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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="E1:86"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="oposet_1.html#V11" target="_self">QuasiOrdered</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="E2:86"/><i><font color="Green">E50</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="orders_2.html#V3">transitive</a> )
 
<b>by </b><i><a class="txt" href="oposet_1.html#E1:86"><i><font color="Green">E49</font></i></a>, <a class="ref" href="oposet_1.html#D22" target="_self">Def22</a></i>;<br/>
<a NAME="E3:86"/><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#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="ref" href="orders_2.html#D4">ORDERS_2:def 4</a></i>;<br/>
<a NAME="E4:86"/><b>then </b><i><font color="Green">E51</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#V1">reflexive</a> &amp;  <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" target="_self">Th7</a></i>;<br/>
<a NAME="E5:86"/>
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> 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#E2:86"><i><font color="Green">E50</font></i></a>, <a class="ref" href="orders_2.html#D5">ORDERS_2:def 5</a></i>;<br/>
<a NAME="E6:86"/><b>then </b><i><font color="Green">E52</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#V8">transitive</a>
 
<b>by </b><i><a class="ref" href="oposet_1.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E7:86"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="oposet_1.html#V4" target="_self">SubReFlexive</a>
 
<b>by </b><i><a class="txt" href="oposet_1.html#E4:86"><i><font color="Green">E51</font></i></a>, <a class="ref" href="oposet_1.html#D9" target="_self">Def9</a></i>;<br/>
<a NAME="E8:86"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="oposet_1.html#V9" target="_self">SubTransitive</a>
 
<b>by </b><i><a class="txt" href="oposet_1.html#E6:86"><i><font color="Green">E52</font></i></a>, <a class="ref" href="oposet_1.html#D19" target="_self">Def19</a></i>;<br/>
<b>hence </b><a NAME="E9:86"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="oposet_1.html#V10" target="_self">SubQuasiOrdered</a>
 <b>by </b><i><a class="txt" href="oposet_1.html#E7:86"><i><font color="Green">E53</font></i></a>, <a class="ref" href="oposet_1.html#D21" target="_self">Def21</a></i>;<br/>


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