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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#NM7">sup-Semilattice</a>;<br/><b>let </b><font color="Maroon">A</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">B</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="E1:6"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">A</font> <a href="yellow_4.html#K2">"\/"</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">y</font></span>)</span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="yellow_4.html#D3">YELLOW_4:def 3</a></i>;<br/>
<a NAME="E2:6"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">A</font> <a href="yellow_4.html#K2">"\/"</a> <font color="Maroon">B</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="E3:6"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">A</font> <a href="yellow_4.html#K2">"\/"</a> <font color="Maroon">B</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">A</font> <a href="yellow_4.html#K2">"\/"</a> <font color="Maroon">B</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D9">YELLOW_0:def 9</a></i>;<br/>
<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a><br/>

<b>assume </b><a NAME="E4:6"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<b>consider </b><font color="Maroon">b</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">B</font>;<br/>
<a NAME="E6:6"/>
<font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="yellow_4.html#K2">"\/"</a> <font color="Maroon">B</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:6"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</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">A</font> <a href="yellow_4.html#K2">"\/"</a> <font color="Maroon">B</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/>
<a NAME="E8:6"/>
 ex <font color="Olive">xx</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">xx</font> &amp; <font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">xx</font> &amp; ( for <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">c</font> &amp; <font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">c</font> holds <br/><font color="Olive">xx</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">c</font> ) )
 
<b>by </b><i><a class="ref" href="lattice3.html#D10">LATTICE3:def 10</a></i>;<br/>
<a NAME="E9:6"/><b>then </b>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D13">LATTICE3:def 13</a></i>;<br/>
<b>hence </b><a NAME="E10:6"/>
<font color="Maroon">x</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">A</font> <a href="yellow_4.html#K2">"\/"</a> <font color="Maroon">B</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a></i>;<br/>


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