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<b>consider </b><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">u</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c3">G</font> =  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">u</font> #) as  <a href="lattices.html#V3" title="LATTICES:attr.3">strict</a> <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a> <b>by </b><i><a class="txt" href="lattices.html#E18"><i><font color="Green" title="E16">Lm4</font></i></a></i>;<br/>
<a NAME="E3:43_1_1"/><i><font color="Green" title="E25">A1</font></i>: 
<font color="Maroon" title="c3">G</font> is <a href="lattices.html#V16" title="LATTICES:attr.16">complemented</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">G</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D19" target="_self" title="LATTICES:def.19">LATTICES:def 19</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">G</font> st <font color="Olive" title="b1">a</font> <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <font color="Maroon" title="c4">b</font></span><br/>

<b>take </b>
<font color="Maroon" title="c4">b</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">b</font> <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <font color="Maroon" title="c4">b</font></span><br/>

<b>thus </b><a NAME="E1:43_1_1_1"/>
( <font color="Maroon" title="c4">b</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c3">G</font> &amp; <font color="Maroon" title="c4">b</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c3">G</font> )
 <b>by </b><i><a class="txt" href="lattices.html#E4"><i><font color="Green" title="E2">Lm1</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D18" target="_self" title="LATTICES:def.18">LATTICES:def 18</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">b</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c3">G</font> &amp; <font color="Maroon" title="c4">b</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c3">G</font> )</span><br/>

<b>thus </b><a NAME="E2:43_1_1_1"/>
( <font color="Maroon" title="c4">b</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c3">G</font> &amp; <font color="Maroon" title="c4">b</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c3">G</font> )
 <b>by </b><i><a class="txt" href="lattices.html#E4"><i><font color="Green" title="E2">Lm1</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>
<a NAME="E4:43_1_1"/>
( <font color="Maroon" title="c3">G</font> is   <a href="lattices.html#NM4" title="LATTICES:NM.4">0_Lattice</a> &amp; <font color="Maroon" title="c3">G</font> is   <a href="lattices.html#NM5" title="LATTICES:NM.5">1_Lattice</a> )
 
<b>by </b><i><a class="txt" href="lattices.html#E19"><i><font color="Green" title="E17">Lm5</font></i></a>, <a class="txt" href="lattices.html#E20"><i><font color="Green" title="E18">Lm6</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">G</font> = <font color="Maroon" title="c3">G</font> as   <a href="lattices.html#NM7" title="LATTICES:NM.7">C_Lattice</a> <b>by </b><i><a class="txt" href="#E3:43_1_1"><i><font color="Green" title="E25">A1</font></i></a></i>;<br/>
<a NAME="E6:43_1_1"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">G</font> holds  <font color="Olive" title="b1">x</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b3">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b3">z</font></span>)</span>
 
<b>by </b><i><a class="txt" href="lattices.html#E4"><i><font color="Green" title="E2">Lm1</font></i></a></i>;<br/>
<a NAME="E7:43_1_1"/><b>then </b>
<font color="Maroon" title="c4">G</font> is <a href="lattices.html#V11" title="LATTICES:attr.11">distributive</a>
 
<b>by </b><i><a class="ref" href="lattices.html#D11" target="_self" title="LATTICES:def.11">Def11</a></i>;<br/>
<a NAME="E8:43_1_1"/><b>then </b>
<font color="Maroon" title="c4">G</font> is <a href="lattices.html#V17" title="LATTICES:attr.17">Boolean</a>
 
;<br/>
<b>hence </b><a NAME="E9:43_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being  <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="lattices.html#V17" title="LATTICES:attr.17">Boolean</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="lattices.html#V3" title="LATTICES:attr.3">strict</a> )
 ; <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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