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

<span class="kw">let </span><font color="Maroon" title="c1">GF</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>   <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a>   <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a>   <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a>   <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>   <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>   <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>   <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">M</font> being non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>   <a href="vectsp_1.html#V8" title="VECTSP_1:attr.8">vector-distributive</a>   <a href="vectsp_1.html#V9" title="VECTSP_1:attr.9">scalar-distributive</a>   <a href="vectsp_1.html#V10" title="VECTSP_1:attr.10">scalar-associative</a>   <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">scalar-unital</a>   <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>   <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>   <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>   <a href="vectsp_1.html#L1" title="VECTSP_1:struct.1">VectSpStr</a> of <font color="Maroon" title="c1">GF</font> holds   <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="Olive" title="b1">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp_5.html#K5" title="VECTSP_5:func.5">SubJoin</a> <font color="Olive" title="b1">M</font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp_5.html#K6" title="VECTSP_5:func.6">SubMeet</a> <font color="Olive" title="b1">M</font></span>)</span> #) is   <a href="lattices.html#NM5" title="LATTICES:NM.5">1_Lattice</a></span><br/><span class="kw">let </span><font color="Maroon" title="c2">M</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>   <a href="vectsp_1.html#V8" title="VECTSP_1:attr.8">vector-distributive</a>   <a href="vectsp_1.html#V9" title="VECTSP_1:attr.9">scalar-distributive</a>   <a href="vectsp_1.html#V10" title="VECTSP_1:attr.10">scalar-associative</a>   <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">scalar-unital</a>   <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>   <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>   <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>   <a href="vectsp_1.html#L1" title="VECTSP_1:struct.1">VectSpStr</a> of <font color="Maroon" title="c1">GF</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <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">M</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">M</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">M</font></span>)</span> #) is   <a href="lattices.html#NM5" title="LATTICES:NM.5">1_Lattice</a></span><br/>



<span class="kw">set </span><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">M</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">M</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">M</font></span>)</span> #);<br/>
<a NAME="E1:89"/>
 ex <font color="Olive" title="b1">C</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <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">M</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">M</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">M</font></span>)</span> #) st <br/>for <font color="Olive" title="b2">A</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <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">M</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">M</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">M</font></span>)</span> #) holds <br/> ( <font color="Olive" title="b1">C</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">C</font> &amp; <font color="Olive" title="b2">A</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">C</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c4">W9</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">M</font><span class="kw"> such that </span><br/><a NAME="E2:89_1"/><span class="lab"><font color="Green" title="E52">A1</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">W9</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">M</font></span>)</span>
 ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">C</font> = <font color="Maroon" title="c4">W9</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <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">M</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">M</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">M</font></span>)</span> #) <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_5.html#D3" target="_self" title="VECTSP_5:def.3">Def3</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">C</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <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">M</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">M</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">M</font></span>)</span> #) holds <br/> ( <font color="Maroon" title="c5">C</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">C</font> &amp; <font color="Olive" title="b1">A</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c5">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">C</font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">A</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <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">M</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">M</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">M</font></span>)</span> #); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">C</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">C</font> &amp; <font color="Maroon" title="c6">A</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c5">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">C</font> )</span><br/>

<span class="kw">consider </span><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">M</font><span class="kw"> such that </span><br/><a NAME="E5:89_1"/><span class="lab"><font color="Green" title="E53">A2</font></span>: 
<font color="Maroon" title="c7">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_5.html#D3" target="_self" title="VECTSP_5:def.3">Def3</a></span>;<br/>
<a NAME="E6:89_1"/><span class="lab"><font color="Green" title="E54">A3</font></span>: 
<font color="Maroon" title="c5">C</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">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="vectsp_5.html#E21"><span class="lab"><font color="Green" title="E15">Lm6</font></span></a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c5">C</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c6">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">M</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">C</font>,<font color="Maroon" title="c6">A</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D1" title="LATTICES:def.1">LATTICES:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c4">W9</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c7">W</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:89_1"><span class="lab"><font color="Green" title="E53">A2</font></span></a>, <a class="ref" href="vectsp_5.html#D7" target="_self" title="VECTSP_5:def.7">Def7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c2">M</font></span>)</span> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c7">W</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:89_1"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="txt" href="vectsp_5.html#E20"><span class="lab"><font color="Green" title="E14">Lm5</font></span></a></span>
<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">M</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">M</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c2">M</font>,the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c2">M</font> #)
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_5.html#T15" target="_self" title="VECTSP_5:th.15">Th15</a></span>
<br/>.= 
<font color="Maroon" title="c5">C</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:89_1"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="txt" href="#E6:89_1"><span class="lab"><font color="Green" title="E54">A3</font></span></a>, <a class="ref" href="vectsp_4.html#T39" title="VECTSP_4:th.39">VECTSP_4:39</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">A</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c5">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">C</font></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c6">A</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c5">C</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">M</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c5">C</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D1" title="LATTICES:def.1">LATTICES:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c7">W</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c4">W9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:89_1"><span class="lab"><font color="Green" title="E53">A2</font></span></a>, <a class="ref" href="vectsp_5.html#D7" target="_self" title="VECTSP_5:def.7">Def7</a></span>
<br/>.= 
<font color="Maroon" title="c7">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">M</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:89_1"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="txt" href="vectsp_5.html#E20"><span class="lab"><font color="Green" title="E14">Lm5</font></span></a></span>
<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">M</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">M</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c2">M</font>,the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c2">M</font> #)
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_5.html#T15" target="_self" title="VECTSP_5:th.15">Th15</a></span>
<br/>.= 
<font color="Maroon" title="c5">C</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:89_1"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="txt" href="#E6:89_1"><span class="lab"><font color="Green" title="E54">A3</font></span></a>, <a class="ref" href="vectsp_4.html#T39" title="VECTSP_4:th.39">VECTSP_4:39</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:89"/>
 <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">M</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">M</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">M</font></span>)</span> #) is   <a href="lattices.html#NM5" title="LATTICES:NM.5">1_Lattice</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_5.html#T75" target="_self" title="VECTSP_5:th.75">Th75</a>, <a class="ref" href="lattices.html#D14" title="LATTICES:def.14">LATTICES:def 14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
