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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="lattices.html#NM8">B_Lattice</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="filter_0.html#M1">Filter</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>set </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="filter_1.html#K4" target="_self">/\/</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="filter_0.html#K12">equivalence_wrt</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E1:31"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is   <a href="filter_0.html#NM2">I_Lattice</a> &amp; <font color="Maroon">c<sub>1</sub></font> is   <a href="lattices.html#NM4">0_Lattice</a> )
 
<b>by </b><i><a class="ref" href="filter_0.html#T40">FILTER_0:40</a></i>;<br/>
<a NAME="E2:31"/><b>then </b><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="filter_1.html#K4" target="_self">/\/</a> <font color="Maroon">c<sub>2</sub></font> is   <a href="lattices.html#NM4">0_Lattice</a> &amp;  <a href="lattices.html#K5">Bottom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="filter_1.html#K4" target="_self">/\/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lattices.html#K5">Bottom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="filter_1.html#K5" target="_self">/\/</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="filter_1.html#K4" target="_self">/\/</a> <font color="Maroon">c<sub>2</sub></font> is   <a href="lattices.html#NM5">1_Lattice</a> &amp;  <a href="lattices.html#K6">Top</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="filter_1.html#K4" target="_self">/\/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="filter_1.html#K5" target="_self">/\/</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="filter_1.html#T18" target="_self">Th18</a>, <a class="ref" href="filter_1.html#T19" target="_self">Th19</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="filter_1.html#K4" target="_self">/\/</a> <font color="Maroon">c<sub>2</sub></font> as   <a href="lattices.html#NM6">01_Lattice</a> <b>by </b><i><a class="ref" href="lattices.html#D15">LATTICES:def 15</a></i>;<br/>
<a NAME="E4:31"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="lattices.html#V16">complemented</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>5</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D19">LATTICES:def 19</a><br/>

<a NAME="E1:31_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <span class="p2">(<span class="default"><a href="filter_0.html#K12">equivalence_wrt</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default">the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font> <a href="filter_1.html#K3" target="_self">/\/</a> <span class="p2">(<span class="default"><a href="filter_0.html#K12">equivalence_wrt</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default">the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">c<sub>1</sub></font> <a href="filter_1.html#K3" target="_self">/\/</a> <span class="p2">(<span class="default"><a href="filter_0.html#K12">equivalence_wrt</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> #)
 
<b>by </b><i><a class="txt" href="filter_1.html#E1:31"><i><font color="Green">E23</font></i></a>, <a class="ref" href="filter_1.html#D5" target="_self">Def5</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:31_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="filter_0.html#K12">equivalence_wrt</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T45">EQREL_1:45</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="filter_1.html#K5" target="_self">/\/</a> <font color="Maroon">c<sub>2</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>5</sub></font> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>8</sub></font>
;<br/>

<a NAME="E5:31_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="filter_1.html#K5" target="_self">/\/</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="filter_1.html#E1:31"><i><font color="Green">E23</font></i></a>, <a class="txt" href="filter_1.html#E3:31_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="filter_1.html#D6" target="_self">Def6</a></i>;<br/>
<b>hence </b><font color="Maroon">c<sub>8</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>6</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="filter_1.html#K5" target="_self">/\/</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="filter_1.html#E1:31"><i><font color="Green">E23</font></i></a>, <a class="ref" href="filter_1.html#T15" target="_self">Th15</a></i>
<br/>.= 
 <a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="filter_1.html#E2:31"><i><font color="Green">E24</font></i></a>, <a class="ref" href="lattices.html#T48">LATTICES:48</a></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="E7:31_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>thus </b><font color="Maroon">c<sub>8</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>6</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="filter_1.html#K5" target="_self">/\/</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="filter_1.html#E1:31"><i><font color="Green">E23</font></i></a>, <a class="txt" href="filter_1.html#E5:31_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="filter_1.html#T15" target="_self">Th15</a></i>
<br/>.= 
 <a href="lattices.html#K5">Bottom</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="filter_1.html#E2:31"><i><font color="Green">E24</font></i></a>, <a class="ref" href="lattices.html#T47">LATTICES:47</a></i>
;<br/>

<b>hence </b><a NAME="E9:31_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:31"/>
<font color="Maroon">c<sub>5</sub></font> is   <a href="filter_0.html#NM2">I_Lattice</a>
 
<b>by </b><i><a class="txt" href="filter_1.html#E1:31"><i><font color="Green">E23</font></i></a>, <a class="ref" href="filter_1.html#T20" target="_self">Th20</a></i>;<br/>
<b>hence </b><a NAME="E6:31"/>
<font color="Maroon">c<sub>1</sub></font> <a href="filter_1.html#K4" target="_self">/\/</a> <font color="Maroon">c<sub>2</sub></font> is   <a href="lattices.html#NM8">B_Lattice</a>
 <b>by </b><i><a class="txt" href="filter_1.html#E4:31"><i><font color="Green">E25</font></i></a>, <a class="ref" href="lattices.html#D20">LATTICES:def 20</a></i>;<br/>


</div>
