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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="relat_1.html#NM1">Relation</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:10"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="relat_2.html#V2" target="_self">irreflexive</a>
 <b>and </b><br/><a NAME="E2:10"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="relat_2.html#V8" target="_self">transitive</a>
 ;<br/>

<a NAME="E3:10"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R2" target="_self">is_irreflexive_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="relat_2.html#E1:10"><i><font color="Green">E19</font></i></a>, <a class="ref" href="relat_2.html#D10" target="_self">Def10</a></i>;<br/>
<a NAME="E4:10"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R8" target="_self">is_transitive_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="relat_2.html#E2:10"><i><font color="Green">E20</font></i></a>, <a class="ref" href="relat_2.html#D16" target="_self">Def16</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:10_1"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><a NAME="E2:10_1"/><b>then </b>
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="relat_2.html#E3:10"><i><font color="Green">E21</font></i></a>, <a class="ref" href="relat_2.html#D2" target="_self">Def2</a></i>;<br/><b>hence </b><a NAME="E3:10_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> implies  not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="relat_2.html#E4:10"><i><font color="Green">E22</font></i></a>, <a class="txt" href="relat_2.html#E1:10_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="relat_2.html#D8" target="_self">Def8</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:10"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R5" target="_self">is_asymmetric_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="relat_2.html#D5" target="_self">Def5</a></i>;<br/>
<b>hence </b><a NAME="E7:10"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="relat_2.html#V5" target="_self">asymmetric</a>
 <b>by </b><i><a class="ref" href="relat_2.html#D13" target="_self">Def13</a></i>;<br/>


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