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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">O</font> is  <a href="oposet_1.html#V4" title="OPOSET_1:attr.4">QuasiOrdered</a>  implies <font color="Maroon" title="c1">O</font> is  <a href="oposet_1.html#V3" title="OPOSET_1:attr.3">SubQuasiOrdered</a>  )</span><br/>

<span class="kw">set </span><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/>
<span class="kw">set </span><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/>
<span class="kw">assume </span><a NAME="E1:45"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
<font color="Maroon" title="c1">O</font> is  <a href="oposet_1.html#V4" title="OPOSET_1:attr.4">QuasiOrdered</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">O</font> is  <a href="oposet_1.html#V3" title="OPOSET_1:attr.3">SubQuasiOrdered</a> </span><br/>

<a NAME="E2:45"/><span class="kw">then </span>
<font color="Maroon" title="c1">O</font> is  <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="oposet_1.html#D12" target="_self" title="OPOSET_1:def.12">Def22</a></span>;<br/>
<a NAME="E3:45"/><span class="kw">then </span>
 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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="orders_2.html#D3" title="ORDERS_2:def.3">ORDERS_2:def 3</a></span>;<br/>
<a NAME="E4:45"/><span class="kw">then </span>
 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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partit_2.html#T27" title="PARTIT_2:th.27">PARTIT_2:27</a></span>;<br/>
<a NAME="E5:45"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A2</font></span>: 
<font color="Maroon" title="c1">O</font> is  <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="oposet_1.html#D10" target="_self" title="OPOSET_1:def.10">Def19</a></span>;<br/>
<a NAME="E6:45"/>
<font color="Maroon" title="c1">O</font> is  <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="oposet_1.html#D12" target="_self" title="OPOSET_1:def.12">Def22</a></span>;<br/>
<a NAME="E7:45"/><span class="kw">then </span>
 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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="orders_2.html#D2" title="ORDERS_2:def.2">ORDERS_2:def 2</a></span>;<br/>
<a NAME="E8:45"/><span class="kw">then </span>
 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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partit_2.html#T21" title="PARTIT_2:th.21">PARTIT_2:21</a></span>;<br/>
<a NAME="E9:45"/><span class="kw">then </span>
<font color="Maroon" title="c1">O</font> is  <a href="oposet_1.html#V2" title="OPOSET_1:attr.2">SubReFlexive</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="oposet_1.html#D4" target="_self" title="OPOSET_1:def.4">Def9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:45"/>
<font color="Maroon" title="c1">O</font> is  <a href="oposet_1.html#V3" title="OPOSET_1:attr.3">SubQuasiOrdered</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:45"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="oposet_1.html#D11" target="_self" title="OPOSET_1:def.11">Def21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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