title="E27">A74</font></i>: 
<font color="Maroon" title="c10">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">z1</font>,<font color="Maroon" title="c12">z2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="wellset1.html#E2:6_1_6_2_1_1_2"><i><font color="Green" title="E24">A71</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E7:6_1_6_2_1_1_2"/><i><font color="Green" title="E28">A75</font></i>: 
<font color="Maroon" title="c11">z1</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="c5">W1</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E3:6_1"><i><font color="Green" title="E10">A5</font></i></a>, <a class="txt" href="wellset1.html#E8:6_1_6"><i><font color="Green" title="E21">A62</font></i></a>, <a class="txt" href="wellset1.html#E9:6_1_6"><i><font color="Green" title="E22">A63</font></i></a>, <a class="txt" href="wellset1.html#E4:6_1_6_2_1_1_2"><i><font color="Green" title="E25">A72</font></i></a></i>;<br/>
<a NAME="E8:6_1_6_2_1_1_2"/>
( <font color="Maroon" title="c11">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c12">z2</font> or ( <font color="Maroon" title="c11">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">z2</font> &amp; not <font color="Maroon" title="c11">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c12">z2</font> ) )
 
<b>by </b><i><a class="txt" href="wellset1.html#E1:6_1_6_2_1_1_2"><i><font color="Green" title="E23">A70</font></i></a>, <a class="txt" href="wellset1.html#E6:6_1_6_2_1_1_2"><i><font color="Green" title="E27">A74</font></i></a>, <a class="ref" href="wellord1.html#D1" title="WELLORD1:def.1">WELLORD1:def 1</a></i>;<br/>
<a NAME="E9:6_1_6_2_1_1_2"/><b>then </b>
( <font color="Maroon" title="c11">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c12">z2</font> or ( <font color="Maroon" title="c11">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">z2</font> &amp; not <font color="Maroon" title="c11">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c12">z2</font> ) )
 
<b>by </b><i><a class="txt" href="wellset1.html#E5:6_1"><i><font color="Green" title="E12">A7</font></i></a>, <a class="txt" href="wellset1.html#E8:6_1_6"><i><font color="Green" title="E21">A62</font></i></a>, <a class="txt" href="wellset1.html#E9:6_1_6"><i><font color="Green" title="E22">A63</font></i></a>, <a class="txt" href="wellset1.html#E5:6_1_6_2_1_1_2"><i><font color="Green" title="E26">A73</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:6_1_6_2_1_1_2"/>
( <font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">W1</font> &amp; <font color="Maroon" title="c10">z</font> <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="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">y3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">y3</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="wellset1.html#E4:6_1"><i><font color="Green" title="E11">A6</font></i></a>, <a class="txt" href="wellset1.html#E6:6_1_6"><i><font color="Green" title="E20">A61</font></i></a>, <a class="txt" href="wellset1.html#E8:6_1_6"><i><font color="Green" title="E21">A62</font></i></a>, <a class="txt" href="wellset1.html#E9:6_1_6"><i><font color="Green" title="E22">A63</font></i></a>, <a class="txt" href="wellset1.html#E2:6_1_6_2_1_1_2"><i><font color="Green" title="E24">A71</font></i></a>, <a class="txt" href="wellset1.html#E6:6_1_6_2_1_1_2"><i><font color="Green" title="E27">A74</font></i></a>, <a class="txt" href="wellset1.html#E7:6_1_6_2_1_1_2"><i><font color="Green" title="E28">A75</font></i></a>, <a class="ref" href="wellset1.html#T6" target="_self" title="WELLSET1:th.6">Th6</a>, <a class="ref" href="wellord1.html#D1" title="WELLORD1:def.1">WELLORD1:def 1</a></i>;<br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:6_1_6_2_1"/>
( <font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">y3</font></span>)</span> iff <font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">y3</font></span>)</span> )
 <b>by </b><i><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="E2:6_1_6_2"/>
<font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">y3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">y3</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:6_1_6"/>
contradiction
 <b>by </b><i><a class="txt" href="wellset1.html#E3:6_1"><i><font color="Green" title="E10">A5</font></i></a>, <a class="txt" href="wellset1.html#E6:6_1_6"><i><font color="Green" title="E20">A61</font></i></a>, <a class="txt" href="wellset1.html#E8:6_1_6"><i><font color="Green" title="E21">A62</font></i></a>, <a class="txt" href="wellset1.html#E9:6_1_6"><i><font color="Green" title="E22">A63</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:6_1"/><i><font color="Green" title="E18">A76</font></i>: 
( <font color="Maroon" title="c7">C</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="c5">W1</font> implies ( <font color="Maroon" title="c5">W1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">W2</font> &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</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="c5">W1</font> holds <br/><font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b1">x</font> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_7"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_1_7"/><i><font color="Green" title="E19">A77</font></i>: 
<font color="Maroon" title="c7">C</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="c5">W1</font>
 ;<br/>

<a NAME="E2:6_1_7"/>
for <font color="Olive" title="b1">z1</font>, <font color="Olive" title="b2">z2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">z1</font>,<font color="Olive" title="b2">z2</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="c5">W1</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">z1</font>,<font color="Olive" title="b2">z2</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="c6">W2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_7_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">z1</font>, <font color="Maroon" title="c9">z2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><a NAME="E1:6_1_7_1"/><i><font color="Green" title="E20">A78</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">z1</font>,<font color="Maroon" title="c9">z2</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="c5">W1</font>
 ;<br/>

<a NAME="E2:6_1_7_1"/><b>then </b><i><font color="Green" title="E21">A79</font></i>: 
<font color="Maroon" title="c8">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E1:6_1_7"><i><font color="Green" title="E19">A77</font></i></a>, <a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a></i>;<br/>
<a NAME="E3:6_1_7_1"/><i><font color="Green" title="E22">A80</font></i>: 
<font color="Maroon" title="c9">z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E1:6_1_7"><i><font color="Green" title="E19">A77</font></i></a>, <a class="txt" href="wellset1.html#E1:6_1_7_1"><i><font color="Green" title="E20">A78</font></i></a>, <a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a></i>;<br/>
<a NAME="E4:6_1_7_1"/><i><font color="Green" title="E23">A81</font></i>: 
<font color="Maroon" title="c8">z1</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="c6">W2</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E3:6_1"><i><font color="Green" title="E10">A5</font></i></a>, <a class="txt" href="wellset1.html#E2:6_1_7_1"><i><font color="Green" title="E21">A79</font></i></a></i>;<br/>
<a NAME="E5:6_1_7_1"/>
( <font color="Maroon" title="c8">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c9">z2</font> or ( <font color="Maroon" title="c8">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">z2</font> &amp; not <font color="Maroon" title="c8">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c9">z2</font> ) )
 
<b>by </b><i><a class="txt" href="wellset1.html#E1:6_1_7_1"><i><font color="Green" title="E20">A78</font></i></a>, <a class="ref" href="wellord1.html#D1" title="WELLORD1:def.1">WELLORD1:def 1</a></i>;<br/>
<a NAME="E6:6_1_7_1"/><b>then </b>
( <font color="Maroon" title="c8">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c9">z2</font> or ( <font color="Maroon" title="c8">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">z2</font> &amp; not <font color="Maroon" title="c8">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c9">z2</font> ) )
 
<b>by </b><i><a class="txt" href="wellset1.html#E5:6_1"><i><font color="Green" title="E12">A7</font></i></a>, <a class="txt" href="wellset1.html#E3:6_1_7_1"><i><font color="Green" title="E22">A80</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:6_1_7_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">z1</font>,<font color="Maroon" title="c9">z2</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="c6">W2</font>
 <b>by </b><i><a class="txt" href="wellset1.html#E4:6_1"><i><font color="Green" title="E11">A6</font></i></a>, <a class="txt" href="wellset1.html#E4:6_1_7_1"><i><font color="Green" title="E23">A81</font></i></a>, <a class="ref" href="wellset1.html#T6" target="_self" title="WELLSET1:th.6">Th6</a>, <a class="ref" href="wellord1.html#D1" title="WELLORD1:def.1">WELLORD1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:6_1_7"/>
( <font color="Maroon" title="c5">W1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">W2</font> &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</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="c5">W1</font> holds <br/><font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b1">x</font> ) )
 <b>by </b><i><a class="txt" href="wellset1.html#E5:6_1"><i><font color="Green" title="E12">A7</font></i></a>, <a class="txt" href="wellset1.html#E1:6_1_7"><i><font color="Green" title="E19">A77</font></i></a>, <a class="ref" href="relat_1.html#D3" title="RELAT_1:def.3">RELAT_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:6_1"/>
( <font color="Maroon" title="c7">C</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="c6">W2</font> implies ( <font color="Maroon" title="c6">W2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">W1</font> &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</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="c6">W2</font> holds <br/><font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b1">x</font> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_8"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_1_8"/><i><font color="Green" title="E19">A82</font></i>: 
<font color="Maroon" title="c7">C</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="c6">W2</font>
 ;<br/>

<a NAME="E2:6_1_8"/>
for <font color="Olive" title="b1">z1</font>, <font color="Olive" title="b2">z2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">z1</font>,<font color="Olive" title="b2">z2</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="c6">W2</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">z1</font>,<font color="Olive" title="b2">z2</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="c5">W1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_8_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">z1</font>, <font color="Maroon" title="c9">z2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><a NAME="E1:6_1_8_1"/><i><font color="Green" title="E20">A83</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">z1</font>,<font color="Maroon" title="c9">z2</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="c6">W2</font>
 ;<br/>

<a NAME="E2:6_1_8_1"/><b>then </b><i><font color="Green" title="E21">A84</font></i>: 
<font color="Maroon" title="c8">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E1:6_1_8"><i><font color="Green" title="E19">A82</font></i></a>, <a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a></i>;<br/>
<a NAME="E3:6_1_8_1"/><i><font color="Green" title="E22">A85</font></i>: 
<font color="Maroon" title="c9">z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E1:6_1_8"><i><font color="Green" title="E19">A82</font></i></a>, <a class="txt" href="wellset1.html#E1:6_1_8_1"><i><font color="Green" title="E20">A83</font></i></a>, <a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a></i>;<br/>
<a NAME="E4:6_1_8_1"/><i><font color="Green" title="E23">A86</font></i>: 
<font color="Maroon" title="c8">z1</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="c5">W1</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E3:6_1"><i><font color="Green" title="E10">A5</font></i></a>, <a class="txt" href="wellset1.html#E2:6_1_8_1"><i><font color="Green" title="E21">A84</font></i></a></i>;<br/>
<a NAME="E5:6_1_8_1"/>
( <font color="Maroon" title="c8">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c9">z2</font> or ( <font color="Maroon" title="c8">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">z2</font> &amp; not <font color="Maroon" title="c8">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c9">z2</font> ) )
 
<b>by </b><i><a class="txt" href="wellset1.html#E1:6_1_8_1"><i><font color="Green" title="E20">A83</font></i></a>, <a class="ref" href="wellord1.html#D1" title="WELLORD1:def.1">WELLORD1:def 1</a></i>;<br/>
<a NAME="E6:6_1_8_1"/><b>then </b>
( <font color="Maroon" title="c8">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c9">z2</font> or ( <font color="Maroon" title="c8">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">z2</font> &amp; not <font color="Maroon" title="c8">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c9">z2</font> ) )
 
<b>by </b><i><a class="txt" href="wellset1.html#E5:6_1"><i><font color="Green" title="E12">A7</font></i></a>, <a class="txt" href="wellset1.html#E3:6_1_8_1"><i><font color="Green" title="E22">A85</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:6_1_8_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">z1</font>,<font color="Maroon" title="c9">z2</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="c5">W1</font>
 <b>by </b><i><a class="txt" href="wellset1.html#E4:6_1"><i><font color="Green" title="E11">A6</font></i></a>, <a class="txt" href="wellset1.html#E4:6_1_8_1"><i><font color="Green" title="E23">A86</font></i></a>, <a class="ref" href="wellset1.html#T6" target="_self" title="WELLSET1:th.6">Th6</a>, <a class="ref" href="wellord1.html#D1" title="WELLORD1:def.1">WELLORD1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:6_1_8"/>
( <font color="Maroon" title="c6">W2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">W1</font> &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</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="c6">W2</font> holds <br/><font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b1">x</font> ) )
 <b>by </b><i><a class="txt" href="wellset1.html#E5:6_1"><i><font color="Green" title="E12">A7</font></i></a>, <a class="txt" href="wellset1.html#E1:6_1_8"><i><font color="Green" title="E19">A82</font></i></a>, <a class="ref" href="relat_1.html#D3" title="RELAT_1:def.3">RELAT_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E13:6_1"/>
( ( <font color="Maroon" title="c5">W1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">W2</font> &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</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="c5">W1</font> holds <br/><font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b1">x</font> ) ) or ( <font color="Maroon" title="c6">W2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">W1</font> &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</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="c6">W2</font> holds <br/><font color="Maroon" title="c6">W2</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b1">x</font> ) ) )
 <b>by </b><i><a class="txt" href="wellset1.html#E10:6_1"><i><font color="Green" title="E17">A58</font></i></a>, <a class="txt" href="wellset1.html#E11:6_1"><i><font color="Green" title="E18">A76</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">W</font> being  <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">$1,$2</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="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G</font> );<br/>
<b>consider </b><font color="Maroon" title="c5">S</font> being   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a><b> such that </b><br/><a NAME="E6:6"/><i><font color="Green" title="E9">A87</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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">x</font>,<font color="Olive" title="b2">y</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="c5">S</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">D</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">D</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] ) )
 <b>from </b><i><a class="ref" href="relat_1.html#S1" title="RELAT_1:sch.1">RELAT_1:sch 1</a>();<br/></i>
<b>take </b><font color="Maroon" title="c6">R</font> = <font color="Maroon" title="c5">S</font>;<br/>

<a NAME="E7:6"/><i><font color="Green" title="E10">A88</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</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="c6">R</font> holds <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">D</font> &amp;  ex <font color="Olive" title="b2">W</font> being  <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a> st <br/>( <font color="Olive" title="b1">x</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="Olive" title="b2">W</font> &amp; <font color="Olive" title="b2">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:6_2"/>
<font color="Maroon" title="c7">x</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="c6">R</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:6_2"/><i><font color="Green" title="E11">A89</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</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="c6">R</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c7">x</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="c6">R</font> )
 <b>by </b><i><a class="ref" href="wellset1.html#T1" target="_self" title="WELLSET1:th.1">Th1</a></i>;<br/>
<a NAME="E4:6_2"/><i><font color="Green" title="E12">A90</font></i>: 
( ( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">D</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">D</font> &amp;  ex <font color="Olive" title="b1">S</font> being  <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</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="Olive" title="b1">S</font> &amp; <font color="Olive" title="b1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G</font> ) ) or ( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">D</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">D</font> &amp;  ex <font color="Olive" title="b1">S</font> being  <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c7">x</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="Olive" title="b1">S</font> &amp; <font color="Olive" title="b1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G</font> ) ) )
 
<b>by </b><i><a class="txt" href="wellset1.html#E6:6"><i><font color="Green" title="E9">A87</font></i></a>, <a class="txt" href="wellset1.html#E3:6_2"><i><font color="Green" title="E11">A89</font></i></a></i>;<br/>
<b>thus </b><a NAME="E5:6_2"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">D</font>
 <b>by </b><i><a class="txt" href="wellset1.html#E6:6"><i><font color="Green" title="E9">A87</font></i></a>, <a class="txt" href="wellset1.html#E3:6_2"><i><font color="Green" title="E11">A89</font></i></a></i>;<br/>

<b>consider </b><font color="Maroon" title="c9">S</font> being   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a><b> such that </b><br/><a NAME="E7:6_2"/><i><font color="Green" title="E13">A91</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</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="c9">S</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c7">x</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="c9">S</font> )
 <b>and </b><br/><a NAME="E8:6_2"/><i><font color="Green" title="E14">A92</font></i>: 
<font color="Maroon" title="c9">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G</font>
 <b>by </b><i><a class="txt" href="wellset1.html#E4:6_2"><i><font color="Green" title="E12">A90</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c9">S</font>
;<br/>

