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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>2</sub></font> =  <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="rlsub_2.html#K3" target="_self">Subspaces</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="rlsub_2.html#K5" target="_self">SubJoin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="rlsub_2.html#K6" target="_self">SubMeet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) as   <a href="lattices.html#NM6">01_Lattice</a> <b>by </b><i><a class="ref" href="rlsub_2.html#T73" target="_self">Th73</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="lattices.html#NM4">0_Lattice</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="lattices.html#NM5">1_Lattice</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> =  <a href="rlsub_1.html#K1">(0).</a> <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>6</sub></font> =  <a href="rlsub_1.html#K2">(Omega).</a> <font color="Maroon">c<sub>1</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="rlsub_2.html#D3" target="_self">Def3</a></i>;<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>3</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>4</sub></font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be   <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>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as    <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="rlsub_2.html#D3" target="_self">Def3</a></i>;<br/><b>thus </b><font color="Maroon">c<sub>9</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>8</sub></font> = 
<span class="p1">(<span class="default"><a href="rlsub_2.html#K5" target="_self">SubJoin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font>

<br/>.= 
<font color="Maroon">c<sub>10</sub></font> <a href="rlsub_2.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><a href="rlsub_1.html#K2">(Omega).</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlsub_2.html#D7" target="_self">Def7</a></i>
<br/>.= 
 <a href="rlvect_1.html#G2">RLSStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">c<sub>1</sub></font> #)
<b>by </b><i><a class="ref" href="rlsub_2.html#T15" target="_self">Th15</a></i>
<br/>.= 
 <a href="rlsub_1.html#K2">(Omega).</a> <font color="Maroon">c<sub>1</sub></font>

;<br/></div><b>end;</b></div>
<a NAME="E8:83"/><b>then </b><i><font color="Green">E64</font></i>: 
 <a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E103"><i><font color="Green">Lemma62</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be   <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>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as    <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="rlsub_2.html#D3" target="_self">Def3</a></i>;<br/><b>thus </b><font color="Maroon">c<sub>9</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>7</sub></font> = 
<span class="p1">(<span class="default"><a href="rlsub_2.html#K6" target="_self">SubMeet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font>

<br/>.= 
<font color="Maroon">c<sub>10</sub></font> <a href="rlsub_2.html#K2" target="_self">/\</a> <span class="p1">(<span class="default"><a href="rlsub_1.html#K1">(0).</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlsub_2.html#D8" target="_self">Def8</a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="rlsub_2.html#T22" target="_self">Th22</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E10:83"/><b>then </b><i><font color="Green">E65</font></i>: 
 <a href="lattices.html#K5">Bottom</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E102"><i><font color="Green">Lemma61</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as    <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="rlsub_2.html#D3" target="_self">Def3</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>11</sub></font> being  <a href="rlvect_1.html#V2">strict</a>  <a href="rlsub_2.html#M1" target="_self">Linear_Compl</a> of <font color="Maroon">c<sub>10</sub></font>;<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>2</sub></font> <b>by </b><i><a class="ref" href="rlsub_2.html#D3" target="_self">Def3</a></i>;<br/><b>take </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font>;<br/><i><font color="Green">E66</font></i>: <font color="Maroon">c<sub>13</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>9</sub></font> = 
<span class="p1">(<span class="default"><a href="rlsub_2.html#K5" target="_self">SubJoin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>9</sub></font>

<br/>.= 
<font color="Maroon">c<sub>11</sub></font> <a href="rlsub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="ref" href="rlsub_2.html#D7" target="_self">Def7</a></i>
<br/>.= 
 <a href="rlvect_1.html#G2">RLSStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">c<sub>1</sub></font> #)
<b>by </b><i><a class="ref" href="rlsub_2.html#T44" target="_self">Th44</a></i>
<br/>.= 
 <a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="rlsub_2.html#E8:83"><i><font color="Green">E64</font></i></a></i>
;<br/><font color="Maroon">c<sub>13</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>9</sub></font> = 
<span class="p1">(<span class="default"><a href="rlsub_2.html#K6" target="_self">SubMeet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>9</sub></font>

<br/>.= 
<font color="Maroon">c<sub>11</sub></font> <a href="rlsub_2.html#K2" target="_self">/\</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="ref" href="rlsub_2.html#D8" target="_self">Def8</a></i>
<br/>.= 
 <a href="lattices.html#K5">Bottom</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="rlsub_2.html#E10:83"><i><font color="Green">E65</font></i></a>, <a class="ref" href="rlsub_2.html#T45" target="_self">Th45</a></i>
;<br/><b>hence </b><a NAME="E6:83_3"/>
<font color="Maroon">c<sub>13</sub></font> <a href="lattices.html#R2">is_a_complement_of</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="rlsub_2.html#E4:83_3"><i><font color="Green">E66</font></i></a>, <a class="txt" href="rlsub_2.html#E101"><i><font color="Green">Lemma60</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:83"/>
 <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="rlsub_2.html#K3" target="_self">Subspaces</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="rlsub_2.html#K5" target="_self">SubJoin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="rlsub_2.html#K6" target="_self">SubMeet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) is <a href="lattices.html#V16">complemented</a>
 <b>by </b><i><a class="ref" href="lattices.html#D19">LATTICES:def 19</a></i>;<br/>


</div>
