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

<b>let </b><font color="Maroon">F</font> be   <a href="funct_1.html#NM1">Function</a>;<br/><b>let </b><font color="Maroon">D</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:6"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> holds <br/>( not <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">X</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">D</font> )
 ;<br/>

<b>set </b><font color="Maroon">W0</font> =  <a href="zfmisc_1.html#K1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">D</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="relat_1.html#NM1">Relation</a>] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> is <a href="wellord1.html#V2">well-ordering</a> &amp; ( for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">a<sub>1</sub></font> holds <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="wellord1.html#K1">-Seg</a> <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="wellord1.html#K1">-Seg</a> <font color="Olive">y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> ) ) );<br/>
<b>consider </b><font color="Maroon">G</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:6"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">W</font> being  <a href="relat_1.html#NM1">Relation</a> holds <br/> ( <font color="Olive">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> iff ( <font color="Olive">W</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">D</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">W</font>] ) )
 <b>from </b><i><a class="ref" href="wellset1.html#S1" target="_self">WELLSET1:sch 1</a>();<br/></i>
<a NAME="E4:6"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">W1</font>, <font color="Olive">W2</font> being  <a href="relat_1.html#NM1">Relation</a>  st <font color="Olive">W1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">W2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; not <!--HUMANRECFAILED-->( <font color="Olive">W1</font> <a href="tarski.html#R1">c=</a> <font color="Olive">W2</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Olive">W1</font> holds <br/><font color="Olive">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Olive">x</font> ) ) holds <br/>( <font color="Olive">W2</font> <a href="tarski.html#R1">c=</a> <font color="Olive">W1</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Olive">W2</font> holds <br/><font color="Olive">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Olive">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">W1</font> be   <a href="relat_1.html#NM1">Relation</a>;<br/><b>let </b><font color="Maroon">W2</font> be   <a href="relat_1.html#NM1">Relation</a>;<br/>



<b>assume </b><a NAME="E1:6_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">W1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Maroon">W2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W2</font> &amp; <font color="Maroon">W1</font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">W2</font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> );<br/>
<b>consider </b><font color="Maroon">C</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:6_1"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E4:6_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">W1</font> is <a href="wellord1.html#V2">well-ordering</a> &amp; <font color="Maroon">W2</font> is <a href="wellord1.html#V2">well-ordering</a> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:6_1"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> holds <br/><font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Olive">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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:6_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/>

<a NAME="E2:6_1_1"/>
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x</font> iff <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">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">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<a NAME="E1:6_1_1_1"/><i><font color="Green">E53</font></i>: 
 <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="wellord1.html#K2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="wellord1.html#T40">WELLORD1:40</a></i>;<br/>
<a NAME="E2:6_1_1_1"/>
 <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><font color="Maroon">W2</font> <a href="wellord1.html#K2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="wellord1.html#T40">WELLORD1:40</a></i>;<br/>
<b>hence </b><a NAME="E3:6_1_1_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x</font> iff <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x</font> )
 <b>by </b><i>, , </i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E6:6_1"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> holds <br/><font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Olive">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:6_1_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/>

<a NAME="E2:6_1_2"/>
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x</font> holds <br/><font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">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">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:6_1_2_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x</font>
 ;<br/>

<a NAME="E2:6_1_2_1"/><b>then </b><i><font color="Green">E57</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/>
<a NAME="E3:6_1_2_1"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>
<a NAME="E4:6_1_2_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>
<a NAME="E5:6_1_2_1"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x</font>
 
<b>by </b><i>, , , <a class="ref" href="wellord1.html#T38">WELLORD1:38</a></i>;<br/>
<a NAME="E6:6_1_2_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E7:6_1_2_1"/><b>then </b><i><font color="Green">E62</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/>
<a NAME="E8:6_1_2_1"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W2</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>
<a NAME="E9:6_1_2_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W2</font>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>
<a NAME="E10:6_1_2_1"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x</font>
 
<b>by </b><i>, , , <a class="ref" href="wellord1.html#T38">WELLORD1:38</a></i>;<br/>
<a NAME="E11:6_1_2_1"/>
<font color="Maroon">W1</font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">W2</font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">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">E66</font></i>: 
<font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">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">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font> = 
<span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="wellord1.html#K2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font>
<b>by </b><i>, , , <a class="ref" href="wellord1.html#T35">WELLORD1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">W2</font> <a href="wellord1.html#K2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font>
<b>by </b><i>, , , <a class="ref" href="wellord1.html#T35">WELLORD1:35</a></i>
;<br/>
<b>hence </b><a NAME="E2:6_1_2_1_1_1"/>
<font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font>
 ;<br/>


</div><b>end;</b></div>
<font color="Maroon">W1</font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="wellord1.html#K2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i>, <a class="ref" href="wellord1.html#T29">WELLORD1:29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">W2</font> <a href="wellord1.html#K2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">W2</font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i>, <a class="ref" href="wellord1.html#T29">WELLORD1:29</a></i>
;<br/>
<b>hence </b><a NAME="E3:6_1_2_1_1"/>
<font color="Maroon">W1</font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">W2</font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:6_1_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i>, , </i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E7:6_1"/><i><font color="Green">E67</font></i>: 
for <font color="Olive">y1</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font> &amp; not <font color="Olive">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> holds <br/> ex <font color="Olive">y3</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y3</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Olive">y3</font> &amp; not <font color="Olive">y3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">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">y1</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:6_1_3"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font> &amp; not <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 ;<br/>

<b>set </b><font color="Maroon">Y</font> = <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">C</font>;<br/>
<a NAME="E2:6_1_3"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:6_1_3"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E5:6_1_3"/><i><font color="Green">E70</font></i>: 
for <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">C</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font>
 <b>by </b><i>, <a class="ref" href="wellord1.html#T10">WELLORD1:10</a></i>;<br/>
<b>take </b><font color="Maroon">y3</font> = <font color="Maroon">a</font>;<br/>

<a NAME="E6:6_1_3"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">y3</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font> &amp; not <font color="Maroon">y3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E7:6_1_3"/>
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">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">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> iff <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>thus </b><a NAME="E1:6_1_3_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">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">E72</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E2:6_1_3_1_1_1"/><i><font color="Green">E73</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y3</font>
 ;<br/>

<a NAME="E3:6_1_3_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:6_1_3_1_1_1"/><b>then </b><i><font color="Green">E74</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y3</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font>
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E5:6_1_3_1_1_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:6_1_3_1_1_1"/>
<font color="Maroon">y3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">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">y3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> implies <font color="Maroon">y3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">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">y3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 ;<br/>

<a NAME="E2:6_1_3_1_1_1_1_1"/><b>then </b>
<font color="Maroon">y3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:6_1_3_1_1_1_1_1"/>
<font color="Maroon">y3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:6_1_3_1_1_1_1"/>
<font color="Maroon">y3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><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="ref" href="xboole_0.html#D4">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">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y3</font> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">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">E76</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y3</font>
 <b>and </b><br/><a NAME="E2:6_1_3_1_1_2"/><i><font color="Green">E77</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/>

<a NAME="E3:6_1_3_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y3</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font>
 
<b>by </b><i>, <a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/>
<a NAME="E4:6_1_3_1_1_2"/><b>then </b><i><font color="Green">E78</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>
<a NAME="E5:6_1_3_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">C</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">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">[</a><span class="default"><font color="Maroon">y3</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:6_1_3_1_1_2"/>
contradiction
 <b>by </b><i>, , , , </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">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y3</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:6_1_3"/>
( <font color="Maroon">y3</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y3</font> &amp; not <font color="Maroon">y3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:6_1"/><i><font color="Green">E79</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> holds <br/><font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Olive">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:6_1_4"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/>

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

<b>assume </b><a NAME="E2:6_1_4"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x</font>
 ;<br/>

<a NAME="E3:6_1_4"/><b>then </b><i><font color="Green">E82</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/>
<a NAME="E4:6_1_4"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W2</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>
<a NAME="E5:6_1_4"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W2</font>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>
<a NAME="E6:6_1_4"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x</font>
 
