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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/>

<b>assume </b><a NAME="E1:74"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="orders_2.html#V2">reflexive</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="oposet_1.html#V7" target="_self">SubAntisymmetric</a> )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> = the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E2:74"/><i><font color="Green">E44</font></i>: 
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R1">is_reflexive_in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="oposet_1.html#E1:74"><i><font color="Green">E43</font></i></a>, <a class="ref" href="orders_2.html#D4">ORDERS_2:def 4</a></i>;<br/>
<a NAME="E3:74"/><i><font color="Green">E45</font></i>: 
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> is <a href="relat_2.html#V4">antisymmetric</a>
 
<b>by </b><i><a class="txt" href="oposet_1.html#E1:74"><i><font color="Green">E43</font></i></a>, <a class="ref" href="oposet_1.html#D15" target="_self">Def15</a></i>;<br/>
<a NAME="E4:74"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K3">field</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="oposet_1.html#E2:74"><i><font color="Green">E44</font></i></a>, <a class="ref" href="oposet_1.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E5:74"/><b>then </b>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R4">is_antisymmetric_in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="oposet_1.html#E3:74"><i><font color="Green">E45</font></i></a>, <a class="ref" href="oposet_1.html#T10" target="_self">Th10</a></i>;<br/>
<b>hence </b><a NAME="E6:74"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="orders_2.html#V4">antisymmetric</a>
 <b>by </b><i><a class="ref" href="orders_2.html#D6">ORDERS_2:def 6</a></i>;<br/>


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