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

<b>let </b><font color="Maroon">L1</font> be   <a href="lattices.html#NM1">Lattice</a>;<br/><b>let </b><font color="Maroon">L2</font> be   <a href="lattices.html#NM1">Lattice</a>;<br/><b>let </b><font color="Maroon">p1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font>;<br/><b>let </b><font color="Maroon">q1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font>;<br/><b>let </b><font color="Maroon">p2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L2</font>;<br/><b>let </b><font color="Maroon">q2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L2</font>;<br/>











<b>assume </b><a NAME="E1:64"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">L1</font> is   <a href="lattices.html#NM6">01_Lattice</a> &amp; <font color="Maroon">L2</font> is   <a href="lattices.html#NM6">01_Lattice</a> )
 ;<br/>

<b>thus </b><a NAME="E2:64"/>
( <font color="Maroon">p1</font> <a href="lattices.html#R2">is_a_complement_of</a> <font color="Maroon">q1</font> &amp; <font color="Maroon">p2</font> <a href="lattices.html#R2">is_a_complement_of</a> <font color="Maroon">q2</font> implies <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="lattices.html#R2">is_a_complement_of</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">q1</font>,<font color="Maroon">q2</font></span><a href="filter_1.html#K9" target="_self">]</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:64_1"/>
( <font color="Maroon">p1</font> <a href="lattices.html#R2">is_a_complement_of</a> <font color="Maroon">q1</font> &amp; <font color="Maroon">p2</font> <a href="lattices.html#R2">is_a_complement_of</a> <font color="Maroon">q2</font> )
 ;<br/>

<a NAME="E2:64_1"/><b>then </b><i><font color="Green">E69</font></i>: 
( <font color="Maroon">p1</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6">Top</a> <font color="Maroon">L1</font> &amp; <font color="Maroon">p1</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L1</font> &amp; <font color="Maroon">p2</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6">Top</a> <font color="Maroon">L2</font> &amp; <font color="Maroon">p2</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L2</font> )
 
<b>by </b><i><a class="ref" href="lattices.html#D18">LATTICES:def 18</a></i>;<br/>
<b>hence </b><span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="lattices.html#K3">"\/"</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">q1</font>,<font color="Maroon">q2</font></span><a href="filter_1.html#K9" target="_self">]</a></span> = 
<span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><a href="lattices.html#K6">Top</a> <font color="Maroon">L1</font></span>)</span>,<span class="p2">(<span class="default"><a href="lattices.html#K6">Top</a> <font color="Maroon">L2</font></span>)</span></span><a href="filter_1.html#K9" target="_self">]</a></span>
<b>by </b><i/>
<br/>.= 
 <a href="lattices.html#K6">Top</a> <span class="p1"><a href="filter_1.html#K7" target="_self">[:</a><span class="default"><font color="Maroon">L1</font>,<font color="Maroon">L2</font></span><a href="filter_1.html#K7" target="_self">:]</a></span>
<b>by </b><i>, </i>
;<br/> <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="E4:64_1"/>
<span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">q1</font>,<font color="Maroon">q2</font></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="lattices.html#K3">"\/"</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6">Top</a> <span class="p1"><a href="filter_1.html#K7" target="_self">[:</a><span class="default"><font color="Maroon">L1</font>,<font color="Maroon">L2</font></span><a href="filter_1.html#K7" target="_self">:]</a></span>
 ;<br/>

<b>thus </b><span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="lattices.html#K4">"/\"</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">q1</font>,<font color="Maroon">q2</font></span><a href="filter_1.html#K9" target="_self">]</a></span> = 
<span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><a href="lattices.html#K5">Bottom</a> <font color="Maroon">L1</font></span>)</span>,<span class="p2">(<span class="default"><a href="lattices.html#K5">Bottom</a> <font color="Maroon">L2</font></span>)</span></span><a href="filter_1.html#K9" target="_self">]</a></span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="lattices.html#K5">Bottom</a> <span class="p1"><a href="filter_1.html#K7" target="_self">[:</a><span class="default"><font color="Maroon">L1</font>,<font color="Maroon">L2</font></span><a href="filter_1.html#K7" target="_self">:]</a></span>
<b>by </b><i>, </i>
;<br/>

