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<b>let </b><font color="Maroon">R</font> be   <a href="relat_1.html#NM1">Relation</a>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:77"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">x</font> <a href="orders_1.html#R8" target="_self">is_superior_of</a> <font color="Maroon">R</font>
 <b>and </b><br/><a NAME="E2:77"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">R</font> is <a href="relat_2.html#V4">antisymmetric</a>
 ;<br/>

<b>thus </b><a NAME="E3:77"/><i><font color="Green">E20</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>by </b><i>, </i>; <i><font color="Red">:: according to </font></i><a class="ref" href="orders_1.html#D11" target="_self">ORDERS_1:def 11</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="E4:77"/><i><font color="Green">E24</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="E5:77"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E6:77"/><i><font color="Green">E27</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>
 ;<br/>

<a NAME="E7:77"/>
( <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">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#D12">RELAT_2:def 12</a></i>;<br/>
<b>hence </b><a NAME="E8:77"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="relat_2.html#D4">RELAT_2:def 4</a></i>;<br/>


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