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<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">O</font> is  <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a>  implies <font color="Maroon" title="c1">O</font> is  <a href="oposet_1.html#V6" title="OPOSET_1:attr.6">SubAntisymmetric</a>  )</span><br/>

<span class="kw">set </span><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/>
<span class="kw">assume </span><a NAME="E1:54"/>
<font color="Maroon" title="c1">O</font> is  <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">O</font> is  <a href="oposet_1.html#V6" title="OPOSET_1:attr.6">SubAntisymmetric</a> </span><br/>

<a NAME="E2:54"/><span class="kw">then </span>
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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="orders_2.html#D6" title="ORDERS_2:def.6">ORDERS_2:def 6</a></span>;<br/>
<a NAME="E3:54"/><span class="kw">then </span>
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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="oposet_1.html#T11" target="_self" title="OPOSET_1:th.11">Th11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:54"/>
<font color="Maroon" title="c1">O</font> is  <a href="oposet_1.html#V6" title="OPOSET_1:attr.6">SubAntisymmetric</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="oposet_1.html#D15" target="_self" title="OPOSET_1:def.15">Def15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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