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<b>let </b><font color="Maroon">L</font> be   <a href="lattices.html#NM1">Lattice</a>;<br/><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>assume </b><a NAME="E1:115"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">L</font> is   <a href="lattices.html#NM7">C_Lattice</a> &amp; <font color="Maroon">L</font> is <a href="lattices.html#V12">modular</a> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">K</font> =  <a href="filter_2.html#K23" target="_self">latt</a> <font color="Maroon">L</font>,<span class="p1"><a href="filter_2.html#K18" target="_self">(.</a><span class="default"><font color="Maroon">p</font></span><a href="filter_2.html#K18" target="_self">.&gt;</a></span> as  <a href="lattices.html#V15">bounded</a> <a href="lattices.html#NM1">Lattice</a> <b>by </b><i/>;<br/>
<a NAME="E3:115"/>
<font color="Maroon">K</font> is <a href="lattices.html#V16">complemented</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D19">LATTICES:def 19</a><br/>

<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:115_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">a</font> <a href="lattices.html#R2">is_a_complement_of</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="lattices.html#D19">LATTICES:def 19</a></i>;<br/>
<a NAME="E4:115_1"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">a</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6">Top</a> <font color="Maroon">L</font> &amp; <font color="Maroon">a</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">b</font> <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="lattices.html#D18">LATTICES:def 18</a></i>;<br/>
<a NAME="E5:115_1"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">p</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">K</font>) <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_2.html#K18" target="_self">(.</a><span class="default"><font color="Maroon">p</font></span><a href="filter_2.html#K18" target="_self">.&gt;</a></span> )
 
<b>by </b><i>, <a class="ref" href="lattices.html#T23">LATTICES:23</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">p</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">a</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">a'</font>
;<br/>

<a NAME="E7:115_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">b</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">p</font>
 
<b>by </b><i>, </i>;<br/>
<b>thus </b><font color="Maroon">a'</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">b'</font> = 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">a</font></span>)</span> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">b</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">p</font>
<b>by </b><i>, , <a class="ref" href="lattices.html#D12">LATTICES:def 12</a></i>
<br/>.= 
<font color="Maroon">p</font>
<b>by </b><i>, , <a class="ref" href="lattices.html#T43">LATTICES:43</a></i>
<br/>.= 
 <a href="lattices.html#K6">Top</a> <font color="Maroon">K</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="E9:115_1"/>
<font color="Maroon">b'</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6">Top</a> <font color="Maroon">K</font>
 ;<br/>

<b>thus </b><font color="Maroon">a'</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">b'</font> = 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">a</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">b</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">p</font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i>, <a class="ref" href="lattices.html#D7">LATTICES:def 7</a></i>
<br/>.= 
 <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font>
<b>by </b><i>, <a class="ref" href="lattices.html#T40">LATTICES:40</a></i>
<br/>.= 
 <a href="lattices.html#K5">Bottom</a> <font color="Maroon">K</font>
<b>by </b><i>, </i>
;<br/>

<b>hence </b><a NAME="E11:115_1"/>
<font color="Maroon">b'</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">K</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:115"/>
 <a href="filter_2.html#K23" target="_self">latt</a> <font color="Maroon">L</font>,<span class="p1"><a href="filter_2.html#K18" target="_self">(.</a><span class="default"><font color="Maroon">p</font></span><a href="filter_2.html#K18" target="_self">.&gt;</a></span> is   <a href="lattices.html#NM7">C_Lattice</a>
 ;<br/>


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