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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#V2">reflexive</a> <a href="orders_2.html#V3">transitive</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<a NAME="E1:71"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">c<sub>2</sub></font> iff <font color="Olive">b<sub>1</sub></font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a>  <a href="waybel_0.html#K3" target="_self">downarrow</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="waybel_0.html#T31" target="_self">Th31</a></i>;<br/>
<b>hence </b><a NAME="E2:71"/>
(  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> implies  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K3" target="_self">downarrow</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="yellow_0.html#T47">YELLOW_0:47</a></i>;<br/>


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