<b>by </b><i>, , , <a class="ref" href="wellord1.html#T38">WELLORD1:38</a></i>;<br/>
<a NAME="E7:6_1_4"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E8:6_1_4"/><b>then </b><i><font color="Green">E87</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/>
<a NAME="E9:6_1_4"/><b>then </b><i><font color="Green">E88</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>
<a NAME="E10:6_1_4"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>
<a NAME="E11:6_1_4"/><b>then </b><i><font color="Green">E90</font></i>: 
<font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x</font>
 
<b>by </b><i>, , , <a class="ref" href="wellord1.html#T38">WELLORD1:38</a></i>;<br/>
<a NAME="E12:6_1_4"/>
<font color="Maroon">W2</font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">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">E91</font></i>: 
<font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">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">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font> = 
<span class="p1">(<span class="default"><font color="Maroon">W2</font> <a href="wellord1.html#K2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font>
<b>by </b><i>, , , <a class="ref" href="wellord1.html#T35">WELLORD1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="wellord1.html#K2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font>
<b>by </b><i>, , , <a class="ref" href="wellord1.html#T35">WELLORD1:35</a></i>
;<br/>
<b>hence </b><a NAME="E2:6_1_4_1_1"/>
<font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font>
 ;<br/>


</div><b>end;</b></div>
<font color="Maroon">W2</font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">W2</font> <a href="wellord1.html#K2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i>, <a class="ref" href="wellord1.html#T29">WELLORD1:29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="wellord1.html#K2">|_2</a> <span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">W1</font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i>, <a class="ref" href="wellord1.html#T29">WELLORD1:29</a></i>
;<br/>
<b>hence </b><a NAME="E3:6_1_4_1"/>
<font color="Maroon">W2</font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E13:6_1_4"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:6_1"/><i><font color="Green">E92</font></i>: 
for <font color="Olive">y1</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W2</font> &amp; not <font color="Olive">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> holds <br/> ex <font color="Olive">y3</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y3</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W2</font> &amp; <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Olive">y3</font> &amp; not <font color="Olive">y3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">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">y1</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:6_1_5"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W2</font> &amp; not <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 ;<br/>

<b>set </b><font color="Maroon">Y</font> = <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">W2</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">C</font>;<br/>
<a NAME="E2:6_1_5"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">W2</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:6_1_5"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">W2</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E5:6_1_5"/><i><font color="Green">E95</font></i>: 
for <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">W2</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">C</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font>
 <b>by </b><i>, <a class="ref" href="wellord1.html#T10">WELLORD1:10</a></i>;<br/>
<b>take </b><font color="Maroon">y3</font> = <font color="Maroon">a</font>;<br/>

<a NAME="E6:6_1_5"/><i><font color="Green">E96</font></i>: 
( <font color="Maroon">y3</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W2</font> &amp; not <font color="Maroon">y3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E7:6_1_5"/>
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">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">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> iff <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>thus </b><a NAME="E1:6_1_5_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">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">E97</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E2:6_1_5_1_1_1"/><i><font color="Green">E98</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y3</font>
 ;<br/>

<a NAME="E3:6_1_5_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W2</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:6_1_5_1_1_1"/><b>then </b><i><font color="Green">E99</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y3</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font>
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E5:6_1_5_1_1_1"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:6_1_5_1_1_1"/>
( <font color="Maroon">y3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> implies <font color="Maroon">y3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">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">y3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 ;<br/>

<a NAME="E2:6_1_5_1_1_1_1"/><b>then </b>
<font color="Maroon">y3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:6_1_5_1_1_1_1"/>
<font color="Maroon">y3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><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="ref" href="xboole_0.html#D4">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">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y3</font> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">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">E101</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y3</font>
 <b>and </b><br/><a NAME="E2:6_1_5_1_1_2"/><i><font color="Green">E102</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/>

<a NAME="E3:6_1_5_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y3</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font>
 
<b>by </b><i>, <a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/>
<a NAME="E4:6_1_5_1_1_2"/><b>then </b><i><font color="Green">E103</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W2</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>
<a NAME="E5:6_1_5_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">W2</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">C</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">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">[</a><span class="default"><font color="Maroon">y3</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:6_1_5_1_1_2"/>
contradiction
 <b>by </b><i>, , , , </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">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y3</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:6_1_5"/>
( <font color="Maroon">y3</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W2</font> &amp; <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y3</font> &amp; not <font color="Maroon">y3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:6_1"/><i><font color="Green">E104</font></i>: 
( <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font> or <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">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">C</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font>
 ;<br/>

<a NAME="E2:6_1_6"/><b>then </b><i><font color="Green">E105</font></i>: 
 ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; not <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font> ) or ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font> &amp; not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> ) )
 
<b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<b>assume </b><a NAME="E3:6_1_6"/>
not <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W2</font>
 ;<br/>

<a NAME="E4:6_1_6"/><b>then </b><i><font color="Green">E106</font></i>: 
 ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; not <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W2</font> ) or ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W2</font> &amp; not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> ) )
 
<b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<b>consider </b><font color="Maroon">y3</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:6_1_6"/><i><font color="Green">E107</font></i>: 
( <font color="Maroon">y3</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y3</font> &amp; not <font color="Maroon">y3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 <b>by </b><i>, , </i>;<br/>
<b>consider </b><font color="Maroon">y4</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:6_1_6"/><i><font color="Green">E108</font></i>: 
( <font color="Maroon">y4</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W2</font> &amp; <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y4</font> &amp; not <font color="Maroon">y4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 <b>by </b><i>, , </i>;<br/>
<a NAME="E9:6_1_6"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">y3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">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">y3</font> = 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y4</font></span>)</span>
<b>by </b><i>, , , </i>
<br/>.= 
<font color="Maroon">y4</font>
<b>by </b><i>, , </i>
;<br/>
<b>hence </b><a NAME="E2:6_1_6_1"/>
<font color="Maroon">y3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y4</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E10:6_1_6"/>
<font color="Maroon">W1</font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y3</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">W2</font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">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">z</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y3</font></span>)</span> iff <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">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">z</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<a NAME="E1:6_1_6_2_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y3</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> iff ( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y3</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</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">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y3</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> implies ( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y3</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</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">E110</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font>
 <b>and </b><br/><a NAME="E2:6_1_6_2_1_1_1"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y3</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<b>consider </b><font color="Maroon">z1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">z2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:6_1_6_2_1_1_1"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y3</font>
 <b>and </b><br/><a NAME="E5:6_1_6_2_1_1_1"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">z2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y3</font>
 <b>and </b><br/><a NAME="E6:6_1_6_2_1_1_1"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z1</font>,<font color="Maroon">z2</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E7:6_1_6_2_1_1_1"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">z1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W2</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E8:6_1_6_2_1_1_1"/>
( <font color="Maroon">z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">z2</font> or ( <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z2</font> &amp; not <font color="Maroon">z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">z2</font> ) )
 
<b>by </b><i>, , <a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/>
<a NAME="E9:6_1_6_2_1_1_1"/><b>then </b>
( <font color="Maroon">z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">z2</font> or ( <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z2</font> &amp; not <font color="Maroon">z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">z2</font> ) )
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E10:6_1_6_2_1_1_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y3</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i>, , , , , , , , <a class="ref" href="wellord1.html#D1">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">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y3</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> implies ( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y3</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</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">E116</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font>
 <b>and </b><br/><a NAME="E2:6_1_6_2_1_1_2"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y3</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<b>consider </b><font color="Maroon">z1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">z2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:6_1_6_2_1_1_2"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y3</font>
 <b>and </b><br/><a NAME="E5:6_1_6_2_1_1_2"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">z2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y3</font>
 <b>and </b><br/><a NAME="E6:6_1_6_2_1_1_2"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z1</font>,<font color="Maroon">z2</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E7:6_1_6_2_1_1_2"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">z1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font>
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E8:6_1_6_2_1_1_2"/>
( <font color="Maroon">z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">z2</font> or ( <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z2</font> &amp; not <font color="Maroon">z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">z2</font> ) )
 
