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<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="waybel_0.html#V24" title="WAYBEL_0:attr.24">up-complete</a> <a href="waybel_0.html#NM7" title="WAYBEL_0:NM.7">sup-Semilattice</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">A</font>, <font color="Olive" title="b2">B</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> holds   <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="yellow_4.html#K2" title="YELLOW_4:func.2">"\/"</a> <font color="Olive" title="b2">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Olive" title="b2">B</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> <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#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="yellow_4.html#K2" title="YELLOW_4:func.2">"\/"</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c3">B</font></span>)</span></span><br/>



<a NAME="E1:7"/><i><font color="Green" title="E4">A1</font></i>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c2">A</font> <a href="yellow_4.html#K2" title="YELLOW_4:func.2">"\/"</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T75" title="WAYBEL_0:th.75">WAYBEL_0:75</a></i>;<br/>
<a NAME="E2:7"/><i><font color="Green" title="E5">A2</font></i>: 
(  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c1">L</font> &amp;  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c1">L</font> )
 
<b>by </b><i><a class="ref" href="waybel_0.html#T75" title="WAYBEL_0:th.75">WAYBEL_0:75</a></i>;<br/>
<a NAME="E3:7"/><b>then </b>
( <font color="Maroon" title="c2">A</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c3">B</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c3">B</font> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T30" title="YELLOW_0:th.30">YELLOW_0:30</a></i>;<br/>
<a NAME="E4:7"/><b>then </b>
<font color="Maroon" title="c2">A</font> <a href="yellow_4.html#K2" title="YELLOW_4:func.2">"\/"</a> <font color="Maroon" title="c3">B</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c3">B</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_4.html#T27" title="YELLOW_4:th.27">YELLOW_4:27</a></i>;<br/>
<a NAME="E5:7"/><b>then </b><i><font color="Green" title="E6">A3</font></i>: 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="yellow_4.html#K2" title="YELLOW_4:func.2">"\/"</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c3">B</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:7"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="yellow_0.html#D9" title="YELLOW_0:def.9">YELLOW_0:def 9</a></i>;<br/>
<a NAME="E6:7"/>
( <font color="Maroon" title="c2">A</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="yellow_4.html#K2" title="YELLOW_4:func.2">"\/"</a> <font color="Maroon" title="c3">B</font></span>)</span> &amp; <font color="Maroon" title="c3">B</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="yellow_4.html#K2" title="YELLOW_4:func.2">"\/"</a> <font color="Maroon" title="c3">B</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="waybel_2.html#T3" target="_self" title="WAYBEL_2:th.3">Th3</a></i>;<br/>
<a NAME="E7:7"/><b>then </b>
(  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c2">A</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="yellow_4.html#K2" title="YELLOW_4:func.2">"\/"</a> <font color="Maroon" title="c3">B</font></span>)</span> &amp;  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c3">B</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="yellow_4.html#K2" title="YELLOW_4:func.2">"\/"</a> <font color="Maroon" title="c3">B</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E2:7"><i><font color="Green" title="E5">A2</font></i></a>, <a class="ref" href="yellow_0.html#T30" title="YELLOW_0:th.30">YELLOW_0:30</a></i>;<br/>
<a NAME="E8:7"/><b>then </b><i><font color="Green" title="E7">A4</font></i>: 
<span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="yellow_4.html#K2" title="YELLOW_4:func.2">"\/"</a> <font color="Maroon" title="c3">B</font></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="yellow_4.html#K2" title="YELLOW_4:func.2">"\/"</a> <font color="Maroon" title="c3">B</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_3.html#T3" title="YELLOW_3:th.3">YELLOW_3:3</a></i>;<br/>
<a NAME="E9:7"/>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="yellow_4.html#K2" title="YELLOW_4:func.2">"\/"</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="yellow_4.html#K2" title="YELLOW_4:func.2">"\/"</a> <font color="Maroon" title="c3">B</font></span>)</span>
 
;<br/>
<a NAME="E10:7"/><b>then </b>
<span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="yellow_4.html#K2" title="YELLOW_4:func.2">"\/"</a> <font color="Maroon" title="c3">B</font></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="yellow_4.html#K2" title="YELLOW_4:func.2">"\/"</a> <font color="Maroon" title="c3">B</font></span>)</span></span>)</span> <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"><font color="Maroon" title="c2">A</font> <a href="yellow_4.html#K2" title="YELLOW_4:func.2">"\/"</a> <font color="Maroon" title="c3">B</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T24" title="YELLOW_0:th.24">YELLOW_0:24</a></i>;<br/>
<b>hence </b><a NAME="E11:7"/>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="yellow_4.html#K2" title="YELLOW_4:func.2">"\/"</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c3">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E5:7"><i><font color="Green" title="E6">A3</font></i></a>, <a class="txt" href="#E8:7"><i><font color="Green" title="E7">A4</font></i></a>, <a class="ref" href="orders_2.html#T25" title="ORDERS_2:th.25">ORDERS_2:25</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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