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<b>let </b><font color="Maroon">R</font> be   <a href="relat_1.html#NM1">Relation</a>;<br/>

<b>assume </b><a NAME="E1:13"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">R</font> is <a href="wellord1.html#V2" target="_self">well-ordering</a> &amp;  <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="E2:13"/><b>then </b><i><font color="Green">E20</font></i>: 
( <font color="Maroon">R</font> is <a href="relat_2.html#V6">connected</a> &amp; <font color="Maroon">R</font> is <a href="relat_2.html#V4">antisymmetric</a> &amp; <font color="Maroon">R</font> is <a href="relat_2.html#V1">reflexive</a> &amp; ( for <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Y</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> &amp; <font color="Olive">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/> 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="Olive">Y</font> &amp; ( 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> <font color="Olive">Y</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">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">R</font> ) ) ) )
 
<b>by </b><i>, </i>;<br/>
<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E3:13"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 ;<br/>

<b>given </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E4:13"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">b</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">b</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a</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">Z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:13"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> iff ( <font color="Olive">c</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">S<sub>1</sub></font>[<font color="Olive">c</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E7:13"/>
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> <font color="Maroon">Z</font> holds <br/><font color="Olive">b</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="wellord1.html#E1"><i><font color="Green">Lemma17</font></i></a></i>;<br/>
<a NAME="E8:13"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">Z</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="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E9:13"/>
<font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="txt" href="wellord1.html#E1"><i><font color="Green">Lemma17</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">d</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:13"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; ( for <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">d</font>,<font color="Olive">c</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="txt" href="wellord1.html#E2"><i><font color="Green">Lemma18</font></i></a>, </i>;<br/>
<b>take </b>
<font color="Maroon">d</font>
;<br/>

<b>thus </b><a NAME="E12:13"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">d</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="wellord1.html#E1"><i><font color="Green">Lemma17</font></i></a>, <a class="txt" href="wellord1.html#E1:13"><i><font color="Green">E19</font></i></a></i>;<br/>

<a NAME="E13:13"/><i><font color="Green">E40</font></i>: 
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">d</font>,<font color="Maroon">a</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="txt" href="wellord1.html#E1"><i><font color="Green">Lemma17</font></i></a>, <a class="txt" href="wellord1.html#E1:13"><i><font color="Green">E19</font></i></a></i>;<br/>
<a NAME="E14:13"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">d</font>
 
<b>by </b><i><a class="txt" href="wellord1.html#E1:13"><i><font color="Green">E19</font></i></a></i>;<br/>
<b>hence </b><a NAME="E15:13"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</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="txt" href="wellord1.html#E2:13"><i><font color="Green">E20</font></i></a>, <a class="txt" href="wellord1.html#E3"><i><font color="Green">Lemma21</font></i></a>, </i>;<br/>

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

<b>assume </b><a NAME="E16:13"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c</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="Maroon">a</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 ;<br/>

<b>assume </b><a NAME="E17:13"/>
<font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font>
 ;<br/>

<a NAME="E18:13"/><b>then </b>
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">a</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="txt" href="wellord1.html#E4"><i><font color="Green">Lemma22</font></i></a>, </i>;<br/>
<a NAME="E19:13"/><b>then </b>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 
<b>by </b><i><a class="txt" href="wellord1.html#E1"><i><font color="Green">Lemma17</font></i></a>, <a class="txt" href="wellord1.html#E4"><i><font color="Green">Lemma22</font></i></a></i>;<br/>
<b>hence </b><a NAME="E20:13"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">d</font>,<font color="Maroon">c</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="txt" href="wellord1.html#E1:13"><i><font color="Green">E19</font></i></a></i>;<br/>


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