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<b>let </b><font color="Maroon" title="c1">R</font> be   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a>;<br/><b>let </b><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:77"/><i><font color="Green" title="E47">A1</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="orders_1.html#R8" title="ORDERS_1:pred.8">is_superior_of</a> <font color="Maroon" title="c1">R</font>
 <b>and </b><br/><a NAME="E2:77"/><i><font color="Green" title="E48">A2</font></i>: 
<font color="Maroon" title="c1">R</font> is <a href="relat_2.html#V4" title="RELAT_2:attr.4">antisymmetric</a>
 ;<br/>

<b>thus </b><a NAME="E3:77"/><i><font color="Green" title="E49">A3</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="orders_1.html#E1:77"><i><font color="Green" title="E47">A1</font></i></a>, <a class="ref" href="orders_1.html#D13" target="_self" title="ORDERS_1:def.13">Def13</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="orders_1.html#D11" target="_self" title="ORDERS_1:def.11">ORDERS_1:def 11</a><br/>

<b>let </b><font color="Maroon" title="c3">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E4:77"/><i><font color="Green" title="E50">A4</font></i>: 
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font>
 <b>and </b><br/><a NAME="E5:77"/><i><font color="Green" title="E51">A5</font></i>: 
<font color="Maroon" title="c3">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font>
 <b>and </b><br/><a NAME="E6:77"/><i><font color="Green" title="E52">A6</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font>
 ;<br/>

<a NAME="E7:77"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c2">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c1">R</font> <a href="relat_2.html#R4" title="RELAT_2:pred.4">is_antisymmetric_in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font> )
 
<b>by </b><i><a class="txt" href="orders_1.html#E1:77"><i><font color="Green" title="E47">A1</font></i></a>, <a class="txt" href="orders_1.html#E2:77"><i><font color="Green" title="E48">A2</font></i></a>, <a class="txt" href="orders_1.html#E4:77"><i><font color="Green" title="E50">A4</font></i></a>, <a class="txt" href="orders_1.html#E5:77"><i><font color="Green" title="E51">A5</font></i></a>, <a class="ref" href="orders_1.html#D13" target="_self" title="ORDERS_1:def.13">Def13</a>, <a class="ref" href="relat_2.html#D12" title="RELAT_2:def.12">RELAT_2:def 12</a></i>;<br/>
<b>hence </b><a NAME="E8:77"/>
contradiction
 <b>by </b><i><a class="txt" href="orders_1.html#E3:77"><i><font color="Green" title="E49">A3</font></i></a>, <a class="txt" href="orders_1.html#E4:77"><i><font color="Green" title="E50">A4</font></i></a>, <a class="txt" href="orders_1.html#E5:77"><i><font color="Green" title="E51">A5</font></i></a>, <a class="txt" href="orders_1.html#E6:77"><i><font color="Green" title="E52">A6</font></i></a>, <a class="ref" href="relat_2.html#D4" title="RELAT_2:def.4">RELAT_2:def 4</a></i>;<br/>


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