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

<b>let </b><font color="Maroon">L</font> be  <a href="yellow_0.html#V3">bounded</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/>

<b>thus </b><a NAME="E1:44"/>
( <font color="Maroon">L</font> is <a href="waybel_1.html#V10">complemented</a> implies <font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a>  is <a href="waybel_1.html#V10">complemented</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:44_1"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  ex <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <font color="Olive">y</font> <a href="waybel_1.html#R6">is_a_complement_of</a> <font color="Olive">x</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D24">WAYBEL_1:def 24</a><br/>

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D24">WAYBEL_1:def 24</a><br/>

<b>consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:44_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">y</font> <a href="waybel_1.html#R6">is_a_complement_of</a>  <a href="lattice3.html#K9">~</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="yellow_7.html#T2" target="_self">Th2</a></i>;<br/>
<b>take </b>
<font color="Maroon">y</font> <a href="lattice3.html#K8">~</a> 
;<br/>

<a NAME="E4:44_1"/>
(  <a href="lattice3.html#K9">~</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">x</font></span>)</span> <a href="lattice3.html#K8">~</a>  <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K9">~</a> <font color="Maroon">x</font> )
 
;<br/>
<b>hence </b><a NAME="E5:44_1"/>
<font color="Maroon">y</font> <a href="lattice3.html#K8">~</a>  <a href="waybel_1.html#R6">is_a_complement_of</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="yellow_7.html#T4" target="_self">Th4</a>, </i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:44"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span>  ex <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span> st <font color="Olive">y</font> <a href="waybel_1.html#R6">is_a_complement_of</a> <font color="Olive">x</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D24">WAYBEL_1:def 24</a><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="waybel_1.html#D24">WAYBEL_1:def 24</a><br/>

<b>consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span><b> such that </b><br/><a NAME="E4:44"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">y</font> <a href="waybel_1.html#R6">is_a_complement_of</a> <font color="Maroon">x</font> <a href="lattice3.html#K8">~</a> 
 <b>by </b><i><a class="ref" href="yellow_7.html#T2" target="_self">Th2</a></i>;<br/>
<b>take </b>
 <a href="lattice3.html#K9">~</a> <font color="Maroon">y</font>
;<br/>

<a NAME="E5:44"/>
(  <a href="lattice3.html#K9">~</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K8">~</a>  <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K9">~</a> <font color="Maroon">y</font> )
 
;<br/>
<b>hence </b><a NAME="E6:44"/>
 <a href="lattice3.html#K9">~</a> <font color="Maroon">y</font> <a href="waybel_1.html#R6">is_a_complement_of</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="yellow_7.html#T4" target="_self">Th4</a>, </i>;<br/>


</div>
