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<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a> <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</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"> for <font color="Olive" title="b1">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>  st  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Olive" title="b1">X</font>,<font color="Maroon" title="c1">L</font> holds <br/> <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Olive" title="b1">X</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">L</font> implies  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Maroon" title="c2">X</font></span>)</span> )</span><br/>



<a NAME="E1:71"/>
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c2">X</font> iff <font color="Olive" title="b1">x</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a>  <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Maroon" title="c2">X</font> )
 
<b>by </b><i><a class="ref" href="waybel_0.html#T31" target="_self" title="WAYBEL_0:th.31">Th31</a></i>;<br/>
<b>hence </b><a NAME="E2:71"/>
(  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">L</font> implies  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Maroon" title="c2">X</font></span>)</span> )
 <b>by </b><i><a class="ref" href="yellow_0.html#T47" title="YELLOW_0:th.47">YELLOW_0:47</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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