<b>thus </b><a NAME="E9:6_2"/>
( <font color="Maroon" title="c7">x</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="c9">S</font> &amp; <font color="Maroon" title="c9">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G</font> )
 <b>by </b><i><a class="txt" href="wellset1.html#E7:6_2"><i><font color="Green" title="E13">A91</font></i></a>, <a class="txt" href="wellset1.html#E8:6_2"><i><font color="Green" title="E14">A92</font></i></a>, <a class="ref" href="wellset1.html#T1" target="_self" title="WELLSET1:th.1">Th1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:6"/><b>then </b>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</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="c6">R</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">D</font>
 
;<br/>
<b>hence </b><a NAME="E9:6"/>
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">D</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>

<a NAME="E10:6"/><i><font color="Green" title="E11">A93</font></i>: 
for <font color="Olive" title="b1">W</font> being  <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a>  st <font color="Olive" title="b1">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G</font> holds <br/> <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Olive" title="b1">W</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="c6">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">W</font> be   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a>;<br/>

<b>assume </b><a NAME="E1:6_3"/><i><font color="Green" title="E12">A94</font></i>: 
<font color="Maroon" title="c7">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:6_3"/>
<font color="Maroon" title="c8">x</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="c7">W</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:6_3"/><i><font color="Green" title="E13">A95</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</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="c7">W</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c8">x</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="c7">W</font> )
 <b>by </b><i><a class="ref" href="wellset1.html#T1" target="_self" title="WELLSET1:th.1">Th1</a></i>;<br/>
<a NAME="E5:6_3"/><i><font color="Green" title="E14">A96</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</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="c7">W</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</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="c6">R</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_3_1"/><i><font color="Green" title="E15">A97</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</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="c7">W</font>
 ;<br/>

<a NAME="E2:6_3_1"/>
<font color="Maroon" title="c7">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</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"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">D</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="wellset1.html#E3:6"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="wellset1.html#E1:6_3"><i><font color="Green" title="E12">A94</font></i></a></i>;<br/>
<a NAME="E3:6_3_1"/><b>then </b>
 ex <font color="Olive" title="b1">z1</font>, <font color="Olive" title="b2">z2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">D</font> &amp; <font color="Olive" title="b2">z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">D</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">z1</font>,<font color="Olive" title="b2">z2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 
<b>by </b><i><a class="txt" href="wellset1.html#E1:6_3_1"><i><font color="Green" title="E15">A97</font></i></a>, <a class="ref" href="zfmisc_1.html#T103" title="ZFMISC_1:th.103">ZFMISC_1:103</a></i>;<br/>
<b>hence </b><a NAME="E4:6_3_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</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="c6">R</font>
 <b>by </b><i><a class="txt" href="wellset1.html#E6:6"><i><font color="Green" title="E9">A87</font></i></a>, <a class="txt" href="wellset1.html#E1:6_3"><i><font color="Green" title="E12">A94</font></i></a>, <a class="txt" href="wellset1.html#E1:6_3_1"><i><font color="Green" title="E15">A97</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:6_3"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c8">x</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="c7">W</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c8">x</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="c6">R</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_3_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_3_2"/><i><font color="Green" title="E15">A98</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c8">x</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="c7">W</font>
 ;<br/>

<a NAME="E2:6_3_2"/>
<font color="Maroon" title="c7">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</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"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">D</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="wellset1.html#E3:6"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="wellset1.html#E1:6_3"><i><font color="Green" title="E12">A94</font></i></a></i>;<br/>
<a NAME="E3:6_3_2"/><b>then </b>
 ex <font color="Olive" title="b1">z1</font>, <font color="Olive" title="b2">z2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">D</font> &amp; <font color="Olive" title="b2">z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">D</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c8">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">z1</font>,<font color="Olive" title="b2">z2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 
<b>by </b><i><a class="txt" href="wellset1.html#E1:6_3_2"><i><font color="Green" title="E15">A98</font></i></a>, <a class="ref" href="zfmisc_1.html#T103" title="ZFMISC_1:th.103">ZFMISC_1:103</a></i>;<br/>
<b>hence </b><a NAME="E4:6_3_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c8">x</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="c6">R</font>
 <b>by </b><i><a class="txt" href="wellset1.html#E6:6"><i><font color="Green" title="E9">A87</font></i></a>, <a class="txt" href="wellset1.html#E1:6_3"><i><font color="Green" title="E12">A94</font></i></a>, <a class="txt" href="wellset1.html#E1:6_3_2"><i><font color="Green" title="E15">A98</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:6_3"/>
<font color="Maroon" title="c8">x</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="c6">R</font>
 <b>by </b><i><a class="txt" href="wellset1.html#E4:6_3"><i><font color="Green" title="E13">A95</font></i></a>, <a class="txt" href="wellset1.html#E5:6_3"><i><font color="Green" title="E14">A96</font></i></a>, <a class="ref" href="wellset1.html#T1" target="_self" title="WELLSET1:th.1">Th1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:6"/><i><font color="Green" title="E12">A99</font></i>: 
<font color="Maroon" title="c6">R</font> is <a href="wellord1.html#V2" title="WELLORD1:attr.2">well-ordering</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_4"><b>proof </b></a><div class="add">

<a NAME="E1:6_4"/><i><font color="Green" title="E13">A100</font></i>: 
<font color="Maroon" title="c6">R</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_4_1"><b>proof </b></a><div class="add">

<a NAME="E1:6_4_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</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="c6">R</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b1">x</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="c6">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_4_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:6_4_1_1"/><i><font color="Green" title="E14">A101</font></i>: 
<font color="Maroon" title="c7">x</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="c6">R</font>
 ;<br/>

<a NAME="E2:6_4_1_1"/><b>then </b><i><font color="Green" title="E15">A102</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">D</font> &amp;  ex <font color="Olive" title="b1">W</font> being  <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a> st <br/>( <font color="Maroon" title="c7">x</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="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G</font> ) )
 
<b>by </b><i><a class="txt" href="wellset1.html#E7:6"><i><font color="Green" title="E10">A88</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">W</font> being   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a><b> such that </b><br/><a NAME="E4:6_4_1_1"/><i><font color="Green" title="E16">A103</font></i>: 
( <font color="Maroon" title="c7">x</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="c8">W</font> &amp; <font color="Maroon" title="c8">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G</font> )
 <b>by </b><i><a class="txt" href="wellset1.html#E7:6"><i><font color="Green" title="E10">A88</font></i></a>, <a class="txt" href="wellset1.html#E1:6_4_1_1"><i><font color="Green" title="E14">A101</font></i></a></i>;<br/>
<a NAME="E5:6_4_1_1"/>
<font color="Maroon" title="c8">W</font> is <a href="wellord1.html#V2" title="WELLORD1:attr.2">well-ordering</a>
 
<b>by </b><i><a class="txt" href="wellset1.html#E3:6"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="wellset1.html#E4:6_4_1_1"><i><font color="Green" title="E16">A103</font></i></a></i>;<br/>
<a NAME="E6:6_4_1_1"/><b>then </b>
<font color="Maroon" title="c8">W</font> <a href="wellord1.html#R2" title="WELLORD1:pred.2">well_orders</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c8">W</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#T8" title="WELLORD1:th.8">WELLORD1:8</a></i>;<br/>
<a NAME="E7:6_4_1_1"/><b>then </b>
<font color="Maroon" title="c8">W</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c8">W</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#D5" title="WELLORD1:def.5">WELLORD1:def 5</a></i>;<br/>
<a NAME="E8:6_4_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c7">x</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="c8">W</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E4:6_4_1_1"><i><font color="Green" title="E16">A103</font></i></a>, <a class="ref" href="relat_2.html#D1" title="RELAT_2:def.1">RELAT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E9:6_4_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c7">x</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="c6">R</font>
 <b>by </b><i><a class="txt" href="wellset1.html#E6:6"><i><font color="Green" title="E9">A87</font></i></a>, <a class="txt" href="wellset1.html#E2:6_4_1_1"><i><font color="Green" title="E15">A102</font></i></a>, <a class="txt" href="wellset1.html#E4:6_4_1_1"><i><font color="Green" title="E16">A103</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:6_4_1"/>
<font color="Maroon" title="c6">R</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font>
 <b>by </b><i><a class="ref" href="relat_2.html#D1" title="RELAT_2:def.1">RELAT_2:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:6_4"/><i><font color="Green" title="E14">A104</font></i>: 
<font color="Maroon" title="c6">R</font> <a href="relat_2.html#R8" title="RELAT_2:pred.8">is_transitive_in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_4_2"><b>proof </b></a><div class="add">

<a NAME="E1:6_4_2"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</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="c6">R</font> &amp; <font color="Olive" title="b2">y</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="c6">R</font> &amp; <font color="Olive" title="b3">z</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="c6">R</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</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="c6">R</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</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="c6">R</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">z</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="c6">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_4_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">y</font>, <font color="Maroon" title="c9">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:6_4_2_1"/>
<font color="Maroon" title="c7">x</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="c6">R</font>
 <b>and </b><br/><a NAME="E2:6_4_2_1"/>
<font color="Maroon" title="c8">y</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="c6">R</font>
 <b>and </b><br/><a NAME="E3:6_4_2_1"/>
<font color="Maroon" title="c9">z</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="c6">R</font>
 <b>and </b><br/><a NAME="E4:6_4_2_1"/><i><font color="Green" title="E15">A105</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</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="c6">R</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c9">z</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="c6">R</font> )
 ;<br/>

<a NAME="E5:6_4_2_1"/><i><font color="Green" title="E16">A106</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">D</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">D</font> &amp;  ex <font color="Olive" title="b1">W</font> being  <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</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="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G</font> ) )
 
<b>by </b><i><a class="txt" href="wellset1.html#E6:6"><i><font color="Green" title="E9">A87</font></i></a>, <a class="txt" href="wellset1.html#E4:6_4_2_1"><i><font color="Green" title="E15">A105</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">W1</font> being   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a><b> such that </b><br/><a NAME="E7:6_4_2_1"/><i><font color="Green" title="E17">A107</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</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="c10">W1</font> &amp; <font color="Maroon" title="c10">W1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G</font> )
 <b>by </b><i><a class="txt" href="wellset1.html#E6:6"><i><font color="Green" title="E9">A87</font></i></a>, <a class="txt" href="wellset1.html#E4:6_4_2_1"><i><font color="Green" title="E15">A105</font></i></a></i>;<br/>
<a NAME="E8:6_4_2_1"/><i><font color="Green" title="E18">A108</font></i>: 
( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">D</font> &amp; <font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">D</font> &amp;  ex <font color="Olive" title="b1">W</font> being  <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c9">z</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="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G</font> ) )
 
<b>by </b><i><a class="txt" href="wellset1.html#E6:6"><i><font color="Green" title="E9">A87</font></i></a>, <a class="txt" href="wellset1.html#E4:6_4_2_1"><i><font color="Green" title="E15">A105</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">W2</font> being   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a><b> such that </b><br/><a NAME="E10:6_4_2_1"/><i><font color="Green" title="E19">A109</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c9">z</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="c11">W2</font> &amp; <font color="Maroon" title="c11">W2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G</font> )
 <b>by </b><i><a class="txt" href="wellset1.html#E6:6"><i><font color="Green" title="E9">A87</font></i></a>, <a class="txt" href="wellset1.html#E4:6_4_2_1"><i><font color="Green" title="E15">A105</font></i></a></i>;<br/>
<a NAME="E11:6_4_2_1"/>
 ex <font color="Olive" title="b1">W</font> being  <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</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="Olive" title="b1">W</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c9">z</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="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_4_2_1_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon" title="c12">W</font> = <font color="Maroon" title="c11">W2</font>;<br/>

<a NAME="E1:6_4_2_1_1"/>
( <font color="Maroon" title="c8">y</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="c10">W1</font> &amp; <font color="Maroon" title="c8">y</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="c12">W</font> )
 
<b>by </b><i><a class="txt" href="wellset1.html#E7:6_4_2_1"><i><font color="Green" title="E17">A107</font></i></a>, <a class="txt" href="wellset1.html#E10:6_4_2_1"><i><font color="Green" title="E19">A109</font></i></a>, <a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a></i>;<br/>
<a NAME="E2:6_4_2_1_1"/><b>then </b><i><font color="Green" title="E20">A110</font></i>: 
<font color="Maroon" title="c10">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">W</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">y</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E4:6"><i><font color="Green" title="E8">A3</font></i></a>, <a class="txt" href="wellset1.html#E7:6_4_2_1"><i><font color="Green" title="E17">A107</font></i></a>, <a class="txt" href="wellset1.html#E10:6_4_2_1"><i><font color="Green" title="E19">A109</font></i></a></i>;<br/>
<a NAME="E3:6_4_2_1_1"/>
( not <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">y</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</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="c12">W</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_4_2_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_4_2_1_1_1"/><i><font color="Green" title="E21">A111</font></i>: 
not <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">y</font>
 ;<br/>

<a NAME="E2:6_4_2_1_1_1"/><i><font color="Green" title="E22">A112</font></i>: 
( <font color="Maroon" title="c7">x</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="c10">W1</font> &amp; <font color="Maroon" title="c8">y</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="c10">W1</font> )
 
<b>by </b><i><a class="txt" href="wellset1.html#E7:6_4_2_1"><i><font color="Green" title="E17">A107</font></i></a>, <a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a></i>;<br/>
<a NAME="E3:6_4_2_1_1_1"/><i><font color="Green" title="E23">A113</font></i>: 
<font color="Maroon" title="c10">W1</font> is <a href="wellord1.html#V2" title="WELLORD1:attr.2">well-ordering</a>
 
<b>by </b><i><a class="txt" href="wellset1.html#E3:6"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="wellset1.html#E7:6_4_2_1"><i><font color="Green" title="E17">A107</font></i></a></i>;<br/>
<a NAME="E4:6_4_2_1_1_1"/><b>then </b><i><font color="Green" title="E24">A114</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c7">x</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="c10">W1</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E1:6_4_2_1_1_1"><i><font color="Green" title="E21">A111</font></i></a>, <a class="txt" href="wellset1.html#E2:6_4_2_1_1_1"><i><font color="Green" title="E22">A112</font></i></a>, <a class="ref" href="wellset1.html#T6" target="_self" title="WELLSET1:th.6">Th6</a></i>;<br/>
<a NAME="E5:6_4_2_1_1_1"/>
<font color="Maroon" title="c10">W1</font> <a href="wellord1.html#R2" title="WELLORD1:pred.2">well_orders</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c10">W1</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E3:6_4_2_1_1_1"><i><font color="Green" title="E23">A113</font></i></a>, <a class="ref" href="wellord1.html#T8" title="WELLORD1:th.8">WELLORD1:8</a></i>;<br/>
<a NAME="E6:6_4_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">W1</font> <a href="relat_2.html#R4" title="RELAT_2:pred.4">is_antisymmetric_in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c10">W1</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#D5" title="WELLORD1:def.5">WELLORD1:def 5</a></i>;<br/>
<a NAME="E7:6_4_2_1_1_1"/><b>then </b><i><font color="Green" title="E25">A115</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E7:6_4_2_1"><i><font color="Green" title="E17">A107</font></i></a>, <a class="txt" href="wellset1.html#E2:6_4_2_1_1_1"><i><font color="Green" title="E22">A112</font></i></a>, <a class="txt" href="wellset1.html#E4:6_4_2_1_1_1"><i><font color="Green" title="E24">A114</font></i></a>, <a class="ref" href="relat_2.html#D4" title="RELAT_2:def.4">RELAT_2:def 4</a></i>;<br/>
<a NAME="E8:6_4_2_1_1_1"/><i><font color="Green" title="E26">A116</font></i>: 
<font color="Maroon" title="c8">y</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="c12">W</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E10:6_4_2_1"><i><font color="Green" title="E19">A109</font></i></a>, <a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a></i>;<br/>
<a NAME="E9:6_4_2_1_1_1"/>
<font color="Maroon" title="c12">W</font> is <a href="wellord1.html#V2" title="WELLORD1:attr.2">well-ordering</a>
 
