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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="vectsp_1.html#NM6">VectSp</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> =  <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="vectsp_5.html#K3" target="_self">Subspaces</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp_5.html#K5" target="_self">SubJoin</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp_5.html#K6" target="_self">SubMeet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> #) as   <a href="lattices.html#NM6">01_Lattice</a> <b>by </b><i><a class="ref" href="vectsp_5.html#T78" target="_self">Th78</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="lattices.html#NM4">0_Lattice</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="lattices.html#NM5">1_Lattice</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> =  <a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">c<sub>2</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="vectsp_5.html#D3" target="_self">Def3</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E6:81"/><i><font color="Green">E57</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="vectsp_5.html#D3" target="_self">Def3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</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/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>5</sub></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">c<sub>11</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>5</sub></font>;<br/><b>consider </b><font color="Maroon">c<sub>12</sub></font> being  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E2:81_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E3:81_1"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> is    <a href="vectsp_4.html#M1">Subspace</a> of  <a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="vectsp_1.html#G4">VectSpStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">c<sub>2</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="vectsp_5.html#E21"><i><font color="Green">Lemma15</font></i></a></i>;<br/><b>thus </b><font color="Maroon">c<sub>11</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>10</sub></font> = 
<span class="p1">(<span class="default"><a href="vectsp_5.html#K5" target="_self">SubJoin</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="ref" href="lattices.html#D1">LATTICES:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>12</sub></font> <a href="vectsp_5.html#K1" target="_self">+</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="vectsp_5.html#E2:81_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="vectsp_5.html#D7" target="_self">Def7</a></i>
<br/>.= 
<font color="Maroon">c<sub>12</sub></font> <a href="vectsp_5.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="vectsp_5.html#E6:81"><i><font color="Green">E57</font></i></a>, <a class="txt" href="vectsp_5.html#E20"><i><font color="Green">Lemma14</font></i></a></i>
<br/>.= 
 <a href="vectsp_1.html#G4">VectSpStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">c<sub>2</sub></font> #)
<b>by </b><i><a class="ref" href="vectsp_5.html#T15" target="_self">Th15</a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="vectsp_5.html#E6:81"><i><font color="Green">E57</font></i></a>, <a class="txt" href="vectsp_5.html#E3:81_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="vectsp_4.html#T39">VECTSP_4:39</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E11:81"/><b>then </b><i><font color="Green">E58</font></i>: 
 <a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="rlsub_2.html#T85">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">c<sub>11</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font>;<br/><b>consider </b><font color="Maroon">c<sub>12</sub></font> being  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E2:81_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#D3" target="_self">Def3</a></i>;<br/><b>thus </b><font color="Maroon">c<sub>11</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>9</sub></font> = 
<span class="p1">(<span class="default"><a href="vectsp_5.html#K6" target="_self">SubMeet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="ref" href="lattices.html#D2">LATTICES:def 2</a></i>
<br/>.= 
<font color="Maroon">c<sub>12</sub></font> <a href="vectsp_5.html#K2" target="_self">/\</a> <span class="p1">(<span class="default"><a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="vectsp_5.html#E2:81_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="vectsp_5.html#D8" target="_self">Def8</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="ref" href="vectsp_5.html#T25" target="_self">Th25</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E13:81"/><b>then </b><i><font color="Green">E59</font></i>: 
 <a href="lattices.html#K5">Bottom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="rlsub_2.html#T84">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">c<sub>11</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font>;<br/><b>consider </b><font color="Maroon">c<sub>12</sub></font> being  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E2:81_3"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#D3" target="_self">Def3</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="vectsp_5.html#M1" target="_self">Linear_Compl</a> of <font color="Maroon">c<sub>12</sub></font>;<br/><b>consider </b><font color="Maroon">c<sub>14</sub></font> being  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E5:81_3"/><i><font color="Green">E61</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="vectsp_5.html#E19"><i><font color="Green">Lemma13</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="vectsp_5.html#D3" target="_self">Def3</a></i>;<br/><b>take </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></font>;<br/><a NAME="E7:81_3"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> is    <a href="vectsp_4.html#M1">Subspace</a> of  <a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="vectsp_1.html#G4">VectSpStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">c<sub>2</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="vectsp_5.html#E21"><i><font color="Green">Lemma15</font></i></a></i>;<br/><i><font color="Green">E63</font></i>: <font color="Maroon">c<sub>16</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>11</sub></font> = 
<span class="p1">(<span class="default"><a href="vectsp_5.html#K5" target="_self">SubJoin</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="ref" href="lattices.html#D1">LATTICES:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>14</sub></font> <a href="vectsp_5.html#K1" target="_self">+</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="vectsp_5.html#E2:81_3"><i><font color="Green">E60</font></i></a>, <a class="ref" href="vectsp_5.html#D7" target="_self">Def7</a></i>
<br/>.= 
<font color="Maroon">c<sub>13</sub></font> <a href="vectsp_5.html#K1" target="_self">+</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="vectsp_5.html#E5:81_3"><i><font color="Green">E61</font></i></a>, <a class="txt" href="vectsp_5.html#E20"><i><font color="Green">Lemma14</font></i></a></i>
<br/>.= 
 <a href="vectsp_1.html#G4">VectSpStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">c<sub>2</sub></font> #)
<b>by </b><i><a class="ref" href="vectsp_5.html#T49" target="_self">Th49</a></i>
<br/>.= 
 <a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="vectsp_5.html#E6:81"><i><font color="Green">E57</font></i></a>, <a class="txt" href="vectsp_5.html#E11:81"><i><font color="Green">E58</font></i></a>, <a class="txt" href="vectsp_5.html#E7:81_3"><i><font color="Green">E62</font></i></a>, <a class="ref" href="vectsp_4.html#T39">VECTSP_4:39</a></i>
;<br/><font color="Maroon">c<sub>16</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>11</sub></font> = 
<span class="p1">(<span class="default"><a href="vectsp_5.html#K6" target="_self">SubMeet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="ref" href="lattices.html#D2">LATTICES:def 2</a></i>
<br/>.= 
<font color="Maroon">c<sub>14</sub></font> <a href="vectsp_5.html#K2" target="_self">/\</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="vectsp_5.html#E2:81_3"><i><font color="Green">E60</font></i></a>, <a class="ref" href="vectsp_5.html#D8" target="_self">Def8</a></i>
<br/>.= 
<font color="Maroon">c<sub>13</sub></font> <a href="vectsp_5.html#K2" target="_self">/\</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="vectsp_5.html#E5:81_3"><i><font color="Green">E61</font></i></a>, <a class="txt" href="vectsp_5.html#E30"><i><font color="Green">Lemma21</font></i></a></i>
<br/>.= 
 <a href="lattices.html#K5">Bottom</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="vectsp_5.html#E13:81"><i><font color="Green">E59</font></i></a>, <a class="ref" href="vectsp_5.html#T50" target="_self">Th50</a></i>
;<br/><b>hence </b><a NAME="E10:81_3"/>
<font color="Maroon">c<sub>16</sub></font> <a href="lattices.html#R2">is_a_complement_of</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="vectsp_5.html#E8:81_3"><i><font color="Green">E63</font></i></a>, <a class="ref" href="lattices.html#D18">LATTICES:def 18</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E15:81"/>
 <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="vectsp_5.html#K3" target="_self">Subspaces</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp_5.html#K5" target="_self">SubJoin</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp_5.html#K6" target="_self">SubMeet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> #) is   <a href="lattices.html#NM7">C_Lattice</a>
 <b>by </b><i><a class="ref" href="lattices.html#D19">LATTICES:def 19</a></i>;<br/>


</div>
