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

<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="waybel_1.html#V11">Boolean</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>





<b>thus </b><a NAME="E1:42"/>
 <a href="waybel_1.html#K7">'not'</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">a</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">b</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>proof </b></a><div class="add">

<i><font color="Green">E11</font></i>: <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p2">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">b</font></span>)</span></span>)</span> = 
<font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p3">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattice3.html#K13">"\/"</a> <span class="p3">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattice3.html#K13">"\/"</a> <span class="p3">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattice3.html#K13">"\/"</a> <span class="p3">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="waybel_1.html#T5">WAYBEL_1:5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></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#K13">"\/"</a> <font color="Maroon">b</font>
<b>by </b><i/>
<br/>.= 
 <a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="waybel_1.html#T5">WAYBEL_1:5</a></i>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p2">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">b</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p2">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">a</font></span>)</span></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">b</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T16">LATTICE3:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">b</font></span>)</span></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">b</font></span>)</span>
<b>by </b><i><a class="ref" href="waybel_1.html#D3">WAYBEL_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">b</font></span>)</span></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">b</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">b</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">b</font></span>)</span>
<b>by </b><i><a class="ref" href="waybel_1.html#T4">WAYBEL_1:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattice3.html#K12">"/\"</a> <span class="p2">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T16">LATTICE3:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="waybel_1.html#T4">WAYBEL_1:4</a></i>
;<br/>
<a NAME="E3:42_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">a</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">b</font></span>)</span> <a href="waybel_1.html#R6">is_a_complement_of</a> <font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="yellow_5.html#T6" target="_self">Th6</a>, <a class="ref" href="waybel_1.html#D23">WAYBEL_1:def 23</a></i>;<br/>
<b>hence </b><a NAME="E4:42_1"/>
 <a href="waybel_1.html#K7">'not'</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">a</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">b</font></span>)</span>
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E2:42"/>
 <a href="waybel_1.html#K7">'not'</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">b</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2"><b>proof </b></a><div class="add">

<i><font color="Green">E16</font></i>: <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">b</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="waybel_1.html#T5">WAYBEL_1:5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="waybel_1.html#T5">WAYBEL_1:5</a></i>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">b</font></span>)</span></span>)</span> = 
<font color="Maroon">a</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattice3.html#K12">"/\"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p3">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T16">LATTICE3:16</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattice3.html#K12">"/\"</a> <span class="p3">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattice3.html#K12">"/\"</a> <span class="p3">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">b</font></span>)</span></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">a</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattice3.html#K12">"/\"</a> <span class="p3">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattice3.html#K12">"/\"</a> <span class="p2">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="waybel_1.html#T4">WAYBEL_1:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K12">"/\"</a> <span class="p2">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="lattice3.html#T16">LATTICE3:16</a></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#K12">"/\"</a> <font color="Maroon">b</font>
<b>by </b><i/>
<br/>.= 
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="waybel_1.html#T4">WAYBEL_1:4</a></i>
;<br/>
<a NAME="E3:42_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">b</font></span>)</span> <a href="waybel_1.html#R6">is_a_complement_of</a> <font color="Maroon">a</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="yellow_5.html#T6" target="_self">Th6</a>, <a class="ref" href="waybel_1.html#D23">WAYBEL_1:def 23</a></i>;<br/>
<b>hence </b><a NAME="E4:42_2"/>
 <a href="waybel_1.html#K7">'not'</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K7">'not'</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i/>;<br/>


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


</div>
