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<b>let </b><font color="Maroon" title="c1">L</font> be   <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a>;<br/><b>let </b><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/>





<b>assume </b><a NAME="E1:116"/><i><font color="Green" title="E50">A1</font></i>: 
( <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> &amp; <font color="Maroon" title="c2">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">q</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c4">K</font> =  <a href="filter_2.html#K23" title="FILTER_2:func.23">latt</a> <font color="Maroon" title="c1">L</font>,<span class="p1"><a href="filter_2.html#K22" title="FILTER_2:func.22">[#</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="filter_2.html#K22" title="FILTER_2:func.22">#]</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> <b>by </b><i><a class="ref" href="filter_2.html#T84" target="_self" title="FILTER_2:th.84">Th84</a></i>;<br/>
<a NAME="E3:116"/>
<font color="Maroon" title="c4">K</font> is <a href="lattices.html#V16" title="LATTICES:attr.16">complemented</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">b'</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">K</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D19" title="LATTICES:def.19">LATTICES:def 19</a><br/>

<b>reconsider </b><font color="Maroon" title="c6">b</font> = <font color="Maroon" title="c5">b'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="filter_2.html#T69" target="_self" title="FILTER_2:th.69">Th69</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E3:116_1"/><i><font color="Green" title="E51">A2</font></i>: 
<font color="Maroon" title="c7">a</font> <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="txt" href="filter_2.html#E1:116"><i><font color="Green" title="E50">A1</font></i></a>, <a class="ref" href="lattices.html#D19" title="LATTICES:def.19">LATTICES:def 19</a></i>;<br/>
<a NAME="E4:116_1"/><i><font color="Green" title="E52">A3</font></i>: 
( <font color="Maroon" title="c7">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c6">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> &amp; <font color="Maroon" title="c7">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c6">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> )
 
<b>by </b><i><a class="txt" href="filter_2.html#E3:116_1"><i><font color="Green" title="E51">A2</font></i></a>, <a class="ref" href="lattices.html#D18" title="LATTICES:def.18">LATTICES:def 18</a></i>;<br/>
<a NAME="E5:116_1"/>
<font color="Maroon" title="c7">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">q</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">q</font>
 
<b>by </b><i><a class="ref" href="lattices.html#T23" title="LATTICES:th.23">LATTICES:23</a></i>;<br/>
<a NAME="E6:116_1"/><b>then </b><i><font color="Green" title="E53">A4</font></i>: 
( <font color="Maroon" title="c2">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</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="c7">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">q</font></span>)</span> &amp; <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="c7">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">q</font> &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">K</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_2.html#K22" title="FILTER_2:func.22">[#</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="filter_2.html#K22" title="FILTER_2:func.22">#]</a></span> )
 
<b>by </b><i><a class="txt" href="filter_2.html#E1:116"><i><font color="Green" title="E50">A1</font></i></a>, <a class="ref" href="filter_2.html#T73" target="_self" title="FILTER_2:th.73">Th73</a>, <a class="ref" href="filter_0.html#T6" title="FILTER_0:th.6">FILTER_0:6</a>, <a class="ref" href="lattices.html#T22" title="LATTICES:th.22">LATTICES:22</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">a'</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="c7">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">q</font></span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">K</font> <b>by </b><i><a class="txt" href="filter_2.html#E1:116"><i><font color="Green" title="E50">A1</font></i></a>, <a class="ref" href="filter_2.html#T63" target="_self" title="FILTER_2:th.63">Th63</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">a'</font>
;<br/>

<a NAME="E8:116_1"/><i><font color="Green" title="E54">A5</font></i>: 
( <font color="Maroon" title="c6">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">q</font> &amp; <font color="Maroon" title="c2">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">b</font> )
 
