<?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="Red">::  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#NM3" title="LATTICES:NM.3">M_Lattice</a> holds <br/> <a href="filter_0.html#K8" title="FILTER_0:func.8">latt</a> <span class="p1"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Olive" title="b1">p</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</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="Red">::  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#NM3" title="LATTICES:NM.3">M_Lattice</a> implies  <a href="filter_0.html#K8" title="FILTER_0:func.8">latt</a> <span class="p1"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</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="E1:89"/><span class="lab"><font color="Green" title="E56">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="E2:89"/><span class="lab"><font color="Green" title="E57">A2</font></span>: 
<font color="Maroon" title="c1">L</font> is   <a href="lattices.html#NM3" title="LATTICES:NM.3">M_Lattice</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="filter_0.html#K8" title="FILTER_0:func.8">latt</a> <span class="p1"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</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">B</font> = <font color="Maroon" title="c1">L</font> as   <a href="lattices.html#NM7" title="LATTICES:NM.7">C_Lattice</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:89"><span class="lab"><font color="Green" title="E56">A1</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">M</font> =  <a href="filter_0.html#K8" title="FILTER_0:func.8">latt</a> <span class="p1"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span> as   <a href="lattices.html#NM6" title="LATTICES:NM.6">01_Lattice</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:89"><span class="lab"><font color="Green" title="E56">A1</font></span></a>, <a class="ref" href="filter_0.html#T73" target="_self" title="FILTER_0:th.73">Th73</a></span>;<br/>
<a NAME="E5:89"/>
<font color="Maroon" title="c4">M</font> is  <a href="lattices.html#V16" title="LATTICES:attr.16">complemented</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">r9</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">M</font>; <span class="comment"><font color="Red">:: 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="Red">::  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="c4">M</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="c5">r9</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">r</font> = <font color="Maroon" title="c5">r9</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span> <span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T63" target="_self" title="FILTER_0:th.63">Th63</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">p1</font> = <font color="Maroon" title="c2">p</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">B</font> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">q</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="E4:89_1"/><span class="lab"><font color="Green" title="E58">A3</font></span>: 
<font color="Maroon" title="c8">q</font> <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <font color="Maroon" title="c6">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:89"><span class="lab"><font color="Green" title="E56">A1</font></span></a>, <a class="ref" href="lattices.html#D19" title="LATTICES:def.19">LATTICES:def 19</a></span>;<br/>
<a NAME="E5:89_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="filter_0.html#K8" title="FILTER_0:func.8">latt</a> <span class="p2"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T63" target="_self" title="FILTER_0:th.63">Th63</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">q9</font> = <font color="Maroon" title="c2">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c8">q</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">M</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T19" target="_self" title="FILTER_0:th.19">Th19</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c9">q9</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">q9</font> <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <font color="Maroon" title="c5">r9</font></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c9">q9</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c5">r9</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c8">q</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c6">r</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T64" target="_self" title="FILTER_0:th.64">Th64</a></span>
<br/>.= 
<font color="Maroon" title="c2">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c6">r</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D5" title="LATTICES:def.5">LATTICES:def 5</a></span>
<br/>.= 
<font color="Maroon" title="c7">p1</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c3">B</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:89_1"><span class="lab"><font color="Green" title="E58">A3</font></span></a>, <a class="ref" href="lattices.html#D18" title="LATTICES:def.18">LATTICES:def 18</a></span>
<br/>.= 
 <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="ref" href="lattices.html#T44" title="LATTICES:th.44">LATTICES:44</a></span>
<br/>.= 
 <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c4">M</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:89"><span class="lab"><font color="Green" title="E56">A1</font></span></a>, <a class="ref" href="filter_0.html#T72" target="_self" title="FILTER_0:th.72">Th72</a></span>
; <span class="comment"><font color="Red">:: 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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">r9</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c9">q9</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="c4">M</font> &amp; <font color="Maroon" title="c9">q9</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c5">r9</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="c4">M</font> &amp; <font color="Maroon" title="c5">r9</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c9">q9</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="c4">M</font> )</span><br/>

<span class="kw">hence </span><a NAME="E8:89_1"/>
<font color="Maroon" title="c5">r9</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c9">q9</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="c4">M</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">q9</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c5">r9</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="c4">M</font> &amp; <font color="Maroon" title="c5">r9</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c9">q9</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="c4">M</font> )</span><br/>

<a NAME="E9:89_1"/>
<font color="Maroon" title="c2">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T18" target="_self" title="FILTER_0:th.18">Th18</a></span>;<br/>
<a NAME="E10:89_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c8">q</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c6">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c6">r</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:89"><span class="lab"><font color="Green" title="E57">A2</font></span></a>, <a class="ref" href="lattices.html#D12" title="LATTICES:def.12">LATTICES:def 12</a></span>;<br/>
<span class="kw">hence </span><font color="Maroon" title="c9">q9</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">r9</font> = 
<font color="Maroon" title="c2">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c6">r</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T64" target="_self" title="FILTER_0:th.64">Th64</a></span>
<br/>.= 
<font color="Maroon" title="c7">p1</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c3">B</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:89_1"><span class="lab"><font color="Green" title="E58">A3</font></span></a>, <a class="ref" href="lattices.html#D18" title="LATTICES:def.18">LATTICES:def 18</a></span>
<br/>.= 
<font color="Maroon" title="c2">p</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T39" title="LATTICES:th.39">LATTICES:39</a></span>
<br/>.= 
 <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c4">M</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T71" target="_self" title="FILTER_0:th.71">Th71</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">r9</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c9">q9</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="c4">M</font></span><br/>

<span class="kw">hence </span><a NAME="E12:89_1"/>
<font color="Maroon" title="c5">r9</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c9">q9</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="c4">M</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:89"/>
 <a href="filter_0.html#K8" title="FILTER_0:func.8">latt</a> <span class="p1"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</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="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
