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

<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   <a href="latstone.html#V3" title="LATSTONE:attr.3">Stone</a>  <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">(<span class="default"><a href="latstone.html#K3" title="LATSTONE:func.3">Skeleton</a> <font color="Maroon" title="c1">L</font></span>)</span>) is  <a href="lattices.html#V16" title="LATTICES:attr.16">complemented</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">P</font> =  <a href="latstone.html#K3" title="LATSTONE:func.3">Skeleton</a> <font color="Maroon" title="c1">L</font>;<br/>
<a NAME="E1:43"/><span class="lab"><font color="Green" title="E22">A0</font></span>: 
(  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="latstone.html#K3" title="LATSTONE:func.3">Skeleton</a> <font color="Maroon" title="c1">L</font> &amp;  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="latstone.html#K3" title="LATSTONE:func.3">Skeleton</a> <font color="Maroon" title="c1">L</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="latstone.html#T21" target="_self" onmouseover="rs('latstone/T21')" onmouseout="rh()">Lm1</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">LL</font> =  <a href="filter_2.html#K10" title="FILTER_2:func.10">latt</a> (<font color="Maroon" title="c1">L</font>,<span class="p1">(<span class="default"><a href="latstone.html#K3" title="LATSTONE:func.3">Skeleton</a> <font color="Maroon" title="c1">L</font></span>)</span>);<br/>
<a NAME="E2:43"/>
 for <font color="Olive" title="b1">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="filter_2.html#K10" title="FILTER_2:func.10">latt</a> (<font color="Maroon" title="c1">L</font>,<span class="p2">(<span class="default"><a href="latstone.html#K3" title="LATSTONE:func.3">Skeleton</a> <font color="Maroon" title="c1">L</font></span>)</span>)</span>)</span>  ex <font color="Olive" title="b2">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="filter_2.html#K10" title="FILTER_2:func.10">latt</a> (<font color="Maroon" title="c1">L</font>,<span class="p2">(<span class="default"><a href="latstone.html#K3" title="LATSTONE:func.3">Skeleton</a> <font color="Maroon" title="c1">L</font></span>)</span>)</span>)</span> st <font color="Olive" title="b2">a</font> <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <font color="Olive" title="b1">b</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="43_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="filter_2.html#K10" title="FILTER_2:func.10">latt</a> (<font color="Maroon" title="c1">L</font>,<span class="p2">(<span class="default"><a href="latstone.html#K3" title="LATSTONE:func.3">Skeleton</a> <font color="Maroon" title="c1">L</font></span>)</span>)</span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="filter_2.html#K10" title="FILTER_2:func.10">latt</a> (<font color="Maroon" title="c1">L</font>,<span class="p2">(<span class="default"><a href="latstone.html#K3" title="LATSTONE:func.3">Skeleton</a> <font color="Maroon" title="c1">L</font></span>)</span>)</span>)</span> st <font color="Olive" title="b1">a</font> <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <font color="Maroon" title="c4">b</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">bb</font> = <font color="Maroon" title="c4">b</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" onmouseover="rs('filter_2/T68')" onmouseout="rh()">FILTER_2:68</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">aa</font> = <font color="Maroon" title="c5">bb</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a> ;<br/>
<a NAME="E2:43_1"/>
<font color="Maroon" title="c5">bb</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="latstone.html#K3" title="LATSTONE:func.3">Skeleton</a> <font color="Maroon" title="c1">L</font>
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">a</font> = <font color="Maroon" title="c5">bb</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a>  as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="filter_2.html#K10" title="FILTER_2:func.10">latt</a> (<font color="Maroon" title="c1">L</font>,<span class="p2">(<span class="default"><a href="latstone.html#K3" title="LATSTONE:func.3">Skeleton</a> <font color="Maroon" title="c1">L</font></span>)</span>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="filter_2.html#T72" onmouseover="rs('filter_2/T72')" onmouseout="rh()">FILTER_2:72</a></span>;<br/>
<a NAME="E4:43_1"/>
<font color="Maroon" title="c4">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="filter_2.html#K10" title="FILTER_2:func.10">latt</a> (<font color="Maroon" title="c1">L</font>,<span class="p2">(<span class="default"><a href="latstone.html#K3" title="LATSTONE:func.3">Skeleton</a> <font color="Maroon" title="c1">L</font></span>)</span>)</span>)</span>
 
;<br/>
<a NAME="E5:43_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">z1</font></span>: 
