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<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="waybel_1.html#V11" title="WAYBEL_1:attr.11">Boolean</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds <br/> ( <font color="Olive" title="b2">y</font> <a href="waybel_1.html#R6" title="WAYBEL_1:pred.6">is_a_complement_of</a> <font color="Olive" title="b1">x</font> iff <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_1.html#K7" title="WAYBEL_1:func.7">'not'</a> <font color="Olive" title="b1">x</font> )</span><br/><b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">y</font> <a href="waybel_1.html#R6" title="WAYBEL_1:pred.6">is_a_complement_of</a> <font color="Maroon" title="c2">x</font> iff <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_1.html#K7" title="WAYBEL_1:func.7">'not'</a> <font color="Maroon" title="c2">x</font> )</span><br/>



<a NAME="E1:11"/><i><font color="Green" title="E8">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds   <a href="waybel_1.html#K7" title="WAYBEL_1:func.7">'not'</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K7" title="WAYBEL_1:func.7">'not'</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font>
 
<b>by </b><i><a class="ref" href="waybel_1.html#T90" title="WAYBEL_1:th.90">WAYBEL_1:90</a></i>;<br/>
<a NAME="E2:11"/><b>then </b>
 <a href="waybel_1.html#K7" title="WAYBEL_1:func.7">'not'</a> <font color="Maroon" title="c2">x</font> <a href="waybel_1.html#R6" title="WAYBEL_1:pred.6">is_a_complement_of</a> <font color="Maroon" title="c2">x</font>
 
<b>by </b><i><a class="ref" href="waybel_1.html#T89" title="WAYBEL_1:th.89">WAYBEL_1:89</a></i>;<br/>
<a NAME="E3:11"/><b>then </b><i><font color="Green" title="E9">A2</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K7" title="WAYBEL_1:func.7">'not'</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c1">L</font> &amp; <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K7" title="WAYBEL_1:func.7">'not'</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font> )
 
<b>by </b><i><a class="ref" href="waybel_1.html#D23" title="WAYBEL_1:def.23">WAYBEL_1:def 23</a></i>;<br/>
<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_1.html#K7" title="WAYBEL_1:func.7">'not'</a> <font color="Maroon" title="c2">x</font> implies <font color="Maroon" title="c3">y</font> <a href="waybel_1.html#R6" title="WAYBEL_1:pred.6">is_a_complement_of</a> <font color="Maroon" title="c2">x</font> )</span>
<div class="add"><b>assume </b><a NAME="E1:11_1"/>
<font color="Maroon" title="c3">y</font> <a href="waybel_1.html#R6" title="WAYBEL_1:pred.6">is_a_complement_of</a> <font color="Maroon" title="c2">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_1.html#K7" title="WAYBEL_1:func.7">'not'</a> <font color="Maroon" title="c2">x</font></span><br/><a NAME="E2:11_1"/><b>then </b><i><font color="Green" title="E10">A3</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c1">L</font> &amp; <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font> )
 <b>by </b><i><a class="ref" href="waybel_1.html#D23" title="WAYBEL_1:def.23">WAYBEL_1:def 23</a></i>;<br/><a NAME="E3:11_1"/><i><font color="Green" title="E11">A4</font></i>: 
 <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">y</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K7" title="WAYBEL_1:func.7">'not'</a> <font color="Maroon" title="c2">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T44" title="YELLOW_0:th.44">YELLOW_0:44</a></i>;<br/><a NAME="E4:11_1"/>
 <a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c1">L</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a>  <a href="waybel_1.html#K7" title="WAYBEL_1:func.7">'not'</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T45" title="YELLOW_0:th.45">YELLOW_0:45</a></i>;<br/><b>then </b><i><font color="Green" title="E12">A5</font></i>:  <a href="waybel_1.html#K7" title="WAYBEL_1:func.7">'not'</a> <font color="Maroon" title="c2">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K7" title="WAYBEL_1:func.7">'not'</a> <font color="Maroon" title="c2">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E2:11_1"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="yellow_0.html#T25" title="YELLOW_0:th.25">YELLOW_0:25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p2">(<span class="default"><a href="waybel_1.html#K7" title="WAYBEL_1:func.7">'not'</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p2">(<span class="default"><a href="waybel_1.html#K7" title="WAYBEL_1:func.7">'not'</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="waybel_1.html#D3" title="WAYBEL_1:def.3">WAYBEL_1:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c3">y</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K7" title="WAYBEL_1:func.7">'not'</a> <font color="Maroon" title="c2">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E3:11"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="#E3:11_1"><i><font color="Green" title="E11">A4</font></i></a>, <a class="ref" href="yellow_0.html#T24" title="YELLOW_0:th.24">YELLOW_0:24</a></i>
;<br/><a NAME="E6:11_1"/>
 <a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c1">L</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c3">y</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T45" title="YELLOW_0:th.45">YELLOW_0:45</a></i>;<br/><b>then </b><font color="Maroon" title="c3">y</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p2">(<span class="default"><a href="waybel_1.html#K7" title="WAYBEL_1:func.7">'not'</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">y</font>
<b>by </b><i><a class="txt" href="#E3:11"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="yellow_0.html#T25" title="YELLOW_0:th.25">YELLOW_0:25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p2">(<span class="default"><a href="waybel_1.html#K7" title="WAYBEL_1:func.7">'not'</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="waybel_1.html#D3" title="WAYBEL_1:def.3">WAYBEL_1:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c3">y</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K7" title="WAYBEL_1:func.7">'not'</a> <font color="Maroon" title="c2">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E2:11_1"><i><font color="Green" title="E10">A3</font></i></a>, <a class="txt" href="#E3:11_1"><i><font color="Green" title="E11">A4</font></i></a>, <a class="ref" href="yellow_0.html#T24" title="YELLOW_0:th.24">YELLOW_0:24</a></i>
;<br/><b>hence </b><a NAME="E8:11_1"/>
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_1.html#K7" title="WAYBEL_1:func.7">'not'</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="txt" href="#E5:11_1"><i><font color="Green" title="E12">A5</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E5:11"/>
( <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_1.html#K7" title="WAYBEL_1:func.7">'not'</a> <font color="Maroon" title="c2">x</font> implies <font color="Maroon" title="c3">y</font> <a href="waybel_1.html#R6" title="WAYBEL_1:pred.6">is_a_complement_of</a> <font color="Maroon" title="c2">x</font> )
 <b>by </b><i><a class="txt" href="#E1:11"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="waybel_1.html#T89" title="WAYBEL_1:th.89">WAYBEL_1:89</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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