<b>by </b><i>, , <a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/>
<a NAME="E9:6_1_6_2_1_1_2"/><b>then </b>
( <font color="Maroon">z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">z2</font> or ( <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z2</font> &amp; not <font color="Maroon">z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">z2</font> ) )
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E10:6_1_6_2_1_1_2"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y3</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i>, , , , , , , , <a class="ref" href="wellord1.html#D1">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">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y3</font></span>)</span> iff <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y3</font></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">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">W1</font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y3</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">W2</font> <a href="wellord1.html#K2">|_2</a> <span class="p1">(<span class="default"><font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y3</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:6_1_6"/>
contradiction
 <b>by </b><i>, , , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:6_1"/><i><font color="Green">E122</font></i>: 
( <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font> implies ( <font color="Maroon">W1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W2</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font> holds <br/><font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Olive">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">E123</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font>
 ;<br/>

<a NAME="E2:6_1_7"/>
for <font color="Olive">z1</font>, <font color="Olive">z2</font> being   <a href="hidden.html#M1">set</a>   st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">z1</font>,<font color="Olive">z2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">z1</font>,<font color="Olive">z2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">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">z1</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">z2</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:6_1_7_1"/><i><font color="Green">E124</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z1</font>,<font color="Maroon">z2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font>
 ;<br/>

<a NAME="E2:6_1_7_1"/><b>then </b><i><font color="Green">E125</font></i>: 
<font color="Maroon">z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>
<a NAME="E3:6_1_7_1"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">z2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 
<b>by </b><i>, , <a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>
<a NAME="E4:6_1_7_1"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">z1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W2</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:6_1_7_1"/>
( <font color="Maroon">z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">z2</font> or ( <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z2</font> &amp; not <font color="Maroon">z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">z2</font> ) )
 
<b>by </b><i>, <a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/>
<a NAME="E6:6_1_7_1"/><b>then </b>
( <font color="Maroon">z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">z2</font> or ( <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z2</font> &amp; not <font color="Maroon">z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">z2</font> ) )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E7:6_1_7_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z1</font>,<font color="Maroon">z2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font>
 <b>by </b><i>, , , <a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:6_1_7"/>
( <font color="Maroon">W1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W2</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font> holds <br/><font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Olive">x</font> ) )
 <b>by </b><i>, , <a class="ref" href="relat_1.html#D3">RELAT_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:6_1"/>
( <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W2</font> implies ( <font color="Maroon">W2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W1</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W2</font> holds <br/><font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Olive">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">E128</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W2</font>
 ;<br/>

<a NAME="E2:6_1_8"/>
for <font color="Olive">z1</font>, <font color="Olive">z2</font> being   <a href="hidden.html#M1">set</a>   st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">z1</font>,<font color="Olive">z2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">z1</font>,<font color="Olive">z2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">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">z1</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">z2</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:6_1_8_1"/><i><font color="Green">E129</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z1</font>,<font color="Maroon">z2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font>
 ;<br/>

<a NAME="E2:6_1_8_1"/><b>then </b><i><font color="Green">E130</font></i>: 
<font color="Maroon">z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>
<a NAME="E3:6_1_8_1"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">z2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 
<b>by </b><i>, , <a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>
<a NAME="E4:6_1_8_1"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">z1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:6_1_8_1"/>
( <font color="Maroon">z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">z2</font> or ( <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z2</font> &amp; not <font color="Maroon">z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">z2</font> ) )
 
<b>by </b><i>, <a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/>
<a NAME="E6:6_1_8_1"/><b>then </b>
( <font color="Maroon">z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">z2</font> or ( <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z2</font> &amp; not <font color="Maroon">z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">z2</font> ) )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E7:6_1_8_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z1</font>,<font color="Maroon">z2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font>
 <b>by </b><i>, , , <a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:6_1_8"/>
( <font color="Maroon">W2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W1</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W2</font> holds <br/><font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Olive">x</font> ) )
 <b>by </b><i>, , <a class="ref" href="relat_1.html#D3">RELAT_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E13:6_1"/>
( ( <font color="Maroon">W1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W2</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font> holds <br/><font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Olive">x</font> ) ) or ( <font color="Maroon">W2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W1</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W2</font> holds <br/><font color="Maroon">W2</font> <a href="wellord1.html#K1">-Seg</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Olive">x</font> ) ) )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">W</font> being  <a href="relat_1.html#NM1">Relation</a> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <font color="Olive">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> );<br/>
<b>consider </b><font color="Maroon">S</font> being   <a href="relat_1.html#NM1">Relation</a><b> such that </b><br/><a NAME="E6:6"/><i><font color="Green">E133</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">D</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">D</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] ) )
 <b>from </b><i><a class="ref" href="relat_1.html#S1">RELAT_1:sch 1</a>();<br/></i>
<b>take </b><font color="Maroon">R</font> = <font color="Maroon">S</font>;<br/>

<a NAME="E7:6"/><i><font color="Green">E134</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">D</font> &amp;  ex <font color="Olive">W</font> being  <a href="relat_1.html#NM1">Relation</a> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Olive">W</font> &amp; <font color="Olive">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:6_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:6_2"/><i><font color="Green">E135</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 <b>by </b><i/>;<br/>
<a NAME="E4:6_2"/><i><font color="Green">E136</font></i>: 
( ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">D</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">D</font> &amp;  ex <font color="Olive">S</font> being  <a href="relat_1.html#NM1">Relation</a> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">S</font> &amp; <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> ) ) or ( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">D</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">D</font> &amp;  ex <font color="Olive">S</font> being  <a href="relat_1.html#NM1">Relation</a> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">S</font> &amp; <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> ) ) )
 
<b>by </b><i>, </i>;<br/>
<b>thus </b><a NAME="E5:6_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">D</font>
 <b>by </b><i>, </i>;<br/>

<b>consider </b><font color="Maroon">S</font> being   <a href="relat_1.html#NM1">Relation</a><b> such that </b><br/><a NAME="E7:6_2"/><i><font color="Green">E137</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 <b>and </b><br/><a NAME="E8:6_2"/><i><font color="Green">E138</font></i>: 
<font color="Maroon">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">S</font>
;<br/>

<b>thus </b><a NAME="E9:6_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">S</font> &amp; <font color="Maroon">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> )
 <b>by </b><i>, , </i>;<br/>


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

<a NAME="E10:6"/><i><font color="Green">E139</font></i>: 
for <font color="Olive">W</font> being  <a href="relat_1.html#NM1">Relation</a>  st <font color="Olive">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> holds <br/> <a href="relat_1.html#K3">field</a> <font color="Olive">W</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">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">W</font> be   <a href="relat_1.html#NM1">Relation</a>;<br/>

<b>assume </b><a NAME="E1:6_3"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 ;<br/>

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

<b>assume </b><a NAME="E2:6_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:6_3"/><i><font color="Green">E141</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> )
 <b>by </b><i/>;<br/>
<a NAME="E5:6_3"/><i><font color="Green">E142</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 
<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">E143</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 ;<br/>

<a NAME="E2:6_3_1"/>
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">D</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:6_3_1"/><b>then </b>
 ex <font color="Olive">z1</font>, <font color="Olive">z2</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">z1</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">D</font> &amp; <font color="Olive">z2</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">D</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">z1</font>,<font color="Olive">z2</font></span><a href="tarski.html#K4">]</a></span> )
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T103">ZFMISC_1:103</a></i>;<br/>
<b>hence </b><a NAME="E4:6_3_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:6_3"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 
<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">E144</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 ;<br/>

<a NAME="E2:6_3_2"/>
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">D</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:6_3_2"/><b>then </b>
 ex <font color="Olive">z1</font>, <font color="Olive">z2</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">z1</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">D</font> &amp; <font color="Olive">z2</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">D</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">z1</font>,<font color="Olive">z2</font></span><a href="tarski.html#K4">]</a></span> )
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T103">ZFMISC_1:103</a></i>;<br/>
<b>hence </b><a NAME="E4:6_3_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:6_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:6"/><i><font color="Green">E145</font></i>: 
<font color="Maroon">R</font> is <a href="wellord1.html#V2">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">E146</font></i>: 
<font color="Maroon">R</font> <a href="relat_2.html#R1">is_reflexive_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 
<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">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:6_4_1_1"/><i><font color="Green">E147</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 ;<br/>