<b>by </b><i><a class="txt" href="filter_2.html#E1:116"><i><font color="Green" title="E50">A1</font></i></a>, <a class="txt" href="filter_2.html#E6:116_1"><i><font color="Green" title="E53">A4</font></i></a>, <a class="ref" href="filter_2.html#T63" target="_self" title="FILTER_2:th.63">Th63</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c8">a'</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c5">b'</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c6">b</font>
<b>by </b><i><a class="ref" href="filter_2.html#T74" target="_self" title="FILTER_2:th.74">Th74</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">q</font></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D5" title="LATTICES:def.5">LATTICES:def 5</a></i>
<br/>.= 
<font color="Maroon" title="c6">b</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">q</font></span>)</span>
<b>by </b><i><a class="txt" href="filter_2.html#E8:116_1"><i><font color="Green" title="E54">A5</font></i></a>, <a class="ref" href="lattices.html#D3" title="LATTICES:def.3">LATTICES:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">q</font>
<b>by </b><i><a class="txt" href="filter_2.html#E1:116"><i><font color="Green" title="E50">A1</font></i></a>, <a class="txt" href="filter_2.html#E4:116_1"><i><font color="Green" title="E52">A3</font></i></a>, <a class="txt" href="filter_2.html#E8:116_1"><i><font color="Green" title="E54">A5</font></i></a>, <a class="ref" href="lattices.html#D12" title="LATTICES:def.12">LATTICES:def 12</a></i>
<br/>.= 
<font color="Maroon" title="c3">q</font>
<b>by </b><i><a class="txt" href="filter_2.html#E1:116"><i><font color="Green" title="E50">A1</font></i></a>, <a class="ref" href="lattices.html#T43" title="LATTICES:th.43">LATTICES:43</a></i>
<br/>.= 
 <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c4">K</font>
<b>by </b><i><a class="txt" href="filter_2.html#E1:116"><i><font color="Green" title="E50">A1</font></i></a>, <a class="ref" href="filter_2.html#T84" target="_self" title="FILTER_2:th.84">Th84</a></i>
; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D18" title="LATTICES:def.18">LATTICES:def 18</a><br/>

<b>hence </b><a NAME="E10:116_1"/>
<font color="Maroon" title="c5">b'</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c8">a'</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">K</font>
 ;<br/>

<b>thus </b><font color="Maroon" title="c8">a'</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">b'</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c6">b</font>
<b>by </b><i><a class="ref" href="filter_2.html#T74" target="_self" title="FILTER_2:th.74">Th74</a></i>
<br/>.= 
<font color="Maroon" title="c2">p</font> <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="c7">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c6">b</font></span>)</span>
<b>by </b><i><a class="txt" href="filter_2.html#E1:116"><i><font color="Green" title="E50">A1</font></i></a>, <a class="txt" href="filter_2.html#E8:116_1"><i><font color="Green" title="E54">A5</font></i></a>, <a class="ref" href="lattices.html#D12" title="LATTICES:def.12">LATTICES:def 12</a></i>
<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="c3">q</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p2">(<span class="default"><a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="filter_2.html#E4:116_1"><i><font color="Green" title="E52">A3</font></i></a>, <a class="ref" href="lattices.html#D7" title="LATTICES:def.7">LATTICES:def 7</a></i>
<br/>.= 
<font color="Maroon" title="c2">p</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="c1">L</font></span>)</span>
<b>by </b><i><a class="txt" href="filter_2.html#E1:116"><i><font color="Green" title="E50">A1</font></i></a>, <a class="ref" href="lattices.html#T40" title="LATTICES:th.40">LATTICES:40</a></i>
<br/>.= 
<font color="Maroon" title="c2">p</font>
<b>by </b><i><a class="txt" href="filter_2.html#E1:116"><i><font color="Green" title="E50">A1</font></i></a>, <a class="ref" href="lattices.html#T39" title="LATTICES:th.39">LATTICES:39</a></i>
<br/>.= 
 <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c4">K</font>
<b>by </b><i><a class="txt" href="filter_2.html#E1:116"><i><font color="Green" title="E50">A1</font></i></a>, <a class="ref" href="filter_2.html#T84" target="_self" title="FILTER_2:th.84">Th84</a></i>
;<br/>

<b>hence </b><a NAME="E12:116_1"/>
<font color="Maroon" title="c5">b'</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c8">a'</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">K</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:116"/>
 <a href="filter_2.html#K23" title="FILTER_2:func.23">latt</a> <font color="Maroon" title="c1">L</font>,<span class="p1"><a href="filter_2.html#K22" title="FILTER_2:func.22">[#</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="filter_2.html#K22" title="FILTER_2:func.22">#]</a></span> is   <a href="lattices.html#NM7" title="LATTICES:NM.7">C_Lattice</a>
 ;<br/>


</div>
