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

<b>let </b><font color="Maroon" title="c1">F</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>;<br/><b>let </b><font color="Maroon" title="c2">V</font> be   <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c1">F</font>;<br/>



<b>reconsider </b><font color="Maroon" title="c3">S</font> =  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="vectsp_5.html#K3" title="VECTSP_5:func.3">Subspaces</a> <font color="Maroon" title="c2">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp_5.html#K5" title="VECTSP_5:func.5">SubJoin</a> <font color="Maroon" title="c2">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp_5.html#K6" title="VECTSP_5:func.6">SubMeet</a> <font color="Maroon" title="c2">V</font></span>)</span> #) as   <a href="lattices.html#NM6" title="LATTICES:NM.6">01_Lattice</a> <b>by </b><i><a class="ref" href="vectsp_5.html#T78" target="_self" title="VECTSP_5:th.78">Th78</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">S0</font> = <font color="Maroon" title="c3">S</font> as   <a href="lattices.html#NM4" title="LATTICES:NM.4">0_Lattice</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c5">S1</font> = <font color="Maroon" title="c3">S</font> as   <a href="lattices.html#NM5" title="LATTICES:NM.5">1_Lattice</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c6">Z</font> =  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">S</font> <b>by </b><i><a class="ref" href="vectsp_5.html#D3" target="_self" title="VECTSP_5:def.3">Def3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">W'</font> being  <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>  <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font><b> such that </b><br/><a NAME="E6:74"/><i><font color="Green" title="E53">A1</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">W'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c2">V</font></span>)</span>
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c8">I</font> = <font color="Maroon" title="c7">W'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">S</font> <b>by </b><i><a class="ref" href="vectsp_5.html#D3" target="_self" title="VECTSP_5:def.3">Def3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">Z0</font> = <font color="Maroon" title="c6">Z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">S0</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c10">I1</font> = <font color="Maroon" title="c8">I</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S1</font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">A</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S1</font>;<br/><b>consider </b><font color="Maroon" title="c12">W</font> being  <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>  <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font><b> such that </b><br/><a NAME="E2:74_1"/><i><font color="Green" title="E54">A2</font></i>: 
<font color="Maroon" title="c12">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">A</font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#D3" target="_self" title="VECTSP_5:def.3">Def3</a></i>;<br/><a NAME="E3:74_1"/><i><font color="Green" title="E55">A3</font></i>: 
( <font color="Maroon" title="c7">W'</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c2">V</font> &amp;  <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">VectSpStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">U6</a> of <font color="Maroon" title="c2">V</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <font color="Maroon" title="c2">V</font>,the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c2">V</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c2">V</font> )
 <b>by </b><i><a class="txt" href="vectsp_5.html#E21"><i><font color="Green" title="E15">Lm6</font></i></a></i>;<br/><b>thus </b><font color="Maroon" title="c11">A</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c10">I1</font> = 
<span class="p1">(<span class="default"><a href="vectsp_5.html#K5" title="VECTSP_5:func.5">SubJoin</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c10">I1</font>
<b>by </b><i><a class="ref" href="lattices.html#D1" title="LATTICES:def.1">LATTICES:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c12">W</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c7">W'</font>
<b>by </b><i><a class="txt" href="vectsp_5.html#E2:74_1"><i><font color="Green" title="E54">A2</font></i></a>, <a class="ref" href="vectsp_5.html#D7" target="_self" title="VECTSP_5:def.7">Def7</a></i>
<br/>.= 
<font color="Maroon" title="c12">W</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <span class="p1">(<span class="default"><a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c2">V</font></span>)</span>
<b>by </b><i><a class="txt" href="vectsp_5.html#E6:74"><i><font color="Green" title="E53">A1</font></i></a>, <a class="txt" href="vectsp_5.html#E20"><i><font color="Green" title="E14">Lm5</font></i></a></i>
<br/>.= 
 <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">VectSpStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">U6</a> of <font color="Maroon" title="c2">V</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <font color="Maroon" title="c2">V</font>,the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c2">V</font> #)
<b>by </b><i><a class="ref" href="vectsp_5.html#T15" target="_self" title="VECTSP_5:th.15">Th15</a></i>
<br/>.= 
<font color="Maroon" title="c7">W'</font>
<b>by </b><i><a class="txt" href="vectsp_5.html#E6:74"><i><font color="Green" title="E53">A1</font></i></a>, <a class="txt" href="vectsp_5.html#E3:74_1"><i><font color="Green" title="E55">A3</font></i></a>, <a class="ref" href="vectsp_4.html#T39" title="VECTSP_4:th.39">VECTSP_4:39</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E11:74"/><b>then </b><i><font color="Green" title="E54">A4</font></i>: 
 <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c3">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">I</font>
 
