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

<b>let </b><font color="Maroon" title="c1">L1</font>, <font color="Maroon" title="c2">L2</font> be   <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a>;<br/>



<b>thus </b><a NAME="E1:65"/>
( <font color="Maroon" title="c1">L1</font> is   <a href="lattices.html#NM7" title="LATTICES:NM.7">C_Lattice</a> &amp; <font color="Maroon" title="c2">L2</font> is   <a href="lattices.html#NM7" title="LATTICES:NM.7">C_Lattice</a> implies <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span> is   <a href="lattices.html#NM7" title="LATTICES:NM.7">C_Lattice</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:65_1"/><i><font color="Green" title="E43">A1</font></i>: 
( <font color="Maroon" title="c1">L1</font> is   <a href="lattices.html#NM7" title="LATTICES:NM.7">C_Lattice</a> &amp; <font color="Maroon" title="c2">L2</font> is   <a href="lattices.html#NM7" title="LATTICES:NM.7">C_Lattice</a> )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c3">L</font> = <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span> as   <a href="lattices.html#NM6" title="LATTICES:NM.6">01_Lattice</a> <b>by </b><i><a class="ref" href="filter_1.html#T42" target="_self" title="FILTER_1:th.42">Th42</a></i>;<br/>
<a NAME="E3:65_1"/>
<font color="Maroon" title="c3">L</font> is <a href="lattices.html#V16" title="LATTICES:attr.16">complemented</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">L</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/>

<b>consider </b><font color="Maroon" title="c5">p1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>, <font color="Maroon" title="c6">p2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font><b> such that </b><br/><a NAME="E2:65_1_1"/><i><font color="Green" title="E44">A2</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c6">p2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9" title="DOMAIN_1:th.9">DOMAIN_1:9</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">q1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font><b> such that </b><br/><a NAME="E4:65_1_1"/><i><font color="Green" title="E45">A3</font></i>: 
<font color="Maroon" title="c7">q1</font> <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <font color="Maroon" title="c5">p1</font>
 <b>by </b><i><a class="txt" href="filter_1.html#E1:65_1"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="lattices.html#D19" title="LATTICES:def.19">LATTICES:def 19</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">q2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font><b> such that </b><br/><a NAME="E6:65_1_1"/><i><font color="Green" title="E46">A4</font></i>: 
<font color="Maroon" title="c8">q2</font> <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <font color="Maroon" title="c6">p2</font>
 <b>by </b><i><a class="txt" href="filter_1.html#E1:65_1"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="lattices.html#D19" title="LATTICES:def.19">LATTICES:def 19</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">b</font> = <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c8">q2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">L</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c9">b</font>
;<br/>

<b>thus </b><a NAME="E8:65_1_1"/>
<font color="Maroon" title="c9">b</font> <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <font color="Maroon" title="c4">a</font>
 <b>by </b><i><a class="txt" href="filter_1.html#E1:65_1"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="filter_1.html#E2:65_1_1"><i><font color="Green" title="E44">A2</font></i></a>, <a class="txt" href="filter_1.html#E4:65_1_1"><i><font color="Green" title="E45">A3</font></i></a>, <a class="txt" href="filter_1.html#E6:65_1_1"><i><font color="Green" title="E46">A4</font></i></a>, <a class="ref" href="filter_1.html#T45" target="_self" title="FILTER_1:th.45">Th45</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:65_1"/>
<span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span> is   <a href="lattices.html#NM7" title="LATTICES:NM.7">C_Lattice</a>
 ;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:65"/><i><font color="Green" title="E43">A5</font></i>: 
<span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span> is   <a href="lattices.html#NM7" title="LATTICES:NM.7">C_Lattice</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c3">C1</font> = <font color="Maroon" title="c1">L1</font>, <font color="Maroon" title="c4">C2</font> = <font color="Maroon" title="c2">L2</font> as   <a href="lattices.html#NM6" title="LATTICES:NM.6">01_Lattice</a> <b>by </b><i><a class="ref" href="filter_1.html#T42" target="_self" title="FILTER_1:th.42">Th42</a></i>;<br/>
<a NAME="E4:65"/>
<font color="Maroon" title="c3">C1</font> is <a href="lattices.html#V16" title="LATTICES:attr.16">complemented</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">p1'</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">C1</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/>