<a NAME="E2:6_4_1_1"/><b>then </b><i><font color="Green">E148</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">D</font> &amp;  ex <font color="Olive">W</font> being  <a href="relat_1.html#NM1">Relation</a> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Olive">W</font> &amp; <font color="Olive">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> ) )
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">W</font> being   <a href="relat_1.html#NM1">Relation</a><b> such that </b><br/><a NAME="E4:6_4_1_1"/><i><font color="Green">E149</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W</font> &amp; <font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E5:6_4_1_1"/>
<font color="Maroon">W</font> is <a href="wellord1.html#V2">well-ordering</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:6_4_1_1"/><b>then </b>
<font color="Maroon">W</font> <a href="wellord1.html#R2">well_orders</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#T8">WELLORD1:8</a></i>;<br/>
<a NAME="E7:6_4_1_1"/><b>then </b>
<font color="Maroon">W</font> <a href="relat_2.html#R1">is_reflexive_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#D5">WELLORD1:def 5</a></i>;<br/>
<a NAME="E8:6_4_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 
<b>by </b><i>, <a class="ref" href="relat_2.html#D1">RELAT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E9:6_4_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i>, , </i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E2:6_4"/><i><font color="Green">E150</font></i>: 
<font color="Maroon">R</font> <a href="relat_2.html#R8">is_transitive_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">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">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:6_4_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 <b>and </b><br/><a NAME="E2:6_4_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 <b>and </b><br/><a NAME="E3:6_4_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 <b>and </b><br/><a NAME="E4:6_4_2_1"/><i><font color="Green">E151</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 ;<br/>

<a NAME="E5:6_4_2_1"/><i><font color="Green">E152</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">D</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">D</font> &amp;  ex <font color="Olive">W</font> being  <a href="relat_1.html#NM1">Relation</a> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <font color="Olive">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> ) )
 
<b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">W1</font> being   <a href="relat_1.html#NM1">Relation</a><b> such that </b><br/><a NAME="E7:6_4_2_1"/><i><font color="Green">E153</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">W1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E8:6_4_2_1"/><i><font color="Green">E154</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">D</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">D</font> &amp;  ex <font color="Olive">W</font> being  <a href="relat_1.html#NM1">Relation</a> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <font color="Olive">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> ) )
 
<b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">W2</font> being   <a href="relat_1.html#NM1">Relation</a><b> such that </b><br/><a NAME="E10:6_4_2_1"/><i><font color="Green">E155</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font> &amp; <font color="Maroon">W2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E11:6_4_2_1"/>
 ex <font color="Olive">W</font> being  <a href="relat_1.html#NM1">Relation</a> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <font color="Olive">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">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">W</font> = <font color="Maroon">W2</font>;<br/>

<a NAME="E1:6_4_2_1_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W</font> )
 
<b>by </b><i>, , <a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>
<a NAME="E2:6_4_2_1_1"/><b>then </b><i><font color="Green">E156</font></i>: 
<font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E3:6_4_2_1_1"/>
( not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">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">E157</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E2:6_4_2_1_1_1"/><i><font color="Green">E158</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font> )
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>
<a NAME="E3:6_4_2_1_1_1"/><i><font color="Green">E159</font></i>: 
<font color="Maroon">W1</font> is <a href="wellord1.html#V2">well-ordering</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:6_4_2_1_1_1"/><b>then </b><i><font color="Green">E160</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E5:6_4_2_1_1_1"/>
<font color="Maroon">W1</font> <a href="wellord1.html#R2">well_orders</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font>
 
<b>by </b><i>, <a class="ref" href="wellord1.html#T8">WELLORD1:8</a></i>;<br/>
<a NAME="E6:6_4_2_1_1_1"/><b>then </b>
<font color="Maroon">W1</font> <a href="relat_2.html#R4">is_antisymmetric_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#D5">WELLORD1:def 5</a></i>;<br/>
<a NAME="E7:6_4_2_1_1_1"/><b>then </b><i><font color="Green">E161</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 
<b>by </b><i>, , , <a class="ref" href="relat_2.html#D4">RELAT_2:def 4</a></i>;<br/>
<a NAME="E8:6_4_2_1_1_1"/><i><font color="Green">E162</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W</font>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>
<a NAME="E9:6_4_2_1_1_1"/>
<font color="Maroon">W</font> is <a href="wellord1.html#V2">well-ordering</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:6_4_2_1_1_1"/><b>then </b>
<font color="Maroon">W</font> <a href="wellord1.html#R2">well_orders</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#T8">WELLORD1:8</a></i>;<br/>
<a NAME="E11:6_4_2_1_1_1"/><b>then </b>
<font color="Maroon">W</font> <a href="relat_2.html#R1">is_reflexive_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#D5">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">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>by </b><i>, , <a class="ref" href="relat_2.html#D1">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">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> &amp; <font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> )
 <b>by </b><i>, , <a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">W</font> being   <a href="relat_1.html#NM1">Relation</a><b> such that </b><br/><a NAME="E13:6_4_2_1"/><i><font color="Green">E163</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> &amp; <font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> )
 ;<br/>
<a NAME="E14:6_4_2_1"/>
<font color="Maroon">W</font> is <a href="wellord1.html#V2">well-ordering</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E15:6_4_2_1"/><b>then </b>
<font color="Maroon">W</font> <a href="wellord1.html#R2">well_orders</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#T8">WELLORD1:8</a></i>;<br/>
<a NAME="E16:6_4_2_1"/><b>then </b><i><font color="Green">E164</font></i>: 
<font color="Maroon">W</font> <a href="relat_2.html#R8">is_transitive_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#D5">WELLORD1:def 5</a></i>;<br/>
<a NAME="E17:6_4_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W</font> )
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>
<a NAME="E18:6_4_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 
<b>by </b><i>, , <a class="ref" href="relat_2.html#D8">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">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i>, , , </i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E3:6_4"/><i><font color="Green">E165</font></i>: 
<font color="Maroon">R</font> <a href="relat_2.html#R4">is_antisymmetric_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">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">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:6_4_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 <b>and </b><br/><a NAME="E2:6_4_3_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 <b>and </b><br/><a NAME="E3:6_4_3_1"/><i><font color="Green">E166</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 ;<br/>

<b>consider </b><font color="Maroon">W1</font> being   <a href="relat_1.html#NM1">Relation</a><b> such that </b><br/><a NAME="E5:6_4_3_1"/><i><font color="Green">E167</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">W1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> )
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">W2</font> being   <a href="relat_1.html#NM1">Relation</a><b> such that </b><br/><a NAME="E7:6_4_3_1"/><i><font color="Green">E168</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font> &amp; <font color="Maroon">W2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E8:6_4_3_1"/><i><font color="Green">E169</font></i>: 
( <font color="Maroon">W1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W2</font> implies <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">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">E170</font></i>: 
<font color="Maroon">W1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W2</font>
 ;<br/>

<a NAME="E2:6_4_3_1_1"/><b>then </b><i><font color="Green">E171</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W2</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W2</font> )
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>
<a NAME="E3:6_4_3_1_1"/>
<font color="Maroon">W2</font> is <a href="wellord1.html#V2">well-ordering</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:6_4_3_1_1"/><b>then </b>
<font color="Maroon">W2</font> <a href="wellord1.html#R2">well_orders</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W2</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#T8">WELLORD1:8</a></i>;<br/>
<a NAME="E5:6_4_3_1_1"/><b>then </b>
<font color="Maroon">W2</font> <a href="relat_2.html#R4">is_antisymmetric_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W2</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#D5">WELLORD1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E6:6_4_3_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i>, , , , <a class="ref" href="relat_2.html#D4">RELAT_2:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:6_4_3_1"/>
( <font color="Maroon">W2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W1</font> implies <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">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">E172</font></i>: 
<font color="Maroon">W2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W1</font>
 ;<br/>

