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<b>let </b><font color="Maroon" title="c1">V</font> be   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>;<br/>

<b>set </b><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="E1:75"/>
 ex <font color="Olive" title="b1">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> #) st <br/>for <font color="Olive" title="b2">A</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 <br/> ( <font color="Olive" title="b1">C</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">C</font> &amp; <font color="Olive" title="b2">A</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">C</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c3">C</font> =  <a href="rlsub_1.html#K1" title="RLSUB_1:func.1">(0).</a> <font color="Maroon" title="c1">V</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> #) <b>by </b><i><a class="ref" href="rlsub_2.html#D3" target="_self" title="RLSUB_2:def.3">Def3</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c3">C</font>
;<br/>

<b>let </b><font color="Maroon" title="c4">A</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> #);<br/>

<b>reconsider </b><font color="Maroon" title="c5">W</font> = <font color="Maroon" title="c4">A</font> as    <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="ref" href="rlsub_2.html#D3" target="_self" title="RLSUB_2:def.3">Def3</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c3">C</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">A</font> = 
<span class="p1">(<span class="default"><a href="rlsub_1.html#K1" title="RLSUB_1:func.1">(0).</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="rlsub_2.html#K2" title="RLSUB_2:func.2">/\</a> <font color="Maroon" title="c5">W</font>
<b>by </b><i><a class="ref" href="rlsub_2.html#D8" target="_self" title="RLSUB_2:def.8">Def8</a></i>
<br/>.= 
<font color="Maroon" title="c3">C</font>
<b>by </b><i><a class="ref" href="rlsub_2.html#T22" target="_self" title="RLSUB_2:th.22">Th22</a></i>
;<br/>

<b>hence </b><a NAME="E4:75_1"/>
<font color="Maroon" title="c4">A</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">C</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:75"/>
 <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#V13" title="LATTICES:attr.13">lower-bounded</a>
 <b>by </b><i><a class="ref" href="lattices.html#D13" title="LATTICES:def.13">LATTICES:def 13</a></i>;<br/>


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