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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><a NAME="E1:11"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="relat_2.html#V5" target="_self">asymmetric</a>
 ;<br/>

<a NAME="E2:11"/><b>then </b><i><font color="Green">E19</font></i>: 
<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#D13" target="_self">Def13</a></i>;<br/>
<a NAME="E3:11"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</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> holds <br/> not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>1</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="ref" href="relat_2.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E4:11"/><b>then </b>
<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="ref" href="relat_2.html#D2" target="_self">Def2</a></i>;<br/>
<b>hence </b><a NAME="E5:11"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="relat_2.html#V2" target="_self">irreflexive</a>
 <b>by </b><i><a class="ref" href="relat_2.html#D10" target="_self">Def10</a></i>;<br/>

<a NAME="E6:11"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</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="Olive">b<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; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<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> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</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> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="relat_2.html#E2:11"><i><font color="Green">E19</font></i></a>, <a class="ref" href="relat_2.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E7:11"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R4" target="_self">is_antisymmetric_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#D4" target="_self">Def4</a></i>;<br/>
<b>hence </b><a NAME="E8:11"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="relat_2.html#V4" target="_self">antisymmetric</a>
 <b>by </b><i><a class="ref" href="relat_2.html#D12" target="_self">Def12</a></i>;<br/>


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