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<b>let </b><font color="Maroon">L</font> be  <a href="waybel_0.html#V24">up-complete</a> <a href="waybel_0.html#NM6">Semilattice</a>;<br/>

<b>assume </b><a NAME="E1:63"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">I1</font>, <font color="Olive">I2</font> being  <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> holds  <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I1</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I2</font></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="Olive">I1</font> <a href="yellow_4.html#K4">"/\"</a> <font color="Olive">I2</font></span>)</span>
 ;<br/>

<b>let </b><font color="Maroon">D1</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">L</font>, <font color="Maroon">D2</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">L</font>;<br/>

<a NAME="E2:63"/><i><font color="Green">E32</font></i>: 
(  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">D1</font>,<font color="Maroon">L</font> &amp;  <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">D1</font>,<font color="Maroon">L</font> )
 
<b>by </b><i><a class="ref" href="waybel_0.html#T75">WAYBEL_0:75</a></i>;<br/>
<a NAME="E3:63"/><i><font color="Green">E34</font></i>: 
(  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">D2</font>,<font color="Maroon">L</font> &amp;  <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">D2</font>,<font color="Maroon">L</font> )
 
<b>by </b><i><a class="ref" href="waybel_0.html#T75">WAYBEL_0:75</a></i>;<br/>
<a NAME="E4:63"/><i><font color="Green">E35</font></i>: 
(  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">D1</font> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">D2</font>,<font color="Maroon">L</font> &amp;  <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="waybel_0.html#K3">downarrow</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">D2</font></span>)</span>,<font color="Maroon">L</font> )
 
<b>by </b><i><a class="ref" href="waybel_0.html#T75">WAYBEL_0:75</a></i>;<br/>
<a NAME="E5:63"/><i><font color="Green">E36</font></i>: 
(  <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">D1</font></span>)</span> <a href="yellow_4.html#K4">"/\"</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">D2</font></span>)</span>,<font color="Maroon">L</font> &amp;  <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="waybel_0.html#K3">downarrow</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">D1</font></span>)</span> <a href="yellow_4.html#K4">"/\"</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">D2</font></span>)</span></span>)</span>,<font color="Maroon">L</font> )
 
<b>by </b><i><a class="ref" href="waybel_0.html#T75">WAYBEL_0:75</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D1</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">D1</font></span>)</span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D2</font></span>)</span>
<b>by </b><i><a class="ref" href="waybel_2.html#T3" target="_self">Th3</a>, <a class="ref" href="waybel_0.html#T33">WAYBEL_0:33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">D1</font></span>)</span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">D2</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="waybel_0.html#T33">WAYBEL_0:33</a></i>
<br/>.= 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">D1</font></span>)</span> <a href="yellow_4.html#K4">"/\"</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">D2</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K3">downarrow</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">D1</font></span>)</span> <a href="yellow_4.html#K4">"/\"</a> <span class="p3">(<span class="default"><a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">D2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="waybel_0.html#T33">WAYBEL_0:33</a></i>
<br/>.= 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K3">downarrow</a> <span class="p2">(<span class="default"><font color="Maroon">D1</font> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">D2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="yellow_4.html#T62">YELLOW_4:62</a></i>
<br/>.= 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">D2</font></span>)</span>
<b>by </b><i>, <a class="ref" href="waybel_0.html#T33">WAYBEL_0:33</a></i>
;<br/>


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