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

<b>let </b><font color="Maroon">V</font> be   <a href="bhsp_1.html#NM1">RealUnitarySpace</a>;<br/>

<b>set </b><font color="Maroon">S</font> =  <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="rusub_2.html#K3" target="_self">Subspaces</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" target="_self">SubJoin</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" target="_self">SubMeet</a> <font color="Maroon">V</font></span>)</span> #);<br/>
<a NAME="E1:80"/>
for <font color="Olive">A</font>, <font color="Olive">B</font>, <font color="Olive">C</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="rusub_2.html#K3" target="_self">Subspaces</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" target="_self">SubJoin</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" target="_self">SubMeet</a> <font color="Maroon">V</font></span>)</span> #)  st <font color="Olive">A</font> <a href="lattices.html#R3">[=</a> <font color="Olive">C</font> holds <br/><font color="Olive">A</font> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Olive">B</font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">C</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">A</font> <a href="lattices.html#K3">"\/"</a> <font color="Olive">B</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Olive">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="rusub_2.html#K3" target="_self">Subspaces</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" target="_self">SubJoin</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" target="_self">SubMeet</a> <font color="Maroon">V</font></span>)</span> #), <font color="Maroon">B</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="rusub_2.html#K3" target="_self">Subspaces</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" target="_self">SubJoin</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" target="_self">SubMeet</a> <font color="Maroon">V</font></span>)</span> #), <font color="Maroon">C</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="rusub_2.html#K3" target="_self">Subspaces</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" target="_self">SubJoin</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" target="_self">SubMeet</a> <font color="Maroon">V</font></span>)</span> #);<br/>

<b>assume </b><a NAME="E1:80_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">A</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">C</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">W1</font> = <font color="Maroon">A</font>, <font color="Maroon">W2</font> = <font color="Maroon">B</font>, <font color="Maroon">W3</font> = <font color="Maroon">C</font> as  <a href="bhsp_1.html#V1">strict</a>  <a href="rusub_1.html#M1">Subspace</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="rusub_2.html#T1" target="_self">Th1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">BC</font> = <font color="Maroon">W2</font> <a href="rusub_2.html#K2" target="_self">/\</a> <font color="Maroon">W3</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="rusub_2.html#K3" target="_self">Subspaces</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" target="_self">SubJoin</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" target="_self">SubMeet</a> <font color="Maroon">V</font></span>)</span> #) <b>by </b><i><a class="ref" href="rusub_2.html#T1" target="_self">Th1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">AB</font> = <font color="Maroon">W1</font> <a href="rusub_2.html#K1" target="_self">+</a> <font color="Maroon">W2</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="rusub_2.html#K3" target="_self">Subspaces</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" target="_self">SubJoin</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" target="_self">SubMeet</a> <font color="Maroon">V</font></span>)</span> #) <b>by </b><i><a class="ref" href="rusub_2.html#T1" target="_self">Th1</a></i>;<br/>
<font color="Maroon">W1</font> <a href="rusub_2.html#K1" target="_self">+</a> <font color="Maroon">W3</font> = 
<span class="p1">(<span class="default"><a href="rusub_2.html#K5" target="_self">SubJoin</a> <font color="Maroon">V</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">A</font>,<font color="Maroon">C</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">A</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">C</font>
<b>by </b><i><a class="ref" href="lattices.html#D1">LATTICES:def 1</a></i>
<br/>.= 
<font color="Maroon">W3</font>
<b>by </b><i><a class="ref" href="rusub_2.html#T8" target="_self">Th8</a>, <a class="ref" href="lattices.html#D3">LATTICES:def 3</a></i>
;<br/>
<a NAME="E6:80_1"/><b>then </b><i><font color="Green">E23</font></i>: 
<font color="Maroon">W1</font> is    <a href="rusub_1.html#M1">Subspace</a> of <font color="Maroon">W3</font>
 
<b>by </b><i/>;<br/>
<b>thus </b><font color="Maroon">A</font> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">C</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="rusub_2.html#K5" target="_self">SubJoin</a> <font color="Maroon">V</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">A</font>,<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">C</font></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D1">LATTICES:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rusub_2.html#K5" target="_self">SubJoin</a> <font color="Maroon">V</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">A</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rusub_2.html#K6" target="_self">SubMeet</a> <font color="Maroon">V</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">B</font>,<font color="Maroon">C</font></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D2">LATTICES:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rusub_2.html#K5" target="_self">SubJoin</a> <font color="Maroon">V</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">A</font>,<font color="Maroon">BC</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">W1</font> <a href="rusub_2.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">W2</font> <a href="rusub_2.html#K2" target="_self">/\</a> <font color="Maroon">W3</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="rusub_2.html#K1" target="_self">+</a> <font color="Maroon">W2</font></span>)</span> <a href="rusub_2.html#K2" target="_self">/\</a> <font color="Maroon">W3</font>
<b>by </b><i><a class="ref" href="rusub_2.html#T9" target="_self">Th9</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rusub_2.html#K6" target="_self">SubMeet</a> <font color="Maroon">V</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">AB</font>,<font color="Maroon">C</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="rusub_2.html#K6" target="_self">SubMeet</a> <font color="Maroon">V</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rusub_2.html#K5" target="_self">SubJoin</a> <font color="Maroon">V</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">C</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="rusub_2.html#K6" target="_self">SubMeet</a> <font color="Maroon">V</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">B</font></span>)</span>,<font color="Maroon">C</font>
<b>by </b><i><a class="ref" href="lattices.html#D1">LATTICES:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">B</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">C</font>
<b>by </b><i><a class="ref" href="lattices.html#D2">LATTICES:def 2</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:80"/>
 <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="rusub_2.html#K3" target="_self">Subspaces</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" target="_self">SubJoin</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" target="_self">SubMeet</a> <font color="Maroon">V</font></span>)</span> #) is <a href="lattices.html#V12">modular</a>
 <b>by </b><i><a class="ref" href="lattices.html#D12">LATTICES:def 12</a></i>;<br/>


</div>