<font color="Maroon" title="c4">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="latstone.html#K3" title="LATSTONE:func.3">Skeleton</a> <font color="Maroon" title="c1">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_2.html#T72" onmouseover="rs('filter_2/T72')" onmouseout="rh()">FILTER_2:72</a></span>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">bb</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c5">bb</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">bb</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a></span>)</span> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a></span>)</span> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c5">bb</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43_1"><span class="lab"><font color="Green" title="E23">z1</font></span></a>, <a class="ref" href="latstone.html#T16" target="_self" onmouseover="rs('latstone/T16')" onmouseout="rh()">Th13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">bb</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a></span>)</span> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a></span>)</span> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">bb</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a></span>)</span> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43_1"><span class="lab"><font color="Green" title="E23">z1</font></span></a>, <a class="ref" href="latstone.html#T16" target="_self" onmouseover="rs('latstone/T16')" onmouseout="rh()">Th13</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="latstone.html#D5" target="_self" onmouseover="rs('latstone/D5')" onmouseout="rh()">SatStone</a></span>
;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E24">A1</font></span>: <font color="Maroon" title="c7">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">b</font> = 
 <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="filter_2.html#T73" onmouseover="rs('filter_2/T73')" onmouseout="rh()">FILTER_2:73</a></span>
<br/>.= 
 <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <span class="p1">(<span class="default"><a href="filter_2.html#K10" title="FILTER_2:func.10">latt</a> (<font color="Maroon" title="c1">L</font>,<span class="p2">(<span class="default"><a href="latstone.html#K3" title="LATSTONE:func.3">Skeleton</a> <font color="Maroon" title="c1">L</font></span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E22">A0</font></span></a>, <a class="ref" href="latstone.html#T3" target="_self" onmouseover="rs('latstone/T3')" onmouseout="rh()">ThC</a></span>
;<br/>
<a NAME="E8:43_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">bb</font> <a href="latstone.html#K1" title="LATSTONE:func.1">*</a></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">bb</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="ref" href="latstone.html#T6" target="_self" onmouseover="rs('latstone/T6')" onmouseout="rh()">ThD</a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c7">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">b</font> = 
 <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="ref" href="filter_2.html#T73" onmouseover="rs('filter_2/T73')" onmouseout="rh()">FILTER_2:73</a></span>
<br/>.= 
 <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <span class="p1">(<span class="default"><a href="filter_2.html#K10" title="FILTER_2:func.10">latt</a> (<font color="Maroon" title="c1">L</font>,<span class="p2">(<span class="default"><a href="latstone.html#K3" title="LATSTONE:func.3">Skeleton</a> <font color="Maroon" title="c1">L</font></span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E22">A0</font></span></a>, <a class="ref" href="latstone.html#T2" target="_self" onmouseover="rs('latstone/T2')" onmouseout="rh()">ThB</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E10:43_1"/>
 ex <font color="Olive" title="b1">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="filter_2.html#K10" title="FILTER_2:func.10">latt</a> (<font color="Maroon" title="c1">L</font>,<span class="p2">(<span class="default"><a href="latstone.html#K3" title="LATSTONE:func.3">Skeleton</a> <font color="Maroon" title="c1">L</font></span>)</span>)</span>)</span> st <font color="Olive" title="b1">a</font> <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <font color="Maroon" title="c4">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D18" onmouseover="rs('lattices/D18')" onmouseout="rh()">LATTICES:def 18</a>, <a class="txt" href="#E7:43_1"><span class="lab"><font color="Green" title="E24">A1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E3:43"/>
 <a href="filter_2.html#K10" title="FILTER_2:func.10">latt</a> (<font color="Maroon" title="c1">L</font>,<span class="p1">(<span class="default"><a href="latstone.html#K3" title="LATSTONE:func.3">Skeleton</a> <font color="Maroon" title="c1">L</font></span>)</span>) is  <a href="lattices.html#V16" title="LATTICES:attr.16">complemented</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D19" onmouseover="rs('lattices/D19')" onmouseout="rh()">LATTICES:def 19</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