<b>by </b><i><a class="txt" href="wellset1.html#E3:6"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="wellset1.html#E10:6_4_2_1"><i><font color="Green" title="E19">A109</font></i></a></i>;<br/>
<a NAME="E10:6_4_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c12">W</font> <a href="wellord1.html#R2" title="WELLORD1:pred.2">well_orders</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c12">W</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#T8" title="WELLORD1:th.8">WELLORD1:8</a></i>;<br/>
<a NAME="E11:6_4_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c12">W</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c12">W</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#D5" title="WELLORD1:def.5">WELLORD1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E12:6_4_2_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</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="c12">W</font>
 <b>by </b><i><a class="txt" href="wellset1.html#E7:6_4_2_1_1_1"><i><font color="Green" title="E25">A115</font></i></a>, <a class="txt" href="wellset1.html#E8:6_4_2_1_1_1"><i><font color="Green" title="E26">A116</font></i></a>, <a class="ref" href="relat_2.html#D1" title="RELAT_2:def.1">RELAT_2:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:6_4_2_1_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</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="c12">W</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c9">z</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="c12">W</font> &amp; <font color="Maroon" title="c12">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G</font> )
 <b>by </b><i><a class="txt" href="wellset1.html#E10:6_4_2_1"><i><font color="Green" title="E19">A109</font></i></a>, <a class="txt" href="wellset1.html#E2:6_4_2_1_1"><i><font color="Green" title="E20">A110</font></i></a>, <a class="ref" href="wellord1.html#D1" title="WELLORD1:def.1">WELLORD1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c12">W</font> being   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a><b> such that </b><br/><a NAME="E13:6_4_2_1"/><i><font color="Green" title="E20">A117</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</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="c12">W</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c9">z</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="c12">W</font> &amp; <font color="Maroon" title="c12">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G</font> )
 ;<br/>
<a NAME="E14:6_4_2_1"/>
<font color="Maroon" title="c12">W</font> is <a href="wellord1.html#V2" title="WELLORD1:attr.2">well-ordering</a>
 
<b>by </b><i><a class="txt" href="wellset1.html#E3:6"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="wellset1.html#E13:6_4_2_1"><i><font color="Green" title="E20">A117</font></i></a></i>;<br/>
<a NAME="E15:6_4_2_1"/><b>then </b>
<font color="Maroon" title="c12">W</font> <a href="wellord1.html#R2" title="WELLORD1:pred.2">well_orders</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c12">W</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#T8" title="WELLORD1:th.8">WELLORD1:8</a></i>;<br/>
<a NAME="E16:6_4_2_1"/><b>then </b><i><font color="Green" title="E21">A118</font></i>: 
<font color="Maroon" title="c12">W</font> <a href="relat_2.html#R8" title="RELAT_2:pred.8">is_transitive_in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c12">W</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#D5" title="WELLORD1:def.5">WELLORD1:def 5</a></i>;<br/>
<a NAME="E17:6_4_2_1"/>
( <font color="Maroon" title="c7">x</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="c12">W</font> &amp; <font color="Maroon" title="c8">y</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="c12">W</font> &amp; <font color="Maroon" title="c9">z</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="c12">W</font> )
 
<b>by </b><i><a class="txt" href="wellset1.html#E13:6_4_2_1"><i><font color="Green" title="E20">A117</font></i></a>, <a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a></i>;<br/>
<a NAME="E18:6_4_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">z</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="c12">W</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E13:6_4_2_1"><i><font color="Green" title="E20">A117</font></i></a>, <a class="txt" href="wellset1.html#E16:6_4_2_1"><i><font color="Green" title="E21">A118</font></i></a>, <a class="ref" href="relat_2.html#D8" title="RELAT_2:def.8">RELAT_2:def 8</a></i>;<br/>
<b>hence </b><a NAME="E19:6_4_2_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">z</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="c6">R</font>
 <b>by </b><i><a class="txt" href="wellset1.html#E6:6"><i><font color="Green" title="E9">A87</font></i></a>, <a class="txt" href="wellset1.html#E5:6_4_2_1"><i><font color="Green" title="E16">A106</font></i></a>, <a class="txt" href="wellset1.html#E8:6_4_2_1"><i><font color="Green" title="E18">A108</font></i></a>, <a class="txt" href="wellset1.html#E13:6_4_2_1"><i><font color="Green" title="E20">A117</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:6_4_2"/>
<font color="Maroon" title="c6">R</font> <a href="relat_2.html#R8" title="RELAT_2:pred.8">is_transitive_in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font>
 <b>by </b><i><a class="ref" href="relat_2.html#D8" title="RELAT_2:def.8">RELAT_2:def 8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:6_4"/><i><font color="Green" title="E15">A119</font></i>: 
<font color="Maroon" title="c6">R</font> <a href="relat_2.html#R4" title="RELAT_2:pred.4">is_antisymmetric_in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_4_3"><b>proof </b></a><div class="add">

<a NAME="E1:6_4_3"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</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="c6">R</font> &amp; <font color="Olive" title="b2">y</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="c6">R</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</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="c6">R</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</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="c6">R</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_4_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:6_4_3_1"/>
<font color="Maroon" title="c7">x</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="c6">R</font>
 <b>and </b><br/><a NAME="E2:6_4_3_1"/>
<font color="Maroon" title="c8">y</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="c6">R</font>
 <b>and </b><br/><a NAME="E3:6_4_3_1"/><i><font color="Green" title="E16">A120</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</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="c6">R</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c7">x</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="c6">R</font> )
 ;<br/>

<b>consider </b><font color="Maroon" title="c9">W1</font> being   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a><b> such that </b><br/><a NAME="E5:6_4_3_1"/><i><font color="Green" title="E17">A121</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</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="c9">W1</font> &amp; <font color="Maroon" title="c9">W1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G</font> )
 <b>by </b><i><a class="txt" href="wellset1.html#E6:6"><i><font color="Green" title="E9">A87</font></i></a>, <a class="txt" href="wellset1.html#E3:6_4_3_1"><i><font color="Green" title="E16">A120</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">W2</font> being   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a><b> such that </b><br/><a NAME="E7:6_4_3_1"/><i><font color="Green" title="E18">A122</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c7">x</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="c10">W2</font> &amp; <font color="Maroon" title="c10">W2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G</font> )
 <b>by </b><i><a class="txt" href="wellset1.html#E6:6"><i><font color="Green" title="E9">A87</font></i></a>, <a class="txt" href="wellset1.html#E3:6_4_3_1"><i><font color="Green" title="E16">A120</font></i></a></i>;<br/>
<a NAME="E8:6_4_3_1"/><i><font color="Green" title="E19">A123</font></i>: 
( <font color="Maroon" title="c9">W1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">W2</font> implies <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_4_3_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_4_3_1_1"/><i><font color="Green" title="E20">A124</font></i>: 
<font color="Maroon" title="c9">W1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">W2</font>
 ;<br/>

<a NAME="E2:6_4_3_1_1"/><b>then </b><i><font color="Green" title="E21">A125</font></i>: 
( <font color="Maroon" title="c7">x</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="c10">W2</font> &amp; <font color="Maroon" title="c8">y</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="c10">W2</font> )
 
<b>by </b><i><a class="txt" href="wellset1.html#E5:6_4_3_1"><i><font color="Green" title="E17">A121</font></i></a>, <a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a></i>;<br/>
<a NAME="E3:6_4_3_1_1"/>
<font color="Maroon" title="c10">W2</font> is <a href="wellord1.html#V2" title="WELLORD1:attr.2">well-ordering</a>
 
<b>by </b><i><a class="txt" href="wellset1.html#E3:6"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="wellset1.html#E7:6_4_3_1"><i><font color="Green" title="E18">A122</font></i></a></i>;<br/>
<a NAME="E4:6_4_3_1_1"/><b>then </b>
<font color="Maroon" title="c10">W2</font> <a href="wellord1.html#R2" title="WELLORD1:pred.2">well_orders</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c10">W2</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#T8" title="WELLORD1:th.8">WELLORD1:8</a></i>;<br/>
<a NAME="E5:6_4_3_1_1"/><b>then </b>
<font color="Maroon" title="c10">W2</font> <a href="relat_2.html#R4" title="RELAT_2:pred.4">is_antisymmetric_in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c10">W2</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#D5" title="WELLORD1:def.5">WELLORD1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E6:6_4_3_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font>
 <b>by </b><i><a class="txt" href="wellset1.html#E5:6_4_3_1"><i><font color="Green" title="E17">A121</font></i></a>, <a class="txt" href="wellset1.html#E7:6_4_3_1"><i><font color="Green" title="E18">A122</font></i></a>, <a class="txt" href="wellset1.html#E1:6_4_3_1_1"><i><font color="Green" title="E20">A124</font></i></a>, <a class="txt" href="wellset1.html#E2:6_4_3_1_1"><i><font color="Green" title="E21">A125</font></i></a>, <a class="ref" href="relat_2.html#D4" title="RELAT_2:def.4">RELAT_2:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:6_4_3_1"/>
( <font color="Maroon" title="c10">W2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">W1</font> implies <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_4_3_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_4_3_1_2"/><i><font color="Green" title="E20">A126</font></i>: 
<font color="Maroon" title="c10">W2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">W1</font>
 ;<br/>

<a NAME="E2:6_4_3_1_2"/><b>then </b><i><font color="Green" title="E21">A127</font></i>: 
( <font color="Maroon" title="c7">x</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="c9">W1</font> &amp; <font color="Maroon" title="c8">y</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="c9">W1</font> )
 
<b>by </b><i><a class="txt" href="wellset1.html#E7:6_4_3_1"><i><font color="Green" title="E18">A122</font></i></a>, <a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a></i>;<br/>
<a NAME="E3:6_4_3_1_2"/>
<font color="Maroon" title="c9">W1</font> is <a href="wellord1.html#V2" title="WELLORD1:attr.2">well-ordering</a>
 
<b>by </b><i><a class="txt" href="wellset1.html#E3:6"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="wellset1.html#E5:6_4_3_1"><i><font color="Green" title="E17">A121</font></i></a></i>;<br/>
<a NAME="E4:6_4_3_1_2"/><b>then </b>
<font color="Maroon" title="c9">W1</font> <a href="wellord1.html#R2" title="WELLORD1:pred.2">well_orders</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c9">W1</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#T8" title="WELLORD1:th.8">WELLORD1:8</a></i>;<br/>
<a NAME="E5:6_4_3_1_2"/><b>then </b>
<font color="Maroon" title="c9">W1</font> <a href="relat_2.html#R4" title="RELAT_2:pred.4">is_antisymmetric_in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c9">W1</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#D5" title="WELLORD1:def.5">WELLORD1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E6:6_4_3_1_2"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font>
 <b>by </b><i><a class="txt" href="wellset1.html#E5:6_4_3_1"><i><font color="Green" title="E17">A121</font></i></a>, <a class="txt" href="wellset1.html#E7:6_4_3_1"><i><font color="Green" title="E18">A122</font></i></a>, <a class="txt" href="wellset1.html#E1:6_4_3_1_2"><i><font color="Green" title="E20">A126</font></i></a>, <a class="txt" href="wellset1.html#E2:6_4_3_1_2"><i><font color="Green" title="E21">A127</font></i></a>, <a class="ref" href="relat_2.html#D4" title="RELAT_2:def.4">RELAT_2:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:6_4_3_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font>
 <b>by </b><i><a class="txt" href="wellset1.html#E4:6"><i><font color="Green" title="E8">A3</font></i></a>, <a class="txt" href="wellset1.html#E5:6_4_3_1"><i><font color="Green" title="E17">A121</font></i></a>, <a class="txt" href="wellset1.html#E7:6_4_3_1"><i><font color="Green" title="E18">A122</font></i></a>, <a class="txt" href="wellset1.html#E8:6_4_3_1"><i><font color="Green" title="E19">A123</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:6_4_3"/>
<font color="Maroon" title="c6">R</font> <a href="relat_2.html#R4" title="RELAT_2:pred.4">is_antisymmetric_in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font>
 <b>by </b><i><a class="ref" href="relat_2.html#D4" title="RELAT_2:def.4">RELAT_2:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:6_4"/><i><font color="Green" title="E16">A128</font></i>: 
<font color="Maroon" title="c6">R</font> <a href="relat_2.html#R6" title="RELAT_2:pred.6">is_connected_in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_4_4"><b>proof </b></a><div class="add">

<a NAME="E1:6_4_4"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</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="c6">R</font> &amp; <font color="Olive" title="b2">y</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="c6">R</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> &amp; not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</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="c6">R</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</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="c6">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_4_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:6_4_4_1"/><i><font color="Green" title="E17">A129</font></i>: 
<font color="Maroon" title="c7">x</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="c6">R</font>
 <b>and </b><br/><a NAME="E2:6_4_4_1"/><i><font color="Green" title="E18">A130</font></i>: 
<font color="Maroon" title="c8">y</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="c6">R</font>
 <b>and </b><br/><a NAME="E3:6_4_4_1"/><i><font color="Green" title="E19">A131</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">y</font>
 ;<br/>

<a NAME="E4:6_4_4_1"/><i><font color="Green" title="E20">A132</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">D</font> &amp;  ex <font color="Olive" title="b1">W</font> being  <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a> st <br/>( <font color="Maroon" title="c7">x</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="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G</font> ) )
 
<b>by </b><i><a class="txt" href="wellset1.html#E7:6"><i><font color="Green" title="E10">A88</font></i></a>, <a class="txt" href="wellset1.html#E1:6_4_4_1"><i><font color="Green" title="E17">A129</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">W1</font> being   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a><b> such that </b><br/><a NAME="E6:6_4_4_1"/><i><font color="Green" title="E21">A133</font></i>: 
( <font color="Maroon" title="c7">x</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="c9">W1</font> &amp; <font color="Maroon" title="c9">W1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G</font> )
 <b>by </b><i><a class="txt" href="wellset1.html#E7:6"><i><font color="Green" title="E10">A88</font></i></a>, <a class="txt" href="wellset1.html#E1:6_4_4_1"><i><font color="Green" title="E17">A129</font></i></a></i>;<br/>
<a NAME="E7:6_4_4_1"/><i><font color="Green" title="E22">A134</font></i>: 
( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">D</font> &amp;  ex <font color="Olive" title="b1">W</font> being  <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a> st <br/>( <font color="Maroon" title="c8">y</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="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G</font> ) )
 
<b>by </b><i><a class="txt" href="wellset1.html#E7:6"><i><font color="Green" title="E10">A88</font></i></a>, <a class="txt" href="wellset1.html#E2:6_4_4_1"><i><font color="Green" title="E18">A130</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">W2</font> being   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a><b> such that </b><br/><a NAME="E9:6_4_4_1"/><i><font color="Green" title="E23">A135</font></i>: 
( <font color="Maroon" title="c8">y</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="c10">W2</font> &amp; <font color="Maroon" title="c10">W2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G</font> )
 <b>by </b><i><a class="txt" href="wellset1.html#E7:6"><i><font color="Green" title="E10">A88</font></i></a>, <a class="txt" href="wellset1.html#E2:6_4_4_1"><i><font color="Green" title="E18">A130</font></i></a></i>;<br/>
<a NAME="E10:6_4_4_1"/><i><font color="Green" title="E24">A136</font></i>: 
(  not <font color="Maroon" title="c9">W1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">W2</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</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="c6">R</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c7">x</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="c6">R</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_4_4_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_4_4_1_1"/>
<font color="Maroon" title="c9">W1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">W2</font>
 ;<br/>

<a NAME="E2:6_4_4_1_1"/><b>then </b><i><font color="Green" title="E25">A137</font></i>: 
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c9">W1</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="c10">W2</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T31" title="RELAT_1:th.31">RELAT_1:31</a></i>;<br/>
<a NAME="E3:6_4_4_1_1"/>
<font color="Maroon" title="c10">W2</font> is <a href="wellord1.html#V2" title="WELLORD1:attr.2">well-ordering</a>
 
<b>by </b><i><a class="txt" href="wellset1.html#E3:6"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="wellset1.html#E9:6_4_4_1"><i><font color="Green" title="E23">A135</font></i></a></i>;<br/>
<a NAME="E4:6_4_4_1_1"/><b>then </b>
<font color="Maroon" title="c10">W2</font> <a href="wellord1.html#R2" title="WELLORD1:pred.2">well_orders</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c10">W2</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#T8" title="WELLORD1:th.8">WELLORD1:8</a></i>;<br/>
<a NAME="E5:6_4_4_1_1"/><b>then </b>
<font color="Maroon" title="c10">W2</font> <a href="relat_2.html#R6" title="RELAT_2:pred.6">is_connected_in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c10">W2</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#D5" title="WELLORD1:def.5">WELLORD1:def 5</a></i>;<br/>
<a NAME="E6:6_4_4_1_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</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="c10">W2</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c7">x</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="c10">W2</font> )
 
