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<b>let </b><font color="Maroon" title="c1">O</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/>

<b>set </b><font color="Maroon" title="c2">RO</font> = the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">O</font>;<br/>
<b>assume </b><a NAME="E1:52"/>
<font color="Maroon" title="c1">O</font> is <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a>
 ;<br/>

<a NAME="E2:52"/><b>then </b>
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">O</font> <a href="relat_2.html#R4" title="RELAT_2:pred.4">is_antisymmetric_in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">O</font>
 
<b>by </b><i><a class="ref" href="orders_2.html#D6" title="ORDERS_2:def.6">ORDERS_2:def 6</a></i>;<br/>
<a NAME="E3:52"/><b>then </b>
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">O</font> is <a href="relat_2.html#V4" title="RELAT_2:attr.4">antisymmetric</a>
 
<b>by </b><i><a class="ref" href="oposet_1.html#T11" target="_self" title="OPOSET_1:th.11">Th11</a></i>;<br/>
<b>hence </b><a NAME="E4:52"/>
<font color="Maroon" title="c1">O</font> is <a href="oposet_1.html#V6" title="OPOSET_1:attr.6">SubAntisymmetric</a>
 <b>by </b><i><a class="ref" href="oposet_1.html#D15" target="_self" title="OPOSET_1:def.15">Def15</a></i>;<br/>


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