<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="waybel_0.html#V24">up-complete</a> <a href="waybel_0.html#NM7">sup-Semilattice</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></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 <font color="Maroon">c<sub>1</sub></font>;<br/>



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


</div>
