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

<b>set </b><font color="Maroon">B</font> =  <a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font>;<br/>
<a NAME="E1:8_1_1"/>
(  <a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font> is   <a href="lattices.html#NM4">0_Lattice</a> &amp;  <a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font> is   <a href="lattices.html#NM5">1_Lattice</a> )
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">B</font> =  <a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font> as   <a href="lattices.html#NM6">01_Lattice</a> <b>by </b><i><a class="ref" href="lattices.html#D15">LATTICES:def 15</a></i>;<br/>
<a NAME="E3:8_1_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">B</font> is <a href="lattices.html#V16">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">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D19">LATTICES:def 19</a><br/>

<a NAME="E1:8_1_1_1"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">B</font>) <a href="hidden.html#R1">=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font> )
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font> ;<br/>
<b>take </b>
<font color="Maroon">y</font>
;<br/>

<b>thus </b><font color="Maroon">y</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">x</font> = 
<font color="Maroon">y</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">x</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T39">XBOOLE_1:39</a></i>
<br/>.= 
<font color="Maroon">X</font>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
<br/>.= 
 <a href="lattices.html#K6">Top</a> <font color="Maroon">B</font>
<b>by </b><i/>
; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D18">LATTICES:def 18</a><br/>

<b>hence </b><a NAME="E4:8_1_1_1"/>
<font color="Maroon">x</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6">Top</a> <font color="Maroon">B</font>
 ;<br/>

<a NAME="E5:8_1_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">y</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>
<b>thus </b><font color="Maroon">y</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">x</font> = 
<font color="Maroon">y</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">x</font>
<b>by </b><i/>
<br/>.= 
 <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>
<br/>.= 
 <a href="lattices.html#K5">Bottom</a> <font color="Maroon">B</font>
<b>by </b><i/>
;<br/>

<b>hence </b><a NAME="E7:8_1_1_1"/>
<font color="Maroon">x</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">B</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:8_1_1"/>
<font color="Maroon">B</font> is <a href="lattices.html#V11">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">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font>, <font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D11">LATTICES:def 11</a><br/>

<b>thus </b><font color="Maroon">x</font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">z</font></span>)</span> = 
<font color="Maroon">x</font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">x</font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">y</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">y</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">y</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">y</font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i/>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:8_1_1"/>
 <a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">X</font> is <a href="lattices.html#V17">Boolean</a>
 <b>by </b><i>, <a class="ref" href="lattices.html#D20">LATTICES:def 20</a></i>;<br/>


</div>
