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

<a NAME="E1:29_1_1"/>
<font color="Maroon" title="c2">F</font> <a href="filter_2.html#K3" title="FILTER_2:func.3">.:</a>  is  <a href="lattices.html#V19" title="LATTICES:attr.19">final</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">p9</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a></span>)</span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="lattices.html#D23" title="LATTICES:def.23">LATTICES:def 23</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a></span>)</span>  holds <br/>(  not <font color="Maroon" title="c3">p9</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive">b<sub>1</sub></font> or  not <font color="Maroon" title="c3">p9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="filter_2.html#K3" title="FILTER_2:func.3">.:</a>  or <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="filter_2.html#K3" title="FILTER_2:func.3">.:</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">q9</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">p9</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">q9</font> or  not <font color="Maroon" title="c3">p9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="filter_2.html#K3" title="FILTER_2:func.3">.:</a>  or <font color="Maroon" title="c4">q9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="filter_2.html#K3" title="FILTER_2:func.3">.:</a>  )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:29_1_1_1"/><span class="lab"><font color="Green" title="E10">A1</font></span>: 
<font color="Maroon" title="c3">p9</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">q9</font>
 <span class="kw">and </span><br/><a NAME="E2:29_1_1_1"/><span class="lab"><font color="Green" title="E11">A2</font></span>: 
<font color="Maroon" title="c3">p9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="filter_2.html#K3" title="FILTER_2:func.3">.:</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">q9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="filter_2.html#K3" title="FILTER_2:func.3">.:</a> </span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">p</font> = <font color="Maroon" title="c3">p9</font> <a href="filter_2.html#K1" title="FILTER_2:func.1">.:</a> , <font color="Maroon" title="c6">q</font> = <font color="Maroon" title="c4">q9</font> <a href="filter_2.html#K1" title="FILTER_2:func.1">.:</a>  as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<a NAME="E4:29_1_1_1"/>
<font color="Maroon" title="c6">q</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c5">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_1_1_1"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="ref" href="lattice2.html#T39" title="LATTICE2:th.39">LATTICE2:39</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:29_1_1_1"/>
<font color="Maroon" title="c4">q9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="filter_2.html#K3" title="FILTER_2:func.3">.:</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_1_1_1"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="ref" href="lattices.html#D22" title="LATTICES:def.22">LATTICES:def 22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:29_1_1"/>
 for <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a></span>)</span>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="filter_2.html#K3" title="FILTER_2:func.3">.:</a>  holds <br/><font color="Olive">b<sub>1</sub></font> is  <a href="lattices.html#V19" title="LATTICES:attr.19">final</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