<a NAME="E2:6_4_3_1_2"/><b>then </b><i><font color="Green">E173</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font> )
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>
<a NAME="E3:6_4_3_1_2"/>
<font color="Maroon">W1</font> is <a href="wellord1.html#V2">well-ordering</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:6_4_3_1_2"/><b>then </b>
<font color="Maroon">W1</font> <a href="wellord1.html#R2">well_orders</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#T8">WELLORD1:8</a></i>;<br/>
<a NAME="E5:6_4_3_1_2"/><b>then </b>
<font color="Maroon">W1</font> <a href="relat_2.html#R4">is_antisymmetric_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#D5">WELLORD1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E6:6_4_3_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i>, , , , <a class="ref" href="relat_2.html#D4">RELAT_2:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:6_4_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i>, , , </i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E4:6_4"/><i><font color="Green">E174</font></i>: 
<font color="Maroon">R</font> <a href="relat_2.html#R6">is_connected_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">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">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> &amp; <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> &amp; not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:6_4_4_1"/><i><font color="Green">E175</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 <b>and </b><br/><a NAME="E2:6_4_4_1"/><i><font color="Green">E176</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 <b>and </b><br/><a NAME="E3:6_4_4_1"/><i><font color="Green">E177</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E4:6_4_4_1"/><i><font color="Green">E178</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">D</font> &amp;  ex <font color="Olive">W</font> being  <a href="relat_1.html#NM1">Relation</a> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Olive">W</font> &amp; <font color="Olive">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> ) )
 
<b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">W1</font> being   <a href="relat_1.html#NM1">Relation</a><b> such that </b><br/><a NAME="E6:6_4_4_1"/><i><font color="Green">E179</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">W1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E7:6_4_4_1"/><i><font color="Green">E180</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">D</font> &amp;  ex <font color="Olive">W</font> being  <a href="relat_1.html#NM1">Relation</a> st <br/>( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Olive">W</font> &amp; <font color="Olive">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> ) )
 
<b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">W2</font> being   <a href="relat_1.html#NM1">Relation</a><b> such that </b><br/><a NAME="E9:6_4_4_1"/><i><font color="Green">E181</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W2</font> &amp; <font color="Maroon">W2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E10:6_4_4_1"/><i><font color="Green">E182</font></i>: 
(  not <font color="Maroon">W1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W2</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 
<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">W1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W2</font>
 ;<br/>

<a NAME="E2:6_4_4_1_1"/><b>then </b><i><font color="Green">E183</font></i>: 
 <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W2</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T31">RELAT_1:31</a></i>;<br/>
<a NAME="E3:6_4_4_1_1"/>
<font color="Maroon">W2</font> is <a href="wellord1.html#V2">well-ordering</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:6_4_4_1_1"/><b>then </b>
<font color="Maroon">W2</font> <a href="wellord1.html#R2">well_orders</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W2</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#T8">WELLORD1:8</a></i>;<br/>
<a NAME="E5:6_4_4_1_1"/><b>then </b>
<font color="Maroon">W2</font> <a href="relat_2.html#R6">is_connected_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W2</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#D5">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">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font> )
 
<b>by </b><i>, , , , <a class="ref" href="relat_2.html#D6">RELAT_2:def 6</a></i>;<br/>
<b>hence </b><a NAME="E7:6_4_4_1_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 <b>by </b><i>, , , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:6_4_4_1"/>
(  not <font color="Maroon">W2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W1</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 
<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">W2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W1</font>
 ;<br/>

<a NAME="E2:6_4_4_1_2"/><b>then </b><i><font color="Green">E184</font></i>: 
 <a href="relat_1.html#K3">field</a> <font color="Maroon">W2</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T31">RELAT_1:31</a></i>;<br/>
<a NAME="E3:6_4_4_1_2"/>
<font color="Maroon">W1</font> is <a href="wellord1.html#V2">well-ordering</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:6_4_4_1_2"/><b>then </b>
<font color="Maroon">W1</font> <a href="wellord1.html#R2">well_orders</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#T8">WELLORD1:8</a></i>;<br/>
<a NAME="E5:6_4_4_1_2"/><b>then </b>
<font color="Maroon">W1</font> <a href="relat_2.html#R6">is_connected_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#D5">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">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> )
 
<b>by </b><i>, , , , <a class="ref" href="relat_2.html#D6">RELAT_2:def 6</a></i>;<br/>
<b>hence </b><a NAME="E7:6_4_4_1_2"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 <b>by </b><i>, , , </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">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 <b>by </b><i>, , , </i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E5:6_4"/>
<font color="Maroon">R</font> <a href="wellord1.html#R1">is_well_founded_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">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">Y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="wellord1.html#D3">WELLORD1:def 3</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:6_4_5"/><i><font color="Green">E185</font></i>: 
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 <b>and </b><br/><a NAME="E2:6_4_5"/><i><font color="Green">E186</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>;<br/>
<a NAME="E4:6_4_5"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 
<b>by </b><i>, , <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">W</font> being   <a href="relat_1.html#NM1">Relation</a><b> such that </b><br/><a NAME="E6:6_4_5"/><i><font color="Green">E187</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W</font> &amp; <font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> )
 <b>by </b><i/>;<br/>
<a NAME="E7:6_4_5"/>
<font color="Maroon">W</font> is <a href="wellord1.html#V2">well-ordering</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:6_4_5"/><b>then </b>
<font color="Maroon">W</font> <a href="wellord1.html#R2">well_orders</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#T8">WELLORD1:8</a></i>;<br/>
<a NAME="E9:6_4_5"/><b>then </b><i><font color="Green">E188</font></i>: 
<font color="Maroon">W</font> <a href="wellord1.html#R1">is_well_founded_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#D5">WELLORD1:def 5</a></i>;<br/>
<b>set </b><font color="Maroon">A</font> = <font color="Maroon">Y</font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">W</font></span>)</span>;<br/>
<a NAME="E10:6_4_5"/><i><font color="Green">E189</font></i>: 
<font color="Maroon">Y</font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">W</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E11:6_4_5"/>
<font color="Maroon">Y</font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">W</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E13:6_4_5"/><i><font color="Green">E190</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">W</font></span>)</span>
 <b>and </b><br/><a NAME="E14:6_4_5"/><i><font color="Green">E191</font></i>: 
<font color="Maroon">W</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">a</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Y</font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">W</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="wellord1.html#D3">WELLORD1:def 3</a></i>;<br/>
<a NAME="E15:6_4_5"/>
 ex <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Olive">b</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">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">b</font> = <font color="Maroon">a</font>;<br/>

<b>thus </b><a NAME="E1:6_4_5_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>

<b>assume </b><a NAME="E2:6_4_5_1"/>
not <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">b</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Y</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:6_4_5_1"/><i><font color="Green">E192</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">b</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E5:6_4_5_1"/><i><font color="Green">E193</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, <a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">W1</font> being   <a href="relat_1.html#NM1">Relation</a><b> such that </b><br/><a NAME="E7:6_4_5_1"/><i><font color="Green">E194</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">W1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> )
 <b>by </b><i/>;<br/>
<a NAME="E8:6_4_5_1"/><i><font color="Green">E195</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font> )
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>
<a NAME="E9:6_4_5_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">b</font>
 
<b>by </b><i>, , <a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/>
<a NAME="E10:6_4_5_1"/><b>then </b><i><font color="Green">E196</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">a</font>
 
<b>by </b><i>, , , , , </i>;<br/>
<a NAME="E11:6_4_5_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/>
<a NAME="E12:6_4_5_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>
<a NAME="E13:6_4_5_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">W</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E14:6_4_5_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">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">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Y</font> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:6_4"/><b>then </b>
<font color="Maroon">R</font> <a href="wellord1.html#R2">well_orders</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 
<b>by </b><i>, , , , <a class="ref" href="wellord1.html#D5">WELLORD1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E7:6_4"/>
<font color="Maroon">R</font> is <a href="wellord1.html#V2">well-ordering</a>
 <b>by </b><i><a class="ref" href="wellord1.html#T8">WELLORD1:8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:6"/><i><font color="Green">E197</font></i>: 
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> holds <br/>( <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Olive">y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">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">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:6_5"/><i><font color="Green">E198</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 ;<br/>

