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<b>let </b><font color="Maroon">L</font> be  <a href="waybel_1.html#V11">Boolean</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/>



<a NAME="E1:11"/><i><font color="Green">E11</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds   <a href="waybel_1.html#K7">'not'</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Olive">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">x</font>
 
<b>by </b><i><a class="ref" href="waybel_1.html#T90">WAYBEL_1:90</a></i>;<br/>
<a NAME="E2:11"/><b>then </b>
 <a href="waybel_1.html#K7">'not'</a> <font color="Maroon">x</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="waybel_1.html#T89">WAYBEL_1:89</a></i>;<br/>
<a NAME="E3:11"/><b>then </b><i><font color="Green">E16</font></i>: 
( <font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">x</font></span>)</span> <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> <span class="p1">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">x</font></span>)</span> <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/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:11_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="E2:11_1"/><b>then </b><i><font color="Green">E24</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/><a NAME="E3:11_1"/><i><font color="Green">E25</font></i>: 
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T44">YELLOW_0:44</a></i>;<br/><a NAME="E4:11_1"/>
 <a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="waybel_1.html#K7">'not'</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T45">YELLOW_0:45</a></i>;<br/><b>then </b><i><font color="Green">E26</font></i>:  <a href="waybel_1.html#K7">'not'</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="ref" href="yellow_0.html#T25">YELLOW_0:25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <span class="p2">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="lattice3.html#K12">"/\"</a> <span class="p2">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="waybel_1.html#D3">WAYBEL_1:def 3</a></i>
<br/>.= 
<font color="Maroon">y</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="txt" href="yellow_5.html#E1:11"><i><font color="Green">E11</font></i></a>, <a class="ref" href="yellow_0.html#T24">YELLOW_0:24</a></i>
;<br/><a NAME="E6:11_1"/>
 <a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T45">YELLOW_0:45</a></i>;<br/><b>then </b><font color="Maroon">y</font> = 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="yellow_0.html#T25">YELLOW_0:25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="lattice3.html#K12">"/\"</a> <span class="p2">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="waybel_1.html#D3">WAYBEL_1:def 3</a></i>
<br/>.= 
<font color="Maroon">y</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="txt" href="yellow_5.html#E1:11"><i><font color="Green">E11</font></i></a>, <a class="ref" href="yellow_0.html#T24">YELLOW_0:24</a></i>
;<br/><b>hence </b><a NAME="E8:11_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="waybel_1.html#K7">'not'</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="yellow_5.html#T2" target="_self">Th2</a></i>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E5:11"/>
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="waybel_1.html#K7">'not'</a> <font color="Maroon">x</font> implies <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="waybel_1.html#T89">WAYBEL_1:89</a></i>;<br/>


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