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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>assume </b><b>that </b><br/><a NAME="E1:10"/><i><font color="Green" title="E19">A1</font></i>: 
<font color="Maroon" title="c1">R</font> is <a href="relat_2.html#V2" title="RELAT_2:attr.2">irreflexive</a>
 <b>and </b><br/><a NAME="E2:10"/><i><font color="Green" title="E20">A2</font></i>: 
<font color="Maroon" title="c1">R</font> is <a href="relat_2.html#V8" title="RELAT_2:attr.8">transitive</a>
 ;<br/>

<a NAME="E3:10"/><i><font color="Green" title="E21">A3</font></i>: 
<font color="Maroon" title="c1">R</font> <a href="relat_2.html#R2" title="RELAT_2:pred.2">is_irreflexive_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="relat_2.html#E1:10"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="relat_2.html#D10" target="_self" title="RELAT_2:def.10">Def10</a></i>;<br/>
<a NAME="E4:10"/><i><font color="Green" title="E22">A4</font></i>: 
<font color="Maroon" title="c1">R</font> <a href="relat_2.html#R8" title="RELAT_2:pred.8">is_transitive_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="relat_2.html#E2:10"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="relat_2.html#D16" target="_self" title="RELAT_2:def.16">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" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:10_1"/><i><font color="Green" title="E23">A5</font></i>: 
( <font color="Maroon" title="c2">a</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> &amp; <font color="Maroon" title="c3">b</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> )
 ;<br/><a NAME="E2:10_1"/><b>then </b>
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</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>
 <b>by </b><i><a class="txt" href="relat_2.html#E3:10"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="relat_2.html#D2" target="_self" title="RELAT_2:def.2">Def2</a></i>;<br/><b>hence </b><a NAME="E3:10_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</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> implies  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</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> )
 <b>by </b><i><a class="txt" href="relat_2.html#E4:10"><i><font color="Green" title="E22">A4</font></i></a>, <a class="txt" href="relat_2.html#E1:10_1"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="relat_2.html#D8" target="_self" title="RELAT_2:def.8">Def8</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:10"/><b>then </b>
<font color="Maroon" title="c1">R</font> <a href="relat_2.html#R5" title="RELAT_2:pred.5">is_asymmetric_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="ref" href="relat_2.html#D5" target="_self" title="RELAT_2:def.5">Def5</a></i>;<br/>
<b>hence </b><a NAME="E7:10"/>
<font color="Maroon" title="c1">R</font> is <a href="relat_2.html#V5" title="RELAT_2:attr.5">asymmetric</a>
 <b>by </b><i><a class="ref" href="relat_2.html#D13" target="_self" title="RELAT_2:def.13">Def13</a></i>;<br/>


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