<b>by </b><i><a class="txt" href="wellset1.html#E3:6_4_4_1"><i><font color="Green" title="E19">A131</font></i></a>, <a class="txt" href="wellset1.html#E6:6_4_4_1"><i><font color="Green" title="E21">A133</font></i></a>, <a class="txt" href="wellset1.html#E9:6_4_4_1"><i><font color="Green" title="E23">A135</font></i></a>, <a class="txt" href="wellset1.html#E2:6_4_4_1_1"><i><font color="Green" title="E25">A137</font></i></a>, <a class="ref" href="relat_2.html#D6" title="RELAT_2:def.6">RELAT_2:def 6</a></i>;<br/>
<b>hence </b><a NAME="E7:6_4_4_1_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</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="c6">R</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c7">x</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="c6">R</font> )
 <b>by </b><i><a class="txt" href="wellset1.html#E6:6"><i><font color="Green" title="E9">A87</font></i></a>, <a class="txt" href="wellset1.html#E4:6_4_4_1"><i><font color="Green" title="E20">A132</font></i></a>, <a class="txt" href="wellset1.html#E7:6_4_4_1"><i><font color="Green" title="E22">A134</font></i></a>, <a class="txt" href="wellset1.html#E9:6_4_4_1"><i><font color="Green" title="E23">A135</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:6_4_4_1"/>
(  not <font color="Maroon" title="c10">W2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">W1</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</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="c6">R</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c7">x</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="c6">R</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_4_4_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_4_4_1_2"/>
<font color="Maroon" title="c10">W2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">W1</font>
 ;<br/>

<a NAME="E2:6_4_4_1_2"/><b>then </b><i><font color="Green" title="E25">A138</font></i>: 
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c10">W2</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="c9">W1</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T31" title="RELAT_1:th.31">RELAT_1:31</a></i>;<br/>
<a NAME="E3:6_4_4_1_2"/>
<font color="Maroon" title="c9">W1</font> is <a href="wellord1.html#V2" title="WELLORD1:attr.2">well-ordering</a>
 
<b>by </b><i><a class="txt" href="wellset1.html#E3:6"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="wellset1.html#E6:6_4_4_1"><i><font color="Green" title="E21">A133</font></i></a></i>;<br/>
<a NAME="E4:6_4_4_1_2"/><b>then </b>
<font color="Maroon" title="c9">W1</font> <a href="wellord1.html#R2" title="WELLORD1:pred.2">well_orders</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c9">W1</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#T8" title="WELLORD1:th.8">WELLORD1:8</a></i>;<br/>
<a NAME="E5:6_4_4_1_2"/><b>then </b>
<font color="Maroon" title="c9">W1</font> <a href="relat_2.html#R6" title="RELAT_2:pred.6">is_connected_in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c9">W1</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#D5" title="WELLORD1:def.5">WELLORD1:def 5</a></i>;<br/>
<a NAME="E6:6_4_4_1_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</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="c9">W1</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c7">x</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="c9">W1</font> )
 
<b>by </b><i><a class="txt" href="wellset1.html#E3:6_4_4_1"><i><font color="Green" title="E19">A131</font></i></a>, <a class="txt" href="wellset1.html#E6:6_4_4_1"><i><font color="Green" title="E21">A133</font></i></a>, <a class="txt" href="wellset1.html#E9:6_4_4_1"><i><font color="Green" title="E23">A135</font></i></a>, <a class="txt" href="wellset1.html#E2:6_4_4_1_2"><i><font color="Green" title="E25">A138</font></i></a>, <a class="ref" href="relat_2.html#D6" title="RELAT_2:def.6">RELAT_2:def 6</a></i>;<br/>
<b>hence </b><a NAME="E7:6_4_4_1_2"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</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="c6">R</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c7">x</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="c6">R</font> )
 <b>by </b><i><a class="txt" href="wellset1.html#E6:6"><i><font color="Green" title="E9">A87</font></i></a>, <a class="txt" href="wellset1.html#E4:6_4_4_1"><i><font color="Green" title="E20">A132</font></i></a>, <a class="txt" href="wellset1.html#E6:6_4_4_1"><i><font color="Green" title="E21">A133</font></i></a>, <a class="txt" href="wellset1.html#E7:6_4_4_1"><i><font color="Green" title="E22">A134</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:6_4_4_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</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="c6">R</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c7">x</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="c6">R</font> )
 <b>by </b><i><a class="txt" href="wellset1.html#E4:6"><i><font color="Green" title="E8">A3</font></i></a>, <a class="txt" href="wellset1.html#E6:6_4_4_1"><i><font color="Green" title="E21">A133</font></i></a>, <a class="txt" href="wellset1.html#E9:6_4_4_1"><i><font color="Green" title="E23">A135</font></i></a>, <a class="txt" href="wellset1.html#E10:6_4_4_1"><i><font color="Green" title="E24">A136</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:6_4_4"/>
<font color="Maroon" title="c6">R</font> <a href="relat_2.html#R6" title="RELAT_2:pred.6">is_connected_in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font>
 <b>by </b><i><a class="ref" href="relat_2.html#D6" title="RELAT_2:def.6">RELAT_2:def 6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:6_4"/>
<font color="Maroon" title="c6">R</font> <a href="wellord1.html#R1" title="WELLORD1:pred.1">is_well_founded_in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_4_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="wellord1.html#D3" title="WELLORD1:def.3">WELLORD1:def 3</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:6_4_5"/><i><font color="Green" title="E17">A139</font></i>: 
<font color="Maroon" title="c7">Y</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="c6">R</font>
 <b>and </b><br/><a NAME="E2:6_4_5"/><i><font color="Green" title="E18">A140</font></i>: 
<font color="Maroon" title="c7">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon" title="c8">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">Y</font>;<br/>
<a NAME="E4:6_4_5"/>
<font color="Maroon" title="c8">y</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="c6">R</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E1:6_4_5"><i><font color="Green" title="E17">A139</font></i></a>, <a class="txt" href="wellset1.html#E2:6_4_5"><i><font color="Green" title="E18">A140</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">W</font> being   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a><b> such that </b><br/><a NAME="E6:6_4_5"/><i><font color="Green" title="E19">A141</font></i>: 
( <font color="Maroon" title="c8">y</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="c9">W</font> &amp; <font color="Maroon" title="c9">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G</font> )
 <b>by </b><i><a class="txt" href="wellset1.html#E7:6"><i><font color="Green" title="E10">A88</font></i></a></i>;<br/>
<a NAME="E7:6_4_5"/>
<font color="Maroon" title="c9">W</font> is <a href="wellord1.html#V2" title="WELLORD1:attr.2">well-ordering</a>
 
<b>by </b><i><a class="txt" href="wellset1.html#E3:6"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="wellset1.html#E6:6_4_5"><i><font color="Green" title="E19">A141</font></i></a></i>;<br/>
<a NAME="E8:6_4_5"/><b>then </b>
<font color="Maroon" title="c9">W</font> <a href="wellord1.html#R2" title="WELLORD1:pred.2">well_orders</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c9">W</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#T8" title="WELLORD1:th.8">WELLORD1:8</a></i>;<br/>
<a NAME="E9:6_4_5"/><b>then </b><i><font color="Green" title="E20">A142</font></i>: 
<font color="Maroon" title="c9">W</font> <a href="wellord1.html#R1" title="WELLORD1:pred.1">is_well_founded_in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c9">W</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#D5" title="WELLORD1:def.5">WELLORD1:def 5</a></i>;<br/>
<b>set </b><font color="Maroon" title="c10">A</font> = <font color="Maroon" title="c7">Y</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c9">W</font></span>)</span>;<br/>
<a NAME="E10:6_4_5"/><i><font color="Green" title="E21">A143</font></i>: 
<font color="Maroon" title="c7">Y</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c9">W</font></span>)</span> <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="c9">W</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E11:6_4_5"/>
<font color="Maroon" title="c7">Y</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c9">W</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="wellset1.html#E2:6_4_5"><i><font color="Green" title="E18">A140</font></i></a>, <a class="txt" href="wellset1.html#E6:6_4_5"><i><font color="Green" title="E19">A141</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E13:6_4_5"/><i><font color="Green" title="E22">A144</font></i>: 
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c9">W</font></span>)</span>
 <b>and </b><br/><a NAME="E14:6_4_5"/><i><font color="Green" title="E23">A145</font></i>: 
<font color="Maroon" title="c9">W</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c11">a</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c7">Y</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c9">W</font></span>)</span>
 <b>by </b><i><a class="txt" href="wellset1.html#E9:6_4_5"><i><font color="Green" title="E20">A142</font></i></a>, <a class="txt" href="wellset1.html#E10:6_4_5"><i><font color="Green" title="E21">A143</font></i></a>, <a class="ref" href="wellord1.html#D3" title="WELLORD1:def.3">WELLORD1:def 3</a></i>;<br/>
<a NAME="E15:6_4_5"/>
 ex <font color="Olive" title="b1">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y</font> &amp; <font color="Maroon" title="c6">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b1">b</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c7">Y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_4_5_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon" title="c12">b</font> = <font color="Maroon" title="c11">a</font>;<br/>

<b>thus </b><a NAME="E1:6_4_5_1"/>
<font color="Maroon" title="c12">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y</font>
 <b>by </b><i><a class="txt" href="wellset1.html#E13:6_4_5"><i><font color="Green" title="E22">A144</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>

<b>assume </b><a NAME="E2:6_4_5_1"/>
not <font color="Maroon" title="c6">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c12">b</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c7">Y</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c13">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:6_4_5_1"/><i><font color="Green" title="E24">A146</font></i>: 
( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c12">b</font> &amp; <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E5:6_4_5_1"/><i><font color="Green" title="E25">A147</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c12">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="c6">R</font> &amp; <font color="Maroon" title="c13">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">b</font> )
 
<b>by </b><i><a class="txt" href="wellset1.html#E4:6_4_5_1"><i><font color="Green" title="E24">A146</font></i></a>, <a class="ref" href="wellord1.html#D1" title="WELLORD1:def.1">WELLORD1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">W1</font> being   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a><b> such that </b><br/><a NAME="E7:6_4_5_1"/><i><font color="Green" title="E26">A148</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c12">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="c14">W1</font> &amp; <font color="Maroon" title="c14">W1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G</font> )
 <b>by </b><i><a class="txt" href="wellset1.html#E6:6"><i><font color="Green" title="E9">A87</font></i></a></i>;<br/>
<a NAME="E8:6_4_5_1"/><i><font color="Green" title="E27">A149</font></i>: 
( <font color="Maroon" title="c13">x</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="c14">W1</font> &amp; <font color="Maroon" title="c12">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="c14">W1</font> )
 
<b>by </b><i><a class="txt" href="wellset1.html#E7:6_4_5_1"><i><font color="Green" title="E26">A148</font></i></a>, <a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a></i>;<br/>
<a NAME="E9:6_4_5_1"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c12">b</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E5:6_4_5_1"><i><font color="Green" title="E25">A147</font></i></a>, <a class="txt" href="wellset1.html#E7:6_4_5_1"><i><font color="Green" title="E26">A148</font></i></a>, <a class="ref" href="wellord1.html#D1" title="WELLORD1:def.1">WELLORD1:def 1</a></i>;<br/>
<a NAME="E10:6_4_5_1"/><b>then </b><i><font color="Green" title="E28">A150</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">W</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c11">a</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E4:6"><i><font color="Green" title="E8">A3</font></i></a>, <a class="txt" href="wellset1.html#E6:6_4_5"><i><font color="Green" title="E19">A141</font></i></a>, <a class="txt" href="wellset1.html#E10:6_4_5"><i><font color="Green" title="E21">A143</font></i></a>, <a class="txt" href="wellset1.html#E13:6_4_5"><i><font color="Green" title="E22">A144</font></i></a>, <a class="txt" href="wellset1.html#E7:6_4_5_1"><i><font color="Green" title="E26">A148</font></i></a>, <a class="txt" href="wellset1.html#E8:6_4_5_1"><i><font color="Green" title="E27">A149</font></i></a></i>;<br/>
<a NAME="E11:6_4_5_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c11">a</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="c9">W</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#D1" title="WELLORD1:def.1">WELLORD1:def 1</a></i>;<br/>
<a NAME="E12:6_4_5_1"/><b>then </b>
<font color="Maroon" title="c13">x</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="c9">W</font>
 
<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="E13:6_4_5_1"/><b>then </b>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c9">W</font></span>)</span>
 
<b>by </b><i><a class="txt" href="wellset1.html#E4:6_4_5_1"><i><font color="Green" title="E24">A146</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E14:6_4_5_1"/>
contradiction
 <b>by </b><i><a class="txt" href="wellset1.html#E14:6_4_5"><i><font color="Green" title="E23">A145</font></i></a>, <a class="txt" href="wellset1.html#E10:6_4_5_1"><i><font color="Green" title="E28">A150</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E16:6_4_5"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y</font> &amp; <font color="Maroon" title="c6">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c7">Y</font> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:6_4"/><b>then </b>
<font color="Maroon" title="c6">R</font> <a href="wellord1.html#R2" title="WELLORD1:pred.2">well_orders</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E1:6_4"><i><font color="Green" title="E13">A100</font></i></a>, <a class="txt" href="wellset1.html#E2:6_4"><i><font color="Green" title="E14">A104</font></i></a>, <a class="txt" href="wellset1.html#E3:6_4"><i><font color="Green" title="E15">A119</font></i></a>, <a class="txt" href="wellset1.html#E4:6_4"><i><font color="Green" title="E16">A128</font></i></a>, <a class="ref" href="wellord1.html#D5" title="WELLORD1:def.5">WELLORD1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E7:6_4"/>
<font color="Maroon" title="c6">R</font> is <a href="wellord1.html#V2" title="WELLORD1:attr.2">well-ordering</a>
 <b>by </b><i><a class="ref" href="wellord1.html#T8" title="WELLORD1:th.8">WELLORD1:8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:6"/><i><font color="Green" title="E13">A151</font></i>: 
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</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="c6">R</font> holds <br/>( <font color="Maroon" title="c6">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font> &amp; <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b1">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:6_5"/><i><font color="Green" title="E14">A152</font></i>: 
<font color="Maroon" title="c7">y</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="c6">R</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">W</font> being   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a><b> such that </b><br/><a NAME="E3:6_5"/><i><font color="Green" title="E15">A153</font></i>: 
( <font color="Maroon" title="c7">y</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="c8">W</font> &amp; <font color="Maroon" title="c8">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G</font> )
 <b>by </b><i><a class="txt" href="wellset1.html#E7:6"><i><font color="Green" title="E10">A88</font></i></a></i>;<br/>
<a NAME="E4:6_5"/><i><font color="Green" title="E16">A154</font></i>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">D</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E7:6"><i><font color="Green" title="E10">A88</font></i></a>, <a class="txt" href="wellset1.html#E1:6_5"><i><font color="Green" title="E14">A152</font></i></a></i>;<br/>
<a NAME="E5:6_5"/><i><font color="Green" title="E17">A155</font></i>: 
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c8">W</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="c6">R</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E10:6"><i><font color="Green" title="E11">A93</font></i></a>, <a class="txt" href="wellset1.html#E3:6_5"><i><font color="Green" title="E15">A153</font></i></a></i>;<br/>
<a NAME="E6:6_5"/>
<font color="Maroon" title="c8">W</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c7">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_5_1"><b>proof </b></a><div class="add">

<a NAME="E1:6_5_1"/><i><font color="Green" title="E18">A156</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">W</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c7">y</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c7">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_5_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:6_5_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">W</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c7">y</font>
 ;<br/>

<a NAME="E2:6_5_1_1"/><b>then </b><i><font color="Green" title="E19">A157</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c7">y</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="c8">W</font> &amp; not <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font> )
 
<b>by </b><i><a class="ref" href="wellord1.html#D1" title="WELLORD1:def.1">WELLORD1:def 1</a></i>;<br/>
<a NAME="E3:6_5_1_1"/><b>then </b>
<font color="Maroon" title="c9">x</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="c8">W</font>
 
