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<b>let </b><font color="Maroon">R</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">D</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">R</font></span>)</span></span>)</span>;<br/>



<a NAME="E1:6"/><i><font color="Green">E32</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">D</font>, <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">R</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T75">WAYBEL_0:75</a></i>;<br/>
<b>reconsider </b><font color="Maroon">UD</font> =  <a href="tarski.html#K3">union</a> <font color="Maroon">D</font> as   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">R</font> <b>by </b><i/>;<br/>
<a NAME="E3:6"/>
<font color="Maroon">UD</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K7">Ids</a> <font color="Maroon">R</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">UD</font> = <font color="Maroon">UD</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">R</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/>
<a NAME="E5:6"/>
( <font color="Maroon">D</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">UD</font> &amp; ( for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">R</font></span>)</span></span>)</span>  st <font color="Olive">b</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">D</font> holds <br/><font color="Maroon">UD</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b</font> ) )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:6"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">D</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#D9">YELLOW_0:def 9</a></i>;<br/>


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