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

<span class="kw">let </span><font color="Maroon" title="c1">B</font> be   <a href="lattices.html#NM8" title="LATTICES:NM.8">B_Lattice</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">FB</font> being  <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <font color="Maroon" title="c1">B</font> holds  <font color="Maroon" title="c1">B</font> <a href="filter_1.html#K4" title="FILTER_1:func.4">/\/</a> <font color="Olive" title="b1">FB</font> is   <a href="lattices.html#NM8" title="LATTICES:NM.8">B_Lattice</a></span><br/><span class="kw">let </span><font color="Maroon" title="c2">FB</font> be   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <font color="Maroon" title="c1">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">B</font> <a href="filter_1.html#K4" title="FILTER_1:func.4">/\/</a> <font color="Maroon" title="c2">FB</font> is   <a href="lattices.html#NM8" title="LATTICES:NM.8">B_Lattice</a></span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">L</font> = <font color="Maroon" title="c1">B</font> <a href="filter_1.html#K4" title="FILTER_1:func.4">/\/</a> <font color="Maroon" title="c2">FB</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">R</font> =  <a href="filter_0.html#K12" title="FILTER_0:func.12">equivalence_wrt</a> <font color="Maroon" title="c2">FB</font>;<br/>
<a NAME="E1:31"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
<font color="Maroon" title="c1">B</font> is   <a href="filter_0.html#NM2" title="FILTER_0:NM.2">I_Lattice</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T40" title="FILTER_0:th.40">FILTER_0:40</a></span>;<br/>
<a NAME="E2:31"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A2</font></span>: 
<font color="Maroon" title="c1">B</font> <a href="filter_1.html#K4" title="FILTER_1:func.4">/\/</a> <font color="Maroon" title="c2">FB</font> is   <a href="lattices.html#NM4" title="LATTICES:NM.4">0_Lattice</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_1.html#T18" target="_self" title="FILTER_1:th.18">Th18</a></span>;<br/>
<a NAME="E3:31"/><span class="lab"><font color="Green" title="E25">A3</font></span>: 
<font color="Maroon" title="c1">B</font> <a href="filter_1.html#K4" title="FILTER_1:func.4">/\/</a> <font color="Maroon" title="c2">FB</font> is   <a href="lattices.html#NM5" title="LATTICES:NM.5">1_Lattice</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="filter_1.html#T19" target="_self" title="FILTER_1:th.19">Th19</a></span>;<br/>
<a NAME="E4:31"/><span class="lab"><font color="Green" title="E26">A4</font></span>: 
 <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">B</font> <a href="filter_1.html#K4" title="FILTER_1:func.4">/\/</a> <font color="Maroon" title="c2">FB</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">B</font></span>)</span> <a href="filter_1.html#K5" title="FILTER_1:func.5">/\/</a> <font color="Maroon" title="c2">FB</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="filter_1.html#T18" target="_self" title="FILTER_1:th.18">Th18</a></span>;<br/>
<a NAME="E5:31"/><span class="lab"><font color="Green" title="E27">A5</font></span>: 
 <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">B</font> <a href="filter_1.html#K4" title="FILTER_1:func.4">/\/</a> <font color="Maroon" title="c2">FB</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">B</font></span>)</span> <a href="filter_1.html#K5" title="FILTER_1:func.5">/\/</a> <font color="Maroon" title="c2">FB</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="filter_1.html#T19" target="_self" title="FILTER_1:th.19">Th19</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">L</font> = <font color="Maroon" title="c1">B</font> <a href="filter_1.html#K4" title="FILTER_1:func.4">/\/</a> <font color="Maroon" title="c2">FB</font> as   <a href="lattices.html#NM6" title="LATTICES:NM.6">01_Lattice</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E25">A3</font></span></a></span>;<br/>
<a NAME="E7:31"/><span class="lab"><font color="Green" title="E28">A6</font></span>: 
<font color="Maroon" title="c5">L</font> is  <a href="lattices.html#V16" title="LATTICES:attr.16">complemented</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">L</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="lattices.html#D19" title="LATTICES:def.19">LATTICES:def 19</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">L</font> st <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <font color="Maroon" title="c6">x</font></span><br/>