<b>by </b><i><a class="ref" href="rlsub_2.html#T85" title="RLSUB_2:th.85">RLSUB_2:85</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">A</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">S0</font>;<br/><b>consider </b><font color="Maroon" title="c12">W</font> being  <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>  <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font><b> such that </b><br/><a NAME="E2:74_2"/><i><font color="Green" title="E55">A5</font></i>: 
<font color="Maroon" title="c12">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">A</font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#D3" target="_self" title="VECTSP_5:def.3">Def3</a></i>;<br/><b>thus </b><font color="Maroon" title="c11">A</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c9">Z0</font> = 
<span class="p1">(<span class="default"><a href="vectsp_5.html#K6" title="VECTSP_5:func.6">SubMeet</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">A</font>,<font color="Maroon" title="c9">Z0</font>
<b>by </b><i><a class="ref" href="lattices.html#D2" title="LATTICES:def.2">LATTICES:def 2</a></i>
<br/>.= 
<font color="Maroon" title="c12">W</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <span class="p1">(<span class="default"><a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font></span>)</span>
<b>by </b><i><a class="txt" href="vectsp_5.html#E2:74_2"><i><font color="Green" title="E55">A5</font></i></a>, <a class="ref" href="vectsp_5.html#D8" target="_self" title="VECTSP_5:def.8">Def8</a></i>
<br/>.= 
<font color="Maroon" title="c9">Z0</font>
<b>by </b><i><a class="ref" href="vectsp_5.html#T25" target="_self" title="VECTSP_5:th.25">Th25</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E13:74"/><b>then </b><i><font color="Green" title="E55">A6</font></i>: 
 <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c3">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Z</font>
 
