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

<b>let </b><font color="Maroon">L</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> <a href="yellow_0.html#V4">full</a> <a href="yellow_0.html#V8">sups-inheriting</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">L</font>;<br/>



<a NAME="E1:38"/>
for <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> holds   <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Olive">X</font>,<font color="Maroon">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>;<br/>

<a NAME="E1:38_1"/><i><font color="Green">E24</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">X</font>,<font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<a NAME="E2:38_1"/><b>then </b>
 <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D19">YELLOW_0:def 19</a></i>;<br/>
<b>hence </b><a NAME="E3:38_1"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">X</font>,<font color="Maroon">S</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T65">YELLOW_0:65</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:38"/><b>then </b>
<font color="Maroon">S</font> is <a href="lattice3.html#V3">complete</a>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T53">YELLOW_0:53</a></i>;<br/>
<b>hence </b><a NAME="E3:38"/>
<font color="Maroon">S</font> is  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>
 <b>by </b><i><a class="ref" href="lattice3.html#T12">LATTICE3:12</a></i>;<br/>


</div>
