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<b>let </b><font color="Maroon" title="c1">r</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">i</font> = the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">r</font>;<br/>
<b>set </b><font color="Maroon" title="c3">c</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">r</font>;<br/>
<b>assume </b><a NAME="E1:23"/><i><font color="Green" title="E11">A1</font></i>: 
( the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">r</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">r</font> &amp; the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">r</font> is <a href="relat_2.html#V4" title="RELAT_2:attr.4">antisymmetric</a> &amp; the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">r</font> is <a href="relat_2.html#V8" title="RELAT_2:attr.8">transitive</a> )
 ;<br/>

<a NAME="E2:23"/><b>then </b><i><font color="Green" title="E12">A2</font></i>: 
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">r</font>
 
<b>by </b><i><a class="ref" href="oposet_1.html#T7" title="OPOSET_1:th.7">OPOSET_1:7</a></i>;<br/>
<a NAME="E3:23"/><b>then </b><i><font color="Green" title="E13">A3</font></i>: 
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">r</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">r</font>
 
<b>by </b><i><a class="txt" href="robbins3.html#E1:23"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="relat_2.html#D12" title="RELAT_2:def.12">RELAT_2:def 12</a></i>;<br/>
<a NAME="E4:23"/>
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">r</font>
 
<b>by </b><i><a class="txt" href="robbins3.html#E1:23"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="relat_2.html#D16" title="RELAT_2:def.16">RELAT_2:def 16</a></i>;<br/>
<b>hence </b><a NAME="E5:23"/>
( <font color="Maroon" title="c1">r</font> is <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a> &amp; <font color="Maroon" title="c1">r</font> is <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a> &amp; <font color="Maroon" title="c1">r</font> is <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a> )
 <b>by </b><i><a class="txt" href="robbins3.html#E1:23"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="robbins3.html#E2:23"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="robbins3.html#E3:23"><i><font color="Green" title="E13">A3</font></i></a>, <a class="ref" href="orders_2.html#D4" title="ORDERS_2:def.4">ORDERS_2:def 4</a>, <a class="ref" href="orders_2.html#D5" title="ORDERS_2:def.5">ORDERS_2:def 5</a>, <a class="ref" href="orders_2.html#D6" title="ORDERS_2:def.6">ORDERS_2:def 6</a></i>;<br/>


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