<b>then consider </b><font color="Maroon">W</font> being   <a href="relat_1.html#NM1">Relation</a><b> such that </b><br/><a NAME="E3:6_5"/><i><font color="Green">E199</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W</font> &amp; <font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> )
 <b>by </b><i/>;<br/>
<a NAME="E4:6_5"/><i><font color="Green">E200</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">D</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:6_5"/><i><font color="Green">E201</font></i>: 
 <a href="relat_1.html#K3">field</a> <font color="Maroon">W</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:6_5"/>
<font color="Maroon">W</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">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">E202</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:6_5_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E2:6_5_1_1"/><b>then </b><i><font color="Green">E203</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 
<b>by </b><i><a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/>
<a NAME="E3:6_5_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>
<a NAME="E4:6_5_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">D</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:6_5_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E6:6_5_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:6_5_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:6_5_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E2:6_5_1_2"/><b>then </b><i><font color="Green">E204</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 
<b>by </b><i><a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">W1</font> being   <a href="relat_1.html#NM1">Relation</a><b> such that </b><br/><a NAME="E4:6_5_1_2"/><i><font color="Green">E205</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">W1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> )
 <b>by </b><i/>;<br/>
<a NAME="E5:6_5_1_2"/><i><font color="Green">E206</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font>
 
<b>by </b><i>, , <a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/>
<a NAME="E6:6_5_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>
<b>hence </b><a NAME="E7:6_5_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font>
 <b>by </b><i>, , , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:6_5_1"/>
<font color="Maroon">W</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:6_5"/>
( <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:6"/>
not  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> <a href="hidden.html#R2">in</a> <font color="Maroon">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">E207</font></i>: 
 <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/>

<b>set </b><font color="Maroon">a0</font> = <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span>;<br/>
<a NAME="E2:6_6"/><i><font color="Green">E209</font></i>: 
not <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 
<b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">W3</font> = <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> as   <a href="relat_1.html#NM1">Relation</a> <b>by </b><i><a class="ref" href="relat_1.html#T6">RELAT_1:6</a></i>;<br/>
<b>reconsider </b><font color="Maroon">W4</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> as   <a href="relat_1.html#NM1">Relation</a> <b>by </b><i><a class="ref" href="relat_1.html#T4">RELAT_1:4</a></i>;<br/>
<a NAME="E5:6_6"/>
 <a href="relat_1.html#K3">field</a> <font color="Maroon">W4</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T32">RELAT_1:32</a></i>;<br/>
<a NAME="E6:6_6"/><b>then </b><i><font color="Green">E212</font></i>: 
 <a href="relat_1.html#K3">field</a> <font color="Maroon">W4</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="enumset1.html#T41">ENUMSET1:41</a></i>;<br/>
<a NAME="E7:6_6"/>
<span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span>,<span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">W3</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">W4</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">W1</font> = <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span>,<span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> as   <a href="relat_1.html#NM1">Relation</a> ;<br/>
<a NAME="E9:6_6"/><i><font color="Green">E213</font></i>: 
 <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span></span><a href="tarski.html#K1">}</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">E214</font></i>: 
(  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  implies  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span></span><a href="tarski.html#K1">}</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">field</a> <font color="Maroon">R</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E2:6_6_1_1"/><b>then </b><i><font color="Green">E215</font></i>: 
 <a href="relat_1.html#K3">field</a> <font color="Maroon">W3</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:6_6_1_1"/>
 <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">W3</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">W4</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T33">RELAT_1:33</a></i>;<br/>
<a NAME="E4:6_6_1_1"/><b>then </b>
 <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, , <a class="ref" href="relat_1.html#T33">RELAT_1:33</a></i>;<br/>
<a NAME="E5:6_6_1_1"/><b>then </b>
 <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font> <a href="hidden.html#R1">=</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">field</a> <font color="Maroon">R</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/>
<a NAME="E6:6_6_1_1"/><b>then </b>
 <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/>
<b>hence </b><a NAME="E7:6_6_1_1"/>
 <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:6_6_1"/>
(  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span></span><a href="tarski.html#K1">}</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">E216</font></i>: 
 <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E2:6_6_1_2"/><i><font color="Green">E217</font></i>: 
 <a href="relat_1.html#K3">field</a> <font color="Maroon">W3</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</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">z3</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K3">field</a> <font color="Maroon">W3</font>;<br/>
<b>assume </b><a NAME="E2:6_6_1_2_1"/>
 <a href="relat_1.html#K3">field</a> <font color="Maroon">W3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E3:6_6_1_2_1"/><b>then </b>
 ex <font color="Olive">z2</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z3</font>,<font color="Olive">z2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W3</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">z2</font>,<font color="Maroon">z3</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W3</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:6_6_1_2_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T113">ZFMISC_1:113</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:6_6_1_2"/>
 <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">W3</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">W4</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T33">RELAT_1:33</a></i>;<br/>
<a NAME="E4:6_6_1_2"/><b>then </b>
 <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, , <a class="ref" href="relat_1.html#T33">RELAT_1:33</a></i>;<br/>
<b>hence </b><a NAME="E5:6_6_1_2"/>
 <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:6_6_1"/>
 <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E10:6_6"/><i><font color="Green">E218</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font> or <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> or <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:6_6_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font>
 ;<br/>

<a NAME="E2:6_6_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:6_6_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span> )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:6_6"/><i><font color="Green">E219</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">D</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> is   <a href="relat_1.html#NM1">Relation</a>
 
<b>by </b><i><a class="ref" href="relat_1.html#T6">RELAT_1:6</a></i>;<br/>
<a NAME="E12:6_6"/><i><font color="Green">E220</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> iff ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W3</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<a NAME="E1:6_6_3"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> iff ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">W3</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W4</font> ) )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E2:6_6_3"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> iff ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W3</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W4</font> ) )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:6_6"/><i><font color="Green">E221</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> holds <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:6_6_4"/><i><font color="Green">E222</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> )
 ;<br/>

<a NAME="E2:6_6_4"/><b>then </b><i><font color="Green">E223</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W3</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W4</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:6_6_4"/><i><font color="Green">E224</font></i>: 
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W3</font>
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T129">ZFMISC_1:129</a></i>;<br/>
<a NAME="E4:6_6_4"/><i><font color="Green">E225</font></i>: 
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">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">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W4</font>
 ;<br/>

<a NAME="E2:6_6_4_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:6_6_4_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">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">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i>, , </i>;<br/>

<b>thus </b><a NAME="E6:6_6_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 <b>by </b><i>, , <a class="ref" href="relat_1.html#T30">RELAT_1:30</a>, <a class="ref" href="zfmisc_1.html#T129">ZFMISC_1:129</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:6_6"/><i><font color="Green">E226</font></i>: 
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> holds <br/><font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Olive">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">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:6_6_5"/><i><font color="Green">E227</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 ;<br/>

<a NAME="E2:6_6_5"/><i><font color="Green">E228</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:6_6_5_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E2:6_6_5_1"/><b>then </b><i><font color="Green">E229</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> )
 
<b>by </b><i><a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/>
<a NAME="E3:6_6_5_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:6_6_5_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:6_6_5"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:6_6_5_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E2:6_6_5_2"/><b>then </b><i><font color="Green">E230</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> )
 
<b>by </b><i><a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/>
<a NAME="E3:6_6_5_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:6_6_5_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:6_6_5"/>
<font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:6_6"/><i><font color="Green">E231</font></i>: 
<font color="Maroon">W1</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">D</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</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">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">D</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:6_6_6_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font>
 ;<br/>

<a NAME="E2:6_6_6_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>
<a NAME="E3:6_6_6_1"/><b>then </b>
( ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span> ) &amp; ( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span> ) )
 
