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

<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</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" title="b1">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Maroon" title="c1">L</font> is   <a href="lattices.html#NM7" title="LATTICES:NM.7">C_Lattice</a> &amp; <font color="Maroon" title="c1">L</font> is  <a href="lattices.html#V12" title="LATTICES:attr.12">modular</a>  holds <br/> <a href="filter_2.html#K10" title="FILTER_2:func.10">latt</a> (<font color="Maroon" title="c1">L</font>,<span class="p1"><a href="filter_2.html#K6" title="FILTER_2:func.6">(.</a><span class="default"><font color="Olive" title="b1">p</font></span><a href="filter_2.html#K6" title="FILTER_2:func.6">.&gt;</a></span>) is   <a href="lattices.html#NM7" title="LATTICES:NM.7">C_Lattice</a></span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <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="c1">L</font> is   <a href="lattices.html#NM7" title="LATTICES:NM.7">C_Lattice</a> &amp; <font color="Maroon" title="c1">L</font> is  <a href="lattices.html#V12" title="LATTICES:attr.12">modular</a>  implies  <a href="filter_2.html#K10" title="FILTER_2:func.10">latt</a> (<font color="Maroon" title="c1">L</font>,<span class="p1"><a href="filter_2.html#K6" title="FILTER_2:func.6">(.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_2.html#K6" title="FILTER_2:func.6">.&gt;</a></span>) is   <a href="lattices.html#NM7" title="LATTICES:NM.7">C_Lattice</a> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:111"/><span class="lab"><font color="Green" title="E46">A1</font></span>: 
<font color="Maroon" title="c1">L</font> is   <a href="lattices.html#NM7" title="LATTICES:NM.7">C_Lattice</a>
 <span class="kw">and </span><br/><a NAME="E2:111"/><span class="lab"><font color="Green" title="E47">A2</font></span>: 
<font color="Maroon" title="c1">L</font> is  <a href="lattices.html#V12" title="LATTICES:attr.12">modular</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="filter_2.html#K10" title="FILTER_2:func.10">latt</a> (<font color="Maroon" title="c1">L</font>,<span class="p1"><a href="filter_2.html#K6" title="FILTER_2:func.6">(.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_2.html#K6" title="FILTER_2:func.6">.&gt;</a></span>) is   <a href="lattices.html#NM7" title="LATTICES:NM.7">C_Lattice</a></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c3">K</font> =  <a href="filter_2.html#K10" title="FILTER_2:func.10">latt</a> (<font color="Maroon" title="c1">L</font>,<span class="p1"><a href="filter_2.html#K6" title="FILTER_2:func.6">(.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_2.html#K6" title="FILTER_2:func.6">.&gt;</a></span>) as   <a href="lattices.html#V15" title="LATTICES:attr.15">bounded</a>  <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:111"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="ref" href="filter_2.html#T81" target="_self" title="FILTER_2:th.81">Th83</a></span>;<br/>
<a NAME="E4:111"/>
<font color="Maroon" title="c3">K</font> is  <a href="lattices.html#V16" title="LATTICES:attr.16">complemented</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="111_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">b9</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">K</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="lattices.html#D19" title="LATTICES:def.19">LATTICES:def 19</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <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 <font color="Maroon" title="c3">K</font> st <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <font color="Maroon" title="c4">b9</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">b</font> = <font color="Maroon" title="c4">b9</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="filter_2.html#T68" target="_self" title="FILTER_2:th.68">Th69</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E3:111_1"/><span class="lab"><font color="Green" title="E48">A3</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <font color="Maroon" title="c5">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:111"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="ref" href="lattices.html#D19" title="LATTICES:def.19">LATTICES:def 19</a></span>;<br/>
<a NAME="E4:111_1"/><span class="lab"><font color="Green" title="E49">A4</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:111_1"><span class="lab"><font color="Green" title="E48">A3</font></span></a>, <a class="ref" href="lattices.html#D18" title="LATTICES:def.18">LATTICES:def 18</a></span>;<br/>
<a NAME="E5:111_1"/><span class="lab"><font color="Green" title="E50">A5</font></span>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">K</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_2.html#K6" title="FILTER_2:func.6">(.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_2.html#K6" title="FILTER_2:func.6">.&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_2.html#T72" target="_self" title="FILTER_2:th.72">Th73</a></span>;<br/>
<a NAME="E6:111_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A6</font></span>: 
<font color="Maroon" title="c5">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_2.html#T28" target="_self" title="FILTER_2:th.28">Th29</a></span>;<br/>
<a NAME="E7:111_1"/>
<font color="Maroon" title="c2">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c6">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T6" title="LATTICES:th.6">LATTICES:6</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">a9</font> = <font color="Maroon" title="c2">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c6">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">K</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:111_1"><span class="lab"><font color="Green" title="E50">A5</font></span></a>, <a class="ref" href="filter_2.html#T28" target="_self" title="FILTER_2:th.28">Th29</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">a9</font>
; <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="c7">a9</font> <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <font color="Maroon" title="c4">b9</font></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c7">a9</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">b9</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c5">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_2.html#T73" target="_self" title="FILTER_2:th.73">Th74</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c2">p</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:111"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="txt" href="#E6:111_1"><span class="lab"><font color="Green" title="E51">A6</font></span></a>, <a class="ref" href="lattices.html#D12" title="LATTICES:def.12">LATTICES:def 12</a></span>
<br/>.= 
<font color="Maroon" title="c2">p</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:111"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E4:111_1"><span class="lab"><font color="Green" title="E49">A4</font></span></a>, <a class="ref" href="lattices.html#T17" title="LATTICES:th.17">LATTICES:17</a></span>
<br/>.= 
 <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c3">K</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_2.html#T79" target="_self" title="FILTER_2:th.79">Th81</a></span>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="lattices.html#D18" title="LATTICES:def.18">LATTICES:def 18</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">b9</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c7">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c3">K</font> &amp; <font color="Maroon" title="c7">a9</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c3">K</font> &amp; <font color="Maroon" title="c4">b9</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c7">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c3">K</font> )</span><br/>

