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

<span class="kw">let </span><font color="Maroon" title="c1">V</font> be   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>; <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="rlsub_2.html#K3" title="RLSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlsub_2.html#K5" title="RLSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlsub_2.html#K6" title="RLSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> #) is   <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a></span><br/>

<span class="kw">set </span><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="rlsub_2.html#K3" title="RLSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlsub_2.html#K5" title="RLSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlsub_2.html#K6" title="RLSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> #);<br/>
<a NAME="E1:82"/><span class="lab"><font color="Green" title="E51">A1</font></span>: 
for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</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="rlsub_2.html#K3" title="RLSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlsub_2.html#K5" title="RLSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlsub_2.html#K6" title="RLSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> #) holds  <font color="Olive" title="b1">A</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">B</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b1">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</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="rlsub_2.html#K3" title="RLSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlsub_2.html#K5" title="RLSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlsub_2.html#K6" title="RLSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</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="c3">A</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c3">A</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">W1</font> = <font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c6">W2</font> = <font color="Maroon" title="c4">B</font> as    <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="rlsub_2.html#D3" target="_self" title="RLSUB_2:def.3">Def3</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c3">A</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">B</font> = 
<font color="Maroon" title="c5">W1</font> <a href="rlsub_2.html#K2" title="RLSUB_2:func.2">/\</a> <font color="Maroon" title="c6">W2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlsub_2.html#D8" target="_self" title="RLSUB_2:def.8">Def8</a></span>
<br/>.= 
<font color="Maroon" title="c6">W2</font> <a href="rlsub_2.html#K2" title="RLSUB_2:func.2">/\</a> <font color="Maroon" title="c5">W1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlsub_2.html#T18" target="_self" title="RLSUB_2:th.18">Th18</a></span>
<br/>.= 
<font color="Maroon" title="c4">B</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c3">A</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlsub_2.html#D8" target="_self" title="RLSUB_2:def.8">Def8</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>
<a NAME="E2:82"/><span class="lab"><font color="Green" title="E52">A2</font></span>: 
for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</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="rlsub_2.html#K3" title="RLSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlsub_2.html#K5" title="RLSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlsub_2.html#K6" title="RLSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> #) holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b2">B</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</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="rlsub_2.html#K3" title="RLSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlsub_2.html#K5" title="RLSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlsub_2.html#K6" title="RLSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> #); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">W1</font> = <font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c6">W2</font> = <font color="Maroon" title="c4">B</font> as   <a href="rlvect_1.html#V1" title="RLVECT_1:attr.1">strict</a>   <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="rlsub_2.html#D3" target="_self" title="RLSUB_2:def.3">Def3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">AB</font> = <font color="Maroon" title="c5">W1</font> <a href="rlsub_2.html#K2" title="RLSUB_2:func.2">/\</a> <font color="Maroon" title="c6">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="rlsub_2.html#K3" title="RLSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlsub_2.html#K5" title="RLSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlsub_2.html#K6" title="RLSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> #) <span class="kw">by</span> <span class="lab"><a class="ref" href="rlsub_2.html#D3" target="_self" title="RLSUB_2:def.3">Def3</a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">B</font> = 
<span class="p1">(<span class="default"><a href="rlsub_2.html#K5" title="RLSUB_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="c7">AB</font>,<font color="Maroon" title="c4">B</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlsub_2.html#D8" target="_self" title="RLSUB_2:def.8">Def8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">W1</font> <a href="rlsub_2.html#K2" title="RLSUB_2:func.2">/\</a> <font color="Maroon" title="c6">W2</font></span>)</span> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <font color="Maroon" title="c6">W2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlsub_2.html#D7" target="_self" title="RLSUB_2:def.7">Def7</a></span>
<br/>.= 
<font color="Maroon" title="c4">B</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="rlsub_2.html#E34"><span class="lab"><font color="Green" title="E22">Lm6</font></span></a>, <a class="ref" href="rlsub_1.html#T38" title="RLSUB_1:th.38">RLSUB_1:38</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>
<a NAME="E3:82"/><span class="lab"><font color="Green" title="E53">A3</font></span>: 
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="rlsub_2.html#K3" title="RLSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlsub_2.html#K5" title="RLSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlsub_2.html#K6" title="RLSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> #) holds  <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#K1" title="LATTICES:func.1">"\/"</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#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b3">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font>, <font color="Maroon" title="c5">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="rlsub_2.html#K3" title="RLSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlsub_2.html#K5" title="RLSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlsub_2.html#K6" title="RLSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</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="c3">A</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c5">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c5">C</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">W1</font> = <font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c7">W2</font> = <font color="Maroon" title="c4">B</font>, <font color="Maroon" title="c8">W3</font> = <font color="Maroon" title="c5">C</font> as    <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="rlsub_2.html#D3" target="_self" title="RLSUB_2:def.3">Def3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">AB</font> = <font color="Maroon" title="c6">W1</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <font color="Maroon" title="c7">W2</font>, <font color="Maroon" title="c10">BC</font> = <font color="Maroon" title="c7">W2</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</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="rlsub_2.html#K3" title="RLSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlsub_2.html#K5" title="RLSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlsub_2.html#K6" title="RLSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> #) <span class="kw">by</span> <span class="lab"><a class="ref" href="rlsub_2.html#D3" target="_self" title="RLSUB_2:def.3">Def3</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c3">A</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c5">C</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="rlsub_2.html#K5" title="RLSUB_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="c3">A</font>,<font color="Maroon" title="c10">BC</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlsub_2.html#D7" target="_self" title="RLSUB_2:def.7">Def7</a></span>
