<?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="E1:78"/>
 ex <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 <br/>for <font color="Olive">A</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> #) holds <br/> ( <font color="Olive">C</font> <a href="lattices.html#K3">"\/"</a> <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">C</font> &amp; <font color="Olive">A</font> <a href="lattices.html#K3">"\/"</a> <font color="Olive">C</font> <a href="hidden.html#R1">=</a> <font color="Olive">C</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">C</font> =  <a href="rusub_1.html#K2">(Omega).</a> <font color="Maroon">V</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>take </b>
<font color="Maroon">C</font>
;<br/>

<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> #);<br/>

<b>reconsider </b><font color="Maroon">W</font> = <font color="Maroon">A</font> as    <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>thus </b><font color="Maroon">C</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">A</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">C</font>,<font color="Maroon">A</font>
<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_1.html#K2">(Omega).</a> <font color="Maroon">V</font></span>)</span> <a href="rusub_2.html#K1" target="_self">+</a> <font color="Maroon">W</font>
<b>by </b><i/>
<br/>.= 
 <a href="bhsp_1.html#G1">UNITSTR</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">V</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font>,the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">V</font>,the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">V</font> #)
<b>by </b><i/>
<br/>.= 
<font color="Maroon">C</font>
<b>by </b><i><a class="ref" href="rusub_1.html#D3">RUSUB_1:def 3</a></i>
;<br/>

<b>hence </b><a NAME="E4:78_1"/>
<font color="Maroon">A</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:78"/>
 <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#V14">upper-bounded</a>
 <b>by </b><i><a class="ref" href="lattices.html#D14">LATTICES:def 14</a></i>;<br/>


</div>