<b>hence </b><a NAME="E6:64_1"/>
<span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">q1</font>,<font color="Maroon">q2</font></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="lattices.html#K4">"/\"</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K5">Bottom</a> <span class="p1"><a href="filter_1.html#K7" target="_self">[:</a><span class="default"><font color="Maroon">L1</font>,<font color="Maroon">L2</font></span><a href="filter_1.html#K7" target="_self">:]</a></span>
 ;<br/>


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

<b>assume </b><a NAME="E3:64"/>
<span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="lattices.html#R2">is_a_complement_of</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">q1</font>,<font color="Maroon">q2</font></span><a href="filter_1.html#K9" target="_self">]</a></span>
 ;<br/>

<a NAME="E4:64"/><b>then </b><i><font color="Green">E76</font></i>: 
( <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="lattices.html#K3">"\/"</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">q1</font>,<font color="Maroon">q2</font></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6">Top</a> <span class="p1"><a href="filter_1.html#K7" target="_self">[:</a><span class="default"><font color="Maroon">L1</font>,<font color="Maroon">L2</font></span><a href="filter_1.html#K7" target="_self">:]</a></span> &amp; <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="lattices.html#K4">"/\"</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">q1</font>,<font color="Maroon">q2</font></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K5">Bottom</a> <span class="p1"><a href="filter_1.html#K7" target="_self">[:</a><span class="default"><font color="Maroon">L1</font>,<font color="Maroon">L2</font></span><a href="filter_1.html#K7" target="_self">:]</a></span> )
 
<b>by </b><i><a class="ref" href="lattices.html#D18">LATTICES:def 18</a></i>;<br/>
<a NAME="E5:64"/>
<span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><a href="lattices.html#K6">Top</a> <font color="Maroon">L1</font></span>)</span>,<span class="p2">(<span class="default"><a href="lattices.html#K6">Top</a> <font color="Maroon">L2</font></span>)</span></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6">Top</a> <span class="p1"><a href="filter_1.html#K7" target="_self">[:</a><span class="default"><font color="Maroon">L1</font>,<font color="Maroon">L2</font></span><a href="filter_1.html#K7" target="_self">:]</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:64"/><b>then </b><i><font color="Green">E78</font></i>: 
<span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><a href="lattices.html#K6">Top</a> <font color="Maroon">L1</font></span>)</span>,<span class="p2">(<span class="default"><a href="lattices.html#K6">Top</a> <font color="Maroon">L2</font></span>)</span></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">q1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">q2</font></span>)</span></span><a href="filter_1.html#K9" target="_self">]</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:64"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">p1</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6">Top</a> <font color="Maroon">L1</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E8:64"/>
<span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><a href="lattices.html#K5">Bottom</a> <font color="Maroon">L1</font></span>)</span>,<span class="p2">(<span class="default"><a href="lattices.html#K5">Bottom</a> <font color="Maroon">L2</font></span>)</span></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K5">Bottom</a> <span class="p1"><a href="filter_1.html#K7" target="_self">[:</a><span class="default"><font color="Maroon">L1</font>,<font color="Maroon">L2</font></span><a href="filter_1.html#K7" target="_self">:]</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:64"/><b>then </b><i><font color="Green">E80</font></i>: 
<span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">q1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">q2</font></span>)</span></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><a href="lattices.html#K5">Bottom</a> <font color="Maroon">L1</font></span>)</span>,<span class="p2">(<span class="default"><a href="lattices.html#K5">Bottom</a> <font color="Maroon">L2</font></span>)</span></span><a href="filter_1.html#K9" target="_self">]</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:64"/><b>then </b>
<font color="Maroon">p1</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L1</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E11:64"/>
<font color="Maroon">p1</font> <a href="lattices.html#R2">is_a_complement_of</a> <font color="Maroon">q1</font>
 <b>by </b><i>, <a class="ref" href="lattices.html#D18">LATTICES:def 18</a></i>;<br/>

<a NAME="E12:64"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">p2</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6">Top</a> <font color="Maroon">L2</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E13:64"/>
<font color="Maroon">p2</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L2</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E14:64"/>
<font color="Maroon">p2</font> <a href="lattices.html#R2">is_a_complement_of</a> <font color="Maroon">q2</font>
 <b>by </b><i>, <a class="ref" href="lattices.html#D18">LATTICES:def 18</a></i>;<br/>


</div>
