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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">O</font> is <a href="oposet_1.html#V10" title="OPOSET_1:attr.10">QuasiOrdered</a> implies <font color="Maroon" title="c1">O</font> is <a href="oposet_1.html#V9" title="OPOSET_1:attr.9">SubQuasiOrdered</a> )</span><br/>

<b>assume </b><a NAME="E1:85"/><i><font color="Green" title="E48">A1</font></i>: 
<font color="Maroon" title="c1">O</font> is <a href="oposet_1.html#V10" title="OPOSET_1:attr.10">QuasiOrdered</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">O</font> is <a href="oposet_1.html#V9" title="OPOSET_1:attr.9">SubQuasiOrdered</a></span><br/>

<b>set </b><font color="Maroon" title="c2">IntRel</font> = the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">O</font>;<br/>
<b>set </b><font color="Maroon" title="c3">CO</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">O</font>;<br/>
<a NAME="E2:85"/><i><font color="Green" title="E49">A2</font></i>: 
( <font color="Maroon" title="c1">O</font> is <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a> &amp; <font color="Maroon" title="c1">O</font> is <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a> )
 
<b>by </b><i><a class="txt" href="#E1:85"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="oposet_1.html#D22" target="_self" title="OPOSET_1:def.22">Def22</a></i>;<br/>
<a NAME="E3:85"/><b>then </b>
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#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">O</font>
 
<b>by </b><i><a class="ref" href="orders_2.html#D4" title="ORDERS_2:def.4">ORDERS_2:def 4</a></i>;<br/>
<a NAME="E4:85"/><b>then </b><i><font color="Green" title="E50">A3</font></i>: 
( 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#V1" title="RELAT_2:attr.1">reflexive</a> &amp;  <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">O</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">O</font> )
 
<b>by </b><i><a class="ref" href="oposet_1.html#T7" target="_self" title="OPOSET_1:th.7">Th7</a></i>;<br/>
<a NAME="E5:85"/>
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#R8" title="RELAT_2:pred.8">is_transitive_in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">O</font>
 
<b>by </b><i><a class="txt" href="#E2:85"><i><font color="Green" title="E49">A2</font></i></a>, <a class="ref" href="orders_2.html#D5" title="ORDERS_2:def.5">ORDERS_2:def 5</a></i>;<br/>
<a NAME="E6:85"/><b>then </b><i><font color="Green" title="E51">A4</font></i>: 
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#V8" title="RELAT_2:attr.8">transitive</a>
 
<b>by </b><i><a class="ref" href="oposet_1.html#T13" target="_self" title="OPOSET_1:th.13">Th13</a></i>;<br/>
<a NAME="E7:85"/><i><font color="Green" title="E52">A5</font></i>: 
<font color="Maroon" title="c1">O</font> is <a href="oposet_1.html#V3" title="OPOSET_1:attr.3">SubReFlexive</a>
 
<b>by </b><i><a class="txt" href="#E4:85"><i><font color="Green" title="E50">A3</font></i></a>, <a class="ref" href="oposet_1.html#D9" target="_self" title="OPOSET_1:def.9">Def9</a></i>;<br/>
<a NAME="E8:85"/>
<font color="Maroon" title="c1">O</font> is <a href="oposet_1.html#V8" title="OPOSET_1:attr.8">SubTransitive</a>
 
<b>by </b><i><a class="txt" href="#E6:85"><i><font color="Green" title="E51">A4</font></i></a>, <a class="ref" href="oposet_1.html#D19" target="_self" title="OPOSET_1:def.19">Def19</a></i>;<br/>
<b>hence </b><a NAME="E9:85"/>
<font color="Maroon" title="c1">O</font> is <a href="oposet_1.html#V9" title="OPOSET_1:attr.9">SubQuasiOrdered</a>
 <b>by </b><i><a class="txt" href="#E7:85"><i><font color="Green" title="E52">A5</font></i></a>, <a class="ref" href="oposet_1.html#D21" target="_self" title="OPOSET_1:def.21">Def21</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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