<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="E4:6_5_1_1"/><b>then </b>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">D</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E7:6"><i><font color="Green" title="E10">A88</font></i></a>, <a class="txt" href="wellset1.html#E5:6_5"><i><font color="Green" title="E17">A155</font></i></a></i>;<br/>
<a NAME="E5:6_5_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c7">y</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="c6">R</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E6:6"><i><font color="Green" title="E9">A87</font></i></a>, <a class="txt" href="wellset1.html#E3:6_5"><i><font color="Green" title="E15">A153</font></i></a>, <a class="txt" href="wellset1.html#E4:6_5"><i><font color="Green" title="E16">A154</font></i></a>, <a class="txt" href="wellset1.html#E2:6_5_1_1"><i><font color="Green" title="E19">A157</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:6_5_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c7">y</font>
 <b>by </b><i><a class="txt" href="wellset1.html#E2:6_5_1_1"><i><font color="Green" title="E19">A157</font></i></a>, <a class="ref" href="wellord1.html#D1" title="WELLORD1:def.1">WELLORD1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:6_5_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c7">y</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">W</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c7">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_5_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:6_5_1_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c7">y</font>
 ;<br/>

<a NAME="E2:6_5_1_2"/><b>then </b><i><font color="Green" title="E19">A158</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c7">y</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="c6">R</font> &amp; not <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font> )
 
<b>by </b><i><a class="ref" href="wellord1.html#D1" title="WELLORD1:def.1">WELLORD1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">W1</font> being   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a><b> such that </b><br/><a NAME="E4:6_5_1_2"/><i><font color="Green" title="E20">A159</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c7">y</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="c10">W1</font> &amp; <font color="Maroon" title="c10">W1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G</font> )
 <b>by </b><i><a class="txt" href="wellset1.html#E6:6"><i><font color="Green" title="E9">A87</font></i></a></i>;<br/>
<a NAME="E5:6_5_1_2"/><i><font color="Green" title="E21">A160</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c7">y</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E2:6_5_1_2"><i><font color="Green" title="E19">A158</font></i></a>, <a class="txt" href="wellset1.html#E4:6_5_1_2"><i><font color="Green" title="E20">A159</font></i></a>, <a class="ref" href="wellord1.html#D1" title="WELLORD1:def.1">WELLORD1:def 1</a></i>;<br/>
<a NAME="E6:6_5_1_2"/>
<font color="Maroon" title="c7">y</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="c10">W1</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E4:6_5_1_2"><i><font color="Green" title="E20">A159</font></i></a>, <a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a></i>;<br/>
<b>hence </b><a NAME="E7:6_5_1_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">W</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c7">y</font>
 <b>by </b><i><a class="txt" href="wellset1.html#E4:6"><i><font color="Green" title="E8">A3</font></i></a>, <a class="txt" href="wellset1.html#E3:6_5"><i><font color="Green" title="E15">A153</font></i></a>, <a class="txt" href="wellset1.html#E4:6_5_1_2"><i><font color="Green" title="E20">A159</font></i></a>, <a class="txt" href="wellset1.html#E5:6_5_1_2"><i><font color="Green" title="E21">A160</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:6_5_1"/>
<font color="Maroon" title="c8">W</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c7">y</font>
 <b>by </b><i><a class="txt" href="wellset1.html#E1:6_5_1"><i><font color="Green" title="E18">A156</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:6_5"/>
( <font color="Maroon" title="c6">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font> &amp; <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c7">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font> )
 <b>by </b><i><a class="txt" href="wellset1.html#E3:6"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="wellset1.html#E3:6_5"><i><font color="Green" title="E15">A153</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:6"/>
not  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_6"/><i><font color="Green" title="E14">A161</font></i>: 
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c7">a0</font> = <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span>;<br/>
<a NAME="E2:6_6"/><i><font color="Green" title="E15">A162</font></i>: 
not <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span> <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="c6">R</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E1:6"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="wellset1.html#E1:6_6"><i><font color="Green" title="E14">A161</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">W3</font> = <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> as   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c9">W4</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a> ;<br/>
<a NAME="E5:6_6"/>
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c9">W4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T32" title="RELAT_1:th.32">RELAT_1:32</a></i>;<br/>
<a NAME="E6:6_6"/><b>then </b><i><font color="Green" title="E16">A163</font></i>: 
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c9">W4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="enumset1.html#T41" title="ENUMSET1:th.41">ENUMSET1:41</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">W1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c6">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span>,<span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a> ;<br/>
<a NAME="E8:6_6"/><i><font color="Green" title="E17">A164</font></i>: 
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c10">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6_1"><b>proof </b></a><div class="add">

<a NAME="E1:6_6_1"/><i><font color="Green" title="E18">A165</font></i>: 
(  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c10">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_6_1_1"/>
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<a NAME="E2:6_6_1_1"/><b>then </b><i><font color="Green" title="E19">A166</font></i>: 
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c8">W3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="wellset1.html#T3" target="_self" title="WELLSET1:th.3">Th3</a></i>;<br/>
<a NAME="E3:6_6_1_1"/>
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c10">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">W3</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c9">W4</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T33" title="RELAT_1:th.33">RELAT_1:33</a></i>;<br/>
<a NAME="E4:6_6_1_1"/><b>then </b>
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c10">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="wellset1.html#E6:6_6"><i><font color="Green" title="E16">A163</font></i></a>, <a class="txt" href="wellset1.html#E2:6_6_1_1"><i><font color="Green" title="E19">A166</font></i></a>, <a class="ref" href="relat_1.html#T33" title="RELAT_1:th.33">RELAT_1:33</a></i>;<br/>
<a NAME="E5:6_6_1_1"/><b>then </b>
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c10">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></i>;<br/>
<a NAME="E6:6_6_1_1"/><b>then </b>
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c10">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></i>;<br/>
<b>hence </b><a NAME="E7:6_6_1_1"/>
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c10">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:6_6_1"/>
(  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c10">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_6_1_2"/><i><font color="Green" title="E19">A167</font></i>: 
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<a NAME="E2:6_6_1_2"/><i><font color="Green" title="E20">A168</font></i>: 
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c8">W3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6_1_2_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c11">z3</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c8">W3</font>;<br/>
<b>assume </b><a NAME="E2:6_6_1_2_1"/>
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c8">W3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<a NAME="E3:6_6_1_2_1"/><b>then </b>
 ex <font color="Olive" title="b1">z2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">z3</font>,<font color="Olive" title="b1">z2</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="c8">W3</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">z2</font>,<font color="Maroon" title="c11">z3</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="c8">W3</font> )
 
<b>by </b><i><a class="ref" href="wellset1.html#T1" target="_self" title="WELLSET1:th.1">Th1</a></i>;<br/>
<b>hence </b><a NAME="E4:6_6_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="wellset1.html#E1:6_6_1_2"><i><font color="Green" title="E19">A167</font></i></a>, <a class="ref" href="zfmisc_1.html#T113" title="ZFMISC_1:th.113">ZFMISC_1:113</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:6_6_1_2"/>
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c10">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">W3</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c9">W4</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T33" title="RELAT_1:th.33">RELAT_1:33</a></i>;<br/>
<a NAME="E4:6_6_1_2"/><b>then </b>
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c10">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="wellset1.html#E6:6_6"><i><font color="Green" title="E16">A163</font></i></a>, <a class="txt" href="wellset1.html#E2:6_6_1_2"><i><font color="Green" title="E20">A168</font></i></a>, <a class="ref" href="relat_1.html#T33" title="RELAT_1:th.33">RELAT_1:33</a></i>;<br/>
<b>hence </b><a NAME="E5:6_6_1_2"/>
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c10">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:6_6_1"/>
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c10">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="wellset1.html#E1:6_6_1"><i><font color="Green" title="E18">A165</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:6_6"/><i><font color="Green" title="E18">A169</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b1">x</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="c10">W1</font> or <font color="Olive" title="b1">x</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="c6">R</font> or <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:6_6_2"/>
<font color="Maroon" title="c11">x</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="c10">W1</font>
 ;<br/>

<a NAME="E2:6_6_2"/><b>then </b>
( <font color="Maroon" title="c11">x</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="c6">R</font> or <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="txt" href="wellset1.html#E8:6_6"><i><font color="Green" title="E17">A164</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:6_6_2"/>
( <font color="Maroon" title="c11">x</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="c6">R</font> or <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span> )
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:6_6"/><i><font color="Green" title="E19">A170</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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">x</font>,<font color="Olive" title="b2">y</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="c10">W1</font> iff ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</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="c6">R</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</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="c8">W3</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</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="c9">W4</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">x</font>, <font color="Maroon" title="c12">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<a NAME="E1:6_6_3"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</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="c10">W1</font> iff ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</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="c6">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">W3</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</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="c9">W4</font> ) )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E2:6_6_3"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</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="c10">W1</font> iff ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</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="c6">R</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</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="c8">W3</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</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="c9">W4</font> ) )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:6_6"/><i><font color="Green" title="E20">A171</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</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="c10">W1</font> &amp; <font color="Olive" title="b2">y</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="c6">R</font> holds <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</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="c6">R</font> &amp; <font color="Olive" title="b1">x</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="c6">R</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">x</font>, <font color="Maroon" title="c12">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><a NAME="E1:6_6_4"/><i><font color="Green" title="E21">A172</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</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="c10">W1</font> &amp; <font color="Maroon" title="c12">y</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="c6">R</font> )
 ;<br/>

<a NAME="E2:6_6_4"/><b>then </b><i><font color="Green" title="E22">A173</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</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="c6">R</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</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="c8">W3</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</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="c9">W4</font> )
 
<b>by </b><i><a class="txt" href="wellset1.html#E10:6_6"><i><font color="Green" title="E19">A170</font></i></a></i>;<br/>
<a NAME="E3:6_6_4"/><i><font color="Green" title="E23">A174</font></i>: 
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</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="c8">W3</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E2:6_6"><i><font color="Green" title="E15">A162</font></i></a>, <a class="txt" href="wellset1.html#E1:6_6_4"><i><font color="Green" title="E21">A172</font></i></a>, <a class="ref" href="zfmisc_1.html#T129" title="ZFMISC_1:th.129">ZFMISC_1:129</a></i>;<br/>
<a NAME="E4:6_6_4"/><i><font color="Green" title="E24">A175</font></i>: 
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</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="c9">W4</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6_4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_6_4_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</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="c9">W4</font>
 ;<br/>

<a NAME="E2:6_6_4_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:6_6_4_1"/>
contradiction
 <b>by </b><i><a class="txt" href="wellset1.html#E2:6_6"><i><font color="Green" title="E15">A162</font></i></a>, <a class="txt" href="wellset1.html#E1:6_6_4"><i><font color="Green" title="E21">A172</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:6_6_4"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</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="c6">R</font>
 <b>by </b><i><a class="txt" href="wellset1.html#E10:6_6"><i><font color="Green" title="E19">A170</font></i></a>, <a class="txt" href="wellset1.html#E1:6_6_4"><i><font color="Green" title="E21">A172</font></i></a>, <a class="txt" href="wellset1.html#E3:6_6_4"><i><font color="Green" title="E23">A174</font></i></a></i>;<br/>

<b>thus </b><a NAME="E6:6_6_4"/>
<font color="Maroon" title="c11">x</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="c6">R</font>
 <b>by </b><i><a class="txt" href="wellset1.html#E2:6_6_4"><i><font color="Green" title="E22">A173</font></i></a>, <a class="txt" href="wellset1.html#E4:6_6_4"><i><font color="Green" title="E24">A175</font></i></a>, <a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a>, <a class="ref" href="zfmisc_1.html#T129" title="ZFMISC_1:th.129">ZFMISC_1:129</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:6_6"/><i><font color="Green" title="E21">A176</font></i>: 
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</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="c6">R</font> holds <br/><font color="Maroon" title="c10">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b1">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:6_6_5"/><i><font color="Green" title="E22">A177</font></i>: 
<font color="Maroon" title="c11">y</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="c6">R</font>
 ;<br/>

<a NAME="E2:6_6_5"/><i><font color="Green" title="E23">A178</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c11">y</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c11">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:6_6_5_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c11">y</font>
 ;<br/>

<a NAME="E2:6_6_5_1"/><b>then </b><i><font color="Green" title="E24">A179</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c11">y</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="c10">W1</font> &amp; <font color="Maroon" title="c12">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">y</font> )
 
<b>by </b><i><a class="ref" href="wellord1.html#D1" title="WELLORD1:def.1">WELLORD1:def 1</a></i>;<br/>
<a NAME="E3:6_6_5_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c11">y</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="c6">R</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E11:6_6"><i><font color="Green" title="E20">A171</font></i></a>, <a class="txt" href="wellset1.html#E1:6_6_5"><i><font color="Green" title="E22">A177</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:6_6_5_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c11">y</font>
 <b>by </b><i><a class="txt" href="wellset1.html#E2:6_6_5_1"><i><font color="Green" title="E24">A179</font></i></a>, <a class="ref" href="wellord1.html#D1" title="WELLORD1:def.1">WELLORD1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:6_6_5"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c11">y</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c11">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6_5_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:6_6_5_2"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c11">y</font>
 ;<br/>

<a NAME="E2:6_6_5_2"/><b>then </b><i><font color="Green" title="E24">A180</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c11">y</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="c6">R</font> &amp; <font color="Maroon" title="c12">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">y</font> )
 
<b>by </b><i><a class="ref" href="wellord1.html#D1" title="WELLORD1:def.1">WELLORD1:def 1</a></i>;<br/>
<a NAME="E3:6_6_5_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c11">y</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="c10">W1</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E10:6_6"><i><font color="Green" title="E19">A170</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:6_6_5_2"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c11">y</font>
 <b>by </b><i><a class="txt" href="wellset1.html#E2:6_6_5_2"><i><font color="Green" title="E24">A180</font></i></a>, <a class="ref" href="wellord1.html#D1" title="WELLORD1:def.1">WELLORD1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:6_6_5"/>
<font color="Maroon" title="c10">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c11">y</font>
 <b>by </b><i><a class="txt" href="wellset1.html#E2:6_6_5"><i><font color="Green" title="E23">A178</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:6_6"/><i><font color="Green" title="E22">A181</font></i>: 
<font color="Maroon" title="c10">W1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</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"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">D</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6_6"><b>proof </b></a><div class="add">

<a NAME="E1:6_6_6"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</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="c10">W1</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</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"><span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">D</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6_6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">x</font>, <font color="Maroon" title="c12">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><a NAME="E1:6_6_6_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</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="c10">W1</font>
 ;<br/>

<a NAME="E2:6_6_6_1"/><b>then </b>
( <font color="Maroon" title="c11">x</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="c10">W1</font> &amp; <font color="Maroon" title="c12">y</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="c10">W1</font> )
 
<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="E3:6_6_6_1"/><b>then </b>
( ( <font color="Maroon" title="c11">x</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="c6">R</font> or <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span> ) &amp; ( <font color="Maroon" title="c12">y</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="c6">R</font> or <font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span> ) )
 
<b>by </b><i><a class="txt" href="wellset1.html#E9:6_6"><i><font color="Green" title="E18">A169</font></i></a></i>;<br/>
<a NAME="E4:6_6_6_1"/><b>then </b>
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">D</font> &amp; <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">D</font> )
 
<b>by </b><i><a class="txt" href="wellset1.html#E1:6"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="wellset1.html#E7:6"><i><font color="Green" title="E10">A88</font></i></a>, <a class="txt" href="wellset1.html#E1:6_6"><i><font color="Green" title="E14">A161</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:6_6_6_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</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"><span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">D</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:6_6_6"/><b>then </b>
<font color="Maroon" title="c10">W1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">D</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#D3" title="RELAT_1:def.3">RELAT_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:6_6_6"/>
<font color="Maroon" title="c10">W1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</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"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">D</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E14:6_6"/><i><font color="Green" title="E23">A182</font></i>: 
<font color="Maroon" title="c10">W1</font> is <a href="wellord1.html#V2" title="WELLORD1:attr.2">well-ordering</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6_7"><b>proof </b></a><div class="add">

<a NAME="E1:6_6_7"/><i><font color="Green" title="E24">A183</font></i>: 
<font color="Maroon" title="c10">W1</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c10">W1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6_7_1"><b>proof </b></a><div class="add">

<a NAME="E1:6_6_7_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</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="c10">W1</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b1">x</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="c10">W1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6_7_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:6_6_7_1_1"/><i><font color="Green" title="E25">A184</font></i>: 
<font color="Maroon" title="c11">x</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="c10">W1</font>
 ;<br/>