<b>by </b><i/>;<br/>
<a NAME="E4:6_6_6_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">D</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">D</font> )
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E5:6_6_6_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">D</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:6_6_6"/><b>then </b>
<font color="Maroon">W1</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">D</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#D3">RELAT_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:6_6_6"/>
<font color="Maroon">W1</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">D</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E16:6_6"/><i><font color="Green">E232</font></i>: 
<font color="Maroon">W1</font> is <a href="wellord1.html#V2">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">E233</font></i>: 
<font color="Maroon">W1</font> <a href="relat_2.html#R1">is_reflexive_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">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">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:6_6_7_1_1"/><i><font color="Green">E234</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font>
 ;<br/>

<a NAME="E2:6_6_7_1_1"/><i><font color="Green">E235</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">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">E236</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 ;<br/>

<a NAME="E2:6_6_7_1_1_1"/>
<font color="Maroon">R</font> <a href="wellord1.html#R2">well_orders</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 
<b>by </b><i>, <a class="ref" href="wellord1.html#T8">WELLORD1:8</a></i>;<br/>
<a NAME="E3:6_6_7_1_1_1"/><b>then </b>
<font color="Maroon">R</font> <a href="relat_2.html#R1">is_reflexive_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#D5">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">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 
<b>by </b><i>, <a class="ref" href="relat_2.html#D1">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">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E3:6_6_7_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">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">E237</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span>
 ;<br/>

<a NAME="E2:6_6_7_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W4</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">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">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font>
 <b>by </b><i>, </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">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font>
 <b>by </b><i>, , </i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E2:6_6_7"/><i><font color="Green">E238</font></i>: 
<font color="Maroon">W1</font> <a href="relat_2.html#R8">is_transitive_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">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">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:6_6_7_2_1"/><i><font color="Green">E239</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font>
 <b>and </b><br/><a NAME="E2:6_6_7_2_1"/><i><font color="Green">E240</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font>
 <b>and </b><br/><a NAME="E3:6_6_7_2_1"/><i><font color="Green">E241</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font>
 <b>and </b><br/><a NAME="E4:6_6_7_2_1"/><i><font color="Green">E242</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> )
 ;<br/>

<a NAME="E5:6_6_7_2_1"/><i><font color="Green">E243</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">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">E244</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 ;<br/>

<a NAME="E2:6_6_7_2_1_1"/><b>then </b><i><font color="Green">E245</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:6_6_7_2_1_1"/><b>then </b><i><font color="Green">E246</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:6_6_7_2_1_1"/>
<font color="Maroon">R</font> <a href="wellord1.html#R2">well_orders</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 
<b>by </b><i>, <a class="ref" href="wellord1.html#T8">WELLORD1:8</a></i>;<br/>
<a NAME="E5:6_6_7_2_1_1"/><b>then </b>
<font color="Maroon">R</font> <a href="relat_2.html#R8">is_transitive_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#D5">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">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 
<b>by </b><i>, , , <a class="ref" href="relat_2.html#D8">RELAT_2:def 8</a></i>;<br/>
<b>hence </b><a NAME="E7:6_6_7_2_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E6:6_6_7_2_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">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">E247</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span>
 ;<br/>

<a NAME="E2:6_6_7_2_1_2"/><i><font color="Green">E248</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">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">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 ;<br/>

<a NAME="E2:6_6_7_2_1_2_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:6_6_7_2_1_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W3</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T129">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">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E3:6_6_7_2_1_2"/>
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">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">E249</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">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">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">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">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W3</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T129">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">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:6_6_7_2_1_2_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">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">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">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">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W4</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">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">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:6_6_7_2_1_2_2"/>
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> )
 <b>by </b><i>, , </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">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font>
 <b>by </b><i>, , </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">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font>
 <b>by </b><i>, , </i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E3:6_6_7"/><i><font color="Green">E250</font></i>: 
<font color="Maroon">W1</font> <a href="relat_2.html#R4">is_antisymmetric_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">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">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:6_6_7_3_1"/><i><font color="Green">E251</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font>
 <b>and </b><br/><a NAME="E2:6_6_7_3_1"/><i><font color="Green">E252</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font>
 <b>and </b><br/><a NAME="E3:6_6_7_3_1"/><i><font color="Green">E253</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> )
 ;<br/>

<a NAME="E4:6_6_7_3_1"/><i><font color="Green">E254</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:6_6_7_3_1"/><i><font color="Green">E255</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> implies <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">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">E256</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 ;<br/>

<a NAME="E2:6_6_7_3_1_1"/><b>then </b><i><font color="Green">E257</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:6_6_7_3_1_1"/><b>then </b><i><font color="Green">E258</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:6_6_7_3_1_1"/>
<font color="Maroon">R</font> <a href="wellord1.html#R2">well_orders</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 
<b>by </b><i>, <a class="ref" href="wellord1.html#T8">WELLORD1:8</a></i>;<br/>
<a NAME="E5:6_6_7_3_1_1"/><b>then </b>
<font color="Maroon">R</font> <a href="relat_2.html#R4">is_antisymmetric_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#D5">WELLORD1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E6:6_6_7_3_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i>, , <a class="ref" href="wellset1.html#T1" target="_self">Th1</a>, <a class="ref" href="relat_2.html#D4">RELAT_2:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:6_6_7_3_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> implies <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">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">E259</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 ;<br/>

<a NAME="E2:6_6_7_3_1_2"/><b>then </b><i><font color="Green">E260</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:6_6_7_3_1_2"/><b>then </b><i><font color="Green">E261</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:6_6_7_3_1_2"/>
<font color="Maroon">R</font> <a href="wellord1.html#R2">well_orders</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 
<b>by </b><i>, <a class="ref" href="wellord1.html#T8">WELLORD1:8</a></i>;<br/>
<a NAME="E5:6_6_7_3_1_2"/><b>then </b>
<font color="Maroon">R</font> <a href="relat_2.html#R4">is_antisymmetric_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#D5">WELLORD1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E6:6_6_7_3_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i>, , <a class="ref" href="wellset1.html#T1" target="_self">Th1</a>, <a class="ref" href="relat_2.html#D4">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">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i>, , , </i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E4:6_6_7"/><i><font color="Green">E262</font></i>: 
<font color="Maroon">W1</font> <a href="relat_2.html#R6">is_connected_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">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">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font> &amp; <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> &amp; not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:6_6_7_4_1"/><i><font color="Green">E263</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font>
 <b>and </b><br/><a NAME="E2:6_6_7_4_1"/><i><font color="Green">E264</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font>
 <b>and </b><br/><a NAME="E3:6_6_7_4_1"/><i><font color="Green">E265</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E4:6_6_7_4_1"/><i><font color="Green">E266</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> &amp; not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">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">E267</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> )
 ;<br/>

<a NAME="E2:6_6_7_4_1_1"/>
<font color="Maroon">R</font> <a href="wellord1.html#R2">well_orders</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 
<b>by </b><i>, <a class="ref" href="wellord1.html#T8">WELLORD1:8</a></i>;<br/>
<a NAME="E3:6_6_7_4_1_1"/><b>then </b>
<font color="Maroon">R</font> <a href="relat_2.html#R6">is_connected_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#D5">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">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 
<b>by </b><i>, , <a class="ref" href="relat_2.html#D6">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">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E5:6_6_7_4_1"/><i><font color="Green">E268</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">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">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 ;<br/>

<a NAME="E2:6_6_7_4_1_2"/><b>then </b><i><font color="Green">E269</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:6_6_7_4_1_2"/><i><font color="Green">E270</font></i>: 
(  not <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">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">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">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">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W4</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">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">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E4:6_6_7_4_1_2"/>
(  not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">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">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 ;<br/>

<a NAME="E2:6_6_7_4_1_2_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W3</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T129">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">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> )
 <b>by </b><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">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> )
 <b>by </b><i>, , <a class="ref" href="wellset1.html#T1" target="_self">Th1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:6_6_7_4_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">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">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 ;<br/>

<a NAME="E2:6_6_7_4_1_3"/><b>then </b><i><font color="Green">E271</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:6_6_7_4_1_3"/><i><font color="Green">E272</font></i>: 
(  not <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">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">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">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">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W4</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">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">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E4:6_6_7_4_1_3"/>
(  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">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">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 ;<br/>

