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

<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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">VectSp-like</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#NM3" title="LATTICES:NM.3">M_Lattice</a></span><br/><b>let </b><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">VectSp-like</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:()"><i><font color="Red">::  thesis: </font></i></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#NM3" title="LATTICES:NM.3">M_Lattice</a></span><br/>



<b>set </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">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="E1:73"/>
for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font>, <font color="Olive" title="b3">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 <font color="Olive" title="b1">A</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Olive" title="b3">C</font> holds <br/><font color="Olive" title="b1">A</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">B</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b3">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b2">B</font></span>)</span> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b3">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">A</font>, <font color="Maroon" title="c5">B</font>, <font color="Maroon" title="c6">C</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">A</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c6">C</font> implies <font color="Maroon" title="c4">A</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c6">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c5">B</font></span>)</span> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c6">C</font> )</span><br/>

<b>assume </b><a NAME="E1:73_1"/><i><font color="Green" title="E53">A1</font></i>: 
<font color="Maroon" title="c4">A</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c6">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">A</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c6">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c5">B</font></span>)</span> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c6">C</font></span><br/>

<b>consider </b><font color="Maroon" title="c7">W1</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><b> such that </b><br/><a NAME="E3:73_1"/><i><font color="Green" title="E54">A2</font></i>: 
<font color="Maroon" title="c7">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">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="c8">W2</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><b> such that </b><br/><a NAME="E5:73_1"/><i><font color="Green" title="E55">A3</font></i>: 
<font color="Maroon" title="c8">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">B</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="c9">W3</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><b> such that </b><br/><a NAME="E7:73_1"/><i><font color="Green" title="E56">A4</font></i>: 
<font color="Maroon" title="c9">W3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">C</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="c10">BC</font> = <font color="Maroon" title="c8">W2</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c9">W3</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> #) <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="c11">AB</font> = <font color="Maroon" title="c7">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c8">W2</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> #) <b>by </b><i><a class="ref" href="vectsp_5.html#D3" target="_self" title="VECTSP_5:def.3">Def3</a></i>;<br/>
<font color="Maroon" title="c7">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c9">W3</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="c4">A</font>,<font color="Maroon" title="c6">C</font>
<b>by </b><i><a class="txt" href="#E3:73_1"><i><font color="Green" title="E54">A2</font></i></a>, <a class="txt" href="#E7:73_1"><i><font color="Green" title="E56">A4</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="c4">A</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c6">C</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="c9">W3</font>
<b>by </b><i><a class="txt" href="#E1:73_1"><i><font color="Green" title="E53">A1</font></i></a>, <a class="txt" href="#E7:73_1"><i><font color="Green" title="E56">A4</font></i></a>, <a class="ref" href="lattices.html#D3" title="LATTICES:def.3">LATTICES:def 3</a></i>
;<br/>
<a NAME="E11:73_1"/><b>then </b><i><font color="Green" title="E57">A5</font></i>: 
<font color="Maroon" title="c7">W1</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c9">W3</font>
 
<b>by </b><i><a class="ref" href="vectsp_5.html#T12" target="_self" title="VECTSP_5:th.12">Th12</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c4">A</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c6">C</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> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c6">C</font></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D1" title="LATTICES:def.1">LATTICES:def 1</a></i>
<br/>.= 
<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="c4">A</font>,<span class="p1">(<span class="default"><span class="p2">(<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 href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c6">C</font></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D2" title="LATTICES:def.2">LATTICES:def 2</a></i>
<br/>.= 
<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="c4">A</font>,<font color="Maroon" title="c10">BC</font>
<b>by </b><i><a class="txt" href="#E5:73_1"><i><font color="Green" title="E55">A3</font></i></a>, <a class="txt" href="#E7:73_1"><i><font color="Green" title="E56">A4</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="c7">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">W2</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c9">W3</font></span>)</span>
<b>by </b><i><a class="txt" href="#E3:73_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/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c8">W2</font></span>)</span> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c9">W3</font>
<b>by </b><i><a class="txt" href="#E11:73_1"><i><font color="Green" title="E57">A5</font></i></a>, <a class="ref" href="vectsp_5.html#T36" target="_self" title="VECTSP_5:th.36">Th36</a></i>
<br/>.= 
<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 href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">AB</font>,<font color="Maroon" title="c6">C</font>
<b>by </b><i><a class="txt" href="#E7:73_1"><i><font color="Green" title="E56">A4</font></i></a>, <a class="ref" href="vectsp_5.html#D8" target="_self" title="VECTSP_5:def.8">Def8</a></i>
<br/>.= 
<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 href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<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="c4">A</font>,<font color="Maroon" title="c5">B</font></span>)</span>,<font color="Maroon" title="c6">C</font>
<b>by </b><i><a class="txt" href="#E3:73_1"><i><font color="Green" title="E54">A2</font></i></a>, <a class="txt" href="#E5:73_1"><i><font color="Green" title="E55">A3</font></i></a>, <a class="ref" href="vectsp_5.html#D7" target="_self" title="VECTSP_5:def.7">Def7</a></i>
<br/>.= 
<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 href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c5">B</font></span>)</span>,<font color="Maroon" title="c6">C</font>
<b>by </b><i><a class="ref" href="lattices.html#D1" title="LATTICES:def.1">LATTICES:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c5">B</font></span>)</span> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c6">C</font>
<b>by </b><i><a class="ref" href="lattices.html#D2" title="LATTICES:def.2">LATTICES:def 2</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:73"/>
 <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#NM3" title="LATTICES:NM.3">M_Lattice</a>
 <b>by </b><i><a class="ref" href="vectsp_5.html#T75" target="_self" title="VECTSP_5:th.75">Th75</a>, <a class="ref" href="lattices.html#D12" title="LATTICES:def.12">LATTICES:def 12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