<a NAME="E2:6_6_7_1_1"/><i><font color="Green" title="E26">A185</font></i>: 
( <font color="Maroon" title="c11">x</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="c6">R</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c11">x</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="c10">W1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6_7_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_6_7_1_1_1"/><i><font color="Green" title="E27">A186</font></i>: 
<font color="Maroon" title="c11">x</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="c6">R</font>
 ;<br/>

<a NAME="E2:6_6_7_1_1_1"/>
<font color="Maroon" title="c6">R</font> <a href="wellord1.html#R2" title="WELLORD1:pred.2">well_orders</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E11:6"><i><font color="Green" title="E12">A99</font></i></a>, <a class="ref" href="wellord1.html#T8" title="WELLORD1:th.8">WELLORD1:8</a></i>;<br/>
<a NAME="E3:6_6_7_1_1_1"/><b>then </b>
<font color="Maroon" title="c6">R</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#D5" title="WELLORD1:def.5">WELLORD1:def 5</a></i>;<br/>
<a NAME="E4:6_6_7_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c11">x</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="c6">R</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E1:6_6_7_1_1_1"><i><font color="Green" title="E27">A186</font></i></a>, <a class="ref" href="relat_2.html#D1" title="RELAT_2:def.1">RELAT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E5:6_6_7_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c11">x</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="c10">W1</font>
 <b>by </b><i><a class="txt" href="wellset1.html#E10:6_6"><i><font color="Green" title="E19">A170</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:6_6_7_1_1"/>
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c11">x</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="c10">W1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6_7_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_6_7_1_1_2"/><i><font color="Green" title="E27">A187</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span>
 ;<br/>

<a NAME="E2:6_6_7_1_1_2"/>
<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="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span></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="c9">W4</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:6_6_7_1_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c11">x</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="c10">W1</font>
 <b>by </b><i><a class="txt" href="wellset1.html#E10:6_6"><i><font color="Green" title="E19">A170</font></i></a>, <a class="txt" href="wellset1.html#E1:6_6_7_1_1_2"><i><font color="Green" title="E27">A187</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:6_6_7_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c11">x</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="c10">W1</font>
 <b>by </b><i><a class="txt" href="wellset1.html#E9:6_6"><i><font color="Green" title="E18">A169</font></i></a>, <a class="txt" href="wellset1.html#E1:6_6_7_1_1"><i><font color="Green" title="E25">A184</font></i></a>, <a class="txt" href="wellset1.html#E2:6_6_7_1_1"><i><font color="Green" title="E26">A185</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:6_6_7_1"/>
<font color="Maroon" title="c10">W1</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c10">W1</font>
 <b>by </b><i><a class="ref" href="relat_2.html#D1" title="RELAT_2:def.1">RELAT_2:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:6_6_7"/><i><font color="Green" title="E25">A188</font></i>: 
<font color="Maroon" title="c10">W1</font> <a href="relat_2.html#R8" title="RELAT_2:pred.8">is_transitive_in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c10">W1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6_7_2"><b>proof </b></a><div class="add">

<a NAME="E1:6_6_7_2"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</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="c10">W1</font> &amp; <font color="Olive" title="b2">y</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="c10">W1</font> &amp; <font color="Olive" title="b3">z</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="c10">W1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</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="c10">W1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</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="c10">W1</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">z</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="c10">W1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6_7_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">x</font>, <font color="Maroon" title="c12">y</font>, <font color="Maroon" title="c13">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:6_6_7_2_1"/><i><font color="Green" title="E26">A189</font></i>: 
<font color="Maroon" title="c11">x</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="c10">W1</font>
 <b>and </b><br/><a NAME="E2:6_6_7_2_1"/><i><font color="Green" title="E27">A190</font></i>: 
<font color="Maroon" title="c12">y</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="c10">W1</font>
 <b>and </b><br/><a NAME="E3:6_6_7_2_1"/><i><font color="Green" title="E28">A191</font></i>: 
<font color="Maroon" title="c13">z</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="c10">W1</font>
 <b>and </b><br/><a NAME="E4:6_6_7_2_1"/><i><font color="Green" title="E29">A192</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</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="c10">W1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c13">z</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="c10">W1</font> )
 ;<br/>

<a NAME="E5:6_6_7_2_1"/><i><font color="Green" title="E30">A193</font></i>: 
( <font color="Maroon" title="c13">z</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="c6">R</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">z</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="c10">W1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6_7_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_6_7_2_1_1"/><i><font color="Green" title="E31">A194</font></i>: 
<font color="Maroon" title="c13">z</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="c6">R</font>
 ;<br/>

<a NAME="E2:6_6_7_2_1_1"/><b>then </b><i><font color="Green" title="E32">A195</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c13">z</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="c6">R</font> &amp; <font color="Maroon" title="c12">y</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="c6">R</font> )
 
<b>by </b><i><a class="txt" href="wellset1.html#E11:6_6"><i><font color="Green" title="E20">A171</font></i></a>, <a class="txt" href="wellset1.html#E4:6_6_7_2_1"><i><font color="Green" title="E29">A192</font></i></a></i>;<br/>
<a NAME="E3:6_6_7_2_1_1"/><b>then </b><i><font color="Green" title="E33">A196</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</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="c6">R</font> &amp; <font color="Maroon" title="c11">x</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="c6">R</font> )
 
<b>by </b><i><a class="txt" href="wellset1.html#E11:6_6"><i><font color="Green" title="E20">A171</font></i></a>, <a class="txt" href="wellset1.html#E4:6_6_7_2_1"><i><font color="Green" title="E29">A192</font></i></a></i>;<br/>
<a NAME="E4:6_6_7_2_1_1"/>
<font color="Maroon" title="c6">R</font> <a href="wellord1.html#R2" title="WELLORD1:pred.2">well_orders</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E11:6"><i><font color="Green" title="E12">A99</font></i></a>, <a class="ref" href="wellord1.html#T8" title="WELLORD1:th.8">WELLORD1:8</a></i>;<br/>
<a NAME="E5:6_6_7_2_1_1"/><b>then </b>
<font color="Maroon" title="c6">R</font> <a href="relat_2.html#R8" title="RELAT_2:pred.8">is_transitive_in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#D5" title="WELLORD1:def.5">WELLORD1:def 5</a></i>;<br/>
<a NAME="E6:6_6_7_2_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">z</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="c6">R</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E1:6_6_7_2_1_1"><i><font color="Green" title="E31">A194</font></i></a>, <a class="txt" href="wellset1.html#E2:6_6_7_2_1_1"><i><font color="Green" title="E32">A195</font></i></a>, <a class="txt" href="wellset1.html#E3:6_6_7_2_1_1"><i><font color="Green" title="E33">A196</font></i></a>, <a class="ref" href="relat_2.html#D8" title="RELAT_2:def.8">RELAT_2:def 8</a></i>;<br/>
<b>hence </b><a NAME="E7:6_6_7_2_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">z</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="c10">W1</font>
 <b>by </b><i><a class="txt" href="wellset1.html#E10:6_6"><i><font color="Green" title="E19">A170</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:6_6_7_2_1"/>
( <font color="Maroon" title="c13">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">z</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="c10">W1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6_7_2_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_6_7_2_1_2"/><i><font color="Green" title="E31">A197</font></i>: 
<font color="Maroon" title="c13">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span>
 ;<br/>

<a NAME="E2:6_6_7_2_1_2"/><i><font color="Green" title="E32">A198</font></i>: 
( <font color="Maroon" title="c12">y</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="c6">R</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">z</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="c10">W1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6_7_2_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_6_7_2_1_2_1"/>
<font color="Maroon" title="c12">y</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="c6">R</font>
 ;<br/>

<a NAME="E2:6_6_7_2_1_2_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</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="c6">R</font> &amp; <font color="Maroon" title="c11">x</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="c6">R</font> )
 
<b>by </b><i><a class="txt" href="wellset1.html#E11:6_6"><i><font color="Green" title="E20">A171</font></i></a>, <a class="txt" href="wellset1.html#E4:6_6_7_2_1"><i><font color="Green" title="E29">A192</font></i></a></i>;<br/>
<a NAME="E3:6_6_7_2_1_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">z</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="c8">W3</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E1:6_6_7_2_1_2"><i><font color="Green" title="E31">A197</font></i></a>, <a class="ref" href="zfmisc_1.html#T129" title="ZFMISC_1:th.129">ZFMISC_1:129</a></i>;<br/>
<b>hence </b><a NAME="E4:6_6_7_2_1_2_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">z</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="c10">W1</font>
 <b>by </b><i><a class="txt" href="wellset1.html#E10:6_6"><i><font color="Green" title="E19">A170</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:6_6_7_2_1_2"/>
( <font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">z</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="c10">W1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6_7_2_1_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:6_6_7_2_1_2_2"/><i><font color="Green" title="E33">A199</font></i>: 
( <font color="Maroon" title="c11">x</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="c6">R</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">z</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="c10">W1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6_7_2_1_2_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_6_7_2_1_2_2_1"/>
<font color="Maroon" title="c11">x</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="c6">R</font>
 ;<br/>

<a NAME="E2:6_6_7_2_1_2_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">z</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="c8">W3</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E1:6_6_7_2_1_2"><i><font color="Green" title="E31">A197</font></i></a>, <a class="ref" href="zfmisc_1.html#T129" title="ZFMISC_1:th.129">ZFMISC_1:129</a></i>;<br/>
<b>hence </b><a NAME="E3:6_6_7_2_1_2_2_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">z</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="c10">W1</font>
 <b>by </b><i><a class="txt" href="wellset1.html#E10:6_6"><i><font color="Green" title="E19">A170</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:6_6_7_2_1_2_2"/>
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">z</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="c10">W1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6_7_2_1_2_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_6_7_2_1_2_2_2"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span>
 ;<br/>

<a NAME="E2:6_6_7_2_1_2_2_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">z</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="c9">W4</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E1:6_6_7_2_1_2"><i><font color="Green" title="E31">A197</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:6_6_7_2_1_2_2_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">z</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="c10">W1</font>
 <b>by </b><i><a class="txt" href="wellset1.html#E10:6_6"><i><font color="Green" title="E19">A170</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:6_6_7_2_1_2_2"/>
( <font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">z</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="c10">W1</font> )
 <b>by </b><i><a class="txt" href="wellset1.html#E9:6_6"><i><font color="Green" title="E18">A169</font></i></a>, <a class="txt" href="wellset1.html#E1:6_6_7_2_1"><i><font color="Green" title="E26">A189</font></i></a>, <a class="txt" href="wellset1.html#E1:6_6_7_2_1_2_2"><i><font color="Green" title="E33">A199</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:6_6_7_2_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">z</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="c10">W1</font>
 <b>by </b><i><a class="txt" href="wellset1.html#E9:6_6"><i><font color="Green" title="E18">A169</font></i></a>, <a class="txt" href="wellset1.html#E2:6_6_7_2_1"><i><font color="Green" title="E27">A190</font></i></a>, <a class="txt" href="wellset1.html#E2:6_6_7_2_1_2"><i><font color="Green" title="E32">A198</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:6_6_7_2_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">z</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="c10">W1</font>
 <b>by </b><i><a class="txt" href="wellset1.html#E9:6_6"><i><font color="Green" title="E18">A169</font></i></a>, <a class="txt" href="wellset1.html#E3:6_6_7_2_1"><i><font color="Green" title="E28">A191</font></i></a>, <a class="txt" href="wellset1.html#E5:6_6_7_2_1"><i><font color="Green" title="E30">A193</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:6_6_7_2"/>
<font color="Maroon" title="c10">W1</font> <a href="relat_2.html#R8" title="RELAT_2:pred.8">is_transitive_in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c10">W1</font>
 <b>by </b><i><a class="ref" href="relat_2.html#D8" title="RELAT_2:def.8">RELAT_2:def 8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:6_6_7"/><i><font color="Green" title="E26">A200</font></i>: 
<font color="Maroon" title="c10">W1</font> <a href="relat_2.html#R4" title="RELAT_2:pred.4">is_antisymmetric_in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c10">W1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6_7_3"><b>proof </b></a><div class="add">

<a NAME="E1:6_6_7_3"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</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="c10">W1</font> &amp; <font color="Olive" title="b2">y</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="c10">W1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</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="c10">W1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</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="c10">W1</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6_7_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">x</font>, <font color="Maroon" title="c12">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:6_6_7_3_1"/><i><font color="Green" title="E27">A201</font></i>: 
<font color="Maroon" title="c11">x</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="c10">W1</font>
 <b>and </b><br/><a NAME="E2:6_6_7_3_1"/><i><font color="Green" title="E28">A202</font></i>: 
<font color="Maroon" title="c12">y</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="c10">W1</font>
 <b>and </b><br/><a NAME="E3:6_6_7_3_1"/><i><font color="Green" title="E29">A203</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</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="c10">W1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c11">x</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="c10">W1</font> )
 ;<br/>

<a NAME="E4:6_6_7_3_1"/><i><font color="Green" title="E30">A204</font></i>: 
( <font color="Maroon" title="c12">y</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="c6">R</font> or <font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="wellset1.html#E9:6_6"><i><font color="Green" title="E18">A169</font></i></a>, <a class="txt" href="wellset1.html#E2:6_6_7_3_1"><i><font color="Green" title="E28">A202</font></i></a></i>;<br/>
<a NAME="E5:6_6_7_3_1"/><i><font color="Green" title="E31">A205</font></i>: 
( <font color="Maroon" title="c11">x</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="c6">R</font> implies <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6_7_3_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_6_7_3_1_1"/><i><font color="Green" title="E32">A206</font></i>: 
<font color="Maroon" title="c11">x</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="c6">R</font>
 ;<br/>

<a NAME="E2:6_6_7_3_1_1"/><b>then </b><i><font color="Green" title="E33">A207</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c11">x</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="c6">R</font> &amp; <font color="Maroon" title="c12">y</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="c6">R</font> )
 
<b>by </b><i><a class="txt" href="wellset1.html#E11:6_6"><i><font color="Green" title="E20">A171</font></i></a>, <a class="txt" href="wellset1.html#E3:6_6_7_3_1"><i><font color="Green" title="E29">A203</font></i></a></i>;<br/>
<a NAME="E3:6_6_7_3_1_1"/><b>then </b><i><font color="Green" title="E34">A208</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</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="c6">R</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E11:6_6"><i><font color="Green" title="E20">A171</font></i></a>, <a class="txt" href="wellset1.html#E3:6_6_7_3_1"><i><font color="Green" title="E29">A203</font></i></a></i>;<br/>
<a NAME="E4:6_6_7_3_1_1"/>
<font color="Maroon" title="c6">R</font> <a href="wellord1.html#R2" title="WELLORD1:pred.2">well_orders</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E11:6"><i><font color="Green" title="E12">A99</font></i></a>, <a class="ref" href="wellord1.html#T8" title="WELLORD1:th.8">WELLORD1:8</a></i>;<br/>
<a NAME="E5:6_6_7_3_1_1"/><b>then </b>
<font color="Maroon" title="c6">R</font> <a href="relat_2.html#R4" title="RELAT_2:pred.4">is_antisymmetric_in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#D5" title="WELLORD1:def.5">WELLORD1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E6:6_6_7_3_1_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">y</font>
 <b>by </b><i><a class="txt" href="wellset1.html#E1:6_6_7_3_1_1"><i><font color="Green" title="E32">A206</font></i></a>, <a class="txt" href="wellset1.html#E2:6_6_7_3_1_1"><i><font color="Green" title="E33">A207</font></i></a>, <a class="txt" href="wellset1.html#E3:6_6_7_3_1_1"><i><font color="Green" title="E34">A208</font></i></a>, <a class="ref" href="relat_2.html#D4" title="RELAT_2:def.4">RELAT_2:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:6_6_7_3_1"/>
( <font color="Maroon" title="c12">y</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="c6">R</font> implies <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6_7_3_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_6_7_3_1_2"/><i><font color="Green" title="E32">A209</font></i>: 
<font color="Maroon" title="c12">y</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="c6">R</font>
 ;<br/>

<a NAME="E2:6_6_7_3_1_2"/><b>then </b><i><font color="Green" title="E33">A210</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</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="c6">R</font> &amp; <font color="Maroon" title="c11">x</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="c6">R</font> )
 
