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

<span class="kw">let </span><font color="Maroon" title="c1">R</font> be   <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">V</font> being   <a href="vectsp_2.html#NM7" title="VECTSP_2:NM.7">RightMod</a> of <font color="Maroon" title="c1">R</font> holds   <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="rmod_3.html#K3" title="RMOD_3:func.3">Submodules</a> <font color="Olive" title="b1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rmod_3.html#K5" title="RMOD_3:func.5">SubJoin</a> <font color="Olive" title="b1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rmod_3.html#K6" title="RMOD_3:func.6">SubMeet</a> <font color="Olive" title="b1">V</font></span>)</span> #) is   <a href="lattices.html#NM3" title="LATTICES:NM.3">M_Lattice</a></span><br/><span class="kw">let </span><font color="Maroon" title="c2">V</font> be   <a href="vectsp_2.html#NM7" title="VECTSP_2:NM.7">RightMod</a> of <font color="Maroon" title="c1">R</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="rmod_3.html#K3" title="RMOD_3:func.3">Submodules</a> <font color="Maroon" title="c2">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rmod_3.html#K5" title="RMOD_3:func.5">SubJoin</a> <font color="Maroon" title="c2">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rmod_3.html#K6" title="RMOD_3:func.6">SubMeet</a> <font color="Maroon" title="c2">V</font></span>)</span> #) is   <a href="lattices.html#NM3" title="LATTICES:NM.3">M_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="rmod_3.html#K3" title="RMOD_3:func.3">Submodules</a> <font color="Maroon" title="c2">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rmod_3.html#K5" title="RMOD_3:func.5">SubJoin</a> <font color="Maroon" title="c2">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rmod_3.html#K6" title="RMOD_3:func.6">SubMeet</a> <font color="Maroon" title="c2">V</font></span>)</span> #);<br/>
<a NAME="E1:61"/>
 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="rmod_3.html#K3" title="RMOD_3:func.3">Submodules</a> <font color="Maroon" title="c2">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rmod_3.html#K5" title="RMOD_3:func.5">SubJoin</a> <font color="Maroon" title="c2">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rmod_3.html#K6" title="RMOD_3:func.6">SubMeet</a> <font color="Maroon" title="c2">V</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="61_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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="rmod_3.html#K3" title="RMOD_3:func.3">Submodules</a> <font color="Maroon" title="c2">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rmod_3.html#K5" title="RMOD_3:func.5">SubJoin</a> <font color="Maroon" title="c2">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rmod_3.html#K6" title="RMOD_3:func.6">SubMeet</a> <font color="Maroon" title="c2">V</font></span>)</span> #); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">assume </span><a NAME="E1:61_1"/><span class="lab"><font color="Green" title="E44">A1</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">consider </span><font color="Maroon" title="c7">W1</font> being   <a href="vectsp_2.html#V2" title="VECTSP_2:attr.2">strict</a>   <a href="rmod_2.html#M1" title="RMOD_2:mode.1">Submodule</a> of <font color="Maroon" title="c2">V</font><span class="kw"> such that </span><br/><a NAME="E3:61_1"/><span class="lab"><font color="Green" title="E45">A2</font></span>: 
<font color="Maroon" title="c7">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rmod_3.html#D3" target="_self" title="RMOD_3:def.3">Def3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">W3</font> being   <a href="vectsp_2.html#V2" title="VECTSP_2:attr.2">strict</a>   <a href="rmod_2.html#M1" title="RMOD_2:mode.1">Submodule</a> of <font color="Maroon" title="c2">V</font><span class="kw"> such that </span><br/><a NAME="E5:61_1"/><span class="lab"><font color="Green" title="E46">A3</font></span>: 
<font color="Maroon" title="c8">W3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rmod_3.html#D3" target="_self" title="RMOD_3:def.3">Def3</a></span>;<br/>
<font color="Maroon" title="c7">W1</font> <a href="rmod_3.html#K1" title="RMOD_3:func.1">+</a> <font color="Maroon" title="c8">W3</font> = 
<span class="p1">(<span class="default"><a href="rmod_3.html#K5" title="RMOD_3: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="c4">A</font>,<font color="Maroon" title="c6">C</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="txt" href="#E5:61_1"><span class="lab"><font color="Green" title="E46">A3</font></span></a>, <a class="ref" href="rmod_3.html#D6" target="_self" title="RMOD_3:def.6">Def6</a></span>
<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>
<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="c8">W3</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_1"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="txt" href="#E5:61_1"><span class="lab"><font color="Green" title="E46">A3</font></span></a>, <a class="ref" href="lattices.html#D3" title="LATTICES:def.3">LATTICES:def 3</a></span>
;<br/>
<a NAME="E7:61_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A4</font></span>: 
<font color="Maroon" title="c7">W1</font> is    <a href="rmod_2.html#M1" title="RMOD_2:mode.1">Submodule</a> of <font color="Maroon" title="c8">W3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rmod_3.html#T8" target="_self" title="RMOD_3:th.8">Th12</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">W2</font> being   <a href="vectsp_2.html#V2" title="VECTSP_2:attr.2">strict</a>   <a href="rmod_2.html#M1" title="RMOD_2:mode.1">Submodule</a> of <font color="Maroon" title="c2">V</font><span class="kw"> such that </span><br/><a NAME="E9:61_1"/><span class="lab"><font color="Green" title="E48">A5</font></span>: 
