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<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V2">reflexive</a> <a href="orders_2.html#V4">antisymmetric</a> <a href="waybel_0.html#V24">up-complete</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/>

<b>assume </b><a NAME="E1:66"/><i><font color="Green">E29</font></i>: 
 <a href="waybel_2.html#K4" target="_self">inf_op</a> <font color="Maroon">L</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 ;<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/>

<b>set </b><font color="Maroon">X</font> = <span class="p1"><a href="yellow_3.html#K10">[:</a><span class="default"><font color="Maroon">D1</font>,<font color="Maroon">D2</font></span><a href="yellow_3.html#K10">:]</a></span>;<br/>
<b>set </b><font color="Maroon">f</font> =  <a href="waybel_2.html#K4" target="_self">inf_op</a> <font color="Maroon">L</font>;<br/>
<a NAME="E2:66"/><i><font color="Green">E32</font></i>: 
 <a href="waybel_2.html#K4" target="_self">inf_op</a> <font color="Maroon">L</font> <a href="waybel_0.html#R4">preserves_sup_of</a> <span class="p1"><a href="yellow_3.html#K10">[:</a><span class="default"><font color="Maroon">D1</font>,<font color="Maroon">D2</font></span><a href="yellow_3.html#K10">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#D37">WAYBEL_0:def 37</a></i>;<br/>
<a NAME="E3:66"/><i><font color="Green">E34</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1"><a href="yellow_3.html#K10">[:</a><span class="default"><font color="Maroon">D1</font>,<font color="Maroon">D2</font></span><a href="yellow_3.html#K10">:]</a></span>,<span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">L</font>,<font color="Maroon">L</font></span><a href="yellow_3.html#K3">:]</a></span>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T75">WAYBEL_0:75</a></i>;<br/>
<a NAME="E4:66"/><i><font color="Green">E35</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> <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/>
<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#K11">"/\"</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="waybel_2.html#K4" target="_self">inf_op</a> <font color="Maroon">L</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D1</font></span>)</span>,<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/>
<br/>.= 
<span class="p1">(<span class="default"><a href="waybel_2.html#K4" target="_self">inf_op</a> <font color="Maroon">L</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p2"><a href="yellow_3.html#K10">[:</a><span class="default"><font color="Maroon">D1</font>,<font color="Maroon">D2</font></span><a href="yellow_3.html#K10">:]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="waybel_2.html#T4" target="_self">Th4</a>, <a class="ref" href="yellow_3.html#T43">YELLOW_3:43</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_2.html#K4" target="_self">inf_op</a> <font color="Maroon">L</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p2"><a href="yellow_3.html#K10">[:</a><span class="default"><font color="Maroon">D1</font>,<font color="Maroon">D2</font></span><a href="yellow_3.html#K10">:]</a></span></span>)</span>
<b>by </b><i>, , <a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</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#K3">"/\"</a> <font color="Maroon">D2</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel_2.html#E1:66"><i><font color="Green">E29</font></i></a></i>
;<br/>


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