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

<b>let </b><font color="Maroon">L</font> be   <a href="lattices.html#NM1">Lattice</a>;<br/>

<a NAME="E1:76"/><i><font color="Green">E14</font></i>: 
( ( <font color="Maroon">L</font> is <a href="lattices.html#V15">bounded</a> implies ( <font color="Maroon">L</font> is <a href="lattices.html#V13">lower-bounded</a> &amp; <font color="Maroon">L</font> is <a href="lattices.html#V14">upper-bounded</a> ) ) &amp; ( <font color="Maroon">L</font> is <a href="lattices.html#V13">lower-bounded</a> &amp; <font color="Maroon">L</font> is <a href="lattices.html#V14">upper-bounded</a> implies <font color="Maroon">L</font> is <a href="lattices.html#V15">bounded</a> ) &amp; ( <font color="Maroon">L</font> <a href="lattice2.html#K1">.:</a>  is <a href="lattices.html#V15">bounded</a> implies ( <font color="Maroon">L</font> <a href="lattice2.html#K1">.:</a>  is <a href="lattices.html#V13">lower-bounded</a> &amp; <font color="Maroon">L</font> <a href="lattice2.html#K1">.:</a>  is <a href="lattices.html#V14">upper-bounded</a> ) ) &amp; ( <font color="Maroon">L</font> <a href="lattice2.html#K1">.:</a>  is <a href="lattices.html#V13">lower-bounded</a> &amp; <font color="Maroon">L</font> <a href="lattice2.html#K1">.:</a>  is <a href="lattices.html#V14">upper-bounded</a> implies <font color="Maroon">L</font> <a href="lattice2.html#K1">.:</a>  is <a href="lattices.html#V15">bounded</a> ) &amp; ( <font color="Maroon">L</font> is <a href="lattices.html#V13">lower-bounded</a> implies <font color="Maroon">L</font> <a href="lattice2.html#K1">.:</a>  is <a href="lattices.html#V14">upper-bounded</a> ) &amp; ( <font color="Maroon">L</font> <a href="lattice2.html#K1">.:</a>  is <a href="lattices.html#V14">upper-bounded</a> implies <font color="Maroon">L</font> is <a href="lattices.html#V13">lower-bounded</a> ) &amp; ( <font color="Maroon">L</font> is <a href="lattices.html#V14">upper-bounded</a> implies <font color="Maroon">L</font> <a href="lattice2.html#K1">.:</a>  is <a href="lattices.html#V13">lower-bounded</a> ) &amp; ( <font color="Maroon">L</font> <a href="lattice2.html#K1">.:</a>  is <a href="lattices.html#V13">lower-bounded</a> implies <font color="Maroon">L</font> is <a href="lattices.html#V14">upper-bounded</a> ) )
 
<b>by </b><i><a class="ref" href="lattice2.html#T63">LATTICE2:63</a>, <a class="ref" href="lattice2.html#T64">LATTICE2:64</a>, <a class="ref" href="lattices.html#D15">LATTICES:def 15</a></i>;<br/>
<b>thus </b><a NAME="E2:76"/>
( <font color="Maroon">L</font> is   <a href="lattices.html#NM7">C_Lattice</a> implies <font color="Maroon">L</font> <a href="lattice2.html#K1">.:</a>  is   <a href="lattices.html#NM7">C_Lattice</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:76_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">L</font> is   <a href="lattices.html#NM7">C_Lattice</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p'</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lattice2.html#K1">.:</a> </span>)</span>;<br/><b>consider </b><font color="Maroon">q</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E2:76_1_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">q</font> <a href="lattices.html#R2">is_a_complement_of</a>  <a href="filter_2.html#K6" target="_self">.:</a> <font color="Maroon">p'</font>
 <b>by </b><i><a class="ref" href="filter_2.html#T3" target="_self">Th3</a>, <a class="ref" href="lattices.html#D19">LATTICES:def 19</a></i>;<br/><b>take </b><font color="Maroon">r</font> = <font color="Maroon">q</font> <a href="filter_2.html#K5" target="_self">.:</a> ;<br/><a NAME="E3:76_1_1"/>
( <font color="Maroon">L</font> is <a href="lattices.html#V13">lower-bounded</a> &amp; <font color="Maroon">L</font> is <a href="lattices.html#V14">upper-bounded</a> &amp;  <a href="filter_2.html#K6" target="_self">.:</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="filter_2.html#K5" target="_self">.:</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> &amp; <font color="Maroon">q</font> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><a href="filter_2.html#K6" target="_self">.:</a> <font color="Maroon">p'</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6">Top</a> <font color="Maroon">L</font> &amp; <font color="Maroon">q</font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><a href="filter_2.html#K6" target="_self">.:</a> <font color="Maroon">p'</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font> )
 <b>by </b><i><a class="ref" href="filter_2.html#T3" target="_self">Th3</a>, , <a class="ref" href="lattices.html#D18">LATTICES:def 18</a></i>;<br/><a NAME="E4:76_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="filter_2.html#K5" target="_self">.:</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">p'</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6">Top</a> <font color="Maroon">L</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="filter_2.html#K5" target="_self">.:</a> </span>)</span> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">p'</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font> &amp;  <a href="lattices.html#K6">Top</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lattice2.html#K1">.:</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font> &amp;  <a href="lattices.html#K5">Bottom</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lattice2.html#K1">.:</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6">Top</a> <font color="Maroon">L</font> )
 <b>by </b><i><a class="ref" href="lattice2.html#T78">LATTICE2:78</a>, <a class="ref" href="lattice2.html#T79">LATTICE2:79</a></i>;<br/><b>hence </b><a NAME="E5:76_1_1"/>
<font color="Maroon">r</font> <a href="lattices.html#R2">is_a_complement_of</a> <font color="Maroon">p'</font>
 <b>by </b><i><a class="ref" href="lattices.html#D18">LATTICES:def 18</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:76_1"/>
<font color="Maroon">L</font> <a href="lattice2.html#K1">.:</a>  is   <a href="lattices.html#NM7">C_Lattice</a>
 <b>by </b><i>, <a class="ref" href="filter_2.html#T3" target="_self">Th3</a>, <a class="ref" href="lattices.html#D19">LATTICES:def 19</a></i>;<br/>


