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<b>let </b><font color="Maroon" title="c1">V</font> be   <a href="bhsp_1.html#NM1" title="BHSP_1:NM.1">RealUnitarySpace</a>;<br/>

<b>reconsider </b><font color="Maroon" title="c2">S</font> =  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="rusub_2.html#K3" title="RUSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" title="RUSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" title="RUSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">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="rusub_2.html#T57" target="_self" title="RUSUB_2:th.57">Th57</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c3">S0</font> = <font color="Maroon" title="c2">S</font> as   <a href="lattices.html#NM4" title="LATTICES:NM.4">0_Lattice</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c4">S1</font> = <font color="Maroon" title="c2">S</font> as   <a href="lattices.html#NM5" title="LATTICES:NM.5">1_Lattice</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c5">Z</font> =  <a href="rusub_1.html#K1" title="RUSUB_1:func.1">(0).</a> <font color="Maroon" title="c1">V</font>, <font color="Maroon" title="c6">I</font> =  <a href="rusub_1.html#K2" title="RUSUB_1:func.2">(Omega).</a> <font color="Maroon" title="c1">V</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="ref" href="rusub_2.html#D3" target="_self" title="RUSUB_2:def.3">Def3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">Z0</font> = <font color="Maroon" title="c5">Z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">S0</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c8">I1</font> = <font color="Maroon" title="c6">I</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">S1</font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">A</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">S1</font>;<br/><b>reconsider </b><font color="Maroon" title="c10">W</font> = <font color="Maroon" title="c9">A</font> as    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="ref" href="rusub_2.html#D3" target="_self" title="RUSUB_2:def.3">Def3</a></i>;<br/><b>thus </b><font color="Maroon" title="c9">A</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c8">I1</font> = 
<span class="p1">(<span class="default"><a href="rusub_2.html#K5" title="RUSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">A</font>,<font color="Maroon" title="c8">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="c10">W</font> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <span class="p1">(<span class="default"><a href="rusub_1.html#K2" title="RUSUB_1:func.2">(Omega).</a> <font color="Maroon" title="c1">V</font></span>)</span>
<b>by </b><i><a class="ref" href="rusub_2.html#D7" target="_self" title="RUSUB_2:def.7">Def7</a></i>
<br/>.= 
 <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <font color="Maroon" title="c1">V</font>,the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">V</font>,the <a href="rlvect_1.html#U2" title="RLVECT_1:sel.2">Mult</a> of <font color="Maroon" title="c1">V</font>,the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> #)
<b>by </b><i><a class="ref" href="rusub_2.html#T11" target="_self" title="RUSUB_2:th.11">Th11</a></i>
<br/>.= 
 <a href="rusub_1.html#K2" title="RUSUB_1:func.2">(Omega).</a> <font color="Maroon" title="c1">V</font>
<b>by </b><i><a class="ref" href="rusub_1.html#D3" title="RUSUB_1:def.3">RUSUB_1:def 3</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E8:81"/><b>then </b><i><font color="Green" title="E60">A1</font></i>: 
 <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">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="81_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">A</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">S0</font>;<br/><b>reconsider </b><font color="Maroon" title="c10">W</font> = <font color="Maroon" title="c9">A</font> as    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="ref" href="rusub_2.html#D3" target="_self" title="RUSUB_2:def.3">Def3</a></i>;<br/><b>thus </b><font color="Maroon" title="c9">A</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c7">Z0</font> = 
<span class="p1">(<span class="default"><a href="rusub_2.html#K6" title="RUSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">A</font>,<font color="Maroon" title="c7">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="c10">W</font> <a href="rusub_2.html#K2" title="RUSUB_2:func.2">/\</a> <span class="p1">(<span class="default"><a href="rusub_1.html#K1" title="RUSUB_1:func.1">(0).</a> <font color="Maroon" title="c1">V</font></span>)</span>
