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

<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> =  <a href="filter_2.html#K12" target="_self">.:</a> <font color="Maroon">c<sub>2</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E2:36_1_1"/>
<font color="Maroon">c<sub>3</sub></font> is    <a href="filter_2.html#M1" target="_self">Filter</a> of <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="filter_0.html#D1">FILTER_0:def 1</a><br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<a NAME="E1:36_1_1_1"/>
(  <a href="filter_2.html#K12" target="_self">.:</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font> <a href="lattice2.html#K1">.:</a> ) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font>) &amp; <font color="Maroon">c<sub>4</sub></font> <a href="filter_2.html#K5" target="_self">.:</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="filter_2.html#K5" target="_self">.:</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="filter_2.html#K5" target="_self">.:</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="filter_2.html#K5" target="_self">.:</a> </span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="filter_2.html#K5" target="_self">.:</a> </span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="filter_2.html#K5" target="_self">.:</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>5</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="lattice2.html#K1">.:</a> </span>)</span> holds <br/>( ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> ) iff <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> ) ) )
 
<b>by </b><i><a class="ref" href="filter_2.html#D3" target="_self">Def3</a></i>;<br/>
<b>hence </b><a NAME="E2:36_1_1_1"/>
( (  not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> or  not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) &amp; (  not <font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> or ( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) ) )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:36_1_1"/>
 <a href="filter_2.html#K8" target="_self">.:</a> <font color="Maroon">c<sub>2</sub></font> is    <a href="filter_2.html#M1" target="_self">Filter</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>


</div>
