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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="lattices.html#NM1">Lattice</a>;<br/>



<b>thus </b><a NAME="E1:65"/>
( <font color="Maroon">c<sub>1</sub></font> is   <a href="lattices.html#NM7">C_Lattice</a> &amp; <font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">E44</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is   <a href="lattices.html#NM7">C_Lattice</a> &amp; <font color="Maroon">c<sub>2</sub></font> is   <a href="lattices.html#NM7">C_Lattice</a> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1"><a href="filter_1.html#K7" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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><a class="ref" href="filter_1.html#T42" target="_self">Th42</a></i>;<br/>
<a NAME="E3:65_1"/>
<font color="Maroon">c<sub>3</sub></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">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></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">c<sub>5</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>6</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E2:65_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>4</sub></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">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>7</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:65_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#R2">is_a_complement_of</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="filter_1.html#E1:65_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="lattices.html#D19">LATTICES:def 19</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E6:65_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="lattices.html#R2">is_a_complement_of</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="filter_1.html#E1:65_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="lattices.html#D19">LATTICES:def 19</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></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">c<sub>3</sub></font> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>9</sub></font>
;<br/>

<b>thus </b><a NAME="E8:65_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="lattices.html#R2">is_a_complement_of</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="filter_1.html#E1:65_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="filter_1.html#E2:65_1_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="filter_1.html#E4:65_1_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="filter_1.html#E6:65_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="filter_1.html#T45" target="_self">Th45</a></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">E44</font></i>: 
<span class="p1"><a href="filter_1.html#K7" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font> = <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="lattices.html#NM6">01_Lattice</a> <b>by </b><i><a class="ref" href="filter_1.html#T42" target="_self">Th42</a></i>;<br/>
<a NAME="E4:65"/>
<font color="Maroon">c<sub>3</sub></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">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></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">c<sub>6</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">E45</font></i>: 
<font color="Maroon">c<sub>9</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="filter_1.html#K9" target="_self">]</a></span>
 <b>by </b><i><a class="txt" href="filter_1.html#E2:65"><i><font color="Green">E44</font></i></a>, <a class="ref" href="lattices.html#D19">LATTICES:def 19</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>11</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E7:65_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>9</sub></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">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></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">c<sub>12</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>12</sub></font>
;<br/>

<b>thus </b><a NAME="E9:65_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="lattices.html#R2">is_a_complement_of</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="filter_1.html#E5:65_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="filter_1.html#E7:65_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="filter_1.html#T45" target="_self">Th45</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:65"/>
<font color="Maroon">c<sub>1</sub></font> is   <a href="lattices.html#NM7">C_Lattice</a>
 ;<br/>

<a NAME="E6:65"/>
<font color="Maroon">c<sub>4</sub></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">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></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">c<sub>6</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">E45</font></i>: 
<font color="Maroon">c<sub>9</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="filter_1.html#K9" target="_self">]</a></span>
 <b>by </b><i><a class="txt" href="filter_1.html#E2:65"><i><font color="Green">E44</font></i></a>, <a class="ref" href="lattices.html#D19">LATTICES:def 19</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>11</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E7:65_3"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>9</sub></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">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></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">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>12</sub></font>
;<br/>

<b>thus </b><a NAME="E9:65_3"/>
<font color="Maroon">c<sub>12</sub></font> <a href="lattices.html#R2">is_a_complement_of</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="filter_1.html#E5:65_3"><i><font color="Green">E45</font></i></a>, <a class="txt" href="filter_1.html#E7:65_3"><i><font color="Green">E46</font></i></a>, <a class="ref" href="filter_1.html#T45" target="_self">Th45</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:65"/>
<font color="Maroon">c<sub>2</sub></font> is   <a href="lattices.html#NM7">C_Lattice</a>
 ;<br/>


</div>