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

<b>assume </b><a NAME="E3:76"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">L</font> <a href="lattice2.html#K1">.:</a>  is   <a href="lattices.html#NM7">C_Lattice</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>consider </b><font color="Maroon">q'</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lattice2.html#K1">.:</a> </span>)</span><b> such that </b><br/><a NAME="E2:76_2"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">q'</font> <a href="lattices.html#R2">is_a_complement_of</a> <font color="Maroon">p</font> <a href="filter_2.html#K5" target="_self">.:</a> 
 <b>by </b><i><a class="ref" href="filter_2.html#T3" target="_self">Th3</a>, <a class="ref" href="lattices.html#D19">LATTICES:def 19</a></i>;<br/><b>take </b><font color="Maroon">r</font> =  <a href="filter_2.html#K6" target="_self">.:</a> <font color="Maroon">q'</font>;<br/><a NAME="E3:76_2"/>
( <font color="Maroon">L</font> is <a href="lattices.html#V13">lower-bounded</a> &amp; <font color="Maroon">L</font> is <a href="lattices.html#V14">upper-bounded</a> &amp;  <a href="filter_2.html#K6" target="_self">.:</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="filter_2.html#K5" target="_self">.:</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">q'</font> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="filter_2.html#K5" target="_self">.:</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6">Top</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lattice2.html#K1">.:</a> </span>)</span> &amp; <font color="Maroon">q'</font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="filter_2.html#K5" target="_self">.:</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K5">Bottom</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lattice2.html#K1">.:</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="filter_2.html#T3" target="_self">Th3</a>, , <a class="ref" href="lattice2.html#T63">LATTICE2:63</a>, <a class="ref" href="lattice2.html#T64">LATTICE2:64</a>, <a class="ref" href="lattices.html#D18">LATTICES:def 18</a></i>;<br/><a NAME="E4:76_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="filter_2.html#K6" target="_self">.:</a> <font color="Maroon">q'</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6">Top</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lattice2.html#K1">.:</a> </span>)</span> &amp; <span class="p1">(<span class="default"><a href="filter_2.html#K6" target="_self">.:</a> <font color="Maroon">q'</font></span>)</span> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K5">Bottom</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lattice2.html#K1">.:</a> </span>)</span> &amp;  <a href="lattices.html#K6">Top</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lattice2.html#K1">.:</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font> &amp;  <a href="lattices.html#K5">Bottom</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lattice2.html#K1">.:</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6">Top</a> <font color="Maroon">L</font> )
 <b>by </b><i><a class="ref" href="lattice2.html#T78">LATTICE2:78</a>, <a class="ref" href="lattice2.html#T79">LATTICE2:79</a></i>;<br/><b>hence </b><a NAME="E5:76_2"/>
<font color="Maroon">r</font> <a href="lattices.html#R2">is_a_complement_of</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="lattices.html#D18">LATTICES:def 18</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:76"/>
<font color="Maroon">L</font> is   <a href="lattices.html#NM7">C_Lattice</a>
 <b>by </b><i>, <a class="ref" href="filter_2.html#T3" target="_self">Th3</a>, <a class="ref" href="lattices.html#D19">LATTICES:def 19</a></i>;<br/>


</div>
