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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>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/>
<b>assume </b><a NAME="E1:23"/><i><font color="Green">E11</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> &amp; 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> &amp; the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> is <a href="relat_2.html#V8">transitive</a> )
 ;<br/>

<a NAME="E2:23"/><b>then </b><i><font color="Green">E12</font></i>: 
 <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> <a href="hidden.html#R1">=</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="ref" href="oposet_1.html#T7">OPOSET_1:7</a></i>;<br/>
<a NAME="E3:23"/><b>then </b><i><font color="Green">E13</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#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="ref" href="relat_2.html#D12">RELAT_2:def 12</a>, <a class="txt" href="robbins3.html#E1:23"><i><font color="Green">E11</font></i></a></i>;<br/>
<a NAME="E4:23"/>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R8">is_transitive_in</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="ref" href="relat_2.html#D16">RELAT_2:def 16</a>, <a class="txt" href="robbins3.html#E1:23"><i><font color="Green">E11</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:23"/>
( <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="orders_2.html#V4">antisymmetric</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="orders_2.html#V3">transitive</a> )
 <b>by </b><i><a class="txt" href="robbins3.html#E3:23"><i><font color="Green">E13</font></i></a>, <a class="txt" href="robbins3.html#E1:23"><i><font color="Green">E11</font></i></a>, <a class="txt" href="robbins3.html#E2:23"><i><font color="Green">E12</font></i></a>, <a class="ref" href="orders_2.html#D5">ORDERS_2:def 5</a>, <a class="ref" href="orders_2.html#D6">ORDERS_2:def 6</a>, <a class="ref" href="orders_2.html#D4">ORDERS_2:def 4</a></i>;<br/>


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