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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> 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="E1:79"/><i><font color="Green" title="E57">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#NM3" title="LATTICES:NM.3">M_Lattice</a> )
 ;<br/>

<b>then reconsider </b><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> ;<br/>
<b>reconsider </b><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> <b>by </b><i><a class="txt" href="filter_0.html#E1:79"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="filter_0.html#T73" target="_self" title="FILTER_0:th.73">Th73</a></i>;<br/>
<a NAME="E4:79"/>
<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="79_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">r'</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">M</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/>

<a NAME="E1:79_1"/><i><font color="Green" title="E58">A2</font></i>: 
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>
 
<b>by </b><i><a class="ref" href="filter_0.html#T63" target="_self" title="FILTER_0:th.63">Th63</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">r</font> = <font color="Maroon" title="c5">r'</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> <b>by </b><i><a class="ref" href="filter_0.html#T63" target="_self" title="FILTER_0:th.63">Th63</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">q</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="E4:79_1"/><i><font color="Green" title="E59">A3</font></i>: 
<font color="Maroon" title="c7">q</font> <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <font color="Maroon" title="c6">r</font>
 <b>by </b><i><a class="txt" href="filter_0.html#E1:79"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="lattices.html#D19" title="LATTICES:def.19">LATTICES:def 19</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">q'</font> = <font color="Maroon" title="c2">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c7">q</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">M</font> <b>by </b><i><a class="txt" href="filter_0.html#E1:79_1"><i><font color="Green" title="E58">A2</font></i></a>, <a class="ref" href="filter_0.html#T19" target="_self" title="FILTER_0:th.19">Th19</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">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/>
<b>take </b>
<font color="Maroon" title="c8">q'</font>
;<br/>

<b>thus </b><font color="Maroon" title="c8">q'</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c5">r'</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="c7">q</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c6">r</font>
<b>by </b><i><a class="ref" href="filter_0.html#T64" target="_self" title="FILTER_0:th.64">Th64</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="c7">q</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c6">r</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="c9">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>
<b>by </b><i><a class="txt" href="filter_0.html#E4:79_1"><i><font color="Green" title="E59">A3</font></i></a>, <a class="ref" href="lattices.html#D18" title="LATTICES:def.18">LATTICES:def 18</a></i>
<br/>.= 
 <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font>
<b>by </b><i><a class="ref" href="lattices.html#T44" title="LATTICES:th.44">LATTICES:44</a></i>
<br/>.= 
 <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c4">M</font>
<b>by </b><i><a class="txt" href="filter_0.html#E1:79"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="filter_0.html#T72" target="_self" title="FILTER_0:th.72">Th72</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="E8:79_1"/>
<font color="Maroon" title="c5">r'</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c8">q'</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>
 ;<br/>

<a NAME="E9:79_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>
 
<b>by </b><i><a class="ref" href="filter_0.html#T18" target="_self" title="FILTER_0:th.18">Th18</a></i>;<br/>
<a NAME="E10:79_1"/><b>then </b>
<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="c7">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="c7">q</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c6">r</font></span>)</span>
 
<b>by </b><i><a class="txt" href="filter_0.html#E1:79"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="lattices.html#D12" title="LATTICES:def.12">LATTICES:def 12</a></i>;<br/>
<b>hence </b><font color="Maroon" title="c8">q'</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">r'</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">q</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c6">r</font></span>)</span>
<b>by </b><i><a class="ref" href="filter_0.html#T64" target="_self" title="FILTER_0:th.64">Th64</a></i>
<br/>.= 
<font color="Maroon" title="c9">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>
<b>by </b><i><a class="txt" href="filter_0.html#E4:79_1"><i><font color="Green" title="E59">A3</font></i></a>, <a class="ref" href="lattices.html#D18" title="LATTICES:def.18">LATTICES:def 18</a></i>
<br/>.= 
<font color="Maroon" title="c2">p</font>
<b>by </b><i><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">M</font>
<b>by </b><i><a class="ref" href="filter_0.html#T71" target="_self" title="FILTER_0:th.71">Th71</a></i>
;<br/><br/>

<b>hence </b><a NAME="E12:79_1"/>
<font color="Maroon" title="c5">r'</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c8">q'</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>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:79"/>
 <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>
 ;<br/>


</div>