<b>by </b><i><a class="ref" href="rlsub_2.html#T84" title="RLSUB_2:th.84">RLSUB_2:84</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">A</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">S</font>;<br/><b>consider </b><font color="Maroon" title="c12">W</font> being  <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>  <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font><b> such that </b><br/><a NAME="E2:74_3"/><i><font color="Green" title="E56">A7</font></i>: 
<font color="Maroon" title="c12">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">A</font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#D3" target="_self" title="VECTSP_5:def.3">Def3</a></i>;<br/><b>consider </b><font color="Maroon" title="c13">L</font> being    <a href="vectsp_5.html#M1" title="VECTSP_5:mode.1">Linear_Compl</a> of <font color="Maroon" title="c12">W</font>;<br/><b>consider </b><font color="Maroon" title="c14">W''</font> being  <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>  <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font><b> such that </b><br/><a NAME="E5:74_3"/><i><font color="Green" title="E57">A8</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c14">W''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c13">L</font>
 <b>by </b><i><a class="txt" href="vectsp_5.html#E19"><i><font color="Green" title="E13">Lm4</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c15">B'</font> = <font color="Maroon" title="c14">W''</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">S</font> <b>by </b><i><a class="ref" href="vectsp_5.html#D3" target="_self" title="VECTSP_5:def.3">Def3</a></i>;<br/><b>take </b><font color="Maroon" title="c16">B</font> = <font color="Maroon" title="c15">B'</font>;<br/><a NAME="E7:74_3"/><i><font color="Green" title="E58">A9</font></i>: 
( <font color="Maroon" title="c8">I</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c2">V</font> &amp;  <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">VectSpStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">U6</a> of <font color="Maroon" title="c2">V</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <font color="Maroon" title="c2">V</font>,the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c2">V</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c2">V</font> )
 <b>by </b><i><a class="txt" href="vectsp_5.html#E21"><i><font color="Green" title="E15">Lm6</font></i></a></i>;<br/><i><font color="Green" title="E59">A10</font></i>: <font color="Maroon" title="c16">B</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c11">A</font> = 
<span class="p1">(<span class="default"><a href="vectsp_5.html#K5" title="VECTSP_5:func.5">SubJoin</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c16">B</font>,<font color="Maroon" title="c11">A</font>
<b>by </b><i><a class="ref" href="lattices.html#D1" title="LATTICES:def.1">LATTICES:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c14">W''</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c12">W</font>
<b>by </b><i><a class="txt" href="vectsp_5.html#E2:74_3"><i><font color="Green" title="E56">A7</font></i></a>, <a class="ref" href="vectsp_5.html#D7" target="_self" title="VECTSP_5:def.7">Def7</a></i>
<br/>.= 
<font color="Maroon" title="c13">L</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c12">W</font>
<b>by </b><i><a class="txt" href="vectsp_5.html#E5:74_3"><i><font color="Green" title="E57">A8</font></i></a>, <a class="txt" href="vectsp_5.html#E20"><i><font color="Green" title="E14">Lm5</font></i></a></i>
<br/>.= 
 <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">VectSpStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">U6</a> of <font color="Maroon" title="c2">V</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <font color="Maroon" title="c2">V</font>,the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c2">V</font> #)
<b>by </b><i><a class="ref" href="vectsp_5.html#T49" target="_self" title="VECTSP_5:th.49">Th49</a></i>
<br/>.= 
 <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c3">S</font>
<b>by </b><i><a class="txt" href="vectsp_5.html#E6:74"><i><font color="Green" title="E53">A1</font></i></a>, <a class="txt" href="vectsp_5.html#E11:74"><i><font color="Green" title="E54">A4</font></i></a>, <a class="txt" href="vectsp_5.html#E7:74_3"><i><font color="Green" title="E58">A9</font></i></a>, <a class="ref" href="vectsp_4.html#T39" title="VECTSP_4:th.39">VECTSP_4:39</a></i>
;<br/><font color="Maroon" title="c16">B</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c11">A</font> = 
<span class="p1">(<span class="default"><a href="vectsp_5.html#K6" title="VECTSP_5:func.6">SubMeet</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c16">B</font>,<font color="Maroon" title="c11">A</font>
<b>by </b><i><a class="ref" href="lattices.html#D2" title="LATTICES:def.2">LATTICES:def 2</a></i>
<br/>.= 
<font color="Maroon" title="c14">W''</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c12">W</font>
<b>by </b><i><a class="txt" href="vectsp_5.html#E2:74_3"><i><font color="Green" title="E56">A7</font></i></a>, <a class="ref" href="vectsp_5.html#D8" target="_self" title="VECTSP_5:def.8">Def8</a></i>
<br/>.= 
<font color="Maroon" title="c13">L</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c12">W</font>
<b>by </b><i><a class="txt" href="vectsp_5.html#E5:74_3"><i><font color="Green" title="E57">A8</font></i></a>, <a class="txt" href="vectsp_5.html#E30"><i><font color="Green" title="E20">Lm8</font></i></a></i>
<br/>.= 
 <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c3">S</font>
<b>by </b><i><a class="txt" href="vectsp_5.html#E13:74"><i><font color="Green" title="E55">A6</font></i></a>, <a class="ref" href="vectsp_5.html#T50" target="_self" title="VECTSP_5:th.50">Th50</a></i>
;<br/><b>hence </b><a NAME="E10:74_3"/>
<font color="Maroon" title="c16">B</font> <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <font color="Maroon" title="c11">A</font>
 <b>by </b><i><a class="txt" href="vectsp_5.html#E8:74_3"><i><font color="Green" title="E59">A10</font></i></a>, <a class="ref" href="lattices.html#D18" title="LATTICES:def.18">LATTICES:def 18</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E15:74"/>
 <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="vectsp_5.html#K3" title="VECTSP_5:func.3">Subspaces</a> <font color="Maroon" title="c2">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp_5.html#K5" title="VECTSP_5:func.5">SubJoin</a> <font color="Maroon" title="c2">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp_5.html#K6" title="VECTSP_5:func.6">SubMeet</a> <font color="Maroon" title="c2">V</font></span>)</span> #) is   <a href="lattices.html#NM7" title="LATTICES:NM.7">C_Lattice</a>
 <b>by </b><i><a class="ref" href="lattices.html#D19" title="LATTICES:def.19">LATTICES:def 19</a></i>;<br/>


</div>