<font color="Maroon" title="c9">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">B</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rmod_3.html#D3" target="_self" title="RMOD_3:def.3">Def3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">AB</font> = <font color="Maroon" title="c7">W1</font> <a href="rmod_3.html#K1" title="RMOD_3:func.1">+</a> <font color="Maroon" title="c9">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="rmod_3.html#K3" title="RMOD_3:func.3">Submodules</a> <font color="Maroon" title="c2">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rmod_3.html#K5" title="RMOD_3:func.5">SubJoin</a> <font color="Maroon" title="c2">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rmod_3.html#K6" title="RMOD_3:func.6">SubMeet</a> <font color="Maroon" title="c2">V</font></span>)</span> #) <span class="kw">by</span> <span class="lab"><a class="ref" href="rmod_3.html#D3" target="_self" title="RMOD_3:def.3">Def3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">BC</font> = <font color="Maroon" title="c9">W2</font> <a href="rmod_3.html#K2" title="RMOD_3:func.2">/\</a> <font color="Maroon" title="c8">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="rmod_3.html#K3" title="RMOD_3:func.3">Submodules</a> <font color="Maroon" title="c2">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rmod_3.html#K5" title="RMOD_3:func.5">SubJoin</a> <font color="Maroon" title="c2">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rmod_3.html#K6" title="RMOD_3:func.6">SubMeet</a> <font color="Maroon" title="c2">V</font></span>)</span> #) <span class="kw">by</span> <span class="lab"><a class="ref" href="rmod_3.html#D3" target="_self" title="RMOD_3:def.3">Def3</a></span>;<br/>
<span class="kw">thus </span><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="rmod_3.html#K5" title="RMOD_3: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="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>)
<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/>.= 
<span class="p1">(<span class="default"><a href="rmod_3.html#K5" title="RMOD_3: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="c4">A</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rmod_3.html#K6" title="RMOD_3: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="c5">B</font>,<font color="Maroon" title="c6">C</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D2" title="LATTICES:def.2">LATTICES:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="rmod_3.html#K5" title="RMOD_3: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="c4">A</font>,<font color="Maroon" title="c11">BC</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:61_1"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="txt" href="#E5:61_1"><span class="lab"><font color="Green" title="E46">A3</font></span></a>, <a class="ref" href="rmod_3.html#D7" target="_self" title="RMOD_3:def.7">Def7</a></span>
<br/>.= 
<font color="Maroon" title="c7">W1</font> <a href="rmod_3.html#K1" title="RMOD_3:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">W2</font> <a href="rmod_3.html#K2" title="RMOD_3:func.2">/\</a> <font color="Maroon" title="c8">W3</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="ref" href="rmod_3.html#D6" target="_self" title="RMOD_3:def.6">Def6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">W1</font> <a href="rmod_3.html#K1" title="RMOD_3:func.1">+</a> <font color="Maroon" title="c9">W2</font></span>)</span> <a href="rmod_3.html#K2" title="RMOD_3:func.2">/\</a> <font color="Maroon" title="c8">W3</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:61_1"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="ref" href="rmod_3.html#T31" target="_self" title="RMOD_3:th.31">Th36</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="rmod_3.html#K6" title="RMOD_3: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="c10">AB</font>,<font color="Maroon" title="c6">C</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:61_1"><span class="lab"><font color="Green" title="E46">A3</font></span></a>, <a class="ref" href="rmod_3.html#D7" target="_self" title="RMOD_3:def.7">Def7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="rmod_3.html#K6" title="RMOD_3: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> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rmod_3.html#K5" title="RMOD_3: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="c4">A</font>,<font color="Maroon" title="c5">B</font>)</span>)</span>,<font color="Maroon" title="c6">C</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="txt" href="#E9:61_1"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="ref" href="rmod_3.html#D6" target="_self" title="RMOD_3:def.6">Def6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="rmod_3.html#K6" title="RMOD_3: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> (<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>)
<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/>.= 
<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 class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D2" title="LATTICES:def.2">LATTICES:def 2</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:61"/>
 <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="rmod_3.html#K3" title="RMOD_3:func.3">Submodules</a> <font color="Maroon" title="c2">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rmod_3.html#K5" title="RMOD_3:func.5">SubJoin</a> <font color="Maroon" title="c2">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rmod_3.html#K6" title="RMOD_3:func.6">SubMeet</a> <font color="Maroon" title="c2">V</font></span>)</span> #) is   <a href="lattices.html#NM3" title="LATTICES:NM.3">M_Lattice</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rmod_3.html#T47" target="_self" title="RMOD_3:th.47">Th63</a>, <a class="ref" href="lattices.html#D12" title="LATTICES:def.12">LATTICES:def 12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