<a NAME="E1:31_1"/>
<font color="Maroon" title="c5">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <span class="p2">(<span class="default"><a href="filter_0.html#K12" title="FILTER_0:func.12">equivalence_wrt</a> <font color="Maroon" title="c2">FB</font></span>)</span></span>)</span>,<span class="p1">(<span class="default">the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">B</font> <a href="filter_1.html#K3" title="FILTER_1:func.3">/\/</a> <span class="p2">(<span class="default"><a href="filter_0.html#K12" title="FILTER_0:func.12">equivalence_wrt</a> <font color="Maroon" title="c2">FB</font></span>)</span></span>)</span>,<span class="p1">(<span class="default">the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c1">B</font> <a href="filter_1.html#K3" title="FILTER_1:func.3">/\/</a> <span class="p2">(<span class="default"><a href="filter_0.html#K12" title="FILTER_0:func.12">equivalence_wrt</a> <font color="Maroon" title="c2">FB</font></span>)</span></span>)</span> #)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="filter_1.html#D5" target="_self" title="FILTER_1:def.5">Def5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">B</font><span class="kw"> such that </span><br/><a NAME="E3:31_1"/><span class="lab"><font color="Green" title="E29">A7</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="filter_0.html#K12" title="FILTER_0:func.12">equivalence_wrt</a> <font color="Maroon" title="c2">FB</font></span>)</span>,<font color="Maroon" title="c7">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#T45" title="EQREL_1:th.45">EQREL_1:45</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">y</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span> <a href="filter_1.html#K5" title="FILTER_1:func.5">/\/</a> <font color="Maroon" title="c2">FB</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">L</font> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c8">y</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">y</font> <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <font color="Maroon" title="c6">x</font></span><br/>

<a NAME="E5:31_1"/><span class="lab"><font color="Green" title="E30">A8</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> <a href="filter_1.html#K5" title="FILTER_1:func.5">/\/</a> <font color="Maroon" title="c2">FB</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="txt" href="#E3:31_1"><span class="lab"><font color="Green" title="E29">A7</font></span></a>, <a class="ref" href="filter_1.html#D6" target="_self" title="FILTER_1:def.6">Def6</a></span>;<br/>
<span class="kw">hence </span><font color="Maroon" title="c8">y</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c6">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c7">a</font></span>)</span> <a href="filter_1.html#K5" title="FILTER_1:func.5">/\/</a> <font color="Maroon" title="c2">FB</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="filter_1.html#T15" target="_self" title="FILTER_1:th.15">Th15</a></span>
<br/>.= 
 <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c5">L</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:31"><span class="lab"><font color="Green" title="E27">A5</font></span></a>, <a class="ref" href="lattices.html#T48" title="LATTICES:th.48">LATTICES:48</a></span>
;<br/> <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="lattices.html#D18" title="LATTICES:def.18">LATTICES:def 18</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">x</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c8">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="c5">L</font> &amp; <font color="Maroon" title="c8">y</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c6">x</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="c5">L</font> &amp; <font color="Maroon" title="c6">x</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c8">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="c5">L</font> )</span><br/>

<span class="kw">hence </span><a NAME="E7:31_1"/>
<font color="Maroon" title="c6">x</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c8">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="c5">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">y</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c6">x</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="c5">L</font> &amp; <font color="Maroon" title="c6">x</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c8">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="c5">L</font> )</span><br/>

<span class="kw">thus </span><font color="Maroon" title="c8">y</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c6">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c7">a</font></span>)</span> <a href="filter_1.html#K5" title="FILTER_1:func.5">/\/</a> <font color="Maroon" title="c2">FB</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="txt" href="#E5:31_1"><span class="lab"><font color="Green" title="E30">A8</font></span></a>, <a class="ref" href="filter_1.html#T15" target="_self" title="FILTER_1:th.15">Th15</a></span>
<br/>.= 
 <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c5">L</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E26">A4</font></span></a>, <a class="ref" href="lattices.html#T47" title="LATTICES:th.47">LATTICES:47</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">x</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c8">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="c5">L</font></span><br/>

<span class="kw">hence </span><a NAME="E9:31_1"/>
<font color="Maroon" title="c6">x</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">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="c5">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:31"/>
<font color="Maroon" title="c5">L</font> is   <a href="filter_0.html#NM2" title="FILTER_0:NM.2">I_Lattice</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="filter_1.html#T20" target="_self" title="FILTER_1:th.20">Th20</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:31"/>
<font color="Maroon" title="c1">B</font> <a href="filter_1.html#K4" title="FILTER_1:func.4">/\/</a> <font color="Maroon" title="c2">FB</font> is   <a href="lattices.html#NM8" title="LATTICES:NM.8">B_Lattice</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:31"><span class="lab"><font color="Green" title="E28">A6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
