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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V3">Abelian</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="group_1.html#V4">associative</a> <a href="vectsp_1.html#V7">distributive</a> <a href="vectsp_1.html#V8">left_unital</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V3">Abelian</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="vectsp_1.html#V12">VectSp-like</a>  <a href="vectsp_1.html#L4">VectSpStr</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="vectsp_5.html#K3" target="_self">Subspaces</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp_5.html#K5" target="_self">SubJoin</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp_5.html#K6" target="_self">SubMeet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> #);<br/>
<a NAME="E1:77"/>
ex <font color="Olive">b<sub>1</sub></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="vectsp_5.html#K3" target="_self">Subspaces</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp_5.html#K5" target="_self">SubJoin</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp_5.html#K6" target="_self">SubMeet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> #) st <br/>for <font color="Olive">b<sub>2</sub></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="vectsp_5.html#K3" target="_self">Subspaces</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp_5.html#K5" target="_self">SubJoin</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp_5.html#K6" target="_self">SubMeet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> #) holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> =  <a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">c<sub>2</sub></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="vectsp_5.html#K3" target="_self">Subspaces</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp_5.html#K5" target="_self">SubJoin</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp_5.html#K6" target="_self">SubMeet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> #) <b>by </b><i><a class="ref" href="vectsp_5.html#D3" target="_self">Def3</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></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="vectsp_5.html#K3" target="_self">Subspaces</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp_5.html#K5" target="_self">SubJoin</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp_5.html#K6" target="_self">SubMeet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> #);<br/>

<b>consider </b><font color="Maroon">c<sub>6</sub></font> being  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:77_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#D3" target="_self">Def3</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c<sub>5</sub></font> = 
<span class="p1">(<span class="default"><a href="vectsp_5.html#K6" target="_self">SubMeet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
<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="vectsp_4.html#K1">(0).</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="vectsp_5.html#K2" target="_self">/\</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="vectsp_5.html#E3:77_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="vectsp_5.html#D8" target="_self">Def8</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="vectsp_5.html#T25" target="_self">Th25</a></i>
;<br/>

<b>thus </b><font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><a href="vectsp_5.html#K6" target="_self">SubMeet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="lattices.html#D2">LATTICES:def 2</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="vectsp_5.html#K2" target="_self">/\</a> <span class="p1">(<span class="default"><a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="vectsp_5.html#E3:77_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="vectsp_5.html#D8" target="_self">Def8</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="vectsp_5.html#T25" target="_self">Th25</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:77"/>
 <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="vectsp_5.html#K3" target="_self">Subspaces</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp_5.html#K5" target="_self">SubJoin</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp_5.html#K6" target="_self">SubMeet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> #) is   <a href="lattices.html#NM4">0_Lattice</a>
 <b>by </b><i><a class="ref" href="vectsp_5.html#T75" target="_self">Th75</a>, <a class="ref" href="lattices.html#D13">LATTICES:def 13</a></i>;<br/>


</div>