<span class="kw">hence </span><a NAME="E10:111_1"/>
<font color="Maroon" title="c4">b9</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c7">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c3">K</font>
 ; <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="c7">a9</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c3">K</font> &amp; <font color="Maroon" title="c4">b9</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c7">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c3">K</font> )</span><br/>

<a NAME="E11:111_1"/><span class="lab"><font color="Green" title="E52">A7</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:111_1"><span class="lab"><font color="Green" title="E48">A3</font></span></a>, <a class="ref" href="lattices.html#D18" title="LATTICES:def.18">LATTICES:def 18</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c7">a9</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">b9</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_2.html#T73" target="_self" title="FILTER_2:th.73">Th74</a></span>
<br/>.= 
<font color="Maroon" title="c2">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:111_1"><span class="lab"><font color="Green" title="E52">A7</font></span></a>, <a class="ref" href="lattices.html#D7" title="LATTICES:def.7">LATTICES:def 7</a></span>
<br/>.= 
 <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:111"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="ref" href="lattices.html#T15" title="LATTICES:th.15">LATTICES:15</a></span>
<br/>.= 
 <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c3">K</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:111"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="ref" href="filter_2.html#T80" target="_self" title="FILTER_2:th.80">Th82</a></span>
; <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">b9</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c7">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c3">K</font></span><br/>

<span class="kw">hence </span><a NAME="E13:111_1"/>
<font color="Maroon" title="c4">b9</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c7">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c3">K</font>
 ; <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="E5:111"/>
 <a href="filter_2.html#K10" title="FILTER_2:func.10">latt</a> (<font color="Maroon" title="c1">L</font>,<span class="p1"><a href="filter_2.html#K6" title="FILTER_2:func.6">(.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_2.html#K6" title="FILTER_2:func.6">.&gt;</a></span>) is   <a href="lattices.html#NM7" title="LATTICES:NM.7">C_Lattice</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>