<a NAME="E2:6_6_7_4_1_3_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W3</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T129">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">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> )
 <b>by </b><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">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> )
 <b>by </b><i>, , <a class="ref" href="wellset1.html#T1" target="_self">Th1</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">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> )
 <b>by </b><i>, </i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E5:6_6_7"/>
<font color="Maroon">W1</font> <a href="wellord1.html#R1">is_well_founded_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">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">Y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="wellord1.html#D3">WELLORD1:def 3</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:6_6_7_5"/><i><font color="Green">E273</font></i>: 
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font>
 <b>and </b><br/><a NAME="E2:6_6_7_5"/><i><font color="Green">E274</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E3:6_6_7_5"/>
<font color="Maroon">R</font> <a href="wellord1.html#R2">well_orders</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 
<b>by </b><i>, <a class="ref" href="wellord1.html#T8">WELLORD1:8</a></i>;<br/>
<a NAME="E4:6_6_7_5"/><b>then </b><i><font color="Green">E275</font></i>: 
<font color="Maroon">R</font> <a href="wellord1.html#R1">is_well_founded_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#D5">WELLORD1:def 5</a></i>;<br/>
<a NAME="E5:6_6_7_5"/><i><font color="Green">E276</font></i>: 
( <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> implies  ex <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Olive">a</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">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">E277</font></i>: 
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 ;<br/>

<b>then consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:6_6_7_5_1"/><i><font color="Green">E278</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">b</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Y</font> )
 <b>by </b><i>, , <a class="ref" href="wellord1.html#D3">WELLORD1:def 3</a></i>;<br/>
<b>take </b>
<font color="Maroon">b</font>
;<br/>

<b>thus </b><a NAME="E4:6_6_7_5_1"/>
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">b</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Y</font> )
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:6_6_7_5"/>
( not <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> implies  ex <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Olive">a</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">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">Y</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 ;<br/>

<a NAME="E2:6_6_7_5_2"/><i><font color="Green">E279</font></i>: 
( <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies  ex <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Olive">a</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">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">E280</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>;<br/>
<a NAME="E3:6_6_7_5_2_1"/><i><font color="Green">E281</font></i>: 
not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:6_6_7_5_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font>
 
<b>by </b><i>, , <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E5:6_6_7_5_2_1"/><b>then </b><i><font color="Green">E282</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="wellset1.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E6:6_6_7_5_2_1"/>
<font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">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">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:6_6_7_5_2_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:6_6_7_5_2_1_1"/><b>then </b><i><font color="Green">E283</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/>
<a NAME="E3:6_6_7_5_2_1_1"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>
<b>hence </b><a NAME="E4:6_6_7_5_2_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 <b>by </b><i>, <a class="txt" href="wellset1.html#E1:6"><i><font color="Green">E37</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">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="wellset1.html#T3" target="_self">Th3</a>, <a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/>
<a NAME="E8:6_6_7_5_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>
<a NAME="E9:6_6_7_5_2_1"/><b>then </b>
<font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<b>hence </b><a NAME="E10:6_6_7_5_2_1"/>
 ex <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Olive">a</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Y</font> )
 <b>by </b><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">field</a> <font color="Maroon">R</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies  ex <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Olive">a</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">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">E284</font></i>: 
not <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>set </b><font color="Maroon">X</font> = <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font>;<br/>
<a NAME="E2:6_6_7_5_2_2"/><i><font color="Green">E285</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:6_6_7_5_2_2"/><i><font color="Green">E286</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font> <a href="xboole_0.html#R1">misses</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font> )
 <b>by </b><i>, , <a class="ref" href="wellord1.html#D3">WELLORD1:def 3</a></i>;<br/>
<a NAME="E5:6_6_7_5_2_2"/><i><font color="Green">E287</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="wellset1.html#T3" target="_self">Th3</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:6_6_7_5_2_2"/><i><font color="Green">E288</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="wellset1.html#T3" target="_self">Th3</a>, <a class="ref" href="xboole_0.html#D7">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">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">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">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:6_6_7_5_2_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font>
 ;<br/>

<a NAME="E2:6_6_7_5_2_2_1"/><b>then </b><i><font color="Green">E289</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:6_6_7_5_2_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/>
<a NAME="E4:6_6_7_5_2_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>
<a NAME="E5:6_6_7_5_2_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="txt" href="wellset1.html#E4:6"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E6:6_6_7_5_2_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font></span>)</span>
 <b>by </b><i><a class="txt" href="wellset1.html#E4:6"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xboole_0.html#D3">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">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="wellset1.html#E3:6"><i><font color="Green">E38</font></i></a>, <a class="ref" href="xboole_1.html#T3">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">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="wellset1.html#T1" target="_self">Th1</a>, <a class="ref" href="wellset1.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E10:6_6_7_5_2_2"/><b>then </b>
<font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<b>hence </b><a NAME="E11:6_6_7_5_2_2"/>
 ex <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Olive">a</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Y</font> )
 <b>by </b><i><a class="txt" href="wellset1.html#E1:6"><i><font color="Green">E37</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">a</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Olive">a</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Y</font> )
 <b>by </b><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">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Y</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E6:6_6_7"/><b>then </b>
<font color="Maroon">W1</font> <a href="wellord1.html#R2">well_orders</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font>
 
<b>by </b><i>, , , , <a class="ref" href="wellord1.html#D5">WELLORD1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E7:6_6_7"/>
<font color="Maroon">W1</font> is <a href="wellord1.html#V2">well-ordering</a>
 <b>by </b><i><a class="ref" href="wellord1.html#T8">WELLORD1:8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E17:6_6"/>
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font> holds <br/>( <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Olive">y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">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">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:6_6_8"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font>
 ;<br/>

<a NAME="E2:6_6_8"/><b>then </b><i><font color="Green">E290</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:6_6_8"/><i><font color="Green">E291</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> implies <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">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">E292</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 ;<br/>

<a NAME="E2:6_6_8_1"/><i><font color="Green">E293</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:6_6_8_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E2:6_6_8_1_1"/><b>then </b><i><font color="Green">E294</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 
<b>by </b><i><a class="ref" href="wellord1.html#D1">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">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">W3</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W4</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E4:6_6_8_1_1"/><b>then </b><i><font color="Green">E295</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W3</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W4</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E5:6_6_8_1_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W4</font> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:6_6_8_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font>
 <b>by </b><i>, , , , <a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a>, <a class="ref" href="zfmisc_1.html#T129">ZFMISC_1:129</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:6_6_8_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:6_6_8_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E2:6_6_8_1_2"/><b>then </b><i><font color="Green">E296</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 
<b>by </b><i><a class="ref" href="wellord1.html#D1">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">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">W3</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">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">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E5:6_6_8_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:6_6_8_1"/>
<font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:6_6_8"/>
( <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">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">W1</font> <a href="wellord1.html#K1">-Seg</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">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">E297</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:6_6_8_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:6_6_8_2_1_1"/><b>then </b><i><font color="Green">E298</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/>
<a NAME="E3:6_6_8_2_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>
<b>hence </b><a NAME="E4:6_6_8_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:6_6_8_2_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:6_6_8_2_1_2"/><i><font color="Green">E299</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 ;<br/>

<a NAME="E2:6_6_8_2_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W3</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T129">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">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">W3</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">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">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E5:6_6_8_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:6_6_8_2_1"/>
<font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:6_6_8_2"/>
( <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:6_6_8"/>
( <font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:6_6"/><b>then </b>
<font color="Maroon">W1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E19:6_6"/><b>then </b><i><font color="Green">E300</font></i>: 
 <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 
<b>by </b><i/>;<br/>
<a NAME="E20:6_6"/><i><font color="Green">E301</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">W1</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E21:6_6"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E22:6_6"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K3">field</a> <font color="Maroon">R</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W1</font>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>
<b>hence </b><a NAME="E23:6_6"/>
contradiction
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E14:6"/>
( <font color="Maroon">R</font> is <a href="wellord1.html#V2">well-ordering</a> &amp; not  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; ( for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> holds <br/>( <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Olive">y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> ) ) )
 <b>by </b><i>, </i>;<br/>


</div>
