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<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>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:43_1"/>
<font color="Maroon">y</font> <a href="waybel_1.html#R6">is_a_complement_of</a> <font color="Maroon">x</font>
 ;<br/><a NAME="E2:43_1"/><b>then </b><i><font color="Green">E32</font></i>: 
( <font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font> &amp; <font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font> )
 <b>by </b><i><a class="ref" href="waybel_1.html#D23">WAYBEL_1:def 23</a></i>;<br/><b>thus </b><a NAME="E3:43_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> <a href="lattice3.html#K8">~</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1"><b>proof </b></a><div class="add">

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K8">~</a> </span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="lattice3.html#K8">~</a> </span>)</span> = 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K8">~</a> 
<b>by </b><i>, </i>
<br/>.= 
 <a href="yellow_0.html#K4">Top</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span>
<b>by </b><i/>
; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D23">WAYBEL_1:def 23</a><br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K8">~</a> </span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="lattice3.html#K8">~</a> </span>)</span> = 
<span class="p1">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K8">~</a> 
<b>by </b><i>, </i>
<br/>.= 
 <a href="yellow_0.html#K3">Bottom</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span>
<b>by </b><i><a class="ref" href="yellow_7.html#T1" target="_self">Th1</a></i>
;<br/>


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

<b>assume </b><a NAME="E2:43"/><i><font color="Green">E33</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K8">~</a> </span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="lattice3.html#K8">~</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K4">Top</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K8">~</a> </span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="lattice3.html#K8">~</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <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#D23">WAYBEL_1:def 23</a><br/>

<b>thus </b><font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y</font> = 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K8">~</a> </span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="lattice3.html#K8">~</a> </span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K8">~</a> 
<b>by </b><i>, <a class="ref" href="yellow_7.html#T1" target="_self">Th1</a></i>
<br/>.= 
 <a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font>

; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D23">WAYBEL_1:def 23</a><br/>

<b>thus </b><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font> = 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K8">~</a> </span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="lattice3.html#K8">~</a> </span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K8">~</a> 
<b>by </b><i>, </i>
<br/>.= 
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font>

;<br/>


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