<b>consider </b><font color="Maroon" title="c6">p2'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">C2</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">p1</font> = <font color="Maroon" title="c5">p1'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c8">p2</font> = <font color="Maroon" title="c6">p2'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font> ;<br/>
<b>consider </b><font color="Maroon" title="c9">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span><b> such that </b><br/><a NAME="E5:65_2"/><i><font color="Green" title="E44">A6</font></i>: 
<font color="Maroon" title="c9">b</font> <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span>
 <b>by </b><i><a class="txt" href="filter_1.html#E2:65"><i><font color="Green" title="E43">A5</font></i></a>, <a class="ref" href="lattices.html#D19" title="LATTICES:def.19">LATTICES:def 19</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">q1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>, <font color="Maroon" title="c11">q2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font><b> such that </b><br/><a NAME="E7:65_2"/><i><font color="Green" title="E45">A7</font></i>: 
<font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c10">q1</font>,<font color="Maroon" title="c11">q2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9" title="DOMAIN_1:th.9">DOMAIN_1:9</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">q1'</font> = <font color="Maroon" title="c10">q1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">C1</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c12">q1'</font>
;<br/>

<b>thus </b><a NAME="E9:65_2"/>
<font color="Maroon" title="c12">q1'</font> <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <font color="Maroon" title="c5">p1'</font>
 <b>by </b><i><a class="txt" href="filter_1.html#E5:65_2"><i><font color="Green" title="E44">A6</font></i></a>, <a class="txt" href="filter_1.html#E7:65_2"><i><font color="Green" title="E45">A7</font></i></a>, <a class="ref" href="filter_1.html#T45" target="_self" title="FILTER_1:th.45">Th45</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:65"/>
<font color="Maroon" title="c1">L1</font> is   <a href="lattices.html#NM7" title="LATTICES:NM.7">C_Lattice</a>
 ;<br/>

<a NAME="E6:65"/>
<font color="Maroon" title="c4">C2</font> is <a href="lattices.html#V16" title="LATTICES:attr.16">complemented</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">p2'</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">C2</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/>

<b>consider </b><font color="Maroon" title="c6">p1'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">C1</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">p1</font> = <font color="Maroon" title="c6">p1'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c8">p2</font> = <font color="Maroon" title="c5">p2'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font> ;<br/>
<b>consider </b><font color="Maroon" title="c9">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span><b> such that </b><br/><a NAME="E5:65_3"/><i><font color="Green" title="E44">A8</font></i>: 
<font color="Maroon" title="c9">b</font> <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span>
 <b>by </b><i><a class="txt" href="filter_1.html#E2:65"><i><font color="Green" title="E43">A5</font></i></a>, <a class="ref" href="lattices.html#D19" title="LATTICES:def.19">LATTICES:def 19</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">q1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>, <font color="Maroon" title="c11">q2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font><b> such that </b><br/><a NAME="E7:65_3"/><i><font color="Green" title="E45">A9</font></i>: 
<font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c10">q1</font>,<font color="Maroon" title="c11">q2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9" title="DOMAIN_1:th.9">DOMAIN_1:9</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">q2'</font> = <font color="Maroon" title="c11">q2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">C2</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c12">q2'</font>
;<br/>

<b>thus </b><a NAME="E9:65_3"/>
<font color="Maroon" title="c12">q2'</font> <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <font color="Maroon" title="c5">p2'</font>
 <b>by </b><i><a class="txt" href="filter_1.html#E5:65_3"><i><font color="Green" title="E44">A8</font></i></a>, <a class="txt" href="filter_1.html#E7:65_3"><i><font color="Green" title="E45">A9</font></i></a>, <a class="ref" href="filter_1.html#T45" target="_self" title="FILTER_1:th.45">Th45</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:65"/>
<font color="Maroon" title="c2">L2</font> is   <a href="lattices.html#NM7" title="LATTICES:NM.7">C_Lattice</a>
 ;<br/>


</div>
