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

<b>set </b><font color="Maroon" title="c2">B</font> =  <a href="lattice3.html#K1" title="LATTICE3:func.1">BooleLatt</a> <font color="Maroon" title="c1">X</font>;<br/>
<a NAME="E1:8_1_1"/>
(  <a href="lattice3.html#K1" title="LATTICE3:func.1">BooleLatt</a> <font color="Maroon" title="c1">X</font> is   <a href="lattices.html#NM4" title="LATTICES:NM.4">0_Lattice</a> &amp;  <a href="lattice3.html#K1" title="LATTICE3:func.1">BooleLatt</a> <font color="Maroon" title="c1">X</font> is   <a href="lattices.html#NM5" title="LATTICES:NM.5">1_Lattice</a> )
 
<b>by </b><i><a class="ref" href="lattice3.html#T3" target="_self" title="LATTICE3:th.3">Th3</a>, <a class="ref" href="lattice3.html#T4" target="_self" title="LATTICE3:th.4">Th4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c3">B</font> =  <a href="lattice3.html#K1" title="LATTICE3:func.1">BooleLatt</a> <font color="Maroon" title="c1">X</font> as   <a href="lattices.html#NM6" title="LATTICES:NM.6">01_Lattice</a> <b>by </b><i><a class="ref" href="lattices.html#D15" title="LATTICES:def.15">LATTICES:def 15</a></i>;<br/>
<a NAME="E3:8_1_1"/><i><font color="Green" title="E6">A1</font></i>: 
<font color="Maroon" title="c3">B</font> is <a href="lattices.html#V16" title="LATTICES:attr.16">complemented</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">B</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D19" title="LATTICES:def.19">LATTICES:def 19</a><br/>

<a NAME="E1:8_1_1_1"/><i><font color="Green" title="E7">A2</font></i>: 
( <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c4">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">B</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font> )
 
<b>by </b><i><a class="ref" href="lattice3.html#D1" target="_self" title="LATTICE3:def.1">Def1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">y</font> = <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c4">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">B</font> <b>by </b><i><a class="ref" href="lattice3.html#D1" target="_self" title="LATTICE3:def.1">Def1</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">y</font>
;<br/>

<b>thus </b><font color="Maroon" title="c5">y</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">x</font> = 
<font color="Maroon" title="c5">y</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">x</font>
<b>by </b><i><a class="ref" href="lattice3.html#T1" target="_self" title="LATTICE3:th.1">Th1</a></i>
<br/>.= 
<font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">x</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T39" title="XBOOLE_1:th.39">XBOOLE_1:39</a></i>
<br/>.= 
<font color="Maroon" title="c1">X</font>
<b>by </b><i><a class="txt" href="lattice3.html#E1:8_1_1_1"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>
<br/>.= 
 <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c3">B</font>
<b>by </b><i><a class="ref" href="lattice3.html#T4" target="_self" title="LATTICE3:th.4">Th4</a></i>
; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D18" title="LATTICES:def.18">LATTICES:def 18</a><br/>

<b>hence </b><a NAME="E4:8_1_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c5">y</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">B</font>
 ;<br/>

<a NAME="E5:8_1_1_1"/><i><font color="Green" title="E8">A3</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">x</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c5">y</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">x</font> = 
<font color="Maroon" title="c5">y</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">x</font>
<b>by </b><i><a class="ref" href="lattice3.html#T1" target="_self" title="LATTICE3:th.1">Th1</a></i>
<br/>.= 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="txt" href="lattice3.html#E5:8_1_1_1"><i><font color="Green" title="E8">A3</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>
<br/>.= 
 <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c3">B</font>
<b>by </b><i><a class="ref" href="lattice3.html#T3" target="_self" title="LATTICE3:th.3">Th3</a></i>
;<br/>

<b>hence </b><a NAME="E7:8_1_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c5">y</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">B</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:8_1_1"/>
<font color="Maroon" title="c3">B</font> is <a href="lattices.html#V11" title="LATTICES:attr.11">distributive</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font>, <font color="Maroon" title="c6">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">B</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D11" title="LATTICES:def.11">LATTICES:def 11</a><br/>

<b>thus </b><font color="Maroon" title="c4">x</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c6">z</font></span>)</span> = 
<font color="Maroon" title="c4">x</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c6">z</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T1" target="_self" title="LATTICE3:th.1">Th1</a></i>
<br/>.= 
<font color="Maroon" title="c4">x</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">z</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T1" target="_self" title="LATTICE3:th.1">Th1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">z</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">z</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T1" target="_self" title="LATTICE3:th.1">Th1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c6">z</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T1" target="_self" title="LATTICE3:th.1">Th1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c6">z</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T1" target="_self" title="LATTICE3:th.1">Th1</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:8_1_1"/>
 <a href="lattice3.html#K1" title="LATTICE3:func.1">BooleLatt</a> <font color="Maroon" title="c1">X</font> is <a href="lattices.html#V17" title="LATTICES:attr.17">Boolean</a>
 <b>by </b><i><a class="txt" href="lattice3.html#E3:8_1_1"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="lattices.html#D20" title="LATTICES:def.20">LATTICES:def 20</a></i>;<br/>


</div>