<b>by </b><i><a class="txt" href="wellset1.html#E11:6_6"><i><font color="Green" title="E20">A171</font></i></a>, <a class="txt" href="wellset1.html#E3:6_6_7_3_1"><i><font color="Green" title="E29">A203</font></i></a></i>;<br/>
<a NAME="E3:6_6_7_3_1_2"/><b>then </b><i><font color="Green" title="E34">A211</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c11">x</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="c6">R</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E11:6_6"><i><font color="Green" title="E20">A171</font></i></a>, <a class="txt" href="wellset1.html#E3:6_6_7_3_1"><i><font color="Green" title="E29">A203</font></i></a></i>;<br/>
<a NAME="E4:6_6_7_3_1_2"/>
<font color="Maroon" title="c6">R</font> <a href="wellord1.html#R2" title="WELLORD1:pred.2">well_orders</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E11:6"><i><font color="Green" title="E12">A99</font></i></a>, <a class="ref" href="wellord1.html#T8" title="WELLORD1:th.8">WELLORD1:8</a></i>;<br/>
<a NAME="E5:6_6_7_3_1_2"/><b>then </b>
<font color="Maroon" title="c6">R</font> <a href="relat_2.html#R4" title="RELAT_2:pred.4">is_antisymmetric_in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#D5" title="WELLORD1:def.5">WELLORD1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E6:6_6_7_3_1_2"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">y</font>
 <b>by </b><i><a class="txt" href="wellset1.html#E1:6_6_7_3_1_2"><i><font color="Green" title="E32">A209</font></i></a>, <a class="txt" href="wellset1.html#E2:6_6_7_3_1_2"><i><font color="Green" title="E33">A210</font></i></a>, <a class="txt" href="wellset1.html#E3:6_6_7_3_1_2"><i><font color="Green" title="E34">A211</font></i></a>, <a class="ref" href="relat_2.html#D4" title="RELAT_2:def.4">RELAT_2:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:6_6_7_3_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">y</font>
 <b>by </b><i><a class="txt" href="wellset1.html#E9:6_6"><i><font color="Green" title="E18">A169</font></i></a>, <a class="txt" href="wellset1.html#E1:6_6_7_3_1"><i><font color="Green" title="E27">A201</font></i></a>, <a class="txt" href="wellset1.html#E4:6_6_7_3_1"><i><font color="Green" title="E30">A204</font></i></a>, <a class="txt" href="wellset1.html#E5:6_6_7_3_1"><i><font color="Green" title="E31">A205</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:6_6_7_3"/>
<font color="Maroon" title="c10">W1</font> <a href="relat_2.html#R4" title="RELAT_2:pred.4">is_antisymmetric_in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c10">W1</font>
 <b>by </b><i><a class="ref" href="relat_2.html#D4" title="RELAT_2:def.4">RELAT_2:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:6_6_7"/><i><font color="Green" title="E27">A212</font></i>: 
<font color="Maroon" title="c10">W1</font> <a href="relat_2.html#R6" title="RELAT_2:pred.6">is_connected_in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c10">W1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6_7_4"><b>proof </b></a><div class="add">

<a NAME="E1:6_6_7_4"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</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="c10">W1</font> &amp; <font color="Olive" title="b2">y</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="c10">W1</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> &amp; not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</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="c10">W1</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</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="c10">W1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6_7_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">x</font>, <font color="Maroon" title="c12">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:6_6_7_4_1"/><i><font color="Green" title="E28">A213</font></i>: 
<font color="Maroon" title="c11">x</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="c10">W1</font>
 <b>and </b><br/><a NAME="E2:6_6_7_4_1"/><i><font color="Green" title="E29">A214</font></i>: 
<font color="Maroon" title="c12">y</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="c10">W1</font>
 <b>and </b><br/><a NAME="E3:6_6_7_4_1"/><i><font color="Green" title="E30">A215</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">y</font>
 ;<br/>

<a NAME="E4:6_6_7_4_1"/><i><font color="Green" title="E31">A216</font></i>: 
( <font color="Maroon" title="c11">x</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="c6">R</font> &amp; <font color="Maroon" title="c12">y</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="c6">R</font> &amp; not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</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="c10">W1</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c11">x</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="c10">W1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6_7_4_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_6_7_4_1_1"/><i><font color="Green" title="E32">A217</font></i>: 
( <font color="Maroon" title="c11">x</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="c6">R</font> &amp; <font color="Maroon" title="c12">y</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="c6">R</font> )
 ;<br/>

<a NAME="E2:6_6_7_4_1_1"/>
<font color="Maroon" title="c6">R</font> <a href="wellord1.html#R2" title="WELLORD1:pred.2">well_orders</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E11:6"><i><font color="Green" title="E12">A99</font></i></a>, <a class="ref" href="wellord1.html#T8" title="WELLORD1:th.8">WELLORD1:8</a></i>;<br/>
<a NAME="E3:6_6_7_4_1_1"/><b>then </b>
<font color="Maroon" title="c6">R</font> <a href="relat_2.html#R6" title="RELAT_2:pred.6">is_connected_in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#D5" title="WELLORD1:def.5">WELLORD1:def 5</a></i>;<br/>
<a NAME="E4:6_6_7_4_1_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</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="c6">R</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c11">x</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="c6">R</font> )
 
<b>by </b><i><a class="txt" href="wellset1.html#E3:6_6_7_4_1"><i><font color="Green" title="E30">A215</font></i></a>, <a class="txt" href="wellset1.html#E1:6_6_7_4_1_1"><i><font color="Green" title="E32">A217</font></i></a>, <a class="ref" href="relat_2.html#D6" title="RELAT_2:def.6">RELAT_2:def 6</a></i>;<br/>
<b>hence </b><a NAME="E5:6_6_7_4_1_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</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="c10">W1</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c11">x</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="c10">W1</font> )
 <b>by </b><i><a class="txt" href="wellset1.html#E10:6_6"><i><font color="Green" title="E19">A170</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:6_6_7_4_1"/><i><font color="Green" title="E32">A218</font></i>: 
( <font color="Maroon" title="c11">x</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="c6">R</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</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="c10">W1</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c11">x</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="c10">W1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6_7_4_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_6_7_4_1_2"/>
not <font color="Maroon" title="c11">x</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="c6">R</font>
 ;<br/>

<a NAME="E2:6_6_7_4_1_2"/><b>then </b><i><font color="Green" title="E33">A219</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span>
 
<b>by </b><i><a class="txt" href="wellset1.html#E9:6_6"><i><font color="Green" title="E18">A169</font></i></a>, <a class="txt" href="wellset1.html#E1:6_6_7_4_1"><i><font color="Green" title="E28">A213</font></i></a></i>;<br/>
<a NAME="E3:6_6_7_4_1_2"/><i><font color="Green" title="E34">A220</font></i>: 
(  not <font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</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="c10">W1</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c11">x</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="c10">W1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6_7_4_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_6_7_4_1_2_1"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span>
 ;<br/>

<a NAME="E2:6_6_7_4_1_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</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="c9">W4</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E2:6_6_7_4_1_2"><i><font color="Green" title="E33">A219</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:6_6_7_4_1_2_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</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="c10">W1</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c11">x</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="c10">W1</font> )
 <b>by </b><i><a class="txt" href="wellset1.html#E10:6_6"><i><font color="Green" title="E19">A170</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:6_6_7_4_1_2"/>
(  not <font color="Maroon" title="c12">y</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="c6">R</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</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="c10">W1</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c11">x</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="c10">W1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6_7_4_1_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_6_7_4_1_2_2"/>
<font color="Maroon" title="c12">y</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="c6">R</font>
 ;<br/>

<a NAME="E2:6_6_7_4_1_2_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c11">x</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="c8">W3</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E2:6_6_7_4_1_2"><i><font color="Green" title="E33">A219</font></i></a>, <a class="ref" href="zfmisc_1.html#T129" title="ZFMISC_1:th.129">ZFMISC_1:129</a></i>;<br/>
<b>hence </b><a NAME="E3:6_6_7_4_1_2_2"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</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="c10">W1</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c11">x</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="c10">W1</font> )
 <b>by </b><i><a class="txt" href="wellset1.html#E10:6_6"><i><font color="Green" title="E19">A170</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:6_6_7_4_1_2"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</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="c10">W1</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c11">x</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="c10">W1</font> )
 <b>by </b><i><a class="txt" href="wellset1.html#E9:6_6"><i><font color="Green" title="E18">A169</font></i></a>, <a class="txt" href="wellset1.html#E2:6_6_7_4_1"><i><font color="Green" title="E29">A214</font></i></a>, <a class="txt" href="wellset1.html#E3:6_6_7_4_1_2"><i><font color="Green" title="E34">A220</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:6_6_7_4_1"/>
( <font color="Maroon" title="c12">y</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="c6">R</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</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="c10">W1</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c11">x</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="c10">W1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6_7_4_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_6_7_4_1_3"/>
not <font color="Maroon" title="c12">y</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="c6">R</font>
 ;<br/>

<a NAME="E2:6_6_7_4_1_3"/><b>then </b><i><font color="Green" title="E33">A221</font></i>: 
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span>
 
<b>by </b><i><a class="txt" href="wellset1.html#E9:6_6"><i><font color="Green" title="E18">A169</font></i></a>, <a class="txt" href="wellset1.html#E2:6_6_7_4_1"><i><font color="Green" title="E29">A214</font></i></a></i>;<br/>
<a NAME="E3:6_6_7_4_1_3"/><i><font color="Green" title="E34">A222</font></i>: 
(  not <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c11">x</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="c10">W1</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</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="c10">W1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6_7_4_1_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_6_7_4_1_3_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span>
 ;<br/>

<a NAME="E2:6_6_7_4_1_3_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c11">x</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="c9">W4</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E2:6_6_7_4_1_3"><i><font color="Green" title="E33">A221</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:6_6_7_4_1_3_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c11">x</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="c10">W1</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</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="c10">W1</font> )
 <b>by </b><i><a class="txt" href="wellset1.html#E10:6_6"><i><font color="Green" title="E19">A170</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:6_6_7_4_1_3"/>
(  not <font color="Maroon" title="c11">x</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="c6">R</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c11">x</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="c10">W1</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</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="c10">W1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6_7_4_1_3_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_6_7_4_1_3_2"/>
<font color="Maroon" title="c11">x</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="c6">R</font>
 ;<br/>

<a NAME="E2:6_6_7_4_1_3_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</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="c8">W3</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E2:6_6_7_4_1_3"><i><font color="Green" title="E33">A221</font></i></a>, <a class="ref" href="zfmisc_1.html#T129" title="ZFMISC_1:th.129">ZFMISC_1:129</a></i>;<br/>
<b>hence </b><a NAME="E3:6_6_7_4_1_3_2"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c11">x</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="c10">W1</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</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="c10">W1</font> )
 <b>by </b><i><a class="txt" href="wellset1.html#E10:6_6"><i><font color="Green" title="E19">A170</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:6_6_7_4_1_3"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</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="c10">W1</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c11">x</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="c10">W1</font> )
 <b>by </b><i><a class="txt" href="wellset1.html#E9:6_6"><i><font color="Green" title="E18">A169</font></i></a>, <a class="txt" href="wellset1.html#E1:6_6_7_4_1"><i><font color="Green" title="E28">A213</font></i></a>, <a class="txt" href="wellset1.html#E3:6_6_7_4_1_3"><i><font color="Green" title="E34">A222</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:6_6_7_4_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</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="c10">W1</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c11">x</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="c10">W1</font> )
 <b>by </b><i><a class="txt" href="wellset1.html#E4:6_6_7_4_1"><i><font color="Green" title="E31">A216</font></i></a>, <a class="txt" href="wellset1.html#E5:6_6_7_4_1"><i><font color="Green" title="E32">A218</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:6_6_7_4"/>
<font color="Maroon" title="c10">W1</font> <a href="relat_2.html#R6" title="RELAT_2:pred.6">is_connected_in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c10">W1</font>
 <b>by </b><i><a class="ref" href="relat_2.html#D6" title="RELAT_2:def.6">RELAT_2:def 6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:6_6_7"/>
<font color="Maroon" title="c10">W1</font> <a href="wellord1.html#R1" title="WELLORD1:pred.1">is_well_founded_in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c10">W1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6_7_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="wellord1.html#D3" title="WELLORD1:def.3">WELLORD1:def 3</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:6_6_7_5"/><i><font color="Green" title="E28">A223</font></i>: 
<font color="Maroon" title="c11">Y</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="c10">W1</font>
 <b>and </b><br/><a NAME="E2:6_6_7_5"/><i><font color="Green" title="E29">A224</font></i>: 
<font color="Maroon" title="c11">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<a NAME="E3:6_6_7_5"/>
<font color="Maroon" title="c6">R</font> <a href="wellord1.html#R2" title="WELLORD1:pred.2">well_orders</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E11:6"><i><font color="Green" title="E12">A99</font></i></a>, <a class="ref" href="wellord1.html#T8" title="WELLORD1:th.8">WELLORD1:8</a></i>;<br/>
<a NAME="E4:6_6_7_5"/><b>then </b><i><font color="Green" title="E30">A225</font></i>: 
<font color="Maroon" title="c6">R</font> <a href="wellord1.html#R1" title="WELLORD1:pred.1">is_well_founded_in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#D5" title="WELLORD1:def.5">WELLORD1:def 5</a></i>;<br/>
<a NAME="E5:6_6_7_5"/><i><font color="Green" title="E31">A226</font></i>: 
( <font color="Maroon" title="c11">Y</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="c6">R</font> implies  ex <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Y</font> &amp; <font color="Maroon" title="c10">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b1">a</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c11">Y</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6_7_5_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_6_7_5_1"/><i><font color="Green" title="E32">A227</font></i>: 
<font color="Maroon" title="c11">Y</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="c6">R</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:6_6_7_5_1"/><i><font color="Green" title="E33">A228</font></i>: 
( <font color="Maroon" title="c12">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Y</font> &amp; <font color="Maroon" title="c6">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c12">b</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c11">Y</font> )
 <b>by </b><i><a class="txt" href="wellset1.html#E2:6_6_7_5"><i><font color="Green" title="E29">A224</font></i></a>, <a class="txt" href="wellset1.html#E4:6_6_7_5"><i><font color="Green" title="E30">A225</font></i></a>, <a class="ref" href="wellord1.html#D3" title="WELLORD1:def.3">WELLORD1:def 3</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c12">b</font>
;<br/>

<b>thus </b><a NAME="E4:6_6_7_5_1"/>
( <font color="Maroon" title="c12">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Y</font> &amp; <font color="Maroon" title="c10">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c12">b</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c11">Y</font> )
 <b>by </b><i><a class="txt" href="wellset1.html#E12:6_6"><i><font color="Green" title="E21">A176</font></i></a>, <a class="txt" href="wellset1.html#E1:6_6_7_5_1"><i><font color="Green" title="E32">A227</font></i></a>, <a class="txt" href="wellset1.html#E3:6_6_7_5_1"><i><font color="Green" title="E33">A228</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:6_6_7_5"/>
( not <font color="Maroon" title="c11">Y</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="c6">R</font> implies  ex <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Y</font> &amp; <font color="Maroon" title="c10">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b1">a</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c11">Y</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6_7_5_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_6_7_5_2"/>
not <font color="Maroon" title="c11">Y</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="c6">R</font>
 ;<br/>

<a NAME="E2:6_6_7_5_2"/><i><font color="Green" title="E32">A229</font></i>: 
( <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c11">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  ex <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Y</font> &amp; <font color="Maroon" title="c10">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b1">a</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c11">Y</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6_7_5_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_6_7_5_2_1"/><i><font color="Green" title="E33">A230</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c11">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon" title="c12">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c11">Y</font>;<br/>
<a NAME="E3:6_6_7_5_2_1"/><i><font color="Green" title="E34">A231</font></i>: 
not <font color="Maroon" title="c12">y</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="c6">R</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E2:6_6_7_5"><i><font color="Green" title="E29">A224</font></i></a>, <a class="txt" href="wellset1.html#E1:6_6_7_5_2_1"><i><font color="Green" title="E33">A230</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="E4:6_6_7_5_2_1"/>
<font color="Maroon" title="c12">y</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="c10">W1</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E1:6_6_7_5"><i><font color="Green" title="E28">A223</font></i></a>, <a class="txt" href="wellset1.html#E2:6_6_7_5"><i><font color="Green" title="E29">A224</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E5:6_6_7_5_2_1"/><b>then </b><i><font color="Green" title="E35">A232</font></i>: 
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span>
 