<br/>.= 
<font color="Maroon" title="c6">W1</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">W2</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <font color="Maroon" title="c8">W3</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlsub_2.html#D7" target="_self" title="RLSUB_2:def.7">Def7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">W1</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <font color="Maroon" title="c7">W2</font></span>)</span> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <font color="Maroon" title="c8">W3</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlsub_2.html#T10" target="_self" title="RLSUB_2:th.10">Th10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlsub_2.html#K5" title="RLSUB_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">AB</font>,<font color="Maroon" title="c5">C</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlsub_2.html#D7" target="_self" title="RLSUB_2:def.7">Def7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c5">C</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlsub_2.html#D7" target="_self" title="RLSUB_2:def.7">Def7</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>
<a NAME="E4:82"/><span class="lab"><font color="Green" title="E54">A4</font></span>: 
for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</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="rlsub_2.html#K3" title="RLSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlsub_2.html#K5" title="RLSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlsub_2.html#K6" title="RLSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> #) holds  <font color="Olive" title="b1">A</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</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="rlsub_2.html#K3" title="RLSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlsub_2.html#K5" title="RLSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlsub_2.html#K6" title="RLSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</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="c3">A</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">W1</font> = <font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c6">W2</font> = <font color="Maroon" title="c4">B</font> as   <a href="rlvect_1.html#V1" title="RLVECT_1:attr.1">strict</a>   <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="rlsub_2.html#D3" target="_self" title="RLSUB_2:def.3">Def3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">AB</font> = <font color="Maroon" title="c5">W1</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <font color="Maroon" title="c6">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="rlsub_2.html#K3" title="RLSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlsub_2.html#K5" title="RLSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlsub_2.html#K6" title="RLSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> #) <span class="kw">by</span> <span class="lab"><a class="ref" href="rlsub_2.html#D3" target="_self" title="RLSUB_2:def.3">Def3</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c3">A</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">B</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="rlsub_2.html#K6" title="RLSUB_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="c3">A</font>,<font color="Maroon" title="c7">AB</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlsub_2.html#D7" target="_self" title="RLSUB_2:def.7">Def7</a></span>
<br/>.= 
<font color="Maroon" title="c5">W1</font> <a href="rlsub_2.html#K2" title="RLSUB_2:func.2">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">W1</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <font color="Maroon" title="c6">W2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlsub_2.html#D8" target="_self" title="RLSUB_2:def.8">Def8</a></span>
<br/>.= 
<font color="Maroon" title="c3">A</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="rlsub_2.html#E36"><span class="lab"><font color="Green" title="E23">Lm7</font></span></a>, <a class="ref" href="rlsub_1.html#T38" title="RLSUB_1:th.38">RLSUB_1:38</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>
<a NAME="E5:82"/><span class="lab"><font color="Green" title="E55">A5</font></span>: 
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="rlsub_2.html#K3" title="RLSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlsub_2.html#K5" title="RLSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlsub_2.html#K6" title="RLSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> #) holds  <font color="Olive" title="b1">A</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</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#K2" title="LATTICES:func.2">"/\"</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="82_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font>, <font color="Maroon" title="c5">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="rlsub_2.html#K3" title="RLSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlsub_2.html#K5" title="RLSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlsub_2.html#K6" title="RLSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</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="c3">A</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c5">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c5">C</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">W1</font> = <font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c7">W2</font> = <font color="Maroon" title="c4">B</font>, <font color="Maroon" title="c8">W3</font> = <font color="Maroon" title="c5">C</font> as    <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="rlsub_2.html#D3" target="_self" title="RLSUB_2:def.3">Def3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">AB</font> = <font color="Maroon" title="c6">W1</font> <a href="rlsub_2.html#K2" title="RLSUB_2:func.2">/\</a> <font color="Maroon" title="c7">W2</font>, <font color="Maroon" title="c10">BC</font> = <font color="Maroon" title="c7">W2</font> <a href="rlsub_2.html#K2" title="RLSUB_2: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="rlsub_2.html#K3" title="RLSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlsub_2.html#K5" title="RLSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlsub_2.html#K6" title="RLSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> #) <span class="kw">by</span> <span class="lab"><a class="ref" href="rlsub_2.html#D3" target="_self" title="RLSUB_2:def.3">Def3</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c3">A</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c5">C</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="rlsub_2.html#K6" title="RLSUB_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="c3">A</font>,<font color="Maroon" title="c10">BC</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlsub_2.html#D8" target="_self" title="RLSUB_2:def.8">Def8</a></span>