<b>by </b><i><a class="ref" href="rusub_2.html#D8" target="_self" title="RUSUB_2:def.8">Def8</a></i>
<br/>.= 
<font color="Maroon" title="c7">Z0</font>
<b>by </b><i><a class="ref" href="rusub_2.html#T18" target="_self" title="RUSUB_2:th.18">Th18</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E10:81"/><b>then </b><i><font color="Green" title="E61">A2</font></i>: 
 <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">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="81_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">A</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</font>;<br/><b>reconsider </b><font color="Maroon" title="c10">W</font> = <font color="Maroon" title="c9">A</font> as    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="ref" href="rusub_2.html#D3" target="_self" title="RUSUB_2:def.3">Def3</a></i>;<br/><b>consider </b><font color="Maroon" title="c11">L</font> being  <a href="bhsp_1.html#V1" title="BHSP_1:attr.1">strict</a>  <a href="rusub_2.html#M1" title="RUSUB_2:mode.1">Linear_Compl</a> of <font color="Maroon" title="c10">W</font>;<br/><b>reconsider </b><font color="Maroon" title="c12">B'</font> = <font color="Maroon" title="c11">L</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="ref" href="rusub_2.html#D3" target="_self" title="RUSUB_2:def.3">Def3</a></i>;<br/><b>take </b><font color="Maroon" title="c13">B</font> = <font color="Maroon" title="c12">B'</font>;<br/><i><font color="Green" title="E62">A3</font></i>: <font color="Maroon" title="c13">B</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c9">A</font> = 
<span class="p1">(<span class="default"><a href="rusub_2.html#K5" title="RUSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c13">B</font>,<font color="Maroon" title="c9">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="c11">L</font> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <font color="Maroon" title="c10">W</font>
<b>by </b><i><a class="ref" href="rusub_2.html#D7" target="_self" title="RUSUB_2:def.7">Def7</a></i>
<br/>.= 
 <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <font color="Maroon" title="c1">V</font>,the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">V</font>,the <a href="rlvect_1.html#U2" title="RLVECT_1:sel.2">Mult</a> of <font color="Maroon" title="c1">V</font>,the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> #)
<b>by </b><i><a class="ref" href="rusub_2.html#T36" target="_self" title="RUSUB_2:th.36">Th36</a></i>
<br/>.= 
 <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c2">S</font>
<b>by </b><i><a class="txt" href="rusub_2.html#E8:81"><i><font color="Green" title="E60">A1</font></i></a>, <a class="ref" href="rusub_1.html#D3" title="RUSUB_1:def.3">RUSUB_1:def 3</a></i>
;<br/><font color="Maroon" title="c13">B</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c9">A</font> = 
<span class="p1">(<span class="default"><a href="rusub_2.html#K6" title="RUSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c13">B</font>,<font color="Maroon" title="c9">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="c11">L</font> <a href="rusub_2.html#K2" title="RUSUB_2:func.2">/\</a> <font color="Maroon" title="c10">W</font>
<b>by </b><i><a class="ref" href="rusub_2.html#D8" target="_self" title="RUSUB_2:def.8">Def8</a></i>
<br/>.= 
 <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c2">S</font>
<b>by </b><i><a class="txt" href="rusub_2.html#E10:81"><i><font color="Green" title="E61">A2</font></i></a>, <a class="ref" href="rusub_2.html#T37" target="_self" title="RUSUB_2:th.37">Th37</a></i>
;<br/><b>hence </b><a NAME="E6:81_3"/>
<font color="Maroon" title="c13">B</font> <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <font color="Maroon" title="c9">A</font>
 <b>by </b><i><a class="txt" href="rusub_2.html#E4:81_3"><i><font color="Green" title="E62">A3</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="E12:81"/>
 <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="rusub_2.html#K3" title="RUSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" title="RUSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" title="RUSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> #) is <a href="lattices.html#V16" title="LATTICES:attr.16">complemented</a>
 <b>by </b><i><a class="ref" href="lattices.html#D19" title="LATTICES:def.19">LATTICES:def 19</a></i>;<br/>


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