<b>by </b><i><a class="txt" href="wellset1.html#E9:6_6"><i><font color="Green" title="E18">A169</font></i></a>, <a class="txt" href="wellset1.html#E3:6_6_7_5_2_1"><i><font color="Green" title="E34">A231</font></i></a></i>;<br/>
<a NAME="E6:6_6_7_5_2_1"/>
<font color="Maroon" title="c10">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span></span>)</span> <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="c6">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6_7_5_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:6_6_7_5_2_1_1"/>
<font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:6_6_7_5_2_1_1"/><b>then </b><i><font color="Green" title="E36">A233</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">z</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span></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="c10">W1</font> &amp; <font color="Maroon" title="c13">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="wellord1.html#D1" title="WELLORD1:def.1">WELLORD1:def 1</a></i>;<br/>
<a NAME="E3:6_6_7_5_2_1_1"/><b>then </b>
<font color="Maroon" title="c13">z</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="c10">W1</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a></i>;<br/>
<b>hence </b><a NAME="E4:6_6_7_5_2_1_1"/>
<font color="Maroon" title="c13">z</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="c6">R</font>
 <b>by </b><i><a class="txt" href="wellset1.html#E9:6_6"><i><font color="Green" title="E18">A169</font></i></a>, <a class="txt" href="wellset1.html#E2:6_6_7_5_2_1_1"><i><font color="Green" title="E36">A233</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:6_6_7_5_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c12">y</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c11">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="wellset1.html#E1:6_6_7_5_2_1"><i><font color="Green" title="E33">A230</font></i></a>, <a class="txt" href="wellset1.html#E5:6_6_7_5_2_1"><i><font color="Green" title="E35">A232</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a>, <a class="ref" href="xboole_1.html#T26" title="XBOOLE_1:th.26">XBOOLE_1:26</a></i>;<br/>
<a NAME="E8:6_6_7_5_2_1"/><b>then </b>
<font color="Maroon" title="c10">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c12">y</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c11">Y</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<b>hence </b><a NAME="E9:6_6_7_5_2_1"/>
 ex <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Y</font> &amp; <font color="Maroon" title="c10">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b1">a</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c11">Y</font> )
 <b>by </b><i><a class="txt" href="wellset1.html#E2:6_6_7_5"><i><font color="Green" title="E29">A224</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:6_6_7_5_2"/>
( not <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c11">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  ex <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Y</font> &amp; <font color="Maroon" title="c10">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b1">a</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c11">Y</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6_7_5_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_6_7_5_2_2"/><i><font color="Green" title="E33">A234</font></i>: 
not <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c11">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>set </b><font color="Maroon" title="c12">X</font> = <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c11">Y</font>;<br/>
<a NAME="E2:6_6_7_5_2_2"/><i><font color="Green" title="E34">A235</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c11">Y</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="c6">R</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c13">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:6_6_7_5_2_2"/><i><font color="Green" title="E35">A236</font></i>: 
( <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c11">Y</font> &amp; <font color="Maroon" title="c6">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c13">y</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c11">Y</font> )
 <b>by </b><i><a class="txt" href="wellset1.html#E4:6_6_7_5"><i><font color="Green" title="E30">A225</font></i></a>, <a class="txt" href="wellset1.html#E1:6_6_7_5_2_2"><i><font color="Green" title="E33">A234</font></i></a>, <a class="ref" href="wellord1.html#D3" title="WELLORD1:def.3">WELLORD1:def 3</a></i>;<br/>
<a NAME="E5:6_6_7_5_2_2"/><i><font color="Green" title="E36">A237</font></i>: 
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Y</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E4:6_6_7_5_2_2"><i><font color="Green" title="E35">A236</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="E6:6_6_7_5_2_2"/><i><font color="Green" title="E37">A238</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c13">y</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c11">Y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="wellset1.html#E4:6_6_7_5_2_2"><i><font color="Green" title="E35">A236</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E7:6_6_7_5_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c13">y</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c11">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c13">y</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c11">Y</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6_7_5_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:6_6_7_5_2_2_1"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c13">y</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c11">Y</font>
 ;<br/>

<a NAME="E2:6_6_7_5_2_2_1"/><b>then </b><i><font color="Green" title="E38">A239</font></i>: 
( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c13">y</font> &amp; <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Y</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:6_6_7_5_2_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c13">y</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="c6">R</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#D1" title="WELLORD1:def.1">WELLORD1:def 1</a></i>;<br/>
<a NAME="E4:6_6_7_5_2_2_1"/><b>then </b>
<font color="Maroon" title="c14">x</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="c6">R</font>
 
<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="E5:6_6_7_5_2_2_1"/><b>then </b>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c11">Y</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E2:6_6_7_5_2_2_1"><i><font color="Green" title="E38">A239</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E6:6_6_7_5_2_2_1"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c13">y</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c11">Y</font></span>)</span>
 <b>by </b><i><a class="txt" href="wellset1.html#E2:6_6_7_5_2_2_1"><i><font color="Green" title="E38">A239</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>
<a NAME="E8:6_6_7_5_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c13">y</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c11">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="wellset1.html#E6:6_6_7_5_2_2"><i><font color="Green" title="E37">A238</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/>
<a NAME="E9:6_6_7_5_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c13">y</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c11">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="wellset1.html#E12:6_6"><i><font color="Green" title="E21">A176</font></i></a>, <a class="txt" href="wellset1.html#E2:6_6_7_5_2_2"><i><font color="Green" title="E34">A235</font></i></a>, <a class="txt" href="wellset1.html#E4:6_6_7_5_2_2"><i><font color="Green" title="E35">A236</font></i></a></i>;<br/>
<a NAME="E10:6_6_7_5_2_2"/><b>then </b>
<font color="Maroon" title="c10">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c13">y</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c11">Y</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<b>hence </b><a NAME="E11:6_6_7_5_2_2"/>
 ex <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Y</font> &amp; <font color="Maroon" title="c10">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b1">a</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c11">Y</font> )
 <b>by </b><i><a class="txt" href="wellset1.html#E5:6_6_7_5_2_2"><i><font color="Green" title="E36">A237</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:6_6_7_5_2"/>
 ex <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Y</font> &amp; <font color="Maroon" title="c10">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b1">a</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c11">Y</font> )
 <b>by </b><i><a class="txt" href="wellset1.html#E2:6_6_7_5_2"><i><font color="Green" title="E32">A229</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:6_6_7_5"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Y</font> &amp; <font color="Maroon" title="c10">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c11">Y</font> )
 <b>by </b><i><a class="txt" href="wellset1.html#E5:6_6_7_5"><i><font color="Green" title="E31">A226</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:6_6_7"/><b>then </b>
<font color="Maroon" title="c10">W1</font> <a href="wellord1.html#R2" title="WELLORD1:pred.2">well_orders</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c10">W1</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E1:6_6_7"><i><font color="Green" title="E24">A183</font></i></a>, <a class="txt" href="wellset1.html#E2:6_6_7"><i><font color="Green" title="E25">A188</font></i></a>, <a class="txt" href="wellset1.html#E3:6_6_7"><i><font color="Green" title="E26">A200</font></i></a>, <a class="txt" href="wellset1.html#E4:6_6_7"><i><font color="Green" title="E27">A212</font></i></a>, <a class="ref" href="wellord1.html#D5" title="WELLORD1:def.5">WELLORD1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E7:6_6_7"/>
<font color="Maroon" title="c10">W1</font> is <a href="wellord1.html#V2" title="WELLORD1:attr.2">well-ordering</a>
 <b>by </b><i><a class="ref" href="wellord1.html#T8" title="WELLORD1:th.8">WELLORD1:8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:6_6"/>
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</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="c10">W1</font> holds <br/>( <font color="Maroon" title="c10">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font> &amp; <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b1">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:6_6_8"/>
<font color="Maroon" title="c11">y</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="c10">W1</font>
 ;<br/>

<a NAME="E2:6_6_8"/><b>then </b><i><font color="Green" title="E24">A240</font></i>: 
( <font color="Maroon" title="c11">y</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="c6">R</font> or <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="wellset1.html#E9:6_6"><i><font color="Green" title="E18">A169</font></i></a></i>;<br/>
<a NAME="E3:6_6_8"/><i><font color="Green" title="E25">A241</font></i>: 
( <font color="Maroon" title="c11">y</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="c6">R</font> implies <font color="Maroon" title="c10">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c11">y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6_8_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_6_8_1"/><i><font color="Green" title="E26">A242</font></i>: 
<font color="Maroon" title="c11">y</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="c6">R</font>
 ;<br/>

<a NAME="E2:6_6_8_1"/><i><font color="Green" title="E27">A243</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c11">y</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c11">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6_8_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:6_6_8_1_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c11">y</font>
 ;<br/>

<a NAME="E2:6_6_8_1_1"/><b>then </b><i><font color="Green" title="E28">A244</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c11">y</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="c10">W1</font> &amp; not <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y</font> )
 
<b>by </b><i><a class="ref" href="wellord1.html#D1" title="WELLORD1:def.1">WELLORD1:def 1</a></i>;<br/>
<a NAME="E3:6_6_8_1_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c11">y</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="c6">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">W3</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c11">y</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="c9">W4</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E4:6_6_8_1_1"/><b>then </b><i><font color="Green" title="E29">A245</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c11">y</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="c6">R</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c11">y</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="c8">W3</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c11">y</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="c9">W4</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E5:6_6_8_1_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c11">y</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="c9">W4</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c11">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:6_6_8_1_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c11">y</font>
 <b>by </b><i><a class="txt" href="wellset1.html#E2:6_6"><i><font color="Green" title="E15">A162</font></i></a>, <a class="txt" href="wellset1.html#E1:6_6_8_1"><i><font color="Green" title="E26">A242</font></i></a>, <a class="txt" href="wellset1.html#E2:6_6_8_1_1"><i><font color="Green" title="E28">A244</font></i></a>, <a class="txt" href="wellset1.html#E4:6_6_8_1_1"><i><font color="Green" title="E29">A245</font></i></a>, <a class="ref" href="wellord1.html#D1" title="WELLORD1:def.1">WELLORD1:def 1</a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a>, <a class="ref" href="zfmisc_1.html#T129" title="ZFMISC_1:th.129">ZFMISC_1:129</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:6_6_8_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c11">y</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c11">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6_8_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:6_6_8_1_2"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c11">y</font>
 ;<br/>

<a NAME="E2:6_6_8_1_2"/><b>then </b><i><font color="Green" title="E28">A246</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c11">y</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="c6">R</font> &amp; not <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y</font> )
 
<b>by </b><i><a class="ref" href="wellord1.html#D1" title="WELLORD1:def.1">WELLORD1:def 1</a></i>;<br/>
<a NAME="E3:6_6_8_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c11">y</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="c6">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">W3</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E4:6_6_8_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c11">y</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="c10">W1</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E5:6_6_8_1_2"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c11">y</font>
 <b>by </b><i><a class="txt" href="wellset1.html#E2:6_6_8_1_2"><i><font color="Green" title="E28">A246</font></i></a>, <a class="ref" href="wellord1.html#D1" title="WELLORD1:def.1">WELLORD1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:6_6_8_1"/>
<font color="Maroon" title="c10">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c11">y</font>
 <b>by </b><i><a class="txt" href="wellset1.html#E2:6_6_8_1"><i><font color="Green" title="E27">A243</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:6_6_8"/>
( <font color="Maroon" title="c10">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font> &amp; <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6_8_2"><b>proof </b></a><div class="add">

<a NAME="E1:6_6_8_2"/>
<font color="Maroon" title="c10">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span></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> <font color="Maroon" title="c6">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6_8_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:6_6_8_2_1"/><i><font color="Green" title="E26">A247</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span></span>)</span> holds <br/><font color="Olive" title="b1">x</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="c6">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6_8_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:6_6_8_2_1_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:6_6_8_2_1_1"/><b>then </b><i><font color="Green" title="E27">A248</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span></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="c10">W1</font> &amp; not <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="wellord1.html#D1" title="WELLORD1:def.1">WELLORD1:def 1</a></i>;<br/>
<a NAME="E3:6_6_8_2_1_1"/><b>then </b>
<font color="Maroon" title="c12">x</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="c10">W1</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a></i>;<br/>
<b>hence </b><a NAME="E4:6_6_8_2_1_1"/>
<font color="Maroon" title="c12">x</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="c6">R</font>
 <b>by </b><i><a class="txt" href="wellset1.html#E9:6_6"><i><font color="Green" title="E18">A169</font></i></a>, <a class="txt" href="wellset1.html#E2:6_6_8_2_1_1"><i><font color="Green" title="E27">A248</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:6_6_8_2_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</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="c6">R</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_6_8_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:6_6_8_2_1_2"/><i><font color="Green" title="E27">A249</font></i>: 
<font color="Maroon" title="c12">x</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="c6">R</font>
 ;<br/>

<a NAME="E2:6_6_8_2_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span></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="c8">W3</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T129" title="ZFMISC_1:th.129">ZFMISC_1:129</a></i>;<br/>
<a NAME="E3:6_6_8_2_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span></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="c6">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">W3</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E4:6_6_8_2_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span></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="c10">W1</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E5:6_6_8_2_1_2"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="wellset1.html#E2:6_6"><i><font color="Green" title="E15">A162</font></i></a>, <a class="txt" href="wellset1.html#E1:6_6_8_2_1_2"><i><font color="Green" title="E27">A249</font></i></a>, <a class="ref" href="wellord1.html#D1" title="WELLORD1:def.1">WELLORD1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:6_6_8_2_1"/>
<font color="Maroon" title="c10">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span></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> <font color="Maroon" title="c6">R</font>
 <b>by </b><i><a class="txt" href="wellset1.html#E1:6_6_8_2_1"><i><font color="Green" title="E26">A247</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:6_6_8_2"/>
( <font color="Maroon" title="c10">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font> &amp; <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span> )
 <b>by </b><i><a class="txt" href="wellset1.html#E1:6_6"><i><font color="Green" title="E14">A161</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:6_6_8"/>
( <font color="Maroon" title="c10">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font> &amp; <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">W1</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y</font> )
 <b>by </b><i><a class="txt" href="wellset1.html#E12:6"><i><font color="Green" title="E13">A151</font></i></a>, <a class="txt" href="wellset1.html#E2:6_6_8"><i><font color="Green" title="E24">A240</font></i></a>, <a class="txt" href="wellset1.html#E3:6_6_8"><i><font color="Green" title="E25">A241</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:6_6"/><b>then </b>
<font color="Maroon" title="c10">W1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E3:6"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="wellset1.html#E13:6_6"><i><font color="Green" title="E22">A181</font></i></a>, <a class="txt" href="wellset1.html#E14:6_6"><i><font color="Green" title="E23">A182</font></i></a></i>;<br/>
<a NAME="E17:6_6"/><b>then </b><i><font color="Green" title="E24">A250</font></i>: 
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c10">W1</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="c6">R</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E10:6"><i><font color="Green" title="E11">A93</font></i></a></i>;<br/>
<a NAME="E18:6_6"/><i><font color="Green" title="E25">A251</font></i>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">W1</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E19:6_6"/>
<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="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span></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="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E20:6_6"/><b>then </b>
<font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font></span>)</span> <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="c10">W1</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E18:6_6"><i><font color="Green" title="E25">A251</font></i></a>, <a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a></i>;<br/>
<b>hence </b><a NAME="E21:6_6"/>
contradiction
 <b>by </b><i><a class="txt" href="wellset1.html#E1:6"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="wellset1.html#E1:6_6"><i><font color="Green" title="E14">A161</font></i></a>, <a class="txt" href="wellset1.html#E17:6_6"><i><font color="Green" title="E24">A250</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E14:6"/>
( <font color="Maroon" title="c6">R</font> is <a href="wellord1.html#V2" title="WELLORD1:attr.2">well-ordering</a> &amp; not  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font> &amp; ( for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</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="c6">R</font> holds <br/>( <font color="Maroon" title="c6">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font> &amp; <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b1">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</font> ) ) )
 <b>by </b><i><a class="txt" href="wellset1.html#E11:6"><i><font color="Green" title="E12">A99</font></i></a>, <a class="txt" href="wellset1.html#E12:6"><i><font color="Green" title="E13">A151</font></i></a></i>;<br/>


</div>