<br/>.= 
<font color="Maroon" title="c6">W1</font> <a href="rlsub_2.html#K2" title="RLSUB_2:func.2">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">W2</font> <a href="rlsub_2.html#K2" title="RLSUB_2:func.2">/\</a> <font color="Maroon" title="c8">W3</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlsub_2.html#D8" target="_self" title="RLSUB_2:def.8">Def8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">W1</font> <a href="rlsub_2.html#K2" title="RLSUB_2:func.2">/\</a> <font color="Maroon" title="c7">W2</font></span>)</span> <a href="rlsub_2.html#K2" title="RLSUB_2:func.2">/\</a> <font color="Maroon" title="c8">W3</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlsub_2.html#T19" target="_self" title="RLSUB_2:th.19">Th19</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlsub_2.html#K6" title="RLSUB_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">AB</font>,<font color="Maroon" title="c5">C</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlsub_2.html#D8" target="_self" title="RLSUB_2:def.8">Def8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c5">C</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlsub_2.html#D8" target="_self" title="RLSUB_2:def.8">Def8</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>
<a NAME="E6:82"/>
for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</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="rlsub_2.html#K3" title="RLSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlsub_2.html#K5" title="RLSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlsub_2.html#K6" title="RLSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> #) holds  <font color="Olive" title="b1">A</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">B</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b1">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</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="rlsub_2.html#K3" title="RLSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlsub_2.html#K5" title="RLSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlsub_2.html#K6" title="RLSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</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="c3">A</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c3">A</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">W1</font> = <font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c6">W2</font> = <font color="Maroon" title="c4">B</font> as    <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="rlsub_2.html#D3" target="_self" title="RLSUB_2:def.3">Def3</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c3">A</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">B</font> = 
<font color="Maroon" title="c5">W1</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <font color="Maroon" title="c6">W2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlsub_2.html#D7" target="_self" title="RLSUB_2:def.7">Def7</a></span>
<br/>.= 
<font color="Maroon" title="c6">W2</font> <a href="rlsub_2.html#K1" title="RLSUB_2:func.1">+</a> <font color="Maroon" title="c5">W1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="rlsub_2.html#E10"><span class="lab"><font color="Green" title="E6">Lm1</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c4">B</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c3">A</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlsub_2.html#D7" target="_self" title="RLSUB_2:def.7">Def7</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>
<a NAME="E7:82"/><span class="kw">then </span>
(  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="rlsub_2.html#K3" title="RLSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlsub_2.html#K5" title="RLSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlsub_2.html#K6" title="RLSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> #) is  <a href="lattices.html#V4" title="LATTICES:attr.4">join-commutative</a>  &amp;  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="rlsub_2.html#K3" title="RLSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlsub_2.html#K5" title="RLSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlsub_2.html#K6" title="RLSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> #) is  <a href="lattices.html#V5" title="LATTICES:attr.5">join-associative</a>  &amp;  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="rlsub_2.html#K3" title="RLSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlsub_2.html#K5" title="RLSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlsub_2.html#K6" title="RLSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> #) is  <a href="lattices.html#V8" title="LATTICES:attr.8">meet-absorbing</a>  &amp;  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="rlsub_2.html#K3" title="RLSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlsub_2.html#K5" title="RLSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlsub_2.html#K6" title="RLSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> #) is  <a href="lattices.html#V6" title="LATTICES:attr.6">meet-commutative</a>  &amp;  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="rlsub_2.html#K3" title="RLSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlsub_2.html#K5" title="RLSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlsub_2.html#K6" title="RLSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> #) is  <a href="lattices.html#V7" title="LATTICES:attr.7">meet-associative</a>  &amp;  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="rlsub_2.html#K3" title="RLSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlsub_2.html#K5" title="RLSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlsub_2.html#K6" title="RLSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> #) is  <a href="lattices.html#V9" title="LATTICES:attr.9">join-absorbing</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:82"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="txt" href="#E2:82"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="txt" href="#E1:82"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="txt" href="#E5:82"><span class="lab"><font color="Green" title="E55">A5</font></span></a>, <a class="txt" href="#E4:82"><span class="lab"><font color="Green" title="E54">A4</font></span></a>, <a class="ref" href="lattices.html#D4" title="LATTICES:def.4">LATTICES:def 4</a>, <a class="ref" href="lattices.html#D5" title="LATTICES:def.5">LATTICES:def 5</a>, <a class="ref" href="lattices.html#D6" title="LATTICES:def.6">LATTICES:def 6</a>, <a class="ref" href="lattices.html#D7" title="LATTICES:def.7">LATTICES:def 7</a>, <a class="ref" href="lattices.html#D8" title="LATTICES:def.8">LATTICES:def 8</a>, <a class="ref" href="lattices.html#D9" title="LATTICES:def.9">LATTICES:def 9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:82"/>
 <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="rlsub_2.html#K3" title="RLSUB_2:func.3">Subspaces</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlsub_2.html#K5" title="RLSUB_2:func.5">SubJoin</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlsub_2.html#K6" title="RLSUB_2:func.6">SubMeet</a> <font color="Maroon" title="c1">V</font></span>)</span> #) is   <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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