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<div class="add">

<b>let </b><font color="Maroon">L1</font> be   <a href="lattices.html#NM1">Lattice</a>;<br/><b>let </b><font color="Maroon">L2</font> be   <a href="lattices.html#NM1">Lattice</a>;<br/>



<b>thus </b><a NAME="E1:65"/>
( <font color="Maroon">L1</font> is   <a href="lattices.html#NM7">C_Lattice</a> &amp; <font color="Maroon">L2</font> is   <a href="lattices.html#NM7">C_Lattice</a> implies <span class="p1"><a href="filter_1.html#K7" target="_self">[:</a><span class="default"><font color="Maroon">L1</font>,<font color="Maroon">L2</font></span><a href="filter_1.html#K7" target="_self">:]</a></span> is   <a href="lattices.html#NM7">C_Lattice</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:65_1"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">L1</font> is   <a href="lattices.html#NM7">C_Lattice</a> &amp; <font color="Maroon">L2</font> is   <a href="lattices.html#NM7">C_Lattice</a> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">L</font> = <span class="p1"><a href="filter_1.html#K7" target="_self">[:</a><span class="default"><font color="Maroon">L1</font>,<font color="Maroon">L2</font></span><a href="filter_1.html#K7" target="_self">:]</a></span> as   <a href="lattices.html#NM6">01_Lattice</a> <b>by </b><i/>;<br/>
<a NAME="E3:65_1"/>
<font color="Maroon">L</font> is <a href="lattices.html#V16">complemented</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D19">LATTICES:def 19</a><br/>

<b>consider </b><font color="Maroon">p1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font>, <font color="Maroon">p2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L2</font><b> such that </b><br/><a NAME="E2:65_1_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="filter_1.html#K9" target="_self">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/>
<b>consider </b><font color="Maroon">q1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font><b> such that </b><br/><a NAME="E4:65_1_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">q1</font> <a href="lattices.html#R2">is_a_complement_of</a> <font color="Maroon">p1</font>
 <b>by </b><i>, <a class="ref" href="lattices.html#D19">LATTICES:def 19</a></i>;<br/>
<b>consider </b><font color="Maroon">q2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L2</font><b> such that </b><br/><a NAME="E6:65_1_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">q2</font> <a href="lattices.html#R2">is_a_complement_of</a> <font color="Maroon">p2</font>
 <b>by </b><i>, <a class="ref" href="lattices.html#D19">LATTICES:def 19</a></i>;<br/>
<b>reconsider </b><font color="Maroon">b</font> = <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">q1</font>,<font color="Maroon">q2</font></span><a href="filter_1.html#K9" target="_self">]</a></span> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<b>take </b>
<font color="Maroon">b</font>
;<br/>

<b>thus </b><a NAME="E8:65_1_1"/>
<font color="Maroon">b</font> <a href="lattices.html#R2">is_a_complement_of</a> <font color="Maroon">a</font>
 <b>by </b><i>, , , , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:65_1"/>
<span class="p1"><a href="filter_1.html#K7" target="_self">[:</a><span class="default"><font color="Maroon">L1</font>,<font color="Maroon">L2</font></span><a href="filter_1.html#K7" target="_self">:]</a></span> is   <a href="lattices.html#NM7">C_Lattice</a>
 ;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:65"/><i><font color="Green">E79</font></i>: 
<span class="p1"><a href="filter_1.html#K7" target="_self">[:</a><span class="default"><font color="Maroon">L1</font>,<font color="Maroon">L2</font></span><a href="filter_1.html#K7" target="_self">:]</a></span> is   <a href="lattices.html#NM7">C_Lattice</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon">C1</font> = <font color="Maroon">L1</font>, <font color="Maroon">C2</font> = <font color="Maroon">L2</font> as   <a href="lattices.html#NM6">01_Lattice</a> <b>by </b><i/>;<br/>
<a NAME="E4:65"/>
<font color="Maroon">C1</font> is <a href="lattices.html#V16">complemented</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p1'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">C1</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D19">LATTICES:def 19</a><br/>

<b>consider </b><font color="Maroon">p2'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">C2</font>;<br/>
<b>reconsider </b><font color="Maroon">p1</font> = <font color="Maroon">p1'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font> ;<br/>
<b>reconsider </b><font color="Maroon">p2</font> = <font color="Maroon">p2'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L2</font> ;<br/>
<b>consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1"><a href="filter_1.html#K7" target="_self">[:</a><span class="default"><font color="Maroon">L1</font>,<font color="Maroon">L2</font></span><a href="filter_1.html#K7" target="_self">:]</a></span><b> such that </b><br/><a NAME="E5:65_2"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">b</font> <a href="lattices.html#R2">is_a_complement_of</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="filter_1.html#K9" target="_self">]</a></span>
 <b>by </b><i>, <a class="ref" href="lattices.html#D19">LATTICES:def 19</a></i>;<br/>
<b>consider </b><font color="Maroon">q1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font>, <font color="Maroon">q2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L2</font><b> such that </b><br/><a NAME="E7:65_2"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">q1</font>,<font color="Maroon">q2</font></span><a href="filter_1.html#K9" target="_self">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">q1'</font> = <font color="Maroon">q1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">C1</font> ;<br/>
<b>take </b>
<font color="Maroon">q1'</font>
;<br/>

<b>thus </b><a NAME="E9:65_2"/>
<font color="Maroon">q1'</font> <a href="lattices.html#R2">is_a_complement_of</a> <font color="Maroon">p1'</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:65"/>
<font color="Maroon">L1</font> is   <a href="lattices.html#NM7">C_Lattice</a>
 ;<br/>

<a NAME="E6:65"/>
<font color="Maroon">C2</font> is <a href="lattices.html#V16">complemented</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p2'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">C2</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D19">LATTICES:def 19</a><br/>

<b>consider </b><font color="Maroon">p1'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">C1</font>;<br/>
<b>reconsider </b><font color="Maroon">p1</font> = <font color="Maroon">p1'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font> ;<br/>
<b>reconsider </b><font color="Maroon">p2</font> = <font color="Maroon">p2'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L2</font> ;<br/>
<b>consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1"><a href="filter_1.html#K7" target="_self">[:</a><span class="default"><font color="Maroon">L1</font>,<font color="Maroon">L2</font></span><a href="filter_1.html#K7" target="_self">:]</a></span><b> such that </b><br/><a NAME="E5:65_3"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">b</font> <a href="lattices.html#R2">is_a_complement_of</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="filter_1.html#K9" target="_self">]</a></span>
 <b>by </b><i>, <a class="ref" href="lattices.html#D19">LATTICES:def 19</a></i>;<br/>
<b>consider </b><font color="Maroon">q1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font>, <font color="Maroon">q2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L2</font><b> such that </b><br/><a NAME="E7:65_3"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">q1</font>,<font color="Maroon">q2</font></span><a href="filter_1.html#K9" target="_self">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">q2'</font> = <font color="Maroon">q2</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">C2</font> ;<br/>
<b>take </b>
<font color="Maroon">q2'</font>
;<br/>

<b>thus </b><a NAME="E9:65_3"/>
<font color="Maroon">q2'</font> <a href="lattices.html#R2">is_a_complement_of</a> <font color="Maroon">p2'</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:65"/>
<font color="Maroon">L2</font> is   <a href="lattices.html#NM7">C_Lattice</a>
 ;<br/>


</div>
